Valleytronics of Quantum Dots of Topological Materials

VALLEYTRONICS OF QUANTUM DOTS OF TOPOLOGICAL MATERIALS

by Mohammadhadi Azari

B.Sc., Shiraz University, 2014

Thesis Submitted in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

in the Department of Physics Faculty of Science

c Mohammadhadi Azari 2020 SIMON FRASER UNIVERSITY Spring 2020

Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval

Name: Mohammadhadi Azari

Degree: Doctor of Philosophy (Physics)

Title: VALLEYTRONICS OF QUANTUM DOTS OF TOPOLOGICAL MATERIALS

Examining Committee: Chair: Erol Girt Professor

George Kirczenow Senior Supervisor Professor

Malcolm Kennett Supervisor Associate Professor

Brad Johnson Supervisor Professor, Western Washington University

David Broun Internal Examiner Associate Professor

Sergio Ulloa External Examiner Professor, Ohio University

Date Defended: April 8, 2020

ii Abstract

The local minima (maxima) in the conduction (valence) band of crystalline materials are referred to as valleys. Similar to the role of spin in spintronics, the manipulation of the electron’s valley degree of freedom may lead to technological applications of the new field of research called valleytronics. Those crystalline solids that have two or more degenerate but well separated valleys in their band structure are considered to be potential valleytronic systems. This thesis presents a theoretical investigation of the valley degree of freedom of electrons in quantum dots of two-dimensional topological materials such as monolayer and bilayer graphene and monolayer bismuthene on SiC. To this end, a method for the calculation of the valley polarization of electrons induced by the electric current flowing through nanostructures was developed in this thesis. The method is based on a projection technique applied to states calculated by solving the Lippmann-Schwinger equation within Landauer-Büttiker theory. Applying the proposed method, this thesis addresses several valleytronic problems of current interest, including: the valley currents, valley polarization, and non-local resistances of four-terminal bilayer graphene quantum dots in the insulating regime, a valley filtering mechanism in monolayer graphene quantum dots decorated by double lines of hydrogen atoms, and the valley polarization of the edge and bulk states in quantum dots of monolayer bismuthene on SiC, a candidate for a high-temperature two-dimensional topological insulator. Keywords: Valleytronics; quantum dot; monolayer graphene; bilayer graphene; valley polarization; valley filter; two-dimensional topological insulators

iii This thesis is dedicated to my beloved parents, Fatemeh and Hamid. I am deeply thankful for their endless support and encouragement.

iv Acknowledgements

It is my pleasure to thank all the wonderful people who helped me through my graduate study, a journey started in 2014. Firstly, I owe many thanks to my supervisor George Kirczenow for his helpful suggestions, endless patience, countless ideas, and keen insights. I especially would like to thank him for his invaluable guidance in fostering my professional skills as a graduate student. He is not only a great teacher but also an outstanding supervisor who helped me in all aspects of my graduate life. I also would like to thank the members of my supervisory committee Malcolm Kennett and Brad Johnson for their helpful questions and enlightening discussions during my committee meetings. I would like to express my appreciation to Alireza Saffarzadeh who helped me to begin my research project as a new member of the nanophysics theory group and his continual support throughout my graduate study. The graduate students in the physics department of SFU and especially my officemates Aliakbar Mehdizadeh, Lavisha Jindal, Matthew Fitz- patrik, and Pau Farré deserve special acknowledgement for making our office an enjoyable place to work in. Thanks also to the amazing friends that I have made outside of SFU, who supported me in all aspects of my life while in Vancouver. Lastly, I would like to thank my family and especially my parents for their continual support, endless encouragement, and unconditional love. Without their support this journey could not be completed.

v Table of Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgements v

Table of Contents vi

List of Tables ix

List of Figures x

1 Introduction 1

2 The electronic structure of graphene 7 2.1 Introduction ...... 7 2.2 Carbon atom ...... 7 2.3 The electronic structure of monolayer graphene ...... 9 2.3.1 Tight-binding Hamiltonian and band structure of pristine monolayer graphene ...... 10 2.4 The electronic structure of bilayer graphene ...... 14 2.4.1 Tight-binding Hamiltonian and band structure of bilayer graphene . 14 2.5 Valley degree of freedom ...... 17 2.6 Breaking inversion symmetry in monolayer and bilayer graphene ...... 17 2.7 Berry phase and Berry curvature ...... 20 2.7.1 Anomalous velocity and semi-classical theory of electron transport . 22 2.8 Interpretation of non-local resistance as a signature of topological currents in monolayer and bilayer graphene ...... 26

3 Bismuthene on SiC: A high-temperature two-dimensional topological insulator 29 3.1 Introduction ...... 29

vi 3.2 Time reversal symmetry ...... 30 3.2.1 Time reversal of a spinless state ...... 31 3.2.2 Time reversal of a spin 1/2 state ...... 32 3.3 Kramer’s theorem ...... 33 3.4 Eﬀect of time reversal operator on Bloch states ...... 34 3.4.1 Spinless Bloch state ...... 34 3.4.2 Spin 1/2 Bloch state ...... 35 3.5 Quantum Hall and quantum spin Hall systems ...... 35 3.5.1 Quantum Hall state ...... 35 3.5.2 Quantum spin Hall state ...... 36 3.6 Experimental observation of the edge states in honeycomb bismuthene on SiC 40 3.7 Minimal tight-binding Hamiltonian ...... 41 3.7.1 Low-energy band structure of monolayer layer bismuthene on SiC . . 43

4 Theory of electron transport through a quantum dot 46 4.1 Introduction ...... 46 4.2 Two terminal conductance: Landauer theory of electron transport ...... 46 4.2.1 Electrodes supporting multiple conducting channels ...... 50 4.3 Multi-terminal conductor: Büttiker-Landauer theory ...... 51 4.4 Electron transmission through the nanostructures ...... 52 4.5 Valley-projected States ...... 57 4.5.1 Monolayer graphene quantum dots ...... 57 4.5.2 Bilayer graphene quantum dots ...... 58 4.5.3 Monolayer bismuthene on SiC substrate quantum dots ...... 59 4.6 Valley accumulations ...... 60 4.7 Valley velocity ...... 60

5 Gate-tunable valley-dependent transport properties of electrons in bilayer graphene nanostructures 63 5.1 Introduction ...... 63 5.1.1 Model ...... 66 5.2 Non-local resistance ...... 66 5.3 Valley currents ...... 68 5.4 Gate-tunabiliy of the non-local resistance and valley current ...... 70 5.5 Scaling relation between the local and non-local resistances ...... 71 5.6 Spatial distribution of valley accumulations and valley currents ...... 71 5.7 Summary ...... 74

6 Graphene valley ﬁlter, accumulator and switch 76 6.1 Introduction ...... 76

vii 6.2 Model ...... 77 6.3 Two-terminal conductance calculations ...... 79 6.4 The role of the band structure ...... 81 6.5 Valley ﬁltering and valley accumulation ...... 87 6.6 Summary ...... 89

7 Valley and spin polarizations of the edge and bulk states in quantum dots of the topological insulator monolayer bismuthene on SiC 92 7.1 Introduction ...... 92 7.2 Model ...... 93 7.3 Valley polarization of the edge states ...... 94 7.3.1 Valley polarization reversal of the edge states ...... 97 7.4 Spin polarization of the edge states ...... 100 7.5 Eﬀect of a nonuniform electrostatic potential on the valley and spin polarizations of the edge and bulk states ...... 102 7.6 Valley and spin correlation ...... 106 7.7 Summary ...... 108

8 Conclusions and Outlook 110

Bibliography 113

viii List of Tables

Table 3.1 Nearest neighbour parameters Hiα,i0α0 . The ﬁtting values are Σ = −0.81 eV, Σ0 = −1.00 eV, Σ00 = −1.57 eV, Π = 0.55 eV. The matrix elements depend on the x and y coordinates of i and i0 near- −1 0 est neighbour Bi atoms through θ = sin [(y − y)/dii0 ] where dii0 = [(x0 − x)2 + (y0 − y)2]1/2. Adapted from Ref.[71] ...... 43 Table 3.2 Matrix elements of the atomic spin-orbit Hamiltonian matrix [Eq.3.43]. The matrix elements are zero when α, α0 = s. Adapted from Ref.[71] . 43 R Table 3.3 The matrix elements Hαs,α0s0 with ﬁtting parameter value R = 0.395 eV. Adapted from Ref.[71] ...... 44

ix List of Figures

Figure 1.1 (a) The honeycomb lattice structure of monolayer graphene. The lattice is composed of two triangular sublattices. The carbon atoms belonging to sublattice A (B) are represented by red (blue) dots. (b) The lattice structure of bilayer graphene in the AB stacking. The top (bottom) layer is shown in blue (black) and the carbon atoms (not shown) are located on the vertices of hexagons in both top and bottom layers. Inset: Side view of the four carbon atoms in the unit- cell of bilayer graphene in AB stacking...... 3

Figure 2.1 The electronic conﬁguration of the (a) ground state and (b) excited state of carbon atom. (c), (d), and (e) show the sp3, sp2, and sp hybridization processes in carbon atom, respectively. (f), (g), and (h) show the molecular geometry of methane, ethylene, and acetylene, respectively...... 8 Figure 2.2 (a) The hexagonal lattice structure of monolayer graphene with rhombic unit-cell (green rhombus) that contains two carbon atoms. The carbon atoms belonging to sublattice A (B) are shown by red (blue)

dots. a1 and a2 are the primitive lattice vectors and δ1, δ2, and

δ3 are the nearest neighbour vectors of the monolayer graphene lattice structure. (b) Reciprocal lattice structure of monolayer graphene

with hexagonal ﬁrst Brillouin zone shaded in blue. b1 and b2 are the primitive reciprocal lattice vectors...... 9 Figure 2.3 The electronic band structure of inﬁnite pristine monolayer graphene within the nearest neighbour approximation. Dotted red line shows the Fermi energy of charge neutral monolayer graphene...... 13 Figure 2.4 (a) Lattice structure of bilayer graphene for AB stacking with rhombic unit-cell (green rhombus) that contains four carbon atoms. Top

(bottom) layer is shown in blue (black). a1 and a2 are the primitive lattice vectors. (b) Hexagonal Brillouin zone of bilayer graphene. . . 16 Figure 2.5 The electronic band structure of inﬁnite pristine AB-stacked bilayer graphene within the nearest neighbour approximation...... 16

x Figure 2.6 The electronic band structure of inﬁnite monolayer graphene with broken inversion symmetry. A gap of the order of 2∆ is opened between the valence and conduction bands at point K. Inset: Schematic representation of the mechanism of breaking inversion symmetry in monolayer graphene...... 18 Figure 2.7 The electronic band structure of inﬁnite bilayer graphene with bro-

ken inversion symmetry. A gap of the order of Vg is opened between the valence and conduction bands at point K. Inset: Side view of the mechanism of breaking inversion symmetry in bilayer graphene. . . 19 Figure 2.8 The k-space contour used for the calculation of the line integral in Eq.2.58...... 24 Figure 2.9 (a) Non-local resistance (red curve) and longitudinal resistance (black curve) measured in monolayer graphene on hBN superlattices. Left inset: Schematic representation of the electronic band structure of monolayer graphene supperlattices with Berry curvature hot spots near the opened gap between the valence and conduction bands. Right Inset: Valley Hall conductivity modeled for graphene with broken inversion symmetry. Adapted from Ref.[30]. (b) Schematic representation of the multi-terminal monolayer graphene superlattice with

the geometry appropriate for non-local resistance RNL measurements. 27

Figure 3.1 States |±i (not shown) have spins tilted along the positive and negative nˆ direction...... 32 Figure 3.2 Schematic representation of a quantum Hall system with spatially separated edge channels propagating in the upward (red) and downward (blue) directions. The backscattering caused by disorder (yellow disks) is suppressed in these chiral channels...... 35 Figure 3.3 Top inset: Schematic representation of a quantum spin Hall system with two conducting channels (red in the upward and blue in the downward directions) with opposite spins propagating at each edge. The yellow arrow shows the direction in which the electric current is ﬂowing. Bottom inset: Copies of each band at the two edges representing the gapless edge states in the bulk energy band gap schematically. Solid (dotted) lines represent occupied (unoccupied) states at zero temperature. The net electric current ﬂowing in the upward direction is carried out by the solid red band...... 37

xi Figure 3.4 Disordered (grey) region characterized by the Hamiltonian V sandwiched between two disorder-free (blue) regions. The edge states with spin up (down) propagating in opposite directions along each region of the system are shown by red (green) arrows for N = 1...... 38 Figure 3.5 (a) Left inset: Planar honeycomb lattice structure of inﬁnite two- dimensional monolayer bismuthene (blue) on SiC (not shown). Right inset: Hexagonal ﬁrst Brillouin zone of bismuthene on SiC. (b) Low- energy band structure of two-dimensional monolayer bismuthene on SiC calculated within the minimal tight-binding Hamiltonian [Eq.3.42]. Energies are measured from the maximum of the valence band E = 0. Adapted from Ref.[71] ...... 42 Figure 3.6 Low-energy band structure of (a) zigzag (b) armchair bismuthene nanoribbons on SiC. The gapless edge states in the bulk band gap are shown in red. Adapted from Ref.[71] ...... 44

Figure 4.1 The schematic representation of a mesoscopic one-dimensional ballistic conductor of length L connected to the electron source and drain

electrodes with electrochemical potentials µs and µd, respectively. s(d),in Im is the longitudinal electric current carried by the transverse mth mode of the conductor injected from the source (drain) electrode. The diﬀerence between the electric currents injected from the source and drain electrodes gives the net longitudinal electric current m Inet ﬂowing through the conductor...... 47 Figure 4.2 Schematic representation of a single mode in a conductor and the occupied states that carry the electric current injected from the source and drain electrodes at zero temperature with electrochemical po-

tentials of µs and µd, respectively. The states with wave vector k

between k and ks (kd), the green (purple) segment of the band, carry the electric current through the conductor injected from the source (drain) electrode. The net electric current is carried by the occupied

states with energies in the range µd < E < µs (red segment of the band)...... 49 Figure 4.3 Ballistic conductor of length L connected to the source and drain electrodes modeled as groups of ideal semi-inﬁnite one-dimensional leads (wavy orange lines)...... 50 Figure 4.4 Ballistic conductor connected to four contact (each at its own elec-

trochemical potential µi) modeled as groups of ideal semi-inﬁnite one-dimensional leads (wavy orange lines)...... 51

xii Figure 4.5 (a)The hexagonal and rhombic Brillouin zone of monolayer graphene, bilayer graphene, and planar bismuthene on SiC. (b) The rhombic Brillouin zone is divided into two parts. Mesh of k-points belonging to valley K (K0) represented by blue (red) dots...... 58

Figure 5.1 Four terminal bilayer graphene nanostructure with armchair edges. The bottom (top) layer is shown in black (blue). Each contact is composed of 40 semi-infinite one-dimensional ideal leads (shown by red wavy lines) that are attached to both layers and connect the nanostructure to the reservoirs. The electric current flows through current contacts 1 and 2 (C.C.1 and C.C.2), while there is no net electric current entering or leaving the voltage contacts 3 and 4 (V.C.3 and V.C.4). In non-local resistance studies the potential difference is measured between contacts 3 and 4. Upper right inset: Two types of first Brillouin zone of bilayer graphene, hexagonal (solid) and rhombic (dotted). Lower right inset: Side view of the four carbon atoms of a unit cell in bilayer graphene in AB stacking. The inversion symmetry point is shown by a red dot...... 65

Figure 5.2 Calculated non-local resistance RNL [Eq.5.2] of the nanostructure of Fig.5.1 in the linear response regime at zero temperature for diﬀerent

values of the gate voltage Vg = 0 eV (green), Vg = 0.3 eV (red), and

Vg = 0.5 eV (blue) as a function of the Fermi energy EF ...... 67 Figure 5.3 (a) Schematic representation of the bilayer graphene nanostructure val with the directions of the ordinary vy and valley vx velocities of electrons flowing through the quantum dot. (b) Normalized valley val velocity vx /vy of the bilayer graphene nanostructure for different values of the gate voltage as a function of the Fermi energy when the net electric current flows between current contacts 1 and 2...... 68 Figure 5.4 Comparison of the x and y components of the normalized valley

velocity as function of the Fermi energy (a) at Vg = 0.5 eV and (b)

Vg = 0.3 eV...... 69 val Figure 5.5 The normalized valley velocity (vx /vy) (purple line) and non-local resistance RNL (green line) of the bilayer graphene nanostructure as

a function of gate voltage at zero Fermi energy EF = 0...... 70 Figure 5.6 The scaling relation between the local and non-local resistance as the gate voltage varies from 0 to 0.2 eV. Red line is the power law α RNL ∼ RL fitted to the simulation data. Upper left inset: The configuration of non-local resistance measurements. Lower right inset: The configuration of local resistance measurements...... 72

xiii v Figure 5.7 The calculated on-site valley polarization Pn of electrons belonging to the bottom layer of the bilayer graphene nanostructure when the

net electric current ﬂows between current contact 1 and 2 at Vg =

0.15 eV and EF = 0 eV. The on-site valley polarization at each atomic site is shown by blue (red) disks when it is positive (negative). 72 Figure 5.8 The calculated valley velocity (green arrow) and unit cell-averaged

valley polarization Pav(x, y) of the bottom layer of bilayer graphene

nanostructure at Vg = 0.15 eV and EF = 0 eV. The unit cell-averaged valley polarization at each unit is shown by blue (red) disks when the it is positive (negative)...... 73

Figure 6.1 (a) Two-terminal monolayer graphene quantum dot with armchair edges. The source and drain contacts are modeled by groups of semi- inﬁnite one-dimensional leads (orange wavy lines). Each contact is composed of 46 leads. Two lines of hydrogen atoms (yellow disks) are adsorbed on top of the monolayer graphene and divide the quantum dot into two equal parts. (b) Relaxed geometry of a single adsorbed hydrogen atom on graphene. The hydrogen and the carbon atom to which hydrogen binds are placed 1.47 and 0.32 Å above the graphene plane. (c) Hexagonal and rhombic representations of the Brillouin zone of monolayer grpahene. Parts (a) and (b) are reproduced from Ref.[89] ...... 78 Figure 6.2 Calculated two-terminal conductance G [Eq.4.12] of the graphene quantum dot in Fig.6.1 with (blue curve) and without (red curve) hydrogen lines in the linear response regime at zero temperature for the symmetric case (∆ = 0), vs. Fermi energy. Inset: The ﬁne structure of the calculated pronounced peak...... 80 Figure 6.3 Calculated two-terminal conductance G [Eq.4.12] of the graphene quantum dot in Fig.6.1 with (blue curve) and without (red curve) hydrogen lines in the linear response regime at zero temperature for

the asymmetric case (∆r = −∆l), vs. Fermi energy. Insets: The fine structures of the calculated pronounced peaks...... 82 Figure 6.4 Infinite zigzag graphene nanoribbon with adsorbed double line of hydrogen atoms (yellow disks). The lines of hydrogen atoms on the ribbon are infinite and run along the center of the ribbon in the y-direction...... 83

xiv Figure 6.5 Band structure of zigzag graphene nanoribbon with adsorbed lines of hydrogen atoms in Fig.6.4 and symmetric on-site energies of the carbon atoms (∆ = 0). States localized near the hydrogen lines (H- band) are shown in red and other graphene states (G-bands) are shown in taupe. The region belonging to valley K (K0) is shaded in blue (red). The valleys K and K0 are indicated at the top of their corresponding regions...... 85 Figure 6.6 Band structure of zigzag graphene nanoribbon with adsorbed lines of hydrogen atoms in Fig.6.4 and asymmetric on-site energies of the

carbon atoms (∆r = −∆l). States localized near the hydrogen lines (H-bands) are shown in red and blue. Other graphene states (G- bands) are shown in taupe. The region belonging to valley K (K0) is shaded in blue (red). The valleys K and K0 are indicated at the top of their corresponding regions...... 86 Figure 6.7 (a) Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed

lines of hydrogen atoms (Fig.6.1) for the symmetric case at EF = −0.0065 eV . Calculated electric current-induced on-site valley accu- 0 K (K) 0 mulations of electrons An in valley K (K) near hydrogen lines shown as red and (much smaller blue) disks. Diameters of the red K0(K) (blue) disks are proportional the calculated An . (b) Calculated square amplitude (shown in red) of a representative electron H-band eigenstate (red band in Fig.6.5) of the zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulations of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown...... 87 Figure 6.8 (a) Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed

lines of hydrogen atoms (Fig.6.1) for the asymmetric case at EF = 0.0502 eV . Calculated electric current-induced on-site valley accu- 0 K (K) 0 mulations of electrons An in valley K (K) near hydrogen lines shown as red and (much smaller blue) disks. Diameters of the red K0(K) (blue) disks are proportional the calculated An . (b) Calculated square amplitude (shown in red) of a representative electron H-band eigenstate (red band in Fig.6.6) of zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulations of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown...... 88

xv Figure 6.9 Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed

lines of hydrogen atoms (Fig.6.1) for the asymmetric case at EF = −0.0638 eV . Calculated electric current-induced on-site valley accu- 0 K (K) 0 mulations of electrons An in valley K (K) near hydrogen lines shown as red (blue) disks. Diameters of the red (blue) disks are pro- K0(K) portional the calculated An . (b) Calculated square amplitude (shown in red) of a representative electron H-band eigenstate (blue band in Fig.6.6) of zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulation of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown...... 90

Figure 7.1 (a) The bismuthene planar honeycomb lattice (black) of the two- terminal monolayer bismuthene quantum dot with contacts attached to the armchair edges. Each contact is composed of 38 ideal semi- infinite one-dimensional leads (shown by orange wavy lines). (b) Two alternative representations of the first Brillouin zone of monolayer bismuthene, hexagonal (solid line) and rhombic (dotted line). This figure is reproduced from Ref.[92] ...... 93 v Figure 7.2 Spatial map of the on-site valley polarization Pn calculated at EF = 0.10 eV. The on-site valley polarization is shown by blue disks when v the calculated Pn is negative. The diameters of the disks are propor- v tional to the magnitude of Pn . Note that the valence (conduction) band edge is at 0 (0.86)eV...... 95 v Figure 7.3 Spatial map of the on-site valley polarization Pn calculated at EF = 0.16 eV. The on-site valley polarization is shown by red disks when v the calculated Pn is positive. The diameters of the disks are propor- v tional to the magnitude of Pn . Note that the valence (conduction) band edge is at 0 (0.86)eV...... 96 Figure 7.4 (a) The on-site valley polarization calculated on the chain of Bi atoms shown in part (b), for different values of the Fermi energy. (b) Chain of Bi atoms that extends in the x-direction from the left zigzag edge to the center of the quantum dot...... 97 Figure 7.5 The strongest on-site valley polarization of the zigzag edge states as a function of the Fermi energy. The blue (red) arrow locates the top of the valence band at zero eV (bottom of the conduction band at v Pmax 0.86 eV). Positive (negative) values of ∆µ correspond to transport of electrons belonging to valley K(K0)...... 98

xvi Figure 7.6 The band structure of the bismuthene on SiC nanoribbon with zigzag edges. Adapted from Ref.[71]. The gapless edge states in the bulk energy band gap with positive velocity are shown by the straight blue and red lines so that the states polarized in valley K0 are shown in blue and the states with K valley polarization are shown in red. 99 Figure 7.7 Spatial map of the on-site out-of-plane component (z-direction) of s s the spin polarization Pnz calculated at EF = 0.10 eV. Pnz is shown by red (blue) disks if it is positive (negative). The diameters of the s disks are proportional to the magnitude of Pnz...... 101 Figure 7.8 Spatial map of the on-site out-of-plane component (z-direction) of s the current induced spin polarization Pnz calculated at EF = −0.09 s eV (Fermi level inside the valence band). Pnz is shown by red (blue) disks if it is positive (negative). The diameters of the disks are pro- s portional to the magnitude of Pnz...... 103 Figure 7.9 (a) The proﬁle of the potential energy increasing in the y-direction (parallel to the zigzag edge). (b) Schematic representation of the top of valence band (olive green line) and bottom of the conduction band (purple line) when the variable potential energy is included in the model. Energies are measured from the top of the valence band at y = −85.6 Å where V (y) = 0. The orange horizontal line locates the Fermi energy for the cases considered in Figs.7.10 and 7.11. . . . . 104 Figure 7.10 The spatial map of the out-of-plane component of the current in-

duced spin polarization hSzi when the model potential energy [Eq.7.2]

is applied for EF = 0.435 eV. Electron flow is from the source to the drain. The region where the Fermi energy lies in (out of) the bulk energy band gap is represented by the orange (black) coloring. Note that the spin polarizations in the orange and black regions are plotted on different scales for clarity. The diameters of the disks plotted in the orange region representing the spin polarizations are scaled up by a factor 5 relative to those in black region...... 105 v Figure 7.11 The spatial map of the current induced valley polarization Pn when the model potential energy [Eq.7.2] is applied for EF = 0.435 eV. Electron flow is from the source to the drain. The region where the Fermi energy lies in (out of) the bulk energy band gap is represented by the orange (black) coloring. Note that the valley polarizations in the orange and black regions are plotted on different scales for clarity. The diameters of the disks plotted in the orange region representing the valley polarizations are scaled up by a factor 7 relative to those in black region...... 107

xvii Figure 7.12 (a) Expectation values of the out-of-plane component of the spin of electrons occupying valence band Bloch states of the inﬁnite 2D crystal of bismuthene on SiC along a line in k space. K, K0, and M are as in Fig.7.1(a). The expectation values are evaluated at the two Bi atomic sites in the crystal unit-cell. The black dashed line separates the regions of k belonging to valley K and K0. (b) Schematic representation of the unit-cells of the crystal (green rhombi) showing the out-of-plane atomic spin polarizations from part (a) (olive green and purple arrows shown in the top rhombi) for electrons in valley K and K0. The corresponding valley polarizations at the atomic sites are indicated by the red and blue disks shown in the bottom rhombi, respectively...... 108

xviii Chapter 1

Introduction

Dimensionality plays a decisive role in affecting the characteristics of crystalline materials. Using synthetic procedures to arrange chemical compounds in 0D, 1D, 2D, or 3D crystal structures illustrates the fact that dimensionality modifies the properties of crystalline materials dramatically [1, 2, 3, 4, 5, 6, 7]. As a consequence, investigating the properties of a chemical compound in different dimensions has benefits due to the fact that in each dimension the same chemical compound offers unique properties that are not accessible in other dimensions. An example is the high carrier mobility at room temperature in monolayer graphene [8] which is a 2D form of graphite. Furthermore, the monolayer form of the molybdenite mineral (MoS2 and MoSe2) becomes a direct band gap material [9] that can efficiently absorb and emit light due to increased light-matter interaction in the 2D form of this crystalline solid [10, 11]. Two dimensional materials are a group of single layer substances which usually are exfoliated from layered structures using chemical or mechanical methods. The physics of the 2D materials is interesting due to the confinement of electrons which gives rise to dramatically different electronic and optical properties that are not exhibited by the same chemical compound in the bulk form [9, 10, 11]. Furthermore, most of the 2D crystalline solids are exfoliated from layered structures with strong in-plane chemical bonds and weak van der Waals coupling between the layers. So, the absence of the interlayer interactions (despite being quite weak in 3D parent materials) alters the electronic band structure of the 2D materials significantly. In 2004, Andre Geim and Konstantin Novoselov succeeded in isolating a single layer of carbon atoms by means of mechanical cleavage [12] and they were awarded the 2010 physics Nobel prize. They also showed that this one atom-thick layer of carbon atoms called graphene is stable under ambient conditions. Carbon atom is the sixth element of the periodic table and has four valence electrons which allows the carbon to form different bonding types and react with a variety of atomic species to form diverse chemical compounds. The isolation of a single layer of carbon atoms from bulk graphite via mechanical cleavage opened a new chapter in condensed matter physics. Owing to its peculiar electronic band structure, graphene offers a wide variety of

