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CHAPTER - I
INTRODUCTION: William Bardforded in California and obtained
Ph.D. in 1936 from the Massachusetts Institute of Technology, USa.
Shickley’s research had been centred in many areas of semiconductor
Physics, e.g., energy bands in semiconductors i order and disorder an alloys i
theory of vaccum theory on ferromagnetic domains i and various topics in
tranistor Physics.
He was awarded the Nobel Prize an Physics in 1956 for his researches
on semiconductors and discovery of transistor effect along with John Bard
een (1908- 1991) and walter Houser Brattain (1902- 1987).
This work revolutionized the world of electronics wide energy Brand
Gap Electronics Devices definitions. Fan Ren (University of Florida, USA)
& Johnc Zolper (DARPA, USA) his book provides a summary of the current
state of the art in sic and GAN and identify future areas of development. The
remarkable improvements an material quality and device peroformance in
the last few years show the promise of these technologies for areas that si
cannot operate because of gt’s smaller band gap. We feel that this collection
of chapters provides and excellent introduction to the field and is an
1
outstanding reference for those peforming research on wide bandgap
semiconductors.
In this book, we bring together numerous experts in the field to review
progression sic and GAN electronic devices and noval detectors. Professor
Morkoc reviews the growth and characterization of nitrides followed by
chapters from professor Karmalkar and Professor Gaska on High Electron
Mobility Transisters Professor Pearton and Co-workers on ultro-high
breakdown voltage GAN based rectifier and the group of professor.
Abernathy on emerging MOS devices in the nitride system Dr. Baca from
Sandia National Laboratories and Dr. Chang from Agilent review the use of
mixed group V-Nitride as the base layer in novel Hetereojunction Biopolar
Transistors. There are 3 chapter on sic, including professor skowronski on
growth and characterization. Professor chow on power Schottky and pin
rectifiers and professor cooper on power MOSFETS. Professor Dupuis and
professor Campbell give an overview of short wavelength nitride based
detectors. Finally Jihyun kim and co-workers describe recent progress in
wide bandgap semiconductor spintronics where one can obtain room
temperature ferromagnetism and exploit the spin of the electron an addition
to its charge.
2
We thank W.de Heer, CT White, F.Liu, S.G. Louie M. Hybertsen, K.
Bolotin and P. Jarrillo- Herrero for helpful discussion. This work is
supported by ONR (No. N000150610138), FENA, DOE (No. DEFG02-
05ER46215), NSF CAREER (No. DMR- 0349232), NSEC(No. CHE-
0117752) and the New York Office of Science Technology and Academic
Research (NYSTAR). M. Han is supported by the National Science
Foundation.
We investigate electronic transport in lithographically patterned
graphically ribbon structures where the lateral confinement of charge carries
creats on energy gap near the charge neutrality point. Individual graphically
layer are contacted with metal electrodes & patterned into ribbons of varying
width and different crystallographic orientations. The temperature dependent
conductance measurement show layers energy gap opening narrower
ribbons. The size of these energy gaps are investigated by measuring the
conductance in the non-linear response regime at low temperatures. We find
that the energy gap scales inversely with the ribbon width, thus
demonstrating the ability to engineer the band gap of graphically
nanostructures by lithographic processes. In solid state physics & related
applied field, a band gap, also called an energy gap or band is an energy,
range in a solid where no electron states exist for insulator & semiconductor
3
the band gap generally refers to energy difference (in electron volts) between
top of valence band and the bottom of the conduction band; it is the amount
of energy required to free an outer shel electron from its orbit about the
nucleus to a free state.
In semiconductor Physics a material with a small, but not null or
negative, band gap (arbitrarily defined as <3ev, although some definitions
place the upper limit 4ev) is referred to as a semiconductor.
4
CHAPTER - II
THE ENERGY BAND GAPS SOME DEFINITIONS:
Band:
A range of some physical variable, as of radiation wavelength or
frequency. A range of very closely spaced electron energy levels in
solids, the distribution and nature of which determine the electrical
properties of a material.
Conduction:
The transmission or conveying of something through a medium or
passage, esp. of electric charge or heat through a conducting medium
without perceptible motion of the medium itself.
Dopant:
A small quantity of a substance, such as phosphorus, added to another
substance, such as a semiconductor, to alter the latter's properties.
Energy:
The work that a physical system is capable of doing in changing from
its actual state to a specified reference state, the total including, in
5
general, contributions of potential energy, kinetic energy, and rest
energy.
Gap:
A suspension of continuity; hiatus. A conspicuous difference;
disparity.
Model:
A tentative description of a system or theory that accounts for all of its
known properties.
Valence:
The capacity of an atom or group of atoms to combine in specific
proportions with other atoms or groups of atoms. A valence electron is
in the outer or next outer shell of an atom and can participate in
forming chemical bonds with other atoms.
THE BAND GAP ENERGY:
The band gap energy is the energy needed to break a bond in the crystal.
When a bond is broken, the electron has absorbed enough energy to leave
the valence band and "jump" to the conduction band. The width of the band
gap determines the type of material (conductor, semiconductor, insulator)
you are working with.
6
Band gap:
In condensed matter physics and related applied fields, a band gap, also
called an energy gap or bandgap, is an energy range in a solid where no
electron states exist. For insulator the band gap generally refers to the energy
difference (in electron volts) between the top of the valence band and the
bottom of the conduction band; it is the amount of energy required to free an
outer shell electron from its orbit about the nucleus to a free state.
7
CHAPTER - III
ENERGY BANDS:
Energy bands consisting of a large number of closely spaced energy levels
exist in crystalline materials. The bands can be thought of as the collection
of the individual energy levels of electrons surrounding each atom. The
wavefunctions of the individual electrons, however, overlap with those of
electrons confined to neighboring atoms. The Pauli exclusion principle does
not allow the electron energy levels to be the same so that one obtains a set
of closely spaced energy levels, forming an energy band. The energy band
model is crucial to any detailed treatment of semiconductor devices. It
provides the framework needed to understand the concept of an energy
bandgap and that of conduction in an almost filled band as described by the
empty states.
