ABSTRACT

LI, XIAODONG. Modeling and Simulation of Electron Transport Properties of Novel TwoDimensional Electron Systems. (Under the direction of Prof. Ki Wook Kim.)

In the past decade, a lot of research work have been focusing on the two-dimensional materi-

als, such graphene, silicene, atomic thin layer transitional metal dichalcogenides (MoS2, MoSe2,

WS2, etc), and topological insulators (Bi2Se3, Bi2Te3, etc), due to their novel properties in

condense-matter physics, and potential applications in electronics and optics. In this work, the

electron transport properties of selected materials are investigated theoretically, together with

the analysis of the effects on the performance of semiconductor devices. Graphene and its deriva-

tive, bilayer graphene, are chosen to be our first two research objects. A full band Monte Carlo

simulation program is developed to investigate the electron transport properties, including var-

ious scattering mechanisms and the influence of the substrates. Secondly, the electron-phonon

interaction in monolayer silicene and MoS2 are studied from first principles as well as the intrin-

sic electron transport. Then, for three dimensional topological insulators, the exotic properties

of the spin-polarized surface states have drawn huge attentions among the scientists. In partic-

ular, for our work, the electron wave guiding phenomena on the surface within the proximate of

the magnetic layer are investigated with the proposal of several potential applications in logic

device and interconnect circuits.

In order to study the electron transport properties, the full band Monte Carlo simulation

tool is developed and utilized to solve the Boltzmann transport equation in semiclassical regime,

with the usage of the intrinsic electron-phonon scattering rates obtained from first principles

calculation. Effects of other scattering mechanisms, such as electron-electron scattering, surface

polar phonon scattering from substrate, and impurity scattering, are also included and carefully

studied. For monolayer graphene, substrate dependent Joule self-heating effect is studied. Two

important heat transfer mechanisms are included and compared for several technologically im-

portant substrate materials (SiC, hexagonal BN, SiO2 and CVD diamond), direct heat transfer

through interfacial thermal conducting and energy transferring to substrate by surface polar

phonon scattering. Considering the overall characteristics, BN appears to compare favorably

against the other substrate choices for graphene electronic applications. For bilayer graphene,

the influence of opened band gap on the transport characteristics is also investigated through

simulation. We found out that the opened band gap substantially degrades the mobility while

has negligible effect on the saturation velocities, for both with and without substrate cases.

In addition, the electron-phonon interaction and related transport properties are investi-

gated in monolayer silicene and MoS2 by using a density functional theory calculation combined

with a full-band Monte Carlo analysis. In the case of silicene, the results illustrate that the out-

of-plane acoustic phonon mode may play the dominant role unlike its close relative, graphene.

The small energy of this phonon mode, originating from the weak sp2 π bonding between Si

atoms, contributes to the high scattering rate and significant degradation in electron transport.

In MoS2, the longitudinal acoustic phonons show the strongest interaction with electrons. The

key factor in this material appears to be the Q valleys located between the Γ and K points

in the first Brillouin zone as they introduce additional intervalley scattering. The analysis also

reveals the potential impact of extrinsic screening by other carriers and/or adjacent materi-

als. Finally, the effective deformation potential constants are extracted for all relevant intrinsic

electron-phonon scattering processes in both materials.

Finally, the possibility of electron wave guiding is explored on the surface of a topological in-

sulator under proximity interaction with a magnetic material. The electronic band modification

induced by the exchange coupling at the interface defines the path of electron propagation that

is analogous to the optical fiber for photons. Simulation results indicate the guiding efficiency

much higher than that in a waveguide formed by an electrostatic potential barrier such as p-n

junctions. Further, it is found that beam steering and flux control can be achieved by altering

the magnetization direction of the magnetic layer. In particular, the feasibility of large-angle

turn and high on/off ratio is illustrated under realistic conditions. Potential implementation to

logic and interconnect applications is examined.

© Copyright 2013 by Xiaodong Li

All Rights Reserved

Modeling and Simulation of Electron TransportProperties of Novel Two Dimensional

Electron Systems

byXiaodong Li

A dissertation submitted to the Graduate Faculty ofNorth Carolina State University

in partial fulfillment of therequirements for the Degree of

Doctor of Philosophy

Electrical Engineering

Raleigh, North Carolina

2013

APPROVED BY:

Prof. David Aspnes Prof. Veena Misra

Prof. John F. Muth Prof. Ki Wook KimChair of Advisory Committee

DEDICATION

This dissertation is dedicated to my dear father Tingjie Li, and mother, Li Ma, who have

loved and supported me throughout my life;

to my beloved and beautiful wife, Huiying Lu, who has always been supporting me, inspiring

me, and bringing happiness to me.

ii

BIOGRAPHY

Xiaodong Li was born on March 23, 1986, in the city of Zhengzhou, in Henan Province, China.

He graduated with the B.S. degree from Institute of Microelectronics, Peking University in 2008.

In the August of the same year, he enrolled in the Ph.D. program in the Department of Electrical

and Computer Engineering at North Carolina State University. During his graduate studies,

he worked as a research assistant under the direction of Professor Ki Wook Kim in Nanoscale

Quantum Engineering group, as well as a teaching assistant for the first two years. He conducted

the scientific research focusing on the nanoelectronics, optoelectronics, semiconductor physics

and device. After graduation, he will work as a senior R&D engineer for Synopsys at Austin.

iii

ACKNOWLEDGEMENTS

First of all, I would like to express my deepest gratitude to my advisor Professor Kim. Without

his direction and education, this Ph.D. can not be completed. He has set a great example as

an excellent academic scientist in many aspects, such as very broad and in-depth knowledge,

academic integrity and judgement. He has given me a lot of very insightful opinions and advice

about the research, the writing, and career development. I would be feeling very lucky and

grateful to have such a great advisor for my whole life.

I am greatly honoured to have Professor Aspnes, Professor Misra, and Professor Muth in

my committee.

I would like to extend my appreciation to my colleagues, Dr Barry, Dr Semenov, Dr Kong, Dr

Borysenko, Rui Mao, Xiaopeng Duan and Zhenghe Jin. They are very kind and smart people.

They have been very good friends and excellent co-workers during my graduate study. I will

keep those wonderful times in my mind as well as our friendship.

At last, I would like to express my sincerest thanks to my family, including my parents, my

wife, and my sisters, for supporting and encouraging me all the time throughout my life.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Electrical Transport in Monolayer Graphene . . . . . . . . . . . . . . 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Electron-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Surface polar phonon dominated transport . . . . . . . . . . . . . . . . . . . . . . 132.4 Joule self heating effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Chapter 3 Electron Transport Properties of Bilayer Graphene . . . . . . . . . . 363.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Relevant Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Electron Transport in BLG vs. MLG . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Transport in BLG with interlayer bias . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 4 Intrinsic Electrical Transport Properties of Monolayer Silicene andMoS2 from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.1 Monolayer silicene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.2 Monolayer MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Deformation Potential Model . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 5 Controlling Electron Wave Propagation on a Topological InsulatorSurface via Proximity Interactions . . . . . . . . . . . . . . . . . . . . . 81

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Principles and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.1 FDTD simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3.2 NEGF simulation for blocking operation . . . . . . . . . . . . . . . . . . . 875.3.3 Potential Applications in Logic and Interconnect circuits . . . . . . . . . . 88

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 6 Summary and Future Research . . . . . . . . . . . . . . . . . . . . . . . 95

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

v

LIST OF TABLES

Table 4.1 Phonon energies (in units of meV) at the symmetry points for monolayersilicene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Table 4.2 Phonon energies (in units of meV) for TA, LA, TO(E′), LO(E′), and A1 (orhomopolar) modes at the Γ, K, M and Q points in the FBZ of monolayerMoS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Table 4.3 Extracted deformation potential constants for electron-phonon interactionin silicene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Table 4.4 Extracted deformation potential constants for electron-phonon interactionin MoS2 for electrons in the K valley [see also Fig. 4.6(f)]. . . . . . . . . . 69

Table 4.5 Extracted deformation potential constants for electron-phonon interactionin MoS2 for electrons in the Q1 valley [see also Fig. 4.7(f)]. Multiple equiv-alent valleys for the final state specify the degeneracy factor gd larger thanone in Eq. (4.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

vi

LIST OF FIGURES

Figure 2.1 Schematic representation of the initial (k1,k2) and final (k′1,k

′2) electron

pair states which satisfy energy and momentum conservation in the linearband structure of graphene. β is defined as the angle between the longaxis and the line OC and β′ between the long axis and the line OC′. . . . 24

Figure 2.2 Electron-electron scattering rate for electron concentrations n = 1 ×1012 cm−2 (circle), n = 5×1012 cm−2 (square), and n = 1×1013 cm−2(triangle).The electron-phonon scattering rate is also plotted for comparison (dashedline), [1] where the details at low energies are not discernible due to thelinear scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 2.3 Drift velocity for n = 1 × 1012 cm−2, n = 5 × 1012 cm−2, and n =1× 1013 cm−2 with (dashed line) and without (solid line) e-e scattering. . 26

Figure 2.4 Energy distribution function for n = 1×1012 cm−2 and n = 1×1013 cm−2

with (dashed line) and without (solid line) e-e scattering. . . . . . . . . . 27Figure 2.5 Calculated Ediff/Etotal (solid line) and vx/vf (dashed line) as a function

of angle β′ assuming that the long axis (x) of the ellipse is along thedirection of the electric field. Etotal = E(k′

1) +E(k′2) and vf is the Fermi

velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.6 Surface polar phonon scattering rate of graphene electron on SiC, SiO2,

and HfO2, for the electron density n of 1 × 1012 cm−2 at 300 K. Alsoplotted is the intrinsic graphene optical phonon scattering rate at 300 K(”intrinsic”) obtained from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . 29

Figure 2.7 Electron drift velocity in graphene on SiC, SiO2, and HfO2. The intrinsiccase without substrate is also shown. The electron density is 1×1012 cm−2

at 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.8 Electron distribution function with different substrate conditions: intrin-

sic (dash-dotted), SiC (solid), SiO2 (dashed), and HfO2 (dotted), for theky = 0 cross-section at 20 kV/cm. The box-like function correspondsto the Fermi-Dirac distribution displaced by the SiO2 SPP energy (60meV) in a simple, metal-like approximation (i.e., EF ≫ kBT ) with n =1× 1012 cm−2. For comparison, the equilibrium Fermi-Dirac distributionat 300 K is also plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 2.9 Electrical resistivity vs. temperature with different substrate conditions:intrinsic (circle), SiC (square), SiO2 (triangle), and HfO2 (diamond), withn = 1× 1012 cm−2. The slope of the straight lines are used to extract theacoustic deformation potential. . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.10 Drift velocities vs. electric field for graphene on different substrates. Thegraphene sample is assumed to be 1 µm × 0.5 µm with a carrier densityof 1×1012 cm−2. The impurity density is 5× 1011 cm−2. As shown, Jouleheating leads to a significant velocity degradation for use of SiO2 substrate. 33

vii

Figure 2.11 (a) Graphene lattice temperature Tg and (b) temperature difference Tg −Ts between the graphene lattice and the top surface of the substrate asa function of driving electric field. The conditions are the same as inFigure 2.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 2.12 Saturation velocity vsat vs. electron density in graphene on different sub-strates. In the calculations, it is assumed that the graphene film of 1 µm× 1 µm is under an electric field of 30 kV/cm. The impurity density is5× 1011 cm−2. The experimental data for the case of SiO2 substrate arefrom Refs. [2] and [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.1 Intrinsic electron scattering rates in MLG and BLG calculated from thefirst principles for the dominant (a) acoustic and (b) optical phonons. [1, 4]The sudden increases shown in (b) are due to the onset of optical phononemission. They correspond to the phonon frequencies at the points of highsymmetry in the first Brillouin zone: ωΓ ≈ 200 meV and ωK ≈ 160 meV. . 45

Figure 3.2 Electron-hole propagator Π(q) normalized to N0 (= 2m∗/π~2) in MLGand BLG with the graphene electron density of 5× 1011 cm−2. For BLG,the calculation considers different interlayer biases with the induced en-ergy gap Eg of 0 eV (no bias), 0.10 eV, 0.18 eV, and 0.24 eV, respectively. 46

Figure 3.3 Electron drift velocity versus electric field in (a) MLG and (b) BLG,with different substrate conditions: intrinsic/no substrate (circle), SiO2

(triangle), SiO2 with impurities (reverse triangle), h-BN (square), and h-BN with impurities (diamond). The electron density is 5× 1011 cm−2 at300 K. The impurities on the surface of the substrate (d=0.4 nm) havethe density 5× 1011 cm−2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 3.4 Cross sectional view (ky=0) of electron distribution functions in (a) MLGand (b) BLG at 20 kV/cm, with different substrate conditions: intrin-sic/no substrate (short dashed line), SiO2 (long dashed line), SiO2 withimpurities (solid line), h-BN (dashed-dotted line), and h-BN with impu-rities (dotted line). The conditions are the same as specified in Fig. 3.3. . 48

Figure 3.5 Density of states in BLG for different interlayer biases with the inducedenergy gap Eg of 0 eV (solid line), 0.10 eV (dashed line), 0.18 eV (dottedline), and 0.24 eV (dashed-dotted line), respectively. The inset shows thecorresponding band structures. . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 3.6 Electron mobility versus bias-induced bandgap in BLG at 300 K, withdifferent substrate conditions. The electron density is 5× 1011 cm−2. Theimpurities on the surface of the substrate (d=0.4 nm) have the density5× 1011 cm−2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 3.7 Electron drift velocity versus electric field in BLG with a bias-inducedgap of Eg of (a) 0.1 eV, (b) 0.18 eV, and (c) 0.24 eV, under differentsubstrate conditions: intrinsic/no substrate (circle), SiO2 (triangle), SiO2

with impurities (reverse triangle), h-BN (square), and h-BN with impu-rities (diamond). The conditions are the same as specified in Fig. 3.6. . . 51

viii

Figure 4.1 Electronic and phononic band structures of monolayer silicene along thesymmetry directions in the FBZ. The Dirac point serves as the referenceof energy scale for electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Figure 4.2 (Color online) Electron-phonon interaction matrix elements |gvk+q,k| (inunits of eV) from the DFPT calculation in silicene for k at the conduction-band minimum K point [i.e., (4π/3a, 0)] as a function of phonon wavevector q for all six modes v. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 4.3 (Color online) Electron scattering rates in silicene via (a) emission and(b) absorption of phonons calculated at room temperature. The electronwave vector k is assumed to be along the K -Γ axis. . . . . . . . . . . . . . 73

Figure 4.4 (Color online) Drift velocity versus electric field in monolayer siliceneobtained from a Monte Carlo simulation at different temperatures: 50K (square), 100 K (triangle), 200 K (diamond), and 300 K (circle). Theresults in (a) consider the scattering by ZA phonons, while those in (b)do not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 4.5 Electronic and phononic band structures of monolayer MoS2 along thesymmetry directions in the FBZ. The conduction-band minimum at theK point serves as the reference of energy scale for electrons. . . . . . . . . 75

Figure 4.6 (Color online) (a)-(e) Electron-phonon interaction matrix elements |gvk+q,k|(in units of eV) from the DFPT calculation in MoS2 for k at the conduction-band minimum K point [i.e., (4π/3a, 0)] as a function of phonon wavevector q. Only the branches with significant contribution are plotted; i.e.,TA, LA, TO(E ′), LO(E ′), and A1 (or homopolar) modes. (f) Schematicillustration of intervalley scattering for electrons in the K valley. . . . . . 76

Figure 4.7 (Color online) (a)-(e) Electron-phonon interaction matrix elements |gvk+q,k|(in units of eV) from the DFPT calculation in MoS2 for k at the Q point[i.e., Q1 ≈ (2π/3a, 0)] as a function of phonon wave vector q. Only thebranches with significant contribution are plotted; i.e., TA, LA, TO(E ′),LO(E ′), and A1 (or homopolar) modes. (f) Schematic illustration of in-tervalley scattering for electrons in the Q valleys. . . . . . . . . . . . . . . 77

Figure 4.8 (Color online) Scattering rates of K -valley electrons in MoS2 via (a) emis-sion and (b) absorption of phonons calculated at room temperature. Theelectron wave vector k is assumed to be along the K -Γ axis. . . . . . . . . 78

Figure 4.9 (Color online) Scattering rates of Q-valley electrons in MoS2 via (a) emis-sion and (b) absorption of phonons calculated at room temperature. TheQ-K separation energy EQK (= 70 meV) denotes the onset of curves asthe K -valley minimum serves as the reference (zero) of energy scale. Theelectron wave vector k is assumed to be along the Q-Γ axis. . . . . . . . . 79

Figure 4.10 (Color online) Drift velocity versus electric field in monolayer MoS2 ob-tained from a Monte Carlo simulation at different temperatures withEQK = 70 meV. When electron transfer to the Q valleys is not considered,the mobility increases to approx. 320 cm2/Vs at 300 K. . . . . . . . . . . 80

ix

Figure 5.1 Principles of electron guiding in TI. The electron wave can propagatealong the direction of Dirac cone shifting, while be blocked along theperpendicular direction, as indicated by the green arrows. . . . . . . . . . 90

Figure 5.2 Device schematic for waveguiding: (a) blocking the electron wave; (b)steering the electron wave by 90. . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 5.3 The electron wave density from FDTD simulation for electron guided by:(a) gate voltage; (b) proximity effect with FMI. The electron wave densityis normalized by the maximum value. The width of the waveguide is 100nm. 92

Figure 5.4 The electron wave density from FDTD simulation for: (a) blocking theelectron wave; (b) steering the electron wave by 90.The electron wavedensity is normalized by the maximum value. The width of the waveguideis 100nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 5.5 NEGF simulation results for the blocking operation. (a) Calculated Trans-mission as a function of electron momentum kx and energy E for in-planeoff state. (b) Calculated Transmission as a function of electron momen-tum kx and energy E for out-of-plane off state. In (a) and (b), the dashedwhite lines depicts the Dirac cone in the waveguide, while the solid greenlines depicts the Dirac cone in the control unit.(c) The on/off ratio ofthe conductance versus Fermi level at 0 K. (c) The on/off ratio of theconductance versus Fermi level at 300 K. . . . . . . . . . . . . . . . . . . 94

x

Chapter 1

Introduction

The searching for new materials used in novel semiconductor devices and integrated circuits has

been driven by the increasing demand of computing power. Lots of material properties have to

be carefully examined for particular applications, such as size of band gap, mobility, saturation

velocity, scalability, surface and edge properties and so on. Particularly, two dimensional mate-

rials, such as graphene, [5, 6, 7] silicene, [8, 9, 10, 11] transitional metal dichalcogenides, [12, 13]

and topological insulators, [14, 15, 16, 17] have attracted huge interests recently, due to their

intriguing electrical and optical properties, perfect candidate to explore fundamental physics,

and potentials in novel applications.

