phonon mean free path in few layer, two-dimensional...

PHONON MEAN FREE PATH IN FEW LAYER, TWO-DIMENSIONALHEXAGONAL STRUCTURES

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

HAMED GHOLIVAND

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

MECHANICAL ENGINEERING

JANUARY 2017

Approval of the thesis:


submitted by HAMED GHOLIVAND in partial fulfillment of the requirements forthe degree of Master of Science in Mechanical Engineering Department, MiddleEast Technical University by,

Prof. Dr. Gülbin Dural ÜnverDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Tuna BalkanHead of Department, Mechanical Engineering

Assist. Prof. Dr. Nazlı DönmezerSupervisor, Mechanical Engineering Department, METU

Examining Committee Members:

Assoc. Prof. Dr. Almıla Güvenç YazıcıoğluMechanical Engineering Department, METU

Assist. Prof. Dr. Nazlı DönmezerMechanical Engineering Department, METU

Assoc. Prof. Dr. Hakan ErtürkMechanical Engineering Department, Boğaziçi University

Assist. Prof. Dr. Sezer ÖzerinçMechanical Engineering Department, METU

Assist. Prof. Dr. Batur ErcanMetallurgical and Materials Department, METU

Date:

I hereby declare that all information in this document has been obtained andpresented in accordance with academic rules and ethical conduct. I also declarethat, as required by these rules and conduct, I have fully cited and referenced allmaterial and results that are not original to this work.

Name, Last Name: HAMED GHOLIVAND

Signature :

iv

ABSTRACT


GHOLIVAND, HAMEDM.S., Department of Mechanical Engineering

Supervisor : Assist. Prof. Dr. Nazlı Dönmezer

January 2017, 82 pages

Two-dimensional materials such as graphene and few layer hexagonal Boron Nitride(h-BN) have been the center of attention in the last decade. These materials pro-vide anisotropic and exclusive properties making them ideal candidates for the mod-ern electronic and optoelectronic applications. With the enhancements in fabricationtechniques and the ability to separate thin layers their popularity is continuously in-creasing. Understanding the thermal properties of these materials is necessary tomake better devices. Phonon mean free path (MFP) is one of the most thermal prop-erties that determines the limits of ballistic-diffusive thermal transport in micro andnanoscale domains. In this study thermal properties, specifically the phonon MFP be-havior, of few layer graphene, h-BN, and composite graphene/h-BN structures werestudied after obtaining the phonon dispersion of each material. After, finding thethermal properties of discrete phonon modes, a plot of accumulated thermal conduc-tivity with respect to phonon MFP of phonons is obtained to understand the ballistic-diffusive limits of in such structures. Bulk structures of graphene and h-BN were alsoanalyzed for comparison purposes and it was observed that single layer structures willexperience ballistic effects more since they have considerably higher MFP than theirbulk counterparts.

v

Keywords: Graphene, hexagonal Boron Nitride (h-BN), 2D Materials, Phonon, PhononLifetime, Thermal Conductivity, Phonon Mean Free Path (MFP)

vi

ÖZ

İKİ BOYUTLU, HEKSAGONAL YAPILARDA FONON ORTALAMASERBEST YOLU

GHOLIVAND, HAMEDYüksek Lisans, Makina Mühendisliği Bölümü

Tez Yöneticisi : Yrd. Doç. Dr. Nazlı Dönmezer

Ocak 2017 , 82 sayfa

Grafit ve Heksagonal Bor Nitrür (h-BN) gibi iki boyutlu yapılar son on yılın ilgi odağıolmuştur. Bu malzemeler yöne bağlı, eşsiz özelliklerinden dolayı elektronik ve opto-elektronik uygulamalar için kusursuz birer adaydır. Üretim tekniklerindeki gelişmelerile ince katmanları ayırmak daha da mümkün oldukça bu malzemelerin kullanımı dagiderek yaygınlaşmaktadır. Bu tür malzemelerin ısıl özelliklerini anlamak daha iyiaygıtlar yapmak için gereklidir. Fonon ortalama serbest yolu (OSY) mikro ve nanoboyutlu malzemelerde balistik-yayınım ısı transfer limitlerini belirleyen en önemliısıl özelliklerdendir. Bu çalışmada az tabakalı grafen, h-BN ve kompozit h-BN/grafenyapıların ısıl özellikleri, özellikle OSY araştırılmıştır. Ayrı fonon modlarının her biri-nin ısıl özellikleri bulunduktan sonra balistik-yayılım ısı transfer limitlerini anlamakiçin ısıl iletkenliğin fonon OSY’na göre yığılma grafiği elde edilmiştir. Çok katmanlıgrafen ve h-BN yapıları da karşılaştırma amaçlı analiz edilmiş ve tek katmanlı yapıla-rın OSY’nun çok katmanlı yapılara göre çok daha uzun olduğu dolayısıyla da balistiketkilerden daha erken etkileneceği anlaşılmıştır.

Anahtar Kelimeler: Grafen, Heksagonal Bor Nitrür, İki Boyutlu Yapılar, Fonon, Isılİletkenlik, Fonon Ortalama Serbest Yolu (OSY)

vii

To My Family

viii

ACKNOWLEDGMENTS

First and formost, I would like to express my deepest gratitude to my research advisor

and mentor, Dr. Nazlı Dönmezer, for her valuable guidance, patience, and commit-

ment to my growth as a researcher. It has been an honor to be her first graduate student

and I am glad to have had the opprtunity to work under her supervision. I would also

like to thank Dr. Hakan Ertük, for reviewing my work and providing helpful advices.

I cannot thank my beloved family enough, for their endless support, encouragement,

and faith in me. I could have not achieved any of my goals without their love and

support.

Finally, I want to thank my friends, who have made this place amazing while I have

been here.

ix

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Graphene and h-BN . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thermal Transport Mechanism in Crystals . . . . . . . . . . 7

1.4.1 Lattice, Crystal Structure, Primitive Cell, Recip-rocal Lattice, and Brillouin Zone . . . . . . . . . . 7

1.4.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.3 Phonon Scattering . . . . . . . . . . . . . . . . . 11

x

1.4.4 Phonon Mean Free Path . . . . . . . . . . . . . . 13

1.5 Previous Research . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Objectives and Research Strategy . . . . . . . . . . . . . . . 17

2 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Ab-initio Calculations . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Brillouin Zone Sampling . . . . . . . . . . . . . . 20

2.1.2 ABINIT . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.3 Phonon Dispersion Relation . . . . . . . . . . . . 22

2.2 Phonon Properties . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Specific Heat . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . 24

2.2.3 Phonon Lifetime . . . . . . . . . . . . . . . . . . 24

2.2.4 Thermal Conductivity . . . . . . . . . . . . . . . . 25

2.2.5 Phonon Mean Free Path . . . . . . . . . . . . . . 26

3 GRAPHENE-BASED STRUCTURES . . . . . . . . . . . . . . . . . 27

3.1 Single Layer Graphene . . . . . . . . . . . . . . . . . . . . 28

3.2 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 H-BN STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Single Layer h-BN . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Bilayer h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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4.3 Bulk h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 GRAPHENE/H-BN STRUCTURE . . . . . . . . . . . . . . . . . . . 63

6 CONCLUSIONS AND FUTURE RESEARCH DIRECTION . . . . . 73

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xii

LIST OF TABLES

TABLES

Table 2.1 Coordinates of the symmetry points in the hexagonal Brillouin zone 23

Table 3.1 Average group velocity and lifetime of phonon modes in graphene . 31

Table 3.2 Average group velocity and lifetime of phonon modes in bilayergraphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 3.3 Average group velocity and lifetime of phonon modes in graphite . . 42

Table 4.1 Average group velocity and lifetime of phonon modes in single layerh-BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Table 4.2 Average group velocity and lifetime of phonon modes in bilayer h-BN 54

Table 4.3 Average group velocity and lifetime of phonon modes in bulk h-BN. 60

Table 5.1 Average group velocity and lifetime of phonon modes in bilayergraphene/h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Table 5.2 Total thermal conductivity and percentage of the phonon mode con-tribution to the total thermal conductivity at T=300 K for graphene, bilayergraphene, graphite, single layer h-BN, bilayer h-BN, bulk h-BN, and com-posite h-BN/graphene structures. . . . . . . . . . . . . . . . . . . . . . . 70

Table 5.3 Minimum MFP of phonons contributing to top 50% of thermal con-ductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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LIST OF FIGURES

FIGURES

Figure 1.1 Graphene as a basic element of buckyballs, carbon nanotubes, andgraphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Figure 1.2 Schematic view of graphene transistors (a) a top gated graphenetransistor and (b) a high speed graphene transistor . . . . . . . . . . . . . 3

Figure 1.3 Schematic of a h-BN/SL-WS2/h-BN field-effect transistor . . . . . 4

Figure 1.4 Combination of 2D materials to produce novel heterostructures . . 4

Figure 1.5 Schematic of (a) graphene FET on h-BN and (b) Fully 2D FET . . 5

Figure 1.6 (a) Schematic of few layer graphene/graphite heat spreader con-nected to the drain contact and (b) Schematic of the device in which fewlayer graphene/graphite is used as heat spreader . . . . . . . . . . . . . . 6

