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NANOMATERIALS
1
Advanced nanomaterials
Cours support
This text is also partially used for the cours
“Introduction to Nanomaterial”
H.Hofmann
Powder Technology Laboratory
IMX
EPFL
Version 1 Sept 2009
NANOMATERIALS
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Contents
1 INTRODUCTION 6
1.1 WHAT ARE NANOMATERIALS? 6
1.1.1 CLASSIFICATION OF NANOSTRUCTURED MATERIALS 6
1.1.2 WHY SO MUCH INTEREST IN NANOMATERIALS? 8
1.1.3 INFLUENCE ON PROPERTIES BY "NANO-STRUCTURE INDUCED EFFECTS" 9
1.2 SOME PRESENT AND FUTURE APPLICATIONS OF NANOMATERIALS 10
1.3 WHAT ARE THE FUNDAMENTAL ISSUES IN NANOMATERIALS? 13
2 ATOMS, CLUSTERS AND NANOMATERIALS 15
3 NANOCOMPOSITES SYNTHESIS AND PROCESSING 19
3.1 INTRODUCTION 19
3.2 INORGANIC NANOTUBES 20
3.2.1 INTRODUCTION 20
3.2.2 GENERAL SYNTHETIC STRATEGIES 26
3.3 FUNCTIONAL MATERIALS BASED ON SELF-ASSEMBLY OF POLYMERIC SUPRAMOLECULES 28
3.4 MOLECULAR BIOMIMETICS: NANOTECHNOLOGY THROUGH BIOLOGY 32
3.4.1 SELECTION OF INORGANIC-BINDING PROTEINS THROUGH DISPLAY TECHNOLOGIES 34
3.4.2 CHEMICAL SPECIFICITY OF INORGANIC-BINDING POLYPEPTIDES 36
3.4.3 PHYSICAL SPECIFICITY OF PEPTIDE BINDING 37
3.4.4 PEPTIDE-MEDIATED NANOPARTICLE ASSEMBLY 38
3.5 REFERENCES 40
4 MECHANICAL PROPERTIES 41
4.1 INTRODUCTION 41
4.2 METALS 41
4.2.1 GRAIN SIZE EFFECTS IN PLASTICITY AND CREEP 42
4.2.2 METAL PLASTIC DEFORMATION: A COMPARISON BETWEEN CU AND NI NANOPHASE SAMPLES
49
4.2.3 HARDNESS 53
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4.3 CERAMICS / NANOCOMPOSITES 56
4.3.1 DENSITY 56
4.3.2 FRACTURE STRENGTH 56
4.3.3 STRENGTHENING AND TOUGHENING MECHANISMS 58
4.3.4 REDUCTION IN PROCESSING FLAW SIZE 60
4.3.5 CRACK HEALING (ANNEALING TREATMENT) 60
4.3.6 TOUGHENING (K-MECHANISMS) 61
4.3.7 GRAIN BOUNDARY STRENGTHENING MECHANISMS 63
4.3.8 THERMAL EXPANSION MISMATCH (SELSING MODEL) 63
4.3.9 AVERAGE INTERNAL STRESSES 64
4.3.10 LOCAL STRESS DISTRIBUTION 67
4.4 FINAL REMARKS ON STRENGTHENING AND TOUGHENING MECHANISMS 67
4.5 REFERENCES 69
5 THERMAL CONDUCTIVITY IN NANOSTRUCTURED MATERIAL 70
5.1 THERMAL CONDUCTIVITY OF THERMAL BARRIER COATINGS 70
5.1.1 THERMAL CONDUCTIVITY 70
5.1.2 LATTICE WAVES 71
5.1.3 INTERACTION PROCESSES 72
5.2 HIGH-TEMPERATURE THERMAL CONDUCTIVITY OF POROUS AL2O3 NANOSTRUCTURES 76
5.2.1 THEORY 76
5.2.2 EXPERIMENT 82
5.2.3 RESULTS AND DISCUSSION 83
5.3 REFERENCES 88
6 THERMODYNAMIC 89
6.1 NANOTHERMODYNAMICS 89
6.1.1 HILL ’S THEORY 90
6.1.2 TSALLIS’ GENERALIZATION OF ORDINARY BOLTZMANN-GIBBS THERMOSTATICS 96
6.1.3 THERMODYNAMICS OF METASTABLE PHASE NUCLEATION ON NANOSCALE 100
6.1.4 NANOTHERMODYNAMIC ANALYSES OF CVD DIAMOND NUCLEATION 112
6.2 THERMODYNAMICS OF MELTING AND FREEZING IN SMALL PARTICLES 117
6.2.1 THEORY – VANFLEET AND AL. MODEL 117
6.3 PHASE DIAGRAMS 128
6.3.1 GOVERNING EQUATIONS 128
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6.3.2 MATHEMATICAL DESCRIPTION FOR NANO-PHASES OF SN-BI ALLOYS 130
6.3.3 PHASE DIAGRAM FOR ISOLATED NANO-PHASES OF SN-BI ALLOYS 133
6.4 CRYSTAL-LATTICE INHOMOGENEOUS STATE 135
6.5 CONCENTRATIONAL INHOMOGENEITY 137
6.6 REFERENCES 140
7 ELECTRONIC AND OPTICAL PROPERTIES OF NANOMATERIALS 141
7.1 INTRODUCTION 141
7.2 METALS 143
7.2.1 INTRODUCTION 143
7.2.2 ELECTRICAL CONDUCTIVITY 154
7.2.3 SURFACE PLASMONS 162
7.3 CARBON NANOTUBES 171
7.3.1 ELECTRONIC STRUCTURE 172
7.3.2 QUANTUM TRANSPORT PROPERTIES 174
7.3.3 NANOTUBE JUNCTIONS AND DEVICES 178
7.4 SEMICONDUCTOR 181
7.4.1 INTRODUCTION 181
7.4.2 BAND GAP MODIFICATION 185
7.4.3 ELECTRICAL PROPERTIES 190
7.4.4 OPTICAL PROPERTIES 212
7.5 REFERENCES 217
8 MAGNETISM 219
8.1 INTRODUCTION 219
8.1.1 CONCEPT 219
8.1.2 PHENOMENA 220
8.2 MAGNETIC PROPERTIES OF SMALL ATOMIC CLUSTERS 222
8.2.1 INTRODUCTION 222
8.2.2 SIZE DEPENDENCE 223
8.2.3 THERMAL BEHAVIOUR 225
8.2.4 RARE EARTH CLUSTERS 226
8.3 SMALL PARTICLE MAGNETISM 226
8.3.1 CLASSIFICATIONS OF MAGNETIC NANOMATERIAL 226
8.3.2 ANISOTROPY 230
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8.3.3 SINGLE DOMAIN PARTICLES 232
8.3.4 SUPERPARAMAGNETISM 239
8.4 MAGNETOELECTRONICS SPINS 252
8.4.1 SPIN-POLARIZED TRANSPORT AND MAGNETORISISTIVE EFFECTS 252
8.4.2 SPIN INJECTION 256
8.4.3 SPIN POLARIZATION 257
8.5 GIANT MAGNETORESISTANCE (GMR) 259
8.6 STORAGE DEVICES 264
8.6.1 MAGNETIC DATA STORAGE : 264
8.6.2 SENSORS: 265
8.7 REFERENCES 266
9 APPLICATIONS 267
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1 INTRODUCTION
1.1 What are nanomaterials?
1.1.1 Classification of Nanostructured Materials
Nanomaterials are materials which are characterised by an ultra fine grain size
(< 50 nm) or by a dimensionality limited to 50 nm. Nanomaterials can be created with
various modulation dimensionalities as defined by Richard W. Siegel: zero (atomic
clusters, filaments and cluster assemblies), one (multilayers), two (ultrafine-grained
overlayers or buried layers), and three (nanophase materials consisting of equiaxed
nanometer sized grains) as shown in the above Figure 1-1.
Figure 1-1 : Definition of nanomaterials following Siegeli
Nanomaterials consisting of nanometer sized crystallites or grains and
interfaces may be classified according to their chemical composition and shape
(dimensionality), as discussed above. According to the shape of the crystallites or
grains we can broadly classify nanomaterials into four categories:
1. clusters or powders (MD=0)
2. Multilayers (MD=1)
3. ultrafine grained overyaers or buried layers (where the layer thickness or
the rod-diameters are <50 nm) (MD=2)
4. nanomaterials composed of equiaxed nanomter-sized grains (MD=3)
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Figure 1-2 : Classification schema for nanomaterials according to their chemical composition and the
dimensionality (shape)of the crystallites (structural elements) forming the nanomaterial. The boundary
regions of the first and second family of nanomaterials are indicated in black to emphasize the
different atomic arrangements in the crystallites and in the boundaries. The chemical composition of
the (black) boundary regions and the crystallites is identical in the first family. In the second family, the
(black) boundaries are the regions where two crystals of different chemical composition are joined
together causing a steep concentration gradient.
The latter three categories can be further grouped into four families as shown in
Figure 1-2.
• In the most simple case (first family in the Figure 1-2), all grains and interfacial
regions have the same chemical composition. Eg. Semicrystalline polymers
(consisting of stacked lamellae separated by non-crystalline region),
multilayers of thin film crystallites separated by an amorphous layer (a-
Si:N:H/nc-Si)iietc.
• As the second case, we classify materials with different chemical composition
of grains. Possibly quantum well structures are the best example of this family.
• In the third family includes all materials that have a different chemical
composition of its forming matter (including different interfaces) eg. Ceramic of
alumina with Ga in its interface.iii
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• The fourth family includes all nanomaterials formed by nanometer sized grains
(layers, rods or equiaxed crystallites) dispersed in a matrix of different
chemical composition. Precipitation hardened alloys typically belong to this
family. Eg. Nanometer sized Ni3Al precipitates dispersed in a nickel matrix-
generated by annealing a supersaturated Ni-Al solid solution- are an example
of such alloys. Most high-temperature materials used in modern jet engines
are based on precipitation-hardened Ni3Al/Ni alloys.
A large part of this definition has been described in an article by Gleiter.iv,v
1.1.2 Why so much interest in nanomaterials?
These materials have created a high interest in recent years by virtue of their
unusual mechanical, electrical, optical and magnetic properties. Some examples
are given below:
• Nanophase ceramics are of particular interest because they are more ductile
at elevated temperatures as compared to the coarse-grained ceramics.
• Nanostructured semiconductors are known to show various non-linear optical
properties. Semiconductor Q-particles also show quantum confinement effects
which may lead to special properties, like the luminescence in silicon powders
and silicon germanium quantum dots as infrared optoelectronic devices.
Nanostructured semiconductors are used as window layers in solar cells.
• Nanosized metallic powders have been used for the production of gas tight
materials, dense parts and porous coatings. Cold welding properties combined
with the ductility make them suitable for metal-metal bonding especially in the
electronic industry.
• Single nanosized magnetic particles are mono-domains and one expects that
also in magnetic nanophase materials the grains correspond with domains,
while boundaries on the contrary to disordered walls. Very small particles have
special atomic structures with discrete electronic states, which give rise to
special properties in addition to the super-paramagnetism behaviour. Magnetic
nano-composites have been used for mechanical force transfer (ferrofluids),
for high density information storage and magnetic refrigeration.
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• Nanostructured metal clusters and colloids of mono- or plurimetallic
composition have a special impact in catalytic applications. They may serve as
precursors for new type of heterogeneous catalysts (Cortex-catalysts) and
have been shown to offer substantial advantages concerning activity,
selectivity and lifetime in chemical transformations and electrocatalysis (fuel
cells). Enantioselective catalysis were also achieved using chiral modifiers on
the surface of nanoscale metal particles.
• Nanostructured metal-oxide thin films are receiving a growing attention for the
realisation of gas sensors (NOx, CO, CO2, CH4 and aromatic hydrocarbons)
with enhanced sensitivity and selectivity. Nanostructured metal-oxide (MnO2)
find application for rechargeable batteries for cars or consumer goods. Nano-
crystalline silicon films for highly transparent contacts in thin film solar cell and
nano-structured titanium oxide porous films for its high transmission and
significant surface area enhancement leading to strong absorption in dye
sensitized solar cells.
• Polymer based composites with a high content of inorganic particles leading to
a high dielectric constant are interesting materials for photonic band gap
structure produced by the LIGA.
1.1.3 Influence on properties by "nano-structure in duced effects"
For the synthesis of nanosized particles and for the fabrication of
nanostructured materials, laser or plasma driven gas phase reactions, evaporation-
condensation mechanisms, sol-gel-methods or other wet chemical routes like inverse
micelle preparation of inorganic clusters have been used, that will be discussed later.
Most of these methods result in very fine particles which are more or less
agglomerated. The powders are amorphous, crystalline or show a metastable or an
unexpected phase, the reasons for which is far from being clear. Due to the small
sizes any surface coating of the nano-particles strongly influences the properties of
the particles as a whole. Studies have shown that the crystallisation behaviour of
nano-scaled silicon particles is quite different from micron-sized powders or thin films.
It was observed that tiny polycrystallites are formed in every nano-particle, even at
moderately high temperatures.
Roughly two kinds of "nano-structure induced effects" can be distinguished:
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• First the size effect, in particular the quantum size effects where the normal
bulk electronic structure is replaced by a series of discrete electronic levels,
• and second the surface or interface induced effect, which is important because
of the enormously increased specific surface in particle systems.
While the size effect is mainly considered to describe physical properties, the
surface or interface induced effect, plays an eminent role for chemical processing, in
particular in connection with heterogeneous catalysis. Experimental evidence of the
quantum size effect in small particles has been provided by different methods, while
the surface induced effect could be evidenced by measurement of thermodynamic
properties like vapour pressure, specific heat, thermal conductivity and melting point
of small metallic particles. Both types of size effects have also been clearly separated
in the optical properties of metal cluster composites. Very small semiconductor (<10
nm), or metal particles in glass composites, and semiconductor/polymer composites
show interesting quantum effects and non-linear electrical and optical properties.
The numerous examples, which are not complete, by far, indicate that these
materials will most probably gain rapidly increasing importance in the near future. In
general, properties, production and characterisation methods and their inter-relations
are however not yet satisfactorily understood. Hence, efforts need to be made to
enable the directed tailoring of nanophase, nanoscopic and nanocomposite materials
needed for future technical and industrial applications.
1.2 Some present and future applications of nanomat erials
Here we list some of the present and future applications of nanomaterials that
has been reported in recent literature:
Table 1-1 : Some typical properties of nano-structured materials and possibilities of future applications
Property Application
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Bulk
Single magnetic domain
Small mean free path of electrons in a solid
Size smaller than wavelength
High & selective optical absorption of metal particles
Formation of ultra fine pores due to
superfine agglomeration of particles
Uniform mixture of different kinds of superfine particles
Grain size too small for stable dislocation
Magnetic recording
Special conductors
Light or heat absorption, Scattering
Colours, filters, solar absorbers,
photovoltaics, photographic
material, phototropic material
Molecular Filters
R&D of New Materials
High strength and hardness of
metallic materials
Surface/ Interface
Large specific surface area
Catalysis, sensors
Large surface area, small heat capacity
Heat-exchange materials
Combustion Catalysts
Lower sintering temperature
Specific interface area, large boundary area
Superplastic behaviour of ceramics
Cluster coating and metallization
Multi-shell particles
Sintering accelerators
Nano-structured materials
ductile ceramics
Special resistors, temperature sensors
Chemical activity of catalysts
Tailored Optical elements
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Table 1-2 : Some examples of present and potential applications with significant technological
impact:vi
Technology Present Impact Potential Impact
Dispersions and
Coatings
Thermal barriers
Optical (visible and UV) barriers
Imaging enhancement
Ink-jet materials
Coated abrasive slurries
Information-recording layers
Enhanced thermal barriers
Multifunctional nanocoatings
Fine particle structure
Super absorbant materials (Ilford
paper)
Higher efficiency and lower
contamination
Higher density information storage
High Surface
Area
Molecular sieves
Drug delivery
Tailored catalysts
Absorption/desorption materials
Molecule-specific sensors
Particle induced delivery
Energy storage (fuel cells, batteries)
Grätzel-type solar cells, Gas sensors
Consolidated
Materials
Low-loss soft magnetic materials
High hardness, tough WC/Co
cutting tools
Nanocomposite cements
Superplastic forming of ceramics
Materials
Ultrahigh-strength, tough structural
materials
Magnetic refrigerants
Nanofilled polymer composites
Ductile cements
Bio-medical
aspects
Functionalised nanoparticles Cell labelling by fluorescent
nanoparticles
Local heating by magnetic
nanoparticles
Nanodevices
GMR read heads
Terabit memory and microprocessing
Single molecule DNA sizing and
sequencing
Biomedical sensors
Low noise, low threshold lasers
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Nanotubes for high brightness displays
1.3 What are the fundamental issues in nanomaterial s?
The fundamental issues in this domain of nanomaterials are:
(1) ability to control the scale (size) of the system,
(2) ability to obtain the required composition -
not just the average composition - but details such as defects, concentration
gradients, etc.,
(3) ability to control the modulation dimensionality,
(4) during the assembly of the nano-sized building blocks, one should be able to
control the extent of the interaction between the building blocks as well as the
architecture of the material itself.
More specifically the following issues have to be considered for the future
development of nanomaterials:
• Development of synthesis and/or fabrication methods for raw materials
(powders) as well as for the nanostructured materials.
• Better understanding of the influence of the size of building blocks in nano
structured materials as well as the influence of microstructure on the physical,
chemical and mechanical properties of this material.
• Better understanding of the influence of interfaces on the properties of nano-
structured material.
• Development of concepts for nanostructured materials and in particular their
elaboration.
• Investigation of catalytic applications of mono- and plurimetallic nanomaterials
• Transfer of developed technologies into industrial applications including the
development of the industrial scale of synthesis methods of nanomaterials and
nanostructured systems.
In the following chapters we will review the various developments that have been
revolutionising the application of nanomaterials. We will attempt to correlate the
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improvements in the material properties that are achieved due to the fine
microstructures arising from the size of the grains and/or dimensionality.
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2 ATOMS, CLUSTERS AND NANOMATERIALS
atoms
?
?
?
?
?
?
??
?
?
??
molecules cluster nanocrystallites
Figure 2-1 : Schematic representation of various st ates of matter
At the beginning of last century, increasing attention was focused on the physical
chemistry of colloidal suspensions. By referring to them as "the world of neglected
dimensions", Oswald was the first to realize that nanoscale particles should display
novel and interesting properties largely dependant on their size and shape. vii
However, it is only in the last two decades that significant interest has been devoted
to inorganic particles consisting of a few hundred or a few dozen atoms, called
clusters. This interest has been extended to a large variety of metals and
semiconductors and is due to the special properties exhibited by these nanometer-
sized particles, which differ greatly from those of the corresponding macrocrystalline
material.
Matter that is constituted of atoms and molecules as such, has been widely classified
and satisfactorily explained. However, an ensemble of atoms, or molecules forming
the so-called ‘Clusters’ are far from being properly understood. Elemental clusters
are held together by various forces depending on the nature of the constituting
atoms:
Inert gas clusters are weakly held together by van-der-waals interactions, eg. (He)n
Semiconductor clusters are held with strong directional covalent bonds, eg. (Si)n
Metallic clusters are fairly strongly held together by delocalised non-directional
bonding, eg. (Na)n
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No. Of Shells 1 2 3 4 5
No. of atoms M13 M55 M147 M309 M561
Percentage of
atoms
92% 76% 63% 52% 45%
Figure 2-2 : Idealized representation of hexagonal close packed full-shell ‘magic number’ clusters.viii
Note that as the number of atoms increases, the percentage of surface atoms decreases.
1
10
100
1000
10000
100000
0 10 20 30
Cluster size (nm)
Tot
al n
o. o
f ato
ms
(1)
0
20
40
60
80
100
120
0 5 10 15 20 25
Size of cluster (nm)
Sur
face
ato
ms
(%)
(2)
Figure 2-3 : (1) Total Number of atoms with size of the cluster. (2) Number of surface atoms for a
hypothetical model sphere of diameter 0.5 nm and density 1000 Kgm-3 with a mass of 6.5 10-26 Kg
occupying a volume of about 6.5 x 10-29 m3 with a geometrical cross-section of 2 x 10-19 m2 (in terms of
atomic mass the sphere is considered to have a mass of 40 amu, where 1 amu = 1.67 x 10-27 Kg).
Calculated by (a) considering dense structures (Square),ix and (b) method suggested by Preining (dark
circle).x
Either elemental clusters or a mixture of clusters of different elements constitute the
vast expanding field of materials sciences called ‘nanomaterials’. One has to be clear
right at this stage that clusters are not a fifth state of matter, as sometimes believed,
but they are simply intermediate between atoms on one hand, and solid or liquid
state of matter on the other, with widely varying physical and chemical properties.
Depending on the number of atoms forming the cluster determines the percentage of
atoms that are exposed on the surface of the cluster. An example of such an
ensemble of metal atoms show the decreasing number of surface atoms with
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increasing size of the cluster as shown in fig: . When an ensemble of atoms add up to
form a few nanometer sized clusters, they form what we call ‘nanoparticles’, since
only a few atoms forming clusters are called ‘molecular clusters’. Agglomeration of a
few atoms have been studied in great details by physicists working with molecular
beams. Today, the mystery related to larger ensemble of atoms (in other words
‘nanomaterial’) are getting clearer due to active research being carried over across
the world over the last decade or two.
Table 2-1 : Idealized representation of the variation of cross section, total mass, number of molecules
and the effective surface atoms in clusters. Note (a) considering dense structures (Square),ix and (b)
method suggested by Preining (dark circle).x
Size
Cross
section
Mass
No. Of
molecules
Fraction of molecules at surface
(%)
(nm) (10-18 m2) (10-25 Kg) a b
0,5 0,2 0,65 1
1 0,8 5,2 8 100 99
2 3,2 42 64 90 80
5 20 650 1000 50 40
10 80 5200 8000 25 20
20 320 42000 64000 12
In the table, we see that the smaller particles contain only a few atoms, practically all
at the surface. As the particle size increases from 1-10 nm, cross-section increases
by a factor of 100 and the mass number of molecules by a factor of 1000. Meanwhile,
the proportion of molecules at the surface falls from 100% to just 25%. For particles
of 20nm size, a little more than 10% of the atoms are on the surface.
Of course this is an idealized hypothetical case. If particles are formed by macro
molecules (that are larger than the present example), number of molecules per
particle will decrease and their surface fraction increase. The electronic properties of
these ensemble of atoms or molecules will be the result of their mutual interactions
so that the overall chemical behaviour of the particles will be entirely different from
the individual atoms or molecules that they are constituted of. They will also be
different from their macroscopic bulk state of the substance in question under the
same conditions of temperature and pressure.
Table 2-2 : Particle size, surface area and surface energy of CaCo3.xi (the surface energy of bulk
CacO3 (calcite) is 0.23 Jm-2.)
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Size
(nm)
Surface area
(m2 mol -1)
Surface energy
(J mol -1)
1 1.11 x 109 2.55 x 104
2 5.07 x 108 1.17 x 104
5 2.21 x 108 5.09 x 103
10 1.11 x 108 2.55 x 103
20 5.07 x 107 1.17 x 103
102 1.11 x 107 2.55 x 102
103 (1 µ m) 1.11 x 106 2.55 x 10
The idea of tailoring properly designed atoms into agglomerates has brought in new
fundamental work in the search for novel materials with uncharacteristic properties.
Among various types of nanomaterials, cluster assembled materials represent an
original class of nanostructured solids with specific structures and properties. In
terms of structure they could be classified in between amorphous and crystalline
materials. In fact, in such materials the short-range order is controlled by the grain
size and no long-range order exists due to the random stacking of nanograins
characteristic of cluster assembled materials. In terms of properties, they are
generally controlled by the intrinsic properties of the nanograins themselves and by
the interactions between adjacent grains. Cluster assembled films are formed by the
deposition of these clusters onto a solid substrate and are generally highly porous
with densities as low as about one half of the corresponding bulk materials densities
and both the characteristic nanostructured morphology and a possible memory effect
of the original free cluster structures are at the origin of their specific properties.xii
From recent developments in the cluster source technologies (thermal, laser
vaporisation and sputtering),xiii,xiv it is now possible to produce intense cluster beams
of any materials, even the most refractory or complex systems (bimetallic,xv oxides
and so on), for a wide range of size from a few atoms to a few thousands of atoms.
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3 NANOCOMPOSITES SYNTHESIS AND PROCESSING
3.1 Introduction
A major goal of material science is to produce hierarchical materials that are
ordered on all length scales, from the molecular (1–100 Å) via the nano (10–100 nm)
to the meso (1–100 mm) scale. In these materials, the larger-scale properties can be
controlled by choosing the appropriate molecular characteristics.
“Nanocomposites” are a special class of materials originating from suitable
combinations of two or more such nanoparticles or nanosized objects in some
suitable technique, resulting in materials having unique physical properties and wide
application potential in diverse areas that can be formed into a useful object which
can be subsequently used. Novel properties of nanocomposites can be derived from
the successful combination of the individual characteristics of parent constituents into
a single material. To exploit the full potential of the technological applications of the
nanomaterials, it is thus extremely important to endow them with good processability.
Nanocomposites are either prepared in a host matrix of inorganic materials (glass,
porous ceramics etc) or by using conventional polymers as one component of the
nanocomposites. The second type of nanocomposites which are a special class of
hybrid materials are termed “polymeric nanocomposites ”. These materials are
intimate combinations (up to almost the molecular level) of one or more inorganic
materials (nanoparticles , eg.) with a polymer so that unique properties of the former
can be mixed with the existing qualities of the polymer to result in a totally new
material suitable for novel applications. In most of the cases such combinations
require blending or mixing of the components, taking the polymer in solution or in
melt form. xvi Resulting nanocomposites have found successful applications in
versatile fields viz. battery cathodes, xvii microelectronics, xviii nonlinear optics, xix
sensors,xx etc.
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3.2 Inorganic Nanotubes xxi
3.2.1 Introduction
In 1991, Iijima observed some unusual structures of carbon under the transmission
electron microscope wherein the graphene sheets had rolled and folded onto
themselves to form hollow structures. Iijima called them nanotubes of carbon which
consisted of several concentric cylinders of grapheme sheets. Graphene sheets are
hexagonal networks of carbon and these layers get stacked one above the other in
the c-direction to form bulk graphite. Following the initial discovery, intense research
has been carried out on carbon nanotubes (CNTs). The nanotubes can be open-
ended or closed by caps containing five-membered rings. They can be multi-
(MWNTs) or singlewalled (SWNTs). We show a typical high-resolution electron
microscope (HREM) image of a multi-walled nanotube in Figure 3-1.
Figure 3-1 : A typical TEM image of a closed, multi -walled carbon nanotube. The separation
between the graphite layers is 0.34 nm. [C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans.,
2003, p.p. 1-24]
Depending on the way the graphene sheets fold, nanotubes are classified as
armchair, zigzag or chiral as shown in Figure 3-2. The electrical conductivity of the
nanotubes depends on the nature of folding.
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Figure 3-2 : Schematic representation of the foldin g of a graphene sheet into (a) zigzag, (b)
armchair and (c) chiral nanotubes. [C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans., 2003,
p.p. 1-24]
Several layered inorganic compounds possess structures comparable to the structure
of graphite, the metal dichalcogenides being important examples. The metal
dichalcogenides, MX2 (M = Mo, W, Nb, Hf; X = S, Se) contain a metal layer
sandwiched between two chalcogen layers with the metal in a trigonal pyramidal or
octahedral coordination mode. The MX2 layers are stacked along the c-direction in
ABAB fashion. The MX2 layers are analogous to the single graphene sheets in the
graphite structure (Figure 3-3).
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Figure 3-3 : Comparison of the structures of (a) graphite and inorganic layered compounds such as
(b) NbS2/TaS2; (c) MoS2; (d) BN. In the layered dichalcogenides, the metal is in trigonal prismatic
(TaS2) or octahedral coordination (MoS2). [C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans.,
2003, p.p. 1-24]
When viewed parallel to the c-axis, the layers show the presence of dangling bonds
due to the absence of an X or M atom at the edges. Such unsaturated bonds at the
edges of the layers also occur in graphite. The dichalcogenide layers are unstable
towards bending and have a high propensity to roll into curved structures. Folding in
the layered transition metal chalcogenides (LTMCs) was recognized as early as
1979, well before the discovery of the carbon nanotubes. Rag-like and tubular
structures of MoS2 were reported by Chianelli who studied their usefulness in
catalysis.
The folded sheets appear as crystalline needles in low magnification transmission
electron microscope (TEM) images, and were described as layers that fold onto
themselves (Figure 3-4).
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Figure 3-4 : Low-magnification TEM images of (a) highly folded MoS2 needles and (b) a rolled sheet
of MoS2 folded back on itself. [C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans., 2003, p.p. 1-
24]
These structures indeed represent those of nanotubes. Tenne et al xxii first
demonstrated that Mo and W dichalcogenides are capable of forming nanotubes
(Figure 3-5 a).
Figure 3-5 : TEM images of (a) a multi-walled nanotube of WS2 and (b) hollow particles (inorganic
fullerenes) of WS2. [C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans., 2003, p.p. 1-24]
Closed fullerene-type structures (inorganic fullerenes) also formed along with the
nanotubes (Figure 3-5 b). The dichalcogenide structures contain concentrically
nested fullerene cylinders, with a less regular structure than in the carbon nanotubes.
Accordingly, MX2 nanotubes have varying wall thickness and contain some
amorphous material on the exterior of the tubes. Nearly defect-free MX2 nanotubes
are rigid as a consequence of their structure and do not permit plastic deformation.
The folding of a MS2 layer in the process of forming a nanotube is shown in the
schematic in Figure 3-6.
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Figure 3-6 : Schematic illustration of the bending of a MoS 2 layer. [C.N.R. Rao, M. Nath, Inorganic
nanotubes, Dalton Trans., 2003, p.p. 1-24]
Considerable progress has been made in the synthesis of the nanotubes of Mo and
W dichalcogenides in the last few years (Table 3-1 and Table 3-2).
Table 3-1 : Synthetic strategies for various chalco genide nanotubes [C.N.R. Rao, M. Nath,
Inorganic nanotubes, Dalton Trans., 2003, p.p. 1-24]
Table 3-2 : Synthetic strategies for various chalco genide nanotubes [C.N.R. Rao, M. Nath,
Inorganic nanotubes, Dalton Trans., 2003, p.p. 1-24]
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There has been some speculation on the cause of folding and curvature in the
LTMCs. Stoichiometric LTMC chains and layers such as those of TiS2 possess an
inherent ability to bend and fold, as observed in intercalation reactions.
The existence of alternate coordination and therefore of stoichiometry in the LTMCs
may also cause folding. Lastly, a change in the stoichiometry within the material
would give rise to closed rings.
Transition metal chalcogenides possess a wide range of interesting physical
properties. They are widely used in catalysis and as lubricants. They have both
semiconducting and superconducting properties (see paragraph 7). With the
synthesis and characterization of the fullerenes and nanotubes of MoS2 and WS2, a
wide field of research has opened up enabling the successful synthesis of nanotubes
of other metal chalcogenides. It may be recalled that the dichalcogenides of many of
the Group 4 and 5 metals have layered structures suitable for forming nanotubes.
Curved structures are not only limited to carbon and the dichalcogenides of Mo and
W. Perhaps the most well-known example of a tube-like structure with diameters in
the nm range is formed by the asbestos mineral (chrysotil) whose fibrous
characteristics are determined by the tubular structure of the fused tetrahedral and
octahedral layers. The synthesis of mesoporous silica with well-defined pores in the
2–20 nm range was reported by Beck and Kresgexxiii. The synthetic strategy involved
the self-assembly of liquid crystalline templates. The pore size in zeolitic and other
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inorganic porous solids is varied by a suitable choice of the template. However, in
contrast to the synthesis of porous compounds, the synthesis of nanotubes is
somewhat more difficult.
Nanotubes of oxides of several transition metals as well as of other metals have been
synthesized employing different methodologies. Silica nanotubes were first produced
as a spin-off product during the synthesis of spherical silica particles by the
hydrolysis of tetraethylorthosilicate (TEOS) in a mixture of water, ammonia, ethanol
and D,L-tartaric acid. Since selfassembly reactions are not straightforward with
respect to the desired product, particularly its morphology, templated reactions have
been employed using carbon nanotubes to obtain nanotube structures of metal
oxides. Oxides such as V2O5 have good catalytic activity in the bulk phase. Redox
catalytic activity is also retained in the nanotubular structure. There have been efforts
to prepare V2O5 nanotubes by chemical methods as well.
Boron nitride (BN) crystallizes in a graphite-like structure and can be simply viewed
as replacing a C–C pair in the graphene sheet with the iso-electronic B–N pair. It can,
therefore, be considered as an ideal precursor for the formation of BN nanotubes.
Replacement of the C–C pairs partly or entirely by the B–N pairs in the hexagonal
network of graphite leads to the formation of a wide array of two-dimensional phases
that can form hollow cage structures and nanotubes. The possibility of replacing C–C
pairs by B–N pairs in the hollow cage structure of C60 was predicted and verified
experimentally. BN-doped carbon nanotubes have been prepared. Pure BN
nanotubes have been generated by employing several procedures, yielding
nanotubes with varying wall thickness and morphology. It is therefore quite possible
that nanotube structures of other layered materials can be prepared as well. For
example, many metal halides (e.g., NiCl2), oxides (GeO2) and nitrides (GaN)
crystallize in layered structures. There is considerable interest at present to prepare
exotic nanotubes and to study their properties.
3.2.2 General synthetic strategies
Several strategies have been employed for the synthesis of carbon nanotubes. They
are generally made by the arc evaporation of graphite or by the pyrolysis of
hydrocarbons such as acetylene or benzene over metal nanoparticles in a reducing
atmosphere. Pyrolysis of organometallic precursors provides a one-step synthetic
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method of making carbon nanotubes. In addition to the above methods, carbon
nanotubes have been prepared by laser ablation of graphite or electron-beam
evaporation. Electrochemical synthesis of nanotubes as well as growth inside the
pores of alumina membranes have also been reported. The above methods broadly
fall under two categories. Methods such as the arc evaporation of graphite employ
processes which are far from equilibrium. The chemical routes are generally closer to
equilibrium conditions. Nanotubes of metal chalcogenides and boron nitride are also
prepared by employing techniques similar to those of carbon nanotubes, although
there is an inherent difference in that the nanotubes of inorganic materials such as
MoS2 or BN would require reactions involving the component elements or
compounds containing the elements. Decomposition of precursor compounds
containing the elements is another possible route.
Nanotubes of dichalcogenides such as MoS2, MoSe2 and WS2 are also obtained by
employing processes far from equilibrium such as arc discharge and laser ablation.
By far the most successful routes employ appropriate chemical reactions. Thus,
MoS2 and WS2 nanotubes are conveniently prepared starting with the stable oxides,
MoO3 and WO3. The oxides are first heated at high temperatures in a reducing
atmosphere and then reacted with H2S. Reaction with H2Se is used to obtain the
selenides. Recognizing that the trisulfides MoS3 and WS3 are likely to be the
intermediates in the formation of the disulfide nanotubes, the trisulfides have been
directly decomposed to obtain the disulfide nanotubes. Diselenide nanotubes have
been obtained from the metal triselenides. The trisulfide route is indeed found to
provide a general route for the synthesis of the nanotubes of many metal disulfides
such as NbS2 and HfS2. In the case of Mo and W dichalcogenides, it is possible to
use the decomposition of the precursor ammonium salt, such as (NH4)2MX4 (X = S,
Se; M = Mo, W) as a means of preparing the nanotubes. Other methods employed
for the synthesis of dichalcogenide nanotubes include hydrothermal methods where
the organic amine is taken as one of the components in the reaction mixture (Table
3-1 and Table 3-2).
The hydrothermal route has been used for synthesizing nanotubes and related
structures of a variety of other inorganic materials as well. Thus, nanotubes of
several metal oxides (e.g., SiO2, V2O5, ZnO) have been produced hydrothermally.
Nanotubes of oxides such as V2O5 are also conveniently prepared from a suitable
metal oxide precursor in the presence of an organic amine or a surfactant.
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Surfactant-assisted synthesis of CdSe and CdS nanotubes has been reported. Here
the metal oxide reacts with the sulfidizing/selenidizing agent in the presence of a
surfactant such as TritonX.
Sol–gel chemistry is widely used in the synthesis of metal oxide nanotubes, a good
example being that of silica and TiO2. Oxide gels in the presence of surfactants or
suitable templates form nanotubes. For example, by coating carbon nanotubes
(CNTs) with oxide gels and then burning off the carbon, one obtains nanotubes and
nanowires of a variety of metal oxides including ZrO2, SiO2 and MoO3. Sol–gel
synthesis of oxide nanotubes is also possible in the pores of alumina membranes. It
should be noted that MoS2 nanotubes are also prepared by the decomposition of a
precursor in the pores of an alumina membrane.
Boron nitride nanotubes have been obtained by striking an electric arc between HfB2
electrodes in a N2 atmosphere. BCN and BC nanotubes are obtained by arcing
between B/C electrodes in an appropriate atmosphere. A greater effort has gone into
the synthesis of BN nanotubes starting with different precursor molecules containing
B and N. Decomposition of borazine in the presence of transition metal nanoparticles
and the decomposition of the 1 : 2 melamine–boric acid addition compound yield BN
nanotubes. Reaction of boric acid or B2O3 with N2 or NH3 at high temperature in the
presence of activated carbon, carbon nanotubes or catalytic metal particles has been
employed to synthesize BN nanotubes.
3.3 Functional Materials Based on Self-Assembly of
Polymeric Supramolecules xxiv
Here, we describe some possibilities for preparing functional polymeric materials
using the "bottom-up" route, based on self-assembly of polymeric supramolecules.
Directed assembly leads to the control of structure at several length scales and
anisotropic properties. The physical bonds within the supramolecules allow controlled
cleavage of selected constituents. The techniques constitute a general platform for
constructing materials that combine several properties that can be tuned separately.
To achieve enhanced functionalities, the principal periodicity is at ~10 to 2000 Å.
There are established ways to accomplish this by using various architectures of block
copolymers, in which the structure formation is based on self-organization, that is, on
the repulsion between the chemically connected blocks. Depending on the
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architecture, block length, and temperature, it is possible to obtain lamellar,
cylindrical, spherical, gyroid, or more complicated structures in the 100 to 2000 Å
range. Also, rodlike moieties within the block copolymers can be used to further tailor
the structures in terms of shape persistency. However, self-organization renders only
the local structures. To fully realize the opportunities offered by the symmetry of the
self-organized structures to prepare materials with a strongly directional variation of
properties, additional mechanisms and interactions have to be invoked to obtain
macroscale order. This may be achieved by flow, by electric or magnetic fields, or by
using topographically patterned surfaces. One can further extend the structural
complexity by mixing block copolymers with additional polymers and inorganic
additives, thereby increasing the self-organization periods into the photonic band gap
regime. Block copolymers have also been used as templates for the synthesis of
inorganic materials, even allowing the creation of separate ceramic nano-objects.
To achieve even greater structural complexity and functionality, we can combine
recognition with self-organization. Lehn elaborated on the concept of recognition in
synthetic materials, whereby two molecules with molecularly matching
complementary interactions and shapes recognize each other and form a receptor-
substrate supramolecule. To achieve sufficient bonding, synergism of several
physical interactions is often required. Homopolymerlike supramolecules have been
constructed based on a combination of four hydrogen bonds and through
coordination. Supramolecules can spontaneously assemble or self-organize to form
larger structures.
A general framework for forming complex functional materials emerges. Molecules
are constructed that recognize each other in a designed way. The subsequent
supramolecules in turn form assemblies or self-organize, possibly even forming
hierarchies. The overall alignment of the local structures can be additionally improved
by electric or magnetic fields, by flow, or by patterned surfaces.
To illustrate recognition-driven supramolecule formation in polymers and the
subsequent self-organization and preparation of functional materials and nano-
objects, we focus on the comb-shaped architecture (Figure 3-7) encouraged by the
enhanced solubility of socalled hairy-rod polymers.
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Figure 3-7 : Comb-shaped supramolecules and their hierarchical self-organization, showing primary
and secondary structures. Similar schemes can, in principle, be used both for flexible and rodlike
polymers. In the first case, simple hydrogen bonds can be sufficient, but in the latter case a synergistic
combination of bondings (recognition) is generally required to oppose macrophase separation
tendency. In (A through C), the self-organized structures allow enhanced processibility due to
plastization, and solid films can be obtained after the side chains are cleaved (D). Self-organization of
supramolecules obtained by connecting amphiphiles to one of the blocks of a diblock copolymer (E)
results in hierarchically structured materials. Functionalizable nanoporous materials (G) are obtained
by cleaving the side chains from a lamellae-within-cylinders structure (F). Disk-like objects (H) may be
prepared from the same structure by crosslinking slices within the cylinders, whereas nano rods (I)
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result from cleaving the side chains from a cylinder-within-lamellae structure. Without loss of
generality, (A) is shown as a flexible polymer, whereas (B) and (C) are shown as rodlike chains. [O.
Ikkala, G.T. Brinke, Functional Materials Based on Self-assembly of polymeric supramolecules,
Science, New Series, Vol. 295, No. 5564 (Mar. 29, 2002), pp. 2407-2409]
The simplest case is a flexible polymer having bonding sites along its backbone
(Figure 3-7 A). Therefore, the backbone is typically polar, and repulsive nonpolar side
groups can be connected by complementary bonds, leading to comb-shaped
supramolecules, which in turn self-organize. We have extensively used hydrogen
bonding or coordination to bond side chains to the polymer backbone. Antonietti et
al.xxv have used ionic interactions in polyelectrolyte-surfactant complexes to form
comb-shaped polyelectrolyte surfactant complexes. The resulting self-organized
multidomain structures may be aligned, using, for example, flow, in order to approach
monodomains. One can also tune the properties by tailoring the nature of the side
chains. For example, if the side chains are partly fluorinated, low surface energy
results, which allows for applications that lead to reduced friction. In another case,
the backbone consists of the double helix of DNA, and self-organization is achieved
by ionically bonding cationic liposomes or cationic surfactants to the anionic
phosphate sites. This allows for materials design beyond the traditional scope of
biochemical applications. For example, dyes can be intercalated into the helices,
suppressing their aggregation tendency and leading to promising properties as
templates for photonic applications. In such a structure, the polymer backbone may
contain two or even more kinds of binding sites where different additives can be
bonded (Figure 3-7 B). Side chains can also have two separate functions. For
example, in addition to providing a repulsive side chain required for self-organization,
the side chains may contain an acidic group that acts as a dopant for a conjugated
polymer such as polyaniline, which leads to electronic conductivity. To introduce
further degrees of freedom in tailoring the self-organized phases and their
processing, polyaniline may first be doped by a substance such as camphor
sulphonic acid and subsequently connected to hexyl resorcinol molecules using their
two hydrogen bonds (Figure 3-7 C). The alkyl chains of the hydrogen-bonded hexyl
resorcinol molecules act as plasticizers, leading to thermoplastic processibility of the
otherwise infusible polymer. They enforce self-organization where camphor sulfonic
acid-doped polyaniline chains are confined in nanoscale conducting cylinders,
leading to increased conductivity. The concept can be applied even to rodlike
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polymers, such as polypyridine, which consists of para-coupled heteroaromatic rings.
Its optical properties can be tuned based on camphor sulphonic acid. Subsequent
hydrogen bonding with alkyl resorcinol creates comb-shaped supramolecules, which
self-organize in lamellae in such a way that the material is fluid even without
additional solvents. Such a fluid state incorporating rigid polymeric rods is uncommon
and allows processing toward monodomains where the rods are aligned. Ultimately,
the plasticizing hydrogen-bonded alkyl resorcinol molecules can be removed by
evaporation in a vacuum oven, thus interlocking the chains in solid stable films
(Figure 3-7 D). In this way, efficient polarized luminance has been achieved.
To increase complexity, one can incorporate structural hierarchies. This can be
accomplished by applying within a single material different self-organization and
recognition mechanisms operating at different length scales. For example, block
copolymeric self-organization at the 100 to 2000 Å length scale and polymer-
amphiphile self-organization at the 10 to 60 Å length scale can be combined (Figure
3-7 E). After selective doping of one block, conductivity can be switched based on a
sequence of phase transitions.
3.4 Molecular biomimetics: Nanotechnology through
biology xxvi
Molecular biomimetics. This is the marriage of materials science engineering and
molecular biology for development of functional hybrid systems, composed of
inorganics and inorganic-binding proteins. The new approach takes advantage of
DNA-based design, recognition,and self-assembly characteristics of biomolecules.
Traditional materials science engineering produces materials (for example, medium-
carbon steels depicted in the bright- and dark-field TEM images), that have been
successfully used over the last century. Molecular biology focuses on structure–
function relations in biomacromolecules, for example, proteins.
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Figure 3-8 : In molecular biomimetics, inorganic-binding proteins could potentially be used as (i)
linkers for nanoparticle immobilization; (ii) functional molecules assembled on specific substrates; and
(iii) heterobifunctional linkers involving two (or more) binding proteins linking several nanoinorganic
units. (I1: inorganic-1,I2: Inorganic-2, P1 and P2: inorganic specific proteins, LP:linker protein, FP:
fusion protein). [Sarikaya, C. Tamerler, A.K.Y. Jen, K. Schulten F. Baneyx, Molecular biomimetics:
nanotechnology through biology, Nature Materials, vol 2, 2003, p.p. 577-585]
In molecular biomimetics, a marriage of the physical and biological fields, hybrid
materials could potentially be assembled from the molecular level using the
recognition properties of proteins (Figure 3-8) under the premise that inorganic
surface-specific polypeptides could be used as binding agents to control the
organization and specific functions of materials.Molecular biomimetics simultaneously
offers three solutions to the development of heterofunctional nanostructures.
• The first is that protein templates are designed at the molecular level through
genetics. This ensures complete control over the molecular structure of the
protein template (that is, DNA-based technology).
• The second is that surface-specific proteins can be used as linkers to bind
synthetic entities, including nanoparticles, functional polymers, or other
nanostructures onto molecular templates (molecular and nanoscale
recognition).
• The third solution harnesses the ability of biological molecules to self- and co-
assemble into ordered nanostructures. This ensures a robust assembly
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process for achieving complex nano-, and possibly hierarchical structures,
similar to those found in nature (self-assembly).
The current knowledge of protein-folding predictions and surface-binding chemistries
does not provide sufficiently detailed information to perform rational design of
proteins. To circumvent this problem, massive libraries of randomly generated
peptides can be screened for binding activity to inorganic surfaces using phage and
cell-surface display techniques. It may ultimately be possible to construct a
‘molecular erector’ set, in which different types of proteins, each designed to bind to a
specific inorganic surface, could assemble into intricate, hybrid structures composed
of inorganics and proteins. This would be a significant leap towards realizing
molecularly designed, genetically engineered technological materials.
3.4.1 Selection of inorganic-binding proteins throu gh display
technologies
There are several possible ways of obtaining polypeptide sequences with specific
affinity to inorganics. A number of proteins may fortuitously bind to inorganics,
although they are rarely tested for this purpose. Inorganic-binding peptides may be
designed using a theoretical molecular approach similar to that used for
pharmaceutical drugs. This is currently impractical because it is time consuming and
expensive. Another possibility would be to extract biomineralizing proteins from hard
tissues followed by their isolation, purification and cloning. Several such proteins
have been used as nucleators, growth modifiers, or enzymes in the synthesis of
certain inorganics. One of the major limitations of this approach is that a given hard
tissue usually contains many proteins, not just one, all differently active in
biomineralization and each distributed spatially and temporally in complex ways.
Furthermore, tissue-extracted proteins may only be used for the regeneration of the
inorganics that they are originally associated with, and would be of limited practical
use. The preferred route, therefore, is to use combinatorial biology techniques. Here,
a large random library of peptides with the same number of amino acids, but of
different sequences, is used to mine specific sequences that strongly bind to a
chosen inorganic surface.
Since their inception, well-established in vivo combinatorial biology protocols (for
example, phage display (PD) and cell-surface display (CSD)) have been used to
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identify biological ligands and to map the epitope (molecular recognition site) of
antibodies. Libraries have also been screened for various biological activities, such
as catalytic properties or altered affinity and specificity to target molecules in many
applications including the design of new drugs, enzymes, antibodies, DNA-binding
proteins and diagnostic agents. The power of display technologies relies on the fact
that an a priori knowledge of the desired amino acid sequence is not necessary, as it
can simply be selected and enriched if a large enough population of random
sequences is available. In vitro methods, such as ribosomal and messenger RNA
display technologies, have been developed for increased library size (1015) compared
to those of in vivo systems (107–10).
Combinatorial biology protocols can be followed in molecular biomimetics to select
polypeptide sequences that preferentially bind to the surfaces of inorganic
compounds chosen for their unique physical properties in nano- and biotechnology.
Libraries are generated by inserting randomized oligonucleotides within certain
genes encoded on phage genomes or on bacterial plasmids (step 1 in Figure 3-9).
Figure 3-9 : Phage display and cell-surface display. Principles of the protocols used for selecting
polypeptide sequences that have binding affinity to given inorganic substrates. [Sarikaya, C. Tamerler,
A.K.Y. Jen, K. Schulten F. Baneyx, Molecular biomimetics: nanotechnology through biology, Nature
Materials, vol 2, 2003, p.p. 577-585]
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This leads to the incorporation of a random polypeptide sequence within a protein
residing on the surface of the organism (for example, the coat protein of a phage or
an outer membrane or flagellar protein of a cell; step 2). The eventual result is that
each phage or cell produces and displays a different, but random peptide (step 3). At
this stage, a heterogeneous mixture of recombinant cells or phages are contacted
with the inorganic substrate (step 4).Several washing cycles of the phages or the
cells eliminate non-binders by disrupting weak interactions with the substrate (step
5). Bound phages or cells are next eluted from the surfaces (step 6). In PD, the
eluted phages are amplified by reinfecting the host (step 7). Similarly in CSD, cells
are allowed to grow (steps 7, 8). This step completes a round of biopanning.
Generally, three to five cycles of biopanning are repeated to enrich for tight binders.
Finally, individual clones are sequenced (step 9) to obtain the amino acid sequence
of the polypeptides binding to the target substrate material.
3.4.2 Chemical specificity of inorganic-binding pol ypeptides
A genetically engineered polypeptide for inorganics (GEPI) defines a sequence of
amino acids that specifically and selectively binds to an inorganic surface. The
surface could be well defined, such as a single crystal or a nanostructure. It might
also be rough, or totally non-descriptive, such as a powder. Researchers have
focused on using materials that can be synthesized in aqueous environments under
physiological conditions (biocompatible) and that exhibit fairly stable surface
structures and compositions. These include noble metals (Pt and Pd) as well as
oxide semiconductors (Cu2O and ZnO) that were biopanned using either PD or
flagellar display (both studies unpublished). Some of the identified binders as well as
sequences selected by other researchers are listed in Table 3-3.
Table 3-3 : Examples of polypeptide sequences exhib iting affinity for various inorganics.
[Sarikaya, C. Tamerler, A.K.Y. Jen, K. Schulten F. Baneyx, Molecular biomimetics: nanotechnology
through biology, Nature Materials, vol 2, 2003, p.p. 577-585]
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3.4.3 Physical specificity of peptide binding
Ideally, selection of sequences should be performed using an inorganic material of
specific morphology, size, crystallography or surface stereochemistry. In practice,
however, powders of various sizes and morphologies have been used for selection.
The sequence space should be largest for powders, as peptides can attach to
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surfaces with various morphological features. On the other hand, because powders
are non-descriptive, the selected polypeptides may exhibit little or no homology and
‘solve’ the binding problem through different strategies. Binders selected for a given
size, morphology, crystallography or stereochemistry may share a higher degree of
homology. For example, a GEPI binding to a material of a certain size may also bind
to a smaller particle of the same material, but less strongly. Similarly, a GEPI binding
strongly to a specific crystallographic surface may bind with an altered affinity to
another surface of the same material. Finally, a GEPI strongly binding to a material of
composition A may bind less strongly to a material B with a different composition but
having similar structure (for example, perovskites). Therefore, if one seeks highly
specific binders, the physical and chemical characteristics of the material must be
known. An alternative approach is that, once a relatively large number of binders
have been identified by panning on powders, a subset specific for morphology, size
or surface could be identified on well-defined materials.