1 unique optical and electronic properties. Among all the carbon allotropes, graphene is the main building block of a wide range of carbon substances like graphite (3D) [13], carbon nanotubes [14, 15] (1D), and fullerenes (0D) [16]. Several extraordinary properties such as the excellent electrical and thermal conductivities at room temperature [12, 17, 18, 19, 20], transparency [21], high room temperature electron mobility [22, 8], and complete imperme- ability to gases [23] have persuaded the scientific community to consider graphene as one of the most useful functional materials. Furthermore, in graphene samples the charge carriers can be controlled readily by applying external electric and magnetic fields, or changing the geometry of the sample which may lead to diverse potential technological applications such as chemical sensors [47], optical devices [48], DNA sequencing [46], and high-speed electronics [45]. Monolayer graphene is composed of a hexagonal lattice structure so that the carbon atoms are placed at the vertices of hexagons as represented in Fig.1.1 (a). Layered structures of carbon atoms can be synthesized by stacking single layers of graphene. For instance, bilayer graphene consists of two electronically coupled single graphene layers as is shown in Fig.1.1 (b). While the bonding between the two layers of carbon atoms is weak relative to the intra-layer σ bonds, it nevertheless gives rise to distinct differences between the band structures of monolayer and bilayer graphene. Topological insulators are another group of paradigm shifting materials which has im- pacted the field of condensed matter physics significantly. This group of materials is insulating in the bulk but conducting at the edges or surfaces, depending on whether they are 2D or 3D, respectively. In 1980, von Klitzing et al. [24] discovered a quantum state belonging to a topological class called the quantum Hall (QH) state, which emerges when the time reversal (TR) symmetry is broken by a strong magnetic field. In this state, the electric current flows along the edges of the 2D samples while the material is insulating in the bulk. In the recent years, a new class of topological quantum states that are invariant under time reversal has been predicted theoretically [25, 27, 28] and observed experimentally [26]. This state has a bulk energy band gap generated by strong spin-orbit coupling (SOC) together with topologically protected gapless edge states or surface states which distinguishes topological insulators from the ordinary insulators. The 2D topological insulators are called quantum spin Hall (QSH) systems since they carry spin currents along the edges. QSH insulator was first predicted theoretically by Kane and Mele [27] and Bernevig et al. [25] independently. The unique electronic properties of the gapless edge states such as dissipationless edge currents [27, 28], quantized conductance G = 2e2/h [27, 28], and robustness against TR invariant disorder [27, 28] make QSH materials potential candidates for practical applications in spintronics and valleytronics. The present day semiconductor technology manipulates the eletcronic charge and and spin as a degrees of freedom to store, encode, and process information. Another degree of freedom with which an electron can be endowed is called the valley degree of freedom. In

2 Figure 1.1: (a) The honeycomb lattice structure of monolayer graphene. The lattice is composed of two triangular sublattices. The carbon atoms belonging to sublattice A (B) are represented by red (blue) dots. (b) The lattice structure of bilayer graphene in the AB stacking. The top (bottom) layer is shown in blue (black) and the carbon atoms (not shown) are located on the vertices of hexagons in both top and bottom layers. Inset: Side view of the four carbon atoms in the unit-cell of bilayer graphene in AB stacking. crystalline materials, the band structure governs the relationship between the energy of an electron and its momentum. The local minima (maxima) in the conduction (valence) band are referred to as valleys. Similar to spin in spintronics, the manipulation of the electron’s valley degree of freedom may lead to technological applications of the new field of research called valleytronics. Those crystalline solids that have two or more degenerate but well separated valleys in their band structure are considered to be promising valleytronic systems. In recent years, the idea of exploiting the valley degree of freedom as a further degree of freedom for the electrons in 2D topological materials has attracted intense attention in the field of condensed matter physics both theoretically [44, 37, 34, 50, 52, 105, 35, 107, 108, 87, 109] and experimentally [49, 51, 30, 31, 32, 101, 102, 103, 104, 106, 36]. Examples of substances that satisfy the requirement of this field - two or more degenerate but well separated valleys in their band structure - are monolayer and bilayer graphene, transition metal dichalcogenides (TMDs), and the recently discovered bismuthene on a SiC substrate, a high temperature 2D topological insulator [29]. Recently Gorbachev et al. [30] have carried out an experiment measuring the non-local four terminal resistances RNL of monolayer graphene on hexagonal boron nitride (hBN) substrate. They measured a striking enhancement in RNL as the Fermi energy passes the energy of Dirac point. According to the experimental results reported by Gorbachev et al., the presence of the hBN substrate opens a band gap in the band structure of monolayer graphene as well as breaking the inversion symmetry of the sample. While Gorbachev et al. did not measure valley currents directly, they interpreted the strong enhancement in the

3 measured RNL as a signature of the presence of valley currents in the sample based on a semi-classical theory of electron transport. A disadvantage of the application of monolayer graphene in valleytronics is the requirement of breaking the inversion symmetry by precise alignment of graphene and hBN. This issue has been resolved in bilayer graphene by applying top and back gates which leads to control of the Fermi energy as well as breaking of the inversion symmetry by the electric field induced by the gates. Sui et al. [31] and Shimazaki et al. [32] have carried out experiments measuring the RNL of bilayer graphene where the inversion symmetry is broken by applying an electric field perpendicular to the plane of bilayer graphene. Similar to the case of monolayer graphene, a strong enhancement in RNL was observed as the Fermi energy passed the Dirac point. Shimazaki et al. have interpreted their experimental results based on the same semi-classical theory of electron transport proposed by Gorbachev et al.. However, using the semi-classical theory of electron transport in this way is questionable in both the monolayer and bilayer graphene cases due to the fact that the strong enhancement in RNL was observed when the Fermi energy lies in the band gap opened during the process of breaking inversion symmetry in monolayer and bilayer graphene by using hBN substrate and applying gate voltages, respectively. Consequently, the transport mechanism of electrons is quantum tunneling and application of this semi- classical theory of electron transport in the insulating regime is questionable. Hence, I have investigated the RNL in four-terminal bilayer graphene nanostructures in the insulating regime from the perspective of quantum scattering theory. This thesis attempts to provide a comprehensive understanding of the non-local transport coefficients of electrons in bilayer graphene in the linear response regime, the valley filtering mechanisms in monolayer graphene decorated by adsorbed lines of hydrogen atoms, and valley polarization and spin polarization of the edge and bulk states in monolayer bismuthene on SiC substrates, a promising candidate for a high temperature topological insulator. I have developed a novel approach to study valley polarization of the scattering quantum states calculated for different nanostructures. To this end, I calculate the scattering states of electrons through the nanostructure by solving the Lippman-Schwinger equation and projecting them on the Bloch subspaces of electrons belonging to valley K or K0. In order to study the non-local transport of electrons flowing through the nanostructure, I have used the Büttiker-Landauer theory of electron transport which is a fully quantum mechanical approach. Moreover, in this thesis novel valley filtering and switching mechanisms are suggested in monolayer graphene quantum dots decorated by adsorbed lines of hydrogen atoms, which compare favorably with previously suggested mechanisms [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Furthermore, I have generalized my previously developed theoretical method of the calculation of the valley polarization of the electron scattering states to multi-orbital topological insulator nanostructures. The remainder of this thesis is organized in 7 chapters. Chapter 2 summarizes the electronic properties of ideal monolayer and bilayer graphene. It begins with a brief description

4 of the general properties of carbon atom, the sixth chemical element of the periodic table. Then, the tight binding Hamiltonians which describe monolayer and bilayer graphene and their unique electronic band structures close to K and K0 points in the first Brillouin zone are explained in detail. Furthermore, the topological concept of the Berry phase as a geometrical phase acquired by the quantum state and its effect on the velocity of electrons from the perspective the semi-classical theory of electron transport is also covered in chapter 2. Finally, this chapter ends with a summary of three experiments which have been carried on monolayer and bilayer graphene nanostructures [30, 31, 32] investigating the presence of topological valley currents when the inversion symmetry of the sample is broken. Chapter 3 begins with a review of the introductory concepts required for understanding QSH properties, such as the time reversal symmetry, Kramer’s theorem, and suppression of backscattering of edge states due to time-reversal symmetry. Then, I discuss the experiment [29] that has been carried out on bismuthene on SiC and showed that this system can be regarded as a high temperature 2D topological insulator. At the end of this chapter I review a minimal tight-binding Hamiltonian that captures the key properties of the low-energy band structure of bismuthene on SiC quantitatively. This minimal tight-binding Hamiltonian was developed in Ref.[71] and has been used in this thesis to investigate the valley and spin polarizations of the edge and bulk states of the quantum dots of high temperature 2D topological insulator bismuthene on SiC. In chapter 4, I describe the methodology which has been used in order to study the local and non-local transport of electrons, valley polarization and valley currents in monolayer and bilayer graphene, and the valley polarization of the edge states in bismuthene on SiC substrate. To this end, the first section of this chapter begins with a detailed explanation of the solution of the Lippmann-Schwinger equation, the application of two-terminal Landauer theory of electron transport, and its generalization to four-terminal calculations which is referred to as Büttiker-Landauer theory. The second section contains the description of the method of the valley projection of the calculated scattering states in graphene quantum dots and its generalization to the multi-orbital bismuthene on SiC substrate nanostructure. The fifth chapter includes my results of the calculation of non-local four terminal resistances of bilayer graphene quantum dots in the insulating regime. Furthermore, a method of the calculation of the valley currents in monolayer graphene nanostructures developed in Ref.[44] has been exploited and extended to the bilayer graphene nanostructure to investigate the valley currents of electrons in bilayer graphene quantum dots. In order to investigate the correlation between the valley polarization of the scattering states and the valley currents flowing through the bilayer nanostructure, I have developed a method for calculating the valley polarization which has not been done in previous studies in this area of research. Mapping the average valley polarization calculated in the bilayer graphene nanostructure with broken inversion symmetry reveals a dipolar distribution that results

5 from the presence of valley currents so that the average valley polarization increases in the direction of valley currents. In chapter 6, I present my results for the valley filtering and switching mechanisms in monolayer graphene nanostructures decorated by adsorbed lines of hydrogen atoms. Here, I have calculated the two terminal conductance of a decorated monolayer graphene quantum dot (the lines of hydrogen atoms are adsorbed on the middle of the quantum dot) which is connected to the source and drain contacts. The plotted band structure of a zigzag graphene nanoribbon which is decorated by lines of hydrogen atoms in the same way as monolayer graphene quantum dot is decorated demonstrates that the presence of the lines of hydrogen atoms leads to a conducting channel close to the energy of the Dirac point. It is shown that the conducting channel induced by lines of hydrogen atoms offers a novel valley filtering mechanism. Importantly, this mechanism is realistic and this nanostructure can be fabricated in the laboratory. The results of the calculation of the valley polarization of the edge and bulk states in bismuthene on SiC are presented in chapter 7. Recently, Reis et al. have suggested monolayer bismuthene on SiC to be a promising high temperature quantum spin Hall system due to the large bulk energy band gap of 0.8 eV that this system possesses [29]. My calculations reveal that the gapless edge states of this system are valley polarized. I have shown that the valley polarization of the edge states switches from valley K0 to K as the Fermi energy increases from the top of the valence band to the bottom of the conduction band in the bulk energy band gap. Furthermore, I have investigated the relation between the valley and spin degree of freedom of electrons in this quantum spin Hall system in the linear response regime. According to my calculations, while the spin polarization of the edge states is ferromagnetic (characteristic of quantum spin Hall systems), the spin polarization of the bulk scattering states exhibits antiferromagnetic character. The origin of the antiferromagnetic character of the bulk scattering states will be explained in detail in this chapter. This thesis is concluded with a summary of what has been learned and a discussion of possible directions that can be taken for future research.

6 Chapter 2

The electronic structure of graphene

2.1 Introduction

This chapter reviews the electronic structures of pristine monolayer and bilayer graphene and the experiments that have been recently carried out on monolayer [30] and bilayer [32, 31] graphene in order to detect the valley currents when the inversion symmetry of the sample is broken. It begins with a brief description of the carbon atom as the sixth element of the periodic table. Then, it introduces the nearest neighbour tight-binding Hamiltonian and the electronic band structure of pristine monolayer and bilayer graphene. The concept of Berry phase, its eﬀect on the electron’s velocity, and the methods of breaking the inversion symmetry are also discussed. Finally, this chapter is concluded with a summary of three experiments carried out on monolayer and bilayer graphene and the semi-classical theory of electron transport which has been used to interpret the enhancement in the measured non-local resistance as a signature of the presence of valley currents in these systems.

2.2 Carbon atom

Carbon is the sixth chemical element of the periodic table and has two stable isotopes 12C and 13C. The diversity of the elements that can form chemical bonds with carbon atom, gives rise to many chemical compounds with favourable distinct physical properties. The carbon atom has six electrons and its ground state electronic conﬁguration is 1s22s22p2 as represented in Fig.2.1 (a). For the carbon atom, only those electrons occupying the orbitals of the 2s and 2p shells participate in chemical bond formation. Although at ﬁrst sight it seems that a carbon atom has two valence electrons (2p2) that can form chemical bonds, carbon has the ability to form more than two chemical bonds due to the formation of the hybridized sp states that are energetically favorable. According to the fact that chemical bond formation decreases the energy of the systems, the atoms participating in the process of chemical bond formation rearrange their valence electrons and form hybridized orbitals

7 Figure 2.1: The electronic configuration of the (a) ground state and (b) excited state of carbon atom. (c), (d), and (e) show the sp3, sp2, and sp hybridization processes in carbon atom, respectively. (f), (g), and (h) show the molecular geometry of methane, ethylene, and acetylene, respectively. in order to maximize the number of bonds that are formed and minimize the energy of the system. In the case of the carbon atom, one electron may be promoted so that it occupies the empty orbital of the 2p (Fig.2.1 (b)) shell which leads to the formation of sp3, sp2, and sp hybrid orbitals as represented in Fig.2.1 (c), (d), and (e), respectively. In the first hybridization process which leads to the formation of the sp3 hybrid orbitals as depicted in Fig.2.1 (b), the four atomic orbitals of 2s and 2p shells are mixed and each sp3 orbital is filled with only one electron. In this case, the carbon atom forms σ bonds with its neighbouring atoms in a tetrahedral geometry. Consequently, all other atoms are ◦ arranged at 109.5 bond angles. Methane (CH4) which is shown in Fig.2.1 (f) and diamond are typical examples of the sp3 hybridization process. In the case of the sp2 hybridization process [Fig.1.1 (d)] only two orbitals of the 2p shell are mixed with the 2s orbital to form three sp2 hybrid orbitals. To minimize the repulsion, the sp2 hybrid orbitals optimize their spatial distance so that the angle between the orbitals is 120◦. As a consequence of the sp2 hybridization, the carbon atom forms strong σ bonds with its neighbouring atoms resulting in trigonal planar geometry. The remaining p atomic orbital which is not mixed can participate in formation of a π bond with other carbon atoms giving rise to carbon-carbon double bonds. Typical examples of this hybridization scheme are ethylene (C2H4) which is shown in Fig.2.1 (g) and benzene (C6H6) molecules. Graphite is a three dimensional crystalline material made of stacked layers of carbon atoms. In each layer the every carbon atom is connected to the other three carbon atoms through the σ

8 Figure 2.2: (a) The hexagonal lattice structure of monolayer graphene with rhombic unit-cell (green rhombus) that contains two carbon atoms. The carbon atoms belonging to sublattice A (B) are shown by red (blue) dots. a1 and a2 are the primitive lattice vectors and δ1, δ2, and δ3 are the nearest neighbour vectors of the monolayer graphene lattice structure. (b) Reciprocal lattice structure of monolayer graphene with hexagonal ﬁrst Brillouin zone shaded in blue. b1 and b2 are the primitive reciprocal lattice vectors. bonds resulting from the sp2 hybridized orbitals, and the adjacent layers are coupled by means of weak van der Waals interactions. In the last possible hybridization process only one orbital of the 2p shell is mixed with the 2s orbital such that the angle between the sp hybrid orbitals is 180◦. Here, the two hybrid orbitals lead to the formation of σ bonds and the remaining unmixed p type orbitals participate in the π bonds with other nearest neighbour carbon atoms. As an illustration, each carbon atom of acetylene (C2H2) shown in Fig.2.1 (h) forms two σ bonds, one with the hydrogen atom and the other with the neighbour carbon atom, and two π bonds with the neighbouring carbon atom which lead to a carbon-carbon triple bond.

2.3 The electronic structure of monolayer graphene

Graphene is a two dimensional (2D) crystalline solid made out of carbon atoms arranged in a hexagonal structure. The carbon atoms in graphene possess sp2 hybridized orbitals. These sp2 orbitals lead to strong σ bonds between the carbon atoms in a monolayer of graphene. The lattice structure of graphene is triangular and its unit cell (shaded green rhombus) contains two carbon atoms as represented in Fig.2.2 (a). The primitive lattice vectors are given by

a √ a √ a = (3, 3), a = (3, − 3), (2.1) 1 2 2 2

9 where a=1.42 A˚is the distance between the nearest neighbour carbon atoms. The triangular lattice structure of graphene is composed of two sublattices, A and B, such that the nearest neighbours of each carbon atom in sublattice A belong to sublattice B. Here, the carbon atoms belonging to sublattice A (B) are represented by red (blue) dots. The corresponding ﬁrst Brillouin zone of mononolayer graphene is represented in Fig.2.2 (b) with the primitive reciprocal lattice vectors

2π √ 2π √ b = (1, 3), b = (1, − 3). (2.2) 1 3a 2 3a The wave vectors of special high-symmetry points of hexagonal ﬁrst Brillouin zone are

2π 1 2π 1 2π K = (1, √ ), K0 = (1, √ ), M = (1, 0), (2.3) 3a 3 3a 3 3a where the K and K0 points are placed at the vertices of this hexagon. These two points are called Dirac points due to the fact that the carriers are massless, chiral, Dirac fermions close to K and K0. As a consequence, it is of interest to investigate the physics of graphene close to the Dirac points. As is represented in Fig.2.2 (a), the nearest neighbour vectors of the graphene lattice structure in the real space are given by

a √ a √ δ = (1, 3), δ = (1, − 3), δ = a(1, 0). (2.4) 1 2 2 2 3

2.3.1 Tight-binding Hamiltonian and band structure of pristine monolayer graphene

In 1947, Wallace [53] developed a tight-binding model to describe the electronic structure of graphite and monolayer graphene from the viewpoint of the nearest neighbour approximation for the unmixed pz orbitals. The nearest neighbour tight-binding Hamiltonian of monolayer graphene in the second quantization representation is given by [53]

X † X † H = nanan − tn,m(anbm + h.c.), (2.5) n hn,mi where n is the on-site energy which usually is taken to be zero when the energy reference is † assumed to be the energy of pz orbital, an (an) is the annihilation (creation) operator which th annihilates (creates) an electron at the n atomic site, and tn,m = t = 2.7 eV is the hopping term between the nearest neighbour carbon atoms. The symbol hn, mi in the second term of the right hand side of Eq.2.5 denotes that the summation is restricted to run over the nearest neighbour atomic sites. It should be noted that this model takes one orbital (pz) per atom into account and describes the low energy band structure of monolayer graphene accurately [54].

10 Since the unit-cell of monolayer graphene includes two pz orbitals belonging to diﬀerent sublattices, the ansatz for the electron Bloch states of this system can be written as the linear combination of Bloch sums of pz atomic orbitals belonging to sublattices A and B,

|Ψ(k, r)i = CA(k)|ψA(k, r)i + CB(k)|ψB(k, r)i, (2.6) where

1 X ik·Ri |ψA(k, r)i = √ e |φ(r − rA − Ri)i, (2.7) N i 1 X ik·Ri |ψB(k, r)i = √ e |φ(r − rB − Ri)i. (2.8) N i

Here, |ψA(k, r)i and |ψB(k, r)i are the Bloch sums of pz atomic orbitals belonging to sublattices A and B, respectively. k is the electron wave vector, Ri is the Bravais lattice vector th which locates the i unit-cell, rA (rB) speciﬁes the location of the carbon atom on sub- th lattice A (B) within the i unit-cell, |φ(r)i is the pz atomic orbital, and N is the total number of the unit-cells. By inserting Eq.2.6 into the Schrödinger equation HΨ = EΨ, and neglecting the overlap of the pz atomic orbitals centered at diﬀerent sites, the following eigenvalue problem is obtained,

"H (k) H (k)#"C (k)# "C (k)# AA AB A = E A , (2.9) HBA(k) HBB(k) CB(k) CB(k)

∗ where HAA = HBB, HAB = HBA, and

1 X H (k) = eik·(Ri−Rj )hφ |H|φ i, (2.10) AA N A,Rj A,Ri i,j 1 X H (k) = eik·(Ri−Rj )hφ |H|φ i. (2.11) AB N B,Rj A,Ri i,j

It should be noted that the orthonormality condition of the pz atomic orbitals, required to derive Eqs.2.10 and 2.11 can be written as

hφX |φY i = δXY , (2.12) where X,Y = A, B. The nearest neighbour approximation restricts the hopping of electrons to nearest neighbour atomic sites implying that hopping of electrons within each sublattice is forbidden and it only occurs between the diﬀerent sublattices. Applying the nearest neighbour approximation to Eq.2.11, one gets the oﬀ-diagonal elements of the 2 × 2 Hamiltonian

11 matrix as

−ik.a1 −ik.a2 HAB(k) = hφB,0|H|φA,0i + e hφB,−a1 |H|φA,0i + e hφB,−a2 |H|φA,0i = −t(1 + e−ik.a1 + e−ik.a2 ) (2.13) = −tf(k).

If the energy reference is taken to be equal to the energy of the pz orbital, hφA,0|H|φA,0i = hφB,0|H|φB,0i = 0, the Hamiltonian matrix will be

" 0 −tf(k)# . (2.14) −tf ∗(k) 0

In order to calculate the dispersion relations of the pristine monolayer graphene within the nearest neighbour approximation shown in Fig.2.3, the Hamiltonian matrix [Eq.2.14] should be diagonalized. This leads to two bands given by

E±(k) = ±t|f(k)| v √ u ! (2.15) u 3akx 3aky √ = ±tt3 + 4cos cos + 2cos ak 3 , 2 2 y where the (+) sign stands for empty π∗ band, and (−) is the fully occupied band of charge neutral graphene at zero temperature. According to this dispersion relation [Eq.2.15], the zeros of the function f(k) correspond to the intersection points of the two bands (π and π∗) in the momentum space. It can be easily shown that the K and K0 points [Eq.2.3] (vertices of the hexagonal Brillouin zone) are the crossing points of the two bands (Fig.2.3) due to the fact that f(k = K) = f(k = K0) = 0. Consequently, inﬁnite pristine graphene is a zero gap material such that the conduction and valence band touch each other at the Dirac points. The expansion of the dispersion relation [Eq.2.15] close to the Dirac points, k = K + q (k = K0 + q) with |q| |K|(|K0|), leads to a linear dispersion relation for monolayer graphene given by

E±(q) ' ±v|q|, (2.16) where q is the wave vector measured relative to the Dirac points, and the electron velocity at these points is given by 3a|t| v = . (2.17) 2 Equation 2.16 shows that the band structure of pristine monolayer graphene forms cones close to the K and K0 points that are called Dirac cones or valleys. Furthermore, this linear dispersion relation [Eq.2.16] exhibits the electron-hole symmetry of the monolayer graphene band structure in the vicinity of Dirac points.

12 Figure 2.3: The electronic band structure of inﬁnite pristine monolayer graphene within the nearest neighbour approximation. Dotted red line shows the Fermi energy of charge neutral monolayer graphene.

