a transport study of emerging phenomena in bilayer

The Pennsylvania State University

The Graduate School

Department of Physics

A TRANSPORT STUDY OF EMERGING PHENOMENA IN BILAYER

GRAPHENE NANOSTRUCTURES

A Dissertation in

Physics

by

Jing Li

2017 Jing Li

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2017

The dissertation of Jing Li was reviewed and approved* by the following:

Jun Zhu

Associate Professor of Physics

Dissertation Advisor, Chair of Committee

Chaoxing Liu

Assistant Professor of Physics

Saptarshi Das

Assistant Professor of Engineering Sciences and Mechanics

Nitin Samarth

Professor of Physics

George A. and Margaret M. Downsbrough Department Head

*Signatures are on file in the Graduate School

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ABSTRACT

Since the advent of graphene, it has attracted lots of attention in the scientific

community because of its outstanding mechanical, optical, thermal and electrical transport

properties. Its remarkably high carrier mobility in a wide range of temperature makes it sui

generis, to date the mean free path in h-BN encapsulated graphene devices can be as large

as 10 μm. In addition, Dirac fermions in graphene possess extra pseudospin and valley

degree of freedom, this makes graphene unique compared with conventional

semiconductor 2DEGs. Charge carriers in bilayer graphene are viewed as massive Dirac

fermions, they inherit novel properties, e.g. high mobility, pseudospin and valley degree of

freedom from graphene. Moreover, bilayer graphene supports an up to 250 meV band gap

tunable by a perpendicular electric field. Besides, the electric field serves as an efficient

knob to manipulate the valley degree of freedom, and this makes bilayer graphene an ideal

platform to implement new types of valley-based electronics (valleytronics). This

motivates the experimental studies done in this dissertation.

In this dissertation, six electrical transport studies in bilayer graphene are summarized

in three chapter. (1) The electric field tunable band gap in bilayer graphene are obtained

using thermal activation measurements with high precision and in a large field range. This

provides precise energy scales in other bilayer graphene studies, e.g. calculating Landau

level (LL) energies described in Chapter 5 (in Chapter 2). (2) We carefully measured the

electrons and holes effective mass (m*) in bilayer graphene using temperature-dependent

Shubnikov-de Haas oscillations, and observed a strong suppression in holes m* at low

carrier density. This study reveals a surprising and unusual effect of disorder on m* that is

iv

unique to gapless 2D materials (in Chapter 2). (3) We experimentally demonstrated the

theory predicted valley-momentum locked conducting channels (kink states) in the line

junction of two oppositely gated bilayer graphene using a dual-split-gated device structure

(in Chapter 4). (4) We obtained ballistic kink states (4 e2/h conductance) with improved

sample quality, and further demonstrated a few valleytronics operations, e.g. valley valve,

waveguide and electron beam splitter, by constructing multiple kink states with

controllable helicity in a dual-quad-split-gated bilayer graphene device. This is the first

experimental realization of a valleytronic device with valley based operational capabilities

(in Chapter 4). (5) We built an empirical LL diagram for the E = 0 octet in bilayer graphene

in the presence of both perpendicular electric field and magnetic field, using measured band

gap values and coincident-points D field for the ν = 0 state up to 31 T. This study provides

a unified and intuitive framework which can interpret many experimental observations in

literature and offers a good base for future experimental and calculation studies (in Chapter

5). (6) We demonstrated gate-controlled transmission of quantum Hall edge states in

bilayer graphene, where perfect transmission and sequential pinch-off of edge states were

observed by controlling the tunnel junction potential with a gate. This study is the first

demonstration of controllable transmission of quantum Hall edge states in graphene

systems, and is a starting point for designing more sophisticated structures, e.g.

interferometers, to study exotic quasiparticle statistics in the novel even denominator

fractional quantum Hall states in bilayer graphene (in Chapter 5).

This dissertation also present device fabrication details, e.g. efforts on making clean h-

BN encapsulated graphene stacks, fabricating sub-100 nm nanostructures on h-BN

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substrate and precise alignment of top and bottom gates. We also discuss how COMSOL

simulations can guide the designing of gates in nanostructures (in Chapter 3).

vi

TABLE OF CONTENTS

LIST OF FIGURES ··········································································· xi

LIST OF TABLES ············································································· xxxii

ACKNOWLEGEMENTS ···································································· xxxiii

CHAPTER 1: INTRODUCTION ·························································· 1

1.1 WHAT MAKES GRAPHENE SYSTEMS STAND-OUT ······················ 1

1.2 LATTICE AND ELECTRONIC STRUCTURE OF GRAPHENE ············· 4

1.3 LATTICE AND ELECTRONIC STRUCTURE OF BILAYER GRAPHEN 8

CHAPTER 2: ENGINEER AND PROBE THE BAND STRUCTURE OF BILAYER

GRAPHENE ···················································································· 13

2.1 INTRODUCTION ····································································· 13

2.2 THE ROUTE TOWARDS CLEAN SAMPLES ·································· 15

2.2.1 Substrate matters ································································ 15

2.2.2 PMMA assisted transfer method ·············································· 20

2.2.3 Post transfer/ fabrication cleaning ············································ 23

2.2.4 Van der Waals transfer method ··············································· 26

2.3 MEASURING THE ELECTRIC FIELD INDUCED BAND GAP ············ 29

2.3.1 Tight binding model on the tunable band gap in bilayer graphene ······ 29

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2.3.2 Device fabrication and characterization ····································· 32

2.3.3 Thermal activation measurements of the band gap ························ 33

2.3.4 Determine the D field dependent band gap ································· 39

2.4 EFFECTIVE MASS MEASUREMENTS IN BILAYER GRAPHENE AT LOW

CARRIER DENSITY ······································································ 44

2.4.1 Motivation for probing m* at low carrier density ·························· 44

2.4.2 Experimental setup ····························································· 45

2.4.3 T-dependent SdH measurements and global fitting to determine m* ···· 49

2.4.4 Discussion on the suppression of m* at low carrier density ··············· 53

CHAPTER 3: THE ART OF GATING IN NANOSTRUCTURES ················· 63

3.1 INTRODUCTION ····································································· 63

3.2 EDGE EFFECTS IN GATING ······················································ 64

3.2.1 Enhanced gating efficiency in graphene nano ribbons ····················· 64

3.2.2 Two examples of non-extended top gate ····································· 66

3.3 COMSOL SIMULATION IN GATE DESIGN ··································· 72

3.3.1 Effective gating efficiency ····················································· 72

3.3.2 Potential profile in complex gating structures ······························· 75

3.4 FABRICATION CHALLEGES ····················································· 80

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CHAPTER 4: BALLISTIC KINK STATES AND VALLEYTRONIC

OPERATIONS IN BILAYER GRAPHENE ············································· 83

4.1 INTRODUCTION ····································································· 83

4.1.1 Electronic degree of freedom ················································· 83

4.1.2 Valley-contrasted physical properties ······································· 84

4.1.3 Quantum valley Hall (kink states) ··········································· 89

4.2 EXPERIMENTAL REALIZATION OF KINK STATES ······················ 92

4.2.1 Motivation for realizing gate controlled kink states ······················· 92

4.2.2 Experimental setup and COMSOL simulation ····························· 93

4.2.3 Device fabrication ······························································ 97

4.2.4 Device characterization ························································ 102

4.2.5 Evidence of kink states ························································ 106

4.2.6 Backscattering mechanism of kink states ··································· 110

4.2.7 Towards ballistic kink states in a magnetic field ··························· 115

4.2.8 Section conclusion ······························································ 121

4.3 BALLISTIC KINK STATES AND VALLEYTRONIC DEVICE IN BILAYER

GRAPHENE ················································································ 122

4.3.1 Motivation for building a valleytronic device ······························ 122


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4.3.3 Device fabrication and characterization ····································· 124

4.3.4 Ballistic kink states ····························································· 128

4.3.5 Waveguide operations ························································· 133

4.3.6 Valley valve and electron beam splitter ····································· 136

4.3.7 Comparisons between experiment and theoretic modeling ··············· 141

4.3.8 Quantized transport in high magnetic field ································· 144

4.3.9 Section conclusion ······························································ 150

CHAPTER 5: VALLEY, SPIN AND ORBITAL COMPETEING ZERO ENERGY

LANDAU LEVELS AND GATE-CONTROLLED TRANSMISSION OF

QUANTUM HALL EDGES IN BILAYER GRAPHENE ···························· 151

5.1 INTRODUCTION ····································································· 151

5.1.1 Brief introduction to quantum Hall effect ··································· 151

5.1.2 Quantum Hall effect in bilayer graphene ···································· 154

5.1.3 Tunneling of quantum Hall edge modes ···································· 157

5.2 AN EMPIRICAL DIAGRAM OF THE VALLEY, SPIN AND ORBITAL

ORDERING IN THE 8-FOLD ZERO ENERGY LL IN BILAYER GRAPHENE

································································································· 162

5.2.1 Motivation of this study ······················································· 162

5.2.2 Building an empirical LL diagram based on our measurements ········· 163

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5.2.3 Using our empirical LL diagram to explain measurements in literature 170

5.2.4 Limitation of our empirical LL diagram and section summary ·········· 173

5.3 GATE-CONTROLLED TRANSMISSION OF QUANTUM HALL EDGES IN

BILAYER GRAPHENE ·································································· 174

5.3.1 Motivation of this study ······················································· 174


5.3.3 Perfect and sequential transmission of quantum Hall edges ············· 179

BIBLIOGRAPHY ············································································· 185

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LIST OF FIGURES

Figure 1-1. (a) The hexagonal graphene lattice structure. Straight lines between atom sites indicate

σ bonds. Dashed line area includes one unit cell consists two inequivalent atoms A and B shown

in red and blue colors. The primitive lattice vectors a1 and a2 with length of the lattice constant a

are marked in figure. 1 - 3 mark the vectors to the three nearest neighbor sites. (b) Reciprocal

lattice structure of graphene. Black dots indicate reciprocal lattice sites, the primitive reciprocal

lattice vectors b1 and b2 are marked in figure. The first Brillouin zone is within the grey shade area.

Blue and red dots mark the inequivalent K and Kʹ points. ··············································· 5

Figure 1-2. Schematic illustration of the pseudospin degree of freedom in graphene. (a) and (b)

Electrons solely occupy A (B) sublattice corresponds to pseudospin up (down) state. (c) Electrons

usually occupy both A and B sublattices with equal possibilities. ······································ 8

Figure 1-3. (a) Lattice structure of bilayer graphene which consists two layers of graphene. A1 and

B2 sites are non-stacking sites while A2 and B1 are the stacking atomic sites. The intra-layer and

inter-layer hopping energy γ0 and γ1 are marked in figure. Dashed lines indicate van der Waals

interaction between two graphene layers. (b) Band diagram of bilayer graphene, black and red

color indicate the low and high energy bands respectively. ············································· 9

Figure 1-4. Schematic illustration of the pseudospin degree of freedom in bilayer graphene. (a) and

(b) Electrons solely occupy A1 (B2) sublattice corresponds to pseudospin up (down) state. (c) Low

energy electrons usually occupy both A1 and B2 sublattices with equal possibilities. ············ 11

Figure 1-5. Lattice structure of bilayer graphene with all four hopping energy γ0 and γ1, γ3 and γ4

marked in figure. ··································································································· 12

Figure 2-1. (a) A schematic of a dual-gated bilayer graphene field effect transistor with oxide

dielectric. (b) Top gate dependent resistance at different fixed bottom gate voltages at T = 4.2 K.

(c) and (d) ln(R) at CNP versus T-1 and T-1/3 respectively with different perpendicular electric field

applied. Adapted from Ref. [33]. ··············································································· 16

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Figure 2-2. Spatial maps of chemical potential (STM tip voltage) at the Dirac point of graphene on

SiO2 substrate (a) and h-BN substrate (b). Scale bar is 10 nm. Adapted from Ref. [55]. ········ 17

Figure 2-3. Schematic of graphene on a 500 nm × 500 nm area AFM topography scan of SiO2 (a)

and h-BN substrate (b). The RMS roughness is 2 Å and 0.3 Å for SiO2 and h-BN substrate

respectively. ········································································································· 20

Figure 2-4. (a) Schematics describe PMMA-assisted 2D flakes transfer process. Adapted from Ref.

[66]. (b) A photograph shows the micro meter precision 2D flakes transfer stage in our lab.

························································································································· 021

Figure 2-5. (a) I-V characterization in current annealing procedures on a h-BN supported graphene

Hall bar device (inset shows an optical image of the device). Maximum DC bias sequentially

increases in the current annealing procedures. (b) R versus gate voltage curves. Black curve is

taken on sample as fabricated. Grey curve is taken one week after the current annealing procedure.

Other curves are taken immediately after each run of current annealing. Color scheme follows that

in (a). (c) and (d) are AFM topography scans on the sample after the current annealing procedures.

(d) is a zoomed in scan in the yellow squared region shown in (c). Scale bars are 2 μm.

························································································································· 023

Figure 2-6. Sheet resistance versus carrier density in 3 different bilayer graphene samples. The

field effect mobility is 4,000 cm2/Vs in a typical good quality SiO2 substrate supported sample

(green), and ranges from 20,000 cm2/Vs (blue) to 100,000 cm2/Vs (red) in our h-BN substrate

supported devices. Insets show schematics of devices measured in each measurement.

························································································································· 025

Figure 2-7. Sheet resistance versus gate voltage for a dual gated h-BN encapsulated device

measured at 1.6 K with magnetic field of 8.9 T. Displacement field is 0.2 V/nm. Individual Landau

level filling factors are labelled. ················································································· 025

Figure 2-8. Schematics show the residual free dry transfer procedures used in our lab. This method

was first proposed by Wang et al [26]. ········································································ 26

xiii

Figure 2-9. (a) Sheet resistance versus gate voltage measurements at room temperature (blue)

and 1.8 K (red) on a h-BN encapsulated graphene device fabricated using the resist free dry

transfer method. Inset shows the ultra-low sheet resistance of graphene at high carrier density

regime. (b) Mean free path versus carrier density measured at room temperature (blue) and 1.8 K

(red). Inset shows an optical image of the measured Hall bar device. The scale bar is 2 μm.

························································································································· 028

Figure 2-10. (a) A schematic shows a perpendicular electric field is applied to bilayer graphene. (b)

Theory calculated bilayer graphene band structure with an electric field induced band gap. Adapted

and modified from Ref. [43]. ····················································································· 29

Figure 2-11. (a) Electric field induced band gap versus applied perpendicular displacement field

determined by infrared microspectroscopy. Adapted from Ref. [36]. (b) A schematic shows the

thermal excitation energy is half of the band gap size in transport measurements when Fermi level

is in the middle of the band gap at CNP. ····································································· 31

Figure 2-12. Schematics (a) and optical image (b) of an h-BN encapsulated dual-gated bilayer

graphene device used in our measurements. (c) and (d) are schematics of dual oxide dielectrics

gated bilayer graphene devices. The results from these two devices will be compared and

discussed. ··········································································································· 32

Figure 2-13. Solid curves plot resistance versus bottom gate voltages as the top gate voltages are

fixed at different values in a semi-log scale, and the x axis is converted to the displacement (D)

field at CNP for each curve. The thicker curves are measured using low frequency AC

measurements and the thinner curves are measured with DC measurements. Black dots are

measured at CNP in a large D field with DC measurements. The green dashed line is a guide to

show the CNP resistance increases exponentially with D field. Insets show schematics of band

structure of bilayer graphene with various applied D field. Measurements are done at T = 1.6 K.

························································································································· 034

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Figure 2-14. Resistance at CNP versus D field for four different samples in a semi-log scale. μFE

for h-BN encapsulated devices are 30,000 and 100,000 cm2/Vs for the data in blue and orange

respectively, and μFE for the other two oxide supported devices are a few thousand cm2/Vs. The

data labeled in black and red are adapted from Ref. [47] and [54] respectively. Measurements are

done at T = 1.5 – 1.6 K. ·························································································· 36

Figure 2-15. Continuous sweep of sheet resistance at CNP versus D field for selected temperatures

from 5 K to 300 K. ································································································· 37

Figure 2-16. Resistance at CNP versus T-1 plotted in semi-log scale for selected D field from 0.2

V/nm to 0.7 V/nm. The dashed line is a linear fit for each set of data in the temperature regime from

5 K to 20 K. ·········································································································· 38

Figure 2-17. (a) Sheet resistance (symbols) at CNP versus T plotted in a semi-log scale for D = 0.7

V/nm in an h-BN encapsulated sample from 1.6 K up to 300 K. Dashed curve shows a fit using Eq.

2-8, and the solid curves plot the three components in the equation, thermal excitation of the band

gap term (magenta), nearest-neighbor hopping term (green) and variable-range hopping term (red).

(b) A comparison adapted from Ref. [47] at D = 0.8 V/nm. ·············································· 40

Figure 2-18. Resistance at CNP (symbols) versus T-1 plotted in semi-log scale for selected D field

from 0.4 V/nm to 1.5 V/nm. T ranges from 115 K to 300 K. Curves with corresponding color are

apparent linear fit of the data. ··················································································· 41

Figure 2-19. (a) Band gap values versus D field for two samples. (b) A polynomial fit of band gap

for sample spg23L in the positive D field region. ··························································· 42

Figure 2-20. Comparisons of band gap values among transport measurements (blue), optical

measurements (magenta) from Ref. [36], DFT calculations (red) from Ref. [68], and tight binding

calculation using method described in Ref. [48] with εG=2.4 (olive). ·································· 43

Figure 2-21. Sheet resistance vs carrier density Rsheet (n) for xbn2-52 (solid red), ggate3 (solid blue)

and C (dashed blue). Samples xbn2-52 and ggate3 are supported on h-BN, sample C on SiO2. The

field effect mobility μFE is 30,000 cm2V-1s-1, 22,000 cm2V-1s-1, and 4000 cm2V-1s-1 respectively for

xv

samples xbn2-52, ggate3 and C. T = 1.6 K. Inset: An optical micrograph for sample xbn2-52.

························································································································· 046

Figure 2-22. (a) Calculated band gap D versus hole (blue) and electron (red) density. (b) Calculated

m* versus carrier density including D (symbols) and setting D = 0 (solid lines). Color scheme follows

(a). TB parameters are γ0 = 3.43 eV, γ1 = 0.40 eV, γ3 = 0 and v4 = 0.063 from Ref. [49]. Disorder

broadening G is set to 3.3 meV, which corresponds to tq = 100 fs. ··································· 47

Figure 2-23. Upper: Rxx(B) for hole density nh = 8.3 × 1011 cm-2 at T = 2.3 K together with the

envelopes (red dashed curves) and the calculated background (blue dashed curve). Lower: δRxx(B)

after the background trace is subtracted. Data from sample ggate3. ································· 49

Figure 2-24. Global fitting of m* and τq to SdH oscillations at all temperatures. (a) Three fitting curves

with m* = 0.026 me and τq = 98 fs (dashed curves), m* = 0.0285 me and τq = 107 fs (solid curves)

and m* = 0.032 me and τq = 120 fs (short dashed curves). All three sets fit the T = 2.3 K data well.

Only m*0 = 0.0285 me and τq = 107 fs (solid curves) also fit the T =15 K data. (b) Fits using m*

0 =

0.0285 me and τq = 107 fs describe data at a range of temperatures very well. Hole density nh = 3.0

× 1011 cm-2. Data from sample ggate3. ······································································· 50

Figure 2-25. δRxx/R0T versus T in a semi-log plot for hole density nh = 3.0 × 1011 cm-2 in sample

ggate3 at B = 1.55 T (circles), and at B = 2.11 T (triangles). The dashed curves are fits to Eq. 2-

11 using m* = 0.0283 me for B = 1.55 T and m* = 0.0289 me for B = 2.11 T respectively. τq = 107 fs

for both. ·············································································································· 51

Figure 2-26. (a) T-dependent magnetoresistance Rxx(B) for nh = 4.7 × 1011 cm-2 at selected

temperatures as indicated in the plot. (b) Oscillation amplitude δRxx(B) of data in (a) after

background subtraction. The solid red curve plots Eq. 2-11 with fitting parameters mh* = 0.0347 me

and q = 140 fs. T = 2.3 K. δRxx(B) starts deviating from the fit above B = 3 T. Conventional method

to extract δRxx is illustrated by the blue dashed lines and produces m* = 0.0311(2) me. (c) δRxx(B)

for nh = 3.0 × 1011 cm-2 at T = 2.3 K and T = 15 K. Dashed curves are fits to Eq. 2-11 with mh* =

0.0285 me and q = 107 fs. All from sample ggate3. (d) The quantum scattering time q as a function

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of density for electrons (red symbol) and holes (blue symbol). From sample xbn2-52. q is about 40

fs (dashed grey line) in sample C [49]. ······································································· 52

Figure 2-27. The effective carrier mass mh* and me

* as a function of the carrier density (+ for

electrons, - for holes) in samples xbn2-52 (squares), ggate3 (stars), and C (triangles). Data on C

is from Ref. (Zou). Together, the measurement covers the density range of approximately 1.4 - 41

× 1011 cm-2. The dashed curves plot m* calculated using a 4 × 4 tight-binding Hamiltonian with

hopping parameters 0 = 3.4 eV, 1 = 0.4 eV, 3 = 0, and v4 = 0.063. These values are obtained by

fitting the data in C at high densities Ref. [49]. ····························································· 54

Figure 2-28. (a) Warped Fermi surfaces in momentum space for EF = 30 meV (solid curve) and EF

= 7 meV (dashed curve) for holes (blue) and electrons (red). They correspond to nh = 2 × 1011 cm-

2, ne = 1.5 × 1011 cm-2 (EF = 7 meV) and nh = 9.4 × 1011 cm-2, ne = 6.7 × 1011 cm-2 (EF = 30 meV)

respectively. v3 = 0.1. (b) Calculated electron and hole mass m* using v3 = 0.11 (symbols) and v3 =

0 (solid curves). Other TB parameters are 0 = 3.43 eV, 1 = 0.4 eV, and v4= 0.063. ············· 55

Figure 2-29. Comparison of calculations and experiment at low carrier density (0.2 – 1.3 × 1012 cm-

2). Experimental data follow the symbols used in Fig. 2-27. The olive dashed lines plot the

calculated m* including e-e interaction in a random phase approximation. The black and gray lines

are calculations that further include the effect of potential disorder using E = 5.4 meV obtained

from q and the temperature dependence of the conductance. In both calculations, 0 = 3.08 eV, 1

= 0.36 eV, 3 = 0 and v4 = 0.063. Inset: A schematic illustration of the electron-hole coexistence at

low carrier densities due to disorder and its effect on the cyclotron motion. ························ 57

Figure 2-30. m* calculated using the Thomas-Fermi screened self-energy and the T-F wavevector

qTF = 0.1𝑚∗𝑒2/ℏ2. The legend follows Fig. 2-29. Hopping parameters are: 0 = 3.08 eV, 1 = 0.37

eV, 3 = 0, and v4= 0.063. Dielectric constant of h-BN εBN = 3. ········································· 58

Figure 2-31. m* calculated using the TB parameters of Ref. [49] and including disorder broadening

E = 5.4 meV. ······································································································· 60

xvii

Figure 3-1. (a) A schematic shows COMSOL simulation setup for a doped silicon gated graphene

device. Origin of x axis and simulation parameters are labeled in the figure. (b) Simulated position

dependent gating efficiency of doped silicon gate for graphene nano ribbon with width 100 nm, 150

nm and 250 nm. ···································································································· 65

Figure 3-2. (a) – (f) Optical images show lithography procedures to fabricate a dry transferred

heterostructure into a dual-gated Hall bar device. Scale bar is 5 μm. Dashed curve in (a) and (b)

shows the boundary of bilayer graphene. ···································································· 67

Figure 3-3. (a) Vbg and Vtg relations for n = 0 condition. Inset shows an optical image of device

measured and pin numbers are labeled. Scale bar is 2 μm. (b) A set of resistance between pins 5

and 4 R5-4 versus Vbg at Vtg fixed from 4 V to -4 V in 0.5 V step measured at T = 77.5 K in a semi-

log scale. ············································································································· 68

Figure 3-4. (a) A schematic shows COMSOL simulation setup for a dual-gated bilayer graphene

device fabricated with dry etching techniques. (b) A schematic shows the details on the edge of the

device. Origin of x axis and gate voltages are labeled in the figure. Dielectric constant and thickness

of h-BN is 3 and 20 nm respectively. (c) Simulated carrier density distribution in the device.

························································································································· 69


device fabricated with top gate stops at left device edge. Origin of x axis and simulation parameters

are labeled in the figure. (b) Simulated carrier density distribution in the device at D field of 0.025

V/nm, 0.055V /nm and 0.2 V/ nm, where n = 0 in the bulk. (c) and (d) Simulated carrier density

distribution close to the left (d) and right (d) edges of the device in semi-log scale. ·············· 71

Figure 3-6. (a) A schematic shows COMSOL simulation setup for a dual split gated bilayer

graphene device with an additional back doped silicon gate. Each metallic unit is represented by a

5 nm thick block. Origin of x axis and simulation parameters are labeled in the figure. (b) Simulated

carrier density distribution in and in adjacent to the 70 nm junction area with VSi = 10 V, 20 V and

30 V, and all other gates are grounded. (c) Simulated carrier density distribution in and in adjacent

xviii

to the 70 nm junction area with VRB = -1 V to 2 V, VSi is fixed at 10 V, and all other gates are

grounded. (d) and (e) Simulated position dependent effective gating efficiency (d) and the junction/

bulk gating efficiency ratio (e) for the right bottom gate, doped silicon gate and right top gate within

the 70 nm junction area. ························································································· 73

Figure 3-7. A schematic shows COMSOL simulation setup for a quad-split gated bilayer graphene

device. ················································································································ 75

Figure 3-8. (a) 3D and (b) 2D plot shows the simulated potential profile near the upper layer of

bilayer graphene with a (+, -, -, +) top gate voltage polarity. (c) Detailed potential profile near the

junction intersect. (d) Potential profile of line cuts with corresponding color coding shown in (c).

