Influence of layer stacking on the phonon properties of ...
Transcript of Influence of layer stacking on the phonon properties of ...
Influence of layer stacking on the phonon properties
of bilayer Molybdenum Disulphide (MoS2)
Chan Su-Wen, Philemon
A0110588E
Supervisor: Asst. Prof. Quek Su Ying
Department of Physics
National University of Singapore
2017
A Thesis Submitted in Partial Fulfilment for the Requirements for the Degree of
Bachelor of Science with Honours
2
Abstract
We simulated the Raman Spectra of monolayer and bilayer MoS2, using first-principles
calculations, and report that different stacking patterns of 2H, AB’ and 3R-like have varying
low-frequency (< 50 cm-1) interlayer shear and breathing modes. The high phonon frequencies
associated with the in-plane and out-of-plane Raman-active modes of the bilayer have values
of ~388 cm-1 and ~407 cm-1. These two modes, which are often referred to as bulk E2g and A1g
modes, remain largely consistent across different bilayer stacking patterns. We report that the
thermal conductivity of monolayer MoS2 at 300 K is calculated to be 50.5 Wm-1K-1. The Debye
temperature of the monolayer was computed to be 687 K.
We also present a new analysis, namely to determine the thermal conductivity of bilayer MoS2
for stacking patterns of 2H and AB’ and the values are 45.6 Wm-1K-1 and 43.5 Wm-1K-1
respectively. The Debye temperatures of these two stacking patterns are 681 K and 683 K
respectively. For temperatures larger than Debye temperature, the thermal conductivities of
monolayer and bilayer decrease have a reciprocal temperature dependence. The degradation of
thermal conductivity with increasing temperature can be attributed to the scattering of optical
phonons.
3
Acknowledgement
It has been a unique and meaningful experience working on this Honours project as it has
piqued my interest about the field of computational condensed matter physics. I am grateful to
have access to avant-garde High Performance Cluster (HPC) under Centre for Advanced 2D
Materials of National University of Singapore.
I would like to express my deep gratitude to my supervisor Assistant Professor Quek Su Ying
for her invaluable insights and guidance throughout this research project, and also the
assistance rendered when I encountered problems faced during the course of this project. In
addition, I would like to express my gratitude to Dr. Luo Xin for his guidance with phono3py
and computed results, and to Dr. Keian Noori for the tutorials on Quantum ESPRESSO. I am
also very grateful for the technical assistance given by Dr. Miguel Dias Costa regarding the use
of HPC resources in Graphene Research Centre and National SuperComputing Centre and
update of phono3py and other modules on the cluster. Finally, I would to thank Mr. Wu Yaze
for giving tips on the visualisation and VASP software which were of help to my project.
Finally, I will like to give all glory to God for walking with me through this exhilarating journey
in undergraduate studies in physics. I am very grateful for the tremendous support that my
family has given me throughout the course of four years in university.
4
Contents
Abstract 2
Acknowledgement 3
Contents 4
1 Introduction
1.1 Introduction to 2-D materials 5
1.2 Raman Spectroscopy of MoS2 7
1.3 Phonon Properties and Thermal Conductivity of MoS2 9
2 Theoretical Background
2.1 Background of Condensed Matter Physics 10
2.2 Density Functional Theory (DFT) 12
2.3 Density Functional Perturbation Theory (DFPT) 14
2.4 Phonons in Crystals 17
2.5 Calculating Lattice Thermal Properties 19
3 Computational Methods
3.1 Quantum ESPRESSO for Geometry Optimization 22
3.2 Quantum ESPRESSO for Raman Spectra Simulation 24
3.3 VASP-Phono3py for Lattice Thermal Conductivity Computation 24
4 Results and Discussion
4.1 Simulated Raman Spectra of 1TL MoS2 27
4.2 Simulated Raman Spectra of 2TL MoS2 32
4.3 MoS2 1TL Thermal Conductivity 43
4.4 Thermal Conductivities of 2TL MoS2 51
5 Conclusion 58
6 Appendices 60
7 References 81
5
Chapter 1
“It would be possible to describe everything scientifically, but it would make no sense; it would
be without meaning, as if you described a Beethoven symphony as a variation of wave
pressure.”
Albert Einstein
Introduction
1.1 Introduction to 2D Materials
A two-dimensional (2D) material is a substance made up of a layer of atoms, a few nanometers
thick, and electrons are able to move freely within the layer [1]. An example of a 2D material is
graphene, a single layer of graphite (see Fig. 1) which contains carbon atoms arranged in a
hexagonal lattice structure and is atomically thin [2].
Figure 1: Top down view of graphene rendered in XCrySDen[3] (crystalline and molecular
structure visualisation software).
6
With the isolation of graphene in 2004 in the work by Andre Geim and Konstantin Novoselov,
the field of two-dimensional (2D) materials has been extensively researched for applications
in the semiconductor industry. As current integrated circuits made of silicon are approaching
the boundary of performance, this led to the foray into identifying new materials to make better
performing semiconductor devices [4].
Aside from graphene, a class of 2D materials known transition metal dichalcogenides (TMDs)
has been discovered to possess unique properties. This class of materials have the formula MX2
where M represents a transition metal such as Molybdenum (Mo) and Tungsten (W) and X
represents chalcogens such as Sulphur (S), Selenium (S) and Tellurium (Te) [2] (see Fig. 2).
Transition metal dichalcogenides such as Tungsten Disulphide (WS2) have band gaps, unlike
graphene, and are candidates for optoelectronic devices [5].
Figure 2: Side view of MX2 molecule consisting of a transition metal element (grey)
sandwiched between two chalcogen atoms (yellow).
Molybdenum Disulphide (MoS2), another transition metal dichalcogenide, takes on the role of
a cocatalyst when placed on (Cadmium Sulphide) in the production of hydrogen (H2) [6]. In
addition, MoS2 has been found to have strong photoluminescence which makes it a good
candidate for photodetectors [7]. Furthermore, in photoluminescence spectroscopy of bilayer
MoS2, the twisting of MoS2 monolayer relative to another layer results in different
photoluminescence intensities [8]. Aside from using photoluminescence to differentiate stacking
configurations (different twisted bilayers) of MoS2, the Raman spectra of different stacking
configurations of MoS2 can be studied and is one of the goals of this research.
7
Figure 3: Top down view of 1 tri-layer (1TL or monolayer) of MoS2 from XCrySDen.
Figure 4: Top down view of 2 tri-layer (2TL or bilayer) of 2H-MoS2 from XCrySDen.
1.2 Raman Spectroscopy of MoS2
The main motivation of this research is from the origami experiments performed on exfoliated
monolayer MoS2 done by Shi Wei Wu group of Fudan University. MoS2 bilayers of different
stacking orders (or patterns) can be obtained from origami folding of monolayer MoS2 and
each of these stacking orders have unique photoluminescence spectrum [9]. Besides having a
unique photoluminescence spectrum, the Raman spectrum, specifically modes with low
frequency, are sensitive to changes in stacking order [10]. This has been observed experimentally
and modelled from first-principles. Hence, this research employed a first-principles approach
to simulate the Raman spectrum of monolayer and bilayer MoS2.
Before delving into Raman spectroscopy, one must understand the process of Raman
Scattering. When a high intensity laser of ultraviolet-visible wavelength and of frequency i
8
is used to irradiate a crystal sample, the scattered light has a frequency of mi ( m is the
molecular vibrational frequency). Moreover, Raman scattering can be classified into anti-
Stokes and Stokes scattering. If the frequency of the scattered photon is mi , this is known
as the anti-Stokes process and the photon has lower energy than the incident photon. On the
other hand, the Stokes process occurs when the scattered photon’s frequency is mi and the
photon has higher energy than the incident photon [11].
Furthermore, the difference in energy between incident and scattered photon is known as the
Raman shift (in units of cm-1). By the definitions of the two processes given above, anti-Stokes
scattering results in a negative Raman shift while Stokes scattering yields a positive Raman
shift. If the incoming photon excites the system to a virtual state (unobservable quantum state)
and subsequently de-excites to a final state, this process is known as non-resonant Raman
Scattering [12]. For resonant Raman scattering, the incident photon energy coincides with the
energy of electronic transition of the crystal sample [13].
As Raman spectroscopy is a non-destructive means of studying 2D materials [12], transition
metal dichalcogenides such as bilayer MoS2 have been examined for its Raman spectra [8]. In
Raman spectroscopy experiments, a solid-state laser of wavelength in the visible spectrum is
focused onto a spot via the objective lens of a microscope [10]. Since Raman scattering is weak
compared to Rayleigh scattering, the scattered light signals are collected by a charged-coupled
device (CCD), which is an array of photosensitive cells containing semiconductor material,
and a Raman spectrograph is generated [5].
However, this research’s first focus is the computational simulation of Raman spectra,
specifically non-resonant Raman scattering, via Density Functional Perturbation Theory
(DFPT) which is implemented in Quantum opEn-Source Package for Research in Electronic
Structure, Simulation, and Optimization (ESPRESSO) [14]. In the subsequent sections, the
background of condensed matter physics and the underpinnings of Density Functional Theory
(DFT) and DFPT will be covered.
9
1.3 Phonon Properties and Thermal Conductivity of MoS2
Aside from studying the Raman spectrum of monolayer MoS2, recent studies have reported
that MoS2 is a candidate for field effect transistor (FET) and thermoelectric devices [15]. As
thermoelectric devices work based on the principle of a temperature gradient inducing a
thermoelectric voltage (Seebeck effect), materials such as MoS2 have a high Seebeck
coefficient (ratio of voltage to temperature difference) of between 2104 and 5101 μVK-1[16].
As the efficiency of a thermoelectric material is given by ZT, where Z is the figure of merit
and T is temperature, which is equal to TS )/( 2 [17] (S is Seeback coefficient and is electrical
conductivity) , a low thermal conductivity and high Seebeck coefficient will render the
material a good candidate for thermoelectric devices as its ZT is high.
Hence, this research’s second aim is to calculate the thermal conductivity of monolayer and
bilayer MoS2. As thermal conductivity calculations require the knowledge of phonon-phonon
interactions and phonon properties, we have also incorporated the study of phonon lifetimes
and phonon dispersion graphs. The first-principles calculation of thermal conductivity and
other phonon properties begins with the use of DFT implemented in Vienna ab initio simulation
package (VASP)[18-21] to optimise the MoS2 structure and ends with Boltzmann transport theory
under single-mode relaxation time approximation (SMRT) as implemented in phono3py [22].
Phonon band structure plots and generation of phonon density of states were obtained using
phonopy package[23].
10
Chapter 2
“Any physical theory is always provisional, in the sense that it is only a hypothesis: you can
never prove it... As philosopher of science Karl Popper has emphasized, a good theory is
characterized by the fact that it makes a number of predictions that could in principle be
disproved or falsified by observation.”
Stephen Hawking
Theoretical Background
2.1 Background of Condensed Matter Physics
The development of Schrödinger’s equation and the discovery of Pauli’s exclusion principle
heralded the dawn of quantum mechanics and condensed matter physics [24]. For a system of
many ions and electrons, the Schrödinger equation has the form:
});({});({ iiii rRrR EH (2.1)
In the case of the above, the full Hamiltonian H of the system is given by
JI
JI
jiil
I
i eI I
ezzeez
mMH
JIjiiI
rR
RRrrrRiI
2222
22
2
2
1
2
1
22
(2.2)
11
In 1927, the concept of nuclei being fixed with respect to electron motion was introduced in
the Born-Oppenheimer Approximation. As a consequence of this, electron and nuclear
dynamics of a condensed matter system is considered separately. By re-writing the Hamiltonian
under (2.2) Born-Oppenheimer approximation (removing first term of (2.2)), one can arrive at
JI
JI
jiiI
I
i e
ezzeV
mH
JIji
Iir
RRrrRr
i
222
2
2
1
2
1)(
2
(2.3)
and the last term in (2.3) is the nuclear electrostatic energy 𝐸𝑁({𝑹})[25]. In (2.3), 𝑉𝐼(𝒓𝒊 − 𝑹𝑰)
is the electron-nucleus interaction (pseudopotential) and is given by
)( Ii RrIV
iI rR
2ezI (2.4)
To calculate forces on a nucleus, one must use Hellmann-Feynman theorem to compute the
force on Ith nucleus 𝑭𝐼
}{
}{
}{
})({R
I
R
R
I
IRR
RF
HE
(2.5)
where 𝐸({𝑹}) can be expressed in the form shown in (2.5) by incorporating the variational
principle. 𝜓𝑹 is known as the ground-state wave function of the Hamiltonian 𝐻. In optimizing
the geometry of a lattice system, the force 𝑭𝐼 acting on Ith nucleus should reduce to zero [26].
By substituting the Hamiltonian in (2.3) into (2.5), one can express the Hellmann-Feynman
force on the Ith nucleus as
II
IiI
R
Rr
R
RrrF
})({)()( NI E
dV
n (2.6)
And where 𝑛(𝒓) is the electron charge density expressed as follows [25]
N22R rrrrrr ddNn N ...),...,,()(
2
}{ (2.7)
Furthermore, to obtain the Hessian of the Born-Oppenheimer energy surface, one can
differentiate (2.6) with respect to nuclear coordinates 𝑹𝑱 as shown below [26].
12
JIJI
Ii
I
Ii
JJIJ
I
RR
Rr
RR
Rrrr
R
Rr
R
r
RR
R
R
F
})({)()(
)()(})({ 222NII E
dV
ndVnE
(2.8)
In (2.8), the ground-state charge density )(rn and its linear response to change in nuclear
coordinates JR
r
)(n are required in the calculation of the Hessian matrix
JI RR
R
})({2Eand this is known
as the matrix of interatomic force constants, and will be covered in greater detail in Chapter
2.3. Since ground-state charge density 𝑛(𝒓) and its linear response are essential to the
determination of the matrix of interatomic force constants, a new approach was developed to
express total energy E as a functional of ground-state electron density. This approach is known
as Density Functional Theory.
2.2 Density Functional Theory (DFT)
As particle numbers increase such as in the case of a N-electron condensed matter system, a
new approach was proposed by Pierre Hohenberg, Walter Kohn and Lu Jeu Sham that
considers electron density 𝑛(𝒓) and is termed Density Functional Theory (DFT). It is based on
sound Hohenberg-Kohn theorems which state that external potential 𝑉𝑒𝑥𝑡(𝒓) is determined by
ground-state electron density 𝑛(𝒓). Due to this, total energy can be expressed as a functional of 𝑛(𝒓):
rrrrr dnVnFnE ext
Total )()()]([)]([ (2.9)
In equation (2.9), 𝐹[𝑛(𝒓)] contains the kinetic energy term 𝑇𝑆, coulomb interaction term and
an exchange correlation functional 𝐸𝑋𝐶 and 𝐹[𝑛(𝒓)] can be expressed as follows [27]:
)]([
)()(
2)]([)]([
2
rr'rr'r
r'rrr nEdd
nnenTnF XCS
(2.10)
Finally, by varying electron density 𝛿𝑛(𝒓) and considering the constraint
0)()()( * rrrrr ddn ii (2.11)
13
in variational calculation 0)]1([ i
iii
TotalE yields the following Kohn-Sham
equations for the unperturbed system and the Kohn-Sham Hamiltonian (HKS) is given in
brackets of equation (2.12).
)()())(,(
2
22
rrrrr iiiSCF nVm
(2.12)
In equation (2.12), the effective potential 𝑉𝑆𝐶𝐹, where 𝑉𝑆𝐶𝐹 is also known as the self-consistent
field potential, is defined as [25]
)(
)]([)()())(,( 2
r
rr'
r'r
r'rrr
n
nEd
neVnV
XC
extSCF
(2.13)
In the Kohn-Sham equations as shown above in (2.12), )(ri are known as Kohn-Sham
orbitals. )(rextV in (2.13) is known as the external potential due to ions and the last term of
(2.13) is known as the exchange-correlation potential or 𝑉𝑋𝐶. SCFV must be solved self-
consistently by means of an iteration method [26].
To solve the Kohn-Sham equations (2.12) self-consistently, one can perform the self-consistent
loop until the difference between the input charge density inn and the output charge density
outn is zero [21]. During implementation, one can exit the loop when nin - nout is less than a
tolerance value. At every self-consistent iteration, a new output charge density is computed and
the schematic of this process is shown in Appendix A1. [28]
The system of non-interacting electrons in Kohn-Sham’s approach will have the following
ground-state charge density (each of the N/2 lowest-lying orbitals obey Pauli Exclusion
Principle and the overall system is not magnetic)
2/
1
2)(2)(
N
n
nn rr (2.14)
and the resultant kinetic energy functional as follows:
14
r
r
rr d
mrnT
N
n
nn
S
2/
12
2* )(
)(2
2)]([
(2.15)
By casting (2.15) as the left-hand side of Schrodinger equation, the full resultant Schrodinger
equation containing Kohn-Sham eigenvalues n is
2/
1
2/
12
2* 2
)()(
22
N
n
n
N
n
nn d
m
r
r
rr
(2.16)
Furthermore, by rearranging (2.13) and making )(rextV the subject, one can arrive at
)(
)())(,()( 2
rr'r'r
r'rrr
XC
SCFext Vdn
enVV
(2.17)
Finally, ground state energy of the system can be obtained by combining equations (2.9),
(2.10), (2.16) and (2.17) to yield [12]
rrrrr'rr'r
r'rr dnVnEdd
nnenE XCXC
N
n
n
Total )()()]([)()(
22)]([
22/
1
(2.18)
In all, DFT is a first principles approach to determine ground state energy, the process of
expressing ground state energy TotalE as a functional of charge density )(rn has been delineated
in this project thesis. In this project, DFT was used to determine the coordinates of the relaxed
transition metal dichalcogenide (MoS2) structure (minimum energy configuration) and to do
so, the DFT code in Quantum ESPRESSO had to calculate the final ground state energy at the
end of various self-consistent calculations. The relaxed structure is then passed to a subsequent
Density Functional Perturbation Theory (DFPT) Code in Quantum ESPRESSO to determine
the phonon frequencies. In the subsequent section, the background and foundation of DFPT
will be discussed in greater detail.