In this section, we present the free electron model and the Kronig-Penney
model. Then we discuss the energy bands of semiconductors and present a
simplified band diagram. We also introduce the concept of holes and the
effective mass.
8
FREE ELECTRON MODEL:
The free electron model of metals has been used to explain the photo-
electric effect. This model assumes that electrons are free to move within
the metal but are confined to the metal by potential barriers. The
minimum energy needed to extract an electron from the metal equals
qM, where M is the workfunction. This model is frequently used when
analyzing metals. However, this model does not work well for
semiconductors since the effect of the periodic potential due to the atoms
in the crystal has been ignored.
PERIODIC POTENTIALS:
The Kronig-Penney model: The analysis of periodic potentials is
required to find the energy levels in a semiconductor. This requires
the use of periodic wave functions, called Bloch functions which are
beyond the scope of this text. The result of this analysis is that the
energy levels are grouped in bands, separated by energy band gaps.
The behavior of electrons at the bottom of such a band is similar to
that of a free electron. However, the electrons are affected by the
presence of the periodic potential. The combined effect of the periodic
9
potential is included by adjusting the value of the electron mass. This
mass will be referred to as the effective mass.
ENERGY BANDS OF SEMICONDUCTORS:
Energy band diagrams of common semiconductors: The energy
band diagrams of semiconductors are rather complex. The detailed
energy band diagrams of germanium, silicon and gallium arsenide.
The energy is plotted as a function of the wavenumber, k, along the
main crystallographic directions in the crystal, since the band diagram
depends on the direction in the crystal. The energy band diagrams
contain multiple completely-filled and completely-empty bands. In
addition, there are multiple partially-filled band.
Simple energy band diagram of a semiconductor: The energy band
diagrams shown in the previous section are frequently simplified
when analyzing semiconductor devices. Since the electronic
properties of a semiconductor are dominated by the highest partially
empty band and the lowest partially filled band, it is often sufficient to
only consider those bands. This leads to a simplified energy band
diagram for semiconductors.
Temperature dependence of the energy bandgap: The energy
bandgap of semiconductors tends to decrease as the temperature is
10
increased. This behavior can be understood if one considers that the
interatomic spacing increases when the amplitude of the atomic
vibrations increases due to the increased thermal energy. This effect is
quantified by the linear expansion coefficient of a material. An
increased interatomic spacing decreases the average potential seen by
the electrons in the material, which in turn reduces the size of the
energy bandgap. A direct modulation of the interatomic distance -
such as by applying compressive (tensile) stress - also causes an
increase (decrease) of the bandgap.
The temperature dependence of the energy bandgap, Eg, has been
experimentally determined yielding the following expression for Eg as
a function of the temperature, T:
Doping dependence of the energy bandgap: High doping
densities cause the bandgap to shrink. This effect is explained by the
fact that the wavefunctions of the electrons bound to the impurity
atoms start to overlap as the density of the impurities increase. For
instance, at a doping density of 1018 cm-3, the average distance
between two impurities is only 10 nm. This overlap forces the
energies to form an energy band rather than a discreet level.
11
METALS, INSULATORS AND SEMICONDUCTORS:
Once we know the bandstructure of a given material we still need to find
out which energy levels are occupied and whether specific bands are
empty, partially filled or completely filled.
Empty bands do not contain electrons. Therefore, they are not expected
to contribute to the electrical conductivity of the material. Partially filled
bands do contain electrons as well as available energy levels at slightly
higher energies. These unoccupied energy levels enable carriers to gain
energy when moving in an applied electric field. Electrons in a partially
filled band therefore do contribute to the electrical conductivity of the
material.
Completely filled bands do contain plenty of electrons but do not
contribute to the conductivity of the material. This is because the
electrons cannot gain energy since all energy levels are already filled.
In order to find the filled and empty bands we must find out how many
electrons can be placed in each band and how many electrons are
available. Each band is formed due to the splitting of one or more atomic
12
energy levels. Therefore, the minimum number of states in a band equals
twice the number of atoms in the material. The reason for the factor of
two is that every energy level can contain two electrons with opposite
spin.
To further simplify the analysis, we assume that only the valence
electrons (the electrons in the outer shell) are of interest. The core
electrons are tightly bound to the atom and are not allowed to freely
move in the material.
ELECTRONS AND HOLES IN SEMICONDUCTORS:
As pointed out in metals, insulators and semiconductors,
semiconductors differ from metals and insulators by the fact that they
contain an "almost-empty" conduction band and an "almost-full" valence
band. This also means that we will have to deal with the transport of
carriers in both bands.
To facilitate the discussion of the transport in the "almost-full" valence
band of a semiconductor, we will introduce the concept of holes. It is
important to understand that one could deal with only electrons if one is
willing to keep track of all the electrons in the "almost-full" valence
13
band. After all, electrons are the only real particles available in a
semiconductor.
The concepts of holes is introduced in semiconductors since it is easier to
keep track of the missing electrons in an "almost-full" band, rather than
keeping track of the actual electrons in that band. We will now first
explain the concept of a hole and then point out how the hole concept
simplifies the analysis.
Holes are missing electrons. They behave as particles with the same
properties as the electrons would have when occupying the same states
except that they carry a positive charge.
THE EFFECTIVE MASS CONCEPT:
Electrons with an energy close to a band minimum behave as free
electrons, since the E-k relation can be approximated by a parabola. They
accelerate in an applied electric field just like a free electron in vacuum.
Their wavefunctions are periodic and extend over the size of the material.
The presence of the periodic potential, due to the atoms in the crystal
without the valence electrons, changes the properties of the electrons.
Therefore, the mass of the electron differs from the free electron mass,
m0. Because of the anisotropy of the effective mass and the presence of
14
multiple equivalent band minima, we define two types of effective mass:
1) the effective mass for density of states calculations and 2) the effective
mass for conductivity calculations.
DETAILED DESCRIPTION OF THE EFFECTIVE MASS
CONCEPT:
Introduction: The effective mass of a semiconductor is obtained by
fitting the actual E-k diagram around the conduction band minimum
or the valence band maximum by a paraboloid. While this concept is
simple enough, the issue turns out to be substantially more complex
due to the multitude and the occasional anisotropy of the minima and
maxima. In this section we first describe the different relevant band
minima and maxima, present the numeric values for germanium,
silicon and gallium arsenide and introduce the effective mass for
density of states calculations and the effective mass for conductivity
calculations.