························································································································· 76

Figure 3-9. 2D plots show the simulated potential profile near the upper layer of bilayer graphene

with a (+, -, -, +) electric field polarity, with different sets of gate voltage setup shown in

corresponding schematic. Grey curves indicate the zero potential lines. ···························· 77

Figure 3-10. (a) 2D plot shows the simulated potential profile near the upper layer of bilayer

graphene with a (+, -, +, -) top gate voltage polarity. (b) Potential profile of line cuts with

corresponding color coding shown in (a). ···································································· 78


bilayer graphene with a (+, +, +, +) top gate voltage polarity. (c) Detailed potential profile near the


························································································································· 079

Figure 3-12. (a) SEM image of a graphite bottom gate with splits in both vertical and horizontal

directions. (b) A zoom in view of (a) in one intersect of splits. ·········································· 81

Figure 3-13. SEM images show an example of unsuccessful metal quad-split gate structure on

untreated h-BN (a) and a successful example on treated h-BN (b).··································· 81

xix

Figure 4-1. (a) Brillouin zone of graphene. (b) Band structure of graphene with degenerate but

inequivalent K and Kʹ valleys. ··················································································· 84

Figure 4-2. (a) and (b) Lattice structure of graphene with inversion symmetry (a) and inversion

symmetry is broken by adding different onsite energy to A and B sublattice (b). (c) A band gap is

opened in graphene as inversion symmetry is eliminated. ·············································· 86

Figure 4-3. Berry curvature Ω near K and Kʹ points in momentum space of graphene. The

integration of Ω in each valley gives rise to “Chern number” of ½ sgn (n). Adapted and modified

from Ref. [122]. ····································································································· 87

Figure 4-4. (a) A schematic show valley Hall effect in graphene. (b) Valley Hall effect is probed in

bilayer graphene with broken inversion symmetry using nonlocal measurement. Adapted from Ref.

[126]. ·················································································································· 88

Figure 4-5. (a) A schematic shows a possible quantum valley Hall effect where edge modes travel

on the sample edge are subjected to strong intervalley scattering from atomic defects on sample

edges. (b) A schematic shows a possible quantum valley Hall effect where edge modes travel along

an internal artificial edge. ························································································ 89

Figure 4-6. (a) A schematic shows a domain wall structure in graphene where onsite potential is

mirrored along the dashed line. Filled and open sites carry opposite onsite potential. (b) A dual-

split-gated bilayer graphene device first proposed by Martin et.al. [40] where quantum valley Hall

edge modes (a.k.a. kink states) is predicted to exist at the zero potential line inside the split region.

························································································································· 091

Figure 4-7. (a) Schematic of our dual-split-gated bilayer graphene device. The four split gates

independently control the bulk displacement fields DL and DR on the left and right sides of the

junction. The Si backgate tunes the Fermi energy EF of the line junction. The gating efficiencies of

the split gates are determined using the quantum Hall effect. We determine the gate voltages

corresponding to the D = 0 and n = 0 state on the left and right sides of the junction using the global

minima of the charge neutrality point resistance Rbulk CNP. Subsequent measurements are done at

xx

nL = nR = 0 and constant displacement fields DL and DR. The diagram shows the odd field

configuration that results in the presence of the helical kink states at the line junction. Blue and red

arrows correspond to modes carrying valley index K and K’ respectively. Each one contains four

modes accounting for the spin and layer degeneracy. (b) External electrostatic potential profile near

the top (gray) and bottom (black) graphene layers for the odd (DLDR < 0) and even (DLDR > 0) field

configurations. Potential simulations are performed using the COMSOL package and parameters

of device 1. The crossing of the potentials at V = 0 gives rise to the topological kink states. The red

curve plots the wave function distribution of one such state schematically, with a full width at half

maximum of 22 nm. ······························································································· 93

Figure 4-8 (a) Schematics of device structure constructed in COMSOL. (b) Side view of the five

gates and the bilayer graphene sheet near the splits with dimensions marked in the figure (not

drawn to scale). The external electrostatic potential U computed along the red and blue dashed

lines is shown in Fig. 4-9. ························································································ 95

Figure 4-9. Utop (red) and Ubot (blue) along the line cuts shown in Supplementary Fig. 1 (b) for device

1 ((a) and (b)) and device 2 ((c) and (d)). Vtl = Vtr = 5 V, Vbl = Vbr = -5 V for the even configurations.

Vbl = Vtr = 5 V, Vtl = Vbr = -5 V for the odd configurations. VSi = 0 V. Vgraphene = 0 V.················· 96

Figure 4-10. (a) A false-color SEM image of a device similar to device 2. The bilayer graphene is

shaded and outlined in purple, the top gates and electrodes gold, the bottom multi-layer graphene

split gates black, and the top h-BN dielectric layer gray. The bottom h-BN layer extends beyond the

entire image. (b) A close-up view of the junction area from another device similar to device 2. The

junction is connected to four bilayer graphene electrodes and the measurements use a quasi-four-

terminal geometry as shown in the image to eliminate the electrode resistance. Alignment of the

gates is generally better than 10 nm. ········································································· 98

Figure 4-11. A set of schematics showing fabrication procedures. ···································· 99

xxi

Figure 4-12. (a) SEM image of a pair of bottom split gates made of multi-layer graphene. (b) Close-

up view of the split in (a). (c) SEM image of a pair of Au top split gates with a split size of 50 nm.

························································································································· 100

Figure 4-13. (a) Optical micrograph of a pair of bottom split gates together with the dummy

graphene split (boxed area) aligned to its center. (b) Optical micrograph of the same area after the

fabrication is completed showing the metal splits patterned in step 5 (top three pairs) and step 6

(bottom two pairs) on top of the dummy graphene split (c) SEM micrograph of one metal split

patterned in step 5. Its center (yellow dashed line) is shifted to the right of the center of the dummy

graphene split (red dashed line) by 20 nm. (d) SEM micrograph of a metal split patterned in step 6

showing precise alignment to the dummy graphene split. ··············································· 101

Figure 4-14. (a) The bulk sheet resistance vs carrier density for device 1 (blue) and 2 (olive). The

gating efficiencies are determined from the quantum Hall effect. The inset shows the schematics

of both devices. Measurements are done on the left side of the devices by grounding the bottom

gate Vbl and sweeping the top gate Vtl. Vtr = Vbr = 0. The sheet resistance is calculated from R16, 78

(current from pin 1 to pin 6 and voltage probes 7 - 8) in both cases. (b) R16, 78 vs Vbl in device 1

showing fully resolved integer quantum Hall states at B = 8.9 T. Vtl / Vbl is swept simultaneously to

maintain D = - 0.2 V/nm. Vtr = Vbr = 2V. Inset: similar measurement at B = 3 T, and D = 0 V/nm

showing the 8-fold degenerate N = 0 and 1 Landau levels of bilayer graphene. (c) R16, 78 vs Vtl at

selected Vbl from 1.2 V (leftmost curve) to -0.5 V (rightmost curve) decreasing in 0.1 V step. Open

circles are two-terminal CNP resistance R78, 78 (contact resistances are negligible compared to bulk

resistance) at Vbl = 1.7 V to 1.0 V (left) and at Vbl = -0.6 to -1.0 V (right). Also in 0.1 V step. The

arrow marks the global minimum. The inset shows the CNP resistance near the global minimum.

(d) The n = 0 line on the Vtl-Vbl plot obtained from the CNP positions in (c). The global minimum of

the CNP resistance in (c) corresponds to n = 0 and D = 0. Constant D lines are indicated in the

graph. Data in (b) - (d) are from device 1. ··································································· 102

Figure 4-15. The bulk charge neutrality point resistance Rbulk CNP as a function of the applied

displacement field D for device 1 (solid symbols) and 2 (open symbols) in a semi-log plot (left axis).

xxii

Rbulk CNP rises much more rapidly with the increase of D compared to oxide-supported samples and

is larger than 10 MΩ in the range of measurements below. Also plotted on the right is the D-

dependent bulk band gap Δ obtained from temperature dependence measurements of a device

similar to device 2. ································································································· 104

Figure 4-16. (a) and (b) Contacting schemes used in device 1 (a) and 2 (b). Regions of different

doping are colored on top of SEM images of similar devices. (c) dI/dV vs Vsd at an EF positioned in

the heavily hole doped continuum (black trace) and an EF positioned in the kink regime (red trace,

DL = +0.2 V/nm, DR = -0.2 V/nm and VSi = -52 V). ·························································· 105

Figure 4-17. (a) The junction conductance σj as a function of Vsi at fixed values of DR from –0.4

V/nm to 0.4 V/nm. Upper panel: DL = –0.25 V/nm. Lower panel: DL = +0.25 V/nm. From device 1.

The diagonal bands in the plots correspond to the CNP of the line junction. (b) σj vs VSi along the

yellow dashed lines marked in the upper (blue curve) and lower (red curve) panels of (a). We

estimate the energy range of the bulk band gap Δ here corresponds to roughly 25 V on VSi . The

presence of the kink states in the (+ –) field configuration (red curve) gives rise to high conductance

inside the band gap while σj is low in the (– –) configuration (blue curve). ·························· 106

Figure 4-18. Junction resistance Rj at the CNP of the junction as a function of DL and DR in all four

field polarities showing systematically high resistances in the even configurations and low

resistances in the odd configurations. ········································································ 000

Figure 4-19. (a) A illustration of the gate alignment situation in device 1. (b) - (d) Junction

conductance σj as a function of VSi and DR at fixed DL’s as labeled in the plots. The yellow dashed

lines are fits that track the CNP of the junction. (e) DL vs b using values obtained in (b) - (d).

························································································································· 107

Figure 4-20. (a) Band structures of the junction in device 1 (w = 70 nm) calculated using COMSOL-

simulated potential profiles shown in Fig. 1b. Only the K valley is shown. Non-chiral states bound

at the junction (blue) reside inside the bulk band gap marked by the green dashed lines. Δ = 30

meV. The kink states are shown in magenta. The gray line corresponds to quantum valley-Hall

xxiii

edge states at the zigzag boundary of the numerical setup, which do not survive edge disorder in

realistic samples. (b) Junction conductance σ vs length L calculated at EF = 0 (black), 5 (red), and

14 meV (blue) as marked by the dashed lines in (a). The disorder strength is chosen to be W = 0.6

eV. One non-chiral state is assumed to contribute conductance 4 e2/h at L = 0. Over 30 samples

are averaged for each data point. Error bars are smaller than the symbol size. Fits to Eq. 4-1 yield

MFP of 266, 223, and 141 nm, respectively. The proximity to non-chiral states leads to enhanced

backscattering. (c) An illustration of inter-valley scattering between the kink states of K and K’

valleys. A kink state may be directly scattered to a different valley or scattered via coupling to non-

chiral states. Non-chiral states can also form quantum dots due to Coulomb potential fluctuations

and co-exist with the kink states over a large energy range, as shown schematically. ··········· 111

Figure 4-21. (a) σj (solid black curve) vs VSi. DL = + 0.20 V/nm and DR = - 0.20 V/nm. Blue and pink

lines are guide to the eye with periods of 5.4 V and 3.2 V respectively. (b) Differential conductance

map dI/dV of the same regime as in (a). (c) Temperature-dependent junction resistance Rj at VSi

marked in (b). Also plotted is the T-dependent resistance of the gapped bulk bilayer at the CNP.

From Device 1. ····································································································· 113

Figure 4-22. Magnetoresistance of the kink state in device 1. DL = +0.2 V/nm and DR = -0.3 V/nm,

VSi = -55.5 V. The inset shows the device schematic. R39,310 - R9 is shown in Fig. 4-24a.

························································································································· 115

Figure 4-23. The kink state in a magnetic field. (a) and (b) COMSOL simulated potential profile of

the even (a) and odd (b) field configurations in device 2. w = 110 nm. The light blue shade mark

the bulk gap Δ (30 meV in (c)-(f)) and the light green shade mark the reduced gap Δ’ in all figures.

(c) to (f) The band structure of the junction at B = 0, 2, 4 and 8 T respectively for the even and odd

field configurations as labeled. Only K valley is shown. The non-chiral states below the bulk gap

are colored blue. The kink states magenta. The N = 0 and 1 LLs of the even configurations are

colored green. The quantum Hall edge states and the zigzag edge states of the system are colored

gray. (g) and (h) Measured σj of the even (blue) and odd (black) bias configurations. The red curves

xxiv

are fits of σj in the odd configurations using σj, total = σkink + σeven. Rkink = 1 / σkink obtained from each

fit is indicated in the plot. ························································································· 117

Figure 4-24. (a) Two representative magnetoresistance traces from device 1. DL = +0.2 V/nm and

DR = -0.3 V/nm, VSi = -55.5 V and -39.3 V for the blue and light blue curve respectively). Symbols

are from Device 2. The solid triangles are raw magnetoresistance data of Rj in the (+ -) field

configuration. Rj can be smaller than h/4e2 due to parallel conduction of the non-chiral states. We

use a two-channel model to estimate the resistance of the kink states, using Rj of the (+ +) field

configuration shown in the inset to approximate the resistance of the parallel channel. The open

squares in (a) plot the estimated resistance of the kink states. At large magnetic field, the non-chiral

states become sufficiently insulating that the raw Rj measures directly the kink state resistance.

Black symbols: T = 1.6 K. Red symbols: T = 310 mK in a separate cool-down. |DL| = |DR| = 0.3

V/nm. (b) Rj versus VSi in device 2 at B = 8 T for all four field configurations as marked in the plot.

From dark to light colors: |DL| = |DR| = 0.5, 0.4 and 0.3 V/nm. Inset: Potential profile for even (red)

and odd (black) field configurations. (c) The band structure of device 1 shown in Fig. 4-20a

recalculated at B = 6 T. The olive dashed lines mark the edges of the bulk conduction and valance

bands in Fig. 4-20a. Non-chiral states residing below the band edges are now lifted to higher

energies. The kink states are shown in magenta and the quantum valley Hall edge states are in

gray (not relevant in realistic samples). (d) The calculated magneto-conductance for device 1. See

Fig. 4-20b for parameters used in the calculation. EF = 5 meV. Inset: Wave functions of the K (blue)

and K’ (red) valley kink states at B = 6 T showing a spatial separation of 14 nm due to the Lorentz

force. The wave function separation is zero at EF = 0 and increases with increasing EF. It also

increases with increasing B. ···················································································· 119

Figure 4-25. (a) illustrates the creation of the valley-momentum locked kink states at the junction

of two oppositely gapped bilayer graphene regions. D > 0 corresponds to a positive voltage on the

bottom gate. States from different valleys have opposite chiralities. Including spin and layer isospin,

there are four modes in each valley. The magenta curve is the calculated wave function distribution

of the kink states in a 70-nm wide junction9. States in K and Kʹ overlap in space. (b) Schematics of

xxv

our quad-split-gated valley router device. The four graphite bottom gates are gray. The four top

gates are gold. The blue sheet and dashed lines represent the bilayer graphene sheet. The global

Si backgate is light green. The four gold arcs are Cr/Au side contacts. ······························ 123

Figure 4-26. (a) A scanning electron microscopy (SEM) image of a bottom graphite split gate

structure. (b) An optical micrograph of device 1 before the top gates are made. The dashed lines

outline the edges of the bilayer graphene sheet. Scale bar is 5 μm in both images. ·············· 125

Figure 4-27. (a) Schematics of our quad-split-gated valley router device. The magenta cross

represents the four kink channels. Each is 70 nm wide and 300 nm long. (b) An optical image of

device 1. (c) A false colored scanning electron micrograph of the central region taken on another

device. Scale bars in (b) and (c) are 3 μm and 100 nm respectively. ································· 126

Figure 4-28. Vtg - Vbg relation for the charge neutrality point of the northwest quadrant. Red line is

a linear fit with a slope of -1.8. ·················································································· 127

Figure 4-29. (a) A band diagram of the channel in magnetic field. See Ref.9 for calculations. (b)

Resistance of the east channel as a function of the Si backgate voltage RE (VSi) at different magnetic

fields as labeled in the graph. As the inset shows, RE is obtained by measuring R24 while doping

the left quadrants heavily so that R24 = RE + Rc. Rc ~ 800 W. (c) Resistance of the kink state as a

function of the magnetic field Rkink (B) in the east (black) and south (blue) channels of device 1 and

south (red) channel of device 2. The right axis labels the corresponding transmission coefficient

τkink. Throughout our experiments, the bottom gates are set to ± 3 V with the polarity given in

diagrams unless otherwise specified. The top gates are set accordingly to place the dual-gated

regions at the CNP. ······························································································· 128

Figure 4-30. (a) A schematic of the contacting scheme. (b) R13 (VSi) at selected B-fields from 0 T to

8 T measuring the resistance of the south channel. The measurement scheme is shown in the

inset, where the north quadrants are heavily doped and the south quadrants are gapped with the

bottom gate voltage (VSWb, VSEb) set at (-2 V, +2 V). (c) R13 vs the plateau index 1/n. Points are

taken from the B = 4 T trace on the hole side. A linear fit (red line) to the data yields an intercept Rc

xxvi

= 1140 Ω. (d) Bottom: Resistance of the kink states Rkink + Rc for the south (magenta) and east

(black) channels in device 1 and the south channel (blue) in device 2. Black and blue curves are

shifted in x and y to overlap with the magenta curve. B = 6 T. Top: R13 measured with the top and

bottom gates set to 0 V. The change reflects the VSi -dependence of the bilayer graphene channel,

which includes the access region. Its profile strongly resembles that of the magenta curve in the

kink regime. This led us to conclude that the VSi – dependence of the magenta curve in this regime

is due to Rc ·········································································································· 130

Figure 4-31. (a) A schematic of the two-channel model. (b) and (c), R13 (VSi) at selected B-fields

from 0 T to 8 T. The color scheme follows that of Fig. S3b. Insets show the measurement schemes

and illustrate the parallel conduction paths through the gapped quadrants. Rpara remains the same

in (b) and (c) while the kink states are absent in (c). We use Rpara measured in (c), R13 measured

in (b), together with Rc = (1440 ± 70) Ω to calculate Rkink of the south channel at VSi = -11 V (red

arrow). (VSWb, VSEb) = (+3 V, -3 V) in (b) and (-3 V, -3 V) in (c). From device 1. ···················· 132

Figure 4-32. (a) – (c) illustrate the “through”, “right turn” and “left turn” configurations of the

waveguide respectively. (d) Two-terminal resistance R13 (VSi) corresponding to the resistance of

the north channel RN (blue), the south channel RS (red) and the “through” configuration (black),

respectively. The overlap of all three indicates ballistic transmission through the intersection region

in (a), i.e. τi = 1. B = 6 T. (e) R34 (orange) and R14 (olive) as a function of VSi in the configurations

shown in (b) and (c) respectively. B = 5 T. The kink states are also transmitted through the bends

ballistically. The variation of R in the kink regime is likely due to the change of Rc. (f) The

transmission coefficient τi of the intersection region in (a) as a function of the magnetic field. τi is

determined using the model shown in the inset. ··························································· 134

Figure 4-33. (a) A schematic of the model used to determine the transmission coefficient τi of the

intersection region. (b) The measured R13 in the “through” configuration and the calculated τi as a

function of the magnetic field. Rc = 1075 W. (VSWb, VSEb) = (-3 V, +3 V). (VNWb, VNEb) = (-3 V, +3 V).

From device 1. ······································································································ 135

xxvii

Figure 4-34. (a) illustrates the chiralities of the kink states in this gating configuration and the

blocking of straight transmission, i.e., the valley valve effect, in the absence of inter-valley

scattering. The wave functions of the K and Kʹ states remain overlapped in space in the absence

of a magnetic field or when EF is at the CNP (e = 0). (b) and (c) illustrate the shifts of the wave

function centers in a positive magnetic field for positive (b) and negative (c) energies. ·········· 136

Figure 4-35. (a) Measurements of the normalized percentage current received at terminals 2 - 4

respectively as labeled in the graph while using terminal 1 as the current source. A small I3 in the

entire kink regime demonstrates the valley valve effect. The inset illustrates the six independent

current transmission coefficients used in our S-matrix model. The diagram reflects an

experimentally observed C2 rotational symmetry. B = 8 T. (b) I3 (VSi) at fixed magnetic fields from 0

to 11 T demonstrating the robustness of the valley valve effect. ······································· 137

Figure 4-36. Properties of the beam splitter. (a) Normalized percentage current I2 at selected B -

fields from 1 to 8 T. Dashed lines mark the tunable range of I2 for the 3 T data. (b) The tunable

range of I2 as a function of the magnetic field. (c) Transmission coefficients T0, T2 and T1’ measured

using terminal 2 (solid curves) or 4 (dashed curves) as the current source. The color scheme follows

that of the inset. The excellent agreement between the solid and dashed curves indicates an

approximate C2 rotational symmetry. (VNWb, VNEb, VSWb, VSEb) = (-3 V, +3 V, +3 V, -3 V). B = 8 T.

························································································································· 139

Figure 4-37. (a) Transmission coefficients T0 (orange), T0ʹ (blue), T2 (red), T2ʹ (olive) as a function

of VSi. (b) The measured (magenta) and calculated (black) two-terminal resistance R13. A contact

resistance of Rc = 1 kΩ is added to the theoretical curve to match data. B = 8 T. Small discrepancy

between theory and experiment is attributed to microscopic imperfections of the device, which

cannot be fully captured by the simple model. Gray dashed lines indicate the corresponding

plateaus in T and R13, with the value of R13 labeled for each line. ····································· 141

Figure 4-38. (a) and (b) The S-matrix model describing the transmission of kink states between

different channels. Two sets of parameters are used to describe the scattering amplitude as

xxviii

illustrated in the diagrams. The diagrams are drawn for e > 0 (See Fig. 3b of the text). Blue (red)

arrows denote the kink states at K (Kʹ). ······································································ 142

Figure 4-39. Comparison between theory (black curves) and experiment (magenta and green

curves) on two-terminal resistances R43, R14 and the partner non-local resistances R12,43 and R23,14.

In R43 and R14, a constant Rc is added to the theoretical curves to match data. Diagrams illustrate

how the non-local resistances are measured. ······························································ 143

Figure 4-40. (a) Quantized transmission coefficients T0 and T2 at fractional values marked in the

plot at selected magnetic fields from 6 to 18 T. The centers of the curves are aligned to facilitate

comparison. (b) The measurement setup. ··································································· 145

Figure 4-41. (a) An energy spectrum of the valley kink states in strong magnetic and electric fields

in a guiding center description. (b) - (f) The evolution and quantization of T0 as a function of EF. T0

represents current transmission from terminal 4 to 3, or equivalently K valley states flowing from

terminal 3 to 4. The connections are made based on the physical distance between modes. A

connecting line represents a transmission probability of 1. (g) An idealized plot of T0 and T2 as a

function of EF. Similar analysis applies to T0ʹ and T2

ʹ. Here we have neglected the effect of intervalley

scattering and assumed that transmissions preserve the spin and isospin indices of the valley kink

states. ················································································································· 146

Figure 4-42. (a) Measured transmission coefficients T0, T2, T0ʹ and T2

ʹ (top) and G13 (bottom) as a

function of VSi at B = 16 T. G13=1/(R13-Rc). RC = 1174 Ω. “Glitches” in the middle of a plateau are

likely due to microscopic potential irregularities. (b) Idealized data in (a) demonstrating the working

of Table 4-3. The color scheme follows that of Table 4-3. Experimentally observed plateaus in G13

are shown in bold in Table 4-3. ················································································· 148

Figure 4-43. (a) The energy diagram of the valley kink states shown in Fig. S11a. The blue lines

mark EF in (b) and (c). (b) The transmission scenario for T0 = T0ʹ = 1. This situation mimics that of

a chiral quantum Hall system with G13 = 4 e2/h. (c) T0 = 3/4. T0ʹ = 1/2. The black dashed lines

illustrate the shift of the crossing points towards the NE and SW quadrants, as discussed in Section

xxix

5. This shift can account for the difference between T0 and T0’, as the diagram illustrates. (a) - (f)

Decomposition of (c) into the sum of three subsystems, the conductance of which add up to the

observed 2 e2/h. ···································································································· 149

Figure 5-1. (a) Landau level band diagram. (b) Potential profile along x axis shows the potential

steeply increases towards sample edges. (c) Schematic shows EF is in the middle of two Landau

levels. ················································································································· 153

Figure 5-2. A diagram shows quantum Hall edge modes tunneling in a two zone uni-polar junction.

Black dashed line indicates the junction center.···························································· 157

Figure 5-3. A diagram shows quantum Hall edge modes tunneling in a two zone bi-polar junction.

Black dashed line indicates the junction center.···························································· 159

Figure 5-4. A diagram shows quantum Hall edge modes tunneling in a three zone uni-polar junction

where edge modes fully mixed with each other. Black dashed lines indicate the junction centers.

