Chapter-6

Ab-initio Vibrational Dynamics of Nano-

structure

Ab-initio Vibrational Dynamics of Nanostructure Chapter-6

150

6.1 Introduction

Significant number of consumer products based on nanotechnology is entering

the market and large quantities of nanostructures are being produced annually. We

already know from experiments that the undesirable properties of nanomaterials

depend on, and are moderated by a wide range of physical parameters such as size,

shape, chemical composition or degree of agglomeration [1-2]. It is also known that

many of these dependencies are linked and we must take this knowledge into account

before we make predictions. Fortunately, this is an area where computational materials

scientists have an advantage because, unlike experimental scientists one can control

each of these critical parameters independently and underlying mechanisms. It is also

possible to investigate materials in highly non-equilibrium environments such as

electric or magnetic fields that are not possible experimentally. Besides, computational

materials science is an inter-disciplinary research area of physics, chemistry and

scientific computing [3-4]. In order to understand the underlying properties of real

materials and this approach which is not readily accessible in laboratory experiments,

may prove to be quite useful in microscopic elucidation of material properties. Another

aspect and goal of study in computational materials is to assist in the prediction of new

materials with technologically useful applications. The development of new and

efficient algorithms and analytical method for investigating properties of simple and

complex material forms an equally important research area in computational material

science, With respect to theoretical tools of computational material science; one can

basically divide them into two groups, empirical and ab-initio (first principles). While

empirical methods are based on classical and quantum mechanical modeling using

various functional forms with adjustable parameters fitted to experimental

Ab-initio Vibrational Dynamics of Nanostructure Chapter-6

151

observations, the ab-initio methods employ quantum mechanical modeling with no

adjustable parameters and few well justified and tested approximations, and the atomic

numbers of the constituent atoms as input parameters. The ab-initio methods offer a

high level of accuracy in understanding physical properties of materials, but compared

to empirical methods these are computationally costly but most important class of such

methods are the ab-initio density functional theoretical methods.

In this work we concentrate on electronic and vibrational properties of zinc

oxide (ZnO) nanowire and silver (Ag) clusters using ab-initio density functional

theoretical (DFT) calculations. We have chosen materials from both semiconductor

and metal categories with one being in wire form while other in cluster form. This

work has been motivated by the recent observations of significant different properties

particularly to clarify the exact nature of the low frequency enhancement of the VDOS

remains still unclear in both classes of materials. In addition, this is a preliminary study

to link the unique properties of nanowire and nanocluster to the vibrational properties

discussed in previous chapters using a computational methodology of present days. In

particular, to understand the low frequency enhancement in vibrational density of

states (VDOS) of low dimensional structures, when compared with their coarse-

grained counterparts. It is found that the grain-boundary components of the VDOS

exhibits g(ω) nw dependence rather than the usual quadratic dependence in the low

frequency limit. Density functional theory is an extremely successful approach for the

description of ground state properties of metals, semiconductors, and insulators. The

success of DFT not only encompasses standard bulk materials but also complex

materials such as proteins and carbon nanotubes. The main idea of DFT is to describe

an interacting system of fermions via its density and not via its many-body wave

function. For N electrons in a solid, which obey the Pauli principle and repulse each

Overview of DFT Calculations Chapter-6

152

other via the Coulomb potential, this means that the basic variable of the system

depends only on three - the spatial coordinates x, y, and z - rather than 3*N degrees of

freedom.

The DFT which is based on approximations for the so called exchange

correlation potential, in principle gives a good description of ground state properties.

The exchange-correlation potential describes the effects of the Pauli Exclusion

Principle and the Coulomb potential beyond a pure electrostatic interaction of the

electrons. Possessing the exact exchange-correlation potential means that we solve the

many-body problem exactly, this is clearly not feasible in solids. The DFT is one of the

most popular and versatile quantum mechanical modeling methods to determine the

electronic structure of many-body systems. The name comes from the fact that it uses

the functional of the electron density. DFT has been a popular solid-state physics

calculation method since the 1970s. However, because of the approximations used in

the theory for exchange and correlation interactions, it was not considered accurate

enough, until the 1990s, when better model for the approximations were developed

along with the availability of powerful computer and algorithms.