Most semiconductors can be described as having one band minimum
at k = 0 as well as several equivalent anisotropic band minima at k
0. In addition there are three band maxima of interest close to the
valence band edge.
15
Effective mass for conductivity calculations
The effective mass for conductivity calculation is the mass, which is
used in conduction related problems accounting for the detailed
structure of the semiconductor. These calculations include mobility
and diffusion constants calculations. Another example is the
calculation of the shallow impurity levels using a hydrogen-like
model.
As the conductivity of a material is inversionally proportional to the
effective masses, one finds that the conductivity due to multiple band
maxima or minima is proportional to the sum of the inverse of the
individual masses, multiplied by the density of carriers in each band,
as each maximum or minimum adds to the overall conductivity. For
anisotropic minima containing one longitudinal and two transverse
effective masses one has to sum over the effective masses in the
different minima along the equivalent directions.
16
CHAPTER – IV
ENERGY GAP :
1. In Semi conductor Physics:
Semiconductor Band Structure
A material with a small, but not null or negative, band gap (arbitrarily
defined as < 3 eV, although some definitions place the upper limit at 4 eV) is
referred to as a semiconductor. A material with a large band gap is called an
insulator.
17
In semiconductors and insulators, electrons are confined to a number of
bands of energy, and forbidden from other regions. The term "band gap"
refers to the energy difference between the top of the valence band and the
bottom of the conduction band; electrons are able to jump from one band to
another. In order for an electron to jump from a valence band to a
conduction band, it requires a specific minimum amount of energy for the
transition. The required energy differs with different materials. Electrons can
gain enough energy to jump to the conduction band by absorbing either a
phonon (heat) or a photon (light).
The conductivity of intrinsic semiconductors is strongly dependent on the
band gap. The only available carriers for conduction are the electrons which
have enough thermal energy to be excited across the band gap.
Band gap engineering is the process of controlling or altering the band gap
of a material by controlling the composition of certain semiconductor alloys,
such as GaAlAs, InGaAs, and InAlAs. It is also possible to construct layered
materials with alternating compositions by techniques like molecular beam
epitaxy. These methods are exploited in the design of heterojunction bipolar
transistors (HBTs), laser diodes and solar cells.
The distinction between semiconductors and insulators is a matter of
convention. One approach is to think of semiconductors as a type of
18
insulator with a narrow band gap. Insulators with a larger band gap, usually
greater than 3 eV, are not considered semiconductors and generally do not
exhibit semiconductive behaviour under practical conditions. Electron
mobility also plays a role in determining a material's informal classification.
The band gap energy of semiconductors tends to decrease with increasing
temperature. When temperature increases, the amplitude of atomic
vibrations increase, leading to larger interatomic spacing. The interaction
between the lattice phonons and the free electrons and holes will also affect
the band gap to a smaller extent. The relationship between band gap energy
and temperature can be described by Varshni's empirical expression,
, where Eg(0), α and β are material
constants.
In a regular semiconductor crystal, the band gap is fixed owing to
continuous energy states. In a quantum dot crystal, the band gap is size
dependent and can be altered to produce a range of energies between the
valence band and conduction band. It is also known as quantum confinement
effect.
Band gaps also depend on pressure. Band gaps can be either direct or
indirect, depending on the electronic band structure.
19
1.1 Mathematical interpretation
Classically, the ratio of probabilities that two states with an energy
difference ΔE will be occupied by an electron is given by the Boltzmann
factor:
where:
is the exponential function
is the energy difference
is Boltzmann's constant
is temperature
At the Fermi level (or chemical potential), the probability of a state being
occupied is ½. If the Fermi level is in the middle of a band gap of 1 eV, this
ratio is e -20 or about 2.0•109 at the room-temperature thermal energy of 25.9
meV.
1.2 Photovoltaic cells
The band gap determines what portion of the solar spectrum a photovoltaic
cell absorbs. A luminescent solar converter uses a luminescent medium to
20
down convert photons with energies above the band gap to photon energies
closer to the band gap of the semiconductor comprising the solar cell.
1.3 List of band gaps
Material Symbol Band gap (eV) @ 300K
Silicon Si 1.11
Selenium Se 1.74
Germanium Ge 0.67
Silicon carbide SiC 2.86
Aluminum phosphide AlP 2.45
Aluminium arsenide AlAs 2.16
Aluminium antimonide AlSb 1.6
Aluminium nitride AlN 6.3
Diamond C 5.5
Gallium(III) phosphide GaP 2.26
Gallium(III) arsenide GaAs 1.43
Gallium(III) nitride GaN 3.4
Gallium(II) sulfide GaS 2.5 (@ 295 K)
Gallium antimonide GaSb 0.7
Indium(III) nitride InN 0.7
Indium(III) phosphide InP 1.35
Indium(III) arsenide InAs 0.36
Zinc oxide ZnO 3.37
Zinc sulfide ZnS 3.6
Zinc selenide ZnSe 2.7
21
Zinc telluride ZnTe 2.25
Cadmium sulfide CdS 2.42
Cadmium selenide CdSe 1.73
Cadmium telluride CdTe 1.49
Lead(II) sulfide PbS 0.37
Lead(II) selenide PbSe 0.27
Lead(II) telluride PbTe 0.29
Copper(II) oxide CuO 1.2
2. In photonics and phononics:
In photonics band gaps or stop bands are ranges of photon frequencies
where, if tunneling effects are neglected, no photons can be transmitted
through a material. A material exhibiting this behaviour is known as a
photonic crystal.
Similar physics applies to phonons in a phononic crystal.
22
CHAPTER – V
ENERGY (BAND) GAP: What do we mean by "allowed" and "forbidden
energies" or equivalently what is an "energy (band) gap." See the page entitled
Energy (Band) Gap.
Viewpoint #1: The gap energy is the ionization energy required to generate
two complementary charge carriers - electron and hole.
