by carlos manuel da silva leal - university of toronto … · abstract predicting phonon transport...

Predicting Phonon Transport in Two-dimensional Materials

by

Carlos Manuel Da Silva Leal

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

c© Copyright 2016 by Carlos Manuel Da Silva Leal

Abstract

Predicting Phonon Transport in Two-dimensional Materials

Carlos Manuel Da Silva Leal

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2016

Over the last decade, substantial attention has been paid to novel nanostructures

based on two-dimensional (2D) materials. Among the hundreds of 2D materials that

have been successfully synthesized in recent years, graphene, boron nitride, and molyb-

denum disulfide are the ones that have been intensively studied. It has been demonstrated

that these materials exhibit thermal conductivities significantly higher than those of bulk

samples of the same material. However, little is known about the physics of phonons in

these materials, especially when tensile strain is applied. Properties of these materi-

als are still not well understood, and modelling approaches are still needed to support

engineering design of these novel nanostructures. In this thesis, I use state-of-the-art

atomistic simulation techniques in combination with statistical thermodynamics formu-

lations to obtain the phonon properties (lifetime, group velocity, and heat capacity) and

thermal conductivities of unstrained and strained samples of graphene, boron nitride,

molybdenum disulfide, and also superlattices of graphene and boron nitride. Special

emphasis is given to the role of the acoustic phonon modes and the thermal response of

these materials to the application of tensile strain. I apply spectral analysis to a set of

molecular dynamics trajectories to estimate phonon lifetimes, harmonic lattice dynam-

ics to estimate phonon group velocities, and Bose-Einstein statistics to estimate phonon

heat capacities. These phonon properties are used to predict the thermal conductivity by

means of a mode-dependent equation from kinetic theory. In the superlattices, I study

ii

the variation of the frequency dependence of the phonon properties with the periodicity

and interface configuration (zigzag and armchair) for superlattices with period lengths

within the coherent regime. The results showed that the thermal conductivity decreases

significantly from the shortest period length to the second period length, 13% across the

interfaces and 16% along the interfaces. For greater periods, the conductivity across the

interfaces continues decreasing at a smaller rate of 11 W/mK per period length increase,

driven by changes in the phonon group velocities (coherent effects). In contrast, the con-

ductivity along the interfaces slightly recovers at a rate of 2 W/mK per period, driven

by changes in the phonon relaxation times (diffusive effects).

iii

To my beloved wife Francys and my daughter Sofia

iv

Acknowledgements

I wish to express my deepest gratitude to my supervisor, Professor Cristina Amon, for

giving me opportunity of working with her, and for her unconditional support throughout

my academic career.

I would like to thank Dr. David Romero, Dr. Fernan Saiz, Samuel Huberman, Julia

Sborz, and Andrei Saikouski for their valuable help and direct contributions to this thesis.

I would also like to thank my fellow graduate students at ATOMS Lab, Juan Stockle,

Jim Kuo, Peter Zhang, Francisco Contreras, Sami Yamani, Enrico Antonini, Armin

Taheri and David Guirguis, for their friendship, encouragement, and wonderful moments

that we spent together.

Finally, I express my sincere thanks to the members of my thesis examination commit-

tee, Professors Charles Ward, Chandra Singh, Markus Bussmann, and Dereje Agonafer,

for their expert and valuable guidance.

v

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and Outline of the Thesis . . . . . . . . . . . . . . . . . . . . 3

2 Nanoscale Thermal Transport Models 5

2.1 Phonon Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Bose-Einstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Harmonic Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Phonon Spectral Energy Density . . . . . . . . . . . . . . . . . . 8

2.2 Molecular Dynamics Models . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Equilibrium Molecular Dynamics: Green-Kubo Method . . . . . . 10

2.2.2 Non-equilibrium Molecular Dynamics: Direct Method . . . . . . . 11

3 Phonon Transport: Monolayers and Superlattices 13

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Monolayers: Graphene, Boron Nitride, and Molybdenum Disulfide . . . . 14

3.2.1 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.4 Thermal Conductivities . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Superlattices of Graphene and Boron Nitride . . . . . . . . . . . . . . . . 21

vi

3.3.1 Simulation Model: Spectral Energy Density . . . . . . . . . . . . 23

3.3.2 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.3 Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.4 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.5 Thermal Conductivities . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Equilibrium Molecular Dynamics: Superlattices 37

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Simulation Model: Green-Kubo . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Heat Current Autocorrelation Functions . . . . . . . . . . . . . . . . . . 39

4.4 Thermal Conductivity Evolution . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Superlattice Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . 43

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Non-equilibrium Molecular Dynamics: Strained Superlattices 46

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Simulation Model: Direct Method . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Strained Nanosheets: Graphene and Boron Nitride . . . . . . . . . . . . 50

5.4 Strained Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Phonon Transport: Strained Monolayers and Superlattices 55

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Strained Monolayers: Graphene, Boron Nitride and Molybdenum Disulfide 56

6.3 Strained Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

vii

7 Conclusion 64

7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A Density Functional Theory Simulations 68

A.1 Structural Optimization: Superlattices . . . . . . . . . . . . . . . . . . . 68

A.2 Phonon Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B Spectral Energy Density Algorithm 71

B.1 Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B.2 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Bibliography 78

viii

List of Tables

3.1 Predicted thermal conductivities in W/mK for grpahene, boron nitride,

and molybdenum disulfide . . . . . . . . . . . . . . . . . . . . . . . . . . 21

A.1 Number of atoms, size of the MD simulations domain (LxxLy), number of

replicated unit cells in the x (Nx) and y (Ny) directions, number of atoms

in the unit cell, and equilibrium lattice parameters ax and ay for each

superlattice period in the zigzag configuration. Reprinted with permission

from Ref. [15], Copyright 2016 American Physical Society . . . . . . . . 69

A.2 Number of atoms, size of the MD simulations domain (LxxLy), number

of replicated unit cells in the x (Nx) and y (Ny) directions, number of

atoms in the unit cell, and equilibrium lattice parameters ax and ay for

each superlattice period in the armchair configuration. Reprinted with

permission from Ref. [15], Copyright 2016 American Physical Society . . 70

ix

List of Figures

1.1 Lattice structures of (a) graphene, (b) boron nitride, and (c) molybdenum

disulfide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Isochoric heat capacity as a function of temperature for graphene(C),

boron nitride (BN), and superlattices of graphene and boron nitride with

periods 1x1 and 5x5. Experimental data is taken from several sources com-

pile in Ref. [81]. Experimental data for graphene is taken from graphite

which should be equal above 100 K. Reprinted with permission from Ref.

[67], Copyright 2015 ASME. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Schematic of the non-equilibrium molecular dynamics simulation. . . . . 12

x

3.1 Lattice structures: (a) graphene, (b) boron nitride, and (c) molybdenum

disulfide. Carbon atoms are painted in gray, boron atoms are painted in

light blue, nitrogen atoms are painted in red, Sulfur atoms are painted

in yellow, and molybdenum atoms are painted in dark blue. The lattice

vectors (a1 and a2) and the reciprocal lattice vectors (b1 and b2) are shown,

along with the schematic of the primitive unit cell (rhombus shown in black

solid lines) and the first Brillouin zone (hexagonal shown in black dashed

lines). Phonon dispersion curves of (d) graphene, (e) boron nitride, and

(f) molybdenum disulfide; along the k-space directions Γ-M-K-Γ. All solid

curves represent data from the HLD simulations with an empirical inter-

atomic potential. The black solid curves represent the optical modes, and

the highlighted curves represent the acoustic modes. The dots represent

data from the DFPT simulations. . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Squared phonon group velocities for graphene, boron nitride, and molyb-

denum disulfide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Phonon lifetime distributions for (a) graphene, and (b) boron nitride.Reprinted

with permission from Ref. [15], Copyright 2016 American Physical Society 19

3.4 Phonon lifetime distributions for molybdenum disulfide. . . . . . . . . . . 20

3.5 Thermal conductivity contributions as functions of the phonon frequencies

for graphene, boron nitride, and molybdenum disulfide. . . . . . . . . . . 21

3.6 Superlattice unit cells for (a) the 1x1 zigzag interface, and (b) the 1x1

armchair interface. Atomic structures of the 1x1 superlattices for (c) the

zigzag interface, and (d) the armchair interface. Reprinted with permission

from Ref. [15], Copyright 2016 American Physical Society. . . . . . . . . 23

xi

3.7 Dispersion curves for the 1x1 zigzag superlattice (a) along the k-space

direction [0 1 0], and (b) along the k-space direction [1 0 0]. (c) Phonon

density of states. The solid lines represent data from the HLD simulations.

The black solid lines represent the optical modes, and the highlighted lines

represent the acoustic branches. The dots represent data from the DFPT

simulations. Reprinted with permission from Ref. [15], Copyright 2016

American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 Dispersion curves for the 1x1 armchair superlattice (a) along the k-space

direction [0 1 0], and (b) along the k-space direction [1 0 0]. (c) Phonon

density of states. The solid lines represent data from the HLD simulations.

The black solid lines represent the optical modes, and the highlighted lines

represent the acoustic branches. The dots represent data from the DFPT

simulations. Reprinted with permission from Ref. [15], Copyright 2016

American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 Acoustic dispersion curves (a) along the k-space direction [0 1 0], and (b)

along the k-space direction [1 0 0]. Reprinted with permission from Ref.

[15], Copyright 2016 American Physical Society. . . . . . . . . . . . . . . 28

3.10 Squared phonon group velocities for the zigzag superlattices in (a) the x

direction, and (b) the y direction. Reprinted with permission from Ref.


3.11 Phonon power spectra for the 1x1 zigzag superlattice at wavevector k′ =

[17π/23ax, 0, 0] for the acoustic polarizations (a) out-of-plane ZA, (b) trans-

verse TA, and (c) longitudinal LA. Reprinted with permission from Ref.


3.12 Phonon lifetime distributions for the superlattices (a) 1x1 zigzag, (b) 2x2

zigzag, (c) 5x5 zigzag, and (d) 10x10 zigzag. Reprinted with permission

from Ref. [15], Copyright 2016 American Physical Society. . . . . . . . . 31

xii

3.13 Variation of the thermal conductivity with the superlattice period and

interface structure. Reprinted with permission from Ref. [15], Copyright

2016 American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . 33

3.14 Thermal conductivity contributions as functions of the phonon frequencies

for the superlattices (a) zigzag in the x direction, (b) armchair in the x

direction, (c) zigzag in the y direction, and (d) armchair in the y direc-

tion. Reprinted with permission from Ref. [15], Copyright 2016 American

Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Unit cells of (a) 1x1 and (b) 5x5 zigzag, and (c) 1x1 and (d) 5x5 (d) arm-

chair superlattices. Reprinted with permission from Ref. [67], Copyright

2015 ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Normalized heat current autocorrelation functions for the 1x1 armchair su-

perlattice in (a) the x, (b) y, and (c) z directions, and (d) their summation.

Reprinted with permission from Ref. [67], Copyright 2015 ASME. . . . . 40

4.3 Normalized heat current autocorrelation functions for the 1x1 zigzag su-

perlattice in (a) the x, (b) y, and (c) z directions, and (d) their summation.


4.4 Evolution of the average thermal conductivities in the x (kxx) and y (kyy)

directions calculated using the Green-Kubo methodology for (a) the 1x1

armchair, (b) 5x5 armchair, (c) 1x1 zigzag, and (d) 5x5 zigzag super-

lattices. The error bars are plotted every 50 ps to show the standard

deviations. Reprinted with permission from Ref. [67], Copyright 2015

ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xiii

4.5 Thermal conductivities versus the superlattice period and interface ori-

entation using the Green-Kubo (G-K) method and results from spectral

energy density (SED) analysis discussed in Chapter B. Panel (a) shows

data for the zigzag configuration and panel (b) shows data for the arm-

chair configuration. Reprinted with permission from Ref. [67], Copyright

2015 ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Zigzag boron nitride model along with the schematic of the NEMD simu-

lations. Reprinted with permission from Ref. [16], Copyright 2014 CSME. 48

5.2 Boron nitride unit cell for: (a) zigzag orientation, and (b) armchair orien-

tation. Reprinted with permission from Ref. [16], Copyright 2014 CSME. 49

5.3 Zigzag BN-graphene hybrid model with the heat flux Jx perpendicular to

the BN-C interfaces. Reprinted with permission from Ref. [16], Copyright

2014 CSME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Zigzag BN-graphene hybrid model with the heat flux Jx parallel to the

BN-C interfaces. Reprinted with permission from Ref. [16], Copyright

2014 CSME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 Normalized thermal conductivity (k/k0) results from NEMD simulations

in boron nitride and graphene monolayers along the zigzag and armchair

orientations. Reprinted with permission from Ref. [16], Copyright 2014

CSME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.6 Thermal conductivity for the BN-Graphene hybrid model with the heat

flux perpendicular to the BN-C interfaces along the zigzag and armchair

orientations. Reprinted with permission from Ref. [16], Copyright 2014

CSME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.7 Thermal conductivity for the BN-Graphene hybrid model with the heat

flux parallel to the BN-C interfaces along the zigzag and armchair orien-

tations. Reprinted with permission from Ref. [16], Copyright 2014 CSME. 53

xiv

6.1 (a) Deformation of the lattice structure of graphene when uniaxial tensile

strain is applied in the x direction. The figure shows three levels of defor-

mation: 0%, 10%, and 20%.(b) Variation of the thermal conductivity of

graphene with the percentage of deformation applied in the x direction. . 57

6.2 (a) Acoustic dispersion curves of graphene along the k-space direction

[1 0 0] for three percentages of uniaxial tensile strain: 0%, 6%, and 12%.

Response of the phonon lifetime distribution of graphene to the application

of strain for (b) the ZA modes, (c) TA modes, and (d) LA modes. (e)

Variation of the thermal conductivity contributions of the acoustic modes

of graphene with the percentage of deformation applied in the x direction. 58

6.3 (a) Deformation of the lattice structure of boron nitride when uniaxial

tensile strain is applied in the x direction. The figure shows three levels of

deformation: 0%, 10%, and 20%.(b) Variation of the thermal conductivity

of boron nitride with the percentage of deformation applied in the x direction. 59

6.4 (a) Acoustic dispersion curves of boron nitride along the k-space direction

[1 0 0] for three percentages of uniaxial tensile strain: 0%, 6%, and 12%.

Response of the phonon lifetime distribution of boron nitride to the appli-

cation of strain for (b) the ZA modes, (c) TA modes, and (d) LA modes.

(e) Variation of the thermal conductivity contributions of the acoustic

modes of boron nitride with the percentage of deformation applied in the

x direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.5 (a) Deformation of the lattice structure of molybdenum disulfide when

uniaxial tensile strain is applied in the x direction. The figure shows three

levels of deformation: 0%, 10%, and 20%.(b) Variation of the thermal

conductivity of molybdenum disulfide with the percentage of deformation

applied in the x direction. . . . . . . . . . . . . . . . . . . . . . . . . . . 61

xv

6.6 Acoustic dispersion curves of molybdenum disulfide along the k-space di-

rection [1 0 0] for three percentages of uniaxial tensile strain: 0%, 6%,

and 12%. Response of the phonon lifetime distribution of molybdenum

disulfide to the application of strain for (b) the ZA modes, (c) TA modes,

and (d) LA modes. (e) Variation of the thermal conductivity contribu-

tions of the acoustic modes of molybdenum disulfide with the percentage

of deformation applied in the x direction. . . . . . . . . . . . . . . . . . . 62

6.7 (a) Atomic structure of the 1x1 zigzag superlattice of graphene and boron

nitride showing the direction in which the the uniaxial tensile strain is

applied (x direction). (b) Variation of the thermal conductivity with the

superlattice period and the percentage of deformation applied in the x

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.1 Unit cells for the superlattices: (a) 1x1 zigzag, and (b) 1x1 armchair.