13 2.4 The electronic structure of bilayer graphene

Bilayer graphene consists of two single electronically coupled layers of graphene each having the honeycomb lattice structure. These layers can be stacked in two diﬀerent ways called the AA and AB stacking. In the case of AA stacking, the upper layer is located exactly on top of the bottom layer such that the carbon atoms belonging to sublattice A of the upper layer are located on top of the carbon atoms in sublattice A of the bottom layer. The geometry of the AB stacked bilayer graphene shown in Fig.2.4 (a) is such that the top layer is shifted by δ3 [Eq.2.4] with respect to the bottom layer. As a consequence, the carbon atoms belonging to the B sublattice of the upper layer are located on top of the carbon atoms of A sublattice of the bottom layer. The carbon atoms of the A sublattice of the upper layer and the B sublattice of the bottom layer are aligned with the hollow sites at the centers of the hexagons of the other layer as is represented in Fig.2.4 (a). In the next section I will discuss the tight-binding Hamiltonian of the AB or Bernal stacking of bilayer graphene that has the lowest stacking energy among all possible geometries of bilayer graphene systems.

2.4.1 Tight-binding Hamiltonian and band structure of bilayer graphene

The primitive lattice vectors and the Brillouin zone of bilayer graphene are the same as for the monolayer graphene lattice structure. Although the unit-cell of monolayer graphene contains two carbon atoms belonging to sublattices A and B, the unit-cell of the lattice structure of bilayer graphene contains four carbon atoms so that, two of them labeled A1 and B1 are on the lower layer and A2 and B2 are on the upper layer. The aligned carbon atoms A1 and B2 in the unit-cell are called dimer sites, while the other two carbon atoms are referred to as the non-dimer sites. The tight-binding Hamiltonian of bilayer graphene that will be discussed in this section includes the in-plane nearest neighbour hopping (the same as for monolayer graphene) and the inter-plane coupling of the dimer sites. The nearest neighbour tight-binding Hamiltonian of AB-stacked bilayer graphene is given by

X † X † X † H = nanan − tnm(anam + h.c.) + t⊥(an1 am2 + h.c.), (2.18) n hn,mi hn1,m2i where the ﬁrst two terms resemble the nearest neighbour tight-binding Hamiltonian of monolayer graphene [Eq.2.5], and the last term considers the inter-layer coupling such that the electron hopping only occurs between the atomic sites of the sublattice A in the lower layer and the sublattice B of the top layer.

Since the unit-cell of bilayer graphene contains four carbon atoms and we have one pz atomic orbital per atomic site, the ansatz for the electron Bloch states can be written as,

1 X ik·Ri |ψA1 (k, r)i = √ e |φ(r − rA1 − Ri)i, (2.20) N i 1 X ik·Ri |ψB1 (k, r)i = √ e |φ(r − rB1 − Ri)i, (2.21) N i 1 X ik·Ri |ψA2 (k, r)i = √ e |φ(r − rA2 − Ri)i, (2.22) N i 1 X ik·Ri |ψB2 (k, r)i = √ e |φ(r − rB2 − Ri)i, (2.23) N i are the Bloch sums over the diﬀerent sublattices of each layer, and the parameters are the same as those mentioned previously in the case of monolayer graphene. Applying the same method used in Sec.2.3.1, the k-representation of the bilayer graphene Hamiltonian is given by   0 −tf(k) 0 t⊥   −tf ∗(k) 0 0 0      , (2.24)  0 0 0 −tf(k)   ∗ t⊥ 0 −tf (k) 0 where the terms −tf(k) and −tf ∗(k) are the same as those in monolayer graphene Hamil- tonian and t⊥ = 0.1t eV is the hopping parameter between the dimer sites. Diagonalization of this Hamiltonian [Eq.2.24] leads to four bands given by

s 1 r 1 E(k)(1,4) = ± t2|f(k)|2 + t2 + t2 t2|f(k)|2 + t4 , (2.25) 2 ⊥ ⊥ 4 ⊥ s 1 r 1 E(k)(2,3) = ± t2|f(k)|2 + t2 − t2 t2|f(k)|2 + t4 . (2.26) 2 ⊥ ⊥ 4 ⊥

The electronic band structure of the inﬁnite pristine bilayer graphene is shown in Fig.2.5. By expanding Eqs.2.25 and 2.26, it can be seen that the E(k)(2,3) touch each other close to the Dirac points K and K0. Furthermore, it can be deduced that unlike the band structure of monolayer graphene band, the dispersion of AB-stacked bilayer graphene shown in Fig.2.5 is parabolic in the vicinity of the Dirac points.

15 Figure 2.4: (a) Lattice structure of bilayer graphene for AB stacking with rhombic unit-cell (green rhombus) that contains four carbon atoms. Top (bottom) layer is shown in blue (black). a1 and a2 are the primitive lattice vectors. (b) Hexagonal Brillouin zone of bilayer graphene.

Figure 2.5: The electronic band structure of inﬁnite pristine AB-stacked bilayer graphene within the nearest neighbour approximation.

16 2.5 Valley degree of freedom

Valleys are the local minima (maxima) of the conduction (valence) band, which can be exploited as a degree of freedom for the electrons to store and process the information. Crystalline materials that possess two or more well-separated valleys in their first Brillouin zone and negligible intervalley scattering are regarded as candidate systems for the manipulation of the valley degree of freedom of the electrons. It should be pointed out that unlike the spin which is an intrinsic degree of freedom for the electrons, the valley degree of freedom is a topological feature of the band structure of crystalline solids in momentum space. The field of the study of the valley degree of freedom of electrons and its technological applications is called valleytronics by analogy with spintronics, the field which exploits the spin degree of freedom of electrons in addition to the electric charge for the technological purposes.

2.6 Breaking inversion symmetry in monolayer and bilayer graphene

As will be explained below, in order to detect topological currents experimentally or calculate them theoretically in monolayer and bilayer graphene, the inversion symmetry of the system must be broken. In the case of monolayer graphene, the interaction with the hexagonal boron nitride (hBN) that plays the role of the substrate for the graphene sample breaks the inversion symmetry of the graphene lattice structure. On the other hand, applying an electric ﬁeld that is perpendicular to the plane of bilayer graphene sample breaks the inversion symmetry of that system. From the theoretical point of view, manipulating the on-site energy of the carbon atoms

n in each unit-cell plays the role of breaking inversion symmetry of the system. To break the inversion symmetry of monolayer graphene system, we modulate the on-site energy of atomic sites so that those belonging to sublattice A have n = ∆, and those belonging to sublattice B have n = −∆ as is represented in the inset of Fig.2.6. This modiﬁes the k-representation monolayer graphene Hamiltonian as follows

" ∆ −tf(k)# . (2.27) −tf ∗(k) −∆ It should be pointed out that ∆ = 0.0602 eV is the corresponding theoretical value of the on-site energy which represents the presence of hBN as a substrate in experiment [55]. In order to break the inversion symmetry of bilayer graphene, we modify the on-site energy of the carbon atoms so that n = Vg/2 for the top layer and n = −Vg/2 for the bottom layer atomic sites to model the eﬀect of the perpendicular electric ﬁeld as represented

17 Figure 2.6: The electronic band structure of inﬁnite monolayer graphene with broken inversion symmetry. A gap of the order of 2∆ is opened between the valence and conduction bands at point K. Inset: Schematic representation of the mechanism of breaking inversion symmetry in monolayer graphene. in the inset of Fig.2.7. As a consequence, the form of the nearest neighbour k-representation Hamiltonian of AB-stacked bilayer graphene after inversion symmetry breaking is

  −Vg/2 −tf(k) 0 t⊥   −tf ∗(k) −V /2 0 0   g    . (2.28)  0 0 Vg/2 −tf(k)   ∗ t⊥ 0 −tf (k) Vg/2 Note that, it can be easily shown that the breaking of inversion symmetry also opens a gap between the conduction and valence bands in the band structure of monolayer and bilayer graphene (See dotted red squares in Figs.2.6 and 2.7). As can be proved by the diagonalization of the k-space representation of the Hamilto- nians [Eqs.2.27 and 2.28], breaking the inversion symmetry opens energy band gaps of 2∆

(Fig.2.6) and Vg (Fig.2.7) between the valence and conduction bands in the electronic band structures of the monolayer and bilayer graphene systems, respectively.

18 Figure 2.7: The electronic band structure of inﬁnite bilayer graphene with broken inversion symmetry. A gap of the order of Vg is opened between the valence and conduction bands at point K. Inset: Side view of the mechanism of breaking inversion symmetry in bilayer graphene.

19 2.7 Berry phase and Berry curvature

The Berry phase or geometrical phase is the phase acquired by a quantum state over the course of a cycle when the time-dependent part of the Hamiltonian of the physical system evolves adiabatically[56, 59, 60, 61, 62, 63]. Suppose the Hamiltonian of the system depends on time through a set of parameters R, so that

H = H(R(t)), R = R(t). (2.29)

Then, the time evolution of the quantum state is determined by the time-dependent Schro¨dinger equation

∂|Ψ(R(t))i i = H(R(t))|Ψ(R(t))i, (2.30) ~ ∂t and the instantaneous eigenstates of this system which form an orthonormal basis are spec- iﬁed by

H(R)|n(R)i = En(R)|n(R)i. (2.31)

According to the quantum adiabatic theorem, if the system is initially prepared in one of the eigenstates of Eq.2.31, |n(R(0))i, apart from a phase factor which is smooth and single valued in the parameter space, the system will remain in one of the instantaneous eigenstates |n(R(t))i of Hamiltonian if the parameter R(t) varies suﬃciently slowly throughout the process. Consequently, the adiabatically evolved quantum state at time t can be written as

i R t 0 0 iγn(t) − En(R(t ))dt |Ψn(t)i = e e ~ 0 |n(R(t))i, (2.32) where the second exponential known as dynamical phase results from the time-dependent part of Eq.2.30. Inserting Eq.2.32 into Eq.2.30 and taking its inner product with hn(R(t))| leads to

dγ (t) d n = ihn(R(t))| |n(R(t))i. (2.33) dt dt It can be shown by using the chain rule and computing the time integral, that the phase acquired by the quantum state γn during adiabatic evolution of the system can be expressed as a path integral in the parameter space

Z γn = i dR · hn(R)|∇R|n(R)i, (2.34) C where the gradient is taken in the parameter space and the path connects the two points R(T ) and R(0). Since the integrand in Eq.2.34 is not gauge-invariant, it was assumed prior to Berry’s work that this phase can be canceled out by choosing a suitable gauge [57].

20 In 1984, Berry [56] showed that the phase γn is a gauge-invariant physical quantity when the system evolves adiabatically along a closed path C, where R(T ) = R(0). As a consequence, it cannot be neglected if the phase resulting from the gauge transformation is smooth and single valued. The gauge-invariant Berry phase is given by a closed path integral in the parameter space as

I γn = dR · An(R), (2.35) C where

An(R) = ihn(R)|∇R|n(R)i, (2.36) is called Berry connection. The phase γn is also called the "geometrical phase" according to the fact that it only depends on the path traversed by the quantum state. Using Stokes’ theorem, the closed line integral in Eq.2.35 can be transformed to an integral over a surface S bounded by C so that Z γn = ∇ × An(R) · ds, (2.37) S where

Ωn(R) = ∇ × An(R), (2.38) is the Berry curvature which is also a gauge-invariant physical quantity. A gauge transformation leads to

|n(R)i −→ eiφ(R)|n(R)i (2.39) An(R) −→ An(R) − ∇Rφ(R).

Since the curl of gradient is equal to zero, the Berry curvature and the Berry phase are gauge-invariant physical quantities. In order to extend the concepts of Berry phase and curvature to the Bloch states of electrons, the generic parameter R can be replaced by the electron wave vector k. Therefore, the momentum space is regarded as the parameter space. The Bloch state of electrons in a periodic lattice structure is deﬁned as [58]

ik·r ψnk(r) = e unk(r), (2.40) where n is the band index. Note that unk(r) is the periodic part of the Bloch state such that unk(r + a) = unk(r). Hence, when the wave vector k traverses a closed path in momentum space, the Berry phase and curvature acquired by the Bloch states of electrons are given by I γn = i dk · hunk|∇k|unki, (2.41) C

Ωn(k) = i∇k × hunk|∇k|unki. (2.42)

21 It should be emphasized that, for the Berry phase to be a gauge-invariant quantity, the integral in Eq.2.41 must be taken along a closed path C. However, in the case of Berry curvature Ωn(k) a closed path is not necessary due to the fact that Berry curvature is a local gauge-invariant quantity [59]. The non-zero Berry curvature in the conduction band resulting from the broken inversion symmetry in monolayer graphene is given by [50, 59, 30]

3a2∆t2 Ω(q) = τ ˆz, (2.43) z 2(∆2 + 3q2a2t2)3/2

0 where τz = ±1 labels the two valleys so that +1 (−1) speciﬁes the valley K (K ), a is the lattice constant, ∆ is the on-site potential which has been used to break the inversion symmetry, and t is the hopping parameter between the nearest neighbour carbon atoms. It is assumed that the monolayer graphene is in the xy-plane. Hence, the Berry curvature is in the z-direction perpendicular to the plane of monolayer graphene. τz = ±1 being the valley index reveals that the Berry curvature points in opposite directions close to the two valleys. Time reversal symmetry and spatial inversion symmetry impose constraints on the Berry curvature so that if the system respects time reversal symmetry, the Berry curvature is an odd function Ω(−q) = −Ω(q), (2.44) and if the system has spatial inversion symmetry the Berry curvature is an even function

Ω(−q) = Ω(q). (2.45)

Consequently, in order to obtain a non-zero Berry curvature in the system, time reversal symmetry or spatial inversion symmetry of the system must be broken.

2.7.1 Anomalous velocity and semi-classical theory of electron transport

It is well known that the group velocity of Bloch electrons is proportional to the k-derivative of the energy dispersion relation v (k) = ∂En(k) . When the system is forced to evolve adia- n ~∂k batically in momentum space by means of an electric field, the electron acquires an anomalous velocity which is proportional to the Berry curvature [59, 60, 62, 61, 63]. Consequently, the velocity of the Bloch electrons undergoing an adiabatic evolution includes the usual k- derivative of the energy dispersion and the anomalous velocity term due to the topological properties of the energy band structure. The anomalous velocity, which is perpendicular to the applied electric field, leads to a valley Hall effect in the system. The steps of derivation of the anomalous velocity of electrons are summarized below and follow the treatment given in Ref.[60].

22 Consider the Hamiltonian of a Bloch electron subjected to a weak uniform electric ﬁeld εxˆ, which enters into the Hamiltonian as an electrostatic potential φ(x) so that

2 H(r) = − ~ ∇2 + V (r) − eεx, (2.46) 2m where V (r) is the periodic lattice potential, and −eεx is the electrostatic potential energy due to the applied uniform electric ﬁeld εxˆ. The electrostatic potential energy does not respect the translational symmetry of the lattice structure since it increases monotonically as a function of x, so the Bloch theorem cannot be applied. Using a classical gauge transformation resolves this issue by replacing the x-dependent electrostatic potential energy with a time-dependent vector potential A(t). The classical gauge f(r, t) transforms the vector and scalar electric potentials, respectively according to

A(r, t) −→ A(r, t) + ∇f(r, t), (2.47) ∂f(r, t) φ(r, t) −→ φ(r, t) − , (2.48) ∂t where f(r, t) = −εxt is the required gauge. This gauge transformation yields the time- dependent Hamiltonian

2 et 2 H(r, t) = ~ −i∇ + εxˆ + V (r). (2.49) 2m ~ In order to consider the eﬀect of Berry curvature on the velocity of Bloch electrons, this Hamiltonian H(r, t) should be transformed to momentum space. Thus, the transformation H(k, t) ≡ e−ik.rH(r, t)eik.r leads to

2 H [q(t)] = ~ [−i∇ + q(t)]2 + V (r), (2.50) 2m where the gauge-invariant crystal momentum q(t), is given by

q(t) = k + k(t) et (2.51) = k + εxˆ. ~ Now, the Hamiltonian [Eq.2.50] depends on time through the parameter q (crystal momentum). According to Berry’s theorem [56], the quantum state of this system acquires Berry phase as the system undergoes adiabatic evolution in the k-space by means of the weak uniform electric ﬁeld ε. At t = 0, the periodic parts of the Bloch states unk(r) are the eigenstates of the Hamiltonian H[q(0)], such that

H(k)unk(r) = E(k)unk(r). (2.52)

23 Figure 2.8: The k-space contour used for the calculation of the line integral in Eq.2.58.

It can be shown that

ik·r ik·r H[k(t)][unq(t)e ] = En[q(t)][unq(t)e ], (2.53)

ik·r where unq(t)e are the instantaneous eigenstates of the Hamiltonian H[q(t)] which become the usual Bloch states when t = 0. Note that n is the band index in Eqs. 2.52 and 2.53 which will be suppressed since it does not alter the derivation of the anomalous velocity. Suppose the system is in one of the eigenstates of Eq.2.52 initially. Hence, the state undergoing adiabatic evolution at time t is given by

t ik.r iγ(t) − i R dt0E[q(t0)] ψ(t) = uq(t)e e e ~ 0 (2.54) where the Berry phase γ(t) is

Z q(t) 0 γ(t) = dq · Aq0 . (2.55) q(0) Note that the integral in Eq.2.55 is taken over the trajectory from q(0) to q(t). Now, if we consider the time evolution of a wave packet composed of Bloch states where the crystal momentum of the central Bloch state is denoted by q0, the wave packet at time t is given by

i R t 0 0 iq0 X i∆q·r − dt E[q(t )] iγ(t) Ψ(r, t) = e h(|∆q|)uq(t)(r)e e ~ 0 e , (2.56) q where h(|∆q|) is the envelope function of the wave packet and ∆q = q(t) − q0(t) is the crystal momentum diﬀerence between the central Bloch state of q0(t) and another Bloch state of q(t). Note that ∆q = q(t) − q0(t) = k − k0 is time independent. We know that the group velocity of the wave packet [Eq.2.56] is given by

1 d Z t Z t ∆q · v = dt0E[q(t0)] − dt0E[q (t0)] g dt 0 ~ 0 0 (2.57) 1 = ∆q · ∇qE[q0(t)] . ~

24 To prove that the Berry phase diﬀerence leads to the anomalous velocity, we consider the parallelogram momentum-space path as it is shown in Fig.2.8. Considering the same scenario of the group velocity it can be shown that the anomalous velocity is given by[60]

" # Z q(t) Z q0(t) d 0 0 ∆q · va = dq · Aq0 − dq · Aq0 . (2.58) dt q(0) q0(0)

If ∆q is suﬃciently small, the contribution of the side segments of the path represented in R Fig.2.8 to the Berry phase dq · Aq become negligible. Consequently, the line integrals of Eq.2.58 can be replaced by a closed path C(t) so that

"I # d 0 ∆q · va ≈ dq · Aq0 dt C(t) " # (2.59) d Z = − da · (∇q0 × Aq0 ) , dt a(t) where a(t) is the area bounded by the path C(t). If the top and bottom sides of the path advance by qdt˙ in time dt, then the additional area, the shaded region in Fig.2.8 is

da = (q˙ × ∆q)dt. (2.60)

Inserting Eq.2.60 into Eq.2.59 leads to

∆q · va = (∇q × Aq) · (∆q × q˙ ) (2.61)

= ∆q · (q˙ × ∇q × Aq0 ), where the anomalous velocity, va, of Bloch electron (which depends on the Berry curvature) is given by

va = q˙ × ∇q × Aq 0 (2.62) = q˙ × Ω(q0), where Ω(q0) is the Berry curvature. As a consequence, the total velocity of Bloch electron in a lattice structure with broken inversion or time-reversal symmetry has the form

1 v(k) = ∇kE(k) + k˙ × Ω(k) ~ (2.63) 1 e = ∇kE(k) + εxˆ × Ω(k), ~ ~ where ~k˙ = eεxˆ in the absence of a magnetic field. It has been shown above that the Bloch components of the wave packets take different momentum-space trajectories due to the small uniform electric field. Therefore, the Bloch states acquire a modified Berry phase which leads to the anomalous velocity term in the velocity of electrons. It can be shown that

25 Eq.2.63 is consistent with the constraints imposed by time reversal symmetry and spatial inversion symmetry [Eqs.2.44 and 2.45]. Under time reversal operation the electron velocity v(k) and the crystal momentum k change sign but the electric ﬁeld εxˆ is invariant and the operation of spatial inversion changes the sign of v(k), k, and εxˆ. Then, the velocity equation should be invariant under the operation of time reversal or spatial inversion if the system respects the reversal symmetry or spatial inversion symmetry. Thus, the Berry curvature is an odd function if the system has time reversal symmetry and an even function in the presence of spatial inversion symmetry. Consequently, the Berry curvature vanishes if the system has both time reversal symmetry and inversion symmetry.

2.8 Interpretation of non-local resistance as a signature of topological currents in monolayer and bilayer graphene

Those crystalline materials that possess two or more well-separated valleys in their band structures are regarded as potential valleytronic systems, in which valley currents can be generated. Monolayer and bilayer graphene samples with broken inversion symmetry are ideal examples of valleytronics systems because of their high electronic quality and the two well-separated valleys K and K0. Corresponding to the importance of spin currents in the field of spintronics, the valley currents play a decisive role in potential valley-based technologies. Therefore, the investigation of valley currents flowing in valleytronic devices has attracted a great amount of attention [30, 32, 31, 64, 65, 66]. Recently, Gorbachev et al. [30] have carried out experiments on graphene superlattice samples (graphene is placed on top of a hexagonal boron nitride substrate), measuring the non-local resistances RNL as a function of the applied gate voltage. In this experiment, the crystallographic axes of monolayer graphene and hexagonal boron nitride (hBN) were aligned, resulting in the breaking of the inversion symmetry of the monolayer graphene sample. In order to investigate the non-local transport, the superlattice was fabricated in a Hall bar geometry as depicted in Fig.2.9 (b). The non-local resistance R = ∆V3,9 is NL I8,4 defined to be the ratio of the applied electric current between the contacts 8 and 4 I8,4 and the measured voltage between the contacts 3 and 9 ∆V3,9(the contacts are shown in Fig.2.9 (b)). The central result of their study was the observation of a striking peak in the measured non-local resistance RNL depicted in Fig.2.9 (a) when the Fermi energy passes the energy of Berry curvature hot spots (defined to be the energies where the Berry curvature is non-zero). Although Gorbachev et al. did not measure the valley currents flowing through the sample directly, they interpreted the measured striking enhancement in RNL as a signature of the presence of valley currents in the sample based on the semi-classical theory of electron transport [Eq.2.63] at zero magnetic field which can be explained as follows: The presence of the hBN substrate breaks the global inversion symmetry of the monolayer graphene

26 Figure 2.9: (a) Non-local resistance (red curve) and longitudinal resistance (black curve) measured in monolayer graphene on hBN superlattices. Left inset: Schematic representation of the electronic band structure of monolayer graphene supperlattices with Berry curvature hot spots near the opened gap between the valence and conduction bands. Right Inset: Valley Hall conductivity modeled for graphene with broken inversion symmetry. Adapted from Ref.[30]. (b) Schematic representation of the multi-terminal monolayer graphene superlattice with the geometry appropriate for non-local resistance RNL measurements. since the two carbon atoms in each unit-cell acquire different on-site energies, resulting from binding of carbon atoms to boron and nitride in each unit-cell. While breaking the inversion symmetry of the sample leads to non-zero Berry curvatures close to the valleys K and K0, the time-reversal symmetry of the system requires the Berry curvatures point in opposite directions in two valleys. Consequently, the electrons in valley K and K0 acquire differing velocities due to the anomalous velocity term [Eq.2.63]. As a result of the difference in the velocity of the electrons belonging to the differing valleys, a charge neutral valley current which is transverse to the applied electric field flows through the system. Hence, this transverse valley current gives rise to a non-local electrical response. So, by using the reverse valley Hall effect (VHE), they interpreted the measured pronounced peak in the

RNL as an evidence for the presence of the valley current in their system. The same experiments have been carried out on bilayer graphene samples by Shimazaki et al. [32] and Sui et al. [31]. To measure the RNL, they fabricated the bilayer graphene sample with a Hall bar geometry. As in the case of monolayer graphene, they measured a striking enhancement in the non-local resistance RNL of bilayer graphene samples with broken inversion symmetry when the Fermi level passes the energy of the Dirac point. Then, Shimazaki et al. [32] interpreted their experimental results as a signature of the presence of valley currents using the semi-classical theory of electron transport and reverse VHE. Note that the main distinction between the experiments on monolayer and bilayer graphene is the mechanism of the breaking of the inversion symmetry. While the inversion symmetry of the monolayer graphene sample can be broken by means of the hBN substrate, an electric

27 ﬁeld which is perpendicular to the layers of bilayer graphene breaks the inversion symmetry of the latter system. Thus, the non-local transport of electrons in bilayer graphene is gate tunable. The process of breaking inversion symmetry opens an energy gap between the conduction and valence band in the electronic band structures of monolayer and bilayer graphene.

According to the experimental results [30, 32, 31], the peaks in RNL were observed when the Fermi level is located inside the energy gap. Therefore, the applicability of the semi- classical theory of the electron transport [Eq.2.63] is arguable in the insulating regime due to the fact that within the gap, the mechanism of electron transport is quantum tunneling. A fully quantum mechanical approach for the calculation of non-local resistances, valley currents and valley accumulations is presented in chapter 5.