3.4.4 Peptide-mediated nanoparticle assembly
Organization and immobilization of inorganic nanoparticles in two- or three-
dimensional geometries are fundamental in the use of nanoscale effects.
a)
b)
c)
f)
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d) e)
Figure 3-10 : Effect of GEPI on nanocrystal morphology. a–c, One of the two mutants (RP1) from a
library of goldbinding GEPIs were tested in the formation of flat gold particles, shown in a, similar to
those formed under acidic (b) or boiling (c) conditions. Particles formed in the presence of vector-
encoded alkaline phosphatase and neutral conditions do not result in morphological change of gold
particles (not shown). d-e, The atomic force microscope images show quantum (GaInAs) dots
assembled on GaAs substrate; d, through high-vacuum (molecular beam epitaxy) strain-induced self-
assembly, and e, through 7-repeat GBP1. f, Schematic illustration of e. PS: polystyrene substrate, GA:
glutaraldehyde, GBP: 7- repeat GBP1, and gold: 12-nm-diameter colloidal gold particles. [Sarikaya, C.
Tamerler, A.K.Y. Jen, K. Schulten F. Baneyx, Molecular biomimetics: nanotechnology through biology,
Nature Materials, vol 2, 2003, p.p. 577-585]
For example, quantum dots can be produced using vacuum techniques, such as
molecular beam epitaxy, shown in Figure 3-10 d for the GaInAs/GaAs system.
However, this can only be accomplished under stringent conditions of high
temperature, very low pressures and a toxic environment. A desirable alternative
would be not only to synthesize inorganic nanodots under mild conditions, but also to
immobilize/self-assemble them. Inorganic particles have been functionalized with
synthetic molecules, including thiols and citrates, and with biological molecules, such
as lipids, amino acids, polypeptides and ligand-functionalized DNA. Using the
recognition properties of the coupling agents, novel materials have been generated
and controlled growth has been achieved. These molecules, however, do not exhibit
specificity for a given material. For example, thiols couple gold as well as silver
nanoparticles in similar ways. Likewise, citrate ions cap noble metals indiscriminately.
A desirable next step would be to use GEPIs that specifically recognize inorganics for
nanoparticle assembly. An advantage of this approach is that GEPI can be
genetically or synthetically fused to other functional biomolecular units or ligands to
produce heterobifunctional (or multifunctional) molecular entities. Figure 3-10 e and f
shows the assembly of nanogold particles on GBP1-coated flat polystyrene surfaces,
which resembles the distribution of quantum dots obtained by high-vacuum
deposition techniques (Figure 3-10 d). The homogenous decoration of the surface
with nanogold suggests that proteins may be useful in the production of tailored
nanostructures under ambient conditions and aqueous solutions. Furthermore, the
recognition activity of the protein could provide an ability to control the particle
distribution, and particle preparation conditions could allow size control. This
approach makes it possible to pattern inorganic-binding polypeptides into desirable
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arrays to produce inorganic particles through templating using, for example, dip-pen
lithography.
3.5 References
Sarikaya, C. Tamerler, A.K.Y. Jen, K. Schulten F. Baneyx, Molecular biomimetics:
nanotechnology through biology, Nature Materials, vol 2, 2003, p.p. 577-585
O. Ikkala, G.T. Brinke, Functional Materials Based on Self-assembly of polymeric
supramolecules, Science, New Series, Vol. 295, No. 5564 (Mar. 29, 2002), pp. 2407-
2409
C.N.R. Rao, M. Nath, Inorganic nanotubes, Dalton Trans., 2003, p.p. 1-24
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4 MECHANICAL PROPERTIES
4.1 Introduction
One of the very basic results of the physics and chemistry of solids is the insight that
most properties of solids depend on the microstructure, i.e. the chemical composition,
the arrangement of the atoms (the atomic structure) and the size of a solid in one,
two or three dimensions. In other words, if one changes one or several of these
parameters, the properties of a solid vary. The most well-known example of the
correlation between the atomic structure and the properties of a bulk material is
probably the spectacular variation in the hardness of carbon when it transforms from
diamond to graphite. The important aspects related to structure are:
• atomic defects, dislocations and strains
• grain boundaries and interfaces
• porosity
• connectivity and percolation
• short range order
Defects are usually absent in either metallic or ceramic clusters of
nanoparticles because dislocations are basically unstable or mobile. The stress field
around a dislocation (or the electrostatic potential around charges and currents)
have to satisfy the Laplace equation: ∇2Φ = 0. This sets up an image dislocation
which pulls the defect out of the particle.
When these clusters are assembled under uniaxial pressure into a pellet, for
example, it is found that the individual clusters are packed very tightly into a
polycrystalline solid. Cluster-assembled materials often show close to 100% density.
A fully consolidated nanophase material looks very much like a normal, dense
polycrystalline aggregate, but at a far smaller scale.
4.2 Metals
The microstructure of a material is controlled by the processing steps chosen for its
fabrication. Such microstructural design affects the nature of the phases present,
their topology (i.e. geometrical distribution and interconnection) and their dispersion
NANOMATERIALS
42
(described by relevant “size” parameters). The full characterization of these
parameters is the domain of quantitative metallurgy.
Most of these size effects come about because of the microstructural constraint to
which a particular physical mechanism is subjected. Consider the classic case of
strengthening a metallic matrix by particles or grain boundaries: lattice dislocations
are forced, by the microstructural constraint, to bow out or pile up, which requires an
external stress characteristic of a microstructural parameter. The wall thickness
relative to the size of the microstructural inhomogeneity can control the macroscopic
behaviour.
In general, it is therefore the competition or coupling between two different size
dependencies that determines the properties of a material. One thus has to deal with
the interaction of two length scales: (1) is the dimension characteristic of the physical
phenomenon involved, called the characteristic length. (2) is some microstructural
dimension, denoted as the size parameter.
4.2.1 Grain size effects in plasticity and creep
4.2.1.1 Hall-Petch effects.
Strengthening of polycrystalline materials by grain size refinement is
technologically attractive because it generally does not adversely affect ductility
and toughness. The classical effect of grain size on yield stress (τ ) can, among
other possibilities, be explained by a model invoking a pile-up of dislocations
against grain boundaries, which results in a dependence of the hardening
increment kHP on the square root of the grain size D
D
kHP=τ (4. 1)
Where kHP is a constant. This is the classical Hall-Petch effect.
In Figure 4-1, some of the available data have been plotted in a Hall-
Petch plot.xxvii
NANOMATERIALS
43
Figure 4-1 : Compilation of yield stress data for s everal metallic systems. [R.A. Masumura, P.M.
Hazzledine, C.S. Pande, Yield stress of fine grained materials, 1998, Acta-Materialia]
It is seen that the yield stress-grain size exponent for relatively large grains
appears to be very close to -0.5 and generally this trend continues until the very fine
grain regime (~100 nm) is reached. The reported data show three different regions:
1. A region from single crystal to a grain size of about 1 mm where the classical
Hall-Petch description can be used.
2. A region for grain sizes ranging from about 1 mm to about 30 nm where the
Hall-Petch relation roughly holds, but deviates from the classical -0.5
exponent to a value near zero (to ascertain such behaviour, a wide range of
grain sizes extending into the ultra-fine grain size regime is required).
3. A region beyond a very small critical grain size where the Hall-Petch slope is
essentially zero, with no increase in strength on decreasing grain size or
where the strength actually decreases with decreasing grain size.
There is universal agreement regarding the first region, i.e. relatively large grain
sizes. Early hardness measurements had already established a distinct increase in
hardness as grain sizes decrease as compared to their annealed coarse grained
counterparts, and this increase follows the Hall-Petch relationship reasonably well.
The trend is less well established for finer grains (Region 2). Some of the deviation
from Hall-Petch strengthening could simply be due to pores in the material (as
evidenced by lower densities) leading also to a lower shear stress for the
NANOMATERIALS
44
deformation mode and lower shear modulus. Indeed, the lower modulus has been
ascribed to a decrease in bulk density. Additional complications arise due to
impurities at the grain boundaries such as oxides and impurities inside the grain
such as trapped or diffused gas. In spite of the above difficulties, once the totality of
the data is taken into consideration, it is fairly safe to conclude that the increase in
strength on grain refinement in the middle region is somewhat less than predicted by
the Hall-Petch relation.
The third region is much more controversial and is going to be discussed later.
4.2.1.2 Limits to Hall-Petch behaviour: dislocation curvature vs. grain size.
Whereas many metallic materials obey such a relationship over several
orders of magnitude in grain size, it is inevitable that the reasoning behind equation
D
kHP=τ (4. 1) must break down for very small grains. A clear limit for the
occurrence of dislocation plasticity in a poly-crystal is given by the condition that at
least one dislocation loop must fit into average grain (Figure 4-2).
Figure 4-2 : Grain size strengthening , as explained by pile-ups of dislocation loops against grain
boundaries (a). this mechanism must break down when the diameter d of the smallest loop no longer
fits into a grain of size D (b). [E. Arzt, Size effects in materials due to microstuctural and dimensional
constraints: A comparative review, 1998, Acta Meteriala]
NANOMATERIALS
45
The characteristic length, i.e. the loop diameter (ττ
τ Gb
b
Td d ==
2)( ), must now be
compared with the grain size D as the relevant size parameter
Dd =)(τ (4. 2)
D
Gb
bD
Td ==2τ (4. 3)
Figure 4-3 illustrates schematically this limit on Hall-Petch behaviour:
“conventional” grain size strengthening can be expected only to the right of the heavy
line which signifies the limiting condition Dd =)(τ (4. 2)D
Gb
bD
Td ==2τ (4. 3).
Figure 4-3 : The limiting condition is shown as the heavy line where the shear strength τ is plotted
schemetically as a function of grain size D. Hall-Petch behavior can only be found to the right of this
line; abnormal or inverse behavior may result otherwise. The dotted line reflects schematically the
equation 0
lnr
D
D
Gb≈τ (4. 6). [E. Arzt, Size effects in materials due to microstuctural and
dimensional constraints: A comparative review, 1998, Acta Meteriala]
For Cu, as an example, the critical grain size estimated in this way is about
50nm; this value is in reasonable agreement with experimental results, as shown in
Figure 4-4.
NANOMATERIALS
46
Figure 4-4 : Inverse Hall-Petch behaviour in nanocrystalline Cu (H-H0 denotes the hardness increment,
D the grain size): the classical behaviour breaks down at a grain size of about 50 nm, in agreement with
an estimate based on the loop diameter [equations Dd =)(τ (4. 2) D
Gb
bD
Td ==2τ
(4. 3)].
Replotted after Chokshi et al. xxviii [E. Arzt, Size effects in materials due to microstuctural and
dimensional constraints: A comparative review, 1998, Acta Meteriala]
The plastic behaviour of nanocrystalline materials with grain sizes below the
critical value is not fully clear. It has been argued that because of the viscous
behaviour of amorphous materials (which can be considered the limiting case for grain
refinement) the grain size strengthening effect will have to be reversed once the grain
size D starts to approach the grain-boundary thickness δδδδb. One possible explanation
for such a softening effect comes from a re-consideration of the line tension Td in
equation D
Gb
bD
Td ==2τ (4. 3). The more refined expression
2
1ln4
²
r
rGbTd π
= (4. 4)
contains a lower (r0) and an upper (r1) cut-off distance for the stress field of the
dislocation. In conventional materials r1 generally lies in the micrometer range and
therefore significantly exceeds r0 (for which values between 2 and 10b are commonly
assumed); this justifies replacing the logarithmic term by a constant. However, in
nanocrystalline materials it is reasonable to equate r1 to the grain size, which now
gives r1 ≈≈≈≈ r0 and makes T sensitive to the value of the grain size D. Therefore, we now
have a case in which the characteristic length (d) is a function of the size parameter
(D). The resulting strength increment is given by
NANOMATERIALS
47
02 r
Dn
D
Gb
πτ = (4. 5)
This expression vanishes rapidly as the grain size D approaches the lower cut-
off distance r0. An even more refined expression has been obtained by Scattergood
and Kochxxix.
The dislocation density ρρρρ scales inversely with grain size D, the obstacle
spacing is L~1/√√√√p~√√√√D, which yields
0
lnr
D
D
Gb≈τ (4. 6)
This expression, which is schematically shown as dotted line in Figure 4-3,
reduces correctly to Hall-Petch behavior for D >>r0. It gives a possible interpretation of
grain-boundary softening behavior in nanocrystalline Cu and Pd.
4.2.1.3 Diffusional creep as a size effect
An alternative explanation of grain-boundary softening in very fine-grained materials
can be based on increasing contributions of diffusional creep. Diffusional processes in
a potential gradient [caused in this case by a normal stress gradient,] exhibit a natural
size effect because the length scale affects the magnitude of the gradient.
Figure 4-5 : Diffusional creep is driven by gradients in normal tractions on grain boundaries (a). Fine
arrows delineate the paths for transport of matter. This mechanism ceases to operate (b) once a grain
boundary dislocation loop no longer fits into a grain facet (d > D’). Note the analogy with Figure 4-2 for
lattice dislocations [E. Arzt, Size effects in materials due to microstuctural and dimensional constraints:
A comparative review, 1998, Acta Meteriala]
For maintaining a constant strain rate •ε by diffusion of atoms from grain boundaries
under compression to those under tension, the following shear stress τ is required
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48
Ω=
•
vDC
kTD
1
2ετ (4. 7)
Here Dv is the volume diffusivity, Ω the atomic volume, and C1 a dimensionless
constant of the order of 10. Accounting for grain-boundary diffusion (with diffusivity Db
through a grain boundary with thickness δb) gives
Ω=
•
bb DC
kTD
δετ2
3
(4. 8)
In addition to this, the triple lines in nanocrystalline materials can also act as fast
diffusion paths. Equations Ω
=•
vDC
kTD
1
2ετ (4. 7) andΩ
=•
bb DC
kTD
δετ2
3
(4. 8) reflect grain
size effects which are opposite in direction and far stronger than those of dislocation
plasticity (Hall-Petch effect). They are due to the increase, with finer grain size, in the
volume fraction of ‘disordered' material which can act as short-circuit diffusion path,
and to the higher density of sinks and sources for matter.
It is still a matter of debate whether grain-boundary softening, which has occasionally
been reported for nanocrystalline materials, can be attributed to these effects at room
temperature.
One can note that in very small grains the rate of creep may no longer be controlled
by the diffusion step [as is tacitly assumed in equations Ω
=•
vDC
kTD
1
2ετ (4. 7)
andΩ
=•
bb DC
kTD
δετ2
3
(4. 8)], but by the deposition and removal of atoms at the grain
boundaries. Ashbyxxx have shown that for such ‘interface-controlled’ diffusional creep
the grain size dependence is much weaker
2/1
2
1
4
DDC
kTGb
eff
b
Ω=
•ετ (4. 9)
This result was obtained by modeling the interface reaction as the climb motion of an
array of grain-boundary dislocations. Here Deff is an effective diffusivity, bb the Burgers
vector of a boundary dislocation and C4 another numerical constant. The
NANOMATERIALS
49
D1/2-proportionality, which results from the assumption of a stress-dependent
dislocation density, is in better agreement with the data of Chokshi et al. (Figure 4-4).
However, because of the reduced grain size dependence, an even lower activation
energy (about 40 kJ/mol) for diffusion has to be assumed to predict realistic
deformation rates at room temperature.
Also, the motion of grain boundary dislocations is subject to a similar grain size limit as
for lattice dislocations: models based on their presence must break down once an
average grain facet of diameter D' can no longer accommodate a grain-boundary
dislocation loop [Figure 4-5(b)]. The corresponding limiting condition is, in analogy with
equation D
Gb
bD
Td ==2τ (4. 3), given by
'D
Gbb=τ (4. 10)
The value of bb corresponds to the difference in Burgers vector between two lattice
dislocations and is therefore only a fraction of b. Hence, a stress window will exist in
which plasticity due to lattice dislocations is suppressed or slowed down [at stresses
below that given by equation D
Gb
bD
Td ==2τ (4. 3)], but diffusion creep operates
because grain-boundary dislocations are still present and mobile.
4.2.2 Metal plastic deformation: A comparison betwe en Cu and Ni
nanophase samples
In the previous paragraph we have studied the relation between yield stress and
grain size. Now two questions are still not answered:
1. Whether the grain boundaries in nanocrystalline materials are unusual or
whether they have the short main structure of most grain boundaries found in
conventional polycrystalline materials
2. What is the influence of the grain size on the plastic deformation mechanism
In this paragraph, we will focus on the question of dislocation activity in two different
materials Cu and Ni. Swygenhoven and al.xxxixxxii worked on molecular dynamics
simulations on nanocrystalline Ni and Cu samples in the grain size range 5-12nm.
They studied the interfaces responsible for the plastic deformation, aiming at
NANOMATERIALS
50
providing a structural characterization of them. They also present evidence of two
competing mechanism for plastic deformation:
• At the smallest grain sizes all the deformation is accommodated in the
grain boundaries.
• At larger grain sizes lattice dislocation activity is observed.
For both materials, uniaxial deformation at the smallest loads reveals that Young’s
modulus equals the value for a polycrystalline material when the grain size is 10 nm
or higher. At smaller grains sizes a gradual reduction of the modulus is observed.
At high load, after a transient period following the application of the load, the strain
increases almost linearly with time for all the grain sizes.
Figure 4-6 : Strain rate as a function of mean grai n size for Cu and Ni. [H. Van Swygenhoven, A.
Caro, D. Farkas, A molecular dynamics study of polycrystalline fcc metals at the nanoscale Grain
boundary structure and its influence on plastic deformation, 2001, Materials Science and Engineering
A.]
Figure 4-6 shows the strain rate versus the inverse of the grain size for the Ni and
Cu samples. The strain rates under these conditions are high compared with actual
experimental values, but for these small sample sizes (10–25 nm) any relative
velocity is still four orders of magnitude smaller than the velocity of sound.
At the smallest grain sizes explored the strain rate for a given applied stress
increases with decreasing grain sizes. This behaviour indicates that the Hall–Petch
slope is negative at these very fine grain sizes. An energy balance indicates that at
these sizes the total amount of grain boundary remains constant during deformation.
These observations suggest an important characteristic of plasticity in nanophase
metals under the present conditions: there is no damage accumulation during
deformation, similar to the case of superplasticity. Careful examination of the
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51
samples confirms the absence of intra-grain defects. Swygenhoven and al. [4]
discussed the deformation mechanism in terms of a model based on Grain Boundary
viscosity controlled by a self-diffusion mechanism at the disordered interface,
activated by thermal energy and stress.
When similar load is applied on samples with larger average grain sizes, the
deformation rate is much smaller. These observations indicate a transition to another
deformation mechanism. They analyzed the atomic structure of Ni and Cu samples
with larger grain sizes, deformed at those stress levels that give approximately the
same strain rate as in the sample with the very small grain sizes, and compare the
structures at similar values of plastic deformation. In this way they take into account
the different elastic contribution in very fine-grained samples due to the reduction of
the Young’s modulus.
Figure 4-7 : Slice of the 12 Ni sample, deformed until a plastic deformation level of 1.4%. Open gray
symbols are perfect f.c.c., full gray are good f.c.c., red are h.c.p., green and blue are other 12- and
non-12-coordinated atoms, respectively. [H. Van Swygenhoven, M. Spaczer, A. Caro, Microscopic
description of plasticity in computer generated metallic nanophase samples: A comparison between
Cu and Ni, 1999, Acta Materialia.]
Figure 4-7 shows a section of Ni 12 nm sample, deformed to a total deformation level
of 2.7% which means a plastic deformation level of 1.4% using a tensile load of 2.6
GPa. The stacking fault observed in this section is produced by motion of Shockley
partial dislocations generated and absorbed in opposite grain boundaries. Figure 4-8
shows the Shockley partial in Ni 12 nm just emitted from a triple point.
NANOMATERIALS
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Figure 4-8 : Shockley partial dislocation traveling through a grain in the 12 nm Ni sample. Black atoms
are the hcp atoms forming the intrinsic stacking fault in a (1, 1, −1) plane. [H. Van Swygenhoven, A.
Caro, D. Farkas, A molecular dynamics study of polycrystalline fcc metals at the nanoscale Grain
boundary structure and its influence on plastic deformation, 2001, Materials Science and Engineering
A.]
Its glide plane is (1, 1, −1); the dislocation line at this time is mainly composed of two
segments, aligned approximately along 0,1,1− and 011 directions. The Burgers
vector being a/6 1,2,1− ; two possible additional partials could give perfect a/2 011
or a/2 0,1,1− dislocations. Both segments in the figure are of mixed character. The
dislocation velocity estimated from sections made at different deformation times is 4
A /psec, a tenth of the speed of sound. Due to the orientation of the particular grain
relative to the strain direction, the two slip directions for the perfect dislocations 011
and 0,1,1− have both a Schmid factor of 0.37, which is the largest possible value in
this grain.
Evidences of stacking faults inside the grains in Ni are observed at a slightly higher
grain size (11 nm) compared to Cu (8 nm), probably due to the higher stacking fault
energy. In Cu, Swygenhoven and al. have observed partial dislocations which glide
on slip systems that are not necessarily those favoured by the Schmid factor.
In Ni and Cu a change in deformation mechanism is observed
• At the smallest grain sizes all deformation is accommodated in the grain
boundaries and grain boundary sliding, a process based on mechanical and
NANOMATERIALS
53
thermally activated single atomic jumps, dominates the contribution to
deformation.
• At large grain sizes, a combination of sliding and intra-grain dislocation activity
is observed.
4.2.3 Hardness
Normally, one would expect that the introduction of a very high fraction of boundaries,
which is the case of nanostructured polycrystals, will result in a further increase in
hardness since the boundaries are thought to play the role of barriers to dislocation
motion, thus hindering deformation. The aforementioned experimental findings
prompted various researchers to propose models in terms of viscous flow or grain
boundary sliding, different dislocation mechanism taking place for small grain sizes,
or by rule-of-mixtures approaches.
In this following section, we are going to study a modelxxxiii that obtains the total
hardness involving the grain size, the grain boundary width, the hardness
characteristics of coarse grain counterparts and an extra parameter describing the
arrangement of triple junctions.
In the present formulation the grain boundary phase is considered as having similar
mechanical behaviour to the grain interior phase, rather than being amorphous.
Moreover, the effect of triple grain boundary junctions and the associated
disclinations is considered. Such considerations amount, in effect, into assuming that
the strength of the nanostructured polycrystal does not depend only on their volume
fraction but also scales with the grain dimension in a way similar to the grain interior,
with an enhanced fraction of triple junctions providing a different scenario for the
available acting deformation mechanisms.
The total hardness may be assumed to obey a rule of mixtures relation of the form
H=HG(1-f) + HGBf (4. 11)
Where H, HG and HGB are the total, grain interior and grain boundary hardness
respectively, while f denotes the volume fraction of the grain boundaries region.
By assuming a tetrakaidecahedra grain configuration, H=HG(1-f) + HGBf (4. 11) can
be rewritten in the form
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54
GBG Hd
ddH
d
dH
3
33
3
3 )()( δδ −−+−= (4. 12)
Where δ is the grain boundary width and d is the grain size.
The basic assumption of the present model is that a Hall-Petch relationship describes
the hardness of both phases, i.e.
HG=H0G + kGd-1/2
HGB=HOGB + kGBd-1/2 (4. 13)
Where the coefficients have their usual meaning though now assigned to each region
separately. For the dependence of hardness on the grain size d, a power of -1/2 is
used. It may be noted that HG is the hardness contribution of the bulk due to the
existence of the grain boundaries necessary to form dislocation pile-ups, while HGB is
an analogous contribution of the boundaries due to the existence of triple junctions
which act as sources and sinks of dislocations and may be thought to possess the
properties of disclination dipoles.
To find kGB the Hall-Petch approach (see paragraph 4.2.1.2) is used and we can find
=
0
1
0
1
ln
ln
r
rr
r
kkc
GGB (4. 14)
Where r1c is the critical transition grain size. For grain sizes D<Dc the movement of
the glide dislocation is no longer prohibited by the dislocation line, a situation known
as “Orowan bypassing”.
Thus, the relation predicting the total hardness of the nanocrystalline materials, when
taking into account the special contribution of the boundaries, is written in its final
form as
2/1
0
1
0
1
3
33
3
3
ln
ln)()( −
−−+−+= d
r
rr
r
d
dd
d
dkHH
cGOG
δδ (4. 15)
NANOMATERIALS
55
It is worth noting that equation 2/1
0
1
0
1
3
33
3
3
ln
ln)()( −
−−+−+= d
r
rr
r
d
dd
d
dkHH
cGOG
δδ
(4. 15) bears some resemblance with a similar expression obtain for the yield
stress by using interphase disclination arguments.
−≈
H
lG
αθ
υπωσ log
)1(2 (4. 16)
Where G is the shear modulus, υ the Poisson’s ratio, ω the disclination power, Hα
the boundary length and θ a numerical factor estimated close to 0.1.
The hardness dependence on grain size for metals are given in Figure 4-9 a) and b)
and for intermetallics in Figure 4-9 c) and d).
Figure 4-9 : Hardness of coarse grained and nanocry stalline metals [(a),(b)] and intermetallics
NANOMATERIALS
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[(c),(d)]. [D.A. Konstantinidis, E.C. Aifantis, On the ‘anomalous’ hardness of nanocrystalline materials,
1998, Nanostructured Materials]
Further development of the model may be possible through the incorporation of a
different dislocation pile-up mechanism which may be active for very small grain
sizes. Another interesting issue is to study the influence of phase boundaries in
addition to grain boundaries so as to elaborate better the differences between pure
metals and intermetalics.
Finally, it is noted that the elastic properties of the grain boundaries could also be
considered different than those of the bulk due to the presence of the free boundary
volume. This free volume may be viewed as a kind of nanoporosity that could have
an effect on the properties of the grain boundaries and of the nanocrystalline material
as a whole.
4.3 Ceramics / Nanocomposites
4.3.1 Density
Nanocomposites with considerably improved mechanical properties have, to date,
only been achieved in hot-pressed materials. The strength depends on relative
density and decreased with increasing porosity. This might explain why Zhao et
al.xxxiv found almost no strength increase in presureless sintered Al2O3/SiC 5vol%
nanocomposites with a density of 98.3% whereas in hot-pressed materials with a
density of 99.9% considerable increase in strength was observed.
4.3.2 Fracture strength
Figure 4-10 shows a comparison of strength and fracture toughness of alumina-
based nanocomposites at room temperature as a function of SiC content as reported
by several groups. The strength of the monolithic alumina used as a reference varied
from 350 to 560 MPa. Therefore, a given relative improvement in strength of a
ceramic nanocomposite can be misleading. Following Niihara’s original work xxxv-, xxxvi , xxxvii , xxxviii , xxxix , xl , xli , xlii the addition of only 5 vol% nanosize SiC increased the
strength to 1050 MPa. A further increase of SiC content lowers the strength to a
constant value of approximately 800 MPa. However, Niihara explains the decrease of
strength value for higher SiC contents due to agglomeration problems.
NANOMATERIALS
57
3
1000
800
600
400
5
4
0 10 20 30
SiC content (vol %)
Str
engt
h (
MP
a)To
ugh
ness
(M
Pax
m1
/2)
Figure 4-10 : Strength and toughness of Al 2O3/SiC nanocomposites as a function of SiC
content (adapted from various works): (•) Niihara & Nakahiraxxxix measured by three-point bend test
and Vickers indentation; ( ) Borsa et al.xliii by four-point bend test; (∆) Zhao et al.xxxiv by four-point bend
test and indentation-strength method; (V) Davidge et al. xliv 58 by three-point bend test and notched
beams
Figure 4-11 shows strength and fracture toughness for various Si3N4/SiC
nanocomposites as a function of SiC volume fraction. All data were obtained from
different publications of Niihara and his co-workers. Nanocomposites derived from
the classical powder route are represented by solid symbols and nanocomposites
fabricated from an amorphous Si-C-N powder by open symbols. With few exceptions,
all data point follows a defined trend. The toughness increased from approximately
5MPa√m for mopnolithic Si3N4 to 6 MPa√m for manocomposites with ≥ 10 vol% SiC.
This increase is accompanied by a modest increase in strength from 900 MPa to
1100 MPa.
NANOMATERIALS
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Figure 4-11 : Strength and fracture toughness for v arious Si 3N4/SiC nanocomposites as a
function of SiC volume fraction
4.3.3 Strengthening and Toughening Mechanisms
The evidence is that Ai2O3/SiC nanocomposites show an explicit increase in strength
accompanied by a modest increase in toughness. Furthermore, grain boundaries are
strengthened in nanocomposites as manifested by the transcrystalline fracture mode
as well as by the increased resistance to wear and creep.
4.3.3.1 Critical flaw size reduction (c-mechanism): Zener grain size boundary
pinning
One of the main features of nanocomposites is that matrix becomes refined on
adding nanosize SiC. Following the Hall-Petch relation a refinement of the grain size
leads to higher strength. Furthermost, abnormal grain growth is reduced in
nanocomposites, thus leading to a narrower grain size distribution. The effect of grain
boundary pinning by small inclusions is described by Smith xlv 107 after a semi
quantitative approach by Zener:
NANOMATERIALS
59
3
4 f
rR
V≅ (4. 17)
Where R is the average matrix grain boundary radius of curvature, depending on the
radius (r) and the volume fraction (Vf) of spherical inclusions. Equation 3
4 f
rR
V≅
(4. 17) is
often given in the form
D∝r/Vf (4. 18)
Where D is the average matrix grain diameter.
Figure 4-12 : Experimental data where R is plotted as a function of 1/ Vf.
In Figure 4-12 the Zener model (equation 3
4 f
rR
V≅ (4. 17)) is compared with
experimental data where R is plotted as a function of 1/Vf. the radius of SiC
NANOMATERIALS
60
nanoparticles used in the calculation was taken to be 150 nm, which is typical for the
materials studied. The model agrees well with the experimental data but large SiC
volume fractions. However, one has to consider that SiC particles are not completely
inert during sintering.
It has been found that α-SiC particles change their morphology form as received
irregular fragments to spherical after sintering. Furthermore, for high SiC volume
fractions SiC particles can form agglomerations or sinter together forming large
particles. This has to be taken in account if better agreement is to be obtained
between the model and experimental results at high SiC volume fraction because
equation 3
4 f
rR
V≅ (4. 17) predicts larger matrix grains for larger SiC particle
sizes.
However, the reduction of the matrix grain size is obvious when taken into
account that monolithic alumina sintered under the same conditions possess a much
larger grain size of about 20 µm.
4.3.4 Reduction in processing flaw size
A further explanation for the increased strength of the nanocomposites is a
reduction in the size of processing flaws. Fractographical studies on broken four-point
bend test beams have shown that the strength-determining processing flaws change
in size and in morphology from large-volume pores in alumina to crack-like flaws due
to SiC agglomerations in Al2O3/SiC nanocomposites.xlvi 36 The different processing
flaw type results from the specific nanocomposite processing rather than from an
intrinsic nanocomposite effect. In Al2O3/SiC nanocomposite powder mixtures, hard
SiC agglomerates. The latter commonly cause large processing flaws such as voids
in alumina ceramics.
4.3.5 Crack healing (annealing treatment)
Zhao et al.xxxiv 55 suggest that SiC particles only indirectly influence the strength by
enabling the compressive stress induced by the grinding process to be retained in the
surface region of the test specimens. Another theory is that cracks in nanocomposites
can heal during annealing. Alumina and Al2O3/SiC nanocomposites are indented with
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a Vicker’s piramid to generate radial cracks. After annealing at 1300 ο C in Ar for 2 h
the material behave completely differently.xlvii 110 Whereas cracks in alumina grow,
cracks in nanocomposites close, thus explaining the strength increase of annealing
nanocomposites.
Possible reasons for the crack healing have not yet found and more systematic work
is needed. However, by taking into account internal stresses, one has to distinguish
between the stresses introduced by the Vicker’s indentation (comparable to
machining-introduced stresses), stresses introduced by the thermal expansion
mismatch of Al2O3 and SiC and, finally, stresses due to the thermal anisotropy of
alumina grains. Fang et al.xlvii 110 observed that residual stresses introduced by
Vicker’s indentation fully relax in alumina after an annealing procedure whereas
compressive stresses are still present in nanocomposites.
4.3.6 Toughening ( K-mechanisms)
4.3.6.1 R-curve effects
Concerning R – curve effects one has to distinguish between mechanisms acting on
the crack wedge behind the crack tip and mechanism acting directly at or in front of
the crack tip. It is well known that manolithic alumina ceramics exhibits R – curve
behaviour due to crack bridging mechanisms. The fracture mode is intercrystalline
with partially connected grains acting as ligaments.xlviii 111 In nanocomposites any
toughening effects acting on the crack wedge behind the crack tip are unlikely
because of the lack of bridging elements. This is supported by the transgranular
fracture mode. Therefore, only mechanisms acting directly at or in front of the crack
tip can be assumed to be applicable to nanocomposites. Such micro-toughening
mechanisms do not necessarily lead to an increase in the toughness plateau value
but they can result in a steep rise of the R – curve for very short crack lengths. This
would explain a higher strength, as schematically shown in Figure 4-13 where the
crack resistance is plotted as a function of crack length. The slope of the tangents on
the R – curves represents the strength. xlviii 111 By assuming the same initial flaw sizes
and plateau toughness values, a higher strength can be achieved for a sharply rising
R-curve.
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Figure 4-13 : Schematical dependence of the toughne ss.
4.3.6.2 Crack Deflection
The interactions of a crack front with second-phase inclusions, such as spherical
particles in nanocomposites, depend on the differences in the thermoelastic
properties of the matrix and inclusions. If there are no differences, the planar crack
front will not be influenced.xlix 112 Following Faber and Evans,l 113 in the case of such
differences, the crack front will be deflected from planarity by a single particle or
twisted between two neighbouring particles. The toughening of the composite is a
result of diminishing the stress intensity directly at the tip of the deflected crack. The
extend of toughening increase can be obtained by calculating the local stress
intensities at the crack front. The toughness increase depends on the shape, volume
fraction and interparticle spacing of the reinforcement. As discussed later, crack
deflection can also explain the change in fracture mode.
Niihara has proposed a toughening effect due to crack deflection caused by
compressive residual stresses around the SiC particles.xxxv One requirement of the
crack deflection theory is strong interface between the SiC particles and the matrix.
Investigations of SiC/Al2O3 interfaces revealed that the boundary is free of second
phases. 114.28 An alignment of the SiC particles with the Al2O3 matrix can be
observed in some cases.li 114 These examinations suggest strong adhesion between
the SiC particles and the matrix. Concerning SiC/Al2O3 interfaces at SiC particles
located at Al2O3/Al2O3 grain boundaries, Jiao et al.li 114 have estimated that the
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interfacial fracture energy between SiC and alumina is twice that of the alumina grain
boundary fracture energy.
4.3.7 Grain boundary strengthening mechanisms
Several authors lii 35, 55, 59 have found that grain pullout is significantly reduced in
Al2O3/SiC nanocomposites during polishing, machining or abrasive wear.
Furthermore, the fracture mode changes to transcrystalline. All these observations
indicate that matrix grain boundaries are strengthened in nanocomposites. This effect
is probably the most important difference between nanocomposites and the pure
matrix. Conceivable reasons for the grain boundary strengthening are:
(1) deflection of a crack running along a grain boundary at an SiC particle into the
grain.
(2) strengthening of the grain boundaries due to local internal stresses.
Therefore, internal stresses are discussed in detail in the following sections.
4.3.8 Thermal expansion mismatch (Selsing model)
Since the microstructures of nanocomposite ceramics are formed during sintering at
high temperatures, differences in the thermal expansion coefficient of the matrix
(αmatrix) and of the nano-particles (αparticle) cause strains during cooling. These thermal
expansion misfit strains,⟨α∗⟩, can be calculate by an integration over temperature.
The upper limit is taken as the temperature below which plastic deformation is
insignificant (Tplastic) and the lower limit is the room temperature.
( )0
* plasticT
T particle matrix dTα α α= −∫ (4. 19)
The thermal expansion misfit stress, σσσσT, inside a single spherical inclusion in infinite
matrix can be described by the following expression after Selsing:liii 118
*
1 212
Tpm
m p
vv
E E
ασ = −+ +
(4. 20)
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E and υ are Young’s modulus and Poisson’s ratio of the matrix (m) and the particles
(p). The tangential, σT, and the radial, σT, stress distributions in the matrix around the
particle are given by:
3 3
2T
T Tr T
r r
x x
σσ σ σ = − = (4. 21)
Where r denotes the radius of the inclusion and x is the radial distance from the
inclusion surface. Assuming Tplastic as 1500oC and the room temperature
thermoelastic data given in Table 4-1, equation (
*
1 212
Tpm
m p
vv
E E
ασ = −+ +
(4. 20) leads
to a compressive hydrostatic stress inside the SiC particle of 2.0 GPa for the
Al2O3/SiC and 4.3 GPa for the MgO/SiC system. Tensile hydrostatic stresses of
500MPa and 600MPa can be calculated for the Si3N4/SiC and the Al2O3/TiN systems,
respectively. Evidently, a change in fracture mode occurs only in systems with high
compressive stresses within the nano-particles such as Al2O3/SiC and MgO/SiC.
Table 4-1 : Thermoelastical data for matrix and nanophase
E[Mpa] ν α[10-6 K-1]
Al2O3 400 0.23 8.3
Si3N4 300 0.27 3.2
MgO 300 0.18 14
TiN ~470 ~0.25 9.4
SiC 480 0.17 4.4
4.3.9 Average internal stresses
A model developed by Taya et al. liv 119 incorporates the influence of the particle
volume fraction on the average residual microstress in the matrix for a composite
material with spherical inclusions showing a thermal expansion mismatch. Their
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model is based on Eshelby’s equivalent inclusions approach referring to the inelastic
eigenstrains. The average residual microstresses inside the matrix and the particles,
⟨σm⟩ and ⟨σp⟩, are given by the following expressions:
*
*
2(1 )
(1 )( 2)(1 ) 3 (1 )
2
(1 )( 2)(1 ) 3 (1 )
fp
m f m f m
fm
m f m f m
V
E V v V v
V
E V v V v
β ασβ β
β ασβ β
− −=
− + + + −
=− + + + −
(4. 22)
where 1
.1 2
pm
p m
Ev
v Eβ +=
−
In the case of αm>αp, the average thermal stresses are compressive inside the
particles and tensile in the matrix. For Vf=0, equation
(
*
*
2(1 )
(1 )( 2)(1 ) 3 (1 )
2
(1 )( 2)(1 ) 3 (1 )
fp
m f m f m
fm
m f m f m
V
E V v V v
V
E V v V v
β ασβ β
β ασβ β
− −=
− + + + −
=− + + + −
(4. 22) provides the same value for the
stresses inside the particles as the Selsing model.liii 118 Another boundary condition is
given by mechanical equilibrium with ⟨σm⟩(1−Vf)+⟨σp⟩Vf=0.
Figure 4-14 shows experimental results together with a prediction using equation
below. A stress-free temperature of 1500oC, below which plastic deformation is
insignificant, is assumed. (By considering that internal stresses at the surface of a
nanocomposite are not relieved after annealing at 1300oC, as observed by Fang et
al.,xlvii 110 the stress-free temperature must be somewhere between 1300 and 1550oC
which is the minimum hot-pressing temperature to get fully dense nanocomposites).
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1000
500
0
-500
-1000
-1500
-2000
-25000.0 0.1 0.2 0.3
Volume fraction V f
Ave
rage
resi
dual
str
ess
[MP
a]
<σm><σp>
Figure 4-14. Average residual microstresses in the matrix ⟨σm⟩ and nano inclusions ⟨σp⟩ for Al2P3/SiC
nanocomposites. ( , ) adapted from Todd et. al; ( , ) Ref. 120)
It has to be emphasised that all theoritical calculations concerning internal stresses
depend strongly on value used for the thermoelastic properties of the matrix and
inclusions.
Levin et al.lv 123 have presented a model for the influence of the SiC particles on the
fracture toughness of nanocomposites. The average tensile stress field in the matrix
due to the thermal expansion mismatch of Al3O2 and SiC reduces the fracture
toughness. As shown in Figure 4-14, the average microstress in the matrix is
approximately 100 MPa for an Al2O3/5 vol% SiC nanocomposites. SiC submicron
particles within the grains strengthen the grain boundaries because of compressive
radial stress components, thus increasing the fracture toughness via a change in the
fracture mode. It has been claimed that a net increase in toughness can only be
achieved for small SiC volume fractions because the strengthening effect of grain
boundaries is high and only in the case are the average internal stresses small. The
maximum increase of fracture toughness at 5 vol% SiC as shown by Niiharaxxxv
agrees with the model proposed by Levin et al.lv 123 and is plausible when taking into
account the two opposite effects of grain boundary strengthening and average tensile
internal stresses.
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4.3.10 Local stress distribution
For discussion of the grain boundary strengthening effect due to internal stresses it is
helpful to review the stress fields around the SiC particles. For the model presented in
this review, a superposition of stresses in an arrangement of several boundaries
including a grain boundary is assumed. Figure 4-15 shows the configuration assuming
nine particles in a cubic-body-centred arrangement with an average nearest particle
spacing equivalent to a SiC volume fraction of 2.5%. One particle is located in the
plane. The remaining particles are the same distance from that plane. Each particles
causes a stress field which can be calculated using the Selsing equation. The total
stress at each point in the x-y-plane highlighted in Figure 4-15 can easily be calculated
assuming a simple superposition. Obviously, the particle at the grain boundary
generates high tensile stresses immediately around it and the other eight particles
nearby to the plane generate compressive stresses up to 120 MPa. SiC particles
within the Al2O3 grain boundaries strengthen the grain boundaries due to radial
compressive stresses. The influence of SiC particles within grain boundaries is still
unclear, but could be expected to generate countervailing tensions across the
boundary.
Figure 4-15 : Configuration for the stress distribu tion model
4.4 Final remarks on strengthening and toughening
mechanisms
Table 4-2 summarises all the mechanisms discussed in the modeling chapter. The
strength increase observed in nanocomposites can be explained by a decrease in
critical flaw size. As shown above, not only in the size of processing flaws decreased
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but also their morphology changes completely. Furthermore, the matrix grain size is
reduced with a narrow size distribution due to the grain boundary pinning by inert SiC
particles. Dislocation network plays only a minor role.
A clear identifications of the toughening mechanism in nanocomposites remains
difficult because, first, the toughness increase is small or even absent and, second,
there is no single persuasive mechanism (Table 4-2). A balance exists between
mechanisms decreasing toughness, i.e. crack deflection or grain boundary
strengthening. All these mechanisms depend highly on specific processing and
microstructural details. However, it appears that grain boundary strengthening due to
local radial compressive stress components around SiC particles or due to a riveting
effect is the most likely nanocomposite effect. The mechanism explains the transition
in fracture mode as well as the improved resistance to wear and machining damage.
Table 4-2 : Summary of strengthening and toughening mechanisms
Mechanism Comment
Zener grain boundary
pinning mechanism (c-
mechanism)
Matrix grain sizes are drastically reduced
(typical for nanocomposites)
Reduction in processing
flaw size (c-mechanism)
Strength increase can be fully explained
by observed change in processing flaw
type (careful processing is very important)
R-curve effects (K-
mechanism)
Steep rising R-curve behaviour is
proposed (no experimental evidence)
Crack Deflection (K-
mechanism)
Cracks seem to be reflected at SiC
particles (importance for toughening is
unclear)
Thermal expansion
mismatch (grain
boundary strengthening)
Fracture mode is changed to
transcrystalline if particle matrixα α< (eg. For
Al2O3/SiC)
Average internal stresses
(grain boundary
Average tensile stresses in matrix if
particle matrixα α< (toughness is reduced)
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strengthening)
Local stress distribution
(grain boundary
strengthening)
Local compressive stresses in matrix
grain boundaries if particle matrixα α< (can
explain change in fracture mode)
4.5 References
R.A. Masumura, P.M. Hazzledine, C.S. Pande, Yield stress of fine grained materials,
1998, Acta Materialia
E. Arzt, Size effects in materials due to microstuctural and dimensional constraints: A
comparative review, 1998, Acta Meteriala
Chokshi, A. H., Rosen, A., Karch, J. and Gleiter, H., Scripta metall., 1989, 23, 1679
H. Van Swygenhoven, A. Caro, D. Farkas, A molecular dynamics study of
polycrystalline fcc metals at the nanoscale Grain boundary structure and its influence
on plastic deformation, 2001, Materials Science and Engineering A.
H. Van Swygenhoven, M. Spaczer, A. Caro, Microscopic description of plasticity in
computer generated metallic nanophase samples: A comparison between Cu and Ni,
1999, Acta Materialia.
D.A. Konstantinidis, E.C. Aifantis, On the ‘anomalous’ hardness of nanocrystalline
materials, 1998, Nanostructured Materials
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70
5 THERMAL CONDUCTIVITY IN NANOSTRUCTURED
MATERIAL
5.1 Thermal conductivity of thermal barrier coating s lvi
5.1.1 Thermal conductivity
The theory of heat conduction by lattice waves in solids can be used to understand
how the conductivity is influenced by lattice defects, grain boundaries and extended
imperfections. At high temperatures one must also consider conduction by
electromagnetic radiation within the solid, which is governed by analogous
considerations. The thermal conductivity by mobile carriers, whether waves or
particles, can be expressed in general in the form
Cvl3
1=κ (5. 23)
where C is their specific heat per unit volume, v their speed and l their mean free
path. If the carriers are waves (lattice waves or electromagnetic waves) ranging over
a spectrum of frequencies f, this must be generalized to
( ) ( )dffvlfC∫=3
1κ (5. 24)
where C( f )df is the contribution to the specific heat per unit volume from waves in
the frequency interval df , v is the group velocity of the waves and l(f) their
attenuation length, usually a function of frequency. Also, fm denotes an effective
upper limit to the spectrum because either v or l become very small for f\ fm. In the
present discussion fm is the upper frequency limit of the acoustic branch, as
discussed in the next section. The energy content of waves consist of quanta,
phonons or photons respectively, and these quanta can also be considered as
particles which carry heat. The present treatment emphasizes the wave nature of the
carriers of heat.
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5.1.2 Lattice waves
Lattice waves are just elastic or ultrasonic waves, but their spectrum extends to very
high frequencies, where their wavelength λ is of atomic dimensions, and where the
relation between f and the wavelength λ is modified. At low frequencies λ=v/f , but at
high frequencies, due to the discrete atomic structure of the lattice, this relation holds
no longer. The displacement field of the wave has the form of a progressive wave, so
that the displacement u(r,t) varies with position r and time t as
( ) ( )tiriqtru ω−∝ .exp, (5. 25)
where the wave-vector q has magnitude 2π/λ and points in the propagation direction,
and ω is the angular frequency ω = 2π f. At high frequencies ω is then no longer
proportional to q. The relation between frequency and wave-vector for a discrete
lattice is shown schematically in Figure 5-1.
Figure 5-1 : Dependence of frequency ω on the wave-vector q for a discrete lattice in the principal
direction. Curve A is the acoustic branch, the only type in simple lattices. Curve O is an optical branch,
present in complex crystals and describing vibration of atoms within molecular units. [P.G. Klemens,
M. Gell, Thermal conductivity of thermal barrier coatings, Materials Science and Engineering A245
(1998), p.p. 143-149]
The slope of the curve defines the group velocity v=dω/dq , that is the speed by
which a wave transports energy.
The Debye theory of lattice vibrations makes the simplification that v is constant, but
that the lattice wave spectrum is terminated at the Debye frequency fD chosen so that
the number of normal modes agrees with the number of atoms. However, when the
basic structural unit contains N atoms of varying mass, the spectrum divides into
acoustic modes, of the progressive wave form, (curve A in Figure 5-1), and into
optical modes (curve O) vibrations of the N atoms in each structural unit relative to
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each other. Only the former, with upper frequency fm = fD / N⅓, have appreciable
group velocity and transport heat. This approximation is used here. In the limit
T > hfm/k, which is the case of interest here, each wave or normal mode has, in
thermal equilibrium, energy kT, and C(T)∝ f². Here k and h are the Boltzmann and
Planck constants, respectively.
The difference between acoustic and optic vibrations is shown schematically in
Figure 5-2.
Figure 5-2 : In crystals composed of molecular units, such as ceramic oxides, the lattice waves are of
two types: acoustic and optical. [P.G. Klemens, M. Gell, Thermal conductivity of thermal barrier
coatings, Materials Science and Engineering A245 (1998), p.p. 143-149]
The optic vibrations are vibrations of the atoms in the lattice; the term ‘optic’ refers to
their strong interaction with infrared electromagnetic waves in the case of ionic solids.
5.1.3 Interaction Processes
In an ideal solid, one that is perfectly harmonic (obeying linear elasticity), structurally
perfect and without external boundaries the lattice waves would be normal modes.
This means that the energy content of each wave would remain constant. Thermal
equilibrium between the waves could not be established. Real solids depart from this
ideal behavior. The lattice forces have anharmonic components, so that there are
departures from linear elasticity, or terms in the strain energy which are cubic in
strain. Also there are always lattice defects and, of course, boundaries. All these
cause some exchange of energy between the waves, which are no longer true
normal modes. The large number of normal modes results in sufficient randomness
in these interactions, so to establish thermal equilibrium. To treat thermal equilibrium
properties it suffices to know that these interactions exist. To study transport
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properties and estimate the attenuation lengths one must consider these interactions
in detail.
The thermal conductivity is limited by various interaction processes, which transfer
energy between the waves and establish thermal equilibrium. These are
schematically shown in Figure 5-3 and correspond to the different types of departures
from ideal behavior.
Figure 5-3 : Interaction processes between lattice waves (a) due to anharmonic interactions, (b)
scattering of waves by lattice defects, and (c) reflection of waves by boundaries and interfaces. [P.G.
Klemens, M. Gell, Thermal conductivity of thermal barrier coatings, Materials Science and Engineering
A245 (1998), p.p. 143-149]
The intrinsic processes exchange energy between triplets of waves (three-phonon
interactions). These satisfy frequency conservation, as in all non-linear processes, as
well as wave-vector selection rules. The resulting intrinsic attenuation length is of the
form
( ) 12, −−= TBfTfli (5. 26)
where mvfaB 3µ∝ , µ being the shear modulus and a3 the volume per atom. It is thus
inversely proportional to the mean square thermal strain kT/µa3. The intrinsic
conductivity becomes then
( )TfvN Di /4
3 23/2 µκ −= (5. 27)
and this describes, reasonably well, the thermal conductivity of structurally perfect
dielectric crystals near and above their Debye temperature hfD/k. Weak binding,
large atomic masses and structural complexity (large N) all tend to reduce the
intrinsic conductivity. Note that if one writes ∫= dffk )(κ , then in the intrinsic case
the integrand )()( flfCk ii ∝ is independent of f, so that equal frequency intervals
make equal contributions to the intrinsic conductivity. This contrasts to the specific
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heat, which for acoustic modes is mainly due to their highest frequencies, since for
f<fm, 2)( ffC ∝ .This is shown schematically in Figure 5-4.
Figure 5-4 : Spectral contributions from lattice waves as function of frequency to the vibrational
specific heat (a), and to the intrinsic thermal conductivity (b). [P.G. Klemens, M. Gell, Thermal
conductivity of thermal barrier coatings, Materials Science and Engineering A245 (1998), p.p. 143-149]
Lattice imperfections further reduce the effective attenuation length. Different
imperfections scatter with different frequency dependence. Point defects scatter as
the fourth power of frequency, with an inverse attenuation length 4)(/ Affll p = ,
where A depends on their nature and concentration. Grain boundaries scatter
independently of frequency with attenuation length L comparable to the grain size.
The inverse attenuation length 1/l(f) is composed of the sum of these processes.
Since point defects and grain boundaries scatter mainly in different frequency ranges,
their conductivity reductions are approximately additive, as shown in Figure 5-5.
Figure 5-5 : Reduction in the thermal conductivity, which is the area under the curve, at high
frequencies due to point defects (δkp), and at low frequencies due to grain boundaries and other
extended imperfections (δkB). [P.G. Klemens, M. Gell, Thermal conductivity of thermal barrier
coatings, Materials Science and Engineering A245 (1998), p.p. 143-149]
And expessed by
PBi δκδκκκ −−= Eq 5.1
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Where δκB and δκP are the reductions due to boundaries and point defects. Point
defects are small regions of volume a3 in which the value of the wave velocity v is
locally changed by δv.
Since the intrinsic attenuation length l i (f, T) has a different dependence on wave
frequency f than that due to point defects, it is convenient to define a frequency f0 at
which the two attenuation lengths are equal, i.e l i (f0, T) = lp (f0) Similarly one can
define a frequency fB such that l i (fB, T) = L, where L is the grain diameter.