2.3 Density Functional Perturbation Theory (DFPT)
Harmonic Approximation and Real Space Equations of Motion
15
First, one should consider a perturbed system where an atom 𝛼 is displaced by 𝒔𝒏,𝜶 from its
equilibrium position (with reference to origin) at 𝒓𝒏,𝜶 = 𝒓𝒏 + 𝒓𝜶 where 𝑛 denotes the nth unit
cell and 𝒓𝒏 is the displacement of the unit cell from origin and 𝒓𝜶 is the displacement of atom
𝛼 from position 𝒓𝒏 [29]. When lattice vibrations occur, the atoms are displaced from equilibrium
position and thus the potential energy of Ith nuclei can be written as a Taylor Expansion:
JI
JI
I
I
III mn
m,βn,α
n
n,α
n,αnn,α ssrr
Es
r
ErEsrE ,,
2
,,2
1})({})({
(2.19)
In (2.19), one can express the following as a matrix 𝐶𝑛𝛼𝐼𝑚𝛽𝐽
of interatomic force constants (IFCs)
calculated in real space:
Jm
In
m,βn,α
Crr
E
JI
2
(2.20)
By differentiating E with respect to 𝑠𝑛,𝛼𝐼, we arrive at the force
I
JJ
I
n
mm
ns
srE
,
,,
,
})({
F
(2.21)
Based on Newton’s second law, one can arrive at the equation of motion as follows
I
I
JJ
n
n
mmsM
s
srE
,
,
,, })({
(2.22)
By considering the second term on the right-hand side of (2.19) and differentiating with respect
to 𝑠𝑛,𝛼𝐼, the equation of motion in real space can be rewritten as
IJ nm
Jm
In sMsC
,, (2.23)
By recalling equation (2.8), the matrix of interatomic force constants Jm
InC
is given by Hessian
matrix JI RR
R
})({2E [29]. The Hessian matrix contains the linear response term JR
r
)(n in (2.8) is
known as the electron-density response and is central to formulating the procedure known as
Density Functional Perturbation Theory [25].
16
In re-writing equation (2.8) in terms of the parameters Ii R andJj R , ignoring the second
order derivative term containing 𝐸𝑁, one will arrive at the following second order derivative of
ground state energy [26]
r
rrr
rrd
Vnd
VnE
jijiji
)(
)()()( 22
(2.24)
By linearizing (2.14), the change in ground state charge density will have the form
2/
1
* )()(Re4)(N
n
nnn rrr (2.25)
And the change in wave function denoted by ∆𝜓𝑛(𝒓) is given by first-order perturbation theory
according to the Schrödinger equation of the perturbed system as follows
))(())(( nnNNnnSCFSCF VH (2.26)
The final form for the equation of the perturbed system after expansion is as follows and this
is known as the Sternheimer equation[26].
nNSCFnNSCF VH )()( (2.27)
In (2.27), the unperturbed Kohn-Sham Hamiltonian and the associated eigenvalue are given by
𝐻𝑆𝐶𝐹 and N . The unperturbed Kohn-Sham Hamiltonian is written as follows
)(
2 2
2
rr
SCFSCF Vm
H
(2.28)
while SCFV , which is the first order correction to self-consistent potential, (form like (2.13))
is given by [25]
r'
r'
r'
rr'
r'r
r'rr d
n
n
Vd
ne
VV
i
XC
ii
SCF
)(
)(
)(1)()()( 2
(2.29)
In (2.27), N is the first order correction to the Kohn-Sham eigenvalue and is given by
nSCFn V . In order to determine the right-hand-side of (2.25), one has to know the density
17
of the occupied states. Thus, the first-order correction n (variation of Kohn-Sham orbitals)
is given by [26]
nm mn
nSCFm
mn
V
)(r
(2.30)
Finally, by substituting (2.30) into (2.25), the resulting equation for the response of charge-
density is
nm mn
nSCFm
m
N
n
n
Vn
)()(4)(
2/
1
*rrr
(2.31)
The set of equations containing (2.27), (2.28), (2.29), (2.30) and (2.31) must be solved self-
consistently [26]. The system of equations outlined in (2.27) can be solved independently for
each of the N/2 first derivatives of Kohn-Sham orbitals given by IR
r
)(nn
. In each self-
consistent cycle of computation, the charge-density response )(rn in (2.31) and the potential
energy response SCFV in (2.29) are updated. Solving the linear system self-consistently will
yield the first derivatives of the Kohn-Sham eigenvalues as IR
N
N
and the linear potential
response as IR
SCF
SCF
VV [26].
Using density functional perturbation theory (DFPT), one can determine the charge density
response and the self-consistent potential response to a perturbation. The strength of DFPT lies
in its ability to calculate phonon frequencies at some wave vector q efficiently without resorting
to supercell approach. Before delving into the calculation of phonon frequencies, the
subsequent section will cover the concept of phonons in a crystal lattice.
2.4 Phonons in Crystals
Equations of Motion in Fourier Space
18
Phonons are quanta of energy ( 1.0~ eV) associated with the vibrations in a lattice, and these
vibrations propagate through the crystal. By revisiting the real space equations of motion in
(2.23), one can also write the equations of motion in Fourier space via exploiting the periodicity
of the lattice system by incorporating Fourier transforms [25]. By expressing displacements
𝑠𝑛,𝛼𝐼 and 𝑠𝑚,𝛽𝐽
as follows in (2.32) and (2.33) (for a single equation of motion),
)(, )(
1 tin eu
Ms
II
•
nrqq (2.32)
)(
, )(1 ti
mm
JJeu
Ms
•
rqq
(2.33)
and by taking the derivative Ins ,with respect to time, one has the following
)(
, )(ti
n euM
is
II
• nrq
q (2.34)
and by finding the �̈�𝑛,𝛼𝐼, one has the following
)(
2
, )(ti
n euM
sII
• nrq
q (2.35)
The equation of motion one can arrive at by substituting (2.33) and (2.35) into (2.23) is
)(1
)()(2
qq nm rrq
JIueC
MMu
iJm
In
• (2.36)
Thus, the dynamical matrix 𝐷𝛼𝐼𝛽𝐽
is given by [23]
)(1nm rrq •
iJm
In
J
I eCMM
D
(2.37)
Finally, by substituting (2.34) into (2.33) the equation of motion can be rewritten as
0)()( 2 qI
uD J
I
J
I
(2.38)
19
where 𝜔2 are the eigenvalues (square of phonon frequencies) and 𝑢𝛼𝐼(𝒒) are the eigenvectors
(amplitude of lattice distortion). 𝑢𝛼𝐼(𝒒) and 𝜔(𝒒) (phonon frequency) are functions of wave-
vector 𝒒 [25].
As phonons are collective excitations and they carry thermal energy, when they collide with
other phonons in the crystal lattice, the phonon-phonon scattering process will determine the
mean free path of phonons and phonon relaxation time [30]. The calculated phonon mean free
path and the calculated heat capacity are essential for determining the thermal conductivity of
the crystal lattice. The process of doing so will be covered in the subsequent section.
2.5 Calculating Lattice Thermal Properties
Lattice Thermal Conductivity
When calculating thermal conductivity of transition metal dichalcogenides, phonon-phonon
interactions are the main factor in the determination of thermal conductivity. As phonon-
phonon collision is an anharmonic process, the Hamiltonian of the system can be written as
follows [22]
320 HHT (2.39)
In (2.39), the constant potential is given by 0 , the harmonic Hamiltonian is given by
20 HTH , which contains the second-order force constants, and anharmonicity is contained
within 3H . In the calculation of thermal conductivity, we are concerned with three phonon
interactions and these processes must obey conservation of momentum. The third-order
correction term 3H can be expressed as follows [22]
l l l
lulululll'' ""
3 )""()''()()"",'',(6
1
(2.40)
In (2.40), contains the cubic anharmonic force constants in Cartesian indices of
and the atomic displacement operator )( lu of the th atom in the l th unit cell can be
20
expressed in terms of phonon annihilation operator jaqˆ and creation operator
†ˆja q (of the normal
mode of band index j) as follows [22]
),(]ˆˆ[2
)( )(†2/1
2/1
jeaaNm
lu li
j
j
jj qWα
rq
q
q
•
(2.41)
where m is the mass of the th atom, N is the number of unit cells,
2/1
jq (from phonon
frequency) and W are obtained from the equation involving dynamical matrix. The eigenvalue
problem to retrieve phonon harmonic frequency and polarisation eigenvectors is cast as
),(),'(),'( 2
'
jWjWD j qqq q
(2.42)
and the third order potential 3H can be re-expressed as
)ˆˆ)(ˆˆ)(ˆˆ( †
''''
†
''
†
'''
'''3
aaaaaa (2.43)
In (2.43), , ' and '' are the phonons in the three-phonon collision process and their
respective annihilation and creation operators are in the brackets. ''' is explicitly written as
follows [22]
)()"",'',0(
222)","()','(),(
!3
11
)0(")()]0(")"([")]0(")"(["
"'
)]0()''([
""'''''
"'
q"q'qrqrrqrrqrrq'
ilili
ll
li eeeell
mmmWWW
N
(2.44)
Now, we must consider all the fundamental components to calculate lattice thermal
conductivity. First, lattice thermal conductivity κ is defined as the energy transferred per unit
time through a unit area per unit temperature gradient. κ is dependent on (1) mode dependent
heat capacity, (2) phonon relaxation time and (3) group velocity of phonon mode. The lattice
thermal conductivity is as follows [22]
SMRTvvC
NV
0
1
(2.45)
where C is the heat capacity that is dependent on phonon mode λ, 0V is the unit cell volume,
v is the group velocity of phonon mode λ and SMRT
is the single-mode relaxation time. First,
to calculate mode dependent heat capacity, we employ the following
21
2)/(
)/(2
]1[
Tk
Tk
B
BB
B
e
e
TkkC
(2.46)
(which can be derived in the Appendix D). Second, the phonon lifetime can be determined
using
)(2
1
(2.47)
Since phonon self-energy can be divided into real and imaginary parts, the part that is required
to calculate phonon lifetime in (2.47) is the imaginary part )( as follows [22, 31]
)]}()()[()()1{(
18)( ''''''''''''''
'''
'
2
'''2
nnnn
(2.48)
and n is the phonon occupation number modelled using Planck distribution. Since momentum
conservation is observed, 0''' Gqqq and )()( '''''' denotes class 1
where two phonons annihilate, resulting in a third phonon. )( ''' is associated with
the class 2 process where one phonon decays into two phonons of lesser energy [32]. These two
classes of processes are the result of cubic anharmonic terms. After the imaginary part of
phonon self-energy has been calculated, the calculated phonon lifetime is assumed to be
equivalent to the phonon-relaxation time SMRT
[22]. Thus, we have determined C and SMRT in
the thermal conductivity equation in (2.45).
Third, group velocity of phonon mode denoted by v is given by solving the eigenvalue
equation (2.42) for the phonon frequency and differentiating with respect to q in
qv
)( . The resultant group velocity is [22]
'
),'(),'(
),(2
1W
q
DW
q
q
(2.49)
In all, the mode dependent heat capacity, group velocity associated with phonon mode and the
phonon-relaxation time are the key components required to determine lattice thermal
conductivity [31] and can be implemented using phono3py software.
22
Chapter 3
“And even when the apparatus exists, novelty ordinarily emerges only for the man who,
knowing with precision what he should expect, is able to recognize that something has gone
wrong.”
Thomas S. Kuhn
Computational Methods
3.1 Quantum ESPRESSO for Geometry Optimization
Ab-initio, from first principles, calculations are performed using Density Functional Theory
(DFT) as implemented in Quantum opEn-Source Package for Research in Electronic Structure,
Simulation, and Optimization (ESPRESSO). Quantum ESPRESSO uses plane waves basis sets
and pseudopotentials to model electron-ion interactions [14]. Since the lattice systems we deal
with in this study are periodic, and thus we can model the Kohn-Sham orbitals or wave
functions as non-local plane waves
23
G
rGrkGr
iik ece)(
(3.1)
where k is the wave vector and G is a reciprocal lattice vector [28]. Plane waves can be used to
model free electrons and these electrons tend to occupy higher energy levels. Thus, a kinetic
energy cut-off energy must be set to limit the expansion of plane waves basis set [28]. This is
cut-off is denoted as cutE and is defined as
2)(2
1Gk cutE
(3.2)
Aside from modelling the electrons as plane waves, there is also a need to reduce the number
of electrons considered in the calculation. Before introducing the concept of pseudopotentials,
the electrons in the system are split into core and valence electron groups. The core electrons
exist within the region crr while valence electrons exist beyond the core radius cr [28]. By
considering the all-electron wave function to be closest to actual orbitals, for the region crr ,
the pseudized wave function is a smoothen version of the all-electron wave function. The first
and second derivatives of the pseudized and all-electron wave functions must match for the
region of crr .
Finally, both pseudized and all-electron wave functions are the same for region crr [28]. From
the valence electron and pseudized wave function one can then generate the pseudopotentials.
In this project, the norm-conserving pseudopotential that is used in DFT calculation is formed
by the following rule
cc r
AE
r
pp dd
0
2
0
2)()( rrrr
(3.3)
where the pseudo and all-electron charge densities in the core region are the same.
Besides the use of pseudopotentials to model electron-ions interactions, the DFT calculation of
total energy of system is dependent on the exchange correlation functional as mentioned in the
last term of (2.10). The Perdew-Zunger, Scalar-Relativistic, local density approximation
(LDA) was used to approximate the exchange-correlation functional.
In the self-consistent calculation (carried out by program called pw.x (in PWscf package) in
Quantum ESPRESSO), a plane-wave kinetic energy cutoff of 65 Ry or 884eV was used to limit
24
the size of the plane-wave basis set. As the charge density is the square of wave functions, with
a cutoff for charge-density at 550 Ry or roughly 8 times the kinetic energy cutoff. In order to
sample the Brillouin Zone of the monolayer and bilayer MoS2 systems, a Monkhorst-Pack k-
point mesh of 11717 is used and the convergence threshold of self-consistency is set at 10-
10 eV. The above parameters were used to calculate the Kohn-Sham orbitals as well as the
charge-density.
3.2 Quantum ESPRESSO for Raman Spectra Simulation
As the theoretical simulation of Raman Spectra requires phonon calculations, these calculations
require one to first ascertain the ground state electronic and atomic configuration. After
structure optimization (relaxation) of the monolayer and bilayers were obtained via pw.x code,
the relaxed structure is passed on to ph.x code (in PHonon package in Quantum ESPRESSO),
which is an implementation of Density Functional Perturbation Theory (DFPT), to compute
the phonon frequencies at phonon wave vector q = 0. As mentioned in Chapter 2.3, the charge
density response to distortions in the lattice, which is a component of the second-order
derivative of energy, is central in DFPT calculations [1]. The k-point sampling of charge density
response in the Brillouin Zone can be governed by phonon wave vector q. The ph.x code
computes the phonon frequencies and eigenvectors at wave vector q = 0 by computing the
dynamical matrix such as that mentioned in equation (2.37). Furthermore, at q = 0, no
longitudinal optical and traverse (LO-TO) optical mode splitting are observed. A list of the full
input parameters to the program ph.x is included in the Appendix A3 section.
3.3 VASP-Phono3py for Lattice Thermal Conductivity
Computation
Finally, VASP (Vienna Ab initio simulation package) interfaced with phono3py was used to
calculate phonon-phonon interaction as well as related properties using the supercell approach.
By using phono3py, the thermal conductivities of MoS2 monolayer (1TL) and bilayer (2TL)
can be studied. The ground state electronic and atomic configuration can be determined using
Density Functional Theory (DFT) calculations and implemented via VASP. Local density
approximation (LDA) was used approximate the exchange-correlation functional, and
25
projector-augmented-plane-wave method was used to generate pseudopotentials for Mo and S
atoms.
The plane-wave energy cutoff was set at 364 eV in the self-consistent calculations of MoS2
monolayer (1TL) and bilayer (2TL). The atomic coordinates and lattice parameters of the
bilayer MoS2 were obtained from relaxation calculations done previously in Quantum
ESPRESSO. As part of DFT calculations, a Monkhorst-Pack k-point grid of dimensions 17 x
17 x 1 was used. The monolayer and bilayer structures were relaxed with the use of VASP and
this was done until total force on the ions is below the threshold of 0.003eV/Å. Once the
monolayer and bilayer structures are relaxed, the atomic configurations in the POSCAR files
are being used in the calculations of thermal conductivity.
As the method of calculating thermal conductivity requires the determination of second and
third order force constants, such as the cubic anharmonic force constants in (2.40), this requires
the use of supercells of dimensions 3 x 3 x 1 and a 4 x 4 x1 𝚪- centered Monkhorst-Pack k-
point grid was used. A supercell method and finite displacement method were used to calculate
force constants [22]. The second-order, harmonic force constant, is given by
)''()()'',(
2
klulkukllk
(3.4)
And the third-order, cubic anharmonic force constant is as follows
)""()''()()"",'',(
3
kluklulkuklkllk
(3.5)
To approximate the second-order and third-order force constants, the finite difference method
is used [22]. For the second-order force constant,
)(
)](;''[)'',(
lku
lkklFkllk
u
(3.6)
In (2.53), )(lku is the atomic displacement of the kth atom in the lth unit cell. Thus, force
)](;''[ lkklF u is the atomic force measured at )''( klr due to displacement )(lku in a supercell [22].