23
Viewpoint #3: The gap energy is a manifestation of the discrete character of
basic atomic energy states.
As distance between atoms gets smaller --->
24
Distance between atoms
25
Three useful references on this viewpoint from the Georgia State
University's HyperPhysics project:
Energy Bands:Insulators and Semiconductors I
Energy Bands: Insulators and Semiconductors II
Interatomic spacing and semiconductors
26
CHAPTER – VI
ELECTRONIC BAND STRUCTURE :
In solid-state physics, the electronic band structure (or simply band
structure) of a solid describes ranges of energy that an electron is
"forbidden" or "allowed" to have. It is due to the diffraction of the quantum
mechanical electron waves in the periodic crystal lattice. The band structure
of a material determines several characteristics, in particular its electronic
and optical properties.
1. Why bands occur in materials
The electrons of a single isolated atom occupy atomic orbitals, which form a
discrete set of energy levels. If several atoms are brought together into a
molecule, their atomic orbitals split, as in a coupled oscillation. This
produces a number of molecular orbitals proportional to the number of
atoms. When a large number of atoms (of order 1020 or more) are brought
together to form a solid, the number of orbitals becomes exceedingly large,
and the difference in energy between them becomes very small, so the levels
may be considered to form continuous bands of energy rather than the
discrete energy levels of the atoms in isolation. However, some intervals of
27
energy contain no orbitals, no matter how many atoms are aggregated,
forming band gaps.
Within an energy band, energy levels are so numerous as to be a near
continuum. First, the separation between energy levels in a solid is
comparable with the energy that electrons constantly exchange with phonons
(atomic vibrations). Second, it is comparable with the energy uncertainty
due to the Heisenberg uncertainty principle, for reasonably long intervals of
time. As a result, the separation between energy levels is of no consequence.
Several approaches to finding band structure are discussed below:-
2. Basic concepts
Figure 1: Simplified diagram of the electronic band structure of metals, semiconductors, and insulators.
28
Figure 2: First Brillouin zone of FCC lattice showing symmetry labels
Figure 3: More complex representation of band structure in silicon showing wavevector dependence. Going from left to right along the
horizontal, the wavevector is tracing out a particular one-dimensional path through the three-dimensional "first Brillouin zone" (see Figure
2). On the vertical axis is the energies of bands at that wavevector.
29
Any solid has a large number of bands. In theory, it can be said to have
infinitely many bands (just as an atom has infinitely many energy levels).
However, all but a few lie at energies so high that any electron that reaches
those energies escapes from the solid. These bands are usually disregarded.
Bands have different widths, based upon the properties of the atomic orbitals
from which they arise. Also, allowed bands may overlap, producing (for
practical purposes) a single large band.
Figure 1 shows a simplified picture of the bands in a solid that allows the
three major types of materials to be identified: metals, semiconductors and
insulators.
Metals contain a band that is partly empty and partly filled regardless of
temperature. Therefore they have very high conductivity.
The lowermost, almost fully occupied band in an insulator or semiconductor,
is called the valence band by analogy with the valence electrons of
individual atoms. The uppermost, almost unoccupied band is called the
conduction band because only when electrons are excited to the conduction
band can current flow in these materials. The difference between insulators
and semiconductors is only that the forbidden band gap between the valence
band and conduction band is larger in an insulator, so that fewer electrons
30
are found there and the electrical conductivity is lower. Because one of the
main mechanisms for electrons to be excited to the conduction band is due
to thermal energy, the conductivity of semiconductors is strongly dependent
on the temperature of the material.
This band gap is one of the most useful aspects of the band structure, as it
strongly influences the electrical and optical properties of the material.
Electrons can transfer from one band to the other by means of carrier
generation and recombination processes. The band gap and defect states
created in the band gap by doping can be used to create semiconductor
devices such as solar cells, diodes, transistors, laser diodes, and others.
2.1 Symmetry
A more complete view of the band structure takes into account the periodic
nature of a crystal lattice using the symmetry operations that form a space
group. The Schrödinger equation is solved for the crystal, which has Bloch
waves as solutions:
,
where k is called the wavevector, and is related to the direction of motion of
the electron in the crystal, and n is the band index, which simply numbers
the energy bands. The wavevector k takes on values within the Brillouin
31
zone (BZ) corresponding to the crystal lattice, and particular
directions/points in the BZ are assigned conventional names like Γ, Δ, Λ, Σ,
etc. These directions are shown for the face-centered cubic lattice geometry
in Figure 2.
The available energies for the electron also depend upon k, as shown in
Figure 3 for silicon in the more complex energy band diagram at the right. In
this diagram the topmost energy of the valence band is labeled Ev and the
bottom energy in the conduction band is labeled Ec. The top of the valence
band is not directly below the bottom of the conduction band (Ev is for an
electron traveling in direction Γ, Ec in direction X), so silicon is called an
indirect gap material. For an electron to be excited from the valence band to
the conduction band, it needs something to give it energy Ec − Ev and a
change in direction/momentum. In other semiconductors (for example
GaAs) both are at Γ, and these materials are called direct gap materials (no
momentum change required). Direct gap materials benefit the operation of
semiconductor laser diodes.
Anderson's rule is used to align band diagrams between two different
semiconductors in contact.
32
2.2 Band structures in different types of solids
Although electronic band structures are usually associated with crystalline
materials, quasi-crystalline and amorphous solids may also exhibit band
structures. However, the periodic nature and symmetrical properties of
crystalline materials makes it much easier to examine the band structures of
these materials theoretically. In addition, the well-defined symmetry axes of
crystalline materials makes it possible to determine the dispersion
relationship between the momentum (a 3-dimension vector quantity) and
energy of a material. As a result, virtually all of the existing theoretical work
on the electronic band structure of solids has focused on crystalline
materials.