························································································································· 161

Figure 5-5. RCNP (D) for five devices measured at T = 1.6 K. Device numbers are labelled with

corresponding color. ······························································································ 164

Figure 5-6. A qualitative LLs for the E = 0 octet of bilayer graphene at a fixed B field as an evolution

of D field. K and Kʹ valley states are in red-like and blue-like colors respectively. ················· 166

Figure 5-7. (a) Color map of Rxx (Vtg, Vbg) in device 6 at B = 8.9 T. Dashed lines show constant

filling factors ν = 0, 1 and D = 0. The positive D*0, ±1 is also marked in figure. White out region

shows where contacts (also bilayer graphene) reach CNP. (b) and (c), Rxx (D) at selected B-fields

from 10 - 16 T in device 24 (b), and at B = 25 T and 31 T in device 6 (c). (d) An example of ν = -2

gap measurement with method described in Ref. [72]. Arrows mark the gap size. (e) Δ (B) for ν = -

2 (symbols), and a linear fit passes through origin with 0.94 meV/T slope. ························· 167

Figure 5-8. (a) A collection of coincident points D field D* versus B from 5 devices. D*h (D* at low B

field) and D*l for ν = 0 states are shown with squares and circles respectively. Data in devices 24,

xxx

6 and 34 are plotted in red, black and orange colors. Stars show D*-1 in device 43 (blue) and data

adapted from Ref. [182] (olive). Top and bottom black dashed lines plot calculated D*h and D*

l for ν

= 0, dark yellow dashed line plots calculated D*1. (b) E10 versus D*

l, corresponding B fields are

marked at the top axis. (c) E10/B versus √D*l and a linear fit. Data in device 24 and 6 with same

color scheme in (a). ······························································································· 168

Figure 5-9. (a) Two-terminal conductance versus filling factors taken at selected D and B field as

labelled in the figure. In Device 1. (b) Calculated LLs for the E = 0 octet of bilayer graphene at D =

0.2 V/nm as an evolution of B field. (c) Calculated LLs at B = 12 T as an evolution of D field. (d)

Calculated energy gap for ν = ±1 and ±3 for a singly gated device where Dν=0 = 220 mV/nm as an

evolution of B field. ································································································ 169

Figure 5-10. (a) Calculated LLs for the E = 0 octet of bilayer graphene at B = 31 T as an evolution

of D field where D 96 mV/nm. Quantum states with corresponding color are labelled on the right.

(b) A sketch of LL diagram at small D field based on gathered experimental results from literature.

Circles show coincident points for ν = 0 and ±1 states (a) and ν = -2 states (b). Inset in (a), a sketch

shows the e-h asymmetry effect on LL diagrams. Color bars above LL diagrams show five different

D-dependent filling sequences also reported in Ref. [182]. ············································· 172

Figure 5-11. (a) Schematic shows a typical global back gate plus local top gate device. Black curve

shows a smoothly potential profile across the junction area. (b) and (c) Schematic diagrams show

quantum Hall edge modes fully mix in a bi-polar (b) and uni-polar junction (c). Adapted and modified

from Ref. [168]. ····································································································· 175

Figure 5-12. Schematic of a dual-split gate plus global back gate device (a), and an optical image

of such device (b). (c) and (d) shows the dimensions in the device. ·································· 177

Figure 5-13. Schematic shows the measurement setup. ················································ 178

Figure 5-14. (a) RT (VRT) with fixed νL = -4, νR varies from -12 to -4 (black curve) and from -4 to -1

(red curve). Inset shows the definition of RT for the two scenario with corresponding color. (b)

Absolute value of R34 – R12 as an evolution of VRT (solid curve) and calculated curve following (1/νL

xxxi

- 1/νR)h/e2 (dashed curve). Well developed νR is labelled in the figures. B = 18 T, T = 300 mK.

························································································································· 179

Figure 5-15. (a) RT as a function of VSi at B = 8.9 T with fixed νL = -8 and νR = -4. RT undergoes

three stages illustrated with schematics from left to right as VSi varies from -60 V to 0 V, which are

3-zone regime (b), perfect transmission regime (c) and towards pinch-off regime (d). ··········· 180

Figure 5-16. RT as a function of VSi in a semi-log scale measured at B = 18 T with fixed νL = -8 and

νR = -4. DL = -0.2 V/nm, DR varies from -0.36 V/nm to -0.58 V/nm marked in figure. Schematics

show the evolution of the potential profile in the junction area, which leads to sequential pinch off

of edge state. ······································································································· 182

Figure 5-17. (a) and (b) Schematic diagrams show the 2-fold LL degeneracy gets lifted as DR

increases from 0.05 V/nm to more than 0.2 V/nm. (c) RT as a function of VSi measured at B = 18 T

with fixed νL = -4 and νR = -2. DL = -0.2 V/nm, DR is -0.05 V/nm (red), -0.2 V/nm (blue) and -0.3

V/nm (olive). Insets show three different scenarios of edge states mixing in the junction. From left

to right, perfect transmission, one edge mode gets reflected, both edge modes get reflected by the

junction. (d) Schematic shows small misalignment of the top right gate. ···························· 183

xxxii

LIST OF TABLES

Table 4-1. Analogy between valley-contrasted physics and gauge field theories ······ 85

Table 4-2. Device characteristics ······························································ 103

Table 4-3. Possible values of G13 as a combination of M and N ·························· 147

xxxiii

ACKNOWLEGEMENTS

First and foremost, I would like to express my sincere gratitude to my advisor Prof. Jun

Zhu for her continuous support in the past six years. This dissertation would not be possible

without her immense knowledge, inspiring guidance, persistent kindness and patience. Jun

walked me through all stages in my graduate career from a novice to research six years ago

to an experimental physicist as I am now. She showed me how to think and respond to

problems as a physicist. Her enthusiasm and encouragement helped me over difficulties in

life and research. I am very fortunate to have her as my PhD advisor.

My sincere thanks also goes to my dissertation committee: Prof. Chaoxing Liu, Prof.

Saptarshi Das and Prof. Nitin Samarth, for their support and insightful comments.

I would also like to thank my fellow labmates for their supports. Thank you to Ke Zou

for teaching me to start transport measurements and also I learnt my first set of lithography

techniques from you. Thank you to Bei Wang for teaching me using AFM. Thank you to

Junjie Wang for the assistance of Raman measurements and working late in the lab together.

Thank you to Hua Wen, Zhenxi Yin and Eugene Tupikov for the sleepless nights we were

working together during the magnet times in Tallahassee. I am also thankful to the people

I have worked with in friend labs. I thank Simin Feng and Zefang Wang for Raman

assistance and stimulating discussion. I thank Ruixing Zhang for the theory discussion for

the valley valve project.

Substantial part of my dissertation work is done in the National High Magnetic Field

Lab in Tallahassee, Florida. It is a great privilege to be able to work there, and I thank Jan

xxxiv

Jaroszynski, Ju-Hyun Park, Hongwoo Baek and Tim Murphy for their support and

experimental assistance.

My thanks also goes to the supportive staff members at Penn State Nanofab. I thank

Chad Eichfeld and Michael Labella for their valuable suggestions on lithography.

I would like to thank my collaborators outside Penn State. Thank you to Dr. Kenji

Watanabe and Dr. Takashi Taniguchi for their high quality h-BN crystals. Thank you to

Dr. Liang Tan, Prof. Steven Louie, Ke Wang, Prof. Zhenhua Qiao, Prof. Herb Fertig, Prof.

Efrat Shimshoni, Prof. Ganpathy Murthy for the fruitful and stimulating discussion.

Finally, I would like to thank my loving wife for her love, persistent support and

sacrifice. I thank my mother, my uncle and grandparents for their love and support. My

special thank you to my lovely daughter, her smile can erase any tiredness.

1

Chapter 1

Introduction

1.1 What makes graphene systems stand-out

Carbon is one of the most abundant element on earth, and it is also the crucial element

for organic compounds and life. The atoms of carbon can bond together via sp3 and sp2

bonding to form various allotropes from macroscopic materials such as diamond and

graphite to mesoscopic and microscopic materials for instance fullerenes [1], carbon

nanotubes [2] and graphene. Graphene is a single atomic sheet, which is a two dimensional

material (2D), can be the basic building blocks for the family of sp2 bonded carbon

allotropes. Stacking graphene sheets together via van der Waals force forms 3D graphite,

while decreasing the dimensionality by rolling up graphene and sewing a small area of

graphene together can produce the 1D carbon nanotubes and 0D fullerenes respectively

[3].

Graphene was theoretically studied [4] many decades before the first experimental

demonstration of its transport properties by Geim and Novoselov in 2004 using a scotch

tape exfoliation technique [5]. Since then graphene becomes a very hot topic in condensed

matter and material science research, and it is expected to replace silicon and becomes the

solution to overcoming the limit of Moore’s Law because graphene based materials, e.g.

graphene and bilayer graphene, carry lots of stand-out properties.

Firstly, graphene is relatively inexpensive. There are three major methods to produce

graphene: 1. Mechanical exfoliation from graphite (e.g. Kish graphite) can produce single

crystal and defect-free graphene with lateral size on the order of 100 μm [6], which is ideal

2

for laboratory research use. 2. Epitaxial grown graphene can be obtained by heating silicon

carbide to high temperature (more than 1000 C) under low pressure (10-6 Torr) [7-9], this

method can produce relatively high quality on-wafer graphene that is compatible with well-

established semiconductor manufacturing technologies. 3. Graphene can also be grown via

chemical vapor deposition (CVD) on metal (e.g. copper and nickel) surface at high

temperature (beyond 1000 C) [10-12], this method produces relatively high quality [13]

graphene with large size and massive volume, which makes it suitable for industrial

production.

Secondly, graphene carries outstanding mechanical and optical properties partially due

to its atomically thin nature and the strong σ bonds between carbon atoms. Graphene is the

strongest but lightest material, which has an intrinsic tensile strength of 130 GPa [14] while

a square meter of graphene weights even less than 1 mg. The bonds between carbon atoms

are very strong along the in-plane direction, while there is little constrain in the out-of-

plane direction, this makes graphene quite flexible and can be conveniently made into

wearable electronics [15-17]. In addition, graphene is an ideal material for optical

electronics application, e.g. transparent display and touchscreen, because one layer of

graphene absorbs only 2.3% of visible light [18] and it is also highly conductive.

Thirdly, graphene shows excellent electrical transport properties. The carrier type and

density in graphene systems can be continuously controlled by gating. Moreover, graphene

possesses remarkably high carrier mobility partially due to its defect-free crystal structure.

In contrast with conventional two dimensional electron gas (2DEG) in GaAs and Si, the

optical phonons couple to the carriers in graphene very weakly, which maintains the high

carrier mobility in graphene in a large temperature range, e.g. at room temperature [19-22].

3

The carrier mobility in graphene is largely limited by the substrate properties, e.g. the

carriers can be scattered by the charge impurities trapped in the graphene/ oxide interface

[20, 23] and by the remote optical phonons from the substrate [24, 25]. To date, the highest

mobility graphene devices are achieved by encapsulating graphene with h-BN using a dry

transfer method [26]. The carrier mean free path in these devices goes beyond the sample

size at low temperature, which is on the order of 10 μm, and at room temperature the carrier

mobility have reached the theory calculated intrinsic mobility limited by the acoustic

phonons in graphene . The high carrier mobility makes graphene a good candidate for high

frequency application [27, 28].

All the above mentioned properties are quite nice, however there are a few more

characters in the graphene systems make them unique and extremely attractive. Compared

with other high mobility 2DEGs, electrons in graphene obtain extra degrees of freedom,

i.e. pseudospin, which describes the electronic density distribution on the A and B

sublattices, and valley degree of freedom. The pseudospin physics in graphene leads to the

observation of exotic phenomena such as Klein tunneling [29], and makes graphene a

unique material to achieve electron optics (Veselago lensing) [30, 31]. Bilayer graphene

consists two layers of graphene via van der Waals interaction, it inherits all the nice

properties in mono layer graphene, but also carries other unique characters. Intrinsic bilayer

graphene is gapless just like graphene, however a perpendicular electric field can break the

layer symmetry and open up a band gap [32-37]. More interestingly, the band gap grows

with the strength of electric field up to 250 meV. The electric field also serves as a knob to

control the valley degree of freedom in bilayer graphene, and can lead to valley Hall effect

and valley-momentum locked ballistic edge modes [38-42]. In a magnetic field, both the

4

zeroth and the first orbital yield zero energy Landau level (LL), therefore there is an 8-fold

degenerate zero energy LL in bilayer graphene [43], and the competing valley, spin and

orbital ordering produces rich and exotic phases to be explored. All these properties make

bilayer graphene stand-out among many other novel materials and systems, and motivated

the works done in this dissertation. We will start with introductions on the lattice and

electronic structures of graphene and bilayer graphene.

1.2 Lattice and electronic structure of graphene

Graphene is an atomically thin carbon sheet, in which each carbon atom forms three

sp2 hybridized orbitals and gets bonded with three neighbor carbon atoms in-plane via σ

bonds, and leaves the 2pz orbital perpendicular to the carbon plane. The sp2 hybridization

prefers a 120 angle between any two hybrid orbitals, therefore graphene obtains a

hexagonal lattice structure as illustrated in Fig. 1-1a. There are two inequivalent atoms A

and B, colored in red and blue respectively, in each unit cell separated by a carbon-carbon

bond distance of ac-c = 1.42 Å. A set of primitive lattice vectors can be expressed as

following:

2

3,

2 ,

2

3,

221

aaa

aaa

, (1-1)

where o

CC A46.23 aa . Each atom in the unit cell can expand via the primitive vectors

to form a Bravais lattice, we call these Bravais lattices originated from A/ B atom A/ B

sublattice. There are three nearest neighbor B sublattice atoms around each atom from the

A sublattice as shown in Fig. 1-1a, the vectors between them are marked as 1 - 3.

5

Figure 1-1. (a) The hexagonal graphene lattice structure. Straight lines between atom sites indicate

σ bonds. Dashed line area includes one unit cell consists two inequivalent atoms A and B shown

in red and blue colors. The primitive lattice vectors a1 and a2 with length of the lattice constant a

are marked in figure. 1 - 3 mark the vectors to the three nearest neighbor sites. (b) Reciprocal

lattice structure of graphene. Black dots indicate reciprocal lattice sites, the primitive reciprocal

lattice vectors b1 and b2 are marked in figure. The first Brillouin zone is within the grey shade area.

Blue and red dots mark the inequivalent K and Kʹ points.

We can construct the reciprocal lattice by choosing the reciprocal primitive vectors b1

and b2 following the convention 2 ii ba

and 0 ji ba

, and therefore get

aab

aab

3

2,

2 ,

3

2,

221

. (1-2)

Figure 1-1b plots b1 and b2 in the reciprocal lattice, which is also hexagonal, including only

the first Brillouin zone. Among the six points on the corners of the first Brillouin zone,

which is shaded in grey color, there are two inequivalent K and Kʹ points marked with blue

and red color respectively. These coordinates of the K and Kʹ points are

6

0,

3

4

aK

(1-3)

where = +1 and -1 represents for K and Kʹ point respectively. We are specifically

interested in these points in the reciprocal space, because the low energy band in graphene

appears solely around these points and we call energy bands centered at K (Kʹ) points K

(Kʹ) valleys.

In graphene, the conduction are contributed by the electrons in the 2pz orbitals,

therefore the electronic properties of graphene are dominated by the π bands. We can

calculate the band structure with a 2 × 2 Hamiltonian using Tight-binding (TB) method.

We define the self-energy of the 2pz orbitals as the zero energy for the energy band,

therefore the diagonal terms in the Hamiltonian matrix become zero, and the Hamiltonian

can be written as

0*

0

0

0

H (1-4)

where

n

ki nek

)( . We only consider the nearest neighbors, the vectors from an A atom

to the nearest three B atom as marked in Fig. 1-1a are

32,

2 and

32,

2 ,

3,0 321

aaaaa

. (1-5)

By applying the values of 1 - 3 we get 2/cos2)(32/3/

akeeek x

aikaik

n

ik yyn

.

We immediately notice 0)()( KK

. In the low energy limit, we can expand k around

7

K and Kʹ points and define the momentum respect to K and Kʹ as Kkp

, then we

have 2

2

3)( pipp

ak yx

. The Hamiltonian in Eq. 1-4 now becomes

0

0

yx

yx

F ipp

ippvH

(1-6)

where 2/3 0avF is the Fermi velocity in graphene, which is about 1.1 × 106 m/s. The

energy eigenvalues and corresponding eigenstates for the equation

EH are

2/

2//

2 ,

i

irpi

B

A

Fe

eepvE

, (1-7)

where ± correspond to the conduction and valence band, and the angle θ in the eigenstates

is defined as x

y

p

ptan . In the vicinity of the K and Kʹ points, the energy dispersion is

linear with momentum, hence we call the low energy bands in graphene as Dirac cones,

and the zero energy point as Dirac point. In addition the corresponding eigenstates have

two components, which are alike the two components of spin-1

2. However, instead of

physical spin of the electrons, the two components in dictate the relative amplitude of

the wavefunction on the A or B sublattice.

The schematics in Figs. 1-2a and 1-2b illustrate two scenarios, in one case all the

electrons only occupy A sublattice which corresponds to a eigenstate of T0,1 , and

electrons only occupy B sublattice in the other situation where we have eigenstate

T1,0 . These two eigenstates can be viewed as the base for a pseudospin which

8

Figure 1-2. Schematic illustration of the pseudospin degree of freedom in graphene. (a) and (b)

Electrons solely occupy A (B) sublattice corresponds to pseudospin up (down) state. (c) Electrons

usually occupy both A and B sublattices with equal possibilities.

describes the electron density distribution on A and B sublattices. In graphene, electrons

usually occupy both A and B sublattices with equal possibilities as illustrated in Fig. 1-2c.

Furthermore, electrons in graphene are also chiral, i.e. their momentum is connected with

pseudospin. For instance, in the K valley the pseudospin is aligned with the momentum in

the conduction band and is anti-aligned in the valence band. Also if the momentum vector

rotates by angle 2π around the Dirac point, then the pseudospin correspondently rotates

angle π, i.e. the Berry phase in graphene is π.

1.3 Lattice and electronic structure of bilayer graphene

Bernal stacked (which is also known as AB stacked) bilayer graphene is the material

been heavily studied in this dissertation. Bernal stacked bilayer graphene consists two

parallel layers of graphene coupled by van der Waals interaction. The two layers of

graphene have a relative shift respect to each other such that the A sublattice atoms from

the top layer is directly on top of the B sublattice atoms in the bottom layer as illustrated

9

Figure 1-3. (a) Lattice structure of bilayer graphene which consists two layers of graphene. A1 and

B2 sites are non-stacking sites while A2 and B1 are the stacking atomic sites. The intra-layer and

inter-layer hopping energy γ0 and γ1 are marked in figure. Dashed lines indicate van der Waals

interaction between two graphene layers. (b) Band diagram of bilayer graphene, black and red

color indicate the low and high energy bands respectively.

in Fig. 1-3a. Apart from the intra-layer nearest neighbor hopping denoted by parameter γ0,

the inter-layer hopping between the stacked A2 and B1 sites is also substantial due to the

large overlap of out-of-plane 2pz orbitals, which is characterized by hopping energy γ1.

There are four atoms per unit cell in bilayer graphene, therefore the Hamiltonian for TB

calculation becomes a 4 × 4 matrix. Again we define the self-energy of the 2pz orbitals as

the zero energy in our calculation, similar with Eq. 1-6, the Hamiltonian in K and Kʹ valley

can be expressed as

0)(00

)(00

00)(

00)(0

1

1

yxF

yxF

yxF

yxF

ippv

ippv

ippv

ippv

H

(1-8)

10

with corresponding four component eigenstates TBABA

2211 ,,, . By solving the

Schrödinger equation

EH , we obtain the eigenvalues

2

1

22

1 41

2

pvE F (1-9)

where ± correspond to the conduction and valence band, and = +1 and -1 represents for

the higher and lower energy bands denoted by red and black color shown in Fig. 2-3b,

where the separation between the two bands is γ1. In the vicinity of K and Kʹ points where

14

2

1

22

pvF , the lower energy band 1

E can be further simplified as

*

2

1

221

2m

ppvE F

(1-10)

where 2

1

* 2/ Fvm is the approximate effective mass in bilayer graphene.

In low energy approximation, Hamiltonian in Eq. 1-18 can be reduced in A1 – B2

subspace as

0

0

2

12

2

*

yx

yx

ipp

ipp

mH

. (1-11)

Then we have the energy eigenvalues and corresponding eigenstates

i

irpi

B

A

e

ee

m

pE

2

2

/

2

1

,*

2

. (1-12)

Bilayer graphene also carries the pseudospin degree of freedom related to the electron

density distribution on the A1 and B2 sublattice, which can also be translated to the layer

distribution of electron density in the low energy approximation, since A1 and B2

11

sublattice resides on different layers. As illustrated in Fig. 1-4, pseudospin up represents

for all electron density is on the bottom layer graphene while when all electrons only

occupy the top layer graphene the pseudospin state is down. In bilayer graphene, electrons

usually occupy both the top and bottom layer graphene with equal possibilities as illustrated

in Fig. 1-4c, and the pseudospin part of eigenstates are linear combination of pseudospin

up and pseudospin down. Electrons in bilayer graphene are also chiral, however the

chirality differs from that in mono layer graphene. The pseudospin in bilayer graphene is

associated with the momentum direction, however not exactly aligned, e.g. in the K valley,

pseudospin is aligned with a unit vector )2sin,2(cosˆ n in the conduction band and

is anti-aligned with n in the valence band. Moreover, if the momentum vector rotates by

angle 2π around the Dirac point, then the pseudospin also rotates angle 2π, therefore the

Berry phase is twice of that in graphene, i.e. 2π.

Figure 1-4. Schematic illustration of the pseudospin degree of freedom in bilayer graphene. (a) and

(b) Electrons solely occupy A1 (B2) sublattice corresponds to pseudospin up (down) state. (c) Low

energy electrons usually occupy both A1 and B2 sublattices with equal possibilities.

Weaker hopping can also occur between atomic sites further apart as illustrated in Fig.

1-5 where the hopping between A1 and B2 sites is characterized by γ3, while γ4 denotes the

12

Figure 1-5. Lattice structure of bilayer graphene with all four hopping energy γ0 and γ1, γ3 and γ4

marked in figure.

hopping process between B1 and B2 sites. In addition, there is an extra onsite energy Δ for

the stacked B1 and A2 atomic sites. By considering all these parameters we can write the

complete Hamiltonian for intrinsic bilayer graphene as

0*

**

*

*0

043

014

410

340

H , (1-13)

and the eigenvalues can be solved numerically. Interestingly the hopping parameter γ3 is

responsible for the trigonal wrapping effect, where the cross section of the energy band

becomes trigonal geometry and further breaks into four pockets at very low energy. Also

an electron-hole asymmetry in the energy band is introduced by γ4, which makes the holes

slightly “heavier” than the electrons in bilayer graphene.

13

Chapter 2

Engineer and probe the band structure of bilayer graphene

2.1 Introduction

AB-stacked (Bernal-stacked) bilayer graphene is the more energetically favored phase

compared with its AA-stacked counterpart [44], and it widely exists in exfoliated flakes

from natural graphite and chemical vapor deposition (CVD) synthesized samples [45]. We

explore the electronic properties of AB-stacked bilayer graphene in all of our studies. Since

the lattice of bilayer graphene inherits a hexagonal symmetry from mono layer graphene,

in the momentum space, there also exist degenerate K and Kʹ valleys in bilayer graphene.

Although there are similarities between mono layer and bilayer graphene in their electronic

structure, the band structure of bilayer graphene is far more interesting. The bilayer

graphene lattice has an inversion symmetry. In a very crude picture, pristine bilayer

graphene has a gapless parabolic band structure, and this immediately distinguish itself

with mono layer graphene, as the carriers in bilayer graphene are massive. In addition, the

inversion symmetry can be conveniently broken by introducing different potential energy

onto the two graphene layers, and this leads to a tunable band gap [43]. In experiments,

both using chemical doping and applying a perpendicular electric field can break the lattice

inversion symmetry and open up a band gap in bilayer graphene up to ~ 250 meV [33, 35,

36]. The tunable band gap in bilayer graphene was first confirmed by experiments using

optical methods [36, 46]. Using a perpendicular electric field to open up a tunable band

gap in bilayer graphene is especially advantageous and convenient. Unlike chemical

doping, which usually introduces extra scattering centers and degrade sample quality, it is

14

generally easy to make gate structures to layered 2D materials. A pair of dual gates is

sufficient to apply a perpendicular electric field to bilayer graphene, and control the carrier

density independently. There are lots of experimental efforts trying to explore and measure

the electric field tunable band gap in bilayer graphene with transport measurements using

oxide dielectric dual-gated bilayer graphene devices [33, 47]. Although many of them

showed increasing CNP resistance (RCNP) with increasing electric field, which showed the

evidence of gate tunable band gap, most of the devices are not insulating even at their

gating limit [33, 47]. With those experiments, no clear band gap values can be extracted

from temperature dependent measurements on RCNP since these experiments are dominated

by hopping conduction mechanism in most of the measured temperature range [33, 47].