Then
−=
=
0
0 arctan1
arctan
f
f
f
f
f
f
f
f
m
miP
B
m
m
BiB
κδκ
κδκ Eq 5.2
Note that the equation for δκP also holds if f0> fm.
For a substitutional atom of mass ∆M+M instead of M the change in velocity is
M
M
vv
∆=2
1δ . Thus one finds
44332
4)(
fvaM
Mc
fl
l
P
−
∆= π Eq 5.3
where c is the defect concentration per atom. Effective values of ∆M/M can be
obtained for other defects, including vacancies. If κi is known, one can deduce l i (f, T)
and thus obtain δκP; this has been done for many systems, e.g. for cubic zirconia.
Cubic zirconia presents a further difficulty: it only exists if it contains stabilizing
solutes together with the accompanying vacancies needed for charge neutrality. Thus
the intrinsic thermal conductivity and l i must first be estimated in terms of its elastic
and anharmonic parameters. The resulting uncertainty of perhaps 30% in κi
translates into a 15% uncertainty in κ. There is rough agreement for various solutes
and concentrations at room temperatures. At high temperatures the theoretical
values are always less than the data. This was shown for several different solutes
and concentrations. This suggests the presence of a radiative component. Values of
δκB can be calculated for various grain diameters, and the resulting lattice thermal
conductivities are shown for zirconia containing 7wt% yttria in Figure 5-6.
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Figure 5-6 : Theoretical reductions in the thermal conductivity of ZrO2 containing 7 wt% Y2O3 due to
point defects and grain boundary scattering for various grain sizes. [P.G. Klemens, M. Gell, Thermal
conductivity of thermal barrier coatings, Materials Science and Engineering A245 (1998), p.p. 143-149]
The reduction is greatest at low temperatures. At high temperatures any radiative
component present would not depend markedly on grain size.
5.2 High-temperature thermal conductivity of porous Al 2O3
nanostructures lvii
5.2.1 Theory
The two mechanisms important for the heat transport in nonmetallic solids at high (of
the order of Debye or higher) temperatures are:
1. the phonon relates to the movement of atoms in the crystalline lattice
2. radiative mechanisms is due to the electron transitions between the energy
levels in these atoms
If we neglect the relation between the energy levels and spacing between the atoms,
then both mechanisms can be considered independently. This assumption seems to
be reasonable, if the amplitude of the atomic oscillation is considerably lower than the
lattice constant. We can then write
rph κκκ += Eq 5.4
where κ is the thermal conductivity, κph and κr are its phonon and radiative
components, and consider each of them separately.
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77
5.2.1.1 Phonon contribution
To investigate the high temperature phonon transport in nanostructures we assume
that two mechanisms are dominant:
1. scattering at the grain boundaries
2. the phononphonon interaction.
At the same time, for the materials whose thermal conductivity is considerably
reduced due to nanostructuring the phonon scattering at the grain boundaries is of
most importance. This scattering determines the phonon mean free path that
becomes now comparable with the grain size and is independent of the phonon
wavelength. This means that the phonons whose wavelength is of about the lattice
constant determine the thermal conductivity at high temperatures. This distinguishes
nanostructured materials from single crystals or amorphous ones where the phonons
of different wavelength are of the same importance for TC (thermal conductivity).
We adopt the following model. We start with the phonon states that initially were
localized at the individual grains of the nanostructure and then consider their
transport as the phonon hopping between the grains. This allows us to reduce the
problem of the phonon transport to the problem of electrical conductivity of a net of
arbitrary resistors Rij, which correspond to relevant grain boundaries (between the
i-th and j-th grains). This yields the following expression for TC:
( )∫
Φ=θ
πσκ
/1
02
2
)(3
dxxBda
STZtkBph
ℏ Eq 5.5
where
Dx
x
T
Tx
e
exxB =
−−
θθ
θ ,1
)1(2
9)(
2
2
44 Eq 5.6
T is the temperature, TD is the Debye temperature, ћ and kB are the Planck and
Boltzmann constants, a is the lattice constant, d is the grain size, S is the main area
of the intergrain boundary (the “neck” size), and Z is the coordination number.
Deviation of this parameters from its mean value is determined with the parameter of
disorder Φ. It has been estimated as
1221
−
+=Φ
Z
σ Eq 5.7
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78
σ is the mean relative deviation of S . ( )∫
Φ=θ
πσκ
/1
02
2
)(3
dxxBda
STZtkBph
ℏ Eq 5.5 results in
the temperaBture dependence of TC that increases at T<TD, and becomes constant
at T>TD. Thus, for T>TD ( )∫
Φ=θ
πσκ
/1
02
2
)(3
dxxBda
STZtkBph
ℏ Eq 5.5
yields
( )da
SZtTk DBph 2
2
20 ℏπσκ Φ= Eq 5.8
The main difference between ( )∫
Φ=θ
πσκ
/1
02
2
)(3
dxxBda
STZtkBph
ℏ Eq 5.5 from the one
derived in Ref. lviii arises from the factor Z/3π. Indeed, in the absence of porosity the
value of Z S is the total surface area of the grain and therefore it is a constant. In
these experiments, however, it is not so because of complexity of the grain shape
and porosity. The factor Z/3π is valid if Z>>1. In particular, for Z~12.
To consider the phonon-phonon scattering we suppose that some additional resistors
(the value of which is r i = κi d, where κi is the bulk part of TC) are series connected to
the boundary ones (Rij). This leads to the following expression for TC:
∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9
Here κi = κi(T) is the thermal conductivity of the single crystal; an influence of
impurities and other bulk mechanisms of phonon scattering can also be included in
this value. It is important that all parameters of
∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 except t can be measured
experimentally. The t value describes interface transparency for the phonon hopping.
To estimate it, the equations of movement for each atom at the interface have to be
solved. We will not carry out this procedure, but we will consider the t value as an
adjustable parameter. It depends only on the interface structure and is independent
of the grain size and porosity. So, it should be similar for nanostructures prepared
using the same technology. This fact will be tested experimentally in this paper.
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5.2.1.2 Radiative part of thermal conductivity
A radiative contribution to the thermal conductivity arises from photon emission
caused by electron transport. These photons can then be absorbed by the electrons.
If the photon mean free path is considerably less than the size of the specimen, then
the photon gas achieves a local thermodynamic equilibrium with the phonons. This
allows us to consider the energy transport that is accompanied by the photons as a
radiative component of thermal conductivity and estimate it as
( ) ( )∫∞
=03
ωωωε
κ dlCc
Vr Eq 5.10
where c, ω, and l(ω) are the photon speed, frequency, and mean free path, ε is the
dielectric permittivity,
( ) ωωπ
ωω ω deTc
dCTkV
B
−∂∂=
1/
3
32 ℏ
ℏ Eq 5.11
is the heat capacity of the photons of the frequency ω.
Let us estimate the mean free path of the thermal photon ћω~kBT in a perfect
α−Al2O3 single crystal.
( )*/2
~2 2 meN
ccl
eππωσω = Eq 5.12
where */2 meN e τσ = is the electric conductivity, e and m* are the charge and
effective mass of the electron, τ is the electron relaxation time; at high temperatures it
can be estimated as τ = ћ/(kBT), so that
Tkm
eN
B
e
*
2ℏ
=σ Eq 5.13
Ne is the electron density. In a perfect crystal it can be estimated as
−
=
Tk
ETkmN
B
gBe 2
exp)2(
*22
2/3
2ℏπ
π Eq 5.14
where for the band gap and effective mass values we have 7.8≈gE eV, emm 35.0* ≈
(e and me are the charge and mass of the free electron). Then
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80
∫∞
−=
0 2
4
232
34
)1(*/23x
e
Br
ex
dxx
meNc
Tk
ππκ
ℏ Eq 5.15
Using Tkm
eN
B
e
*
2ℏ
=σ Eq 5.13 and ∫∞
−=
0 2
4
232
34
)1(*/23x
e
Br
ex
dxx
meNc
Tk
ππκ
ℏ Eq 5.15 for
T=2000 K yields )/(11 mKWr ≈κ , and cml 14≈ . This means that a perfect Al2O3
crystal shows no resistance to the photon transport. The resistance could arise from
the impurities or intrinsic defects (the energy level of the defect closest to the top of
the valence band is Ei=0.3 eV). The latter can be responsible for heat resistance if
their density is rather large, so that l becomes smaller than thickness of the specimen
L. The effect can be estimated by ∫∞
−=
0 2
4
232
34
)1(*/23x
e
Br
ex
dxx
meNc
Tk
ππκ
ℏ Eq 5.15,
where Ne~Ni exp(−Ei/kBT) and m* is effective mass of the hole, Ni is the defect
density. However, this yields κr that is usually small, if l<<L .
It is easy to understand the reason. The electron density is low at low temperature.
This means long mean free paths both of electrons and photons [Tkm
eN
B
e
*
2ℏ
=σ Eq
5.13]. Increase of the temperature leads to increase of the photon part of the heat
capacity (which is proportional to T3), but also to rapid decrease of the photon mean
free path (which exponentially depends on temperature).
Appreciable contribution to the radiative component could originate from the electron
levels localized less than 0.1 eV from a band edge. They can be the electrons from
impurity or surface levels segregated from the conductive or valence bands
(electrons from dangling bonds). In the latter case, the density of the levels can be
roughly estimated as Da
pN i 2
24
π≤ , where D is the pore size. Such levels become
ionized at T≥1000 °K. Then N e→Ni becomes independent of temperature, so that κr
increases as T3.
Crystal disorder decreases the phonon mean free path and so decreases the
radiative component. To estimate this effect we take into account that wavelength of
the heat photons at T=1000 K is of about 5000 nm, so that it significantly exceeds the
grain size. This allows us to use a continuum approximation: We consider the
electromagnetic wave propagation in the media with the position dependant dielectric
permittivity. In the first approximation (with respect to ratio of the grain size to the
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81
photon wavelength) this effectively change the permittivity, however, in the second
approximation this leads to decay of the electromagnetic waves due to the structural
disorder. The decay length considerably exceeds not only the grain size, but also the
wavelength.
To consider the propagation of electromagnetic waves in a granular material we find
the solution of the Maxwell equations in the media with the dielectric
permittivity )()( rr ξεε += , where ε is the average permittivity, which is independent of
r, and the random function ξ(r) is its deviation. This allows us to express the electric
and magnetic fields of the waves via the correlation function )()'()'( rrrrW ξξ=− ,
which we assume to be equal to )/'exp()'( 2 ℜ−−×=− rrrrW ϑ , whereϑ is the mean
square deviation of the permittivity and d≈ℜ is the correlation length. Unlike the
commonly used Gauss correlation function, this one is more appropriate for the
steplike random values (ε(r) = ε1 = ε in the grains or ε(r) = ε2 = 1 in the pores).
We found that disorder leads to the effective correction to the permittivity
εϑεε 3/~ 2−= and attenuation of the electromagnetic waves 34ℜ∝ ωγ . The effective
mean free path is
4323
)(
1
)(
11−
−−
ℜ=+=clll SCeff
ωϑωω
where 1)( −= γωSCl Eq 5.16
Here, 22 )1)(1( −−= εϑ pp , and p is the porosity
The effective mean free path )(ωeffl substitutes )(ωl in ( ) ( )∫∞
=03
ωωωε
κ dlCc
Vr Eq
5.10. It is apparent this is approximately the smallest value between 2
1
)(−
∝ ωωl and
4)( −= ωωSCl . In the perfect crystal )()( ωω SCll << , and the value of ℏ
TkB=ω is
effectively the upper limit for the integral ( ) ( )∫∞
=03
ωωωε
κ dlCc
Vr Eq 5.10. In the
real crystal the solution ω0 of the equation )()( 00 ωω SCll = can be either larger or
smaller thanℏ
TkB . In the first case the disorder has no appreciable influence on κr, in
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82
the second one the value of ℏ
TkB<0ω becomes the effective upper limit of
( ) ( )∫∞
=03
ωωωε
κ dlCc
Vr Eq 5.10; this decreases the radiative component κr.
5.2.2 Experiment
5.2.2.1 Sample preparation
Commercial high purity transition alumina powder (Baikowski CR125; samples
FD1/1–FD1/3) as well as MgO doped (0.5 wt%) γ-alumina powder (Baikowski
CR11325; samples FD2/1–FD2/3) were used. The powders were attrition milled for
3h with dispersant addition, freeze-dried and cold isostatically pressed. The samples
were consolidated by spark plasma sintering using a heating rate of 200 °C/min up to
the sintering temperatures of 1400 °C and holding t imes ranging from 0 to 5 min. The
pressure was applied from the beginning of the sintering cycle and released during
cooling. These different sintering conditions were chosen to produce different
microstructures (grain size, porosity) (Table 5-1).
Table 5-1 : Experimental data used in calculation. [L. Braginsky, V. Shklover, H. Hofmann, P.
Bowen, High-temperature thermal conductivity of porous Al2O3 nanostructures, Physical review B 70,
134201 (2004)]
5.2.2.2 Characterization of the samples
The porosity, grain size as well as the contact area between the grains were
determined using image analysis methods applied to SEM pictures of polished and
thermally etched samples. Additionally TEM studies of the samples FD1/1, FD1/2,
and FD1/3 sintered at different conditions (Figure 5-7) were performed with TEM
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Philips CM300, operating at 300 kV. The samples were coated with carbon layer to
avoid charging.
Figure 5-7 : TEM images. (a) Smaller grain size, lower density sample FD1/2, scale bar 200 nm. (b)
Larger grain size, higher density sample FD1/3, scale bar 200 nm. (c) Two sintered grains in powder
FD1/2. The interface (indicated with an arrow) has a thickness, 1 nm. Both grains are crystalline, but
oriented differently and the crystalline fringes can be easily seen only on one of them. Scale bar 2 nm.
[L. Braginsky, V. Shklover, H. Hofmann, P. Bowen, High-temperature thermal conductivity of porous
Al2O3 nanostructures, Physical review B 70, 134201 (2004)]
5.2.2.3 Thermal conductivity measurements
The laser flash method was used for measurements of heat diffusivity in a “TC-
3000H/L SINKU-RIKO” device as described in Refslix,lx.
5.2.3 Results and discussion
The results of the TEM study of our samples are presented in Figure 5-7, where the
structure of the interfaces between the α−Al2O3 grains is shown. This is important,
because it influences the phonon propagation in nanopowder. The TEM study shows,
that
1. The interfaces between the a−Al2O3 grains are very thin (~1 nm, Figure 5-7)
2. No epitaxial relation between the neighboring crystalline α−Al2O3 grains was
observed.
Table 5-1 presents results of the structure measurements of our specimens together
with the values of the parameter t for each of them. These values of t ensure the best
fit of ∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 with measured
temperature dependence of TC (Figure 5-9). The value of t does not vary significantly
for the specimens of each set, but change approximately two times for the specimens
of different sets. This means that the parameter t depends on the intergrain interface
structure, but it is independent of the grain size and porosity.
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To understand this, we recall that ~0.5 wt % of MgO was added during the
preparation of FD2/1–FD2/3. This amount is much larger than solubility of MgO in
alumina, therefore most of the added MgO should be located at the grain boundaries
affecting the t value.
Figure 5-8 : (Color online) Temperature dependence of TC of the sample FD1/2: measured (dots) and
calculated (solid line). Different types of the dots label the different series of the measurements.
Dashed line indicates TC of the bulk Al2O3 single crystal. [L. Braginsky, V. Shklover, H. Hofmann, P.
Bowen, High-temperature thermal conductivity of porous Al2O3 nanostructures, Physical review B 70,
134201 (2004)]
Figure 5-8 presents results of our TC measurements for the FD1/2 specimen along
with temperature dependence of Al2O3 thermal conductivity in the bulk that was
measured in Reflxi. It is apparent that our experimental dots do not show significant
temperature dependence. In addition, TC of our structures is much less than that of
the bulk single crystal; this confirms the approximations of our model. Indeed, as a
result of nanostructuring the mean free path of phonons is determined by the grain
boundaries, and it is independent of the phonon frequency and the temperature. This
is the main reason why the short wavelength phonons, which prevail at high
temperatures, are mostly responsible for the heat transport. In the single crystals, on
the contrary, the phonon mean free path, which is determined by the phonon-phonon
interaction, decreases with the frequency as 2−∝ ωl , so that the phonons of any
frequency are of equal importance for TC. This enables one to estimate roughly the t
value from ( )
da
SZtTk DBph 2
2
20 ℏπσκ Φ
= Eq 5.8. The results are close to that of Table 5-1.
Measured values of thermal conductivity as a function of temperature are presented
in Figure 5-9.
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Figure 5-9 : (Color online) Temperature dependence of TC of the smaples: measured (dots) and
calculated (solid line). Different types of the dots indicate the different series of the measurements. [L.
Braginsky, V. Shklover, H. Hofmann, P. Bowen, High-temperature thermal conductivity of porous
Al2O3 nanostructures, Physical review B 70, 134201 (2004)]
We see that theoretical curves as a rule demonstrate more rapid decrease with
temperature than observed. This means that the expression
∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 overestimates the influence of
the phonon-phonon interaction. Indeed, when obtaining
∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 we had assumed this
interaction as independent of scattering at the grain boundaries; we suppose 1/l =
1/lB + 1/lph, where l is the phonon mean free path, lB and lph are the phonon mean
free paths relative to scattering at the grain boundaries and the phonon-phonon
interaction, respectively. Actually, we have to take into account only the interactions
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86
that affect the phonon hopping from one grain into another. Since the in-grain
phonon-phonon scattering affects only the density of the phonon states inside the
grain, but it does not change the thermal conductivity.
The radiative component alone causes an increase of TC at high temperatures. This
follows from ∫∞
−=
0 2
4
232
34
)1(*/23x
e
Br
ex
dxx
meNc
Tk
ππκ
ℏ Eq 5.15, if Ne is independent of
temperature. This is of particular importance for specimens of low porosity due to an
increase of the effective photon mean free path [4
323)(
1
)(
11−
−−
ℜ=+=clll SCeff
ωϑωω
where 1)( −= γωSCl Eq 5.16]. We have observed some increase of TC for the
specimen FD1/3; however, its value is comparable with the experimental errors. The
grain size dependence of TC following from
∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 is close to that estimated with
the Kapitza model.
dGk
i
iph κ
κκ+
=1
Eq 5.17
Where Gk is the Kapitza conductance. Indeed,
dGk
i
iph κ
κκ+
=1
Eq 5.17 results
from ∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9, if we substitute x by its
average value x . Thus,
( )22
2
3
)(
da
xBSZtTkG DB
kℏπ
σΦ= Eq 5.18
In particular, at low temperatures when κi>>κph this expression accepts the form
( )∫
Φ=
θ
πσ /1
022
2
)(3
dxxBda
SZtTkG DB
kℏ
Eq 5.19
Apparently Gk is independent of the grain size, if 2dS ∝ ; however, it depends on
temperature dependence of the factor ∫θ/1
0)( dxxB . This allows us to interpret our
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phenomenological parameter kGt ∝ as the transparency of a grain boundary for the
short wavelength phonons.
Strictly speaking, the average value of x resulting from the integrand
( ∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9) depends on d. Therefore,
B( x ) and so the Kapitza conductance Gk also depend on d. This dependence could
be significant, if κi ≤ κph at low temperatures T≤TD, when the simple expression
(
dGk
i
iph κ
κκ+
=1
Eq 5.17) could be impracticable. The parameter S determines
the influence of the small size pores. Indeed, this value should be equal to 2d in the
absence of pores. If we assume a cubic shape of grains and suppose the pores lie
along the cube edges then for the porosity we obtain
−−
−=d
xd
d
xdp
9
41
4
32π
Eq 5.20
Where x is the “neck” size ( 2xS = ). For the model of spherical grains such an
estimation has been carried outlxii. The main deviation of both estimations is due to
the large size pores, i.e., the pores of about or greater than the grain size. To
estimate their influence, we have to remove some resistors from our net, so that their
resistivity becomes equal to infinity; this means the absence of the neighbours for
some grains due to the large scale porosity. This factor can be related to the
cohesion C of each sample, which can be measured experimentally. Cohesion is an
average ratio of the number of boundaries grain/pore to the number boundaries
grain/grain along some straight line in the TEM pattern. Assuming a homogeneous
distribution of the pores, we can estimate the density of the nonremoved resistors in
the net as
C
C
−=
2η Eq 5.21
Then the expression for TC should be multiplied by η, if η ≈ 1. In the case of large
porosity we have to take into account the problem of “dead ends.” In the more
general case the problem has to be considered with the percolation technique. The
resistivity of the net has an increase (and so the TC of our structure has a decrease)
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at η ≈ 3/(2Z), then the expression for TC should be multiplied by (η ≈ 3/(2Z))1.6. At
smaller values of η the short-wavelength phonons becomes localized, and heat
transport would occur due to long-wavelength phonons only. Such a situation cannot
be realized in porous nanostructures, but it seems to be the case in composite
structures where the nanostructure is embedded into some amorphous media.
Our model ∫ Φ+Φ
=−
θ
κπκκ
/1
0 21 )(3
)(dx
SZtxBTkdak
SZtxBTk
DBiB
iBph
ℏ Eq 5.9 considers only the
short-wavelength phonon transport. Nevertheless, the long-wavelength phonons, i.e.,
phonons that wavelength is about or exceeds the grain size λ>d, also participate in
the heat transfer. For a rough estimation of their effect (κλ) we suppose that the
phonons in each frequency interval within
<d
aDωω make the same contribution to
the TC. This is the case of phonon-phonon interaction when the larger number of the
phonons with larger frequencies compensates the smaller value of their mean free
path. For T>TD this leads to κλ~ κi(a/d). For the smaller temperatures we have to take
into account only the phonons with TkB<ℏω so that κλ~ κi(a/d)(TD/T). There are two
possibilities when this value might be important: at low temperatures and in the case
of the above-mentioned localization of the short-wavelength phonons.
5.3 References
P.G. Klemens, M. Gell, Thermal conductivity of thermal barrier coatings, Materials
Science and Engineering A245 (1998), p.p. 143-149
L. Braginsky, V. Shklover, H. Hofmann, P. Bowen, High-temperature thermal
conductivity of porous Al2O3 nanostructures, Physical review B 70, 134201 (2004)
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6 THERMODYNAMIC
6.1 Nanothermodynamics
With the advancement of techniques of creating and characterizing materials, a huge
of ‘‘small’’ sizes grains (micrometers), nanosystems, molecular magnets, and atomic
clusters, has been formed and displays a variety of interesting physical and chemical
properties. It is important and timely to develop the new theoretical tools to address
experimental findings. On the other hand, the rapid progress in the synthesis and
processing of materials with the structures at the nanometer size has created a
demand for greater scientific understanding of the thermodynamics on nanoscale.
The issue of application of the thermodynamics on nanoscale has been continuously
attracted, since the nucleation reaction was discovered in the early 1930s. For
instance, a good example is the renowned publication of two books with the title
‘‘Thermodynamics of Small Systems’’ by Hill in early 1960s, and recently, the
thermodynamics of small systems is renamed ‘‘nanothermodynamics’’.
Traditionally, thermodynamics of large systems composing many particles has been
well developed. Classical thermodynamics describes the most likely macroscopic
behavior of large systems with the change of macroscopic parameters. The really
large systems of astrophysical objects as well as small systems containing a
relatively small number of constituents (at the nanometer scale) are excluded.
Therefore, there is a great deal of interest and activity in the present day to extend
the macroscopic thermodynamics and statistical mechanics to the nanometer scale
consisting of countable particles below the thermodynamic limit due to the recent
developments in nanoscience and nanotechnology. To generalize the
thermodynamics on scale, we need to well understand the unique properties of
nanosystems. We have seen previously, that one of the characteristic features of
nanosystems is their high surface-to-volume ratio. As results of surface effects
becoming increasingly important with decreasing size, and then, the Gibbs free
energy relatively increases for some thermodynamic equilibrium systems. Therefore,
the behavior of such nanoscopic clusters differs significantly from the usual
thermodynamic limit. On the other hand, it is clearly known that when the system size
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decreases, one has to consider the fluctuations. Based on the nucleation reactions,
the first considerations are on the temperature fluctuations. The quantitative
measurements of temperature fluctuations were realized by superconducting
magnetometers. Interestingly, it is well explained in the following statement by the US
National Initiative on Nanotechnology that the fluctuations play an important role:
‘‘There are also many different types of time scales, ranging from 10-15 s to several
seconds, so consideration must be given to the fact that the particles are actually
undergoing fluctuations in time and to the fact that there are uneven size
distributions. To provide reliable results, researchers must also consider the relative
accuracy appropriate for the space and time scales that are required; however, the
cost of accuracy can be high. The temporal scale goes linearly in the number of
particles N, the spatial scale goes as O(N log N), yet the accuracy scale can go as
high as N7 to N! with a significant prefactor.’’ Therefore, these valuable hints motivate
researchers to pursue the thermodynamic description at the nanometer size for the
nucleation of the metastable phase. Up to date, there are two kinds of fundamental
approaches to open out the thermodynamics on nanoscale based on the microscopic
and macroscopic viewpoints, respectively. One would go back to the fundamental
theorem of the macroscopic thermodynamics and establish the new formalism of the
nanothermodynamics by introducing the new function(s) presenting the fluctuations
or the surface effect of nanosystems. Another one could directly modify the equations
of the macroscopic thermodynamics and establish the new model of the
thermodynamics on nanoscale by incorporating the Laplace–Young or Gibbs–
Thomson relation presenting the density fluctuation of nanosystems in the
corresponding thermodynamic expressions. The fundamental outlines of these
approaches will be given in the following section.
6.1.1 Hill’s Theory
In the early 1960s, Hill addressed the subject of the thermodynamics of small
systems due to his interest in thermodynamics of polymers and macromolecules. In
order to clarify the relationship between the macroscopic thermodynamics and the
nanothermodynamics of Hill, first of all, go back to the fundamental theorem and
recapitulate the foundations of the thermodynamics of macroscopic systems.
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In the case of the equilibrium thermodynamics of a macroscopic system, the
fundamental equation for the internal energy, U, in the absence of an external field is
expressed as
U(S, V, N) = TS - PV + µN (6. 28)
Where S is the entropy (an extensive state function), and it is a function of the
extensive variables (U, V, N) in one-component system, T the absolute temperature,
P the pressure, V the volume, µ the chemical potential, and N is the number of
particles. The differential form of Eq. U(S, V, N) = TS - PV + µN (6. 28) may be
represented as
dU = SdT + TdS – VdP – PdV + Ndµ + µdN (6. 29)
On the other hand, the relationships among U, S, V, N, T, P, and µ can be expressed
as
VSN
U
,
∂∂=µ (6. 30)
VNS
UT
,
∂∂= 6. 31)
NSV
UP
,
∂∂−= (6. 32)
Eq. dU = SdT + TdS – VdP – PdV + Nd µ + µdN (6. 29) will change into the
following form by employing one of the above three equations VSN
U
,
∂∂=µ (6.
30)-NSV
UP
,
∂∂−= (6. 32).
SdT – VdP + Ndµ = 0 (6. 33)
This is the celebrated Gibbs–Duhem relation, and implies that the changes in the
intensive quantities (µ, T, P) are not independent. However, the usual choice (T, P) is
made in the literature, defining an equation of the state for the system. In particular,
the Gibbs–Duhem relation implies that
N
V
P T
=
∂∂µ
(6. 34)
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and
N
S
T P
−=
∂∂µ
(6. 35)
It is well known that three other functions, besides the internal energy, U, are very
useful in applications to the specific physical situations.
The enthalpy is
H (S,P,N) = U (S,V,N) + PV (6. 36)
The Helmholtz free energy
F (T,V,N) = U (S,V,N) – TS (6. 37)
And the Gibbs free energy
G (T,P,N) = U (S,V,N) – TS + PV (6. 38)
Eq. H (S,P,N) = U (S,V,N) + PV (6. 36) in eq. G (T,P,N) = U (S,V,N) – TS + PV
(6. 38), you obtain
G (T,P,N) = H (S,P,N) –TS (6. 39)
According to the dependence relationships of these functions (U, H, F, G) and their
continuity properties in their appropriate variables, four thermodynamic equations
called the Maxwell relations can be yielded as
VS S
P
V
T
∂∂−=
∂∂
(6. 40)
PS S
V
P
T
∂∂=
∂∂
(6. 41)
VT T
P
V
S
∂∂=
∂∂
(6. 42)
and
PT T
V
P
S
∂∂−=
∂∂
(6. 43)
If one combines eq. dU = SdT + TdS – VdP – PdV + Nd µ + µdN (6. 29) with SdT –
VdP + Ndµ = 0 (6. 33) we have the following Gibbs generalized relation
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∑+−=i
ii dNPdVTdSdU µ (6. 44)
Equation ∑+−=i
ii dNPdVTdSdU µ (6. 44) applies to a microscopic system only.
For example, if the system is a small one-component spherical aggregate that has a
nonnegligible surface free energy term proportional to N2/3, equation
∑+−=i
ii dNPdVTdSdU µ (6. 44) is no longer applicable. This equation can be
generalized further to include small systems, by adding another term. But this
addition must be made at the ensemble level rather than at the single-system level.
The reason for this is that an ensemble of B equivalent and noninteracting small
systems is itself a macroscopic system, and hence we can use equation
∑+−=i
ii dNPdVTdSdU µ (6. 44), rewritten for an ensemble, as a reliable
starting point:
∑+−=i
itittt dNPdVTdSdU µ
Ut ≡ BU
St ≡ BS
Vt ≡ BU
Nit ≡ BNi (6. 45)
where t = total (i.e., the whole ensemble of small systems).
The term we add to eq Nit ≡ BN i (6. 45) is EdB :
EdBdNPdVTdSdUi
itittt ++−= ∑µ (6. 46)
Where E ≡ (∂ Ut/ ∂ B)St,Vt,Nit . The motivation is that, for a small system, Ut will also be
a function of B. E is a kind of system (rather than molecule) chemical potential, called
the “sub-division potential.” If the systems of the ensemble are macroscopic and are
subdivided in order to increase B in eq EdBdNPdVTdSdUi
itittt ++−= ∑µ (6.
46) (e.g., in an extreme case, each system is cut in half to double B ), this will not
have a noticeable effect on Ut, St, Vt, Nit are all held constant as in E ≡ (∂ Ut/ ∂ B).
That is, surface effects, edge effects, system rotation and translation, etc., are all
negligible for macroscopic systems, and hence E is essentially zero: the term E d B
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in eq EdBdNPdVTdSdUi
itittt ++−= ∑µ (6. 46) does not contribute appreciably
to the equation. But the effects just mentioned are not negligible if an ensemble of
small systems is subdivided to increase B in eq EdBdNPdVTdSdUi
itittt ++−= ∑µ
(6. 46). This is because such an increase in B implies a decrease in V and Ni
(Vt, and Nit are held constant): unlike macroscopic systems, size effects are
significant in small systems. E may be regarded as a function of T, P, and the µi, as
will be seen explicitly below. Thus, E is a composite “intensive” property at the
ensemble level. Equation EdBdNPdVTdSdUi
itittt ++−= ∑µ (6. 46) can be
integrated from B = 0 to B, holding all small system properties constant, to give
EBNPVTSUi
itittt ++−= ∑µ (6. 47)
Here, new equivalent small systems are added to create the ensemble (subdivision is
not used). Division by B then yields, for a single small system,
ENPVTSUi
iti ++−= ∑µ (6. 48)
The presence of E in this equation is a new feature of nanothermodynamics.
If we return to eq EdBdNPdVTdSdUi
itittt ++−= ∑µ (6. 46), replace dUt by d(BU) =
BdU + UdB , and similarly for dS t, dV t, dN it the terms in dB drop out in view of eq
ENPVTSUi
iti ++−= ∑µ (6. 48). The remaining equation is (for a single small
system)
∑+−=i
ii dNPdVTdSdU µ (6. 49)
This is the same as eq ∑+−=i
ii dNPdVTdSdU µ (6. 44) for a macrosystem: a
small system has the same energy-heat-work relation as in a macrosystem (with the
Ni constant) and also the same chemical potential terms. However, eq
∑+−=i
ii dNPdVTdSdU µ (6. 49) cannot be integrated to give eq
ENPVTSUi
iti ++−= ∑µ (6. 48) because U is no longer (for a small system) a linear
homogeneous function of S, V, and N i.
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If we take differentials in eq ENPVTSUi
iti ++−= ∑µ (6. 48) and cancel the
terms in eq ∑+−=i
ii dNPdVTdSdU µ (6. 49), the remaining relation is
∑−+−=i
ii NdVdPSdTdE µ (6. 50)
The left-hand side is zero in macroscopic thermodynamics. Note that T, P, and the µi
are all independent in eq ∑−+−=i
ii NdVdPSdTdE µ (6. 50). A macrosystem has one
less “degree of freedom” because dE = 0.
Equation ∑−+−=i
ii NdVdPSdTdE µ (6. 50) confirms that E is a function of µi, T,
P. Note that derivatives of E, a composite “intensive” property, with respect to
intensive variables yield extensive variables:
i
i
i
pTii
T
p
EN
P
EV
T
ES
µ
µ
µ
µ,,
,
,
∂∂=
∂∂=
∂∂=−
(6. 51)
This is a rather surprising result. Equations ENPVTSUi
iti ++−= ∑µ (6. 48),
∑+−=i
ii dNPdVTdSdU µ (6. 49) and ∑−+−=i
ii NdVdPSdTdE µ (6. 50)
provide the fundamentals of the generalization of the thermodynamics of Gibbs to
include small systems. Although conventional macrothermodynamics is by far the
more widely known subject, it can be viewed as a limiting case of
nanothermodynamics. This is somewhat reminiscent of classical mechanics (large)
as a limiting case of quantum mechanics (small). In contrast to
macrothermodynamics, the thermodynamic properties of a small system will usually
be different in different “environments.” An example will clarify this statement.
Consider a rather rigid incompressible linear aggregate of N more or less spherical
molecules in an inert constant-temperature bath at T. N might be on the order of 102
or 103. The “environmental variables” here are N and T (P can be ignored because of
the incompressibility). The aggregate will have a certain amount of entropy, S,
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associated with translation, rotation, and the vibrational motion of and within the
molecules of the chain. Now suppose, in contrast, that the bath includes the
individual molecules at chemical potential µ and that these molecules can go on and
off the two ends of the aggregate. At equilibrium, there will be a mean number N of
molecules in the aggregate. The environmental variables here are µ and T. If µ is
chosen so that N in the µ, T case is the same as N in the N, T case above, the
entropy S will be larger in the µ, T case because of fluctuations in N that are not
present in the N, T case. However, this extra entropy becomes negligible as the
aggregate approaches macroscopic size (by adjusting µ).
To study such effects using theoretical models, the applicable set or sets of
environmental variables must play a primary role. For given environmental variables,
the particular statistical mechanical partition function appropriate for these variables
must be chosen in order to deduce the thermodynamic properties associated with
these environmental variables. This is in contrast with the treatment of macroscopic
systems: the partition function used is optional and can be selected merely for
convenience.
It is interesting that the basic equations of nanothermodynamics are the same for all
sets of environmental variables but the thermodynamic properties are often not the
same.
It is worth noting that Chamberlin et al. have extended Hill’s idea by considering the
independent thermal fluctuations inside bulk materials. In detail, they adapt Hill’s
theory to obtain a mean-field model for the energies and size distribution of clusters
in condensed matter. Importantly, the model provides a common physical basis for
many empirical properties, including non-Debye relaxation, non-Arrhenius activation,
and non-classical critical scaling
6.1.2 Tsallis’ generalization of ordinary Boltzmann -Gibbs
thermostatics
The thermodynamic theory is on the basis of the Tsallis’ generalization of the
ordinary Boltzmann–Gibbs thermostatistics by relaxing the additive properties of the
thermodynamic quantities (the entropy, in particular) to include non-extensivity of
nanosystems. As described by Rajagopal et al. [], the nanothermodynamics differs
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from Hill’s approach by considering that each of the nanosystems fluctuates around
the temperature of the reservoir, while nanosystems are coupled to the reservoir.
This means that the Boltzmann–Gibbs distribution has to be averaged over the
temperature fluctuations induced by the reservoir.
The formalism of nonextensive statistical mechanics has been developed over the
past 13 years as a powerful and beautiful generalization of ordinary statistical
mechanics. It is based on the extremization of the Tsallis entropies:
−−
= ∑i
qiq p
qS 1
1
1 (6. 52)
subject to suitable constraints. Here the p i are the probabilities of the physical
microstates, and q is the entropic index. The Tsallis entropies reduce to the
Boltzmann Gibbs (or Shannon) entropy ii
i ppS ∑−= log1 for q → 1, hence ordinary
statistical mechanics is contained as a special case in the generalized formalism.
There is growing experimental evidence that q ≠ 1 yields a correct description of
many complex physical phenomena, including hydrodynamic turbulence, scattering
processes in particle physics, and self-gravitating systems in astrophysics, to name
just a few. The Tsallis entropies are nonextensive. Given two independent
subsystems Ι and ΙΙ with probabilities Ιip and ∐
ip , respectively, the entropy of the
composed system Ι + ΙΙ (with probabilities ΙΙΙΙΙ+Ι = jiij ppp ) satisfies
ΙΙΙΙΙΙΙΙ+Ι −−+= qqqqq SSqSSS )1( (6. 53)
Hence there is additivity for q = 1 only. This property has occasionally lead to
unjustified prejudices, using arguments of the type “entropy must be extensive”. In
the following, however, we will see that the nonadditivity property is not at all a “bad”
property, but rather a natural and physically consistent property for those types of
systems where nonextensive statistical mechanics is expected to work.
Broadly speaking, the formalism with q ≠ 1 has so far been observed to be relevant
for two different classes of systems. One class are systems with long-range
interactions, the other one are systems with fluctuations of temperature or of energy
dissipation. For the first class of systems, the concept of independent subsystems
does not make any sense, since all subsystems are interacting. Hence there is no
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contradiction with the nonadditivity of entropy for independent subsystems since
independent subsystems do no exist for these systems (by definition of the long-
range interaction).
Let us thus concentrate on the other class, systems with fluctuations. Consider a
system of ordinary statistical mechanics with Hamiltonian H. Tsallis statistics with
q > 1 can arise from this ordinary Hamiltonian if one assumes that the temperature
β−1 is locally fluctuating. From the integral representation of the gamma-function one
can easily derive the formula
( )( ) ∫∞
−−
− =−+0
1
1
0 )(11 βββ β dfeHq Hq (6. 54)
where
( ) ( )
−−
−
−Γ
=−
−−
0
11
11
1
0 1exp
1
1
1
1
1)(
βββ
ββ
q
f qq
(6. 55)
is the probability density of the χ2 distribution. The above formula is valid for arbitrary
Hamiltonians H and thus of great significance. The left-hand side of eq.
( )( ) ∫∞
−−
− =−+0
1
1
0 )(11 βββ β dfeHq Hq (6. 54) is just the generalized Boltzmann
factor emerging out of nonextensive statistical mechanics. It can directly be obtained
by extremizing Sq. The right-hand side is a weighted average over Boltzmann factors
of ordinary statistical mechanics. In other words, if we consider a nonequilibrim
system (formally described by a fluctuating β), then the generalized distribution
functions of nonextensive statistical mechanics are a consequence of integrating over
all possible fluctuating inverse temperatures β, provided β is χ2 distributed.
Wilk et al. [] have also suggested a formula similar to eq.
( )( ) ∫∞
−−
− =−+0
1
1
0 )(11 βββ β dfeHq Hq (6. 54) which is valid for
q < 1. However, in that case the distribution function f(β) is more complicated and
different from the simple form ( ) ( )
−−
−
−Γ
=−
−−
0
11
11
1
0 1exp
1
1
1
1
1)(
βββ
ββ
q
f qq
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(6. 55). In particular, for q < 1 one has f(β) = f(β,H), i.e. the probability density
of the fluctuating temperature also depends on the Hamiltonian H. This is physically
less plausible, because usually one assumes that the heat bath is independent of the
Hamiltonian. Hence in the following we will restrict ourselves to the physically more
interesting case q > 1.
For q > 1, the χ2 distribution obtained in eq.
( ) ( )
−−
−
−Γ
=−
−−
0
11
11
1
0 1exp
1
1
1
1
1)(
βββ
ββ
q
f qq
(6. 55) is a universal distribution
that occurs in many very common circumstances. For example, it arises if β is the
sum of squares of n Gaussian random variables, with n = 2/(q−1) . Hence one
expects Tsallis statistics with q > 1 to be relevant in many applications. For fully
developed turbulent hydrodynamic flows, where Tsallis statistics has been observed
to work very well, β−1 is not the physical temperature of the flow but a formal
temperature related to the fluctuating energy dissipation rate times a time scale of the
order of the Kolmogorov time. In the application to scattering processes in collider
experiments, β−1 is a fluctuating inverse temperature near the Hagedorn phase
transition.
The constant β0 in eq. ( ) ( )
−−
−
−Γ
=−
−−
0
11
11
1
0 1exp
1
1
1
1
1)(
βββ
ββ
q
f qq
(6.
55) is the average of the fluctuating β,
∫∞
==0
0)()( βββββ dfE (6. 56)
(E denotes the expectation with respect to f(β)). The deviation of q from 1 can be
related to the relative variance of β. One has
2
22
)(
)()(1
βββ
E
EEq
−=− (6. 57)
We can now give physical meaning to eq. ΙΙΙΙΙΙΙΙ+Ι −−+= qqqqq SSqSSS )1( (6. 53) for q
> 1. Consider two independent subsystems Ι and ΙΙ that are composed into one
system Ι + ΙΙ . In system Ι (as well as ΙΙ ) the fluctuations of temperature T are
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expected to be larger than in Ι + ΙΙ , due to the smaller size of Ι (or ΙΙ ) as compared
to Ι + ΙΙ . Remember that generally entropy is a measure of missing information on
the system. The entropy ΙΙΙ + qq SS is larger than ΙΙ+ΙqS , since in the single systems the
probability distribution of T is broader, thus our missing information on these systems
is larger. Hence there must be a negative correction term to ΙΙΙ + qq SS in eq.
ΙΙΙΙΙΙΙΙ+Ι −−+= qqqqq SSqSSS )1( (6. 53) . It is physically plausible that this correction
term is proportional
• to the relative strength of temperature fluctuations as given by q − 1 and
• to the entropies in the single ΙqS and ΙΙ
qS systems. Hence we end up with eq.
ΙΙΙΙΙΙΙΙ+Ι −−+= qqqqq SSqSSS )1( (6. 53) .
For most physical applications the fluctuations of β (or the relevant value of q) are
observed to be dependent on the spatial scale r. For example, in the turbulence
application detailed measurements of q(r) have been presented in []. q turns out to
be a strictly monotonously decreasing function of the distance r on which the velocity
differences are measured. Similarly, for the application to e+e− annihilation, q(r) turns
out to be again a strictly monotonously decreasing function of the scale r, which in
this case is given by r ∼ ћ/Ecm, where Ecm is the center-of-mass energy of the beam.
6.1.3 Thermodynamics of metastable phase nucleation on
nanoscale
6.1.3.1 Classical nucleation thermodynamics
Before starting the analysis of the thermodynamics of MPNUR (Metastable Phase
Nucleation in the strongly Unstable Region), we need to look back on some
fundamental concepts of the nucleation theory.
Actually, the nucleation refers to the kinetic processes that initiate the first-order
phase transitions in non-equilibrium systems, and the nucleation of a new phase is
largely determined by the nucleation work W. The quantity is equal to spending the
Gibbs free energy having or at least resembling the properties of the new phase
appearance in the parent phase of a density fluctuation and staying in the labile
thermodynamic equilibrium together with the parent phase. With a random acquisition
even of a single molecule of a new phase, the fluctuation may result in the
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spontaneous formation of the critical nucleation of the new phase. For this reason, W
is the energy barrier (critical energy of cluster formation, ∆G*) of the nucleation.
Therefore, the nucleation work plays an important role in the formation of a new
phase. However, it is well known that the initially homogeneous system is also
heterogeneous in characterized by the non-uniform density and pressure. Therefore,
the determination of ∆G* is a hard problem. Namely, the case above makes it
impossible to derive the nucleation work only from the method of the thermodynamics
of uniformly dense phases. In CNT (Classical Nucleation Theory), the critical nucleus
is regarded as a liquid drop with a sharp interface (a dividing surface) that separates
the new and parent bulk phases. Matter within the dividing surface is treated as a
part of a bulk phase whose chemical potential is the same as that of the parent
phase. In the absence of knowledge of the properties of the microscopic clusters
including the surface tension, the bulk thermodynamic properties with several
approximations are used to evaluate the nucleation work in the discussions below.
In 1878, Gibbs published his monumental work with the title ‘‘On the Equilibrium of
Heterogeneous Substances,’’ and his other publications have a special place in
thermodynamics of the phase’s mixture and equilibrium. Concretely, Gibbs extended
the science of thermodynamics in a general form to heterogeneous systems with and
without chemical reactions. Especially, he introduced the method of the dividing
surface (DS) and used it to derive an exact formula for ∆G* in the nucleation of a new
phase in the bulk parent phase. In detail, with the aid of an arbitrarily chosen
spherical DS, he divided the heterogeneous system consisting of the density
fluctuation and the parent phase into two homogeneous subsystems, which are
corresponding to the microscopical and macroscopical subsystems, respectively. The
macroscopically large subsystem equals the parent phase with the uniform density
and pressure before the fluctuation formed. The microscopically small subsystem is
an imaginary particle (nucleus) replaced a reference new phase with the uniform
density and pressure, and surrounded by the large subsystem. The imaginary particle
substitutes for the real nucleus of the new phase, which is created by the density
fluctuation. As Gibbs described the difference between the imaginary particle (globule
by Gibbs definition) and the density fluctuation by the following statement: ‘‘For
example, in applying our formulas to a microscopic globule of water in steam, by the
density or pressure of the interior mass we should understand not the actual density
or pressure at the centre of the globule, but the density of liquid water (in large
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quantities) which has the temperature and potential of the steam.’’ Furthermore, very
recently, Kashchiev detailedly expatiated the difference between the imaginary
particle and the real density fluctuation by the following statement:
I. ‘‘the nucleus size depends on the choice of the DS and may therefore be very
different from the characteristic size of the density fluctuation;
II. the surface layer of the nucleus is represented by the mathematical DS and is
thus with zero thickness, whereas that of the density fluctuation is diffuse and
can extend over scores of molecular diameters;
III. the pressure and molecular density of the nucleus are uniform, and those of
the fluctuation are not and might even be hard to define when ‘at its centre the
matter cannot be regarded as having any phase of matter in mass;
IV. the uniform pressure and density of the nucleus are equal to those of a
reference bulk new phase rather than to those at the center of the fluctuation.’’
Therefore, based on these approaches above, Gibbs showed that the reversible work
W (free energy of nucleation), required to form the critical nucleus of a new phase, is
)(* VlT PPVAG −−=∆ γ (6. 58)
Where A and V are the area and volume of the specific surface energy of a specially
chosen DS, Pl the pressure of the new bulk reference phase at the same chemical
potential as the parent phase, and Pv is the pressure of the parent phase far from the
nucleus. γT is the ‘‘surface of tension,’’ called by Gibbs, of the specific surface energy
of a specially chosen DS, and it is called as the surface tension at the present day.
In the Gibbs’ analysis, he found out that the classical Laplace–Young equation is
valid in his DS, and governs the pressure of droplets across a curved interface. For a
spherical droplet with the critical nucleus radius r*, the Laplace–Young equation
reads
*
2
rPP T
vl
γ=− (6. 59)
Further, for the spherical critical nucleus, Gibbs showed that with eq. *
2
rPP T
vl
γ=−
(6. 59), eq. )(* VlT PPVAG −−=∆ γ (6. 58) becomes
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2
3
)(3
16*
vl
T
PPG
−=∆
γπ (6. 60)
However, the quantity γT could not be obtained by experiment, because it describes
such a surface—an imaginary physical object, i.e., the nucleus characterized by the
surface of tension. Therefore, the dependent relationships of γT and pressure,
temperature, and composition of the parent phase, respectively, are not uncovered.
This limits the application of the nucleation theory to various cases of interest. On the
other hand, in order to describe the thermodynamic characterization of various
practical cases, we have to approximate γT by a real physical quantity. Clearly, in
nearly all nucleation papers that followed Gibbs’ equation, e.g., one used the real
interface energy γ0 between the bulk parent and new phases at the phase
equilibrium, i.e., at their coexistence, to replace the surface tension γT of the
imaginary DS. However, to apply this famous formula of Gibbs, one has to know the
exact interface energy that is related to the radius of droplets and the droplet
reference pressure. Unfortunately, there are tremendous challenges because there is
no simple way to extract the interface energy from the force measurements in theory.
A given interface energy is a function of the many coordinates of the nanoparticles.
Lacking the knowledge of the exact interface energy, the first approximation is to use
the experimental interface energy of a flat interface, i.e., γ0 = γ. Actually, the surface
structure of droplets is different from that in the bulk, so that, strictly speaking, the
boundary surface never coincides with the equimolecular surface. Nevertheless, they
are usually close to each other and are often taken equally as the physical surface of
the droplet. By assuming γ0 = γ, one can obtain the first form of the nucleation work
2
3
)(3
16*
vl PPG
−=∆
γπ (6. 61)
In principle, by adopting the approximation, the validity of the Gibbs’s expression on
the basis of the Laplace–Young equation should be limited to the sufficiently large
nuclei. Interestingly, the applications of the nucleation theorem in the analysis of
experimental data in various cases of nucleation implied that the Laplace–Young
equation could predict well the size of the nuclei built up of less than a few tens of
molecules.
For instance, Tolbert and Alivisatos discussed the elevation of pressure in the solid–
solid structural transformation as the crystallite size decreases in the high-pressure
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system using the Laplace–Young equation. Accordingly, it seems to be recognized
that the Laplace–Young equation could be used to predict well the size of nuclei built
up of less than a few tens of molecules. On the other hand, based on the
thermodynamic identity, we have
∫=−l
v
P
P
mvlll dPVPP )()( µµ (6. 62)
Where Vm is the molar volume of a new phase and )( ll Pµ and )( vl Pµ are the chemical
potential of matter in the new phase at the pressures Pl and Pv. When the critical
droplet and the metastable vapour locate the condition of the unstable equilibrium,
one can obtain
)()( llvv PP µµ = (6. 63)
Furthermore, if we approximate Pl by assuming that the droplet is incompressible,
and assume that Vm is a constant. With Eqs. ∫=−l
v
P
P
mvlll dPVPP )()( µµ (6. 62) and
)()( llvv PP µµ = (6. 63) becomes
mm
llvvvl VV
PPPP
µµµ ∆=−
=−)()(
(6. 64)
Eq. mm
llvvvl VV
PPPP
µµµ ∆=−
=−)()(
(6. 64) turns into Eq. 2
3
)(3
16*
vl PPG
−=∆
γπ
(6. 61), one can obtain the second form of the nucleation work
2
23
)(3
16*
µγπ∆
=∆ mVG (6. 65)
As Obeidat et al. stated [], the form of the nucleation work is most useful if the
chemical potential difference between a new phase and its parent phase can be
obtained. However, the actual performance is quite complicated due to the lack of the
accurate potentials for most substances. In order to obtain the ∆µ, we have to adopt
some necessary approximations. Generally, if we assume the supersaturated and
saturated vapors are the ideal gases and the droplet is an incompressible liquid, the
difference of the chemical potential ∆µ is more commonly derived from the
approximate system. In detail, under the above assumptions, we have
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evl PP = (6. 66)
Where Pev is the equilibrium-vapor pressure. With eq. evl PP = (6. 66),
eq. ∫=−l
v
P
P
mvlll dPVPP )()( µµ (6. 62) becomes
)()()( evvmevvvl PPVPP −+= µµ (6. 67)
When the new bulk phase and the parent phase are in the state of thermodynamic
equilibrium, one can obtain
)()( evvevl PP µµ = (6. 68)
With eqs. )()()( evvmevvvl PPVPP −+= µµ (6. 67) and )()( evvevl PP µµ = (6. 68), eq.
mm
llvvvl VV
PPPP
µµµ ∆=−
=−)()(
(6. 64) becomes
)()()( evvmevvvv PPVPP −−−=∆ µµµ (6. 69)
Under the ideal vapor condition, we can easily obtain
=−
ev
vevvvv P
PkTPP ln)()( µµ (6. 70)
Where k is the Boltzmann constant, T the absolute temperature, and Pv is the actual
pressure. With Eq.