Figure 1.7 Primitive vectors, unit cell, conventional unit cell, and Wigner-Seitz primitive unit cell in a sample 2D lattice . . . . . . . . . . . . . . . 8

Figure 1.8 Hexagonal lattice in real and reciprocal space . . . . . . . . . . . . 9

Figure 1.9 Linear diatomic chain and the corresponding phonon dispersionrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 1.10 Phonon dispersion relation of graphene . . . . . . . . . . . . . . . 10

Figure 1.11 Sketch of the longitudinal optical (LO), transverse optical (TO),and out-of-plane optical (ZO) phonon modes of h-BN at Γ point . . . . . . 11

Figure 1.12 Three phonon interactions . . . . . . . . . . . . . . . . . . . . . . 12

Figure 1.13 Phonon-boundary scattering. . . . . . . . . . . . . . . . . . . . . . 13

Figure 1.14 Calculated thermal conductivities of few layer graphene and h-BNwith contribution of phonon branches to these total thermal conductivities 14

Figure 1.15 Experimental thermal conductivity of (a) few layer graphene and(b) few layer h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

xiv

Figure 1.16 Experimental and reconstructed accumulated thermal conductivityof suspended single layer graphene with respect to mean free path . . . . . 16

Figure 2.1 Flowchart of the methodology. . . . . . . . . . . . . . . . . . . . . 20

Figure 2.2 Evenly spaced q-point grid inside the irreducible Brillouin zone. . . 21

Figure 2.3 Primitive unit cell of a hexagonal structure. . . . . . . . . . . . . . 22

Figure 2.4 Flowchart of the ab-initio calculations. . . . . . . . . . . . . . . . 23

Figure 3.1 Crystal structure of single and bilayer graphene . . . . . . . . . . . 27

Figure 3.2 Phonon dispersion relation of graphene. . . . . . . . . . . . . . . . 28

Figure 3.3 Specific heat of phonons with respect to their frequencies in graphene. 29

Figure 3.4 Group velocity of phonons with respect to their frequencies ingraphene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 3.5 Phonon lifetimes with respect to their frequencies in graphene. . . . 30

Figure 3.6 Thermal conductivity with respect to frequency in graphene. . . . . 32

Figure 3.7 Accumulated thermal conductivity of graphene with respect tophonon mean free path. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 3.8 Phonon dispersion relation of bilayer graphene. . . . . . . . . . . . 34

Figure 3.9 Specific heat of phonons with respect to their frequencies in bilayergraphene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.10 Group velocity of phonons with respect to their frequencies in bi-layer graphene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.11 Phonon lifetimes with respect to their frequencies in bilayer graphene. 36

Figure 3.12 Thermal conductivity with respect to frequency in bilayer graphene. 38

Figure 3.13 Accumulated thermal conductivity of bilayer graphene with re-spect to phonon mean free path. . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.14 Crystal structure of graphite with optimized lattice parameters. . . . 39

Figure 3.15 Phonon dispersion relation of graphite. . . . . . . . . . . . . . . . 40

Figure 3.16 Specific heat of phonons with respect to their frequencies in graphite. 41

xv

Figure 3.17 Group velocity of phonons with respect to their frequencies ingraphite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 3.18 Phonon lifetimes with respect to their frequencies in graphite. . . . 42

Figure 3.19 Thermal conductivity with respect to frequency in graphite. . . . . 43

Figure 3.20 Accumulated thermal conductivity of graphite with respect to phononmean free path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 4.1 Crystal structure of single and bilayer h-BN . . . . . . . . . . . . . 45

Figure 4.2 Phonon dispersion relation of single layer h-BN. . . . . . . . . . . 46

Figure 4.3 Specific heat of phonons with respect to their frequencies in singlelayer h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 4.4 Group velocity of phonons with respect to their frequencies in sin-gle layer h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 4.5 Phonon lifetimes with respect to their frequencies in single layerh-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 4.6 Thermal conductivity with respect to frequency in single layer h-BN. 50

Figure 4.7 Accumulated thermal conductivity of single layer h-BN with re-spect to phonon mean free path. . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 4.8 Phonon dispersion relation of bilayer h-BN. . . . . . . . . . . . . . 52

Figure 4.9 Specific heat of phonons with respect to their frequencies in bilayerh-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.10 Group velocity of phonons with respect to their frequencies in bi-layer h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.11 Phonon lifetimes with respect to their frequencies in bilayer h-BN. 54

Figure 4.12 Thermal conductivity with respect to frequency in bilayer h-BN. . . 55

Figure 4.13 Accumulated thermal conductivity of bilayer h-BN with respect tophonon mean free path. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 4.14 Crystal structure of bulk h-BN with optimized lattice parameters. . 57

Figure 4.15 Phonon dispersion relation of bulk h-BN. . . . . . . . . . . . . . . 58

Figure 4.16 Specific heat of phonons with respect to their frequencies in bulkh-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xvi

Figure 4.17 Group velocity of phonons with respect to their frequencies in bulkh-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 4.18 Phonon lifetimes with respect to their frequencies in bulk h-BN. . . 60

Figure 4.19 Thermal conductivity with respect to frequency in bulk h-BN. . . . 61

Figure 4.20 Accumulated thermal conductivity of bulk h-BN with respect tophonon mean free path. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 5.1 Crystal structure of composite bilayer graphene/h-BN with AAstacking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 5.2 Phonon dispersion relation of bilayer graphene/h-BN . . . . . . . . 64

Figure 5.3 Superposition of phonon dispersion relation of graphene and singlelayer h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 5.4 Specific heat of phonons with respect to their frequencies in bilayergraphene/h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 5.5 Group velocity of phonons with respect to their frequencies in bi-layer graphene/h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 5.6 Phonon lifetimes with respect to their frequencies in bilayer graphene/h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 5.7 Thermal conductivity with respect to frequency in bilayer graphene/h-BN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 5.8 Accumulated thermal conductivity of bilayer graphene/h-BN withrespect to phonon mean free path. . . . . . . . . . . . . . . . . . . . . . . 69

xvii

LIST OF ABBREVIATIONS

BN Boron Nitride

BTE Boltzmann Transport Equation

CNT Carbon Nanotubes

CVD Chemical Vapor Deposition

DDB Derivative Database

DFT Density Functional Theory

FET Field Effect Transistor

h-BN Hexagonal Boron Nitride

HEMT High Electron Mobility Transistor

HFET Heterostructure Filed Effect Transistor

IC Integrated Circuit

IFCs Interatomic Force Constants

LA Longitudinal Acoustic

LED Light Emitting Diode

LO Longitudinal Optical

MFP Mean Free Path

MISHFET Metal-Insulator-Semiconductor Heterostructure Field Effect Tran-sistors

TA Transverse Acoustic

TO Transverse Optical

ZA Out-of-plane Acoustic

ZO Out-of-plane Optical

xviii

CHAPTER 1

INTRODUCTION

1.1 Graphene and h-BN

Graphene is a 2-dimensional crystalline structure formed by the hexagonal alignment

of carbon atoms. Graphene is also the basic element for widely used materials such

as carbon nanotubes (CNTs), fullerenes, and graphite shown in Fig. 1.1.

Figure 1.1: Graphene as a basic element of buckyballs, carbon nanotubes, and

graphite [1].

1

Boron Nitride (BN) consists of equal number of boron and nitrogen atoms and can be

found in various crystalline forms such as cubic, amorphous, and hexagonal. Among

these crystalline forms, hexagonal Boron Nitride (h-BN) is the most thermally and

chemically stable and provides unique properties such as wide indirect electrical

bandgap up to 6 eV [2] and high thermal conductivity compared to other insulators.

1.2 Applications

Today, as the size of the electronic devices decreases and their electronic performance

increases, the thermal management of such devices becomes more important. Ther-

mal management can be improved with introducing novel materials with unique ther-

mal properties into systems as heat spreaders or cooling fluids. There is ongoing

research on finding such materials to improve the thermal transport in electronic sys-

tems. Carbon allotropes, such as carbon nanotubes (CNTs) or graphene, standout

among the other materials due to their extremely high thermal conductivities exceed-

ing 3000W/mK [3].

Graphene has been popular in modern electronic and optoelectronic applications dur-

ing the recent years due to its favorable properties such as extremely high thermal

conductivity and transparency. Due to its many favorable properties graphene is an

ideal candidate for next generation Integrated Circuits (ICs), high power light emit-

ting diodes (LEDs), and Heterostructure Field Effect Transistors (HFETs) [4, 5, 6, 7]

that are shown in Fig. 1.2.

Before the revolutionary exfoliation of single layer graphene by the scientist group in

University of Manchester in 2004 [10], graphene was an exclusive realm of condensed-

matter physicists. After this breakthrough, graphene has attracted many interests by

the electron-device society. Since then, different methods such as mechanical exfo-

liation, epitaxial growth, reduction methods, and Chemical Vapor Deposition (CVD)

has been used to make large scale high quality graphene films. Among these meth-

ods, CVD has shown its potential to produce large scale high quality graphene films

[11]. Within the evolution of fabrication techniques, production of large scale defect

free graphene thin films, thus graphene-based nanoelectronic devices with high elec-

2

Figure 1.2: Schematic view of graphene transistors (a) a top gated graphene transistor

[8] and (b) a high speed graphene transistor [9].

tron mobility for radio-frequency applications became possible [12, 13]. Even though

graphene has an extremely high thermal conductivity reported up to 5000 W/mK in

room temperature [14], its zero electronic bandgap can be considered as a disadvan-

tage for electronic applications.