6.2 Overview of DFT Calculations

In many-body electronic structure calculations, an electronic state is usually described

by a wavefunction Ψ that satisfies the many-electron time-independent Schrӧdinger

equation, provided the Born-Oppenheimer approximation is used to separate at the

ionic coordinates. The Schrӧdinger equation can be expressed as:

ψU+V+T=Eψ=ψH (6.1)

Where Ĥ is the Hamiltonian, E is the total energy, which contains T, the kinetic

energy; V, the local, one-particle potential energy from the external field; U, the

Overview of DFT Calculations Chapter-6

153

electron-electron interaction energy. This many-particle equation cannot be converted

into simpler single-particle equations because of the interaction term U and thus

making hard to be solved. The solving methods very likely consume a huge load of

computational effort. This is where DFT comes to our rescue. DFT is able to

systematically map the many-body problem, with U, onto a single-body problem

without U. The basic grounding of DFT is two Hohenberg-Kohn theorems [5]. The

first one describes that a many-electron system can be uniquely defined by an electron

density n( r

), which means a many-electron wave function Ψ, is a unique functional of

n( r

). The second one says the correct ground state electron density minimizes the

system‘s energy functional. Using Kohn-Sham approach [6], which is the essential part

of DFT, the interacting electrons in a static external potential was reduced to non-

interacting, fictitious particles moving in an effective potential eff , which can be

expressed as:

rφε=rφrυ+ iiieff

2

2

2m, with (6.2)

rυ+rυ+rυ=rυ xcHexteff

(6.3)

rext

is the external potential, rH

is the Hartee (or Coulomb) potential,

and rxc

is the exchange-correlation potential. The last term contains all the

complexities of the many-electron system, and it is the only unknown functional in the

Kohn-Sham approach, which needs to be approximated. A widely-used approximation

method for the exchange-correlation is called local density approximation (LDA) [7],

whose functional depends only on the electron density. The functional can be written

as [7]:

rdnεrn=nE xc

LDA

XC

(6.4)

ZnO Nanowire Chapter-6

154

And we have the relation:

rδn

nδE=rV

LDA

xcxc

(6.5)

In the above equation, xc is the exchange-correlation energy density. The fact that the

approximation method works is largely because that the exchange-correlation energy

only takes as low as 3% of the total energy. However, in calculating semiconductor‘s

band gap using DFT-LDA method, a discrepancy of about 30% to 50% is generally to

be expected.

Another approximation is called generalized gradient approximation (GGA),

which accounts in the gradient of the density. The functional can be written as

following equation:

rdnnrnnE xc

LDA

XC

,

(6.6)

GGA also results in a band gap calculation discrepancy. Nevertheless, they both give much

more accurate ground-state properties, such as the structural properties, where typically the

result is within 1% to 3%.

6.3 ZnO Nanowire

Zinc oxide (ZnO) is an ionic semiconductor that has a wide range of

technological applications ranging from its use as a white pigment to its use in rubber

industry, where it shortens the time of vulcanization, through its applications in

catalysis and gas sensing systems [8]. Hexagonal ZnO, a typical wide bandgap

(Eg=3.37 eV) II-VI semiconductor with lattice spacing a and c =0.325 and 0.521 nm

respectively has recently attracted the most intensive research for many properties and

potential applications in building optical and optoelectronic nanodevices [9-12]. ZnO

nanostructures such as nanowire, nanotubes, nanobelts, nanosheets nanoparticles and

nanorods are promising building blocks for optical and electronic devices because of

Results and Discussion Chapter-6

155

the potential applications in the fields of blue-light emitting short-wavelength laser

diodes solar cells, surface acoustic wave devices and chemical and biological sensors

[12-16]. An important part of the development of many nano-devices is the thermal

design. Thermal properties like heat capacity and thermal conductivity along with

several other properties are thoroughly influenced by the phonon properties

particularly the vibrational density of states. This makes an important area of research

to understand the laws governing the vibrational properties of nanostructured materials

from high technological and fundamental point of view. Further, the low dimensional

structure raises fundamental question to the localized and discrete nature of the

electronic and vibrational states due to increasing surface/volume ratio [17-18]. To our

knowledge, no systematic studies on the dimension and size dependent properties of

ZnO nanowires (NWs) have been made using ab-initio calculations. In particular, the

lattice vibrational modes and vibrational DOS in ZnO NWs are still unclear.