=−
ev
vevvvv P
PkTPP ln)()( µµ (6. 70),
Eq. )()()( evvmevvvv PPVPP −−−=∆ µµµ (6. 69) becomes
)(ln evvmev
v PPVP
PkT −−
=∆µ (6. 71)
In Eq. )(ln evvmev
v PPVP
PkT −−
=∆µ (6. 71), compared with the first term on the
right, the second term on the right is almost extremely small, and it is customary to
neglect it. Therefore, Eq. 2
23
)(3
16*
µγπ∆
=∆ mVG (6. 65) will become the third form of
the nucleation work
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=∆
ev
v
m
P
PkT
VG
ln3
16*
23γπ (6. 72)
Applying the first two forms of the nucleation work requires the knowledge of the
droplet reference pressure or chemical potential. Usually, this information is
unavailable, and the experimental results are, instead, compared with the rates
predicted using the third form, because the supersaturation ratio is readily
determined from the experimental data. Naturally, the size of the critical nucleation,
the critical energy, the phase transition probability, and the nucleation rate would be
obtained by the determined nucleation work.
In summary, from the point of CNT above, one can see that there is an important
approximation, i.e., assuring γT = γ0 = γ. Namely, the surface tension (γT) of a
specially chosen DS, the real interface energy (γ0) between the bulk parent and new
phases at the phase equilibrium, and the experimental interface energy of a flat
interface (γ) are approximated to equal. Furthermore, CNT indicates that the
Laplace–Young equation seems to be capable of predicting well the size of nuclei
built up of less than a few tens of molecules. However, on the other hand, it is well
known that the CNT describes that a stable new phase forms from a metastable
parent phase.
6.1.3.2 Thermodynamics of metastable phase nucleati on in unstable of
thermodynamic equilibrium phase diagram
MPNUR, such as CVD (Chemical Vapor Deposition) diamond, CVD c-BN, HSRC
(Hydrothermal Synthesis and the Reduction of Carbide) diamond, and c-BN (cubic
boron nitride) nucleation, seems to be impossible from the standpoint of
thermodynamics, because the nucleation happens in the strongly unstable region of
the metastable structural state in terms of the thermodynamic equilibrium phase
diagram, and violates the second law of thermodynamics. For this issue, as Hwang
and Yoon stated regarding CVD diamond [], ‘‘Something must be wrong either in
interpreting the experimental observation or in applying thermodynamics.’’ In fact, in
the early 1965s, Garvie pointed out that MPNUR likely arises out of the capillary
pressure built up in the nuclei. Namely, the nanosize-induced additional pressure
could be so large that the high-pressure metastable phase tends to become more
stable than the low-pressure stable phase, as shown in Figure 6-1.
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Figure 6-1 : A sketch map of MPNUR mechanism. A region is the metastable structural state of M
phase, and B region is the new stable state of M phase by the nanosize-induced additional pressure
driving. The inset shown in the sketch map displays the spherical nuclei nucleated on the hetero-
substrate. [C.X. Wang, G.W. Yang, Thermodynamics of metastable phase nucleation, Materials
Science and Engineering, 2005, p.p. 157-202]
Note that, in the following description, a phase is metastable or stable if it is stable or
metastable without the effect of the nanosize-induced additional pressure. In our
theoretical approach, it also is emphasized that the nanosize-induced additional
pressure is reasonably taken into account in the below analysis.
Generally, the Gibbs free energy is an adaptable measure of the energy of a state in
phase transformations among competing phases. At the given thermodynamic
condition, both stable and metastable phases can coexist, but only one of the two
phases is stable, with the minimal free energy, and the other must be metastable and
may transform into the stable state. Thermodynamically, the phase transformation is
promoted by the difference of the free energies. The Gibbs free energy of a phase
can be expressed as a function of the pressure–temperature condition, and
determined by a general coordinate or reactive coordinate. According to CNT, the
Gibbs free energy difference arises from the formation of spherical clusters in the
low-pressure gas phase is expressed as a function of radius r, pressure P, and
temperature T
( )sesesnsnnenem
S AAAgV
VTPrG γγγ −++∆×=∆ ),,( (6. 73)
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Where Vs and Vm are the volume of the spherical clusters with the metastable
structural phase and its molar volume, ∆g the Gibbs free energy of molar volume
depending on the pressure P and temperature T in the phase transition, Ane and γne
the interface area and the energy between the spherical clusters of the metastable
structural phase and the environment gas phase, Asn and γsn the interface area and
the energy between the spherical clusters of the metastable phase and the hetero-
substrate, and Ase and γse are the interface area and the energy between the hetero-
substrate and the environment gas phase. The formation of the spherical clusters
with the metastable structural phase produces two interfaces, i.e., the interface Ane
between the spherical clusters and the environment gas phase and the interface Asn
between the spherical clusters and the hetero-substrate, and makes the original
interface Ase (be equal to Asn) between the hetero-substrate and the environment
gas phase vanish. According to the geometry, the volume Vs of the spherical clusters
of the metastable structural state, the interface area Ane between the spherical
clusters of the metastable state and the environment gas phase, and the interface
area Asn between the spherical clusters and the hetero-substrate are expressed as
( )( )3
12 23 mmrVs
−+= π (6. 74)
( )mrAne −= 12 2π (6. 75)
( )22 1 mrAsn −= π (6. 76)
Where r is the curvature radius of spherical clusters of the metastable structural
phase and m is given by
ne
snsemγ
γγθ −== cos (6. 77)
Where θ is the contact angle between the spherical clusters of the metastable
structural state and the hetero-substrate, as shown in the inset of Figure 6-1. Here,
γne is assumed to be approximately equal to the surface tension value of the
metastable structural phase (γ), γse for the interface energy between the hetero-
substrate and the environment gas phase is taken to be equal to the surface tension
value of the hetero-substrate, and the interface between the spherical clusters and
the hetero-substrate is assumed to be incoherent interface; therefore
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ne
senesn γ
γγγ += (6. 78)
Thus, we can obtain
( ) ( )( )4
124
3
4,,,
223 mm
rV
grTPTrG
m
−+
+∆×=∆ γππ (6. 79)
Where the factor of
( )( )4
12)(
2mmf
−+=θ (6. 80)
is called as the heterogeneous factor, and its value is in the range of 0–1. Especially,
when the clusters nucleated on the homo-substrate, its value is 1.
According to thermodynamics, we have
( )V
P
PTg
T
∆=
∂
∆∂ , (6. 81)
Then the difference of the Gibbs free energy per mole can be defined by
( ) ( ) ∫ ∆×∆=−∆≈∆=∆−∆P
P
O PVPPVVdPPTgPTg0
)(,, 0 (6. 82)
Where ∆V is the mole volume difference between the metastable and the stable
phase. When the conditions are near the equilibrium line, one can approximately
have ∆g(T, P0) = 0. Thus, Eq. ( ) ( ) ∫ ∆×∆=−∆≈∆=∆−∆P
P
O PVPPVVdPPTgPTg0
)(,, 0
(6. 82) would be defined as
( ) PVPTg ∆×∆=∆ , (6. 83)
On the other hand, due to the nanosize-induced additional pressure ∆Pn, the clusters
enduring pressure will increase by the same amount. Under the assumptions of
spherical and isotropic clusters, the nanosize-induced additional pressure is denoted
by the Laplace–Young equation, i.e.
rP n γ2=∆ (6. 84)
Furthermore, as mentioned above, the nanosize-induced additional pressure can
drive the metastable phase regions into the stable phase region near the boundary
line of the high-pressure phase in the equilibrium phase diagram. Therefore, one can
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obtain the size-dependent equilibrium phase boundary line between the metastable
and the stable phases, and it can be approximately defined as
rPP b γ2−= (6. 85)
Where Pb is the equilibrium phase boundary equation between the metastable and
the stable phases. From Figure 6-1, one can see that the equilibrium phase boundary
between the metastable and the stable phases can be expressed by
00 bTkP b += (6. 86)
Where k0 and b0 are the slope and intercept in the P coordinate axis of the
equilibrium phase boundary line between the metastable and the stable phases. With
Eq. 00 bTkP b += (6. 86), Eq. r
PP b γ2−= (6. 85) can be defined as
rbTkP
γ200 −+= (6. 87)
Therefore, ∆P would change into
rbTkPP
γ200 +−−=∆ (6. 88)
With Eq. r
bTkPPγ2
00 +−−=∆ (6. 88), Eq. ( ) PVPTg ∆×∆=∆ , (6. 83) can be
denoted by
( )
+−−×∆=∆r
bTkPVPTgγ2
, 00 (6. 89)
With Eq. r
bTkPPγ2
00 +−−=∆ (6. 88), Eq.
( ) ( )( )4
124
3
4,,,
223 mm
rV
grTPTrG
m
−+
+∆×=∆ γππ (6. 79) can be expressed as
( ) )(4
2
3
4,, 2
003 θγπ
γ
π frV
rbTkPV
rTPrGm
+
+−−×∆×=∆ (6. 90)
When 0)( =
∂∆∂
r
rG, the critical size of the high-pressure phase is obtained as
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PbTk
V
V
r
m
−+
∆
+=
00
* 3
22γ
(6. 91)
Substituing Eq. PbTk
V
V
r
m
−+
∆
+=
00
* 3
22γ
(6. 91) into
Eq. ( ) )(4
2
3
4,, 2
003 θγπ
γ
π frV
rbTkPV
rTPrGm
+
+−−×∆×=∆ (6. 90), the critical
energy of the high-pressure phase nuclei is given by
( ) )(3
22
4
3
23
22
3
4,,
2
00
0000
3
00
* θγ
πγ
π fPbTk
V
V
V
VPbTk
bTkPV
V
PbTk
V
V
TPrG
m
m
m
m
−+
∆
++
∆+
−++−−∆×
−+
∆
+=∆
6. 92)
On the other hand, it is well known that the phase transition is determined by the
probability. We have studied the nanosize effect on the probability of the phase
transformation based on the thermodynamic equilibrium phase diagram. The
probability of the phase transformation from the metastable phase to the stable
phase is related not only to the Gibbs free energy difference ∆g(T, P), but also to an
activation energy (Ea - ∆g(T, P)), which is necessary for the phase transition, as
shown in Figure 6-2.
Figure 6-2 : The schematic diagram of Gibbs free en ergy vs. coordinate. [C.X. Wang, G.W. Yang,
Thermodynamics of metastable phase nucleation, Materials Science and Engineering, 2005, p.p. 157-
202]
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When the two phases are at the equilibrium condition, i.e.,
∆g (T, P) = 0, Ea is the maximum potential energy for both sides with respect to the
general coordinate. The general expression of the probability f of the phase
transformation from the initial states to final states is
−−
∆−−=
RT
E
RT
PTgEf aa exp
),(exp (6. 93)
Where R is the gas constant and ∆g(T, P) is defined by Eq.
( )
+−−×∆=∆r
bTkPVPTgγ2
, 00 (6. 89). Accordingly, we have established the
thermodynamic approach at the nanometer size to quantitatively describe the
nucleation and the phase transition of the metastable phase in the strongly unstable
phase region of the metastable structural state in the thermodynamic equilibrium
phase diagram. In fact, the developed approach is a useful and effective theoretical
tool to address MPNUR, although it looks a little bit simple in thermodynamic.
Importantly, the validity of our thermodynamic theory has been substantively checked
by use in the nucleation of diamond and c-BN.
6.1.4 Nanothermodynamic analyses of CVD diamond nuc leation
Generally, CVD (Chemical Vapor Deposition) diamond is usually a typical quasi-
equilibrium process, and the pressure is in the range of 102–105 Pa and temperature
is in the range of 1000–1300 K. In the carbon phase diagram shown in Figure 6-3,
the general thermodynamic region of the diamond nucleation upon CVD is shown as
G region, which belongs to the strongly unstable or metastable region of the diamond
structural state, i.e., the stable region of graphite structural state.
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Figure 6-3 : Carbon thermodynamic equilibrium phase diagram. The G region means a
metastable phase region of the diamond nucleation upon CVD; the D region means a new stable
phase region of the CVD diamond nucleation with respect to the effect of the nanosize-induced
additional pressure. The inset shows the relationship between the nanosize-induced additional
pressure and the nuclei size. [C.X. Wang, G.W. Yang, Thermodynamics of metastable phase
nucleation, Materials Science and Engineering, 2005, p.p. 157-202]
It is well known that the graphite nucleation would be prior to diamond nucleation in
the G region from the point of view of thermodynamics. Therefore, the diamond
nucleation would not happen unless the graphite nucleation is restrained or stopped.
For the issue, the most popular explanation is that the atomic hydrogen plays an
important role. Atomic hydrogen is an essential factor in CVD diamonds due to its
higher etching rate for the graphite phase and less etching rate for the diamond
phase. Unfortunately, some researchers have reported that diamond films are grown
upon CVD with a hydrogen-free environment. Further, Gruen [] concluded that CVD
diamonds do not require the reactant gas mixtures consisting primarily of hydrogen,
and the microstructure of diamond films can change continuously from micro- to
nanocrystalline when hydrogen is successively replaced by a noble gas such as
argon. Moreover, they pointed out that a chief function of the atomic hydrogen is to
reduce the secondary nucleation rates. Therefore, these experiments clearly indicate
that, besides enhancing the growth of the diamond nuclei, the atomic hydrogen may
have little function for the diamond primary nucleation. Our question is: would the
diamond nucleation be really in the strongly unstable region of the diamond structural
phase upon CVD?
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The nucleation of CVD diamonds should happen in the D region of Figure 6-3 on the
basis of the nanosize-induced additional pressure. When we assume the surface
tension of diamond is 3.7 J/m2, the dependence relationships of the nanosize-
induced additional pressure based on the Laplace–Young equation and the size of
diamond clusters can be obtained, as shown in the inset of Figure 6-3. From the
inset, one can see that the additional pressure increases with the crystal particle’s
size decreasing. Notably, when the radius is less than 4 nm, the additional pressure
goes up to above 2.0 GPa, which is above the phase equilibrium line shown as the D
region, i.e., the diamond stable region, in Figure 6-3. In other words, the nanosize-
induced additional pressure could drive the metastable region (G region) of the
diamond nucleation into the new stable region (D region) in the thermodynamic
equilibrium phase diagram of carbon. These deductions are supported by the
experimental cases from the CVD diamonds on non-diamond substrates. For
instance, Lee et al. reported that the size of the nuclei of CVD diamonds on Si
substrates is in the range of 2–6 nm []. Consequently, the nanosize-induced
additional pressure of 1–3 nm radius of the diamond nuclei would be enough to drive
the G region into the D region. Therefore, the nucleation of CVD diamonds should
happen in the D region based on our nanothermodynamic approach.
According to Eq. ( ) )(4
2
3
4,, 2
003 θγπ
γ
π frV
rbTkPV
rTPrGm
+
+−−×∆×=∆ (6.
90), γ = 3.7 J/m2, Vm = 3.417x10-6 m3 mol-1, ∆V = 1.77x10-6 m3 mol-1, k0 = 2.01x106,
and b0 = 2.02x109 Pa, one can obtain the relationship curves between the size of the
diamond critical nuclei and the pressure at the temperature of 1300 K upon the CVD
diamond case, and it is displayed in Figure 6-4, in which the inset shows the
dependent relations of the pressure and the critical radius at the given various
temperatures.
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Figure 6-4 : The relationship curves between the radii of the critical nuclei and the pressure at the
temperature of 1300 K upon CVD systems. The inset shows the dependence relation of the pressure
and the critical radii at given various temperatures. One can see that, in faith, the radii of the critical
nuclei of diamond upon CVD for a broad range of pressures and temperatures are less than 5 nm.
Namely, the nucleation of CVD diamond could occur in the stable phase region of diamond. [C.X.
Wang, G.W. Yang, Thermodynamics of metastable phase nucleation, Materials Science and
Engineering, 2005, p.p. 157-202]
Clearly, we can see that the radii of the critical nuclei are less than 5 nm in a broad
range of the pressure and temperature. The diamond nucleation upon CVD seems to
be in the stable phase region of diamond due to the driving of the nanosize-induced
additional pressure of the diamond nuclei. In addition, from this figure, one also can
see the very weak dependence of the pressure on the critical radius. In other words,
at the given temperatures, the critical radii are hardly changed with the external
pressure change, because the external pressure is quite small compared with the
nanosize-induced additional pressure.
Based on Eq. PbTk
V
V
r
m
−+
∆
+=
00
* 3
22γ
(6. 91), the value of the surface tension of silicon
(1.24 J/m2), and the given parameters above, we display the relationship curves
between the pressure and the critical energy of CVD diamonds at the temperature of
1200 K, as shown in Figure 6-5, in which the inset displays the dependent relations
of the pressure on the critical energy at the given various temperature.
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Figure 6-5 : The relationship curves between the pressure and the critical energy of the nucleation
upon CVD diamond at the temperature of 1200 K, and the inset showing the curves at various given
temperatures. [C.X. Wang, G.W. Yang, Thermodynamics of metastable phase nucleation, Materials
Science and Engineering, 2005, p.p. 157-202]
Obviously, one can see in Figure 6-5 that the critical energy of the diamond nuclei
slowly increases with the pressure increasing at a given temperature, and
approximately remains as unchanged. The case results from the too little external
pressure compared with the nanosize-induced additional pressure. These results
indicate that the critical energy of the diamond nucleation upon CVD is quite low
(10-16 J), suggesting that the heterogeneous nucleation of CVD diamonds does not
require high forming energy.
Apparently, the low forming energy of the heterogeneous nucleation of CVD
diamonds implies that it is not difficult to nucleate diamond, and the diamond
nucleation could happen in the diamond stable region (G region) as shown in Figure
6-3. Based on the analyses above, the diamond nucleation upon CVD would happen
in the diamond stable region in the carbon phase diagram from the point of view of
thermodynamics. In fact, when the size of the crystalline particles is in the nanometer
scale, the additional pressure induced by the curvature of the nanometer-sized
particles is so high as to exceed the equilibrium pressure between diamond and
graphite, i.e., going up to break through the phase equilibrium boundary line, which
means that the additional pressure could drive the thermodynamic phase region of
the diamond nucleation from the metastable to the stable as shown in Figure 6-3.
Therefore, the diamond nucleation upon CVD is not actually the ‘‘thermodynamic
paradox,’’ and ‘‘violating the second law of thermodynamics.’’ These results indicate
that the presence of the atomic hydrogen is not a vital factor to grow diamonds upon
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CVD from the viewpoint of the nanothermodynamics above. However, why did most
experimental studies all show that the atomic hydrogen plays a very important role in
CVD diamonds? In fact, it is recognized experimental evidence that the atomic
hydrogen etching the graphite phase (more etching rate to the graphite phase and
less etching rate to the diamond phase) and helping the sp3 hybridization of carbon
atoms. Naturally, the diamond nucleation could be enhanced in the low-pressure gas,
only when the graphite phase forming is restrained or stopped by the atomic
hydrogen or other factors. Thus, the presence of the atomic hydrogen could increase
the rates of the diamond growth. Accordingly, the effect of the atomic hydrogen on
the diamond growth is much larger than that on the diamond nucleation upon CVD.
In other words, the influence of the atomic hydrogen on the diamond nucleation
would be small from the point of view of the experimental investigations involved in
how to enhance the diamond nucleation upon CVD.
In conclusion, aiming at a clear insight into the nucleation of CVD diamonds, we
studied the diamond nucleation from the point of the view of a nanoscaled
thermodynamics. Notably, these theoretical results show that the diamond nucleation
would happen in the stable phase region of diamond in the thermodynamic
equilibrium phase diagram of carbon, due to the nanosized effect induced by the
curvature–surface tension of the diamond nuclei. In other words, at the nanometer
size, the diamond nucleation is prior to the graphite nucleation in competing growth
of diamond and graphite upon CVD.
6.2 Thermodynamics of melting and freezing in small
particles
6.2.1 Theory – Vanfleet and al. model
Couchman [] developed a thermodynamic framework which Vanfleet and al. [] have
modified with Van Der Veen’s [] ideas to include surface melting. The combination of
their ideas results in a thermodynamic model that can help explain observed results.
Vanfleet and al. [] base their model on a spherical particle surrounded by another
material. This surrounding material will be referred to as the imbedding matrix. This
matrix can be as simple as the particle’s own vapor, or as complicated as a
crystalline or amorphous solid (although certain assumptions will be made shortly
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about its effect on the particle). The particle’s temperature will be held constant, and
we will explore the Helmholtz free energy difference between the particle when it has
a liquid layer with a solid core and when it is fully solid. We assume that both the
solid and liquid phases are incompressible. Our interest is the shift of the melting and
freezing points due to finite size effects, so we must include both the surface energy
(γ) and the surface stress (σ). We will assume that both of these surface quantities
are isotropic and independent of particle size and temperature. We will also assume
that the only effect of the matrix will be to vary the surface energies and surface
stresses of the particle. There can be confusion on the use of the terms surface
energy and surface stress. γ is the surface energy and has units of energy per area.
γ times the area is the energy attributed to the surface; it is the energy required to
make new surface. It can also be thought of as the energy of the unsatisfied bonds
on the surface, but as we will see later, it also includes the surface-induced order or
disorder in the bulk of the material. σ is the surface stress and also has units of
energy per area. This is the energy required to alter a given surface by elastically
deforming that surface, including the energy of the stretched or compressed bonds
within the surface. These two surface quantities are related. σ, the surface stress, is
the change in the total energy of the surface with a small change in its area.
AA
dA
Ad
∂∂+== γγγσ )(
(6. 94)
For the surface of a liquid the molecules are free to move and therefore cannot
sustain a bond stress. Thus, the second term in Eq. A
AdA
Ad
∂∂+== γγγσ )(
(6. 94) is
zero and the surface stress is equal to the surface energy. The final surface will be
just like the original yet there will be more or less of it. Solids can sustain stresses.
Thus, the second term in Eq. A
AdA
Ad
∂∂+== γγγσ )(
(6. 94) can be significant and is
of the same order of magnitude as the surface energy. It may be either positive or
negative. The choice of which surface quantity to use depends upon whether new
surface is being created (γ) or an existing surface is being modified (σ).
The free energy of the fully solid particle is
Fsolid = µ (T, P) Ns – PsVs + γsmAs (6. 95)
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γ is used because the solid matrix surface must be created, and µ (T, P), a function
of the local temperature and pressure, is the chemical potential per particle. With a
spherical geometry assumed, the surface area (As) and the volume (Vs) are just
functions of the number of atoms in the particle (Ns).
Here, and for the rest of the paragraph, a single subscript will refer to the volume
specified by the subscript, and a double subscript will refer to the surface specified by
the two subscripts. For example, in the above equation, the number of atoms in the
solid volume is Ns the surface area of the solid volume is As and the solid-matrix
surface energy is γsm
The local pressure (P) is the externally applied pressure (P0) plus the increase in
pressure due to the surface. The external pressure (P0) is all pressures except those
that arise from the particle itself. The increase in pressure, using the Laplace-Young
equation for a solid or liquid sphere of radius r, is
rPP
σ20 =− (6. 96)
The surface stress (σ) is used because the surface is attempting to compress the
particle and change its area. The chemical potential is analyzed about the bulk
melting point (T0) and the externally applied pressure (P0). We will assume that the
externally applied pressure is small enough that there is no shift in the bulk melting
point from standard values. At this reference point, the chemical potential per particle
is the same for the solid and liquid phases. Only first-order terms are kept, to give
( ) ( ) )()(,, 0000 PPP
TTT
PTPT −∂∂+−
∂∂+= µµµµ (6. 97)
That is
( ) ( ) )()(,, 0000 PPTTsPTPT −+−+= υµµ (6. 98)
Where s is the entropy per particle and υ is the volume per particle. The pressure
due to the surface can be significant, but because of the assumed incompressibility
and small linear expansion coefficient, we can neglect second-order terms in
pressure. The second-order term in temperature is at most a few percent of the first-
order terms and will also be neglected. The final free energy of a crystalline particle is
( ) ssmss
sms
s
smsssssolid AVP
rN
rNTTsNPTF γσσυµ +
+−
+−+= 0000
22)(, (6. 99)
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Where the terms involving the surface stress will cancel.
For the liquid particle with the crystalline core we must split these free energy terms
between the phases. The subscript l will refer to the liquid layer and r l will be the
radius of the liquid-matrix surface. The subscript c will refer to the solid core that is
surrounded by the liquid layer and rc will be its radius.
Fmixed = Fliquid + Fcore (6. 100)
( ) llmll
lml
l
lmllllliquid AVP
rN
rNTTsNPTF γσσυµ +
+−
+−+= 0000
22)(, (6. 101)
( ) cslcc
sl
l
lmc
c
sl
l
lmscsccore AVP
rrN
rrNTTsNPTF γσσσσυµ +
++−
++−+= 0000
2222)(,
(6. 102)
The volumes (υ ) and entropies (s) are labeled for the phase (solid or liquid) to which
they apply. Again, surface stress terms cancel. This cancellation of the surface stress
is a result of the assumed incompressibility of the phases. An incompressible
material can have no net work done by changes in the pressure. Therefore, the
change in the chemical potential is countered by another term to leave no net change
in the free energy due to pressure.
The free energy difference between these two situations is now
∆F = Fmixed - Fsolid (6. 103)
( )( ) ( )( ) ( )ssmcslllm
sclssccllscl
AAA
VVVPTTNsNsNsNNNPTF
γγγµ
−++−+−−−+−−+=∆ 0000 ,
(6. 104)
The total number of atoms is conserved (Nl + NC - Ns) = 0. Entropy terms can be
written as the difference between the liquid and solid entropy per particle and the
number of liquid particles.
( )( ) ( )
−−=−−+−
000 1
T
TNssTTTNsNsNs lslssccll (6. 105)
The latent heat (l0) per atom is (s l - ss)T0
( ) ssmcslllmscll AAAVVVPNT
TlF γγγ −++−+−
−=∆ 0
00 1 (6. 106)
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There is a bulk term due to the temperature difference from the bulk value, a volume
change term, and surface terms that are proportional to the surface areas. From
geometrical considerations given below, the PV terms can be rewritten as a function
of Nl.
( ) lsl
scl NPVVVP
−−=−+−
ρρ11
00 (6. 107)
Where ρ is the density (atoms per volume) of the labeled phase. This term is small
compared to the latent heat term and will be neglected. For example, at atmospheric
pressure this is about 10-4 times the latent heat term, unless the temperature is very
near the bulk melting point.
We would like to take into account the ordering or disordering induced by the surface.
Vanfleet and al. took their cue from Van Der Veen and they derived a correction to
the surface energies, due to surface-induced ordering or disordering. A surface will
attempt to propagate its order into the volume of which it forms the surface. The order
will be related to a correction to the energy for each unit of volume. This effect will be
screened by intervening material and is analogous to a screened potential. Using the
screened potential decay of the interaction we will write that the energy, due to the
surface, at a distance r from an area dA will be
dAr
r
dE
−
∝ξ
exp
(6. 108)
Where ξ is the scale length for the screening. The total contribution, at a point, due to
the surface would then be the integral over the surface of the above. For a flat
surface this integration gives C exp (-t / ξ), where C is the constant of proportionality
(units of energy per unit volume) and t is the perpendicular distance from the surface.
Therefore, for a flat surface, the total contribution to the energy due to the surface is
the integral over the volume of C exp (-t / ξ), plus a surface term.
∫∫
−+
−+=
2 2
22
1 1
11
* expexp dVz
CdVz
CAEsurface ξξγ (6. 109)
The 1’s and 2’s designate which side of the surface is the volume to be integrated. If
we limit our interest to a thickness t on each side of the surface and integrate we
have
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−−+
−−+=
2
222
1
111
* exp1exp1ξ
ξξ
ξγ tC
tCAEsurface (6. 110)
For t’s much larger than the scale lengths, the energy per unit area due to the
surface is 2211* ξξγγ CC ++= . This is the observed surface energy that we find in the
literature. The limit of t going to 0 gives γ*, which we will call the skin energy. If the
presence of the surface increases the order for a short distance into the bulk, the
energy will be lowered, and C will be negative. Increasing the disorder of the material
will raise the energy and C will be positive. If we now ask what is the energy per area
due to the surfaces, for non-interacting solid-liquid liquid-matrix surfaces a distance t
apart, where the solid and matrix are considered infinite, we have
ssll
llsmmll
llmsllm Ct
CCt
CA
E ξξ
ξξξ
ξγγ +
−−++
−−++= exp1exp1**
(6. 111)
Van Der Veen stated that there must be no difference between the energy of a solid
surface, and a solid surface with a liquid layer, if the liquid layer has zero thickness.
We will make that assumption but will see shortly that this imposes some conditions
under which this model can be applied. Requiring that the above equation give γsm in
the limit of t = 0 results in sslmmlsllmsm CC ξξγγγ +++= ** . Letting t now be large
compared to the scale length, the mismatch between the observed values of the
surface energies is
( ) llslmsslllsmmlllmsllmsmlmslsm CCCCCC ξξξξξγγγγγγγ +−=−−−−+−=−−=∆ ** (6. 112)
Because the presence of the surface will impose ordering on the liquid, both Clm and
Cls will be negative and therefore ∆γ is positive. We know that there are materials
where this is not the case. When ∆γ < 0 we cannot impose the constraint that
Eq. ssll
llsmmll
llmsllm Ct
CCt
CA
E ξξ
ξξξ
ξγγ +
−−++
−−++= exp1exp1**
(6. 111)
give γsm , when t = 0. ∆γ < 0 is also the non-wetting condition, therefore, a flat liquid
layer would be unstable in that situation. Thus, for the moment, we will confine our
interest to ∆γ > 0 and assume that Eq.
( ) llslmsslllsmmlllmsllmsmlmslsm CCCCCC ξξξξξγγγγγγγ +−=−−−−+−=−−=∆ ** (6.
112) holds. The energy of these two surfaces separated by a thickness t can now be
written as
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( )
( )
−−+−+=
−−++−+=
−
lsl
t
lmsmlm
llmslsmsm
tAE
tAE
l
ξγγγγ
ξγγγγ
ξ exp1
exp1
exp
(6. 113)
This last version is exactly the correction given by Van Der Veen.
To simplify the implementation with spherical geometry, we will make the
approximation that the corrections due to order or disorder propagating into the solid
or matrix are small compared to the contributions due to the order in the liquid. This
approximation will leave us with surface terms and the bulk liquid terms which will be
written as
γαξγαξ
∆−−=∆−=
)1(lls
llm
C
C (6. 114)
α will be between 0 and 1 and will be approximated for the experiment in question. α
divides ∆γ between the order imposed on the liquid due to the surface with the matrix
and that due to the surface with the solid. If the solid and matrix were very similar, we
would expect α to be near one half. The other extreme would be a very dissimilar
matrix or vapor, in which case α would be small but non zero.
Even if the matrix has no order to impose on the liquid, the surface itself will impose
some order. Therefore, α will never be zero.
Going back to integrating over the surface, but changing to a spherical geometry we
get
( )
+−−
−−∝
ξξrRrR
r
RrE expexp)( (6. 115)
R is the radius of the surface and r is the radius of the point in question. The total
energy due to the surface will again be found by integrating the above equation over
the volume. Using the approximation made above, we will limit our interest to
corrections to the bulk surface energies that involve ∆γ. Using the geometry of a
liquid sphere with a solid core, the energy of the solid-liquid and liquid-matrix
surfaces is
sllmslsllmlm GGAAE +++= γγ (6. 116)
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( ) ( ) ( )
−+−
+−++
−−−+∆=
l
lll
l
cllc
l
cllclllm
rr
rrr
rrrrG
ξξ
ξξ
ξξξγαπ 2
exp)(
exp)(
exp4
(6. 117)
( ) ( ) ( )
−++
+−+−
−−−+−∆−=
l
clc
l
slll
l
sllllcsl
rr
rrr
rrrrG
ξξ
ξξ
ξξξγαπ 2
exp)(
exp)(
exp)1(4
(6. 118)
This is more complicated than the simple form given by Van Der Veen, but it takes
into account the disappearance of the solid-liquid surface’s correction when the solid
core disappears.
Using the new correction to the solid-liquid liquid- matrix surface energies results in
the following free energy difference.
sllmssmcslllml GGAAANT
TlF ++−++
−=∆ γγγ
00 1 (6. 119)
This can be written as a function of the number of atoms in the liquid layer by using
the total number of atoms, the solid and liquid densities, and the known geometry to
get
3
1
4
3
=
s
ss
Nr
ρπ (6. 120)
3
1
4
3
−=
s
lsc
NNr
ρπ (6. 121)
3
1
4
3
+
−=
l
l
s
lsl
NNNr
ρρπ (6. 122)
The thickness of the liquid layer, the volumes, and the areas of each surface follows
normally from these radii.
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Figure 6-6 : Free energy difference from crystalline state as function of fraction of particles in melted
layer. Data are for a lead particle r = 5.0 nm at three different temperatures. (A) T=480 K,
(B) T=502 K and (C)T=540 K. [R.R. Vanfleet, J.M. Mochel, Thermodynamics of melting and freezing in
small particles, Surface Science, 1995, p.p. 40-50]
Figure 6-6 shows the free energy difference versus the fraction of the atoms in the
liquid layer for three different temperatures. The data are for an r = 5 nm lead particle
and the temperatures are as follows: (A) is T=480 K, (B) is 502 K, and (C) is 540 K.
For (A) and (B), there is a minimum in the energy for a liquid layer of finite thickness.
(B) corresponds to the layer being one monolayer in thickness. This minimum arises
from the propagation of the surface order into the liquid, and corresponds to surface
melting as predicted by Van Der Veen et al. []. (A) and (B) also show another local
minimum with all atoms melted, at the extreme right. This second minima is
separated from the surface melted state by an energy barrier. For (C), all atoms in
the liquid state is the minimum of energy. For a given material, both the presence or
the absence of these minima and the height of the barrier are functions of the size of
the particle, the temperature, the surface energies, the scale length, and the dividing
parameter α.
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Figure 6-7 : Melting point (M), freezing point (F), and temperature at which there is a monolayer of
surface melt (S) for lead, as a function of radius of the particle. G is the region below which a cluster
once disordered will remain disordered. Arrows indicate direction of temperature change when
transition occurs. [R.R. Vanfleet, J.M. Mochel, Thermodynamics of melting and freezing in small
particles, Surface Science, 1995, p.p. 40-50]
We have used bulk parameters in Figure 6-7 to calculate melting point, freezing
point, and temperature where the surface melting is equal to one monolayer in
thickness, as functions of particle radius for lead. Figure 6-7 also shows the region
where a disordered particle, once created, would remain disordered. This is the
“frozen in” region. The arrows indicate the direction of the temperature change for
which the transition would be expected to occur. The curves were calculated with a
simple approach of choosing a particle size and then adjusting the temperature until
the melting point, freezing point, surface melt and “frozen in” temperatures were
found. All these points were then strung together to form the curves in Figure 6-7 and
Figure 6-8.
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Figure 6-8 : Melting point (M), freezing point (F), and temperature at which there is a monolayer of
surface melt (S) for platinum, as a function of radius of the particle. G is the region below which a
cluster once disordered will remain disordered. Arrows indicate direction of temperature change when
transition occurs. [R.R. Vanfleet, J.M. Mochel, Thermodynamics of melting and freezing in small
particles, Surface Science, 1995, p.p. 40-50]
For a given size and temperature, the free energy as a function of the number of
atoms in the liquid layer is calculated. The position and depth of the surface melt
minimum is found and the position is compared to the number of atoms in a single
monolayer. Based on this comparison, the temperature is either raised or lowered
and the calculation is repeated. This process is continued until the surface melting
temperature is found to within a step size. The melting point is based upon the
difference between the surface melt minimum and the maximum free energy.
The main features of Figure 6-7 are a difference between the melting and freezing
points, and the disappearance of this difference for small enough particles. In the
region of differing melting and freezing points, the dominant factor in determining
these critical temperatures is the height of the energy barrier. A lower temperature is
required for the liquid-to-solid transition to occur than for the solid-to-liquid transition.
The difference between melting and freezing disappears when the barrier becomes
so small that it can be easily overcome in either direction. At this point the absolute
stability becomes the important factor leading to identical melting and freezing points.
Another feature of Figure 6-7, is the lack of surface melting for the smallest particles.
This is where the melting barrier is surmountable before a surface melt layer will
form. The line labeled “G” in Figure 6-7, indicates the temperature below which a
particle once disordered would stay disordered due to a high free energy barrier.
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6.3 Phase Diagrams
In order to determine the phase diagram of a small system whose thermodynamic
equilibrium is a metastable one determined in part by the size of the system, isolated
particles will be considered. The transition solid-liquid results in an equilibrium
between the solid and its melt. Further, in view of the experimental observation that
many liquids have a zero contact angle with their solids, this wetting feature of the
system will be incorporated in the model by having the solid in contact with its melt.
Even though the isolated particle will likely be crystalline and hence faceted, as a
simplification, spherical shapes will be assumed. With these initial assumptions and
conditions, the geometry of the system to be modeled is an isolated solid sphere
transforming on heating to a specific temperature to a liquid layer surrounding a
spherical solid core. The initial state of the particle before melting is a solid sphere.
Upon melting the solid sphere develops a sheath of liquid which will completely
surround a small solid core. This case is illustrated in Figure 6-9.
Figure 6-9 : Schematic drawing of isolated particle before melting (complete solid) and after melting
(liquid sheath exhibiting zero contact angle with the solid). [W.A. Jesser, G.J. Shiflet, G.L. Allen, J.L.
Crawford, Equilibrium phase diagrams of isolated nano-phases, Mat. Res. Innovat, 1999, p.p. 211-
216]
6.3.1 Governing equations
First, the thermodynamic potential that is a minimum at equilibrium is considered. For
an infinite reservoir in contact with a small, spherical particle through a wall that is
rigid (no change in volume of the reservoir), diathermal (heat exchanges may occur
with the reservoir), and impermeable (no exchange of matter between the system
and reservoir), it has been shown (see previous section) that the Helmholtz potential
is minimized at equilibrium. For the more realistic case of an infinite reservoir in
contact with the particle through a wall that is flexible, diathermal, and impermeable,
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the Helmholtz potential is minimized at equilibrium in this case too, provided the
reservoir pressure is zero. Accordingly, thermodynamic equations for the Helmholtz
potential will be applied to the model of a sphere of solid isothermally melting to form
a spherical shell of liquid surrounding a solid core.
The approach is to express in analytical form the difference ∆F in Helmholtz energy
between the final state, a liquid sheath of thickness t surrounding a solid core of
radius Rs, and the initial state, a solid sphere of radius R0. The initial and final states
can be written in terms of the chemical potential µ at the actual pressure P,
temperature T, and composition Xi in the small particle of volume V and area A and
the number of atoms in the spherical particle Ni, where the subscript i refers to a
particular component, and the surface energies of the solid, liquid, and solid-liquid
interface are γs, γl, γsl respectively.
The Helmholtz energy F0 of the initial state can be written
ssss
i
si
si AVPNF 000000 γµ +−=∑ (6. 123)
Where the subscript 0 refers to the initial state of a solid sphere and the superscripts
s or l refer to the solid, or liquid phases. The Helmholtz energy for the final state of a
liquid sheath of thickness t surrounding a solid core of radius Rs can be written as
ssss
i
si
si
llll
i
li
lif AVPNAVPNF γµγµ +−++−= ∑∑ (6. 124)
The Helmholtz energy difference between the final and initial states, ∆F is plotted
against the composition and fraction of liquid for constant materials parameters in
order to determine whether and where any minima exist on the energy surface. It will
be assumed that a regular solution model is adequate to represent the bulk chemical
potentials and that a linear relationship exists between the surface and interfacial
energies and composition.
Using a regular solution approach one can incorporate interaction parameters ωl for
the liquid and ωs for the solid. These parameters are taken as temperature, pressure
and composition independent. The final and simplified expression for the Helmholtz
free-energy difference ∆F can be written as
( ) ( )( ) ( )
( )[ ]slssl
BAss
BsA
sllB
lA
lBBAA
sB
sB
sA
sA
slB
lB
lA
lAB
lBA
lA
l
AA
NXXNXXNXXXkTNXkTN
XXXXkTNXXXXCXCXkTNF
γγγωω
+−+
−++−−
+++++=∆
0000000 lnln
lnlnlnlnlnln
(6. 125)
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Where
ssB
sB
ssA
sA
llB
lB
llA
lA NNXNNXNNXNNX /...,/...,/...,/... (6. 126)
With
lB
lA
l NNN += and sB
sA
s NNN += (6. 127)
and
ssB
llBB
ssA
llAA NXNXNNXNXN ++ ...,... 00 (6. 128)
For convenience ( )bulkllss
BA XXC ΓΓ /..., represents the bulk compositions and activity
coefficients for components A and B respectively for the case of the bulk phase
diagram.
The relationship between interaction parameter ω and activity coefficient Γ is
( ) lsi
lsi kTX ,, ln21 Γ=− ω (6. 129)
6.3.2 Mathematical description for nano-phases of S n-Bi alloys
To find local minima in the Helmholtz free-energy difference between a spherical
solid particle and a liquid sheath surrounding a solid core, equation
( ) ( )( ) ( )
( )[ ]slssl
BAss
BsA
sllB
lA
lBBAA
sB
sB
sA
sA
slB
lB
lA
lAB
lBA
lA
l
AA
NXXNXXNXXXkTNXkTN
XXXXkTNXXXXCXCXkTNF
γγγωω
+−+
−++−−
+++++=∆
0000000 lnln
lnlnlnlnlnln
(6.
125) is plotted as a function of composition of the solid core and fraction of liquid
present in the two-phase state thereby generating an energy surface. The calculation
requires a number of terms to be specified. Typically one should have data for the
dependence of interface and surface energies as a function of composition and even
the interaction parameter ω used in the regular solution approximation can be limited
to a fixed composition range. In this section the calculations for a binary phase
diagram of tin and bismuth will be given. The solidus and liquidus lines of the Bi-rich
and Sn-rich sides of the bulk phase diagram provide the reference state of the bulk
phase diagram from which the small particle system of Bi-Sn is calculated. For the
Sn-rich side of the bulk phase diagram the equation of the solidus is closely
approximated by
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883.01062.6²1073.5 36 −×+×−= −− TTX sSn (6. 130)
Table 6-1 : Values of parameters employed for the B i-Sn binary phase diagram. [W.A. Jesser,
G.J. Shiflet, G.L. Allen, J.L. Crawford, Equilibrium phase diagrams of isolated nano-phases, Mat. Res.
Innovat, 1999, p.p. 211-216]
and that of the liquidus by
70.11023.61076.1 326 −×+×−= −− TTX lSn (6. 131)
while for the Bi-rich side of the phase diagram the equation of the solidus is taken as
639.11001.3 3 +×−= − TX sBi (6. 132)
and that of the liquidus as
06559.010204.1 4 +×−= − TX lBi (6. 133)
The values of the parameters used in equation 2 may be found in Table 6-1.
When the above parameters are substituted into equation
( ) ( )( ) ( )
( )[ ]slssl
BAss
BsA
sllB
lA
lBBAA
sB
sB
sA
sA
slB
lB
lA
lAB
lBA
lA
l
AA
NXXNXXNXXXkTNXkTN
XXXXkTNXXXXCXCXkTNF
γγγωω
+−+
−++−−
+++++=∆
0000000 lnln
lnlnlnlnlnln
(6.
125), a ∆F surface versus composition and fraction liquid may be plotted and
analyzed for the presence of local minima. As an illustrative example of a typical ∆F
surface Figure 6-10 is presented for a binary bismuth alloy with lead for two
temperatures.
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Figure 6-10 : Helmholtz free energy surface plotted for the case of a binary bismuth alloy as a function
of composition and fractional amount of liquid for two temperatures. a) intermediate temperature
corresponding to a metastable solid-liquid equilibrium, b) elevated temperature showing a stable
equilibrium corresponding to complete liquid. [W.A. Jesser, G.J. Shiflet, G.L. Allen, J.L. Crawford,
Equilibrium phase diagrams of isolated nano-phases, Mat. Res. Innovat, 1999, p.p. 211-216]
Here one finds a relative minimum whose location is indicated by the broad dot and
an absolute minimum located at the point where only liquid exists. These two minima
are separated by a saddle-point maximum that prevents the system from jumping this
energy barrier. Accordingly, the temperature of this plot corresponds to a metastable
solid-liquid equilibrium. At significantly lower temperatures no local minimum exists
and complete solid is the stable state. At significantly higher temperatures, again no
local minimum exists and the saddle-point maximum disappears so that the free
energy surface is one that continuously decreases as one moves toward the point of
minimum free energy which corresponds to complete liquid. Figure 6-10 b)
represents this latter condition in which the minimum is indicated by the broad dot.
There is no progression of the local minimum continuously to the condition of
complete liquid. Rather there is a discontinuous, abrupt shift of the minimum from a
location away from the point of complete liquid to the point of complete liquid once
the saddle-point maximum disappears. Physically this corresponds to heating the
two-phase particle to a temperature at which a discontinuous transition occurs from
the state of a liquid sheath surrounding a solid core to the state of complete liquid.
Such an abrupt transition is experimentally observed for this binary alloy as well as
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for the solid elements tin and bismuth both of which exhibit no stable solid-liquid
equilibrium.
6.3.3 Phase diagram for isolated nano-phases of Sn- Bi alloys
A temperature-composition phase diagram for isolated nano-spheres of Sn-Bi alloys
can be constructed as a function of size of the isolated spherical particle. This is
accomplished by tracking of the position of the local minimum in the Helmholtz free
energy surface as a function of temperature for several values of radius of the initial
solid sphere. The resultant phase diagram for the liquid-solid phase equilibria, i.e.,
the location of the liquidus and solidus lines are shown in Figure 6-11 for the case of
a spherical particle 40 nm in diameter.
Figure 6-11 : Phase diagram for the binary alloy Sn -Bi. The bulk phase diagram is shown
superimposed on the phase diagram for isolated spheres of alloy 40 nm in diameter. The nano-particle
phase diagram is the shaded region as given by the thermodynamic calculations and the data points
for each of eight compositions as determined from in-situ TEM measurements. [W.A. Jesser, G.J.
Shiflet, G.L. Allen, J.L. Crawford, Equilibrium phase diagrams of isolated nano-phases, Mat. Res.
Innovat, 1999, p.p. 211-216]
This Figure also includes the phase diagram for bulk Sn-Bi alloys and the
experimentally measured liquidus and solidus lines for the isolated spherical
particles. The experimental data reported here were measured inside a transmission
electron microscope modified for ultra high vacuum so that in-situ deposition of the
isolated nanospheres could be performed. These new data show that for the Sn-rich
side of the phase diagram, the liquid-solid two-phase field pinches off to a line
thereby indicating no stable equilibrium between the liquid and solid phases in this
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region. There was no detectable pinching off observed on the Bi-rich side of the
diagram.
In addition to the pinching off of the two-phase field near the composition of pure tin,
there are two other noteworthy features in the phase diagram. First, one notices that
there is a substantial increase in solubility for the nano-particles as compared to that
in the bulk alloy. The solubility limit of tin in bismuth increased from 1.5% (bulk) to
about 43% in particles of radius 20 nm. A strong increase also occurred in the tin-rich
phase.
Second, because the initial melting event is observed to be discontinuous (the liquid
sheath forms with a minimum finite thickness of about 2 nm) and the final melting
event is also a discontinuous event with the isothermal jump from a thick liquid
sheath surrounding a solid core to a completely liquid sphere, the location on the
phase diagram of the locus of the temperatures at which these discontinuous
transitions occur do not represent liquidus and solidus lines that can be used to
construct the tie lines. The conventional tie lines that describe mass conservation and
are useful for lever-rule calculations extend isothermally beyond the two phase
regions. If a tie line is constructed between the liquidus and solidus lines shown on
the phase diagram, its ends do not correspond to either the solid composition or the
liquid sheath composition. In order to account for these compositions a longer tie line
is required. As the temperature is increased, the tie line changes its length. The two
ends of the tie line demark the compositions of the liquid and solid phases and
sweep out two areas on the phase diagram both of which are detached from the
drawn liquidus and solidus lines that demark the extent of the two phase field. This
feature is shown in Figure 6-12.
Figure 6-12 : Sn-rich segment of a Sn-Bi phase diagram for an isolated sphere of radius 20 nm
showing that the areas swept out by the ends of the tie lines are detached from the two-phase region.
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Note that the two-phase regions and the ends of the tie lines pinch off at about 5% Bi. [W.A. Jesser,
G.J. Shiflet, G.L. Allen, J.L. Crawford, Equilibrium phase diagrams of isolated nano-phases, Mat. Res.
Innovat, 1999, p.p. 211-216]
One can incorporate the size-dependence of the phase diagram by making a three-
dimensional plot of liquidus and solidus temperatures as a function of both
composition and size. One can incorporate a size variable in the phase diagram by
recognizing that the Helmholtz free energy depends upon pressure which exhibits a
discontinuity ∆P across a curved surface of
∆P = 2γ / R (liquids) (6. 134)
Accordingly the equilibrium lines shift approximately linearly with reciprocal radius.
The axes of the three-dimensional phase diagram then become temperature,
reciprocal radius, and composition. Figure 6-13 shows a 3-D phase diagram for the
Sn-Bi alloy system.
Figure 6-13 : 3-D phase diagram for an isolated tin-bismuth alloy sphere showing the size
dependence of the liquidus and solidus lines as well as the eutectic isotherm over the range of
reciprocal radius 0 nm–1 to 0.05 nm–1. Over this size range the eutectic temperature changes from 412
to 320 K and the solubility of Sn in Bi increased from 1.5% to 43%. The dashed lines on the left margin
are reference lines parallel to the 1/R axis. The solid lines on the left margin are projections of the
melting and eutectic temperatures. The solid lines on the bottom margin are projections of the
solubility limits and the eutectic composition. [W.A. Jesser, G.J. Shiflet, G.L. Allen, J.L. Crawford,
Equilibrium phase diagrams of isolated nano-phases, Mat. Res. Innovat, 1999, p.p. 211-216]
6.4 Crystal-lattice inhomogeneous state
In nanoparticles and nanocrystallities the fraction of ‘surface’ (or
‘interface’) atoms is large. Then, appreciable distinctions between nano-objects
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and massive bodies in lattice parameters and the types of atomic structures are
plausible. These difference manifest themselves in normal and tangential
relaxations of nanoparticles.(54) An atom on the surface has fewer neighbours
than one in the volume, all the neighbours of the former being located on one
side only. This breaks the equilibrium of the interatomic forces and gives rise to
both the change of interatomic spacing (normal relaxation) and tensile
deformations which ‘smooth’ verticles and edges (tangential relaxation).
A rough estimate of the average distortion of the interatomic distances a
in nanoparticles caused by surface tension is shown below: (55)
'2 ra
a γκθ=∆ (6. 135)
where θθθθ2 is the numerical coefficient of the order of 1 (in the case of isotropic
spherical particles it may be estimated as 2/3), κκκκ is the volume compressibility
coefficient. According to the estimate, the lattice construction is about several
tenths of a percent at typical values of parameters κκκκ ~ 10-11 m3/J, γγγγ ∼∼∼∼ 1111 J/m 2 for
nanoparticles with radius r ~ 10 nm .
Here it should be mentioned that parameters entering this formula are, in
turn, size-dependent. The surface energy of nanoparticles, for example,
depends on r as follows: (56)*
+
+−= ...1'2
43 r
a
r
a θθγγ (6. 136)
where numerical coefficient κκκκ ∼∼∼∼ 11110000−−−−11111111m3/J, γγγγ ∼∼∼∼ 1111 J/m 2 . According to this estimate
the surface energy of nanoparticles decreases as r decreases which is
intuitively clear, since both the number of bonds formed by the surface atoms
and their bond energy are reduced as size decreases. Size-dependent
corrections are, in a general case, functions of temperature.(59) The difference in
the lattice parameters of contacting nanoparticles may seriously influence the
process of interparticle sliding.
For several decades contradictory experimental results were
calling into question eq. '2 ra
a γκθ=∆ (6. 135) for the contraction of
interatomic spacing. Some of these ambigious results are caused by
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experimental artifacts and difficulties, while others are associated with the
complicated spatial distribution of lattice parameter distortions. In the absence of
oxide layers,(60) absorbed impurities, etc., there is an average overall contraction
in the lattice parameter of a nanoparticle. However, interatomic spacing in the
outer part of nanoparticles may be expanded despite the contracted interior.(61)
Moreover, the degree of distortion varies with crystallographic as a result of the
anisotropy of the volume compressibility coefficient.(62,63) In summary, the
experimental and theoretical results show that the average contraction of the
lattice parameter in the nanoparticles is inversely proportional to nanoparticles
radius r.