Since the experimental realization of graphene in 2004, [5] its extraordinary electric prop-

erties have led to near exponentially growing attention in possible applications. [6, 7, 18, 19,

20, 21, 22] In particular, the linear band structure in the vicinity of the Dirac points, and

resultant massless fermions, have excited interest in both its applicability for electronic and

optoelectronic devices and as a testbed for quantum electrodynamics phenomena. [23, 24, 25]

Lots of Graphene’s interesting properties originate from the chirality of the linear dispersion of

electrons. Electrons with low energy in graphene are also called Dirac fermions since the Hamil-

tonain can be well described by the Dirac equation, H = ~vFΩσ · k, where vF ≈ 1× 108 cm/s

is the Fermi velocity, k is the electron momentum and σ is the 2D Pauli matrix. The eigenvalue

1

has the linear dispersion, E = ~vF |k|, while more interestingly, the eigenfunctions have two

components corresponding to the two sublattices, which give the quasiparticle the Berry phase

of π. [26, 27] This two-component part of the wavefunction is usually referred as pseudospin.

Due to the conservation of this quantity, graphene demonstrates many exotic phenomena, such

as chiral quantum Hall effect, [28, 29, 30] chiral tunneling and klein paradox, [20, 31, 32], weak

localization, [33, 34, 35] and universal conductance fluctuations. [36, 37]

In addition to the tremendous research on the fundamental physics of graphene, the most

attracting ability of graphene is its amazing transport properties, which not only endow the

value of research but also have the practical significance for electronic and optoelectronic semi-

conductor devices. Graphene has shown the ultra-high mobility and large saturation velocity,

which allows it to bare six time larger current than Cu. Experimentally, the measured mobil-

ity of graphene have the range from 103 ∼ 106 cm2/Vs, [38, 39, 40, 41, 42, 43, 44] while the

mobility for Silicon is 1350 cm2/Vs, for GaAs is 4600 cm2/Vs, and for 2DEG in AlGaN/GaN

is 1500 ∼ 2000 cm2/Vs. [22] At the same time, the carrier saturation velocity in graphene is

over 4×107 cm/s, much higher than that in conventional semiconductor: Silicon ∼1×107 cm/s,

GaAs ∼2×107 cm/s and InP ∼0.5×107 cm/s. [23] Higher carrier saturation velocity provides

the capability for the larger saturation current. Thus, in integrated circuits, the transistor can

have smaller width while maintaining the fan-out ability. These properties of graphene are de-

termined by the various scattering mechanisms from its own phonons and extrinsic scattering

sources. The mobility is strongly dependent on the substrates, due to the charged impurity scat-

tering, surface polar phonon scattering, and other defect scattering, such as vacancies, ripples

and cracks.

Taking advantage of the transport properties of graphene, there have been a lot of effort

working on the specific device configurations. Due to the lack of bandgap, graphene field effect

transistor can not provide sufficient current on/off ratio. According to ITRS, the current on/off

ratio are necessary to be at least 104 for the logic device. However, graphene FET typically shows

a on/off ratio less than 10. [24, 25, 45] This current switching is obtained through controlling

2

the density of states and scattering time around Fermi level. Despite of the usability in logic

level, graphene has demonstrate remarkable performance in high speed analog devices. [46, 45,

47, 48, 49, 50] The graphene transistors showed good scalability of the gate length, and the

potential performance with higher cut off frequency. But one of the drawbacks is the weak

current saturation of the transistor, which is caused by the zero bandgap and conducting of

the two type of carriers. [47, 46, 48] The unsaturated current is not favored in RF applications,

since it may result to the very small, or even negative, transconductance at high drain-source

voltage.

Compared to the monolayer graphene, bilayer graphene, consisting of two layers of carbon

films with each having two sublattices, has one apparent advantage that the bandgap can be

induced by breaking the inversion symmetry between two layers. [30, 51, 52, 53] One common

way is to apply perpendicular electric field, which can achieve a finite band gap of hundreds

of meV and form a band structure with a shape of ’Mexican hat’. [51, 54] Another practical

route is to dope top layer graphene with adsorption atoms, e.g. potassium, in order to induce

the internal electric field to break the symmetry. [55] In particular, a current on/off ratio of 100

can be obtained at room temperature, giving the implication that bilayer graphene may be the

better choice for logic device. [56]

Besides the extensively study and investigations on the electrical properties in graphene,

graphene also shows the great potential in photonics and optoelectronics. [57, 24, 22] For a mono-

layer graphene, due to the linear dispersion of electrons, the optical absorbance is a constant

for a wide frequency range from ultraviolet to terahertz, πα ≈ 2.3%, where α = e2/(4πε0~c) is

called fine structure constant. [58, 59] The transmittance is T = 1 − πα ≈ 97.7%, which also

means the very low reflectance. Together with the high mobility, flexibility and environmental

stability , graphene is a very promising choice for transparency electrode for optoelectronic de-

vices. [60, 61, 62, 63] Since photons interact with the two dimensional electron gas in graphene,

the absorbance for a few layer of graphene is almost proportional to the number of layers.

Graphene is also a good candidate for photodetector active material. For a graphene photode-

3

tector, the bandwidth is largely improved compared to those materials with bandgaps, since

photons with energy lower than the bandgap can penetrate the materials without being ab-

sorbed. The ultra high mobility is also an advantage for ultra fast photodetector based on

graphene. [64, 65, 66] For example, Ref. [66] reported a maximum external photoresponsivity of

6.1 mA·W−1 can be achieved at a wavelength of 1.55 µm, and possible operational wavelength

from 300nm to 6 µm could be realized in future. Moreover, for on-chip optical communications,

graphene can be used as the ultrafast optical modulator due to its tunability of the Fermi level,

compactness of footprint, and the broadband of absorbance. [67]

The ideal physical properties of graphene and its derivatives make them the very promising

candidates for electrical and optoelectronics devices. Its extreme high mobility gives ultrafast

speed to the devices, the nature of chirality of Dirac fermions brings in the possibilities for

new type of devices, such as BISFET and pseudospin valve, [68, 69] and the zero bandgap and

strong electron-photon interaction can be utilized in optical devices enabling the response for

a wide range of frequency. Recently, lots of effort have been spent on producing large area and

high quality graphene with low cost, [61, 70, 71] which will speed up the commercialization of

graphene based applications in future.

Besides graphene, the attention to other low-dimensional materials has expanded widely. [72,

12, 13, 73] In particular, silicene [8, 9, 10, 11], on type of group IV materials with atomic thick-

ness, and molybdenum disulfide [74, 75, 76, 77], one type of transitional metal dichalcogenides

(TMDs), have gained much interest due to their unique properties in electronics, optoelectron-

ics, and magnetics. Silicene is expected to share certain superior properties of graphene due

to its structural similarity and the close position in the periodic table. More importantly, it is

compatible with the current silicon-based technology and can be grown on a number of different

substrates. [78, 79, 80, 81, 82]

Silicene forms hexagonal structure through sp2 hybridization possessing π bands, similar

with graphene. However, due to the weak bonding compared with graphene, the planar structure

of silicene is not stable, as well as the high buckled structure which is confirmed by examining

4

the phonon energy through first principles calculation. [10] Only low-buckled structure for free

standing silicene can exist and show positive phonon frequency in the first principles calculation.

The electron dispersion shows Dirac cone formed by the π and π∗ bands, with the Fermi velocity

about half of that in graphene. When putting silicene on subtrates, the Si layer is reconstructed

to a more complicated triangular structures. Correspondingly, in presence of the substrate,

the band structure shows a band gap, which is verified by both ARPES experiment and DFT

calculations. [11, 80] The opening of the band gap resembles the case with graphene on substrate.

For instance, large band gap about ∼0.74 eV is observed for graphene on Ir with Na adsorption

as well as band gop of 0.26 eV for graphene on SiC substrate. [83, 84] For silicene on substrate,

two factors would induce the opening gap, the interaction between silicene and the substrate,

and the atomistic bulking by itself. The DFT calculation in Ref. [80] confirmed that the band

gap is induced primarily due to the latter factor. This discover also indicates the potential band

structure engineering for epitaxial silicene through strain engineering, such as self-doping and

semimetal-metal transition. [85, 86, 87, 88]

Also, the silicene nanoribbon shows very interesting electronic and magnetic properties,

strongly depending on the width and orientation. For example, the armchair nanoribbons have

oscillatory band gaps with decreasing width. Zigzag nanoribbons exhibit both metallic ferro-

magnetic and antiferromagnetic states, whose energy can be different if the edge is passivated

by hydrogen. . [10] The magnetoresistance for zigzag silicene nanoribbons is estimated to reach

the order of 106%. [89] The mobility based on the deformation potential theory for silicene

nanoribbon is preditected to oscillate with the width, which is on the same order of magni-

tude of that in graphene nanoribbon. [85] Additionally, the silicene nanoribbon is predicted to

be a very promising thermoelectric materials. Recent theoretical study using nonequilibrium

Green’s function method and nonequilibrium molecular dynamics simulations predicts that the

ZT value for silicene nanoribbon can achieve as high as 4.9 when the doping level is optimized,

which is very promising in thermoelectric devices. [90]

On the other hand, atomically thin MoS2 is a semiconductor with a finite band gap that

5

ranges from approximately 1.3 to 1.9 eV depending on the thickness. [74] It has been used as

the channel material in field effect transistors with promising results, such as channel mobility

measured from 200 up to 1000 cm2/Vs, room temperature current on/off ratio up to 108, and

subthreshold swing 74 mV/dec. [75, 91] In addition, monolayer MoS2 offers the possibilities

of interesting spin and valley physics utilizing the strong spin-orbit coupling. [92, 93, 94, 95]

Due to the combination of inversion symmetry breaking and spin-orbital coupling, electrons in

the valence band edge can have very long relaxation time for both spin and valley index, since

the flipping violates the symmetry. When shining light to monolayer MoS2, only the certain

interband transitions can happen depending on the polarization and frequency of the light

source. Through this way, Ref. [95] demonstrates a complete valley polarization in monolayer

MoS2, with the retention time longer than 1 ns.

In addition to atomic thin film TMDs and group IV materials, three-dimensional topological

insulators, such as Bi2Se3, Bi2Te3, and Sb2Te3, have also become one of the most popular topics.

It is a fascinating new state of quantum matter, with metallic states on its surface but insulating

in its bulk. [14, 15, 16, 17] The surface electrons can be described by the Dirac equation, as in

graphene, leading to a linear dispersion and zero band gap that is protected by the time-reversal

symmetry. [96, 97] Instead of the pseudospin in graphene, the surface states of a TI are spin-

polarized, [98, 99] with the spin direction locked perpendicular to the momentum, suppressing

the electron backscattering. [100, 101] Since these peculiar properties of are protected by the

time-reversal symmetry, breaking this symmetry can modify the surface states and change

the band structure, leading to many interesting quantum phenomena and applications in both

spintronics and quantum computing. [102, 103, 104, 105] For example, the proximity effect of

TI induced by gated ferromagnetic insulator (FMI) has been studied from the point of electric

transport. [106, 107, 108, 109] In particular Ref. [109] proposed a TI switching device showing

a giant magnetoresistance as large as 800% at room temperature with the proximate exchange

energy of 40 meV. Also, the photogalvanic effects on the surface of TI enables the potential

applications in spintronics and optoelectronics.

6

Moreover, the photogalvanic effect in topological insulator has also been investigated, which

is strongly dependent on the polarization of the light due to the spin polarization of the electron

states on the surface. [110, 111, 112] Shining circular polarized electromagnetic (EM) wave to

the surface of TI can induce a DC photocurrent whose direction can be flipped by reversing the

helicity of the EM wave. [113] Due to the similar linear dispersion with graphene, topological

insulator is also a promising candidate for high performance optoelectronics applications, such

as transparent flexible electrodes, [114] terahertz generation and detection. [115, 116]

7

Chapter 2

Electrical Transport in Monolayer

Graphene

2.1 Introduction

Accurately modeling and calculating the electrical properties of graphene, such as mobility and

saturation velocity, are essential for utilizing this material in practical usage. The mobility and

saturation velocity are limited by various conditions, including different source of scattering,

temperature, Joule self heating, substrate and so on. In this chapter, we adopt Monte Carlo

simulation method to study the electrical transport properties in monolayer graphene. Several

scattering source are included in the simulation. Their influence on the transport properties are

carefully studied and discussed. As an important issue in modern integrated circuits, Joule self

heating effect is very crucial to the electron transport, and strongly depending on the choice of

substrate. In section 2.4, we will discuss this effect particularly.

2.2 Electron-electron scattering

An interesting consequence of the linear energy dispersion is that the ensemble average velocity

is not necessarily conserved upon an electron-electron (e-e) scattering event. Accordingly, inter-

8

carrier collisions deserve careful consideration in the determination of the transport properties.

Das Sarma et al. [117, 118] found the inelastic e-e scattering rate and mean free path in graphene

through the analysis of the quasiparticle self-energies. The scattering rate, calculated for electron

densities from 1 ∼ 10×1012 cm−2, was found to be of the same order of magnitude for electron-

phonon scattering rates evaluated in the deformation potential approximation. [119] Several

authors have considered the electronic transport properties of graphene based on approaches

such as the Monte Carlo simulation (see, for example, Ref. [120]); however, the effects of e-e

scattering have yet to be addressed. On the other hand, those utilizing the perturbative Green’s

function approach accounted for the many-body effect of e-e interactions in graphene but only

at very low (zero) temperatures and without electron-phonon relaxation. [121] In the present

analysis, we examine the influence of this (e-e) interaction mechanism in intrinsic monolayer

graphene at room temperature. A full-band ensemble Monte Carlo method is used for accurate

analysis of the distribution function as well as its macroscopic manifestations, particularly, the

electron low-field mobilities and drift velocities.

In both bulk and two-dimensional conventional semiconducting systems, e-e scattering has

been well studied as is documented in the literature. [122, 123, 124] During an e-e scattering

event, both the energy and momentum are conserved. In a parabolic band structure, common

in conventional semiconductors, momentum conservation directly leads to the conservation of

velocity. This can be readily shown by multiplying the momentum conservation equation by

~/m, which gives v1 + v2 = v′1 + v′

2. It is then clear that, in a material with a parabolic

band structure, e-e scattering has no direct effect on the drift velocity (which is an ensemble

averaged quantity). The situation becomes different in graphene, where the energy dispersion

in the region of the inequivalent Dirac points is linear, therefore removing the constraint that

momentum conservation leads directly to velocity conservation.