Amorphous oxide dielectric has nanometer scale surface topography variations and lots of

dangling bond, this provide charge traps and cause electron-hole (e-h) puddles in bilayer

graphene samples [47]. The electron-hole puddles degrade the sample quality, and provide

hopping conduction channels in a wide temperature range. In the Section 2.1 of this

chapter, I will discuss the efforts on improving sample quality by using hexagonal boron

nitride (h-BN) as dielectric. Then I will discuss our experiments on measuring electric field

dependent band gap values in bilayer graphene with transport measurements in Section 2.2

of this chapter.

Tight binding (TB) method in general describe the band structure for graphene systems

very well in the regime where electron-electron (e-e) interaction is weak. In bilayer

graphene, TB method utilizes 4 hopping parameters, γ0, γ1, γ3 and γ4 to calculate the band

structure. The term γ4 introduces an e-h asymmetry in the band structure of bilayer

graphene. This asymmetry was first observed in experiments with optical methods [48],

15

then in our own group Zou et al also revealed the e-h asymmetry by showing an asymmetry

in measured electrons and holes effective mass [49]. Effective mass is an important

electronic parameter for materials, which can capture the change of Fermi surface area. In

two-dimensional electron gas systems (2DEGS), e-e interaction always renormalizes Fermi

liquid parameters, such as effective mass, spin susceptibility, and compressibility. The

effective mass renormalization is a good tool to probe e-e interaction. For example, in

GaAs, the effective mass increases in low carrier densities due to e-e interaction [50]. In

Zou’s experiment, the relatively low sample quality in the oxide dielectric samples limited

his measurements to the carrier density (n) around 1 × 1012 cm-2. In the Section 2.3 of this

chapter I will discuss our effective mass measurements in bilayer graphene at carrier

densities down to 1 × 1011 cm-2 in h-BN encapsulated samples.

2.2 The route towards clean samples

2.2.1 Substrate matters

SiO2/ Si substrate is an idea substrate for graphene study in the beginning as these wafer

scaled substrates can be easily obtained commercially, and the SiO2 dielectric is usually

guaranteed to provide high quality gating, which benefits from the rich experience of the

matured silicon industry. Also mechanical exfoliated graphene can be easily visualized on

110 nm or 290 nm thick SiO2 substrates, and the number of layers can be easily identified

by the color contrast [51]. Take the 290 nm thick SiO2 dielectric as an example, the

dielectric constant εr is around 4, and the gate break down voltage can be as high as 100 V.

This gate alone can provide ~ 7 × 1012 cm-2 carrier doping into graphene systems. In order

to study the electric field induced band gap in bilayer graphene, a pair of top and bottom

gates is required to control the carrier doping and electric field independently. In the early

16

studies, oxides such as SiO2, Al2O3 and HfO2 are grown directly on top of bilayer graphene

using e-beam evaporation or atomic layer deposition (ALD) techniques [25, 52, 53]. As

top gate dielectrics, the thickness of these oxides are a few 10s nm, and the break down

voltages are on the 10 V order, therefore they can provide a decent amount of carrier doping

and also create a relatively large displacement field (D field). In the case of dual HfO2

gated graphene samples, one can obtain a large D field up to ~ 5 V/nm [54]. However,

these grown amorphous oxides have huge disadvantages, as described earlier, they bring

in extra Coulomb scattering sources and introduce lots of e-h puddles into graphene

systems [47].

Figure 2-1. (a) A schematic of a dual-gated bilayer graphene field effect transistor with oxide

dielectric. (b) Top gate dependent resistance at different fixed bottom gate voltages at T = 4.2 K.

(c) and (d) ln(R) at CNP versus T-1 and T-1/3 respectively with different perpendicular electric field

applied. Adapted from Ref. [33].

17

An example of transport measurements on dual oxide dielectric gated bilayer graphene

device is shown in Fig. 2-1. A global minimum point, which is at n = 0 and D = 0, is at the

CNP (where the resistance peaks) in the Vbg = 0 V curve. RCNP increases as the magnitude

of D field increases in both positive and negative directions as shown in Fig. 2-1a. This

shows the evidence of an electric field induced band gap Δ (D), however the sample is far

from insulating even at the highest gating limit. Temperature dependent measurements on

RCNP at fixed D field plotted in Fig. 2-1 c and d suggest the conduction in the sample

actually comes from a hopping conduction mechanism which follows 𝑅 ∝ 𝑒1

𝑇1/3 relation

instead of being caused by thermal activation of a band gap which dictates 𝑅 ∝ 𝑒1

𝑇. In this

case the properties of band gap in bilayer graphene are concealed by extrinsic effects such

as hopping conduction induced by e-h puddles. The e-h puddles in oxide supported

graphene systems are confirmed with scanning tunneling microscopy (STM) techniques

[55], an example is shown in Fig. 2-2a.

Figure 2-2. Spatial maps of chemical potential (STM tip voltage) at the Dirac point of graphene on

SiO2 substrate (a) and h-BN substrate (b). Scale bar is 10 nm. Adapted from Ref. [55].

18

In order to study the intrinsic properties of graphene systems, one needs to improve the

device qualities. The crystal quality of graphene is near perfect in mechanical exfoliated

samples, and this has been shown in various studies [56]. However, it is the substrate where

the surface of graphene interacts with requires improvements. Graphene is an atomically

thin material, and the electronic properties are greatly affected by the surface condition as

any scattering sources on the substrate are extremely close to the carriers in graphene. One

method to study the intrinsic properties of graphene system is simply removing the

substrate by making the graphene suspended in vacuum [57, 58]. Suspension of graphene

devices requires complicated lithography procedures, and in order to remove the resist

residual from the device fabrication process one needs to carefully execute the current

annealing procedures [57, 58]. Current annealing utilizes the Joule heating produced by the

large current density flow in the suspended graphene sample to heat up graphene locally

and remove the resist residual. In suspended graphene samples, the field effect mobility

(μFE) is enhanced by nearly two orders of magnitude compared with SiO2 supported

samples and reaches a few hundred thousand cm2/Vs [57, 58]. In these ultra-clean graphene

devices, e-e interaction induced Dirac cone reformation and fractional quantum Hall effects

have been reported [59]. The many body ground state with an intrinsic e-e interaction

induced gap was also observed in suspended bilayer graphene devices [60]. The suspension

idea opens up many opportunities to study the exotic and emerging phenomena in graphene

systems, however it has some limitations. On one hand, it is very challenging to fabricate

more complicated structures such as dual-suspended-gate structures. Although with many

efforts, some groups managed to fabricate those more complicated structures [61], the gate

voltage can be applied to these systems are very limited as the graphene layers can be easily

19

attracted towards the gates and get destroyed. On the other hand, the current annealing

process is not quite effective in multi-terminal devices, and most of the measurements on

suspended samples are two-probe measurements. There were lots of efforts trying to

explore many other substrates such as mica, polymethylmethacrylate (PMMA), MBE

grown oxide etc. to substitute SiO2 substrate in order to improve the mobility of graphene

samples [62, 63]. Although these substrates come with large area and graphene can be

mechanical exfoliated onto them, the results are not satisfying and the device quality in

these samples are comparable with that of SiO2 supported samples. The discovery of using

h-BN as dielectric for graphene systems together of introducing a PMMA assisted

micrometer-precision transfer technique pioneered by the Colombia University brought in

a new era of the graphene research.

Similar to graphite, h-BN is also a layered van der Waals material which can be

mechanically exfoliated onto a substrate from single layer form to bulk form. With boron

and nitrogen atoms substitute A and B sites respectively in a graphene lattice, there is no

A, B sub-lattice symmetry and a large band gap ~ 5 eV exists in h-BN [64]. The dielectric

constant is similar compared with SiO2, and the break down voltage is also comparable

with SiO2 scaled with dielectric thickness [65]. Furthermore, h-BN is atomically flat and

there are no dangling bonds, also it cooperates well in device assembling as a layered 2D

material. A spatial map of chemical potential at the Dirac point of graphene supported by

h-BN measured using STM technique is plotted in Fig. 2-2b. The standard deviation of

chemical potential measured in h-BN is less than 1/10 of that been measured in SiO2

supported samples [55], and this suggests lots of improvement on the e-h puddles problem.

Figure 2-3 shows a comparison of the surface roughness between SiO2 and h-BN substrate

20

Figure 2-3. Schematic of graphene on a 500 nm × 500 nm area AFM topography scan of SiO2 (a)

and h-BN substrate (b). The RMS roughness is 2 Å and 0.3 Å for SiO2 and h-BN substrate

respectively.

using atomic force microscope (AFM) topography scan. While the root mean square (RMS)

roughness of our commercially available SiO2 (Nova Electronics Materials) is 2 Å, which

is already pretty good, the measured RMS roughness of h-BN substrate is basically down

to our AFM noise level. However, the size of mechanical exfoliated h-BN is comparable

with the size of exfoliated graphene, which is in the 10 to 100 μm length scale, in order to

utilize h-BN as dielectric material for graphene devices we follow the transfer technique

introduced in Ref. [66].

2.2.2 PMMA assisted transfer method

21

The transfer process is described in Fig. 2-4a, and a photograph of our transfer setup is

shown in Fig. 2-4b. As DI water is involved in the process, we call this transfer technique

wet transfer to be distinguished from another transfer technique described later. We

perform the wet transfer with the following steps.

Figure 2-4. (a) Schematics describe PMMA-assisted 2D flakes transfer process. Adapted from Ref.

[66]. (b) A photograph shows the micro meter precision 2D flakes transfer stage in our lab.

1. Spin coat hexamethyldisilizane (HMDS), polyvinyl alcohol (PVA) and PMMA

sequentially onto a clean silicon substrate to form a PMMA/PVA/Si stack. HDMS is used

to promote adhesion of PVA to silicon substrate. The combined thickness for PVA and

22

PMMA is ~ 300 nm. Cut a 5 mm × 5 mm size of substrate and make sure there is no edge

has PMMA hangs over. Exfoliate graphene onto the prepared substrate and identify wanted

flakes. Drop the stack onto DI water surface, and the stack will float due to water surface

tension.

2. PVA is water-soluble, and the graphene/PMMA stack should float on the water

surface once detached from the silicon substrate. Carefully fish out the graphene/PMMA

stack using a slide with a hole, and make sure the wanted graphene flake is within the hole.

Flip the slide and hang to dry naturally.

3. The stack may be wrinkled after dry. Gently heat up the stack by putting it close to

a 90 °C heater will make it flatten. Align the wanted graphene flake to the target flake, e.

g. one h-BN flake exfoliated on silica substrate, under a long working distance 60 ×

objective optical microscope using a micrometer 3-axis manipulator. Once the to-be-

transferred flake and the target flake are precisely aligned, put the two objects into an

intimately close position and raise the heater temperature to 120 °C. Finally further lower

the stack such that graphene flake touches the target flake. PMMA layer will melt onto the

target flake together with the graphene flake been transferred.

4. Raines the transferred flake with acetone and Isopropyl alcohol (IPA) sequentially

to remove the PMMA residual, and this completes the one layer transfer process.

It is critical to completely remove the PMMA residual introduced after each transfer

steps if there are more than one layer transfer in the device fabrication procedure in order

to achieve high quality samples. We anneal the transferred stack in a tube furnace with a

500 sccm flow of mixed gas (10 % hydrogen balanced in argon) at 450 °C for 4 hours to

23

remove the PMMA residual introduced in the transfer process. If graphene is not protected

by h-BN encapsulation, a tube furnace anneal step after e-beam lithography is necessary to

obtain high mobility devices as current annealing in the cryostat is not efficient to remove

the resist residual introduced in lithography process for substrate supported graphene

samples.

2.2.3 Post transfer/ fabrication cleaning

Figure 2-5. (a) I-V characterization in current annealing procedures on a h-BN supported graphene

Hall bar device (inset shows an optical image of the device). Maximum DC bias sequentially

increases in the current annealing procedures. (b) R versus gate voltage curves. Black curve is

taken on sample as fabricated. Grey curve is taken one week after the current annealing procedure.

Other curves are taken immediately after each run of current annealing. Color scheme follows that

in (a). (c) and (d) are AFM topography scans on the sample after the current annealing procedures.

(d) is a zoomed in scan in the yellow squared region shown in (c). Scale bars are 2 μm.

24

Figure 2-5 plots an example of current annealing process on an as fabricated graphene/

h-BN device. After transferring graphene onto an exfoliated h-BN using transfer method

described above, standard e-beam lithography are utilized to pattern the device into Hall

bar geometry and make Ti/ Au contacts. An optical image of the device is shown in the

inset of Fig. 2-5a. The as fabricated device is wire bonded and cooled down in a He-4

cryostat. In the beginning, the device is heavily hole doped with a relatively low mobility

of μFE ~ 3000 cm2/Vs. The CNP shifts more towards zero gate voltage as larger current

annealing bias is applied to the device, however there is little mobility increment. One week

after the current annealing process, the mobility of the device increases to 9,000 cm2/Vs,

but it is still much lower than expected. AFM topography scans on the current annealed

device show that although most PMMA residual is removed, it is not uniformly removed.

There are small dot shape PMMA clusters formed all over the device and a line structure

of PMMA is formed across the Hall bar. This accounts for the low mobility of the current

annealed device, and suggests current annealing on our h-BN supported graphene devices

are not efficient. We believe there are following few reasons causing the inefficiency. 1. It

is very challenging to pass current uniformly through the whole sample and heat up the

sample evenly as there are multiple terminals. 2. The multiple Ti/ Au contacts can dissipate

heat rather efficiently compared with two terminal devices. 3. Unlike suspended samples,

the substrate in our device dissipate majority of the Joule heating.

It turns out tube furnace annealing can remove the PMMA residual introduced by

lithography quite well, and by doing so we have achieved very high mobility h-BN

25

Figure 2-6. Sheet resistance versus carrier density in 3 different bilayer graphene samples. The

field effect mobility is 4,000 cm2/Vs in a typical good quality SiO2 substrate supported sample

(green), and ranges from 20,000 cm2/Vs (blue) to 100,000 cm2/Vs (red) in our h-BN substrate

supported devices. Insets show schematics of devices measured in each measurement.

Figure 2-7. Sheet resistance versus gate voltage for a dual gated h-BN encapsulated device

measured at 1.6 K with magnetic field of 8.9 T. Displacement field is 0.2 V/nm. Individual Landau

level filling factors are labelled.

26

supported bilayer graphene samples. After furnace annealing treatment, the device mobility

in our h-BN supported bilayer graphene samples usually ranges from 20,000 to 100,000

cm2/Vs, and this is an order of magnitude higher than μFE of SiO2 support samples which

is typically a few thousand cm2/Vs [47, 49]. The comparisons of sheet resistance versus

carrier density measurements on h-BN and SiO2 supported samples are plotted in Fig. 2-6.

The 8-fold Landau level degeneracy [67] is fully resolved at a moderate magnetic field B

= 8.9 T at T = 1.6 K in these high quality h-BN encapsulated bilayer graphene devices as

shown in Fig. 2-7.

2.2.4 Van der Waals dry transfer method

More recently Wang et al proposed a resist free, which is also contamination free,

transfer method taking advantage of van der Waals force to pick up 2D layered materials

[26]. Since DI water is no longer involved in this transfer process, in contrary with the

previously mentioned PMMA assisted transfer method, we call this method dry transfer

process. The dry transfer process contains the following steps and is illustrated in Fig. 2-8.

Figure 2-8. Schematics show the residual free dry transfer procedures used in our lab. This method

was first proposed by Wang et al [26].

27

1. At T ~ 43 °C, an exfoliated h-BN flake (top h-BN) is picked up by a transparent

polypropylene carbonate (PPC)/ polydimethylsiloxane (PDMS) stamp.

2. The h-BN/ PPC/ PDMS stamp is precisely aligned with a graphene flake under a

microscope with a micrometer manipulator. The graphene flake is picked up by the stamp

also at T ~ 43 °C.

3. The graphene/ h-BN/ PPC/ PDMS stamp is then precisely aligned with another h-

BN flake (bottom h-BN).

4. The bottom h-BN is then picked up by the stamp to form an h-BN/ graphene/ h-BN

sandwich structure at T ~ 43 °C.

5. The h-BN/ graphene/ h-BN sandwich structure is finally dropped onto a target, which

can be a local bottom gate structure, at T ~ 120 °C. Since PPC loses its stickiness at this

high temperature, the PPC layer would either detach from the PDMS layer and melt onto

the target substrate or detach from the sandwich structure depend on the thickness of the

PPC. The PPC will be transferred together with the sandwich structure as shown in Fig. 2-

8 if it is thin (a few hundred nanometer). In this case, the PPC can be removed using anisole

and IPA.

In the dry transfer method, graphene is transferred onto h-BN and meanwhile get

encapsulated with top and bottom h-BN flakes. A non-vertical etching together with Cr/

Au contacts, which is referred as 1D edge contact technique [26], is used to make ohmic

contacts to graphene. The inset of Fig. 2-9b shows a graphene device made using dry

transfer and edge contact techniques. The dry transfer method has the following advantages

compared with the wet transfer method. 1. As no resist is in direct contact with graphene/

28

Figure 2-9. (a) Sheet resistance versus gate voltage measurements at room temperature (blue)

and 1.8 K (red) on a h-BN encapsulated graphene device fabricated using the resist free dry

transfer method. Inset shows the ultra-low sheet resistance of graphene at high carrier density

regime. (b) Mean free path versus carrier density measured at room temperature (blue) and 1.8 K

(red). Inset shows an optical image of the measured Hall bar device. The scale bar is 2 μm.

h-BN interface, it introduces no contamination to start. And this ensures the high quality

of the device. 2. Since there is no drying process or resist residual to be annealed in the

furnace, it is less time consuming especially in the case one needs to do multiple layer

transfers. Figure 2-9 plots the transport characterization for one of our graphene device

made with this dry transfer method. As phonons in graphene are frozen at T = 1.6 K, the

carrier mean free path reaches 2 μm which is only limited by the width of the Hall bar. In

other words, one can make a ballistic graphene device with dimension up to ~ 10 μm with

this suite of techniques [26]. However there is also limitations for this dry transfer method

such as it is very challenging to make complicated dual-gated structures, which will be

further discussed in Chapter 3.

29

2.3 Measuring the electric field induced band gap

2.3.1 Tight binding model on the tunable band gap in bilayer graphene

Figure 2-10. (a) A schematic shows a perpendicular electric field is applied to bilayer graphene. (b)

Theory calculated bilayer graphene band structure with an electric field induced band gap. Adapted

and modified from Ref. [43].

As shown in Fig. 2-10a, when a perpendicular electric field is applied through a bilayer

graphene, the top and bottom graphene layer gains a potential difference of V. This breaks

the inversion symmetry of bilayer graphene lattice and a band gap is opened. Tight binding

calculation can capture this band gap, and a “Mexican hat” shape band structure from TB

result is plotted in Fig. 2-10b. With a different potential energy on two graphene layers,

the Hamiltonian of bilayer graphene described in Chapter 1 becomes

2*

*2

*

2*

*2

043

014

410

340

V

V

V

V

H

. (2-1)

30

The value of V as a function of n or D field can be calculated self-consistently by the

electrostatics of the system. Here we adapt the method used in Ref. [48] to estimate the

value of V. In a one gate model, the gate dopes carrier density n into bilayer graphene and

also create a potential difference between the two graphene layers. The electrostatics gives,

bt nnn (2-2)

and cnne

V bt

r

24 (2-3)

where nt and nb are the carrier densities in the top and bottom layer respectively, εr is the

effective dielectric constant for bilayer graphene and c is the distance between the two

graphene layers. When the chemical potential is much smaller than γ1, and consider there

is no initial charge imbalance in the two graphene layers at zero gate voltage, then in

approximation

01

1 ,

ln2

1 n

nX

XX

XnnV

. (2-4)

The characteristic carrier density n0 is given by2-13

22

2

10 cm 107.3

vn

, where v = 1.0

× 108 m/s is the Fermi velocity in mono layer graphene, and the dimensionless parameter

10

2 / rcne describes the interlayer screening strength. Take the disorder broadening

effect into consideration the interlayer potential is further modified as

nnnV , (2-5)

31

1

022 2sign ,sign

nnnnnnn

. (2-6)

The disorder broadening energy Γ can be determined using quantum scattering time τq

measured in experiments as Γ = ħ / 2 τq.

Figure 2-11. (a) Electric field induced band gap versus applied perpendicular displacement field

determined by infrared microspectroscopy. Adapted from Ref. [36]. (b) A schematic shows the

thermal excitation energy is half of the band gap size in transport measurements when Fermi level

is in the middle of the band gap at CNP.

The electric field induced band gap in bilayer graphene was first determined using

infrared microspectroscopy by Zhang et al [36], and the result is plotted in Fig. 2-11a. Their

result shows the band gap increases with D field, and the gap starts to saturate above 200

meV when D = 3 V/nm. However, the uncertainty of their measurement is large. We

determine the electric field induced band gap with temperature dependent measurements

of resistance at CNP for various D field. The schematics in Fig. 2-11b shows at a fixed D

field, we park the Fermi level of bilayer graphene at the CNP and measure the thermal

excitation energy E1 which is half the size of the band gap Δ.

32

2.3.2 Device fabrication and characterization

Figure 2-12. Schematics (a) and optical image (b) of an h-BN encapsulated dual-gated bilayer

graphene device used in our measurements. (c) and (d) are schematics of dual oxide dielectrics

gated bilayer graphene devices. The results from these two devices will be compared and

discussed.

The device fabrication starts with exfoliating a few-layer graphite flake onto SiO2/ Si

substrate as bottom gate electrode, then bottom h-BN and bilayer graphene are sequentially

transferred onto the graphite flake using wet transfer method mentioned in Section 2.1.

Bilayer graphene is then etched into a Hall bar geometry with e-beam lithography. Ti/ Au

electrodes and metal top gates are defined with standard e-beam lithography after the top

h-BN flake is transferred onto bilayer graphene. The thickness of h-BN dielectrics ranges

from 15 to 40 nm in most of our devices. The stacks are annealed in Ar/ H2 environment

with conditions mentioned in Section 2.1 before each transfer step. Figures 2-12a and b

show a schematics of the side view and an optical image of a typical device used in our

measurements. Schematics of the side view of a HfO2/ bilayer graphene / SiO2 device and

33

a HfO2/ trilayer graphene/ HfO2 device are shown in Fig. 2-12 c and d. These two types of

devices are measured in Ref. [47] and [54], and we will compare our results with theirs.

In a dual-gated device the carrier density 00 bbbttt VVVVn , where Vt/ Vb

is the applied top/ bottom gate voltage and Vt0/ Vb0 is the gate voltage required to

compensate the unexpected doping in the top/ bottom layer of graphene. The top/ bottom

gating efficiency is defined as bt

btbt

ed /

/0/

, where e is the electron charge, ε0 is the

vacuum permittivity, εt/ εb is the dielectric constant for the top/ bottom gate dielectric and

dt/ db is the dielectric thickness for the top/ bottom gate. The average displacement field

applied through bilayer graphene is defined as

00

2

1tt

t

tbb

b

b VVd

VVd

D

.

Quantum oscillations allow us to determine the gating efficiency in our devices, and the

thickness of h-BN can be measured with AFM. In our h-BN encapsulated devices, we

measure εt = εb = 2.9. There is only one pair of top and bottom gate voltage, which is (Vt,

Vb) = (Vt0, Vb0), makes n = 0 and D = 0 simultaneously. We call this gating configuration

global minimum condition since this set of gating configuration gives the lowest RCNP in a

device.

2.3.3 Thermal activation measurements of the band gap

Figure 2-13 plots a series of resistance versus bottom gate voltage curves which are

taken at various fixed top gate voltages in a semi-log scale for one of our h-BN

encapsulated bilayer graphene devices (device spg23L). D field for CNP, which is the

highest peak, in each curve is calculated and labelled on the x axis. A band gap is induced

in bilayer graphene by a perpendicular electric field, and RCNP increases nearly

34

exponentially with increasing D field. The increasing trend of RCNP appears to follow

straight lines on both directions of D field in a semi-log scale. The dashed olive curve

Figure 2-13. Solid curves plot resistance versus bottom gate voltages as the top gate voltages are

fixed at different values in a semi-log scale, and the x axis is converted to the displacement (D)

field at CNP for each curve. The thicker curves are measured using low frequency AC

measurements and the thinner curves are measured with DC measurements. Black dots are

measured at CNP in a large D field with DC measurements. The green dashed line is a guide to

show the CNP resistance increases exponentially with D field. Insets show schematics of band

structure of bilayer graphene with various applied D field. Measurements are done at T = 1.6 K.

shown in Fig. 2-13, which captures the increasing trend of RCNP, is basically a “V” shape.