6.5 Concentrational inhomogeneity
The spatial distribution of a second component over the bulk of
heterogeneous nanoparticles and nanocrystallities may contrast with that for
analogous macrocrystals. A non-uniform distortion of the crystal lattice in
nanovolumes (see the previous section) and the comparability of the width of
segregated layers ∆∆∆∆s, with the radius r may both bring about an inhomogeneous
distribution of the second component.
One may estimate the characteristic of a nano-object when the
segregated layer occupies half its volume. A simple geometrical consideration
gives the following relationship:(65)
ssr ∆=∗5θ (6. 137)
Where the numerical coefficient θθθθ5 is of the order of unity. For spherical
nanoparticles (nanocrystallites) it is close to 5. Using ∆∆∆∆s ≈≈≈≈ 3a as typical width of
the segregated layers the corresponding diameter of nanoparticles
(nanocrystallites) is d ~10nm . Thus, under some conditions the multicomponent
nanocrystals with grains of the size may be considered ‘massive’ segregated
layers.
If the main contribution of driving force for segregation is the mismatch in
atomic radii of the matrix and foreign atoms, ∆∆∆∆s and r*s may be estimated for a
heterophase nanocrystal. The mismatch may be described by the relative
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difference δδδδs = (af / a – 1) where af is the effective radius of a foreign atom. If the
nanocrystallites in a nanocomposite the components of which have shear
moduli G and G’ are elastically isotropic and possess non-slipping boundaries,
the characteristic length l*sl is as follows:(65)
( ) 3/132
6
'
−=Ι kT
GGaal s
s
δθ (6. 138)
Where the numerical coefficient θθθθ6 ≈≈≈≈ 3.5, k is the Boltzman constant, and
T is temperature. This estimate was obtained by equation kT to the energy of
the elastic interaction of the foreign atom with the non-slipping interface.
Provided that other driving forces for segregation (heat of mixing, the difference
between surface energies of the component) are negligible or cancel each
other, when a nanocrystallite has a shear modulus G less than that G’ of the
neighbouring crystallites segregation in that nanocrystallite becomes impossible.
So, the concentration of foreign atoms in the vicinity of interfaces diminishes in
elastically softer nanocrystallites.
Segregation effects change the composition of the interior of
nanoparticles. It can be shown (66) that the average concentration c of the
second component for a two-component nanoparticle varies with its size,
following the relationship:
( )r
cccc sbsb
∆−+= 8θ (6. 139)
where the numerical coefficient θθθθ8888 ≈≈≈≈ 3, cs and cb are concentrations in the
surface layer of width ∆∆∆∆s and in bulk of nanoparticles, respectively. If
concentrations cs and cb were independent of r over the whole range of sizes,
this relationship would be of a general character. When size r is not too small,
one may estimate the change in the concentration of foreign atoms in the
smaller the size r, the stronger the concentrational inhomogeneity which may
reach as much as 10%. This conclusion agrees with the numerical simulations
of small-atomic clusters (67) and experiments (68) with CuNi nanoparticles.
Certain conclusions about the character of the independence of surface
concentration cs on nanoparticle (nanocrystallite) radius r can be drawn on the
basis of simple mass balance considerations. In the limit of small nanoparticles,
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when all the atoms belong to the surface, the surface composition in an
ensemble of nanoparticles would be identical to the average composition in the
massive alloy. (69) Such a decrease of the surface segregation has been
observed experimentally. (70) In the opposite of very large particles they will
behave like semi-infinite crystals, and the surface composition would be similar
to that obtained in microscopic samples. In the intermediate range of sizes,
where the number of surface sites is not a negligibly small fraction of the total
number of sites available in the particle volume, there exists a regime where
segregation to the surface tends to decrease the bulk concentration of the
segregant.
The depletion of the segregated atoms, which would follow from
elementary mass-balance considerations, may be impossible if the physical
and/or chemical properties are size-dependent. For instance, the change in the
surface energy (see eq.
+
+−= ...1'2
43 r
a
r
a θθγγ (6. 136)) may modify the
segregation effect. The lattice contraction (eq. '2 ra
a γκθ=∆ (6. 135)) and the
external pressure also affect segregation phenomena.
In their effects on segregation, applying external pressure and
contracting the crystal lattice of nano-objects are to a certain degree equivalent
to the difference in the atomic radii of the segregant and matrix atoms, (71) which
is described by the relative size mismatch δδδδs. In fact, external pressure P is
connected with the change in the relative size mismatch δδδδs by the simple
relationship (cf. eq. '2 ra
a γκθ=∆ (6. 135)):
( )213κκδ −=∆ P
s (6. 140)
where κκκκ1 and κκκκ2 are the volume compressibility of the segregant and the
solvent, respectively. The hydrostatic stress include by surface tension,
approximated as 2γγγγ / l, may, thus, change the relative size mismatch by ∆∆∆∆δδδδs.
According to the well-known Hume-Rothery rule (if the relative size mismatch
exceeds the 15% limit, solid solubility will be restricted (72)) a critical size l*sol
should exist, above which the solid solubility remains qualitatively the same:
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( )s
sol δκκγθ
−−
=∆∗
15.021
2 (6. 141)
6.6 References
C.X. Wang, G.W. Yang, Thermodynamics of metastable phase nucleation, Materials
Science and Engineering, 2005, p.p. 157-202
T. L. Hill, A Different Approach to Nanothermodynamics, Nano letters, 2001 Vol.1,
N°5, p.p. 273-275
C. Beck, Non-additivity of Tsallis entropies and fluctuations of temperature,
Europhysics Letters, 2002, p.p. 329-333
R.R. Vanfleet, J.M. Mochel, Thermodynamics of melting and freezing in small
particles, Surface Science, 1995, p.p. 40-50
W.A. Jesser, G.J. Shiflet, G.L. Allen, J.L. Crawford, Equilibrium phase diagrams of
isolated nano-phases, Mat. Res. Innovat, 1999, p.p. 211-216
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7 ELECTRONIC AND OPTICAL PROPERTIES OF
NANOMATERIALS
7.1 Introduction
Energy
Forbidden bandgap HOMO
LUMO
Molecule
Nanoparticle
Semiconductor Metal
Crystalline
Atom
Cluster
Figure 7-1: Size quantization effect. Electronic state transition from bulk metal/semiconductor to small
cluster.
Nanocrystalline particles represent a state of matter in the transition region
between bulk solid and single molecule. As a consequence, their physical and
chemical properties gradually change from solid state to molecular behaviour with
decreasing particle size. The reasons for this behaviour can be summarised as two
basic phenomena.
First, as we have already discussed, owing to their small dimensions, the
surface-to-volume ratio increases, and the number of surface atoms may be similar
to or higher than those located in the crystalline lattice core, and the surface
properties are no longer negligible. When no other molecules are adsorbed onto the
nanocrystallites, the surface atoms are highly unsaturated and their electronic
contribution to the behaviour of the particles is totally different from that of the inner
atoms. These effects may be even more marked when the surface atoms are
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ligated. lxiii This leads to different mechanical and electronic transport properties,
which account for the catalytic properties of the nanocrystalline particles.lxiv
The second phenomenon, which only occurs in metal and semiconductor
nanoparticles, is totally an electronic effect. The band structure gradually evolves with
increasing particle size, i. e., molecular orbital convert into delocalised band states.
Figure above, shows the size quantization effect responsible for the transition
between a bulk metal or semiconductor, and cluster species. In a metal, the quasi-
continuous density of states in the valence and the conduction bands splits into
discrete electronic levels, the spacing between these levels and the band gap
increasing with decreasing particle size.lxv,lxvi
In the case of semiconductors, the phenomenon is slightly different, since a
band gap already exists in the bulk state. However, this band gap also increases
when the particle size is decreased and the energy bands gradually convert into
discrete molecular electronic levels.lxvii If the particle size is less than the De Broglie
wavelength of the electrons, the charge carriers may be treated quantum
mechanically as "particles in a box", where the size of the box is given by the
dimensions of the crystallites. lxviii In semiconductors, the quantization effect that
enhances the optical gap is routinely observed for clusters ranging from 1 nm to
almost 10 nm. Metal particles consisting of 50 to 100 atoms with a diameter between
1 and 2 nm start to loose their metallic behaviour and tend to become
semiconductors.lxix Particles that show this size quantization effect are sometimes
called Q-particles or quantum dots that we will discuss later in this chapter.
Certain particle sizes, determined by the so-called magic numbers, are more
frequently observed than others. This is not only due to the thermodynamic stability
of certain structures but is also related to the kinetics of particle growth. In
semiconductor nanoparticles such as CdS, the growth of the initially formed smallest
particles with an agglomeration number k occurs by combination of the particles.
Thus, particles so formed would have the agglomeration number of 2k, 3k and so
on. lxx Metals have a cubic or hexagonal close-packed structure consisting of one
central atom, which is surrounded in the first shell by 12 atoms, in the second shell by
42 atoms, or in principle by 10n2+2 atoms in the nth shell.lxxi For example, one of the
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most famous ligand stabilized metal clusters is a gold particle with 55 atoms (Au55)
first reported by G. Schmid in 1981.lxxii
7.2 Metals
7.2.1 Introduction
In drive towards nanotechnology, there is tremendous interest in understanding the
electric and thermal transport through nanowires. The questions that need to be
answered pertain to how the electric and heat transport laws that govern the
macroscopic systems get modified at the level of a few molecules or atoms. Wires
and contacts having average radius R, in the range of the Fermi wavelength λF, can
indeed be fabricated and have shown unusual mechanical and electronic properties.
The effects of reduced size and dimensionality, and their quantum properties have
been investigated actively in the last decade. Ultrathin nanowires and even
monatomic chains suspended between two metal electrodes have been produced.
The discovery of scanning tunnelling microscopy (STM) and later atomic force
microscopy (AFM) has sparked a tremendous recent research effort on nanoparticles
because many properties have become ‘directly’ observable. The tunnelling
microscope has also made it possible to produce atomic size contacts and atomic
wires only several angstroms in length. Electrical conduction and force have been
measured concomitantly in the course of stretching of nanowires. Initially, much of
the work was carried out on GaAs–AlGaAs heterostructures which were grown to
contain a thin conducting layer at the interface. The conducting layer is treated as a
two-dimensional (2D) electron gas in which a narrow constriction of desired width w,
and length l, can be created by applying a negative gate voltage.
The conductivity σ, which relates the electric current density to the electric field by
j = σE, is expressible in terms of the areal charge density ρS for 2D electron gas of
effective mass m∗∗∗∗ through the equation
*
2
m
eS τρσ = (7. 142)
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The experimental quantity of interest, however, is the conductance G = I /V, which is
the ratio of the total current I to the voltage drop V across the sample of length l in
the direction of current flow. For 2D, since I = wj , one can also write
l
wG σ= (7. 143)
For a 3D conductor, this relationship is valid provided that w is replaced by the cross
sectional area A orthogonal to the current flow direction. Similar expressions are also
valid for the thermal transport of energy. The thermal conductance related to the
energy (or heat) current Jx through a sample between two reservoirs is given by
Kx = Jx ∆T , where ∆T is the temperature difference. Depending on whether the
energy is carried by electrons (x = e) or by phonons (x = p ) the thermal conductance
is identified as electronic Ke or phononic Kp. Here our focus is on quantum transport
through materials of very small dimensions.
Novel size-dependent effects emerge as w and l are reduced towards atomic
dimensions in the nanometre range. The relationship expressed by equation l
wG σ=
(7. 143) holds in the diffusive transport regime where both w and l are greater
than the mean free path. As the width of the constriction decreases, there comes a
point where quantum mechanics makes its presence known. The quantum
confinement of a carrier in a strip of width w leads to the discretization of energy
levels given by
єn = n2h2/(8m∗∗∗∗w2). The number of these w-dependent transverse modes, which are
occupied, determines the conductance. Thus, rather than a simple linear
dependence of G on w, quantum mechanics forces this ‘indirect’ dependence on w.
As w is altered, the energy spectrum changes and so does the number of occupied
modes below the Fermi energy and hence the conductance.
Simple physical considerations show that the number of transverse occupied states,
N ∼∼∼∼ 2w/λF, increases with the width of the constriction. Since all these modes can
contribute to the conductance, one still expects the conductance to increase linearly
with w even in the nanodomain. This is almost the case, except with one important
distinction. The width w can change continuously, at least in principle. But the
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number of modes, being an integer, can only vary in discrete steps. The concept
becomes physically more transparent if we rewrite n
~∈
FEnN . The highest occupied
єn = EF gives N from a simple counting of the number of discrete eigenvalues. Since
it is a rare coincidence for an eigenvalue to exactly align with the Fermi energy, N is
taken to be the integer which corresponds to the highest occupied level just below
EF. Thus the effect of quantum mechanics due to the reduced dimension w is to
cause the conductance to change in discrete steps in a staircase fashion. We have
yet to find the step height. This simple view is necessarily modified when other
factors explained later start to play a crucial role. Obviously, one then requires a
more detailed analysis.
The characteristic length that comes into play is λF, since only those electrons having
energy close to the Fermi energy carry current at low temperatures. If w >> λF, the
number of conducting modes is very large and so is the conductance. This is typical
in metals where λF ∼∼∼∼ 0.2 nanometres is very small and consequently the observation
of discrete conductance variation requires w ∼∼∼∼ λF. That is why the discrete
conductance behaviour was first observed in semiconductor heterojunctions having
very low electron density and λF two orders of magnitude larger than in metals.
The effect of reduced length l on the conductance is even more striking. If the ohmic
regime were to hold in equation l
wG σ= (7. 143), one would expect G to
increase without limit (or resistance to reach zero) as l was reduced towards zero.
We shall see that there would always be finite residual resistance. Heisenberg’s
uncertainty principle defines a natural characteristic length as the mean free path le. If
l < l e, carriers can propagate without losing their initial momentum, and this domain is
referred to as the ballistic transport regime. Landauer pointed out that ‘conduction is
transmission’. If we follow the standard definition, that the conductance is a measure
of current through a sample divided by the voltage difference,
VG
∆Ι= (7. 144)
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The quantum of electrical conductance occurs naturally as a consequence of
Heisenberg’s uncertainty principle. The current is given by the rate of charge flow
I = ∆Q/ttransit . Now charge is quantized in units of elementary charge, e. Hence in
the extreme quantum limit, setting ∆Q = e and recognizing that the transit time
should then be at least in the range of time implied by Heisenberg’s uncertainty
principle, we get
t
e
∆=Ι (7. 145)
Combining equation V
G∆
Ι= (7. 144) and equation t
e
∆=Ι (7. 145), and
using the fact that the potential difference, ∆V, is equal to electrochemical potential
difference ∆E divided by the electronic charge e, one gets
tE
eG
∆∆= ²
(7. 146)
Next invoking Heisenberg’s uncertainty principle, ∆E ∆t >> h , the expression for
ballistic conductance including the spin degeneracy becomes G = 2e2/h in the ideal
case. It has a maximum value of 8×10−5 Ω−1. This is the step height, the conductance
per transverse mode.
The corresponding resistance h/2e2 has a value of 12.9 kΩ (it suffices to call it
12345Ω for ease of remembrance) and is attributed to the resistance at the contacts
where the conductor is attached to the electrodes or electron reservoirs. Thus
classically G ∝∝∝∝ w. Quantum mechanically it increases in discrete steps. It jumps by
2e2/h as w increases enough to permit one more transverse mode to be occupied
and hence available for conduction. More recently, the electronic and transport
properties of metallic point contacts and wires having average dimension in the range
of metallic λF produced by STM and also by mechanical break junctions have
displayed various quantum effects. The two-terminal electrical conductance, G, of the
wire showed a stepwise variation with the stretch. The sudden jumps of conductance
in the course of the stretch have been taken as the realization of quantized
conductance at room temperature and thus have created much popular interest.
Metallic carbon nanotubes providing two conductance channels at the Fermi level are
considered almost perfect one-dimensional (1D) conductors. These tubes, which can
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be grown to several micrometres in length with a variety of diameters and chirality,
display unique electromechanical properties and dimensionality effects. They can be
either semiconductor or metal depending on their chirality and diameter. The
nanotubes are flexible, and at the same time are very strong with high yield strength.
Their strength far exceeds that of any other fibre. These properties are being actively
explored with a view to novel applications in nanotechnology. It has been
demonstrated experimentally that an individual nanotube can sustain very high
current density; first the current increases with the bias voltage and it eventually
saturates at a value which is very high for a nanowire. Recent studies have shown
that the electronic properties of semiconducting carbon nanotubes can be modified
variably and reversibly by radial deformation, a feature that may lead to some device
applications.
The thermal conductance may exhibit quantum features depending on the size of the
nanostructure that transfers the phononic energy. Owing to the finite level spacing of
vibrational frequencies, the phononic energy transfer through an electrically non-
conducting nano-object (i.e. a molecule, atomic chain, or a single atom) between two
reservoirs shows behaviour similar to the ballistic electron transport. Recent
theoretical studies indicate that the thermal conductance of an acoustic branch at low
temperature is independent of any material property, and is linearly dependent on
temperature. At low temperature the discrete vibrational frequency spectrum of a
‘chain’ gives rise to a sudden increase of the thermal conductance.
Study of quantum effects in 1D conductors has seen a tremendous explosion
because of growing interest in nanoscience and the quest for novel nanodevices. In
this section we will discuss some theoretical methods used in the calculation of
quantum transport then about the electrical conductance through metallic nanowires.
After a brief description of the structure and electronic properties, paragraph 7.3
shows recent experimental and theoretical investigation on the electronic transport
through singlewall carbon nanotubes (SWNTs).
Quantum transport of electrons
In 1957, Landauer introduced a novel way of looking at conduction. He taught us to
view the conduction as transmission and gave the famous formula, which has been a
breakthrough in the conductance phenomena and in the physics of mesoscopic
systems. According to Landauer, the conduction is a scattering event, and the
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transport is the consequence of the incident current flux. On the basis of the self-
consistency arguments for reflections and transmissions, he derived his famous
formula for a one-dimensional conductor yielding the conductance
G = (2e²/h)Τ /R (7. 147)
Where T and R = 1 − T are transmission and reflection coefficients, respectively.
Later, Sharvin investigated the electron transport through a small contact between
two free-electron metals. Since the length of the contact is negligible, i.e. l → 0, the
scattering in the contact was absent and hence T ∼∼∼∼ 1. By using a semiclassical
approach he found an expression for the conductance,
GS = (2e2/h)(Ak2F /4π) (7. 148)
Which has come to be known as Sharvin’s conductance.
The original Landauer formula, i.e. equation G = (2e²/h)Τ /R (7. 147), and
Sharvin’s conductance have seemed to be at variance, since the former yields G →
∞ as R → 0 in the absence of scattering. The confusion in the literature has been
clarified by recognizing the fact that the original formula is just the conductance of a
barrier in a 1D conductor. Engquist and Anderson pointed out an interesting feature
by arguing that the conductance has a close relation with the type of measurement.
Landauer’s formula was extended to the conductance measured between the two
outside reservoirs within which the finite-length conductor is placed. The
corresponding two-terminal conductance formula, G = (2e2/h)T , incorporates the
resistance due to the contacts to the reservoirs. Accordingly, for perfect transmission,
T ∼∼∼∼ 1, the conductance is still finite and equal to 2e2/h. The corresponding resistance
R = G−1 = h/2e2 is attributed to the resistance arising from the reflections at the
contacts where the finite-length conductor is connected to the reservoirs. This leaves
unanswered the question of how the contact resistance can be independent of the
type of contact. Also the infinite conductance within the conductor at perfect
transmission requires an in-depth discussion. Whether the finite conductance of an
ideal channel, 2e2/h, is due to the contact or due to the capacity of the current-
transporting state seems to be a matter of interpretation.
Büttiker developed the multi-probe generalization of the theory. During the last
decade, important progress has been made in mesoscopic physics; the quantum
transport of electrons through constrictions has been a subject of many in-depth
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studies. In particular, the variation of the current through a point contact created by
an STM tip and the measurements of the conductance G through a narrow
constriction between two reservoirs of 2D electron gas in a high-mobility GaAs–
GaAlAs heterostructure showing step behaviour with a step height of 2e2/h have led
to basic confirmation of Landauer’s fundamental ideas.
The step behaviour of G has been treated in several studies. Here we provide a
simple and formal explanation by using a 1D idealized uniform constriction having a
width w. The electrons are confined in the transverse direction and have states with
quantized energy єi . They propagate freely along the length of the constriction. The
propagation constant for an electron with єi < EF is
( )2/1
²
*2
∈−= iFi Em
ℏγ (7. 149)
Whenever the width of the constriction increases by λF /2, a new subband with
energy єi + ћ²γ²i/2m dips below the Fermi level and contributes to the current under
the small bias voltage ∆V. The current is
( ) ( )[ ]∑=
−∆+=Ιj
iFiFii EDVeEDen
i1
2 γϑ (7. 150)
Here j is the index of the highest subband that lies below the Fermi level, i.e.
єi ≤ EF + e∆V and єj+1 > EF + e∆V, and n i is the degeneracy of the state i. By
assuming perfect transmission in the absence of any contact resistance and barrier
inside the constriction, i.e. T = 1, and by expressing the group velocity iγϑ and the
density of states Di (є) in terms of the subband energy є = єi + ћ²γ²i/2m* and dividing
I by ∆V we obtain
∑=
=j
iin
h
eG
1
²2 (7. 151)
Accordingly, each current-transporting state with energy in the range
EF < є < EF + e∆V contributes to G an amount 2e²n i/h.
For a uniform, infinite wall constriction, the states are non-degenerate, i.e. n i = 1, and
hence the increase of w by λF /2 causes G to jump by 2e²/h. As a result, the G(w)
curve exhibits a staircase structure. Since the level spacing in the transverse
quantization is rather small for the low electron density in the 2D electron gas system,
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and ∆є ∼∼∼∼ λF-2, the sharp step structure is likely to be smeared out at T ∼∼∼∼ 10 K or at
finite bias voltage.
For a finite-length constriction the scattering at the contacts to the reservoirs (i.e. the
contact resistance) and at the non-uniformities or the potential barriers inside the
constriction affect the transmission. Then, the conductance of a single channel
expressed as
G = (2e²/h)T (7. 152)
may deviate from the perfect quantized values. As a result, the effects, such as the
contact and potential barrier, surface roughness, impurity scattering, cause the sharp
step structure of G(w) to smear out. The local widening of the constriction or impurity
potential can give rise to quasi-bound (0D) states in the constriction and to resonant
tunnelling effects.
The length of the constriction, l, is another important parameter. In order to get sharp
step structure, l has to be greater than λF , but smaller than the electron mean free
path le; G(w) is smoothed out in a short constriction (l < λF ). Therefore, G(w) exhibits
sharp step structure if the constriction is uniform, and w ∼∼∼∼λF and λF << l < le . On the
other hand, the theoretical studies predict that the resonance structures occur on the
flat plateaus due to the interference of waves reflected from the abrupt connections
to the reservoirs. The stepwise variation of G with w or EF has been identified as the
quantization of conductance. This is, in fact, the reflection of the quantized
constriction states in the electrical conductance.
We now extend the above discussion to analyse the ballistic electron conductance
through a point contact or a nanowire, in which the electronic motion is confined in
two dimensions, but freely propagates in the third dimension. The point contact (or
quantum contact) created by the indentation of the STM, a nanowire (or a connective
neck) that is produced by retracting the tip from an indentation and also a metallic
SWNT are typical systems of interest. Nanowires created by STM are expected to be
round (though not perfectly cylindrical) and have radius R ∼∼∼∼λF at the neck. The neck
is connected to the electrodes by horn-like ends, and hence the radius increases as
one goes away from the neck. An extreme case for l → 0 is Sharvin’s conductance,
GS = (2e²/h)(πR/ λF )², with contact radius R, where the step structure is almost
smeared out and plateaus disappear. In the quantum regime, where the cross
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section A ∼∼∼∼λF², GS should vanish when A is smaller than a critical cross section set
by the uncertainty principle. As l increases, a stepwise behaviour for GS develops.
The real point contacts and metallic nanowires differ from the perfect uniform
constriction in the way that the electrons are quantized in the neck. The size and form
of the neck, and also its electronic potential, influence the quantization of the
electronic states and the level spacing. The form of the neck may be irregular if the
cross section is large and comprises few atoms, but shows cylindrical symmetry for
an undeformed SWNT or metallic nanowire having a single atom at the neck. The
cylindrical symmetry becomes apparent with the degeneracy n i = 1, 2, 2, 1, . . for
i = 1, 2, 3, 4, . . .. For the most general constriction the expression for the two-
terminal conductance in equation V
G∆
Ι= (7. 144) is generalized to the form
)(²2
ttTrh
eG += (7. 153)
Where the elements of the transmission matrix t, t ij, are the amplitudes of
transmission from the incoming channels to the outgoing ones. In actual electron
transport through a contact or wire, the electrons of the electrodes with proper
representation (say in Bloch form) enter in the constriction, and pass to the other
electrode after multiply scattering from the walls or from other scattering centres in
the constriction. In the steady state the potential of the wire can be different from that
in the case of a self-consistently calculated potential under zero-current conditions.
Therefore, full calculation of the conductance based on the density functional theory
has to start with the actual states of the electrodes and evaluate the t-matrix for the
self-consistent potential under the steady-state current flow. Various methods at
different levels of sophistication have been applied to calculate the conductance.
The simplest model for representing a nanowire or point contact is a cylindrical or
circular hard-wall potential with varying radius, R(z). The axis of the constriction is
along the z direction; the x- and y-axes are in the transverse plane. An approach
alternative to the circular cross section is the rectangular cross section with sides
Lx(z) and Ly(z), both varying smoothly with z. Many studies employed these models
to show the step behaviour of the conductance. The potential of the constriction
corresponding to a given atomic structure which was omitted in these model studies
has been taken into account first by performing SCF electronic structure calculations,
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or by obtaining the potential from the linear combination of atomic potentials. Then,
the calculated potential is approximated by a potential having cylindrical symmetry:
²)()(),( ραφρ zzzV lll += (7. 154)
throughout the constriction. Here l is the length of the constriction, and
ρ = (x² + y²) 1/2. The saddle point potential Φl(z) is an essential ingredient which was
omitted in the calculations using the hard-wall potential. The current-transporting
state Ψki (ρ, z) corresponding to the incident plane wave eiki ·r of energy E = ћ²k² i/2m*
is written as a linear combination of the longitudinal and transverse states:
[ ] ),())(()(),( )()( zezzBezAz nn
zzink
zzinkk
n
i
n
iiρρ γγ Φ+=Ψ ∑ − (7. 155)
Where the transverse state zn ,(ρΦ ) is the 2D harmonic oscillator solution for a given
αl(z) with n = n x + ny and energy [ ] 2/1, */)(²2)1()( mznz lln αℏ+=∈ . The propagation
constant is given by
[ ])()(²
2)( , zzE
mz lnln ∈−Φ−=
ℏγ (7. 156)
The conductance of the wire is calculated by using the transfer-matrix method, where
Vl (ρ, z) is divided into discrete segments of equal widths, and in each segment
zj < z < zj+1, the average values are used. The coefficients Anki and Bnki are
determined from the multiple boundary matching conditions. The total conductance is
calculated by integrating the expectation value of the current operator over the Fermi
surface.
A constriction having more general geometry and potential can be treated by
expressing the current-transporting state in the Laue expansion
∑=Ψn
jniG
j zfez n )(),( ,ρρ (7. 157)
in terms of transverse reciprocal-lattice vectors,Gi . This method is convenient when
using the SCF potentials obtained from supercell calculations. Furthermore, the
conductance of rather long constrictions calculated by using the recursion scheme do
not diverge. The Schrödinger equation for jΨ is discretized in the segments
zj < z < zj+1, and converted to a matrix equation. The details of the calculations can
be found elsewhere.
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The Green’s function method has been used extensively to study the electrical
conduction through nanowires and also SWNTs. In the self-consistent calculation,
where the wave functions of the bare electrodes, 0Ψ , are evaluated from a realistic
Hamiltonian, the wave functions of the complete system including the constriction,
CΨ , are put into the Lippmann–Schwinger form:
∫ Ψ+Ψ=Ψ )()'','()',(''')()( 00 rrrVrrGdrdrrr CC δ (7. 158)
Where δV is the potential difference between the complete and bare systems, and G0
is the Green’s function obtained for the bare system. In the work by Lang the
electrodes are treated in the jellium approximation, and the potentials at the
constriction due to metal atoms are expressed by the pseudopotentials. Once 0Ψ
and CΨ are expressed in the plane-wave representation, the currents between the
electrodes in the presence and in the absence of the constriction are obtained from
the expectation value of the current operator by summing over all the states of the
jellium electrodes under the bias e δV. The calculations using the Laue-type
expansion have indicated that the division of the current-transporting state in the
electrode by jellium and in the wire by pseudopotential, which essentially neglects the
binding structure at the entrance to the reservoirs, may not fully describe the real
physical system.
In the Green’s function method within the tight-binding representation using empirical
energy parameters, a conducting nanowire or a nanotube is divided into three parts.
These are the left (L) , right (R) and central (C) regions ; the left and right regions
are coupled to two semi-infinite leads. The central region is the computationally
important part which may include tube junctions, defects or vacancies. Partitioning
the Green’s function into submatrices due to the left, right and central regions, one
can obtain the Green’s function for the central region as
1)( −Σ−Σ−−∈= RLCrC Hς (7. 159)
The self-energy terms, LΣ and RΣ , describe the effect of semi-infinite leads on the
central region. The functions for the coupling can be obtained as
][ ,,,a
RLr
RLRL i Σ−Σ=Γ in terms of the retarded (r) and advanced (a) self-energies. An
important part of the problem is calculating the self-energy terms. The surface
Green’s function matching method or computational algorithms are used to calculate
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the Green’s functions of semi-infinite leads and the self-energy terms. These terms
can be obtained by using wave functions of ideal leads also. The transmission
function T, that represents the probability of transmitting an electron from one end of
the conductor to the other end, can be calculated using Green’s functions of the
central region and couplings to the leads:
)( aCR
rCLTr ςς ΓΓ=Τ (7. 160)
Here arC
,ς are the retarded and advanced Green’s functions of the centre, and RL,Γ are
functions for couplings to the leads. This approach is used efficiently to treat
nanotube junctions and nanodevices. Note that the success of the approach depends
on the transferability of the empirical energy parameters.
7.2.2 Electrical Conductivity
An atomic size contact and connective neck first created by Gimzewski andMöller by
using a STM tip exhibited abrupt changes in the variation of the conductance with the
displacement of the tip. Initially, the observed behaviour of the conductance was
attributed to the quantization of conductance. At that time, from calculating the
quantized conductance of a perfect but short connective neck, Ciraci and Tekman
concluded that the observed abrupt changes of the conductance can be related to
the discontinuous variation of the contact area. Later, Todorov and Sutton performed
an atomic scale simulation of indentation based on the classical molecular dynamics
(MD) method and calculated the conductance for the resulting atomic structure using
the s-orbital tight-binding Green’s function method. They showed that sudden
changes of conductance during indentation or stretching are related to the
discontinuous variation of the contact area. Recently, nanowires of better quality
have been produced using STM and also by using the mechanical break junction.
The two-terminal electrical conductance of these wires showed abrupt changes with
the stretching.
The radius of the narrowest cross section of the wire prior to the break is only a few
ångströms; it has the length scale of λF , where the discontinuous (discrete) nature of
the metal dominates over its continuum description. Since the level spacing ∆є is in
the region of ∼∼∼∼1 eV at this length scale, the peaks of the density of states D(є) of the
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connective neck become well separated and hence the transverse quantization of
states becomes easily resolved even at room temperature. Furthermore, any change
in the atomic structure or the radius induces significant changes in the level spacing
and in the occupancy of states. This, in turn, leads to detectable changes in the
related properties. Therefore, the ballistic electron transport through a nanowire
should be closely related to its atomic structure and radius at its narrowest part. On
the other hand, irregularities of the atomic structure and electronic potential enhance
scattering that destroys the regular staircase structure. In the following section, we
summarize results that emerged from recent experimental and theoretical studies on
stretched nanowires.
7.2.2.1 Atomic structure and mechanical properties
Computer simulations of the atomic structure of connective necks created by STM
were first performed in the seminal works by Sutton and Pethica [], and Landman et
al []. The deformation (or elongation) generally occurred in two different and
consecutive stages that repeat while the wire is stretched.
1. In the first stage, which was identified as quasi-elastic, the stored strain energy
and average tensile force increase with increasing stretch s, while the atomic
layers are maintained. The variation of the applied tensile force, Fz(s), in this
stage is approximately linear, but it deviates from linearity as the number of
atoms in the neck decreases. Fluctuations in Fz(s) can occur possibly due to
displacement and relocation of atoms within the same layer or atom exchange
between adjacent layers. Also, intralayer and interlayer atom relocations can
give rise to conductance fluctuations.
2. The second stage that follows each quasi-elastic stage is called the yielding
stage. A wire can yield by different mechanisms depending on its diameter.
The motion of the dislocation and/or the slips on the glide planes are generally
responsible for the yielding if the wire maintains an ordered (crystalline)
structure and has a relatively large cross section. The type of the ordered
structure and its orientation relative to the z-axis (or stretching direction) are
expected to affect the yielding. On the other hand yielding can occur by order–
disorder transformation and single-atom exchange if the cross section of the
wires is relatively small. Once the elongation reaches approximately the
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interlayer spacing at the end of the quasi-elastic stage, the structure becomes
disordered.
But after further stretching, it recovers with the formation of a new layer. In the
yielding stage, |Fz| decreases abruptly; the cross section of the layer formed at the
end of the yielding stage is abruptly reduced by a few atoms. As a typical example,
the force variation and atomic structure calculated by the MD method are illustrated
in Figure 7-2.
Figure 7-2: The variation of the tensile force Fz (in nanonewtons) with the strain or elongation s along
the z-axis of the nanowire having Cu(001) structure. The stretch s is realized in m discrete steps. The
snapshots show the atomic positions at relevant stretch steps m. The MD simulations are performed
at T = 300 [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal transport
through nanowires, 2001, Journal of Physics Condensed Matter]
When the neck becomes very narrow (having 3–4 atoms), the yielding is realized,
however, by a single atom jumping from one of the adjacent layers to the interlayer
region.
Under certain circumstances, atoms at the neck form a pentagon that becomes
staggered in different layers. The interlayer atoms make a chain passing through the
centre of the pentagon rings. In this new phase, the elastic and yielding stages are
intermixed and elongation, which is by more than one interlayer distance, can be
accommodated. As the cross section of the wire is further decreased the pentagons
are transformed to a triangle. In the initial stage of pulling off, the single-atom
process, in which individual atoms also migrate from central layers towards the end
layers, can give rise to a small and less discontinuous change in the cross section.
The tendency to minimize the surface area, and hence to reduce the strain energy of
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the system, is the main driving force for this type of neck formation. While quasi-
elastic and yielding stages are distinguishable initially, the force variation becomes
more complex and more dependent on the migration of atoms for atomic necks.
Atomic migrations have important implications such as dips in the variation of
conductance with stretch. The simulation studies on metal wires with diameters ∼∼∼∼λF
at the neck, but increasing gradually as one goes away from the neck region, have
shown structural instabilities. Such wires displayed spontaneous thinning of the
necks even in the absence of any tensile strain.
Under prolonged stretching, shortly before the break, the cross section of the neck is
reduced to include only 2–3 atoms. In this case, the hollow-site registry may change
to the top-site registry. This leads to the formation of a bundle of atomic chains
(rope) or of a single atomic chain (see Figure 7-3). We consider this a dramatic
change in the atomic structure of the wire that has important implications. For
example, first-principles calculations indicated that a chain of single Al atoms has an
effective Young modulus stronger than that of the bulk.
Figure 7-3: The top view of three layers at the neck showing atomic positions and their relative
registry at different levels of stretch. m = 15 occurs before the first yielding stage. The atomic positions
in the layers 2, 3 and 4 are indicated by +, • and ◊, respectively. m = 38 and m = 41 occur after the
second yielding stage. m = 46, 47 and m = 49 show the formation of bundle structure (or strands). In
the panels for m = 38–49 the positions of atoms in the third, fourth and fifth (central) layers are
indicated by +, • and ◊, respectively. Atomic chains in a bundle are highlighted by square boxes. [K. S.
Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal transport through nanowires,
2001, Journal of Physics Condensed Matter]
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Using an ultrahigh-vacuum electron microscope, Ohnishi et al [] observed ‘strands’
and a single chain of gold atoms suspended between two gold electrodes. They
deduced interatomic distances (as large as 3.5 to 5 Å) much larger than the bulk
value. Recently, the structure and the stability of gold chains between two electrodes
have been the subject of extensive calculations. The first-principles density functional
calculations by Portal et al [] found that a zigzag structure of seven gold atoms
between two gold electrodes is energetically favourable and can explain the field-
emission images. On the other hand, calculations by Häkkienen et al [] favour the
dimerization of the linear chain. It appears that the atomic arrangements of the
electrodes where the chain is connected are crucial in determining the atomic
structure of the monatomic chain. The zigzag structure forming equilateral triangles is
favoured for a 1D infinite metallic chain since the number of nearest neighbours
increases from two (corresponding to an undimerized linear chain) to four.
Nevertheless, several questions remain to be clarified in future studies:
I. Why does a metal chain form a zigzag structure?
II. Are there other lower energy structures indigenous to one-dimensional
systems?
III. Why is the gold chain stable while other metal monatomic chains appear to be
unstable under strain?
IV. How does the conductance depend on the structure of the chain?
V. How does the current flow affect the stability of the chain?
Recent computer simulations have suggested that ultrathin lead wires should develop
exotic, non-crystalline stable atomic structures once their diameter decreases below
a critical size of the order of a few atomic spacings. The new structures, whose
details depend upon the material and the wire thickness, may be dominated by
icosahedral packings. Helical, spiral-structured wires with multi-atom pitches are also
predicted. Recently, helical multishell gold nanowires were observed by high-
resolution electron microscopy. The phenomenon, analogous to the appearance of
icosahedral and other noncrystalline shapes in small clusters, can be rationalized in
terms of surface energy anisotropy and optimal packing. Moreover, the electronic
structure of the wire might carry a measurable imprint of its exotic shape.
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7.2.2.2 Electrical conductance of nanomires
The overall features of electrical conductance have been obtained from a statistical
analysis of the results of many consecutive measurements to deduce how frequently
a measured conductance value occurs. Many distribution curves exhibited a peak
near G0 = 2e²/h for metal nanowires, and a relatively broad peak near 3G0, and
almost no significant structure for larger values of conductance. It appears that the
cross section of a connective neck can be reduced down to a single atom just before
the wire breaks. In certain situations, the connective neck can be a monatomic chain
comprising of a few atoms arranged in a row. In this case, a transverse state
quantized in the neck at the Fermi level is coupled to the states of the electrodes and
forms a conducting channel yielding a plateau in the G(s) curve, and hence a peak in
the statistical distribution curve. The step structure for large necks comprising of a
few atoms cannot occur at integer multiples of G0 due to the scattering from
irregularities, and hence due to the channel mixing.
It should be noted that in the course of stretching, elastic and yielding stages repeat;
the surface of the nanowire roughens and deviates strongly from circular symmetry.
The length of the narrowest part of the neck is usually only one or two interlayer
separations, and is connected to the horn-like ends. Under these circumstances, the
quantization is not complete and the contribution of the tunnelling is not negligible.
Results of atomic simulations point to the fact that neither adiabatic evolution of
discrete electronic states, nor perfect circular symmetry can occur in the neck.
Consequently the expected quantized sharp structure shall be smeared out by
channel mixing and tunnelling.
Contradicting these arguments, the changes in the conductance are, however,
abrupt. This controversial situation has been explained by the combined
measurements of the conductance and force variation with the stretch. The abrupt
changes of conductance in the course of stretching coincide with the sudden release
(or relief) of the measured tensile force. The tensile stress is released suddenly
following the yielding stage, whereby the cross section of the wire is reduced
discontinuously, and hence the electronic structure and the electronic level spacing
undergo an abrupt change near the neck. The variation of the conductance, i.e. G(s),
calculated for stretching the nanowire is presented as an example in Figure 7-4.
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Figure 7-4: Variation of the conductance calculated for the stretching nanowire described in
Figure 7-2. [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal transport
through nanowires, 2001, Journal of Physics Condensed Matter]
Calculations of G(s) corresponding to the atomic structure generated from classical
MD simulations are using the Green’s function method within the s-orbital tight-
binding approximation. Barnett and Landman [] performed similar calculations on the
Na nanowires by using MD based on the ab initio electronic structure calculations.
They found that the conductances of these nanowires exhibit dynamical thermal
fluctuations on a subpicosecond timescale owing to rearrangements of the metal
atoms explained in section 7.2.2.1. The giant conductance fluctuation, i.e.
Gmax/Gmin ∼∼∼∼ 10–15, is rather surprising, and perhaps is due to the artifact of applying
Kubo–Greenwood formalism for a finite system.
The density of states of a perfect 1D atomic wire can be expressed as
2/1))(()( −∈−∈∈−∈=∈ ∑ ii
iD θ (7. 161)
It has peaks at the quantized energies of transversely confined states, i.e. when
є = єi. The occupancy of these states becomes strongly dependent on the size and
the geometry of the wire. Sudden reduction in the ballistic conductance will occur
whenever a channel is closed (i.e. a current-transporting state moves above the
Fermi energy, EF). Also, whenever the tensile force along the axis of the wire |Fz|
shows a sudden fall, it is possible that the actual R (or the narrowest cross section,
A) experiences a change only at certain positions of єi relative to EF. Such a
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possibility was pointed out on the basis of earlier works indicating quantum size
effects. In fact, it is known that the surface energy, work function etc for a 2D electron
system show discrete changes as the relevant size (or thickness) changes. This is
explained by noting that the increase occurs when an empty band gets occupied as a
result of size change (explain later). Similar effects have been reported for clusters
treated in the jellium approximation. Stafford et al [] showed that in fact abrupt
changes in conductance and diameter of a uniform wire are correlated, so certain
values of R or A are energetically favourable as if the size is ‘quantized’.
7.2.2.3 Electrical conductance through a single ato m
The conductance through a single atom, a nanowire with the ultimate small cross
section, has provided fundamental understanding. On the basis of ab initio
calculations it has been shown that the conductance depends on the valence states
as well as the site where the single atom is bound to the electrode. The coupling to
electrodes and hence the transmission coefficients are expected to depend on the
binding structure. There is a direct link between valence orbitals and the number of
conduction channels in the conductance through a single atom. Lang [] calculated the
conductance through a single Na atom, as well as a monatomic chain comprising
two, three and four Na atoms between two jellium electrodes by using the Green’s
function formalism. He found an anomalous dependence of the conductance on the
‘length’ of the wire. The conductance through a single atom was low (∼∼∼∼G0/3), but
increased by a factor of two in going from a single atom to the two-atom wire. This
behaviour was explained as the incomplete valence resonance of a single Na atom
interacting with the continuum of states of the jellium electrodes. Each additional Na
atom modifies the electronic structure and shifts the energy levels. The closer a state
is to the Fermi level, the higher is its contribution to the electrical conductance.
According to the Kalmeyer–Laughlin theory [], a resonance with the maximum DOS
at the Fermi level makes the highest contribution to the transmission; the
conductance decreases as the maximum shifts away from the Fermi level.
This explanation is valid for a single atom between two macroscopic electrodes
forming a neck with length l < λF. The situation is, however, different for long
monatomic chains. Since the electronic energy structure of a short monatomic chain
varies with the number of atoms, the results are expected to change if one goes
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beyond the jellium approximation and considers the details of the coupling of chain
atoms to the electrode atoms. It is also expected that the conductance will depend on
where the monatomic chain (represented by the pseudopotentials) ends and where
the jellium edge begins.
7.2.3 Surface Plasmons
Gold colloidal nanoparticles are responsible for the brilliant reds seen in stained glass
windows. Silver particles are typically yellow. These properties have been of interest
for centuries, and scientific research on metal nanoparticles dates at least to Michael
Faraday. It was therefore one of the great triumphs of classical physics when in 1908,
Mie presented a solution to Maxwell’s equations that describes the extinction spectra
(extinction = scattering + absorption) of spherical particles of arbitrary size. Mie’s
solution remains of great interest to this day, but the modern generation of metal
nanoparticle science, including applications to medical diagnostics and nanooptics,
has provided new challenges for theory.
7.2.3.1 Origin of surface plasmon resonance in nobl e metal nanoparticles
The free electrons in the metal (d electrons in silver and gold) are free to travel
through the material. The mean free path in gold and silver is ~50 nm, therefore in
particles smaller than this, no scattering is expected from the bulk. Thus, all
interactions are expected to be with the surface. When the wavelength of light is
much larger than the nanoparticle size it can set up standing resonance conditions as
represented in Figure 7-5.
Figure 7-5: Schematic of plasmon oscillation for a sphere, showing the displacement of the
conduction electron charge cloud relative to the nuclei. [K. Lance Kelly, E. Coronado, L.L. Zhao, G.C.
Schatz, The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric
Environment, 2003, Journal of Physical Chemistry]
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Light in resonance with the surface plasmon oscillation causes the free-electrons in
the metal to oscillate. As the wave front of the light passes, the electron density in the
particle is polarized to one surface and oscillates in resonance with the light’s
frequency causing a standing oscillation. The resonance condition is determined from
absorption and scattering spectroscopy and is found to depend on the shape, size,
and dielectric constants of both the metal and the surrounding material. This is
referred to as the surface plasmon resonance, since it is located at the surface. As
the shape or size of the nanoparticle changes, the surface geometry changes
causing a shift in the electric field density on the surface. This causes a change in the
oscillation frequency of the electrons, generating different cross-sections for the
optical properties including absorption and scattering.
Changing the dielectric constant of the surrounding material will have an effect on the
oscillation frequency due to the varying ability of the surface to accommodate
electron charge density from the nanoparticles. Changing the solvent will change the
dielectric constant, but the capping material is most important in determining the shift
of the plasmon resonance due to the local nature of its effect on the surface of the
nanoparticle. Chemically bonded molecules can be detected by the observed change
they induce in the electron density on the surface, which results in a shift in the
surface plasmon absorption maximum. This is the basis for the use of noble metal
nanoparticles as sensitive sensors. Mie originally calculated the surface plasmon
resonance by solving Maxwell’s equations for small spheres interacting with an
electromagnetic field. Gan was able to extend this theory to apply to ellipsoidal
geometries. Modern methods using the discrete dipole approximation (DDA) allow
one to calculate the surface plasmon resonance absorption for arbitrary geometries.
Calculation of the longitudinal plasmon resonance for gold nanorods generates an
increase in the intensity and wavelength maximum as the aspect ratio (length divided
by width) increases. Thus, the plasmon resonance can be tuned across the visible
region by changing the aspect ratio. The increase in the intensity of the surface
plasmon resonance absorption leads to an enhancement of the electric field, as
exploited in many applications.
7.2.3.2 Dipole plasmon resonanaces
When a small spherical metallic nanoparticle is irradiated by light, the oscillating
electric field causes the conduction electrons to oscillate coherently (Figure 7-5).
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When the electron cloud is displaced relative to the nuclei, a restoring force arises
from Coulomb attraction between electrons and nuclei that results in oscillation of the
electron cloud relative to the nuclear framework. The oscillation frequency is
determined by four factors: the density of electrons, the effective electron mass, and
the shape and size of the charge distribution. The collective oscillation of the
electrons is called the dipole plasmon resonance of the particle (sometimes denoted
“dipole particle plasmon resonance” to distinguish from plasmon excitation that can
occur in bulk metal or metal surfaces). Higher modes of plasmon excitation can
occur, such as the quadrupole mode where half of the electron cloud moves parallel
to the applied field and half moves antiparallel. For a metal like silver, the plasmon
frequency is also influenced by other electrons such as those in d-orbitals, and this
prevents the plasmon frequency from being easily calculated using electronic
structure calculations. However, it is not hard to relate the plasmon frequency to the
metal dielectric constant, which is a property that can be measured as a function of
wavelength for bulk metal.
To relate the dipole plasmon frequency of a metal nanoparticle to the dielectric
constant, we consider the interaction of light with a spherical particle that is much
smaller than the wavelength of light. Under these circumstances, the electric field of
the light can be taken to be constant, and the interaction is governed by electrostatics
rather than electrodynamics. This is often called the quasistatic approximation, as we
use the wavelength-dependent dielectric constant of the metal particle, єi, and of the
surrounding medium, є0, in what is otherwise an electrostatic theory.
Let’s denote the electric field of the incident electromagnetic wave by the vector Eo.
We take this constant vector to be in the x direction so that Eo = Eo x , where x
is a
unit vector. To determine the electromagnetic field surrounding the particle, we solve
LaPlace’s equation (the fundamental equation of electrostatics), ϕ2∇ = 0, where φ is
the electric potential and the field E is related to φ by E = -∇φ. In developing this
solution, we apply two boundary conditions: (i) that φ is continuous at the sphere
surface and (ii) that the normal component of the electric displacement D is also
continuous, where D = є E.
It is not difficult to show that the general solution to the LaPlace equation has angular
solutions which are just the spherical harmonics. In addition, the radial solutions are
of the form r l and r-(l +1), where l is the familiar angular momentum label (l = 0, 1, 2,...)
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of atomic orbitals. If we restrict our considerations for now to just the l = 1 solution
and if Eo is in the x direction, the potential is simply φ = A r sin θ cosΦ inside the
sphere (r < a) and φ = (-Eor + B/r²) sin θ cosΦ outside the sphere (r > a), where A
and B are constants to be determined. If these solutions are inserted into the
boundary conditions and the resulting φ is used to determine the field outside the
sphere, Eout , we get
++−−= )(3
53zzyyxx
r
x
r
xExEE ooout
α (7. 162)
Where R is the sphere polarizability and x , y , and z
are the usual unit vectors. We
note that the first term in eq
++−−= )(3
53zzyyxx
r
x
r
xExEE ooout
α (7. 162) is
the applied field and the second is the induced dipole field (induced dipole moment =
αEo) that results from polarization of the conduction electron density.
For a sphere with the dielectric constants indicated above, the LaPlace equation
solution shows that the polarizability is
3ag d=α (7. 163)
with
0
0
2∈+∈∈−∈
=i
idg (7. 164)
Although the dipole field in eq
++−−= )(3
53zzyyxx
r
x
r
xExEE ooout
α (7. 162) is
that for a static dipole, the more complete Maxwell equation solution shows that this
is actually a radiating dipole, and thus, it contributes to extinction and Rayleigh
scattering by the sphere. This leads to extinction and scattering efficiencies given by
)Im(4 dext gxQ = (7. 165)
24
3
8dsca gxQ = (7. 166)
where x = 2πa(є0)1/2/λ. The efficiency is the ratio of the cross-section to the
geometrical cross-section πa². Note that the factor gd from eq 0
0
2∈+∈∈−∈
=i
idg (7.
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164) plays the key role in determining the wavelength dependence of these cross-
sections, as the metal dielectric constant єi is strongly dependent on wavelength.
7.2.3.3 Quadrupole plasmon Resonances
For larger particles, higher multipoles, especially the quadrupole term (l=2) become
important to the extinction and scattering spectra. Using the same notation as above
and including the l=2 term in the LaPlace equation solution, the resulting field outside
the sphere, Eout , now can be expressed as
( ) ( ) ( )
++−+−++−−++= zxzyyxxr
z
r
zzxxEzzyyxx
r
x
r
xEzzxxikExEE ooooout
²²
537553
βα
(7. 167)
and the quadrupole polarizability is
5ag d=α (7. 168)
0
0
2/3 ∈+∈∈−∈
=i
idg (7. 169)
Note that the denominator of eq 0
0
2/3 ∈+∈∈−∈
=i
idg (7. 169) contains the factor 3/2
while in eq 0
0
2∈+∈∈−∈
=i
idg (7. 164) the corresponding number is 2. These factors
arise from the exponents in the radial solutions to LaPlace’s equation, e.g., the
factors r l and r -(l +1) that were discussed above. For dipole excitation, we have l=1,
and the magnitude of the ratio of the exponents is (l + 1)/l = 2 , while for quadrupole
excitation (l + 1)/l = 3/2 . Higher partial waves work analogously.
Following the same derivation, we get the following quasistatic (dipole + quadrupole)
expressions for the extinction and Rayleigh scattering efficiencies:
( )
−∈++= 130
²
12
²Im4 iqdext
xg
xgxQ (7. 170)
−∈++= 24
24
24 19002403
8iqdext
xg
xgxQ (7. 171)
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7.2.3.4 Extinction for silver spheres
We now evaluate the extinction cross-section using the quasistatic expressions, eqs
4, 5 and 9,10 as well as the exact (Mie) theory. We take dielectric constants for silver
that are plotted in Figure 7-6 a and the external dielectric constant is assumed to be 1
(i.e., a particle in a vacuum). The resulting efficiencies for 30 and 60 nm spheres are
plotted in Figure 7-6 b,c, respectively.