The problem of e-e scattering in nonparabolic bands was previously treated by Bonno et

al. [125] Here we use a similar approach, making it specific to the linear band structure of

graphene. The transition probability for two electrons, with wavevectors k1 and k2, is given by

9

Fermi’s golden rule

s(k1,k2;k′1,k

′2) =

2π

~|M |2δ[E(k′

1) + E(k′2)− E(k1)− E(k2)] , (2.1)

where E(k) is the electron energy dispersion relation and k′1 and k′

2 are the wavevectors of

the final states after scattering. The matrix element of interaction |M |, when accounting for

exchange scattering (i.e., the indistinguishability of electron pairs with like spin after collision),

is given by [126]

|M |2 = 1

2[|V (q)|2 + |V (q′)|2 − V (q)V (q′)] , (2.2)

where the Coulomb scattering matrix V (q) between two electrons (k1 → k′1; k2 → k′

2) is

V (q) =2πe2

ϵ(q)qA

1 + cos(θk1k′1)

2

1 + cos(θk2k′2)

2(2.3)

with q = |k1 − k′1|. In the expression, ϵ(q) is the static dielectric function, θkk′ is the angle

between the wavevectors k and k′, and A is the normalization area. The corresponding element

V (q′) (k1 → k′2; k2 → k′

1) can be expressed likewise with q′ = |k1 − k′2|. The dielectric

function is considered in the random-phase approximation, which is valid for densities n ≥

1× 1012 cm−2. [118] A detailed expression of ϵ(q) used in the study can be found in Eqs. (20)-

(22) of Ref. [118]. The scattering rate, including the effect of degeneracy, is then obtained as

the double sum over all the final states k′1 and the partner electrons k2 (with k′

2 determined

by momentum conservation)

1

τee(k1)=

∑k2

∑k′1

s(k1,k2;k′1,k

′2)f(k2)[1− f(k′

1)][1− f(k′2)] , (2.4)

where the f ’s are the occupation probabilities.

In graphene, the energy dispersion of the conduction band is linear in the vicinity of the

Dirac points; i.e., E = ~vf |k|, with the Fermi velocity vf (≃ 108 cm/s). Therefore, the electrons

behave as massless particles and can not be analyzed by the center of mass concept. [127]

10

However, by considering the equations of momentum and energy conservation, it can be shown

that all the possible final states after scattering lie on an ellipse defined by k1+k2 = k′1+k′

2 and

|k1|+ |k2| = |k′1|+ |k′

2|. As shown in Figure 2.1, the initial point (i.e., the coordinate centrum

at the K or K ′ point in the Brillouin zone) and the terminal point of the vector k1+k2 specify

the two foci of the ellipse (i.e., F1 and F2), with the length of long axis of the ellipse determined

by |k1|+ |k2|.

Our Monte Carlo model applied in the simulation includes the full electronic band structure

as calculated in the tight binding approximation. The electron-phonon scattering rates and

phonon dispersion relations, previously obtained from a first principles approach based on

density functional perturbation theory, are included for all six branches. [1] All electron-phonon

transition possibilities, including intravalley and intervalley, are accounted for as well as the

e-e scattering. Finally, since our simulations include large electron densities we account for

degeneracy, where Pauli exclusion is non-negligible, by implementation of the rejection technique

in the selection of the final state after scattering. [128] The non-equilibrium electron distribution

function is obtained self-consistently through simulation.

Figure 2.2 shows the calculated e-e scattering rate at 300 K, assuming a Fermi-Dirac distri-

bution of initial states and that all final states are available. [129] The scattering rate is found

to have a maximum of approximately 40−80 ps−1, for the energy range and electron densities

considered, which is significantly larger than the total electron-phonon scattering rate (plotted

in the dashed line). [1] The effect of dominant e-e scattering on the drift velocity is examined

in Figure 2.3. As shown, a couple of points are evident from the results. Namely, pair-wise

collisions can considerably reduce the electron drift velocity in graphene and the impact is the

most conspicuous at the lowest carrier density under consideration. For instance, a degradation

larger than a factor of 2 is estimated for low-field mobility along with an approximately 15 %

drop in the saturation drift velocity at n = 1×1012 cm−2. The relatively larger effect in the low

field regime indicates the dominance of e-e scattering over the interaction with phonons when

the electron energy remains small. [117, 118] Partial mitigation of its influence at higher electric

11

fields coincides with the emergence/prevalence of optical phonon scattering via emission. This

suggests that the presence of other competing interaction mechanisms (i.e., substrate dependent

surface polar phonons, remote impurities, etc.) may further wash out the e-e induced effect.

To examine the origin of velocity degradation, the electron energy distribution is analyzed

in Figure 2.4 at E = 1 kV/cm. Just as in conventional semiconductors, e-e scattering results

in an extension of the thermal tail of the distribution function, while simultaneously shifting

its peak backward (e.g., see the curves with n = 1 × 1012 cm−2). This large split in the final

state energies E(k′1) and E(k′

2) tends to decrease the average drift velocity in graphene with a

linear dispersion. As indicated in Figure 2.5, selection of large Ediff (= |E(k′1)−E(k′

2)|) for the

final electron pair clearly coincides with a small ensemble velocity vx (= v′1x + v′2x) along the

direction of the field (i.e., the x axis) after scattering. At larger densities, the lowest energy states

are already mostly occupied, thereby further preventing transition to the energy states (i.e.,

large Ediff) which most greatly reduce the drift velocity. In other words, the degeneracy factor in

Eq. 2.4 (which was not accounted for in Figure 2.2) significantly reduces these scattering events.

Accordingly, the e-e induced effect is largely suppressed for n = 1× 1013 cm−2 (see Figures 2.3

and 2.4). Note that the e-e scattering should also approach to zero as the electron density

becomes negligible. This means that the predicted degradation may be the most significant at

moderate densities (say, ≈ 1011 − 1012 cm−2). However, n < 1012 cm−2 is beyond the range

of this investigation owing to the limitation imposed by the random-phase approximation. A

more sophisticated model for screening is needed to further quantify the impact.

Our results clearly indicate that due to e-e scattering, there is an electron density depen-

dent degradation of the low-field mobility and saturation drift velocity in intrinsic monolayer

graphene. This effect is most evident at moderate electron densities, while being suppressed

at higher densities due to the unavailability of low energy states. For moderate electron den-

sities, it is predicted that the low-field mobility and saturation drift velocity are reduced by

approximately a factor of 2 and 15 %, respectively.

12

2.3 Surface polar phonon dominated transport

When a graphene layer is in close proximity to a polar substrate, inelastic carrier scattering

with surface polar phonons (SPPs) can result in significant reduction in the low-field mobility

of graphene. [41, 130, 131, 132] Due to the inelastic nature of SPP, they also provide pathway to

current saturation, in conjunction with, or as a substitute for, intrinsic optical phonons. While

a number of studies generally suggested the negative influence of reduced saturation velocities

(due to the relative small energies of SPPs),[2, 133, 134] conflicting reports exist that predict

a very different picture (including enhanced velocities) based on the analysis of Boltzmann

transport equation. [132, 135]

In this section, we investigate the influence of carrier scattering due to SPPs on the electronic

transport properties of monolayer graphene, via a full-band ensemble Monte Carlo method. Par-

ticularly, the low-field electron mobility and saturation velocity are calculated in the presence of

three different substrates (SiC, SiO2, and HfO2) and compared with those of intrinsic graphene

at room temperature. Furthermore, we examine the impact of SPPs on the low temperature

electrical resistivity, with attention to its implication on the experimental determination of the

acoustic phonon deformation potential constant.

The Monte Carlo model adopted in the calculation utilizes the complete electron and phonon

spectra in the first Brillouin zone. While a tight-binding band is used for the electronic energy

structure, all six branches of the graphene phonon spectra are considered with the phonon dis-

persion and electron-phonon scattering rates obtained from first principles calculations. [1] The

effect of degeneracy is accounted for by the rejection technique, after final state selection. [136]

The distribution function is obtained self-consistently from the ensemble simulation. The SPP

scattering rate is introduced by following [130, 131]

1

τS(ki)=

2π

~∑q

e2F2

2ε(q)2e−2qd

q(1 + cos θ)(nq +

1

2± 1

2)δ(Ef − Ei ± ~ωS) , (2.5)

where q = |kf − ki| is the SPP momentum, Ef (Ei) the final (initial) electron energies, d

13

the distance between monolayer graphene and the substrate (0.4 nm), ωS the SPP energy,

and F2 = ~ωS2Aε0

( 1κ∞S +1 − 1

κ0S+1

) the square of Frohlich coupling constant. The (1 + cos θ) factor

originates from the overlap integral of the pseudospin part of the electron wavefunction and

ε(q) is the dielectric function in graphene. In addition, F contains the dependence on high

(low) frequency dielectric constant of the polar substrate κ∞S (κ0S) along with the normalized

area A. The scattering by ionized impurities in the substrate is not included in the effort to

clearly identify the role of SPPs. The values for relevant substrate parameters can be found in

Refs. [132, 130, 131].

Figure 2.6 shows the SPP scattering rates calculated at 300 K for three substrates, SiC,

SiO2, and HfO2. For comparison, the strength of electron-optical phonon interaction inherent

in graphene (via the deformation potential) is also plotted from a first-principles analysis. [1]

The results clearly illustrate the dominance of SPPs over intrinsic optical phonons in graphene

for the entire electron energy under consideration. Furthermore, this enhancement in scattering

is more pronounced for the substrates with small ωS that is apparent from the onset energy of

emission process (e.g., HfO2 - 19.4 meV; SiO2 - 60.0 meV; SiC - 116 meV). Due to the Coulombic

nature, the SPP scattering is a function of electron density n and subsequent screening in

graphene. Throughout the calculation, we assume n = 1 × 1012 cm−2 along with the static

screening function ε(q) in the random-phase approximation. [118] The intrinsic scattering via

deformation potential interaction has no dependence on n.

Figure 2.7 shows the electron drift velocity vs. the electric field at T=300 K, for intrin-

sic graphene and graphene on different substrates. As expected, the addition of electron-SPP

interactions lead to general decrease of electron drift velocities in the low-field region. Conse-

quently, the low-field mobilities are reduced for all three substrate with the largest decrease in

HfO2. However, the difference with the intrinsic case becomes progressively smaller with the

increasing field (thus, the increasing average electron energy) and, in the case of SiO2 and SiC,

the velocity appears to saturate at a higher value. No degradation is observed even for HfO2.

Clearly, this behavior does not follow the saturation velocity model suggested by Ref. [2] based

14

on the onset of optical phonon emission, intrinsic phonon or otherwise (i.e., SPPs). Rather, it

is in general agreement with more recent studies that solved the Boltzmann transport equation

numerically assuming a displaced Fermi-Dirac distribution function. [132, 135]

The apparent inconsistency of increased total scattering rate and absence of velocity degra-

dation in the high-field region may be explained by considering the linear energy dispersion and

the characteristics of the Frohlich interactions. As it is well known, a change in electron energy

(say, via an inelastic scattering) does not automatically relax the drift velocity in monolayer

graphene near the K and K ′ points. What matters is the direction of the final momentum.

Due to the Coulombic nature, the Frohlich interactions, including those by SPPs, prefer small

angle events. When a SPP scattering occurs, the electron will likely emit (absorb) a SPP, with

the phonon momentum pointing along the radial direction toward (away from) the Dirac point.

Consequently, the velocity of the scattered electron may not change significantly. In contrast,

optical phonon scattering of intrinsic graphene is via deformation potential interactions and,

thus, randomize/relax effectively the direction of the electron momentum/drift velocity. Ignor-

ing the difference between the Frohlich and deformation potential interactions [133] can lead to

inaccurate depiction of transport properties.

When the electron-SPP interactions are taken into account along with other scattering

mechanisms, they provide an additional channel for energy and momentum relaxation. Accord-

ingly, the average electron energy becomes substantially lower at a given electric field. As the

applied field increases, it means that the shift in the distribution function (along the direction

of the electric field) would be smaller in energy along with a shorter tail compared to that in

intrinsic graphene. This is clearly visible in Figure 2.8 plotted for the ky = 0 cross-section at

20 kV/cm. The observed shorter tails in the negative kx space (i.e., with negative velocities) as

well as in the high energy region with nonlinear dispersion [132] appear to more than compen-

sate the additional momentum relaxation of SPP scattering on the SiO2 and SiC substrates,

leading to larger saturation velocities shown in Figure 2.7. In the case of HfO2, the distribu-

tion is much less heated with a significant population in the negative half space. Due to the

15

very strong inelastic scattering by surface phonons, the drift velocity may not have reached

the saturation point at 20 kV/cm and could grow further. Cooling of electron distributions via

surface phonon interactions may also reduce the self-heating as the generated heat (i.e., SPPs)

is on the substrate and, thus, can be more readily dissipated. Another interesting point to note

from Figure 2.8 is that the electron distribution function in graphene does not resemble that of

a highly degenerate case, certainly not when subject to an appreciable electric field. A simple

approximation of a displaced box-like function [2, 134] cannot describe transport properties

accurately.

The strong influence of the substrate on low-field mobility observed at 300 K raises a pos-

sibility that the SPP scattering may be an efficient mechanism even at low temperatures. This

is a distinct possibility due to the small phonon energies ωS . Figure 2.9 plots the electrical

resistivity ρ in graphene vs. temperature for all four cases under consideration. Indeed, the

resistivity is substantially affected by the substrate conditions at T & 50 K and the effect of

SPP scattering can not be immediately separated from those of other mechanisms (most impor-

tantly, the acoustic phonon scattering). The acoustic deformation potential is often determined

experimentally utilizing the assumption that the low temperature resistivity is linearly depen-

dent on temperature with a slope that is proportional to the square of acoustic deformation

potential. [41, 137] However, the calculation results indicate that this may not be possible; the

extracted deformation potential (Dac) is not an intrinsic quantity but an ”effective” parameter

that includes substrate effects and screening (i.e., the charge density in graphene) among others.

Concerning the specific results, the intrinsic case gives Dac ≈ 6.8 eV with a well-defined

linear region. This (i.e., the strong linearity) is due to the negligible contribution of optical

phonon scattering at low temperatures with relatively large energies (~ωop ∼ 160 meV). The

small deviation of Dac from the deformation potential constant obtained directly from the

acoustic phonon scattering rate (≈ 4.5 eV) [1] may be attributed to a degenerate electron

density considered in the present calculation (n = 1 × 1012 cm−2). It is also interesting to

note that this value (6.8 eV) is actually rather close to the experimentally extracted on a non-

16

polar substrate (7.8 eV). [138] For the SiC substrate, we estimate Dac ≈ 7.1 eV similar to the

intrinsic result. As T & 150 K, however, there is a substantial increase in SPP scattering and

the difference between the SiC and intrinsic cases becomes more discernible. When SiO2 is used

as the substrate, the slope of ρ is further increased and becomes nonlinear earlier due to the

small SPP energy. The deduced value in the linear region (T . 125 K) gives the appearance of

Dac ≈ 13.2 eV, which is not unlike 16−18 eV estimated experimentally on SiO2. [41] Finally,

HfO2 has the largest effect among the three substrates. The resistivity is greatly increased with

no apparent linear region in the temperature range under consideration. Accordingly, it is not

plausible to determine Dac. The relevance of SPP scattering even in the T . 150 K regime

can explain, at least in part, the very disparate results for the magnitude of acoustic phonon

deformation potential in graphene. [41, 132, 1, 138]

In summary, we investigate the effects of substrate-induced SPP scattering on the electronic

transport properties in monolayer graphene by using a full-band ensemble Monte Carlo simu-

lation. It is found that the electron velocity-field characteristics are highly dependent on the

choice of substrate at room temperature. Specifically, the Frohlich nature of the interaction

appears to be crucial for correctly describing the saturation of drift velocity. The simulation

also shows that the SPP scattering remains an efficient mechanism even at low temperatures

(T & 50 K), substantially affecting the electrical resistivity. This makes it difficult to experi-

mentally determine the acoustic deformation potential for intrinsic graphene in close proximity

to a substrate.

2.4 Joule self heating effect

Superior electronic properties are one of the main attractions of graphene in device applica-

tions. The massless Dirac Fermions originating from a linear dispersion relation imply high

electron mobilities and drift velocities − an ideal trait for devices in high-frequency/high-speed

operation. [24] Nevertheless, carrier motions are subject to scattering by perturbations such as

lattice vibrations or impurities in most realistic conditions. Particularly, the electron interaction

17

with the lattice is essentially a thermalization process from a system point of view. As electrons

gain energy from an external source (such as an electrical bias), a part of the excess energy is

transferred to the lattice via phonon emission. Subsequent increase in the lattice temperature

(i.e., the Joule heating) acts as a counter weight to limit further energy gain from the source by

causing degradation in the electronic transport. Eventually, a balance is reached and the system

approaches the steady state. Thus, the details of heat dissipation including the properties of its

primary path (i.e., the substrate) could have a major influence. This is even more so in graphene

based structures, [139] where the two-dimensional (2D) nature dictates a large interface with

the substrate compared to the volume.

In this Letter, we theoretically investigate the effect of Joule heating in graphene. Specifi-

cally, the impact of different substrates on graphene electron transport properties are examined

with four technologically important candidate materials: SiO2, SiC, hexagonal BN (h-BN),

and diamond. While SiO2 is the most commonly used substrate due to its compatibility with

conventional technology, [2, 3] SiC has seen its use in producing large-area graphene by solid

state graphitization. [140] Recently, h-BN has drawn attention for its structural similarity to

graphene − a much desired condition for high quality samples. [141] On the other hand, dia-

mond can also be a possibility. Beside the anticipated affinity with graphene, it is one of the

most thermally conductive materials. In the analysis, we consider a model problem, where a

small graphene sample is placed on a 2D plane of relatively thick dielectric or substrate (300

nm), which is in turn on top of a bulk Si layer. The graphene film is subject to a uniform electric

field, generating excess heat that must be dispersed through the layers underneath. To obtain

the self-consistent solution across the structure, we solve simultaneously the 3D heat transfer

equation (including the estimated interfacial thermal resistance) and the Monte Carlo electron

dynamics in graphene.