B.1 Flowchart of the spectral energy density algorithm. . . . . . . . . . . . . 72

xvi

Chapter 1

Introduction

1.1 Motivation

The phonon transport problem in nanomaterials is relevant to current and future

advances in high-power density nanoelectronic and optoelectronic devices [7]. The success

of these devices will be strongly influenced by the ability of designers to manipulate heat

dissipation. Thermal management issues have become the major limiting factor that has

slowed down the miniaturization trend of transistors [55], and satisfying Moore’s Law in

the near future will require the exploration of new device architectures, nanostructures

and materials [76]. As the number of transistors per unit area increases, energy is gener-

ated at too high a rate for it to be safely transported out of the device, creating localized

hot spots and high temperature gradients, both of which have a detrimental effect on

performance and reliability [72, 62]. This challenge is compounded by the exponential

growth in the number of material interfaces on the nanoscale, which makes energy trans-

port across them the most significant determinant of electrical and thermal performance

[28, 7, 41]. Accordingly, a theoretical understanding of nanoscale thermal transport phe-

nomena is paramount for the development of the next generation of field-effect transistors

[63, 80], interconnects [59, 52], composite substrates for power electronics [68, 38], and

data storage systems [85, 5]; required for the continued evolution of the information age.

1

Chapter 1. Introduction 2

Figure 1.1: Lattice structures of (a) graphene, (b) boron nitride, and (c) molybdenumdisulfide.

Miniaturization is no longer the only option pursued to improve the performance of

electronic devices. Promising new nanostructures such as nanowires [51], nanoribbons

[87], and superlattices [14, 15, 32] are being actively investigated for their mechanical,

electronic, and thermal properties. Over the last decade, substantial attention has been

paid to novel nanostructures based on 2D materials, i.e., materials whose thickness is only

one atom. Among the hundreds of 2D materials that have been successfully synthesized in

recent years, graphene [86], boron nitride [42], and molybdenum disulfide [84] are the ones

that have been intensively studied. It has been demonstrated that these materials exhibit

thermal conductivities significantly higher than those of bulk samples of the same material

[2]. Graphene (Fig. 1.1a), with measured conductivities of up to several thousand W/mK

[86, 1], has probably received the most attention, but graphene is unsuitable as the sole

building block of transistors because it lacks of an electronic bandgap. In contrast, 2D

layers of boron nitride (Fig. 1.1b) and molybdenum disulfide (Fig. 1.1c), typically exhibit

a bandgap and unique thermal properties, placing them at the forefront of the next

electronics revolution [13]. The physical properties of 2D materials have been engineered

by applying a variety of techniques such as strain [17], doping [11], lattice defects [23],

and electric fields [57]. In addition, superlattices made of alternating layers of these

nanosheets have been proven to be effective ways to control the electronic and thermal

properties. These layers can be vertically stacked to form 3D superlattices [26] or placed

periodically in plane to form 2D superlattices [46]. For example, the electronic band

structure of 2D superlattices of graphene and boron nitride can be tuned in such a way


that the hybrid material undergoes a transition from a semiconductor at the shortest

period length to a metal at higher period lengths [78]. However, little is known about the

physics of phonons (thermal energy carriers) in these short-period superlattices, especially

regarding their propagation and scattering at interfaces.

1.2 Objectives and Outline of the Thesis

The main objective of this thesis is to predict phonon properties and thermal conductivi-

ties of 2D materials using state-of-the-art atomistic simulation techniques in combination

with statistical thermodynamics formulations. Special emphasis is given to the role of the

acoustic phonon modes and the thermal response of these materials to the application

of tensile strain. The phonon transport problem is addressed on a mode-by-mode basis

to investigate the behavior of phonons across monolayers of graphene, boron nitride and

molybdenum disulfide; and also across 2D superlattices of graphene an boron nitride.

These superlattices represent excellent test scenarios to study the coherent transport

of phonons at interfaces of 2D materials. Promoting coherent transport of phonons is a

promising strategy for controlling thermal transport in nanostructures and an alternative

to traditional methods based on structural defects. Coherent transport is particularly

relevant in short-period heterostructures with smooth interfaces and long wavelength

heat-carrying phonons, such as those of graphene and boron nitride.

The thermal conductivities of these monolayers and superlattices are estimated by

means of a mode-dependent equation from kinetic theory. This equation requires the

previous calculation of the phonon properties: relaxation times, group velocities and

heat capacities. Phonon relaxation times are predicted from spectral analysis applied to

a set of molecular dynamics (MD) trajectories, phonon group velocities are predicted from

harmonic lattice dynamics, and phonon heat capacities are predicted from Bose-Einstein

statistics. Another objective of this thesis is to compare the thermal conductivity results


from this phonon transport approach to those obtained from standard MD simulation

techniques such as Green-Kubo method and Direct method. In Addition, independent

first-principles simulations based on Density Functional Perturbation Theory (DFPT)

were conducted to validate the empirical interatomic potential used to model the force

field of the superlattices.

After this introductory chapter, the rest of the thesis is organized as follows:

In Chapter 2, the phonon thermal transport model is introduced, including a detailed

description of the simulation techniques used to predict the phonon properties.

In Chapter 3, the phonon thermal transport model is applied on unstrained samples of

the monolayers and superlattices. This chapter presents the predicted phonon dispersion

curves, group velocities, relaxation times and thermal conductivities. Special emphasis

is given to the role of the acoustic phonon modes and the sensitivity of the phonon

properties to the superlattice period and interface configuration. The methodologies to

assemble the simulation domains are also described in this section.

In Chapter 4, equilibrium MD simulations (Green-Kubo method) are applied to pre-

dict the thermal conductivities of the superlattices as functions of the period length

and interface configuration. The validity of this method is discussed and the thermal

conductivity results are compared to those obtained from phonon spectral analysis.

In Chapter 5, non-equilibrium MD simulations (Direct method) are applied to study

the thermal response of graphene, boron nitride, and the superlattices to tensile strain.

In Chapter 6, the thermal response to uniaxial tensile strain is investigated from a

phonon transport perspective, with focus on the effect of strain on the relative contribu-

tions of the acoustic phonon modes in the monolayers and superlattices.

In Chapter 7, the major contributions of this research are summarized. Future re-

search directions are also discussed in this chapter.

Chapter 2

Nanoscale Thermal Transport

Models

2.1 Phonon Thermal Transport

The ultimate goal of this thesis is to calculate thermal conductivities using phonon prop-

erties. To this end, a mode-dependent equation from kinetic theory is implemented [90],

to obtain the phonon thermal conductivity in the αth direction as

Kα =∑k,ν

Cph(k, ν) v2α(k, ν) τ(k, ν), (2.1)

where Cph(k, ν) is the phonon heat capacity, vα(k, ν) is the αth component of the phonon

group velocity, and τ(k, ν) is the phonon lifetime or relaxation time. Bose-Einstein statis-

tics is employed to estimate the heat capacities and harmonic lattice dynamics (HLD)

calculations to predict the group velocities. The relaxation times are extracted from

molecular dynamics (MD) simulations using a frequency domain normal mode analysis.

The MD simulations naturally include four- and higher-order intrinsic scattering pro-

cesses and the extrinsic effects of boundaries and interfaces [22]. This approach contrasts

with other methods such as the standard single mode relaxation time approximation

5

Chapter 2. Nanoscale Thermal Transport Models 6

[56] and the iterative solution to the Boltzmann transport equation [45], where only

three-order scattering processes are considered and extrinsic effects are only empirically

included.

The phonon properties are functions of the wave vectors k and the polarization

branches ν. These wave vectors are specified within the first Brillouin zone (BZ). The

first BZ is, in all cases studied in this thesis, a rectangular prism with reciprocal lattice

vectors bα given by 2π/aα, where aα is the lattice vector in real space. Based on the num-

ber of replicated unit cells Nα, the wave vectors are specified as k = bα(nα/Nα), where

nα is an integer with allowed values in the range of −Nα/2 and Nα/2. The symmetry

of the BZ is imposed by computing the phonon properties only for the irreducible wave

vectors in the first quadrant of this zone, that is 0 ≤ nx ≤ Nx/2 and 0 ≤ ny ≤ Ny/2.

2.1.1 Bose-Einstein Statistics

The thermal conductivities are estimated at room temperature (T = 300 K), well below

the Debye temperature (classic limit) in graphene and boron nitride (∼ 2100 K) [64], as

shown in Fig. A.1. In MD simulations, the heat capacities are typically estimated by

means of the equipartition theorem, which assigns equal amount of energy to all phonons.

However, quantum effects become important in 2D materials at room temperature, and

applying the equipartition theorem is no longer valid. Therefore, quantum effects are

incorporated here by applying Bose-Einstein statistics to estimate phonon heat capacities

Cph(k, ν) as functions of phonon frequencies w(k, ν) as

Cph(k, ν) =kB x(k, ν)2 ex(k,ν)

(ex(k,ν) − 1)2 (2.2)

where kB is the Boltzmann constant, and x(k, ν) = hw(k, ν)/kBT , with h being the

reduced Planck’s constant and T the system’s temperature. Figure A.1 illustrates the

normalized heat capacities from 0 K to 1500 K in graphene, boron nitride, and super-


lattices of both materials, calculated by solving Eq. 2.2 in the harmonic approximation.

The heat capacities are in good agreement with those obtained by experiments. For

example, the theoretical and experimental heat capacities in boron nitride diverge by

11.8 % at 300 K and 9.16 % at 1400 K. On the other hand, these results reveal that the

heat capacities of the superlattices are practically unaffected by the change of periodicity

and interface orientation. Consequently, the focus in this thesis is on the variation of

phonon group velocities and relaxation times. The results in Fig. A.1 also show that the

superlattices’ heat capacities lay approximately in the average of those of graphene and

boron nitride above 25 K, with a discrepancy less than 5.3 %. At 300 K the difference

between the values for 1x1 armchair with respect to graphene are 0.06 % and 0.07 % with

respect to boron nitride. Such difference begins to shrink for temperatures above 1000 K

and is negligible at 1500 K, which suggests that the in-plane Debye temperature of the

superlattices should be similar for all their configurations and close to that of graphene.

2.1.2 Harmonic Lattice Dynamics

The HLD calculations solve the eigenvalue problem for the atoms in the lattice structure,

to obtain the harmonic phonon frequencies w(k, ν) and the normal mode eigenvectors.

The frequencies are used to build the dispersion curves and then to obtain the phonon

group velocities by calculating the slope of the curves with central differences. The eigen-

vectors are used to isolate individual phonon modes from the phonon spectrum, and to

precisely locate the acoustic phonon modes. The acoustic modes are characterized by

having all atoms moving in phase [20], being the lattice vibrational modes that exhibit

the greatest contributions to the thermal conductivity in 2D materials. These HLD

simulations are conducted with the General Utility Lattice Program (GULP) [24] using

empirically approximated interatomic potentials. A Tersoff-type [79] interatomic poten-

tial developed by Kinaci et al. [36] is implemented to describe the lattice vibrational

properties of single-layer graphene, boron nitride, and superlattices of both materials.


Figure 2.1: Isochoric heat capacity as a function of temperature for graphene(C), boronnitride (BN), and superlattices of graphene and boron nitride with periods 1x1 and 5x5.Experimental data is taken from several sources compile in Ref. [81]. Experimental datafor graphene is taken from graphite which should be equal above 100 K. Reprinted withpermission from Ref. [67], Copyright 2015 ASME.

On the other hand, the vibrational properties of single-layer molybdenum disulfide are

described by a Stillinger-Weber-type potential [77] recently developed by Kandemir et

al. [35].

2.1.3 Phonon Spectral Energy Density

The phonon relaxation times are computed using a normal mode decomposition (NMD)

approach based on phonon spectral energy density [40, 32]. This approach is applied

to a set of MD trajectories generated at room temperature with the LAMMPS Package

[61]. Once the MD simulations are completed, the atomic velocities, equilibrium atomic

positions, and phonon mode eigenvectors are used to construct the time derivative of the

normal mode coordinates q(k, ν, t) as [20, 40]


q(k, ν, t) =3,n,N∑l,b,α

√mb

Nuα(l, b, t) e∗(k, ν, b, α) × exp[ik · ro(l, b)], (2.3)

where n is the number of atoms in the unit cell, N is the number of unit cells, mb is the

atomic mass of atom b, uα(l, b, t) is the αth component of the velocity of atom b at the

unit cell l and time instant t, and e∗(k, ν, b, α) is the complex conjugate of the eigenvector

associated with atom b and direction α.

Fast Fourier transforms (FFT) are then applied to the autocorrelation of the time

derivative given in Eq.2.3 to generate the phonon power spectrum T (k, ν, w) as [40]

T (k, ν, w) = limτ0→∞

1

2τ0

∣∣∣∣∣ 1√2π

∫ τ0

0q(k, ν, t) exp(−iwt) dt

∣∣∣∣∣2

, (2.4)

where τ0 is the time of the simulation sampling window. The power spectrum of an

individual phonon at wave vector k and polarization ν is averaged over results from

four sampling windows and ten MD simulations with different initial conditions, totaling

40 independent calculations. A complete description of the computational algorithm is

provided in Appendix B.

According to anharmonic theory [48], the phonon lifetimes are finally predicted by

fitting the averaged power spectra to the Lorentzian function

T (k, ν, w) ≈ I(k, ν)[wA(k,ν)−w

Γ(k,ν)

]2+ 1

, (2.5)

where the fitting parameters are the intensity of the peak I(k, ν), the anharmonic angular

phonon frequency wA(k, ν) at the centre of the peak, and the half width at half maximum

of the peak Γ(k, ν). The phonon relaxation time τ(k, ν) is calculated as the inverse of

the full width at half maximum, 1/2Γ(k, ν). Equation 2.5 is fitted to the discrete points

yielded by Eq. 2.4.The fitted data includes all the points whose abscissa is higher than

an arbitrary threshold whose value is equal to the maximum point of the spectrum

divided by 10,000. This threshold is a reasonable choice because it provides an excellent


accuracy while it requires a tolerable computational effort; using smaller thresholds does

not increase the fitting accuracy. This fit is made with an iteratively reweighted least

squares (IRLS) method with a Cauchy weight function [31]. The fit’s precision is very

sensitive to the initial guess for Γ(k, ν), which has to be carefully chosen at an arbitrary

frequency. In this work, the process is automated by conducting the fitting in two

steps. In the first step, the intensity I(k, ν) is set equal to the maximum energy of the

phonon spectrum, and the angular frequency wA(k, ν) is set equal to the frequency at the

maximum energy. Then, the data is fitted to Eq. 2.5 with Γ(k, ν) as the only adjusted

parameter. The initial guesses for Γ(k, ν) are chosen according to the frequency range.

In the second fitting step, three parameters are adjusted and the outputs from the first

step are taken as the initial guesses.

2.2 Molecular Dynamics Models

The thermal conductivity of nanomaterials is typically computed at the system level,

using equilibrium and non-equilibrium MD simulations techniques. These techniques

does not provide insight into the phonon transport problem; however, they have proven

to be qualitatively effective for studying trends and the relative effects of interfaces and

lattice defects. In this thesis, these simulations are conducted for validations purposes.