In the present work, we present ab-initio calculation of the phonon properties

such as dispersion curves, VDOS and specific heat for a thin ZnO NW and compared

with its bulk counterpart. For the sake of completeness, we also present the electronic

band structure and try to link with phonons.

6.3.1 Results and Discussion

We have performed the phonon and electronic properties calculations for the

ZnO NW using density functional theory with the GGA using Perdew–Burke–

Ernzerhof parameterization of exchange correlation functional as implemented in the

Quantum Espresso code [19]. These calculations are performed in a super cell

structures using a plane-wave basis [20].

Result and Discussion Chapter-6

156

The vacuum region between neighboring NWs is chosen to be about ~ 1 nm in

order to avoid spurious interaction, and the plane wave energy cutoff is 40 Ry. For the

Brillouin zone (BZ) integration we use a 10x10x10 sampling mesh. Phonon

calculations are performed using density functional perturbation theory (DFPT) [21].

The computational parameters considered in the present calculations were sufficient in

leading to well converged total energy, geometrical configurations, elastic moduli and

phonons. All structures are fully relaxed before further calculations.

First, we construct the geometrical configuration of ZnO nanowires. In our

case, ZnO nanowires have infinite length along the [001] direction the c axial. ZnO

nanowires are placed in unit cells where the inter wire distance is larger than 5 Å,

Figure-6.1: Relaxed structure of ZnO NWs with diameter (a) 3.2 (b) 5.6 Å

Result and Discussion Chapter-6

157

-6

-3

0

3

6

9

HLMKHA

En

erg

y (

eV

)

Ef

which effectively prevents the interaction effect from neighboring cells.Note that, since

these nanostructures are often synthesized at the high temperatures, the surface

passivation is not considered in our calculations. Figure 6.1 depicts the relaxed

structures of a ZnO NWs along the [001] direction with a diameter of 3.2 and 5.6 Å. In

order to test the accuracy of the present ab-initio calculation to the electronic structures

of the ZnO structure, we firstly calculate the optimized structure and self consistent

band structure of wurzite ZnO crystal. We found that the optimized lattice constants a=

3.359Å and c= 5.21 Å for ZnO crystal agree well with the experimental values

(a=3.249 Å and c= 5.204 Å). The band structure of bulk ZnO were calculated along

lines connecting high symmetry points in the Brillouin zone (BZ) and displayed in Fig

6.2. Our results show that both the top of the valence band and the bottom of the

conduction band are located at the Γ- point (k=0), which indicates the presence of

direct band gap in wurzite ZnO. The band gap of ZnO in the present study is 0.8eV,

which is smaller

than that by

experiments but

better than the

energy gap 0.88

eV obtained by

LDA [22].

It is an

established fact

that the both

GGA and LDA calculations always underestimate the energy gap. The calculated band

structure of two ZnO nanowires are shown in Fig 6.3. For the ZnO NW of 5.6 Å we

Figure- 6.2: Electronic band structure for Bulk ZnO. Fermi

level is set to zero

Result and Discussion Chapter-6

158

plot band structure along only along Γ-X line of the BZ. Clearly, the ZnO NWs are

semiconductor due to the existence of an energy gap between the valence and

conduction bands. The total DOS of the two ZnO NWs with bulk ZnO is plotted in Fig

6.4. An analysis of DOS alongwith the band structure indicates that conduction bands

originate mainly from the contribution of oxygen atoms. The valence band above -10

eV and -17 eV in the case of 3.2 and 5.6 Å ZnO NWs come mainly from the Zn 3d

orbitals. The band structure shows that these ZnO NWs have direct band gap similar to

the ZnO bulk. However, the band gap increases with the decrease in diameter of

nanowires. This is due to the increase in surface atoms, which have main contribution

from oxygen 2p line dangling bonds [12]. In addition the interactions among electric

charges result in the delocalized characters and the electrons have greater mobility

along these surfaces.

Figure-6.3: Electronic band structure along the selected line for

ZnO-NW with diameter 3.21 Å (b) ZnO-NW with diameter 5.6 Å.

Fermi level is set to zero in both the case.

Result and Discussion Chapter-6

159

-30 -20 -10 0 10 20

Energy (eV)

5.6Å

bulk

D

OS

(a

.u.)