Although, there has been a lot of research on the integration of carbon allotropes

into electronic devices, little has been done about integration of h-BN to such de-

vices. Despite relatively lower conductivity of h-BN compared to graphene, due to

its favorable properties such as high thermal/chemical stability, dielectric nature, and

the anisotropic thermal conductivity, which allows effective in-plane heat spreading,

h-BN can still be a good candidate for electronic devices. Even more, recently the

in-plane thermal conductivity of single and few layer h-BN films have been inves-

tigated and found to exhibit even higher thermal conductivities than the bulk h-BN

[15]. These exclusive properties makes h-BN also an ideal candidate for high elec-

tron mobility devices. Figure 1.3 shows an example of high electron mobility FET

made using single layer h-BN.

Synthesis and fabrication of single layer materials such as graphene and single layer

h-BN, black phosphorus, and molybdenum disulfide (MoS2) and their integration as

shown in Fig. 1.4 has opened new doors for use of new heterostructure materials in

electronic applications. For example graphene with null electronic bandgap can be

combined with large bandgap material, e.g., h-BN in order to be used in transistors.

3

Figure 1.3: Schematic of a h-BN/SL-WS2/h-BN field-effect transistor [16].

Figure 1.4: Combination of 2D materials to produce novel heterostructures.

In addition, h-BN with same crystalline structure and matching lattice constant to

that of graphene [17] has been proved to be good substrate for high quality graphene

[18, 19] providing smooth surface. Therefore, graphene grown on h-BN substrate has

less crystal defects and shows high carrier mobility [20]. Figure 1.5 is the schematic

of a graphene field effect transistor (FET) formed by the h-BN/graphene bilayer [21].

Since these types of devices are used for high frequency and power applications,

high electron mobility is always desired. Figure 1.5(b) shows the first fully 2D FET

introduced by Roy et al. [22]. In this FET, h-BN has been used as the top-gate

dielectric and graphene as the source/drain and top-gate contacts. Unlike conventional

4

silicon transistors, it has shown a high electron mobility even at high gate voltages

without degradation. Knowing the fact that this device has been built based on van

der Waals bonding, a promising future for van der Waals heterostructure devices can

be predicted. Finally, few layer h-BN was also used as a gate dielectric in graphene

heterostructure field effect transistors (HFETs) [18] and AlGaN/GaN metal-insulator-

semiconductor heterostructure field effect transistors(MISHFETs) [23].

Figure 1.5: Schematic of (a) graphene FET on h-BN [21] and (b) Fully 2D FET [22].

1.3 Motivation

In all of the above electronic applications, thin layers of h-BN and/or graphene are

located near the regions where the maximum heat dissipation occurs. Heat dissipa-

tion, i.e. Joule heating, occurs as a result of electron movement in these devices and is

always an issue since high temperatures cause performance and physical degradation

in devices. Since few layer h-BN and/or graphene layers are very close to heat dis-

sipation regions it is very important to understand their thermal transport properties

for two reasons. First, if enough information about the thermal transport properties of

h-BN and/or graphene is known then these thin layers can be modeled realistically in

thermal simulations and more accurate device temperature predictions can be made.

Second, with the knowledge of the thermal properties, these layers can be better de-

signed to serve the thermal management needs of such devices. For example, better

graphene/graphite heat spreaders can be designed such as the ones used for the ther-

mal management of AlGaN/GaN HFETs as illustrated in Fig. 1.6 [24]. Moreover,

dielectric h-BN heat spreaders can be designed and used as an alternative to elec-

trically conductive graphene heat spreaders which might cause shorting in devices

[24].

5

Figure 1.6: (a) Schematic of few layer graphene/graphite heat spreader connected

to the drain contact and (b) Schematic of the device in which few layer graphene/-

graphite is used as heat spreader [24].

To perform thermally conscious device designs or design thermal management struc-

tures, such as heat spreaders using few layer h-BN and/or graphene, the anisotropic

thermal properties of single and few layer h-BN and/or graphene has to be well

understood. Moreover, not only the individual thermal properties of few layer h-

BN/graphene, but also the effects of the neighboring materials (such as GaN, SiO2,

metal, etc.) on the thermal properties of h-BN and the thermal interface resistances

have to be understood as well.

Energy in solids is carried by electrons and lattice vibrations (phonons). Heat con-

duction in metals is mainly transported by electrons, while in semiconductors and

insulators phonons are mainly responsible for the energy transport [25, 26]. As the

device size decreases, thermal analysis must be inspected at the energy carrier scales,

i.e., phonons in semiconductors and insulators and electrons in metals. Therefore,

recent studies has inspected the energy transport regime on the scale of phonon mean

free path (MFP) [27, 28]. Relative magnitude of phonon MFP and characteristic di-

mension of the device or material strongly effects the thermal transport mechanism

[29, 30]. If the characteristic length of a material is much larger than phonon MFP,

thermal transport can be calculated using heat diffusion equation and bulk thermal

conductivity of the material. If the characteristic length of a material is in the scale

of phonon MFP or smaller than that, then thermal conductivity reduce, and boundary

scatterings should be taken into account for thermal transport calculations. Therefore,

not only understanding the thermal conductivity but also the MFP characteristics in

6

such materials is necessary to characterize the thermal properties. This is the aim of

this study.

1.4 Thermal Transport Mechanism in Crystals

Before explaining the methodology of this study, there are some concepts that need

to be explained. Understanding these concepts will facilitate the comprehension of

next section and chapter.

1.4.1 Lattice, Crystal Structure, Primitive Cell, Reciprocal Lattice, and Bril-

louin Zone

A perfect bulk crystal structure is constructed of repetition of identical unit arrange-

ment of atoms in space. Lattice is a set of periodic atomless mathematical points that

represent the same periodicity as actual crystal structure. Actual crystal structures

consist of a lattice and a basis which define the type and the number of atoms, ions,

or molecules. Basis is not necessarily on the lattice points. In fact, it can take any

position relative to lattice points, provided that, it keeps the same relative position

with respect to all the lattice points.

Bravais lattices can be constructed by set of vectors called primitive lattice vectors.

For a three dimensional lattice, set of primitive lattice vectors can be defined by lattice

constants (a1, a2, a3) so that from any lattice point, all other lattice points could be

expressed by a proper choice of integers through the following equation:

R = n1a1 + n2a2 + n3a3 (n1, n2, n3 are integers) (1.1)

The parallelepiped defined by the primitive lattice vectors is called the primitive unit

cell. Primitive unit cell is the smallest region which can span the entire lattice without

overlap, when translated by any chosen set of primitive vectors. Unlike conventional

unit cell which may consist of a number of lattice points, primitive unit cell consists

of only one lattice point. The choice of primitive vectors and unit cell is not unique.

One method to obtain a primitive unit cell is called Wigner-Seitz primitive unit cell,

7

which is constructed by bisecting planes perpendicular to the connection lines of a

randomly chosen point to neighboring points as shown in Fig. 1.7. The smallest

space enclosed by all the bisecting lines is called Wigner-Seitz primitive cell.

Figure 1.7: Primitive vectors, unit cell, conventional unit cell, and Wigner-Seitz prim-

itive unit cell in a sample 2D lattice [31].

Periodic nature of the crystal structure in real space leads to the definition of a re-

ciprocal space. Reciprocal space is the spatial Fourier transform of the real space.

Reciprocal primitive lattice vectors (b1, b2, b3) are defined using real space primitive

lattice vector (a1, a2, a3) as:

b1 = 2π(a2 × a3)/V, b2 = 2π(a3 × a3)/V, b3 = 2π(a1 × a2)/V (1.2)

where V is the volume of the primitive unit cell in real space and can be found as:

V = a1 • (a2 × a3) (1.3)

Wigner-Seitz cell represented in the reciprocal space is the Brillouin zone. The small-

est Wigner-Seitz cell or Wigner-Seitz primitive unit cell is called the first Brillouin

zone. Hexagonal lattice structure in real and reciprocal space are shown in the Fig.

1.8. By definition the volume enclosed by high symmetry lines (red lines in Fig. 1.8)

is called the irreducible Brillouin zone.

8

Figure 1.8: Hexagonal lattice in (a) real space and (b) reciprocal space [32].

1.4.2 Phonons

Unlike liquids and gases, atoms in solids cannot move. They just vibrate around their

equilibrium positions. These vibrations and free electrons are responsible of energy

transport in solids. Fee electrons are the main energy carriers in metals and lattice

vibrations or phonons are the main energy carriers in insulators and semiconductors.

Phonons of a crystal have unique wavevector and frequency (ω). Direction of wavevec-

tor (k = 2π/λ) is usually (not always) the direction of wave propagation and its

magnitude is called wavenumber .

Figure 1.9 shows the linear diatomic chain and two types of phonon modes in its

corresponding dispersion relation. Equation 1.4 shows the relation of wavevector and

frequency in diatomic linear chain [26]:

ω2 = K

(1

m1+

1

m2

)±K

[(1

m1+

1

m2

)2− 4sin

2(ka/2)

m1m2

]1/2(1.4)

where the plus and minus sign correspond to the upper branch (optical) and lower

branch (acoustic) of dispersion relation (Fig. 1.9(b)), respectively.