2.2.1 Equilibrium Molecular Dynamics: Green-Kubo Method

The Green-Kubo method, an equilibrium MD approach based on the fluctuation-dissipation

theorem, is used to compute the thermal conductivity tensor Kα as [70]

Kα =1

V kBT 2limτ0→∞

∫ τ

0〈Sα(t)Sα(0)〉dt, (2.6)

where V is the volume of the simulation cell, T is the equilibrium temperature of the

system, kB is the Boltzmann constant, S(t) is the heat current vector, 〈Sα(t)Sα(0)〉 is the


heat current autocorrelation function, and τ is the time needed for such function to decay

to zero. This decay occurs because the heat flow in a system of particles in equilibrium

fluctuates around zero. Equation 2.6 is discretized to compute the components of the

thermal conductivity tensor as [70]

Kα =∆t

V kBT 2

M∑m=1

1

Ns −m

M−m∑n=1

Sα(m+ n)Sα(n), (2.7)

where ∆t is the simulation timestep, Ns is the total number of time steps of the simulation

after the system has been equilibrated, and M is the number of time steps for the corre-

lation of heat current vectors (M∆t = τ). The correlation time must be sufficiently long

to calculate thermal conductivities for several hundreds of picoseconds, which requires

running simulations for a few nanoseconds [70].

2.2.2 Non-equilibrium Molecular Dynamics: Direct Method

The direct method, whose schematic is shown in Figure 2.2, is a non-equilibrium molec-

ular dynamics (NEMD) simulation technique in which an artificial heat flux Jx is forced

into the system by adding and subtracting energy at the hot and cold regions, respec-

tively. This heat flux generates a temperature gradient that is later used to estimate the

thermal conductivity when steady state conditions are reached [70]. The temperature

profile is averaged over a given period of time, and the thermal conductivity Kx can be

calculated by means of the Fourier’s Law of conduction

Jx =Q

A= Kx

∂T

∂x, (2.8)

where, A is the cross-section area LyxLz, with Ly being the width of the atomic layer,

and Lz its thickness; ∂T∂x

is the temperature gradient along the x axis, and Q represents

the heat rate at which energy is being added and subtracted. In the MD simulation, the

domain is divided into thin bins along the x direction. The average temperature for the


atoms within a bin at position x 〈T (x)〉M is calculated as [70]

〈T (x)〉M =1

M

M−1∑m=0

Tn−m(x), (2.9)

where, M is the number of timesteps used for averaging, and n is the total number of

timesteps in the MD simulation. The value of M has to be smaller than (n−1). Transient

effects are minimized when the average is computed as described by Eq. 2.9.

Figure 2.2: Schematic of the non-equilibrium molecular dynamics simulation.

Chapter 3

Phonon Transport: Monolayers and

Superlattices

3.1 Motivation

Nanostructured materials do not transport heat as well as bulk materials [21, 71, 82], but

the situation is different in strictly 2D nanomaterials where the the interlayer phonon

scattering mechanism is absent [2, 74]. The thermal transport in few-layer graphene is

highly anisotropic due to the strong in-plane covalent binding between light-weight car-

bon atoms and the weak out-of-plane van der Waals interactions with other graphene

layers or substrates [55, 54]. This behavior gives graphene, and related 2D nanomateri-

als, their unique thermal properties, mainly characterized by exceptionally high in-plane

thermal conductivity and limited heat transport for the out-of-plane direction. Two-

dimensional materials can reach in-plane thermal conductivities significantly higher than

bulk or thin-film samples of the same material. This fact is extremely important in high

power density applications where heat dissipation significantly limits the performance of

electronic devices. Recent studies about heat conduction in these materials have revealed

the unique behavior of thermal phonons (lattice vibrations). However, many intriguing

13

Chapter 3. Phonon Transport: Monolayers and Superlattices 14

questions remains unanswered about phonon transport in 2D materials, especially about

the contribution of different phonon modes, and the role of interfaces. Understanding

phonon physics is crucial for thermal management in nanoelectronics, and this is the main

focus in this chapter. The phonon transport problem is addressed here in a mode-by-

mode basis in order to elucidate the behavior of phonons as they cross isolated material

layers of graphene, boron nitride, and molybdenum disulfide; and also superlattices of

graphene and boron nitride.

3.2 Monolayers: Graphene, Boron Nitride, and Molyb-

denum Disulfide

Graphene, a single layer of carbon atoms hexagonally arranged (see Fig. 3.1a), was suc-

cessfully exfoliated from graphite in 2004. Since then, it has been the focus of attention

in many research fields due to its excellent properties and unique 2D structure. The

extensive research done on graphene in the last decade, and certainly the advances in

synthesis techniques, have led to the development of the new research field of 2D nano-

materials, where hundreds of others materials, only theoretically defined in the past, has

been synthesized and suggested as promising competitors of graphene [6, 13]. Hexagonal

boron nitride is just one of these nanomaterials formed by the combination of boron and

nitrogen atoms arranged in a honeycomb lattice (see Figure 3.1b). Boron nitride is a

semiconductor whose bandgap is too high to be switched in a practical way. For this

reason, it is commonly used as an electrical insulator. It has been proposed to be used

in combination with graphene to compensate the absent of the natural bandgap. From a

thermal point of view, boron nitride in-plane thermal conductivity (∼ 400 W/mK [73])

has been reported to be much smaller than the one for graphene (as high as 3000 W/mK

[1, 10]). However, it is still comparable with the thermal conductivity of metals such as

silver and cooper.


Figure 3.1: Lattice structures: (a) graphene, (b) boron nitride, and (c) molybdenumdisulfide. Carbon atoms are painted in gray, boron atoms are painted in light blue,nitrogen atoms are painted in red, Sulfur atoms are painted in yellow, and molybdenumatoms are painted in dark blue. The lattice vectors (a1 and a2) and the reciprocal latticevectors (b1 and b2) are shown, along with the schematic of the primitive unit cell (rhombusshown in black solid lines) and the first Brillouin zone (hexagonal shown in black dashedlines). Phonon dispersion curves of (d) graphene, (e) boron nitride, and (f) molybdenumdisulfide; along the k-space directions Γ-M-K-Γ. All solid curves represent data fromthe HLD simulations with an empirical interatomic potential. The black solid curvesrepresent the optical modes, and the highlighted curves represent the acoustic modes.The dots represent data from the DFPT simulations.


Another important group of 2D nanomaterials are the transition metal dichalco-

genides, which combine a transition metal with one of three of the following chalcogen

materials: Sulfur (S), Tellurium (Te) or Selenium (Se). Several combinations of these

materials have been studied so far, being molybdenum disulfide (MoS2) one of the most

well-documented [53, 6]. Molybdenum disulfide is a combination of one molybdenum

atom with two sulfur atoms (see Fig. 3.1c). The three atoms are hexagonally arranged

in three different planes forming a trigonal prism. In contrast with graphene, single-layer

molybdenum disulfide posses a direct bandgap that makes it suitable for logic applica-

tions. Information about thermal transport properties of molybdenum disulfide is still

very limited, as it is the case for most 2D nanomaterials, except for graphene.

3.2.1 Dispersion Curves

The lattice structure of graphene is shown in Fig. 3.1a, the primitive rhombic unit cell

(black solid line) contains two atoms, which means that there are six phonon modes,

one for each degree of freedom. Three phonon modes behave as acoustic (A) waves and

the other three, as optical (O). These phonon modes, also called polarization branches,

describe dispersion relations along certain directions in reciprocal space, as those shown

for graphene in Fig. 3.1d along the k-space directions Γ-M-K-Γ of the first Brillouin

zone. The same number of polarization branches apply for boron nitride, but not for

molybdenum disulfide (see Figure 3.1c), which has three atoms in the primitive unit

cell instead of two. Therefore, molybdenum disulfide will have nine phonon modes,

three acoustic modes and six optical modes. These numbers of polarization branches

correspond to the primitive unit cells. For other unit cell configurations, these numbers

will be different.

According to the direction in which atoms are being displaced, the acoustic (A) and

optical (O) branches can be longitudinal (LA and LO), traverse (TA and TO) or out-of-

plane (ZA and ZO). Regarding these branches, there are some particularities to highlight


about 2D materials. First, the group velocities of acoustic modes are substantially higher

than the ones of optical modes, accounting for a higher contribution of the acoustic modes,

especially at low frequencies where the dispersion tends to be linear for the LA and TA

modes, as seen in Figs. 3.1d, 3.1e, and 3.1f. Second, the out-of-plane acoustic phonons

(ZA), usually called flexural phonons, can be easily distinguished from the TA and LA

branches. This pattern is unusual in 3D crystals, where the ZA and TA branches are

generally degenerate due to the symmetrical disposition of the atoms in the unit cell.

Finally, the mean free path of acoustic phonons is significantly high, sometimes in the

order of microns, as it is the case for graphene. This high mean free path of acoustic

phonons is responsible for the strong dependency of the intrinsic thermal conductivity

on the size of the material.

3.2.2 Group Velocities

The phonon group velocities of graphene, boron nitride, and molybdenum disulfide are

discussed in this section. Figure 3.2 shows the variation of the average squared values of

these group velocities with the phonon frequency. According to Eq. 2.1, the phonon group

velocity contributes to the thermal conductivity as v2α; therefore, showing the variation

of the squared value is more suitable to study the contribution of this phonon property.

The phonon spectrum of molybdenum disulfide (0-14 THz) is much shorter than that of

graphene and boron nitride (0-50 THz). The group velocities in molybdenum disulfide

are significantly smaller than those in graphene and boron nitride. In all cases, the group

velocities are dominated by the acoustic branches, which show the steepest slopes in

the phonon dispersion (see Figs. 3.1d, 3.1e, and 3.1f). The longitudinal acoustic (LA)

branches exhibit the greatest group velocities, 20.7 nm/ps in graphene, 18.2 nm/ps in

boron nitride, and 6.3 nm/ps in molybdenum disulfide. The group velocities of graphene

and boron nitride are higher over a broader frequency range because the linear dispersion

of the acoustic branches can reach frequencies as high as 40 THz. In molybdenum


disulfide, the acoustic branches only reach frequencies of about 6 THz, with slopes that

are one order of magnitude smaller than those in graphene and boron nitride.

Figure 3.2: Squared phonon group velocities for graphene, boron nitride, and molybde-num disulfide.

3.2.3 Relaxation Times

Figure 3.3 shows the phonon lifetime distributions for graphene and boron nitride. These

lifetimes are calculated for a simulation domain of approximately 20 nm x 20 nm (∼ 15000

atoms), with the same four-atom unit cell structure used for the 1x1 zigzag superlattice

(see Section 3.3.1). Thus, direct comparisons can be made between the lifetimes for the

superlattices and the lifetimes for the bulk materials. Despite the similarities in atomic

masses and lattice constants between graphene and boron nitride, the magnitude and

spectral dependence of their lifetimes are substantially different. The lifetimes of acoustic

modes in graphene exhibit a weak dependence on frequency, in qualitative agreement with

previous works [23, 65]. However, it is shown here that they are distributed within shorter

ranges. The lifetimes of the ZA modes are in the range of 8-16 ps up to 14 THz, and

the lifetimes of the TA and LA modes are in the range of 4-10 ps up to 26 THz. In

contrast, Qiu and Ruan [65] obtained lifetimes for the ZA modes in the range of 10-40

ps, and Feng et al. [23] in the range of 10-30 ps. These discrepancies are attributed

to differences in the resolution of the power spectra and the sensitivity of the fitting

procedure, especially of those peaks with the highest intensities at low frequencies. The


lifetimes of acoustic modes in boron nitride show a stronger dependence on frequency

(∼ w−0.2), but still in disagreement with the expected w−2 scaling at low frequency [8].

The absence of a w−2 scaling in these samples of graphene and boron nitride indicates

that the phonon-phonon scattering (intrinsic effect) is not dominant at low frequencies,

as it is the case for bulk argon [83] and silicon [29]. The verification of this scaling in

2D materials will require much bigger simulation domains; nonetheless, going beyond the

size adopted in this work becomes computationally prohibitive for spectral analyses. In

the frequency range between 4 and 16 THz, the average lifetime of ZA, TA and O modes

in boron nitride (∼ 6 ps) is approximately half of that in graphene (∼ 16 ps). In this

same range, it is also notable the smaller lifetimes of the LA modes (∼ 3 ps) in boron

nitride compared to those of the TA modes (∼ 6 ps). At intermediate frequencies (16-22

THz), the lifetime distribution in boron nitride develop a peak at 16 THz for the TA and

O modes, and at 20 THz for the LA mode, coinciding with the location of the second

peak in the phonon DOS (∼ 18 THz).

Figure 3.3: Phonon lifetime distributions for (a) graphene, and (b) boron ni-tride.Reprinted with permission from Ref. [15], Copyright 2016 American Physical Soci-ety

The phonon lifetimes of molybdenum disulfide shown in Fig. 3.4 reveal a completely

different lifetime distribution for this material. All acoustic phonons in molybdenum

disulfide present a linear scaling at low frequency, with lifetimes of up to 60 ps, four

times higher than those obtained in graphene and boron nitride. Interestingly, the ZA


modes exhibit the smallest lifetimes among the acoustic modes. In graphene and boron

nitride, the ZA modes exhibit the highest lifetimes. It is noteworthy that lifetimes in

molybdenum disulfide have a more predominant role in the thermal conductivity than

group velocities. In graphene and boron nitride occurs the opposite, group velocities are

much more relevant than lifetimes.

Figure 3.4: Phonon lifetime distributions for molybdenum disulfide.

3.2.4 Thermal Conductivities

Figure 3.5 shows the thermal conductivity contributions as functions of the phonon fre-

quencies for graphene, boron nitride, and molybdenum disulfide. The total thermal

conductivities are proportional to the areas under these curves. The predicted thermal

conductivities in the x and y directions are provided in Table 3.1. Overall, the ther-

mal conductivities in these materials are highly isotropic. The greatest difference was

obtained in graphene, where the conductivity in the x direction was found to be 5.8%

higher than that in the y direction. The curves shown in Fig. 3.5 correspond to the

thermal conductivities contributions in the x direction. These results show that 90% of

the contributions to the thermal conductivities come from phonons frequencies up to 28.0

THz in graphene, 22.9 THz in boron nitride and 5.8 THz in molybdenum disulfide.


Figure 3.5: Thermal conductivity contributions as functions of the phonon frequenciesfor graphene, boron nitride, and molybdenum disulfide.

Table 3.1: Predicted thermal conductivities in W/mK for grpahene, boron nitride, andmolybdenum disulfide

Kx Ky

Graphene 570.58 539.25Boron Nitride 261.79 260.14Molybdenum Disulfide 104.89 102.83

3.3 Superlattices of Graphene and Boron Nitride

The primitive lattice vectors of the honeycomb lattices of graphene and boron nitride

are nearly the same, enabling the synthesis of superlattices with smooth interfaces [46].

These interfaces favor the specular scattering of phonons [47], making this combination

of materials an excellent test scenario to evaluate the isolated effect of the period on the

phonon transport in 2D superlattices. From a thermal point of view, superlattices have

been mainly investigated for thermoelectric applications, where a minimum thermal con-

ductivity is preferred. As the superlattice period increases, the thermal conductivity first

decreases until it reaches a minimum value, and later it increases until diffusive effects

dominate the thermal transport [47]. This minimum, representing the crossover between


coherent (wave-like) and incoherent (particle-like) phonon transport, has been verified

experimentally in 3D superlattices [66], and predicted theoretically in 2D superlattices

[89]. In the coherent regime, where wave interference effects dominate the thermal trans-

port [47], the thermal conductivity of 3D superlattices of silicon and germanium has

been shown to surpass the thermal conductivities of its constituent materials if the pe-

riod is the shortest possible [25], broadening the range of potential applications for these

nanostructures. A rigorous phonon thermal transport analysis at this short-period limit

is presented here to verify this thermal behavior in 2D superlattices.