28 Chapter 3

Bismuthene on SiC: A high-temperature two-dimensional topological insulator

3.1 Introduction

Quantum spin Hall (QSH) materials are two-dimensional topological insulators that support the transmission of electrons through gapless conducting states at the edges of the system. The quantum spin Hall state is regarded as a topological quantum state of matter characterized by a bulk band gap and gapless edge states. The QSH state was first predicted theoretically by Kane and Mele in graphene [27, 28], and Bernevig et al. in HgTe quantum wells [25]. In 2007, König et al. [26] experimentally confirmed the presence of the QSH state in HgTe/CdTe quantum wells. The unique electronic properties of the gapless edge states in topological insulator materials, such as their spin polarization [27, 28], quantized two-terminal conductance G = 2e2/h [28, 27, 71], dissipationless edge currents [27, 28], and robustness against time-reversal-invariant disorder [27, 28], may support future technological applications. The small bulk band gaps (less than 30 meV) in the first QSH systems to be discovered, have been a serious obstacle to exploiting the potential advantages of the edge states propagating in these systems. Hence, topological insulators with bulk band gaps of the order of ∼ 0.5 eV or larger are needed, in order to take advantage of the aforementioned electronic properties of the edge states at room temperature, i.e. a large bulk energy band gap would protect the edge currents carried by the edge states from the detrimental contributions of the bulk states resulting from the thermal excitations of the electrons. Using scanning tunneling spectroscopy, Reis et al. [29] have detected a bulk band gap of ∼ 0.8 eV and conductive edge states consistent with theory [90, 29] in monolayer bismuthene grown on an insulating silicon carbide substrate. Therefore, they suggested this system as a promising candidate for a high temperature two-dimensional topological insulator. The valleytronic properties of this material are one of the topics considered in this thesis.

29 This chapter begins with a review of the preliminary concepts required for understanding QSH properties, such as time reversal symmetry, Kramer’s theorem, and suppression of backscattering of edge states due to time-reversal symmetry. Then, I discuss the experiment [29] that has been carried out on bismuthene on SiC that showed that this system can be regarded as a high temperature 2D topological insulator. At the end of this chapter I review a minimal tight-binding Hamiltonian that captures the key properties of the low-energy band structure of the bismuthene on SiC quantitatively. This minimal tight-binding Hamiltonian was developed in Ref.[71] and has been used in this thesis to investigate the valley and spin polarizations of the edge and bulk states of the quantum dots of the high temperature 2D topological insulator bismuthene on SiC.

3.2 Time reversal symmetry

According to its deﬁnition, the time reversal operator Θ is an antiunitary operator that can be written as

Θ = UT K, (3.1) where UT is unitary operator and K is the complex conjugation operator, i.e. Ki = −iK. The time reversal operator acts on a state |φi as

Θ|φi = |φ˜i, (3.2) where |φ˜i is the time-reversed state of |φi. By considering the time evolution of the time- reversed state |φ˜i, it can be shown that the time reversal operator Θ must be an antiunitary operator. To this end, consider a physical system described by the state |φi with its time- reversed state |φ˜i at t = 0. Then, after an inﬁnitesimal time interval (t = δt) the time- reversed state |φ˜i is evolved as

|φ˜(t = δt)i = U(δt)|φ˜(t = 0)i iHδt iHδt (3.3) = 1 − |φ˜(t = 0)i = 1 − Θ|φ(t = 0)i. ~ ~ Here, the Hamiltonian of the system enters the unitary time evolution operator as U(t) = e−iHt/~, that can be expanded for an inﬁnitesimal time evolution (t = δt) as U(t) ' 1 − iHδt/~. If this system respects the time reversal symmetry, it is expected for |φ˜(t = δt)i to be equal to the time-reversed replica of the state |φ(t = −δt)i that has the form

iH Θ|φ(t = −δt)i = Θ 1 − (−δt) |φ(t = 0)i. (3.4) ~

30 By equating Eqs.3.3 and 3.4 we have

iHδt iH 1 − Θ|φ(t = 0)i = Θ 1 − (−δt) |φ(t = 0)i (3.5) ~ ~ which leads to

− iHΘ|φ(t = 0)i = ΘiH|φ(t = 0)i. (3.6)

Since |φ(t = 0)i is considered to be any generic state describing a physical system the above equation is true for a physical system with time reversal symmetry. Now it can be shown that Θ cannot be unitary if the operation of the time reversal operator on a physical system is to make sense. By deﬁnition of the time reversal operator we have

ΘPΘ−1 = −P, (3.7) where P is the momentum operator. If we assume that the time reversal operator is unitary, then i can be canceled legitimately on both sides of Eq.3.6 and leads to

− HΘ = ΘH, (3.8) which is in a direct contradiction to the deﬁnition of Eq.3.7. To show this we can consider P2 the Hamiltonian of a free particle that is quadratic in momentum (Hfp = 2m ). According −1 to Eq.3.7, Hfp is invariant under time reversal (ΘHfpΘ = Hfp). However, Eq.3.8 is −1 p2 equivalent to ΘHfpΘ = −Hfp = − 2m , that contradicts Eq.3.7 directly. Therefore, the time reversal operator is antiunitary and from Eq.3.6 it can be deduced that for a system with time reversal symmetry the Hamiltonian of the system H and the time reversal operator Θ commute as

HΘ = ΘH. (3.9)

3.2.1 Time reversal of a spinless state

Consider the expansion of the state |ψi as

Z |ψi = d3r |rihr|ψi. (3.10)

The time reversal operator acts on the state |ψi as

31 Figure 3.1: States |±i (not shown) have spins tilted along the positive and negative nˆ direction.

Θ(hα|βi) = hα|βi∗. (3.12)

According to Eqs.3.10 and 3.11, it can be deduced that the time reversal operator acts on a spinless state as

Θψ(r) = ψ∗(r). (3.13)

3.2.2 Time reversal of a spin 1/2 state

Consider two states |+i and |−i with spins up and down respectively along the direction nˆ. These states can be obtained by rotating the state |↑i which denotes the spin up state along the z-direction (see Fig.3.1). So, we have

32 |+i = e−iSzφ/~e−iSyθ/~ |↑i (3.14) |−i = e−iSzφ/~e−iSy(θ+π)/~ |↑i .

These states are related by the time reversal operator as

Θ|+i = |−i. (3.15)

Due to the fact that the spin operator S is transformed by the time reversal operator as ΘSΘ−1 = −S, we can re-write Eq.3.15 as

Θ|+i = Θe−iSzφ/~Θ−1Θe−iSyθ/~Θ−1Θ |↑i = e−iSzφ/~e−iSyθ/~ |↓i (3.16) = e−iSzφ/~e−iSy(θ+π)/~ |↑i = |−i.

By comparing the two lines of the above equation and noting that K |↑i = |↑i we can infer 1 that the time reversal operator for the spin 2 system has the form

Θ = e−iSyπ/~K 0 −1! (3.17) = −iσyK = K. 1 0

2 1 Using this equation it can be shown that Θ = −1 for spin 2 states.

3.3 Kramer’s theorem

According to Kramer’s theorem, if a physical system is invariant under time reversal and has half-integer spin, each eigenstate |ψi of the Hamiltonian of the system (H|ψi = ε|ψi) is degenerate and orthogonal to its time-reversed state Θ|ψi so that

HΘ|ψi = ΘH|ψi = εΘ|ψi. (3.18)

The above equation shows that the time-reversed state Θ|ψi is also an eigenstate of the Hamiltonian with eigenvalue ε. Now, there are two possibilities for |ψi and Θ|ψi. First, they can be linearly dependent and describe the same state. Second, they are linearly independent and describe two degenerate states. To show which case occurs in systems with half-integer spin, we can use the property of the time reversal operator in these systems Θ2 = −1 and the identity hα|βi = hβ˜|α˜i where |α˜i and |β˜i are the corresponding time-reversed states of the |αi and |βi, respectively. So, we have

3.4 Eﬀect of time reversal operator on Bloch states

3.4.1 Spinless Bloch state

Consider the Bloch state ψnk(r) that can be written as a plane wave times a periodic function unk(r) as

ik·r ψnk(r) = e unk(r), (3.20) where n denotes the band index and k is the wave vector. Here, unk(r) is a periodic function so that unk(r) = unk(r + R) where R is a lattice vector. Then, the translation TR and the time reversal Θ operators act respectively on a spinless Bloch state as

ik·R TRψnk(r) = ψnk(r + R) = e ψnk(r), (3.21) ∗ Θψnk(r) = ψnk(r). (3.22)

Then, we can infer the eﬀect of the time reversal operator on the spinless Bloch states by ∗ considering the operation of TR on the Bloch states TRψnk(r) and TRψn−k(r) as

∗ ∗ −ik·R ∗ TRψnk(r) = ψnk(r + R) = e ψnk(r), (3.23) −ik·R TRψn−k(r) = ψn−k(r + R) = e ψn−k(r). (3.24)

Comparing Eqs.3.21, 3.23, and 3.24 shows that for non-degenerate levels, the time reversal operator acts on the Bloch states as

∗ Θψnk(r) = ψnk(r) = ψn−k(r). (3.25)

Thus, Kramer’s theorem for a spinless system with TRS implies that the energy bands of the system En(k) are even functions of the wave vector as

En(k) = En(−k). (3.26)

34 Figure 3.2: Schematic representation of a quantum Hall system with spatially separated edge channels propagating in the upward (red) and downward (blue) directions. The backscattering caused by disorder (yellow disks) is suppressed in these chiral channels.

3.4.2 Spin 1/2 Bloch state

1 According to Eq.3.17, the time reversal operator derived for a system with spin 2 , acts on 1 a Bloch state with spin 2 as

ψ ! −ψ∗ ! −ψ ! Θ nk↑ = nk↓ = n−k↓ . (3.27) ∗ ψnk↓ ψnk↑ ψn−k↑ 1 Therefore, it can be concluded that in a spin 2 system with TRS we have

En↑(k) = En↓(−k), (3.28) En↓(k) = En↑(−k).

3.5 Quantum Hall and quantum spin Hall systems

3.5.1 Quantum Hall state

In 1980, von Klitzing et al. [24] discovered a new state of matter called quantum Hall (QH) state. In this state, for two-dimensional systems the electric current is carried by the states propagating along the edges of the system, while the system is insulating in the bulk (shown schematically in Fig.3.2). The quantum Hall eﬀect occurs when a strong magnetic ﬁeld is applied perpendicularly to a two-dimensional system, for instance a two-dimensional electron gas (2DEG) formed by electrons trapped at a semiconductor heterojunction, at

35 low-temperatures. Under these conditions, if the Fermi level is between Landau levels the system turns into an insulator in the bulk while the electric current flowing through the system is carried by the one-dimensional conducting channels (shown in red and blue in Fig.3.2) that extend along the edges of the system. The quantum Hall state is characterized 2 by a Hall conductance Gxy which is quantized in integer or rational multiples of e /h, as manifestations of the integer or fractional QH effect, respectively. The one-dimensional conducting channels induced in the QH system are chiral in the sense that each conducting channel supports the transport of electrons in a single direction as shown in Fig.3.2. As is shown, the conducting channel propagating along the right (left) edge of the system supports the transport of electrons in the upward (downward) direction and this chiralty of the QH edge channels can be reversed by changing the sign of applied magnetic field. This chiral behaviour leads to robustness of the QH edge channels against the backscattering caused by disorder (yellow disks in Fig.3.2). For electrons to be backscattered, their direction of motion must be reversed. However, in a QH system electrons are only allowed to move in a single direction on each edge of the system and consequently are unable to change their direction of motion. This yields suppressed backscattering in the QH edge channels due to their chiral nature.

3.5.2 Quantum spin Hall state

The quantum spin Hall (QSH) insulator, equivalently referred to as the two-dimensional topological insulator, was predicted independently by Kane and Mele [28] and Bernevig et al. [25] in monolayer graphene and HgTe quantum wells, respectively. Then, König et al. [26] experimentally confirmed the existence of QSH state in HgTe/CdTe quantum wells. The QSH state is characterized by a bulk band gap and spin-filtered gapless edge states in the gap. This state resembles two copies of the QH state with both spin up and down at each edge as shown in the top inset of Fig.3.3. The bottom inset of Fig.3.3 schematically represents the gapless edge states in the bulk energy band gap. Similar to the QH state, these states are localized at the edges of the system. Each of the two bands has a degenerate copy which propagates at the opposite edge of the system. These degenerate copies are spin-filtered so that the electric current that is flowing in the upward direction (yellow arrow) is carried by the states (shown in red) with spin up on the right edge and spin down on the left edge of the sample. Based on this reasoning, changing the direction of motion of electrons (downward electric current) switches the transport of electrons to other band (shown in blue) and consequently leads to a reversed spin degree of freedom at each edge. The conducting channels propagating in a QSH system are termed "helical" since the spin is correlated with their direction of propagation. These edge states are robust against time-reversal-invariant disorder. For backscattering to occur in a QSH system, the spin of electrons must be flipped which requires the breaking of time reversal symmetry. Thus, the backscattering of an electron is suppressed in a QSH system if the disorder is time-reversal-

36 Figure 3.3: Top inset: Schematic representation of a quantum spin Hall system with two conducting channels (red in the upward and blue in the downward directions) with opposite spins propagating at each edge. The yellow arrow shows the direction in which the electric current is ﬂowing. Bottom inset: Copies of each band at the two edges representing the gapless edge states in the bulk energy band gap schematically. Solid (dotted) lines represent occupied (unoccupied) states at zero temperature. The net electric current ﬂowing in the upward direction is carried out by the solid red band. invariant. This can be shown mathematically as follows. Suppose there are N edge states propagating in each direction along one edge and time-reversal-invariant disorder is present along this edge of the system. Consider that the disordered region which is characterized by the Hamiltonian V is sandwiched between two disorder-free regions as is shown in Fig.3.4. The incoming edge states propagating in opposite directions in the left and right disorder- free regions are respectively labeled by |n, ↑,Liin and |n, ↓,Riin, with n being the number of the edge state. Note that these edge states are related to the outgoing edge states by the time reversal operator Θ as

37 Figure 3.4: Disordered (grey) region characterized by the Hamiltonian V sandwiched between two disorder-free (blue) regions. The edge states with spin up (down) propagating in opposite directions along each region of the system are shown by red (green) arrows for N = 1.

|n, ↓,Liout = Θ|n, ↑,Liin, (3.29) |n, ↑,Riout = Θ|n, ↓,Riin, (3.30) where |n, ↓,Liout and |n, ↑,Riout are the outgoing edge states propagating in the left and right regions. In the presence of the term V , the edge states are no longer the eigenstates of the Hamiltonian of the system. Hence, the scattering states propagating in the left and right disorder-free regions can be written as

N X in in out in |ΨiL = αnL|n, ↑,Li + βnL Θ|n, ↑,Li , (3.31) n=1 N X in in out in |ΨiR = αnR|n, ↓,Ri + βnR Θ|n, ↓,Ri , (3.32) n=1 where N is the total number of the edge states propagating in each direction along one edge in of the system, αnL(R) are the incoming amplitudes in the left (right) region, and βnL(R) are the outgoing amplitudes in the left (right) region. Here, the incoming and outgoing amplitudes in each region form vectors. For example, the vector formed the incoming amplitudes T in the left region is αL = (α1L, ..., αNL) . Then, these vectors are related by the scattering matrix S (characterizes the disordered region) as

β ! α ! L = S L , (3.33) βR αR where S is a 2N × 2N matrix. The S-matrix is unitary due to particle conservation and current normalization. The scattering matrix can be decomposed into transmission and reﬂection blocks as

38 r t ! , (3.34) t0 r0 where r, r0, t, and t0 are N×N matrices. Applying the time-reversal operator to the scattering states [Eqs.3.31 and 3.32], leads to

β∗ ! α∗ ! SΘ2 L = L . (3.37) ∗ ∗ βR αR Multiplying both sides of the above equation by Θ2S†, yields

β ! α ! L = Θ2ST L , (3.38) βR αR Comparing Eqs.3.33 and 3.38 leads to a constraint on the S-matrix imposed by the time- reversal symmetry of the system as

S = Θ2ST , (3.39) where ST denotes the transpose of S-matrix. Since Θ2 = −1 for a system with half-integer spin, the S-matrix must be antisymmetric so that

S = −ST . (3.40)

If the system possesses a single edge state Kramers pair at each edge (N = 1), Eq.3.40 leads to r = r0 = 0, (3.41) which is equivalent to perfect transmission and suppressed backscattering. The above discussion can be generalized to the two-dimensional topological insulators with odd number (N is odd but not one) of the edge state Kramer pairs [110]. It can be concluded that in a system with properties of Θ2 = −1 and odd number of edge state Kramer pairs, the edge states are robust against time-reversal-invariant disorder and cannot be gapped out,

39 implying that this system is topologically non-trivial. However, this is not true for systems with even numbers of propagating edge state Kramers pairs [110].

3.6 Experimental observation of the edge states in honeycomb bismuthene on SiC

As was mentioned above, the bulk energy band gaps of the initially predicted QSH systems were small (less than 30 meV). Such small band gaps prevent exploiting the unique electronic properties of the edge states propagating in the 2D topological insulators at room temperature. Employing a QSH system with large bulk band gap is one of the effective strategies that can resolve this difficulty. A large bulk band gap is regarded as material- specific and can be characterized by the strength of the spin orbit coupling (SOC), the symmetry of the layer and substrate system, and orbital hybridization [29]. Among these factors that can affect the bulk energy band gap, principal bulk band gaps in the QSH systems are mainly defined by the SOC. Monolayer graphene is the first system in which the QSH state was predicted theoretically [28]. The minute SOC in graphene opens a gap of the order 1µeV [95] which is too small for the room-temperature operations. The strength of the SOC scales as Z4 in the atomic number Z of elements [29]. Therefore, investigating the QSH state in systems made of heavier elements can be a practical approach to generating larger bulk energy band gaps and exploiting the electronic properties of the edge states at room temperature. Although using group IV elements can open a gap of the order 2 − 100 meV [96, 97], these gaps are still small for the purpose of the room temperature operations. It is expected that SOC yields a larger bulk energy band gap for the heavier group V elements of the periodic table, for instance Bi with atomic number Z = 83. Although theoretical studies have predicted the existence of the QSH state in a monolayer bismuthene on silicon and silicon carbide substrates [100, 90], the experimental growth of a hexagonal monolayer bismuthene on Si has not been successful [98, 99]. Recently, Reis et al. [29] have successfuly grown planar honeycomb monolayer bismuthene on SiC with a lattice constant of 5.35 Å. Using scanning tunneling microscopy (STM) they confirmed that the Bi atoms form a honeycomb lattice structure1. In order to obtain the electronic structure of this system, they carried out angle-resolved photoelectron spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). Then they compared the experimental data with the results obtained from density functional theory (DFT). Ac- cording to their DFT calculations with a hybrid exchange-correlation functional, this system possesses an indirect band gap of the order 0.67 eV with the valence band maximum located at the K point and the conduction band minimum at the Γ point of its hexagonal Brillouin

1The lattice structure and first Brillouin zone of bismuthene on SiC are similar to the lattice structure and first Brillouin zone of monolayer graphene discussed in Chapter 2. Note their differing lattice constants.

40 zone. From ARPES they have located the valence band maximum to be at the K point of the Brillouin zone. Moreover, Reis et al. measured a band splitting of the order ∼ 0.43 eV at the maximum of the valence band by means of ARPES, which is in good agreement with the results they obtained from a DFT calculation. They have identified this band splitting as a fingerprint of the Rashba effect resulting from the broken inversion symmetry of the monolayer bismuthene. By decomposing the band structure of this system they have shown that that s, px, and py valence orbitals of the Bi atom govern the low-energy band structure of this system, while the pz valence orbital of the Bi atom is shifted out of the low-energy sector of the band structure due to its hybridization with the SiC substrate. By using scanning tunneling spectroscopy (STS), Reis et al. have studied the local density of states of monolayer bismuthene on SiC. Thus, they showed that this system has a bulk energy band gap of the order ∼ 0.8 eV and that the states propagating at the edges of the sample are gapless and are localized to the vicinity of the edges so that they rapidly decay toward the bulk of the system within one lattice constant.

3.7 Minimal tight-binding Hamiltonian

In this section I review a minimal tight-binding Hamiltonian that correctly describes the low-energy electronic structure of monolayer bismuthene on SiC. This model which was developed in Ref.[71], accurately describes the indirect bulk energy band gap, the Rashba valence band splitting, and the conducting gapless edge states in quantitative agreement with the experimental data obtained from ARPES and STS [29]. According to previous studies [29, 90], Bi atoms of monolayer bismuthene on SiC form a planar honeycomb lattice structure with lattice constant of 5.35 Å, [see Fig.3.5 (a)]. The minimal tight-binding Hamil- tonian that accurately describes the features of the low-energy band structure of bismuthene on SiC has the form [71]

◦ NN SO R Hiαs,i0α0s0 = Hαδii0 δαα0 δss0 + Hiα,i0α0 δss0 + Hαs,α0s0 δii0 + Hαs,α0s0 δii0 . (3.42)

0 0 Here, s and s are the spin indices, and α and α denote s, px, and py valence orbitals of i and i0 Bi atoms, respectively. As was mentioned in the preceding section, the low-energy band structure of this system is governed by s, px, and py valence orbitals of Bi atoms, whereas the pz orbital is shifted out of the low energy region due to the presence of the SiC substrate. ◦ NN The ﬁrst term of the above Hamiltonian Hα represents the atomic orbital energies, H describes the hopping between the α0 orbital of the i0 Bi atom and the α orbital of i that 0 SO R is a nearest neighbour of i , Hαs,α0s0 is the atomic spin-orbit (SO) interaction, and Hαs,α0s0 describes the atomic Rashba eﬀect. The values of the model parameters are adapted from Ref. [71]. ◦ ◦ ◦ In this model, the orbital energies Hα are chosen to be H6s = −10.22 eV and H6px = ◦ NN H6py = 0 eV [71]. The nearest neighbour parameters Hiα,i0α0 are given in Table 3.1. The

41 Figure 3.5: (a) Left inset: Planar honeycomb lattice structure of inﬁnite two-dimensional monolayer bismuthene (blue) on SiC (not shown). Right inset: Hexagonal ﬁrst Brillouin zone of bismuthene on SiC. (b) Low-energy band structure of two-dimensional monolayer bismuthene on SiC calculated within the minimal tight-binding Hamiltonian [Eq.3.42]. En- ergies are measured from the maximum of the valence band E = 0. Adapted from Ref.[71]

42 Table 3.1: Nearest neighbour parameters Hiα,i0α0 . The ﬁtting values are Σ = −0.81 eV, Σ0 = −1.00 eV, Σ00 = −1.57 eV, Π = 0.55 eV. The matrix elements depend on the x and y 0 −1 0 coordinates of i and i nearest neighbour Bi atoms through θ = sin [(y − y)/dii0 ] where 0 2 0 2 1/2 dii0 = [(x − x) + (y − y) ] . Adapted from Ref.[71]

NN 0 0 0 Hiα,i0α0 6s 6px 6py 6s Σ −Σ0cosθ −Σ0sinθ 0 00 2 2 00 6px Σ cosθ Σ cos θ + Πsin θ (Σ − Π)cosθsinθ 0 2 00 00 2 2 6py −Σ sin θ (Σ − Π)cosθsinθ Σ sin θ + Πcos θ atomic spin-orbit interaction term in Eq.3.42 is approximated as

0 SO hCαs|S.L|Cα0 s i Hαs,α0s0 = ζl , (3.43) ~2 where ζl is the strength of the spin-orbit interaction with l being the orbital angular momentum, and Cα is the corresponding cubic harmonic of the atomic orbital α. Here, ζl is considered to be a ﬁtting parameter of the model with value 1.53 eV for the px and py valence orbitals. The matrix elements of the intra-atomic spin-orbit interaction are presented in Table 3.2. Table 3.2: Matrix elements of the atomic spin-orbit Hamiltonian matrix [Eq.3.43]. The matrix elements are zero when α, α0 = s. Adapted from Ref.[71]

0 hCαs|S.L|Cα0 s i 6p0 ↑0 6p0 ↓0 6p0 ↑0 6p0 ↓0 ~2 x x y y

6px ↑ 0 0 −i/2 0 6px ↓ 0 0 0 i/2 6py ↑ i/2 0 0 0 6py ↓ 0 −i/2 0 0

The presence of the SiC substrate breaks the inversion symmetry of the monolayer R bismuthene which leads to a Rashba term Hαs,α0s0 in the minimal tight-binding Hamiltonian. ~ This Rashba term has the form of (2mc)2 σ.∇V (r) × p [71], where ∇V (r) points in the direction perpendicular to the plane of the monolayer bismuthene. The values of the matrix R elements corresponding to Hαs,α0s0 are given in Table 3.3.