RCNP increases over four orders of magnitude from the global minimum to D = 0.5 V/nm,

and the data was taken with three strategies. In Fig. 2-13, the thicker curves are taken with

low frequency alternating current (AC) four probe method using an SR560 preamplifier

(100 MΩ input impedance) together with an SR830 lock-in amplifier as the device

resistance is less than 1 MΩ. The source-drain current used in these measurements varies

35

from 50 nA to 0.2 nA as RCNP increases from a few kΩ to about 1 MΩ to avoid current

heating, and meanwhile the AC frequency was adjusted from 47 Hz to a few Hz to avoid

large capacitive component in the measurements. Two probe direct current (DC)

measurements are used to measure even higher resistance since the contact resistance is

negligible compared with RCNP. We use Yokogawa GS200 to supply a DC voltage bias on

our device and measure the source-drain current with a DL 1211 preamplifier. For each

measurement shown with thinner lines in Fig. 2-13, in order to get rid of the offset

introduced by the instruments, we first apply a positive DC voltage bias V+ and measure

the current I+ for the whole range of the bottom gate, and then a negative DC voltage bias

V- is applied and we measure the current I- again. The resistance of the device is then

calculated as R = (V+ - V-) / (I+ - I-). We adjust the DC voltage bias from 0.2 mV to 2 mV

as RCNP increases from ~ 1 MΩ to ~ 100 MΩ so that we can measure a reasonable current

signal and meanwhile avoid current heating effect. When D is larger than 0.5 V/nm, it

becomes very challenging to precisely measure the resistance in the whole gate voltage

sweep with the protocol mentioned above since at the CNP the measured current falls into

pA regime and the noise level becomes very high. In this case we fix both the top and

bottom gate voltage and only measure RCNP by taking an I – V curve. We fit the slope of

the I – V curve in the linear excitation regime and the reciprocal of the slope is RCNP. Some

examples are plotted as discrete dots in Fig. 2-13. In order to combine the data measured

with both AC and DC methods, we perform both four probe AC measurement and two

probe DC measurement on the device in the ~ 1 MΩ resistance regime and compare the

results. We find the results from both measurements scales well in the whole gate sweep

36

range with a scaling factor close to 1, so it is reasonable to patch up the results from both

AD can DC measurements in the whole D field range.

Figure 2-14. Resistance at CNP versus D field for four different samples in a semi-log scale. μFE

for h-BN encapsulated devices are 30,000 and 100,000 cm2/Vs for the data in blue and orange

respectively, and μFE for the other two oxide supported devices are a few thousand cm2/Vs. The

data labeled in black and red are adapted from Ref. [47] and [54] respectively. Measurements are

done at T = 1.5 – 1.6 K.

Generally speaking bilayer graphene devices with good quality become insulating at a

small D field, and RCNP increases rapidly with increasing D field. In Figure 2-14, we plot

RCNP versus D field for four devices in a semi-log scale. The data of a HfO2/ bilayer

graphene / SiO2 device and a HfO2/ trilayer graphene/ HfO2 device are adapted from Ref.

[47] and [54] for comparison. μFE for two of our h-BN encapsulated devices are 30,000

cm2/Vs (device spg23L) and 100,000 cm2/Vs (device #6) respectively, and it is only a few

thousand cm2/Vs for the other two oxide gated devices. RCNP increases exponentially with

D field for all of the devices, however the slop of the increasing trend for the two h-BN

37

encapsulated devices are much larger than the two oxide gated devices, in addition the

slope is larger for higher μFE. At a moderate field D = 0.5 V/nm, for instance, RCNP has

increased by more than 7 and 3 orders of magnitude device #6 and device spg23

respectively compared with zero field scenarios, and it has only increased by a few times

for the other low mobility devices. Indeed, our h-BN encapsulated bilayer graphene

becomes very insulating (10 – 100 GΩ) at a relatively small D field. We need to perform

temperature dependent measurements on the RCNP to study the band gap size, and our band

gap measurements are carried out on device spg23L and spg23R.

Figure 2-15. Continuous sweep of sheet resistance at CNP versus D field for selected temperatures

from 5 K to 300 K.

We synchronize the ramp rate of the top and bottom gates such that they always follow

the relation 000 bbbttt VVVV . By doing so we are effectively ramping the D

field and meanwhile maintaining the device at CNP. Figure 2-15 plots how the sheet

resistance at CNP changes with D field in a semi-log scale at selected temperatures from 5

38

K to 300 K. Each individual curve here is similar with the dashed olive curve plotted in

Fig. 2-14, and they all appear to be a “V” shape in a semi-log scale. As shown in the figure,

at all D field temperature goes up RCNP decreases. And we can obtain the information of

the band gap at a given D field by analyzing how RCNP decreases with T. At a few fixed D

fields, we take multiple vertical cuts in Fig. 2-15 and the results are plotted as RCNP versus

T-1 in a semi-log scale (similar with an Arrhenius plot) in Fig. 2-16. In the simplest model,

the band gap will introduce a thermal activation behavior of RCNP which follows

Tk

CNPBeRR

2

0

. (2-7)

Figure 2-16. Resistance at CNP versus T-1 plotted in semi-log scale for selected D field from 0.2

V/nm to 0.7 V/nm. The dashed line is a linear fit for each set of data in the temperature regime from

5 K to 20 K.

Here Δ is the size of the band gap, R0 is the high temperature resistance limit and kB is the

Boltzmann constant. If there only exist a clean band gap in our device, then our data would

show a strict linear behavior in an Arrhenius plot. However, that is not the case shown in

Fig. 2-16. There seems to be three different temperature regimes in our plot, at temperature

39

below 5 K there is no clear linear behavior, and there are kind of two different slopes in

the temperature range 5 K – 20 K and temperature above 20 K respectively. In addition,

the slope in T > 20 K regime is much larger than the 5 K – 20 K regime. It turns out the

RCNP versus T relation cannot be completely described by Eq. 2-7 in a real device with some

degree of disorder.

2.3.4 Determine the D field dependent band gap

Similar problem was studied in our lab by Zou et al [47]. In their study, a three-

component conduction model is proposed to account for the temperature dependence of

RCNP which can be expressed as,

3/1

3

3

2

2

1

1

/exp1

)/exp(1

)/exp(1

)(

1TT

RTkE

RTkE

RTRBB

CNP

, (2-8)

where E1, E2 and T3 are the two activation energies and the hopping energy respectively,

and R1, R2, and R3 are the corresponding resistance at high temperature limit. Each term in

Eq. 2-8 shows the conduction contributed by one mechanism at CNP, and each mechanism

dominates RCNP in a specific temperature regimes. At high temperature the thermal

activation to the conduction band edge dominates RCNP (T), and E1 ~ 2

1 which

corresponds to half of the band gap size. In the intermediate temperature regime, RCNP (T)

is dominated by a nearest-neighbor (NN) hopping mechanism, where E2 characterize the

average energy difference between two adjacent localized states and R2 is inversely related

to the average distance of the adjacent localized states. At the lowest temperature the

variable-range (VR) hopping mechanism starts to dominate RCNP (T). Since the latter two

hopping mechanisms contribute conduction much more effectively than the thermal

40

activation from the band gap to conduction band edge process, RCNP is essentially shorted

by the hopping conduction at low temperature. The hopping mechanisms kick in at a higher

temperature in a more inhomogeneous device since the disorder energy scales are larger,

and this explains why the devices are not insulating even at large D field in the earlier

experiments [33, 47]. Also the real band gap values cannot be precisely determined as they

are affected by the large disorder background (tens of meV) in these experiments.

Figure 2-17. (a) Sheet resistance (symbols) at CNP versus T plotted in a semi-log scale for D = 0.7

V/nm in an h-BN encapsulated sample from 1.6 K up to 300 K. Dashed curve shows a fit using Eq.

2-8, and the solid curves plot the three components in the equation, thermal excitation of the band

gap term (magenta), nearest-neighbor hopping term (green) and variable-range hopping term (red).

(b) A comparison adapted from Ref. [47] at D = 0.8 V/nm.

Figure 2-17a plots RCNP versus T in a semi-log scale at D = 0.7 V/nm in device spg23L.

We fit our data following Eq. 2-8 shown with a dashed line, and the three terms in Eq. 2-8

41

are also plotted individually. The fitting parameters are E1 = 46 meV, E2 = 2.7 meV and T3

= 0.5K. It is very clear that the thermal activation behavior of the band gap alone can well

explain our data in the temperature regime T > 80 K. As the temperature decreases, NN

and VR hopping sequentially dominates RCNP (T) in the 5 K < T < 80 K and T < 5 K

temperature regimes, and these two hopping mechanisms slow down the increase of RCNP

at low temperature. Similar data adapted from Ref. [47] at D = 0.8 V/nm is also plotted in

Fig. 2-17b for comparison. Their data can be well fitted using Eq. 2-8, however there is no

temperature regime that can be well described by the thermal activation of the band gap

alone within the measured temperature limit, due to the large potential fluctuation in their

devices. As a contrast, the effects from NN and VR hopping become negligible in our

devices when T > 100 K or so. Therefore we can determine the band gap size Δ (D) very

Figure 2-18. Resistance at CNP (symbols) versus T-1 plotted in semi-log scale for selected D field

from 0.4 V/nm to 1.5 V/nm. T ranges from 115 K to 300 K. Curves with corresponding color are

apparent linear fit of the data.

precisely using the simple model describe in Eq. 2-7. We plot RCNP versus T-1 in a semi-

log scale at selected D field ranges from 0.4 V/nm to 1.5 V/nm in the temperature regime

42

115 K < T < 300 K in Fig. 2-18. In the semi-log scale RCNP grows linearly with T-1 over

more than two decades of data range, and the apparent linear fit fits the data quite well for

all D field. The slope of each fit can be converted to the band gap size at the given D field.

The results of Δ (D) measured in two different devices in the -1.6 < D < 1.6 V/nm range

are plotted in Fig. 2-19a. The results from both devices agree well, and they both show Δ

grows rather linearly with D in the measured D field range. A polynomial fit of Δ (D) is

shown in Fig. 2-19b. Empirically we can use

3825 108.1104113.0 DDDD (2-9)

to estimate the electric field induced band gap in bilayer graphene, where D is in the unit

of mV/nm and Δ is in the unit of meV.

Figure 2-19. (a) Band gap values versus D field for two samples. (b) A polynomial fit of band gap

for sample spg23L in the positive D field region.

In Fig. 2-20 we plot our Δ (D) results together with the results obtained from optical

measurements [36] and theory calculations using density functional theory (DFT) [68] and

TB methods for comparisons. Our results in general agree rather well with the optical

43

Figure 2-20. Comparisons of band gap values among transport measurements (blue), optical

measurements (magenta) from Ref. [36], DFT calculations (red) from Ref. [68], and tight binding

calculation using method described in Ref. [48] with εG=2.4 (olive).

measurements within error bars. Nevertheless our data has much smaller error bars, for

instance, the uncertainty of Δ (D) in our measurements is as small as 0.3 – 3.6 meV, and

this is much smaller compared with the optical results, which has an uncertainty of 10 – 20

meV. The band gap measured in both studies are consistently larger than the DFT

calculation for all D field, and this is not surprising as DFT is known to under estimated

band gaps. However, Δ obtained in our study increases more quickly compared with the

trend described by TB calculation. Our TB calculation is done following the method

described in Ref. [48], and we set the dielectric constant of bilayer graphene εG = 2.4 and

Γ = 3 meV. Since the dielectric constant for graphene is complicated, and some studies

suggest εG also has a D field dependence [69], it is not appropriate to fix εG = 2.4 in the TB

calculation. In fact our results can further provide experimental inputs on the value of εG

(D).

44

2.4 Effective mass measurements in bilayer graphene at low carrier density

2.4.1 Motivation for probing m* at low carrier density

Bilayer graphene is a unique 2DEG system with unusual electronic properties [70]. At

high carrier densities, its hyperbolic bands are well described by a four-band Hamiltonian

[71, 72] given by the TB description [73], where the hopping parameters are determined

by experiments or first-principles calculations [48, 49, 74-77]. Close to the CNP, bilayer

graphene exhibits fascinating e-e interaction driven ground states [35, 60, 78-80]. A natural

question arises: How does the density of states of bilayer graphene near the Fermi energy

evolve as carrier density n decreases continuously? The study of the effective carrier mass

m* is a powerful tool to probe this evolution. Indeed, in conventional 2DEGs, increasing e-

e interaction leads to substantial increase of m* at low carrier densities, long before

predicated many-body instabilities [50, 81-85]. Such studies provide valuable inputs to

advance many-body calculations [86]. In monolayer and bilayer graphene, the proximity

of the conduction and valence bands and their pseudospin characters, play a significant role

in the screening of the Coulomb interaction. This has consequences for the dispersions of

the elementary excitations and the transport properties of these systems [76, 87-89]. In

monolayer graphene, both calculations [90], and measurements of m* [91] report strong

enhancement of the Fermi velocity vF at low carrier densities. In comparison, the situation

in bilayer graphene is much less clear. Existing theoretical predictions vary greatly on the

sign and magnitude of the interaction correction to m* [92-97] while measurements have

been lacking.

In the earlier work of our lab [49], Zou et al reported on the measurements of m* of

bilayer graphene in the density regime of order 1 × 1012 cm-2. A TB description was found

45

to work well, the hopping parameters of which were accurately extracted from data. As the

previous samples rested on oxides, disorder (field effect mobility μFE ~ a few thousand

cm2V-1s-1 and disorder energy δE of a few tens of meV [47, 98]) prevented measurements

at lower densities. In our current h-BN supported samples, μFE reaches 30,000 cm2V-1s-1,

which allows for precise determination of m* down to n = 2 × 1011 cm-2 for both electrons

and holes. Following the conventional definition of the interaction parameter 𝑟𝑠 = 𝑈 𝐸𝐹⁄ ,

where U is the Coulomb interaction energy 𝑒2√𝑛𝜋 (4𝜋휀0휀)⁄ , and ε = 2.9 is the dielectric

constant of h-BN in our case. EF is the Fermi energy, we estimate rs to be 7.5 √𝑛⁄ , where

n is in the unit of 1011 cm-2, using m* = 0.033 me, which is the average value of the measured

electron and hole masses near 1 × 1012 cm-2 in Ref. [49]. In our present studied carrier

density regime (2 – 12 × 1011 cm-2), rs ranges from 2.2 to 5.3, which is quite large compared

to GaAs 2DEG, where the renormalized m* exceeds the band mass by 40% at rs ~ 5 due to

e-e interaction [50]. Here, we find that m*e and m*

h behave very differently as n decreases.

While m*e continues to follow the high-density TB extrapolation, m*

h sharply dives in value

below n = 6 × 1011 cm-2, reaching about 70% of the TB band mass at n = 2 × 1011 cm-2. A

thorough theoretical investigation evaluating the effect of e-e interaction in different

approximations, together with the effect of Coulomb potential disorder, identifies density

inhomogeneity to be a key factor in explaining the experimental observations. This unusual

effect of disorder is unique to 2D semi-metallic systems.

2.4.2 Experimental setup

Bilayer Hall bar devices are made by exfoliating, transferring, stacking and patterning

of multi-layer-graphene bottom gate electrode, 15 – 30nm thick h-BN gate dielectric

(Momentive, Polartherm grade PT110 and NIMS) and bilayer graphene sheet (Kish

46

Graphite) using the wet transfer method described in Section 2.1 [66] and standard e-beam

lithography. Transport experiments are carried out in a variable-temperature, pumped He4

cryostat with a 9 T magnet using standard low-frequency lock-in technique (47 Hz) with

current excitation 50 nA. Figure 2-21 plots the sheet resistance versus carrier density Rsheet

(n) of samples xbn2-52 and ggate3, together with sample C reported in Zou et al [49] for

comparison. The field effect mobility μFE is 30,000 cm2V-1s-1 and 22,000 cm2V-1s-1

respectively in samples xbn2-52 and ggate3, in comparison to μFE = 4,000 cm2V-1s-1 in

sample C, which is supported on SiO2 substrate.

Figure 2-21. Sheet resistance vs carrier density Rsheet (n) for xbn2-52 (solid red), ggate3 (solid blue)

and C (dashed blue). Samples xbn2-52 and ggate3 are supported on h-BN, sample C on SiO2. The

field effect mobility μFE is 30,000 cm2V-1s-1, 22,000 cm2V-1s-1, and 4000 cm2V-1s-1 respectively for

samples xbn2-52, ggate3 and C. T = 1.6 K. Inset: An optical micrograph for sample xbn2-52.

The unintentional doping for both devices are small, and the effect of the displacement

(D) field on the bare band mass is modeled here and shown to be negligible in the density

range studied. An electric field D induced band gap modifies the band structure of a bilayer

graphene and can potentially lead to the enhancement of m* at low carrier densities [70].

47

Since our measurements are not performed at D = 0, the effect of the D-field needs to be

assessed. In sample ggate3, the residual chemical doping from above and below the bilayer

graphene is approximately known, using knowledge from a dual-gated sample in

immediate proximity. Following Ref. [99], we can compute the band gap parameter as a

function of the carrier density induced by the backgate accurately, then use this information

to compute the tight-binding bands using a 4 × 4 Hamiltonian and a full set of hopping

parameters determined by Zou et al [100]. We then calculate the band mass m* using Eq.

2-10.

Figure 2-22. (a) Calculated band gap D versus hole (blue) and electron (red) density. (b) Calculated

m* versus carrier density including D (symbols) and setting D = 0 (solid lines). Color scheme follows

48

(a). TB parameters are γ0 = 3.43 eV, γ1 = 0.40 eV, γ3 = 0 and v4 = 0.063 from Ref. [49]. Disorder

broadening G is set to 3.3 meV, which corresponds to tq = 100 fs.

Figure 2-22a plots the calculated as a function of the backgate-induced carrier density

in the range studied here. Figure 2-22b plots the calculated m* for this sample, together

with calculations corresponding to a strictly gapless bilayer. It is clear from the comparison

that the effect of the finite D field on m* is weak for both electrons and holes. At the lowest

studied carrier density n = 2 × 1011 cm-2, calculations show a slight increase of m* of less

than 2% and 6% for holes and electrons respectively, whereas our experiment shows a

suppression of 30% in m*h. The trend of data clearly cannot be explained by the presence

of a band gap. Furthermore, the relative magnitude of the suppression the measurements

reflect will also not change significantly even when the effect of is included. In Sample

xbn2-52, we do not have enough information to accurately determine the D field in the

sample. Our estimates, using the resistance peak at the CNP of the sample and collective

knowledge on bilayer graphene devices of similar construction, suggest that although some

enhancement of m* due to the presence of should also be present at the lowest carrier

densities, its impact would be similar to what’s shown here. We have therefore neglected

the effect of the D-field in subsequent theoretical calculations.

The effective mass m* as measured in quantum oscillations is given by

FEEdE

EdAm

2

2*

(2-10)

where A(E) is the k-space area enclosed by the contour of constant energy E in the quasi-

particle band structure. To accurately determine m*, we measure the temperature-

49

dependent magneto-resistance Rxx(B) at a fixed carrier density, extract the low-field

Shubnikov de Haas (SdH) oscillation amplitude Rxx (T, B) and perform simultaneous

fitting of the temperature and magnetic field dependence to the Lifshitz-Kosevich

formula[101],

cB

cBth

qc

thxx

Tk

Tk

R

R

/2sinh

/2 , exp4

2

2

0

(2-11)

where ωc = 𝑒𝐵

𝑚∗ is the cyclotron frequency. The effective mass m* and the quantum scattering

time τq are the two fitting parameters.

2.4.3 T-dependent SdH measurements and global fitting to determine m*

Figure 2-23. Upper: Rxx(B) for hole density nh = 8.3 × 1011 cm-2 at T = 2.3 K together with the

envelopes (red dashed curves) and the calculated background (blue dashed curve). Lower: δRxx(B)

after the background trace is subtracted. Data from sample ggate3.

-0.6

0.0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

0.0 0.8 1.6 2.4 3.2 4.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Rxx (

k

)

R

xx (

k

)

B (T)

50

We perform global fittings on our SdH results to obtain m* and τq simultaneously. The

global fitting procedure starts with a background subtraction from SdH oscillations. Figure

2-23 illustrates the background subtraction process. We obtain the upper and lower

envelopes of the SdH oscillations (red dashed curves) by using a spline fit to the oscillation

maxima and minima (black squares) respectively. The average of the two (blue dashed

curve) is subtracted from the Rxx (B) data to obtain δRxx shown in the lower plot. Figure 2-

24 illustrates how we simultaneously determine m* and τq using the SdH oscillations at

Figure 2-24. Global fitting of m* and τq to SdH oscillations at all temperatures. (a) Three fitting curves

with m* = 0.026 me and τq = 98 fs (dashed curves), m* = 0.0285 me and τq = 107 fs (solid curves)

and m* = 0.032 me and τq = 120 fs (short dashed curves). All three sets fit the T = 2.3 K data well.

Only m*0 = 0.0285 me and τq = 107 fs (solid curves) also fit the T = 15 K data. (b) Fits using m*

0 =

0.0285 me and τq = 107 fs describe data at a range of temperatures very well. Hole density nh = 3.0

× 1011 cm-2. Data from sample ggate3.

multiple temperatures. Figure 2-24a shows a few exemplary fits at T = 2.3 K and 15 K for

nh = 3.0 × 1011 cm-2 in sample B. At T = 2.3 K, three combinations: (m*, τq) = (0.0260 me,

98 fs) or (0.0285 me, 107 fs), or (0.0320 me, 120 fs) can all fit data equally well, and the

51

three fits overlap. However, only the pair m* = 0.0285 me and τq = 107 fs can also fit data

at T = 15 K (Fig. 2-24a) and at all other temperatures (Fig. 2-24b). As trial values of m*

deviate from the optimal value m*0, systematic deviation of the fit from data guides us

towards m*0 quickly. The cases for m* > m*

0 and m* < m*0 are illustrated in Fig. 2-24a.

Figure 2-25 illustrates how we estimate the uncertainty of m*. Curves generated using the

optimal values of m* and τq (determined by the global fitting) are plotted together with data,

as shown in Fig. 2-24b. We read off the temperature-dependent oscillation amplitude

δRxx(T) at a fixed magnetic field combing measurement at one end and the global fit on the

other end (see e.g. B = 1.55 T marked by a triangle in Fig. 2-24a. We then plot δRxx/R0T,

where R0 is the zero-field Rxx, as a function of temperature and fit to Eq. 2-11 to obtain m*.

Two examples are shown in Fig. 2-25. We typically do 4 to 6 fittings for each carrier

density, and obtain the average and standard deviation of m*. The standard deviation of m*

varies from ± 0.0002 me to ± 0.004 me from high to low densities.

Figure 2-25. δRxx/R0T versus T in a semi-log plot for hole density nh = 3.0 × 1011 cm-2 in sample

ggate3 at B = 1.55 T (circles), and at B = 2.11 T (triangles). The dashed curves are fits to Eq. 2-

0 5 10 15 2010

-3

10-2

10-1

R

xx/R

0T

(K

-1)

T (K)

B=1.55T, m*=0.0283me

B=2.11T, m*=0.0289me

52

11 using m* = 0.0283 me for B = 1.55 T and m* = 0.0289 me for B = 2.11 T respectively. τq = 107 fs

for both.

Figure 2-26. (a) T-dependent magnetoresistance Rxx(B) for nh = 4.7 × 1011 cm-2 at selected

temperatures as indicated in the plot. (b) Oscillation amplitude δRxx(B) of data in (a) after

background subtraction. The solid red curve plots Eq. 2-11 with fitting parameters mh* = 0.0347 me

and q = 140 fs. T = 2.3 K. δRxx(B) starts deviating from the fit above B = 3 T. Conventional method

to extract δRxx is illustrated by the blue dashed lines and produces m* = 0.0311(2) me. (c) δRxx(B)

for nh = 3.0 × 1011 cm-2 at T = 2.3 K and T = 15 K. Dashed curves are fits to Eq. 2-11 with mh* =

0.0285 me and q = 107 fs. All from sample ggate3. (d) The quantum scattering time q as a function

of density for electrons (red symbol) and holes (blue symbol). From sample xbn2-52. q is about 40

fs (dashed grey line) in sample C [49].

53

Figures 2-26b and 2-26c show examples of our global fitting procedures for two carrier

densities nh = 4.7 and 3.0 × 1011 cm-2. Compared to common practice of approximating

Rxx at a fixed B-field by linearly interpolating adjacent peak heights and analyze its T-

dependence to obtain m*, fits to Eq. 2-11 better represent the oscillation amplitude Rxx,

especially at low carrier densities when only a few oscillations are available (See Fig. 2-

26c for example). It also enables us to discern and avoid using the T-dependent oscillations

of nascent quantum Hall states, the analysis of which can lead to error in m*. An example

is shown in Fig. 2-26b, where the conventional method produces m* = 0.0311(2) me, and

this is 10% smaller than mh* = 0.0347 me obtained from the global fitting. The effective

mass m* obtained using the global fitting procedure is B-independent and best extrapolates

to the density-of-states mass of the bilayer graphene at B = 0, which is expected to be

modified by e-e interactions [92-97].

2.4.4 Discussion on the suppression of m* at low carrier density

The above analysis enables us to accurately determine both the electron and hole

effective mass m*h and m*

e for the approximate carrier density range of 1 - 10 × 1011 cm-2.

The uncertainty of m* varies from ± 0.0002 me to ± 0.004 me from high to low densities.

The high accuracy of the measurements facilitates comparison to theory as interaction

corrections to m* are expected to be typically in the few to tens of percent range [50, 81].

Also plotted in Fig. 2-26d is the quantum scattering time τq for both electrons and holes. τq

is between 100 and 140 ps in xbn2-52, compared to ~ 40 ps in sample C [49]. The increase

of τq attests to the improvement of sample quality. Below n =1 × 1011 cm-2, the SdH

oscillations become increasingly more non-sinusoidal due to density inhomogeneity and

global fits cannot be obtained reliably.