2.3 Density of states
While the density of energy states in a band could be very large for some
materials, it may not be uniform. It approaches zero at the band boundaries,
and is generally highest near the middle of a band. The density of states for
the free electron model in three dimensions is given by,
33
2.4 Filling of bands
Although the number of states in all of the bands is effectively infinite, in an
uncharged material the number of electrons is equal only to the number of
protons in the atoms of the material. Therefore not all of the states are
occupied by electrons ("filled") at any time. The likelihood of any particular
state being filled at any temperature is given by Fermi-Dirac statistics. The
probability is given by the following expression:
where:
kB is Boltzmann's constant,
T is the temperature,
μ is the chemical potential (in semiconductor physics, this quantity is
more often called the "Fermi level" and denoted EF).
The Fermi level naturally is the level at which the electrons and protons are
balanced.
At T=0, the distribution is a simple step function:
34
At nonzero temperatures, the step "smooths out", so that an appreciable
number of states below the Fermi level are empty, and some states above the
Fermi level are filled.
3. Band structure of crystals
3.1 Brillouin zone
Because electron momentum is the reciprocal of space, the dispersion
relation between the energy and momentum of electrons can best be
described in reciprocal space. It turns out that for crystalline structures,
the dispersion relation of the electrons is periodic, and that the Brillouin
zone is the smallest repeating space within this periodic structure. For an
infinitely large crystal, if the dispersion relation for an electron is defined
throughout the Brillouin zone, then it is defined throughout the entire
reciprocal space.
4. Theory of band structures in crystals
The ansatz is the special case of electron waves in a periodic crystal lattice
using Bloch waves as treated generally in the dynamical theory of
diffraction. Every crystal is a periodic structure which can be characterized
by a Bravais lattice, and for each Bravais lattice we can determine the
reciprocal lattice, which encapsulates the periodicity in a set of three
reciprocal lattice vectors ( , , ). Now, any periodic potential
35
which shares the same periodicity as the direct lattice can be expanded out as
a Fourier series whose only non-vanishing components are those associated
with the reciprocal lattice vectors. So the expansion can be written as:
where for any set of integers (m1,m2,m3).
From this theory, an attempt can be made to predict the band structure of a
particular material, however most ab initio methods for electronic structure
calculations fail to predict the observed band gap.
4.1 Nearly-free electron approximation
In the nearly-free electron approximation in solid state physics interactions
between electrons are completely ignored. This approximation allows use of
Bloch's Theorem which states that electrons in a periodic potential have
wavefunctions and energies which are periodic in wavevector up to a
constant phase shift between neighboring reciprocal lattice vectors. The
consequences of periodicity are described mathematically by the Bloch
wavefunction:
where the function is periodic over the crystal lattice, that is,
36
.
Here index n refers to the n-th energy band, wavevector k is related to the
direction of motion of the electron, r is position in the crystal, and R is
location of an atomic site.
4.2 Tight-binding model
The opposite extreme to the nearly-free electron approximation assumes the
electrons in the crystal behave much like an assembly of constituent atoms.
This tight-binding model assumes the solution to the time-independent
single electron Schrödinger equation Ψ is well approximated by a linear
combination of atomic orbitals .
,
where the coefficients are selected to give the best approximate solution
of this form. Index n refers to an atomic energy level and R refers to an
atomic site. A more accurate approach using this idea employs Wannier
functions, defined by:
;
in which is the periodic part of the Bloch wave and the integral is over
the Brillouin zone. Here index n refers to the n-th energy band in the crystal.
37
The Wannier functions are localized near atomic sites, like atomic orbitals,
but being defined in terms of Bloch functions they are accurately related to
solutions based upon the crystal potential. Wannier functions on different
atomic sites R are orthogonal. The Wannier functions can be used to form
the Schrödinger solution for the n-th energy band as:
4.3 KKR model
The simplest form of this approximation centers non-overlapping spheres
(referred to as muffin tins) on the atomic positions. Within these regions, the
potential experienced by an electron is approximated to be spherically
symmetric about the given nucleus. In the remaining interstitial region, the
potential is approximated as a constant. Continuity of the potential between
the atom-centered spheres and interstitial region is enforced.
A variational implementation was suggested by Korringa and by Kohn and
Rostocker, and is often referred to as the KKR model.
4.4 Order-N spectral methods
To quote RP Martin: "The concept of localization can be imbedded directly
into the methods of electronic structure to create algorithms that take
advantage of locality … For large systems, this fact can be used to make
38
"order-N" or O(N) methods where the computational time scales linearly in
the size of the system"
4.5 Density-functional theory
In recent physics literature, a large majority of the electronic structures and
band plots are calculated using density-functional theory (DFT), which is
not a model but rather a theory, i.e., a microscopic first-principles theory of
condensed matter physics that tries to cope with the electron-electron many-
body problem via the introduction of an exchange-correlation term in the
functional of the electronic density. DFT-calculated bands are in many cases
found to be in agreement with experimentally measured bands, for example
by angle-resolved photoemission spectroscopy (ARPES). In particular, the
band shape is typically well reproduced by DFT. But there are also
systematic errors in DFT bands when compared to experiment results. In
particular, DFT seems to systematically underestimate by about 30-40% the
band gap in insulators and semiconductors.
It must be said that DFT is, in principle an exact theory to reproduce and
predict ground state properties (e.g., the total energy, the atomic structure,
etc.). However, DFT is not a theory to address excited state properties, such
as the band plot of a solid that represents the excitation energies of electrons
injected or removed from the system. What in literature is quoted as a DFT
39
band plot is a representation of the DFT Kohn-Sham energies, i.e., the
energies of a fictive non-interacting system, the Kohn-Sham system, which
has no physical interpretation at all. The Kohn-Sham electronic structure
must not be confused with the real, quasiparticle electronic structure of a
system, and there is no Koopman's theorem holding for Kohn-Sham
energies, as there is for Hartree-Fock energies, which can be truly
considered as an approximation for quasiparticle energies. Hence, in
principle, DFT is not a band theory, i.e., not a theory suitable for calculating
bands and band-plots.
4.6 Green's function methods and the ab initio GW approximation
To calculate the bands including electron-electron interaction many-body
effects, one can resort to so-called Green's function methods. Indeed,
knowledge of the Green's function of a system provides both ground (the
total energy) and also excited state observables of the system. The poles of
the Green's function are the quasiparticle energies, the bands of a solid. The
Green's function can be calculated by solving the Dyson equation once the
self-energy of the system is known. For real systems like solids, the self-
energy is a very complex quantity and usually approximations are needed to
solve the problem. One such approximation is the GW approximation, so
called from the mathematical form the self-energy takes as the product Σ =
40
GW of the Green's function G and the dynamically screened interaction W.
