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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
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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