When there are multiple atoms in the primitive unit cell of a crystal, two types of

phonon appear in the phonon dispersion relation; acoustic phonon modes and optical

phonon modes. Acoustic phonons are due to coherent movement (in-phase) of atoms

and optical phonons are are due to out-of-phase vibration of atoms inside the unit cell.

9

Figure 1.9: (a) Linear diatomic chain and (b) corresponding phonon dispersion rela-

tion of diatomic chain [26].

For more complex crystalline structures (2D and 3D), multiple branches of phonons

appear in the dispersion relation. For instance, in the dispersion relation of graphene

shown in Fig. 1.10, there are six branches in the phonon dispersion relation; three

acoustic and three optical modes. These branches are explained in detail in the fol-

lowing. Phonons based on their relative direction of vibration to the direction of wave

Figure 1.10: Phonon dispersion relation of graphene [33].

propagation are divided into three types. Longitudinal modes vibrate parallel to the

direction of wave propagation, transverse modes vibrate perpendicular to the direc-

tion of wave propagation, and third type, vibrate out-of-plane. Sketch of longitudinal

optical (LO), transverse optical (TO), and out-of-plane optical (ZO) phonon modes

of h-BN at Γ point are shown in Fig. 1.11.

10

Figure 1.11: Sketch of phonon modes of h-BN at Γ point (a) out-of-plane optical

(ZO), (b) Transverse optical (TO) and (c) Longitudinal optical (LO) [34].

1.4.3 Phonon Scattering

Phonon scattering including phonon-phonon, phonon-boundary, and phonon-defect

(e.g., impurities, dislocations, or vacancies) scattering dominates, the energy transport

properties of insulators and semiconductors.

Phonon-phonon interactions come from the anharmonic terms of crystal potential

energy expansion and anharmonicity itself is the result of significant deviation of

atoms from their equilibrium position.

Conservation of energy requires at least three phonons to be involved in the scattering

process. There are also phonon-phonon interactions with more than three phonons,

but probability of three-phonon interactions are much larger, specifically at room tem-

perature [35, 26].

Phonon-phonon scattering processes are divided into two categories, normal (or N)

and umklapp (or U) processes. Normal process include the interactions in which all

the three phonons are inside the first Brillouin zone and umklapp process include the

remaining interactions in which the resulting phonon happens to be outside the first

11

Figure 1.12: Three phonon interactions.

Brillouin zone. These phonons are folded back to the first Brillouin zone by the recip-

rocal lattice vector. Unlike normal process which conserves the momentum, umklapp

process does not conserve the momentum. Both processes are important and should

be included in the phonon-phonon interaction calculations. Even though normal pro-

cesses redistribute the phonon occupation, since they conserve the momentum, they

do not affect the thermal conductivity. Thermal resistance to the thermal current

originates from the umklapp processes and therefore reduce the thermal conductivity.

Conservation of energy for three phonon scattering can be written as the following

equation:

ωqs ± ωq′s = ωq′′s (1.5)

where ωqs is the frequency of the phonon corresponding to the wavevector q and

mode s. And conservation of momentum for three phonon interactions can be written

as:

q± q′ = q′′ + G (1.6)

where G is the reciprocal lattice vector. For normal processes G = 0 and for the

umklapp processes G 6= 0.

There are two possible processes in three-phonon interactions. One phonon could

emerge into two new phonons (fission process) (minus sign in Eq. 1.5 and Eq. 1.6). In

the second possibility, two phonons could merge into a new phonon (fusion process)

(plus sign in Eq. 1.5 and Eq. 1.6). These processes are shown in Fig. 1.12.

12

1.4.4 Phonon Mean Free Path

Phonon mean free path (MFP) is the distance a phonon travels between successive

collisions with other phonons. The concept of MFP and its relative importance to

the characteristic length give rise to the definition of the Knudsen number. Knudsen

number is the ratio of the MFP (Λ) to the characteristic length (L), as:

Kn =Λ

L(1.7)

For phonon studies, this number has an important role that describes the thermal

transport mechanism in the crystal structure. When the MFP is much smaller than

the characteristic length (Kn � 1), there will be no boundary scatterings, shown inFig. 1.13, and only the phonon-phonon interactions will be present in the crystal.

When the MFP is in the order of characteristic length (Kn > 1 or Kn 6 1), then

boundary scatterings shown in Fig. 1.13 become important and even dominate the

phonon-phonon scattering. In this study, it is assumed that phonon MFPs are much

smaller than characteristic length (Kn � 1), and therefore only the phonon-phononscatterings are inspected.

Figure 1.13: Phonon-boundary scattering.

1.5 Previous Research

There has been and increasing attention to thermal properties of few layer graphene

and h-BN due to their increasing popularity in electronic applications. Both experi-

mental and theoretical studies have been performed to investigate the thermal prop-

erties of these structures. Perhaps, the most popular of these are the study conducted

13

by Lindsay et al. where they theoretically examined the thermal properties and cal-

culated the thermal conductivity of the single and few layer graphene [35] and single

and few layer h-BN [15] by modeling their lattice vibrations.

Figure 1.14: Calculated thermal conductivities and contribution of phonon branches

to the total thermal conductivity of (a) few layer graphene with respect to thermal

conductivity of single layer graphene [35] and (b) few layer h-BN with respect to

thermal conductivity of single layer h-BN [15].

Figure 1.14 (a) and (b) show the contribution of phonon modes and the effect of the

increasing number of layers to the total thermal conductivity of graphene and h-BN

based structures, respectively. It has been shown that the total thermal conductiv-

ity decreases monotonically and converges to the bulk total thermal conductivity at

around five layers. Total thermal conductivity of graphite and bulk h-BN drop to 66%

and 56% of their single layer counterparts. The significant variation of the thermal

conductivity has been explained by the out of plane vibrations and the interactions

between the layers. It has also been claimed that ZA modes play the main role in

thermal transport, i.e., almost 75% of total energy is carried by ZA phonon modes in

graphene and single layer h-BN.

Figure 1.15(a) shows an experimental study to calculate the thermal conductivity of

suspended few layer graphene. Unlike the expectations based on previous theoreti-

14

cal study, in which thermal conductivity decreases by adding layers and reaches to

its minimum value at bulk structure, it has been observed that 8 layer graphene has

a higher thermal conductivity than 2,3, and 4 layer graphene [36]. Figure 1.15(b)

shows the variation of thermal conductivity of the 5 and 11 layer h-BN samples with

temperature. Although it is expected to observe similar or even higher thermal con-

ductivity with few-layer h-BN compared to bulk h-BN, thermal conductivity of 5 and

11 layer h-BN are observed to be lower than the thermal conductivity of bulk h-BN

[37].

Figure 1.15: Experimental thermal conductivity of (a) few layer graphene [36] and

(b) few layer h-BN [37].

Thermal conductivity of suspended single layer graphene has been also measured and

calculated for different sample lengths [38]. It has been observed that by increasing

the sample length thermal conductivity increases suggesting the effect of boundary

scattering on thermal conductivity. This emphasizes that sample length has a signifi-

cant effect on the value of calculated thermal conductivity.

There are a few studies about phonon MFP of suspended single layer graphene. Ma-

jority of these studies has been conducted over limited sample length. Effect of sam-

ple length over phonon MFP has been studied to some extent [39]. It has been seen

that sample length scale changes the average phonon MFP significantly. For instance,

15

average phonon MFPs of a bulk (in-plane) sheet of graphene is one order of magni-

tude larger than the limited samples [39]. Reported phonon MFP of bulk graphene

covers a very large span of 10 nm up to 10 µm. It has also been observed that 70% of

heat is carried by the phonons with MFPs of larger than 1 µm [39]. Experimentally

and theoretically obtained average phonon MFP values for limited length case are

240 nm and 80 nm, respectively [38]. Comparison of these two studies (one over lim-

ited length scale and the other over bulk sheet) with each other, shows the substantial

effect of sample length over thermal conductivity and MFP.

Figure 1.16: Experimental and reconstructed accumulated thermal conductivity of

suspended single layer graphene with respect to mean free path [39].

All these studies showed that number of layers, being in contact with similar or differ-

ent materials, and sample length significantly affect the thermal conductivity. How-

ever, none of these studies shared the changes in the phonon dispersion and MFP

when number of layers are decreased and/or additional layers are introduced. More-

over, although these studies investigated the effects of boundary scatterings on the

thermal conductivity, they were not able to systematically determine the threshold of

ballistic-diffusive heat transfer since little has been known about the MFP of phonons.

Both the changes in phonon dispersion and the MFPs of dominant heat carriers are

important parameters to understand the heat transfer. While observing the changes in

phonon dispersion will be an indicator of the changing relaxation and the conduction

mechanism in the crystal, obtaining the MFP of phonons can provide useful informa-

16

tion about the impact of the boundary scatterings. As a result, there is a need for a

systematic theoretical study that examines the individual effects of interlayer phonon

scatterings on the phonon dispersion and MFPs. In addition, there is not any research

to study the MFP in few layer h-BN. Finally, neither thermal transport nor MFP of

bilayer graphene/h-BN has been studied so far.