The Monte Carlo simulation developed in this study takes into account the complete elec-

tron and phonon spectra in the first Brillouin zone. Both the graphene phonon dispersion and

its interaction with electrons are obtained from the density functional theory calculations, [1]

18

whereas a tight binding model is used for the electronic energy bands. [19] In addition, graphene

electron interactions with charged impurities (5× 1011 cm−2) on the substrate surface [142] as

well as the surface polar phonons (SPPs) are included. [132] For simplicity, the phonon system

is assumed to reach thermal equilibrium fast enough such that the electron-phonon scatter-

ing rates, for both intrinsic graphene phonons and SPPs, have the temperature dependence

based on the Bose-Einstein distribution: i.e., Nq = 1/[exp(~ωph

kBT )− 1], where ωph is the phonon

frequency, kB the Boltzmann constant, and T the temperature (a function of position).

The electron energy transferred to the lattice vibrational modes increases the lattice temper-

atures of graphene and the substrate. Specifically, the interaction with graphene phonons (e.g.,

emission) leads to the elevation of graphene lattice temperature (Tg), whereas the substrate has

contributions from both the direct excitation of SPPs (i.e., electron-SPP scattering) and the

heat conduction from the graphene lattice. As summarized above, the thermal part of the self-

consistent model utilizes a 3D heat transfer equation in the substrate;∇·[κ(x, y, z)∇T (x, y, z)] =

0, where κ is the thermal conductivity of the material. The values used in the calculations are

κSiO2 = 1.4 Wm−1K−1, κSiC = 370 Wm−1K−1 and κdia = 1800 Wm−1K−1 for SiO2, SiC, and

diamond, respectively. [143, 144] Unlike the first three, the thermal conductivity of h-BN is

anisotropic with a large difference in the in-plane and the out-of-plane components due to the

layered nature; κh-BN∥ ≈ 300 Wm−1K−1 and κh-BN⊥ ≈ 2 Wm−1K−1. [145] The corresponding

details on Si can be found in Ref. [146].

In a heterogeneous system, there is an extra thermal resistance rgs at the interface between

graphene and the substrate (i.e., the so-called Kapitza resistance). An experimental measure-

ment reported rgs ranging from 5.6× 10−9 to 1.2× 10−8 Km2W−1 in the graphene/SiO2 struc-

ture. [147] Interestingly, a first principles calculation conducted very recently also suggests

similar numbers for the graphene interface with h-BN and SiC. [148] As such, a typical value of

8.8× 10−9 Km2W−1 (i.e., the median of the range observed for SiO2) is adopted in this study

for all four substrate materials. Concerning SiC, however, the situation is more complex. When

it is exposed to the air, hydrogen tends to be adsorbed and terminate the dangling bonds of Si

19

lonely atoms in order to form a stable surface. [149] The surface may also be passivated inten-

tionally to reduce the interface states. As there is an indication that this could cause a drastic

increase in rgs, an additional case of fully hydrogen terminated SiC (SiC-H) is considered with

rgs = 7.9 × 10−8 Km2W−1 to gauge the impact. [148] In a real situation, the surface is more

likely to show partial termination (i.e., somewhere between the cases of SiC and SiC-H).

Then, the total power dissipation per unit area across the interface between graphene and

the substrate can be expressed as Pt = (Tg−Ts)/rgs+Pspp, where Ts is the lattice temperature

of the substrate at the interface and Pspp is the power transferred via the direct SPP scattering.

As Pt should be equal to the net power loss by graphene electrons in a steady state, both this

quantity and Pspp can be estimated from the Monte Carlo simulation, providing the necessary

boundary condition for the heat transfer equation. Finally, a self-consistent temperature profile

(including Tg and Ts) is obtained by an iterative process.

Figure 2.10 illustrates the potential impact of Joule heating in graphene on different sub-

strate materials. Two sets of data are provided with the electron density of 1× 1012 cm−2 and

the graphene sample dimension of 1 µm × 0.5 µm. One set of results, shown in lines, represents

the drift velocity versus electric field when Joule heating is ignored by fixing Tg=Ts=300 K. The

other set, in data points, examines the same curves with the Joule heating effect taken into ac-

count. Of the cases under consideration, the most drastic changes appear in the graphene/SiO2

structure. While the deviation is minor in the low-field region, the saturation velocity vsat de-

grades substantially (e.g., by about 20% to 3.9 × 107 cm/s at 30 kV/cm). In comparison, the

velocity-field curves for all other substrates reveal little impact of Joule heating. Only SiC-H

shows a minor influence; the rest including SiC (unpassivated; not provided in Figure 2.10) are

virtually unaffected. Here, it is also interesting to note that the graphene-on-diamond structure

actually has the lowest vsat independent of Joule heating. The origin of this departure is the ab-

sence of SPPs in diamond. Without the SPP scattering, the electrons in graphene lose a major

energy relaxation mechanism and remain more energetic than otherwise. Consequently, it is not

unreasonable to expect a smaller drift velocity on a non-polar substrate (such as diamond) than

20

on a polar counterpart. [150] Indeed, the calculations show that the average graphene electron

energy on the diamond substrate is 0.44 eV at 30 kV/cm, while it is for instance only 0.25 eV

on SiO2.

To better examine the observed impact on electron drift velocities, the temperatures in the

graphene film (Tg) and at the interface directly below (Ts) are provided in Figure. 2.11 as a

function of applied bias. Both Tg and Ts increase with the field but their magnitudes vary widely

depending on the substrate. Clearly, SiO2 shows the extreme case of Joule heating with very high

Tg and Ts consistent with Figure 2.10. Due to the poor thermal conductivity κSiO2 , the excess

heat emitted by graphene electrons cannot be efficiently channeled through the substrate. The

resulting increase in temperature induces stronger electron-phonon scattering and subsequently

degrades the drift velocity. As for h-BN, the rise in Tg and Ts is far more modest despite

the very low out-of-plane thermal conductivity κh-BN⊥, which is in fact almost the same as

κSiO2 . The discrepancy comes from the in-plane thermal conductivity κh-BN∥ that is about two

orders of magnitude larger. Consequently, the transferred heat in h-BN can easily spread in-

plane unlike in SiO2, utilizing a much wider thermal channel. Indeed, the 3D iso-thermal profile

illustrates a laterally extended distribution near the interface − a sign of efficient heat removal.

Two substrates with high thermal conductivities, diamond and SiC (unpassivated; not shown),

indicate even smaller deviations from room temperature as expected.

In the case of SiC-H, the moderate increase in Tg (and the subsequent velocity decay) has a

different origin. While thermal transport in the substrate is excellent (owing to a superior κSiC),

the heat conduction across the interface with a relatively large rgs provides the bottleneck. This

point is clearly illustrated in Figure 2.11(b) by the largest Tg − Ts among those plotted; the

substrate surface temperature Ts stays near 300 K in the graphene/SiC-H structure. Another

interesting observation in Figure 2.11(b) is that the absence of SPP interaction is visible from

the simulation of diamond. The comparatively large Tg − Ts (over those of h-BN and SiO2)

indicates a greater disconnect between the graphene film and the substrate in terms of heat

transfer. Since an identical rgs is assumed (except for SiC-H), the lack of additional heat path

21

via SPP emission appears to be the main reason for the increased Tg − Ts.

Finally, the effect of Joule heating is examined as a function of electron density n along with

the available experimental studies in the literature. Since the detailed heat dissipation (thus,

the extent of Joule heating) depends on such factors as the exact geometry, etc., that not only

vary from sample to sample but also are not fully characterized, it is rather difficult to have

a meaningful one-to-one comparison at the quantitative level. For instance, when the lateral

dimension of the graphene film is much larger than the thickness of substrate dielectric, [139]

heat transfer through the structure could simply be projected to a 1D problem with the out-

come potentially much different from that considered in the current calculation. The 3D effect

such as the lateral heat spread would be absent and h-BN would behave more like SiO2 with

pronounced Joule heating. Moreover, the additional structural details including the location

and the dimension of metal contacts could alter the heat dissipation pattern that are not in-

cluded in the current model. Consequently, we treat the graphene sample size as an effective

parameter that is adjusted to provide a good fit with the experimental data (specifically, those

on SiO2). [2, 3]

Figure 2.12 shows the comparison of vsat vs. n. The results clearly illustrate the increased

prominence of Joule heating at large values of n. This is obvious since more electrons mean

lager heat generation per unit area, which leads to elevated lattice temperature and reduced

vsat. Consistent with the results discussed above, SiO2 suffers the biggest impact and then SiC-H

is the next, while the other substrates (BN, SiC, and diamond) remain largely unaffected in the

considered range (n . 4× 1012 cm−2). One particularly interesting point to note is the slope of

decay. The slope deduced from the simulation becomes steeper in close correlation with the rise

in Tg and reaches the 1/√n dependence for SiO2 that matches well with the experimental data

from Refs. [2] and [3]. The agreement, however, cannot be explained when the Joule heating is

excluded as is evident from the figure (see the solid line). Consequently, it strongly indicates

that the so-called 1/√n decay may not come from the SPP energy of the SiO2 substrate as

originally suggested. [2] Rather, it is a manifestation of Joule heating in its entirety including

22

the influence of SPP scattering characteristics − a case-specific outcome and not a general rule.

Considering the findings thus far, SiO2 may not be a desirable choice for the substrate

material. As for diamond, the lack of SPP scattering results in the drift velocities substantially

smaller than other candidates (such as BN and SiC). However, it is also the most immune from

the Joule heating degradation for its high thermal conductivity and may have an advantage at

very high carrier densities (& 1013 cm−2). While the performance of BN and SiC are generally

comparable, SiC-H shows the sign of elevated temperatures in graphene. This could cause

a significant concern in device breakdown characteristics in addition to the channel velocity

reduction. Since the dangling bonds at the surface tend to be terminated by hydrogen or other

specifies, the realistic structure involving the SiC substrate may be more like SiC-H than the

ideal case without termination. Thus, it appears that h-BN provides the best characteristics

among the studied to interface with graphene in electronic applications.

23

'ββ

2k

1

'k

2

'k

2F

O

C

'C

1F

1k

Figure 2.1: Schematic representation of the initial (k1,k2) and final (k′1,k

′2) electron pair states

which satisfy energy and momentum conservation in the linear band structure of graphene. βis defined as the angle between the long axis and the line OC and β′ between the long axis andthe line OC′.

24

Figure 2.2: Electron-electron scattering rate for electron concentrations n = 1 × 1012 cm−2

(circle), n = 5 × 1012 cm−2 (square), and n = 1 × 1013 cm−2(triangle). The electron-phononscattering rate is also plotted for comparison (dashed line), [1] where the details at low energiesare not discernible due to the linear scale.

25

Figure 2.3: Drift velocity for n = 1 × 1012 cm−2, n = 5 × 1012 cm−2, and n = 1 × 1013 cm−2

with (dashed line) and without (solid line) e-e scattering.

26

Figure 2.4: Energy distribution function for n = 1 × 1012 cm−2 and n = 1 × 1013 cm−2 with(dashed line) and without (solid line) e-e scattering.

27

Figure 2.5: Calculated Ediff/Etotal (solid line) and vx/vf (dashed line) as a function of angleβ′ assuming that the long axis (x) of the ellipse is along the direction of the electric field.Etotal = E(k′

1) + E(k′2) and vf is the Fermi velocity.

28

Figure 2.6: Surface polar phonon scattering rate of graphene electron on SiC, SiO2, and HfO2,for the electron density n of 1 × 1012 cm−2 at 300 K. Also plotted is the intrinsic grapheneoptical phonon scattering rate at 300 K (”intrinsic”) obtained from Ref. [1].

29

Figure 2.7: Electron drift velocity in graphene on SiC, SiO2, and HfO2. The intrinsic casewithout substrate is also shown. The electron density is 1× 1012 cm−2 at 300 K.

30

Figure 2.8: Electron distribution function with different substrate conditions: intrinsic (dash-dotted), SiC (solid), SiO2 (dashed), and HfO2 (dotted), for the ky = 0 cross-section at 20kV/cm. The box-like function corresponds to the Fermi-Dirac distribution displaced by theSiO2 SPP energy (60 meV) in a simple, metal-like approximation (i.e., EF ≫ kBT ) withn = 1× 1012 cm−2. For comparison, the equilibrium Fermi-Dirac distribution at 300 K is alsoplotted.

31

Figure 2.9: Electrical resistivity vs. temperature with different substrate conditions: intrinsic(circle), SiC (square), SiO2 (triangle), and HfO2 (diamond), with n = 1×1012 cm−2. The slopeof the straight lines are used to extract the acoustic deformation potential.

32

Figure 2.10: Drift velocities vs. electric field for graphene on different substrates. The graphenesample is assumed to be 1 µm × 0.5 µm with a carrier density of 1× 1012 cm−2. The impuritydensity is 5×1011 cm−2. As shown, Joule heating leads to a significant velocity degradation foruse of SiO2 substrate.

33

Figure 2.11: (a) Graphene lattice temperature Tg and (b) temperature difference Tg − Ts be-tween the graphene lattice and the top surface of the substrate as a function of driving electricfield. The conditions are the same as in Figure 2.10.

34

Figure 2.12: Saturation velocity vsat vs. electron density in graphene on different substrates.In the calculations, it is assumed that the graphene film of 1 µm × 1 µm is under an electricfield of 30 kV/cm. The impurity density is 5× 1011 cm−2. The experimental data for the caseof SiO2 substrate are from Refs. [2] and [3].

35

Chapter 3

Electron Transport Properties of

Bilayer Graphene

3.1 Introduction

Although graphene is very attractive in many applications, particularly in high speed devices. [5,

28, 46, 19] the gapless spectrum of monolayer graphene (MLG) makes it difficult to turn off the

electrical current due to tunneling. Bilayer graphene (BLG), on the other hand, can provide a

finite band gap up to hundreds of meV, when the inversion symmetry between top and bottom

layers is broken by an applied perpendicular electric field. [51, 30, 52, 53] A current on/off ratio of

about 100 was observed at room temperature, [56] offering a much needed control for nonlinear

functionality. Unfortunately experimental studies have also indicated that the typical mobility of

charge carriers in BLG may be substantially smaller than in MLG. [151] A recent work based on

the first principles calculations has suggested that this discrepancy may start with the intrinsic

transport properties, which results from substantial differences in electron-phonon coupling in

these two materials. [4] Additionally, it is widely accepted that extrinsic factors such as charged

impurities, disorder, and surface polar phonons (SPPs) can significantly alter carrier transport in

graphene on a substrate, a most commonly used configuration. [152, 153, 130] Despite extensive

36

research efforts, a comprehensive understanding of electron transport properties in BLG is still

a work in progress.

The purpose of the present chapter is to address this issue theoretically by taking advantage

of a first principles analysis and the Monte Carlo simulation. Specifically, the impacts of sub-

strate conditions and perpendicular electric fields are examined. The calculation results indicate

that graphene in the bilayer form loses much of its advantage over conventional semiconductors

in the low-field transport, particularly when the band structure is modified to induce non-

zero energy gap. The saturation drift velocity, on the other hand, can remain relatively high.

Due to the screening, electrons in BLG appear less susceptible to the interactions with remote

Coulomb sources, such as SPPs and impurities on the substrates, compared to the monolayer

counterparts. Below, we begin with a brief overview of the models used to estimate the relevant

scattering rates.

3.2 Relevant Scattering Mechanisms

The strength of the electron-phonon coupling is estimated from the first principles by using the

density functional theory. [1, 4] A comparative analysis reveals several qualitative differences

in the intrinsic scattering of MLG and BLG. For one, MLG has six phonon branches with two

carbon atoms in a unit cell, whereas these numbers double in BLG. Then, BLG may need to

consider both the intraband and interband transitions due to the presence of a close second

conduction band π∗2 in addition to the lowest π∗

1 states. At the same time, the optical phonons

in BLG appear to be a relatively weak source of interaction unlike in MLG. Ultimately, the

intrinsic scattering rate in BLG is dominated by the long wave acoustic phonons (intravalley

scattering). Figure 3.1 illustrates this general trend; only the dominant branches are shown for

clarity of presentation. The nomenclature for the phonon modes can be found in Refs. [4] and

[1].