54

Figure 2-27 plots m*h and m*

e obtained in samples xbn2-52 and ggate3, together with data

from sample C in Ref. [49]. In the overlapping density regime, current and previous results

agree very well and are well described by the TB model with hopping parameters 0 = 3.43

eV, 1 = 0.40 eV 3 = 0 and v4 = 4/0 = 0.063, Δ = 0.018 eV, which are determined in Ref.

[49]. The calculated m* are plotted as dashed lines in Fig. 2-27. The electron and hole

branches use the same set of parameters, with their mass differences captured by v4. On the

electron side, the TB parameters continue to describe all the m*e data very well down to the

lowest density measured. On the hole side, however, m*h exhibits a sharp drop from the TB

model as nh is decreased to less than 5 × 1011 cm-2, reaching a large suppression of 30% at

nh = 2 × 1011 cm-2.

Figure 2-27. The effective carrier mass mh* and me

* as a function of the carrier density (+ for

electrons, - for holes) in samples xbn2-52 (squares), ggate3 (stars), and C (triangles). Data on C

is from Ref. (Zou). Together, the measurement covers the density range of approximately 1.4 - 41

× 1011 cm-2. The dashed curves plot m* calculated using a 4 × 4 tight-binding Hamiltonian with

hopping parameters 0 = 3.4 eV, 1 = 0.4 eV, 3 = 0, and v4 = 0.063. These values are obtained by

fitting the data in C at high densities Ref. [49].

55

The inter-layer hopping integral 3 deforms the spherical symmetry of the Fermi surface

as shown in Fig. 2-28a. This effect becomes more pronounced at low carrier densities and

can lead to the breaking up of the Fermi surface into three pockets, i.e. the Lifshitz

transition [67]. Although the density range studied here (0.2 ~ 1.2 × 1012 cm-2) is far above

the Lifshitz transition density, we investigated the role of the warping on m*. Figure 2-28a

plots examples of deformed Fermi surfaces for EF = 7 meV and 30 meV respectively for

Figure 2-28. (a) Warped Fermi surfaces in momentum space for EF = 30 meV (solid curve) and EF

= 7 meV (dashed curve) for holes (blue) and electrons (red). They correspond to nh = 2 × 1011 cm-

2, ne = 1.5 × 1011 cm-2 (EF = 7 meV) and nh = 9.4 × 1011 cm-2, ne = 6.7 × 1011 cm-2 (EF = 30 meV)

respectively. v3 = 0.1. (b) Calculated electron and hole mass m* using v3 = 0.11 (symbols) and v3 =

0 (solid curves). Other TB parameters are 0 = 3.43 eV, 1 = 0.4 eV, and v4= 0.063.

electrons (blue) and holes (red), using a 4 × 4 Hamiltonian (see Fig. 2-22) and the largest

v3 = 3 /00.11, i.e. 3 = 0.38 eV, found in the literature [32, 75, 77] EF = 7 meV

corresponds to the lowest carrier densities we measured. The corresponding m* is plotted

in Fig. 2-28b, together with calculations corresponding to 3 = 0. It is clear from the

comparison that trigonal warping plays a negligible role on m* in our studied density range,

56

despite the deformation of the Fermi surface. We have therefore neglected the trigonal

warping effect and set 3 = 0 in all subsequent calculations.

In existing theoretical studies of bilayer electronic dispersions, the effect of e-e

interaction manifests in two ways, i. e. by renormalizing the hopping parameters within the

TB model at high carrier densities [95] and by causing deviations of m* from the TB

description at low carrier densities, with different trends of m* predicted [92-94, 96, 97].

We begin our calculations with a four-band TB Hamiltonian with non-interacting hopping

parameters and explicitly include e-e interaction with the random phase approximation

(RPA) of the screened exchange self-energy

qkkFq

qVk SS

q

D

,2

(2-12)

using a dielectric function 휀(𝑞) = 휀𝐵𝑁 − 𝑉2𝐷(𝑞)𝜒(𝑞), that includes contributions from both

the bilayer graphene and the h-BN substrate and overlayer. Here 휀𝐵𝑁 = 3.0 is determined

from the gating efficiency of the backgate, and 𝐹𝑠𝑠′ is the pseudospin overlap factor [92,

93]. Eq. 2-12 provides the RPA correction to the bare energy bands 𝐸0(𝑘) obtained from

TB calculation to yield the quasiparticle band structure 𝐸(𝑘) = 𝐸0(𝑘) + Σ(k) . The

effective mass is then computed using Eq. 2-10.

The calculated m*e and m*

h are plotted in Fig. 2-29 in olive dotted lines. Interaction

leads to a slightly faster decrease of m*e and m*

h at low carrier densities, in contrast to the

sudden drop observed in the measured m*h for nh < 5 × 1011 cm-2. Examining the problem

from a different angle, we note that in the RPA model, the dielectric function is well

described by the Thomas-Fermi (TF) screening 휀(𝑞) = 휀𝐵𝑁 +𝑞𝑇𝐹

𝑞 in the small q limit [96].

57

Figure 2-29. Comparison of calculations and experiment at low carrier density (0.2 – 1.3 × 1012 cm-

2). Experimental data follow the symbols used in Fig. 2-27. The olive dashed lines plot the

calculated m* including e-e interaction in a random phase approximation. The black and gray lines

are calculations that further include the effect of potential disorder using E = 5.4 meV obtained

from q and the temperature dependence of the conductance. In both calculations, 0 = 3.08 eV, 1

= 0.36 eV, 3 = 0 and v4 = 0.063. Inset: A schematic illustration of the electron-hole coexistence at

low carrier densities due to disorder and its effect on the cyclotron motion.

Figure 2-30 plots the theoretical fit of the data using Thomas-Fermi screening. The

Thomas-Fermi wavevector qTF is reduced ten-fold from its expected value of 𝑞𝑇𝐹 =

𝑚∗𝑒2/ℏ2 to fit the hole mass data. This ten-fold reduction implies extremely weak

screening, which cannot be justified in our devices. Even so, the agreement with the

electron branch is still poor. Together with the RPA results plotted in Fig. 2-29, these

calculations show e-e interaction effect, at least at the RPA level, appears to be too weak

to account for the experimental observations. In comparison, in monolayer graphene, a

58

large suppression of m* is also observed at low carrier densities and well described by RPA

calculations [91].

Figure 2-30. m* calculated using the Thomas-Fermi screened self-energy and the T-F wavevector

qTF = 0.1𝑚∗𝑒2/ℏ2. The legend follows Fig. 2-29. Hopping parameters are: 0 = 3.08 eV, 1 = 0.37

eV, 3 = 0, and v4= 0.063. Dielectric constant of h-BN εBN = 3.

Can Coulomb potential fluctuation and the resulting density inhomogeneity [47, 98,

102] play a role? The answer is not so intuitive at the first glance. In a conventional

semiconducting 2DEG, density inhomogeneity results in the smearing of m*(n). This effect

does not alter the trend of m*(n) and is typically non-consequential in the carrier density

regime where the SdH oscillations are well-behaved. In Fig. 2-26c, the SdH oscillations at

nh = 3 × 1011 cm-2 appear to be well-behaved, yet the measured m*h is already 14% below

the TB band mass. Here, the gapless nature of the bilayer bands makes a crucial difference

between bilayer graphene and a conventional 2DEG. As the inset of Fig. 2-29 illustrates,

as the Fermi energy EF approaches the disorder energy scale δE, instead of depletion,

carriers of the opposite sign start to appear in parts of the sample. The SdH oscillations of

0 1 2 3 40.02

0.03

0.04

0.05

m*

(me)

Carrier density (1012

cm-2)

59

a minority carrier type have the opposite sign in dA/dE; their presence in some regions of

the sample therefore contributing negatively to the average of m*, resulting in a decrease

in its value. Such cancellation effect does not occur in a conventional semiconductor 2DEG.

This situation can be modeling by defining the overall carrier density and effective mass

as ensemble averages of their local counterparts 𝑛𝑙𝑜𝑐 and 𝑚𝑙𝑜𝑐 respectively:

EnfdnEn locloc (2-13)

EmfdmEm locloc (2-14)

Here, the fluctuation of energy is assumed to have a Gaussian profile f (μ) with standard

deviation δE.

Effective masses calculated using the RPA model and including disorder characterized

by a broadening energy 𝛿𝐸 = 5.4 meV are plotted as solid lines in Fig. 2-29. Evidently, the

combination of e-e interaction and Coulomb potential fluctuations can now quantitatively

reproduce the observed behavior of m*e and m*

h over the entire range of measurement.

Remarkably, the same value for 𝛿𝐸 simultaneously captures the sharp decrease of m*h at

nh < 5 × 1011 cm-2 and the absence of such decrease on the electron side. Our calculations

predict that m*e should also substantially decrease from the TB values at yet lower carrier

densities, just below the range probed in our measurements. The difference arises from a

smaller electron density inhomogeneity due to a smaller m*e. The disorder energy scale 𝛿𝐸

= 5.4 meV corresponds to ne = 1.1 × 1011 cm-2 and nh = 1.6 × 1011 cm-2. These values are

consistent with estimates obtained by locating the crossover density n(h/e)c ~ 1.0 × 1011

cm-2, where the temperature dependence of R(n) changes from that of a metal, i. e. dR/dT >

60

0 to that of an insulator, i. e. dR/dT < 0 [103]. Additionally, it is also in good agreement

with 𝛿𝐸~ℏ/2𝜏𝑞~ 3 meV deduced from the measured quantum scattering time τq shown in

Fig. 2-26d. Furthermore, our calculations also show that interaction renormalizes the inter-

band transition energy 1 from the “bare” value of 0.36 eV (Fig. 2-29) to 0.38 eV, in

excellent agreement with infra-red absorption measurements [48, 75, 76].

Figure 2-31. m* calculated using the TB parameters of Ref. [49] and including disorder broadening

E = 5.4 meV.

In Ref. [49], Zou et al have shown that a set of renormalized TB hopping parameters

can capture m* in the high-density regime very well, without explicitly including e-e

interactions (See dashed lines in Fig. 2-27). Figure 2-31 plots the calculated m*, using the

same set of renormalized TB parameters empirically determined in Ref. [100] and

including disorder broadening E = 5.4 meV, obtained in the best fit in Fig. 2-29. This set

of parameters can also capture the main trend of data, with the diving of m*h at low densities

slightly too abrupt compared to experiment. It should be emphasized however, that this

agreement does not mean the effect of the e-e interaction is unimportant but rather it can

0 1 2 3 40.02

0.03

0.04

0.05

m*

(me)

Carrier density (1012

cm-2)

61

be well captured by renormalized TB parameters in the entire density range (1011 -

1012/cm2) we studied.

The above studies highlight a few remarkable differences between bilayer graphene, a

gapless Dirac Fermi liquid and conventional semiconductor 2DEGs. Firstly, both our

calculations and measurements suggest that the effect of e-e interaction on m* in bilayer

graphene remains weak down to n ~ 2 × 1011 cm-2 (rs = 5.3) while past studies on GaAs

electrons showed an enhancement of more than 40% at this interaction parameter [50].

Secondly, the effect of disorder appears quite different in these two systems. In

conventional semiconducting 2DEGs, disorder leads to localization and therefore the

increase, rather than the decrease of m* at low carrier densities [50]. Here in gapless bilayer

graphene, disorder leads to coexisting electrons and holes and consequently a partial

cancellation effect on m*. In comparison to the well-recognized Klein tunneling effect in

p-n junctions [104, 105], our study exposed a more elusive effect of electron-hole puddle.

Studies of low-carrier-density regimes in Dirac materials thus require a great deal of

caution. For now, samples of yet higher qualities are necessary to elucidate the intrinsic

behavior of m* near the charge neutral point of bilayer graphene.

In summary, we have performed careful measurements of the effective mass m* in high-

quality h-BN supported bilayer graphene samples down to the carrier density regime of 1

× 1011 cm-2 and observed sharp decrease of the hole mass at low carrier densities. Our

calculations show that while the inclusion of electron-electron interaction is necessary to

reach excellent quantitative agreement with data at all carrier densities, Coulomb potential

fluctuations, which result in the co-existence of electron and hole regions and a partial

cancellation of m*, is chiefly responsible for the observed sharp drop in m*h at low densities.

62

This mechanism, which is absent in finite-gap semiconductor two-dimensional systems, is

another manifestation of the unusual consequences of gapless Dirac bands.

63

Chapter 3

The art of gating in nanostructures

3.1 Introduction

Gating is a flexible and powerful tool to study 2DEG systems. It plays two major roles

namely doping carriers into the 2DEG and generating potential profiles to guide or confine

carriers in the 2DEG. In conventional 2DEG such as GaAs, gating are widely used to

partially deplete the sample, i.e. to study the interactions between quantum Hall edges by

constructing a point contact gating geometry [106], and to study quantum dots by depleting

and isolating a small area of 2DEG [107]. In recently discovered and widely studied layered

2D materials such as graphene and transition metal dichalcogenides (TMDs), gating

becomes the essential tool to study the transport and optical properties of these materials

[108, 109]. Since these materials are atomically thin and the dielectric materials are also as

thin as 10s nm, the gating profiles can be much sharper compared with those in GaAs as

the 2DEGs in GaAs are buried 100s nm beneath the gates. There is a great flexibility to

build gating structures in graphene systems, and this makes lots of experiments such as

Klein tunneling [29], gate-controlled electron optics [30, 110], and ultra-clean bilayer

graphene shows strong e-e interactions at low carrier density [60] possible. The gating in

2D layered materials can take the form of conventional metal/ dielectric structure,

suspended metal gate [60] and ionic liquid [111]. The latter two forms have drawbacks

such as limited gating power and difficulties in operations, and this chapter will focus on

the conventional metal/ dielectric gating structures. The gating effect in these structures,

without considering the quantum capacitance of the 2DEGs, can be, in general, understood

64

with classical electrostatic models plus the boundary conditions. Therefore, we can use

finite element analysis software such as COMSOL Multiphysics to simulate the gating

effects in graphene devices to guide the designing of gate structures.

3.2 Edge effects in gating

3.2.1 Enhanced gating efficiency in graphene nano ribbons

In graphene devices, the gating efficiency of a gate can be determined though Hall

measurements or by fitting the filling factors in SdH oscillations and quantum Hall

measurements. The gating efficiency for the doped silicon (Si++) gate in our 290 nm thick

SiO2 supported graphene samples is usually around 7 × 1010 cm-2V-1 when the graphene

size is much larger than the thickness of SiO2. However, in graphene nano ribbon devices

where the width of graphene is on the order of 100 nm, which is comparable or smaller

than the dielectric thickness, we observe an enhancement of the gating efficiency of the

same Si++ gate, i.e. it is 9 × 1010 cm-2V-1 in a 250 nm wide graphene nano ribbon device.

In a parallel plate capacitor, the capacitance in a unit area is d

c r 0 in the center

region of the capacitor, where ε0 and εr are the vacuum permittivity and relative permittivity

of the dielectric material and d is the distance between the two plates. However, this does

not hold on the capacitor edge, and higher density of charge is accumulated on the edge.

Similarly, gating efficiency is only uniform in the center of a bulk sample, while it is much

higher on the sample edge. And this non-uniformity is worse for a global gate configuration

when the size of the gate is much larger than the dimension of gated material. This edge

effects in gating become very important when the sample dimension decreases to a few

times of the dielectric thickness. In a graphene nano ribbon device, the gating efficiency

65

becomes non-uniform over the entire sample, and we can simulate the position dependent

gating efficiency using COMSOL Multiphysics.

Figure 3-1. (a) A schematic shows COMSOL simulation setup for a doped silicon gated graphene

device. Origin of x axis and simulation parameters are labeled in the figure. (b) Simulated position

dependent gating efficiency of doped silicon gate for graphene nano ribbon with width 100 nm, 150

nm and 250 nm.

The schematic in Fig. 3-1a shows a COMSOL simulation setup for our 290 nm thick

SiO2 supported graphene nano ribbon device. Graphene, SiO2 and Si++ are represented by

metal blocks with thickness of 2 nm, 290 nm and 100 nm respectively. The width (along x

direction) of graphene is varied from 100 nm to 250 nm in three simulations, and the width

of SiO2 and Si++ is 10 μm. We set the origin of x axis at the center of graphene, and the

dielectric constant of SiO2 is set to be 3.9. The whole device is then enclosed in a larger

66

block with ε = 1. Graphene is grounded, and VSi = 10 V in all three simulations. Figure 3-

1b plots the simulated gating efficiency variation along the x direction of graphene nano

ribbon devices with 100 nm, 150 nm and 250 nm width. The gating efficiency is minimum

at x = 0. And the change of gating efficiency is also smallest around x = 0, nevertheless

non-zero, and it becomes larger as x deviates further from the center of device. Furthermore,

the gating efficiency at x = 0 also decreases with increasing ribbon width, i.e. 12 × 1010 cm-

2V-1 (100 nm ribbon), 9.7 × 1010 cm-2V-1 (150 nm ribbon) and 9.5 × 1010 cm-2V-1 (250 nm

ribbon), however it is much larger than the bulk device value even at the center of these

nano ribbon devices. The simulation results capture the enhancement of Si++ gate gating

efficiency in graphene nano ribbon devices. In many cases COMSOL simulations can

provide us better understanding of the gating effects in devices, and therefore help us

design gating structures in devices.

3.2.2 Two examples of non-extended top gate

The dry transfer procedure described in Section 2.2 is an ideal method to build ultra

clean h-BN encapsulated bilayer graphene heterostructures, however it is very challenging

to fabricate dual-gated multi-terminal devices with these heterostructures. In a

conventional device fabrication procedure, where heterostructures are stacked with wet

transfer procedures, graphene is etched into Hall bar geometry and contacted (or preserved

to be contacted later) first, and then the top h-BN is transferred and extends over the Hall

bar channel. In this case, it is trivial to pattern a top metal gate that extends over the channel

area. With a dry transferred stack, since the graphene flake is encapsulated in h-BN flakes

without any prior lithography processing, the followed etching procedure would etch the

top h-BN and graphene into the exact same geometry with the edges from two layer aligned.

67

This makes the top gate fabrication very difficult. However, with smart planning we are

able to fabricate the dry transferred heterostructures into dual-gated Hall bar devices with

the lithography procedures illustrated in Fig. 3-2.

Figure 3-2. (a) – (f) Optical images show lithography procedures to fabricate a dry transferred

heterostructure into a dual-gated Hall bar device. Scale bar is 5 μm. Dashed curve in (a) and (b)

shows the boundary of bilayer graphene.

The fabrication includes the following steps.

1. A bubble free dry area on a transferred h-BN/ bilayer graphene/ h-BN stack is

identified, and a rectangle with a “tail” hard etching mask is defined by standard e-beam

68

lithography using Hydrogen silsesquioxane (HSQ)/ PMMA bilayer resist. The “tail”

structure, which is preserved route for top gate electrode, extends through the crystal edge

of bilayer graphene shown in Figs. 3-2a and b.

2. CHF3 and O2 (10 to 1 ratio) plasma mixture etches through the stack to exposes the

edges of bilayer graphene to make Cr/ Au side contacts. (Fig. 3-2c)

3. Cr/ Au (5 nm/ 60 nm) side contacts and top gate electrode are patterned with standard

e-beam lithography. (Fig. 3-2d)

4. HSQ/PMMA and the top gate electrode together act as etching mask to pattern the

device into a Hall bar geometry. CHF3 and O2 (10 to 1 ratio) plasma mixture etches the

excessive part of the stack. (Figs. 3-2e and f)

Figure 3-3. (a) Vbg and Vtg relations for n = 0 condition. Inset shows an optical image of device

measured and pin numbers are labeled. Scale bar is 2 μm. (b) A set of resistance between pins 5

and 4 R5-4 versus Vbg at Vtg fixed from 4 V to -4 V in 0.5 V step measured at T = 77.5 K in a semi-

log scale.

The inset of Fig. 3-3a shows a dual-gated bilayer graphene device fabricated following

the methods described above. Figure 3-3a plots the Vbg and Vtg relation at CNP conditions,

69

and it follows a straight line as expected. Figure 3-3b plots a group of R5-4 (Vbg) curves

taken at fixed Vtg from 4 V to -4 V with a 0.5 V step in a semi-log scale. The global

minimum (D = 0) appears to be at Vbg = 0 V and Vtg = 0 V. This suggests negligible

unintentional doping, which is usually a sign of a clean sample. Also μFE is estimated to be

60,000 cm2V-1s-1 for this device. Devices with similar quality show the CNP resistance

increases exponentially with D field [112]. In this device, the CNP resistance increases

with the magnitude of D field only in a small D field range, and it soon saturates at around

10 kΩ in the negative D field, and deviates from the exponentially increasing trend (marked

with dashed line in Fig. 3-3b) in the positive D field. It seems the sample is shorted at large

D field.


device fabricated with dry etching techniques. (b) A schematic shows the details on the edge of the

device. Origin of x axis and gate voltages are labeled in the figure. Dielectric constant and thickness

of h-BN is 3 and 20 nm respectively. (c) Simulated carrier density distribution in the device.

70

We suspect the short comes from the improperly designed top gate. The size of the

metal top gate in a device, which is fabricated following the procedures mentioned in this

section, is the same compared with the size of the Hall bar channel, since the metal top gate

is used as an etching mask. Furthermore, the etching process creates non-vertical edges as

illustrated in Fig. 3-4a. This gating configuration makes the carrier density (n) on the

sample edges differ a lot compared with that in the bulk area, and this effect is confirmed

by COMSOL simulation.

As shown in Fig. 3-4b, in the COMSOL simulation we set the thickness and dielectric

constant of h-BN to be 20 nm and 3 respectively. A 60 degree of etching angle is set on the

edge of h-BN. Graphene is grounded, and we assign – 2 V and 2 V to the top and bottom

gate respectively. This mimics the n = 0 and D = 0.3 V/nm condition in the measurement

shown in Fig. 3-3. The simulated n distribution near the sample edge is plotted in Fig. 3-

4c, where n approaches 0 around 30 nm away from the edge and n is larger than 1011 cm-2

within an area of 20 nm width from the sample edge. This suggests there is a conducting

ribbon around the whole sample edge, and it effectively shorts out the insulating bulk. The

etching angle depends on the etching conditions, i.e. gas pressure and gas mixture ratio,

and it can be as small as ~ 45 degree. If the conducting edge is solely caused by the non-

vertical edge profile, then one can fine tune the etching recipe to avoid this issue. However,

we find out if the top gate is not properly extended over the sample then it is likely to have

a conducting edge when the bulk is insulating.

We consider another situation illustrated in Fig. 3-5a, where bilayer graphene is dual-

gated by two gates with the top and bottom dielectric thickness of 30 nm and 120 nm

respectively, and there is no non-vertical edge of the h-BN dielectric. This setup can

71


device fabricated with top gate stops at left device edge. Origin of x axis and simulation parameters

are labeled in the figure. (b) Simulated carrier density distribution in the device at D field of 0.025

V/nm, 0.055V /nm and 0.2 V/ nm, where n = 0 in the bulk. (c) and (d) Simulated carrier density

distribution close to the left (d) and right (d) edges of the device in semi-log scale.

effectively simulate an h-BN encapsulated bilayer graphene device gated with a local top

metal gate and a global Si++ gate. In order to compare the edge gating effects between

non-extended and properly extended gating schemes, we deliberately match the left edge

of the top gate and bilayer graphene while extend the top gate 1 μm over the right edge of

bilayer graphene. In the simulation we apply positive and negative voltage on the Si++ and

top metal gate respectively to create positive D field while maintain n = 0 in the bilayer

72

graphene bulk. The simulated variation of n along the x direction of the device at three

different D field is plotted in Figs. 3-5b – d, where Figs. 3-5c and d plots n near the left and

right edge of the device respectively in a semi-log scale. A few hundred nm away from the

edges of bilayer graphene, n is 0 as expected. There is a very large electron doping near

the left edge of bilayer graphene, and the doping level also increases with D field. Although

there is also excessive doping near the right edge of bilayer graphene, the doping level is

more than two orders of magnitude smaller compared with that near the left edge. We know

empirically the gating scheme on the right edge would not cause shorting problems in our

measurements. However, at D = 0.2 V/nm there is an appreciable width of 30 nm near the

left edge where n is on the order of 1011 cm-2, and this is comparable with the situation

shown in the previous non-vertical h-BN edge example.

These simulation results suggest extending top gate over both edges of the device

channel is a necessity in a dual-gated bilayer graphene device to avoid excessive doping

on the sample edge, which can short the bulk effect in measurements. This is true for most

of the device fabrication in reality where global gate such as Si++ gate is widely used, and

it is rare to have identical top and bottom gate dielectrics, i.e. same thickness and dielectric

constant. However, the device fabrication procedure described in this section is not

worthless. One can adapt the procedure to fabricate devices where the relatively conducting

edge with a small width is not a concern, for instance to fabricate a dual-gated h-BN

encapsulated WSe2 device where the goal is to have sufficient doping into the bulk.

3.3 COMSOL simulation in gate design

3.3.1 Effective gating efficiency

73

Figure 3-6. (a) A schematic shows COMSOL simulation setup for a dual split gated bilayer

graphene device with an additional back doped silicon gate. Each metallic unit is represented by a

5 nm thick block. Origin of x axis and simulation parameters are labeled in the figure. (b) Simulated

carrier density distribution in and in adjacent to the 70 nm junction area with VSi = 10 V, 20 V and

30 V, and all other gates are grounded. (c) Simulated carrier density distribution in and in adjacent

to the 70 nm junction area with VRB = -1 V to 2 V, VSi is fixed at 10 V, and all other gates are

grounded. (d) and (e) Simulated position dependent effective gating efficiency (d) and the junction/

bulk gating efficiency ratio (e) for the right bottom gate, doped silicon gate and right top gate within

the 70 nm junction area.