1.6 Objectives and Research Strategy

Objectives of this study can be categorized into four. First goal is to inspect the

variation of thermal transport in few layer graphene and few layer h-BN to understand

the effect of additional layers on thermal transport mechanism. Second is to compare

the transport phenomena in graphitic and h-BN structures. Since graphene and h-BN

are having almost same lattice parameters, average atomic masses, and bondings, it is

expected to show similar behavior. The validity of this statement will be inspected and

in case of being false, the cause of behavior variation will be searched. Third goal is

to calculate the accumulated thermal conductivity with respect to phonon MFP for all

structures, to study the effect of number of layers and compare the difference between

graphitic and h-BN structures. Finally, thermal transport and phonon MFP in bilayer

graphene/h-BN structure will be compared to those of bilayer graphene and bilayer

h-BN to understand how graphene and h-BN layers effect each other.

For this purpose, using ab-initio calculation frequency of discrete phonons in the irre-

ducible Brillouin zone will be calculated for single layer graphene, bilayer graphene,

graphite, single layer h-BN, bilayer h-BN, bulk h-BN, and bilayer graphene/h-BN

structures. Then, using these frequencies, thermophysical properties of these struc-

tures such as phonon lifetimes, group velocities, specific heat of each discrete phonon

will be obtained. Finally, using these parameters thermal conductivity and phonon

MFP of each structure will be calculated. Phonon MFP spectra will be used to under-

stand the limits of ballistic-diffusive heat transfer in these structures.

Chapter 2 will provide the methodology used in this study. Chapter 3 will summarize

the procedure and result of geometry optimization, phonon dispersion, thermophysi-

cal properties, thermal conductivity and accumulated thermal conductivity for single

17

layer graphene, bilayer graphene, and graphite. Chapters 4 and 5 will provide the

same information as chapter 3 for h-BN (single layer, bilayer, and bulk) and bilayer

graphene/h-BN structures, respectively. In addition, Chapter 5 will summarize all the

result obtained in chapters 3,4, and 5 and will compare the results obtained for differ-

ent structures. Chapter 6 will provide final remarks of this study and will suggest the

further improvements that can be made on this study.

18

CHAPTER 2

METHODOLOGY

In this chapter the methodology of this study is explained. Methodology of this study

is divided into two sections. In the first section, wavevectors are defined and using

ab-initio calculations frequency of individual phonons inside the irreducible Brillouin

zone are calculated. In the second section, obtained frequency of phonons are used to

calculate the thermophysical properties of phonons using MATLAB.

2.1 Ab-initio Calculations

In order to characterize the heat transport mechanism in crystalline structures, phonon

vibration properties and their interactions must be clarified. In order to obtain accu-

rate result, first the geometry optimization should be performed for structure. As

previously explained, phonons are defined by their frequency and wavevector. In ad-

dition to defining wavevectors, frequency of phonons inside the whole irreducible

Brillouin zone is needed, to calculate the phonon interactions. For this purpose, first

an evenly spaced q-points (wavevector) grid inside the irreducible Brillouin zone is

created and frequency of phonons in these q-points are calculated using an ab-initio

software. Finally, obtained frequencies are used to calculate specific heat, group ve-

locity, and lifetime of individual phonons. These thermophysical properties will be

used to calculate the thermal conductivities and phonon MFPs of phonons at room

temperature. Flowchart of the methodology is shown in Fig. 2.1.

19

Figure 2.1: Flowchart of the methodology.

2.1.1 Brillouin Zone Sampling

As discussed previously, phonons are defined by their wavevector (q-point) and fre-

quencies. In order to characterize the phonon properties inside the whole Brillouin

zone, an evenly spaced q-point mesh inside the irreducible Brillouin zone (the vol-

ume enclosed with red lines in Fig. 1.8 (b)) is created. Using ab-initio calculations

frequency of phonons at these q-points for different modes (polarizations) are cal-

culated. Different density of meshes were inspected to ensure the result would be

independent of mesh size. Figure 2.2 shows the created q-point grid.

2.1.2 ABINIT

After obtaining wavevectors, ab-initio calculations were performed using ABINIT

[40, 41]. ABINIT is an open-source software package which has the ability to cal-

culate the total energy, the charge density, and the electronic structure of materials

within Density Functional Theory (DFT). ABINIT can also do geometry optimiza-

tion of crystalline structures using DFT forces and stresses.

Initial step of the phonon calculations is the geometry optimization. This step is im-

20

Figure 2.2: Evenly spaced q-point grid inside the irreducible Brillouin zone.

plemented to ensure that the atoms are in their most relaxed equilibrium positions.

This step prevents the instabilities and results in calculation of more accurate energy

derivatives. Before starting the ab-initio calculations, primitive unit cell of the struc-

tures should be defined. Figure 2.3 shows the defined primitive cell for the hexagonal

structure that represents both graphene and h-BN. In Fig. 2.3 a and x are the lattice

constant and the interatomic distance, respectively. Atoms assigned with numbers 1

and 2 are both carbon, or boron and nitrogen atoms in case of graphene-based and h-

BN structures, respectively. Primitive unit vectors in real space are defined as follows:

a1 =

√3a

2î+

a

2ĵ, a2 = −

√3a

2î+

a

2ĵ, a3 = ck̂ (2.1)

Using Eq. 1.2, primitive unit vectors of the hexagonal structure in reciprocal space

can be found as:

b1 =2π√3a

k̂x +2π

ak̂y, b2 = −

2π√3a

k̂x +2π

ak̂y, b3 = ck̂z (2.2)

Geometry optimization for the bulk structures are performed in two steps. First, op-

timized lattice parameters are found and then the atomic coordinates are set based on

the obtained lattice parameters.

Second step of the ab-initio calculation is to obtain the derivative databases (DDB).

Derivative databases contain derivatives of the total energy inside the Brillouin zone.

These derivatives are generated with the phonon, electric field, and stress perturba-

21

Figure 2.3: Primitive unit cell of a hexagonal structure.

tions. DDBs can be further processed and analyzed to obtain the desired properties,

such as interatomic force constants. Then, obtained DDBs are combined into a single

file which contains the dynamical matrices for all the calculated q-points. Further

analyzing of this file at created q-points grid (explained in the following section),

will provide frequency of phonons inside the irreducible Brillouin zone. Ab-initio

calculations procedure is summarized in the flowchart shown in Fig. 2.4.

2.1.3 Phonon Dispersion Relation

Following the calculation of frequency of phonons inside the irreducible Brillouin

zone, phonon dispersion of structures can be obtained. Phonon dispersion relation

can be plotted using frequency of phonons along the high symmetry lines (shown with

red lines in Fig. 1.8 (b)) with respect to wavevector. Coordinates of high symmetry

points which were used for both sampling the irreducible Brillouin zone and plotting

the phonon dispersion relation are summarized in Table 2.1.

22

Figure 2.4: Flowchart of the ab-initio calculations.

Table 2.1: Coordinates of the symmetry points in the hexagonal Brillouin zone

Point Coordinate

Γ 0 0 0

M 12

0 0

K 13

13

0

A 0 0 12

L 12

0 12

H 13

13

12

23

2.2 Phonon Properties

After obtaining frequency of discrete phonons inside the Brillouin zone, thermophys-

ical properties of phonons can be obtained using the following procedure.

2.2.1 Specific Heat

Total specific heat of the structures can be found through summation of specific heat

of discrete phonon modes using the following equation [42]:

C =∑qs

Cqs =∑qs

kB

[~ωqskBT

]2 exp [~ωqskBT

][exp

[~ωqskBT

]− 1]2 (2.3)

Cqs is the specific heat of the phonon mode qs, kB is the Boltzmann constant, ~ is

the reduced Planck’s constant, ωqs is the frequency of the phonon mode qs, and T is

the lattice temperature.

2.2.2 Group Velocity

Group velocity of discrete phonon modes (qs) is the gradient of phonon dispersion

curve (spatial derivative of frequency with respect to wavevector):

vqs =∂ω

∂k(2.4)

2.2.3 Phonon Lifetime

Phonon lifetimes can be found using the three phonon interactions proposed by Sri-

vasatava [43]. Using this approach intrinsic transition of phonon probability of the

fusion process in which qs + q′s′ → q′′s′′, can be found by the following equation[44]:

P̄q′′s′′

qs,q′s′ =

√π~

ρN0V

ωqsωq′′s′′

σF 2n̄qsn̄q′s′ (n̄q′′s′′ + 1) δq+q′+q′′,Gexp

[−(

(ωq′′s′′ − ωqs − ωq′s′)σωqs

)2](2.5)

24

and intrinsic transition probability of the fission process in which qs→ q′s′ + q′′s′′,can be found by the following equation [44]:

P̄q′s′,q′′s′′

qs =

√π~

ρN0V

ωq′s′ωq′′s′′

σF 2n̄q′s′ n̄q′′s′′ (n̄qs + 1) δq+q′+q′′,Gexp

[−(

(ωqs − ωq′s′ − ωq′′s′′)σωqs

)2](2.6)

Here ρ is the solid density,N0 is the number of unit cells, V is the volume per unit cell,

σ is the broadening parameter of the Gaussian peak, δq+q′+q′′,G is the delta function

that conserves the energy, G is the reciprocal lattice vector, and exponential function

is the Gaussian peak which conserves the momentum.