In the presence of a polar substrate, the graphene electron interaction with SPPs can play

a significant role as the earlier studies in MLG have illustrated comprehensively. [130, 131, 150]

37

A similar treatment can be extended to BLG. By assuming that the electrons are equally

distributed in the two layers of BLG, we can derive the corresponding scattering rate as

1

τS(ki)=

2π

~∑q

e2F2

ε(q)2

[e−2qd + e−2q(d+c)

2q

]

×(nq +

1

2± 1

2

)|gs,s

′

ki(q)|2δ(Ef − Ei ± ~ωS) , (3.1)

where q = |kf − ki| is the magnitude of the SPP momentum, Ef (Ei) is the final (initial)

electron energy, d is the distance between the first layer and the substrate (0.4 nm), c is the

interlayer distance (0.34 nm), ωS is the SPP energy, nq is the phonon occupation number,

and ε(q) is the dielectric function. Additionally, F2 = ~ωS2Aε0

( 1κ∞S +1 − 1

κ0S+1

) is the square of

Frohlich coupling constant, where κ∞S (κ0S) is the high (low) frequency dielectric constant of

the substrate. The term |gs,s′

k (q)|2 = 12 [1 + ss′ cosαk cosαk+q + ss′ sinαk sinαk+q cos 2θ] comes

from the overlap of the electron wave functions of the initial and final states with the scattering

angle θ; [154] s and s′ are the band indices whose product is +1 for the intraband (for example,

π∗1-π

∗1) and −1 for the interband (π∗

1-π∗2) transitions. [155] For intrinsic BLG, αk = π/2 for an

arbitrary k. When an interlayer bias of u is applied, it is modified to satisfy tanαk = ~2k2/m∗u,

where m∗ is the effective mass of unbiased (or intrinsic) BLG at the K or K ′ point. [154] For

MLG, |gs,s′

k (q)|2 = 12 [1+ ss′ cos θ]. The specific values for the relevant substrate parameters can

be found in the literature. [130, 131, 132] As for the remote impurity scattering, the charged

impurities are assumed to be located on the surface of the substrate, in which case the scattering

rate is given by

1

τim(ki)=

2πni

A~∑q

[e2

2ε0κε(q)q

]2 [e−2qd + e−2q(d+c)

2

]

× |gs,s′

k (q)|2δ(Ef − Ei) , (3.2)

where κ = (κ0S+1)/2 is the background dielectric constant and ni is the impurity concentration.

The other parameters are the same as defined in Eq. (3.1).

38

In the random phase approximation, the graphene dielectric function can be expressed

as [137, 154] ε(q) = 1 − vqΠ(q) with the bare Coulomb interaction vq = e2/2ε0q and the

electron-hole propagator

Π(q) = 2∑s,s′,k

|gs,s′

k (q)|2fs′k+q − f s

k

Es′k+q − Es

k

. (3.3)

Here, fsk is the electron distribution function in band s and the factor of 2 takes into account

the spin degeneracy. Figure 3.2 shows the numerically obtained propagators for MLG and BLG

with the graphene electron density of n = 5×1011 cm−2 at 300 K. In this calculation [i.e., Π(q)],

electron occupation in the π∗2 states is ignored for its negligible contribution. Compared to MLG,

the screening in BLG appears to be much stronger due mainly to the large density of state at the

bottom of the conduction band (π∗1). [137] Additionally, the impact of the interlayer potential

on the electron screening in BLG can be substantial and is a strong function of temperature.

3.3 Electron Transport in BLG vs. MLG

A full-band ensemble Monte Carlo calculation is used for evaluating electron transport charac-

teristics self-consistently. The model takes into account the complete electron and phonon spec-

tra in the first Brillouin zone. Specifically, both the graphene phonon dispersion and its interac-

tion with electrons are obtained from the first principles calculations as discussed above, [1, 4]

whereas analytical expressions from the tight binding approximation are used for the electronic

energy bands in MLG and BLG (with the nearest-neighbor hopping energy γ0 = 3.3 eV and

the interlayer hopping energy γ1 = 0.4 eV). [156] The effect of degeneracy in the electronic

system is taken into account by the rejection technique, after the final state selection. Electron

scattering with the SPPs and remote impurities are also considered in the calculation as de-

scribed above whenever necessary. Throughout the calculation, we assume an electron density

n = 5× 1011 cm−2 and T = 300 K.

Figure 3.3 shows the electron drift velocity as a function of the electric field in MLG and

39

BLG (with no interlayer bias). It provides a comparison of all examined cases: namely, intrinsic

graphene and graphene on two different substrates, SiO2 and hexagonal BN (h-BN), for which

a charged impurity density of ni = 5 × 1011 cm−2 is considered along with the SPPs. As

illustrated, intrinsic MLG (i.e., no substrate) shows remarkable transport properties, [136, 150]

with the mobility and the saturation velocity as high as 1.0×106 cm2/Vs and 4.3×107 cm/s. An

analogous calculation for BLG gives a substantially lower mobility of 1.2× 105 cm2/Vs and the

saturation velocity of 1.8× 107 cm/s. These results for the intrinsic drift velocity are consistent

with the scattering rates in Fig. 3.1; they demonstrate how acoustic and optical phonons affect

the charge carriers in MLG and BLG at various electric fields.

The lower mobility in BLG can be readily explained by the larger acoustic phonon scattering

rates, as well as the lower electron group velocity near the bottom of the conduction band, where

all electrons tend to congregate at low electric fields. To understand the smaller saturation

velocity in BLG, on the other hand, it is convenient to examine the distribution function at

high electric field plotted in Fig. 3.4. As the electrons gain energy in the applied field, the

distribution function shifts in the k-space along the direction of the drift. At the same time,

increased scattering with the long-wave acoustic phonons leads to a further broadening of the

electron distribution. The result of this interplay between the displacement and the broadening

is the saturation of the drift velocity in intrinsic BLG, where the acoustic phonons dominate the

scattering (particularly, momentum relaxation). In MLG, however, the velocity curve appears to

demonstrate another pattern, which points to the presence of a competing intrinsic scatterer.

Indeed, as we discussed earlier, the inelastic scattering via optical phonons is strong unlike

in BLG, providing efficient energy relaxation for hot electrons. Consequently, the distribution

in MLG is prevented from shifting further towards higher energies, which in turn effectively

curtails the momentum relaxing interactions and results in a higher saturation velocity. Unlike

in the conventional semiconductors, the loss of electron energy does not bring about the reduced

group velocity (i.e., the slope) due to the linear dispersion.

A similar consideration applies when an additional source of optical phonon scattering comes

40

into play. That is, the saturation velocity may be actually enhanced when graphene electrons

are subject to the SPP scattering, as it provides another path for hot electron energy relax-

ation. Moreover, the SPPs prefer small angle interactions (i.e., small momentum relaxation)

due to their Coulombic nature. Figure 3.3 clearly demonstrates this effect. In MLG, the SPP

scattering can increase the saturation velocity up to 6.5× 107 cm/s if SiO2 or h-BN is used for

the substrate material. A similar, positive impact of the substrate on the saturation velocity

was also suggested in recent studies. [132, 135, 150] In BLG, the saturation velocity can reach

2.9 × 107 cm/s, which is about as 1.5 times large as the intrinsic value. Apparently, the en-

hancement of drift velocity is still prominent despite the stronger screening in BLG leading to

smaller SPP scattering rates. This is due partly to the fact that the competing relaxation mech-

anism (i.e., intrinsic optical phonon scattering) is relatively weak in BLG as discussed earlier.

Nonetheless, interactions with optical phonons (both intrinsic and SPP) provide the dominant

energy relaxation process even in BLG; other mechanisms not included in the current study

such as emission of photons [157] are not expected to play a significant role in the field range

under consideration.

When the charged impurity scattering is taken in account, the electron drift velocities in

MLG and BLG are substantially reduced at low electric fields (see Fig. 3.3). If h-BN is used

as the substrate, the low-field mobility is 1.9× 104 cm2/Vs in MLG, and 1.2× 104 cm2/Vs in

BLG. The drift velocity at the electric field of 20 kV/cm in MLG decreases to 4.8× 107 cm/s,

while in BLG it stays nearly the same as the case without impurity scattering, 2.8× 107 cm/s.

Clearly, the impact of ionized impurity scattering is much less severe in BLG. It is because

the BLG electrons on average are more separated from the impurities (i.e., the surface of the

substrate) and experience stronger screening. A similar trend is observed in the calculation

with SiO2 as well. In this case, the result for MLG also appears to be in good agreement with

the available experimental data; [135] the discrepancy may be attributed to the presence of

additional sources of interaction such as neutral scatters on the substrate, ripples and other

defects in graphene crystal lattice. [19] On a related note, it is important to point out that the

41

samples of graphene on h-BN frequently show much higher mobility measurements than those

on SiO2. This is because the graphene/h-BN interface tends to be naturally of higher quality

due to their structural similarity (for example, no dangling bonds). Consequently, one can

expect fewer surface impurities/defects, better stability, and reduced roughness (than graphene

on SiO2), leading to superior transport characteristics. [158] However, the interface quality

varies from sample to sample, making a meaningful comparison rather difficult as it requires

a thorough characterization of each interface. Our calculations, on the other hand, assume an

identical impurity concentration on both substrates to elucidate the fundamental impact of this

scattering mechanism in a direct one-to-one analysis.

3.4 Transport in BLG with interlayer bias

BLG with an interlayer bias offers the advantage of a tunable bandgap with potential applica-

tions to transistors, tunable photo-detectors and lasers. [30, 56] The band structure in this case

changes to, [51]

E2k =

γ212

+u2

4+ ~2v2Fk2 ±

√γ414

+ ~2v2Fk2(γ21 + u2) , (3.4)

where u is the difference between on-site energies in the two layers, vF = (√3/2)aγ0/~ ≈ 1×108

cm/s is the Fermi velocity (a = 0.246 nm), and the minus and plus signs correspond the π∗1

and π∗2 conduction bands, respectively. However, this ability to tune the band gap comes at the

expense of the material’s intrinsic transport properties. As the bandgap opens, the bottom of

the lowest conduction band changes its shape from a hyperbola to a so-called ”Mexican hat”

and the density of states exhibits a Van Hove singularity as shown in Fig. 3.5. Consequently,

the increased density of states leads to stronger quasi-elastic electron interaction with the long-

wavelength acoustic phonons at low electron energies. Figure 3.6 provides the dependence of

the mobility on the size of the bandgap. When the bandgap reaches 0.24 eV, the mobility drops

to as low as 1.2× 104 cm2/Vs even without any external scattering mechanisms, which is one

42

order of magnitude smaller than in the unbiased case. The reduction in the mobility becomes

even more pronounced when the ionized impurity scattering is taken into account. The Van

Hove singularity also enhances the impact of electron coupling with the remote impurities.

On the other hand, it appears that the velocities at high fields are mostly unaffected by the

gap. Figure 3.7 shows the drift velocity for u = 0.1 V, 0.2 V, and 0.3 V, which correspond to the

bandgap of 0.1 eV, 0.18 eV, and 0.24 eV, respectively. As the phenomenon of velocity saturation

is associated with hot electrons, it is relatively immune from the changes at the bottom of the

energy dispersion. Similarly, the impact of the SPP scattering on the drift velocity, as it is

felt primarily via the high energy electrons, is not affected by the gap in the electron energy

spectrum. Accordingly, no appreciable difference is observed in the saturation velocities for the

three considered values of u. When the ionized impurity scattering is included, the velocity

saturation is progressively pushed to a higher field due to the reduction in the mobility (i.e.,

the slope) that was discussed above.

Finally, it may be worth pointing out that the electron-electron scattering could potentially

be significant for accurate calculation of low-field transport properties in biased BLG. Due to the

diverging density of states at the band minima, electrons tend to congregate in the low-energy

states with small momentum vectors, causing stronger inter-carrier interactions. Accordingly,

it could limit the accuracy of the obtained mobility values even at the assumed relatively low

density of 5× 1011 cm−3. This is not a concern in MLG or BLG without the interlayer bias as

their density of states is finite even at zero energy.

3.5 Summary

Electron transport properties of BLG are studied under realistic conditions in the presence of

the SPPs and charged impurities. Overall, BLG has a lower mobility and saturation velocity

than MLG, due to the stronger acoustic phonon scattering, weaker optical phonon scattering,

and nonlinear dispersion at the bottom of the conduction band. It is also shown that SPPs

can improve the saturation velocity in BLG by effectively dissipating the electron energy. The

43

impurity scattering has a considerable effect in decreasing the drift velocities in both MLG

and BLG. However, BLG appears more resistant to impurity scattering than MLG, due to a

stronger screening and larger effective distance between electrons and the impurities. In BLG

with a interlayer bias, the changes in the band structure result in drastically reduced mobilities,

particularly in the presence of charged impurities. This may have a negative consequence in the

potential device application.

44

Figure 3.1: Intrinsic electron scattering rates in MLG and BLG calculated from the first princi-ples for the dominant (a) acoustic and (b) optical phonons. [1, 4] The sudden increases shown in(b) are due to the onset of optical phonon emission. They correspond to the phonon frequenciesat the points of high symmetry in the first Brillouin zone: ωΓ ≈ 200 meV and ωK ≈ 160 meV.

45

Figure 3.2: Electron-hole propagator Π(q) normalized to N0 (= 2m∗/π~2) in MLG and BLGwith the graphene electron density of 5×1011 cm−2. For BLG, the calculation considers differentinterlayer biases with the induced energy gap Eg of 0 eV (no bias), 0.10 eV, 0.18 eV, and 0.24 eV,respectively.

46

Figure 3.3: Electron drift velocity versus electric field in (a) MLG and (b) BLG, with differ-ent substrate conditions: intrinsic/no substrate (circle), SiO2 (triangle), SiO2 with impurities(reverse triangle), h-BN (square), and h-BN with impurities (diamond). The electron densityis 5× 1011 cm−2 at 300 K. The impurities on the surface of the substrate (d=0.4 nm) have thedensity 5× 1011 cm−2.

47

Figure 3.4: Cross sectional view (ky=0) of electron distribution functions in (a) MLG and (b)BLG at 20 kV/cm, with different substrate conditions: intrinsic/no substrate (short dashedline), SiO2 (long dashed line), SiO2 with impurities (solid line), h-BN (dashed-dotted line), andh-BN with impurities (dotted line). The conditions are the same as specified in Fig. 3.3.

48

Figure 3.5: Density of states in BLG for different interlayer biases with the induced energy gapEg of 0 eV (solid line), 0.10 eV (dashed line), 0.18 eV (dotted line), and 0.24 eV (dashed-dottedline), respectively. The inset shows the corresponding band structures.

49

Figure 3.6: Electron mobility versus bias-induced bandgap in BLG at 300 K, with differentsubstrate conditions. The electron density is 5 × 1011 cm−2. The impurities on the surface ofthe substrate (d=0.4 nm) have the density 5× 1011 cm−2.

50

Figure 3.7: Electron drift velocity versus electric field in BLG with a bias-induced gap of Eg

of (a) 0.1 eV, (b) 0.18 eV, and (c) 0.24 eV, under different substrate conditions: intrinsic/nosubstrate (circle), SiO2 (triangle), SiO2 with impurities (reverse triangle), h-BN (square), andh-BN with impurities (diamond). The conditions are the same as specified in Fig. 3.6.

51

Chapter 4

Intrinsic Electrical Transport

Properties of Monolayer Silicene and

MoS2 from First Principles

4.1 Introduction

Characterization of electronic transport, particularly the intrinsic properties, is critical for

assessing and understanding the potential significance of a material. In the case of silicene,

many of the crucial parameters are presently unknown due to the brief history of this mate-

rial. In comparison, notable advances have been made in MoS2 lately. Experimental investiga-

tion of transistor characteristics claimed the channel mobilities ranging from ∼200 cm2/Vs to

∼1000 cm2/Vs at room temperature, [75, 91] while a theoretical study estimated an intrinsic

phonon-limited value of ∼410 cm2/Vs based on a first-principles calculation of electron-phonon

interaction. [159] However, questions remain regarding the intrinsic electron transport in MoS2.

For instance, those extracted from transistor current-voltage (I -V ) measurements are indirect

accounts and can be strongly affected by extrinsic factors, requiring caution as illustrated in the

latest studies. [160, 161, 162] Similarly, the latter work [159] includes only the electronic states

52

in the conduction band minima at the K points in the first Brillouin zone (FBZ); the impact

of Q valleys located along the Γ-K symmetry directions (which correspond to the minima of

bulk MoS2) were not considered. A detailed investigation is clearly called for.

In this paper, we examine intrinsic transport properties of monolayer silicene and MoS2

by taking advantage of first-principles analysis and full-band Monte Carlo simulation. Along

with the electronic band structure, the phonon spectra and electron-phonon coupling matrix

elements are calculated for all phonon branches within the density functional theory (DFT) for-

malism. [163, 1] The obtained electron scattering rates are subsequently used in the Boltzmann

transport equation to compute the intrinsic velocity-field characteristics with a full-band Monte

Carlo treatment. The calculation results are compared with the available data in the literature

and the key factors affecting electron transport in these materials elucidated. The investigation

also provides the effective deformation potential constants extracted from the first-principles

results.

4.2 Theoretical Model

Both monolayer silicene and MoS2 are hexagonal crystals. To account for their delicate atomic

structures accurately, the calculations are performed in the DFT framework, as it is implemented

in the QUANTUM-ESPRESSO package, [164] using ultrasoft pseudopotentials. A minimum of

35 Ry is used for the energy cut-off in the plane-wave expansion along with the charge truncation

∼15 times larger. The generalized gradient approximation is used for the exchange-correlation

potential for silicene, while the local density approximation is adopted for MoS2. The momentum

space is sampled on a 36×36×1 Monkhorst-Pack grid with no offset (silicene) or on a 18×18×1

grid (MoS2) for electronic band calculation. The simulated cells are optimized in both cases

until the atomic forces decrease to values less than 0.015 eV/A.