3.7.1 Low-energy band structure of monolayer layer bismuthene on SiC

Fig.3.5 (b) shows the low-energy band structure of inﬁnite two-dimensional planar honeycomb monolayer bismuthene on SiC as calculated within the minimal tight-binding model [Eq.3.42]. As is shown, the maximum of the valence band at which E = 0 is located at the K point of the ﬁrst Brillouin zone [shown in Fig.3.5 (a)]. Within this model the size of the indirect and direct band gaps are 0.86 and 1.22 eV, respectively. The splitting of the valence

43 R Table 3.3: The matrix elements Hαs,α0s0 with ﬁtting parameter value R = 0.395 eV. Adapted from Ref.[71]

R 0 0 0 0 0 0 0 0 0 0 0 0 Hαs,α0s0 6s ↑ 6s ↓ 6px ↑ 6px ↓ 6py ↑ 6py ↓ 6s ↑ 0 0 0 R 0 −iR 6s ↓ 0 0 −R 0 −iR 0 6px ↑ 0 −R 0 0 0 0 6px ↓ R 0 0 0 0 0 6py ↑ 0 iR 0 0 0 0 6py ↓ iR 0 0 0 0 0

Figure 3.6: Low-energy band structure of (a) zigzag (b) armchair bismuthene nanoribbons on SiC. The gapless edge states in the bulk band gap are shown in red. Adapted from Ref.[71] band maximum arising from the Rashba term HR is 0.46 eV that is in good agreement with experimental data obtained from ARPES and STS [29]. This model predicts an indirect band gap for the system with the conduction band minimum at the Γ point consistent with the DFT calculations carried out by Reis et al. [29]. As a consequence, this model accurately describes the low-energy band structure of the monolayer bismuthene on SiC, in agreement with the experimentally observed low-energy electronic structure of this system. A quantum spin Hall system is characterized by a bulk energy band gap traversed by gapless edge states. To show these edge states, the band structures of bismuthene on SiC nanoribbons with zigzag and armchair edges calculated within the aforementioned tight- binding Hamiltonian [Eq.3.42], are shown in Fig.3.6 (a) and (b), respectively. Here, the edge states in the bulk energy band gaps are shown in red. There is a single edge-state Kramers pair propagating at each edge of the zigzag and armchair monolayer bismuthene on SiC nanoribbons. This implies that this system is topologically nontrivial. Furthermore, the conductance calculations carried out within the aforementioned minimal tight-binding

44 2e2 Hamiltonian [71], revealed a precisely quantized two-terminal conductance G = h that is a characteristic of two-dimensional topological insulators with a single edge-state Kramers pair propagating at each edge of the system when the Fermi energy lies in the bulk energy band gap. The tight-binding Hamiltonian reviewed in this section is employed in this thesis to investigate the spin and valley polarizations of the edge and bulk states in this two-dimensional topological insulator. This investigation is presented in Chapter 7.

45 Chapter 4

Theory of electron transport through a quantum dot

4.1 Introduction

This chapter is divided into two parts such that the ﬁrst part reviews the quantum mechanical theory of electron transport that is used in this thesis in order to calculate the conductance of two and four terminal nanostructures. It begins with a description of the Landauer theory of electron transport at non-zero temperatures. Then, the zero temperature limit of the conductance of a two-terminal nanostructure in the linear response regime, the so-called Landauer formula and its generalization to the four-terminal nanostructure, the Büttiker-Landauer theory are discussed. This discussion is followed by a detailed explanation of the method of solving the Lippmann-Schwinger equation that is required to calculate the transmission probability of electrons that enters the Landauer and Büttiker- Landauer theories. The second part of this chapter formulates the method of projecting the calculated scattering states onto the Bloch subspaces associated with valleys K and K0, and the calculation of the valley accumulations and currents. Although spin accumulations have been studied previously in the context of spintronic devices, valley accumulations were studied for the ﬁrst time in the work presented in this thesis.

4.2 Two terminal conductance: Landauer theory of electron transport

In 1957, Landauer proposed [68] a quantum mechanical approach for the calculation of the conduction in one-dimensional systems composed of a mesoscopic conductor and two quasi- one-dimensional electrodes. The idea of viewing the conduction in mesoscopic systems as a quantum transmission problem evolved into the Landauer transport formula [69, 70] that is used in this thesis to calculate the two-terminal conductance of monolayer and bilayer graphene nanostructures. The coupling between the quantum dots and electrodes is assumed

46 Figure 4.1: The schematic representation of a mesoscopic one-dimensional ballistic conductor of length L connected to the electron source and drain electrodes with electrochemical s(d),in potentials µs and µd, respectively. Im is the longitudinal electric current carried by the transverse mth mode of the conductor injected from the source (drain) electrode. The difference between the electric currents injected from the source and drain electrodes gives the m net longitudinal electric current Inet ﬂowing through the conductor. to be strong so that Coulomb blockade phenomena are not present in the systems that will be considered. Consider an ideal one-dimensional conductor of length L that is connected to two electrodes (source and drain) shown in Fig.4.1. The quantized electron states in this conductor are distinguished by the longitudinal momenta k along the conductor and an index m which denotes diﬀerent discrete electron states. Thus, the energies of the mth transverse electron state Em(k) in this conductor form a band called subband m. The density of electrons per unit length in the momentum range between k and k+dk corresponding to the mth subband is given by

2 L n (k)dk = f(E (k) − µ)dk, (4.1) m L 2π m where f(Em(k) − µ) is the Fermi distribution function and the factor 2 counts the spin degeneracy of the subband m. The electron source and drain electrodes are assumed to be reﬂectionless (adsorb all the electrons injected from the conductor to the electrodes) having electrochemical potentials µs and µd, respectively. The electrons travel through the one- dimensional ideal conductor ballistically without scattering. Therefore, the electric current carried by the subband m and injected into the ideal one-dimensional conductor from the source electrode is given by

47 Z ∞ Z ∞ s,in fs(Em(k) − µs) Im = − evm(k)nm(k)dk = −2e |vm(k)| dk, (4.2) −∞ −∞ 2π where 1 fs(Em(k) − µs) = , (4.3) e(Em(k)−µs)/kB T + 1

∂Em(k) is the Fermi distribution function for electrons in the source electrode, and vm(k) = ∂(~k) is the electron velocity along the conductor in mode m. In the same way, the electric current originating from the drain electrode and carried by the subband m through the conductor is

Z ∞ Z ∞ d,in fd(Em(k) − µd) Im = − evm(k)nm(k)dk = 2e |vm(k)| dk, (4.4) −∞ −∞ 2π where fd(Em(k)−µd) is the Fermi distribution of electrons in drain electrode. Note that the electron velocity associated with the states carrying the electric current originating from the source (drain) electrode is positive (negative). So, the net electric current carried by conducting channel (subband) m through the ideal one-dimensional conductor is given by

Z ∞ m s,in d,in fs(Em(k) − µs) fd(Em(k) − µd) Inet = Im − Im = −2e |vm(k)| − dk. (4.5) −∞ 2π 2π

dk Changing the variable of the above integral from wave vector k to energy E (dk = dE dE), leads to

Z ∞ m −2e Inet = [fs(Em − µs) − fd(Em − µd)]dEm. (4.6) h −∞

In the zero temperature limit, the Fermi distribution functions fs(Em(k) − µs) and fd(Em(k) − µd) are the step functions as

fs(E − µs) −→ ϑ(µs − E), fd(E − µd) −→ ϑ(µd − E). (4.7)

Using the Eq.4.7 for the zero temperature limit and changing the variable of integration, Eq.4.5 takes the form

Z µs 2 m 2e ∂Em ∂k 2e 2e Inet = − dEm = − (µs − µd) = Vbias, (4.8) 2π µd ∂(~k) ∂Em h h

µs−µd where Vbias = − e is the applied bias voltage between the source and drain electrodes. Fig.4.2 shows the dispersion of a single transverse mode of the conductor schematically. At zero temperature the states with the wave vectors from kd to ks are occupied and carry the electric current inside the conductor. As is represented, the net electric current is carried by

48 Figure 4.2: Schematic representation of a single mode in a conductor and the occupied states that carry the electric current injected from the source and drain electrodes at zero temperature with electrochemical potentials of µs and µd, respectively. The states with wave vector k between k and ks (kd), the green (purple) segment of the band, carry the electric current through the conductor injected from the source (drain) electrode. The net electric current is carried by the occupied states with energies in the range µd < E < µs (red segment of the band).

49 Figure 4.3: Ballistic conductor of length L connected to the source and drain electrodes modeled as groups of ideal semi-inﬁnite one-dimensional leads (wavy orange lines).

the states with the energies lying within the energy channel µs − µd. According to Eq.4.8, the conductance of a single conducting mode in the zero temperature limit is given by

I 2e2 G0 ≡ = , (4.9) Vbias h where the factor 2 counts the spin degeneracy of the mode. As it was assumed that the mode 5 is fully conducting, G0 = 7.75 × 10 S is the maximum conductance of a single conducting channel and is called conductance quantum. In practice, the backscattering of the electrons within the conductor and at its interface with the electrodes decrease the conductance of a channel. Hence, the net electric current [Eq.4.6] ﬂowing through a single channel nanodevice with single channel source and drain electrodes can be generalized as

Z ∞ m −2e Inet = T (E)[fs(Em(k) − µs) − fd(Em(k) − µd)]dE, (4.10) h −∞ where T (E) is the transmission probability for an electron with energy E from the source to the drain electrodes while being scattered by the conductor. In the linear response regime (low applied bias voltage), the conductance of the conducting channel m at zero temperature can be written as

2e2 Gm = T (E ) = G T (E ), (4.11) h F 0 F where EF is the electrode Fermi level.

4.2.1 Electrodes supporting multiple conducting channels

In this thesis the source and drain contacts are modeled as groups of ideal semi-inﬁnite one- dimensional leads as represented in Fig.4.3. Here, each lead supports one spin degenerate conducting channel. In order to calculate the two-terminal conductance of the nanostructures connected to the contacts each supporting multiple conducting channels, Eq.4.11 can be modiﬁed as

50 Figure 4.4: Ballistic conductor connected to four contact (each at its own electrochemical potential µi) modeled as groups of ideal semi-inﬁnite one-dimensional leads (wavy orange lines).

2e2 G = X T (E ), (4.12) h ij F ij

th where Tij(E) is the transmission probability of electrons from the j conducting mode of the source contact to the ith conducting mode of the drain contact. This equation is the Landauer formula for the two-terminal conductance when the source and drain electrodes are supporting multiple conducting channels. The transmission coeﬃcients Tij that enter the Landauer formula are given by

s X ds 2 vi Tij = |tij | d , (4.13) ij vj

∂ where vs(d) = j(i) is the electron velocity in the jth(ith) lead of the source (drain) at j(i) ∂(~k) energy E, and j(i) are the energy eigenvalues of the tight-binding Hamiltonian of the 1D ds th semi-inﬁnite leads. tij is the transmission amplitude of an electron transmitted from the j lead of the source to the ith lead of the drain. In this thesis, the transmission amplitudes are obtained by solving the Lippmann-Schwinger equation.

4.3 Multi-terminal conductor: Büttiker-Landauer theory

The generalization of the Landauer theory of electron transport to a conductor with more than two contacts is called Büttiker-Landauer theory [67]. Consider a conductor which is

51 connected to four contacts each at a speciﬁc electrochemical potential µi (shown in Fig.4.4).

Then, the current Ii in each contact at the zero temperature in the linear response regime can be written as

2e I = (M µ − R µ − X T µ ), (4.14) i h i i ii i ij j j6=i where Mi is the number of modes supported by contact i, µi(j) is the electrochemical th potential of the i(j) contact, Rii is the reflection probability of electrons scattered by the conductor and reflected back into the contact i, and Tij is the transmission probability of electrons from contact j to contact i. The coefficients Tij and Rii have the same form as Eq.4.13 and are obtained by solving the Lippmann-Schwinger equation.

4.4 Electron transmission through the nanostructures

In this section the method of the calculation of the scattering states in a nanostructure coupled to the electrodes that I have used in this thesis is discussed in detail. Although, the method is explained for a two-terminal nanostructure, I have taken the same steps to apply this method to the four-terminal nanostructures as well. The nanostructure system represented in Fig.4.3 can be divided into three parts: the source and drain electrodes and the quantum dot. Hence, the Hamiltonian that describes this system can be written as

H = HQD + Hs + Hd + W s + W d, (4.15) where HQD is the tight-binding Hamiltonian of the quantum dot (the systems used in this thesis are quantum dots made of monolayer and bilayer graphene and bismuthene on SiC). Hs(d) is the Hamiltonian of the source (drain) electrode, and W s(d) is the coupling Hamiltonian between the source (drain) electrode and the quantum dot. In order to apply this method to the quantum dots of monolayer graphene, bilayer graphene and bismuthene on SiC, HQD is replaced with their corresponding tight-binding Hamiltonians. I model the source and drain electrodes as a group of semi-inﬁnite one-dimensional ideal leads represented by wavy orange lines in Fig.4.3. Each lead is considered to be a semi- inﬁnite one-dimensional atomic chain with lattice parameter a and one orbital per atomic site that supports the propagation of a single spin degenerate mode. Using a tight-binding model in the nearest-neighbour approximation, the Hamiltonians which describe the source

52 and drain electrodes can be written respectively as

−1 s X X s s Hl = [l |φlnihφln| + tl (|φlnihφln+1| + h.c.)], (4.16) l n=−∞ ∞ d X X d d Hl = [l |φlnihφln| + tl (|φlnihφln+1| + h.c.)], (4.17) l n=1 where the ﬁrst sums in the above equations run over the leads l making up the source and s(d) drain electrodes, the second sums run over the atomic sites n of each lead, ln is the on-site th th s(d) energy of the n atomic site belonging to the l lead, tl is the nearest-neighbour hopping parameter between the nearest-neighbour atomic sites in each lead, and |φlni is the atomic orbital at site n of the lead l. Each lead is considered to be isolated from the other leads so that no electron hopping occurs between the atomic sites belonging to diﬀerent leads. The coupling Hamiltonian between the electrodes and the quantum dot nanostructure is given by

s(d) X X X ls(d) W = Win (|ciihφln| + h.c.), (4.18) l i n where l denotes the lead in the source (drain) electrode, the second sum runs over the atomic sites i of the quantum dot to which the leads are coupled, n denotes the atomic site of the lth ls(d) th lead coupled to the quantum dot, Win is the hopping parameter between the n atomic site belonging to the lth lead and the ith atomic site of the quantum dot that is coupled to th the l lead, |φlni is the atomic orbital at site n of the lead l, and |cii is the atomic orbital at atomic site i of the quantum dot. Here, I have considered that each lead is coupled to only one atomic orbital of the quantum dot and the hopping only occurs between the last atomic site of the lead (the atomic site denoted by n = −1(1) for the source (drain) electrode that is the closest site to the quantum dot) and the atomic orbital to which the lead is coupled. ls(d) The hopping amplitude Win , is assumed to be the same as the nearest-neighbour hopping parameter in the corresponding tight-Hamiltonians [Eqs.4.16 and 4.17]. Hence, the coupling Hamiltonian between the source and drain electrodes and the quantum dot becomes

s(d) X X ls(d) W = Wi,−1(1)(|ciihφl,−1(1)| + h.c.), (4.19) l i ls(d) s(d) with Wi,−1(1) = tl . The investigation of the conductance of the quantum dot nanostructure coupled to the source and drain electrodes, requires the evaluation of the total transmission probability Tij of an electron which is injected into the system from the contacts. The evaluation of the total transmission probability used in this thesis is equivalent to solving an elastic quantum mechanical scattering problem in which the electron wavefunction injected from the lth lead of the source contact is scattered by the quantum dot so that it is partially transmitted to

53 the mth lead of the drain electrode and partially reﬂected back to the same lead l or other leads of the source electrode. The electron scattering wavefunction |Ψli of the complete coupled system (two contacts and quantum dot) associated with the electron injected from the lth lead can be written as

th where rpl is the reflection amplitude for an electron injected from the l lead and reflected th back into the p lead of the source contact, tml is the transmission amplitude for the injected electron from the lead l in the source contact to the mth lead of the drain contact, i specifies the atomic sites in the quantum dot, Ci is the scattering state amplitude at atomic site i in the quantum dot, and |ni (|ci) is the atomic orbital in the leads (quantum dot). The atomic sites of the leads are labeled so that for a lead in the source (drain) contact, n starts from −∞ (∞) to −1 (1) and n = −1 (1) is considered to be the lead atomic site that is coupled to the nanostructure. To calculate the scattering states of the coupled system |Ψli using the Lippmann- Schwinger equation, the coupled system Hamiltonian [Eq.4.15] can be written as

H = H0 + V, (4.21)

QD s d where H0 = H + H + H is the total Hamiltonian of the decoupled system, and V = W s + W d is the term which couples the ideal leads of the source and drain contacts to the quantum dot. The Hamiltonian of the coupled system satisﬁes

H|Ψli = E|Ψli, (4.22) implying that |Ψli is the eigenstate of the fully coupled system (scattering state). In order to calculate the scattering states of the coupled system associated with the injected electron from the lead l of the source contact |Ψli, I numerically solve the Lippmann- Schwinger equation that can be written as

l l l |Ψ i = |Φ0i + G0(E)V |Ψ i, (4.23)

l th where |Φ0i is the electron wave function in the l lead when it is decoupled from the graphene nanostructure, G0(E) is the Green’s function of the decoupled system, and V is the coupling term between the contacts and quantum dot as has been introduced in Eq.4.18.

54 Within the tight-binding model, the dispersion relation of a semi-inﬁnite one-dimensional ideal lead with one atomic orbital per site can be written as

E(k) = ε + 2τcos(ka), (4.24) where ε is the on-site energy assumed to be equal to the on-site energy of the atomic site of the dot to which the lead is attached, τ is the hopping amplitude between the nearest- neighbour atomic sites of the lead with lattice spacing a. Therefore, the eigenstate of the electron with wave vector kl in the lth lead when it is decoupled from the nanostructure can be written as

N l 1 X iklna −iklna |Φ0kl i = √ (e − e )|ni, (4.25) 2N n=1 where |ni is the atomic orbital in that lead, and N is the total number of the lead’s atomic sites (assumed to be very large to model semi-inﬁnite one dimensional leads).

The decoupled Green’s function of the system at energy E, G0(E), can be written as the sum of the decoupled Green’s functions of the parts of the system when they are decoupled as

s QD d G0(E) = G0(E) + G0 (E) + G0(E), (4.26)

QD where G0 (E) is the Green’s function of the quantum dot when it is decoupled from the s d electrodes. The Green’s functions of the source G0 and drain G0 electrodes can be written as the sum over their corresponding constituent leads

s(d) X l G0 (E) = G0(E), (4.27) l∈s(d) where

|Φl ihΦl | Gl (E) = X 0kl 0kl 0 s(d) l E − (H ) + iδ k l (4.28) X l 0 = (G0)nn0 |nihn |, n,n0

l and l denotes the lead index. The matrix elements of the Green’s function (G0)nn0 are given by

iklna −iklna −ikln0a ikln0a l 1 X (e − e )(e − e ) (G0)nn0 = . (4.29) 2N E − E l + iδ kl k

55 As N is assumed to be a large number the summation over kl can be converted to an integral π P L R a so that kl −→ π , where L = Na is the length of the one-dimensional lead with lattice 2π − a parameter a. Since only the last atomic site of each lead is coupled to the quantum dot, the only matrix elements of the Green’s function required for the calculation of the transmission 0 probability Tij are obtained by setting n, n = 1. So, the matrix elements are given by

Z π 2iyl l 1 Na l 2(1 − e ) (G0)11 = dy l , (4.30) 2Na 2π −π [E − ( + 2τcos(y )) + iδ] where yl = kla. By application of the residue theorem (poles of the integrand satisfy l E− cos(y◦) = 2τ ), the integral [Eq.4.30] can be evaluated as

l 2ik◦a l i 1 − e (G0)11 = l , (4.31) 2τ sin(k◦a)

l l where y◦ = k◦a. Also, the Green’s function of the decoupled quantum dot has the form 1 GQD(E) = 0 E − HQD + iδ |c ihc | (4.32) = X i i , E − + iδ i i

QD QD where i are the energy eigenvalues obtained by diagonalizing H , and H |cii = i|cii are the corresponding eigenstates of the HQD. Inserting Eqs.4.20 and 4.25 and the calculated decoupled Green’s functions Eqs. 4.31 and 4.32 into the Lippmann-Schwinger equation, leads to

−ikma d tm,l = e h1m|Ψ i, s −ikla −ikpa rp,l = h−1p|Ψ i − e δp,l e , (4.34) QD Ci = hci|Ψ i.

56 As only the transmission tm,l and reﬂection rp,l amplitudes entering the Landauer formula and Büttiker-Landauer theory and the scattering amplitude at each site of the quantum dot Ci are required, one can multiply Eq.4.33 by h−1p|, h1m|, hci| yielding

−ikla ikpa s p X s p,i −ikla ikla e δp,l + rp,le − (G0)−1,−1 (W )−1,0Ci,l = (e − e )δp,l, l QD l X s l,p −ikla ikpa QD l X d i,m ikma Ci,l − (G0 )0,0 (W )0,−1[e δp,l − rp,le ] − (G0 )0,0 (W )0,1 tm,le = 0, p m ikma d m X d m,i tm,le − (G0)1,1 (W )1,0 Ci,l = 0. l (4.35)

In this thesis the required amplitudes are evaluated by solving the above ﬁnite set of linearly independent equations.

4.5 Valley-projected States

The method of the calculation of the valley accumulations and their spatial distributions in quantum dots is proposed for the ﬁrst time in this thesis. In order to calculate the valley accumulations, I exploited the method of projecting the scattering states in the quantum dots of monolayer graphene that was ﬁrst proposed in Ref.[44]. Then, to investigate the valley accumulations of electrons in quantum dots of bilayer graphene and bismuthene on SiC, the method of the calculation of the valley-projected states has been generalized to these systems in this thesis. A detailed description of the projection method required for the evaluation of the valley currents, valley accumulations and valley polarizations in the nanostructures, is presented in this section.

4.5.1 Monolayer graphene quantum dots

Each unit-cell of monolayer graphene lattice structure contains two carbon atoms. If only one orbital is assumed per carbon atomic site, the crystal Bloch states of electrons in monolayer graphene can be written as

1 N α X ik·Ri α A α B |ψk i = √ e [cA(k)|pz,ii + cB(k)|pz,ii], (4.36) N i=1 where, α = 1, 2 denotes the diﬀerent Bloch states with wave vector k, N is the total number of unit-cells in monolayer graphene quantum dot, k is the wave vector of Bloch state, the A(B) set of Ri are the Bravais lattice vectors of monolayer graphene, and |pz,i i are the carbon th α atomic orbitals in the i unit-cell. The k-dependent coeﬃcients cA(B)(k) are evaluated analytically by diagonalizing the tight-binding Hamiltonian of monolayer graphene in the k-representation. In order to project the calculated scattering states of electrons onto the

57 Figure 4.5: (a)The hexagonal and rhombic Brillouin zone of monolayer graphene, bilayer graphene, and planar bismuthene on SiC. (b) The rhombic Brillouin zone is divided into two parts. Mesh of k-points belonging to valley K (K0) represented by blue (red) dots.

Bloch states, the rhombic Brillouin zone of monolayer graphene shown in Fig.4.5(a) is α divided into two parts. Then, the Bloch states |ψk i are calculated on a mesh of k points represented in Fig.4.5(b). To separate the Bloch states belonging to diﬀerent valleys, it is assumed that each Bloch state belongs to valley K (K0) if its wave vector lies in the upper (lower) triangle. Therefore, the valley-projected states corresponding to valley K and K0 are respectively written as

N where A = number of mesh points is the normalization factor. Since the Bloch states are deﬁned on a continuum in momentum space for a periodic inﬁnite crystal, the number of mesh points is chosen to be large enough for the results of calculations of the valley currents and accumulations to have converged with increasing numbers of mesh points.

4.5.2 Bilayer graphene quantum dots

In this thesis the method explained above is generalized to bilayer graphene and bismuthene on SiC quantum dots by calculating their corresponding Bloch states.

58 In the case of bilayer graphene quantum dots, each unit-cell of bilayer graphene includes four carbon atoms belonging to the sublattices A1 (A2) and B1 (B2) of the bottom (top) layer. So, the Bloch states of electrons (assuming one orbital per atomic site and suppressing the spin index) can be written as

N 1 X α √ ik·Ri α A1 α B1 α A2 α B2 |ψk i = e [cA1 (k)|pz,i i + cB1 (k)|pz,i i + cA2 (k)|pz,i i + cB2 (k)|pz,i i], (4.38) N i=1 where α = 1, ..., 4 denotes the diﬀerent Bloch states with wave vector k and the rest of parameters are the same as those explained in the case of monolayer graphene. The k- dependent coeﬃcients are calculated analytically by diagonalizing the tight-binding Hamil- tonian of bilayer graphene in the k-representation. Then, the valley-projected states in bilayer graphene quantum dots are calculated by replacing the Bloch states of monolayer graphene with the Bloch states of bilayer graphene in Eq.4.37.

4.5.3 Monolayer bismuthene on SiC substrate quantum dots

The Bi atoms in monolayer bismuthene on SiC form a planar hexagonal lattice structure with a lattice constant of 5.35 A˚. This system also has the same hexagonal or rhombic ﬁrst Brillouin zone represented in Fig.4.5(a). Therefore, the method of the calculation of the valley-projected states is applicable to the bismuthene monolayer on SiC. The unit-cell of the lattice structure of monolayer bismuthene includes two Bi atoms each having six valence orbitals 6s, 6px, 6py with spin up and down. Hence, the tight-binding Bloch states of electrons in this lattice structure can be written as

1 N 6 2 α X ik·Ri X X α |ψk i = √ e Cijs(k)|ji,si. (4.39) N i=1 j=1 s=1

Here, α = 1, ..., 12 enumerates diﬀerent Bloch states with wave vector k, jis denotes the atomic orbitals in the ith unit-cell with spin s. As in Eqs. 4.36 and 4.38, N is chosen to be the total number of unit-cells in the monolayer bismuthene, and Ri are the Bravais lattice α vectors. In this case, the k-dependent coeﬃcients Cijs(k) are calculated numerically for each k point in the mesh, by diagonalizing the k-representation tight-binding Hamiltonian. The tight-binding parameters that describe the key properties of the low energy band structure of the monolayer bismuthene on SiC are provided in Ref.[71]. In order to calculate the valley projected states in bismuthene quantum dots, the Bloch states of monolayer graphene in Eq.4.37 are replaced with the corresponding Bloch states of bismuthene [Eq.4.39].