F is the temperature dependent fitting parameter that is used to match the thermal

conductivity of graphene and bulk h-BN, and can be expressed as [45]:

F =γ

c̄(2.7)

Here γ is averaged Grüneisen parameter which shows the effect of temperature vari-

ation on phonon frequencies [45] and c̄ is the average acoustic phonon speed.

n̄qs is the equilibrium occupation number of phonon mode qs, and using Bose-

Einstein can be described as [43]:

n̄qs =1

exp[

~ωqskBTqs

]− 1

(2.8)

Tqs is the lattice temperature of mode qs.

Finally, using scattering matrices, lifetime of discrete phonons can be found as:

1/τqs = P̄q′′s′′

qs,q′s′ + 1/2P̄q′s′,q′′s′′

qs (2.9)

2.2.4 Thermal Conductivity

Following the calculation of specific heat, group velocity, and lifetime of phonons, to-

tal thermal conductivity of the crystal structure can be found by summation of thermal

conductivity of individual phonon modes. Total thermal conductivity of individual

25

phonon modes can be found using the kinetic theory:

k =∑

kqs =∑ 1

3c2qsCqsτqs (2.10)

For 3D materials. The 1/3 factor becomes 1/2 for 2D materials. Here cqs = |vqs| isthe magnitude of group velocity.

2.2.5 Phonon Mean Free Path

Using group velocity and phonon lifetime, mean free path of phonons can be found

as:

Λqs = vqsτqs (2.11)

In the following three chapters, based on the methodology explained in this chapter,

results for graphitic, h-BN, and composite structures will be provided, respectively.

26

CHAPTER 3

GRAPHENE-BASED STRUCTURES

In this chapter results for graphene, bilayer graphene, and graphite will be provided

and a comparison between the aforementioned structures will be made.

Graphite is a bulk structure that consists of honeycomb layers bonded with weak van

der Waals forces together. Inside the layer, atoms are bonded with strong covalent

bonds. Yet, using the nature of such bondings in graphite single layers from graphite

can be exfoliated [1, 10]. These single sheets of graphite are called graphene. Crystal

structure of single and bilayer graphene are shown in Fig. 3.1.

Figure 3.1: Crystal structure of (a) single layer graphene and (b) bilayer graphene

with AA stacking.

In the following sections, obtained dispersion relations and calculated thermophysical

properties, thermal conductivity, and accumulated thermal conductivity with respect

to MFP of single layer, bilayer graphene, and graphite obtained through the method-

ology presented in Chapter 2, will be provided and discussed.

27

3.1 Single Layer Graphene

Geometry optimization was done by obtaining the minimum total potential energy

of structure and the optimum in-plane lattice parameter found as a = 2.455 Å. Ob-

tained phonon dispersion of graphene plotted along the high symmetry lines shown in

Fig. 3.2 is in very good agreement with previous experimental and theoretical stud-

ies [46, 47]. Since primitive unit cell of graphene has two carbon atoms, there are

six branches in the phonon dispersion of graphene. These are Longitudinal Acous-

tic (LA), Transverse Acoustic (TA), out-of-plane acoustic (ZA), Longitudinal Optical

(LO), Transverse Optical (TO), and out-of-plane optical (ZO). Due to presence of

atoms with same atomic masses (carbon atoms) there is no bandgap between acoustic

and optical modes. Also for the same reason, two out-of-plane modes (ZA and ZO)

degenerate at K point of the Brillouin zone. Lower frequency of out-of-plane optical

mode (ZO) compared to other optical modes (TO and LO) proves the weak nature of

van der Waals forces.

Figure 3.2: Phonon dispersion relation of graphene.

28

Using dispersion information of graphene, its thermophysical properties such as group

velocity, specific heat, and phonon lifetime were obtained. For this purpose, first dis-

crete and total volumetric specific heat was obtained using Eq. 2.3. Total volumetric

specific heat of graphene was found as 8.2435× 105 J/m3K.

From the scattering of phonon specific heat with respect to their frequencies shown in

Fig. 3.3, it can be concluded that phonons with lower frequencies which are mainly

acoustic modes have higher specific heat. Specific heat of discrete phonons decrease

monotonically by the increase of their frequencies. Figure 3.4 shows the scattering

of phonons group velocities. It can be observed that LA and TA modes have higher

group velocity than other modes. Group velocity of optical modes is lower than that

of LA and TA modes but are not negligible. Based on the scattering of phonon life-

times shown in Fig. 3.5, it can be deduced that phonons with ZA mode have the

highest lifetimes. Unlike the acoustic modes, optical modes have short lifetimes.

Figure 3.3: Specific heat of phonons with respect to their frequencies in graphene.

29

Figure 3.4: Group velocity of phonons with respect to their frequencies in graphene.

Figure 3.5: Phonon lifetimes with respect to their frequencies in graphene.

30

Average group velocity and lifetime of phonons in graphene were calculated using

the methodology presented in Chapter 2. Results are summarized in Table 3.1. Based

on the information provided in Table 3.1, it can be concluded that LA and TA phonon

modes are the fastest, and optical phonon modes (ZO, TO, LO) have very short life-

times. Therefore, it is expected that optical phonons would have very short MFPs.

Table 3.1: Average group velocity and lifetime of phonon modes in graphene

Phonon mode Average group velocity (m/s) Average phonon lifetime (ps)

ZA 6258 189.8

TA 9926 38.5

LA 13166 25.09

ZO 4499 2.8

TO 1496 0.2

LO 5089 0.1

Following the calculation of essential properties, thermal conductivity of graphene

can be found using the Eq. 2.10 with the 1/2 factor. For this purpose, fitting pa-

rameter (Eq. 2.7) were used to match the bulk thermal conductivity of graphene to

the reported value in the literature (2000 W/m.K) [39, 48]. Thermal conductivity of

bilayer graphene and graphite were calculated using the same fitting parameter.

Although the total thermal conductivity is matched to the bulk thermal conductivity,

due to difference in thermal properties, thermal conductivity of phonon modes are

different. To understand this, scattering of thermal conductivity of discrete phonon

modes in graphene shown in Fig. 3.6 is obtained. Figure 3.6 shows that optical

phonon modes does not play significant role in thermal transport mechanism. Contri-

bution of ZA, TA, and LA modes to the total thermal conductivity of were calculated

as 18.9%, 30.2%, and 50.5%, respectively. Therefore, most of the heat is transported

by LA mode and among acoustic modes, and ZA modes has the lowest contribution

to the total thermal conductivities. Contribution of optical modes to the total thermal

conductivity is less than 1%. The small role of optical modes in thermal transport at

31

room temperature is also reported in [15]. Even though, optical modes have group ve-

locities comparable to that of ZA mode, having very short lifetimes and lower specific

heats, result in their negligible role in the thermal conductivity of graphene.

Figure 3.6: Thermal conductivity with respect to frequency in graphene.

Finally MFP of phonons were calculated using Eq. 2.11. Figure 3.7 shows the accu-

mulated thermal conductivity of graphene with respect to phonon MFPs. Mean free

path of phonons in graphene covers a large span from 10nm up to 16µm and at least

%50 of thermal conductivity is attributed to phonons with MFPs larger than 3826nm.

Our results agree with previous study that reports %70 of heat is transported by the

phonons with MFPs larger than 1µm [39]. In this study this parameter was calculated

as 1.69µm. Difference can be explained through the fact that an experimental study

were used in [39] to reconstruct the accumulated thermal conductivity of suspended

bulk graphene.

Figure 3.7 defines the ballistic-diffusive limit of graphene. For instance, for samples

larger than 16µm, total thermal conductivity of graphene could be used for thermal

simulations. However, when the sample length reaches to 3826nm, thermal conduc-

tivity of graphene will drop to %50 of the total thermal conductivity.

32

Figure 3.7: Accumulated thermal conductivity of graphene with respect to phonon

mean free path.

3.2 Bilayer Graphene

Optimized lattice parameters of bilayer graphene were obtained by optimizing the to-

tal potential energy. Interlayer lattice parameter was found as 3.411 Å and calculated

interatomic distance is almost same as single layer graphene. Phonon dispersion of

bilayer graphene shown in Fig. 3.8 is in very good agreement with the reported dis-

persion in literature [33]. In primitive unit cell of bilayer graphene and graphite there

are four carbon atoms. Therefore, unlike graphene there are 12 branches in phonon

dispersion of bilayer graphene and graphite. Due to weak van der Waals forces almost

all of the branches except out-of-plane with frequency of less than 400 cm−1 doubly

degenerate. ZO′ phonon mode is due the oscillation of the two carbon atoms in the

adjacent layers with a phase difference of π [46, 49]. ZA and ZO′ modes degener-

ate between M and K point of the irreducible Brillouin zone where the interlayer

interactions are not effective [50].

33

Figure 3.8: Phonon dispersion relation of bilayer graphene.

Volumetric specific heat of bilayer graphene was calculated as 1.6493× 106 J/m3K.Figure 3.9 shows the distribution of specific heat of discrete phonons with respect to

their frequencies. Figure 3.9 shows a similar distribution to that of graphene. Opti-

cal modes have the lowest specific heat in the scattering. In addition, a similar trend

to graphene was observed in the distribution of phonon group velocities of bilayer

graphene (shown in Fig. 3.10). LA and ZA modes have the highest group veloci-

ties, and among the optical modes LO and ZO have group velocities comparable to

that of ZA mode. Scattering of phonon lifetimes with respect to their frequency in

bilayer graphene is shown in Fig. 3.11. Based on this scattering, it can be concluded

that acoustic modes have significantly longer lifetimes compared to optical modes.