3.2Å

Figure-6.4: Density of states of bulk ZnO along with the

ZnO NWs with diameters 3.2 and 5.6 Å

0

100

200

300

400

500

600

LAK

Figure-6.5: Phonon dispersion curves for Bulk ZnO in

wurzite structure

Result and Discussion Chapter-6

160

ZnO NWs shows strong ionic characters rather than a covalent character. The

pd hybridizations present in ZnO NWs, indicating that the ZnO NWs are the mixed

bonding semiconductor material with ionic bond much stronger than the covalent

bond. We can also see that there is only a very weak overlap of electron density

between outer layer and inner layer atoms, but the overlap of the same layer atoms is

very strong. From the density of states, the highest occupied state is mainly composed

of O-2p states, while the lowest unoccupied state is mostly Zn-4s states. Contributions

of the Zn-3d states to the valence top maximum can be seen.

Now, we turn our attention to the phonon properties of ZnO NWs. For this

initially we have calculated the phonon dispersion curves for bulk ZnO and presented

them in Fig 6.5. All general features of phonon dispersion curves of ZnO in wurzite

structure are present and there is a good agreement with experimental [23-24] and

theoretical data [25]. The phonon frequencies throughout the BZ are positive and

confirm the high quality phonon calculation and dynamical stability of the considered

0

200

400

600

0

20

40

60

80

100

200

300

400

500

Fre

qq

ue

ncy (

cm

-1)

q

ZnO-NW-1

q

q

Figure-6.6: Phonon dispersion curve for ZnO NW with diameter

3.2 Å

Result and Discussion Chapter-6

161

structure. The wurzite ZnO belongs to the space group P63mc or C6v4 with two formula

units in the primitive cell. Each primitive cell of ZnO has four atoms. There is an

overall good agreement with the available experimental data [23-24] except at few

points such the phonon frequencies corresponding to the longitudinal optic modes at Γ,

the splitting between them LOE1 and the

LOA1 . Fig. 6.6 presents the phonon dispersion

curve of ZnO NW of diameter 3.2 Å with 20 ZnO molecules in a unit cell. As there are

20 atoms in the unit cell there are 60 phonon branches. For clarity of the behavior of

phonon branches they are enlarged and shown in the right panels of Fig 6.6. Since the

ZnO NW has a lower symmetry than that of bulk ZnO, the degeneracy of many

phonon modes is lifted. In addition, quantum confinement introduces modes that do

not exist in the bulk. One can observe that the vibrational modes resulting from

confinement normally lies between the middle region of the acoustic and optical

phonon regions but increases the gap.

0.00

0.05

0.10

0 100 200 300 400 500 6000.0

0.1

0.2

0.3

0.4

PD

OS

(a

rb.

un

its

)

bulk

Frequency (cm-1)

112.45 cm-1

ZnO-20

117.35 cm-1

Figure-6.7: Vibrational density of state of bulk and ZnO NW with

diameter 3.2 Å

Result and Discussion Chapter-6

162

Fig 6.7 presents the vibrational density of states (VDOS) of ZnO NW of 3.2 Å. For

comparison the VDOS of bulk is also presented alongwith the VDOS of ZnO

nanowire, compared with the bulk counterpart, several obvious changes could be

observed. Essentially three major features can be observed, namely the red shifting of

low frequency region, weakening and broadening of several peaks and leading to

asymmetrical tail in optical region. The enhancement of the modes at lower frequency

which can be directly attributed to the features observed in acoustic phonon region of

the phonon dispersion curves. We observe a clear broadening of the peak at 150 cm-1

.

The gap between acoustic and optical region increases in the case of NW. The peaks

show discrimination in the case of NW. In the high frequency range the VDOS of ZnO

NW show distribution of peaks in the range 400-575 cm-1

with enhanced intensity.

The temperature variation of constant-volume lattice-specific heat Cv presented

in Fig. 6.8 depicts an increase of Cv with temperature until 700 K and then it gets

0 200 400 600 800

0

10

20

30

40

50

60

Temperature(K)

zno-20

Sp

ec

ific

he

at

(Cv)

R bulk

Figure-6.8: Lattice Specific heat for ZnO bulk and ZnO

NW of diameter 3.2 Å.

