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

qq

•

(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

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