This approach is more pertinent when addressing the calculation of band
plots (and also quantities beyond, such as the spectral function) and can also
be formulated in a completely ab initio way. The GW approximation seems
to provide band gaps of insulators and semiconductors in agreement with
experiment, and hence to correct the systematic DFT underestimation.
4.7 Mott insulators
Although the nearly-free electron approximation is able to describe many
properties of electron band structures, one consequence of this theory is that
it predicts the same number of electrons in each unit cell. If the number of
electrons is odd, we would then expect that there is an unpaired electron in
each unit cell, and thus that the valence band is not fully occupied, making
the material a conductor. However, materials such as CoO that have an odd
number of electrons per unit cell are insulators, in direct conflict with this
result. This kind of material is known as a Mott insulator, and requires
inclusion of detailed electron-electron interactions (treated only as an
averaged effect on the crystal potential in band theory) to explain the
discrepancy. The Hubbard model is an approximate theory that can include
these interactions.
41
4.8 Others
Calculating band structures is an important topic in theoretical solid state
physics. In addition to the models mentioned above, other models include
the following:
The Kronig-Penney Model, a one-dimensional rectangular well model
useful for illustration of band formation. While simple, it predicts
many important phenomena, but is not quantitative.
Bands may also be viewed as the large-scale limit of molecular orbital
theory. A solid creates a large number of closely spaced molecular
orbitals, which appear as a band.
Hubbard model
The band structure has been generalised to wavevectors that are complex
numbers, resulting in what is called a complex band structure, which is of
interest at surfaces and interfaces.
Each model describes some types of solids very well, and others poorly. The
nearly-free electron model works well for metals, but poorly for non-metals.
The tight binding model is extremely accurate for ionic insulators, such as
metal halide salts (e.g. NaCl).
42
CHAPTER – VII
THE ENERGY BAND MODEL:
Material Classification: Crystalline materials can be classified according to
their bandgap.
An insulator is a poor conductor since it requires a lot of energy, 5-8 eV, to
excite the electrons enough to get to the conduction band. We can say that
the width of the band gap is very large, since it requires that much energy to
traverse the band gap, and draw the band diagram respectively.
A metal is an excellent conductor because, at room temperature, it has
electrons in its conduction band constantly, with little or no energy being
applied to it. This may be because of its narrow or nonexistent band gap, the
conduction band may be overlapping the valence band so they share the
electrons. The band diagram would be drawn with Ec and Ev very close
together, if not overlapping.
The reason semiconductors are so popular is because they are a medium
between a metal and an insulator. The band gap is wide enough to where
current is not going through it at all times, but narrow enough to where it
does not take a lot of energy to have electrons in the conduction band
creating a current.
43
CHAPTER – VII
ENERGY BAND-GAP ENGINEERING OF GRAPHENE
NANORIBBONS:
The recent discovery of graphene, a single atomic sheet of graphite,
has ignited intense research activities to elucidate the electronic
properties of this novel twodimensional (2D) electronic system.
Charge transport in graphene is substantially different from that of
conventional 2D electronic systems as a consequence of the linear
energy dispersion relation near the charge neutrality point
(Dirac point) in the electronic band structure. This unique band
structure is fundamentally responsible for the distinct electronic
properties of carbon nanotubes (CNTs) .
When graphene is patterned into a narrow ribbon, and the
carriers are confined to a quasi-one-dimensional (1D) system, we
expect the opening of an energy gap. Similar to CNTs, this energy
gap depends on the width and crystallographic orientation of the
graphene nanoribbon (GNR). However, despite numerous recent
theoretical studies, the energy gap in GNRs has yet to be
investigated experimentally.
44
In this Letter, we present electronic transport measurements of
lithographically patterned GNR structures where the lateral
confinement of charge carriers creates an energy gap. More than
two dozen GNRs of different widths and crystallographic orientations
were measured. We find that the energy gap depends strongly on
the width of the channel for GNRs in the same crystallographic
direction, but no systematic crystallographic dependence is
observed. The GNR devices discussed here are fabricated from
single sheets of graphene which have been mechanically extracted
from bulk graphite crystals onto a SiO2=Si substrate. Graphene
sheets with lateral
sizes of _20 _m are contacted with Cr=Au (3=50 nm) metal
electrodes. Negative tone e-beam resist, hydrogen silsesquioxane
(HSQ), is then spun onto the samples and patterned to form an etch
mask defining nanoribbons with widths ranging from 10–100 nm
and lengths of 1–2 _m. An oxygen plasma is introduced to etch
away the unprotected graphene, leaving the GNR protected beneath
the HSQ mask [Fig. (a)].
In this Letter, we study two different types of device sets:
device sets P1-P4 each contain many ribbons of varying width
running parallel [Fig. (b)], and a device sets D1 and D2 have ribbons
of uniform width and varying relative orientation [Fig. (c)]. In either
45
case, each device within a given set is etched from the same sheet
of graphene, so that the relative orientation of the GNRs within a
given set is known.
We remark that each GNR connects two blocks of wider (_0:5 _m)
graphene, which are in turn contacted by metal electrodes. Thus,
unlike CNTs, Schottky barrier formation by the metal electrodes is
absent in our GNR devices. Furthermore, multiple contacts on the
wider block of grapheme allow for four-terminal measurements in
order to eliminate the residual contact resistance (_1 k_). A heavily
doped silicon substrate below the 300 nm thick SiO2 dielectric layer
serves as a gate electrode to tune the carrier density in the GNR.
The width (W) and the length of each GNR were measured using a
scanning electron microscope (SEM) after the transport
measurements were performed.
46
Figure: (color online). (a) Atomic force microscope image of GNRs in set P3 covered by a protective HSQ
etch mask. (b) SEM image of device set P1 with parallel GNRs of varying width. (c) SEM image of
device set D2 containing GNRs in different relative crystallographic directions with uniform width (e)– (f ) Conductance of GNRs in device set P1 as a function of gate voltage measured at different temperatures. The
width of each GNR is designated in each panel.