The third-order, cubic anharmonic force constant is determined as follows
26
)''()(
)]''(),(;""[)"",'',(
klulku
kllkklFklkllk
uu
(3.7)
The pairs of atoms in the supercells are displaced from their equilibrium positions by )(lku and
)''( klu , and the resultant force determined at )""( klr is computed from first-principles. The
displacement amplitude used for calculations is 0.09 Å for both second and third-order force
constants calculations. Phono3py generates the structures with a pair of displaced atoms in each
displaced configuration in the supercell, and VASP was used to calculate forces in supercells.
Once the VASP calculations are complete, phono3py software collects the forces and generates
second and third-order force constants.
Finally, the tetrahedron method was used to integrate within the Brillouin zone with 21x21x1
q-point mesh being used. The use of q-point sampling mesh is to enable discrete sampling
within Brillouin zone. Integration within the Brillouin zone is required to compute the
imaginary part of the self-energy in (2.48), and the phonon lifetime can be determined as stated
in equation (2.47).
27
Chapter 4
“The aim of argument, or of discussion, should not be victory but progress.”
Karl R. Popper
Results and Discussion
4.1 Simulated Raman Spectrum of 1TL MoS2
One trilayer (1TL), also known as monolayer, of molybdenum disulphide has a unit cell that
contains 3 atoms, where 1 molybdenum atom is sandwiched between 2 sulphur atoms. By
looking at the structure of MoS2 closely, one can use XCrysDen[3] software to visualise the
monolayer structure.
28
Figure 5: Visualisation of MoS2 monolayer using XCrysDen software in the xz (left) and xy-
plane (right)
Monolayer molybdenum disulphide exhibits a D3h point-group symmetry and it belongs to the
26mp
space group. By utilising density functional perturbation theory (DFPT), we can simulate
9 normal modes (3 acoustic and 6 optical modes) of the single MoS2 layer. We can further
reduce the 9 modes to 6 modes as 3 modes are doubly degenerate modes. Prior to calculating
the normal modes, the 6 irreducible representations associated with the D3h point-group (in
Mulliken notation) corresponding to 6 modes (2 acoustic and 4 optical modes) measured at 𝚪-
point associated with the monolayer are as follows [33]
'2"2"' 2112 EAEA
MoSTL
(4.1)
If the number of MoS2 layers are odd, where N=1,3,5…, the irreducible representation can also
be given by the following formula in (2.58) [12, 34].
'')"'"'(
2
)13(2211
2 EAEEAANMoS
TL
(4.2)
For single layer MoS2, the value of N is 1 and the resultant irreducible representation is given
by '"
2"'"
2'11 )(2 EAEEAA
MoSTL . Out of the 6 normal modes of MoS2 single layer,
the Raman-active modes are of the irreducible representation of A1’, E’ and E”. Moreover, the
E symbol represents vibrational modes that are doubly degenerate and the A symbol represents
vibrational modes that are non-degenerate. In the case of the MoS2 system, the E modes are in-
29
plane modes and the A modes are out-of-plane modes. The subscripts u and g represent modes
that are symmetric or anti-symmetric to inversion.
Using Quantum ESPRESSO to perform Density Functional Perturbation Theory (DFPT)
calculations for monolayer MoS2, the phonon frequencies (in-plane and out-of-plane modes)
and the associated irreducible representations calculated for 1TL (monolayer) MoS2 are
presented in Table 1 below.
In-plane modes 1TL 0 287.57 389.32
(0) (0) (0.006)
[0] [0] [0.006]
E’ [I + R] E”[R] E’[I + R]
Out-of-plane
modes
1TL 0 407.17 473.33
(0) (0.04) (0)
[0] [0] [0]
A2” [I] A1’ [R] A2” [I]
Table 1: Table of calculated phonon frequencies from Quantum ESPRESSO for 1TL MoS2
In Table 1, [I] denotes infrared active modes, [R] denotes Raman active modes and [I + R]
denotes phonon modes that exhibit both infrared and Raman activity. From Table 1, the Raman
active modes E’, E” and A1’ correspond to the modes shown in the Character Table E1 in
Appendix E. Furthermore, the relative Raman intensities Rxx and Rxy are given by curved
brackets () and square brackets [] respectively. The subscripts xx and xy denote the parallel-
polarized and cross-polarized configurations.
The Raman intensity measured in Raman spectroscopy studies is proportional to
2~
si ee R
where ie is the polarization vector of the incident laser light and se is the polarization vector
of the scattered laser light [12,13]. The reason for studying the polarization configurations of xx
is that vibration modes can be measured when polarizations of incident and scattered light are
both in x-direction, and are parallel to trilayer plane (xy-plane in Figure 5) [33]. If the incident
30
light is polarised in x-direction and the scattered light is polarised in y-direction, this cross-
polarized configuration (xy) can also be measured in experiment.
Furthermore, ~
R is a 3x3 Raman tensor and thus by calculating Rxx and Rxy intensities, these
values correspond, but are not equal, to tensor components of ~
R . The 3x3 Raman tensor is ~
R
and its components are given by ij , where i and j can take on x, y and z configurations as
follows [12]
z
y
x
zyxR
zzzyzx
yzyyyx
xzxyxx
)(~
si ee
(4.3)
For a phonon mode to be observed in experiment,
2~
si ee R must be non-zero and this is known
as the Raman selection rule. Since the Raman tensor is indicative of the crystal symmetry, we
can examine the Raman tensor of E’ and A1’ modes. The Raman tensor of the E’ mode is as
follows [12]
000
0
0
:' ac
ca
E (4.4)
From (4.4), we can see that yyxx and
yxxy . From the phg.dyn.out file in Appendix
B1, the Raman intensity values of 510491.5 yyxx RR reflect the properties of the Raman
tensor for E’ that yyxx . In addition, the Raman intensity value of 169.1xyR is non-zero
which is consistent with value cxy . This is also consistent with quadratic notation listed at
the end of the E’ row of the D3h character Table E1 in Appendix E.
Next, the Raman tensor of the A1’ is as follows [12]
c
a
a
A
00
00
00
:'1
(4.5)
Based on (4.5), we can see that yyxx and czz . From the phg.dyn.out file in Appendix
B1, the Raman intensities 04058.0 yyxx RR which reflects the properties of the Raman tensor
31
for E’ that yyxx . In addition, the Raman intensity 0xyR corroborates with the fact that
0xy . This is also consistent with quadratic notation listed at the end of the A1’ row of the D3h
character table in Appendix E. Thus, we can conclude that the Raman tensor reflects the MoS2
monolayer crystal symmetry.
By comparing the vibration modes of the MoS2 monolayer obtained from computation and the
modes recorded in literature (as shown in Table 2), the percentage difference between
calculated frequencies associated with E’’ [R], E’ [I + R] and A1’ [R] and their respective
literature values are as follows in Table 2.
Irreducible
Representation
Frequencies (cm-1) Literature
Frequencies (cm-1)
Percentage
Difference (%)
E’ (I + R) 0.00 0.00 [34] 0
A2’’(I) 0.00 0.00 [34] 0
E’’ (R) 287.57 287.38 [34] 0.1
E’ (I + R) 389.32 389.00 [34] 0.1
A1’ (R) 407.17 406.07 [34] 0.3
A2’’(I) 473.33 474.52 [34] 0.3
Table 2: Table of calculated phonon frequencies from theoretical literature [34] for MoS2
From Table 2, the percentage difference between computed values in literature and values
obtained from our calculations are in close agreement. Moreover, we should examine the
computed values with values obtained from Raman Spectroscopy experiments. Thus, by
comparing the experiment values with values obtained from computation in Quantum
ESPRESSO, one can arrive at Table 3.
Irreducible
Representation
Raman Shift
(cm-1)
Computed
Frequencies (cm-1)
Percentage
Difference (%)
E’ (I + R) 384.7 [30] 389.32 1
A1’ (R) 406.1 [30] 407.17 0.3
Table 3: Table of calculated phonon frequencies from experiment literature for MoS2
32
The Raman shift (frequencies) of the two modes E’ and A1’ are usually measured in
experimental studies due to its frequencies being close to the bulk E12g and A1g modes of
MoS2[35, 36]. Based on Table 3, the computed and experimental frequencies are close to within
a percentage difference of 1%. The consistency between our computed values and those
reported in theoretical and experimental literature indicates the appropriate use of DFPT in the
theoretical study of MoS2 Raman frequencies.
Aside from studying the Raman frequencies of the monolayer, the displacement representations
associated with the vibration modes are as follows in Figure 6. The modes with symbol E are
usually referred to in-plane or shear modes. On the other hand, the modes with symbol A are
often referred to as out-of-plane or breathing modes.
Figure 6: Eigenvectors associated with the phonon modes as seen from xz-plane.
The displacement vectors of 3 atoms in (a) and (b) of Figure 6 are all pointing in the same
direction. This is a characteristic of acoustic modes as the molybdenum and sulphur atoms
move in phase with each other. In (d), (e) and (f), the movement of molybdenum and sulphur
atoms are not in phase with each other and are known as optical modes. These modes can be
categorised into longitudinal (LO) and traverse optical modes (TO); (c) and (d) are longitudinal
optical modes, while (e) and (f) are out-of-plane optical modes [36].
4.2 Simulated Raman Spectra of 2TL MoS2
Having studied the Raman spectrum of monolayer molybdenum disulphide, we also studied
the Raman spectra of bilayer MoS2. The very first motivation for this project was to determine
the effect of different bilayer stackings of MoS2 on the phonon frequencies. We determined the
phonon frequencies associated with the MoS2 2TL (bilayer) for the following stacking
configurations using Quantum ESPRESSO and they are presented in Tables 4 and 5.
Prior to discussion of 2TL MoS2 simulation results, the space and point group of the 2H bilayer
should be discussed. The 2H stacking pattern of the bilayer belongs to the 13mp
space group
(a) E’[I+R] (b) A2” [I] (c) E” [R] (d) E’[I+R] (e) A
1’ [R] (f) A
2” [I]
33
and D3d point-group. There is a total of 6 irreducible representations associated with the D3d
point-group as shown in Table E2 of Appendix E. There is a total of 18 normal modes simulated
at 𝚪-point, of which 6 are degenerate. This reduces to 12 modes and their respective irreducible
representations are as follows [33]
ugug
MoSH EEAA 3333 212
2 (4.6)
Since the number of MoS2 layers is even, where N=2,4,6…, the irreducible representation can
also be given by the following formula in (4.4) [12, 34].
)(
2
3212
2ugug
MoSH EEAA
N
(4.7)
For bilayer MoS2, the number of layers is given by N = 2 and the resultant irreducible
representation is given by ugugMoSH EEAA 3333 212
2 . Aside from the irreducible
representation being consistent with the D3d point-group character Table E2 of Appendix E,
these 12 phonon modes can also be simulated within density functional perturbation theory as
implemented in Quantum ESPRESSO. The Raman intensities of the phonon modes associated
with the bilayer MoS2 are presented in the tables that follow. First, we study the 2H bilayer
configuration.
4.2.1 Raman Intensities of 2H Bilayer Configuration
The Raman intensities associated with the 2H bilayer are presented in Table 4.
In-plane modes 2H 0 24.81 286.35 287.70 387.07 387.43
(0) (0.001) (0) (0.0003) (0.02) (0)
[0] [0.001] [0] [0.0003] [0.02] [0]
Eu[I] Eg[R] Eu[I] Eg[R] Eg[R] Eu[I]
Out-of-plane
modes
2H 0 40.86 405.42 407.49 470.21 470.98
(0) (0.05) (0) (0.11) (0) (0.000078)
[0] [0] [0] [0] [0] [0]
A2u [I] A1g [R] A2u [I] A1g [R] A2u[I] A1g [R]
Table 4: Table of calculated phonon frequencies associated with in-plane and out-of-plane
modes for MoS2 bilayer 2H configuration
34
Like the case of monolayer MoS2 in Chapter 4.1, the values in curved brackets are associated
with relative Raman intensity (Rxx) and the square brackets are associated with the Raman
intensity (Rxy) in Table 4 and subsequent tables for Raman intensities of the bilayer.
By looking at the in-plane modes (in Table 4) of the MoS2 2H bilayer, the Raman intensities
Rxx and Rxy of the three Eg modes are non-zero, this is consistent with the D3d point group Table
E2 in Appendix E. Under the quadratic functions column of E2, x2 and xy are present for the Eg
mode. For the 2H polytype, since N is even for bilayer, the Raman tensors of the Eg modes are
as follows [12]
0
:
fd
fac
dca
Eg
(4.8)
For the Eg modes such as mode 4 in the phg.dyn.out file in Appendix B2, the Raman intensities
are 710796.1 yyxx RR . This is consistent with the Raman components in (4.8) such that
ayyxx . Furthermore, the calculated Raman intensity of zzR is negligible and this reflects
the component 0zz . The values of Rxy, Ryz and Rxz are 310924.1 , 710886.1 and 31002.2
respectively and these correspond to the fact that xy ,
yz and xz in (4.8) are non-zero.
Next, we look at the out-of-plane modes associated with the bilayer 2H polytype. The Raman
intensities Rxx of the three A1g modes are non-zero. This is consistent with the D3d point group
Table E2 in Appendix E; under the quadratic functions column of E2, x2 is present for the A1g
mode. For the 2H polytype, the Raman tensor for the A1g mode is as follows
c
a
a
A g
00
00
00
:1
(4.9)
For the A1g modes such as mode 6 in the phg.dyn.out file in Appendix B2, the Raman intensities
are 05235.0 yyxx RR . As Raman intensities are proportional to the Raman tensor components,
the Raman components in (4.9) such that ayyxx reflects the property of the Raman
intensities of the xx and yy polarization configurations. Furthermore, the calculated Raman
intensity of zzR is 410894.4 and this reflects the component non-zero property of the
35
component czz . The values of Rxy, Ryz and Rxz are all negligible and these correspond to the
fact that xy ,
yz and xz in (4.8) are all zero.
Having verified that the Raman intensity results are consistent with the symmetry group of 2H,
the next step is to compare the results with computational studies in literature and experiments.
In literature, it is common to refer to the high frequency Raman-active modes as E2g and A1g
and these are the Mulliken notation for the bulk configuration of MoS2[36].
Stacking
Configuration
E2g Frequency (cm-1) A1g Frequency (cm-1)
Computed Literature Expt. Computed Literature Expt.
2H 387.07 388.99[37] 383[38] 407.49 411.91[37] 408[38]
Table 5: Table of calculated phonon frequencies, literature frequencies and experiment
frequencies
By comparing the computed and literature frequencies for E2g and A1g modes, the percentage
differences are 0.5% and 1% respectively. Similarly, if we compare the computed and
experiment E2g and A1g modes, the percentage differences are 1% and 0.1%. The computed
values are in close agreement with values obtained from experimental and computational
studies found in literature.
As low frequency Raman modes are known to defer from one stacking configuration to another
[37,44], it is imperative to study the low frequency Raman-active modes. In literature, it is
common to refer to the low frequency Raman-active modes as S and B modes, and their
associated frequencies are represented in the table below.
Stacking
Configuration
S Frequency (cm-1) B Frequency (cm-1)
Computed Literature Expt. Computed Literature Expt.
2H 24.81 28.89[37] 22.54[39] 40.86 42.55[37] 40.03[39]
Table 6: Table of calculated phonon frequencies, literature frequencies and experiment
frequencies
By comparing the computed and literature frequencies for S and B modes, the percentage
differences are 10% and 4% respectively. Similarly, if we compare the computed and
experiment S and B modes, the percentage differences are 10% and 2%. The computed values
agree with values obtained from experimental and computational studies found in literature.
36
Aside from studying the low and high frequencies of 2H configuration, the displacement
representations associated with the vibration modes are as follows in Figure 7 and 8.
Figure 7: Eigenvectors associated with the in-plane vibration modes as seen from xz-plane
The displacement vectors of 6 atoms in (a) of Figure 7 are all pointing in the same direction.
This is a signature of acoustic modes as all atoms move in phase with each other. (b) to (f) of
Figure 7 are optical modes, and this characteristic is similar for Figure 8.
Figure 8: Eigenvectors associated with the out-of-plane vibration modes as seen from xz-plane
4.2.2 Raman Intensities of AB’ Bilayer Configuration
Having examined the 2H, configuration, we move on to study the AB’ configuration. AB’ is a
high-symmetry stacking configuration and the resultant intensity. According to the output from
Quantum ESPRESSO, the point group for AB’ is D3d and 13mp
space group. The Raman
intensities associated with the AB’ bilayer are in Table 7.
In-plane modes AB’ 0 20.47 286.35 287.3 387.35 388.04
(0) (0.06) (0) (0.0006) (0) (0.01)
[0] [0.06] [0] [0.0006] [0] [0.01]
Eu[I] Eg[R] Eu[I] Eg[R] Eg[R] Eu[I]
Out-of-plane
modes
AB’ 0 32.45 405.82 408.13 470.58 472.80
(0) (0.25) (0) (0.12) (0) (0.00096)
[0] [0] [0] [0] [0] [0]
A2u [I] A1g [R] A2u [I] A1g [R] A2u[I] A1g [R]
a) Eu[I] b) E
g[R] c) E
u[I] d) E
g[R] e) E
g[R] f) E
u[I]
a) A2u
[I] b) A1g
[R] c) A2u
[I] d) A1g
[R] e) A2u
[I] f) A1g
[R]
37
Table 7: Table of calculated phonon frequencies associated with in-plane and out-of-plane
modes for MoS2 bilayer AB’ configuration
By looking at the in-plane modes (in Table 7) of the MoS2 AB’ bilayer, the Raman intensities
Rxx and Rxy of the three Eg modes are non-zero, this is consistent with the D3d point group Table
E2 in Appendix E; under the quadratic functions column of E2, x2 and xy are present for the Eg
mode. For the 2H polytype, since N is even for bilayer, the Raman tensors of the Eg modes are
as follows [12]
0
:
fd
fac
dca
Eg
(4.10)
For the Eg modes such as mode 4 in the phg.dyn.out file in Appendix B3, the Raman intensities
are 1059.0 yyxx RR . This is consistent with the Raman components in (4.10) such that
ayyxx . Furthermore, the calculated Raman intensity of zzR is negligible and this reflects
the component 0zz . The values of Rxy, Ryz and Rxz are 310655.6 , 310922.7 and 410976.4
respectively and these correspond to the fact that xy ,
yz and xz in (4.10) are non-zero.