59 4.6 Valley accumulations

The novel method of the calculation of the spatial distribution of the valley accumulations of the electrons in quantum dots is developed in the studies that are reported in this thesis. The current induced accumulations of electrons at atomic site n in valleys K and K0 of the band structure of the quantum dots connected to two or four contacts each having electrochemical potential µi are deﬁned to be

1 ∂ζl AK = X |hn|Ψl i|2 ∆µ , n 2π K ∂E i i,l (4.40) l K0 1 X l 2 ∂ζ A = |hn|Ψ 0 i| ∆µ n 2π K ∂E i i,l where i denotes the contact number, l runs over the constituent leads of the ith contact, l QD 0 and |ΨK(K0)i is the scattering state |Ψl i projected onto the valley K (K ). In a two- terminal quantum dot ∆µi is the electrochemical potential difference between the source and drain terminals, while in a four-terminal quantum dot, if the lowest of the electrochemical potentials of the terminals is µmin, then ∆µi is defined to be ∆µi = µi − µmin. In this ∂ζl definition, ∂E is interpreted as the density of states of electrons at the Fermi energy in the lth lead from which the electron is injected to the nanostructure. This factor influences the K (K0) dependence of the valley accumulations An on the energy. Here, the scattering state |Ψli [Eq.4.20] with energy eigenvalue E within the lth lead can be written as

l l hm|Ψli = eiζ m + rle−iζ m (4.41) at atomic site m. rl is the reflection probability of the electron reflected by the dot to the lth lead. Note that in this thesis the on-site valley accumulations are calculated in the linear ∆µi response regime where the bias voltage applied between the contacts Vbias = e is assumed to be small. The imbalance of the valley accumulations [Eq.4.40] between valley K and K0 (the so-called on-site valley polarization) is defined to be the difference between the valley accumulations of the electrons at site n and can be written as

v K K0 Pn = An − An . (4.42)

4.7 Valley velocity

To investigate the valley currents in bilayer graphene quantum dots, it is necessary to derive the velocity operator for electrons within these nanostructures. To this end, the η-component of the velocity operator is

60 1 vˆη = [ˆη, HˆBLG], (4.43) i~ P where ηˆ = n ηn|nihn| is the η coordinate operator (i.e. ηˆ =x, ˆ yˆ) and ηn is the η coordinate of the atomic site n. Inserting the ηˆ operator and the tight-binding Hamiltonian of bilayer graphene HˆBLG into Eq.4.43, the η-component of the velocity operator for electrons in bilayer graphene nanostructures can be written as

t vˆ = X [(η − η )|nihm| + (η − η )|mihn|]. (4.44) η i m n n m ~ hn,mi where the sum runs over the in-plane nearest neighbour atomic sites n and m (note that interlayer terms do not contribute to the velocity). Then, the η-component of the velocity of the electrons in bilayer graphene quantum dots described by the tight-binding Hamiltonian

HˆBLG is calculated as the expectation value of the velocity operator [Eq.4.44] with respect QD to the scattering states of electrons within the bilayer graphene quantum dot |Ψl i = P l n Cn|ni, so that

QD QD vη = hΨl |vˆη|Ψl i it (4.45) = X (η − η )(Cl∗Cl − Cl Cl∗), 2 m n m n m n ~ hn,mi

l QD th where Cm = hm|Ψl i and |mi is the 2pz carbon atomic orbitals of the m atomic site. K K0 0 The valley velocities vη and vη of electrons accumulating in valley K and K are deﬁned QD by replacing the scattering states of electrons |Ψl i in Eq.4.45 with the valley-projected l l states |ΨK i and |ΨK0 i, so that

K l l vη = hΨK |vˆη|ΨK i, (4.46) K0 l l vη = hΨK0 |vˆη|ΨK0 i.

Then, the η-component of the net valley velocity of electrons in the nanostructure are defined to be val K K0 vη = vη − vη , (4.47) which is the difference between the valley velocities of electrons accumulating in valleys K and K0. If the system is a nanostructure connected to multiple terminals i each at its own electrochemical potential µi, the ordinary vη velocity of an electron is calculated as a weighted QD average over the velocities associated with the scattering states |Ψl i injected from different leads l belonging to terminal i. Hence, the weighted average ordinary velocity can be written as

61 P QD QD i,lhΨl |vˆη|Ψl i∆µi vη = P , (4.48) i,l ∆µi where ∆µi = µi − µmin with µmin is the lowest electrochemical potential among all of the terminals. All of the leads li that make up each terminal are considered to have the same QD electrochemical potential. By replacing the scattering states of electrons |Ψl i with the l valley-projected states |ΨK(K0)i in Eq.4.48, the average valley velocity of electrons in the nanostructure associated with the electrons injected from the leads l belonging to terminal i can be written as P hΨl |vˆ |Ψl i∆µ K(K0) i,l K(K0) η K(K0) i vη = P . (4.49) i,l ∆µi The spatial distributions of the valley currents in the quantum dots are investigated using the following method: The expectation value of the velocity operator [Eq.4.45] is a sum of pairs (m, n) of the nearest neighbour carbon atoms. Thus, each term of this sum can be interpreted as the value of the valley velocity that is assigned to the midpoint

xm+xn ym+yn ( 2 , 2 ) of the line connecting the (m, n) pairs. Then, at the midpoint of each pair of nearest neighbour carbon atoms the assigned value of the electrons’ valley velocity can be written as

l l it l∗ l l l∗ hΨK(K0)|vη(x, y)|ΨK(K0)i = (ηm − ηn)(ψK(K0) ψK(K0) − ψK(K0) ψK(K0) ), (4.50) 2~ m n m n where

l X l |Ψ 0 i = ψ 0 |ni, (4.51) K(K ) K(K )n n l l is the valley-projected state, and ψ 0 = hn|Ψ 0 i with |ni representing the atomic K(K )n K(K ) orbitals of the atoms in the quantum dot. The net valley velocity for each pair is given by

val K K0 vη (x, y) = vη (x, y) − vη (x, y). (4.52)

Note that the valley velocity for each pair is also calculated as a weighted average [Eq.4.49] if the quantum dot is connected to multiple terminals.

62 Chapter 5

Gate-tunable valley-dependent transport properties of electrons in bilayer graphene nanostructures

5.1 Introduction

This chapter presents the results of calculations of the gate-tunable valley-dependent transport properties of electrons in bilayer graphene nanostructures from a quantum mechanical point of view that have been published in Ref.[72]. The spatial distribution of the valley accumulations of electrons and the correlation between the unit-cell averaged valley accumulations and the valley currents flowing through the nanostructure are investigated for the first time. Using the Büttiker-Landauer theory of electron transport in the linear response regime, the non-local resistances of broken inversion symmetry bilayer graphene nanostructures are investigated as a function of the Fermi energy at zero temperature. By applying the method of projection of the scattering states that is described in the preceding chapter, the magnitudes of the valley currents relative to the electric currents flowing through the system are calculated in the quantum tunneling regime as the Fermi energy varies from the valence band to the conduction band. The results of the calculation of the non-local resistance RNL and valley current as the gate voltage varies show that these quantities scale differently with applied gate voltage. Furthermore, it is shown that a power law governs the scaling relation between the local and non-local resistances when the gate voltage varies at zero Fermi energy. Experimental measurements of the non-local four-terminal resistances in monolayer graphene [30] and bilayer graphene [32, 31] in a Hall bar geometry with broken inversion symmetry, showed a striking enhancement of RNL as the Fermi energy passes the energy of Dirac point. Although the valley currents were not measured directly, the enhancement of

RNL was considered as a signature for the presence of valley currents ﬂowing through the samples. A brief description of the reasoning used by Gorbachev et al. [30] and Shimazaki et al. [32] to interpret the pronounced peak of the RNL as a signature of the ﬂowing val-

63 ley currents transverse to the applied electric current is as follows1: The broken inversion symmetry of the sample leads to a non-zero Berry curvature pointing in opposite directions in the K and K0 valleys. Then, different anomalous velocities (second term on the right- hand side in Eq.2.63) of electrons belonging to different valleys generate valley currents that are transverse to the electric current. Thus, these valley currents flow between the voltage probes and consequently lead to a potential difference and pronounced peak of the non-local resistance RNL. The application of the semi-classical theory [Eq.2.63] to the interpretation of the enhancement of non-local resistance as a signature of the presence of valley currents in the monolayer and bilayer samples is questionable, due to fact that inversion symmetry breaking mechanisms open a spectral gap in the monolayer and bilayer graphene samples

[30, 32, 31, 73] and the striking enhancement of the RNL was observed when the Fermi energy was in the energy gap (insulating regime). Thus, within the gap the electron transport mechanism is quantum tunneling that has no classical analog. Furthermore, in the linear response regime the electric field that generates the electric current through the system approaches zero and does not affect the transmission probabilities of electrons Tij that determine the transport coefficients in Büttiker-Landauer theory. Hence, the non-local resistances RNL are not influenced by the anomalous velocity term in Eq.2.63 in the linear response regime because that term is proportional to the electric field. Thus, interpreting the striking enhancement of RNL as a signature of the presence of valley currents based on anomalous velocity term in Eq.2.63 is open to criticism. I have taken a fully quantum mechanical approach to evaluate the non-local resistances

RNL and valley currents in the bilayer graphene nanostructure. My results obtained using the Büttiker-Landauer theory of electron transport in the linear response regime at zero temperature showed that a striking enhancement in RNL occurs and also valley currents become several times stronger than the electric current ﬂowing within the inversion symmetry-broken nanostructure when the Fermi energy passes the energy of Dirac point. The inversion symmetry of bilayer graphene can be broken by an electric ﬁeld which is perpendicular to the graphene layers that make up the bilayer graphene. This mechanism of breaking inversion symmetry opens up the possibility of the investigation of gate-tunable valley-dependent transport properties in bilayer graphene. Regarding the potential technological applications in future valleytronics, the mechanism of breaking inversion symmetry in bilayer graphene makes this system more favorable than monolayer graphene that employs an aligned hexagonal boron nitride substrate to break its inversion symmetry, a mechanism that is not readily tunable.

1A detailed discussion of the interpretation of the experimental results is presented in Sec.2.8

64 Figure 5.1: Four terminal bilayer graphene nanostructure with armchair edges. The bottom (top) layer is shown in black (blue). Each contact is composed of 40 semi-infinite one- dimensional ideal leads (shown by red wavy lines) that are attached to both layers and connect the nanostructure to the reservoirs. The electric current flows through current contacts 1 and 2 (C.C.1 and C.C.2), while there is no net electric current entering or leaving the voltage contacts 3 and 4 (V.C.3 and V.C.4). In non-local resistance studies the potential difference is measured between contacts 3 and 4. Upper right inset: Two types of first Brillouin zone of bilayer graphene, hexagonal (solid) and rhombic (dotted). Lower right inset: Side view of the four carbon atoms of a unit cell in bilayer graphene in AB stacking. The inversion symmetry point is shown by a red dot.

65 5.1.1 Model

Fig.5.1 represents the four-terminal nanostructure that was used in the present work to investigate the gate-tunable non-local transport. The bilayer graphene nanostructure with armchair edges, and the two current contacts (C.C.1 and C.C.2), and two voltage contacts (V.C.3 and V.C.4) are the constituent components of this system. The single graphene layers of the bilayer graphene quantum dot are stacked in the AB geometry (Bernal stacking). In this geometry the top layer (shown in blue in Fig.5.1) is shifted by the nearest neighbour distance of a = 1.42 Å in the positive x-direction with respect to the bottom layer (shown in black in Fig.5.1). The current contacts and the voltage contacts are modeled as groups of semi-inﬁnite one-dimensional leads shown by red wavy lines in Fig.5.1. As was discussed in Chapter 2, the nearest neighbour tight-binding Hamiltonian that is employed to describe the bilayer graphene nanostructure has the form

X † X † X † HBLG = nanan − tnm(anam + H.c.) + tn1m2 (an1 am2 + H.c.), (5.1) n hn,mi hn1,m2i where n is the on-site energy, tnm = t = 2.7 eV is the in-plane hopping amplitude between † pz orbitals of the nearest neighbour atomic sites, and an (an) is the creation (annihilation) operator which creates (annihilates) electrons at atomic site n. tn1m2 is the hopping amplitude between the pz orbitals of the nearest neighbour atomic sites belonging to different layers (dimer sites). The last term on the right-hand side of the Hamiltonian [Eq.5.1] couples the single graphene layers and is responsible for the difference between the electronic band structures of monolayer and bilayer graphene. To model the perpendicular electric field used in the experiments [31, 32] to break the inversion symmetry of bilayer graphene sample, the on-site energies of the carbon atoms are modified. Hence, n = +Vg/2 for the atoms of the top graphene layer (blue color) and

n = −Vg/2 for the atoms of the bottom graphene layer (black color). The lower right inset of Fig.5.1 shows the side view of the four carbon atoms of a unit-cell in bilayer graphene and the red dot represents the inversion symmetry point in the unit-cell in the absence of the perpendicular electric ﬁeld.

5.2 Non-local resistance

In order to calculate the four-terminal non-local resistance RNL of a bilayer graphene quantum dot (shown in Fig.5.1), the transmission probabilities of electrons Tij are calculated at the Fermi energy EF by solving the Lippmann-Schwinger equation that is discussed in Sec.4.4 of the previous chapter. Afterwards, the Büttiker equations [Eq.4.14] are solved and the non-local resistance is calculated as

66 Figure 5.2: Calculated non-local resistance RNL [Eq.5.2] of the nanostructure of Fig.5.1 in the linear response regime at zero temperature for diﬀerent values of the gate voltage Vg = 0 eV (green), Vg = 0.3 eV (red), and Vg = 0.5 eV (blue) as a function of the Fermi energy EF .

∆V3,4 RNL = , (5.2) I1,2 where the I1,2 is the electric current flowing through the nanostructure in Fig.5.1 between the current contacts 1 and 2, and ∆V3,4 is the potential difference between the voltage contacts 3 and 4. The relationship between the potential difference ∆V and the electrochemical ∆µ potential of the contacts which enter the Büttiker-Landauer theory is given by ∆V = e . The results of the calculation of the non-local resistance RNL [Eq.5.2] of the four-terminal bilayer graphene nanostructure in the linear response regime at zero temperature are shown in Fig.5.2. The non-local resistance is evaluated as a function of the Fermi energy for different values of the gate voltage Vg. As is shown, greatly enhanced peaks occur in the non-local resistance as the Fermi energy passes the energy of Dirac point (EF = 0) if the gate voltage is non-zero. In other words, breaking the inversion symmetry (Vg 6= 0) of the bilayer graphene nanostructure leads to a striking enhancement in non-local resistance. A comparison of the maximum values of the non-local resistances RNL (red and blue curves) which occur at

EF = 0 calculated for diﬀerent values of the gate voltage Vg, shows that RNL is enhanced

67 Figure 5.3: (a) Schematic representation of the bilayer graphene nanostructure with the val directions of the ordinary vy and valley vx velocities of electrons flowing through the val quantum dot. (b) Normalized valley velocity vx /vy of the bilayer graphene nanostructure for different values of the gate voltage as a function of the Fermi energy when the net electric current flows between current contacts 1 and 2.

by increasing the Vg. Thus, the non-local transport in the bilayer graphene nanostructure is gate-tunable.

5.3 Valley currents

In order to investigate the valley current flowing through the four-terminal bilayer graphene nanostructure, the scattering state |Ψli associated with lead l from which the electron is injected is calculated at all atomic sites of the nanostructure by solving the Lippmann- Schwinger equation. Then, the valley-projected states required for the calculation of the valley velocity of electrons are evaluated as described in Sec.4.5 as per Eq.4.37. The η- val K K0 K K0 component of the weighted valley velocity is defined as, vη = vη − vη , where vη and vη are the weighted average velocity of electrons in valley K and K0, respectively. val The weighted average ordinary and valley velocities of electrons, vη and vη respectively, were calculated in the x and y directions. As is expected, the computed weighted average ordinary velocity of electrons in the x-direction vx is nearly zero within numerical error due to the fact that the main electric current flows in the y-direction between current contacts 1 and 2 (schematically represented in Fig.5.3 (a)). The valley currents and valley velocities were not measured experimentally or computed theoretically for the bilayer graphene quantum dot in previous studies. In order to help gain insight as to how the magnitude of the valley currents with respect to the electric current changes when the Fermi energy EF and

68 Figure 5.4: Comparison of the x and y components of the normalized valley velocity as function of the Fermi energy (a) at Vg = 0.5 eV and (b) Vg = 0.3 eV.

gate voltage Vg vary, the evaluated valley velocities in the x and y-directions are normalized val to the ordinary velocity of electrons in the y-direction (vη /vy). val The results of the calculation of the normalized valley velocity in the x-direction (vx /vy) as a function of the Fermi energy EF for diﬀerent values of the gate voltage Vg are shown in

Fig.5.3 (b). As can be seen here, in the absence of the inversion symmetry breaking (Vg = 0) the valley current in the x-direction (green curve) is negligible and becomes zero at EF = 0. However, the magnitude of valley current in the x-direction (red and blue curves) becomes several times greater than the electric current ﬂowing through the nanostructure when the

Fermi energy lies in the energy gap around EF = 0 in the presence of inversion symmetry breaking (Vg 6= 0). According to Fig.5.3 (b), the maximum value of the valley current in the x-direction increases when the gate voltage Vg increases. It can also be concluded that the valley current in the broken inversion symmetry bilayer graphene quantum dot is gate- tunable. Based on the results of the computed normalized valley velocities in the x and y-direction shown in Fig.5.4, the valley current in the y-direction approaches zero as the

Fermi energy EF passes the energy of Dirac point. To summarize, the valley current ﬂows mainly in the x-direction (transverse to applied electric current) when EF lies in the gap and inversion symmetry of the bilayer graphene quantum dot is broken. However, the x and y components of the normalized valley velocities are relatively small in the presence or absence of the inversion symmetry breaking when EF is well away from the energy of the

Dirac point. Note that the enhancement of the valley velocity near EF = 0 is a quantum eﬀect since it occurs primarily in the gap of the density of states of the nanostructure. This gap is due in part to the quantum conﬁnement of electrons in the bilayer graphene nanostructure and in part to the application of the gate voltage Vg.

69 val Figure 5.5: The normalized valley velocity (vx /vy) (purple line) and non-local resistance RNL (green line) of the bilayer graphene nanostructure as a function of gate voltage at zero Fermi energy EF = 0.

The non-local resistance RNL and normalized valley velocity were also calculated for a bilayer graphene quantum dot with dimensions one half of those of the quantum dot considered in Fig.5.1 and whose inversion symmetry was broken by Vg = 0.3 eV, to investigate the eﬀect of the size of the system on the non-local transport of electrons. The simulation results showed that the RNL of the smaller quantum dot to be smaller by a factor of ∼ 16 at EF = 0, while its valley current in the x-direction was smaller by a factor of ∼ 2.5.

5.4 Gate-tunabiliy of the non-local resistance and valley current

As has been mentioned earlier, an advantage of the bilayer graphene over the monolayer graphene quantum dots is the possibility of gate tunability of the valley currents. So, the

70 evaluation of the gate-tunability of non-local resistance and valley currents in valleytronic devices is of great importance due to its potential technological application in this field. To this end, the non-local resistance and normalized valley current flowing in the x-direction val (vx /vy) were investigated at EF = 0 as the gate-voltage Vg varies. The results of the gate- tunability of the non-local resistance and valley currents are shown in Fig.5.5. According to the computational results, the valley current (purple line) increases from zero at Vg = 0 (when the inversion symmetry is not broken) to a value of five times greater than the electric current. For the low values of the gate voltage (i.e., 0 < Vg < 0.1 eV) the valley current increases linearly with increasing Vg. However, for low values of the gate voltage the results of the computed non-local resistance RNL (green curve) reveal quadratic behaviour as a function of Vg. This shows clearly that measurements of the non-local resistance do not provide a direct measure of the valley current. Note, that deviations from these scaling relations for the valley current and non-local resistance are expected for higher values of the gate voltage (i.e.,Vg > 0.1 eV).

5.5 Scaling relation between the local and non-local resistances

Fig.5.6 shows the connection between the local and non-local resistances of this four-terminal bilayer graphene quantum dot. The local resistance RL is calculated as the ratio of the potential difference between contact 1 and 2, and the electric current which is flowing between the same contacts (RL = ∆V1,2/I1,2). According to Fig.5.6, the relationship between RNL α and RL satisfies a power-law relation RNL ∼ RL with α = 2.19. The reported values based on the experimental results for α are 3 [32] and 2.77 [31]. It has been suggested in Ref.[31] that the value of α varies between different samples because of the disorder in the samples. So, deviation of the value of α reported in this thesis from the experimental results may be due to the difference between ballistic transport system investigated in this thesis and diffusive transport in the experiments.

5.6 Spatial distribution of valley accumulations and valley currents

From the technological applications point of view, it is important to investigate the spatial distribution of the valley accumulations of electrons, valley currents, and their possible correlation in the valleytronic devices. Valley accumulations and their correlations with valley currents are studied for the ﬁrst time in this thesis. To this end, the scattering states of electrons associated with the injection lead l, |Ψli, are projected onto the Bloch subspaces of electrons which leads to the calculation of the valley-projected states [Eq.4.37]. Having l l evaluated the valley-projected states |ΨK i and |ΨK0 i, the on-site valley accumulations and

71 Figure 5.6: The scaling relation between the local and non-local resistance as the gate α voltage varies from 0 to 0.2 eV. Red line is the power law RNL ∼ RL fitted to the simulation data. Upper left inset: The configuration of non-local resistance measurements. Lower right inset: The configuration of local resistance measurements.

v Figure 5.7: The calculated on-site valley polarization Pn of electrons belonging to the bottom layer of the bilayer graphene nanostructure when the net electric current ﬂows between current contact 1 and 2 at Vg = 0.15 eV and EF = 0 eV. The on-site valley polarization at each atomic site is shown by blue (red) disks when it is positive (negative).

72 Figure 5.8: The calculated valley velocity (green arrow) and unit cell-averaged valley polarization Pav(x, y) of the bottom layer of bilayer graphene nanostructure at Vg = 0.15 eV and EF = 0 eV. The unit cell-averaged valley polarization at each unit is shown by blue (red) disks when the it is positive (negative). polarizations of electrons are evaluated using Eq.4.40 and Eq.4.42, respectively. Then, the v calculated valley polarizations Pn of electrons and generated valley currents are mapped onto the lattice structure of the bilayer graphene quantum dot to obtain a clear picture of the possible correlation between the valley polarizations of electrons and the valley currents. v The map of the on-site valley polarization Pn of the electrons belonging to the bottom layer of the bilayer graphene nanostructure is represented in Fig.5.7. The electrons flow through the quantum dot from contact 1 and 2 represented in Fig.5.1. Here, the positive v on-site valley polarizations (Pn > 0) are represented by the blue disks and the negative on-site valley polarizations are depicted by the red disks. The radius of the depicted disk v at each atomic site is proportional to the magnitude of its corresponding Pn . According to the definition of the on-site valley polarization [Eq.4.42] as the difference between the K K0 0 on-site valley accumulations of electrons (An and An ) in valley K and K at atomic site n, a blue (red) disk at atomic site n shows that the electron is polarized in valley K (K0). Fig.5.7 shows that the on-site valley polarizations are stronger on the sites close to the contact (current contact 1) from which the electrons are injected to the bilayer graphene nanostructure. It should be pointed out that the on-site valley polarization mapped onto the top graphene layer which is not shown, also exhibits the same spatial distribution. v As can be seen in Fig.5.7, the on-site valley polarizations Pn often have differing signs at the different atomic sites in the unit-cell (considering the rhombic unit-cell of monolayer graphene for each layer). Therefore, to better characterize the spatial distribution of the valley polarizations, I define a cell-averaged valley polarization by

P v + P v P (x, y) = n m , (5.3) av 2

73 v v where Pn and Pm are the on-site valley polarizations calculated at the atomic sites n and m, respectively. The calculated Pav(x, y) is mapped on the lattice structure of bilayer graphene at (x, y) = [(xn + xm)/2, (yn + ym)/2]. Inspection of spatial distribution of the cell-averaged valley polarization helps to gain insight about the relationship between valley accumulations and valley currents. Fig.5.8 shows the spatial distribution of cell-averaged valley polarizations Pav (red and blue disks) and the weighted average valley velocities [Eq.4.49] (green arrows) at Vg = 0.3 eV . Following the same convention as is used in Fig.5.7, the positive average valley polarizations are depicted by blue (red) disks and the radius of each disk is proportional to the magnitude of its corresponding average valley polarization. As is shown in Fig.5.8, the calculated valley velocity is chiral and directed from left to right near the lower boundary of the nanostructure. However, the chiral valley velocity near the opposite boundary (not shown) travels in the opposite direction from right to left. The valley currents flowing along the upper boundary are weaker, attributed to the quantum tunneling nature of the electron transport in the insulating regime. Changing the sign of the gate voltage Vg along the layers of bilayer graphene (i.e., n = −Vg/2 on the atoms of the top layer and n = +Vg/2 on the atoms of the bottom layer) leads to reversing the overall directions of the valley velocities along the lower and upper boundaries of the bilayer graphene quantum dot. If a few exceptional sites near the ends of contacts 1 and 3 are excluded, the calculated cell-averaged valley polarization Pav (red and blue disks in Fig.5.8) exhibits a dipolar distribution, being negative (red) on the left and positive (blue) on the right region of the quantum dot. This dipolar character not only has physical significance, but also can have potentially practical applications in future valleytronics. It indicates that the valley currents flowing through the nanostructure from left to right, transport the valley degree of freedom which results in a considerable valley imbalance between the left and right regions of the nanostructure. The on-site and cell-averaged valley polarizations are negligible when the gate voltage

Vg is zero (absence of inversion symmetry breaking) and grow as the gate voltage increases. According to similar calculations for the top layer of the bilayer graphene, the spatial distribution of the on-site valley polarization, cell-averaged valley polarization, and the valley current display the same behaviour for the top layer as those calculated in the bottom layer that are shown in Fig.5.8.