In this section, the interplay between phonon group velocities and phonon relaxation

times is investigated, along with the role of the acoustic phonon modes in the thermal

transport in these short-period 2D superlattices of graphene and boron nitride. Thermal

conductivities and phonon properties are estimated in the directions across and along

the interfaces, for superlattices with zigzag and armchair interfaces. Special emphasis is

given to the variation of these properties with superlattice periods within the coherent

phonon transport regime, starting at the short-period limit. This knowledge is key to

develop the capability of designing superlattices to improve the thermal performance of

2D materials. The phonon group velocities, phonon frequencies, and eigen-displacements

are estimated using harmonic lattice dynamics (HLD) simulations. Density functional

theory (DFT) simulations are used to relax the unit cells of the hybrid structures, and

density functional perturbation theory (DFPT) simulations to validate the empirically-

approximated phonon dispersion curves. The phonon relaxation times are extracted from

molecular dynamics (MD) simulations by collecting atomic velocities that are later post-

processed using a normal mode decomposition (NMD) approach. Finally, the properties

of individual phonon modes are used to estimate the thermal conductivity with a mode-

dependent equation from kinetic theory (Eq. 2.1).


3.3.1 Simulation Model: Spectral Energy Density

The unit cells of the superlattices with the shortest period, denoted as 1x1 zigzag (Fig.

3.6a) and 1x1 armchair (Fig. 3.6b), are formed by one irreducible block of boron nitride

and one irreducible block of graphene. Superlattices with both interfaces are investigated

here, with periods 1x1, 2x2, 3x3, 4x4, 5x5, 7x7, and 10x10. These unit cells are initially

assembled with a uniform bond length of 0.143 nm, which is the average value of those

for the honeycomb lattices of boron nitride (0.145 nm) and graphene (0.141 nm) [36].

The equilibrium lattice parameters of these cells are then obtained via variable-cell opti-

mization with the ab-initio Quantum ESPRESSO package [27], following the simulation

setup presented in Appendix A.1. After this relaxation, the unit cells are replicated in

the x and y directions to generate the superlattices of alternated zigzag (Fig. 3.6c) or

armchair (Fig. 3.6d) in-plane layers of graphene and boron nitride. Tables A.1 and A.2

collect the equilibrium lattice vectors ax, number of replicated unit cells Nα, and simu-

lation lengths Lα. Further relaxation is conducted with the interatomic potential within

both HLD and MD simulations.

Figure 3.6: Superlattice unit cells for (a) the 1x1 zigzag interface, and (b) the 1x1armchair interface. Atomic structures of the 1x1 superlattices for (c) the zigzag interface,and (d) the armchair interface. Reprinted with permission from Ref. [15], Copyright 2016American Physical Society.


The simulation domains are rectangular prisms with widths Lx and lengths Ly of

approximately 20 nm and a fixed depth Lz of 1.5 nm that contain around 15000 atoms.

The depth is sufficiently long to preclude self-interaction of atoms in the z direction as

periodic boundary conditions are imposed in all directions of the simulation cell. Once

the simulation cell is built, the superlattices are first relaxed at constant temperature

of 300 K and pressure of 0 bar for 1200 ps with a timestep of 0.2 fs. The superlattices

are next equilibrated at constant volume and a temperature of 300 K for 800 ps. The

equations of motion are then integrated at constant volume and energy for 420 ps (221

steps). In this final integration, the atomic velocities are stored to disk in windows of

105 ps (219 steps) every 6.4x10−3 ps (25 steps). These atomic velocities are later post-

processed to extract the phonon relaxation times, following the NMD approach discussed

in Section 2.1.3.

3.3.2 Dispersion Curves

In this section, the phonon dispersion curves are presented as functions of wavevectors

along the k-space directions [0 1 0] and [1 0 0] for the superlattices 1x1 zigzag (Fig.

3.7) and 1x1 armchair (Fig. 3.8). The density of states (DOS) is plotted beside the

dispersion curves. The dispersion curves exhibit 12 polarization branches for the zigzag

interface and 24 branches for the armchair interface, in correspondence with the number

of degree of freedoms in the zigzag unit cell (4x3=12) and armchair unit cell (8x3=24).

All solid lines are obtained from HLD simulations. The optical (O) modes are shown

in black, out-of-plane acoustic (ZA) in blue, transversal acoustic (TA) in green, and

longitudinal acoustic (LA) in red. For clarity, the results from DFPT simulations (dots)

are shown only for the acoustic branches. For the 1x1 zigzag superlattice, these branches

are responsible for 70.7 % of the thermal transport in the y direction and 57.6 % in the x

direction. This validation with results from DFPT simulations was necessary because the

dispersion curves were not included in the original work where the parametrization of the


potential was developed [36]. There is an excellent agreement between HLD and DFPT

results, confirming that the Tersoff potential used in the HLD simulations reproduces the

main features of the dispersion curves of these short-period superlattices. This agreement

extends to higher-frequency optical branches (not shown), especially those with higher

slopes [14]. These DFPT results are shown here only for validation purposes. The group

velocities are estimated from the HLD curves to be consistent with the lifetime estimation

approach, which uses the same Tersoff potential [36].

Figure 3.7: Dispersion curves for the 1x1 zigzag superlattice (a) along the k-space direc-tion [0 1 0], and (b) along the k-space direction [1 0 0]. (c) Phonon density of states.The solid lines represent data from the HLD simulations. The black solid lines representthe optical modes, and the highlighted lines represent the acoustic branches. The dotsrepresent data from the DFPT simulations. Reprinted with permission from Ref. [15],Copyright 2016 American Physical Society.

The superlattice structure disrupts the continuity of the acoustic branches in the [0 1

0] direction, as illustrated by the segmented branches highlighted in Fig. 3.7a and 3.8a for

the TA and LA modes. This disruption has also been observed in the in-plane phonon

dispersion of vertically-stacked superlattices [3], caused by the spatial confinement of

phonons. Overall, the segmented branches resemble the behavior of the acoustic phonons

in pure graphene [43] and boron nitride [44]. The most distinct feature of these dispersion


curves lies on the frequencies reached by the acoustic branches in each direction, which

is determined by the configuration of the unit cell. The unit cell of the 1x1 armchair

superlattice is approximately squared, causing the acoustic branches to reach similar

frequencies (up to 20 THz) in both directions. However, the unit cell for the 1x1 zigzag

superlattice is rectangular, with ax approximately twice ay, causing the acoustic branches

to reach higher frequencies (up to 30 THz) in the [0 1 0] direction than in the [1 0 0]

direction. These differences are responsible for the anisotropic behavior of the thermal

conductivity in these superlattices. The frequencies reached by the acoustic branches are

reflected on the phonon DOS. The DOS of the 1x1 zigzag superlattice exhibits its highest

peak at 48 THz, in agreement with the DOS of graphene and boron nitride [14]. The

second-highest peak emerges at 20 THz, and the third-highest peak at 10 THz. These

lower-intensity peaks appear around the maximum frequencies reached by the LA and

TA modes in the [1 0 0] direction. This pattern is also found in the DOS of graphene

and boron nitride [14], but with the peaks shifted according to the maximum frequencies

of the acoustic modes: 8 THz and 18 THz for boron nitride and 14 THz and 26 THz for

graphene. The DOS of superlattices with greater periods (not shown) exhibit a similar

shape, but with a greater concentration at 48 THz and a more evenly distributed DOS

at lower frequencies, without an energy gap in the phonon dispersion.

3.3.3 Group Velocities

The effect of the superlattice period on the group velocities of individual acoustic modes

and average group velocities is discussed in this section. Figure 3.9 shows the acoustic

branches for zigzag superlattices with periods 1, 2 and 5, plotted alongside the acoustic

branches of graphene and boron nitride. The length of the first BZ in the [1 0 0] direction

decreases as the period increases, as indicated by the vertical dotted line for the 2x2

superlattice, and the vertical dashed line for the 5x5 superlattice. The length of the

first BZ in the [0 1 0] direction remains unchanged because the lattice vector in this


Figure 3.8: Dispersion curves for the 1x1 armchair superlattice (a) along the k-spacedirection [0 1 0], and (b) along the k-space direction [1 0 0]. (c) Phonon density of states.The solid lines represent data from the HLD simulations. The black solid lines representthe optical modes, and the highlighted lines represent the acoustic branches. The dotsrepresent data from the DFPT simulations. Reprinted with permission from Ref. [15],Copyright 2016 American Physical Society.

direction is the same for all periods. For the sake of comparing the acoustic branches

of the superlattices with those of graphene and boron nitride, the segmented curves in

the [0 1 0] direction are adjusted in Fig. 3.9a to fit the linear dispersion of TA and LA

modes at low frequency. In all cases, the curves corresponding to the superlattices are in

between those of graphene and boron nitride. The TA modes in both directions exhibit

the greatest differences in slope (group velocities) between graphene and boron nitride.

The group velocities of the TA modes in graphene (14.91 nm/ps) are approximately 34 %

higher than those in boron nitride (11.12 nm/ps). This difference is reduced to 15 % for

the LA modes. All the ZA branches show a parabolic dispersion with approximately zero

group velocity at the gamma point. In the [1 0 0] direction, the acoustic branches are

truncated at smaller frequencies as the period increases; however, their group velocities

are preserved and they have approximately the same value as those for boron nitride.

In the [0 1 0] direction, the group velocities for the 1x1 and 2x2 superlattices are closer


to the average group velocities of graphene and boron nitride, and they decrease as the

period increases.

Figure 3.9: Acoustic dispersion curves (a) along the k-space direction [0 1 0], and (b)along the k-space direction [1 0 0]. Reprinted with permission from Ref. [15], Copyright2016 American Physical Society.

Figure 3.10 shows the variation of the average squared group velocity with the phonon

frequency and superlattice period in the x and y directions for the zigzag superlattices.

The squared group velocities are predominantly higher at period 1 in both directions. At

this period, the greatest averages are observed at 12 THz in the x direction (50 nm2/ps2)

and 14 THz in the y direction (75 nm2/ps2). These peaks represent the frequency level at

which the interplay between acoustic and optical branches maximizes the average group

velocities, which occurs at wavevectors around the center of the irreducible Brillouin zone

(k/kmax = 0.5) in both directions, as observed in Fig. 3.7. On the other hand, the wells

at 10 THz account for the combined effect of steep acoustic branches at the center of

the zone and flat optical branches at the borders. In the x direction, the velocities are

consistently reduced as the period increases. The average velocity in the frequency range

between 4 THz and 20 THz drops from 33.08 nm2/ps2 to 29.61 nm2/ps2 from period 1

to period 2, a decreasing ratio that is more than twice higher than that from period 2 to


period 5 (1.30 nm2/ps2/period). In the y direction, the average velocity in this frequency

range drops from 48.74 nm2/ps2 to 40.21 nm2/ps2 from period 1 to period 2, and then

remains practically unchanged for higher periods.

Figure 3.10: Squared phonon group velocities for the zigzag superlattices in (a) the xdirection, and (b) the y direction. Reprinted with permission from Ref. [15], Copyright2016 American Physical Society.

3.3.4 Relaxation Times

This section start by presenting a sample of the fitted power spectra, from which the

relaxation times are extracted. Figure 3.11 shows the discrete power spectra (circles) and

fitting curves (solid lines) as functions of the anharmonic phonon frequencies for three

isolated peaks, which correspond to the acoustic polarizations at the wave vector k′ =

[17π/23ax, 0, 0] for the 1x1 zigzag superlattice (shown in Fig. 3.7b). By incorporating

the harmonic eigen-displacements in the spectral analysis, the lifetimes are estimated

from fully isolated peaks. These peaks are obtained without any signature from other

frequencies, meaning that the harmonic eigen-displacements are an excellent description

of the anharmonic modes at room temperature. This mode-by-mode analysis allows

fitting the spectra considering all data points within five orders of magnitude below the


point with the maximum energy. This range of data minimizes the root mean squared

error (RMSE), whose averaged value for the 1x1 zigzag superlattice (353) is fairly low

compared to the range of energy values (100 − 106). The uncertainty due to fitting is

estimated with a cross-validation approach by varying the range of the data points used

for the fitting, considering data points within 3 to 5 orders of magnitude of the maximum

data value. The average uncertainty for all phonon lifetimes increases from 2.75 % at

period one to 4.62 % at period 10. The peak intensity is the greatest for the ZA mode

and decreases for higher-frequency modes (TA and LA), and so does the phonon lifetime.

This trend is replicated at all wave vectors. In addition, the anharmonic frequencies for

these acoustic modes are fairly close to the corresponding harmonic frequencies, only

shifted by 2.83 % for the ZA mode, 1.13 % for the TA mode, and 1.12 % for the LA

mode. The average shift in frequency is below 2 % for all superlattice periods; therefore,

it is reasonable to calculate the group velocities from HLD simulations.

Figure 3.11: Phonon power spectra for the 1x1 zigzag superlattice at wavevector k′ =[17π/23ax, 0, 0] for the acoustic polarizations (a) out-of-plane ZA, (b) transverse TA, and(c) longitudinal LA. Reprinted with permission from Ref. [15], Copyright 2016 AmericanPhysical Society.

The lifetime distributions for the zigzag superlattices are shown in Fig. 3.12 for


periods 1, 2, 5 and 10. The dimensions of the simulation cell are kept approximately

constant for all periods (∼ 20 nm x 20 nm), such that lifetimes are not affected by the size

of the samples and the effect of the period can be addressed independently. The lifetimes

for the 1x1 superlattice resemble the scaling observed in boron nitride, but with average

values in between those of graphene and boron nitride. The resolution of the acoustic

modes decreases at higher periods because there are less number of replicated unit cells.

At the same time, there are more optical modes due to branch folding; therefore, the whole

resolution of the BZ remains approximately constant. Overall, lifetimes show a weak

dependence on the superlattice period, indicating that the phonon transport is dominated

by coherent effects, i.e., long-wavelength phonons traveling across the interfaces without

scattering [17]. However, it is noticeable that the maximum lifetimes of optical modes

in the frequency range of 10-20 THz and 24-40 THz increase from approximately 9 ps at

period 1 to 12 ps at period 10. Phonons experiencing this increase come from flat optical

branches at intermediate frequencies in the phonon dispersion, with very small group

velocities and short wavelengths. Thus, these phonons are expected to scatter diffusively

at the interfaces [47]. This increase is not as significant for flat optical branches at higher

frequencies (40-50 THz) due to the much higher phonon DOS.

Figure 3.12: Phonon lifetime distributions for the superlattices (a) 1x1 zigzag, (b) 2x2zigzag, (c) 5x5 zigzag, and (d) 10x10 zigzag. Reprinted with permission from Ref. [15],Copyright 2016 American Physical Society.