Among the acoustic modes, phonons of ZA mode have the longest lifetime since they

are freer to move in the out-of-plane direction.

34

Figure 3.9: Specific heat of phonons with respect to their frequencies in bilayer

graphene.

Figure 3.10: Group velocity of phonons with respect to their frequencies in bilayer

graphene.

35

Figure 3.11: Phonon lifetimes with respect to their frequencies in bilayer graphene.

Average group velocity and lifetime of discrete phonons are summarized in Table

3.2. By comparing the information provided in Table 3.1 and 3.2 it can be concluded

that average phonon group velocity of bilayer graphene is same as that of graphene.

This is due to the fact that group velocity is the gradient phonon dispersion relation

with respect to wavevector and phonon dispersions of graphene and bilayer graphene

with the exception of ZO′ mode are almost the same. On the other hand, lifetime

of phonons in bilayer graphene are almost half of that of the graphene. This was an

expected result since in bilayer graphene phonons now interact with phonons of the

adjacent layer which was absent in the graphene. This increasing rate of interactions

decreases the phonon lifetime significantly.

Using Eq. 2.10 and the fitting parameter used for graphene, thermal conductivity of

bilayer graphene was obtained 1571 W/m.K. The ratio of thermal conductivity of

bilayer graphene to that of single layer graphene was obtained as 0.78. This ratio

was reported as 0.73 in [35]. Contribution of ZA, TA, and LA modes to the total

conductivity of bilayer graphene was calculated as 19.1%, 26.4%, and 54%, respec-

tively. Having highest average group velocity and relatively high phonon lifetime and

specific heat, LA mode has the highest contribution to the total thermal conductivity

36

Table 3.2: Average group velocity and lifetime of phonon modes in bilayer graphene


ZA 6268 89.63

TA 9918 20.16

LA 13177 12.29

ZO 4452 1.33

TO 1608 0.05

LO 5656 0.06

of bilayer graphene. To understand the contribution discrete phonons, distribution of

phonons to the thermal conductivity of bilayer graphene with respect to their frequen-

cies is plotted and grouped into different branches in Fig. 3.12. Acoustic modes have

the highest thermal conductivities, and among the acoustic modes LA and TA modes

has the highest thermal conductivities, respectively. Optical modes with very short

lifetimes and relatively lower specific heats, have a negligible contribution to the total

thermal conductivity of bilayer graphene.

With the same approach used for graphene, accumulated thermal conductivity of bi-

layer graphene with respect to phonon MFPs was obtained (shown in Fig. 3.13). As

highlighted in Fig. 3.13, 50% of heat in bilayer graphene is transported by phonons

with the MFPs of larger than 1905nm. Reduction of this parameter compared to sin-

gle layer graphene was expected since the average phonon lifetimes was decreased

in bilayer graphene and considering the fact that average phonon group velocities

remained almost the same.

37

Figure 3.12: Thermal conductivity with respect to frequency in bilayer graphene.

Figure 3.13: Accumulated thermal conductivity of bilayer graphene with respect to

phonon mean free path.

38

3.3 Graphite

Geometry optimization of graphite structure was done using the vc-relax (variable

cell) option of ABINIT. First, optimum lattice constants is found, and then equi-

librium position of atoms inside the structure is defined to minimize the total po-

tential energy in the crystalline structure. Optimized lattice constants was found as

a = 2.454 Å and c = 6.67 Å . These values are in very good agreement with previous

theoretical and experimental studies [33, 51, 52]. Crystal structure of graphite with

optimized interatomic and interlayer distances is shown in Fig. 3.14.

Figure 3.14: Crystal structure of graphite with optimized lattice parameters.

Phonon dispersion of graphite along the high symmetry lines is shown in Fig. 3.15.

Obtained phonon dispersion of graphite shows similar attributes to that of single and

bilayer graphene. This is due to the weak van der Waals bonding between layers

compared to strong covalent bonds between two in-plane atoms. Dispersion relations

of bilayer graphene and graphite barely differ from each other. There are slight dif-

ferences between the ZO′ mode of bilayer graphene and graphite. As atoms are freer

to move in out-of-plane in bilayer graphene than in graphite, their frequencies are

smaller. In addition, for the same reason ZO′ and ZA mode splits are more obvious

in graphite than in bilayer graphene.

39

Figure 3.15: Phonon dispersion relation of graphite.

Volumetric specific heat of graphite was obtained as 1.6495 × 106 J/m3K at roomtemperature. Using optimized lattice parameters and atomic mass of carbon, density

of graphite calculated as 2293 kg/m3. Converting volumetric specific heat using the

obtained density, mass-based specific heat of graphite was calculated as 719.4 J/kg.K.

Experimental specific heat of graphite was reported as 706.9 J/kg.K in [53]. Calcu-

lated total volumetric specific heat of bilayer graphene and graphite are almost same

and are almost twice as that of graphene. This is reasonable since the total specific

heat depends on the number and type of atoms in the primitive cell.

Scattering of specific heat of discrete phonons in graphite is shown in Fig. 3.16.

Specific heat of phonons decrease monotonically by the increase of their frequencies.

Therefore, acoustic modes which have lower frequencies have higher specific heats in

contrast to optical modes. Distribution of phonons group velocities and lifetimes with

respect to their frequencies are shown in Fig. 3.17 and Fig. 3.18, respectively. Acous-

tic modes have higher phonon group velocities and lifetimes, and optical modes have

group velocities comparable to those of acoustic modes but much shorter lifetimes.

40

Figure 3.16: Specific heat of phonons with respect to their frequencies in graphite.

Figure 3.17: Group velocity of phonons with respect to their frequencies in graphite.

Summary of average group velocity and lifetime of phonons is provided in Table

3.3. Average group velocities are similar to those of previous two structures as ex-

41

Figure 3.18: Phonon lifetimes with respect to their frequencies in graphite.

pected. Phonon lifetime of TA and LA modes are almost the same as those of bilayer

graphene. However, average phonon lifetime of ZA mode is less than that of the bi-

layer graphene suggesting the increase of out-of-plane interactions due to presence of

more layers in graphite.

Table 3.3: Average group velocity and lifetime of phonon modes in graphite


ZA 6198 54.8

TA 9911 20.49

LA 13162 13.02

ZO 4407 0.97

TO 1462 0.01

LO 5100 0.09

Equation 2.10 with the 1/3 factor and the fitting parameter used for graphene in Eq.

42

2.7, were used to obtain the thermal conductivity of graphite which is 1395 W/m.K.

The ratio of thermal conductivity of graphite to that of single layer graphene was

obtained as 0.69. This ratio was reported as 0.65 in [35]. It was also observed that

thermal conductivity of bilayer graphene is slightly higher than that of graphite. This

behavior was also reported in [35]. Contribution of ZA, TA, and LA modes to the

total thermal conductivity in graphite was calculated as 19.1%, 24.4%, and 56.1%,

respectively. Scattering of thermal conductivity of discrete phonons in graphite is

shown in Fig. 3.19.

Figure 3.19: Thermal conductivity with respect to frequency in graphite.

From the accumulated thermal conductivity of graphite shown in Fig. 3.20, it can be

concluded that at least 50% of thermal transport is carried by the phonons with MFPs

of larger than 1402nm. This parameter is smaller than that of bilayer graphene due to

increase of phonon interactions and decrease in phonon lifetimes.

43

Figure 3.20: Accumulated thermal conductivity of graphite with respect to phonon

mean free path.

44

CHAPTER 4

H-BN STRUCTURES

In this chapter results of single, bilayer, and bulk h-BN will be provided and differ-

ences of heat transport mechanism in these structures will be inspected.

Hexagonal boron nitride (h-BN) is a chemical compound composed of equal number

of boron and nitrogen atoms. Comparing the energy transport mechanism in h-BN

and graphite seems interesting since they share almost the same lattice constants and

the unit cell masses [54]. Even though, unlike graphite nature of bondings in h-BN is

not purely covalent and is partially ionic, layers in both structures are bonded together

by weaker van der Waals forces. Therefore, they are both ideal for so called van der

Waals heterostructures. Crystal structure of single and bilayer h-BN are shown in Fig.

4.1.

Figure 4.1: Crystal structure of (a) single layer h-BN (b) bilayer h-BN with AA stack-

ing.

In the following sections, thermal transport mechanism in single, bilayer, and bulk

h-BN based on the methodology provided in Chapter 2, will be inspected.

45

4.1 Single Layer h-BN

Calculating the minimum total potential energy of single layer h-BN, in-plane lattice

constant was obtained as a = 2.494 Å. Figure 4.2 shows the calculated phonon dis-

persion relation of single layer h-BN which is in very good agreement with previous

theoretical and experimental studies [55, 56, 57]. Similar to graphene, presence of

two atoms in primitive cell of single layer h-BN gives rise to 6 phonon branches in

the dispersion relation. Similarity of lattice constants, atomic masses, and bonding

types in h-BN and graphene, results in a similar phonon dispersion. However, slight

difference between atomic masses of boron and nitrogen atoms (which is not the case

in graphene), leads in to a small bandgap between acoustic and optical phonons. For

the same reason, degeneracy of ZA and ZO phonon modes at K point of the Brillouin

zone is lifted.