5.7 Summary

In this chapter, the results of calculations of the non-local resistance, valley current, and on- site valley polarization of four-terminal bilayer graphene nanostrutures from the perspective of a quantum mechanical approach in the insulating regime are presented. The results show that a striking enhancement occurs in the non-local resistance of the quantum dot with

74 broken inversion symmetry as the Fermi energy passes the energy of Dirac point. The valley currents are also maximal (several times greater than the electric current flowing through the nanostructure) when the inversion symmetry of the bilayer graphene quantum dot is broken (Vg 6= 0) and the Fermi energy is close to the energy of the Dirac point. The calculations of the valley current and non-local resistance have been carried out when the effect of the electric field that drives the electrons has been sent to zero (the linear response ˙ regime). Consequently, the underlying physics of the anomalous velocity term (k × Ωk) for the interpretation of the generated valley velocity is not applicable in the linear response regime since k˙ is proportional to the electric field. The calculated valley currents discussed in this chapter result from the non-adiabatic injection of electrons to the bilayer graphene nanostructure but not the acceleration of electrons in an electric field in the presence of non-zero Berry curvature. Furthermore, mapping the average valley polarization calculated in the bilayer graphene nanostructure with broken inversion symmetry reveals a dipolar distribution that results from the presence of valley currents so that the average valley polarization increases in the direction of the valley current.

75 Chapter 6

Graphene valley ﬁlter, accumulator and switch

6.1 Introduction

Despite the fact that spintronics and valleytronics exploit different degrees of freedom for technological applications, there is correspondence between their principal elements. An analogy between these two fields acknowledges the potential importance of the investigation of the valley filtering mechanisms in valleytronics, in view of the decisive role played by spin filters in spintronics. The valley filtering mechanisms have attracted considerable attention recently. Hence, many mechanisms have been proposed based on graphene line defects [86, 85, 36, 35], disorder [37], electrostatic potentials [87, 88], and transport through nano- constrictions [34]. However, it should be emphasized that difficulties in achieving control and reproducibility of the valley filters based on many of these proposed mechanisms pose an obstacle in the path of further progress. In this chapter, I propose a mechanism of valley filtering in monolayer graphene quantum dots decorated by double lines of adsorbed hydrogen atoms. The valley filtering mechanism in monolayer graphene by means of adsorbed species has not been investigated in previous studies either experimentally or theoretically. In this thesis, the mechanism of filtering electrons according to their valley degree of freedom by means of adsorbed lines of hydrogen atoms has been proposed for the first time. The results for the proposed valley filtering mechanism are published in Ref.[89]. Adsorbed hydrogen atoms change the electronic properties of graphene [75, 76, 77, 78] and endow it with functionalities that may have technological applications [79, 80, 81]. Since the positioning of a hydrogen atom on monolayer graphene with atomic precision has been demonstrated experimentally (using STM) [74], the valley filtering mechanism in monolayer graphene quantum dots using adsorbed lines of hydrogen atoms has the potential to be experimentally reproducible and realizable. Hydrogen atoms adsorbed on monolayer graphene induce scattering resonances referred to as Dirac point resonances at energies close to the energy of the Dirac point [82, 83].

76 These resonant scattering states are due to the coupling between the discrete states of the adsorbed hydrogen atoms and the monolayer graphene continuum. The valley filtering mechanism introduced in this chapter exploits these scattering resonances. It is found that the Dirac point resonances induced by the hydrogen atoms fall within the energy gap of the band structure of the monolayer graphene quantum dot and form a narrow conducting channel. The aforementioned band gap is induced due to the quantum confinement resulting from the finite size of the monolayer graphene quantum dot. As will be demonstrated below, the electrons belonging to the K and K0 valleys of the graphene quantum dot, travel in opposite directions along the induced conducting channel. Hence, the electric current flowing in a given direction along the induced one-dimensional conductor is carried by the electrons belonging to one of the two graphene valleys. Monolayer graphene quantum dot decorated by adsorbed double line of hydrogen atoms can also function as a valley switch so that, if the direction of the electric current is reversed, the conduction is mediated by the electrons belonging to the other valley. Furthermore, the conducting channel induced by the hydrogen atoms is strongly localized to the vicinity of the hydrogen lines. Hence the electric current flowing through this one-dimensional conductor induces strong accumulation of electrons belonging to only one valley near the hydrogen lines. Since the conducting channel is localized exponentially to the vicinity of the adsorbed lines of hydrogen, the efficiency of the proposed valley filter, switch, and accumulator is not affected by impurities and other imperfections located several lattice spacings away from the hydrogen lines.

6.2 Model

Monolayer graphene quantum dot decorated by a double line of hydrogen atoms is represented in Fig.6.1 (a). The hydrogen atoms are depicted by the yellow disks and the lines of hydrogen are oriented in the y-direction. The source and drain contacts (orange wavy lines) connected to the monolayer graphene quantum dot (shown in black) are modeled as groups of semi-inﬁnite one-dimensional leads. The properties of these leads and the methodology of coupling them to the quantum dot are described in detail in Chapter 4. The relaxed geometry of an adsorbed hydrogen atom on monolayer graphene is shown in Fig.6.1 (b). The hydrogen and carbon atom to which hydrogen binds are placed 1.47 and 0.32 Å above the graphene plane, respectively. The nanostructure shown in Fig.6.1 (a) is modeled by a tight-binding Hamiltonian that describes the Dirac point resonances of hydrogen on monolayer graphene. This Hamiltonian has the form

X † X † X † X † H = ∆n(anan) − tn,m(anam + H.c.) + αdαdα + γα,n(dαan + H.c.), (6.1) n hn,mi α α,n

77 Figure 6.1: (a) Two-terminal monolayer graphene quantum dot with armchair edges. The source and drain contacts are modeled by groups of semi-inﬁnite one-dimensional leads (orange wavy lines). Each contact is composed of 46 leads. Two lines of hydrogen atoms (yellow disks) are adsorbed on top of the monolayer graphene and divide the quantum dot into two equal parts. (b) Relaxed geometry of a single adsorbed hydrogen atom on graphene. The hydrogen and the carbon atom to which hydrogen binds are placed 1.47 and 0.32 Å above the graphene plane. (c) Hexagonal and rhombic representations of the Brillouin zone of monolayer grpahene. Parts (a) and (b) are reproduced from Ref.[89]

78 where the ﬁrst two terms describe the well known nearest neighbour tight-binding Hamilto- nian of monolayer graphene explained in Chapter 2. ∆n is the on-site energy of the carbon atoms in the monolayer graphene quantum dot. The third term considers the on-site energy of the adsorbed hydrogen atoms with α = −0.0383 eV. The last term is the hopping between the hydrogen atom and the carbon atom to which hydrogen binds with hopping amplitude γα,n = 2.219 eV. This tight-binding model of Dirac point resonances due to adsorbed hydrogen atoms [Eq.6.1] and its corresponding parameters are taken from Ref.[82].

The parameters α and γα,n in Ref.[82] were chosen so as to accurately describe the scattering of the π electrons in graphene by the sp3 bonded hydrogen/carbon complex. The effects of the px, py, and s valence orbitals of carbon atoms were included so that the model agreed with the results of the DFT calculations presented in Ref.[84]. To investigate the effect of the left-right mirror symmetry with respect to the hydrogen lines on the valley filtering mechanism in monolayer graphene quantum dot shown in Fig.6.1

(a), I investigate two cases by modifying the on-site energies of the carbon atoms (∆n). In the ﬁrst case that will be referred to as "symmetric", the hydrogen atoms are adsorbed on pristine graphene (the case ∆n = 0). The second case or asymmetric case, considers the adsorbed hydrogen atoms on graphene with broken inversion symmetry in each unit- cell (as in graphene on hexagonal boron nitride). In this case the absolute value of the on-site energies of the carbon atoms is |∆n| = 0.0602 eV with diﬀering signs for the nearest- neighbour carbon atoms in each unit-cell. This method of breaking inversion symmetry also breaks the left-right mirror symmetry of nanostructure with respect to the double line of hydrogen atoms. Thus, ∆l and ∆r, the on-site energies for carbon atoms on the right-and left-hand side of the lines of hydrogen atoms have opposite signs.

6.3 Two-terminal conductance calculations

I have carried out two-terminal conductance calculations to explore the eﬀect of the lines of hydrogen atoms on the electron transport in the nanostructure represented in Fig.6.1 (a). Here, the electron current ﬂows through the nanostructure from the source to the drain contact in the y-direction. The two-terminal electrical conductance G of this system was calculated using the Landauer formula of electron transport at zero temperature in the linear response regime, [Eq.4.12]. The calculated two-terminal conductances of the monolayer graphene quantum dot with and without the double line of hydrogen atoms (Fig.6.1 (a)) for the symmetric case (∆n = 0) are shown in Fig.6.2. The blue (red) curve represents the results of the calculated conductance when the lines of hydrogen are present (absent). As can be seen, when the lines of hydrogen are present (blue curve) a striking enhancement occurs in conductance (G ∼ 4e2/h) when the Fermi energy lies in the range −0.0157 eV < EF <−0.005 eV. For the same tight- binding parameters discussed in Eq.6.1, the Dirac point resonance due to a single hydrogen

79 Figure 6.2: Calculated two-terminal conductance G [Eq.4.12] of the graphene quantum dot in Fig.6.1 with (blue curve) and without (red curve) hydrogen lines in the linear response regime at zero temperature for the symmetric case (∆ = 0), vs. Fermi energy. Inset: The ﬁne structure of the calculated pronounced peak.

80 atom on an inﬁnite two-dimensional graphene sheet is centered at −0.00702eV [82] which is in this range. This fact suggests that the pronounced peak near EF = 0 in the conductance of the decorated quantum dot can be attributed to the transmission of electrons from the source to the drain by the Dirac point resonances of the individual hydrogen atoms which together induce a conducting channel. The inset in Fig.6.2 shows the ﬁne structure of this conductance enhancement. The series of conductance maxima and minima in the range

−0.0157 eV < EF <−0.005 eV shown in the inset, are due to the multiple reflections of electrons in the induced conducting channel at the ends of the lines of hydrogen. In the absence of the adsorbed hydrogen atoms quantum confinement that results from the finite size of the nanostructure opens an energy gap in the band structure of the monolayer graphene quantum dot. The much smaller but non-zero conductance (G ∼ 0.8e2/h) near zero Fermi energy in the absence of the lines of hydrogen atoms (shown in red in Fig.6.2) is due to the quantum tunneling of the elecrons in this energy gap. In order to investigate the effect of the left-right mirror symmetry with respect to the double line of hydrogen atoms on the conductance of the induced channel, I have also carried out the two-terminal conductance calculations when the left-right mirror symmetry of the quantum dot is broken with respect to the lines of hydrogen atoms by modifying the on-site energies of the carbon atoms (∆n 6= 0) as dicussed above. These evaluated conductances of the quantum dot with and without the lines of hydrogen atoms are shown by the blue and red colors respectively in Fig.6.3. As is shown by the blue curve, if the mirror symmetry of the system is broken two peaks occur near EF ∼ 0.05 and EF ∼ −0.06 eV . The fine structures of these two peaks shown in the insets of Fig.6.3, are not similar and their series of the maxima and minima can be attributed to the reflection of electrons in the conducting channel at the end of the lines of hydrogen atoms. A comparison of the red curves in Fig.6.2 and Fig.6.3 (the calculated conductances of monolayer graphene quantum dot without lines of hydrogen for the symmetric and asymmetric cases, respectively) sheds light on the fact that the much smaller conductances in the absence of hydrogen atoms are due to the quantum tunneling of electrons in the opened energy gap. The energy gap of the asymmetric case that is opened due to the symmetry breaking and the finite size of the sample is larger than the energy gap opened by finite size of the sample in the symmetric case. Hence, according to the nature of the quantum tunneling a lower conductance near the zero Fermi energy is expected in the asymmetric case due to its larger energy gap.

6.4 The role of the band structure

To clarify the underlying physics of the pronounced peaks that occur in the two-terminal conductances, I have plotted the band structures of inﬁnite uniform graphene nanoribbons with and without a double line of hydrogen atoms. Here, inﬁnite lines of hydrogen atoms

81 Figure 6.3: Calculated two-terminal conductance G [Eq.4.12] of the graphene quantum dot in Fig.6.1 with (blue curve) and without (red curve) hydrogen lines in the linear response regime at zero temperature for the asymmetric case (∆r = −∆l), vs. Fermi energy. Insets: The ﬁne structures of the calculated pronounced peaks.

82 Figure 6.4: Inﬁnite zigzag graphene nanoribbon with adsorbed double line of hydrogen atoms (yellow disks). The lines of hydrogen atoms on the ribbon are inﬁnite and run along the center of the ribbon in the y-direction.

83 on the ribbons run along the centers of the ribbons. As is shown in Fig.6.4 the ribbons have zigzag edges and the lines of hydrogen atoms are oriented in the y-direction.

The presence of the two lines of hydrogen atoms in the symmetric case (∆n = 0) results in a fourfold degenerate narrow band close to the zero energy which is shown in red in Fig.6.5. The part of the band structure of the graphene ribbon corresponding to states not localized near the hydrogen lines is plotted in the taupe color. Note that the ﬂat band at zero energy is due to the zigzag edges of the ribbon. According to the Fig.6.2 and Fig.6.5 (comparing the energy at which the pronounced peak occurs in Fig.6.2 and the energy of the induced conducting channel in Fig.6.5), it is evident that the pronounced peak in the conductance results from the states of the hydrogen-induced band close to the Dirac point energy. Here, it should be emphasized that there are no features due to the zigzag edges in the two terminal conductance calculations since they have been carried out for the monolayer graphene quantum dot with only armchair edges (Fig.6.1 (a)). The band structure of the graphene ribbon (Fig.6.4) with broken mirror symmetry

(∆r = −∆l) is shown in Fig.6.6. As can be seen, breaking the mirror symmetry opens a band gap of ∼ 2|∆| = 0.12 eV around zero energy. It also splits the conducting channel induced due to the lines of hydrogen atoms into two bands (red and blue). Then, in the asymmetric case these split channels give rise to the pronounced conductance peaks near the zero Fermi energy in Fig.6.3. In Figs.6.5 and 6.6 the electronic states of electrons induced due to the lines of hydrogen atom have positive (negative) group velocities v = ∂E for negative (positive) values of ∂(~k) wave vector k. The positive and negative values of k in the band structure of the ribbons can be regarded as the projections of the valley K and K0 that are defined in the rhombic representation of the Brillouin zone of monolayer graphene (Fig.6.1 (c)) on the ky axis. Hence, the positive (negative) k values in Figs.6.5 and 6.6 are labeled by valley K (K0) indicated at the top of the figures. This suggests that the electronic states induced by the hydrogen lines should support the transport of electrons belonging to valley K in the negative y-direction and electrons belonging to valley K0 in the positive y-direction shown in Fig.6.1 (a). Consequently, the Dirac point resonance states due to the hydrogen atoms function as a valley filter if the electric current is injected into the quantum dot in the positive or negative y-direction. Based on the same reasoning, reversing the source and the drain contacts (changing the direction of the electric current) switches the conduction to the other valley. Hence, this monolayer graphene quantum dot decorated by the double line of adsorbed hydrogen atoms is predicted to also function as a valley switch. Furthermore, an electric current mediated by the Dirac point resonances (i.e., when EF lies in the conducting channel induced by lines of hydrogen atoms) can induce a strong valley polarization in the vicinity of the adsorbed lines of hydrogen atoms.

84 Figure 6.5: Band structure of zigzag graphene nanoribbon with adsorbed lines of hydrogen atoms in Fig.6.4 and symmetric on-site energies of the carbon atoms (∆ = 0). States localized near the hydrogen lines (H-band) are shown in red and other graphene states (G- bands) are shown in taupe. The region belonging to valley K (K0) is shaded in blue (red). The valleys K and K0 are indicated at the top of their corresponding regions.

85 Figure 6.6: Band structure of zigzag graphene nanoribbon with adsorbed lines of hydrogen atoms in Fig.6.4 and asymmetric on-site energies of the carbon atoms (∆r = −∆l). States localized near the hydrogen lines (H-bands) are shown in red and blue. Other graphene states (G-bands) are shown in taupe. The region belonging to valley K (K0) is shaded in blue (red). The valleys K and K0 are indicated at the top of their corresponding regions.

86 Figure 6.7: (a) Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed lines of hydrogen atoms (Fig.6.1) for the symmetric case at EF = −0.0065 eV . Calculated electric current-induced on-site valley 0 K (K) 0 accumulations of electrons An in valley K (K) near hydrogen lines shown as red and (much smaller blue) disks. Diameters of the red (blue) disks are proportional the calcu- K0(K) lated An . (b) Calculated square amplitude (shown in red) of a representative electron H-band eigenstate (red band in Fig.6.5) of the zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulations of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown.

6.5 Valley ﬁltering and valley accumulation

In order to validate the idea of valley ﬁltering in monolayer graphene by means of adsorbed lines of hydrogen atoms described in the previous section, I have calculated the spatial 0 K K0 distribution of the on-site valley accumulations [Eq.4.40] in valleys K and K , An and An , respectively. To this end, the projection method explained in chapter 4 was used to evaluate the valley-projected states [Eq.4.37]. The on-site valley accumulations were calculated for Fermi energies lying in the induced narrow H-bands for the symmetric and asymmetric cases. The valley accumulations of electrons in valley K (K0) are shown in Fig.6.7 by blue K(K0) (red) disks such that the diameters of the disks are proportional to the An . In Fig.6.7 (a) a representative example of the spatial distribution of the current-induced on-site valley accumulations of electrons in the monolayer graphene quantum dot decorated by lines of hydrogen atoms (Fig.6.1) with mirror symmetry (∆ = 0) is shown. As predicted by the above heuristic reasoning based on the band structure of the ribbon (Fig.6.5) with mirror symmetry, when the electrons ﬂow through the nanostructure in the positive y-

87 Figure 6.8: (a) Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed lines of hydrogen atoms (Fig.6.1) for the asymmetric case at EF = 0.0502 eV . Calculated electric current-induced on-site valley 0 K (K) 0 accumulations of electrons An in valley K (K) near hydrogen lines shown as red and (much smaller blue) disks. Diameters of the red (blue) disks are proportional the calculated K0(K) An . (b) Calculated square amplitude (shown in red) of a representative electron H-band eigenstate (red band in Fig.6.6) of zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulations of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown. direction, the electrons accumulate mainly in the K0 valley (red disks). However, the much smaller accumulation of electrons in valley K (blue disks) attributed to the partial reflection of electrons by the boundaries (at the ends of the hydrogen lines). The reflected electrons travel in the negative y-direction and consequently accumulate in valley K. Fig.6.7 (b) shows the electron H-band eigenstate distribution over a fragment of the unit-cell of the zigzag graphene nanoribbon (Fig.6.4) with hydrogen atoms. This distribution confirms that the hydrogen-induced states are localized exponentially to the vicinity of the lines of hydrogen atoms in a similar way to its corresponding on-site valley accumulations shown in Fig.6.7 (a). Figs.6.8 (a) and 6.9 (a) present the calculated on-site valley accumulations of electrons for the asymmetric case (∆r = −∆l) when the Fermi energy lies in the upper (red) and the lower (blue) induced H-bands, respectively. As can be seen, due to the fact that the electrons are flowing in the positive y-direction, the valley accumulation of the electrons is overwhelmingly in valley K0. Unlike the symmetric case, the split H-bands resulting from

88 the broken mirror symmetry allow the electrons to travel only on one side of the hydrogen lines. So that, the upper (lower) H-band mediates the electron transport on the right (left) side of the hydrogen lines. This fact is confirmed by the results of the evaluated spatial distributions of the H-band eigenstates of the graphene nanoribbons with broken mirror symmetry shown in Figs.6.8 (b) and 6.9 (b). In order to quantify the functionality of the valley filtering mechanism discussed above, I define the efficiency of the valley filter as the current-induced valley polarization = 0 P AK (K) n n . Then, the efficiency of the symmetric valley filter is found to be 90.8% at P (AK +AK0 ) n n n EF = −0.0065eV and the calculated efficiencies of the asymmetric filters at EF = 0.0502eV

(upper H-band) and EF = −0.0638 eV (lower H-band) are 91.8 and 95.1%, respectively. The efficiency of the proposed valley filtering mechanism can be tuned by modifying the conditions so as to vary the reflection probability of electrons. For instance, if the Fermi energy matches the energy of a broader conductance peak (shown in the insets of Figs.6.2 and 6.3), the reflection probability of electron decreases and leads to minimized accumulation of electrons in valley K, effectively increasing the efficiency of the accumulation for valley K0.

6.6 Summary

This chapter revealed an efficient and experimentally realizable valley filtering mechanism which exploits the Dirac point resonances due to a double row of hydrogen atoms adsorbed on a graphene quantum dot. I have shown that the adsorbed lines of hydrogen atoms induce an electrically conducting channel close to the Dirac point energy within the energy gap resulting from quantum confinement in the pristine monolayer graphene nanostructure. Ac- cording to the calculated band structure of the zigzag graphene nanoribbon, it is predicted that an electric current passing through the induced channel in a given direction is carried by electrons of only one of the two graphene valleys. Consequently, the proposed nanostructure made of the monolayer graphene quantum dot with adsorbed lines of hydrogen atoms functions as a valley filter if the Fermi energy is tuned so as to lie in the induced conducting channel. Furthermore, if the direction of the flowing electric current is reversed (swapping the source and drain contacts), the conduction switches to the electrons belonging to the other valley. Hence, the proposed decorated quantum dot of monolayer graphene can also function as a valley switch. Additionally, the effect of breaking the mirror symmetry of the decorated quantum dot with respect to the lines of hydrogen atoms by a hexagonal boron nitride substrate was investigated. It has been shown that the broken mirror symmetry splits the induced conducting channel. Then, each conducting channel accumulates electrons only on one side of the hydrogen lines. The accumulations of electrons in the symmetric and asymmetric cases are localized exponentially to the vicinity of the hydrogen lines. This suggests that the functionality of the nanostructure as a valley filter, accumulator, or switch is

89 Figure 6.9: Spatial distribution of the on-site valley accumulations [Eqs.4.40] in the armchair monolayer graphene quantum dot with adsorbed lines of hydrogen atoms (Fig.6.1) for the asymmetric case at EF = −0.0638eV . Calculated electric current-induced on-site valley 0 K (K) 0 accumulations of electrons An in valley K (K) near hydrogen lines shown as red (blue) K0(K) disks. Diameters of the red (blue) disks are proportional the calculated An . (b) Calcu- lated square amplitude (shown in red) of a representative electron H-band eigenstate (blue band in Fig.6.6) of zigzag graphene nanoribbon near lines of hydrogen atoms corresponding to the valley accumulation of (a). In order to show that these eigenstates are exponentially localized, a ribbon unit-cell fragment is also shown.

90 not expected to be affected by impurities located more than a few lattice spacings from the hydrogen lines. Finally, I have shown that the efficiency of this valley filtering mechanism increases if the reflection of electrons at the end of the conducting channel is minimized.

91 Chapter 7

Valley and spin polarizations of the edge and bulk states in quantum dots of the topological insulator monolayer bismuthene on SiC

7.1 Introduction

The bulk energy band gap in typical topological insulator materials is small (less than 30 meV) [25, 26, 93, 94]. For such small band gaps, thermal excitations of electrons lead to competition between the edge currents and currents due to the bulk energy bands except at very low temperatures. Therefore, one serious obstacle in the path of progress towards exploiting the potential technological applications of topological insulators is the small bulk band gap. Systems that support the quantum spin Hall phase and possess large bulk energy gaps can in principle overcome this diﬃculty. According to the recent theoretical and experimental studies that have been carried out on monolayer bismuthene on SiC, this system can be regarded as a promising candidate for a high-temperature topological insulator system due its large bulk energy band gap (∼ 0.86 eV) [90, 29, 71, 91]. In this chapter I present the valleytronic properties of quantum dots of the topological insulator bismuthene on SiC which have not been explored previously either theoretically or experimentally. Within the tight-binding model introduced in Chapter 3, the transmission probability and associated scattering states of electrons traveling through the quantum dot are calculated. Using the valley-projection method,1 the valley and spin polarizations of the conducting channels propagating at the zigzag edges of the system are investigated. A valley polarization reversal of the edge states is predicted as the Fermi energy varies in the gap from the top of the valence band to the bottom of the conduction band, and an intuitive explanation of this novel eﬀect is provided. The results obtained from the investigation

1The valley-projection method for the multiorbital systems is presented in Chapter 4.

92 Figure 7.1: (a) The bismuthene planar honeycomb lattice (black) of the two-terminal monolayer bismuthene quantum dot with contacts attached to the armchair edges. Each contact is composed of 38 ideal semi-infinite one-dimensional leads (shown by orange wavy lines). (b) Two alternative representations of the first Brillouin zone of monolayer bismuthene, hexagonal (solid line) and rhombic (dotted line). This figure is reproduced from Ref.[92] of the valleytronic properties of the edge and bulk states of quantum dots of monolayer bismuthene on SiC that are presented in this chapter have been published in Ref.[92] I have also modeled a potential energy increasing in the direction parallel to the zigzag edges of the nanostructure, to explore the correlation of the spin and valley degrees of freedom of electrons in this quantum spin Hall system. In this model, I compare two differing cases: First case, when the conduction of electrons is only mediated by the edge states (Fermi energy lies in the bulk energy gap) and the second case, when both the bulk and edge states conduct the electrons through the quantum dot (Fermi energy lies in the valence band).