Each phonon is treated as a perturbation of the self-consistent potential, created by all

electrons and ions of the system, within the linear response [i.e., density functional perturbation

theory (DFPT)]. [163] The calculation of the potential change due to this perturbation gives

53

the value of the electron-phonon interaction matrix element [1]:

g(i,j)vq,k =

√~

2Mωv,q⟨j,k+ q|∆V v

q,SCF|i,k⟩, (4.1)

where |i,k⟩ is the Bloch electron eigenstate with the wave vector k, band index i, and energy

Ei,k; ∆V vq,SCF is the derivative of the self-consistent Kohn-Sham potential [163] with respect

to atomic displacement associated with the phonon from branch v with the wave vector q

and frequency ωv,q; and M is the atomic mass. Numerical calculations of lattice dynamics are

conducted on a 18×18×1 Monkhorst-Pack grid. Indices i, j are dropped hereinafter as only the

first (lowest) conduction band is considered.

With the electron-phonon interaction matrix from the first-principles calculation, the scat-

tering rate of an electron at state |k⟩ can be obtained by using Fermi’s golden rule,

1

τk=

2π

~∑q,v

|gvq,k|2[Nv,qδ(Ek+q − ~ωv,q − Ek) + (Nv,q + 1)δ(Ek−q + ~ωv,q − Ek)], (4.2)

where Nv,q = [exp(~ωv,q/kBT ) + 1]−1 is the phonon occupation number, kB the Boltzmann

constant, and T the temperature. As we are interested in the intrinsic scattering probability

that is not limited to a specific carrier distribution (and thus, the Fermi level), our formulation

assumes that all final electronic states are available for transition (i.e., nondegenerate) in the

bands under consideration.

For transport properties, a Monte Carlo approach with full-band treatment is adopted. All

of the details described above, including the wave vector (k,q) dependence of the scattering

matrix elements [e.g., Eq. (4.1)], are accounted for. This allows solution of the Boltzmann

transport equation beyond the conventional relaxation time approximation.

54

4.3 Results and Discussion

4.3.1 Monolayer silicene

Earlier first-principles studies have shown that the stable structure for monolayer silicene has

a low-buckled configuration. [73, 10] While planar and high-buckled cases lead to imaginary

phonon frequencies around the Γ point, indicating an unstable structure, the low-buckled con-

struction provides well-separated phonon branches and positive frequencies. The origin of buck-

led geometry in silicene is the weakened π bonding of the electrons in the outer shell. Compared

with graphene, which has very strong π bonding and planar geometry, the interatomic bonding

distance is much larger in silicene, which decreases the overlap of pz orbitals and dehybridizes

the sp2 states. Accordingly, the planar geometry can not be maintained. In our analysis, the

lattice constant a is optimized to be 3.87 A, with the buckling distance of 0.44 A, in good

agreement with Ref. [73].

Figure 4.1 shows the outcome of electronic and phononic band calculation in monolayer

silicene. The Fermi velocity extracted from the Dirac cone is around 5.8 × 107 cm/s that is

roughly one half of that in graphene [see bands π and π∗ in Fig. 4.1(a)]. While this result is

in agreement with other theoretical predictions, [165, 166] a value as high as 1× 108 cm/s was

also claimed in the literature. [10] As for phonons in Fig. 4.1(b), six branches are identified

with two atoms per unit cell. The transverse acoustic (TA) and longitudinal acoustic (LA)

phonon dispersion relations are well approximated by sound velocities in the long-wavelength

limit; vTA = 5.4 × 105 cm/s and vLA = 8.8 × 105 cm/s. Although the out-of-plane acoustic

(ZA) phonon exhibits an approximate q2 dependence near the center of the Brillouin zone, its

sound velocity can also be estimated; vZA = 6.3× 104 cm/s. An interesting point to note in the

phonon dispersion is the discontinuities in the frequency derivative of the highest optical branch

that, similar to graphene, appear at the high symmetry points, Γ and K. These discontinuities

are referred to as Kohn anomalies, [167, 165] induced by unusual screening of lattice vibra-

tions by conduction electrons. Sharp cusps typically indicate strong electron-phonon coupling.

55

Calculated phonon energies at the Γ, M, and K points in the FBZ are summarized in Table 4.1.

The electron-phonon interaction matrix elements obtained for the electron at the Dirac

point [i.e., k = (4π/3a, 0)] are plotted in Fig. 4.2 as a function of phonon wave vector q.

Kohn anomalies, illustrated by the three peaks at three equivalent K points in the transverse

optical (TO) mode and another at the zone center for the longitudinal optical (LO) branch,

are not as distinct as those observed in graphene. [1] Overall, coupling of optical phonons

with electrons appears to be relatively weak. In comparison, the acoustic phonons show much

stronger interaction. Particularly striking is the large strength of ZA phonon coupling, unlike

in graphene. Due to the buckled geometry (originating from the weak π bonding mentioned

earlier), silicene does not maintain certain key characteristics of ideal planar lattice, especially

the reflection symmetry with respect to the atomic plane. As such, the symmetry consideration,

in which only the in-plane phonons can couple linearly to two-dimensional (2D) electrons, [168]

no longer applies. An increased role of ZA phonons is clearly expected.

The scattering rates calculated at room temperature (T = 300 K) are shown in Fig. 4.3.

The result is plotted specifically for electrons with wave vector k along the K -Γ direction. Since

the integration in Eq. (4.2) is over the entire FBZ, both intravalley (K → K) and intervalley

(K → K′) transition events are included. As the interaction matrix elements illustrated above

suggest, acoustic phonons have much larger scattering rates than optical modes. Specifically,

the ZA branch provides the dominant contribution, which can be attributed to the observed

large coupling strength as well as the small phonon energy near the zone center (i.e., a large

occupation number NZA,q). This also indicates that the scattering rates are very sensitive to

the phonon energies (or equivalently the value of vZA). Since an accurate description of ZA

dispersion in the long wavelength limit requires a well converged calculation with a sufficiently

dense grid, care must be taken when evaluating accuracy of the data in the literature. [166, 10]

Figure 4.4(a) provides the drift velocity versus electric field at different temperatures ob-

tained by full-band Monte Carlo simulations. The intrinsic mobility estimated from the figure is

approximately 1200 cm2/Vs and the saturation velocity (defined at 100 kV/cm) 3.9×106 cm/s at

56

300 K. When the temperature decreases, both the mobility and the saturation velocity enhance

due to the suppression of phonon excitation; the respective values at 50 K are 3.0×104 cm2/Vs

and 6.2 × 106 cm/s. The drift velocities show a slight negative slope at high fields that be-

comes more pronounced at low temperatures. This phenomenon (i.e., the negative differential

resistance) can be explained, at least in part, by the nonlinear band dispersion at high electron

energies as in graphene. [169]

The calculation results discussed above demonstrate the intrinsic properties of silicene.

When this material is synthesized or placed on a substrate, however, additional scattering

sources such as surface polar phonons and impurities must be considered, which could degrade

the performance further. A topic that may need additional attention is the role of ZA phonons

in the presence of a supporting material. As recent measurement of graphene in-plane thermal

conductivity attests, [170] even a weak binding between a 2D crystal and the substrate could

dampen the ZA vibrations substantially. Moreover, it is reasonable to anticipate that the extent

of this suppression would be dependent on the detailed interaction between two materials. Since

ZA phonons provide the dominant role in the electron-phonon interaction in silicene, it (i.e.,

the damped vibration) could actually lead to sizable reduction in the scattering rate. To gauge

the impact, transport characteristics are also studied without the ZA scattering. As shown in

Fig. 4.4(b), the mobility experiences an increase of greater than threefold (3900 cm/s), while the

saturation velocity goes up more modestly (5.6× 106 cm/s). This estimate may be considered

an upper limit for silicene on a substrate.

4.3.2 Monolayer MoS2

In the present DFT calculation for monolayer MoS2, the optimized lattice constant is 3.13 A,

consistent with other theoretical studies. [171, 159, 172] Furthermore, this value is in good

agreement with 3.15 A determined experimentally in bulk MoS2. [173] The resulting electronic

and phononic band dispersion is depicted in Fig. 4.5. As shown, monolayer MoS2 is a semicon-

ductor with a direct gap of 1.86 eV at the K point − a number within a few percent from a

57

recent measurement of 1.9 eV. [74] Our calculation also predicts the presence of second energy

minima only about 70 meV higher. These so-called Q valleys are located along the Γ-K axes at

approximately the half-way points [e.g., Q=(0.34× 2π/a, 0) versus K=(4π/3a, 0)]. At present,

the energy separation EQK between the K and Q valleys is unsettled with the estimates ranging

from 50 to 200 meV. [171, 159] Since this is a crucial parameter for electron transport, a more

extended discussion is provided later in the paper in relation to intervalley scattering. The band

dispersion relations around the energy minima are nearly quadratic and can be well described

by the effective mass approximation. For the K valleys (i.e., the energy minima at the equiva-

lent K points), the extracted longitudinal and transverse effective masses are almost identical,

mlK = mt

K = 0.50m0. On the other hand, the Q valleys yield mlQ = 0.62m0, m

tQ = 1.0m0 with

the longitudinal direction defined along the Γ-K axis. m0 denotes the electron rest mass.

Monolayer MoS2 has the symmetry of the point group D3h, with nine branches of phonons.

The irreducible representations associated with each phonon mode, together with the polariza-

tion (longitudinal or transverse), help denote all of the vibrational modes, [172, 174] as plotted

in Fig. 4.5. The E ′′ modes are degenerate at the Γ point. These two modes are the in-plane

optical vibrations, with two S atoms moving out of phase and Mo atom static. The E ′ modes

are polar LO and TO phonons, with Mo atom and two S atoms moving out of phase. Due to the

coupling with the macroscopic electric field, there is LO-TO separation at the Γ point, which

slightly lifts the LO(E ′) mode upward on energy scale. A non-analytical part is added to the dy-

namic matrix resulting in a small splitting of about 0.3 meV [not visible due to the energy scale

of Fig. 4.5(b)]. The A1 and A′′2 branches are two out-of-plane optical phonon modes. A1 is also

referred to as the homopolar mode, with two S atoms moving out of phase and Mo atom static.

In the A′′2 mode, Mo atom and two S atoms vibrate out of phase. The three lowest branches are

LA, TA, and ZA modes, with sound velocities of vLA = 6.6× 105 cm/s, vTA = 4.3× 105 cm/s.

The phonon energies at different symmetric points are summarized in Table 4.2.

Figures 4.6 and 4.7 show the electron-phonon interaction matrix elements for the initial

electron state at k = K [=(4π/3a, 0)] and k = Q [≈ (2π/3a, 0)], respectively, for TA, LA,

58

TO(E ′), LO(E ′), and A1 (or homopolar) phonon modes. The contribution from the remaining

four branches is found to be negligible due to the weak coupling. The matrix elements for

k = K demonstrate a threefold rotational symmetry (i.e., 120), while those of k = Q show

the reflection symmetry with respect to the qx axis. As expected, the LO(E ′) phonons near

the Γ point possess the characteristics of Frohlich coupling through the induced macroscopic

electric field typical of polar materials. Since the relative electronic potential is not periodic in

the long wavelength limit, [163] DFPT does not yield a correct value to the electron-phonon

interaction matrix. For an approximation, the matrix element of LO(E ′) is interpolated at Γ

by using the values from the nearby q points. This [i.e., LO(E ′)] and A1 are the only two

modes that have non-zero scattering matrix as q → 0 (intravalley scattering); in the other

three branches, the matrix elements only have first-order components, |gq,k| ∼ q, leading to

|gq→0,k| → 0. With regard to intervalley scattering that requires large q phonons, a number

of different transition processes are possible as shown in Figs. 4.6(f) and 4.7(f). For instance,

Fig. 4.6 indicates strong electron-phonon interaction at the symmetry points M in the phonon

momentum space (denoted as q = M for simplicity) for all modes except LO(E ′); these phonons

can induce electron transition from K to Q ′ valleys. Another example is the phonons at q = K′

for all five modes in Fig. 4.7, which can be associated with electron scattering from Q1 to Q4.

The electron-phonon scattering rates are calculated as a function of electron energy using

Fermi’s golden rule. Figure 4.8 gives the rates for electrons in the K valleys at room temperature,

while the result for Q-valley electrons is shown in Fig. 4.9. Similarly to silicene, the wave vector

k of the initial electronic state is chosen along the K -Γ or Q-Γ axis, respectively. As can be

seen from the figures, the LA mode provides the largest scattering rates consistent with its

large coupling strength. The discontinuities or steps in the curves represent either the onset of

optical phonon emission or intervalley scattering. For instance, the abrupt increase observed in

the rate of LA phonons at ∼100 meV in Fig. 4.8(a) can be attributed to the above mentioned

strong K → Q′ transition via emission of a LA phonon with q = M. Since this phonon energy

is approximately 30 meV (see Fig. 4.5), the final state energy of ∼70 meV indeed matches

59

to the Q-K separation EQK. Similarly, the onset of transition via absorption can be found

around 40 meV in Fig. 4.8(b). The large density of states in the Q valleys (evident from the

large effective masses) makes the contribution of this scattering even more prominent. If, on

the other hand, all of the final states in the Q valleys are excluded, the total scattering rate

for the K -valley electron reduces drastically to approximately 2× 1013 s−1, which is consistent

with the prediction of an earlier first-principles calculation. [159] The observed difference of an

order of magnitude clearly illustrates the strong dependence of the scattering rates on EQK.

The inconsistency of this value in the recent publications [171, 159] adds difficulty to accurately

evaluating the role of Q valleys.

Utilizing the scattering rates, the velocity versus field relation is obtained by a full-band

Monte Carlo simulation at four different temperatures. As shown in Fig. 4.10, the extracted

mobility decreases from 4000 cm2/Vs at 50 K to about 130 cm2/Vs at room temperature while

the saturation velocity changes from 7.6× 106 cm/s to 3.4× 106 cm/s. The small mobility and

saturation velocity can be attributed to strong electron-phonon scattering as well as the heavy

effective masses. With massive electrons that hinder acceleration and many states to scatter

into (e.g., K, K ′, Q, Q ′ valleys), this is an expected outcome.

Compared to a recent theoretical estimation [159] of 410 cm2/Vs, however, our mobility is

significantly smaller, requiring a careful analysis of the discrepancy. One factor that could yield

at least a partial explanation is the issue surrounding the Q-K separation. With inconsistencies

reported in several first-principles results on this sensitive quantity (see the discussion above),

it is not unreasonable to imagine that our DFT calculation may have also experienced similar

inaccuracies. If EQK proves to be substantially larger than the estimated 70 meV, then the Q

valleys would have a limited influence on the low-field mobility and can be neglected in the

calculation as in Ref. [159] (with 200 meV). In this case, our simulation estimates the K -valley

dominated mobility of 320 cm2/Vs that is essentially in agreement with the earlier prediction

(410 cm2/Vs). [159] Clearly, both first-principles models produce a consistent picture ofK -valley

electron dynamics including intrinsic scattering with relevant phonon modes. The difference is

60

the relative significance of Q valleys (e.g., 70 meV vs. 200 meV). As such, further clarification

of intrinsic mobility in monolayer MoS2 may need to be preceded by accurate experimental

determination of EQK.

Even when the influence of Q valleys becomes negligible, a sizable disparity remains between

the theoretically obtained mobility and the highest value claimed experimentally (e.g., ∼1090

cm2/Vs). [91] One reason could be inaccuracy in the measurement due to external artifacts as

mentioned earlier. Indeed the latest analysis revealed that the attempts based on the two-point

method in the thin-film MoS2 transistor channel ignore the coupling capacitance between top

and bottom gates, leading to an overestimation of the mobility by approximately a factor of

14. [160, 161] When accounting for this effect, the re-calibrated data from the experiments indi-

cate mobility values substantially smaller than the theoretical prediction (< 130−410 cm2/Vs).

A subsequent Hall measurement also yielded 63.7 cm2/Vs at 260 K. [162] Considering the in-

fluence of extrinsic scattering sources such as impurities, our calculation is actually consistent

with the revised experimental estimate and provides a reasonable upper limit for the intrinsic

mobility.

An additional point to note when interpreting the experimental data is that the screening

by degenerate electrons can affect the bare scattering rates substantially (thus, the mobility) in

a low-dimensional system. [175, 176] This effect can be included by renormalizing the electron-

phonon interaction through the dielectric function; i.e., gvq,k → gvq,k/ϵ(q). To gauge the potential

significance in monolayer MoS2, we adopt a simple model for the dielectric function based on

Thomas-Fermi screening of only the K-valley electrons: i.e., ϵ(q) = 1+ qTF/q, where the screen-

ing wave vector qTF = 4mKe2/~2κ. Here, the factor of 4 accounts for the spin and valley

degeneracies, e is the electron charge, and κ is the background dielectric constant. Subsequent

calculation with a rough estimate of ϵ(q) shows that the scattering rates can experience a de-

crease of about an order of magnitude through screening. Thus, it is evident that the effect (i.e.,

screening) must be taken into account accurately when the carrier density becomes degenerate

in MoS2. A detailed analysis of ϵ(q) is, however, beyond the scope of this investigation as our

61

focus is on the properties of intrinsic electron-phonon interaction (i.e., non-degenerate).