Next, we look at the out-of-plane modes associated with the bilayer AB’. The Raman intensities
Rxx of the three A1g modes are non-zero, this is consistent with the D3d point group Table E2 in
Appendix E; under the quadratic functions column of E2, x2 is present for the A1g mode. For
the 2H polytype, the Raman tensor for the A1g mode is as follows
c
a
a
A g
00
00
00
:1
(4.11)
For the A1g modes such as mode 6 in the phg.dyn.out file in Appendix B3, the Raman intensities
are 2530.0 yyxx RR . As Raman intensities are proportional to the Raman tensor components,
the Raman components in (4.11) such that ayyxx reflects the property of the Raman
intensities of the xx and yy polarization configurations. Furthermore, the calculated Raman
intensity of zzR is 410512.5 and this reflects the component non-zero property of the
component czz . The values of Rxy, Ryz and Rxz are all negligible and these correspond to the
fact that xy ,
yz and xz in (4.11) are all zero.
38
Having verified that the Raman intensity results are consistent with the symmetry group of 2H,
the next step is to compare the results with computational studies in literature and experiments.
In literature, it is common to refer to the high frequency Raman-active modes as E2g and A1g
and these are the Mulliken notation for the bulk configuration of MoS2[36].
Stacking
Configuration
E2g Frequency (cm-1) A1g Frequency (cm-1)
Computed Literature % Diff Computed Literature % Diff
AB’ 387.35 390.22[37] 0.7 408.13 412.36[37] 1
Table 8: Table of calculated phonon frequencies and literature frequencies
By comparing the computed and literature frequencies for E2g and A1g modes, the percentage
differences are 0.7% and 1% respectively. The computed values are in close agreement with
values obtained from computational studies found in literature. Next, we study the low
frequency Raman-active modes, the S and B modes, of AB’ stacking configuration. The results
are shown in table below.
Stacking
Configuration
S Frequency (cm-1) B Frequency (cm-1)
Computed Literature % Diff Computed Literature % Diff
AB’ 20.47 25.10[33] 20 32.45 35.80[33] 10
Table 9: Table of calculated phonon frequencies, literature frequencies and experiment
frequencies
By comparing the computed and literature frequencies for S and B modes, the percentage
differences are 20 % and 10% respectively. As the frequencies computed in literature were
done in VASP with different pseudopotentials, on the other hand, this project was done in
Quantum ESPRESSO with different pseudopotentials, that results in the discrepancy between
computed and literature values. Aside from studying the low and high frequencies of AB’
configuration, the displacement representations associated with the vibration modes are as
follows in Figure 7 and 8.
Figure 9: Eigenvectors associated with the in-plane vibration modes as seen from xz-plane.
a) Eu[I] b) E
g[R] c) E
u[I] d) E
g[R] e) E
g[R] f) E
u[I] f) E
u[I]
39
The displacement vectors of 6 atoms in (a) of Figure 9 are all in same direction and is a
signature of acoustic modes. (b) to (f) of Figure 9 are optical modes. The displacement
representation of out-of-plane modes are shown in Figure 10.
Figure 10: Eigenvectors associated with the out-of-plane vibration modes as seen from xz-
plane
4.2.3 Raman Intensities of 3R-like Bilayer Configuration
Next, we studied the 3R-like bilayer configuration and 3R bulk polytype belongs to the C3v
point group and R3m space group [40]. Before analysing the Raman intensities associated with
this configuration, the irreducible representation of this stacking configuration should be
ascertained. There is a total of 3 irreducible representations associated with the C3v point-group
as shown in Table E3 of Appendix E. There is a total of 18 normal modes simulated at 𝚪-point,
of which 6 are E modes (degenerate) and 6 are A1 modes which are non-degenerate. This
reduces to 12 modes and their respective irreducible representations are as follows
13 662 AE
MoSlikeR
(4.12)
For 3R-like bilayer MoS2, the resultant representation is 12 662 AEMoSTL . Aside from the
irreducible representation being consistent with the C3v point-group character Table E3 of
Appendix E, these 12 phonon modes can also be simulated within density functional
perturbation theory as implemented in Quantum ESPRESSO. The Raman intensities of the
phonon modes of the 3R-like bilayer MoS2 are presented in the Table 10.
In-plane
modes
3R-
like
0 26.68 285.98 288.85 387.24 388.21
(0) (0.01) (0.000004) (0.0004) (0.02) (0.00001)
[0] [0.01] [0.000004] [0.0004] [0.02] [0.00001]
f) Eu[I] a) A
2u[I] b) A
1g[R] c) A
2u[I] d) A
1g[R] e) A
2u[I] f) A
1g[R]
40
E[I + R] E[I + R] E[I + R] E[I + R] E[I + R] E[I + R]
Out of
plane
3R-
like
0 38.80 405.52 408.23 469.59 471.95
(0) (0.16) (0.0022) (0.12) (0.000044) (0.00076)
[0] [0] [0] [0] [0] [0]
A1[I + R] A1[I + R] A1[I + R] A1[I + R] A1[I + R] A1[I + R]
Table 10: Table of calculated phonon frequencies associated with in-plane and out-of-plane
modes for MoS2 bilayer 3R-like configuration
Examining the in-plane modes (in Table 10) of the MoS2 3R-like bilayer, the Raman intensities
Rxx and Rxy of the five E modes are all non-zero, this is consistent with the C3v point group,
without inversion symmetry, Table E3 in Appendix E; under the quadratic functions column
of E3, x2 and xy are present for the E mode.
As for the out-of-plane A1 modes, only the Rxx values are non-zero, this is also in agreement
with the Table E3 where the x2 is present in the quadratic functions column of row A1 while xy
is not present. Thus, the results are consistent with the symmetry of the C3v point group. The
Raman tensor associated with the E representation is as follows [41,42,43]
0
:
dd
dcc
dcc
E (4.13)
For the E modes such as mode 4 (26.68 cm-1) in the phg.dyn.out file in Appendix B4, the
Raman intensities are 310628.7 yyxx RR . This is consistent with the Raman components in
(4.13) such that cyyxx . In addition, the calculated Raman intensity of zzR is negligible
and this reflects the component 0zz . The values of Rxy, Ryz and Rxz are 0.01467, 41036.9
and 31080.1 respectively and these correspond to the fact that xy ,
yz and xz in (4.13) are non-
zero.
On the other hand, the Raman tensor associated with the A1 is [41,42,43]
b
a
a
A
00
00
00
:1
(4.14)
41
For the A1 modes such as mode 6 in the phg.dyn.out file in Appendix B4, the Raman intensities
are 1565.0 yyxx RR . As Raman intensities are proportional to the Raman tensor components,
the Raman components in (4.14) such that ayyxx reflects the property of the Raman
intensities of the xx and yy polarization configurations. Furthermore, the calculated Raman
intensity of zzR is 410036.5 and this reflects the component non-zero property of the
component bzz . The values of Rxy, Ryz and Rxz are all negligible and these correspond to the
fact that xy ,
yz and xz in (4.14) are all zero. From the Raman intensities collected, we have
verified that the results are consistent with the symmetry of the 3R-like MoS2 bilayer.
Having determined that the Raman intensity results are consistent with the symmetry group of
3R, we compare the results with computational studies in literature and experiments. In
literature, the Raman-active modes as E2g and A1g, Mulliken notation for the bulk configuration
of MoS2[36], are often used as benchmarks for comparison between theoretical and experiment
studies. This comparison is presented in Table 11.
Stacking
Configuration
E2g Frequency (cm-1) A1g Frequency (cm-1)
Computed Literature Expt. Computed Literature Expt.
3R-like 388.21 389.44[37] 385.00[44] 408.23 411.91[37] 407.23[44]
Table 11: Table of calculated phonon (high) frequencies, literature frequencies and experiment
frequencies
By comparing the computed and literature high frequencies for E2g and A1g modes, the
percentage differences are 0.3% and 0.9% respectively. Similarly, if we compare the computed
and experiment E2g and A1g modes, the percentage differences are 0.8% and 0.2% respectively.
The computed values are in close agreement with values obtained from experimental and
computational studies found in literature. Aside from the high frequency modes, the low
frequency Raman modes are tabulated as follows.
Stacking
Configuration
S Frequency (cm-1) B Frequency (cm-1)
Computed Literature Expt. Computed Literature Expt.
AB’ 26.68 28.89[37] 18.31[45] 32.45 35.80[37] 33.03[45]
Table 12: Table of calculated phonon (low) frequencies, literature frequencies and experiment
frequencies
42
By comparing the computed and literature values of the S and B frequencies, we obtained 8%
and 2% percentage differences respectively. On the other hand, the percentage differences of
between computed and experimental values for the two frequency modes are 37% and 16%
respectively. Despite the larger difference between computed and experimental values, the
presence of two peaks (one in plane and another out-of-plane) within the low frequency range
of 0 to 50 cm-1 shows that our computation agrees with experiment and computational literature.
Aside from studying the frequencies associated with the Raman-active modes, the
displacement representations associated with the various phonon modes are depicted in Figures
11 and 12.
Figure 11: Displacement representation of in-plane vibration modes of 3R-like MoS2 stacking
configuration
Figure 12: Displacement representation of out-of-plane vibration modes of 3R-like MoS2
stacking configuration.
The S and B modes reported in computational and experimental literature are associated with
the vibrations that we see in (b) E [I + R] and (b) A1[I + R] in Figures 11 and 12 respectively.
As for the high frequency E2g and A1g vibration modes reported in computational and
experimental literature, the frequencies of these modes are similar to the computed frequencies
associated with the displacement representations found in (f) E [I + R] and (d) A1 [I+R] of
Figures 11 and 12 respectively. It is coincidental that the A modes are all out-of-plane
vibrational modes and the E modes are all in-plane vibrational modes for the case of monolayer
and bilayer MoS2.
a) E[I+R] b) E[I+R] c) E[I+R] d) E[I+R] e) E[I+R] f) E[I+R]
a) A1[I+R] b) A
1[I+R] c) A
1[I+R] d) A
1[I+R] e) A
1[I+R] f) A
1[I+R]
43
4.3 Thermal Conductivities of 1TL MoS2
4.3.1 Temperature Dependence of 1TL MoS2 Thermal
Conductivity
The lattice thermal conductivity of 1TL MoS2 can be determined by studying the phonon-
phonon interaction processes at varying temperatures. Using the phono3py software package,
the lattice thermal conductivity of 1TL MoS2 at temperatures ranging from 0 to 1000 K in
increments of 10 K can be simulated. The calculated lattice thermal conductivity as a function
of temperature was determined with the help of phono3py and the following plot was obtained
as depicted in Figure 13. Prior to applying a logarithmic scale, the plot of lattice thermal
conductivity against temperature shows a rapid decline of thermal conductivity values for the
temperature range of 10 K to 1000 K.
Figure 13: Graph of Lattice Thermal Conductivity (W/mK) against Temperature (K)
The thermal conductivity at a temperature of 300 K is calculated to have a value of 50.484
W/mK. Based on literature in computational condensed matter studies, the values of thermal
conductivity of MoS2 monolayer calculated at room temperature are varied, and they can take
on values of 23.2 W/mK [46], 26.2 W/mK[47], 83 W/mK[48] and 103 W/mK[49].
In experimental studies of monolayer MoS2 thermal conductivity, the thermal conductivity
values are reported to be (34.5±4) W/mK[50]. The experimental determination of MoS2
monolayer thermal conductivity is smaller and in agreement with the computed value of 50.484
W/mK within a difference of 37%. Having determined the thermal conductivity values of the
44
monolayer at room temperature, we proceeded to study the temperature dependence of thermal
conductivity.
For the low-temperature regime, we will use Debye’s Model to approximate the thermal
conductivity trend observed across temperatures. The Debye temperature is given by the
following [30]
B
DD
k
(4.15)
The Debye temperature calculated is 687 K and this is based on the maximum phonon
frequency cut-off value of 14.3 THz calculated every temperature ranging from 0 to 1000 K at
intervals of 10 K. By plotting the accumulated thermal conductivity at 300 K against frequency,
the resultant graph is shown in Figure 14.
Figure 14: Graph of Accumulated Thermal Conductivity (W/mK) against Frequency (THz)
First, we examined the temperatures below Debye temperature ( D = 687K) and the graph and its
best-fit line (in green) are shown in Figure 15.
45
Figure 15: Graph of ln κ against ln T for temperatures below Debye temperature D .
The equation of the straight line is determined to be of the following
0517.12ln3915.1ln T
(4.16)
and from the gradient of the graph we obtain the exponent of T in the relation that governs the
temperature dependence of κ. We can rewrite (4.16) as κ being a function of temperature T as
3915.1
171390
T
(4.17)
Aside from studying the temperature dependence of thermal conductivity in the regime of
temperatures being less than Debye temperature D , we also studied the temperature
dependence of κ at temperatures much smaller than D to test the following relation from
literature [30].
T
To
e~ (4.18)
From literature, phonon lifetime is proportional to T
To
e where To is of the order of Debye
temperature[30]. If T
To
e~ and ~ , thus we can deduce that T
To
e~ which is shown in (4.18).
46
Using a non-linear model fit to the computed thermal conductivity for temperatures equal to
and below 90 K, we obtained the following plot.
Figure 16: Graph of κ against T for temperature less and equal to 90 K (T<< D ).
The coefficients of the model fit of T
b
ae to the computational data are 564.908a and
1991.23b . Although there is clear non-linear decreasing of with increasing temperature (for
temperatures DT ), the estimate of T0 of 31.23 K is one order of magnitude different from
the calculated Debye temperature. This discrepancy is because T
b
ae is inadequate to model
the temperature dependence of thermal conductivity at low temperatures. A better model may
take the form of this T
b
aeT [23] and this requires better algorithm to fit this model (nonlinear
least-squares algorithm in gnuplot does not converge for this model). Still, thermal conductivity
decays in the low temperature regime.
Next, we studied the temperature dependence of thermal conductivity for temperatures greater
than Debye temperature. By plotting ln κ against ln T, the following graph was obtained.
47
Figure 17: Graph of ln κ against ln T for temperatures above Debye temperature D .
The equation of the straight line is determined to be of the following
71325.9ln02489.1ln T (4.19)
and from the gradient of the graph we obtain the exponent of T in the relation that governs the
temperature dependence of κ. We can rewrite (4.19) as κ being a function of temperature T as
follows
02489.1
25.16535
T
(4.20)
Therefore, we noticed that κ ~ xT
1where x takes on the value of 1.02489 which is between 1
and 2, and this agrees with literature [30]. At temperatures greater than Debye temperature, the
phonon mode specific heat is not dependent on temperature and it follows the Dulong and Petit
law Bv NkC 3[51]. As temperature is high, phonon collision rate increases, relaxation time
decreases, so does thermal conductivity. In the case of simulations at high temperature, the
Umklapp process, three-phonon processes where total phonon momentum is not conserved, is
dominant or only process that we consider, then thermal conductivity and phonon lifetime is
approximately proportional to 1/T which is consistent with equation (3) of [47] and equation
(25.33) of [30].
48
4.3.2 Study of phonon lifetimes of 1TL MoS2
Aside from temperature influencing thermal conductivity, other phonon properties have a role
to play in determining thermal conductivity. We looked at the distribution of phonon lifetimes
at 300 K as a function of phonon frequency with the use of Seaborn Python visualisation
library. The frequency dependence of phonon lifetimes is plotted in the figure below.
Figure 18: Phonon lifetime measured in picoseconds (ps) as a function of phonon frequency
measured in THz
From Figure 18, the phonon lifetimes range from 2.85 ps to 50.9 ps and a cluster of phonon
modes centred at around 11.6 THz with a phonon lifetime of about 6.98 ps can be observed.
By magnifying the plot, we can generate a second phonon lifetime distributions plot below.
Figure 19: Phonon lifetime measured in ps as a function of phonon frequency measured in THz
49
In Figure 19, the phonon modes are represented by black dots on the coloured background.
Regions with high density of phonon modes are represented by red or orange. Based on the
above plot of phonon lifetime distributions, it is evident that there are 3 clusters. The first
cluster of phonon modes is centred about the point where phonon frequency is 12.4 THz and
the phonon lifetime is 4.75 ps. The second cluster has a centroid with a phonon frequency of
approximately 11.4 THz and phonon lifetime of 9.05 ps. Finally, the third cluster is centred
about the point 9.30 THz with a lifetime of 6.16 ps.
To identify the phonon modes that play a role in affecting thermal conductivity, one must first
examine the phonon density of states and dispersion plots of the 1TL MoS2 system. The phonon
density of states is plotted as follows in Figure 20.
Figure 20: (a) Phonon dispersion and (b) density of states plot for monolayer MoS2
From the density of states plot, the 3 highest peak frequencies are located at frequencies of 10.1
THz, 11.7 THz and 12.5 THz, these three peaks coincide with the cluster centroids that are
located at 9.30 THz, 11.4 THz and 12.4 THz in the phonon lifetime distribution plot of Figure
19. As phonon-phonon interaction occurs in these dense regions as shown in red or orange in
Figure 19, the Umklapp processes result in the relatively shorter lifetimes (~ 6.98ps) of phonons
that are in these clusters. Having determined the key phonon frequencies that are associated
with the phonons that participate in Umklapp scattering, we can now proceed to identify the
phonons that play a key role in the degradation of thermal conductivity, since they have low
lifetimes. The phonon band structure associated with the 1TL MoS2 system is shown in Figures
20 and 21.