47
Figure: (color online). Conductance vs width of parallel GNRs (set P4) measured at Vg _ VDirac __50 V at three representative temperatures. The square and
triangle symbols correspond to T _ 300, and 1.6 K, respectively. Dashed lines represent the linear fits at each temperature. The insets show the conductivity (upper) and the inactive GNR width (lower) obtained
from the slope and x intercept of the linear fit at
48
varying temperatures. Dashed curves are shown in the insets as a guide to the eye.
Since the HSQ protective layer was not removed from the GNR for
this imaging, this measurement provides an upper bound to the true
width of the GNR.
The conductance G of the GNRs was measured using a
standard lock-in technique with a small applied ac voltage ( < 100
_V@8 Hz). Figure (d)–1(f) shows the measured G of three
representative GNR devices of varying width (W _ 24 _ 4, 49 _ 5, and
71 _ 6 nm) and uniform length (L _ 2 _m) as a function of gate
voltage Vg at different temperatures. All curves exhibit a region of
depressed G with respect to Vg. In ‘‘bulk‘‘ (i.e., unpatterned)
graphene, this dip in G is well understood and corresponds to the
minimum conductivity _4e2=h at the charge neutrality point, Vg _
VDirac, where e and h are the electric charge and Planck constant,
respectively. At room temperature, our GNRs exhibit qualitatively
similar G_Vg_ behaviors, showing a minimum conductance Gmin on
the order of 4e2=h_W=L_. Unlike the bulk case, GNRs with width
W<100 nm show a decrease in Gmin of more than an order of
magnitude at low temperatures. The narrowest GNRs show the
greatest suppression of Gmin. For example, for the GNR with W _ 24
_ 4 nm [Fig. 1(d)], a large ‘‘gap’’ region appears for 25<Vg < 45 V,
49
where Gmin is below our detection limits (<10_8 __1). This strong
temperature dependence of G_Vg_ in GNRs is in sharp contrast to
that of the ‘‘bulk’’ graphene samples where Gmin changes less than
30% in the temperature range 30 mK–300 K. The suppression of G
near the charge neutrality point suggests the opening of an energy
gap. We observe [Fig. 1(d)–1(f)] stronger temperature dependence
of G for a broader range of Vg values in narrower GNRs, suggesting
larger energy gaps in narrower GNRs.
Outside of the ‘‘gap’’ region near the Dirac point, the
conductance scales with the width of the GNR. Figure: shows the
conductance of a set of parallel GNRs, with widths ranging from 14–
63 nm, measured at two temperatures, T _ 1:6 and 300 K. The gate
voltage is fixed at Vg _ VDirac _ 50 V, which corresponds to a hole
carrier density of n _ 3:6 _ 1012 cm_2. The conductance is well
described by the linear fit G _ __W _ W0_=L (dashed line). Here _ and
W _ W0 can be interpreted as the GNR sheet conductivity and the
active GNR width participating in charge transport, respectively. The
sheet conductivity is _1:7 mS and decreases with decreasing
temperature, reaching _75% of the room temperature value at T _
1:6 K . The inactive GNR width W0 increases from 10 nm at room
temperature to 14 nm at 1.6 K. A reduced active channel width was
initially reported in GNRs fabricated on epitaxial multilayer
50
graphene films, where much larger inactive edges (W0 _ 50 nm)
were estimated than for our GNR samples. We suggest two possible
explanations for the finite W0 measured in our experiment: (i)
contribution from localized edge states near the GNR edges due to
structural disorder caused by the etching process and (ii) inaccurate
width determination due to over-etching underneath the HSQ etch
mask. To investigate this, we removed the HSQ etch mask from
several GNRs and found that the actual GNR is often _10 nm
narrower than the HSQ protective mask. This suggests that the
inactive region due to the localized edge states is small (<2 nm) at
room temperature and spreads to as much as _5 nm at low
temperatures.
We now discuss the quantitative scaling of the energy gap as
a function of GNR width. By examining the differential conductance
in the nonlinear response regime as a function of both the gate and
bias voltage, we can directly measure the size of the energy gap.
Figure (a) shows a schematic energy band diagram for a GNR with
source and drain electrodes. As the bias voltage, Vb, increases, the
source and drain levels approach the conduction and valence band
edges, respectively. When conduction (valence) band edge falls into
the bias window between the source and drain electrodes, electrons
(holes) are injected from source (drain) and the current I rises
51
sharply. The gate voltage adjusts the position of the gap relative to
the source-drain levels. Figure (b), (d) shows the conductance
versus Vg and Vb for three representative GNR devices of different
width measured at T _ 1:6 K. The color indicates conductivity on a
logarithmic scale, with the large dark area in each graph
representing the turnedoff region in the Vg-Vb plane where the both
band edges are outside of the bias windows. The diamond shape of
this region indicates that both Vb and Vg adjust the position of the
band edges relative to the source and drain energy levels,
analogous to nonlinear transport in quantum dots . As designated
by the arrows, the GNR band gap Egap can be directly obtained from
the value of Vb at the vertex of the diamond.
In order to obtain the quantitative scaling of Egap with respect
to W, we now plot E_1 gap against W in Fig.(e) for a set of 13 parallel
GNRs. The dashed line indicates a linear fit to the data,
corresponding to Egap _ _=_W _ W__, where we obtained _ _ 0:2
eVnm and W_ _ 16 nm from the fit. Recent density functional theory
studies predict that the energy gap of a GNR scales inversely with
the channel width, with a corresponding _value ranging between
0:2–1:5 eVnm, which is consistent with this observation. We also
note that W_ W0, in good agreement with the independent
estimation of the GNR edge effects above.
52
A similar scaling behavior holds even across GNR device sets
running in different crystallographic directions. Figure shows the
overall scaling of Egap as a function of W for six different device sets.