4.3.3 Deformation Potential Model

For practical applications, it would be convenient to approximate the ab initio results for

electron-phonon coupling by a simple analytical model. Particularly useful in the present case

is the deformation potential approximation. Under this treatment, the coupling matrix ⟨j,k+

q| V vq,SCF|i,k⟩ in Eq. (4.1) can be expressed in the first order (D1q), or in the zeroth order

(D0). [177] The first-order deformation potential constant (D1) is adopted to represent the

coupling matrices for the acoustic phonon modes in the long wavelength limit (i.e., intravalley

scattering). In comparison, those involving the near zone-edge acoustic phonons (i.e., intervalley

scattering) are treated by using the zeroth-order deformation potential (D0) in a manner anal-

ogous to the optical modes. In the latter case (D0), the phonon energy is assumed independent

of the momentum for simplicity. The obtained analytical expressions of the scattering rates

are then matched to the first-principles results by fitting the effective deformation potential

constants.

For silicene, the intravalley scattering rate by acoustic mode v (= LA, TA, ZA) is obtained

as

1

τ(1)k,v

∣∣∣∣∣∣Si

=D2

1kBT

~3v2Fρv2vEk . (4.3)

Here, ρ is the mass density (= 7.2× 10−8 g/cm2) and vv denotes the sound velocity, for which

we can take the value of vZA = 6.3×104 cm/s, vTA = 5.4×105 cm/s, and vLA = 8.8×105 cm/s,

respectively, as discussed earlier. On the other hand, the rate of optical phonon scattering (both

intravalley and intervalley transitions) as well as the intervalley acoustic phonon scattering can

be expressed by the following form:

1

τ(2)k,v

∣∣∣∣∣∣Si

=D2

0

2~2v2Fρωv[(Ek + ~ωv)Nv + (Ek − ~ωv)(Nv + 1)Θ(Ek − ~ωv)], (4.4)

62

where Θ(x) is the Heaviside step function and Nv = [exp(~ωv/kBT ) + 1]−1 for phonon mode

v. Since the phonon dispersion is treated constant in Eq. (4.4), the intravalley optical phonon

scattering approximates the zone-center (i.e., Γ) phonon energies for ~ωv (v = LO, TO, ZO). In

the case of intervalley scattering via acoustic or optical phonons, ~ωv takes the respective phonon

energy at the zone-edge K point corresponding to electron transition K ↔ K′. The specific

values used in the calculation can be found in Table 4.1. When matched to the first-principles

rates, the effective deformation potential constants can be extracted for each scattering process

as summarized in Table 4.3. A particularly interesting point to note from the result is that

all three acoustic phonons show generally comparable values of D1 and D0 despite the large

scattering rate of ZA mode (see also Fig. 4.3). Clearly, this mode (ZA) couples strongly with

electrons but not enough to prevail over other acoustic branches; its dominant contribution is

due mainly to the small phonon energy as discussed earlier.

The electron-phonon scattering processes in MoS2 are much more complicated as the defor-

mation potentials need to be determined for both K - and Q-valley electrons. While feasible,

it is not practically useful to define the effective interaction constants individually based on

the mode of involved phonons and the transition types. Accordingly, we adopt a simplified de-

scription by combining the appropriate contributions into just two modes, acoustic and optical,

respectively.

Using the effective mass approximation for the band structure near the valley minima, the

scattering rate for intravalley acoustic phonon scattering (i.e., K → K or Q1 → Q1 by both LA

and TA phonons; see Figs. 4.6 and 4.7) is given by

1

τ(1)k

∣∣∣∣∣MoS2

=m∗

pD21kBT

~3ρv2s, (4.5)

where ρ = 3.1 × 10−7 g/cm2 for MoS2 and m∗p is the density-of-states effective mass for the

K or Q valley (final state), m∗p =

√ml

pmtp (p = K,Q). For electrons in the Q valleys, strictly

speaking, an additional factor of Θ(Ek − EQK) is multiplied to the left side of Eq. (4.5) to

63

account for the non-zero energy at the bottom of the Q valley. By taking the sound velocity

vs = vLA (= 6.6 × 105 cm/s), the value of D1 is estimated to be 4.5 eV and 2.8 eV in the K

and Q valleys, respectively.

The analytical expression that describes intravalley and intervalley optical phonon scattering

as well as intervalley acoustic phonon scattering rate is obtained as

1

τ(2)k,v

∣∣∣∣∣∣MoS2

= gdm∗

pD20

2~2ρωv[Nv1 + (Nv + 1)2], (4.6)

where gd is the valley degeneracy for the final electron states, and 1 and 2 denote the

onset of scattering for phonon absorption and emission, respectively. For instance, 1 = 1 and

2 = Θ(Ek − ~ωv) for electron transitions between the K valleys, whereas 1 = Θ(Ek −EQK)

and 2 = Θ(Ek−~ωv−EQK) for transitions between the Q valleys. The factors corresponding

to intervalley transfer between K and Q valleys can be constructed accordingly, where EQK may

be treated as a potentially adjustable parameter. Tables 4.4 and 4.5 summarize the initial/final

electron states, the phonon momentum that is involved (in the form of its location in the

momentum space), and the extracted deformation potential constants for each transition process

considered in the investigation. For a given phonon momentum, the actual value ~ωv used

in the analytical calculation is the average of the relevant phonon modes. Specifically, the

acoustic (optical) phonon energy is obtained as the average of LA and TA [TO(E ′), LO(E ′),

and A1] modes at the respective symmetry points given in Table 4.2. An additional point to

note is that the estimate of Dop0 at the Γ point includes the effect of Frohlich scattering by

the LO(E ′) mode. [159] While this is a mechanism physically distinct from the deformation

potential interaction and must be handled separately, its impact is relatively modest, at least

for the electrons in the K valley. Accordingly, the present treatment is considered adequate.

Further simplification of the model may also be possible judging from the narrow range of values

in Dac0 and Dop

0 (mostly in the low to mid 108 eV/cm).

64

4.4 Summary

We have performed a first-principles calculation together with a full-band Monte Carlo analysis

to examine electron-phonon interaction and the intrinsic transport properties in monolayer

silicene and MoS2. The results clearly elucidate the role of different branches as well as the

intravalley/intervalley scattering. The predicted intrinsic mobility for silicene is approximately

1200 cm2/Vs, with saturation velocity of 3.9 × 106 cm/s at room temperature. In the case of

MoS2, the K -valley-dominated mobility gives approximately 320 cm2/Vs, while the intrinsic

value reduces to about 130 cm2/Vs when the energy separation of 70 meV is used between

the K and Q minima. The estimated saturation velocity is 3.7 × 106 cm/s. The investigation

also illustrates the significance of extrinsic screening, particularly in numerical evaluation of

transport characteristics. The extracted deformation potential constants may prove to be useful

in further studies of these materials.

65

Table 4.1: Phonon energies (in units of meV) at the symmetry points for monolayer silicene.

Phonon modes Γ K MZA 0 13.2 13.0TA 0 23.7 13.4LA 0 13.2 13.5ZO 22.7 50.6 50.7TO 68.8 50.6 56.7LO 68.8 61.7 64.4

66

Table 4.2: Phonon energies (in units of meV) for TA, LA, TO(E′), LO(E′), and A1 (or ho-mopolar) modes at the Γ, K, M and Q points in the FBZ of monolayer MoS2.

Phonon modes Γ K M QTA 0 23.1 19.2 17.9LA 0 29.1 29.2 23.6

TO(E ′) 48.6 46.4 48.2 48.0LO(E ′) 48.9 42.2 44.3 44.2

A1 50.9 51.9 50.1 52.2

67

Table 4.3: Extracted deformation potential constants for electron-phonon interaction in sil-icene.

Phonon mode Intravalley Intervalley

ZA 2.0 eV 6.1×107 eV/cmTA 8.7 eV 1.4×108 eV/cmLA 3.2 eV 4.2×107 eV/cmZO 6.3×107 eV/cm 4.3×107 eV/cmTO 1.8×108 eV/cm 1.4×108 eV/cmLO 1.9×108 eV/cm 1.7×108 eV/cm

68

Table 4.4: Extracted deformation potential constants for electron-phonon interaction in MoS2for electrons in the K valley [see also Fig. 4.6(f)].

Phonon Electron DeformationMomentum Transition potentials

Dac1 =4.5 eV

Γ K → KDop

0 =5.8× 108 eV/cm

Dac0 =1.4× 108 eV/cm

K′ K → K′Dop

0 =2.0× 108 eV/cm

Dac0 =9.3× 107 eV/cm

Q′ K → QDop

0 =1.9× 108 eV/cm

Dac0 =4.4× 108 eV/cm

M K → Q′Dop

0 =5.6× 108 eV/cm

69

Table 4.5: Extracted deformation potential constants for electron-phonon interaction in MoS2for electrons in the Q1 valley [see also Fig. 4.7(f)]. Multiple equivalent valleys for the final statespecify the degeneracy factor gd larger than one in Eq. (4.6).

Phonon Electron DeformationMomentum Transition potentials

Dac1 =2.8 eV

Γ Q1 → Q1 Dop0 =7.1× 108 eV/cm

Dac0 =2.1× 108 eV/cm

Q3(Q5) Q1 → Q2(Q6) Dop0 =4.8× 108 eV/cm

Dac0 =2.0× 108 eV/cm

M3(M4) Q1 → Q3(Q5) Dop0 =4.0× 108 eV/cm

Dac0 =4.8× 108 eV/cm

K′ Q1 → Q4 Dop0 =6.5× 108 eV/cm

Dac0 =1.5× 108 eV/cm

Q1 Q1 → KDop

0 =2.4× 108 eV/cm

Dac0 =4.4× 108 eV/cm

M2(M5) Q1 → K′Dop

0 =6.6× 108 eV/cm

70

-4

-2

0

2

4

K M K M

Ele

ctro

n en

ergy

(eV

)

EF

0

20

40

60

LATA

ZAPho

non

ener

gy (

meV

)

ZO

TO LO

Figure 4.1: Electronic and phononic band structures of monolayer silicene along the symmetrydirections in the FBZ. The Dirac point serves as the reference of energy scale for electrons.

71

Figure 4.2: (Color online) Electron-phonon interaction matrix elements |gvk+q,k| (in units ofeV) from the DFPT calculation in silicene for k at the conduction-band minimum K point [i.e.,(4π/3a, 0)] as a function of phonon wave vector q for all six modes v.

72

0.0 0.1

(b)

TO

ZO

LO

LA

TA

ZA

1010

1011

1012

1013

Sca

tterin

g ra

te (s

-1)

Electron energy (eV)

1014

ZA

TA

LA

ZO

LO

TO (a)

0.0 0.1 0.2

Figure 4.3: (Color online) Electron scattering rates in silicene via (a) emission and (b) absorp-tion of phonons calculated at room temperature. The electron wave vector k is assumed to bealong the K -Γ axis.

73

0

3

6

9

(a) with ZA

0 25 50 75 1000

4

8

12

Vel

ocity

(106 cm

/s)

Electric field (kV/cm)=90 cm2/Vs

(b) without ZA

Figure 4.4: (Color online) Drift velocity versus electric field in monolayer silicene obtainedfrom a Monte Carlo simulation at different temperatures: 50 K (square), 100 K (triangle), 200K (diamond), and 300 K (circle). The results in (a) consider the scattering by ZA phonons,while those in (b) do not.

74

-4

-2

0

2

0

20

40

60

Ele

ctro

n en

ergy

(eV

)

70meV1.86eV

A2"

E'

LATA

Pho

non

ener

gy (m

eV)

ZA

E"

A1

Q K M K M

Figure 4.5: Electronic and phononic band structures of monolayer MoS2 along the symmetrydirections in the FBZ. The conduction-band minimum at the K point serves as the referenceof energy scale for electrons.

75

Figure 4.6: (Color online) (a)-(e) Electron-phonon interaction matrix elements |gvk+q,k| (inunits of eV) from the DFPT calculation in MoS2 for k at the conduction-band minimum Kpoint [i.e., (4π/3a, 0)] as a function of phonon wave vector q. Only the branches with significantcontribution are plotted; i.e., TA, LA, TO(E ′), LO(E ′), and A1 (or homopolar) modes. (f)Schematic illustration of intervalley scattering for electrons in the K valley.

76

Figure 4.7: (Color online) (a)-(e) Electron-phonon interaction matrix elements |gvk+q,k| (inunits of eV) from the DFPT calculation in MoS2 for k at the Q point [i.e., Q1 ≈ (2π/3a, 0)]as a function of phonon wave vector q. Only the branches with significant contribution areplotted; i.e., TA, LA, TO(E ′), LO(E ′), and A1 (or homopolar) modes. (f) Schematic illustrationof intervalley scattering for electrons in the Q valleys.

77

0.0 0.1 0.2

1014

1013

1012

Sca

tterin

g ra

te (s

-1) LA

TA

A1

TO(E')

LO(E') (a)1011

0.0 0.1 0.2

1013

1012

1011(b)

LA

Electron energy (eV)

TAA1

TO(E')

LO(E')

Figure 4.8: (Color online) Scattering rates of K -valley electrons in MoS2 via (a) emissionand (b) absorption of phonons calculated at room temperature. The electron wave vector k isassumed to be along the K -Γ axis.

78

0.10 0.15 0.20

(a)

TO(E')

Sca

tterin

g ra

te (s

-1) LA

TA

LO(E')

A1

1012

1013

1012

1013

0.10 0.15 0.20

(b)

Electron energy (eV)

LA

TA

A1

TO(E')

LO(E')

Figure 4.9: (Color online) Scattering rates of Q-valley electrons in MoS2 via (a) emission and(b) absorption of phonons calculated at room temperature. The Q-K separation energy EQK

(= 70 meV) denotes the onset of curves as the K -valley minimum serves as the reference (zero)of energy scale. The electron wave vector k is assumed to be along the Q-Γ axis.

79

0 25 50 75 1000

2

4

6

8=4000 cm2/Vs

=1900 cm2/Vs

=460 cm2/Vs

Vel

ocity

(106 cm

/s)

Electric field (kV/cm)

50K 100K 200K 300K

=130 cm2/Vs

Figure 4.10: (Color online) Drift velocity versus electric field in monolayer MoS2 obtainedfrom a Monte Carlo simulation at different temperatures with EQK = 70 meV. When electrontransfer to the Q valleys is not considered, the mobility increases to approx. 320 cm2/Vs at 300K.

80

Chapter 5

Controlling Electron Wave

Propagation on a Topological

Insulator Surface via Proximity

Interactions

5.1 Introduction

One of the possible applications for topological insulator is realizing the electron waveguiding

analogy to optical systems, which have gained much interest in both graphene and TIs; notable

examples include electronically reconfigurable wiring, [178] low-power nanoscale device, [179]

electron waveguide networks, [180] and spin and momentum filter. [181, 182]

In graphene, the electron confinement in the waveguide can be realized by forming electro-

static potential barriers using gate voltage. Depending on the carrier types, two basic electron

guiding have been investigated: bipolar p-n junction guiding and unipolar fibre-optic guid-

ing. [178] Due to the similar band structures of TI, the same electron guiding using electro-

static potential can be realized too. Moreover, the nature of spin-momentum locking gives

81

the additional controllability over the electronic properties using ferromagnetic insulator layer

(FMI). [183, 108, 106] The proximity effect due to the exchange interaction between TI and

FMI can break the time-reversal symmetry and change the band structure. When putting the

FMI on top of TI surface, the direction of the magnetization in FMI strongly affects the electron

transmission through the channel, resulting in the nonzero magnetoresistance. [184, 109, 185] In

stead of using FMI as the gate material, we propose the electron waveguide using the proximity

effect between TI and FMI to confine electrons, and investigate the guiding operations and

possible applications under this scenario.

First, the Hamiltonian including the exchange interaction between TI and FMI is intro-

duced. Then, the principles of the electron waveguide based on the proximity effect is discussed

with the analysis of two basic operations on the electron propagation: blocking and steering.

After that, the simulation results from the finite-difference-time-domain (FDTD) [186] and

Non-equilibrium Green’s Function (NEGF) are demonstrated, [187] in order to show the elec-

tron guiding phenomena in the waveguide and quantitatively estimate the on/off ratio of elec-

tron blocking. Finally, the possible applications combing the electrical control of magnetization

switching and electron waveguide are discussed.