Metallic Silver Nanocluster Chapter-6

163

saturated at around 55 J/mol-c-k (6 NK Where N is Avogadro number and K is the

Boltzmann’s constant) the classical value as per Dulong–Petit’s law. This is due to the

anharmonic approximation of the Debye model. However, at higher temperatures the

anharmonic effect on Cv is suppressed. In comparison with the corresponding bulk

ZnO, the specific heat of ZnO nanowire is lower which can be attributed to the change

in VDOS in low frequency region and blue shifting of optical phonon.

6.4 Metallic Silver Nanocluster

The clusters of atoms and molecules are somewhat intermediate in several

aspects (number of constituent atoms N, size, basic properties etc.) between simple

atoms and molecules on one side and macroscopic aggregates (bulk solids) on the other

side. The advent of clusters as stable or at least reproducible nanosystems composed of

a few atoms, having an average size up to a few nanometers, so that quite often termed

as nanoparticles and took place gradually with the parallel development and

refinements to techniques to master and study their properties. When an increasing of

atoms are progressively aggregated to form clusters, the evolution from atomic like to

bulk like behavior is ascertained to have happened in many different ways, depending

on the kind of atoms. It is established that space confinement plays a key role in

modeling and tuning the physical properties of nanocrystals.

Metal nanocluster is a rapidly growing field of research due to attractive idea of

tailoring material properties by acting on the morphology of structure. The optical

properties which depend on the diameter of nanocrystals or quantum dots are well

understood both theoretically and experimentally. However, the vibration of clusters

which is an unique property of metallic clusters particularly at room temperature is still

unclear [30]. Despite the fact that the understanding of the vibrational properties of

Results and Discussion Chapter-6

164

nanocluster is essential [31-36] a serious effort is still lacking. Though literature

consists some of the pioneer work on experimental studies [37] followed by the model

[38-39] and molecular dynamical [40-41] calculations, but there is yet to come a

first principles study on vibrational properties of metal clusters. There is some

scattered first principles study on electronic properties [41-43]. Recently, the first

principles study on the vibrational properties of some semiconductor nanoclusters is

reported and VDOS is analyzed in terms of special features such as low frequency

DOS enhancement, surface acoustic and optical modes. In the present work, we study

the vibrational and electronic properties of metallic silver nano clusters of different

sizes. However, we present vibrational density of states only for two cluster sizes due

to costly computation. While, the electronic densities of states are presented for all

considered six silver nano-clusters.

6.4.1 Results and Discussion

The ab-initio calculations for the metallic silver nano clusters have been

performed using density functional theory within the GGA as implemented in the

Quantum Espresso code [19], similar to the bulk ZnO and ZnO NWs work presented in

section 6.3. The calculations are performed in a larger cell structure avoiding the other

interactions, using a plane-wave code. The plane wave energy cutoff is 40 Ry. For the

Brillouin zone (BZ) integration we use an 8x8x8 sampling mesh. For the exchange-

correlation functional we have employed the generalized gradient approximation

(GGA) functional developed by Perdew et al (PBE) [21], since, it is known that the

GGA gives better results than the simpler local density approximation (LDA) when

describing the structural properties of transition metals and its compounds [19, 21].

Before starting the calculation on ground state and linear response calculations, a set of

convergence tests have been performed in order to choose correctly the mesh of k-

Results and Discussion Chapter-6

165

ag-bulk

ag-3

ag-5

ag-6

ag-7

ag-4

ag-9

ag-8

-8 -6 -4 -2 0 2 4

ag10

Energy (eV)

points and cut-off kinetic energy for the plane waves. Convergence tests prove that the

BZ sampling and the kinetic energy cut-off are sufficient to guarantee an excellent

convergence. Phonon calculations are performed using density functional perturbation

theory (DFPT) [21].The convergence of the total energy of around 0.0001 Ry and the

phonon frequencies by 4 cm-1

is ensured.