(a) (b) ZO1
ZO2
LO2
TO2
TO1
LO1
ZA
TA LA
50
Figure 21: Phonon dispersion of monolayer MoS2 with categorization of phonon branches
From the band structure plot in Figure 21, the phonon frequencies associated with the optical
modes measured at 𝚪 point are 8.65 THz, 11.7 THz, 12.3 THz and 14.3 THz. The phonon band
structure between 𝚪 and M points is such that the acoustic phonon branches are split into LA,
TA and ZA branches. The optical phonon branches are split into longitudinal optical (LO) and
transverse optical (TO) modes. The out-of-plane optical modes (ZO) measured at 𝚪 point do
not split between 𝚪 and M points. Moreover, the phonon dispersion observed from Figure 21
is consistent with the phonon dispersion of monolayer MoS2 in theoretical literature [46,52].
By comparing the phonon frequencies in Figure 19, 20 and 21, the optical phonons with
frequencies ranging from 8.64 THz to 14.3 THz have short lifetimes (~2 to 10 ps). As thermal
conductivity is proportional to phonon lifetimes, the Umklapp scattering of optical phonons
results in relatively short lifetimes, these phonons contribute significantly to the degradation of
thermal conductivity with increasing temperature as observed in section 4.3.1.
ZA
TA LA
TO1
LO1
TO2
LO2
A1: ZO2
A2”: ZO1
E’
E”
51
4.4 Thermal Conductivities of 2TL MoS2
4.4.1 Temperature Dependence of 2TL MoS2 Thermal
Conductivity
Besides studying the thermal conductivity of monolayer MoS2, we have simulated the thermal
conductivity of the 2H and AB’ stacking configuration of bilayer (2TL) MoS2. The thermal
conductivity calculated at 300 K for these two stacking configurations are 45.611 W/mK and
43.506 W/mK respectively. According the first principles calculations in literature, the thermal
conductivity of MoS2 is 83 W/mK[31]. In experiments, the lattice thermal conductivity is
measured to have a value of (77 ± 25) W/mK[53] and 52 W/mK[54]. Although the computational
and experimental thermal conductivities are varied in literature, our computed value is still in
agreement with that reported in literature.
Next, we plotted the lattice thermal conductivity as a function of temperature for the above two
stacking configurations. The plots are shown in the figure below.
Figure 22: Lattice thermal conductivity as a function of temperature graphs for (a) 2H and (b)
AB’ bilayer MoS2
By comparing Figure 22 with Figure 13 (1TL MoS2), the graphs have a similar profile, and for
temperatures of approximately greater than 100 K, the natural logarithm of lattice thermal
conductivity is negatively correlated (linear) with the natural logarithm of temperature. Prior
to studying the temperature dependence of thermal conductivity of MoS2 bilayer, we plotted
the accumulated thermal conductivity at 300 K for 2H and AB’ stacking configurations and
they are shown in Figure 23.
(a) (b)
52
Figure 23: Accumulated thermal conductivity plots of (a) 2H and (b) AB’ bilayer MoS2
After plotting the accumulated thermal conductivity at 300 K for the above two stacking
configurations, we determined that the exact calculated maximum phonon frequencies are
14.196120 THz and 14.240817 THz for the 2H and AB’ stacking configurations respectively.
From these maximum phonon frequencies, we can then compute the Debye temperatures for
the respective configurations based on equation (4.15). The computed Debye temperatures of
the 2H and AB’ stackings are 681 K and 683 K respectively.
Having determined the Debye temperatures, we can now examine the temperature dependence
of thermal conductivity. From Figure 22 (a) and (b), there is clearly a non-linear relationship
between the natural logarithms of thermal conductivity and temperature. A linear model fit will
not work for temperatures DT . Thus, we tried to fit the model in (4.18) to our computed
thermal conductivity data for 2H, and a cubic model to fit the data for AB’.
Figure 24: Lattice thermal conductivity plots of (a) 2H and (b) AB’ bilayer MoS2 using (4.18)
as model to fit data
The coefficients for the model T
b
ae are 771.364a and 745.12b , and since b = To = 12.745,
we realise that this is much lower than the order of Debye temperature for the 2H configuration.
Similarly, the coefficients for the model of AB’ configuration results are 618.359a and
(a) (b)
(a) (b)
53
11933.2b . Both model fits of the 2H and AB’ results are show the inadequacy of (4.18) as a
model to describe the temperature dependence of thermal conductivity and phonon lifetime for
the low temperature regime of DT .
Instead of fitting a theoretical model such as the relation (25.40) in [30] to the computational
results, we can find the approximate temperature dependence of thermal conductivity by trying
out different non-linear models.
Figure 25: Lattice thermal conductivity plots of (a) 2H and (b) AB’ bilayer MoS2 using cubic
model to fit data.
The coefficients for the cubic model dcTbTaT 23 for 2H are 00311915.0a and
603559.0b , 3369.44c and 16.1587d . For the AB’ configuration, the coefficients of the
cubic model are 00310228.0a and 562878.0b , 2628.26c and 201.156d . From these
coefficients, we noticed that linear temperature dependence is dominant and due to the
coefficient c being largest amongst the weights a , b and c . However, the graphs of thermal
conductivity in Figure 25 have cubic temperature dependence as the coefficient a is non-zero.
Aside from temperature dependence of phonon lifetime, mode-dependent specific heat is also
a component of and is proportional to thermal conductivity according to (2.45). Moreover, the
observed T3 dependence of the thermal conductivity graphs in Figure 25, despite being less
significant compared to linear T dependence, is consistent with Debye T3 law. Low temperature
specific heat Cv has T3 dependence according to (23.27) of [23], this may explain the existence
of cubic nature of the thermal conductivity graphs plotted in Figure 25.
(a) (b)
54
4.4.2 Study of phonon lifetimes of 2TL MoS2
Having studied the temperature dependence of thermal conductivity on 2TL MoS2, we
examined the phonon lifetime distributions of the MoS2 bilayer at a temperature of 300 K. The
frequency dependence of phonon lifetimes is of the 2H and AB’ bilayer is as follows.
Figure 26: Phonon lifetime distribution plots of (a) 2H and (b) AB’ bilayer MoS2
The cluster centroid of the phonon modes of 2H bilayer MoS2 is located at a frequency of 11.7
THz and a lifetime of 8.38 ps. Similarly, the cluster of low lifetime phonons is centred around
the frequency of 11.7 THz and 7.34 ps. Both phonon lifetime distribution plots are similar to
the distribution plot obtained for monolayer MoS2 with phonon modes centred around higher
frequencies and lower lifetimes.
If we were to magnify both distribution plots in Figure 26, then the resultant plots are shown
in the figure below.
Figure 27: Phonon lifetime distribution plots of (a) 2H and (b) AB’ bilayer MoS2 showing three
prominent phonon clusters
(b) (a)
(a) (b)
55
Based on the above two plots of phonon lifetime distributions, it is evident that there are 3
clusters for each plot. The first cluster of phonon modes for 2H and AB’ bilayers is centred
about the point where phonon frequency is 12.4 THz and 12.4 THz respectively, and the
respective phonon lifetimes are 5.29 ps and 4.74 ps. The second cluster for 2H and AB’ bilayers
has a centroid with phonon frequencies of approximately 11.2 THz and 11.1 THz, and phonon
lifetimes of 11.2 ps and 9.43 ps respectively. Finally, the third cluster is centred about the
frequency points of 9.26 THz and 9.24 THz respectively, with a lifetime of 6.97 ps and 6.23
ps. Relatively shorter phonon lifetimes are the result of phonon-phonon scattering, and these
are the main processes occurring at these dense regions.
By looking at the phonon density of states (DOS), we can better understand the phonon
frequencies and the phonon modes responsible for limiting the thermal conductivity. The
phonon density of states of the 2H and AB’ bilayer is presented below.
Figure 28: Phonon density of states plots of (a) 2H and (b) AB’ bilayer MoS2
The phonon frequencies associated with the 4 DOS peaks of 2H configuration in Figure 28 (a)
are 10.1 THz, 10.7 THz, 11.4 THz and 12.5 THz respectively. From Figure 28 (b), the phonon
frequencies associated with the DOS peaks of AB’ are 10.0 THz, 10.7 THz, 11.5 THz and 12.4
THz. These density of states peaks located at the above frequencies correspond the same
frequencies obtained in the phonon lifetime distribution plots of Figure 27. The simulated
phonon band gap of the 2H and AB’ stacking configurations are 1.47 THz and 1.44 THz
respectively.
Next, we examined the phonon dispersion of the 2H and AB’ stacking configurations and the
obtained dispersion plots are as follows.
(a) (b)
56
Figure 29: Phonon dispersion plots of (a) 2H and (b) AB’ bilayer MoS2
The obtained phonon dispersion plot 29 (a) is consistent with plot in literature [47]. As both 2H
and AB’ configuration belong to the same 13mp
space group and D3d point-group, the phonon
band structures are largely similar. The optical phonon modes measured at 𝚪 point of 2H
stacking have associated phonon frequencies of 8.68 THz (E2g), 11.7 THz, 12.3 THz (A1g) and
14.3 THz. Similarly for the AB’ configuration, the associated frequencies of the optical phonon
modes are 8.64 THz, 11.7 THz, 12.4 THz and 14.3 THz. The band structure is also consistent
with the phonon density of states plot.
By matching the DOS and phonon dispersion plots of 2H bilayer MoS2 (see Fig. 30) using the
same vertical axis scale, the 3 clusters of low phonon lifetime phonons in Figure 27(a) are
identified to be optical phonon modes. The crossing of ZO1 and ZO2 out-of-plane phonon
branches correspond to the DOS peak observed at 12.5 THz and the cluster centred at 12.4 THz
and 5.29 ps in Figure 27(a). The crossing of LO2 and ZO1 phonon branches coincide with DOS
peak at 11.4 THz and the cluster centroid at 11.2 THz and 11.2 ps of Figure 27(a). Finally, the
combination of LO1 and TO1 phonon branches corroborates with the DOS peak at 10 THz and
the cluster centroid at 9.26 THz and 6.97 ps.
Figure 30: (a) Phonon dispersion and (b) density of states plot for 2H bilayer MoS2
ZA
TA LA
E2g
ZA
TA LA
E2g
ZA
TA LA
ZO1
A1g: ZO2
LO2
TO2
TO1
LO1
ZO1
A1g: ZO2
LO2
TO2
TO1
LO1
ZO1
A1g: ZO2
LO2
TO2
TO1
LO1 E2g
(a) (b)
(a) (b)
57
Again, we matched the DOS and phonon dispersion plots of AB’ bilayer MoS2 and conclude
that the 3 clusters of low phonon lifetime phonons in Figure 27(b) are optical phonon modes.
Figure 31: (a) Phonon dispersion and (b) density of states plot for AB’ bilayer MoS2
The crossing of ZO1 and ZO2 out-of-plane phonon branches (see Fig. 31a) correspond to the
DOS peak observed at 12.4 THz and the cluster centred at 12.4 THz and 4.74 ps in Figure
27(b). The crossing of LO2 and ZO1 phonon branches (see Fig. 31a) coincide with DOS peak
at 11.5 THz and the cluster centroid at 11.1 THz and 9.43 ps of Figure 27(b). Finally, the
combination of LO1 and TO1 phonon branches (see Fig. 31a) corroborates with the DOS peak
at 10 THz and the cluster centroid at 9.24 THz and 6.23 ps. In all, the high frequency optical
phonon modes have the shortest lifetimes, as they participate most in the phonon-phonon
scattering processes. As thermal conductivity is proportional to phonon lifetime, a skewed
distribution towards shorter phonon lifetimes will drastically reduce thermal conductivity.
Having examined the monolayer, bilayer 2H and AB’ configurations, we report that the optical
phonon modes have the shortest phonon lifetimes and the finite intrinsic thermal conductivity
can be attributed to the Umklapp scattering of these phonons. As temperature of the MoS2
system increases, the number of phonons increases with temperature (phonon occupation
number is proportional to temperature:
TkBkn )( )[23]. Thus, more optical phonons can undergo
scattering and proportion of phonons with short lifetimes increases. As the lifetime is
proportional to thermal conductivity, a high density of short lifetime optical phonons will
greatly decrease thermal conductivity. Hence, we conclude that the scattering of optical
phonons degrades the thermal conductivity.
ZO1
A1g: ZO2
LO2
TO2
TO1
LO1 E2g
(a) (b)
58
Chapter 5
“Time is the best appraiser of scientific work, and I am aware that an industrial discovery
rarely produces all its fruit in the hands of its first inventor.”
Louis Pasteur
Conclusion
We simulated the Raman spectra of monolayer molybdenum disulphide (MoS2) and 2H, AB’
and 3R-like stacking configurations of the MoS2 bilayer and report that the low-frequency ( <
50 cm-1) phonon modes are unique to each of the 3 stacking patterns. Low frequency Raman-
active modes are sensitive to changes in stacking order. By examining the high frequency
Raman-active modes of the monolayer and 3 bilayers, the reported frequencies are ~388 cm-1
and ~407 cm-1.
Furthermore, we have verified the symmetry of the 3 stacking configurations based on the
calculated Raman intensities obtained for different polarisation configurations We noticed that
the Raman tensors associated with the different phonon modes are also consistent with the
calculated Raman intensities. In all, the computed Raman intensities of the 3 stacking
59
configurations of the bilayer reflect their respective symmetry groups stated in literature and is
corroborated by other experimental studies on twisted bilayer Raman spectra.
We also studied the thermal conductivity of monolayer Molybdenum disulphide (MoS2) and
the thermal conductivity calculated yields a value of 50.484 W/mK. Theoretical studies of
MoS2 thermal conductivity yields varied range of thermal conductivity values; our reported
value is larger than the experimental value of 34.5±4 W/mK [50]. We have looked at the
temperature dependence of thermal conductivity and can be modelled by the reciprocal of
temperature T for temperatures above Debye temperature of 687K.
We also determined the thermal conductivity of bilayer MoS2 for stacking patterns of 2H and
AB’ and the values are 45.6 Wm-1K-1 and 43.5 Wm-1K-1 respectively. The Debye temperatures
of these two stacking patterns are 681 K and 683 K respectively. For temperatures larger than
Debye temperature, the thermal conductivities of monolayer and bilayer decrease with
increasing temperature according to a T
1~ model. The degradation of thermal conductivity
with increasing temperature can be attributed to the scattering of optical phonons.