3.3.5 Thermal Conductivities

The thermal conductivities are predicted from the phonon properties discussed in pre-

vious sections and presented in Figure 3.13 as functions of the superlattice period for

both directions and interface configurations. The uncertainties due to the statistical er-

ror (not shown) are less than 1 % in all cases, estimated by systematically removing

one MD simulation from the average power spectra. The high resolution of the power

spectra (∆w = 0.06rad/ps) and the improved fitting procedure adopted here allowed

to capture peaks with higher intensities, revealing new physics in the relaxation time

distribution. The first focus is on the highest thermal conductivities, predicted at period

1 for the zigzag configuration. At this period, the thermal conductivity in the x direction

(227.15 W/mK) is approximately 13 % smaller than that of a same-size sample of boron

nitride (261.79 W/mK), and 60 % smaller than that of graphene (570.58 W/mK). The

thermal conductivity in the y direction (340.82 W/mK) is closer to the average conduc-

tivity of graphene and boron nitride. These results contrast with those found by Gard et

al. [25] in 3D superlattices of silicon and germanium with perfect interfaces, where the

thermal conductivities at the shortest period were found to be higher than those of the

constituent materials. The authors showed how the significant difference in the atomic

masses of silicon (28.09 amu) and germanium (72.64 amu) induces an energy gap in the

phonon dispersion, causing a substantial increase of the relaxation times at this limit.

The formation of such a gap in these superlattices is precluded due to the similarities

between the masses of carbon (12.01 amu), boron (10.81 amu) and nitrogen (14.01 amu)

atoms.

The response of the thermal conductivities to an increase in the periodicity follows

the same trend in both interface configurations, being the conductivities for the zigzag

superlattices always higher than those for the armchair superlattices, on average 7 %

higher in the x direction and 19 % in the y direction. Note that the greatest decrease in

the conductivities occurs between period 1 and 2, approximately a 13 % decrease in the


Figure 3.13: Variation of the thermal conductivity with the superlattice period andinterface structure. Reprinted with permission from Ref. [15], Copyright 2016 AmericanPhysical Society.

x direction and 16 % in the y direction. In the x direction, the thermal conductivities

monotonically decrease with increasing periodicity, which is consistent with a phonon

transport regime dominated by coherent effects [47]. The rate of decrease from period 1

to 2 (∼ 30 W/mK/period) is approximately three times higher than that from period 2

to 10 (∼ 11 W/mK/period). This monotonic decrease up to period 10 (ax = 4.32 nm) is

in qualitative agreement with results from non-equilibrium molecular dynamics (NEMD)

simulations [89], where coherent effects were found dominant for period lengths smaller

than 6 nm, regardless of the sample size. In the y direction, the thermal conductivities

first decrease from period 1 to 2 (∼ 50 W/mK/period), and then they slightly recover for

periods higher than 2 (∼ 2 W/mK/period). This recover indicates that diffusive effects

are dominant in the y direction, i.e., the thermal conductivity increases with increas-

ing periodicity because there are less sites for the diffuse scattering of short-wavelength

phonons at the interfaces.

Figure 3.14 shows the contribution of each phonon frequency to the thermal conduc-

tivity for the zigzag and armchair superlattices with periods 1, 2, 5 and 10. The total


Figure 3.14: Thermal conductivity contributions as functions of the phonon frequenciesfor the superlattices (a) zigzag in the x direction, (b) armchair in the x direction, (c)zigzag in the y direction, and (d) armchair in the y direction. Reprinted with permissionfrom Ref. [15], Copyright 2016 American Physical Society.

thermal conductivity is proportional to the areas under these curves. For both interface

configurations, 90% of the contributions come from phonon frequencies up to approxi-

mately 23 THz in the x direction and 26 THz in the y direction, regardless of the period.

In all cases, the contributions at period 1 significantly deviate from those at larger peri-

ods. This deviation extends over the frequency range between 6 THz and 14 THz in the

x direction, and between 6 THz and 26 THz in the y direction. For periods higher than

2, the contributions within these ranges are consistently reduced in the x direction, and

remain practically unchanged in the y direction. These findings are in direct correlation

with the group velocity variations shown in Fig. 3.10, denoting the more relevant role

of these phonon properties in the anisotropic behavior of the thermal transport. Inter-

estingly, the maximum peaks in the contributions at period 1 emerge at approximately

10 THz, coinciding with the location of the first well in the group velocities. However,

at 10 THz also emerges the first peak in the phonon DOS. Therefore, these maxima are

driven by the higher concentration of states at this frequency. Similarly, the second peak


in the contributions appears at approximately 20 THz and is more significant in the y

direction for the 1x1 zigzag superlattice, coinciding with the location of the second well

in the group velocities and the second peak in the phonon DOS. In Fig. 3.14c, it is

noteworthy the slight increase in the contributions for periods higher 2 in the frequency

range 10-20 THz, confirming the origin of the increase in the thermal conductivities in

the y direction, coming from the increase in the relaxation times reported within this

range, as discussed in Section 3.3.4. Finally, the contributions from frequencies higher

than 26 THz in all superlattices converge and ultimately decay to zero, due the combined

effect of the low group velocities, relaxation times and specific heats of high-frequency

phonons.

3.4 Summary

A normal mode decomposition (NMD) approach was implemented to predict the phonon

properties and thermal conductivities at 300 K of molybdenum disulfide, graphene, boron

nitride and seven short-period superlattices of boron nitride and graphene, with zigzag

and armchair interfaces. The simulations have been conducted on fully relaxed squared

samples of these layered materials. A rigorous description of the dispersion curves and

density of states was provided, including a comparison between the acoustic branches in

the superlattices with those in the constituent materials. It was found in the superlat-

tices that 90 % of the thermal conductivity is contributed from phonons with frequencies

up to 23 THz in the x direction and 26 THz in the y direction, regardless of the period

length. The mode-by-mode analysis has revealed the dominant role of group velocities

(coherent effects) in the x direction, causing a monotonic decrease in the thermal con-

ductivity as the period increases. Notably, the rate of this decrease from the first to the

second period (∼ 30 W/mK/period) is three times higher than that at greater periods

(∼ 11 W/mK/period). It was also detected the less relevant role of diffusive scatter-


ing (incoherent effects), particularly evident in the y direction, where the increase in

the relaxation times of short wavelength phonons causes a slight increase in the thermal

conductivities (∼ 2 W/mK/period).

Chapter 4

Equilibrium Molecular Dynamics:

Superlattices

4.1 Motivation

In this chapter, the thermal conductivities of the two-dimensional superlattices of graphene

and boron nitride are investigated with equilibrium MD simulations, following the Green-

Kubo formalism described in Section 2.2.1. This method has a number of advantages.

First, the formulae to calculate the heat current vector and the correlation functions are

straightforward to implement. Second, the phonon scattering on the domain edges is

practically precluded by applying periodic boundary conditions. These conditions allow

reducing the systems size and hence the computational effort. Third, MD simulations

capture anharmonicity effects on phonon transport such as thermal expansion, temper-

ature dependence of elastic constants and phonon frequencies, and phase transitions.

Fourth, in contrast to the direct method, a single Green-Kubo simulation is able to yield

the whole thermal conductivity tensor. This ability is a powerful feature to investigate

anisotropic effects on thermal transport. Conversely, the Green-Kubo method requires

a significant number of simulations to reduce the statistical error and provide accurate

37

Chapter 4. Equilibrium Molecular Dynamics: Superlattices 38

qualitative values of the thermal properties of interest.

4.2 Simulation Model: Green-Kubo

The superlattices of graphene and boron nitride studied in this chapter have period

lengths of one, two, three, four, and five atoms. These periods will be hereinafter referred

to as 1x1, 2x2, 3x3, 4x4, and 5x5. Once the unit cell is built, the superlattice is assembled

by replicating the unit cell in the x and y direction. The correspondent lengths and widths

of the 1x1 superlattices are 20.00 nm x 19.87 nm in the armchair configuration, and 20.04

nm x 19.86 nm in the zigzag configuration. The depth of the simulation cells is set to 1.5

nm for all cases, which is long enough to avoid self-interactions [16]. Tables A.1 and A.2

collect the lengths and number of repetitions of the unit cells in x and y directions for each

configuration. Figure 4.1 illustrates the 1x1 and 5x5 superlattices in which carbon atoms

are painted in green, boron atoms are painted in blue, and nitrogen atoms are painted in

red. The grey frames depict every unit cell which is constituted by four carbon atoms,

two boron atoms, and two nitrogen atoms in the 1x1 armchair configuration and two

carbon atoms, two boron atoms, and two nitrogen atoms 1x1 zigzag configuration.

The simulation timestep is chosen at 0.10 fs to conserve energy per atom to the sixth

significant figure, and periodic boundary conditions are imposed in all MD simulations

to prevent phonon scattering across the edges of the computational domain. Thirty

simulations are made with new initial velocities for each superlattice period and interface

orientation to reduce the statistical error. Each system is relaxed by integrating the

equations of motion at constant number of atoms, pressure of 0 bar, and temperature

of 300 K for 2.5x106 time steps with a damping temperature constant of 20 fs and a

damping pressure constant of 200 fs. During the equilibration, the velocity of the center

of mass of the system is subtracted in each direction every 1000 steps. This subtraction

maintains a zero center of mass velocity that minimizes the divergence of the heat current


Figure 4.1: Unit cells of (a) 1x1 and (b) 5x5 zigzag, and (c) 1x1 and (d) 5x5 (d) armchairsuperlattices. Reprinted with permission from Ref. [67], Copyright 2015 ASME.

auto-correlation functions and thus unphysical values of thermal conductivity. After the

relaxation, the system is equilibrated for another 2.5x106 time steps at constant number

of atoms, volume, and energy. The simulation is then run for 1.5x107 time steps (1.5 ns).

These long simulations are required to collect the contribution of all phonon frequencies

to thermal transport.

4.3 Heat Current Autocorrelation Functions

Figure 4.2 illustrates the decay of the averaged signals of the thirty normalized heat

current autocorrelation functions for the 1x1 armchair superlattice in the x, y, and z

directions and their summation. Each signal is normalized with their initial value. The

insets show the high oscillations of the signals within the first 0.5 ps of correlation time.

These oscillations behave differently in each direction. After their initial drop in the x

and y directions, the crests height are higher than the valleys. After the first valley,

the highest crests have amplitudes of 0.62 at 0.08 ps in the y-direction and 0.51 at

0.16 ps in the x-direction, while the deepest valleys have amplitudes of -0.16 at 0.11


ps in the y-direction and -0.28 at 0.05 ps in the x direction. In contrast, the signal in

the z direction follows a more symmetric pattern. Given that the oscillations in the

autocorrelation functions result from the rapid transport of energy back and forth over

interatomic distances [39], this symmetric pattern suggests that the out-of-plane thermal

transport is negligible. This negligible transport is caused by the weak van der Waals

forces in the z direction in two-dimensional materials.

Figure 4.2: Normalized heat current autocorrelation functions for the 1x1 armchair su-perlattice in (a) the x, (b) y, and (c) z directions, and (d) their summation. Reprintedwith permission from Ref. [67], Copyright 2015 ASME.

The decay of the correlation functions is affected by the interface orientation. Figure

4.3 shows the averaged autocorrelation functions for the 1x1 zigzag superlattice. Al-

though the correlations functions keep exhibiting similar oscillating decays in this orien-

tation, the in-plane signals behave differently in each direction. The deepest valleys have

amplitudes of -0.39 at 0.15 ps in the x-direction and -0.12 at 0.13 ps in the y-direction.

Conversely, the height of the crests are not significantly affected. After the first valley,

the highest crests have amplitudes of 0.47 at 0.13 ps in the x-direction and 0.56 at 0.04


ps in the y-direction. Moreover, the correlation functions plotted in Fig. 4.2 and 4.3

show a much faster decay in the in-plane directions than in the out-of-plane direction.

The correlation functions are described by a number of damped oscillation functions of

the form exp(−αt)cos(w0t), where α is the attenuation factor, w0 is oscillation frequency,

and the distance between crests or wavelength is 2π/w0. The heat current correlation

functions should exhibit a decay that can be reasonably fitted by an exponential func-

tion.Oscillating heat current correlation functions, as those shown in Fig. 4.2 and 4.3,

have been also observed in other materials such as silica [49, 50], GaN [34], and diamond

[19] and appear to be dependent on the functional form of the interatomic potential.

Figure 4.3: Normalized heat current autocorrelation functions for the 1x1 zigzag super-lattice in (a) the x, (b) y, and (c) z directions, and (d) their summation. Reprinted withpermission from Ref. [67], Copyright 2015 ASME.


4.4 Thermal Conductivity Evolution

Figure 4.4 illustrates the evolution of the average thermal conductivities in the x and y

directions obtained by direct integration of the autocorrelation functions for the 1x1 and

5x5 zigzag superlattices. Conductivities in the z directions are not presented because

they are close to zero. The evolution of the conductivities are similar in all cases in

which the signal surges during the first 15 ps of correlation time, reaching a value that

varies little for longer times. At 20 ps, the signals stabilize except for the 1x1 armchair

case which weakly increases. Given that the average signals are well stabilized above 200

ps, the thermal conductivities in this work are taken as the average values between this

time and 400 ps.The correspondent conductivities in the x and y directions are 204.9

W/mK and 361.2 W/m K for the 1x1 armchair case, and 315.0 W/m K and 369.4 W/m

K for the 1x1 zigzag case. In addition, the thermal conductivities in the x and y direc-

tions are 143.3 W/m K and 245.4 W/m K for the 5x5 armchair case, and 172.1 W/m

K and 319.4 W/m K for the 5x5 zigzag case. On the other hand, the superlattice peri-

odicity and orientation have a relevant effect on the standard deviations of the averaged

conductivities. For instance, the deviations increase with increasing correlation time,

reaching values similar to the averages. This high statistical variance confirms that the

Green-Kubo method produces results with high dispersions in systems with high thermal

conductivities. High dispersions have been also reported when predicting thermal con-

ductivities of pure graphene [88]. In contrast, the standard deviations are considerably

lower in the x direction of the 1x1 armchair superlattices. For example, at 350 ps the

standard deviation of the 1x1 armchair signal in the X direction is 161.6 W/m K with

an average of 189.2 W/m K , whereas in the y direction, the standard deviation is 483.8

W/m K with an average of 366.9 W/m K.The high standard deviations are caused by

two reasons: first, in a few runs thermal conductivities reach very low values of a few

W/m K but not reaching negative values; and second, some runs exhibit extremely high

thermal conductivities surpassing 1000 W/m K.


Figure 4.4: Evolution of the average thermal conductivities in the x (kxx) and y (kyy)directions calculated using the Green-Kubo methodology for (a) the 1x1 armchair, (b)5x5 armchair, (c) 1x1 zigzag, and (d) 5x5 zigzag superlattices. The error bars are plottedevery 50 ps to show the standard deviations. Reprinted with permission from Ref. [67],Copyright 2015 ASME.

4.5 Superlattice Thermal Conductivity

Figure 4.5 shows the predicted thermal conductivities as a function of the superlattice

period and interface orientation. Thermal conductivities generally decrease with the

superlattice period. Except for the armchair geometry in the x direction, the highest

decrease occurs when the periodicity rises from one to two. For example, conductivities

shrink by up to 26.5% along the y direction in the 1x1 armchair superlattice. This

decrease is less significant for periodicities above two. In addition, the highest thermal

conductivities are predicted in the zigzag chirality along the y direction (parallel to the

interface), with a maximum value of 369.4 W/m K. These values are then followed by

those predicted in the zigzag chirality in the x direction, excluding the periodicity of

three. Finally, the lowest thermal conductivities are found in the armchair chirality in


Figure 4.5: Thermal conductivities versus the superlattice period and interface orienta-tion using the Green-Kubo (G-K) method and results from spectral energy density (SED)analysis discussed in Chapter B. Panel (a) shows data for the zigzag configuration andpanel (b) shows data for the armchair configuration. Reprinted with permission fromRef. [67], Copyright 2015 ASME.

the x direction. Although these values are subject to a high statistical variation, they

reveal that the interface orientation can be used to tune the thermal conductivities of

two-dimensional superlattices of graphene and boron nitride .