Figure 4.2: Phonon dispersion relation of single layer h-BN.

46

Using the calculated frequency of phonons, thermophysical properties were obtained.

Volumetric specific heat of single layer h-BN was calculated as 9.1058×105 J/m3K.Distribution of specific heat of discrete phonons in single layer h-BN with respect

to their frequencies are shown in Fig. 4.3. It can be deduced that specific heat of

phonons has an inverse relation with their frequencies. Acoustic modes with lower

frequencies have higher specific heats than optical modes with higher frequencies.

From the scattering of group velocity of discrete phonons shown in Fig. 4.4, it can

be concluded that LA and TA modes have the highest group velocities, respectively.

Group velocity of optical modes are comparable to that of acoustic modes. Finally,

from the scattering of phonon lifetimes shown in Fig. 4.5, it can be observed that ZA

and TA modes have the longest phonon lifetimes.

Figure 4.3: Specific heat of phonons with respect to their frequencies in single layer

h-BN.

47

Figure 4.4: Group velocity of phonons with respect to their frequencies in single layer

h-BN.

Figure 4.5: Phonon lifetimes with respect to their frequencies in single layer h-BN.

48

Average phonon group velocities and lifetimes of single layer h-BN are summarized

in Table 4.1. Based on the information provided in this table, it can concluded that

optical phonon modes are scattered before they can participate in thermal transport.

Among acoustic modes, ZA and TA modes have the highest lifetimes. Average group

velocity of ZA mode in single layer graphene is significantly higher than of that

of single layer h-BN. Average group velocity of ZA and LA mode in single layer

graphene found 51% and 15% higher than in single layer h-BN, respectively. Aver-

age phonon lifetimes in single layer graphene is notably higher than in single layer

h-BN. This is an indicator of stronger phonon interactions in single layer h-BN com-

pared to graphene. This behavior was also reported by Lindsay et al. [15, 58].

Table 4.1: Average group velocity and lifetime of phonon modes in single layer h-BN


ZA 4126 23.09

TA 9986 21.49

LA 11409 7.71

ZO 3166 0.55

TO 1451 0.01

LO 5977 0.11

To calculate the thermal conductivity of single layer h-BN, a fitting parameter used to

match the bulk thermal conductivity of bulk h-BN to the reported value in literature,

and using the same parameter, thermal conductivity of single layer was calculated

579 W/m.K. To calculate the thermal conductivity of single layer h-BN, 1/2 factor

was used in Eq. 2.10 instead of 1/3 due to the 2D nature of single layer h-BN. The ra-

tio of thermal conductivity of bulk h-BN to single layer was calculated as 0.67 which

agrees well with [58]. Scattering of thermal conductivity of phonons with respect to

frequency is shown in Fig. 4.6. Finally, contribution of ZA, TA, and LA modes to

the total thermal conductivity of single layer h-BN was obtained as 7.9%, 41.5%, and

50.3%, respectively.

49

Figure 4.6: Thermal conductivity with respect to frequency in single layer h-BN.

Using group velocity and lifetime of phonon in Eq. 2.11, MFP of phonons were

calculated. Normalized thermal conductivity of discrete phonons with respect to their

MFPs in single layer h-BN is shown in Fig. 4.7. It was observed that half of the

bulk thermal conductivity of single layer h-BN is related to the phonons with MFPs

of larger than 1274nm.

50

Figure 4.7: Accumulated thermal conductivity of single layer h-BN with respect to


4.2 Bilayer h-BN

Interlayer distance of bilayer h-BN was obtained as 3.392 Å considering the mini-

mum energy of the structure. Calculated in-plane lattice constant of bilayer h-BN

was found almost same as its single layer counterpart. Phonon dispersion relation of

bilayer h-BN is shown in Fig. 4.8. As there are four atoms in the primitive cell of

bilayer h-BN, there are 12 phonon branches in the dispersion relation. Due to weak

van der Waals forces between layers, most of the branches doubly degenerate with

exception of out-of-plane modes with the frequency of below 300 cm−1. Unlike bi-

layer graphene, degeneracy of ZO mode in bilayer h-BN is more obvious. This is

expected to be because of interaction of boron and nitrogen atoms in the neighboring

layers.

51

Figure 4.8: Phonon dispersion relation of bilayer h-BN.

Total volumetric specific heat of bilayer h-BN was calculated as 1.8115×106 J/m3Kusing Eq. 2.3. Behavior of specific heat, group velocity, and lifetime of discrete

phonons with respect to their frequencies are shown as of scattering plots in Fig. 4.9,

4.10, and 4.11, respectively. Similar behavior to previous structure was also observed

for the phonons specific heat in bilayer h-BN which is shown in Fig. 4.9. Phonons

specific heat decreases monotonically by the increase of their frequencies. From the

Fig. 4.10, it can be concluded that LA and TA modes have the highest group veloci-

ties in the distribution. Finally, in the distribution of phonon lifetimes shown in Fig.

4.10, acoustic modes have significantly longer lifetimes than optical modes.

52

Figure 4.9: Specific heat of phonons with respect to their frequencies in bilayer h-BN.

Figure 4.10: Group velocity of phonons with respect to their frequencies in bilayer

h-BN.

53

Figure 4.11: Phonon lifetimes with respect to their frequencies in bilayer h-BN.

Calculated average phonon group velocities and lifetimes are summarized in Table

4.2. Decrease in phonon lifetimes is an indicator of increased scattering events com-

pared to single layer h-BN. Similar to previous structures, ZA mode has the highest

phonon lifetime among acoustic modes, and optical modes are being scattered very

fast. Average group velocity of phonons in bilayer graphene do not vary much from

those of single layer h-BN.

Table 4.2: Average group velocity and lifetime of phonon modes in bilayer h-BN


ZA 4256 12.01

TA 9031 7.07

LA 11428 3.72

ZO 3145 0.28

TO 1106 0.01

LO 5190 0.04

54

With the same fitting parameter used to match the total thermal conductivity of bulk

h-BN, thermal conductivity of bilayer h-BN was calculated 401 W/m.K, which is

slightly higher than bulk h-BN. Similar behavior was also reported in [15]. Contri-

bution of ZA, TA, and LA modes found to be 7.9%, 40.3%, and 51.5%, respectively.

It can be concluded that LA and TA modes are the dominant heat carriers in bilayer

h-BN. This is mostly because of superior group velocity of LA and TA modes which

effects the thermal conductivity with the power of 2 in Eq. 2.10. Scattering of thermal

conductivity of phonons with respect to their frequencies is shown in Fig. 4.12.

Figure 4.12: Thermal conductivity with respect to frequency in bilayer h-BN.

Accumulated thermal conductivity with respect to MFP of bilayer h-BN is shown in

Fig. 4.13. It is shown that 50% of heat in bilayer h-BN is carried by phonons with

MFPs larger than 392nm. This is significantly lower than that of single layer h-BN.

The reason can be explained by the increase of phonon interactions caused by the

additional layer which decrease the phonon lifetime.

55

Figure 4.13: Accumulated thermal conductivity of bilayer h-BN with respect to


56

4.3 Bulk h-BN

Optimized lattice parameters and atomic positions of bulk h-BN was calculated us-

ing vc-relax option of ABINIT. Optimized lattice parameters of crystal structure of

graphite was found as a = 2.492 Å and c = 6.63 Å. Crystal structure of graphite with

optimized interatomic and interlayer distances are shown in Fig. 4.14. Optimized

lattice constants are in a very good agreement with previous theoretical and experi-

mental studies [59, 60, 61].

Figure 4.14: Crystal structure of bulk h-BN with optimized lattice parameters.

Obtained phonon dispersion relation of bulk h-BN shown in Fig. 4.15 closely fol-

lows the dispersion relation of single and bilayer h-BN. This is due to the fact that

van der Waals bonding between layers are much weaker than covalent bonds and

therefore less effective. For the same reason out-of-plane optical mode (ZO) in h-BN

structures have considerably less energy (frequency) than other optical (TO and LO)

modes. Due to additional layers in crystal structure of bulk h-BN compared to bilayer

h-BN, the ZO′ mode has higher energy in bulk h-BN, and also the split in ZO modes

occurs with a higher frequency difference. ZA and ZO′ modes degenerate betweenM

andK points, where the interlayer interactions is switched off in the edge of Brillouin

zone.

57

Figure 4.15: Phonon dispersion relation of bulk h-BN.

Using calculated frequency of phonons in Eq. 2.3, total volumetric specific heat of

bulk h-BN was obtained which is 1.8185 × 106 J/m3K at room temperature. Usingoptimized lattice parameters and atomic masses of boron and nitrogen, density of

bulk h-BN calculated as 2311 kg/m3, therefore specific heat of bulk h-BN can also

be stated as 786.9 J/kg.K. Total volumetric specific heat of bilayer and bulk h-BN

are almost same and are twice as of that single layer h-BN. This was expected since

there are half number of atoms in the primitive cell of single layer h-BN as bilayer

and bulk h-BN.

phonon mean free path in few layer, two-dimensional...

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