74

We usually refer gating efficiency as the gating power of a gate on the center region of

a bulk sample, but in a situation where multiple gates are adjacent to a gated sample, the

effective gating power for each gate on the sample deviates a lot from its conventional

value. COMSOL simulation can provide useful information for example the effective

gating efficiency of individual gate in a complicated gating configuration such as the one

shown in Fig. 3-6a, which contains five gates, two pairs of split local gates with 70 nm split

size and one global Si++ gate. The simulation parameters are shown in Fig. 3-6a, and they

are set to make the conventional gating efficiency of the local top gate, local bottom gate

and Si++ gate to be 8.04 × 1011 cm-2V-1, 6 × 1011 cm-2V-1 and 7 × 1010 cm-2V-1 respectively.

At the first glance, one would assume the two pairs of split gates only control the doping

level in the left and right bulk regions next to the center 70 nm split region, while the doping

level in the split region is solely controlled by the Si++ gate. This is actually not true, in

reality each one of the gate has gating contribution to the split region and the gating

efficiency also depends largely on the position inside the split region. Figure 3-6b plots n

versus position in the vicinity of the split center with VSi set at 10 V, 20 V and 30 V while

graphene and all the other gates are grounded. The carrier density peaks at the center of the

split region, and also distribute symmetrically around the center. The doping effect from

Si++ gate becomes negligible around 60 nm away from the center, this suggests the Si++

gate is not fully screened by the bottom split gates within a ~25 nm wide region on either

side of the bulk region, and this dimension is comparable with the bottom h-BN thickness.

Figure 3-6c plots n (x) around the center of the split region with VSi fixed at 10 V and VRB

varied from -1 V to 2 V. The carrier density saturates at expected values, i.e. 6 × 1011 VRB

cm-2 around 60 nm away from the split center, and within the split region n is both

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controlled by VSi and VRB. We find n (x) grows linearly with individual gate voltage at any

given position, and therefore we can calculate the effective gating efficiency for the right

top, right bottom and Si++ gate, and the results are plotted in Fig. 3-6d. We compare the

effective gating efficiency with the conventional gating efficiency and plot the ratio of the

two (η) in Fig. 3-6e for the three gates. The effective gating efficiency for Si++ gate is only

a small fraction of the conventional value, and it peaks at the center of the split with 35%

of the conventional gating efficiency. Interestingly η (x) for the top and bottom split gate

are very similar, however it is consistently larger for the bottom gate, since the thickness

of the top and bottom h-BN differs. This suggests when the bulk, for instance the right

side bulk, is at CNP condition, the effective gating from the right top and right bottom gate

to the split region does not completely cancel and result in a net effective gating from the

right bottom gate to the split region. The simulation results provide us guidance on how to

control or design measurements in the split region.

3.3.2 Potential profile in complex gating structures

Figure 3-7. A schematic shows COMSOL simulation setup for a quad-split gated bilayer graphene

device.

In complex structures such as all gate controlled valley valve devices [113] and gate

confined quantum dots in bilayer graphene [61], COMSOL simulation is utilized to

illustrate the potential profiles in these devices. In this section, we take a dual-quad-split-

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gate bilayer graphene device as an example to show how COMSOL simulation helps in

understanding the potential profiles in the device. A schematic of such device is shown in

Fig. 3-7, where four pairs of gates define four quadrant namely northwest (NW), northeast

(NE), southwest (SW) and southeast (SE) quadrant, and four 1D channels are also created

between any two quadrants. Origin is defined at the intersection of the four 1D channels.


bilayer graphene with a (+, -, -, +) top gate voltage polarity. (c) Detailed potential profile near the


Gate split size is 70 nm, and both top and bottom dielectric material are 20 nm thick h-BN

with dielectric constant of 3. Bilayer graphene is within a square region where | x | and | y

| both are ≤ 300 nm. In the following simulations, bilayer graphene is grounded, while the

top and bottom gate with the same index are set to be opposite values, e.g. VNWt = -VNWb,

this creates a large D field while maintains n = 0 in each quadrant. We examine the potential

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at 0.1 nm above the upper layer of bilayer graphene at various gating configurations and

the results are plotted in Figs. 3-8 – 3-11.

Figure 3-9. 2D plots show the simulated potential profile near the upper layer of bilayer graphene

with a (+, -, -, +) electric field polarity, with different sets of gate voltage setup shown in

corresponding schematic. Grey curves indicate the zero potential lines.

In Fig. 3-8, we set the top gate voltage (VNWt, VNEt, VSWt, VSEt) = (5 V, -5 V, -5 V, 5 V)

to create a valley valve and electron beam splitter configuration [113]. A saddle potential

is created at the center of the device, and the potential profiles cut along the two diagonal

directions are plotted in olive and magenta color in Fig. 3-8d. The orange curve in Fig. 3-

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8d plots the potential profile cut along the x direction through the two north quadrants, in

the north channel region (| x | < 35 nm) the potential changes nearly linearly from positive

to negative values.

Another interesting problem is to control the zero potential line in the valley valve and

electron beam splitter configuration, since theory predicts kink states exist at the zero

potential lines [40]. In Figure 3-9, we fix VNWt = VSEt = 5 V while change VNEt and VSWt

from -5 V to -10 V, this effectively increases the imbalance of band gap (Δ) on the two

diagonal directions, where there is no imbalance in Fig. 3-9a and there is a large imbalance

in Fig. 3-9c. The grey line outlines the simulated zero potential lines in all three figures.

In Fig. 3-9a the zero potential lines in all four channels intersect strictly at the center of the

Figure 3-10. (a) 2D plot shows the simulated potential profile near the upper layer of bilayer

graphene with a (+, -, +, -) top gate voltage polarity. (b) Potential profile of line cuts with

corresponding color coding shown in (a).

device where there Δ is the same in all four quadrants. The zero potential lines are

connected into two groups, i.e. they are connected from the west to the north channels and

from the south to the east channels, and the two groups of zero potential lines “repels” each

other at the center of the device. The separation between the two groups grows with the

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imbalance of Δ. The simulation results evaluate the tunability of our devices and are

extremely useful for designing experiments.

In Fig. 3-10, we set the top gate voltage (VNWt, VNEt, VSWt, VSEt) = (5 V, -5 V, 5 V, -5

V) to create a kink state transmission through configuration [113] in the north to south

direction. In general, the potential varies from positive to negative values as x changes from

negative to positive and the potential is always 0 at x = 0. The potential profile cut at y =

150 nm in the two north quadrants is identical with the orange curve shown in Fig. 3-8d,

however the potential profile cut within the west and east channel region (| y | < 35 nm)

differs a lot. The potential profile cut at y = 0 is plotted in Fig. 3-10b in black color, the

slope of the potential change is much smaller compared with that in the orange curve, and

this suggests a weaker potential confinement in the center region of the device.


bilayer graphene with a (+, +, +, +) top gate voltage polarity. (c) Detailed potential profile near the


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Another example of gating configuration where we set (VNWt, VNEt, VSWt, VSEt) = (5 V,

5 V, 5 V, 5 V) is shown in Fig. 3-11. This gating configuration kind of simulate a quantum

dot confinement in the center region (less than a circle with ~ 350 nm radius) of the device.

This is not a strict quantum dot device for the reason that each 1D channel is also confined

and gapped, and it becomes hard to contact the center quantum dot. Nevertheless, the

potential profile cut through the center quantum dot from various directions, e.g. diagonal

direction and x direction (Fig. 3-11d) can provide information such as the approximate size

of the quantum dot. We can empirically estimate the cut off potential for bilayer graphene

to become insulating, together with the range of applied gate voltage we can design a

quantum dot with approximately expected dimension by controlling the actual gate

dimensions.

3.4 Fabrication challenges

The local bottom gates in our complex devices are usually fabricated with multiple

layer graphite flakes (6 – 12 layers of graphene). Compared with metal gates, these graphite

gates have three major advantages. Firstly, graphite gates are very thin and chemically

stable in ambient environment, the thickness of which ranges from 2 to 4 nm, and this

makes the successive device layers flat and easy to be transferred. Metal gate with the same

thickness, for instance a 2 nm gold film deposited using physical vapor deposition (PVD)

method, does not conduct well since it is too thin to be a continuous film. Secondly,

graphite are atomically smooth and the combination of atomically smooth graphite and h-

BN ensures the highest quality of the sample. Empirically, similar h-BN encapsulated

devices gated by graphite show higher quality than those gated by metal gates [114]. The

possible reason for this may due to the lower level of homogeneity of gating by the metal

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gates, because of the finite surface roughness. Finally, graphite, as a layered two

dimensional material, can also be incorporated into transfer procedures, therefore devices

with multiple layers of gates can be made with graphite.

Figure 3-12. (a) SEM image of a graphite bottom gate with splits in both vertical and horizontal

directions. (b) A zoom in view of (a) in one intersect of splits.

Figure 3-13. SEM images show an example of unsuccessful metal quad-split gate structure on

untreated h-BN (a) and a successful example on treated h-BN (b).

An example of our multiple split bottom graphite gate with split size of 70 nm is shown

in the scanning electron microscopy (SEM) in Fig. 3-12. In order to achieve high quality

small features, we use ZEP520 as e-beam resist to define etching mask. To ensure high

resolution of the etching mask, we use ice-bathed cold development after e-beam writing.

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Then we employ reactive ion etching with oxygen plasma to etch graphite flakes into

designed gate structures.

If top gate is the final layer of the device, we usually use metal top gate for its simplicity

in fabrication, since we do not need to worry about the thickness of the top gate. Fabricating

metal top gates with sharp edges or small features on h-BN is challenging. Collapse of

features often happen due to the strain between e-beam resist and h-BN surface. An

example of failed quad-split-gate with split size of 60 nm is shown in Fig. 3-13b. We can

solve this problem with two methods. One is to apply an adhesion promoting layer, i.e.

Surpass 2000 onto h-BN before e-beam resist is coated. This method improves the yield of

fabricating 60 nm split features to about 50 %. Another strategy is to very gently ash the

h-BN flake with oxygen plasma (at most one layer of graphene can be etched with this

condition) before e-beam resist is coated. This method improves the yield of fabricating 60

nm split features to a 100 % yield. A successful example of a dual-quad-split gate with 65

split size is shown in the SEM image in Fig. 3-13b.

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Chapter 4

Ballistic kink states and valleytronic operations in bilayer graphene

nanostructures

4.1 Introduction

4.1.1 Electronic degree of freedom

It has been a long history that people are utilizing the charge degree of freedom (DOF)

of electrons to perform high speed logic calculations. The building block of such logic

devices, i.e. metal–oxide–semiconductor field-effect transistor (MOSFET), is turned on/

off by controlling the amount of the carriers through a metal gate. The successful

miniaturization and integration of these transistors into integrated circuits have greatly

improved the calculation power, therefore brought us the digital revolution and

convenience in life. The density of those transistors have already reached a few billion per

square inch in a dense integrated circuit, and it is still growing following the Moore’s law,

which approximately doubles in every two year. The large power consumption and heating

dissipation issues become serious challenges for the semiconductor microprocessor

industry. Overcoming the limit of Moore’s law requires pursuing new technologies which

take advantages of other electronic DOFs of electrons such as spin. Devices which function

on spin current and zero net charge current are ideal candidates for low power consumption

devices [115]. The magnetic tunnel junction (MTJ) [116, 117] is a successful example of

utilizing the spin degree of freedom of electrons, and they have already been widely used

in various memory devices. However, the implementation of gate controlled spin

transistors [118] are still at the ongoing laboratory research stage.

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4.1.2 Valley-contrasted physical properties

The recent advance in two dimensional layered materials with hexagonal lattice

structures including graphene systems and a class of TMDs, e.g. WSe2 and MoS2 [119],

lead us to a new aspect to create low power consumption devices. In addition to charge and

spin, electrons in these materials also possess valley DOF. One can manipulate the valley

DOF and create a new paradigm of electronics called valleytronics. Valley index

Figure 4-1. (a) Brillouin zone of graphene. (b) Band structure of graphene with degenerate but

inequivalent K and Kʹ valleys.

distinguishes the degenerate but inequivalent energy bands which have a local minimum

and maximum, e.g. there are two valleys namely K and Kʹ in the momentum space of

graphene as shown in Fig. 4-1b, and the corresponding K and Kʹ points at the corners of

the first Brillouin zone are marked in Fig. 4-1a. The valley index is associated with an

intrinsic magnetic momentum (ℳ (k)) in the momentum space, which can be expressed as

ℳ (k) = χτz, where τz = +/ - 1 for K and Kʹ valleys respectively and χ is a coefficient

characterizing the material [120]. τz changes sign under spatial inversion while ℳ (k) does

not, therefore χ is always zero unless inversion symmetry is broken. The breaking of

inversion symmetry leads to many valley-contrasted physical properties in the momentum

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Table 4-1. Analogy between valley-contrasted physics and gauge field theories.

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space, and they have close analogies to gauge field theories in the real space [120] as listed

in Table 4-1. For instance, Berry curvature 𝛺(𝒌), which has a simple relation with ℳ (k)

as ℳ (k) = (𝑒/ℎ)휀(𝒌)𝛺(𝒌), can be understood as a pseudo magnetic field in the momentum

space and it is the curl of a Berry connection. The Berry phase, which is an integral of

Berry curvature within a certain k area is analogous to the Aharonov–Bohm phase

accumulated in the real space. Furthermore the integration of Berry curvature in a closed k

area, i.e. Chern number, which is always an integer, is similar to the Dirac monopole idea.

Although Dirac monopole is not experimentally verified yet, Chern numbers have already

been used to explain quantized Hall plateaus in quantum Hall effect [121]. These analogies

help us to establish an intuitive picture of the valley-contrasted physics.

Figure 4-2. (a) and (b) Lattice structure of graphene with inversion symmetry (a) and inversion

symmetry is broken by adding different onsite energy to A and B sublattice (b). (c) A band gap is

opened in graphene as inversion symmetry is eliminated.

Let’s take graphene as an example. Intrinsic graphene lattice which consists of A and

B sublattices has a spatial inversion symmetry as shown in Fig. 4-2a. The Hamiltonian of

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an intrinsic graphene can be written as yyxxzF kkvH ˆˆˆ where vF is the Fermi

velocity in graphene and σ is the Pauli matrix. This leads to a gapless band structure as

shown in Fig. 4-2c and all the pseudo-spin are strictly in the kx - ky plane. In order to obtain

nonzero 𝛺(𝒌) in graphene, one has to break the inversion symmetry of its lattice. One can

apply onsite potential 2

/

to A and B sublattice sites respectively to break the inversion

symmetry as shown in Fig. 4-2b. The Hamiltonian now becomes

zyyxxzF kkvH ˆ

2ˆˆˆ

and a band gap is also opened in its band structure.

Meanwhile the pseudo-spin also gains an additional out of kx - ky plane component, and

this leads to non-zero 𝛺(𝒌). The analytical expression for the Berry curvature can be

written as 2/32222

22

4

2

F

Fz

vq

vq

[122], where q is the momentum measured from K

Figure 4-3. Berry curvature Ω near K and Kʹ points in momentum space of graphene. The

integration of Ω in each valley gives rise to “Chern number” of ½ sgn (n). Adapted and modified

from Ref. [122].

or Kʹ valley centers. Figure 4-3 plots 𝛺 around the two valley centers along the K - Kʹ cut.

𝛺 peaks at the two valley centers then decreases and quickly vanishes away from K/ Kʹ

point, also the Berry curvature carries opposite signs in K and Kʹ valleys. The integration

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of 𝛺 in each valley gives a not-so-strict definition of “Chern number” for the individual

valley as marked in Fig. 4-3.

Figure 4-4. (a) A schematic show valley Hall effect in graphene. (b) Valley Hall effect is probed in

bilayer graphene with broken inversion symmetry using nonlocal measurement. Adapted from Ref.

[126].

A charged particle in a magnetic field experiences a Lorentz force perpendicular to the

travel direction where Brk , and this is the origin of the Hall effect. In analogy to the

Hall effect, electrons/ holes carry nonzero Berry curvature also acquire an anomalous

velocity in the transverse direction which is proportional to 𝛺 and kr , when an in

plane electric field is applied to accelerate the carriers. Carriers in the K and Kʹ valleys

have opposite Berry curvature, therefore the anomalous velocity acquired by carriers from

different valleys also has opposite directions as shown in Fig. 4-4a. Carriers from K and Kʹ

valleys are bent towards opposite side of the sample and this is the valley Hall effect where

there is finite valley current while the net charge current is zero. Since there is no charge

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current involved, the idea of utilizing valley current in logic calculation has the advantage

of very low power consumption. Valley Hall effect has been predicted [122, 123] and also

detected in various materials such as monolayer and bilayer graphene, monolayer and

bilayer MoS2 with both transport measurements and optical methods [39, 124-127]. Figure

4-4b shows an example of detecting valley Hall effect in an inversion-symmetry-broken

bilayer graphene using a non-local measurement setup [126].

4.1.3 Quantum valley Hall (kink states)

It’s well known that the Hall effect and spin Hall effect both has its quantized version

with either chiral or helical edge modes conducting on the sample boundary. The natural

question to be asked is does the quantized version of the valley Hall effect exist. In analogy

to quantum spin Hall effect, one would imagine helical quantum valley Hall edge modes

travel on the boundary of the sample as illustrate in Fig. 4-5a, where carriers with different

Figure 4-5. (a) A schematic shows a possible quantum valley Hall effect where edge modes travel

on the sample edge are subjected to strong intervalley scattering from atomic defects on sample

edges. (b) A schematic shows a possible quantum valley Hall effect where edge modes travel along

an internal artificial edge.

valley index travel oppositely while their wavefunction overlap in space. In the quantum

spin Hall system the ballisticity of the edge modes are protected by the difficulty in spin-

flip backscattering in absence of magnetic impurities. However, there are lots of atomic

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defects on the sample boundary which provide large momentum transfer, and intervalley

scattering is quite effective. Therefore edge modes illustrated in Fig. 4-5a is not protected.

Furthermore, Li and collaborators also point out even with an ideal crystal boundary

condition, i.e. zigzag sample edge without defects, the valley-contrasted edge modes in

general do not hold on the graphene – vacuum interface [128], because the Chern number,

which is the integral of 𝛺 in the whole Brillouin zone eventually vanishes although the

integration of 𝛺 for each valley is finite. However, the valley-momentum locked edge

states do hold at domain wall structures, which is an artificial boundary established by

flipping the sign of 𝛺. Figure 4-5b shows an example of domain wall structure in graphene,

where the sign of the onsite potential changes across the dashed line. In this case the “Chern

number” calculated based on each valley changes by 1 across the boundary, therefore one

gapless edge mode per valley can hold at the domain wall boundary.

The domain wall structure in graphene requires a staggered onsite energy which also

has a reflection symmetry around the dashed line shown in Fig. 4-6a, and it is very

challenging to be realized in experiments. The good news is that the domain wall structure

is much more practical in bilayer graphene. The inversion symmetry in bilayer graphene

comes from the layer symmetry, as mentioned in Chapter 2 we can easily break the

inversion symmetry by applying a perpendicular D field. Also we can conveniently reverse

the sign of Δ by changing the direction of applied D field. Figure 4-6b illustrates a dual-

split-gated bilayer graphene device proposed in Ref. [40] to create a domain wall structure,

where the direction of D field is reversed across a narrow line junction, and the change of

“Chern number” for each valley is two across the junction. Hence two valley-momentum

locked edge modes (a.k.a. kink states) per valley are predicted to exist at the zero potential

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line within the potential defined junction. Consider the two fold spin degeneracy, the

conductance of the kink states is 4 e2/h in absence of intervalley scattering. Similar domain

wall structures are also proposed [41] and experimentally verified [129] in a uniformly

gated bilayer graphene with an AB/ BA stacking domain boundary. In the following

sections of this chapter we will discuss our efforts on establishing the kink states using

scalable gate defined structures [112] and furthermore realize an all-electric valley valve

and electron beam splitter [113] in bilayer graphene.

Figure 4-6. (a) A schematic shows a domain wall structure in graphene where onsite potential is

mirrored along the dashed line. Filled and open sites carry opposite onsite potential. (b) A dual-

split-gated bilayer graphene device first proposed by Martin et.al. [40] where quantum valley Hall

edge modes (a.k.a. kink states) is predicted to exist at the zero potential line inside the split region.

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4.2 Experimental realization of kink states

4.2.1 Motivation for realizing gate controlled kink states

Two-dimensional materials with hexagonal symmetry are characterized by a valley

DOF in momentum space, the manipulation of which can lead to new types of valley-based

electronics (valleytronics), in analogy to the role played by electron spin [38, 120, 125]. In

inversion-symmetry-broken materials, such as an electrically gated bilayer graphene [33,

36, 43, 47], the nonzero Berry curvature carries opposite signs in the two inequivalent

valleys K and K′. Reversing the sign of the Berry curvature along an internal boundary of

the material gives rise to counter-propagating one-dimensional conducting modes encoded

with opposite valley indices, in analogy to the quantum spin Hall edge states [130]. These

metallic wires are topologically protected against backscattering in the absence of

intervalley scattering, and thus can carry current ballistically [40, 42, 120, 131-133]. A

Berry curvature reversal requires inverting the interlayer potential difference in electrically

gated bilayer graphene. This can occur at the AB/ BA bilayer graphene stacking domain

boundary[41, 129, 134], or at the line junction of two oppositely gated bilayer graphene

[40], with the latter scheme being particularly attractive for valleytronic functionalities

such as valves and waveguides, since the creation and manipulation of the valley-polarized

wires are entirely through gating and potentially scalable [132, 135]. The implementation

of this scheme is however technically challenging. In this section we will demonstrate the

fabrication of a dual-split-gate structure in bilayer graphene and show experimental

evidence for the presence of 1D conducting channels in the gating configurations where

valley-polarized conducting states are expected.

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Exploiting the valley degree of freedom in honeycomb offers an alternative pathway to

achieving low-power-consumption electronics. Experiments have shown that a net valley

polarization in the material can be induced by the use of circularly polarized light [127,

136-138] or a net bulk current [39, 126, 139]. However the use of light is not always

desirable in electronics and valleytronic device proposals using bulk valley polarization

often put stringent requirements on the size and edge orientation of the active area [38].

Alternatively, valley-polarized topological conducting channels can offer a robust platform

to realize valleytronic operations [40-42, 120, 129, 131-135]. The all-electrical creation in

a high-mobility two-dimensional material, bilayer graphene [40], is particularly appealing

from the viewpoint of scaling and electronics development.

4.2.2 Experimental setup and COMSOL simulation

Figure 4-7. (a) Schematic of our dual-split-gated bilayer graphene device. The four split gates

independently control the bulk displacement fields DL and DR on the left and right sides of the

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junction. The Si backgate tunes the Fermi energy EF of the line junction. The gating efficiencies of

the split gates are determined using the quantum Hall effect. We determine the gate voltages

corresponding to the D = 0 and n = 0 state on the left and right sides of the junction using the global

minima of the charge neutrality point resistance Rbulk CNP. Subsequent measurements are done at

nL = nR = 0 and constant displacement fields DL and DR. The diagram shows the odd field

configuration that results in the presence of the helical kink states at the line junction. Blue and red

arrows correspond to modes carrying valley index K and K’ respectively. Each one contains four

modes accounting for the spin and layer degeneracy. (b) External electrostatic potential profile near

the top (gray) and bottom (black) graphene layers for the odd (DLDR < 0) and even (DLDR > 0) field

configurations. Potential simulations are performed using the COMSOL package and parameters

of device 1. The crossing of the potentials at V = 0 gives rise to the topological kink states. The red

curve plots the wave function distribution of one such state schematically, with a full width at half

maximum of 22 nm.

Figure 4-7a illustrates the dual-split-gate structure proposed by Martin et.al. [40],

where an AB-stacked bilayer graphene (BLG) sheet is controlled by two pairs of top and

bottom gates separated by a line junction. The device operates in the regime where both

the left and the right regions of the BLG sheet are insulating due to a bulk band gap

generated by the independently applied displacement fields DL and DR as shown in Fig. 4-

7a. When DL and DR point in opposite directions (DLDR < 0, i.e. “odd” configuration),

theory predicts the existence of eight counter-propagating conducting channels whose

wave functions center at the line of interlayer potential difference V = 0 as illustrated in

Fig. 4-7b. Since they exist in the interior of the sample, these modes are referred to as the

“kink” states. There are four chiral modes (2 due to spin degeneracy, 2 due to layer number)

in each valley. The modes from different valleys overlap in real space but are orthogonal

in the absence of short-range disorder or valley coherence. In such cases, backscattering is

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forbidden and the junction is expected to exhibit a quantized conductance of 4 e2/h. In

contrast, the junction is expected to be insulating in the “even” field configuration (DLDR

> 0 as shown in Fig. 4-7b) due to the absence of the kink states. This sharp contrast thus

enables a clear and convincing demonstration of the existence of the kink states.

Figure 4-8 (a) Schematics of device structure constructed in COMSOL. (b) Side view of the five

gates and the bilayer graphene sheet near the splits with dimensions marked in the figure (not

drawn to scale). The external electrostatic potential U computed along the red and blue dashed

lines is shown in Fig. 4-9.