7.2 Model

To describe the bismuthene on SiC nanostructure shown schematically in Fig.7.1 (a), I have used the minimal tight-binding Hamiltonian reviewed in Chapter 3. The presence of SiC as a substrate in this nanostructure stabilizes the two-dimensional planar monolayer

93 bismuthene and shifts the pz valence orbital of the Bi atoms out of the low-energy sector of the band structure of the system [29]. So, the low energy band structure of monolayer bismuthene on SiC is governed by the s, px, and py valence orbitals of the Bi atoms. As was mentioned in Chapter 3, this tight-binding model predicts correctly the key properties of the low-energy band structure of bismuthene on SiC, such as the magnitude of the bulk energy gap (∼ 0.86 eV) and the Rashba valence band splitting (∼ 0.43 eV) consistent with the experimental data [29]. Throughout this chapter, the electrons are traveling between the source and drain contacts through the nanostructure in the y-direction (shown in Fig.7.1 (a)). Here, each contact is modeled as a group of semi-inﬁnite one dimensional leads (orange wavy lines in Fig.7.1 (a)), as has been discussed in Chapter 4.

7.3 Valley polarization of the edge states

In order to investigate the valley polarization of the states propagating at the edges of the quantum dots of bismuthene on SiC, I exploited the valley-projection method explained v K K0 in Chapter 4. Then, the on-site valley polarization Pn = An − An is defined to be the K K0 difference between the calculated on-site valley accumulations of electrons, An and An , in valley K and valley K0 at atomic site n [Eq.4.42]. Figs.7.2 and 7.3 show representative v examples of the calculated spatial distributions of the on-site valley polarizations Pn for edge states with Fermi energies inside the bulk band gap. In these figures, the calculated on-site valley polarizations are represented by red (blue) disks if they are positive (negative), i.e. valley K (K0) predominates. The diameters of the disks are proportional to the magnitude of the calculated on-site valley polarizations. If the Fermi energy lies in the bulk band gap of the topological insulator, conduction is expected to be mediated by the edge states localized to the edges of the quantum spin Hall system. Therefore, any valley polarization induced by the electric current is also expected to be localized in the vicinity of the edges. Fig.7.2 exhibits the localized valley predominantly K0 polarization (blue disks) of the edge states at EF = 0.10 eV. Note that the valence (conduction) band edge is at 0 (0.86) eV. However,

Fig.7.3 reveals that the edge states are polarized in valley K (red disks) at EF = 0.16 eV. This suggests that the valley polarization of the edge states of the quantum spin Hall system propagating along the zigzag edges can be reversed by varying the Fermi energy inside the bulk band gap from the top of the valence band to the bottom of the conduction band. To quantify the polarization of the edge states and compare it at different Fermi energies, 0 P AK(K ) the valley filter efficiency can be defined as n n . Then, I have found efficiencies of P AK +AK0 n n n 91.8% and 90.2% in Figs.7.2 and 7.3, respectively. Fig.7.4 (a) shows the spatial profiles of the current-induced valley polarization calculated on a chain of Bi atoms extending from the left zigzag edge to the center of the quantum dot. This chain of Bi atoms is depicted in Fig.7.4 (b). As can be seen, the valley polarizations calculated at different values of Fermi energies inside the bulk energy band gap are well

94 v Figure 7.2: Spatial map of the on-site valley polarization Pn calculated at EF = 0.10 eV. v The on-site valley polarization is shown by blue disks when the calculated Pn is negative. v The diameters of the disks are proportional to the magnitude of Pn . Note that the valence (conduction) band edge is at 0 (0.86) eV

95 v Figure 7.3: Spatial map of the on-site valley polarization Pn calculated at EF = 0.16 eV. v The on-site valley polarization is shown by red disks when the calculated Pn is positive. v The diameters of the disks are proportional to the magnitude of Pn . Note that the valence (conduction) band edge is at 0 (0.86) eV

96 Figure 7.4: (a) The on-site valley polarization calculated on the chain of Bi atoms shown in part (b), for diﬀerent values of the Fermi energy. (b) Chain of Bi atoms that extends in the x-direction from the left zigzag edge to the center of the quantum dot. localized to the vicinity of the zigzag edges. This strong localization is a consequence of the rapidly decaying nature of the edge states.

7.3.1 Valley polarization reversal of the edge states

In order to analyze the reversal of the valley polarization of the edge states between Figs.7.2 v and 7.3, I have investigated the variation of the strongest on-site valley polarization Pmax as a function of the Fermi energy. As can be seen in Fig.7.5, if the Fermi energy is close to the top of the valence band (indicated by the blue arrow) the edge states conduct electrons that accumulate primarily in valley K0 and, therefore, their valley polarization is negative. As the Fermi energy increases the strength of the valley polarization decreases and the intersection of the solid black line and dotted orange line indicates the Fermi energy EFs ∼ 0.13 eV at which the polarization of edge states switches to valley K. Then, the edge states support the v transport of electrons accumulating preferentially in valley K (positive Pn ) when the Fermi energy lies between the EFs and bottom of the conduction band (shown by the red arrow). It should also be pointed out that the valley polarization of the edge states can be switched

97 Figure 7.5: The strongest on-site valley polarization of the zigzag edge states as a function of the Fermi energy. The blue (red) arrow locates the top of the valence band at zero eV v Pmax (bottom of the conduction band at 0.86 eV). Positive (negative) values of ∆µ correspond to transport of electrons belonging to valley K(K0).

98 Figure 7.6: The band structure of the bismuthene on SiC nanoribbon with zigzag edges. Adapted from Ref.[71]. The gapless edge states in the bulk energy band gap with positive velocity are shown by the straight blue and red lines so that the states polarized in valley K0 are shown in blue and the states with K valley polarization are shown in red. by reversing the source and drain contacts shown in Fig.7.1 (a). Hence, the quantum dot made of topological insulator bismuthene on SiC can function as a valley switch based on gate control (varying the Fermi energy) or reversing the contacts. By considering Fig.7.6 which shows the dispersion of the zigzag edge states of a bismuthene on SiC nanoribbon in the bulk energy gap, the reversal of the valley polarization of zigzag edge states can be understood as follows: The group velocities of the edge states that carry the net electric current in Figs.7.2 and 7.3 both point in the positive y-direction (see Fig.7.1 (a)) from the source to the drain contacts. Then, as the Fermi energy varies from the top of the valence band to the bottom of the conduction band, the dispersion of the edge states with the positive slope, and equivalently, positive velocity (shown in red and blue in Fig.7.6) crosses from the left-hand half of the Brillouin zone where the edge states are in the K0 valley to the right-hand half of the Brillouin zone where the edge states are in valley K. The corresponding energy of the crossing point is ∼ 0.13 eV that is close to the Fermi energy at which the reversal of the valley polarization of the edge states occurs

(EFs shown in Fig.7.5). Hence, the reversal of the valley polarization of the zigzag edge

99 states shown in Fig.7.5 is clearly due to the dispersion crossing the center of the Brillouin 0 zone Γ, from valley K to valley K at EF = EFs . As a consequence, I predict that the aforementioned reversal of the valley polarization to be a general phenomenon occurring at the zigzag edges of all quantum spin Hall systems with honeycomb lattice structures and qualitatively similar edge state dispersions. I have also evaluated the valley polarization for armchair edge states by attaching the leads to the zigzag edges of the quantum dot instead of to the armchair edges as in Fig.7.1 (a). Based on my calculations, unlike the zigzag edge states, the conducting channels propagating at the armchair edges of the quantum dot support no well-deﬁned valley-polarized transport of electrons in the system. This result can be interpreted according to the following intuitive reasoning. As can be deduced from comparing Figs.7.1 (a) and 7.1 (b), the vector K − K0 is perpendicular to the armchair edges. Because of this, the armchair edge scatters electrons strongly between K and K0 valleys. Thus, there can be no well-deﬁned valley polarization for the armchair edge states. Since the vector K − K0 is parallel to the zigzag edges, this argument does not apply to the zigzag edges states, consistent with the results presented above.

7.4 Spin polarization of the edge states

s I have calculated the on-site spin polarization of the electrons Pnj, in the monolayer bismuthene on SiC quantum dot, in order to investigate the possible correlation between the valley and spin polarizations of the edge states. To this end, I have evaluated the expectation value of the spin operator S in the scattering states |Ψli. As a reminder, the scattering states of electrons traveling in the nanostructure are obtained by solving the Lippmann-Schwinger equation.2 Then, by analogy with the valley accumulation expression [Eq.4.40] the current- induced on-site spin polarization of the electrons travelling through the quantum dot from source to drain has the form

1 ∂ζl P s = Xhψl |S |ψl i ∆µ , (7.1) nj 2π n j n ∂E i l,i

l where j = x, y, z denotes the components of the spin operator S, and ψn are the calculated scattering states of electrons at atomic site n. Note that the other terms of this equation K(K0) are as deﬁned in Eq.4.40 for the on-site valley accumulation An . A representative example of the spatial map of the out-of-plane component of the on-site s spin polarization Pnz when the Fermi energy lies in the bulk band gap (EF = 0.10 eV) is s shown in Fig.7.7. The calculated Pnz are represented by red (blue) disks if they are positive (negative). Following the same convention as is used in the representation of the on-site

2The method of solving the Lippmann-Schwinger equation is presented in Chapter 4.

100 Figure 7.7: Spatial map of the on-site out-of-plane component (z-direction) of the spin s s polarization Pnz calculated at EF = 0.10 eV. Pnz is shown by red (blue) disks if it is s positive (negative). The diameters of the disks are proportional to the magnitude of Pnz.

101 v valley polarizations Pn , the diameter of each disk is proportional to the magnitude of the spin polarization calculated at the corresponding atomic site. As can be seen in Fig.7.7, the edge states propagating at the opposite edges of the quantum dot have opposite spin polarizations which is characteristic of quantum spin Hall materials. I have also investigated the spin polarization of the electronic states when the Fermi energy lies in the valence band (EF = −0.9 eV). If the Fermi energy lies in the valence band, conduction is mediated by both edge and bulk states. As is shown in Fig.7.8, although the spin polarization of the states at each edge is ferromagnetic, the spin polarization of the bulk states shows antiferromagnetic order. Comparing the results of the calculations of all the components of the spin polarization (x,y, and z) reveals that the spin polarization of the zigzag edge states is dominated by the out-of-plane component while for bulk valence band states this is true only at the valence band maxima.

7.5 Eﬀect of a nonuniform electrostatic potential on the valley and spin polarizations of the edge and bulk states

The switching of the valley polarization of the edge states between the valleys (shown in Figs. 7.2, 7.3, and 7.5) due to the variation of the Fermi energy inside the bulk band gap, suggests considering a complementary case in which the energies of the top of the valence band and bottom of the conduction band depend on the position in the quantum dot. Therefore, I have considered a model in which the electron potential energy depends on its position (y-coordinate) in the quantum dot. For simplicity I have considered a potential energy function that varies linearly as a function of the y-coordinate of the electron so that

V (yn) = V0(ymax + yn), (7.2)

1 th where V0 = 171.2 eV/Å, ymax = 85.6 Å, and yn is the y-coordinate of the n atomic site. The profile of the potential energy increasing in the y-direction parallel to the zigzag edges is shown in Fig.7.9 (a). V (y) increases from zero for the bottommost atomic sites to 1 eV for the topmost atomic sites in the y-direction. This model will also be used to examine the behaviour of the out-of-plane component of the spin polarization and the valley polarization when electrons flow from a region where the transport is mediated by the edge states (Fermi energy lies in the bulk band gap) to a region where not only the edge states, but also the bulk states mediate the electron transport (Fermi energy lies in the valence band). The effect of the increasing potential on the top of the valence band (shown in olive green) and bottom of the conduction band (shown in purple) is shown in Fig.7.9 (b). In this model, the locations of the bulk band edges relative to the Fermi energy are varying as a function of the y-coordinate of the atomic site. The Fermi energy lies in the bulk band gap for the points where it is between the maximum of the valence band and minimum of the conduction band. The spatial maps of the out-of-plane component of the spin polarization and the

102 Figure 7.8: Spatial map of the on-site out-of-plane component (z-direction) of the current s induced spin polarization Pnz calculated at EF = −0.09 eV (Fermi level inside the valence s band). Pnz is shown by red (blue) disks if it is positive (negative). The diameters of the s disks are proportional to the magnitude of Pnz.

103 Figure 7.9: (a) The proﬁle of the potential energy increasing in the y-direction (parallel to the zigzag edge). (b) Schematic representation of the top of valence band (olive green line) and bottom of the conduction band (purple line) when the variable potential energy is included in the model. Energies are measured from the top of the valence band at y = −85.6 Å where V (y) = 0. The orange horizontal line locates the Fermi energy for the cases considered in Figs.7.10 and 7.11.

valley polarization at EF = 0.435 eV when the potential energy is applied are shown in Figs.7.10 and 7.11, respectively. As depicted in Fig.7.9 (b), the Fermi energy coincides with the maximum of the valence band at y = yg. Therefore, in the region of the system with yn < yg (shown in orange in Figs.7.10 and 7.11) the electron transport is mediated by the edge states and in the remaining region (shown in black in Figs.7.10 and 7.11) both bulk and edge states propagate through the quantum dot. As can be seen in Fig.7.10, when the Fermi level is in the valence band, the electric current ﬂowing through the dot results in antiferromagnetic order for the out-of-plane component of the spin polarizations induced on adjacent bismuth atoms in the interior of the s quantum dot (Pnz have opposite signs on the nearest neighbour bismuth atoms). This is in s contrast to the ferromagnetic order of Pnz within each zigzag edge of the quantum that is typical of the edge states in the quantum spin Hall systems. As is shown in Fig.7.11, the valley polarization of the edge states in the orange region switches from valley K (at the bottom of the orange region in Fig.7.11) to K0 (at the boundary between the orange and black regions in Fig.7.11) due to the variation of the location of bulk band gap relative to the Fermi level as a function of the y-coordinate of the atomic sites. As is seen in Fig.7.11, the edge states propagating at the bottommost sites of the quantum dot in the orange region are polarized in valley K due to the fact that EF is greater than EFs (Fig.7.5) for these sites. As yn increases the maximum of the valence band

104 Figure 7.10: The spatial map of the out-of-plane component of the current induced spin polarization hSzi when the model potential energy [Eq.7.2] is applied for EF = 0.435 eV. Electron ﬂow is from the source to the drain. The region where the Fermi energy lies in (out of) the bulk energy band gap is represented by the orange (black) coloring. Note that the spin polarizations in the orange and black regions are plotted on diﬀerent scales for clarity. The diameters of the disks plotted in the orange region representing the spin polarizations are scaled up by a factor 5 relative to those in black region.

105 approaches the Fermi level. Consequently, the edge states propagating at the atomic sites in the orange region close to the boundary between the two regions are polarized in valley 0 K due to the fact that EF lies below EFs . Note that increasing the Fermi energy when the potential energy is applied, leads to the enhancement of yg (shown in the Fig.7.9 (b)), and consequently a larger orange region where on the edge states are propagating. Interestingly, it is shown in Fig.7.11 that a position-dependent potential can reverse the current-induced valley polarization of the edge states and that the valley polarization of an electron can reverse as it travels along a zigzag edge of the quantum dot.

7.6 Valley and spin correlation

In order to investigate the origin of antiferromagnetic order of the out-of-plane spin polarization seen in the interior of the quantum dots in Figs.7.8 and 7.10, I have calculated the expectation values of the z-component of the spin on the two Bi atoms in the unit-cell of inﬁnite 2D bismuthene on SiC as

X j j hSz(k, m)i = hψk(m)|Sz|ψk(m)i, (7.3) j where m = 1, 2 denotes the two Bi atoms in the rhombic unit-cell of planar bismuthene, j j = 1, ..., 12 stands for diﬀerent Bloch states |ψk(m)i. The sum in Eq.7.3 is over the valence j band Bloch states |ψk(m)i of inﬁnite 2D bismuthene on SiC in the absence of any applied bias.

The result of the calculation of hSzi is shown in Fig.7.12 (a). The k values on the horizontal axis in Fig.7.12 (a) parametrize the wave vectors along the straight line passing through the valley vectors K0 and K in the Brillouin zone. As is seen in Fig.7.12 (a), the

Sz expectation values for the left (shown in olive green) and right (shown in purple) Bi 0 atoms in the unit-cell have opposite signs for k in valley K or K . Hence, the signs of hSzi reverse if k switches between the valleys, i.e., in valley K (shown in Fig.7.12 (a)) the value of hSzi is positive for the left atom and negative for the right atom and these signs reverse as k enters valley K0. A schematic representation of the nature of the correlation of the out-of-plane spin and valley polarizations is depicted in Fig.7.12 (b). As is seen, the left Bi atom in the rhombic unit-cell (shaded in green) has positive (green olive) and the right Bi atom has negative (purple) out-of-plane spin polarization, if the valence band electrons are polarized in valley K. However, if the valence band electrons are polarized in valley K0, the out-of-plane spin polarizations of the Bi atoms in the unit-cell are reversed. Hence, I have shown that the antiferromagnetic order seen in the interior of the dots in Figs.7.8 and 7.10 is a direct consequence of the properties of the valence band Bloch states. It should be added that the conduction band Bloch states show the opposite antiferromagnetic order. Namely, if the conduction band electrons are polarized in valley K (K0), then the left Bi atom in the

106 v Figure 7.11: The spatial map of the current induced valley polarization Pn when the model potential energy [Eq.7.2] is applied for EF = 0.435 eV. Electron ﬂow is from the source to the drain. The region where the Fermi energy lies in (out of) the bulk energy band gap is represented by the orange (black) coloring. Note that the valley polarizations in the orange and black regions are plotted on diﬀerent scales for clarity. The diameters of the disks plotted in the orange region representing the valley polarizations are scaled up by a factor 7 relative to those in black region.

107 Figure 7.12: (a) Expectation values of the out-of-plane component of the spin of electrons occupying valence band Bloch states of the inﬁnite 2D crystal of bismuthene on SiC along a line in k space. K, K0, and M are as in Fig.7.1(a). The expectation values are evaluated at the two Bi atomic sites in the crystal unit-cell. The black dashed line separates the regions of k belonging to valley K and K0. (b) Schematic representation of the unit-cells of the crystal (green rhombi) showing the out-of-plane atomic spin polarizations from part (a) (olive green and purple arrows shown in the top rhombi) for electrons in valley K and K0. The corresponding valley polarizations at the atomic sites are indicated by the red and blue disks shown in the bottom rhombi, respectively. unit-cell has negative (positive) and the right Bi atom has positive (negative) out-of-plane spin polarization. Although in general the Bloch states of inﬁnite 2D monolayer bismuthene on SiC may have nonzero in-plane (x and y) as well as out-of-plane (z) spin components, I have found that the in-plane spin components vanish at the valence band maxima.

7.7 Summary

In this chapter I investigated the electric current-induced valley and spin polarizations of the edge and bulk states in the quantum dots of bismuthene on SiC that is regarded as a high temperature topological insulator due to its large bulk energy band gap. It was shown that the conducting channels propagating at the edges of the quantum dot are strongly valley polarized. The results of the calculation of the valley polarization of the edge states as a function of the Fermi energy revealed that the valley polarization of the edge states switches between valleys K and K0 as the Fermi level varies from the top of the valence band to the bottom of the conduction band in the bulk band gap. It is predicted that this valley polarization reversal should occur in all honeycomb lattice topological insulators that have edge state dispersions qualitatively similar to that of bismuthene on SiC. The evaluated spin polarization of the conducting channels propagating at the zigzag edges showed that the spin polarizations of the edge states are predominantly in the out-of-plane direction. It was also shown that the spin polarizations within each conducting channel propagating at the zigzag

108 edge of the quantum dot is ferromagnetic. Furthermore, I predicted that if the Fermi energy lies in the valence band, the spin polarization of the scattering states propagating through the interior atomic sites of the quantum dot has antiferromagnetic order, in contrast to the ferromagnetic order at the edge atomic sites of the system. Finally, it was illustrated that in the presence of a position-dependent bias, it is possible for opposite valley polarizations to be present simultaneously at diﬀerent positions in the same edge. Hence, the valley polarization of an electron can be reversed as it travels through the quantum dot in the presence of a position-dependent potential energy.

109 Chapter 8

Conclusions and Outlook

This thesis has explored the valleytronic properties of two-dimensional topological materials. To this end, a valley projection method developed for systems with one orbital per atomic site has been adapted and extended to systems with multiple orbitals per atomic site. Furthermore, the method of calculations of the spatial distributions of on-site valley polarization has been developed for the first time in this thesis. From the technological point of view, the method of the calculation of the on-site valley polarization introduced here can be helpful due to the fact that it provides the possibility of the spatial investigation of the valley degree of freedom of electrons throughout nanostructure. Moreover, by exploiting this proposed method, the effect of the external factors that modify the electronic properties of the nanostructure, for instance a gate voltage applied to a specific region of the nanostructure or randomly distributed disorder, can be investigated. Motivated by several experiments [30, 31, 32], I have investigated the non-local resistances and valley currents of four-terminal bilayer graphene nanostructures with broken inversion symmetry by employing the Büttiker-Landauer formulation of transport theory. The calculations of non-local resistances and valley currents have been carried out in the linear response regime where the electric field that drives the electrons between the current contacts in a four-terminal nanostructure approaches zero. Hence, it has been shown that the semiclassical interpretation of the valley currents by application of the anomalous velocity term that arises from the Berry curvature is not appropriate in the insulating regime or in the linear response regime. Using the valley projection method, the valley currents of electrons, their spatial distributions, and their relative magnitude with respect to the flowing electric current in a four-terminal bilayer graphene nanostructure have been studied theoretically for the first time in this thesis. It has been revealed that the valley currents become several times larger than the electric current when the Fermi energy lies in the spectral gap close to the energy of Dirac point and mainly flow in a direction transverse to the flowing electric current. Furthermore, I have demonstrated that the spatial map of the unit-cell averaged valley polarization in a bilayer graphene nanostructure with broken

110 inversion symmetry shows a dipolar distribution that results from the presence of valley currents so that the average valley polarization increases in the direction of valley currents. The method of the calculation of the on-site valley polarization introduced in this thesis can be readily extended to different promising valleytronic candidates in future research. Moreover, the investigation of the effect of single or multiple randomly distributed defects on the non-local topological currents is another instructive path that can be taken to pursue the study of the non-local topological transport of electrons in valleytronic systems. This investigation can be beneficial due to the fact that the challenge of fabricating a defect-free nanostructure is still formidable. An advantage of the proposed method for calculating the valley polarization is that the investigation of the valley polarization of electrons induced by defects or probes in different regions of valleytronic systems is now feasible. Hence, the investigation of the effect of the local gate voltages on the valley polarization of electrons that would be enlightening from both theoretical and experimental points of view, can be carried out readily by exploiting this method. In this thesis I have proposed a valley filtering mechanism in monolayer graphene quantum dots decorated by double lines of hydrogen atoms. Since adsorbing a hydrogen atom on monolayer graphene with atomic precision has been demonstrated experimentally, this valley filtering mechanism has the potential to be experimentally realizable and reproducible. In terms of the correspondence between the principal elements of spintronics and valleytronics, the potential importance of valley filters in valleytronics is evident, in view of the decisive role of spin filters in spintronics. I have shown that adsorbed double lines of hydrogen atoms induce an electrically conducting channel close to the Dirac point energy within the energy gap resulting from the quantum confinement in pristine monolayer graphene quantum dots. My valley polarization calculations revealed that this induced conducting channel filters electrons according to their valley degree of freedom so that an electric current flowing in a given direction is carried almost entirely by the electrons belonging to one valley if the Fermi energy is tuned so as to lie in the induced conducting channel. Addition- ally, this mechanism can function as a valley switch if the direction of the flowing electric is reversed (swapping the source and drain contacts). Although there have been previous studies proposing several differing valley filtering mechanisms, the investigation of valley filtering by means of patterned adsorbates on graphene has been presented for the first time in this thesis. The study of valley filtering mechanisms in monolayer graphene and other valleytronic systems decorated by different adsorbates (other than hydrogen) is still an open question. For instance, the tight-binding model (developed in Ref.[82]) which describes the Dirac point resonances due to adsorbed O, F, and OH on monolayer graphene can be empoloyed to study their potential effects on the functionality of the valley filtering mechanism which exploits the Dirac point resonances due to these adsorbed atoms and molecules on monolayer graphene. As was mentioned in this thesis, the efficiency of the proposed valley filter can be enhanced by decreasing the

111 reflection of electrons at the ends of the conducting channel. Hence, the study of filtering mechanisms exploiting adsorbed lines of hydrogen atoms on monolayer graphene quantum dots can be extended by the investigation of possible methods for minimizing the reflection of electrons at the ends of the conducting channel induced by the adsorbed double line of hydrogen atoms. Furthermore, the proposed method of the calculation of the valley polarization has been exploited in this thesis to investigate the valley polarizations of the edge and bulk states in bismuthene on SiC quantum dots. Bismuthene on SiC has been confirmed experimentally and theoretically to be a two-dimensional topological insulator with a large band gap of the order of 0.8 eV. The present work has revealed that the valley polarization of the edge states switches between valleys K and K0 as the Fermi level varies from the top of the bismuthene valence band to the bottom of the conduction band. It was predicted that this valley polarization reversal should occur in all honeycomb lattice two-dimensional topological insulators that have edge state dispersions qualitatively similar to that of bismuthene on SiC. It has been shown that if the Fermi energy lies in the valence band, the spin polarization of the scattering states propagating through the interior of the quantum dot has antiferromagnetic order in contrast to the ferromagnetic order of the edge states (see Sec.7.6). Although, the topological character of the edge states in bismuthene on SiC has been investigated theoretically by calculation of the quantized two-terminal conductance that is a characteristic of the edge states in quantum spin Hall systems, its experimental confirmation has yet to be obtained. Moreover, the effect of different types of applied gate voltages (other than those studied in this thesis) on the spin and valley polarizations of the edge and bulk states in two- and four-terminal quantum dots of monolayer bismuthene on SiC is still an open topic for further investigations.

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