Figure 7-6: (a) Real and imaginary part of silver dielectric constants as function of wavelength. (b)
Extinction efficiency, i.e., the ratio of the extinction cross-section to the area of the sphere, as obtained
from quasistatic theory for a silver sphere whose radius is 30 nm. (c) The corresponding efficiency for
a 60 nm particle, including for quadrupole effects, and correcting for finite wavelength effects. In b and
c, the exact Mie theory result is also plotted. [K. Lance Kelly, E. Coronado, L.L. Zhao, G.C. Schatz,
The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric
Environment, 2003, Journal of Physical Chemistry]
The cross-section in Figure 7-6 b shows a sharp peak at 367nm, with a good match
between quasistatic and Mie theory. This peak is the dipole surface plasmon
resonance, and it occurs when the real part of the denominator in eq 0
0
2∈+∈∈−∈
=i
idg
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(7. 164) vanishes, corresponding to a metal dielectric constant whose real part
is -2. For particles that are not in a vacuum, the plasmon resonance condition
becomes
Re єi/є0= -2, and because the real part of the silver dielectric constant decreases with
increasing wavelength (Figure 7-6 a), the plasmon resonance wavelength for є0 > 1 is
longer than in a vacuum. The plasmon resonance wavelength also gets longer if the
particle size is increased above 30 nm. This is due to additional electromagnetic
effects that will be discussed later.
Figure 7-6 c presents the l = 2 quasistatic (dipole + quadrupole) cross-section as well
as the full Mie theory result for a radius 60 nm sphere. The quasistatic result also
includes a finite wavelength correction. We see that the dipole plasmon wavelength
has shifted to the red, and there is now a distinct quadrupole resonance peak at 357
nm. This quadrupole peak occurs when the real part of the denominator in eq 8
vanishes, corresponding to a metal dielectric constant whose real part is -3/2. For a
sphere of this size, there are clear differences between the quasistatic and the Mie
theory results; however, the important features are retained. Although Mie theory is
not a very expensive calculation, the quasistatic expressions are convenient to use
when only qualitative information is needed.
7.2.3.5 Electromanetic fields for spherical particl es
So far, we have emphasized the calculation of extinction and Rayleigh scattering
cross-section; however, for certain properties, such as SERS and hyperRaman
scattering (HRS) intensities, it is the electromagnetic field at or near the particle
surfaces that determines the measured intensity. Thus, if E(ω) is the local field for
frequency ω, then the SERS intensity is determined by 22
)'()( ωω EE where ω’ is
the Stokes-shifted frequency and the brackets are used to denote an average over
the particle surface. The HRS intensity is similarly (but approximately) determined by
24)2()( ωω EE . Also, when one makes an aggregate or array of metal
nanoparticles, the interaction between the particles is determined by the polarization
induced in each particle due to the fields E arising from all of the other particles.
At the dipole (dipole + quadrupole) level, the field outside a particle is given by eq
++−−= )(3
53zzyyxx
r
x
r
xExEE ooout
α (7. 162) and
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( ) ( ) ( )
++−+−++−−++= zxzyyxxr
z
r
zzxxEzzyyxx
r
x
r
xEzzxxikExEE ooooout
²²
537553
βα
(7. 167). These expressions determine the near-fields at the particle surfaces quite
accurately for small enough particles; however, the field beyond 100 nm from the
center of the particle exhibits radiative contributions that are not contained in these
equations. To describe these, we need to replace the dipole or quadrupole field by its
radiative counterpart. In the case of the dipole field, this is given by
( )53
)(3²)1(²
r
PrrPrikre
r
prrekE ikrikr
dipole
•−−+××= (7. 172)
where P is the dipole moment. Note that this reduces to the static field in eq
++−−= )(3
53zzyyxx
r
x
r
xExEE ooout
α (7. 162) in the limit k→0 where only the term
in square brackets remains. However, at long range, the first term becomes dominant
as it falls off more slowly with r than the second.
Figure 7-7: E-field contours for radius 30 and 60 nm Ag spheres in a vacuum. Two cross-sections are
depicted for each sphere. (a, b) The plane containing the propagation and polarization axes and (c, d)
the plane perpendicular to the propagation axis. The 30 nm sphere refers to 369 nm light, the main
extinction peak for this size, whereas the larger sphere is for 358 nm light, the quadrupole peak for this
size. Labeled points 1 and 2 illustrate locations for Figure 7-8. [K. Lance Kelly, E. Coronado, L.L.
Zhao, G.C. Schatz, The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and
Dielectric Environment, 2003, Journal of Physical Chemistry]
Figure 7-7 presents contours of the electric field enhancement |E|² around 30 and 60
nm radius silver spheres, based on a Mie theory calculation in which all multipoles
are included. Two planes are chosen for these plots, the xz plane that is formed by
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the polarization and k vectors and the yz plane that is perpendicular to the
polarization vector. The wavelength chosen for the 30 nm particle is the dipole
plasmon peak, so since the dipole field dominates, we see the characteristic p-orbital
shape around the sphere in Figure 7-7 a,c. Note that a small quadupole component
to the field makes the p-orbital lobes slightly asymmetrical. At long range, the
radiative terms in eq ( )
53
)(3²)1(²
r
PrrPrikre
r
prrekE ikrikr
dipole
•−−+××= (7. 172)
become more important, and then, the field has a characteristic spherical wave
appearance.
The wavelength for the 60 nm particle has been chosen to be that for the peak in the
quadrupole resonance, and as a result, the field contours close to the particle in
Figure 7-7 b look like a dxz-orbital (slightly distorted by a small dipole component that
is also present). In addition, Figure 7-7 d, which is a nodal plane for the dxz-orbital,
only shows the weak dipolar component. Note that the peak magnitude of the field for
the 30 nm particle occurs at the particle surface, along the polarization direction. This
peak is over 50 times the size of the applied field, while that for the 60 nm particle is
over 25 times larger. This is responsible for the electromagnetic enhancements that
are seen in SERS, and they also lead to greatly enhanced HRS.
Figure 7-8: Comparison of extinction efficiency, surface-averaged E-field enhancement, and E-field
enhancement for specific points for radius 30 nm (top) and 60 nm (bottom) Ag spheres in a vacuum.
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The two points chosen are point 1, along the polarization direction, and point 2, at a 45° angle relat ive
to the polarization direction and in the xz plane. [K. Lance Kelly, E. Coronado, L.L. Zhao, G.C. Schatz,
The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric
Environment, 2003, Journal of Physical Chemistry]
Figure 7-8 plots the surface-averaged E-field enhancement for the 30 and 60 nm
spheres as a function of wavelength, along with the extinction efficiency. In addition,
the figure includes the E-field enhancement associated with two points on the
sphere: (a) along the polarization direction (point 1) and (b) rotated 45 degrees away
from the polarization direction (point 2). The E-field enhancement associated with
specific points on the surface would be appropriate for understanding a single
molecule SERS experiment, should this be possible for a spherical particle. For both
sphere sizes, the field enhancement due to the dipole resonance peaks to the red of
the extinction. Quasistatic theory predicts that both peaks should occur at the same
wavelength; however, the finite wavelength corrections to the quasistatic result lead
to depolarization of the plasmon excitation on the blue side of the extinction peak,
resulting in a smaller average field and a red-shifted peak.
For the smaller sphere (top panel of Figure 7-8), the E-field enhancement associated
with point 1 is about three times larger than the surface-averaged value, and the
lineshapes are the same. Point 2 shows a smaller enhancement, and it peaks toward
the blue, indicating the influence of a weak quadrupole resonance. For the larger
sphere (bottom), the maximum for point 1 is about 3.5 times greater than the surface
average for the dipole peak. For point 2, we see a maximum at the quadrupole
resonance wavelength, and the enhancement is about three times greater than for
the quadrupole peak in the surface-averaged result. Thus, for the larger sphere, it is
possible for the largest SERS enhancement to be at a location on the surface that is
not along the polarization direction.
7.3 Carbon Nanotubes
Nanotubes were discovered by Iijima in the form of multiple coaxial carbon fullerene
shells (multi-wall nanotubes, MWNTs). Later, in 1993 single fullerene shells (single-
wall nanotubes, SWNTs) were synthesized using transition metal catalysts. A
nanotube can be simply described as a sheet of graphite (or graphene) coaxially
rolled to create a cylindrical surface (as shown in Figure 7-9(a)).
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Figure 7-9: (a) A single-wall nanotube is graphene wrapped on a cylinder surface. (b) Nanotubes are
described by a set of two integers (n, m) which indicate the graphite lattice vector components. A
chiral vector can be defined as C = na1 + ma2. Tubes are called ‘zigzag’ if either one of the integers is
zero (n, 0) or called ‘armchair’ if both integers are equal (n, n) . [K. S. Ciraci, A. Buldum, I.P. Batra,
Quantum effects in electrical and thermal transport through nanowires, 2001, Journal of Physics
Condensed Matter]
In this way the 2D hexagonal lattice of grapheme is mapped onto a cylinder of radius
R. The mapping can be realized with different helicities resulting in different
nanotubes. Each nanotube is characterized by a set of two integers (n, m) indicating
the components of the chiral vector C = na1 + ma2 in terms of the 2D hexagonal
Bravais lattice vectors of graphene, a1 and a2, as illustrated in Figure 7-9(b). The
chiral vector is a circumferential vector and the tube is obtained by folding the
graphene such that the two ends of C are coincident. The radius of the tube is given
in terms of (n, m) through the relation π2/nm m² n²a R 0 ++= , where |a1| = |a2| = a0.
When C involves only a1 (corresponding to (n, 0)) the tube is called ‘zigzag’, and if C
involves both a1 and a2 with n = m (corresponding to (n, n) ) the tube is called
‘armchair’. The chiral (n, n) vector is rotated by 30 relative to that of the zigzag (n, 0)
tube. SWNTs are found in the form of nanoropes, each rope consisting of up to a few
hundred nanotubes arranged in a hexagonal lattice structure.
7.3.1 Electronic structure
As a nanotube is in the form of a wrapped sheet of graphite, its electronic structure is
analogous to the electronic structure of a graphene. Graphene has the lowest π∗∗∗∗-
conduction band and the highest π-valence band, which are separated by a gap in
the entire hexagonal Brillouin zone (BZ) except at its K corners where they cross. In
this respect, graphene lies between a semiconductor and a metal with Fermi points at
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the corners of the BZ. You can imagine an unrolled, open form of nanotube, which is
graphene subject to periodic boundary conditions on the chiral vector. This in turn
imposes quantization on the wave vector. This is known as zone folding, whereby the
BZ is sliced with parallel lines of wave vectors, leading to subband structure. A
nanotube’s electronic structure can thus be viewed as a zone-folded version of the
electronic band structure of the graphene. When these parallel lines of nanotube
wave vectors pass through the corners, the nanotube is metallic. Otherwise, the
nanotube is a semiconductor with a gap of about 1 eV, which is reduced as the
diameter of the tube increases. Within this simple approach, (n, m) nanotubes are
metallic if n − m = 3× integer . Consequently, all armchair tubes are metallic. The
conclusion that you draw from the above paragraph is that the electronic structures of
nanotubes are determined by their chirality and diameter, i.e. simply by their chiral
vectors C.
The first theoretical calculations were performed and the above simple understanding
was provided much earlier than the first conclusive experiments were carried out . In
these early calculations, a simple one-band π-orbital tight-binding model was used.
However, different calculations have been at variance on the values of the band gap.
For example, while the σ∗–π∗ hybridization due to the curvature can be treated well
by calculations, simple tight-binding methods may have limitations for small-radius
nanotubes.
Figure 7-10: (a) The band structure, (b) density of states and (c) conductance of a (10, 10) nanotube.
The tight-binding model is used to derive the electronic structure. The conductance is calculated using
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the Green’s function approach with the Landauer formalism. [K. S. Ciraci, A. Buldum, I.P. Batra,
Quantum effects in electrical and thermal transport through nanowires, 2001, Journal of Physics
Condensed Matter]
In Figure 7-10(a) and (b) the band structure and density of states (DOS) of a (10, 10)
tube are given, based on a tight-binding calculation. Samples prepared by laser
vaporization consist predominantly of (10, 10) metallic armchair SWNTs. As can be
seen in Figure 7-10 (a), band crossing is allowed, and the bonding π- and
antibonding π∗-states cross the Fermi level at kz = 2π/3.
In Figure 7-10 (b) the density of states is plotted for the (10, 10) tube. The E−1/2-
singularities which are typical for 1D energy bands appear at the band edges. STM
spectra of nanotubes show densities of states with similar singularities to those
obtained by band calculations. One-dimensional metallic wires are generally unstable
and have Peierls distortion.
The energy gain after Peierls distortion is found to be very small for nanotubes and
the Peierls energy gap can be neglected. The curvature of nanotubes introduces
hybridization also between sp2 and sp3 orbitals, but these effects are small when the
radius of a SWNT is large. However, the π∗- and σ∗-state mixing was enhanced for
small-radius zigzag SWNTs. Recently, it has been shown that the electronic
properties of a SWNT can undergo dramatic changes owing to the elastic
deformations. For example, the band gap of a semiconducting SWNT can be
reduced or even closed by the elastic radial deformation. The gap modification and
the eventual strain-induced metallization seem to offer new alternatives for reversible
and tunable quantum structures and nanodevices.
7.3.2 Quantum transport properties
A nanotube can be an ideal quantum wire for electronic transport; two subbands
crossing at the Fermi level should nominally give rise to two conducting channels.
Under ideal conditions each channel can carry current with unit quantum
conductance 2e2/h; the total resistance of an individual SWNT would be h/4e2 or ∼6
kΩ. The contribution of each subband to the total conductance is clearly seen in
Figure 7-10 (c) illustrating the calculated conductance of the (10, 10) nanotube.
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The first electronic transport measurements of nanotubes were carried out using
MWNTs. These measurements found MWNTs to be highly resistive due to defect
scattering and weak localization. The first electronic transport measurements of
individual SWNTs and nanoropes were performed by Tans et al [] and Bockrath et al
[]. In these measurements nanotubes or nanoropes were placed on an insulating
(oxidized silicon) substrate containing metallic electrodes.
(a) An AFM image of a carbon nanotube on top
of a Si/SiO2 substrate with Pt electrodes. A gate
voltage Vg is applied to the third electrode in the
upper left corner to vary the electrostatic potential
of the nanotube.
(b) Current–voltage curves of nanotubes for
different Vg values (A: 88.2 mV, B: 104.1 mV and
C: 120.0 mV).
Figure 7-11: a) and b) [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal
transport through nanowires, 2001, Journal of Physics Condensed Matter]
In Figure 7-11 (a) an AFM image of an individual SWNT on a silicon dioxide substrate
is shown with two Pt electrodes. A gate voltage Vg is applied to the third electrode in
the upper left corner to shift the electrostatic potential of the nanotube. The
measurements were performed at 5 mK and step-like features are observed in
current–voltage curves shown in Figure 7-11 (b). Note that the voltage scale is mV
and these steps are not due to the quantized increase of conductance with subbands
shown in Figure 7-10 (c). These steps are due to resonant tunnelling of electrons to
the states of a finite nanotube. The presence of metallic electrodes introduces
significant contact resistances and changes the bent part of the nanotube into a
quantum-dot-like structure. The same phenomena were seen in individual ropes of
nanotubes with oscillations in the conductance at low temperatures.
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Figure 7-12: Variation of the conductance with the gate voltage at temperatures from 1.5 K to 282 K
for a nanorope with metallic tubes. [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical
and thermal transport through nanowires, 2001, Journal of Physics Condensed Matter]
Figure 7-12 shows conductance versus gate voltage for a nanorope for various
temperatures. The average conductance drops with T. Conductance oscillations at
low temperatures are due to the Coulomb blockade effect. The energy levels of the
rope are quantized and the single molecular level spacing ∆ε exceeds the thermal
energy, kBT. On changing the gate voltage, single electrons tunnel to the quantized
molecular levels. Recently, metallic contacts with low resistances have been
achieved which do not show Coulomb blockade oscillations down to 1.7 K. Another
important behaviour in Figure 7-12 is the monotonic decrease of conductance with
decrease of temperature which is a signature of correlated electron liquids. It is well
known that the electrons in one-dimensional systems may form not Fermi liquids but
the so-called Luttinger liquids with electron correlation effects. An important feature of
Luttinger liquids is the separation of spin and charge by formation of quasi-particles.
Power-law dependences of the conductance on voltage (G ∝∝∝∝Vα) have been observed
and also on temperature (G ∝∝∝∝ Tα), which are typical for Luttinger liquids.
An interesting transport experiment was performed by Frank et al []. MWNTs were
dipped into liquid metal with the help of a scanning probe microscope tip and the
conductance was measured simultaneously. Figure 7-13 (a) and Figure 7-13 (b)
show the nanotube contact used in the measurements.
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Figure 7-13: (A) A transmission electron micrograph (TEM) image of the end of a nanotube fibre
which consists of carbon nanotubes and small graphitic particles. (B) A schematic diagram of the
experimental set-up. Nanotubes are lowered under SPM control to a liquid metal surface. (C) Variation
of conductance with nanotube fibre position. Plateaus are observed corresponding to additional
nanotubes coming into contact with the liquid metal. [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum
effects in electrical and thermal transport through nanowires, 2001, Journal of Physics Condensed
Matter]
The nanotubes were straight with lengths of 1 to 10 µm. As the nanotubes were
dipped into the liquid metal one by one, the conductance increased in steps of 2e2/h
as shown in Figure 7-13(c). Each step corresponds to an additional nanotube coming
into contact with liquid metal. The electronic transport is found to be ballistic, since
the step heights do not depend on the different lengths of nanotubes coming into
contact with the metal.
On the theory side, different techniques are used to calculate quantum effects on the
conductance of nanotubes. These techniques are based on linear response theory
and use the Landauer formalism.
1. The tip–nanotube–substrate system in STM.
2. Investigation of the Aharonov–Bohm effect in nanotubes.
3. Study of tunnelling conductance of nanotube junctions which are joined by a
connecting region with a pentagon–heptagon pair.
4. Effective-mass theory with envelope functions for similar junctions…
Among these methods, Green’s function methods are found to be the most effective
and hence are widely used for nanotubes with local basis sets.
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1. Study of nanotubes having vacancies and other defects by using a Green’s
function technique with a surface Green’s function matching method.
2. Carbon nanotubes with disorder using a similar Green’s function technique
with an efficient numerical procedure.
3. Surface Green’s function matching method. Iterative calculations of transfer
matrices are combined with the Landauer formula to calculate the
conductance. In these techniques, electronic structures of nanotubes are
derived from the π-orbital tight-binding model.
7.3.3 Nanotube junctions and devices
Current trends in microelectronics are to produce smaller and faster devices. Owing
to the novel and unusual mechanical and electronic properties, carbon nanotubes
appear to be potential candidates for meeting the demands of nanotechnologies.
There are already nanodevices which use nanotubes. Single-electron transistors
were produced by using metallic tubes; the devices formed therefrom have operated
at low temperatures. For example, a field-effect transistor that consists of a
semiconductor nanotube and operates at room temperature. The nanotube placed on
two metal electrodes and a Si substrate which is covered with SiO2 is used as a
back-gate. The nanotube is switched from a conducting to an insulating state by
applying a gate voltage.
One approach used in making a nanodevice was based on the fact that the defects in
a nanotube can form an intermolecular junction from an individual tube. It was shown
that topological defects like pentagon–heptagon pair defects can change the helicity
and hence the electronic structure of the nanotube. This raises the possibility of
fabricating metal–metal (M–M), metal–semiconductor (M–S) and semiconductor–
semiconductor (S–S) junctions on a single nanotube.
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a) Atomic structure of a
semiconductor–metal
junction ((8, 0)–(7, 1)) with a
pentagon–hexagon pair
defect. Grey balls denote the
atoms forming the defect.
b) The local density of states (LDOS) at cells 1, 2 and 3. Cell 3 is on
the (8, 0) side of the tube and cell 1 is at the interface.
Figure 7-14: a) and b) [K. S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal
transport through nanowires, 2001, Journal of Physics Condensed Matter]
Figure 7-14 (a), the atomic structure of a M–S junction is shown. An (8, 0) tube
(semiconductor) is joined to a (7, 1) tube (metal) with a pentagon–hexagon defect
(highlighted in grey). The local density of states (LDOS) based on a π-orbital tight-
binding model is presented in Figure 7-14 (b). The LDOS is distorted close to the
interface, but recovers the DOS of a perfect tube away from the interface on both
sides. There is a conductance gap in M–S and S–S junctions and diode-like
behaviour can be achieved. However, in M–M junctions, the conductance depends
on the arrangement of the defects. The junction is insulating for symmetric
arrangement, but conducting for asymmetric arrangement. Kink structures which
consist of such defects have been observed experimentally. M–S junction behaves
like a rectifying diode with non-linear transport characteristics and the conductance
of the M–M junction was suppressed. Other intermolecular junctions include
nanotube–silicon nanowire junctions and Y-junctions which exhibit rectification.
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Two or more nanotubes can form nanoscale junctions with unique properties. A
simple intermolecular nanotube junction can be formed by bringing two tube ends
together. Figure 7-15 (a) illustrates such a junction with two semi-infinite (10, 10)
tubes in parallel and pointing in opposite directions.
Figure 7-15: (a) An intermolecular nanotube junction formed by bringing two semi-infinite (10, 10)
tubes together. l is the contact length. (b) The conductance, G, of a (10, 10)–(10, 10) junction as a
function of energy, E, for l = 64 Å . Interference of electron waves yields resonances in conductance.
(c) Current–voltage characteristics of a (10, 10)–(10, 10) junction at l = 46 Å. [K. S. Ciraci, A. Buldum,
I.P. Batra, Quantum effects in electrical and thermal transport through nanowires, 2001, Journal of
Physics Condensed Matter]
These junctions have high conductance values and exhibit negative differential
resistance behaviour. Interference of electron waves reflected and transmitted at the
tube ends gives rise to the resonances in conductance shown in Figure 7-15 (b). The
current–voltage characteristics of this junction presented in Figure 7-15 (c) show a
negative differential resistance effect, which may have applications in high-speed
switching, memory and amplification devices.
A four-terminal junction can be constructed by placing one nanotube perpendicular to
another and forming a cross-junction (Figure 7-16 (a)).
Figure 7-16: (a) A four-terminal cross-junction with two nanotubes perpendicular to each other. (b)
The resistance of an (18, 0)–(10, 10) junction as a function of tube rotation. The rotation angle, Θ, and
terminal indices are shown in the inset. The tube which is labelled by 2 and 4 is rotated by Θ. The
contact region structure is commensurate at Θ = 30, 90, 150.[K. S. Ciraci, A. Buldum, I.P. Batra,
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Quantum effects in electrical and thermal transport through nanowires, 2001, Journal of Physics
Condensed Matter]
The M–M and S–S junctions had high conductance values (∼∼∼∼0.1e2/h) and the M–S
junction formed a rectifying Schottky barrier. The conductance of such intermolecular
junctions strongly depends on the atomic structure in the contact region.
Conductance between the tubes was found to be high when the junction region
structure was commensurate and conductance was low when the junction was
incommensurate. Figure 7-16 (b) shows the variation of the resistance with the
rotation of one of the tubes. The junction structure is commensurate at angles 30,
90 and 150, and hence the resistance values are lower. Similar variation of the
resistance with the atomic structure in the contact region is observed in nanotube–
surface systems. This significant variation of transport properties with atomic scale
registry was found also in mechanical/frictional properties of nanotubes. Recently,
Rueckes et al introduced a random-access memory device for molecular computing
which is based on cross-junctions. They showed that crossjunctions can have
bistable, electrostatically switchable on/off states and an array of such junctions can
be used as an integrated memory device.
7.4 Semiconductor
7.4.1 Introduction
The enormous range of physical properties afforded by size-tuning of semiconductor
nanocrystals, in a class of materials with so many established applications in
electronics, optics, and sensors, has drawn the attention of scientists from diverse
disciplines, from synthetic and physical chemists to materials scientists, condensed
matter physicists, and electrical engineers. A direct consequence of the
interdisciplinary character of this problem is the diversity of alternative visions for the
“ideal” semiconductor nanocrystal. Since none of these ideals have been fully
realized, it is of great importance for those working in the field of semiconductor
nanocrystals to understand the competing visions, as well as the realities of the
samples themselves.
A compositionally pure collection of atoms, mass-selected, isolated in the gas phase,
and thermally annealed, is one of the most compelling ideals for a semiconductor
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nanocrystal. It is in this form that the influence of size alone might be most directly
observed. The ability to prepare clusters in the gas phase by laser vaporization must
be considered one of the great achievements of cluster science and led to an
explosion of work in the field. In the case of semiconductors, early work on bare
clusters in the 3-50 atoms range has shown that remarkable changes occur in the
electronic structure in this regime. As there is no clearly identifiable interior in clusters
of this size, it is not too surprising that unique bonding geometries, distinct from those
of the bulk solid, are assumed. As the size produced and studied in the gas phase
continues to increase into the nanocrystal regime, it remains true that this form of
cluster is one in which much fundamental science remains to be done. The inability to
prepare large quantities of mass selected nanocrystals by laser vaporization and the
difficulties associated with direct measurements of structural properties in the gas
phase have also led cluster scientists to pursue other forms of nanocrystals in
parallel with the gas phase studies.
Despite the seeming perfection of a pure cluster in the gas phase, from the point of
view of semiconductor physics, it is in many ways a highly defective system. At the
surface of a pure semiconductor, substantial reconstructions in the atomic positions
occur, and there invariably lie energy levels within the energetically forbidden gap of
the bulk solid. These surface states act as traps for electrons or holes and degrade
the electrical and optical properties of the material. “Passivation” is the chemical
process by which these surface atoms are bonded to another material of a much
larger band gap, in such a way as to eliminate all the energy levels inside the gap.
The ideal termination naturally removes the structural reconstructions, leaving no
strain, and simply produces an atomically abrupt jump in the chemical potential for
electrons or holes at the interface. The termination of Si with SiO2 and that of
Al1-xGaxAs with GaAs are probably the best known examples of successful
passivation. In the first case the passivation is achieved with a disordered material
that can accommodate its local bonding geometry to that of the underlying
semiconductor; in the second case the crystal structures and bond lengths of the two
materials are matched, so that passivation is achieved epitaxially.
Over decades, the ability to control the surfaces of semiconductors with near atomic
precision has led to a further idealization of semiconductor structures: quantum wells,
wires, and dots. Ignoring for a moment the detailed atomic level structure of the
material, it is possible to imagine simple geometric objects of differing dimensionality
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(2,1, and 0), in each case made out of homogeneous semiconductor material and
with perfect surface termination. Such structures should exhibit the idealized
variations in density of electronic states predicted by simple particle in a box type
models of elementary quantum mechanics, with the continuous levels of the 3D case
evolving into the discrete states of the 0-dimensional case (Figure 7-17).
Figure 7-17: Idealized density of states for one band of a semiconductor structure of 3, 2, 1, and “0”
dimensions. In the 3D case the energy levels are continuous, while in the “0D’” or molecular limit the
levels are discrete. [A.P. Alivisatos, Perspectives on the physical chemistry of semiconductor
nanocrystals, 1996, Journal of Physical Chemistry]
Phenomenal success has been achieved in making quantum well films of nanometer
thick layers of alternating GaAs and Al1-xGaxAs, using the techniques of molecular
beam epitaxy. By manipulating steps or defects on a substrate, these same methods
can be employed to form wires or, even most recently, dots. One of the most
powerful features of these methods is the fact that, at the end, the low-dimensional
semiconductor structure is completely embedded inside another material with larger
gap, providing a high degree of surface passivation.
A chemist immediately recognizes the zero-dimensional quantum dot as a rather
large molecule. Indeed, the third ideal of the nanocrystal derives from a long history
of synthetic inorganic cluster chemistry. Chemists have long prepared ever larger
inorganic cluster compounds: collections of inorganic atoms bound to each other and
surrounded by organic ligands that confer solubility and prevent agglomeration.
These clusters are only considered well characterized when they have been
crystallized and their structures determined by X-ray diffraction. Unlike any other form
of nanocrystal, in this case, the precise atomic composition and the location of each
atom are known. In the past decade, chemists have achieved great success in
increasing the number of atoms in the inorganic cores of these cluster compounds, to
the point where they extend clearly into the nanocrystal regime (Figure 7-18). It is of
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considerable interest to note that this form of cluster will naturally assume high-
symmetry shapes, such as tetrahedral or hexagonal prism.
Figure 7-18: Ball and stick model of a Cd32S55 molecule recently synthesized and structurally
characterized. The organic ligands are omitted for clarity. This molecule is a fragment of the CdS zinc
blende lattice. [A.P. Alivisatos, Perspectives on the physical chemistry of semiconductor nanocrystals,
1996, Journal of Physical Chemistry]
From the perspective of solid state physics and materials science, it is perhaps
surprising that nanocrystals of inorganic solids, coated with organic ligands, may
prove one of the most diverse and powerful ideals of a “quantum dot.” This is so
because, as a molecule, the nanocrystal can now be considered not just a
component embedded in the surface of a solid state device, but rather a chemical
reagent. In this form the nanocrystal may be dissolved in a fluid, spun into a polymer,
attached to an electrical circuit, bound to other nanocrystals as dimers, trimers, etc.
or someday perhaps bound to biological molecules. It is important to realize that, in
the construction of optical and electronic materials using components of nanometer
size, it is not only the physical properties of matter that change but also the chemical
methods by which the materials are constructed.
It is not yet established one way or the other whether organic ligands, which turn
nanocrystals into chemical reagents, also act as good passivating layers. One
difficulty arises in matching the large, bulky ligands required to confer solubility with
the compact packing of atoms on the surfaces of the crystallites. Bulky ligands with
many torsional modes confer solubility entropically by mixing with the solvent.
However, their size ensures that invariably some surface sites are unterminated.
Inorganic passivation of nanocrystals still is possible, however. This is clearly
demonstrated in the case of SiO2-coated Si nanocrystals and in the case of oxidized
InP nanocrystals. It is also seen in nanocrystals that are grown inside glass. Indeed,
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there are even analogs to quantum wells, in which layers of inorganic solids are
grown in successive shells around one another, epitaxially. The best documented
example of this is the CdS/HgS/CdS quantum dot quantum well. In each case, the
final product is still small enough that a final layer of ligands bound to the outer
surface is sufficient to confer solubility and open the use of these solid state materials
to the world of chemical synthesis.
7.4.2 Band Gap modification
Independent of the large number of surface atoms, semiconductor nanocrystals with
the same interior bonding geometry as a known bulk phase often exhibit strong
variations in their optical and electrical properties with size. These changes arise
through systematic transformations in the density of electronic energy levels as a
function of the size of the interior, known as quantum size effects. Nanocrystals lie in
between the atomic and molecular limit of discrete density of electronic states and
the extended crystalline limit of continuous bands (Figure 7-19).
a) The Fermi level is in the center of a
band in a metal, and so kT will exceed the
level spacing even at low temperatures and
small sizes.
b) In contrast, in semiconductors, the Fermi
level lies between two bands, so that the
relevant level spacing remains large even at
large sizes. The HOMO-LUMO gap increases
in semiconductor nanocrystals of smaller size.
Figure 7-19: Density of states in metal (a) and sem iconductor (b) nanocrystals. In each case, the
density of states is discrete at the band edges. [A.P. Alivisatos, Perspectives on the physical chemistry
of semiconductor nanocrystals, 1996, Journal of Physical Chemistry]
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7.4.2.1 Quantum Size Effects
The most striking property of semiconductor nanocrystals is the massive changes in
optical properties as a function of the size. As size is reduced, the electronic
excitations shift to higher energy, and there is concentration of oscillator strength into
just a few transitions. These basic physical phenomena of quantum confinement
arise by changes in the density of electronic states and can be understood by
considering the relationship between position and momentum in free and confined
particles:
2/ℏ≥∆∆ xp (7. 173)
For a free particle or a particle in a periodic potential, the energy and the crystal
momentum ћk may both be precisely defined, while the position is not. As a particle
is localized, the energy may still be well-defined; however, the uncertainty in position
decreases, so that momentum is no longer well-defined. The energy eigenfunctions
of the particle may then be viewed as superpositions of bulk k states. In the extended
case, there is a relationship between energy and momentum, and to a first
approximation, the change in energy as a function of the size can be estimated
simply by realizing that the energy of the confined particle arises by superposition of
bulk k states of differing energy.
For a free particle, the dependence of energy on wavevector is quadratic:
mkE 2/2ℏ= (7. 174)
In the effective mass approximation, this relationship is assumed to hold for an
electron or hole in the periodic potential of the semiconductor, with a reduced mass
which is inversely proportional to the width of the band. Given the relationship
between confinement in space and momentum superposition, this leads directly to
the approximate dependence of energy on size as 1/r², as expected for a simple
particle in a box. For large sizes, the approximation is nearly correct but breaks down
for even moderately sized because energy does not depend quadratically on k in real
crystallites.
To gain a physical understanding of the variation of E with k, it is at this point useful
to switch to a molecular picture of bonding in the solid (Figure 7-20) and of the
quantum confinement process.
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Figure 7-20: (A) A simplified MO
diagram for the electronic structures of
zinc blende CdSe and diamond structure
Si. (B) A comparison of the HOMO-LUMO
transitions for CdSe and Si. In CdSe the
transition is dipole allowed, while in Si it is
not. [A.P. Alivisatos, Perspectives on the
physical chemistry of semiconductor
nanocrystals, 1996, Journal of Physical
Chemistry]
The single-particle wave functions for electrons and holes in the extended solid can
be viewed as linear combination of unit cell atomic orbitals, multiplied by phase
factors between the unit cells. When all the cells are in phase, the wavevector,
k=(2π/λ), is equal to 0; when adjacent cells are out of phase, k takes on its maximum
value, π/a. In a simple one-dimensional single band tight binding model, the
dependence of E on k is
)cos(2 kaE βα += (7. 175)
where α, the energy of the linear combination of atomic orbitals inside the unit cell,
determines the center position of the band in energy. 2β gives the width of the band
and is directly related to the strength of nearest-neighbor coupling and inversely
proportional to the effective mass. An expansion of the cos(ka) term for small k
yields a quadratic term as its first term, so that one can see why the effective mass
approximation only describes the band well near either its minimum or its maximum.
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Considering now a real binary semiconductor, such as CdSe, the single particle
states can be viewed as products of unit cell atomic orbital combinations and phase
factors between unit cells (Figure 7-20 A). For example, the highest occupied
molecular orbital or top of the valence band may be viewed as arising from Se 4p
orbitals, arranged to be in phase between unit cells. This will be the maximum of the
valence band, since adjacent p orbitals in phase are σ antibonding (and π bonding).
Similarly, the lowest unoccupied molecular orbital will be comprised of Cd 5s atomic
orbitals, also in phase between unit cells. This is the minimum of the conduction
band, since s orbitals in phase constructively interfere to yield a bonding level. In this
case, the minimum of the conduction band and the maximum of the valence band
have the same phase factor between unit cells.
How does optical absorption arise in the extended solid? The optical absorption
matrix element is given by
)exp()exp(45 ,, rikrikpSeersCd vcvibivibf ψψµ = (7. 176)
where exp(ikr) , the envelope functions, denote the phase factors between unit cells,
and the vibrational overlap between ground and excited vibrational states is
determined by Franck-Condon factors, just as for allowed transitions in molecules.
Note that in this case the electronic transition between the unit cell functions is dipole
allowed. The radiative rate is determined from
∫ +−=f ikfkphotonif dEEEEEEw ),()(2 2 ρδµπℏ
(7. 177)
where ρ denotes the joint density of valence and conduction band electronic states
with the same k. The matrix element is factored so that the dipole operator acts on
the electronic wave function within the unit cell, and the phase factors between unit
cells, exp(ik vr) and exp(ik cr), must overlap for the transition to be allowed (kv = kc).
In this case, the transition is dipole allowed within each unit cell. Further, the
transition dipole moment points in the same direction from unit cell to unit cell.
Thus, the electric field of light, which is very long wavelength compared to the size of
the unit cell, can drive all the transition dipoles in phase, and overall the transition is
allowed. In the language of solid state physics, this is a vertical transition, with ∆k=0,
and hence CdSe is called a direct band gap material. At low temperature, the
electronic transition will be accompanied by vibrational excitation of totally symmetric
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modes, in accordance with the Franck-Condon factors. Past the absorption threshold,
the unit cell matrix element for absorption is largely unchanged, and the absorption
efficiency rises steeply at higher incident photon energies because of the large
increase in the joint density of states for vertical transitions. Numerous experiments
show that quantum confinement dominantly affects the absorption spectrum by
changes in the envelope functions and the density of states as a function of the
energy, with relatively less effect on the intrinsic, unit cell based, matrix element for
absorption.
7.4.2.2 Quantization and energy level spacing
As the size is reduced, the electronic states may be viewed as superpositions of bulk
states. Hence, there is a shift to higher energy, the development of discrete features
in the spectra, and concentration of the oscillator strength into just a few transitions.
Qualitatively, all of these effects can be readily observed in the spectra of Figure
7-21, which show data for CdSe.
Figure 7-21: Optical absorption vs size for CdSe nanocrystals shows the shift to higher energy in
smaller sizes, as well as the development of discrete structure in the spectra and the concentration of
oscillator strength into just a few transitions. [A.P. Alivisatos, Perspectives on the physical chemistry of
semiconductor nanocrystals, 1996, Journal of Physical Chemistry]
The quantitative analysis of these spectra remains a difficult subject, for several
reasons:
• The foregoing picture is a single-particle one and does not include the
substantial effects of correlation. In molecules this is analogous to trying to use
the highly approximate molecular orbital theory, instead of more advanced
quantum chemistry methods. Regrettably, the nanocrystals are too large to
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describe using even moderately advanced methods that are routinely applied
to small molecules.
• Further, in CdSe at least, the large atomic number of the Se ensures that the
coupling between spin and orbital momenta is very strong in the valence
bands (p bands). This coupling is in the j-j, and not the Russell-Saunders, L-S,
coupling regime. When translational symmetry is removed, the mixing of k
vectors can also result in different bands mixing together.
• The shape of the crystallites, which is regular (tetrahedral, hexagonal prisms),
or spherical, or ellipsoidal, will determine the symmetry of the nanocrystals
and will influence the relative spacing of the levels.
• Finally, surface energy levels are completely excluded from this simple
quantum confinement picture. Yet it seems apparent that surface states near
the gap can mix with interior levels to a substantial degree, and these effects
may also influence the spacing of the energy levels.
Theoretical approaches that directly include the influence of the surface, as well as
electronic correlation, are being developed rapidly.
7.4.3 Electrical properties
We talk about single-electronics whenever it is possible to control the
movement and position of a single or small number of electrons. To understand how
a single electron can be controlled, one must understand the movement of electric
charge through a conductor. An electric current can flow through the conductor
because some electrons are free to move through the lattice of atomic nuclei. The
current is determined by the charge transferred through the conductor. Surprisingly
this transferred charge can have practically any value, in particular, a fraction of the
charge of a single electron. Hence, it is not quantized.
This, at first glance counterintuitive fact, is a consequence of the displacement
of the electron cloud against the lattice of atoms. This shift can be changed
continuously and thus the transferred charge is a continuous quantity (see left side of
Figure 7-22).
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+ + + + + +- - - - - -
+ + + + + +- - - - - -
+ + + + + +- - - - - -
+ + +
+
+++
+ +
+ + +
+
+++
+ +
tunn
el ju
nctio
n
Figure 7-22 : The left side shows, that the electron cloud shift against the lattice of atoms is not
quantized. The right side shows an accumulation of electrons at a tunnel junction.
If a tunnel junction interrupts an ordinary conductor, electric charge will move
through the system by both a continuous and discrete process. Since only discrete
electrons can tunnel through junctions, charge will accumulate at the surface of the
electrode against the isolating layer, until a high enough bias has built up across the
tunnel junction (see right side of Figure 7-22). Then one electron will be transferred.
Likharev has coined the term `dripping tap' as an analogy of this process. In other
words, if a single tunnel junction is biased with a constant current I, the so called
Coulomb oscillations will appear with frequency f = I/e, where e is the charge of an
electron (see Figure 7-23).
I
I
conduction current
Figure 7-23 : Current biased tunnel junction showing Coulomb osc illations.
Charge continuously accumulates on the tunnel junction until it is energetically
favorable for an electron to tunnel. This discharges the tunnel junction by an
elementary charge e. Similar effects are observed in superconductors. There, charge
carriers are Cooper pairs, and the characteristic frequency becomes f = I/2e, related
to the so called Bloch oscillations.
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Figure 7-24 : The electron-box can be filled with a precise number of electrons.
The current biased tunnel junction is one very simple circuit, that shows the
controlled transfer of electrons. Another one is the electron-box (see Figure 7-24).
A particle is only on one side connected by a tunnel junction. On this side electrons
can tunnel in and out. Imagine for instance a metal particle embedded in oxide, as
shown in Figure 7-25.
oxide
metalparticle
Figure 7-25 : Metal particle embedded in oxide. Tunneling is only possible through the thin top layer
of oxide.
The top oxide layer is thin enough for electrons to tunnel through. To transfer
one electron onto the particle, the Coulomb energy EC = e2/2C, where C is the
particles capacitance, is required. Neglecting thermal and other forms of energy, the
only energy source available is the bias voltage Vb. As long as the bias voltage is
small enough, smaller than a threshold Vth = e/C, no electron can tunnel, because
not enough energy is available to charge the island. This behavior is called the
Coulomb blockade . Raising the bias voltage will populate the particle with one, then
two and so on electrons, leading to a staircase-like characteristic.
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It is easily understandable, that these single-electron phenomena, such as
Coulomb oscillations and Coulomb blockade , only matter, if the Coulomb energy
is bigger than the thermal energy. Otherwise thermal fluctuations will disturb the
motion of electrons and will wash out the quantization effects. The necessary
condition is,
2
2c B
eE k T
C= > (7. 178)
Where kB is Boltzmann's constant and T is the absolute temperature. This means
that the capacitance C has to be smaller than 12 nF for the observation of charging
effects at the temperature of liquid nitrogen and smaller than 3 nF for charging effects
to appear at room temperature. A second condition for the observation of charging
effects is, that quantum fluctuations of the number of electrons on an island must be
negligible. Electrons need to be well localized on the islands. If electrons would not
be localized on islands one would not observe charging effects, since islands would
not be separate particles but rather one big uniform space. The charging of one
island with an integer number of the elementary charge would be impossible,
because one electron is shared by more than one island. The Coulomb blockade
would vanish, since no longer would a lower limit of the charge, an island could be
charged with, exist. This leads to the requirement that all tunnel junctions must be
opaque enough for electrons in order to confine them on islands. The `transparency'
of a tunnel junction is given by its tunnel resistance RT which must fulfill the following
condition for observing discrete charging effects, where h is Planck's constant. This
should be understood as an order-of-magnitude measure, rather than an exact
threshold.
225813T
hR
e> = Ω (7. 179)
Therefore, these effects are experimentally verifiable only for very small high-
resistance tunnel junctions, meaning small particles with small capacitances and/or
very low temperatures. Advanced fabrication techniques, such as the production of
granular films with particle sizes down to 1 nm, and deeper physical understanding
allow today the study of many charging effects at room temperature.
Based on the Coulomb blockade many interesting devices are possible, such as
precise current standard, lxxiii very sensitive electrometers ,lxxiv logic gates lxxv and
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memories lxxvi with ultra low power consumption, down-scalability to atomic
dimensions, and high speed of operation. Altogether, single-electronics will bring new
and novel devices and is a very promising candidate to partly replace MOS
technology in the near future.
7.4.3.1 Characteristic Energies
Charging an island with a small number of electrons is associated with two
kinds of energies:
1. One kind arises from electron-electron interaction
2. and the other is linked to the spatial confinement of electrons.
These energies are only significant for very small structures in the nanometer
regime. Otherwise they are masked by thermal fluctuations.
We discuss two other important characteristic energies, the work done by the voltage
sources and Helmholtz's free energy, and the role they play in single-electron device
behavior.
Electron Electron Interaction
An entirely classical model for electron-electron interaction is based on the
electrostatic capacitive charging energy. The interaction arises from the fact, that for
every additional charge dq which is transported to a conductor, work has to be done
against the field of already present charges residing on the conductor. Charging a
capacitor with a charge q requires energy, where C is the total capacitance of the
capacitor. A first simple model for an island is a conducting sphere which has a self-
capacitance of (see the next table)
2
4c
qE
dπε= (7. 180)
Where ε is the permittivity of the surrounding dielectric material and d is the
diameter of the sphere. Inserting 2
2c B
eE k T
C= > (7. 178) in
2
4c
qE
dπε= (7.
180)) yields :
2C dπε= (7. 181)
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An indirect proportionality to the size of the particle. Figure 7-26 to Figure 7-28
compare this electrostatic energy to the other characteristic energies involved in
charging an island with a small number of electrons.
Change of E F for 1st additional electron in Si Electrostatic charging energy e 2 /2c
thermal energy at 300 k
10 -1 10 0 10 1 10 2 10 -3
10 - 2
10 -1
10 1
10 0
Particle diameter [nm]
Change of E F for 2nd additional electron in Si Change of E F for 3rd additional electron in Si
Ene
rgy
[eV
]
Figure 7-26 : Comparison of the change in Fermi energy for the addition of one, two, and three
electrons in Si to the electrostatic charging energy.
10 1
Change of E F for 1st additional electron in Si Electrostatic charging energy e 2 /2c
10 -1 10 0 10 1 10 2 10 -3
10 - 2
10 -1
10 0
Particle diameter [nm]
Change of E F for 2nd additional electron in Si
Ene
rgy
[eV
]
Figure 7-27 : Comparison of the change in Fermi energy for the addition of one electron in Si and Al.
The values below 1 nm particle diameter are questionable (see later)
For the sake of simplicity, the relative permittivity ε was assumed to be 10. A more
accurate model for the capacitance of an island is the capacitance of two or more
spheres in a line. A calculation for characteristic spatial dimensions shows an
increase of about 15% of the total capacitance.
For a system of N conductors the charge on one conductor may be written as
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1
N
i ij jj
q C ϕ=
=∑ ( 1,2,3,......)i = (7. 182)
Where the ijC denote the elements of the capacitance matrix. The diagonal elements
of the capacitance matrix ijC are the total capacitances of the conductors i and the
off-diagonal elements ijC are the negative capacitances between conductor i and j.
The electrostatic charging energy is in analogy to 2
2c B
eE k T
C= > (7. 178)
( )1
1 1 1
1 1
2 2
N N N
c i i i jiji i j
E q C q qϕ −
= = =
= =∑ ∑∑ (7. 183)
where C-1 is the inverse of the capacitance matrix.
Systems with sufficiently small islands are not adequately described with this
model. They exhibit a second electron-electron interaction energy, namely the
change in Fermi energy, when charged with a single electron. One must distinguish
between metals and semiconductors, because of their greatly different free carrier
concentrations and the presence of a band gap in the case of semiconductors.
Metals have typically a carrier concentration of several 1022/cm3 and undoped Si
about 1010/cm3 at room temperature. The low intrinsic carrier concentration for
semiconductors is in reality not achievable, because of the ubiquitous impurities
which are ionized at room temperature. These impurities supply free charge carriers
and may lift the concentration, in the case of Si, to around 1014/cm3, still many orders
of magnitude smaller than for metals. Clearly, doping increases the carrier
concentration considerably, and can lead in the case of degenerated semiconductors
to metal like behavior. We are considering only undoped semiconductors to
emphasize the difference to metals.
The dependence of the Fermi energy EF on carrier concentration n can be expressed
as below (N. W. Ashcroft and N. D. Mermin, Solid State Physics, W. B. Saunders
Company, international edition, 1976). For metals it is given by
( )2
2 32*
32FE n
mπ= ℏ (7. 184)
and for semiconductors it is
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,2sinh F F inet
i B
E En
n k T
−= (7. 185)
Where nnet is the net carrier concentration defined as nnet= n-p . Figure 7-26
compares the change in the Fermi energy for the addition of one, two, and three
electrons in Si to the electrostatic charging energy.
The first electron causes the biggest change in Fermi energy. The second and
third electron contribute a change smaller than the thermal energy for room
temperature. Hence, for single-electron devices one will try to build structures with
few free carriers, to achieve a high change in energy when charging an island.
Figure 7-27 compares Si with Al. It is clearly visible that the change is much
bigger in Si. This is caused by two reasons. First, the difference in free carrier
concentration. In metals much more carriers are available and therefore the addition
of one electron has not such a big impact anymore. Second in metals an additional
electron finds a place slightly above the Fermi level. In semiconductors due to the
energy gap the electron needs to be inserted considerably above the Fermi level.
Quantum Confinement Energies
With decreasing island size the energy level spacing of electron states
increases indirectly proportional to the square of the dot size. Taking an infinite
potential well as a simple model for a quantum dot, one calculates by solving
Schrödinger's equation the energy levels to
2
*
1
2N
NE
m d
π =
ℏ (7. 186)
Because of the lower effective mass in Si compared to Al, the quantum confinement
energy is bigger in Si (see Figure 7-28).
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Confinement energy for SiConfinement energy for metalsElectrostatic energy e 2/2c
thermal energy at 300 k
thermal energy at 77 k
10-1 100 101 10210-3
10-2
10-1
101
100
Particle diameter [nm]
Figure 7-28 : Comparison of the quantum confinement energy in Si and Al.
For very small particle diameters (around 1 nm) the formulas
( )2
2 32*
32FE n
mπ= ℏ (7. 184), ,2sinh F F inet
i B
E En
n k T
−= (7. 185) and
2
*
1
2N
NE
m d
π =
ℏ
(7. 186) cease to be valid. The concept of the effective mass is based on a
periodic lattice and starts to break down for small islands. One would have to apply
cluster theory to calculate the energy levels more precisely. In reality slightly smaller
confinement energies than predicted in 2
*
1
2N
NE
m d
π =
ℏ (7. 186) are observed,
which may partly be attributed to the model of an infinite well. The energy levels
slightly decrease in the case of a finite well. Furthermore, the lattice constant for
metals is about 0.4 nm whereas for semiconductors it is almost 0.6 nm. Thus the
concept of carrier densities is not valid anymore.
Room temperature operation is achievable with structures smaller than 10 nm.
Cooling single-electron devices with liquid nitrogen reduces thermal fluctuations to
some extent, but relaxes the size constrains only by a factor of two.
7.4.3.2 Tunneling
In 1923 de Broglie introduced a new fundamental hypothesis that particles
may also have the characteristics of waves. Schrödinger expressed this hypothesis
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1926 in a definite form which is now known as the Schrödinger wave equation. The
continuous nonzero nature of its solution, the wave function which represents an
electron or particle, implies an ability to penetrate classically forbidden regions and a
probability of tunneling from one classically allowed region to another. Fowler and
Nordheim explained in 1928 the main features of electron emission from cold metals
by high external electric fields on the basis of tunneling through a triangular potential
barrier. Conclusive experimental evidence for tunneling was found by L. Esaki in
1957 and by I. Giaever in 1960. Esaki's tunnel diode had a large impact on the
physics of semiconductors, leading to important developments such as the tunneling
spectroscopy, and to increased understanding of tunneling phenomena in solids. L.
Esaki , I. Giaever and B. Josephson received 1973 the Nobel prize for their work
about tunneling in semiconductors, superconductors and theoretical predictions of
the properties of a supercurrent through a tunnel barrier, respectively. The concept of
resonant tunneling in double barriers was first introduced by R. Davis and H. Hosack.
At about the same time C. Neugebauer and M. Webb and some years later H. Zeller
and I. Giaever and J. Lambe and R. Jaklevic studied granular films. They observed a
current suppression at low bias voltage, which is today known as the Coulomb
blockade. It took almost two decades until in 1985 D. Averin and K. Likharev
formulated the `orthodox' theory of single-electron tunneling or short SET, which
quantitatively describes important charging effects such as Coulomb blockade and
Coulomb oscillations. The orthodox theory makes the following assumptions: It
neglects dimensions and shapes of islands. Thus it is a zero-dimensional model. The
tunnel process is assumed to be instantaneous. Actually the tunnel time, the duration
an electron spends below a barrier, is of the magnitude of 10-14 s. Charge
redistributions after a tunnel event are also assumed to be instantaneous. In addition
energy spectra in leads and islands are taken to be continuous. The main result of
the orthodox theory is that the rate of a tunnel event strongly depends on the change
in free energy the event causes.