5.2 Principles and Methods

The exchange interaction between TI and FMI is described as a perturbation added to the

Hamiltonian for the low-energy TI surface electrons, [97, 96]

H = ~vF [σ × k] · z+ U + αM · σ, (5.1)

where k = (kx, ky) is the electron momentum, z is the unit vector normal to the TI surface, vF =

4.28×107cm/s is the Fermi velocity, [96] U is the potential energy induced by the gate voltage,

and σ = (σx, σy, σz) is the vector Pauli matrices for the electron spin. The exchange interaction

between FMI and TI is described by the last term in Eq. 5.1, αM · σ, where M = (Mx,My,Mz)

82

is the magnetization in the FMI and α is proportional to the exchange integral. [184, 109] If the

magnetization of the FMI is in z direction, the proximity effect opens up a band gap, 2αMz,

which can be easily derived by solving the eigenvalue problem. For the in-plane magnetization

cases, the proximity effect can be simply regarded as a momentum renormalization for the first

term in Eq. 5.1, with ky → ky +αMx/~vF and kx → kx−αMy/~vF . In other words, the Dirac

cone is shifted along the direction perpendicular to the magnetization.

The proposed waveguide is defined by FMI strip as shown in Figure5.1. In order to con-

fine the electron in the waveguide, the edge of the waveguide should be perpendicular to the

magnetization. Specifically, the waveguide along y direction requires the FMI strp with mag-

netization in x direction, and vice versa. The confinement of the electrons in the waveguide

can be understood by examining the momentum conservation along the boundary direction.

For the waveguide along y direction, the Dirac cone is shifted along y direction, leading to

the mismatch of momentum ky across the edge of the waveguide. When the electron tries to

go across the boundary, the momentum ky should be conserved due to the invariance along

y direction. Due to the mismatch caused by the proximity effect, the conservation can not be

satisfied. As a result, the electron is reflected and confined in the waveguide.

Given that the guiding direction is determined by the magnetization, we propose two ba-

sic operations for controlling electron propagation in the waveguide: blocking and steering, as

shown in Figure 5.2 (a) and (b), respectively. In Figure 5.2 (a), a separate control unit of FMI

is put in the waveguide, with the ability to switch the magnetization direction by 90. When

the waveguide is in ”on” state, the magnetization of the control unit is aligned with the ad-

jacent FMI’s magnetization. For ”off” state, the magnetization in the control unit is switched

by 90, either in-plane or out-of-plane. The in-plane switch, Mx → My, induces the Dirac cone

shifting perpendicular to the waveguide, while the out-of-plane switch, Mx → Mz, opens up

a band gap, both of which block the electron propagation. Figure 5.2(b) shows the schematic

where another branch waveguide along x direction is connecting to the main waveguide along y

direction through a triangular shape control unit. By aligning the magnetization of the control

83

unit to either of the adjacent FMI’s magnetization, the electron wave can propagate through

the straight line or be steered to a 90 turn going into the branch waveguide. The triangular

shape for the control unit is critical for steering the electron wave, which may need a little

more analysis to understand. When the magnetization of the control unit is aligned parallel to

the main waveguide magnetization, the band mismatch between branch and main waveguide

will dissatisfy the momentum conservation across the boundary, the side edge ”1” shown in

Figure 5.2 (b). The electron will keep propagating along the main waveguide. When the con-

trol unit steers the electron propagation, the magnetization is aligned parallel to the branch

waveguide magnetization, the boundary becomes the hypotenuse of the triangular control unit,

edge ”3”. Since on both sides of edge ”3”, the amount of Dirac cone shifting projected on the

boundary direction are the same, the momentum along the edge ”3” can be easily satisfied.

The electron wave can easily transmit into the control unit and go into the branch waveguide.

At the same time, edge ”2” blocks the electron propagation to the main waveguide. Overall,

the electron is steered by a 90 turn through the triangular control unit.

To clearly demonstrate the electron guiding phenomena, we adopt the finite-difference-time-

domain (FDTD) method to solve time dependent Dirac equation including the proximity effect.

This method has been used to successfully investigate the electron behaviour for the optical-like

dynamics, such as Klein tunnelling and Goos-Hanchen effect in graphene. [186] The simulation

is performed in the 500 nm× 300 nm region with the waveguide lying in y direction. The

absorbing boundary condition is adopted by adding decaying layers surrounding the simulation

region. The mesh size in simulation is 1×1 nm. The width of the waveguide is set to be 100 nm.

The injected electron energy is chosen to be 75 meV, with wavelength ∼ 24 nm. In order to

excite the electron wave propagating in all directions, the point source of continuous sinusoidal

wave is used.

The FDTD simulation gives very direct and intuitive picture of electron guiding and the

basic operations. However, there are some difficulties to obtain the accurate data through this

method, concerning the mesh size and reflection error from the boundary. [186] In order to

84

quantify the block on/off ratio, we use the NEGF formalism to calculate the transmission

function for the off states. [187] The calculation is done for one dimensional case, assuming the

width of the waveguide is wide enough to have negligible effect to the final results. The length

of the control unit is assumed to be 100 nm.

G(E, ky) = [E −H + iη − Σ1(E, ky)− Σ2(E, ky)]−1, (5.2)

where E is the electron energy, and η is an infinitesimal small positive number. H is the Hamil-

tonian defined in Eq. 1, with different proximity effects in the waveguide and the control unit.

Σ1 and Σ2 are the self-energies of the semi-infinite leads on the left and right, calculated using

the Sancho-Rubio iterative method. [188, 189] The transmission function can be calculated as,

T (E, ky) = trace[Γ1(E, ky)G(E, ky)Γ2(E, ky)G(E, ky)+], (5.3)

where Γ1,2 = i(Σ1,2 − Σ+1,2) is the broadening of the contact. With the calculated transmission

function, the current can be calculated using Landauer formula,

I =e

~

∫∫dE

2π

dky2π

T (E, ky)(f1 − f2), (5.4)

where f1 and f2 are the Fermi-Dirac distribution functions of the contacts, and e is the electron

charge. With the transmission probability, the channel conductance can be calculated using the

Landauer formular,

G =e2

~

∫∫dE

2π

dkx2π

T (E, kx)(−∂f0(E)/∂E)E=EF, (5.5)

where e is the electron charge, f0 is the Fermi-Dirac distribution function, and EF is the Fermi

level at the contacts.

85

5.3 Results and Discussion

5.3.1 FDTD simulation

Figure 5.3 shows the FDTD simulation results in two different waveguides formed by: (a) the

potential barrier induced by the gate voltage with height 150 meV, and (b) the proximity effect

with exchange interaction energy 150 meV. The normalized electron wave density is plotted

when the whole system gets stable after long enough simulation time. Notice that the color bar

is in logarithmic scale. The point source is located near the left end of the waveguide. Due to the

confinement of the potential barrier, the electron wave propagates to the +y direction. However,

shown in Figure 5.3(a), very strong leakage appears outside of the waveguide, indicating that

the guiding efficiency is quite lousy. Also, the y component of the leakage wave turns around to

the −y direction, which resembles optical refraction in Veselago lens. Since the barrier potential

is chosen just twice as the electron energy, the equivalent index is -1. When the electron cross

the edge of the waveguide, it goes from conduction band to valence band, which shows the

pattern similar in Graphene PN junction.[190] Due to the absence of band gap, the electron

is not completely confined by the potential barrier. Since the electron transmission is strongly

dependent on the type of the regions, guiding efficiency changes with the different potential

barrier. [178] However, in our simulation (not shown here), the leakage of electron wave is

still quite prominent with other choices of barrier heights. On the other hand, the proximity

effect induced by the exchange energy between TI and FMI can well confine the electron wave

along the magnetization direction, as shown in Figure 5.3(b). The magnitude of the leakage

wave outside the waveguide is several orders of magnitude smaller. When the electron hits the

boundary, it will see the ”vacuum” outside the waveguide, due to the mismatch of the bands.

Thus, the electron wave is reflected back into the waveguide, producing much better guiding

efficiency.

For the cases of blocking and steering electron wave propagation, the results are plotted

in Figure 5.4 (a) and (b), respectively. The blocking is realized by turning the magnetization

86

90 in-plane, from Mx to My. The length of the blocking region is set to be 100 nm, the same

as the waveguide width. A large portion of the electron wave is reflected due to the mismatch

of the band structure, while there is still very small portion is transmitted into the blocking

unit and leaks outside the waveguide region. However, behind the block unit, the wave density

is negligible showing very good blocking ratio. In Figure 5.4 (b), the triangular control unit is

used connecting the two waveguides together, with the magnetization aligned parallel with the

branch waveguide magnetization. A large portion of the wave goes in to the branch waveguide.

However, there is still small fraction leaking to the right part of the main waveguide through

the lower right corner of the control unit. In order to eliminate this portion, another blocking

unit can be added behind the control unit.

5.3.2 NEGF simulation for blocking operation

For the blocking operation, it is easy to see that the transmission probability is 1 for on states,

since there is no barrier when the control unit magnetization is aligned parallel to the waveguide

magnetization. Figure 5.5 (a) and (b) show the calculated transmission probability T (E, kx) as

a function of electron momentum kx and energy E, for the off states when the magnetization of

the control unit is My and Mz, referred as in-plane and out-of-plane, respectively. In Figure 5.5

(a), compared with the dispersion in the waveguide region, plotted in white dashed lines, the

Dirac cone in the control unit region, plotted in solid green lines, is shifted along the x direction.

In Figure 5.5 (b), the band gap induced by the proximity effect from the control unit is shown.

The overlap of these band structures determines the states with nonzero transmission coefficient.

At zeros temperature, the conductance, shown in Figure 5.5 (c), is proportional to the

transmission function for E = EF . When the Fermi level is near the Dirac cone, the conductance

is reduced by several orders of magnitude due to the mismatch of the bands. The on/off ratio

between the conductance can achieve as high as 1010 for out-of-plane switch and 107 for in-

plane switch. However, at room temperature, the effect of the thermal broadening needs to be

included. Averaging the transmission function over a few kBT energy range near the Fermi

87

level, the conductance is obtained, as shown in Figure 5.5 (d). Due to the narrow range of the

band mismatch, ∼ 75 meV for in-plane switch and ∼ 150 meV for out-of-plane switch, after

averaging, the off conductance is substantially increased compared with the zero temperature

case. The maximum on/off ratio is largely reduced, about 260 when the Fermi level lies at zero

energy. In order to increase the on/off ratio of the conductance at room temperature, we can

put another block unit behind applied with a gate voltage. Then, the band mismatch for the

second control unit is shifted in the energy axis. Two control units in series will provide the the

larger energy range for blocking the transmission, resulting in more effective shutting down.

5.3.3 Potential Applications in Logic and Interconnect circuits

We have investigated the principles of the basic operations for the electron waveguide via prox-

imity effect. In order to further consider the realistic applications, it is imperative to examine

the critical part in the control unit, switching the magnetization. This can be realized using

the magnetoelectric effect, such as spin-torque transfer, [191, 192, 193] and multiferroic mate-

rials/structures. [194, 195, 196, 197] Here, instead of going into details of these methods, we

simply assume the direction of magnetization can be contolled by the gate voltage: (1) at low

gate voltage, the direction of the magnetization, Mlow, can be either x or y, determined by the

fabrication; (2) at high gate voltage, the magnetization, Mhigh, is switched 90 either in-plane

or out-of-plane. Under this assumption, the waveguide can be well fitted into several electronic

applications. For example, the waveguide, (magnetization in FMI is Mguide), with the block

unit can be regarded as a transistor, while Mlow determines the type of the transistor, either

”n-type” ( Mlow ⊥ Mguide) or ”p-type” (Mlow ∥ Mguide). Then, an inverter can be directly built

by fabricating two block units separately on one waveguide strip. Other logic gates, such as

NAND and NOR, can be built similarly. As to the steering operation, the waveguides can real-

ize the multiplexing/demultiplexing function and reconfigurable interconnect, since the signal

path is chosen between two waveguides. Moreover, it is a natural combination of arbitration and

multiplexing if we connect multiple branches of waveguides to the single main waveguide. The

88

multiple input signals can go into the branch waveguides separately, while the output signal

comes from the main waveguide. The branch waveguides nearer to the output have higher prior-

ities, because once the control unit starts the steering operation, it not only puts the associated

input signal to the main waveguide, but also blocks the input signal behind.

5.4 Summary

In summary, we have investigated the electron guiding phenomena on the surface of topological

insulator via proximity effect by exchange interaction with the FMI. The high guiding efficiency

is compared with the gate-controlled guiding through the FDTD simulation. The leakage out-

side the waveguide is about several orders of magnitude smaller. By taking the advantage of

the magnetization switching, we also proposed two basic operations for the electron guiding:

blocking and steering. These two operations are also demonstrated using FDTD simulation.

Moreover, NEGF method is adopted to estimate the block ratio. Several electronic applications

are envisioned, such as CMOS logic, interconnect, and multiplexing/demultiplexing.

89

Figure 5.1: Principles of electron guiding in TI. The electron wave can propagate along thedirection of Dirac cone shifting, while be blocked along the perpendicular direction, as indicatedby the green arrows.

90

Figure 5.2: Device schematic for waveguiding: (a) blocking the electron wave; (b) steering theelectron wave by 90.

91

-5

-4

-3

-2

-1

0

Figure 5.3: The electron wave density from FDTD simulation for electron guided by: (a) gatevoltage; (b) proximity effect with FMI. The electron wave density is normalized by the maximumvalue. The width of the waveguide is 100nm.

92

-5

-4

-3

-2

-1

0

Figure 5.4: The electron wave density from FDTD simulation for: (a) blocking the electronwave; (b) steering the electron wave by 90.The electron wave density is normalized by themaximum value. The width of the waveguide is 100nm.

93

Figure 5.5: NEGF simulation results for the blocking operation. (a) Calculated Transmissionas a function of electron momentum kx and energy E for in-plane off state. (b) CalculatedTransmission as a function of electron momentum kx and energy E for out-of-plane off state.In (a) and (b), the dashed white lines depicts the Dirac cone in the waveguide, while the solidgreen lines depicts the Dirac cone in the control unit.(c) The on/off ratio of the conductanceversus Fermi level at 0 K. (c) The on/off ratio of the conductance versus Fermi level at 300 K.

94

Chapter 6

Summary and Future Research

To sum up, the electron transport properties in monolayer graphene and bilayer graphene are in-

vestigated through the full band Monte Carlo simulation, with intrinsic electron-phonon scatter-

ing rates calculated from first-principles. Due to the untraditional band structure of monolayer

graphene, specifically speaking, the zero effective mass, the electron-electron scattering can re-

duce the low-field mobility and saturation drift velocity by approximately a factor of 2 and 15 %,

respectively. For both monolayer and bilayer graphene, it is found that the electron velocity-

field characteristics are highly dependent on the choice of substrate at room temperature. For

monolayer graphene, the simulation also shows that the SPP scattering can potentially make

it difficult to experimentally measure the acoustic deformation potential. Additionally, bilayer

graphene shows lower mobility and saturation velocity than that in monolayer graphene, which

can be attributed to the stronger electron-acoustic phonon interaction and weaker electron-

optical phonon interaction. Although, the the ability of opening band gap in bilayer graphene

makes it a very promising material for device applications, the mobility decreases drastically

with the increase of band gap, particularly in the presence of charged impurities. This may

hinder the potential applications for bilayer graphene.

Using the same method, the intrinsic electron transport properties for monolayer silicene

and MoS2 is studied in the same procedure, with the first-principles analysis of the electron-

95

phonon interaction. The intrinsic mobility and saturation velocity of these two materials are

estimated, providing the guild lines for device applications. Moreover, the deformation potentials

for electron-phonon interaction are extracted by fitting the scattering rates, which can be further

utilized for device modelling and simulation

Besides MoS2, other transitional metal dichalcogenides (TMDs) have also drawn a lot of

attentions, such as MoSe2, WS2, WSe2. These materials have direct band gaps in monolayer

structure, while indirect band gaps in multilayer structure. [12] Recent first-principles study also

predicts that the mixed MoS2/MoSe2/MoTe2 structure are thermodynamically stable at room

temperautre. [198] The band gaps can be continuously tuned in these materials. These properties

make these materials very good candidates for electronic and optoelectronic applications. [12,

199] In future research, it is very interesting to examine the electron-phonon interaction and

electron transport properties in these materials using the same methodology develop in this

work.

The electron guiding phenomena based on proximity effect by the exchange interaction be-

tween topological insulator and ferromagnetic insulator is demonstrated using finite-difference-

time-domain (FDTD) method. We show that the guiding efficiency is much higher than tradi-

tional electron waveguide formed by electrostatic potential. The leakage electron wave density

outside the waveguide is about several orders of magnitude smaller. By taking the advantage of

the magnetization switching, the electron beam steering and flux control can be realized, which

are also demonstrated in the FDTD simulation. In particular, the on/off ratio for the electron

flux control is calculated using NEGF method. Together with the magnetization swithcing, the

electron guiding can be utilized in several potential electronic applications, such as CMOS logic,

interconnect, and multiplexing/demultiplexing.

In future research, the performance of the waveguide will be analysed more quantitatively

using NEGF method, including the calculation for two dimensional electron beam waveguiding,

steering and flux control. Moreover, together with the modelling of magnetization switching,

compact models for the proposed devices can be derived, which can be used in the circuit level

96

modelling. [200]

97

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[2] Inanc Meric, Melinda Y Han, Andrea F Young, Barbaros Ozyilmaz, Philip Kim, andKenneth L Shepard. Current saturation in zero-bandgap, top-gated graphene field-effecttransistors. Nature nanotechnology, 3(11):654–9, November 2008.

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