60
Appendix A1 (Input Files for Monolayer Structure Optimization)
generate )(rinn
Evaluate HKS
Solve Kohn-Sham Equations
)(
)()(
r
rr
i
iiiKS
new
H
)(routn
Calculate forces and update ion positions
Total Energy E[n(r)]
Schematic of Density Functional Theory Self-Consistent Iteration
vcrelax.in file for MoS2 1TL
&control
outdir='./tmp1' //temporary files/output files location
prefix='mos2tl1' //prepended to I/O filenames
calculation='vc-relax' //variable-cell relaxation
restart_mode='from_scratch', //from scratch, normal pwscf
pseudo_dir = './' //location of pseudopotential files
/
&system
nat= 3 //number of atoms in unit cell
ibrav = 4, //Bravais-lattice index – Hexagonal and Trigonal P
celldm(1) = 5.971914879 //a: in-plane lattice parameter
celldm(3) = 4.182263 //c/a: ratio of c to a
ntyp= 2, //2 types of atoms per unit cell
ecutwfc = 65.0, ecutrho = 550, //KE cut-off (Ry) and Charge-density cut-off
occupations='fixed', //for insulators with a gap
/
&electrons
conv_thr = 1.0e-10 //convergence threshold for self-consistency (est E error)
mixing_beta = 0.1 //mixing factor for self-consistency
/
&ions
/
&CELL
cell_dynamics = 'bfgs' //Broyden-Fletcher-Goldfarb-Shanno quasi-newton
algorithm (default)
press = 0.00 //Target pressure [KBar] in a variable-cell md or
relaxation run
cell_factor=1.8 //should exceed maximum linear contraction of cell
/
ATOMIC_SPECIES
Mo 95.94 Mo.pz-n-nc.UPF
Converged nin - nout < tol
Not Converged
Not Converged Converged
61
S 32.065 S.pz-n-nc.UPF
ATOMIC_POSITIONS crystal //crystal/fractional coordinates (relative coordinates of
the primitive lattice)
Mo 0.33333333 0.66666666 0.50000000000000
S 0.66666666 0.33333333 0.38096223300000
S 0.66666666 0.33333333 0.61903776700000
K_POINTS automatic //generate uniform grid of K-points: in Monkhorst Pack
grid
17 17 1 0 0 0
62
Appendix A2 (Input Files for Bilayer Structure Optimization)
vcrelax.in file for MoS2 2TL in 2H Configuration &control
outdir='./tmp1' //temporary files/output files location
prefix='mos2tl1' //prepended to I/O filenames
calculation='vc-relax' //variable-cell relaxation
restart_mode='from_scratch', //from scratch, normal pwscf
pseudo_dir = './' //location of pseudopotential files
forc_conv_thr = 1.0d-4 //
/
&system
nat= 6 //number of atoms in unit cell
ibrav = 4, //Bravais-lattice index – Hexagonal and Trigonal P
celldm(1) =5.986069782 //In-plane lattice parameter (a)
celldm(3) =7.961252964 //ratio of c to a (c/a)
ntyp= 2 //2 types of atoms per unit cell
ecutwfc = 65.0 //KE cutoff, ecutrho is default 4 * ecutwfc
ecutrho = 550 //KE cutoff for charge density
occupations = 'fixed', //For insulators with gap
/
&electrons
electron_maxstep= 450 //Maximum number of iterations in a scf step
mixing_beta = 0.1 //mixing factor for self-consistency
conv_thr = 1.0d-10 //convergence threshold for self-consistency (est E error)
/
&ions
/
&CELL
cell_dynamics = 'bfgs' //Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-
newton algorithm (default)
press = 0.00 //Target pressure [KBar] in a variable-cell md or
relaxation run
cell_factor=1.8 //should exceed maximum linear contraction of cell
/
ATOMIC_SPECIES
Mo 95.94 Mo.pz-n-nc.UPF
S 32.065 S.pz-n-nc.UPF
ATOMIC_POSITIONS crystal //crystal/fractional coordinates (relative coordinates of
the primitive lattice)
S 0.333333330 0.666666670 0.682485836
Mo 0.666666670 0.333333330 0.619712926
S 0.333333330 0.666666670 0.556795882
S 0.666666670 0.333333330 0.443204118
Mo 0.333333330 0.666666670 0.380287074
S 0.666666670 0.333333330 0.317514164
K_POINTS (automatic)
17 17 1 0 0 0 //Monkhorst-Pack K-point mesh
63
Appendix A3 (Input Files for Phonon Frequencies and Dynamical
Matrix Calculations )
phg.dyn.in file for MoS2 2TL in 2H Configuration
&input
fildyn='mos2tl2.dyn',
asr='crystal'
q(1)=0.0 , q(2)=0.0 , q(3)=0.0
/
phg.in file for MoS2 2TL in 2H Configuration
mos2tl2
&inputph
outdir='./tmp1' //directory containing input, output and scratch files
prefix='mos2tl2' //prepended to input and output filenames
fildyn='mos2tl2.dyn', //file where dynamical matrix is written
alpha_mix(1)=0.2 // mixing factor for updating the scf potential
tr2_ph=1.0d-19, //threshold for self-consistency
lraman=.true., //if true, calculate non-resonant Raman coefficients
epsil=.true., //if true, in a q=0 calculation for a non-metal the macroscopic
dielectric constant of the system is computed
amass(1)=95.94,
amass(2)=32.065,
/
0.0 0.0 0.0
64
Appendix B1 (Output Files for Monolayer MoS2 Raman
Intensities)
Monolayer Molybdenum Disulphide
Program DYNMAT v.5.0.2 (svn rev. 9392) starts on 16Mar2017 at 7:36:52
This program is part of the open-source Quantum ESPRESSO suite
for quantum simulation of materials; please cite
"P. Giannozzi et al., J. Phys.:Condens. Matter 21 395502 (2009);
URL http://www.quantum-espresso.org",
in publications or presentations arising from this work. More details at
http://www.quantum-espresso.org/quote.php
Parallel version (MPI), running on 16 processors
R & G space division: proc/nbgrp/npool/nimage = 16
Reading Dynamical Matrix from file mos2tl2.dyn
...Force constants read
...epsilon and Z* read
...Raman cross sections read
Acoustic Sum Rule: || Z*(ASR) - Z*(orig)|| = 0.109213E-02
Acoustic Sum Rule: ||dyn(ASR) - dyn(orig)||= 0.173313E-03
A direction for q was not specified:TO-LO splitting will be absent
Polarizability (A^3 units)
multiply by 0.410743 for Clausius-Mossotti correction
55.828864 0.000000 0.000000
0.000000 55.828864 0.000000
0.000000 0.000000 5.792164
IR activities are in (D/A)^2/amu units
Raman activities are in A^4/amu units
multiply Raman by 0.168710 for Clausius-Mossotti correction
mo [cm-1] [THz] IR Rxx_inten Rxy_inten Ryz_inten Rxz_inten Ryy_inten Rzz_inten
1 0.00 0.0000 0.0000 0.8680E+11 0.6251E+12 0.1507E-19 0.1553E-19 0.8680E+11 0.0000E+00
2 0.00 0.0000 0.0000 0.7361E+11 0.3811E+09 0.3569E-21 0.1546E-20 0.7361E+11 0.7009E-20
3 0.00 0.0000 0.0000 0.6856E+11 0.3032E+11 0.3241E-20 0.1545E-20 0.6856E+11 0.5504E-20
4 287.57 8.6212 0.0000 0.2532E-32 0.6322E-34 0.2207E-06 0.8970E-04 0.2062E-32 0.6103E-34
5 287.57 8.6212 0.0000 0.1414E-32 0.1044E-32 0.8970E-04 0.2207E-06 0.1143E-32 0.2349E-36
6 389.32 11.6716 1.1568 0.5491E-04 0.1169E-01 0.9847E-36 0.6271E-35 0.5491E-04 0.1181E-32
7 389.32 11.6716 1.1568 0.1169E-01 0.5491E-04 0.4687E-35 0.5164E-37 0.1169E-01 0.3170E-34
8 407.17 12.2066 0.0000 0.4058E-01 0.1729E-31 0.3878E-38 0.4773E-35 0.4058E-01 0.7729E-03
9 473.33 14.1902 0.0114 0.1747E-31 0.4861E-33 0.4518E-39 0.1427E-35 0.1747E-31 0.2730E-33
DYNMAT : 0.00s CPU 0.21s WALL
This run was terminated on: 7:36:52 16Mar2017
=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=
65
Appendix B2 (Output Files for Bilayer 2H MoS2 Raman
Intensities)
Bilayer Molybdenum Disulphide of 2H Configuration
Program DYNMAT v.5.0.2 (svn rev. 9392) starts on 14Mar2017 at 9:13:46
This program is part of the open-source Quantum ESPRESSO suite
for quantum simulation of materials; please cite
"P. Giannozzi et al., J. Phys.:Condens. Matter 21 395502 (2009);
URL http://www.quantum-espresso.org",
in publications or presentations arising from this work. More details at
http://www.quantum-espresso.org/quote.php
Parallel version (MPI), running on 16 processors
R & G space division: proc/nbgrp/npool/nimage = 16
Reading Dynamical Matrix from file mos2tl2.dyn
...Force constants read
...epsilon and Z* read
...Raman cross sections read
Acoustic Sum Rule: || Z*(ASR) - Z*(orig)|| = 0.162862E-03
Acoustic Sum Rule: ||dyn(ASR) - dyn(orig)||= 0.410148E-03
A direction for q was not specified:TO-LO splitting will be absent
Polarizability (A^3 units)
multiply by 0.393836 for Clausius-Mossotti correction
114.653670 0.000000 0.000000
0.000000 114.653670 0.000000
0.000000 0.000000 12.265689
IR activities are in (D/A)^2/amu units
Raman activities are in A^4/amu units
multiply Raman by 0.155107 for Clausius-Mossotti correction
mode [cm-1] [THz] IR Rxx_inten Rxy_inten Ryz_inten Rxz_inten Ryy_inten Rzz_inten
1 -0.00 -0.0000 0.0000 0.1031E-15 0.2630E-18 0.1984E-17 0.1888E-19 0.5186E-16 0.8524E-18
2 0.00 0.0000 0.0000 0.9685E-17 0.2626E-16 0.8912E-19 0.2533E-16 0.5840E-17 0.6504E-19
3 0.00 0.0000 0.0000 0.7232E-16 0.4502E-18 0.2713E-16 0.1125E-18 0.2434E-17 0.1256E-18
4 24.81 0.7437 0.0000 0.1796E-06 0.1924E-02 0.1886E-06 0.2020E-02 0.1796E-06 0.1054E-32
5 24.81 0.7437 0.0000 0.1924E-02 0.1796E-06 0.2020E-02 0.1886E-06 0.1924E-02 0.5387E-31
6 40.86 1.2249 0.0000 0.5235E-01 0.3177E-31 0.3147E-31 0.3328E-32 0.5235E-01 0.4894E-03
7 286.35 8.5844 0.0000 0.7590E-30 0.1909E-30 0.3734E-30 0.9268E-31 0.6537E-30 0.8176E-35
8 286.35 8.5844 0.0000 0.2083E-31 0.4773E-31 0.1135E-31 0.8505E-32 0.2231E-31 0.3818E-36
9 287.70 8.6250 0.0000 0.2617E-03 0.4052E-03 0.1093E-03 0.1692E-03 0.2617E-03 0.7954E-35
10 287.70 8.6250 0.0000 0.4052E-03 0.2617E-03 0.1692E-03 0.1093E-03 0.4052E-03 0.4131E-36
11 387.07 11.6041 0.0000 0.2067E-01 0.1686E-01 0.2106E-05 0.1718E-05 0.2067E-01 0.8760E-33
12 387.07 11.6041 0.0000 0.1686E-01 0.2067E-01 0.1718E-05 0.2106E-05 0.1686E-01 0.1024E-33
13 387.43 11.6147 2.2822 0.1969E-28 0.3526E-27 0.1794E-32 0.3643E-31 0.1623E-28 0.6725E-33
14 387.43 11.6147 2.2822 0.1320E-27 0.2345E-27 0.1357E-31 0.2595E-31 0.1386E-27 0.3428E-33
15 405.42 12.1543 0.0007 0.1176E-27 0.1884E-32 0.8978E-36 0.1310E-36 0.1109E-27 0.1711E-29
16 407.49 12.2163 0.0000 0.1098E+00 0.1732E-31 0.1080E-35 0.1786E-35 0.1098E+00 0.1650E-02
17 470.21 14.0966 0.0338 0.3120E-29 0.8737E-33 0.1301E-37 0.9584E-37 0.3049E-29 0.3447E-32
18 470.98 14.1196 0.0000 0.7758E-04 0.1581E-33 0.8755E-37 0.1556E-37 0.7758E-04 0.1007E-06
DYNMAT : 0.01s CPU 0.18s WALL
This run was terminated on: 9:13:46 14Mar2017
=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=
66
Appendix B3 (Output Files for Bilayer AB’ MoS2 Raman
Intensities)
Bilayer Molybdenum Disulphide of AB’ Configuration
Program DYNMAT v.5.0.2 (svn rev. 9392) starts on 23Mar2017 at 20:21:59
This program is part of the open-source Quantum ESPRESSO suite
for quantum simulation of materials; please cite
"P. Giannozzi et al., J. Phys.:Condens. Matter 21 395502 (2009);
URL http://www.quantum-espresso.org",
in publications or presentations arising from this work. More details at
http://www.quantum-espresso.org/quote.php
Parallel version (MPI), running on 16 processors
R & G space division: proc/nbgrp/npool/nimage = 16
Reading Dynamical Matrix from file mos2tl2.dyn
...Force constants read
...epsilon and Z* read
...Raman cross sections read
Acoustic Sum Rule: || Z*(ASR) - Z*(orig)|| = 0.259099E-03
Acoustic Sum Rule: ||dyn(ASR) - dyn(orig)||= 0.335382E-03
A direction for q was not specified:TO-LO splitting will be absent
Polarizability (A^3 units)
multiply by 0.391316 for Clausius-Mossotti correction
115.930361 0.000000 0.000000
0.000000 115.930361 0.000000
0.000000 0.000000 12.279040
IR activities are in (D/A)^2/amu units
Raman activities are in A^4/amu units
multiply Raman by 0.153128 for Clausius-Mossotti correction
mode [cm-1] [THz] IR Rxx_inten Rxy_inten Ryz_inten Rxz_inten Ryy_inten Rzz_inten
1 -0.00 -0.0000 0.0000 0.4750E-14 0.1391E-17 0.2050E-16 0.1079E-18 0.1287E-14 0.7582E-17
2 0.00 0.0000 0.0000 0.1509E-13 0.1749E-14 0.1551E-14 0.1285E-15 0.2630E-13 0.4656E-18
3 0.00 0.0000 0.0000 0.4326E-17 0.8208E-15 0.3035E-19 0.4992E-16 0.5732E-16 0.3156E-18
4 20.47 0.6136 0.0000 0.1059E+00 0.6655E-02 0.7922E-02 0.4976E-03 0.1059E+00 0.2009E-32
5 20.47 0.6136 0.0000 0.6655E-02 0.1059E+00 0.4976E-03 0.7922E-02 0.6655E-02 0.3795E-31
6 32.45 0.9728 0.0000 0.2530E+00 0.5023E-30 0.3525E-31 0.3291E-31 0.2530E+00 0.5512E-03
7 286.35 8.5845 0.0000 0.8274E-30 0.1496E-30 0.4024E-30 0.6836E-31 0.8408E-30 0.1344E-35
8 286.35 8.5845 0.0000 0.6554E-30 0.1526E-31 0.3795E-30 0.1056E-31 0.8143E-30 0.4280E-34
9 287.30 8.6131 0.0000 0.2229E-03 0.9460E-03 0.1150E-03 0.4882E-03 0.2229E-03 0.1794E-34
10 287.30 8.6131 0.0000 0.9460E-03 0.2229E-03 0.4882E-03 0.1150E-03 0.9460E-03 0.4116E-34
11 387.35 11.6124 2.2129 0.1320E-28 0.9624E-29 0.2324E-31 0.1551E-31 0.1681E-28 0.9739E-33
12 387.35 11.6124 2.2129 0.8906E-28 0.2966E-27 0.1277E-30 0.4429E-30 0.8188E-28 0.5534E-33
13 388.04 11.6332 0.0000 0.9582E-02 0.1662E-01 0.1445E-04 0.2507E-04 0.9582E-02 0.6593E-33
14 388.04 11.6332 0.0000 0.1662E-01 0.9582E-02 0.2507E-04 0.1445E-04 0.1662E-01 0.1146E-32
15 405.82 12.1662 0.0003 0.4113E-28 0.4829E-32 0.2963E-36 0.1004E-34 0.4027E-28 0.7215E-30
16 408.13 12.2354 0.0000 0.1193E+00 0.2658E-31 0.7870E-35 0.2209E-35 0.1193E+00 0.2114E-02
17 470.58 14.1076 0.0289 0.3450E-29 0.9543E-33 0.6620E-37 0.1937E-36 0.3450E-29 0.7877E-33
18 472.80 14.1741 0.0000 0.9606E-03 0.3171E-33 0.6548E-35 0.2273E-35 0.9606E-03 0.2920E-07
DYNMAT : 0.01s CPU 0.33s WALL
This run was terminated on: 20:21:59 23Mar2017
=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=
67
Appendix B4 (Output Files for 3R-like Bilayer MoS2 Raman
Intensities)
Bilayer Molybdenum Disulphide of 3R-like Configuration
Program DYNMAT v.5.0.2 (svn rev. 9392) starts on 15Mar2017 at 15:35:16
This program is part of the open-source Quantum ESPRESSO suite
for quantum simulation of materials; please cite
"P. Giannozzi et al., J. Phys.:Condens. Matter 21 395502 (2009);
URL http://www.quantum-espresso.org",
in publications or presentations arising from this work. More details at
http://www.quantum-espresso.org/quote.php
Parallel version (MPI), running on 16 processors
R & G space division: proc/nbgrp/npool/nimage = 16
Reading Dynamical Matrix from file mos2tl2.dyn
...Force constants read
...epsilon and Z* read
...Raman cross sections read
Acoustic Sum Rule: || Z*(ASR) - Z*(orig)|| = 0.221301E-03
Acoustic Sum Rule: ||dyn(ASR) - dyn(orig)||= 0.593970E-03
A direction for q was not specified:TO-LO splitting will be absent
Polarizability (A^3 units)
multiply by 0.389900 for Clausius-Mossotti correction
116.593113 0.000000 0.000000
0.000000 116.593113 0.000000
0.000000 0.000000 12.410076
IR activities are in (D/A)^2/amu units
Raman activities are in A^4/amu units
multiply Raman by 0.152022 for Clausius-Mossotti correction
mode [cm-1] [THz] IR Rxx_inten Rxy_inten Ryz_inten Rxz_inten Ryy_inten Rzz_inten
1 -0.00 -0.0000 0.0000 0.1522E+12 0.6716E+09 0.1674E+08 0.7345E+05 0.1538E+12 0.3486E+04
2 0.00 0.0000 0.0000 0.2251E+10 0.5213E+12 0.2501E+06 0.5701E+08 0.2323E+10 0.4693E+03
3 0.00 0.0000 0.0000 0.2343E+11 0.1887E+06 0.1206E+04 0.2064E+02 0.2144E+11 0.7441E+08
4 26.68 0.7999 0.0002 0.7628E-02 0.1467E-01 0.9360E-03 0.1800E-02 0.7628E-02 0.2553E-32
5 26.68 0.7999 0.0002 0.1467E-01 0.7628E-02 0.1800E-02 0.9360E-03 0.1467E-01 0.4581E-33
6 38.80 1.1631 0.0001 0.1565E+00 0.1923E-27 0.8640E-34 0.1624E-32 0.1565E+00 0.5036E-03
7 285.98 8.5734 0.0001 0.4915E-05 0.3723E-05 0.1662E-04 0.1259E-04 0.4915E-05 0.4263E-36
8 285.98 8.5734 0.0001 0.3723E-05 0.4915E-05 0.1259E-04 0.1662E-04 0.3723E-05 0.2758E-34
9 288.85 8.6596 0.0000 0.6655E-03 0.2105E-03 0.3752E-03 0.1187E-03 0.6655E-03 0.2637E-34
10 288.85 8.6596 0.0000 0.2105E-03 0.6655E-03 0.1187E-03 0.3752E-03 0.2105E-03 0.8358E-35
11 387.24 11.6093 2.1994 0.1900E-01 0.1329E-01 0.1522E-06 0.1065E-06 0.1900E-01 0.1839E-33
12 387.24 11.6093 2.1994 0.1329E-01 0.1900E-01 0.1065E-06 0.1522E-06 0.1329E-01 0.4097E-33
13 388.21 11.6383 0.0216 0.9520E-05 0.1033E-04 0.1488E-04 0.1615E-04 0.9520E-05 0.2073E-33
14 388.21 11.6383 0.0216 0.1033E-04 0.9520E-05 0.1615E-04 0.1488E-04 0.1033E-04 0.2296E-33
15 405.52 12.1571 0.0007 0.2165E-02 0.2485E-29 0.3713E-35 0.2786E-35 0.2165E-02 0.2843E-04
16 408.23 12.2385 0.0001 0.1217E+00 0.1510E-27 0.2256E-35 0.3596E-35 0.1217E+00 0.1678E-02
17 469.59 14.0779 0.0305 0.4407E-04 0.5578E-31 0.3088E-38 0.2637E-37 0.4407E-04 0.2529E-06
18 471.95 14.1486 0.0064 0.7573E-03 0.9147E-30 0.4565E-36 0.5370E-38 0.7573E-03 0.3439E-07
DYNMAT : 0.01s CPU 0.07s WALL
This run was terminated on: 15:35:17 15Mar2017
=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=
68
Appendix C1
NEW DOCUMENTATION
Running VASP-Phono3py for MoS2 Monolayer
Adapted from: https://docs.it4i.cz/salomon/software/chemistry/phono3py/
1) In same directory of POTCAR, POSCAR, INCAR and KPOINTS, load phono3py
module using the following (always load module at start of any phono3py calculation):
2) On GRC cluster:
$ module load phono3py/1.11.7.20-intel-2016.01-Python-
2.7.12
On NSCC cluster:
$ source /home/projects/c2dmatproj/easybuild/env.sh
$ module load phono3py/1.11.7.20-imodule load
$ phonopy/1.11.8-intel-2017a-Python-2.7.12ntel-2017a-
Python-2.7.12
3) To locate phono3py module: $ module avail
phono3py/1.11.7.20-intel-2016.01-Python-2.7.12
4) Create supercell size of 3x3x1 and displacement amplitude of 0.09Å (second-order and
third-order force constants of same supercell size):
$ phono3py -d --amplitude="0.09" --dim="3 3 1" --dim_fc2="3 3 1" -c POSCAR If FC2 is not required: $ phono3py -d --amplitude="0.09" --dim="3 3 1" -c POSCAR
5) Check for number of displacements (POSCAR-XXXXX files), for MoS2 monolayer is
365 and 5 additional, (POSCAR_FC2-XXXXX), displacements
69
6) Write Bash script, prepare.sh, for 365 displacements:
#!/bin/bash P=`pwd` # number of displacements poc=9 for i in `seq 1 $poc `; do cd $P mkdir disp-0000"$i" cd disp-0000"$i" cp ../KPOINTS . cp ../INCAR . cp ../POTCAR . cp ../POSCAR-0000"$i" POSCAR echo $i done poc=99 for i in `seq 10 $poc `; do cd $P mkdir disp-000"$i" cd disp-000"$i" cp ../KPOINTS . cp ../INCAR . cp ../POTCAR . cp ../POSCAR-000"$i" POSCAR echo $i done poc=365 for i in `seq 100 $poc `; do cd $P mkdir disp-00"$i" cd disp-00"$i" cp ../KPOINTS . cp ../INCAR . cp ../POTCAR . cp ../POSCAR-00"$i" POSCAR echo $i done
7) Prepare directories containing POSCAR, POTCAR, INCAR KPOINTs of 365
displacements in disp-XXXXX (folder name)
$ ./prepare.sh
8) If permission is denied for running script:
70
$ chmod +x ./prepare.sh
9) Ensure have the following files in each disp-XXXXX directory before running VASP:
1) CONTCAR 2) POTCAR 3) INCAR 4) KPOINTS
10) Write BASH script, submit.sh, to handle all 365 job submissions.