Four device sets (P1–P4) have parallel GNRs with W ranging from
15–90 nm, and two device sets (D1, D2) have GNRs with similar W
but different crystallographic directions. The energy gap behavior of
all devices is well described by the scaling Egap _ _=_W _ W__ as
discussed above, indicated by the dashed line. Remarkably, energy
gaps as high as _200 meV are achieved by engineering GNRs as
narrow as W _ 15 nm. Based on the empirical scaling determined
here, a narrower GNR may show an even larger band gap, making
the use of GNRs for semiconducting device components in ambient
conditions a possibility.
Finally, we remark on the crystallographic directional
dependence of Egap. The inset to Fig. 4 shows Egap versus the
relative orientation angle _ for two sets of GNRs. In principle, we
expect Egap___ for each set to be periodic in _, provided all GNRs in
the set have similar edge structures. However, experimental
observation shows randomly scattered values around the average Egap
corresponding to W with no sign of crystallographic directional dependence.
This suggests that the detailed edge structure plays a more important role
than the overall crystallographic direction in determining the properties of
53
the GNRs. Indeed, theory for ideal GNRs predicts that Egap depends
sensitively on the boundary conditions at the edges. The lack of directional
dependence indicates that at this point our device fabrication process does
not give us the atomically precise control of the GNR edges necessary to
reveal this effect. The interplay between the precise width, edge orientation,
edge structure, and chemical termination of the edges in GNRs remains a
rich area for future research.
Figure: (color online). (a) Schematic energy band diagram of a GNR with bias voltage Vb applied. The current Iis controlled by both source-drain bias Vb and gate voltage Vg. (b)–(d) The differential conductance (dI=dVb) of three representative GNRs from set P4 with W _ 22, 36, and
48 nm as a function of Vb and Vg measured at T _1:6 K. The light
54
(dark) color indicates high (low) conductance as designated by the color map. The horizontal arrows represent Vb _ Egap=e. (e) E_1 gap vs W obtained from similar analysis as (b)– (d), with a linear fit of the data.
55
In conclusion, we demonstrate that the energy gap in patterned graphene
nanoribbons can be tuned during fabrication with the appropriate choice of
ribbon width. An understanding of ribbon dimension and orientation as
control parameters for the electrical properties of grapheme structures can be
seen as a first step toward the development of graphene-based electronic
devices.
Figure: (color online). Egap vs W for the 6 device sets considered in this study: four (P1–P4) of the parallel type and two (D1,D2) with varying
orientation. The inset shows Egap vs relative angle _ for the device sets D1 and D2. Dashed lines in the inset show the value of Egap as
predicted by the empirical scaling of Egap vs W.
56
RESULTS
Figure : Semiconductor Band Structure
57
Figure: As distance between atoms gets smaller --->
58
Figure: Distance between atoms
59
Figure : Simplified diagram of the electronic band structure of metals, semiconductors, and insulators.
60
Figure: First Brillouin zone of FCC lattice showing symmetry labels
Figure: More complex representation of band structure in silicon showing wavevector dependence. Going from left to right along the
horizontal, the wavevector is tracing out a particular one-dimensional path through the three-dimensional "first Brillouin zone" (see Figure
2). On the vertical axis is the energies of bands at that wavevector.
61
Figure: (color online). (a) Atomic force microscope image of GNRs in set P3 covered by a protective HSQ
etch mask. (b) SEM image of device set P1 with parallel GNRs of varying width. (c) SEM image of
device set D2 containing GNRs in different relative crystallographic directions with uniform width (e)– (f ) Conductance of GNRs in device set P1 as a function of
62
gate voltage measured at different temperatures. The width of each GNR is designated in each panel.
63
Figure: (color online). Conductance vs width of parallel GNRs (set P4) measured at Vg _ VDirac __50 V at three representative temperatures. The square and
triangle symbols correspond to T _ 300, and 1.6 K, respectively. Dashed lines represent the linear fits at each temperature. The insets show the conductivity (upper) and the inactive GNR width (lower) obtained
from the slope and x intercept of the linear fit at varying temperatures. Dashed curves are shown in
the insets as a guide to the eye.
64
Figure: (color online). (a) Schematic energy band diagram of a GNR with bias voltage Vb applied. The current Iis controlled by both source-drain bias Vb and gate voltage Vg. (b)–(d) The differential conductance (dI=dVb) of three representative GNRs from set P4 with W _ 22, 36, and
48 nm as a function of Vb and Vg measured at T _1:6 K. The light (dark) color indicates high (low) conductance as designated by the color
map. The horizontal arrows represent Vb _ Egap=e. (e) E_1 gap vs W obtained from similar analysis as (b)– (d), with a linear fit of the data.
65
Figure: (color online). Egap vs W for the 6 device sets considered in this study: four (P1–P4) of the parallel type and two (D1,D2) with varying
orientation. The inset shows Egap vs relative angle _ for the device sets D1 and D2. Dashed lines in the inset show the value of Egap as
predicted by the empirical scaling of Egap vs W.
66
REFERENCES
Hilmi, Unlu. A Thermodynamic Model for Determining Pressure and
Temperature Effects on the Bandgap Energies and other Properties of
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a b c d e f g h i j k l m n o p q r s t Streetman, Ben G.; Sanjay Banerjee (2000).
Solid State electronic Devices (5th edition ed.). New Jersey: Prentice
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J. Wu, W. Walukiewicz, H. Lu, W. Schaff, et. al.; Unusual Properties of
the Fundamental Bandgap of InN; Appl. Phys. Lett., 80, 3967 (2002).
Madelung, Otfried (1996). Semiconductors - Basic Data (2nd rev. ed.
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60883-4.
P. Ordejon Order-N tight-binding methods for electronic-structure and
molecular dynamics\
Because of difficulties in removing the HSQ etch mask and
imaging the underlying ribbon without damage, we were
67
able to measure the actual width of only a few GNRs
studied in this experiment.
See, e.g. L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen,
S. Tarucha, R. M. Westervelt, and N. S. Wingreen, in
Mesoscopic Electron Transport (Plenum, New York, 1997).
Although the relative crystallographic directions between
GNRs in a same graphene flake can be controlled by the
proper lithography processes, the absolute
crystallographic direction relative to the graphene lattice
is unknown.
68