The thermal conductivities obtained with the Green-Kubo method are compared in

Fig. 4.5 with those from the SED analysis discussed in Chapter B.The Green-Kubo

results are qualitatively in good agreement with the SED results. Despite the high stan-

dard deviation of the Green-Kubo method, these simulations are able to reproduce the

decrease in thermal conductivity from period one to two, followed by a plateau for higher

periods. In contrast, the main discrepancies are that Green-Kubo conductivities are gen-

erally higher than the SED counterparts, and the behavior of the averaged conductivities

in the x direction for the armchair configuration, which do not decrease as the superlat-

tice period increases. In addition, the statistical error of the SED results is significantly

lower than those of Green-Kubo, with a maximum standard deviation of only 3.71 W/m

K. Therefore, this results show that the SED approach is more appropriate to study

thermal transport of high-conductivity systems such as two-dimensional superlattices of

graphene and boron nitride.


4.6 Summary

The Green-Kubo method has been employed to compute thermal conductivities at room

temperature of two-dimensional superlattices of graphene and boron nitride as a func-

tion of the superlattice periodicity and interface orientation. For all interface geometries

thermal conductivities generally decrease with increasing superlattice period. Such de-

crease is more intense when the periodicity increases from one to two, with a reduction

of up to 26.5%. In addition, the Green-Kubo method predicts time-domain thermal

conductivities with high standard deviations. These deviations are as large as the aver-

aged values, which makes the Green-Kubo method a less efficient option for computing

thermal conductivities than the SED approach.

Chapter 5

Non-equilibrium Molecular

Dynamics: Strained Superlattices

5.1 Motivation

Two-dimensional (2D) boron nitride-graphene strip superlattices can now be synthesized

with strip sizes that can be precisely controlled [46]. This discovery has led to an in-

creased interest on these hybrid materials, opening opportunities for doing research on

more complex nanostructures with enhanced electronic and thermal properties. Boron

nitride has been tested in combination with graphene as a supporting substrate [18],

and also embedded in the graphene layer forming 2D superlattices [12, 46]. Dean et

al. [18] reported significant improvements in the electronic properties of graphene when

it is supported on boron nitride instead of silicon dioxide (SiO2). Silicon dioxide is a

dielectric material widely used in current nanoelectronics and has the disadvantage of

having thermal conductivities much smaller than the ones reported for boron nitride at

the few-layer phase [33]. Although the thermal transport across these interfaces needs

to be further investigated, there is certainly a potential for BN to become a main heat

spreader in high power density applications. Recent studies on thermal transport on BN-

46

Chapter 5. Non-equilibrium Molecular Dynamics: Strained Superlattices 47

graphene superlattices [36, 75] show that the thermal conductivity parallel to the strips

interfaces is more sensitive to the superlattice composition and greater than the thermal

conductivity perpendicular to the interface. However, there is a lack of understanding

of how stretching, for instant, which is a technique commonly used to tune the thermal

conductivity, affects the thermal transport in these nanostructures. In this chapter, non-

equilibrium molecular dynamics (NEMD) simulations are used to investigate the thermal

transport in boron nitride (BN) nanosheets and BN-graphene superlattices, considering

the effect of uniaxial tensile strain along the two highly symmetric lattice orientations

(zigzag and armchair).

5.2 Simulation Model: Direct Method

Uniaxial tensile strain is applied on samples of boron nitride (BN) nanosheets (Fig. 5.1)

and boron nitride-graphene (BN-C) superlattices (Fig. 5.3 and 5.4) at room temperature

(T=300K). The thermal conductivity is estimated by implementing NEMD simulations

with the so called ‘direct method’, whose schematic is also shown in Fig. 5.1. This

simulation technique requires the generation of an artificial heat flux Jx by adding and

subtracting energy at the hot and cold regions, respectively. Once the steady state is

reached, the temperature profile is averaged over a given period of time, and the thermal

conductivity Kx can be calculated by means of the Fourier’s Law of conduction

Jx =Q

A= Kx

∂T

∂x(5.1)

where, A is the cross-section area LyxLz, with Ly being the width of the atomic layer,

and Lz its thickness. The width Ly was set equal to 20 nm in all cases tested here,

which was considered to be sufficient to replicate infinitely wide nanosheets when periodic

boundaries conditions are applied in the y direction. A thickness Lz of 0.34 nm was

considered for all nanosheets [36, 75]. In Equation 5.1, ∂T∂x

is the temperature gradient


along the x axis, and Q represents the heat rate at which energy is being added and

subtracted, which was chosen to be 2 eV/ps in all the simulations. This value was found

to be in the range where the nonlinearities produced by the heat current are very small,

thus, Fourier’s Law is still valid.

Figure 5.1: Zigzag boron nitride model along with the schematic of the NEMD simula-tions. Reprinted with permission from Ref. [16], Copyright 2014 CSME.

The honeycomb lattices structures of BN and graphene were built base on the nearest

B-N and C-C bond length distances of 0.145 nm and 0.142 nm, respectively [36, 75]. The

BN and graphene nanosheets were generated based on rectangular unit cells containing

four atoms. Two different unit cell orientations were considered with respect to the x

and y axes. These unit cells determines whether the highly symmetric zigzag (Fig. 5.2a)

or armchair (Fig. 5.2b) orientations are obtained along the x direction. For simplicity,

only the zigzag orientations are shown in Fig. 5.1 for BN, and in Fig 5.3 and Fig. 5.4

for BN-C superlattices, however results were also presented for the armchair orientation

in all cases.

The zigzag BN model implemented here is shown in Fig. 5.1, consisting of 960 unit

cells in the longitudinal direction and 46 in the transverse direction, which corresponds

to an area of 240 nm x 20 nm, approximately. In order to generate a similar surface

area with the armchair orientation, 556 and 80 unit cells were required in the longitudi-

nal and transverse direction, respectively. Zigzag and armchair graphene domains with


Figure 5.2: Boron nitride unit cell for: (a) zigzag orientation, and (b) armchair orienta-tion. Reprinted with permission from Ref. [16], Copyright 2014 CSME.

similar surface areas were also generated using the same unit cell configuration. In all

cases, periodic boundaries conditions were implemented in the transverse direction, and

non-periodic in the longitudinal. In consequence, atoms at the two cells adjacent to the

longitudinal edges were fixed, this aiming to prevent them to sublimate from the simu-

lation box at the edges where they are not allowed to move from one side of the box to

the other, how it is the case when periodic boundary conditions are used.

The BN-Graphene hybrid structures consist of rectangular domains with dimensions

Lx x Ly equal to 100 nm x 20 nm, and with two kind of strip superlattices: one in which

the heat flux is perpendicular to the BN-C interfaces (Fig. 5.3), and the other where

it is parallel (Fig. 5.4). For the perpendicular case, a superlattice period of 20 nm was

considered, formed by three different configurations given by LBN/LG equal to 1.0, 1.7

and 3.0. For the parallel scenario, superlattice periods of 10 nm were used, with the

same relations LBN/LG considered in the perpendicular case. The C-C interactions were

modeled by the optimized Tersoff potential develop by Lindsay and Broido [43]. The B-B

and B-N-C interactions were modeled by a different set of Tersoff parameters created by

Kinaci et al. [36].

Uniaxial tensile strain was applied along the longitudinal direction (x axis) as shown

in Fig 5.1, 5.3 and 5.4, following a simulation approach similar to the one applied by

Hu et al. [30] on silicene sheets. This approach contemplates successive simulation steps


Figure 5.3: Zigzag BN-graphene hybrid model with the heat flux Jx perpendicular to theBN-C interfaces. Reprinted with permission from Ref. [16], Copyright 2014 CSME.

that include strain application, relaxation, equilibration, and steady-state averaging of

the temperature profile, totaling 11 ns of total run time with an integration timestep of

0.5 fs. First, the tensile strain was applied dynamically by deforming the simulation box

for 1 ns at a constant strain rate. After the strain is applied, the system is relaxed for 1

ns in the NPT ensemble. During this process the edges are kept fixed while the system

is allowed to relax in the y direction. Now, the system is permitted to fully equilibrate

in the NVT ensemble for another 1 ns. Once the system is equilibrated, energy is added

and subtracted at a constant rate of 2 eV/ps for 3 ns, while the system is run in the NVE

ensemble until steady state conditions are reached. Finally, the system is let to evolve for

5 ns in the NVE ensemble and the temperature profile is time-averaged over that time.

5.3 Strained Nanosheets: Graphene and Boron Ni-

tride

Normalized thermal conductivity (K/K0) results are shown in Fig. 5.5 for different

tensile strain levels applied on boron nitride and graphene monolayers along the zigzag


Figure 5.4: Zigzag BN-graphene hybrid model with the heat flux Jx parallel to the BN-Cinterfaces. Reprinted with permission from Ref. [16], Copyright 2014 CSME.

and armchair orientations. These results were normalized with respect to the thermal

conductivities (K0) obtained at 0 % strain, whose values were found to be 447.537 W/mK

and 451.906 W/mK for BN along the zigzag and armchair orientations, respectively. In

the case of graphene, the reference values are 1206.819 W/mK for zig-zag, and 1205.890

for armchair. It is important to notice that, despite the significant similarities in their

lattice structures, the thermal conductivity of BN is substantially smaller (about 60 %)

than the one obtained for graphene. This behavior has been reported in previous works

[75], along with the fact that there are not relevant differences in the thermal conductivity

among both orientations at 0 % strain [9], which is consistent with these findings.

The thermal conductivity of graphene, as can be seen in Fig. 5.5, shows a slight

increase in the zigzag orientation at 2 % of strain and starts dropping dramatically after

4 % of strain, which is in line with previous studies [30]. On the other hand, the thermal

conductivity of boron nitride significantly increases (15-25 %) in both orientations when 6

% of tensile strain is applied. This anomalous thermal response will be addressed in next

chapter from a phonon transport perspective. Another important finding is the difference

in the thermal response of both orientations to the application of tensile strain, obtaining

in all cases higher thermal conductivities for the zigzag orientation, this in contrast with


Figure 5.5: Normalized thermal conductivity (k/k0) results from NEMD simulationsin boron nitride and graphene monolayers along the zigzag and armchair orientations.Reprinted with permission from Ref. [16], Copyright 2014 CSME.

the behavior of BN at 0 % strain, for which the orientation does not seem to be relevant

for the thermal transport.

5.4 Strained Superlattices

Figures 5.6 and 5.7 show results about the thermal response to tensile strain of the BN-

Graphene superlattice models in both orientations when the heat flux is perpendicular

and parallel to the interfaces, respectively. Contrasting Fig. 5.6 and 5.7, it is relevant

to highlight that, first, the thermal conductivities are, in all cases, much smaller in the

direction perpendicular to the interfaces and they keep practically unchanged for the

three LBN/LG configuration tested. This is an expected behavior because the thermal

transport in the perpendicular direction is restricted by the material with the smallest

thermal conductivity, in this case boron nitride. Second, we found particularly relevant

that the thermal conductivities parallel to the interfaces (see Fig. 5.7) are significantly

affected by the LBN/LG configurations, especially in the zigzag orientation. The highest

thermal conductivities were obtained for the smallest LBN/LG relation (1.0), taking


advantage in this case of the higher thermal conductivities associated to wider graphene

strips and also reproducing the thermal response of pristine boron nitride shown in Fig.

5.5, characterized by an increase in the thermal conductivities when strain is applied.

Finally, it was found that the thermal response is much more sensitive to the lattice

orientation (zigzag or armchair) when the heat flux is parallel to BN-C interfaces, as

evidenced by the much higher differences in thermal conductivities shown in Fig. 5.7.

Figure 5.6: Thermal conductivity for the BN-Graphene hybrid model with the heatflux perpendicular to the BN-C interfaces along the zigzag and armchair orientations.Reprinted with permission from Ref. [16], Copyright 2014 CSME.

Figure 5.7: Thermal conductivity for the BN-Graphene hybrid model with the heat fluxparallel to the BN-C interfaces along the zigzag and armchair orientations. Reprintedwith permission from Ref. [16], Copyright 2014 CSME.


5.5 Summary

In this chapter, a non-equilibrium molecular dynamics (NEMD) simulation technique

was used to analyze the effect of uniaxial strain in pristine samples of boron nitride and

boron nitride graphene superlattices. Results show that the thermal conductivity of

boron nitride increases until a 6 % of strain is applied, in contrast with the behavior

of graphene, for which the thermal conductivity drops dramatically in response to small

strains. In all cases, higher thermal conductivities were found for the zigzag orientations.

Results regarding the thermal transport in BN-C superlattices indicate that thermal

conductivities parallel to the interfaces are much higher and also more affected by the

LBN/LG configuration, in contrast with the perpendicular scenario. It was found that

the thermal transport is much more sensitive to the lattice orientation when the heat flux

is parallel to BN-C interfaces. In next chapter, these findings will be further investigated,

using more comprehensive simulations based on phonon transport analysis.

Chapter 6

Phonon Transport: Strained

Monolayers and Superlattices

6.1 Motivation

Strain effects have been extensively studied in bulk materials. Phonons are known to

be ‘stiffened’ when compressive strain is applied, increasing the thermal conductivity. In

contrast, phonons are known to be ‘softened’ when tensile strain is applied, decreasing the

thermal conductivity. Phonon dispersion curves change when the material is subjected

to strain because the lattice structure is physically modified. If the strain is tensile,

phonon branches with lower frequencies are obtained, and the contributions of the group

velocities to the thermal conductivity are consequentially reduced. Another well-known

negative effect of tensile strain in bulk materials is the confinement of energy states in a

narrower space, which increases the probability of phonon scattering, and then decreases

the thermal conductivity. In single-layer materials, the thermal response to strain is

different due to the two-dimensional nature of the lattice structure and the role of out-

of-plane phonon modes (flexural phonons).

55

Chapter 6. Phonon Transport: Strained Monolayers and Superlattices 56

In this chapter, the phonon transport approach described in Section 2.1 is applied

on strained samples of graphene, boron nitride, molybdenum disulfide, and strained su-

perlattices of graphene and boron nitride. The lattice structures are uniaxially stretched

at a constant engineering strain rate of 2.3x10−5 1/ps for 800 ps, which corresponds to

a deformation of 2% with a simulation timestep of 0.2 fs. Once the lattice is stretched

to the desired level, the structure is allowed to relax in the y direction. Then, atomic

velocities are collected to be later postprocessed. Successive deformation steps of 2% are

conducted until observing a conclusive trend in the thermal conductivity response.The

results at 0% strain correspond to those presented in Chapter 3.

6.2 Strained Monolayers: Graphene, Boron Nitride

and Molybdenum Disulfide

The honeycomb lattice structures of graphene, boron nitride, and molybdenum disulfide

are stretched in the x direction in successive steps of 2% of deformation, as shown in Figs.

6.1a, 6.3a, and 6.5a, respectively. For clarity, these figures show only three percentages of

deformation: 0%, 10%, and 20%. The corresponding thermal responses to all percentages

of deformation are shown in Figs. 6.1b, 6.3b, and 6.5b. As discussed in Section 3.2.4, the

in-plane thermal conductivities of all unstrained monolayers (0% strain) are isotropic;

however, they become highly anisotropic as the lattices of graphene and boron nitride

are stretched. In molybdenum disulfide, the isotropic behavior is preserved and the con-

ductivities in both directions drop dramatically when strain is applied (see Fig. 6.5b).