7.4.3.3 Minimum Tunnel Resistance for Single Electr on Charging
The formulation of the Coulomb blockade model is only valid, if electron states are
localized on islands. In a classical picture it is clear, that an electron is either on an
island or not. That is the localization is implicit assumed in a classical treatment.
However a preciser quantum mechanical analyses describes the number of electrons
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localized on an island N in terms of an average value <N> which is not necessarily an
integer. The Coulomb blockade model requires
21N N− ⟨⟨ (7. 187)
Clearly, if the tunnel barriers are not present, or are insufficiently opaque, one can not
speak of charging an island or localizing electrons on a quantum dot, because
nothing will constrain an electron to be confined within a certain volume.
A qualitative argument is to consider the energy uncertainty of an electron
E t h∆ ∆ > (7. 188)
The characteristic time for charge fluctuations is
Tt R C∆ ≅ (7. 189)
The time constant for charging capacitance C through tunnel resistor RT, and the
energy gap associated with a single electron is
2eE
C∆ = (7. 190)
Combining these equations gives the condition for the tunnel resistance
225813T Q
hR R
e> = = Ω (7. 191)
Another line of thought proceeds as follows. The condition2
1N N− ⟨⟨ requires that
the time t which an electron resides on the island be much greater than t∆ , the
quantum uncertainty in this time.
ht t
E>> ∆ ≥
∆ (7. 192)
The current I cannot exceed e/t since for moderate bias and temperature at most one
extra electron resides on the island at any time. The energy uncertainty of the
electron ∆ E is no larger than the applied voltage Vb.
bE eV∆ < (7. 193)
Inserting t=e/I results in
2b
T
V hR
I e= >> (7. 194)
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In fact, more rigorous theoretical studies of this issue have supported this conclusion.
Experimental tests have also shown this to be a necessary condition for observing
single-electron charging effects.lxxvii
7.4.3.4 The Double Tunnel Junction
Consider two tunnel junctions in series biased with an ideal voltage source as shown
in Figure 7-29. The charges on junction one, junction two, and on the whole island
can be written as
1 1 1 2 2 2
2 1 0 0
, ,q C V q C V and
q q q q ne q
= == − + = − +
(7. 195)
respectively, with n1 the number of electrons that tunneled through the first junction
entering the island, n2 the number of electrons that tunneled through the second
junction exiting the island, and n=n1-n2 the net number of electrons on the island.
A background charge q0 produces generally a non-integer charge offset. The
background charge is induced by stray capacitances that are not shown in the circuit
diagram Figure 7-29 and impurities located near the island, which are practically
always present. Using 1 1 1 2 2 2
2 1 0 0
, ,q C V q C V and
q q q q ne q
= == − + = − +
(7. 195) and Vb=V1+V2 gives
2 0 1 01 2
1 2
,
,
b bC V ne q C V ne qV V
C C
where C C CΣ Σ
Σ
+ − − += =
= + (7. 196)
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Figure 7-29 : Two tunnel junctions in series biased with an ideal voltage source. The background
charge q0 is non-integer, and n1 and n2 denote the number of tunneled electrons through junction one
and junction two, respectively.
With 2 0 1 0
1 2
1 2
,
,
b bC V ne q C V ne qV V
C C
where C C CΣ Σ
Σ
+ − − += =
= + (7. 196) the electrostatic energy
stored in the double junction is
( )222 21 2 01 2
1 22 2 2b
C
C C V ne qq qE
C C C∑
+ −= + = (7. 197)
In addition, to get the free energy one must consider, the work done by the
voltage source. If one electron tunnels through the first junction the voltage source
has to replace this electron -e, plus the change in polarization charge caused by the
tunneling electron. V1 changes according to 2 0 1 0
1 2
1 2
,
,
b bC V ne q C V ne qV V
C C
where C C CΣ Σ
Σ
+ − − += =
= + (7.
196) by e C∑− and hence the polarization charge is 1eC C∑−
The charge q1 gets smaller, which means that the voltage source `receives'
polarization charge. The total charge that has to be replaced by the voltage source is
therefore 2eC C∑− and the work done by the voltage source in case electrons tunnel
through junction one and junction two is accordingly
1 21
2 12
b
b
n eV CW
C
and
n eV CW
C
∑
∑
= −
= −
(7. 198)
The free energy of the complete circuit is
( ) ( ) ( ) 221 2 1 2 0 1 2 2 1
1 1,
2c b bF n n E W C C V ne q eV C n C nC∑
= − = + − + + (7. 199)
At zero temperature, the system has to evolve from a state of higher energy to
one of lower energy. At non-zero temperatures transitions to higher energy states are
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possible, but have exponentially reduced probability. The change in free energy for
an electron tunneling through junction one and two is given by
( ) ( ) ( )1 1 2 1 2 2 01, ,2 b
e eF F n n F n n V C ne q
C±
∑
∆ = ± − = ± + − (7. 200)
( ) ( ) ( )2 1 2 1 2 1 0, 1 ,2 b
e eF F n n F n n V C ne q
C±
∑
∆ = ± − = ± − + (7. 201)
The probability of a tunnel event will only be high, if the change in free energy is
negative - a transition to a lower energy state. The leading term in equations above
causes ∆F to be positive until the magnitude of the bias voltage Vb exceeds a
threshold which depends on the smaller of the two capacitances. This is the case for
all possible transitions starting from an uncharged island, n=0 and q0=0. For
symmetric junctions (C1 = C2) the condition becomes
bV e C∑> (7. 202)
This suppression of tunneling for low bias is the Coulomb blockade. The Coulomb
blockade can be visualized with an energy diagram Figure 7-30.
filled energy states empty energy
states
EF ga p
EF
-eVb
Figure 7-30 : Energy diagram of a double tunnel junction without and with applied bias. The Coulomb
blockade causes an energy gap where no electrons can tunnel through either junction. A bias larger
than e/C overcomes the energy gap.
Due to the charging energy of the island, a Coulomb gap has opened, half of which
appears above and half below the original Fermi energy. No electrons can tunnel into
the island from the left or right electrode, or out of the island. Only if the bias voltage is
raised above a threshold can electrons tunnel in and out, and current will flow.
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The same fact, the existence of a Coulomb blockade, is clearly visible in the IV-
characteristic, Figure 7-31.
Figure 7-31 : IV-characteristic of a double tunnel junction. The solid line gives the characteristic for
q0=0 and the dashed line for q0=0.5e. The Coulomb blockade is a direct result of the additional
Coulomb energy, e2/2C, which must be expended by an electron in order to tunnel into or out of the
island.
For low bias no current flows. As soon as Vb exceeds the threshold the junction
behaves like a resistor.
However, the background charge q0 can reduce, or for ( )0 0.5q m e= ± + even
eliminate the Coulomb blockade. This suppression of the Coulomb blockade due to
virtually uncontrollable background charges is one of the major problems of single-
electron devices.
If an electron enters the island via junction one, it is energetically highly favorable for
another electron to tunnel through junction two out of the island. Thus an electron will
almost immediately exit the island after the first electron entered the island. This is a
space-correlated tunneling of electrons (see Figure 7-32). After a varying duration
another electron might first exit the island via junction two and again, immediately a
new electron will tunnel through junction one, entering the island and establishing
charge neutrality on the island. If the transparency of the tunnel junctions is strongly
different, for example 1 2T T TR R R<< = a staircase like IV-characteristic appears, as
shown in Figure 7-33.
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t
t
I
I
junction one
junction two
Figure 7-32 : Space-correlation of tunneling in a double tunnel junction. A tunnel event in junction one
is immediately followed by an event in junction two and vice versa. The duration between those tunnel
pairs is varying.
4
3 2 3 0
0 2 4 6 Vb
I
4
3 2 1 0
2 4 6 Vb
q
-4
-3
2
-2 -1
Figure 7-33: For strongly differing tunnel junctions a staircase like IV-characteristic appears.
Depending on which tunnel junction is more transparent, and the direction in which the charge carriers
flow, the island will be populated or depleted by an integer number of carriers with increasing bias
voltage. If carriers enter the island through the more transparent junction and leave through the
opaque one the island will be populated with excess carriers. If they have to enter through the opaque
junction and leave through the transparent one, the island is depleted of carriers.
Carriers enter the island through the first tunnel junction and are kept from the high
resistance of the second junction from immediately leaving it. Finally the carrier will,
due to the high bias, tunnel out of the island, which quickly triggers another electron
to enter through junction one. For most of the time the island is charged with one
excess elementary charge. If the bias is increased more electrons will most of the
time populate the island. The charge characteristic is shown on the right side of
Figure 7-33. If the asymmetry is turned around and the second junction is more
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transparent then the first one 2 1T T TR R R<< = the island will be de-populated and the
charge on the island shows a descending staircase characteristic. Carriers are
sucked away from the island through the transparent junction and can not be
replenished quickly enough through the opaque one. However, the IV-characteristic
does not change.
7.4.3.5 Single Electron Transistor
Adding to the double tunnel junction a gate electrode Vg which is capacitively coupled
to the island, and with which the current flow can be controlled, a so-called SET
transistor is obtained (see Figure 7-34).
Figure 7-34 : SET transistor.
The first experimental SET transistors were fabricated by Fulton and
Dolan, lxxviii and Kuzmin and Likharev lxxix already in 1987. The effect of the gate
electrode is that the background charge q0 can be changed at will, because the gate
additionally polarizes the island, so that the island charge becomes
( )0 2g gq ne q C V V= − + + − (7. 203)
The formulas derived in Section 7.4.3.3 for the double junction can be modified to
describe the SET transistor. Substituting
( )0 0 2g gq q C V V→ + − (7. 204)
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in 2 0 1 0
1 2
1 2
,
,
b bC V ne q C V ne qV V
C C
where C C CΣ Σ
Σ
+ − − += =
= + (7. 196), the new voltages across the
junctions are
( )21
12
,g b g g o
b g g o
C C V C V ne qV
C
C V C V ne qV
C
∑
∑
+ − + −=
+ − +=
(7. 205)
with 1 2 gC C C C∑ = + +
The electrostatic energy has to include also the energy stored in the gate capacitor,
and the work done by the gate voltage has to be accounted for in the free energy.
The change in free energy after a tunnel event in junctions one and two becomes
( )
1 2 0
1 1 0
2 g b g g
b g g
e eF C C V C V ne q
C
eF V C V C ne q
C
±
∑
±
∑
∆ = ± + − + −
∆ = + − + (7. 206)
At zero temperature only transitions with a negative change in free energy,
1 0F∆ <
Or,
2 0F∆ <
are allowed. These conditions may be used to generate a stability plot in the Vb-Vg
plane, as shown in Figure 7-35.
Figure 7-35 : Stability plot for the SET transistor. The shaded regions are stable regions.
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The shaded regions correspond to stable regions with an integer number of
excess electrons on the island, neglecting any non-zero background charge. If the
gate voltage is increased, and the bias voltage is kept constant below the Coulomb
blockade, bV e C∑< which is equivalent to a cut through the stable regions in the
stability plot, parallel to the x-axis, the current will oscillate with a period of e/Cg. As
opposed to the Coulomb oscillations in a single junction, which were explained in the
introduction, where the periodicity in time of discrete tunnel events is observed, these
are Coulomb oscillations which have a periodicity in an applied voltage, where
regions of suppressed tunneling and space correlated tunneling alternate. Figure
7-36 shows the qualitative shape of the current oscillations
I
1
-1 1 2 0 0
V g
Figure 7-36 : Coulomb oscillations in a SET transistor.
Increasing the bias voltage will increase the line-width of the oscillations,
because the regions where current is allowed to flow grow at the expense of the
remaining Coulomb blockade region. Thermal broadening at higher temperatures or
a discrete energy spectrum change the form of the oscillations considerably.
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7.4.3.6 Coulomb Blockade in a Quantum Dot: Phenomen ological Explanation
DrainSource
Dot
e2/C
EF
Figure 7-37: Quantum dot for Coulomb blockade demon stration.
By using the split-gate techniques, one can form tunnel barriers which isolate
a puddle of electrons, often called a "dot ", from the leads. To add one electron to the
dot costs a "charging energy " e2/C. The total capacitance of a sub-micron-sized dot
to the surrounding gates and leads can be very small, less than 1 fF (10-15 F), so this
energy is greater than the thermal energy for temperatures of a few Kelvin or less.
Thus at such temperatures, transport through the dot is determined by Coulomb
charging. Smaller dots can give even higher temperatures of operation.
For an electron to tunnel onto the dot, when the applied bias is small, there
must be no net energy cost of adding an electron. Thus tunneling can only occur
when the size of the dot is such that it would be neutral if it contained an integer N
plus a half electrons. Since it can only contain an integer number, the net charge on
the dot will be ±e/2, depending on whether there are N or N+1 electrons in the dot.
The charging energy is then (±e/2)2/C, which is the same in both cases. Hence
changing the number of electrons in the dot between N and N+1 costs no energy, so
electrons can flow on and off the dot with ease, and a current can flow. If the size of
the dot is changed slightly, the charging energy in the two cases is no longer the
same, and so the current falls to zero, only rising again when the size of the dot has
changed by one electron. This is called the Coulomb Blockade (CB) of tunnelling .
Sweeping a gate (shown shaded in the figure) changes the size of the dot and thus
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the conductance between the leads (source and drain) has periodic peaks rising from
zero to a value which may be as high as 2e2/h.
This effect can be seen in metal and semiconductor dots. In the latter, the quantum-
mechanical energy levels of the electrons are also important, as their wavelength is
comparable to the size of the dot, thus the term "quantum dot " is used. The number
of electrons in a semiconductor dot can also sometimes be reduced all the way to
zero, close to which the spacing of CB peaks becomes less regular as the energy
spacing of the electronic states, and the mutual Coulomb interactions of the electrons
in the dot, cannot be ignored.
Quantized charge transport
The S.I. definition of the ampere is still "That current which, when flowing
through two parallel infinite wires held one meter apart will induce between the wires
a force of 2 10-7N/m of length". A much better definition would be one based on the
flow of a known number of electrons through a system.
Several results have been published on devices which "pump" one electron at
a time through the circuit, giving a current equal to ef, where e is the charge on an
electron and f is the pumping frequency. All of these devices have relied on the same
principle, that of Coulomb Blockade by an isolated region of electrons (known as a
quantum dot). These are generally defined electrostatically using surface gates, with
electrons able to move in and out of the quantum dot across tunnel barriers.
Lowering the entrance barrier allows a single electron to tunnel onto the dot. The
entrance barrier is then raised, and the exit lowered so an electron may move off the
dot on the other side of the device. The reliance on quantum-mechanical tunneling
limits the frequency of operation of these "electron turnstiles" to about 10MHz. This
corresponds to a current of a few picoamps, which is too small to be measured to the
required level of accuracy (parts per billion) at the moment. However researchers
have demonstrated that the devices are stable to this level, and attempts are being
made to improve the accuracy of the measurements.
Harnessing the Energy of a Single Electron
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Figure 7-38: A schematic of the device structure of the artificial atom consisting of an inverted
heterostructure of a degenerately doped substrate upon which is grown a layer of aluminum gallium
arsenide and a layer of undoped gallium arsenide. (from The Foundations of Next-Generation
Electronics: Condensed-Matter Physics at RLE, RLE Currents Vol. 10, No. 2: Fall 1998; MIT, USA)
As the physical laws related to today’s computer memory and processor
fabrication reach their limits, new approaches such as single-electron technology are
being explored. Single-electron devices are the potential successors to the
conventional technology employed to make Metal-Oxide Semiconductor Field-Effect
Transistors (MOSFETs). This is because they make use of the electron—the smallest
unit of electrical charge—to represent bits of information. While electron tunneling in
MOSFETs limits their smallest integration scale, this same behavior in single-electron
devices may prove to be the ultimate solution.
When electrons are confined to a small particle of metal or a small region of a
semiconductor, both the energy and charge of the system are quantized. In this way,
such nanometer-sized systems behave like artificial atoms. While artificial atoms can
be constructed using metals and semiconductors in various geometries, the physics
of these devices remains the same.
Considered one of the simplest types of artificial atoms, the metallic Single-
Electron Transistor (SET) is a nanoscale device that was developed in 1987. A
metallic SET consists of a metal particle isolated from its leads by two tunnel
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junctions (which are similar to diodes) and capacitively coupled to a common gate
electrode. The tunnel junctions create what is known as a Coulomb island, which the
electrons can enter only by tunneling through one of the insulators. Coulomb
repulsion prohibits more than one extra electron at a time on the island (near the
gate). Thus, electronic circuits can be made to pump or count electrons one at a time.
Because an SET’s electrical resistance is highly sensitive to the electrical fields from
nearby charges, it can easily detect not only single electrons, but also charges as
small as 1 percent of an electron’s electrical field. The current as a function of bias
across the tunnel barriers can also be measured in order to observe the so-called
Coulomb staircase, a stepwise increase of current as electrons are added to the
metal particle.
Because SETs exhibit extremely low power consumption, reduced device
dimensions, excellent current control, and low noise, they promise to reveal new
physics and innovative electronic devices. These features hold the potential for using
SETs in specialized metric-scale applications, such as in current standards and
precision electrometers. Also, possible design applications that exploit the SET’s
reduced dimensions and use a minimum number of devices may result in the
creation of high-density neural networks. However, their use in high-density computer
memory and data-processing systems must overcome problems associated with
quantum charge fluctuations and the SET’s sensitivity to microwave radiation.
7.4.4 Optical Properties
7.4.4.1 Line widths: Size distributions and intrins ic broadening
As a consequence of their being a new type of material, the properties of
semiconductor nanocrystals can be expected to evolve with improvements in sample
preparation. On the face of it, no property appears to have changed more in the past
10 years than the simple optical absorption spectrum. Spectra which at first seemed
featureless and diffuse have gradually acquired definition, with multiple discrete
states apparent in the latest generation of samples. Essentially all of this progress
derives from narrowing the distribution of sizes in the sample. Since the energies of
the transitions depend so strongly on the size, size variation is a special form of
inhomogeneous broadening at work here, which over time has been largely reduced.
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Today it is really the intrinsic, or single particle, line widths that are a matter of greater
concern.
Ignoring for a moment the effects of the size distribution, it is of great importance to
understand just how much oscillator strength can be compressed into a narrow
region of the spectrum by the method of quantum confinement. In typical
semiconductor nanocrystals, the energy level spacing is on the order of 0.15-0.3 eV.
If the integrated oscillator strength over 0.15 eV of the bulk spectrum could be
compressed into lines with widths on the order of 0.1-0.5 meV, nanocrystals would
fulfill an important and new limit in nonlinear optical materials. Recall that the
polarizability scales with volume, so that the sharp, intense (radiative rates on the
order of picoseconds) transitions of the nanocrystals could be readily manipulated by
off-resonant electric fields. A prototype optical switch with gain, for instance, would be
one in which the transmission of a highpower laser beam near the absorption
threshold of the nanocrystals is modulated by a weaker, off-resonant pulse, via the ac
Stark effect.
Many techniques have been employed to measure the average homogeneous
spectra of nanocrystals, despite the presence of inhomogeneous broadening. These
include transient hole burning, luminescence line narrowing, and photoluminescence
excitation. A direct measure of the average homogeneous line width comes from
three-pulse photon echo experiments. In these experiments, two pulses interfere in
the sample to create a spatial grating. The third pulse scatters off this grating. When
the delay between the first and third pulse is changed, the signal oscillates at the
vibrational period of any vibrations which happen to be strongly coupled to the
electronic excitation. For example, in high-quality CdSe nanocrystals, a single mode
at 210 cm-1 is observed in both the photon echo and resonance Raman experiments.
When the first and second pulse are coincident in time, the maximum amplitude
grating is prepared for the third pulse to scatter from. If the second pulse is delayed
by more than the electronic dephasing time with respect to the first, then the
amplitude of the grating will be greatly diminished. Thus, the intensity of the scattered
light from the third pulse as a function of the delay between the first two pulses
provides a measure of the homogeneous line width. This is shown in Figure 7-39.
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Figure 7-39: (A) Three-pulse photon echo experiments
show the strength of the coupling of the electronic
excitation to the relatively ionic nanocrystal lattice. In this
experiment the delay between the first two pulses is fixed
at 33 fs, and the diffraction intensity of the third pulse is
measured as a function of the delay between the first and
third. Strong coupling to a single mode at 210 cm-1 shows
up as a quantum beat. [A.P. Alivisatos, Perspectives on
the physical chemistry of semiconductor nanocrystals,
1996, Journal of Physical Chemistry]
(B) Measurement of the electronic
dephasing time in CdSe nanocrystals by
the three-pulse photon echo technique.
In this case, the time between the first
and third pulse is fixed, and the delay
between the first two pulses is varied.
The dephasing time is extremely fast, on
the order of 100 fs for 22.6 Å diameter
nanocrystals.
The homogeneous line widths are extremely broad, corresponding to dephasing
times on the order of 100 fs, with faster decays observed in smaller crystallites.
It is interesting to note that the coupling to the ionic lattice remains very strong even
in small CdS and CdSe nanocrystals. Simple models predict that as the
semiconductor is reduced in size, coupling to the polar vibrations should be reduced,
because the optically generated electron and hole are spatially coincident. This
clearly does not occur and suggests that other effects, such as the presence of polar
crystallographic faces, may act to separate the optically generated electron-hole pair,
even at the instant of optical excitation.
In these experiments it proved possible to separate out all the contributions to the
average homogeneous line width (Figure 7-40).
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Figure 7-40: Different contributions to the total line width of CdSe nanocrystals, as determined in the
three-pulse photon echo experiments. [A.P. Alivisatos, Perspectives on the physical chemistry of
semiconductor nanocrystals, 1996, Journal of Physical Chemistry]
Three mechanisms were shown to be important.
1. The least important contribution came from lifetime broadening or population
decay, which shows up as a fast recovery of the bleach induced by a single
pulse and which is insensitive to the temperature. This corresponds to decay
from the initially prepared electronic state into some other, for instance, a
lower lying state that is optically forbidden or a surface trap.
2. A somewhat more important mechanism of line broadening is dephasing by
low-frequency acoustic modes (density fluctuations) of the crystallites. The
contribution of this mechanism was identified via the strong temperature
dependence of the dephasing time. This dephasing mechanism is significant,
in that it appears to be intrinsic and not due to sample quality. When the
electronic excitations are localized in a small volume, the coupling to the
acoustic modes increases inevitably, placing a limit on the order of at least of a
few microelectronvolts on the line widths of nanocrystals.
3. Finally, there is a temperature-independent contribution to the dephasing time,
which may be due to scattering off of surfaces or defects and which in
principle at least may be eliminated by improvements in sample preparation.
The homogeneous line widths of nanocrystals remain a major topic of research. Vast
improvements in the structural quality of samples, and in the size distributions, have
resulted in no particular narrowing of the apparent homogeneous spectra.
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Furthermore, there appear to be some intrinsic limits to the narrowness of the lines in
nanocrystals which arise from strong coupling to low-frequency vibrations. Currently,
the spacings between nanocrystal electronic transitions appear to only slightly
exceed the line widths, so that despite the large shifts to higher energy. Further work,
particularly in the area of inorganic passivation, is warranted. Given the multiple
sources of inhomogeneous broadening (size, shape, local fields, defects, etc.), the
advent of single-molecule spectroscopy and near-field scanning probe microscopy
shows great promise as tools to aid in deciphering the nature of the intrinsic
photophysics of nanocrystals.
7.4.4.2 Luminescence and electroluminescence
Narrow band (15-20 nm), size-tunable luminescence, with efficiencies at least of
order 10%, is observed at room temperature from semiconductor nanocrystals. The
origin of this luminescence remains the topic of some controversy. For some time
researchers thought that this luminescence arose from partially surface trapped
carriers. Other experiments strongly suggest that the luminescence in fact arises
from low-lying “dark” states of the nanocrystal interior, and surface modifications only
influence the quantum yield by modulating the non radiative rates. Just as in organic
molecules a singlet state may be optically prepared, followed by rapid relaxation to
triplet states with long decay times, so in semiconductor nanocrystals, where j-j
coupling dominates, an angular momentum allowed state is initially prepared, and on
the time scale of picoseconds or longer, there is a decay to a lower lying angular-
momentum-forbidden state, which decays relatively slowly (nanoseconds to
microseconds). One success of this dark state model is the accurate prediction that
the magnitude of the exchange splitting increases as r-3, thus explaining one long
puzzling feature of the luminescence. As the size is reduced, the shift between the
absorbing and emitting state is observed to increase. The rigorous separation of
interior and surface states is somewhat artificial in nanocrystals in any case, since
substantial mixing may be expected. Indeed, other features of the spectra suggest
that there may well be substantial surface character associated with the emitting
state. For example, electric field modulation of the luminescence yields signals a
factor of 100 or larger than modulation of absorption, indicating that the emitting state
is not as well confined spatially. Further evidence for surface localization of the
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emitting state comes from low-temperature studies of the vibronic coupling of the
emission, which show that there is a well-defined localization temperature.
Independent of the exact origin of the luminescence, it does appear to be one
property which can be manipulated in useful ways. For example, two reports of light-
emitting diodes made with polymers and CdSe nanocrystals have appeared within
the past year. In the first instance, nanocrystals were assembled in layers a few
nanocrystals thick on the surface of PPV, an electroluminescent polymer. The PPV
itself was grown on a layer of indium tin oxide, a transparent hole-injecting contact.
Finally, the nanocrystal layer was coated with a film of Mg/Ag, the electron-injecting
contact. This complete assembly electroluminesces when a voltage is applied. The
recombination of electrons and holes may take place either in the polymer layer
(which emits green light) or in the nanocrystal layer. The nanocrystal emission shifts
with size. Thus, these LEDs provide a variety of means for tuning the output color.
This advance is particularly important, since it constitutes the first example of
electrical, rather than purely optical, investigation of semiconductor nanocrystals.
7.5 References
A.P. Alivisatos, Perspectives on the physical chemistry of semiconductor
nanocrystals, 1996, Journal of Physical Chemistry
S. Ciraci, A. Buldum, I.P. Batra, Quantum effects in electrical and thermal transport
through nanowires, 2001, Journal of Physics Condensed Matter
S. Eustis, M. A. El-Sayed, Why gold nanoparticles are more precious than pretty
gold: Noble metal surface plasmon resonance and its enhancement of the radiative
and nonradiative properties of nanocrystals of different shapes, 2005, The royal
Society of Chemistry
K. Lance Kelly, E. Coronado, L.L. Zhao, G.C. Schatz, The Optical Properties of Metal
Nanoparticles: The Influence of Size, Shape, and Dielectric Environment, 2003,
Journal of Physical Chemistry
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8 MAGNETISM
8.1 Introduction
The phenomenon of magnetism, that is, the mutual attraction of two pieces of
iron or iron ore, was surely known to the antique world. The ancient Greeks have
been reported to experiment with this mysterious force. The name magnetism is said
to be derived from a region in Turkey which was known by the name of Magnesia
and which had plenty of iron ore.
Over the past several decades, amorphous and more recently nanocrystalline
materials have been investigated for applications in magnetic devices requiring
magnetically soft materials such as transformers, inductive devices, etc. Most
recently, research interest in nanocrystalline soft magnetic alloys has dramatically
increased. This is due, in part, to the properties common to both amorphous and
crystalline materials and the ability of these alloys to compete with their amorphous
and crystalline counterparts. The benefits found in the nanocrystalline alloys stem
from their chemical and structural variations on a nanoscale which are important for
developing optimal magnetic properties.
In the following sections, we will first revisit the magnetic properties of matter
and then pass on to discuss about the magnetic nanomaterials.
8.1.1 Concept
The magnetic field parameters at a given point in space are defined to be the
magnetic field strength H, the magnetic flux density or magnetic induction B and the
magnetization of the material M. Inside a magnetic material:
)(0 MHB += µ (8. 207)
µ0 = 4π x 10-7 H/m is the permeability of free space.
We can also define the magnetic field parameters by this way:
B = µ0 H(1 + χ) =µH (8. 208)
Where µµµµ = µ0(1 + χχχχ) is the permeability of the material and χχχχ = M/H is the
susceptibility of the material.
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Finally, we need to define the magnetic moment, µµµµm, through the following equation:
M = µm / V (8. 209)
It needs to be noted that in magnetic theory several unit systems are
commonly in use. Historically, workers in magnetic materials have used the cgs
(centimetre, gram, second) or Gaussian system. More recently attempts have been
made to change over the SI system. As you can see on the next figure, two SI
systems are used; we will focus on the Sommerfield convention.
Table 8-1 : Magnetic units. A is ampere, cm is centimeter, m is meter, emu is electromagnetic unit, B
is magnetic induction, H is magnetic field strength, M is magnetization of a substance per unit volume.
[K. J Klabunde, Nanoscale Materials in Chemistry, Wiley-Interscience, 2001]
8.1.2 Phenomena
8.1.2.1 Atomic origins
Magnetic field is relative motion of an electric charge and the observer. Thus
magnetism is a result of moving charges. From an atomic view of matter, there are
two electronic motions:
- the orbital motion of the electron (revolution of electron around the core)
- and the spin motion of the electron (revolve around its own axis).
These two electron motions are the source of macroscopic magnetic phenomena in
materials.
8.1.2.2 Classification of magnetic phenomena
We use the word magnetism very loosely when implying the mutual magnetic
attraction of pieces of iron. There are, however, several other classes of magnetic
materials that differ in kind and degree in their mutual interaction. We shall
distinguish in the following between:
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- Ferromagnetism: Imagine the external field strength is increased (in a
ferromagnetic solid), then the magnetization rises at first slowly and then more
rapidly. Finally, M levels off and reaches a constant value, called the saturation
magnetization Ms. When H is reduced to zero, the magnetization retains a positive
value, called the remanent magnetization or remanence M. It is this retained
magnetization which is utilized in permanent magnets. The remanent magnetization
can be removed by reversing the magnetic field strength to a value Hc, called the
coercive field.
- Paramagnetism: Without an external magnetic field, orbiting electrons
magnetic moments are randomly oriented and thus they mutually cancel one another.
As a result, the net magnetization is zero. However, when an external field is applied
the individual magnetic vectors tend to turn into the field direction, which is
counteracted by thermal agitation alignment (see the Curie law).
- Diamagnetism: may be explained by postulating that the external
magnetic field induces a change in the magnitude of inner-atomic currents, that is,
the external field accelerates or decelerates the orbiting electrons, in order that their
magnetic moment is in the opposite direction from the external magnetic field.
- Antiferromagnetism: Antiferromagnetic materials exhibit, just as
ferromagnetics a spontaneous alignment of moments below a critical temperature.
However, the responsible neighbouring atoms in antiferromagnetics are aligned in an
antiparallel fashion. Antiferromagnetic materials are paramagnetic above the Neel
temperature TN
- Ferrimagnetism: Ferrimagnets are similar to antiferromagnets in that two
sublattices exist that couple through a superexchange mechanism to create an
antiparallel alignment. However the magnetic moments on the ions of the sublattices
are not equal and hence they do not cancel; rather, a finite difference remains to
leave a net magnetization. This spontaneous magnetization is defeated by the
thermal energy above the Curie temperature, and then the system is paramagnetic
having a nonlinear 1/x versus T relationship.
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8.2 Magnetic Properties of small atomic clusters
8.2.1 Introduction
Metal clusters are composed of a few atoms to hundreds of atoms which
demonstrate properties different from the bulk due to the large surface to volume
ratio compared to its bulk state.lxxx e.g. for a cluster of thousand atoms, one quarter of
the atoms lie on the surface. Even in the very compact icosahedral configuration, a
nickel cluster of 55 atoms has 32 atoms on the surface. This provides a unique
opportunity to study the evolution of the physical properties of bulk solid with the
variation of the number of atoms in the system under investigation. Hence, the
understanding of the properties of the clusters in isolation has become a subject of
considerable interest from the stand point of fundamental research.
Another important aspect of the cluster research is that since the physical
properties of the clusters are size specific, there is possibility that materials with
desired properties can be custom made by changing the size and composition of the
cluster aggregates. e.g. thin films with controlled nanostructure characteristics can be
obtained by low energy cluster beam deposition technique.lxxxi It is however essential
to understand the intrinsic cluster properties in order to control the final material
properties.
Molecular beam technique is ideal for measuring the properties of clusters in
isolation as the clusters are free from any external influences. Metal clusters can be
produced by making a supersonic beam of metal vapour from an oven, typically
known as cluster-source. In the molecular beam experiment a cluster source and one
cluster detector is required. With the development of the pulsed laser vaporization of
metal substrates inside a high pressure pulsed nozzle, it is possible to synthesize
clusters of almost any metal. Cluster detectors are typical ion detectors specially
designed for the particular mode of experiment.
One of the areas of cluster research currently being pursued is the
investigation of the magnetic properties of small atomic clusters. Since large numbers
of atoms of the clusters are on the surface, one expects the clusters of these
elements to exhibit larger magnetic moments. Being small particles, it is possible to
measure the magnetic moment directly by measuring the interaction with a magnetic
field in terms of their deflection from the original trajectories.
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8.2.2 Size dependence
The magnetic moment of transition metals decreases in general with the increase of
number of atoms (size) in the clusters. However, the moment of different transition
metal clusters are found to depend differently on the number of atom per clusters (N).
0
0.5
1
1.5
2
2.5
3
0 100 200 300 400 500 600 700 800
Cluster Size (N) [Number of atoms]
Bulk Fe
Fe-clusters
Co-clusters
Bulk Ni Ni-clusters
Rh-clusters
Bulk Co
Mag
netic
mom
ent p
er a
tom
Figure 8-1: Magnetic moment per atom in units of Bohr magnetron for iron, cobalt, nickel and rodium
clusters as a function of cluster size (number of atom). The enclosed data points show the sizes of the
respective clusters at which they attain bulk moments. Only the maximum error in the data for each
clusters are shown. Iron, cobalt and nickel data are from W. A. de Heer et al.lxxxii and Billas.Erreur ! Signet
non défini. The rodhium data is from Cox et. al.Erreur ! Signet non défini.
Among the ferromagnetic 3d transition metals, Nickel clusters attain the bulk
moment at N=150, whereas Cobalt and Iron clusters reach at the corresponding bulk
limit at N=450 and N=550 respectively (Figure 8-1 ). In icosahedral configuration these
numbers correspond to 3, 4 and 5 shells respectively.lxxxiii However one needs to
confirm the bulk like behaviour only after measuring the temperature dependence of
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magnetic moment for clusters. The spin imbalance at 3d-orbital has been correlated
to the measured moment as:
)(2
)()( Ng
nn xx
µ=−−+ (8. 210)
Where gx is the bulk g-factor for the respective metal. The results show that in the
lower size limit the clusters have high spin majority configuration and the behaviour is
more like an atom, but with the increase of size an overlap of the 3d-band with the
Fermi level occurs, which reduces the moment towards that of the bulk with slow
oscillations.Erreur ! Signet non défini. In analogy with the layer-by-layer magnetic moments
in the studies of surface magnetism, the clusters are assumed to be composed of
spherical layers of atoms forming shells and the value of moment of an atom in a
particular shell is considered to be independent of the size of cluster. The magnetic
moment values of the different shells are optimized to fit with the total moment per
atom maintaining the overall trend of observed decrease with size. It is generally
observed that the per atom moment for the surface atoms in the 1st layer are close to
that of an atom and deep layers correspond the bulk. The intermediate layers show
an overall decreasing trend. For both nickel and cobalt, there are oscillations in the
values at least up to the 5th shell. For nickel, the second layer is found to have a
negative coupling with the surface atoms and the 3rd-4th layers in iron are
magnetically dead. The model gives more or less close agreement with the
experimental results for cobalt and nickel. But in a more realistic model one has to
account for both the geometric shell closing as well as the electronic structure.
However this model provides a good first order picture to understand the magnetic
behaviour of 3d transition metal clusters.
Though the 4d transition metals are non-magnetic, the small clusters of
rhodium are found to be magnetic following the calculations. lxxxiv Within the size
range N=10-20, the moment varies between 0.8 to 0.1µB per atom but the clusters
become non-magnetic within N=100. Due to reduced size, the individual moments on
the atoms in clusters are aligned. First principal calculations show that the ground
state of Rh6 is non-ferromagnetic, while Rh9, Rh13 and Rh19 clusters have non-zero
magnetic moments. Rh43 is found to be non-magnetic as the bulk. Ruthenium,
palladium, Chromium, Vanadium and Aluminium clusters are found to be non-
magnetic.Erreur ! Signet non défini.
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8.2.3 Thermal behaviour
Since ferromagnetic state requires the moments to remain mutually aligned
even at relatively high temperatures, it is important to study the thermal behaviour of
the magnetic clusters. Temperature dependent studies show that up to 300K the
moment remains constant for nickel clusters and then it decreases at higher
temperatures resembling closely the bulk behaviour. This indicates that the
interactions affecting the mutual alignment of moments with temperature are in the
same order of magnitude as in the bulk. Ni550-600 is almost bulk-like. Cobalt clusters
also follow the same trend but the moment increases slightly, between 300K to 500K,
which might be due to a structural phase transition corresponding to the hcp-fcc
phase transition in bulk at 670K, where the moment increases by 1.5%. Clusters of
different size ranges of iron behave differently; Fe50-60 shows a gradual reduction of
moments from 3µB at 120K to 1.53µB at 800K. Up to 400K, Fe120-140 remains with a
constant moment of 3µB and then decreases to finally level off at 700K to 0.7µB. The
size range N=82-92 behaves like Fe120-140 below 400K and like Fe50-60 above 400K.
The moment for the higher size ranges decreases steadily with the increase of
temperature but Fe250-290 levels off at 700K whereas Fe500-600 levels off between 500-
600K to 0.4µB. Comparing with the thermal behaviour of bulk iron, the level-off
temperature can be termed as the critical temperature (TC) for clusters of a particular
size and it decreases with the increase of size of the clusters. But, since the bulk
Curie temperature is 1043K, one realizes clearly that this trend must reverse at
higher sizes to reach the bulk. It is also interesting to note that for Fe120-140 the
moment decreases more rapidly within 600-700K. The specific heat measurement for
both Fe120-140 and Fe250-290 also show a peak at 650K. This might be because iron
clusters undergo a structural phase transition in this temperature range similar to the
bulk iron which undergoes bcc-fcc-bcc phase transition, but only beyond the Curie
temperature. Since the structure of clusters cannot be detected in a molecular beam,
this remains an unsettled question. However, there is a competition between the
different structures for iron clusters,Erreur ! Signet non défini. and it is more probable that a
structural transition occurs at low temperature for increasing cluster size but revealing
it needs a better characterization.
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8.2.4 Rare earth clusters
The magnetic interaction in rare earth solids are RKKY type which is mediated
by the conduction electrons. Since in clusters, structure and filling up of the
conduction band differs from that in the bulk, rare-earth clusters are expected to
exhibit an altogether different magnetism compared to the bulk. Magnetic properties
of gadolinium, terbium and dysprosium clusters show a very size-specific
behaviour.Erreur ! Signet non défini. Gd22 behaves super-paramagnetically (8.3.4) but Gd23
shows a behaviour like a rigid magnetic rotor as if the moment is locked to the cluster
lattice. Gd17 shows super-paramagnetism near room temperature but at low
temperatures it shows locked-moment behaviour. It indicates that there are at least
two groups of clusters sharing the same number of atoms which might correspond to
two different isomers, either magnetic or structural. But it is not clear whether the two
isomers co-exist in the beam or there occurs a phase transition of any kind. The
behaviour of other clusters from N=11-30, is like Gd22 or Gd23 or Gd17. Similar is the
case for Tb and Dy clusters. Tb17, Tb22, Tb27, Dy22 and Dy27 show super-
paramagnetic behaviour but Tb23 and Tb26 show locked-moment behaviour. In fact
Tb26 is temperature dependent like Gd17. Assuming the clusters to be spherical with a
moment of inertia and a body fixed moment which rotates with the cluster,
calculations show a very complex precession and nutation when the rotational kinetic
energy becomes comparable to the magnetic potential energy. With two parameters
viz. the rotational temperature and the magnetic moment, a fit of the experimental
data for Gd with several initial conditions and a rotational temperature of 5K, shows
an agreement at least for sizes lower than N=30.
8.3 Small particle magnetism
8.3.1 Classifications of magnetic nanomaterial
The magnetic behaviour of most experimental realizable systems of is result of
contributions from both interaction and size effects. The correlation between
nanostructure and magnetic properties suggests a classification of nanostructure
morphologies. The following classification is designed to emphasize the physical
mechanisms responsible for the magnetic behaviour. Figure 8-2 schematically
represents the classification. At one extreme are systems of isolated particles with
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nanoscale diameters, which are denoted by type A. These non interacting systems
derive their unique magnetic properties strictly from the reduced size from the
components, with no contribution from interparticle interactions. At the other extreme
are bulk materials with nanoscale structure denoted by type D, in which a significant
fraction (up to 50%) of the sample volume is composed of grain boundaries and
interfaces. In contrast to type A systems, magnetic properties here are dominated by
interactions. The length of the interactions can span many grains and is critically
dependent on the character of interface. Due to this dominance of interaction and
grain boundaries, the magnetic behaviour of type D nanostructures cannot be
predicted simply by applying theories of polycrystalline materials with reduced length
scale. In type B particles, the presence of a shell can help prevent particle
interactions, but often at the cost of interaction between the core and the shell. In
many cases the shells are formed via oxidation and may themselves be magnetic. In
type C materials, the magnetic interactions are determined by the volume fraction of
the magnetic particles and the character of the matrix they are embedded in.
A
B
C
D
Figure 8-2 : Schematic representation of the different types of magnetic nanostructures. Type-A
materials include the ideal ultrafine particle system, with interparticle spacing large enough to
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approximate the particles as noninteracting. Ferrofluids , in which magnetic particles are surrounded
by a surfactant preventing interactions, are a subgroup of Type-A materials. Type-B materials are
ultrafine particles with a core-shell morphology. Type-C nanocomposites are composed of small
magnetic particles embedded in a chemically dissimilar matrix. The matrix may or may not be
magnetoactive. Type-D materials consist of small crystallites dispersed in a noncrystalline matrix. The
nanostructure may be two-phase, in which nanocrystallites are distinct phases from the matrix, or the
ideal case in which both the crystallites and the matrix are made of the same material.
8.3.1.1 Ferrofluid
A ferrofluid is a synthetic liquid that holds small magnetic particles in a colloidal
suspension, with particles held aloft their thermal energy. The particles are
sufficiently small that the ferrofluid retains its liquid characteristics even in the
pressure of a magnetic field, and substantial magnetic forces can be induced which
results in fluid motion.
A ferrofluid has three primary components. The carrier is the liquid element in
which the magnetic particles are suspended. Most ferrofluids are either water based
or oil based. The suspended materials are small ferromagnetic particles such as iron
oxide, on the order of 10 – 20 nm in diameter. The small size is necessary to
maintain stability of the colloidal suspension, as particles significantly larger than this
well precipitate. A surfactant coats the ferrofluid particles to help maintain the
consistency of the colloidal suspension.
The magnetic properties of the ferrofluid are strongly dependent on particle
concentration and on the properties of the applied magnetic field. With an applied
field, the particles align in the direction of the field, magnetizing the fluid. The
tendency for the particles to agglomerate due to magnetic interaction between
particles is opposed by the thermal energy of the particles. Although, particles vary in
shape and size distribution, insight into fluid dynamics can be gained by considering
a simple spherical model of the suspended particles.
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S
2R σ ,σ ,σ ,σ ,µ µ µ µ
Figure 8-3: Representative model of a typical ferrofluid.
The particles are free to move in the carrier fluid under the influence of an applied
magnetic field, but on average the particles maintain a spacing S to nearest
neighbours. In a low density fluid, the spacing S is much larger than the mean
particle radius 2R, and magnetic dipole – dipole interactions are minimal.
Applications for ferrofluids exploit the ability to position and shape the fluid
magnetically. Some applications are:
• rotary shaft seals
• magnetic liquid seals, to form a seal between region of different pressures
• cooling and resonance damping for loudspeaker coils
• printing with magnetic inks
• inertial damping, by adjusting the mixture of the ferrofluid, the fluid viscosity
may be change to critically damp resonances accelerometers,
• level and attribute sensors
• electromagnetically triggered drug delivery
Table 8-2: Material Parameters of a typical ferrofluid:lxxxv
Sample Volume 1.7 x 10-6 m3
Electrical conductivity of particles 6 13 10 ( )f mσ −= × Ω
Electrical conductivity of ferrofluid 7 110 ( )fluid mσ − −< Ω
Volume percentage of particles ≈3% by volume
Initial magnetic permeability of particles 0100µ µ≈
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Particle mean radius R ≈10-8 m
Density of magnetic particles FEρ = 7.8 g/cm3
Fluid density fρ = 1 g/cm3
Fluid viscosity of carrier fluid η =1 cp = 0.01 g/cm. sec
8.3.2 Anisotropy
In many situations the susceptibility of a material will depend on the direction in which
it is measured. Such a situation is called magnetic anisotropy. When magnetic
anisotropy exists, the total magnetization of a ferromagnet Ms will prefer to lie along a
special direction called the easy axis. The energy associated with this alignment is
called the anisotropy energy and in its lowest form is given by
Ea = K sin²θ (8. 211)
Where θ is the angle between Ms and the easy axis. K is the anisotropy constant.
Most materials contain some type of anisotropy affecting the behaviour of the
magnetization. The most common forms of anisotropy are:
• Crystal anisotropy
• Shape anisotropy
• Stress anisotropy
• Externally induced anisotropy
• Exchange anisotropy.
The two most common anisotropies in nanostructure materials are crystalline
and shape anisotropy.
8.3.2.1 Magnetocrystalline anisotropy
Only magnetocrystalline anisotropy, or simply crystal anisotropy, is intrinsic to the
material; all other anisotropies are induced. In crystal anisotropy, the ease of
obtaining saturation magnetization is different for different crystallographic directions.
An example is a single crystal of iron for which Ms is most easily obtained in the [100]
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direction, then less easy for the [110] direction, and most difficult for the [111]
directions. These directions and magnetization curves for iron are given in Figure 8-4.
The [100] direction is called the easy direction, or easy axis, and because the other
two directions have an overall smaller susceptibility, the easy axis is the direction of
spontaneous magnetization when below Tc (Curie Temperature). Both iron and
nickel are cubic and have three different axes, whereas cobalt is hexagonal with a
single easy axis perpendicular to the hexagonal symmetry. Figure 8-4 also gives
magnetization curves for cobalt and nickel.
Figure 8-4 : Magnetization curves for single crystals of iron, cobalt and nickel along different
directions. [K. J Klabunde, Nanoscale Materials in Chemistry, Wiley-Interscience, 2001]
You may now imagine a situation in which the system has spontaneous
magnetization along the easy axis but a field is applied in another direction.
Redirection of the magnetization to be aligned with the applied field requires energy
(through the change in HM
⋅ ), hence the crystal anisotropy must imply a crystal
anisotropy energy given by equation Ea = K sin² θ (8. 211) for a uniaxial material.
This energy is an intrinsic property of the material and is parametrized, to lowest
order, by the anisotropy constant K=K1 which has units of erg per cm3 of material.
Roughly speaking K1 is the energy necessary to redirect the magnetization. For a
uniaxial material with only K1, the field necessary to rotate the magnetization 90°
away from the easy axis is
H = 2K1/Ms (8. 212)
The physical origin of the magnetocrystalline anisotropy is the coupling of the
electron spins, which carry the magnetic moment, to the electronic orbit, which in turn
is coupled to the lattice. Recall it was the strong coupling of the orbit to the lattice via
the crystal field that quenched the orbital angular momentum.
A polycrystalline sample with no preferred grain orientation has no net crystal
anisotropy due to averaging over all orientations.
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8.3.2.2 Shape anisotropy
It is easier to induce a magnetization along a long direction of a nonspherical piece of
material than along a short direction. This is so because the demagnetizing field is
less in the long direction, because the induced poles at the surface are farther apart.
Thus, a smaller applied field will negate the internal, demagnetizing field. For a
prolate spheroid with major axis c greater than the other two and equal axes of
length a, the shape anisotropy constant is
( ) 2
2
1MNNK caS −= (8. 213)
Where Na and Nc are demagnetization factors.
For spheres, Na = Nc because a = c.
It can be shown that 2Na + Nc = 4π; then the limit c>>a, that is, a long rod,
Ks = 2πM².
Thus a long rod of iron with Ms = 1714 A m-1 would have a shape anisotropy constant
of Ks = 1.85 x 107erg cm-3. This is significantly greater than the crystal anisotropy so
we see that shape anisotropy can be very important for nonspherical particles.
8.3.2.3 Other anisotropy
Stress anisotropy result from internal or external stresses that occur due to
rapid cooling, application of external pressure etc. Anisotropy may also be induced by
annealing in a magnetic field, plastic deformation, or by ion beam irradiation.
Exchange anisotropy occurs when a ferromagnet is in close proximity to an
antiferromagnet or ferromagnet. Magnetic coupling at the interface of the two
materials can create a preferential direction in the ferromagnetic phase, which takes
the form of a unidirectional anisotropy. This type of anisotropy is most commonly
observed in type – B particles, when an antiferromagnetic or ferromagnetic oxide
forms around a ferromagnetic core.
8.3.3 Single Domain Particles
The magnetism of small ferromagnetic particles is dominated by two key features.
The first one is the size limit below which the specimen can no longer gain a
favourable energy configuration by breaking up into domains, hence it remains with
one domain. The second one is the thermal energy that can decouple the
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magnetization from the particle itself to give rise to the phenomenon of
superparamagnetism.
8.3.3.1 The role of the particle volume
The most obvious finite-size effect in magnetic fine-particle systems consists in the
fact that each entity (magnetic particle) has a very small volume compared to the
typical sizes of the magnetic domains in the corresponding bulk materials. In a sense,
the volume of a particle is so small (commonly, a few tens or hundreds of nm3) that it
can be considered as a zero-dimension magnetic system, which strongly affects its
magnetic behaviour. In real systems, the size of the particles is usually not uniform
and distributed following a function f(V) which is satisfactorily described by a
logarithmic-linear distribution of the form
−=²2
)/²(lnexp
2
1)( 0
σσπVV
VVf (8. 214)
Where V0 is the most probable particle volume and σ is the standard deviation of
ln(V). The validity of this simple assumption has been experimentally verified in many
systems by transmission electron microscopy (TEM) and other indirect techniques
such as x-ray diffraction and magnetic granulometry.
One fundamental question related to the finite and small volume of the particles is
how such volume determines the internal domain structure. In magnetic bulk
materials, it is well known that there exists a multidomain structure constituted by
regions of uniform magnetization separated by Domain Walls (DWs). The size and
shape of these domains depends on the interplay between the exchange,
magnetostatic and anisotropy energies of the system. As the volume of the magnetic
system decreases, the size of the domains and the width of the walls are reduced,
modifying, at the same time, their inner structure. Below a certain critical volume, the
energy cost to produce a DW is greater than the corresponding reduction in the
magnetostatic energy. Consequently, the system no longer divides in smaller
domains, instead maintaining the magnetic structure of a single domain. This critical
value depends on the saturation magnetization of the particle, anisotropy energy and
exchange interactions between individual spins. For example, for spherical particles
the critical diameter is within 10–800 nm. Typical values for Fe and Co metallic
particles are 15 and 35 nm, respectively, while for SmCo5 it is as large as 750 nm.
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In many fine-particle systems, f(V) is narrowand centred at very small V0 below the
critical value, so that all the particles constituting the system can be considered as a
single domain.
This is the situation that we are going to assume in what follows. In thermal
equilibrium, the resulting magnetization of a particle points in a direction tending to
minimize its total anisotropy energy, which for a single-domain particle can be
considered as proportional to its volume, at least, in first approximation (Figure 8-6).
The total anisotropy of the particle can often be assumed to have uniaxial character,
being determined by a single constant K. Therefore, the anisotropy energy can be
expressed as
E(θ) = kV Sin2θ (8. 215)
where θ is the angle between anisotropy axis and magnetization, and KV is the
anisotropy energy barrier (8.3.2) separating both easy directions for magnetization,
which in zero magnetic field correspond to θ = 0 and θ = π. Taking into account the
spread of volumes f(V) given by equation E(θθθθ) = kV Sin 2θθθθ (8. 215) and
−=²2
)/²(lnexp
2
1)( 0
σσπVV
VVf (8. 214), it is a matter of fact that the magnetization
of the particles will behave as if it were driven by an energy barrier distribution
)( 0BEf , where 0
BE = kV.