We have investigated

the clusters electronic

properties via the

electronic density of

states (DOS). In Fig. 6.9

we present the total

DOS for eight cluster

Ag3, Ag4, Ag5, Ag6,

Ag7, Ag8, Ag9, Ag10

alongwith bulk DOS of

fcc crystal silver using

ab-initio density

functional theoretical

calculation. Generally,

the total DOS is

composed by the

relatively compact d

states and the mere

expanded sp states [43-

44]. In smallest cluster such as Ag3 the states show discrete peaks. As the cluster size

Figure-6.9: Electronic density of states of Ag (3-10) nano

clusters with its relaxed structure.[http://www-

wales.ch.cam.ac.uk/CCD.html]

Results and Discussion Chapter-6

166

increases, the states gradually shift and overlap with each other and finally come into

being electronic band. The DOS of Ag7 and Ag8 still have molecular-like some

discrete peaks but there electronic spectra peaks tend to overlap and form continuous

band.

2 3 4 5 6 7 8 9-650

-600

-550

-500

-450

-400

-350

-300

-250

-200

En

erg

y (

eV

)

Total energy (eV)

No. of atoms

Figure-6.11: Total energy vs cluster

size of Ag-(3-7)

0 50 100 150 200

Ag-bulk

PD

OS

(a

.u.) Ag-4

Frequency (cm-1

)

Ag-6

3.0 3.5 4.0 4.5 5.0 5.5 6.0

2.6

2.8

3.0

3.2

3.4

3.6

Bo

nd

le

ng

th (Å

)

Bond length

Cluster size (Å)

Figure-6.10: Bond-length vs cluster size of

Ag-(3-7)

Figure- 6.12: Vibrational Densities of states for Ag-bulk and Ag-4 and Ag-6

Results and Discussion Chapter-6

167

Figures 6.10 and 6.11 present the cluster size variation of Bond length and total energy.

While bond-length shows initially no variation, but the total energy linearly decreases

with the increase of cluster size. Fig 6.12 presents the vibrational density of states for

two silver nano clusters (Ag-4 and Ag-6). This figure also includes the vibrational

DOS of bulk silver. The vibrational DOS of bulk silver shows all important features of

the phonon dispersion curves of bulk silver [45-46].

The phonon dispersion curve for bulk silver shows that the phonon frequencies

are positive throughout the Brillouin zone which confirms the high quality phonon

calculations significantly. VDOS for bulk silver is spanned through 50 to 175 cm-1

with two main features, one broad peak centered at around 80 cm-1

and other around

150 cm-1

with three sub peaks. Now turning attention to the VDOS of the clusters of 4

and 6 silver atoms and its comparison with its bulk counterparts, a distinctive behavior

of VDOS for cluster of small number of silver atoms from bulk silver is clearly seen.

The most distinctive feature apart from the descritization of DOS is the presence of

VDOS in the low frequency region. The descritization in VDOS for Ag-4 reveals its

more atom-like behavior. There is almost non vanishing VDOS for ω=0. This may

enhance the specific heat and thermal conductivity of the cluster of small size [46]. In

the high frequency region, there is clear evidence of red shifting which may turn into

tail in the case of large number of atoms normally observed in MD calculation of

VDOS for other metallic system. The calculated vibrational spectra have no imaginary

frequencies, implying that the optimized geometry is located at the minimum point of

the potential surface. Furthermore, an analysis of clear size dependent expected blue

shift and red shift is out of scope for present study as it requires many calculations with

large number of atoms which is a very costly computational affair and need longer

Conclusion Chapter-6

168

time. Our purpose was here just to show the usefulness of abinitio calculations of

metallic cluster.

6.5 Conclusion

The present chapter describes the vibrational properties of two different class of

low dimensional structures such as nanowires and clusters of semiconductor and metal

respectively using present state-of –art density functional theoretical calculations. The

geometry of two ZnO nanowires of 3.2 and 5.3 Å and silver nanoclusters of eight

different sizes is optimized and then used for electronic band structure and vibrational

DOS calculations. The energy gap increases with the decrease of diameter of nanowire

in the case of ZnO. There is removal of degeneracy of many phonon modes alongwith

the appearance of several new phonon modes in the case of nanowire in comparison to

the bulk. In the case of silver nanocluster, we observe two different features in the

electronic DOS. The vibrational DOS turns to the descritization alongwith the

significant changes in the low frequency region of the vibrational DOS in the case of

clusters which may affect the specific heat and thermal conductivity. The present

chapter clearly brings out the effect of spatial confinement on electronic and phonon

properties of semiconductor and metal nanostructures.

References Chapter-6

169

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