The conductivities of graphene and boron nitride are significantly higher in the stretch-

ing direction (x direction). In the y direction, the thermal conductivity decreases at a

constant rate of approximately 23 W/mK/%strain in graphene, and 10 W/mK/%strain

in boron nitride. In graphene, the thermal conductivity in the x direction first increases

until a 8% of strain is applied, and then it decreases for higher percentages. In contrast,


the thermal conductivity of boron nitride monotonically increases in the x direction. It is

noteworthy that the maximum increase in boron nitride (∼90%) is substantially higher

than that predicted in graphene (∼30%). These results contrast with those obtained with

the Direct method (see Fig. 5.5), where approximately a 25% maximum increase was

predicted in boron nitride and only a 2% maximum increase in graphene. These discrep-

ancies are attributed to the strong effect imposed by the size of the simulation domain

in the Direct method, which is aggravated as the thermal conductivity increases because

the mean free path of phonons also increases. The phonon transport approach used in

this chapter is more suitable to study this thermal response in 2D materials because it

considers periodic boundary conditions at the boundaries, and it takes into account the

quantum effects. Also, it allows to explore the response of individual phonon modes.

Figure 6.1: (a) Deformation of the lattice structure of graphene when uniaxial tensilestrain is applied in the x direction. The figure shows three levels of deformation: 0%,10%, and 20%.(b) Variation of the thermal conductivity of graphene with the percentageof deformation applied in the x direction.

The mode-by-mode analysis of the phonon properties and thermal conductivities of

graphene, boron nitride, and molybdenum disulfide is shown in Figs. 6.2, 6.4, and 6.6,

respectively. These results reveal that the anomalous thermal response to tensile strain

in 2D layers of graphene and boron nitride is mainly caused by the higher contributions


Figure 6.2: (a) Acoustic dispersion curves of graphene along the k-space direction [1 0 0]for three percentages of uniaxial tensile strain: 0%, 6%, and 12%. Response of the phononlifetime distribution of graphene to the application of strain for (b) the ZA modes, (c)TA modes, and (d) LA modes. (e) Variation of the thermal conductivity contributionsof the acoustic modes of graphene with the percentage of deformation applied in the xdirection.

of out-of-plane phonon modes (ZA), which are in turn caused by the linearization of the

dispersion curves of ZA modes (higher group velocities) and the increase in lifetimes of

low-frequency phonons (up to ∼8THz). As shown in Fig. 6.2e, the maximum thermal

conductivity predicted for graphene at 8% of strain (see Fig. 6.1b) is driven by the

interplay between the contributions of in-plane acoustic modes (TA and LA) and out-of-

plane acoustic modes (ZA). Overall, the increase in lifetimes of low-frequency phonons

also causes a slight increase in the contributions of in-plane acoustic modes up to 8%

of strain is applied. However, the contributions of these in-plane modes in graphene

drop significantly at percentages of strain higher than 8%, due to the combined effect

of smaller group velocities and the reduction in lifetimes of higher-frequency phonons.

Figures 6.2c and 6.2d show how the lifetimes of TA and LA are reduced in the frequency

range 10-26 THz when a 12% of strain is applied.


Figure 6.3: (a) Deformation of the lattice structure of boron nitride when uniaxial tensilestrain is applied in the x direction. The figure shows three levels of deformation: 0%, 10%,and 20%.(b) Variation of the thermal conductivity of boron nitride with the percentageof deformation applied in the x direction.

Figure 6.4e shows that the contributions of ZA modes in boron nitride also increase

significantly when the lattice is stretched; however, the contributions of the in-plane

modes remains practical unchanged for percentages of strain higher that 8%. Conse-

quently, the thermal response of boron nitride shown in Fig. 6.3b does not exhibit an

inflection point as the one predicted for graphene, instead it grows monotonically until

18% of strain is applied. It is noteworthy that the relaxation time of higher-frequency

phonons (10-26 THz) are not reduced in the same magnitude as those in graphene, in-

stead they are practically unaffected by the application of strain. In addition, it is also

relevant to highlight that the group velocities of TA moves (see Fig. 6.4a) slightly increase

when the structure is stretched in the x direction.

The response of ZA modes in molybdenum disulfide diverges significantly from those

of graphene and boron nitride, as shown in Fig. 6.6e. In this case, the contributions

of all in-plane phonon modes are consistently reduced when the lattice is subjected to

different percentages of strain. This response is caused by the reduction of both group

velocities and relaxation times, which is consistent with the expected response in bulk


Figure 6.4: (a) Acoustic dispersion curves of boron nitride along the k-space direction[1 0 0] for three percentages of uniaxial tensile strain: 0%, 6%, and 12%. Response ofthe phonon lifetime distribution of boron nitride to the application of strain for (b) theZA modes, (c) TA modes, and (d) LA modes. (e) Variation of the thermal conductivitycontributions of the acoustic modes of boron nitride with the percentage of deformationapplied in the x direction.

materials. Single-layer molybdenum disulfide is not strictly two-dimensional because

the sulfur atoms are located below and above the molybdenum atoms, creating a quasi

two-dimensional lattice structure. The unique response of ZA modes in graphene and

boron nitride is not observed in molybdenum disulfide because the out-of-plane scattering

mechanisms are not absent in the quasi two-dimensional lattice.

6.3 Strained Superlattices

The phonon transport analysis was also applied to investigate the variation of the thermal

conductivities of the superlattices of graphene and boron nitride when uniaxial tensile

strain is applied in the direction perpendicular to the interfaces (x direction), as shown

in Fig. 6.7a.


Figure 6.5: (a) Deformation of the lattice structure of molybdenum disulfide when uniax-ial tensile strain is applied in the x direction. The figure shows three levels of deformation:0%, 10%, and 20%.(b) Variation of the thermal conductivity of molybdenum disulfidewith the percentage of deformation applied in the x direction.

Figure 6.7b shows the thermal conductivities as functions of the superlattice period

and the percentage of strain. The thermal respond to strain in the superlattices resem-

bles the response observed in graphene: the conductivity first increases until a maximum

value is reached, and then it decreases for higher percentages of strain. The maximum

thermal conductivity is always obtained at 6% of strain, regardless the superlattice pe-

riod. It is relevant to mention that the applied strain has a stronger effect on the thermal

conductivities as the superlattice period increases. For example, the thermal conductiv-

ity at 6% of strain for the 1x1 superlattice is 45% higher than that at 0% of strain, and

the thermal conductivity at 6% of strain for the 10x10 superlattice is 65% higher than

that at 0% of strain. The maximum percentages of increase caused by the tensile strain

in the superlattices (45-65%) are in between those found in graphene(30%) and boron

nitride(90%).


Figure 6.6: Acoustic dispersion curves of molybdenum disulfide along the k-space direc-tion [1 0 0] for three percentages of uniaxial tensile strain: 0%, 6%, and 12%. Responseof the phonon lifetime distribution of molybdenum disulfide to the application of strainfor (b) the ZA modes, (c) TA modes, and (d) LA modes. (e) Variation of the ther-mal conductivity contributions of the acoustic modes of molybdenum disulfide with thepercentage of deformation applied in the x direction.

Figure 6.7: (a) Atomic structure of the 1x1 zigzag superlattice of graphene and boronnitride showing the direction in which the the uniaxial tensile strain is applied (x di-rection). (b) Variation of the thermal conductivity with the superlattice period and thepercentage of deformation applied in the x direction.


6.4 Summary

In this chapter, the thermal response of the monolayers and superlattices to uniaxial

tensile strain was investigated from a phonon transport perspective. The mode-by-mode

analysis reveals that the thermal conductivities of graphene, boron nitride, and molyb-

denum disulfide respond differently when their lattices are subjected to tensile strain,

despite the similarities of their lattice structures. On the one hand, the thermal con-

ductivity of boron nitride monotonically increases until a 18% of strain is applied, ap-

proximately doubling the conductivity of an unstrained sample. On the other hand, the

thermal conductivity of graphene first increases by approximately 30% until 8% of strain

is applied, and then it sharply decreases for higher percentages of strain. In contrast,

the conductivity of molybdenum disulfide drops dramatically in response to percentages

of strain as small as 2%. These thermal responses were addressed in the context of the

phonon properties, with particular attention on the role of the acoustic phonon modes.

Chapter 7

Conclusion

7.1 Contributions

The theoretical approach applied in this research is a valuable tool for providing insight

into material properties, applications, and further areas of study, partially eliminating

the need for expensive, tedious experimental testing on the nanoscale. This thesis pro-

vides a fundamental understanding about the physics of phonons in 2D materials. This

knowledge is key to designing material structures at the atomic level to improve the

thermal properties of layered materials, with the corresponding impact on performance

of next-generation electronic devices.

In this thesis, I investigated the phonon properties and thermal conductivities of un-

strained and strained monolayers of graphene, boron nitride, and molybdenum disulfide,

and also of short-period 2D superlattices of graphene and boron nitride. I implemented a

robust simulation approach that combine Bose-Einstein statistics, lattice dynamics, and

molecular dynamics. The main advantage of this approach over other simulation tech-

niques is the treatment of phonons on a mode-by-mode basis. The eigendisplacements

of the lattice vibrational modes were calculated and used to isolate individual phonon

modes from the phonon spectrum, and to precisely locate the acoustic phonon modes.

64

Chapter 7. Conclusion 65

The natural inclusion of four- and higher-order scattering phonon processes and the in-

corporation of quantum effects are also important considerations of this approach that

are of particular relevance for thermal analysis of 2D materials. It is important to high-

light that molecular dynamics analysis of 2D materials available in the literature have

been conducted mainly at the system level with the Green-Kubo and Direct methods,

without insight into the phonon properties.

Significant amount of computational resources were used in this thesis to extract

phonon relaxation times from high-resolution power spectra. This high resolution re-

vealed interesting physics in the relaxation time distributions and it is, to the best of

my knowledge, the highest resolution used for these types of studies so far in graphene.

Also, the phonon relaxation time distributions obtained with phonon spectral analysis

in boron nitride, molybdenum disulfide and the superlattices were reported by the first

time in this thesis.

Predicting coherent transport of phonons in 2D superlattices of graphene and boron

nitride was a significant contribution of this thesis, no such predictions had been con-

ducted before in 2D superlattices.The rigorous mode-by-mode analysis applied to these

superlattices revealed the interplay between diffusive and coherent effects. I also inves-

tigated the role of acoustic phonon modes in the thermal transport and how they are

affected by the period length and interface configuration of the superlattices. I predicted

the variation of the phonon properties with the phonon frequency and how coherent

effects are responsible for the trends observed in the thermal conductivity of this short-

period 2D superlattices.

The anomalous thermal response to uniaxial tensile strain in graphene and boron

nitride represents another important finding reported in this thesis. Interestingly, the

mode-by-mode phonon analysis revealed a significant increase in the thermal conductiv-

ities of these monolayers when small percentages of strain are applied, driven by changes

in both relaxation times and group velocities.


7.2 Future Research Directions

The main challenge associated with the implementation of this phonon transport analysis,

besides the computational cost of the simulations, is the need for accurate interatomic

potentials. Graphene and boron nitride have been extensively studied in recent years,

and there are several potentials available that accurately reproduce their vibrational

properties. However, that is not the case for hundreds of other 2D materials that have

been synthesized in recent years. Developing new interatomic potentials is paramount to

apply this methodology in other 2D materials. An alternative would be conducting the

phonon transport analysis at the electronic level, using density functional theory (DFT)

simulations. These simulations have the advantage of being exceptionally accurate and

parameter-free, meaning that they do not relay on empirically approximated parameters.

This simulations are computationally expensive but offer no limitation in terms of the

materials and nanostructures that can be tested.

Theoretical and experimental works are still needed to continue investigating effective

strategies for controlling the thermal transport in periodic heterostructures of 2D mate-

rials using specular scattering of phonons at material interfaces. Heat conduction can be

controlled via wave-interference effects when phonons cross multiple interfaces specularly.

This approach represents a promising alternative to traditional methods in which phonons

are scattered diffusively by incorporating lattice defects such as impurities, vacancies, and

disorder. Periodic heterostructures of 2D materials are of particular interest because they

usually exhibit high-quality interfaces and long wavelength heat-carrying phonons, cru-

cial conditions for promoting specular scattering and wave interference. Since phonons

that transport heat are distributed across a wide range of wavelengths, the challenge is

to design heterostructures that maximize the number of phonons contributing to thermal

transport via wave interference.

The results presented in this thesis demonstrated that the thermal transport in 2D

materials becomes highly anisotropic when these materials are periodically arranged in


the form of superlattices or subjected to tensile strain. These lattice modifications rep-

resent promising strategies for thermal management in nanostructured devises, allowing

to either reduce or increase the thermal conductivity along desired directions. In this

regard, it is interesting to further investigate how this anisotropic behavior is affected

when 2D materials are subjected to uniaxial strain in different directions, biaxial strain,

in-plane shear, and torsion.

Another attractive research avenue will be to transfer physically-accurate information

from these atomistic 2D models to macroscale models. This hierarchical approach will

enable accurate simulations of heat transfer in electronic devices without substantial

increase in computational cost, by decoupling the atomistic models from the macroscale

ones. The ultimate objective of this simulation approach is to conduct system-level

thermal analysis, creating transition regions between the molecular dynamics domains

and the finite element domains.

Investigating the in-plane and out-of-plane size effects of these 2D materials is also

crucial to understand the phonon physics in nanostructures. For example, it is interesting

to study how the anomalous thermal response of the out-of-plane acoustic mode reacts

when the structure evolves from a single-layer configuration to a two- or three-layer

configuration.

Appendix A

Density Functional Theory

Simulations

A.1 Structural Optimization: Superlattices

Figure A.1: Unit cells for the superlattices: (a) 1x1 zigzag, and (b) 1x1 armchair.Reprinted with permission from Ref. [14], Copyright 2015 ASME.

The Quantum ESPRESSO package [27] have been employed to optimize the hybrid

honeycomb lattice structures of boron nitride and graphene. The unit cells of these

structures are initially constructed by taking the average value of the B-N (0.145 nm) and

C-C (0.141 nm) nearest bond lengths [36]. The equilibrium lattice parameters of these

cells are then obtained via viable-cell optimization. We have used a projector augmented

68

Appendix A. Density Functional Theory Simulations 69

wave (PAW) pseudopotential [4, 37] and a Perdew-Burke-Ernzerhof generalized gradient

approximation (PBE-GGA) [60] for the exchange-correlation functional, with a cutoff

energy of 80 Ry for the plane wave expansion and 500 Ry for the charge density. The

integration in reciprocal space is conducted over a uniformly spaced Monkhorst-Pack

grid [58] of 14x18x1 points for 1x1 zigzag, and 14x16x1 points for 1x1 armchair. The

resolution of this grid in the x direction is reduced as the period increases, up to 2x18x1

points for 10x10 zigzag, and 2x16x1 points for 10x10 armchair. Convergence is achieved

when changes in total energy between consecutive self-consistent steps are less than

1x10−8 Ry and all forces are smaller than 1x10−4 Ry/au. After this relaxation, the unit

cells are replicated in the x and y directions to generate the superlattices of alternated

zigzag or armchair layers of graphene and boron nitride. Tables A.1 and A.2 collects the

equilibrium lattice vectors aα, number of replicated unit cells Nα, and simulation lengths

Nα. Further relaxation is conducted with the interatomic potential within both HLD

and MD simulations.