Under the application of a magnetic field H forming an angle ψ with the anisotropy
axis, the energy of a particle is modified as
)cossinsincos(cos²sin)( ϕψθψθθθ ++−= SHMKVE (8. 216)
Where Ms is the spontaneous magnetization of the particle, and the axis system is
defined in Figure 8-5.
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Figure 8-5 : Definition of the axis system for a fine particle. The uniaxial-anisotropy axis is along the z-
axis. [X. Batle, A. Labarta, Finite-size effects in fine particles: magnetic and transport properties,
Journal of Physics, 2002, ppR15-R42].
Under zero field, the equilibrium direction for the magnetization vector M coincides
with the anisotropy axis, while when a magnetic field is applied, it rotates away from
the anisotropy axis towards the field direction, forming an angle ψ−θ with H, and the
stability problem can be reduced to a two-dimensional one (ϕ = 0). In this case, and
if H is lower than a certain critical value known as switching field at zero temperature, 0SWH , E(θ) has two minima at different heights which are separated by an energy
barrier. When the field is applied opposite to the magnetization, the unstable
minimum is separated from the stable one by an energy barrier EB, which can be
approximated by the following expression:
κ
−=
00 1)(
SWBB H
HEHE (8. 217)
Where κκκκ is a phenomenological exponent that depends on ψψψψ. Consequently, 0SWH is
the minimum value of the field at zero T, at which EB vanishes, and the magnetization
inverts its orientation. When H is applied along the anisotropy axis and in the
opposite direction to magnetization, equation κ
−=
00 1)(
SWBB H
HEHE (8. 217) is
exact with κκκκ = 2. In this case, M irreversably rotates at zero T when H = 0SWH = Ha =
2K/Ms, where Ha is the anisotropy field. If H is applied at an arbitrary angle ψψψψ, there is
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not a simple analytical expression. However, equation κ
−=
00 1)(
SWBB H
HEHE (8.
217) works quite well with κκκκ ≈≈≈≈ 1.5, provided that the applied field forms an angle of a
few degrees with the anisotropy axis. On the other hand, Pffeifer gave the following
phenomenological approximation for any angle ψψψψ
a
SW
H
H 014.186.0 +=κ (8. 218)
A method frequently used to characterize the nature and strength of the particle
anisotropy consists in the determination of 0SWH as a function of ψψψψ. The resulting
curve turns to be the so-called Stoner Wohlfarth (SW) astroid. In fact, the SW astroid
represents the set of fields (applied opposite to the particle magnetization) at which
irreversible jumps occur. For uniaxial anisotropy, 0SWH can be evaluated from
equation )cossinsincos(cos²sin)( ϕψθψθθθ ++−= SHMKVE (8. 216) with the
conditions dE/d θθθθ = d²E/d θθθθ² = 0, since irreversible jumps occur when the energy
barrier vanishes and the maximum in E(θθθθ) is substituted by an inflection point. The
obtained expression is
2/33/23/20
)cos(sin ψψ += a
SW
HH (8. 219)
Many groups have attempted to study the reversal process in single particles but the
experimental precision did not yield quantitative information.
Single domain
Multi domain
Super Paramagnetic
H c
D c D
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Figure 8-6 : Qualitative illustration of the behaviour of the coercivity in ultrafine particle systems are
the particles size changes.
As the particles size continues to decrease (towards some critical particle diameter,
Dc) below the single domain value, the spins get increasing affected by thermal
fluctuations and the system becomes superparamagnetic. Particles with sufficient
shape anisotropy can remain single domain to much larger dimensions than their
spherical counterparts.
8.3.3.2 Single-Domain Characteristics:
In a granular magnetic solid with a low volume fraction, one has a collection of single
domain particles each with a magnetic axis along which all the moments are aligned.
In the absence of a magnetic field, parallel and anti parallel orientations along the
magnetic axis are energetically equivalent but separated by an energy barrier of CV,
where C is the total anisotropy per volume, and V is the particle volume. Since the
size of each single domain remains fixed, under an external field, only the magnetic
axes rotate. Thus the measured magnetization M of a granular magnetic field solid
with a collection of single domain particles is the global magnetization
.cosS
M HM M
Hθ= =
(8. 220)
Where θθθθ is the angle between the magnetic axis of a particle. SM , is the saturation
magnetization, H
is the external field, and the average <cos θθθθ> is taken over many
ferromagnetic particles. The hysterisis loop of a granular solid is thus a signature of
the rotation of the magnetic area of the single-domain particles. This should be
contrasted with the hysterisis loop of a bulk ferromagnet, in which the sizes and the
direction of the domains are altered drastically under an external field.
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M
H
MS
MR
M C
(a)
TBT
H C (c)
M S
M
M S /2
T B T
e c
d
fb
a
T B
(b) (d) χ
T
Figure 8-7 : (a) Hysterisis loop at 5K; (b) temperature dependence of saturation magnetization (MS)
and remnant magnetization (MR=MS/2); (c) temperature dependence of field cooled (FC) and zero-field
cooled (ZFC) susceptibility. At the blocking temperature (TB), MR and HC vanish, whereas the ZFC
susceptibility shows a cusp like feature.
An example of a hysterisis loop of a granular magnetic solid at low temperature is
shown in Figure 8-7a. In the initial unmagnetized state with M = 0 at H = 0, the
magnetic axes of the particles are randomly oriented, each along its own magnetic
axis, which is determine by the total magnetic anisotropy of the particles. The
directions of the giant moments are random and static at low temperatures. A
saturation magnetization (M = MS) is realized under a large field when all the
magnetic axes are aligned. In the remnant state when H is reduced to H = 0, one
observes the remnant magnetization (MR), whose values at low temperature is MR =
MS/2. This is because the magnetic axes are oriented only in one hemisphere due to
the uniaxial anisotropy of the single-domain particle.
It should be noted that the initial M=0 state and the initial magnetization curve if in
Figure 8-7a does not reappear, whereas the field cycle part does. The simples way to
recover the initial M=0 state is to heat the sample above the blocking temperature
and cool the sample in zero field back to low temperatures.
Because of the single domain nature of the magnetic entities, the coercivity
(HC) of the ultra fine particles is much higher than that in bulk material.
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8.3.4 Superparamagnetism
8.3.4.1 The SPM regime
Below the Curie temperature of a ferromagnet or ferrimagnet, all the spins are
coupled together and so cooperate to yield a large moment. This moment is bound
rigidly to the particle by one or more of the variety of anisotropies that we have
discussed (8.3.2). With decreasing particle size, the energy Barrier EB decreases
until the thermal energy kT can disrupt the bonding f the total moment to the particle.
Then this moment is free to move and respond to an applied field independent of the
particle. This moment is the particle moment and is equal to
µp=MSV (8. 221)
It can be quite large, thousands of Bohr magnetons. An applied field would tend to
align this giant (super) moment, but kT would fight he alignment just as it does in a
paramagnet. Thus, this phenomenon is called superparamagnetism.
It is good to notice that if the anisotropy is very weak (zero), one would expect that
the total moment µp could point in any direction, hence the Langevin law would apply.
The phenomenon of superparamagnetism is, in fact, timescale-dependent due to the
stochastic nature of the thermal energy. The anisotropy energy EB represents an
energy barrier to the total spin reorientation; hence the probability for jumping this
barrier is given by Arrhenius law:
=kT
EBexp0ττ (8. 222)
Where 0τ is the attempt time by supposing that the particle spins are rigidly coupled
and reverse as a whole. This pre-factor depends on a large variety of parameters:
temperature, gyromagnetic ratio, saturation magnetization, energy barrier, direction
of the applied field and damping constant. τ 0 is assumed to be constant and
equation
=kT
EBexp0ττ (8. 222), together with this assumption, is known as the
Neel–Brown model. If the characteristic time window of an experiment is much
shorter than τ at a fixed temperature, the particle moment remains blocked during
the observation period (blocked regime), while, in the opposite situation, the rapid
fluctuations of the particle moment mimics the atomic paramagnetism (SPM regime).
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In the intermediate regime, the probability that the magnetization has not switched
after an observation time t is given by
−=τ
ttP exp)( (8. 223)
A method used to verify the Neel–Brown model in real systems is the scaling of the
switching field distributions measured at different T and rates of field ramping.
Thermal activation leads to a switching field distribution from the SW model, the
mean switching field being given by
−= −
κ
κυε
/1
10 ln1
cT
E
TkHH
B
BSWSW (8. 224)
Where )/( 00
BSWB EHkc ατ= , )/1( 0SWHH−=ε and υ is the field sweeping rate. From
the equation above, it is evident that Hsw measured at different T and υ can be
collapsed in a single master curve when they are plotted as a function of the scaling
variable (T ln(cT / υ εκ−1))1/κ . In fact, the collapse is obtained by choosing the
appropriate values of κ and c (obtaining κ = 1.5, as expected for measurements with
the field slightly deviated from the anisotropy axis, see paragraph 8.3.3.1). This
scaling procedure has been successfully applied to analyse data obtained above
about 0.5K from single-particle experiments carried out with different kind of
nanoparticles.
At very low temperatures, (below about 0.5 K) strong deviations from the Neel–Brown
model have been reported in this kind of scaling when it was applied to single-particle
experiments. The deviations consisted in a saturation of Hsw and σ (width of the
switching field distribution), and a dependence of Hsw on υ faster than the prediction
of the Neel–Brown model. All these anomalies have been attributed to the
prevalence of a nonthermal process by which the particle magnetization escapes
from the metastable state through the energy barrier by macroscopic quantum
tunnelling (MQT). The theoretical models propose that there is a crossover
temperature TB, below which quantum tunnelling dominates the reversal process,
whereas above TB, the escape rate from the metastable state is given by thermal
activation, and the Neel–Brown model is accomplished. Consequently, TB is the
temperature at which the thermal and quantum rates coincide.
The main parameters determining the particular value of TB for a given system are:
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• the number of spins contained in each magnetic entity (which should be as
small as possible to get a high TB),
• anisotropy energy (which should be as high as possible)
• applied magnetic field.
In a sense, this quantum phenomenon is related to finite-size effects, considering that
TB can only be experimentally achievable in magnetic particles of small enough size
(at most, few nanometers).
Deduced values for α are 0.5 and 0.035 for Co and BaFe12−2xCoxTixO19
nanoparticles, the former larger, as expected, being due to the effect of the
conduction electrons. Little information is available on other fine-particle systems.
Moreover, damping also plays an important role in MQT. In general, dissipation due
to, e.g. conduction electrons, strongly reduces quantum effects, in agreement with
the observation of MQT in BaFe12−2xCoxTixO19 for T < 0.4K, whereas this effect is
absent for Co particles down to 0.2K.
8.3.4.2 Magnetism relaxation
If the total magnetization of a fine-particle system is measured with an observation
time window which for some particle sizes is comparable to their reversal time, τ, the
magnetization will evolve during the experiment, a phenomenon known as magnetic
relaxation. The study of this non-equilibrium phenomenon is a common method to
experimentally characterize the energy spectrum of the system, since it is the
macroscopic signature of the distribution of energy barriers separating local minima
which correspond to different orientations of the particle moments. Magnetic
relaxation experiments are carried out by measuring the time-dependence of the
magnetization after the system has been driven out of thermal equilibrium by the
application or removal of a magnetic field. For instance, starting from the equilibrium
state at a given field and temperature, and then removing the field, the time evolution
of the magnetization for noninteracting particles can be written as follows
−= ∫∞
)()()(
0
0 E
txpeEdEfMtM
τ (8. 225)
Where f (E)dE is the fraction of particles having energy barriers in between E and E
+ dE, and M0 is the initial magnetization. In equation
−= ∫∞
)()()(
0
0 E
txpeEdEfMtM
τ
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(8. 225) and for the sake of simplicity, the particle moment has been assumed
to be volume independent, although, in fact, it is proportional to it. This assumption
does not significantly affect the results, providing that for a logarithmic-linear
distribution, f (E) and Ef (E) (which is equivalent to Vf (V)) are similarly shaped
functions. Let us introduce the function p(t, T,E) , which is defined by
−
−=
kT
EtETtp expexp),,(
0τ (8. 226)
Taking into account that p(t, T,E) for a given t varies abruptly from 0 to 1, as the
energy barrier E increases, we can approximate p(t, T,E) by a step function whose
discontinuity Ec(t, T ) moves towards higher energies as time elapses. Consequently,
the integral is cut off at the lower limit by the value of Ec(t, T ) and can be written as
∫∞
≅Ec
EdEfMtM )()( 0 (8. 227)
Where Ec(t, T ) = kBT ln(t/ ττττ0) is the only time-dependent parameter and signals the
position of the inflection point of p(t, T,E) . In fact, the narrow energy range spanned
by the step at Ec(t, T ) can be understood as the experimental window of the
measurements. The experimental window can be swept over f (E) by varying the
temperature or the observation time. From equation ∫∞
≅Ec
EdEfMtM )()( 0 (8. 227), it
can be deduced that M(t) obtained after integration is a function of the parameter
Ec(t, T ), which acts as a scaling variable. The existence of this scaling variable
implies that measuring the magnetization as a function of the temperature at a given
time is equivalent to measuring the magnetization as a function of ln(t/ ττττ0) at a given
temperature. This time–temperature correspondence is characteristic of activated
processes governed by Arrhenius law. The validity of the scaling hypothesis is only
determined by the validity of the cut-off approximation. Therefore, the scaling will be
accomplished as long as the width of p(t, T,E) , which is approximately given by kBT ,
is small when compared to the width of the energy barrier distribution. Although this
condition seems to be very restrictive, it is the usual situation found in experimental
observations of the magnetic relaxation in fine-particle systems.
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As an example of this scaling method, Figure 8-8 shows the scaling of the relaxation
curves obtained at different temperatures for a sample consisting on Fe0.78C0.22
particles with a mean diameter of 3.6 nm, dispersed in stable dilution with a
hydrocarbon oil. In this sample, the dipolar interaction among particles was estimated
to be very small. Figure 8-8 shows all the relaxation curves corresponding to
different temperatures collapsed into a single master curve when plotted as a
function of the scaling variable T ln(t/ ττττ0), being τ0 = (3.5 ± 5) × 10−11 s, which is an
expected value for ferromagnetic fine particles.
Figure 8-8 : The resulting scaling for a sample composed of FeC nanoparticles dispersed in a
hydrocarbon oil is shown. Open and full circles correspond alternatively to adjoining temperatures,
which are indicated above the corresponding interval (K units). The solid line is the theoretical curve
calculated by fitting the scaling curve to equation ∫∞
≅Ec
EdEfMtM )()( 0 (8. 227) with a
logarithmic-linear distribution of energies. The values of the fitted parameters are: σ = 0.44 ± 0.05 and
E0 = 287 ± 50K (position of the maximum of the energy barrier distribution). [X. Batle, A. Labarta,
Finite-size effects in fine particles: magnetic and transport properties, Journal of Physics, 2002,
ppR15-R42].
One of the most interesting aspects of these results is that, in fact, measuring the
relaxation at a given temperature is completely equivalent to measuring it at different
temperature but shifting the observation time window according to the law
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T ln(t/ ττττ0). In this sense, the method enables us to obtain the relaxation curve at a
given temperature, in a time range that is not experimentally accessible, by simply
dividing the T ln(t/ ττττ0) axis by this temperature. For instance, in the case shown in
Figure 8-8, we can obtain the relaxation curve at the lowest measured temperature of
1.8K at times as large as 10119 s, which is obviously an experimentally inaccessible
time. The validity of the T ln(t/ ττττ0) scaling method to analyse the relaxation behaviour
has been proved in many systems of non or slightly interacting fine particles.
Moreover, from the T ln(t/ ττττ0) scaling of relaxation data, insight can be gained into the
microscopic details of the energy barrier distribution producing the relaxation. If f (E)
is nearly constant in the range of energy barriers which can be overcome during the
observation time, then equation ∫∞
≅Ec
EdEfMtM )()( 0 (8. 227) can be approximated
by
−≈
00 ln)(1)(
τt
ETfkMtM CB (8. 228)
Where Ec is the mean energy barrier within the experimental window. From equation
−≈
00 ln)(1)(
τt
ETfkMtM CB (8. 228)) we can define the viscosity parameter as
TkEft
M
MS BC )(
))(ln(
1
0
=∂
∂−= (8. 229)
Therefore, as Ec is varied, the magnetic viscosity maps the energy barrier distribution
at low enough temperatures for which the width of p(t, T,E) (equation
−
−=
kT
EtETtp expexp),,(
0τ (8. 226)) function around the inflection point is
small compared to the width of the energy distribution and the scaling hypothesis is
fulfilled. As an example of the applicability of the method, in Figure 8-9, we show S/T
derived from the data of Figure 8-8 as a function of T ln(t/ ττττ0). In Figure 8-9, the
differential of the thermoremanence relative to the saturation magnetization with
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respect to the temperature is also shown, the agreement between both sets of data
being very good.
Figure 8-9 : Numerical derivative of the data of Figure 8-8 with respect to the scaling variable (open
circles) and the logarithmic-linear distribution (dashed line) obtained by fitting the data of Figure 8-8 to
the equation ∫∞
≅Ec
EdEfMtM )()( 0 (8. 227). The differential of the thermoremanence relative to
the saturation magnetization versus the temperature is also shown in full circles for comparison. [X.
Batle, A. Labarta, Finite-size effects in fine particles: magnetic and transport properties, Journal of
Physics, 2002, ppR15-R42].
If f (E) is almost constant over a certain range of energies, then S will be linear in T
over the corresponding temperature range; consequently, relaxation as a function of
time will follow a time logarithmic decay at fixed T. This behaviour is only
accomplished in a narrow region around the maximum of a bell-shaped energy
barrier distribution (see for instance, the region around the inflection point of the
master curve shown in Figure 8-8).
T ln(t/ ττττ 0) scaling can be also applied to relaxation experiments measuring the
acquisition of magnetization of an initially demagnetized system under the application
of a magnetic field. In this kind of experiments, the field modifies the energy barriers
that are responsible for the time evolution, as well as the final equilibrium state
towards which the system relaxes. The main effect of the field over f (E) is the
splitting into two sub-distributions which are centred at lower and higher energy
values as the field increases, the former dominating the relaxation of the system. For
a noninteracting system, the relaxation curves collected at different temperatures and
under a fixed field collapse into a single master curve when they are plotted as a
function of T ln(t/ ττττ0) variable and proper normalization of the magnetization data to
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the final equilibrium values is performed. Moreover, for fields smaller than the
switching field corresponding to the smallest barriers, the relaxation curves at
different fields and fixed T can be collapsed into a single curve, in a way similar to T
ln(t/ ττττ0) scaling for curves at fixed field, but in this case, the additional shifts in the T
ln(t/ ττττ0) axis necessary to get field scaling give the field dependence of the mean
relaxing barriers present in the system.
8.3.4.3 Effects of the interparticle interaction
We have reviewed different aspects of the magnetic relaxation of fine-particle
systems with random orientations, volume distribution and negligible magnetic
interactions. In this situation, the system is in an SPM regime, for which the time
evolution of the total magnetization is simply governed by the thermal activation over
the individual energy barriers of each particle. When interparticle interactions (for
instance, dipolar interactions) are non-negligible the behaviour of the system is
substantially more complicated and the problem becomes non-trivial, even if the
spins of the particle are assumed to be coupled to yield a super-spin moment.
The main types of magnetic interactions that can be found in fine-particle assemblies
are:
• dipole–dipole interactions, which always exit
• exchange interactions through the surface of the particles which are in close
contact
• in granular solids, RKKY interactions through a metallic matrix when particles
are also metallic, and super exchange interactions when the matrix is
insulating.
Bearing in mind the anisotropic character of dipolar interactions, which may favour
ferromagnetic or antiferromagnetic alignments of the moments depending on
geometry, fine-particle systems have all the ingredients necessary to give rise to a
spin-glass state, namely, random distribution of easy axes and frustration of the
magnetic interactions. The complex interplay between both sources of magnetic
disorder determines the state of the assembly and its dynamical properties.
Magnetic interactions modify the energy barrier coming from the anisotropy
contributions of each particle and, in the limit of strong interactions, their effects
become dominant and individual energy barriers can no longer be considered, only
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the total energy of the assembly being a relevant magnitude. In this limit, relaxation is
governed by the evolution of the system through an energy landscape with a
complex hierarchy of local minima similar to that of spin-glasses. It is worth noticing
that in contrast with the static energy barrier distribution arising only from the
anisotropy contribution, the reversal of one particle moment may change the energy
barriers of the assembly, even in the weak interaction limit. Therefore, the energy
barrier distribution may evolve as the magnetization relaxes.
Three models have been developed to introduce particle interaction:
- Shtrikmann and Wohlfarth
For a weak interaction limit, the relaxation time is
−=
)(exp
00 TTk
E
BB
Bττ (8. 230)
Where TB is the blocking temperature and T0 is an effective temperature which
accounts for the interaction effects.
- Dormann et al
This model correctly reproduces the variation of the blocking temperature TB, as a
function of the observation time window of the experiment, ττττm, at least in a range of
time covering eight decades. The increase of TB with the strength of the dipolar
interactions (e.g. increasing particle concentration or decreasing particle distances)
has been predicted by this model and also experimentally confirmed.
- Morup and Tronc
For the weak interaction limit, the opposite dependence of TB with the strength of the
interactions is predicted. Morup suggested that two magnetic regimes, governed by
opposite dependencies of TB, occur in interacting fine particles. At high temperatures
and/or for weak interactions, TB signals the onset of a blocked state and TB
decreases as the interactions increase. In contrast, at high temperatures and/or for
strong interactions, a transition occurs from an SPM state to a collective state which
shows most of the features of typical glassy behaviour. In this case, TB is associated
with a freezing process and it increases with the interactions.
Non-equilibrium dynamics, showing ageing effects in the relaxation of the residual
magnetization, have been observed in interacting fine-particle assemblies, but some
important differences with spin-glasses have been established. In the last few years,
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several works demonstrated that for concentrated frozen ferrofluids and assemblies
of nanoparticles dispersed in a polymer, the relaxation depends on the waiting time,
tw, spent at constant temperature before the magnetic field is changed (Figure 8-10).
Figure 8-10 : The relaxation rate S versus log10(t) at different wait times for the concentrated sample
(a) and for the most diluted sample (b). The measurements were performed on a ferrofluid consisting
of closely spherical particles of maghemite with a mean diameter of 7 nm. [X. Batle, A. Labarta, Finite-
size effects in fine particles: magnetic and transport properties, Journal of Physics, 2002, ppR15-R42].
This phenomenon is absent in the most diluted samples, confirming that it is due to
dipole–dipole interactions (Figure 8-10). Ageing effects on the magnetic relaxation
are the fingerprint of the existence of many minima in the phase space. However, a
complex hierarchy of energy minima is not an exclusive feature of spin-glasses and
ageing has also been found in other cluster-glass systems not having a true phase
transition. In fact, the non-equilibrium dynamics of interacting fine particles largely
mimics the corresponding of spin-glasses, including ‘memory experiments’ in which
ageing is studied at different heating or cooling rates and/or cycling the temperature.
The main differences between the dynamics of canonical spin-glasses and that of
fine particles are:
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• the dependence of the magnetic relaxation on tw is weaker than in spin-
glasses
• in the collective state, the relaxation times are widely distributed and strongly
dependent on temperature
• large particles are blocked in all time scales acting as a temporary random
field and not taking part in the collective state of the system.
Another feature that differentiates the low-temperature collective state from the spin-
glass one is its extreme sensitivity to the application of an external magnetic field. In
particular, it has been shown that the collective state of strong interacting particles
can be erased by a field-cooling process with an applied magnetic field of moderate
strength which yields an asperomagnetic state. It has also been demonstrated that
the dynamics of these systems are strongly affected by the initial magnetic moment
configuration, in such a way that the collective state determines the dynamic
behaviour only in low-cooling-field experiments, while at high cooling fields the
dynamics are mostly dominated by the intrinsic energy barriers of the individual
particles.
8.3.4.4 Surface effects
Surface effects result from the lack of translational symmetry at the boundaries of the
particle because of the lower coordination number there and the existence of broken
magnetic exchange bonds which lead to surface spin disorder and frustration.
Surface effects dominates the magnetic properties of the smallest particles since
decreasing the particle size increases the ratio of surface spins to the total number of
spins. For example, for fcc Co with a lattice constant of 0.355 nm, particles having
about 200 atoms will have diameters around 1.6 nm and 60% of the total spins will be
at the surface. This represents more than half of the spins. Consequently, the ideal
model of a superspin formed by all the spins of the particle pointing in the anisotropy
direction and coherently reversing due to thermal activation is no longer valid, since
misalignment of the surface spin yields strong deviations from the bulk behaviour.
This is true, even for particles with strong exchange interactions such as many
ferrimagnetic oxides with magnetically competing sublattices, since broken bonds
destabilize magnetic order giving rise to frustration which is enhanced with the
strength of the interactions. As a consequence of the combination of both finite-size
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and surface effects, the profile of the magnetization is not uniform across the particle
and the magnetization of the surface layer is smaller than that corresponding to the
central spins.
Many experimental results for metallic and oxide particles indicate that the anisotropy
of fine particles increases as the volume is reduced because of the contribution of
what is known as surface anisotropy. For instance, the anisotropy per unit volume
increases by more than one order of magnitude for 1.8 nm fcc Co particles being
3×107 erg cm−3 compared with the bulk value of 2.7×106 erg cm−3. Even an
anisotropy value one order of magnitude larger than the preceding case has been
reported for Co particles embedded in a Cu matrix. In fact, surface anisotropy has a
crystal-field nature and it comes from the symmetry breaking at the boundaries of the
particle. The structural relaxation yielding the contraction of surface layers and the
existence of some degree of atomic disorder and vacancies induce local crystal fields
with predominant axial character normal to the surface, which may produce easy-axis
or easy-plane anisotropies. This can be justified by noting that the axis of the local
crystal field, ^
n , may be evaluated from the dipole moment of the nearest-neighbour
atomic positions with respect to the position of a given surface atom as follows:
∑ −∝nn
jiji PPn )(
^
(8. 231)
Where Pi is the position of the ith atom and the sum extends to the nearest
neighbours of this atom. Since at the surface some of the neighbours are missing, in^
is non-zero and directed approximately normal to the surface. The effect of these
local fields is obtained by adding a term of the form 2ξKS , where Sςςςς is the component
of the spin along a vector normal to the surface and, K < 0 corresponds to the easy-
axis case and K > 0 to the easyplane one. When |K| is comparable to the
ferromagnetic exchange energies, spin configurations like those shown in Figure
8-11 are obtained. These result from the competition of surface anisotropy and
ferromagnetic alignment. Another kind of surface anisotropy was suggested that
treated acicular particles of γ -Fe2O3 as infinite cylinders possessing uniaxial
anisotropy parallel to the surface of the cylinder. In general, the surface anisotropy
makes the surface layer magnetically harder than the core of the particle.
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Figure 8-11 : Surface spin arrangement of a ferromagnetic particle with a surface anisotropy of the
form 2ξKS . Both cases corresponding to K <0 (radial) and K >0 (tangential) are displayed. [X. Batle, A.
Labarta, Finite-size effects in fine particles: magnetic and transport properties, Journal of Physics,
2002, ppR15-R42].
The second contribution due to the existence of boundaries is a consequence of
strains related to lattice deformations occurring at the surface, which through
magnetostriction effects induce an additional surface anisotropy. Strain anisotropy
has been observed in thin films because of stresses induced at the interface between
substrate and film due to non-matching lattice constants. Depending on the structural
deformations and the nature of the thin films, strain anisotropy could yield
perpendicular anisotropy to the film plane, as has been observed in as-prepared thin
films of heterogeneous Co(Fe)–Ag(Cu) alloys. However, for the majority of the
nanoparticle systems, the strain energy is weak, it being difficult to give a general
formulation, and, in particular, for free particles it is negligible.
An effective anisotropy energy per unit volume, Keff, could be obtained by adding the
core (i.e. bulk anisotropy) and surface contributions. For a spherical particle the
following phenomenological expression has been used to account for Keff:
Sbeff Kd
KK6+= (8. 232)
Where Kb is the bulk anisotropy energy per unit volume, Ks is the surface density of
anisotropy energy and d is the diameter of the particle. It is worth noticing that for a
spherical particle and based on symmetry arguments, surface anisotropy (normal to
the surface) would average to zero. However, this is not true for a nanometric particle
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with a few atomic layers. For instance, applying equation Sbeff Kd
KK6+= (8.
232) for a 2 nm particle of fcc Co with Kb = 2.7×106 erg cm−3 and Ks ≈ 1 erg cm−2,
the surface contribution to the total anisotropy, Keff , is about 3 × 107 erg cm−3, which
is one order of magnitude larger than the bulk contribution, and Keff = 3.3 × 107 erg
cm−3.
This example is representative of the major role of surface contribution to the total
anisotropy in fine-particle systems, for which the anisotropy energy is governed by
surface anisotropy.
8.4 Magnetoelectronics spins
Spintronics is a multidisciplinary field whose central theme is the active manipulation
of spin degrees of freedom in solid-state systems. The goal of spintronics is to
understand the interaction between the particle spin and its solid-state environments
and to make useful devices using the acquired knowledge.
In this chapter, we will focus on magnetoelectronics materials. Typically they cover
paramagnetic and ferromagnetic metals and insulators which utilize magnetorisistive
effects, realized, e.g., as magnetic read heads in computer hard drives, nonvolative
magnetic random access.
8.4.1 Spin-polarized transport and magnetorisistive effects
Some Scientifics sought an explanation for an unusual behaviour of resistance in
ferromagnetic metals. They realized that at sufficiently low temperatures, where
magnon scattering becomes vanishingly small, electrons of majority and minority
spin, with magnetic moment parallel and antiparallel to the magnetization of a
ferromagnet, respectively, do not mix in the scattering processes. The conductivity
can then be expressed as the sum of two independent and unequal parts for two
different spin projections, the current in ferromagnets is spin polarized.
8.4.1.1 Tunneling magnetoresistance (TMR)
Tunneling measurements play a key role in work on spin-polarized transport. It is
important to study N/F/N junctions, where N was a nonmagnetic metal and F was an
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Eu-based ferromagnetic semiconductor. These investigations showed that when
unpolarized current is passed across a ferromagnetic semiconductor, the current
becomes spin-polarized.
Ferromagnet / insulator / superconductor (F/I/S) and Ferromagnet / insulator /
Ferromagnet (F/I/F) junctions were studies these last years. F/I/S have proved that
the tunneling current remains spin polarized even outside of the ferromagnetic region
and was used as a detector of spin polarization of conduction electrons. By analysing
the tunnelling conductance from F/I/S to the F/I/F junctions, Jullière (1975) formulated
a model for a change of conductance between the parallel (↑↑) and antiparallel (↑↓)
magnetization in the two ferromagnetic regions F1 and F2, as depicted in Figure
8-12.
Figure 8-12 : Schematic illustration of electron tunneling in F/I/F tunnel junctions: (a) Parallel and (b)
antiparallel orientation of magnetizations with the corresponding spinresolved density of the d states in
ferromagnetic metals that have exchange spin splitting ∆ex . Arrows in the two ferromagnetic regions
are determined by the majority-spin subband. Dashed lines depict spinconserved tunneling. [I. Zutic, J.
Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Reviews of modern physics,
volume 76, 2004, pp 323-410]
The corresponding tunneling magnetoresistance (TMR) in an F/I/F magnetic tunnel
junction (MTJ) is defined as
↑↓
↑↓↑↑
↑↑
↑↑↑↓
↑↑
−=
−=∆=
G
GG
R
RR
R
RTMR (8. 233)
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Where conductance G and resistance R=1/G are labeled by the relative orientations
of the magnetizations in F1 and F2 (it is possible to change the relative orientations,
between ↑↑ and ↑↓, even at small applied magnetic fields ~10 G). TMR is a particular
manifestation of a magnetoresistance that yields a change of electrical resistance in
the presence of an external magnetic field. Due to spin-orbit interaction, electrical
resistivity changes with the relative direction of the charge current (for example,
parallel or perpendicular) with respect to the direction of magnetization. Assuming
constant tunnelling matrix elements and that electrons tunnel without spin flip,
equation ↑↓
↑↓↑↑
↑↑
↑↑↑↓
↑↑
−=
−=∆=
G
GG
R
RR
R
RTMR (8. 233) yields
21
21
1
2
PP
PPTMR
−= (8. 234)
Where the polarization Pi is expressed in terms of the spin-resolved density of states
for majority and minority spin in Fi.
8.4.1.2 Giant magnetoresistance (GMR)
The prediction of Jullière’s model illustrates the spinvalve effect: the resistance of a
device can be changed by manipulating the relative orientation of the magnetizations
M1 and M2, in F1 and F2, respectively. Such orientation can be preserved even in the
absence of a power supply, and the spin-valve effect displays the giant
magnetoresistance (GMR) effect can be used for nonvolatile memory applications.
GMR structures are often classified according to whether the current flows parallel
(CIP, current in plane) or perpendicular (CPP, current perpendicular to the plane) to
the interfaces between the different layers, as depicted in Figure 8-13.
Figure 8-13 : Schematic illustration of (a) the current in plane (CIP), (b) the current perpendicular to
the plane (CPP) giant magnetoresistance geometry. [I. Zutic, J. Fabian, S. Das Sarma, Spintronics:
Fundamentals and applications, Reviews of modern physics, volume 76, 2004, pp 323-410]
Most of the GMR applications use the CIP geometry, while the CPP version is easier
to analyze and relates to the physics of the tunneling magnetoresistance effect. The
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size of magnetoresistance in the GMR structures can be expressed analogously to
Equation ↑↓
↑↓↑↑
↑↑
↑↑↑↓
↑↑
−=
−=∆=
G
GG
R
RR
R
RTMR (8. 233), where parallel and
antiparallel orientations of the magnetizations in the two ferromagnetic regions are
often denoted by ‘‘P’’ and ‘‘AP,’’ respectively (instead of ↑↑ and ↑↓). Realization of a
large roomtemperature GMR enabled a quick transition from basic physics to
commercial applications in magnetic recording.
One of the keys to the success of magnetoresistance-based applications is their
ability to control the relative orientation of M1 and M2. The flow of spin-polarized
current can transfer angular momentum from carriers to ferromagnet and alter the
orientation of the corresponding magnetization, even in the absence of an applied
magnetic field. This phenomenon is known as spin-transfer torque. It was also shown
that the magnetic field generated by passing the current through a CPP giant
magnetoresonance device could produce roomtemperature magnetization reversal.
In the context of ferromagnetic semiconductors additional control of magnetization
was demonstrated optically and electrically to perform switching between the
ferromagnetic and paramagnetic states. More information about GMR are given in
8.5 paragraph.
The challenge remains to preserve such spin polarization above room temperature
and in junctions with other materials, since the surface (interface) and bulk magnetic
properties can be significantly different.
8.4.1.3 Applications
While many existing spintronic applications are based on the GMR effects, the
discovery of large room-temperature TMR has renewed interest in the study of
magnetic tunnel junctions, which are now the basis for the several magnetic random-
access memory prototypes. Future generations of magnetic read heads are expected
to use MTJ’s instead of CIP giant magnetoresonance. To improve the switching
performance of related devices it is important to reduce the junction resistance, which
determines the RC time constant of the MTJ cell. Consequently, semiconductors,
which would provide a lower tunneling barrier than the usually employed oxides, are
being investigated both as the nonferromagnetic region in MTJ’s and as the basis for
an all-semiconductor junction that would demonstrate large TMR at low
temperatures.
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8.4.2 Spin injection
Many materials in their ferromagnetic state can have a substantial degree of
equilibrium carrier spin polarization. However, this alone is usually not sufficient for
spintronic applications, which typically require current flow and/or manipulation of the
nonequilibrium spin (polarization). The importance of generating nonequilibrium spin
is not limited to device applications; it can also be used as a sensitive spectroscopic
tool to study a wide variety of fundamental properties ranging from spin-orbit and
hyperfine interactions to the pairing symmetry of high-temperature superconductors
and the creation of spin-polarized beams to measure parity violation in high-energy
physics.
Nonequilibrium spin is the result of some source of pumping arising from transport,
optical, or resonance methods. Once the pumping is turned off the spin will return to
its equilibrium value. While for most applications it is desirable to have long spin
relaxation times, it has been demonstrated that short spin relaxation times are useful
in the implementation of fast switching.
The Feher effect is based on the principle that the hyperfine coupling between the
electron and nuclear spins, together with different temperatures representing electron
velocity and electron spin populations, is responsible for the dynamical nuclear
polarization.
Figure 8-14 : Pedagogical illustration of the concept of electrical spin injection from a ferromagnet (F)
into a normal metal (N). Electrons flow from F to N: (a) schematic device geometry; (b) magnetization
M as a function of position—nonequilibrium magnetization δM (spin accumulation) is injected into a
normal metal; (c) contribution of different spin-resolved densities of states to both charge and spin
transport across the F/N interface. Unequal filled levels in the density of states depict spin-resolved
electrochemical potentials different from the equilibrium value µ0. [I. Zutic, J. Fabian, S. Das Sarma,
Spintronics: Fundamentals and applications, Reviews of modern physics, volume 76, 2004, pp 323-
410]
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When a charge current flowed across the F/N junction (Figure 8-14), spin-polarized
carriers in a ferromagnet would contribute to the net current of magnetization entering
the nonmagnetic region and would lead to nonequilibrium magnetization δM,
depicted in Figure 8-14(b), with the spatial extent given by the spin diffusion length.
Such a δM, which is also equivalent to a nonequilibrium spin accumulation, was first
measured in metals by Johnson and Silsbee (1985, 1988d). In the steady state δM is
realized as the balance between spins added by the magnetization current and spins
removed by spin relaxation.
8.4.3 Spin Polarization
Spin polarization not only of electrons, but also of holes, nuclei, and excitations can
be defined as
Px = Xs/X (8. 235)
The ratio of the difference Xs = Xλ – X-λ, and the sum Xs = Xλ + X-λ of the spin-
resolved λ components for a particular quantity X. To avoid ambiguity as to what
precisely is meant by spin polarization, both the choice of the spin-resolved
components and the relevant physical quantity X need to be specified.
Conventionally, λ is taken to be ↑ or + (numerical value +1) for spin up, and ↓ or -
(numerical value -1) for spin down, with respect to the chosen axis of quantization. In
ferromagnetic etals it is customary to refer to ↑ (↓) as carriers with magnetic moment
parallel (antiparallel) to the magnetization or, equivalently, as carriers with majority or
minority spin.
The spin polarization of electrical current or carrier density, generated in a
nonmagnetic region, is typically used to describe the efficiency of electrical spin
injection. Silsbee (1980) suggested that the nonequilibrium density polarization in the
N region, or equivalently the nonequilibrium magnetization, acts as the source of spin
electromotive force (emf) and produces a measurable ‘‘spin-coupled’’ voltage
MVS δ∝ . Using this concept, also referred to as spin-charge coupling, Silsbee
proposed a detection technique consisting of two ferromagnets F1 and F2 (Figure
8-15) separated by a nonmagnetic region.
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Figure 8-15 : Spin injection, spin accumulation, and spin detection: (a) two idealized completely
polarized ferromagnets F1 and F2 with parallel magnetizations are separated by the nonmagnetic
region N; (b) density-of-states diagrams for spin injection from F1 into N, accompanied by the spin
accumulation-generation of nonequilibrium magnetization δM. At F2 in the limit of low impedance
(Z=0) electrical spin is detected by measuring the spin-polarized current across the N/F2 interface. In
the limit of high impedance (Z=∞) spin is detected by measuring the voltage Vs~δM developed across
the N/F2 interface; (c) spin accumulation in a device in which a superconductor is occupying the
region between F1 and F2. [I. Zutic, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and
applications, Reviews of modern physics, volume 76, 2004, pp 323-410]
F1 serves as the spin injector (spin aligner) and F2 as the spin detector. This could
be called the polarizer-analyzer method, the optical counterpart of the transmission of
light through two optical linear polarizers.
From Figure 8-15 it follows that the reversal of the magnetization direction in one of
the ferromagnets would lead either to Vs→ -Vs , in an open circuit (in the limit of
large impedance Z), or to the reversal of charge current j→ -j, in a short circuit (at
small Z). Spin injection could be detected through the spin accumulation signal as
either a voltage or a resistance change when the magnetizations in F1 and F2 are
changed from parallel to antiparallel alignment.
The generation of nonequilibrium spin polarization has a long tradition in magnetic
resonance methods. However, transport methods to generate carrier spin
polarization are not limited to electrical spin injection. For example, they also include
scattering of unpolarized electrons in the presence of spin-orbit coupling and in
materials that lack inversion symmetry, adiabatic and nonadiabatic quantum spin
pumping, and proximity effects.
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8.5 Giant Magnetoresistance (GMR)
8.5.1.1 Origin of the GMR effect
For a macroscopic electrical conductor the resistance R is given by R=P/A, which is
independent of shape, cross section and above all the scattering process involved
therein. Especially, electron-electron and electron-proton scattering process bring the
out-of-equilibrium distribution of accelerated electrons back to the equilibrium
distribution thereby contribution to resistance. This usual picture loses its validity
when the dimension of the material becomes of the order of relevant collision-mean-
free path (λ ). Now for an elastic scattering process the conductivity due to spin up
and spin down electron is given by
1 2σ σ σ= + (8. 236)
For an elastic scattering processlxxxvi
For the occurrence of the GMR effect it is crucial that the angles between the
magnetization directions of the ferromagnetic layer can be modified by the application
of an applied magnetic field, the electron scattering probability at the interfaces or
within the bulk of the layers is spin-dependent, and in the case of parallel
magnetizations, the layer averaged electron mean-free path for (at least) one spin
direction is larger than the thickness of the non-magnetic spacer layer.
Now, each layer in a GMR- structure acts as a spin selective valve. In case of
parallel-alignment-layer the contribution towards conductance due to spin up electron
is highly leading to higher valve of total conductance. However, for antiparallel
alignments the magnetic layers results in appreciable scattering for electrons in both
spin direction and hence a lower total conductance. If the F layers are much thinner
than λ↑ the spin up conductance in the parallel magnetization state is limited by
strong spin-dependent scattering in the buffer layer and in the AF layer. The contrast
with spindown conductance can then be enhanced by increasing the F-layer
thickness. It is to be noted that only a part of F-layer which is of the order of λ↑/2 from
the interface responds to the above situation.
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R/R
sat (%
)
free F-layer thickness (nm)
0 20 40 60
8
6
4
2
0
12
10
8
6
4
2 0
(a) 293 K
(b) 5 K
R/R
sat (%
)
Figure 8-16 : Dependence of the MR ratio on the free magnetic layer thickness, measured at 293 K
(a) and at 5K (b), for (X/Cu/X/Fe50Mn50) spin valves, grown on 3nm Ta buffer layer on Si(100)
substracts, with X=Co (squares), X=Ni66Fe16Co18 (traingles) and (X= Ni80Fe20) (+), measured at 5K.
Layer thickness: tCu = 3nm (but 2.5 for F=(X= Ni80Fe20); tF(pinned layer)=5nm (but 6nm for F=
Ni66Fe16Co18)lxxxvii.
Requirements for F/N combinations with a high GMR ratio are:
• the F and N materials possess, for one spin direction, very similar electronic
structures, whereas for the other spin direction the electronic structures are
very dissimilar,
• the electron transmission probability (transmission without diffusive scattering)
through the F/N interfaces is large for the type of electrons for which the
electronic structure in the F and N layers are very similar,
• and the crystal structures of the F and N materials match very well.
8.5.1.2 Thermal stability
A number of factors control the use and processing of exchange-biased spin-
valve layered structures at elevated temperatures:
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• Exchanging biasing field decreases with increasing temperature. For some
exchange biasing materials this effect determines the temperature range
within which sensors can be applied.
• The induced magnetocrystalline anisotropy of the free layer may be affected
by heating the material.
• As multilayers are thermodynamically metastable system the enhanced
diffusion rates during annealing will inversibly degrade the layered structure.
Diffusion may also affect the effective coupling between the layers. There are
three contributions in interlayer coupling:
• Coupling via ferromagnetic bridges between the ferromagnetic layers.
• The mangetostatic Neel-type coupling.
• Interlayer exchange coupling.
8.5.1.3 Factors determining the switching field int erval
8.5.1.3.1 Induced magnetic anisotropy
This is believed to be due to the result of a very small degree of divertional pair order
of atom a in the otherwise random solid solution. The anisotropy field depends upon
the composition, the layer thickness, and the material between which the layers are
sandwiched. In case of a GMR spin valve the anisotropy effect is relevant for the
thickness below 15 nm. The effect is most likely due to a region with a decreased
degree of pair order situated in the ferromagnetic layer close to the interface with the
buffer layer.
The result shows that addition of Co to permalloy is unfavourable as far as switching
field interval is concerned. But one of the advantages is that the range of linear
operation of GMR sensor elements is improved. Annealing the sample can affect the
anisotropy field. It is often necessary to anneal the sample after decomposition in a
magnetic field that is directly perpendicular to field direction during growth, in order to
rotate the exchange direction over an angle of 90o. This leads to cross anisotropy
configuration. Therefore it is suggestive to anneal with the field parallel to the induced
anisotropy before annealing briefly with perpendicular field. On the other hand strain
results in an additional magnetocrystalline anisotropy due to magnetostriction. So for
sensor element coefficient of magnetostriction should be small. Ni66Co18Fe16 alloy is
a good candidate for application in GMR sensor elements.
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8.5.1.3.2 Superimposed AMR effect: lxxxviii
If the current direction is parallel to the easy axis of the free F layer, the
magnetoresistance is given by
2
R(GMR)(1-cos )R(H)= + R(AMR) sin
2
θ θ∆∆ ∆ (8. 237)
The AMR-term represents the resistance change due to the change of angle (2
π θ− )
between the magnetization of the free layer and the current direction. The presence
of AMR effect increases the sensitivity R
HR
∂ ∂ if an operating point between H=0 to
H=+Ha is chosen. Experiment shows that due to AMR effects the sensitivity of the
spin valve is almost 50% higher then expected from GMR effect. It is suggestive for
practical purpose to operate a material around the field H=0, where the element is
magnetically most stable.
8.5.1.3.3 Coupling between the magnetic layer
If we make N layer very thin the free F layer is found to be magnetically
coupled. This coupling leads to an offset of the field around which the free layer
switches to a field H= - Hcouple . The coupling field is due tolxxxix
• Ferromagnetic coupling via pinhole.
• Ferromagnetic Neel-type coupling.
• An oscillatory indirect exchange coupling.
All three contributions are influenced by the microstructure of the material and
hence control of the coupling field requires very good control of the deposition
condition. This issue is very relevant for sensor application. Most of the cases the
coupling field is equal or greater than the anisotropy field. Fortunately, in a sensor
element the resulting shift of the switching field is balanced by:
• Application of a bias magnetic field from a current though an integrated bias
conductor.
• The magnetostatic coupling between the pinned and the free layer which
arises as a result of the stray magnetic field origination from the magnetization
of a pinned layer, which is directly along the long axis of the stripe.
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Interlayer coupling influences the switching field interval due to the following
two effects:
• The magnetization of the pinned layer is not quite fixed upon the rotation of
the free layer. This increases the switching field interval. This effect increases
upon an increases of the ratio between the coupling field and the exchange
biasing field.
• Lateral fluctuations in the coupling field increase the switching field interval.
8.5.1.3.4 Exchange biasing
The interaction energy E between the pinned F layer and the
antiferromagnetic layer is given by,
E=-KcbCosθ (8. 238)
Where θθθθ is the angle between the biasing direction and the magnetization of
the pinned F layer. The exchange biasing field varies inversely with the thickness of
the pinned layer.
To obtain exchange biasing the AF layer thickness should generally exceed
critical valve.
In conclusion, we must say that exchange-biased spin valve layered structures
show a combination of properties that are attractive for the application in low field
sensors, such as read head in hard disk and tape recording. For a yoke-type GMR
head an output gain of a factor 10 has been obtained compared with a barbar-pole
biased AMR head. The following factors contribute to operational gain:
• Larger MR ratio.
• Larger sheet resistance.
• Larger flux efficiency of the head, due to the small thickness of the magnetic
layer.
Future developments are expected to focus on exchange-biased spin valve
materials with a high thermal stability, on the improvement of the effective softness,
of the free magnetic layer while remaining a high GMR ratio, and on relation between
micromagnetic aspects such as domain wall nucleation and propagation and sensor
noise.
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8.6 Storage Devices
The design and fabricatrion of nonstructured film are playing an increasing important
role in modern science and technology. In the case of non-scale magnetic films, a
large part of the interest in driven by information storage and magneto electronic
devices.
8.6.1 Magnetic data storage :
Magnetic non-structure are starting to pay a role in technology, particularly in “non-
volatile magnetic data storage”. New combination with semi-conductor technology
are developing such as magnetic random access memories (MRAMS) where the
storage capacitor of a traditional semiconductor in replaced by a non-volatile
magnetic dot. The storage media to today’s magnetic hard disc drives may be viewed
as an array of magnetic nanoparticles. The magnetic coating of the disk consists of a
ternary Co-Pt-Cr. Mixture which segregates into magnetic Co-Pt grains. These grains
are magnetically separated by Cr at the grain boundaries. Typical grain sizes are 10-
20 NM using about 103 grains/bit at a recording density of 1 Gbit/inch2 for
commercial devices. The grains segregate randomly, which introduces statistical
noise into the read out signal due to the variation in grain size, coercivity and domain
structure. This explains the large number of grains that are required to reduce these
fluctuation in a device.
There is a fundamental limit or the improvement of magnetic storage density. The
limit is controlled by
i) Thermal flipping of the bits as the grains become smaller;
ii) Energy barrier at between the two stable magnetizations along the easy
axis becomes comparable with kT. Eventually superparamagnetic limit
is reached where individual grains stay matnetized, but their orientation
fluctuates thermally.
For typical magnetic storage media, the superparmagnetic limit imposes a minimum
particle size of about 10 nm, that is a maximum recording density of several terabits
per square inch. This is almost four order of magnitude higher than the density of 1
Gbit/inch2 found in top-of-the-line disc drives today. While the current particle size is
already close to the superparamagnetic limit the number of particle / bits is still more
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than 103. There are many signal to noise issues on the way towards reducing this
number and reaching the theoretical limit. Inconsistent switching of different particles
and an irregular domain structure require averaging over many particles. Controlling
coercivity, size, orientation and position of magnetic non-particles will be essential for
reducing the number of particles needed to store a bit. For example a large
crystalline anisotropy can produce a higher switching barrier than the shape
anisotropy above. Single domain nanoparticle with high saturation magnetization and
coercivity are being optimized for this purpose. The orientation of segregated grains
can be controlled using multilayered structures where the first layer acts as seed for
small grains and subsequent layers shape the crystalline orientation for the desired
anisotropy. A further improvement would be the more from longitudinal to
perpendicular recording where demagnetizing field does not destabilize the written
domains.
The ultimate goal in magnetic storage in single-particle-per-bit on quantized
recording. It is aimed at producing single domain particles close to the
superparamagnetic limit with uniform switching properties. Lithography is currently
the method of choice for producing regular arrays of uniform magnetic dots. Dot
arrays with a density of 65 – 250 Gbit/inch2 have been produced by electron beam
lithography.
8.6.2 Sensors:
Nanostructured material has also been used in the development of some
reading head devices. The traditional inductive pick up of the magnetic signal is
replaced by a magnetoresistive sensor in state-of-the-art devices.
Permalloy / (Cu/Co) multilayers are utilized in reading heads. Currently,
reading heads in high-end disc drives are based on the 2% AMR of permalloy. The
resistance is highest for the current parallel to the magnetization and lowest
perpendicular to it, producing a sinusoidal orientational dependence. The magnetic
stray field between adjacent bits with opposite orientation rotates the magnetization
in the permalloy film with respect to the current and thus induces a resistance
change. That is directly connectible into a read-out voltage.
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Present activities with GMR are directed towards, lowering the switching field
while keeping a large magneto-resistance. To obtain the best of both characteristics
one obtains soft permalloy layer for easy switching with a high-spin co-layer that
enhances the magnetoresistance.
Another type of non-volatile magnetic storage device avoid moving parts
altogether at the expense of having to pattern the storage medium. This is a
combination of magnetic memory elements with semi-conductor circuits that sense
and amplify the magnetic state (MRAM).
Further into the future are logic devices based on magnetic nanostructures. A
bipolar spin switch has been demonstrated that acts like a transistor.
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