#!/bin/bash P=`pwd` # number of displacements poc=9 for i in `seq 1 $poc `; do cd $P cd disp-0000"$i" cp ../job.lsf . bsub < job.lsf echo $i done poc=99 for i in `seq 10 $poc `; do cd $P cd disp-000"$i" cp ../job.lsf . bsub < job.lsf echo $i done poc=365 for i in `seq 100 $poc `; do cd $P cd disp-00"$i" cp ../job.lsf . bsub < job.lsf echo $i done
11) To submit and run all 365 jobs:
$ ./submit.sh
12) Repeat steps 5 to 10 for second order force constants (POSCAR_FC2-XXXXX) and
name bash scripts submitfc2.sh and preparefc2.sh.
71
13) Post-processing with phono3py, collection of vasprun.xml files and create
FORCES_FC2 and FORCES_FC3 file:
$ phono3py --cf2 disp_fc2-{00001..00005}/vasprun.xml
$ phono3py --cf3 disp-{00001..00365}/vasprun.xml
14) Ensure POSCAR for unitcell, disp_fc2.yaml, disp_fc3.yaml, FORCES_FC2 and
FORCES_FC3 are present in the directory before proceeding.
15) Create fc3.hdf5 and fc2.hdf5: $ phono3py --dim="3 3 1" --sym_fc3r --sym_fc2 --tsym -c POSCAR
16) Using 21x21x1 sampling mesh, lattice thermal conductivity is calculated by
$ phono3py --dim="3 3 1" -c POSCAR --mesh="21 21 1" --fc2
--fc3 --br
kappa-m21211.hdf5 is written as the result. The lattice thermal conductivity is
calculated as 50.484W/mK at 300 K.
17) Ensure new environment modules and python packages/settings have been loaded:
$ touch ~/.lmod
18) Run ipython: $ ipython
19) In python shell: In [1]: import h5py In [2]: f = h5py.File("kappa-m21211.hdf5") In [3]: f['kappa'][30] Out[3]:ipyth In [4]: exit
20) Accumulated lattice thermal conductivity is calculated with ‘kaccum’ script: $ kaccum --mesh="21 21 1" POSCAR kappa-m21211.hdf5 |tee kaccum21211.dat
21) We use the following script (kaccum21211.p), is in same folder as kaccum21211.dat,
to generate accumulated lattice thermal conductivity plot: # Gnuplot script file for plotting data in file "kaccum21211.dat" # This file is called kaccum21211.p set autoscale # scale axes automatically unset log # remove any log-scaling unset label # remove any previous labels set xtic auto # set xtics automatically set ytic auto # set ytics automatically set title "Accumulated Thermal Conductivity (W/mK) against Frequency (THz)" set xlabel "Frequency (THz)" set ylabel "Accumulated Thermal Conductivity (W/mK)" plot "kaccum21211.dat" i 30 u 1:2 w l, "kaccum21211.dat" i 30 u 1:8 w l
72
22) Generated plot of accumulated thermal conductivity against Frequency:
23) To look at the band density of states requires phonopy and h5py, thus we load them: $ module load phonopy/1.11.6.20-intel-2016.01-Python-2.7.12 $ module load h5py/2.6.0-intel-2017a-Python-2.7.12-HDF5-1.8.18
24) Create new file and run phonopy:
$ cp fc2.hdf5 force_constants.hdf5
$ phonopy --dim="3 3 1" -c POSCAR --mesh="21 21 1" --band="
0 0 0 0.5 0 0 1/3 1/3 0 0 0 0" --hdf5 --readfc --thm -p
25) The file total_dos.dat and band.yaml should be created.
26) To generate bandstructure plot:
$ bandplot band.yaml -o bandstructure
$ bandplot --gnuplot band.yaml -o band
27) To generate data file, we take
28) To generate phonon lifetimes against frequencies plot, we use the following command:
$ kdeplot --nbins=200 kappa-m21211.hdf5
73
29) The resultant plot is saved in lifetime.png and for the monolayer it looks like this
30) To collect phonon lifetimes in a file
In [2]: f = h5py.File("kappa-m21211.hdf5") In [3]: g = f['gamma'][30] In [4]: import numpy as np In [5]: g = np.where(g > 0, g, -1) In [6]: lifetime = np.where(g > 0, 1.0 / (2 * 2 * np.pi * g), 0) In [7]: np.savetxt('lf300K.txt', lifetime)
74
Appendix C2
Guidelines for use of Centre for Advanced 2D Materials and Graphene Research Centre
High Performance Computing resources:
The following are the resources available in the GRC cluster:
32 nodes with 16 cores and 64 GB of RAM each (grc-c{01..32}, lowmem queues),
32 nodes with 20 cores and 256 GB of RAM each (grc-d{01..32}),
24 nodes with 24 cores and 256 GB of RAM each (grc-e{01..32}) and
7 nodes with 32 cores and 374 to 1024 GB of RAM (grc-s{00..06}, bigmem queues).
One should check the memory and time required for each calculation and choose queues that
minimally fits the space and time requirements to reduce wait time and to optimize resources.
For example, if only 10GB memory is required and the run duration is approximately a day,
jobs with these requirements stated above should be submitted to the day_lowmem queue (16
cores and 60GB of memory per node, so that it can run four 4core/10G jobs on each node.
If necessary, one can also use the day queue (20 cores, 250GB per node) for a 10GB job, but
that would also incur memory wastage and may have lower priority in the queue. Week queues
should only be used if the jobs need to run for more than one day each, and only use bigmem
queues if the jobs require more than 250GB of memory per node.
For calculations that involve the use of VASP, the following flags in INCAR file may be used:
LPLANE =.TRUE.
NCORE = number of cores per node (e.g. 4 or 8)
LSCALU = .FALSE.
NSIM = 4
The above flags may be used as GRC cluster is a LINUX cluster linked by Infiniband, modern
multicore machines and these machines are linked by a fast network.
75
Appendix D
Determining mode-dependent thermal conductivity
The heat capacity at constant volume is given by an increase in internal energy of system with
respect to temperature T
V
VT
EC
(D1)
The total energy of phonons at a certain temperature T is given by the following relation
)(q
q
q p
p
pnE (D2)
In equation D2, pnq is the thermal equilibrium occupancy of phonons of wavevector q and from the
Boltzmann distribution, we have the following relation
Tk
n
nBePP
0 (D3)
where P0 is given by the following
0
0
1
n
Tk
n
Be
P
(D4)
Furthermore, the occupancy of phonons can be written as
0
0
0
n
Tk
n
n
Tk
n
n
n
B
B
e
ne
nPn
(D5)
We let TkBex
and thus (D5) becomes
0
0
n
n
n
n
x
nx
n
(D6)
where the denominator is a geometric series and (D6) can be simplified to
1
1
1
1
1
1
1
)1(1
2
TkBexx
x
x
x
x
n
(D7)
76
From total energy of phonons in (D2), we can substitute occupancy of phonons found in (D7)
into (D2) to give
)(
1
1q
q
p
p TkBe
E
(D8)
By considering the number of modes, within the given frequency range from ω to d ω, the
energy of phonons can be rewritten as
p Tk
p
Be
DdE
1
)(
(D9)
Finally, we differentiate energy E with respect to temperature to find the mode-dependent
specific heat.
pTk
Tk
BpBV
B
B
e
eTk
DdkT
EC
2
2
1
)(
(D10)
And the resultant mode-dependent specific heat can be expressed as
2
2
1
Tk
Tk
BB
B
B
e
eTk
kC
(D11)
77
Appendix E CHARACTER TABLES
D3h Point Group[55]
E 2C3 3C'2 σh 2S3 3σv linear,
quadratic
rotations
A'1 1 1 1 1 1 1 x2+y2, z2
A'2 1 1 -1 1 1 -1 Rz
E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1
A''2 1 1 -1 -1 -1 1 z
E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
Table E1: Character Table Associated with D3h point group
D3d Point Group[55]
E 2C3 3C'2 i 2S6 3σd
linear,
rotations quadratic
A1g 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 -1 1 1 -1 Rz
Eg 2 -1 0 2 -1 0 (Rx, Ry) (x2-y2, xy)
(xz, yz)
A1u 1 1 1 -1 -1 -1
A2u 1 1 -1 -1 -1 1 z
Eu 2 -1 0 -2 1 0 (x, y)
Table E2: Character Table Associated with D3d point group
C3v Point Group[55]
E 2C3 (z) 3σv linear,
rotations quadratic
A1 1 1 1 z x2+y2, z2
A2 1 1 -1 Rz
E 2 -1 0 (x, y) (Rx, Ry) (x2-y2, xy)
(xz, yz)
Table E3: Character Table Associated with C3v point group
78
Appendix F
Temperatures below Debye Temperature ML MoS2
Linear Fit with GNU Plot
#gnuplot> f(x) = a*x + b
#gnuplot> fit f(x) "belowD.dat" using 1:2 via a, b
#iter chisq delta/lim lambda a b
# 0 1.1519433013e+08 0.00e+00 7.31e-01 1.000000e+00 1.000000e+00
# * 7.9105383511e+122 1.00e+05 7.31e+00 2.176470e+03 1.338063e+03
# 1 3.5700971146e+07 -2.23e+05 7.31e-01 3.573517e+02 2.558181e+01
# 2 1.2630858745e+07 -1.83e+05 7.31e-02 9.051838e+02 2.022820e+01
# 3 6.8470393615e+06 -8.45e+04 7.31e-03 8.654276e+02 2.425006e+01
# 4 6.5373756482e+06 -4.74e+03 7.31e-04 9.172829e+02 2.308547e+01
# 5 6.5360026243e+06 -2.10e+01 7.31e-05 9.072616e+02 2.321516e+01
# 6 6.5359863041e+06 -2.50e-01 7.31e-06 9.085640e+02 2.319914e+01
#iter chisq delta/lim lambda a b
#After 6 iterations the fit converged.
#final sum of squares of residuals : 6.53599e+006
#rel. change during last iteration : -2.49699e-006
#degrees of freedom (FIT_NDF) : 7
#rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 966.288
#variance of residuals (reduced chisquare) = WSSR/ndf : 933712
#Final set of parameters Asymptotic Standard Error
#======================= ==========================
#a = 908.564 +/- 264.1 (29.06%)
#b = 23.1991 +/- 3.22 (13.88%)
#correlation matrix of the fit parameters:
# a b
#a 1.000
#b -0.948 1.000
Temperatures above Debye Temperature ML MoS2
Linear Fit with GNU Plot
gnuplot> f(x) = a*x + b
gnuplot> fit f(x) "aboveD.dat" using (log($1)):(log($2)) via a, b
iter chisq delta/lim lambda a b
0 7.7649659309e+02 0.00e+00 4.81e+00 1.000000e+00 1.000000e+00
1 8.4962988977e-01 -9.13e+07 4.81e-01 2.951277e-01 8.986026e-01
2 6.2030777104e-01 -3.70e+04 4.81e-02 2.384221e-01 1.204768e+00
3 2.9094782630e-02 -2.03e+06 4.81e-03 -7.513024e-01 7.870617e+00
4 2.5726492353e-06 -1.13e+09 4.81e-04 -1.024137e+00 9.708173e+00
5 2.3515688492e-06 -9.40e+03 4.81e-05 -1.024891e+00 9.713252e+00
6 2.3515688490e-06 -7.22e-06 4.81e-06 -1.024891e+00 9.713252e+00
iter chisq delta/lim lambda a b
After 6 iterations the fit converged.
final sum of squares of residuals : 2.35157e-006
rel. change during last iteration : -7.21704e-011
degrees of freedom (FIT_NDF) : 30
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.000279974
variance of residuals (reduced chisquare) = WSSR/ndf : 7.83856e-008
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = -1.02489 +/- 0.0004491 (0.04382%)
b = 9.71325 +/- 0.003025 (0.03114%)
correlation matrix of the fit parameters:
a b
a 1.000
b -1.000 1.0
79
Temperatures below Debye Temperature 2TL MoS2 2H Configuration
Non-Linear Fit with GNU Plot
#gnuplot> f(x) = a*x**3 + b*x**2 + c*x + d
#gnuplot> fit f(x) "belowD.dat" using 1:2 via a, b,c,d
#iter chisq delta/lim lambda a b c d
# 0 1.0017079743e+12 0.00e+00 1.65e+05 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
# 1 7.3487004621e+08 -1.36e+08 1.65e+04 1.566112e-02 9.877705e-01 9.998442e-01 9.999980e-01
# 2 3.1297285753e+06 -2.34e+07 1.65e+03 -1.158450e-02 9.801737e-01 9.996861e-01 9.999990e-01
# 3 2.3339321187e+06 -3.41e+04 1.65e+02 -8.028978e-03 6.916622e-01 9.982629e-01 1.000479e+00
# 4 2.1522111996e+06 -8.44e+03 1.65e+01 -5.514038e-03 4.783058e-01 1.762073e+00 1.072202e+00
# 5 9.4481786554e+05 -1.28e+05 1.65e+00 3.588051e-03 -7.433711e-01 3.989487e+01 5.259463e+00
# 6 4.7666063905e+05 -9.82e+04 1.65e-01 1.131774e-02 -1.749490e+00 6.830316e+01 1.227032e+02
# 7 2.3749000755e+04 -1.91e+06 1.65e-02 -1.473062e-03 3.355475e-01 -3.152929e+01 1.421186e+03
# 8 1.7856177254e+04 -3.30e+04 1.65e-03 -3.117053e-03 6.032165e-01 -4.432057e+01 1.586946e+03
# 9 1.7856167650e+04 -5.38e-02 1.65e-04 -3.119154e-03 6.035587e-01 -4.433692e+01 1.587158e+03
#iter chisq delta/lim lambda a b c d
#After 9 iterations the fit converged.