Table A.1: Number of atoms, size of the MD simulations domain (LxxLy), number ofreplicated unit cells in the x (Nx) and y (Ny) directions, number of atoms in the unit cell,and equilibrium lattice parameters ax and ay for each superlattice period in the zigzagconfiguration. Reprinted with permission from Ref. [15], Copyright 2016 AmericanPhysical Society

Zigzag

Period Atoms Lx x Ly Nx x Ny Atoms ax ay(nm x nm) (unit cell) (nm) (nm)

1x1 14720 20.04 x 19.86 46 x 80 4 0.4357 0.24822x2 15360 20.84 x 19.86 24 x 80 8 0.8686 0.24823x3 15360 20.81 x 19.86 16 x 80 12 1.3007 0.24834x4 15360 20.79 x 19.86 12 x 80 16 1.7326 0.24835x5 16000 21.64 x 19.87 10 x 80 20 2.1642 0.24847x7 13440 18.16 x 19.88 6 x 80 28 3.0268 0.248510x10 12800 17.28 x 19.88 4 x 80 40 4.3204 0.2485

Appendix A. Density Functional Theory Simulations 70

Table A.2: Number of atoms, size of the MD simulations domain (LxxLy), number ofreplicated unit cells in the x (Nx) and y (Ny) directions, number of atoms in the unit cell,and equilibrium lattice parameters ax and ay for each superlattice period in the armchairconfiguration. Reprinted with permission from Ref. [15], Copyright 2016 AmericanPhysical Society

Armchair

Period Atoms Lx x Ly Nx x Ny Atoms ax ay(nm x nm) (unit cell) (nm) (nm)

1x1 14720 20.00 x 19.87 40 x 46 8 0.5001 0.43192x2 14720 19.99 x 19.82 20 x 46 16 0.9993 0.43093x3 15456 20.96 x 19.83 14 x 46 24 1.4971 0.43104x4 14720 19.95 x 19.82 10 x 46 32 1.9945 0.43095x5 14720 19.95 x 19.82 8 x 46 40 2.4932 0.43087x7 15456 20.93 x 19.82 6 x 46 56 3.4886 0.430810x10 14720 19.92 x 19.81 4 x 46 80 4.9798 0.4307

A.2 Phonon Dispersion Curves

The phonon dispersion curves of the superlattices have been validated with density func-

tional perturbation theory (DFPT) simulations with the Quantum ESPRESSO package

[27]. For these DFPT simulations we have implemented a 16x16x1 grid to obtain the dy-

namical matrices for the phonon calculations. We have used the same pseudopotentials,

cut-offs energies, and k-point sampling employed for the structural optimization.

Appendix B

Spectral Energy Density Algorithm

B.1 Computational Approach

The spectral analysis described in Section 2.1.3 is performed with a parallel FORTRAN

code. The algorithm of the code is shown in Fig. B.1, and the complete source code is

provided in Section B.2. This code is a modified version of a serial code developed in

collaboration with Julia Sborz [69], who worked on this research as part of her undergrad-

uate thesis project. This algorithm performs signal analysis to a set of MD trajectories

to obtain power spectra. For each Fourier sampling window (ifft) and MD simulation

(iseed), an output file is generated with a discrete power spectrum for each phonon mode

(imode) and wavevector (ikslice). The calculation of the time derivative of the normal

mode coordinates (Eq. 2.3) is executed in parallel using shared memory multiprocessing

programming (OpenMP). The autocorrelation of Eq. 2.3 is computed in two steps using

the Wiener-Khichin theorem, and the power spectra are generated by Fourier transform-

ing this autocorrelation. Finally, the power spectra from all independent MD simulations

are averaged and fitted in Matlab to estimate the phonon relaxation times.

71

Appendix B. Spectral Energy Density Algorithm 72

Figure B.1: Flowchart of the spectral energy density algorithm.


B.2 Source Code

program NMD

use, intrinsic :: iso c bindinguse omp libIMPLICIT NONEinclude ‘fftw3.f03’type(C PTR) :: fftw plan Atype(C PTR) :: fftw plan Btype(C PTR) :: fftw plan C

!############### Variables Definition #####################

INTEGER, parameter :: material = MATERIAL ! 1:C, 2:BN, 4:MoS2, 5:CBN

character(len=100) :: path output = ‘PATHOUTPUT’character(len=100) :: path input = ‘PATHINPUT’

INTEGER, parameter :: NUM SEED = NUMSEEDINTEGER, parameter :: T TOTAL = TTOTALINTEGER, parameter :: T FFT = TFFTINTEGER, parameter :: NUM TSTEPS = NUMTSTEPSINTEGER, parameter :: NUM MODES = NUMMODESINTEGER, parameter :: NUM ATOMS = NATOMSINTEGER, parameter :: NUM ATOMS UCELL = NUMATOMSUCELLINTEGER, parameter :: NUM UCELL COPIES = NUMUCELLCOPIESINTEGER, parameter :: NUM KPTS = NUMKPTSINTEGER, parameter :: Nx = NXINTEGER, parameter :: Ny = NYINTEGER, parameter :: Nz = NZREAL, parameter :: amass 1 = AMASS1 ! carbon atomsREAL, parameter :: amass 1 = AMASS2 ! boron atomsREAL, parameter :: amass 3 = AMASS3 ! nitrogen atomsREAL, parameter :: amass 4 = AMASS4 ! molybdenum atomsREAL, parameter :: amass 5 = AMASS5 ! sulfur atomsREAL, parameter :: alat 1 = ALAT1REAL, parameter :: alat 2 = ALAT2REAL, parameter :: alat 3 = ALAT3

!——————– Imaginary UnitCOMPLEX :: j = (0.0,1.0)

!——————– Variables for loopsINTEGER :: iseed, ifft, ikslice, i, iatom, imode, kindex, iks, n, ncount

!——————– Variables for reading atomic positionsREAL, DIMENSION(NUM ATOMS,5) :: NMDx0pos

!——————– Variables for reading eigenvectorsCOMPLEX, DIMENSION(NUM KPTS*NUM MODES,NUM MODES) :: eigvec

!——————– Variables for atomic massesREAL, DIMENSION(NUM ATOMS) :: mass

!——————– Variables for reading K point listINTEGER, DIMENSION(NUM KPTS,3) :: kptlistINTEGER, DIMENSION(3,NUM KPTS) :: kptINTEGER, DIMENSION(NUM KPTS) :: kpt index

!——————– Variables for reading velocity filesCHARACTER(len=2):: str2 ! seed identifierCHARACTER(len=4):: str4 ! fft windows identifierCHARACTER(len=100):: strdump ! input file nameinteger(C INT64 T) :: ntimestep, natomsinteger(C INT) :: triclinic


integer(C INT), dimension(6) :: boundaryreal(C DOUBLE), dimension(6) :: xyzinteger(C INT) :: size n, nchunk, chunkREAL(C DOUBLE), DIMENSION(3*NUM ATOMS) :: filechunkREAL, DIMENSION(NUM ATOMS,NUM TSTEPS) :: velx, vely, velz

!——————– Variables for QDOTCOMPLEX, DIMENSION(NUM ATOMS) :: spatialCOMPLEX, DIMENSION(NUM ATOMS UCELL) :: tempeigx, tempeigy, tempeigz

!——————– Variables for autocorrelationDOUBLE COMPLEX, DIMENSION(2*NUM TSTEPS) :: QDOTCOMPLEX :: norm

!——————– Variables for power spectrumDOUBLE COMPLEX, DIMENSION(NUM TSTEPS) :: KEFFTREAL, DIMENSION(NUM KPTS,(NUM TSTEPS/2),NUM MODES) :: SED

!——————– Other variablesCHARACTER(len=3):: fft, seedCHARACTER(len=100):: strwritesingleREAL :: PIPI = 4.D0*DATAN(1.D0)

!############### Reading input files #####################

!——————– Reading atomic positionsopen(100, FILE=trim(adjustl(path input))//‘NMDx0pos’,action=‘read’,status=‘old’)read(100, *) NMDx0posclose(100)NMDx0pos = reshape(NMDx0pos,(/NUM ATOMS,5/), ORDER=(/2,1/))

!——————– Reading eigenvectorsopen(120, FILE=trim(adjustl(path input))//‘eigvec.dat’,action=‘read’,status=‘old’)read(120, *) eigvecclose(120)eigvec = reshape(eigvec,(/(NUM KPTS)*(NUM MODES),NUM MODES/),ORDER=(/2,1/))

!——————– Assigning atomic massesdo i=1,NUM ATOMS

if (material==1) then ! Cmass(i) = amass 1

end if

if (material==2) then ! BNif (NMDx0pos(i,2)==1) then

mass(i) = amass 2end ifif (NMDx0pos(i,2)==2) then

mass(i) = amass 3end if

end if

if (material==4) then ! MoS2if (NMDx0pos(i,2)==1) then




end if

if (material==5) then ! Superlattices CBNif (NMDx0pos(i,2)==1) then





end if

end do

!——————– Reading K Pointsopen(250, FILE= trim(adjustl(path input))//‘kptlist.dat’,action=‘read’,status=‘old’)read(250, *) kptlistclose(250)kptlist = reshape(kptlist,(/NUM KPTS,3/), ORDER=(/2,1/))

do iks = 1,NUM KPTSdo i = 1,3

kpt(i,iks) = kptlist(iks,i)end dokpt index(iks) = iks

end do

fftw plan A = fftw plan dft 1d(2*NUM TSTEPS,QDOT,QDOT,FFTW FORWARD,FFTW ESTIMATE)fftw plan B = fftw plan dft 1d(2*NUM TSTEPS,QDOT,QDOT,FFTW BACKWARD,FFTW ESTIMATE)fftw plan C = fftw plan dft 1d(NUM TSTEPS,QDOT,KEFFT,FFTW FORWARD,FFTW ESTIMATE)

!################## Loops #########################

do iseed = 1, NUM SEED ! Loop over independent MD seeds

do ifft = 1, (T TOTAL/T FFT) ! Loop over FFT sampling windows

!——————– Reading velocities from input files

write(str2,‘(I2)’) iseedwrite(str4,‘(I4)’) ifftstrdump = trim(adjustl(path input))//‘LAMMPS/dump ’//trim(adjustl(str2))//‘ ’//trim(adjustl(str4))//‘.bin’open(350, file=strdump, access=‘stream’, form=‘unformatted’, action=‘read’)do i = 1,NUM TSTEPS

read(350) ntimestep, natoms, triclinic, boundary, xyz, size n, nchunkncount = 0do n = 1, nchunk

read(350) chunkread(350) filechunk(1:chunk)do iatom = 1,(chunk/3)

velx(iatom+ncount,i)=real(filechunk(iatom*3-2))vely(iatom+ncount,i)=real(filechunk(iatom*3-1))velz(iatom+ncount,i)=real(filechunk(iatom*3))

end doncount = ncount + (chunk/3)

end doend doclose(350)

do ikslice = 1 , NUM KPTS ! Loop over wavevectors

spatial(:) = 2*PI*j*(&(NMDx0pos(:,3))*((kpt(1,ikslice))/(alat 1*Nx))+&(NMDx0pos(:,4))*((kpt(2,ikslice))/(alat 2*Ny))+&(NMDx0pos(:,5))*((kpt(3,ikslice))/(alat 3*Nz)))

kindex = kpt index(ikslice)

do imode=1,NUM MODES ! Loop over phonon modes


!——————– time Derivative of the normal mode coordinate (QDOT)

do i = 1,NUM ATOMS UCELLtempeigx(i) = conjg(eigvec(((NUM ATOMS UCELL*3)*(kindex-1)+1)+3*(i-1),imode))tempeigy(i) = conjg(eigvec(((NUM ATOMS UCELL*3)*(kindex-1)+2)+3*(i-1),imode))tempeigz(i) = conjg(eigvec(((NUM ATOMS UCELL*3)*(kindex-1)+3)+3*(i-1),imode))

end do

QDOT = (0.0,0.0)QDOT(1:NUM TSTEPS) = &

sum(bsxfun2(NUM ATOMS,NUM TSTEPS,((exp(spatial))*(sqrt(mass/NUM UCELL COPIES))), &(bsxfun(NUM ATOMS,NUM TSTEPS,repmat(tempeigx,NUM ATOMS UCELL,NUM UCELL COPIES),velx) + &bsxfun(NUM ATOMS,NUM TSTEPS,repmat(tempeigy,NUM ATOMS UCELL,NUM UCELL COPIES),vely) + &bsxfun(NUM ATOMS,NUM TSTEPS,repmat(tempeigz,NUM ATOMS UCELL,NUM UCELL COPIES),velz))),DIM=1)

!——————– Autocorrelation Computation

call fftw execute dft(fftw plan A, QDOT, QDOT)QDOT(:) = ABS(QDOT(:))**2call fftw execute dft(fftw plan B, QDOT, QDOT)norm = 1./QDOT(1)QDOT(:) = QDOT(:)*norm

!——————– Power spectrum

call fftw execute dft(fftw plan C, QDOT, KEFFT)SED(ikslice,:,imode) = (REAL(KEFFT(1:(NUM TSTEPS/2)))**2 + AIMAG(KEFFT(1:(NUM TSTEPS/2)))**2)

end do ! end of imodeend do ! end of ikslice

call fftw destroy plan(fftw plan A)call fftw destroy plan(fftw plan B)call fftw destroy plan(fftw plan C)

!################ End of main loop ######################

write(seed, ‘(I3)’) iseedwrite(fft, ‘(I3)’) ifftstrwritesingle=trim(adjustl(path output))//‘SED ’//trim(adjustl(seed))//‘ ’//trim(adjustl(fft))//‘.bin’open(400, file = strwritesingle, form=‘unformatted’)write(400) SEDclose(400)

end do ! end of ifftend do ! end of iseed!################## Functions ########################

CONTAINS

function repmat(A,n,M)INTEGER, INTENT(IN) :: n,MCOMPLEX, INTENT(IN) :: A(n)COMPLEX :: repmat(n*M)INTEGER :: i, n11, n12!$OMP PARALLEL DEFAULT(NONE) SHARED(repmat, A, m, n) PRIVATE(i, n11, n12)!$OMP DODO i = 1,M

n11 = 1+(i-1)*nn12 = n11+n-1repmat(n11:n12) = A

END DO!$OMP END DO!$OMP END PARALLELend function repmat

function bsxfun(n,m,st,A)integer, intent(IN) :: n,m


complex, intent(IN) :: st(n)real, intent(IN) :: A(n,m)integer :: hcomplex :: bsxfun(n,m)!$OMP PARALLEL DEFAULT(NONE) SHARED(st,bsxfun,A,m) PRIVATE(h)!$OMP DOdo h = 1,m

bsxfun(:,h) = st(:)*A(:,h)end do!$OMP END DO!$OMP END PARALLELend function bsxfun

function bsxfun2(n,m,st,A)integer, intent(IN) :: n,mcomplex, intent(IN) :: st(n)complex, intent(IN) :: A(n,m)integer :: hcomplex :: bsxfun2(n,m)!$OMP PARALLEL DEFAULT(NONE) SHARED(st,bsxfun2,A,m) PRIVATE(h)!$OMP DOdo h = 1,m

bsxfun2(:,h) = st(:)*A(:,h)end do!$OMP END DO!$OMP END PARALLELend function bsxfun2

END PROGRAM NMD

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by carlos manuel da silva leal - university of toronto … · abstract predicting phonon transport...

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