#final sum of squares of residuals : 17856.2
#rel. change during last iteration : -5.37823e-007
#degrees of freedom (FIT_NDF) : 5
#rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 59.7598
#variance of residuals (reduced chisquare) = WSSR/ndf : 3571.23
#
#Final set of parameters Asymptotic Standard Error
#======================= ==========================
#a = -0.00311915 +/- 0.001583 (50.74%)
#b = 0.603559 +/- 0.2398 (39.74%)
#c = -44.3369 +/- 10.59 (23.9%)
#d = 1587.16 +/- 129.2 (8.141%)
#
#correlation matrix of the fit parameters:
# a b c d
#a 1.000
#b -0.990 1.000
#c 0.944 -0.980 1.000
#d -0.808 0.869 -0.940 1.000
Temperatures above Debye Temperature 2TL MoS2 2H Configuration
Linear Fit with GNU Plot
gnuplot> f(x) = a*x + b
gnuplot> fit f(x) "aboveD.dat" using (log($1)):(log($2)) via a, b
iter chisq delta/lim lambda a b
0 8.5737005531e-02 0.00e+00 9.22e+00 -1.155929e+00 1.046350e+01
1 6.9788319641e-03 -1.13e+06 9.22e-01 -1.152819e+00 1.049140e+01
2 3.5873318713e-03 -9.45e+04 9.22e-02 -1.115021e+00 1.023764e+01
3 3.5745018314e-06 -1.00e+08 9.22e-03 -1.021341e+00 9.606800e+00
4 1.3670095592e-06 -1.61e+05 9.22e-04 -1.018958e+00 9.590754e+00
5 1.3670094164e-06 -1.04e-02 9.22e-05 -1.018958e+00 9.590750e+00
iter chisq delta/lim lambda a b
After 5 iterations the fit converged.
final sum of squares of residuals : 1.36701e-006
rel. change during last iteration : -1.04477e-007
degrees of freedom (FIT_NDF) : 30
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.000213464
variance of residuals (reduced chisquare) = WSSR/ndf : 4.5567e-008
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = -1.01896 +/- 0.0003424 (0.03361%)
b = 9.59075 +/- 0.002306 (0.02404%)
correlation matrix of the fit parameters:
a b
a 1.000
b -1.000 1.000
80
Temperatures below Debye Temperature 2TL MoS2 AB’ Configuration
Non-Linear Fit with GNU Plot
gnuplot> f(x) = a*x**3 + b*x**2 + c*x + d
gnuplot> fit f(x) "belowD.dat" using 1:2 via a, b,c,d
iter chisq delta/lim lambda a b c d
0 4.5462677892e+02 0.00e+00 1.47e+03 3.102281e-03 -5.628776e-01 2.626279e+01 1.562007e+02
* 4.5462677892e+02 1.65e-09 1.47e+04 3.102281e-03 -5.628776e-01 2.626279e+01 1.562007e+02
* 4.5462677892e+02 3.13e-10 1.47e+05 3.102281e-03 -5.628776e-01 2.626279e+01 1.562007e+02
1 4.5462677892e+02 -4.13e-10 1.47e+04 3.102281e-03 -5.628776e-01 2.626279e+01 1.562007e+02
iter chisq delta/lim lambda a b c d
After 1 iterations the fit converged.
final sum of squares of residuals : 454.627
rel. change during last iteration : -4.12609e-015
degrees of freedom (FIT_NDF) : 5
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 9.53548
variance of residuals (reduced chisquare) = WSSR/ndf : 90.9254
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 0.00310228 +/- 0.0002525 (8.141%)
b = -0.562878 +/- 0.03827 (6.799%)
c = 26.2628 +/- 1.691 (6.437%)
d = 156.201 +/- 20.62 (13.2%)
correlation matrix of the fit parameters:
a b c d
a 1.000
b -0.990 1.000
c 0.944 -0.980 1.000
d -0.808 0.869 -0.940 1.000
Temperatures above Debye Temperature 2TL MoS2 AB’ Configuration
Linear Fit with GNU Plot
gnuplot> f(x) = a*x + b
gnuplot> fit f(x) "aboveD.dat" using (log($1)):(log($2)) via a, b
iter chisq delta/lim lambda a b
0 8.1702823218e+02 0.00e+00 4.81e+00 1.000000e+00 1.000000e+00
1 8.3388830335e-01 -9.79e+07 4.81e-01 2.769599e-01 8.958425e-01
2 5.9670526687e-01 -3.97e+04 4.81e-02 2.208482e-01 1.196054e+00
3 2.7986782535e-02 -2.03e+06 4.81e-03 -7.498653e-01 7.733861e+00
4 1.4745688721e-06 -1.90e+09 4.81e-04 -1.017459e+00 9.536120e+00
5 1.2619001649e-06 -1.69e+04 4.81e-05 -1.018198e+00 9.541102e+00
6 1.2619001647e-06 -1.29e-05 4.81e-06 -1.018198e+00 9.541102e+00
iter chisq delta/lim lambda a b
After 6 iterations the fit converged.
final sum of squares of residuals : 1.2619e-006
rel. change during last iteration : -1.29096e-010
degrees of freedom (FIT_NDF) : 30
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.000205093
variance of residuals (reduced chisquare) = WSSR/ndf : 4.20633e-008
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = -1.0182 +/- 0.000329 (0.03231%)
b = 9.5411 +/- 0.002216 (0.02322%)
correlation matrix of the fit parameters:
a b
a 1.000
b -1.000 1.000
81
References [1] Xia, F.; Wang, H.; Xiao, D.; Dubey, M.; Ramasubramaniam, A. Two-dimensional material
nanophotonics. Nat. Photonics 2010, 271, 899.
[2] Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Electronics and
optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnology
2012, 193, 699-700.
[3] Kokalj, A. Computer graphics and graphical user interfaces as tools in simulations of matter
at atomic scale. Com. Mat. Sci. 2003, 28, 155-168.
[4] Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.;
Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science
2004, 306, 666-667.
[5] Tan, H.; Fan, Y.; Zhou, Y.; Chen, Q.; Xu, Wu.; Warner, J. H. Ultrathin 2D Photodetectors
Utilizing Chemical Vapor Deposition Grown WS2 With Graphene Electrodes. ACS Nano 2016,
10, 7866.
[6] Zong, X.; Yan, H.; Wu, G.; Ma, G.; Wen, F.; Wang, L.; Li, C. J. Enhancement of
Photocatalytic H2 Evolution on CdS by Loading MoS2 as Cocatalyst under Visible Light
Irradiation. J. Am. Chem. Soc. 2008, 130, 7176.
[7] Kufer, D; Konstantatos, G. Highly Sensitive, Encapsulated MoS2 Photodetector with Gate
Controllable Gain and Speed. Nano Lett. 2015, 15, 7307.
[8] Huang, S.; Ling, X.; Liang, L.; Kong, J.; Terrones, H.; Meunier, V.; Dresselhaus, M. S.
Probing the Interlayer Coupling of Twisted Bilayer MoS2 Using Photoluminescence
Spectroscopy. Nano Lett. 2014, 14, 5503.
[9] Jiang, T.; Liu, H.; Huang, D.; Zhang, S.; Li, Y.; Gong, X.; Shen, Y.; Liu, W.; Wu, S. Valley
and band structure engineering of folded MoS2 bilayers. Nat. Nanotech. 2014, 9, 825-826.
[10] Huang, S.; Liang, L.; Ling, X.; Puretzky, A. A.; Geohegan, D. B.; Sumpter, D. G.; Kong,
J.; Meunier, V.; Dresselhaus, M. S. Low-Frequency Interlayer Raman Modes to Probe Interface
of Twisted Bilayer MoS2. Nano Lett. 2016, 16, 1435-1444.
82
[11] Ferraro, J.; Nakamoto, K.; Brown, C. W. Introductory Raman Spectroscopy. Elsevier
Science Academic Press, 2003, 15.
[12] Lu, X.; Luo, X.; Zhang, J.; Quek, S. Y.; Xiong, Q. Lattice vibrations and Raman scattering
in two-dimensional layered materials beyond graphene. Nano Res. 2016, 9(12), 3560, 2-7.
[13] Yu, P.; Cardona, M. Fundamentals of Semiconductors: Physics and Materials Properties.
Springer-Verlag, Berlin, 2010, 377-380.
[14] Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.;
Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Corso, A. D.; Gironcoli, S. d.; Fabris, S.; Fratesil,
G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.;
Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.;
Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M.
QUANTUM ESPRESSO: a modular and open-source software project for quantum
simulations of materials. J. Phys.:Condens. Matter 2009, 21 395502, 7-8.
[15] Cai, Y.; Lan, J.; Zhang, G.; Zhang, Y. Lattice vibrational modes and phonon thermal
conductivity of monolayer MoS2. Phys. Rev. B, 2014, 89, 035438, 1-5.
[16] Buscema, M.; Barkelid, Maria. Zwiller, V.; Zant, H. S. J. v. d.; Steele, G. A.; Castellanos-
Gomez, A. Large and Tunable Photothermoelectric Effect in Single-Layer MoS2. Nano Lett.
2013, 13, 358.
[17] Zhao, L.; Lo, S.; Zhang, Y.; Sun, H.; Tan, G.; Uher, C.; Wolverton, C.; Dravid, W. P.;
Kanatzidis, M. G. Ultralow thermal conductivity and high thermoelectric figure of merit in
SnSe crystals. Nature, 2014, 508, 373.
[18] Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993,
47:558.
[19] Kresse, G.; Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal-
amorphous-semiconductor transition in germanium. Phys. Rev. B. 1994, 49:14251.
[20] Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and
semiconductors using a plane-wave basis set. Comput. Mat. Sci. 1996, 6:15.
[21] Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy
calculations using a plane-wave basis set. Phys. Rev. B. 1996, 54:11169
[22] Togo, A.; Chaput, L.; Tanaka, I. Distributions of phonon lifetimes in Brillouin zones.
Phys. Rev. B 2015, 91, 094306,1-3.
83
[23] Togo, A.; Tanaka, I. First principles phonon calculations in materials science. Scr. Mater.,
2015, 108, 1-5.
[24] Kohn, W.; An essay on condensed matter physics in the twentieth century. Rev. Mod.
Phys. 1999, 71, 61.
[25] Giannozzi, P.; Baroni, S. Handbook of Materials Modeling. Springer, Netherlands, 2005.
[26] Baroni, S.; Gironcoli, S. d.; Corso, A. D. Phonons and related crystal properties from
density-functional perturbation theory. Rev. Mod. Phys. 2001, 73, 515-521.
[27] Kaxiras, E.; Atomic and Electronic Structure of Solids. Cambridge University Press, New
York, 2003.
[28] Lee, J. G. Computational Materials Science An Introduction. CRC Press, 2017, 164, 175-
179.
[29] Ibach, H.; Lüth, H.; Solid-State Physics. An Introduction to Theory and Experiment.
Springer-Verlag, Berlin Heidelberg, 1991, 51-65.
[30] Ashcroft, N. W.; Mermin, N. D. Solid State Physics. Cengage Learning, India, 1976, 453,
495-509.
[31] Lindroth, D. O.; Erhart, P. Thermal Transport in van der Waals solids from first-principles
calculations. Phys. Rev. B, 2016, 94, 115205, 1-3.
[32] Shindé, S. L.; Goela, J. S. High Thermal Conductivity Materials. Springer, New York,
2006, 18.
[33] Scheuschner, N.; Gillen, R.; Staiger, M.; Maultzsch, Janina. Newly Observed first-order
resonant Raman modes in few-layer MoS2. arXiv:1503.08980. 2015, 2-3.
[34] Luo, X.; Zhao, Y.; Zhang, J.; Xiong, Q.; Quek, S. Y. Anomalous frequency trends in MoS2
thin films attributed to surface effects. Phys. Rev. B, 2013, 88, 075320, 2-5.
[35] Tongay, S.; Zhou, J.; Ataca, C.; Lo, K.; Matthews, T. S.; Li, J.; Grossman, J. C.; Wu, J.
Thermally Driven Crossover from Indirect toward Direct Bandgap in 2D Semiconductors:
MoSe2 versus MoS2. Nano Lett. 2012, 12, 5576-5580.
[36] Zhang, X.; Qiao, X.; Shi, W.; Wu, J.; Jiang, D.; Tan, P. Phonon and Raman scattering of
two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material.
Chem. Soc. Rev., 2015, 44, 2757-2765.
84
[37] Huang, S.; Liang, L.; Ling, X.; Puretzky, A. A.; Geohegan, D. B.; Sumpter, B. G.; Kong,
J.; Meunier, V.; Dresselhaus, M. S. Low-frequency Interlayer Raman Modes to Probe Interface
of Twisted Bilayer MoS2. Nano Lett., 2016, 16, DOI:10.1021/acs.nanolett.5b05015,1435-
1444.
[38] Livneh, T.; Spanier, J. E. A comprehensive multiphonon spectral analysis in MoS2.
Institute of Physics, 2D Mater. 2, 2015, 035003, 1-3.
[39] Zhao, Y.; Luo, X.; Li, H.; Zhang, J.; Araujo, P. T.; Gan, C. K.; Wu, J.; Zhang, H.; Quek,
S. Y.; Dresselhaus, M. S.; Xiong, Q. Interlayer Breathing and Shear Modes in Few-Trilayer
MoS2 and WSe2. Nano Lett., 2013, 13, 1010.
[40] Ribeiro-Soares, J.; Almeida, R. M.; Barros, E. B.; Araujo, P. T.; Dresselhaus, M. S.;
Cancado, L. G.; Jorio, A. Group Theory Analysis of phonons in two-dimensional transition
metal dichalcogenides. Phys. Rev. B, 2014, 90, 115438, 2-8.
[41] Aroyo, M. I.; Perez-Mato, J. M.; Orobengoa, D.; Tasci, E.; Flor, G. d. l.; Kirov, A.
Crystallography online: Bilbao Crystallographic Server. Bulg. Chem. Commun, 2011, 43(2),
183-197.
[42] Aroyo, M. I.; Perez-Mato, J. M.; Capillas, C.; Kroumova, E.; Ivantchev, S.; Madariaga,
G.; Wondratschek, K.; Wondratschek, H. Bilbao Crystallographic Server I: Databases and
crystallographic computing programs. Z. Krist., 2006, 221, 1, 15-27.
DOI:10.1524/zkri.2006.221.1.15.
[43] Aroyo, M. I.; Kirov, A.; Capillas, C.; Perez-Mato, J. M.; Wondratschek, H. Bilbao
Crystallographic Server II: Representations of crystallographic point groups and space groups.
Acta Cryst. A62, 2006, 115-128. DOI:10.1107/S0108767305040286.
[44] Liu, K.; Zhang, L.; Cao, T.; Jin, C.; Qiu, D.; Zhou, Q.; Zetti, A.; Yang, P.; Louie, S. G.;
Feng, W. Evolution of interlayer coupling in twisted molybdenum disulphide bilayers. Nat.
Comms, 2014, 4996, DOI:10.1038/ncomms5966
[45] O’Brien, M.; McEvoy, N.; Hanlon, D.; Hallam, T.; Coleman, J. N.; Duesberg, G. S.
Mapping of Low-Frequency Raman Modes in CVD-Grown Transition Metal Dichalcogenides:
Layer Number, Stacking Orientation and Resonant Effects. Nat. Sci. Rep., 2016, 19476, 4.
DOI:10.1038/srep19476.
85
[46] Cai, Y.; Lan, J.; Zhang, G.; Zhang, Y. Lattice vibrational modes and phonon thermal
conductivity of monolayer MoS2. Phys. Rev. B, 2014, 89, 035438, 1-5.
[47] Wei, X.; Wang, Y.; Shen, Y.; Xie, G.; Xiao, H.; Zhong, J.; Zhang, G.; Phonon Thermal
Conductivity of monolayer MoS2: A comparison with single layer graphene. Appl. Phys. Lett.,
2014, 105, 103902, 5. DOI: 10.1063/1.4895344.
[48] Li, Wu.; Carrete, J.; Mingo, N. Thermal Conductivity and phonon linewidths of monolayer
MoS2 from first principles. Appl. Phys. Lett., 2013, 103, 253103, 1. DOI: 10.1063/1.4850995.
[49] Peng, B.; Zhang, H.; Shao, H.; Xu, Y.; Zhang, X.; Zhu, H. Thermal Conductivity of
monolayer MoS2, MoSe2 and WS2: interplay of mass effect interatomic bonding and
anharmonicity. RSC Adv., 2016, 6, 5767.
[50] Yan, R.; Simpson, J. R.; Bertolazzi, S.; Brivio, J.; Watson, M.; Wu, X.; Kis, A.; Luo, T.;
Walker, A. R. H.; Xing, H. G. Thermal Conductivity of Monolayer Molybdenum Disulfide
Obtained from Temperature-Dependent Raman Spectroscopy. ACS Nano, 2014, 8, 992.
[51] Kittel, C. Introduction to Solid State Physics. 8th Edition. John Wiley & Sons Inc., 2005,
117, 125-126.
[52] Molina-Sánchez, A.; Wirtz, L. Phonons in single-layer and few-layer MoS2 and WS2.
Phys. Rev. B, 2011, 84, 155413, 3-6.
[53] Zhang, X.; Sun, D.; Li, Y.; Lee, G.; Cui, X.; Chenet, D.; You, Y.; Heinz, T. F.; Hone, J.
C. Measurement of Lateral and Interfacial Thermal Conductivity of Single- and Bilayer MoS2
and MoSe2 Using Refined Optothermal Raman Technique. ACS Appl. Mater. Interfaces, 2015,
7, 25923.
[54] Sahoo, S.; Gaur, A. P. S.; Ahmadi, M.; Guinel, M. J.-F.; Katiyar, R. S. Temperature-
Dependent Raman Studies and Thermal Conductivity of Few-Layer MoS2. J. Phys. Chem. C.,
2013, 117, 9046.
[55] Bishop, D. M. Group Theory and Chemistry. Dover Publications Inc., New York, 1993,
188-191, 284.