We use COMSOL Multiphysics to simulate the external electrostatic potential profile near the

line junction and the effective gating efficiency of the Si backgate. Figure 4-8 shows the setup

in COMSOL and the dimensions of the structure simulated. Split width w = 70 nm (device

1) or 110 nm (device 2). All five gates and the bilayer graphene are represented by 5 nm

thick metal plates. The whole device is enclosed in a dielectric environment of ε = 3 that

describes our h-BN flakes. The SiO2 thickness is set to be 220 nm to account for the smaller

ε used (actual thickness 290 nm and ε ~ 3.9).

Figure 4-9 plots the external electrostatic potential Utop along a line cut of 0.1 nm above

the bilayer (red dashed line in Fig. 4-8b) and Ubot along a line cut of 0.1 nm below the

bilayer (blue dashed line in Fig. 4-8b) in devices 1 and 2 for both even and odd bias

configurations. Without considering the screening of the bilayer explicitly, the spatial

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Figure 4-9. Utop (red) and Ubot (blue) along the line cuts shown in Supplementary Fig. 1 (b) for device

1 ((a) and (b)) and device 2 ((c) and (d)). Vtl = Vtr = 5 V, Vbl = Vbr = -5 V for the even configurations.

Vbl = Vtr = 5 V, Vtl = Vbr = -5 V for the odd configurations. VSi = 0 V. Vgraphene = 0 V.

dependence of Utop and Ubot approximates that of the top and bottom layer graphene

potential respectively across the junction. The bulk band gap Δ is approximately Utop-Ubot.

In the even configuration, both Utop and Ubot approach zero in the junction; the gap shrinks

but remains finite. As the comparison between devices 1 and 2 shows, a narrower split

(device 1) can maintain a large gap in the junction. In the odd configuration, Utop and Ubot

intersect at the zero potential line (through the paper), where the kink states are predicted

to exist. The finite width of the junction causes Utop and Ubot to change smoothly across the

junction. As our calculations show (Fig. 4-9), the smooth potential profile gives rise to

additional non-chiral states bound at the junction, the energy of which extends into the bulk

band gap. The energy proximity of the non-chiral states plays an importance role in the

backscattering of the kink states.

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We have also simulated the gating efficiency of the silicon backgate Si on the junction

by simulating the VSi dependent carrier density in the junction while the split gates are fixed

at voltage described in the caption of Fig. 4-9. Near the center of the junction, it is found

to be 3.5 × 1010 cm-2V-1 in device 1 and 4.8 × 1010 cm-2V-1 in device 2. Both are independent

of the voltages applied to the four split gates as expected from electrostatics. Both are

smaller than the conventional value of 6.9 × 1010 cm-2V-1, consistent with the junction

geometry. Using Si and a density of states value of 2.5 × 1013 cm-2 /eV for bilayer

graphene, we estimate that a range of 25 V on the silicon gate corresponds to an energy

range of ~ 35 meV, in good agreement with the value of the band gap at the D field

strength used in our experiments.

4.2.3 Device fabrication

The realization of the Martin proposal faces a number of challenges. It is

lithographically challenging to precisely match and align the four split gates shown in Fig.

4-7a. The suppression of mid-gap conduction due to disorder [47, 140] also requires high

quality BLG, necessitating hexagonal boron nitride (h-BN) encapsulation. In this work, we

have found ways to overcome these obstacles and present evidence of ballistic conduction

of the kink states. Figure 4-10a shows a false-color Scanning Electron Microscopy (SEM)

image of a device. The devices are made by sequentially stacking h-BN, BLG and h-BN

atop multi-layer graphene split bottom gates supported on a SiO2/doped Si substrate using

a polymer-based transfer method [66]. The SEM image in Fig. 4-10b highlights the

junction area. We can align the top and bottom splits to better than 10 nm in general

following the fabrication protocols described below.

98

Figure 4-10. (a) A false-color SEM image of a device similar to device 2. The bilayer graphene is

shaded and outlined in purple, the top gates and electrodes gold, the bottom multi-layer graphene

split gates black, and the top h-BN dielectric layer gray. The bottom h-BN layer extends beyond the

entire image. (b) A close-up view of the junction area from another device similar to device 2. The

junction is connected to four bilayer graphene electrodes and the measurements use a quasi-four-

terminal geometry as shown in the image to eliminate the electrode resistance. Alignment of the

gates is generally better than 10 nm.

The fabrication of the structure shown in Fig. 4-10a involves six steps as shown in Fig.

4-11. In step 1, we fabricate the bottom split gates made of multi-layer graphene exfoliated

from Kish graphite. Thin flakes (~2 nm) are exfoliated to SiO2 /doped Si substrates with

290 nm of thermal oxide. We use electron beam lithography (EBL) and oxygen plasma

etching (Plasma-Therm Versalock oxygen plasma 14 Watt power for ~30 seconds) to

pattern the bottom split gate. Resist ZEP 520a (300 μC/cm2 dose, developed in n-amyl

acetate, MIBK: IPA 8:1, and IPA at ~4 °C for 30 seconds each) is used as it provides better

resolution than PMMA. An example of a finished bottom split gate is shown in Figs 4-12a

and 4-12b. The sample is then annealed in Ar/H2 and/or treated in a very gentle oxygen

plasma (MetroLine M4L) to remove ZEP residue. In step 2, thin flakes of h-BN (15–30

99

Figure 4-11. A set of schematics showing fabrication procedures.

nm) and bilayer graphene are sequentially transferred to the bottom split gates using a

PMMA/PVA stamp[66]. In step 3, the bilayer graphene flake is shaped into a Hall bar with

leads using standard EBL and oxygen plasma etching (see Fig. 4-10a). In step 4, another

thin flake of h-BN is transferred atop the stack to cover the Hall bar but not the graphene

leads entirely. In step 5, standard EBL and metal deposition (5 nm Ti / 70 nm Au) is used

to make electrical contacts to the graphene leads. The resist is slightly over-developed to

ensure good ohmic contacts. In step 6, we pattern the top split gates using EBL and metal

deposition (5 nm Ti / 20 nm Au). We measure the width of each pair of bottom split gates

using SEM after ZEP removal. To obtain top split gates of matching width, we use

empirical relations between the designed width and the actual width after exposure and lift-

off, e.g. a split designed to be 100 nm comes out to be around 70 nm due to evaporation

angle and the proximity effect of the e-beam dose. Developing the top split gate pattern in

ice bathed developer at ~ 4 °C (with a higher e-beam dose of 450 μC/cm2) provides better

control of the development process and hence the dimension of the top split. Split widths

w down to 50 nm can be made this way, with higher yield for w > 70 nm due to occasional

shorting along the splits. An example of the top split gates is shown in Fig. 4-12c.

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Figure 4-12. (a) SEM image of a pair of bottom split gates made of multi-layer graphene. (b) Close-

up view of the split in (a). (c) SEM image of a pair of Au top split gates with a split size of 50 nm.

To ensure the high quality of the sample, we anneal the h-BN top surfaces in Ar/O2

(90/10%, 500 sccm) at 450 °C for 3 hours and the graphene surfaces in Ar/H2 (90/10%,

500 sccm) at 450 °C for 3 hours before each transfer to remove the polymer residue from

previous transfer or lithography [141]. The layer-by-layer transfer approach ensures that

both the top and bottom gates extend beyond the bilayer sheet, creating a uniform density

profile and ensuring the bulk of the bilayer becomes very insulating in the gapped regime.

We avoid aligning the line junction with a straight flake edge that may suggest either the

zigzag or the armchair orientation.

The alignment of the top and bottom splits is critical to the success of the experiment.

While relying on the same pre-patterned alignment markers (made by GCA 8000 stepper

in our case, squares with 20 μm on the side) suffices for most multi-step EBL fabrications

that require alignment, the misalignment between the two pairs of split gates can be up to

90 nm in random directions even when an advanced e-beam writer with precise stage

movement such as ours (Vistec EBPG5200) is used. The error primarily comes from the

imperfection of the alignment markers made by optical lithography. We have employed a

realignment procedure to address this issue. In step 1 of the lithography, dummy graphene

splits aligned with the bottom split gates were made (boxed area in Fig. 4-13a. In step 5

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Figure 4-13. (a) Optical micrograph of a pair of bottom split gates together with the dummy

graphene split (boxed area) aligned to its center. (b) Optical micrograph of the same area after the

fabrication is completed showing the metal splits patterned in step 5 (top three pairs) and step 6

(bottom two pairs) on top of the dummy graphene split (c) SEM micrograph of one metal split

patterned in step 5. Its center (yellow dashed line) is shifted to the right of the center of the dummy

graphene split (red dashed line) by 20 nm. (d) SEM micrograph of a metal split patterned in step 6

showing precise alignment to the dummy graphene split.

when we pattern the Ti/Au electrodes, we also pattern metal splits designed to overlap with

the dummy graphene splits as shown in Fig. 4-13b. In the same step, we write a second set

of alignment markers that is designed to track the center of the bottom split gates. As Fig.

4-13c shows, the metal splits are slightly shifted from the dummy graphene splits (20 nm

to the right in this case). In step 6 where we pattern the top split gates, we use the alignment

markers made by EBL in step 5, which has more precise dimensions and make corrections

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for the misalignment determined in Fig. 4-13c. As a final check, we make another pair of

dummy metal splits concurrently with the top split gates in step 6. This is shown in Fig. 4-

13d. It is indeed aligned with the dummy graphene split underneath. Using this procedure,

we can align the top and bottom split gates to better than 10 nm reliably. The width of the

top and bottom split gates can be matched to better than 15 nm.

4.2.4 Device characterization

Figure 4-14. (a) The bulk sheet resistance vs carrier density for device 1 (blue) and 2 (olive). The

gating efficiencies are determined from the quantum Hall effect. The inset shows the schematics

of both devices. Measurements are done on the left side of the devices by grounding the bottom

gate Vbl and sweeping the top gate Vtl. Vtr = Vbr = 0. The sheet resistance is calculated from R16, 78

(current from pin 1 to pin 6 and voltage probes 7 - 8) in both cases. (b) R16, 78 vs Vbl in device 1

showing fully resolved integer quantum Hall states at B = 8.9 T. Vtl / Vbl is swept simultaneously to

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maintain D = - 0.2 V/nm. Vtr = Vbr = 2V. Inset: similar measurement at B = 3 T, and D = 0 V/nm

showing the 8-fold degenerate N = 0 and 1 Landau levels of bilayer graphene. (c) R16, 78 vs Vtl at

selected Vbl from 1.2 V (leftmost curve) to -0.5 V (rightmost curve) decreasing in 0.1 V step. Open

circles are two-terminal CNP resistance R78, 78 (contact resistances are negligible compared to bulk

resistance) at Vbl = 1.7 V to 1.0 V (left) and at Vbl = -0.6 to -1.0 V (right). Also in 0.1 V step. The

arrow marks the global minimum. The inset shows the CNP resistance near the global minimum.

(d) The n = 0 line on the Vtl-Vbl plot obtained from the CNP positions in (c). The global minimum of

the CNP resistance in (c) corresponds to n = 0 and D = 0. Constant D lines are indicated in the

graph. Data in (b) - (d) are from device 1.

The experimental results discussed in this section are based on two devices. Their

junction widths w and lengths L are respectively w = 70 nm, L = 1 μm for device 1 and w

= 110 nm, L = 400 nm for device 2. Measurements of the bulk regions of the BLG yield

high carrier mobilities μ of 100,000 cm2V-1s-1 and 22,000 cm2V-1s-1 respectively for devices

1 and 2, in comparison with μ of a few thousand cm2V-1s-1 on oxide-supported samples [36,

47]. Figure 4-14a plots the sheet resistance vs. carrier density in the bilayer bulk for both

devices. The high quality of the devices is further proved in Fig. 4-14b, where the integer

quantum Hall states of device 1 are fully resolved at B = 8.9 T and D = -0.2 V/nm. We use

the quantum Hall effect to determine the gating efficiency of the top and bottom gates

accurately, the results of which are given in Table 4-2. The schematics of the devices are

Table 4-2. Device characteristics

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given in the inset of Fig. 4-14a. Figure 4-14c plots the left bulk resistance Rbulk (R16,78) vs

Vtl at various fixed Vbl in device 1. The CNP resistance starts to rise approximately

exponentially with increasing D at a small onset field of Don = 0.01 V/nm (0.07 V/nm for

device 2), suggesting small Coulomb potential fluctuations of a few meV caused by

electron-hole puddles[47]. Tracking the location of the CNP on the Vtl-Vbl plane allows us

to determine the n = 0 line, the D = 0, n = 0 point and constant D field lines, following the

definite of D in Ref.[47]. These are shown in Fig. 4-14d. We sweep gates simultaneously

to follow constant D field lines while keeping the bulk BLG at the CNP. The line junction

resistance is measured using a quasi-four-terminal geometry for device 2 (R3,9,410) and a

quasi-three-terminal geometry for device 1 (R39,310) (see inset of Fig. 4-14a). The resistance

of electrode 3 is estimated to be 3k and subtracted from the three-terminal data. We use

standard low current excitation lock-in techniques in the low impedance regime (< 1 M).

In the high impedance regime, we source a constant voltage and measure the current in a

two-terminal geometry. T = 1.6 K unless otherwise noted.

Figure 4-15. The bulk charge neutrality point resistance Rbulk CNP as a function of the applied

displacement field D for device 1 (solid symbols) and 2 (open symbols) in a semi-log plot (left axis).

105

Rbulk CNP rises much more rapidly with the increase of D compared to oxide-supported samples and

is larger than 10 MΩ in the range of measurements below. Also plotted on the right is the D-

dependent bulk band gap Δ obtained from temperature dependence measurements of a device

similar to device 2.

The high quality of the devices ensures insulating behavior of the bulk BLG when the

Fermi level EF resides inside the bulk band gap Δ which is a few tens of meV in our

experiment [142]. This is evident in Fig. 4-15, where we plot the displacement-field-

dependent bulk charge neutrality point (CNP) resistance Rbulk CNP. It is more than 10 MΩ

in the displacement field range of our measurements, rendering its contribution to the

measurement of the junction conductance negligible.

Figure 4-16. (a) and (b) Contacting schemes used in device 1 (a) and 2 (b). Regions of different

doping are colored on top of SEM images of similar devices. (c) dI/dV vs Vsd at an EF positioned in

the heavily hole doped continuum (black trace) and an EF positioned in the kink regime (red trace,

DL = +0.2 V/nm, DR = -0.2 V/nm and VSi = -52 V).

Next we discuss the contacting schemes of the kink states and extra access region

resistance. Figs. 4-16a and 4-16b show the two different schemes used to contact the kink

states in device 1 (Fig. 4-16a) and device 2 (Fig. 4-16b) respectively, overlaid on top of

SEM images of devices of each type. Two leads are patterned at each end of the junction,

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forming a Y-shaped connection. The leads are made of BLG and are contiguous with the

bulk BLG and the junction itself. However, region 1 (colored red) is heavily doped to n++

~ 2 × 1012 cm-2 because it is only gated by the back gates and the junction itself is at low

doping nj. In between the two an access region exists, where nac is in between n++ and nj.

In device 1, the top split gates have close to 90° degree bent at the end of the junction and

the change from nac to nj is relatively abrupt. We suspect such sharp transition may have

led to a junction and consequently a small interfacial resistance Rac that is on the order of

a few k. Figure 4-16c plots the differential conductance dI/dV of a highly conductive kink

state (~ 50 k) together with a heavily hole-doped continuum regime. Both show slight

non-linear I-V. The conductance change of the kink state corresponds to ~ 7 k. This may

have been caused by the interface. This resistance was not subtracted from any data

presented. Inspired by practices in studying quantum point contacts, the top split gates in

device 2 tapers gradually into the junction at a 45° angle to create a smoother transition

between nac and nj and a “funnel” effect.

4.2.5 Evidences of kink states

Figure 4-17. (a) The junction conductance σj as a function of Vsi at fixed values of DR from –0.4

V/nm to 0.4 V/nm. Upper panel: DL = –0.25 V/nm. Lower panel: DL = +0.25 V/nm. From device 1.

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The diagonal bands in the plots correspond to the CNP of the line junction. (b) σj vs. VSi along the

yellow dashed lines marked in the upper (blue curve) and lower (red curve) panels of (a). We

estimate the energy range of the bulk band gap Δ here corresponds to roughly 25 V on VSi. The

presence of the kink states in the (+ –) field configuration (red curve) gives rise to high conductance

inside the band gap while σj is low in the (– –) configuration (blue curve).

Figure 4-18. Junction resistance Rj at the CNP of the junction as a function of DL and DR in all four

field polarities showing systematically high resistances in the even configurations and low

resistances in the odd configurations.

Figures 4-17 and 4-18 present the experimental observations of the kink states in device

1. We measure the junction conductance σj = 1/Rj as a function of the silicon backgate

voltage VSi, which controls the Fermi level EF in the junction. This measurement is made

for a series of fixed DL and DR values of both polarities. As an example, Fig. 4-17a plots

the junction conductance as functions of DR and VSi at fixed DL = -0.25 V/nm (upper panel)

and +0.25 V/nm (lower panel). The diagonal features connecting the lower left to the upper

right corners of the panels correspond to the charge neutrality region of the line junction,

whose dependence on DR and DL is attributed to a slight misalignment between the top and

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bottom gates in device 1 (further explained in the following paragraph). It is immediately

clear from the data that σj near the CNP of the junction is high in the odd field configuration

(in white) but low in the even field configuration (in blue). This difference is further

illustrated in Fig. 4-17b, where we plot σj cut along the two yellow dashed lines drawn in

Fig. 4-17a. One can see that σj is high in both configurations when EF is outside the bulk

band gap. However, for EF inside the band gap, σj decreases to less than 1 μS in the even

configuration, but remains high in the range of 10 - 15 μS in the odd configuration,

indicating the presence of additional conducting channels. Such disparate behavior of σj in

the even and odd field configurations is systematically observed. To illustrate this contrast,

we plot in Fig. 4-18 the junction resistance Rj at the CNP of the junction only, taking data

from many panels similar to that shown in Fig. 4-17a and spanning all four polarities of DL

and DR. The clear contrast between the even, i.e. (+ +) and (– –), and odd, i.e. (+ –) and (–

+) quadrants of the graph strongly supports the existence of the kink states in the odd field

configuration, as expected by theory [40].

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Figure 4-19. (a) A illustration of the gate alignment situation in device 1. (b) - (d) Junction

conductance σj as a function of VSi and DR at fixed DL’s as labeled in the plots. The yellow dashed

lines are fits that track the CNP of the junction. (e) DL vs b using values obtained in (b) - (d).

A combination of slight misalignment and mismatch in the location and width of the

top and bottom split gates as illustrated in Fig. 4-19a can cause net doping of the junction

area as a function of changing DL and DR even when the bulk BLG on both sides is kept at

charge neutrality. Consequently the CNP of the junction appears at different VSi for

different DL / DR, giving rise to a slope on the DR-VSi plane.

Figures 4-19b – 4-19d show three more σj plots similar to that of Fig. 4-17a, with DL

fixed at values from -0.25 V/nm to +0.25 V/nm. , We draw three lines of the same slope

through the tip of the blue-colored region (i.e. resistance peak at low DR field in VSi sweep)

and require the lines to pass through the region of the highest resistances at large DR field

(this region corresponds to the black color in the figure but typically contains more than

one peak) in each plot. We manually assess the quality of the fits and adjust the slope until

a value that works satisfactorily for all three plots is obtained. This process yields three

linear fits 𝐷𝑅 = 𝑘(𝑉𝑆𝑖 + 𝑏(𝐷𝐿))with the same slope k = (+0.0066 ± 0.0002) /nm that is

robust from cool down to cool down. The parameter k represents the gating effect of the

right gates on the junction. A positive value indicates the top right gate extends into the

junction as illustrated in Fig. 4-19a. The parameter b represents the gating effect of the left

gates, the change of which with DL yields a slope of -0.0162, indicating the bottom left

gate extends into the junction, but not as much as the top right gate. This means the top

split is slightly narrower than the bottom split (70 nm) in device 1. We note that in device

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2, the kink regime appears at roughly the same range of VSi in spite of the change of DL

and DR, suggesting nearly perfect match of the split gates.

4.2.6 Backscattering mechanism of kink states

In general, Rj of the kink states is found to range from 40 to 100 kΩ, which corresponds

to a mean free path (MFP) of 𝐿𝑘 = 70–200 nm using the Landauer-Büttiker formula

𝑅𝑗 = 𝑅0(1 + 𝐿/𝐿0), (4-1)

where R0 = h/4e2 = 6.5 kΩ is the ballistic resistance limit of the 4-fold degenerate kink

states and L = 1 μm is the junction length in device 1 [143]. Although a small contact

resistance (several kΩ) may originate from the electrode/kink interface (see discussion in

Section 4.2.4), the large value of Rj indicates the presence of significant inter-valley

mixing. A MFP of L0 < 200 nm for the kink states is surprisingly short, given the cleanness

of our samples. In a recent experiment where Ju et al. observed evidence of conducting

domain wall modes at the domain boundary of AB and BA stacked BLG, the MFP of the

domain wall modes was found to be about 400 nm, in spite of a short bulk MFP of L2D ~ 8

nm [129]. The observation of L0 ≫ L2D is well understood, since L2D is dominated by intra-

valley scattering events caused by long-range Coulomb impurities. Backscattering between

the counter-propagating kink states, on the other hand, requires inter-valley scattering with

large momentum transfer, which should occur much less frequently, especially in h-BN

encapsulated clean samples [144]. In our high-quality devices, L2D is a few hundred nm,

yet L0 of the kink states is of similar order. This difference perhaps suggests different

backscattering mechanisms in the two systems. Lattice defects cannot account for our

observations; their rare occurrence is confirmed by the high mobility of the BLG bulk

111

[145]; the large wave function spread (Fig. 4-7b) also renders the kink states rather

insensitive to scattering by point defects [132].

Figure 4-20. (a) Band structures of the junction in device 1 (w = 70 nm) calculated using COMSOL-

simulated potential profiles shown in Fig. 1b. Only the K valley is shown. Non-chiral states bound

at the junction (blue) reside inside the bulk band gap marked by the green dashed lines. Δ = 30

meV. The kink states are shown in magenta. The gray line corresponds to quantum valley-Hall

edge states at the zigzag boundary of the numerical setup, which do not survive edge disorder in

realistic samples. (b) Junction conductance σ vs length L calculated at EF = 0 (black), 5 (red), and

14 meV (blue) as marked by the dashed lines in (a). The disorder strength is chosen to be W = 0.6

eV. One non-chiral state is assumed to contribute conductance 4 e2/h at L = 0. Over 30 samples

are averaged for each data point. Error bars are smaller than the symbol size. Fits to Eq. 4-1 yield

MFP of 266, 223, and 141 nm, respectively. The proximity to non-chiral states leads to enhanced

backscattering. (c) An illustration of inter-valley scattering between the kink states of K and K’

valleys. A kink state may be directly scattered to a different valley or scattered via coupling to non-

112

chiral states. Non-chiral states can also form quantum dots due to Coulomb potential fluctuations

and co-exist with the kink states over a large energy range, as shown schematically.

Although the exact nature of the backscattering mechanisms is unknown at the moment,

we note several features of our experiments that may play a role. Our band structure

calculations show that a smoothly varying electrostatic potential profile such as the one

shown in Fig. 4-7b supports, in addition to the kink states which are chiral in each valley,

non-chiral one-dimensional states whose energies protrude into the bulk band gap Δ, as

shown in Fig. 4-20a [132, 133]. These states are bound by the width of the junction but

delocalized along its length. Their presence effectively reduces the size of the bulk gap Δ,

which is a few tens of meV in our experiment. Coulomb potential fluctuations, as well as

geometrical variations of the lithographically defined junction, can potentially lower the

energy of the non-chiral states further, resulting in their presence even at EF close to the

CNP of the junction, likely in the form of quantum dots. This situation is illustrated in Fig.

4-20c and worsens when the junction width w is large. Co-existing non-chiral states, either

extended or localized, can provide additional inter-valley scattering paths. The formation

of quantum dots by non-chiral states can provide an explanation for the junction

conductance oscillations seen in the traces shown in Fig. 4-17b. The coupling to non-chiral

quantum dots allows the kink states from different valleys to mix via multi-particle

processes. Several such mechanisms have been put forward to explain the backscattering

of the quantum spin hall edge states, including the possibility of a Kondo effect [146-149].

These mechanisms may be of relevance here and will be the subject of future studies.

Next we further discuss the conductance oscillations, their I-V characteristics and the

temperature dependence of the kink states. A general feature of the kink regime is the

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appearance of conductance oscillations on the order of a few μS superimposed onto a

smooth σj background. One example is shown in Fig. 4-21a. Such oscillations are also

Figure 4-21. (a) σj (solid black curve) vs VSi. DL = + 0.20 V/nm and DR = - 0.20 V/nm. Blue and pink

lines are guide to the eye with periods of 5.4 V and 3.2 V respectively. (b) Differential conductance

map dI/dV of the same regime as in (a). (c) Temperature-dependent junction resistance Rj at VSi

marked in (b). Also plotted is the T-dependent resistance of the gapped bulk bilayer at the CNP.

From Device 1.

present in the even field configurations (see Fig. 4-17b). The oscillations typically consist

of two or three prominent periods in VSi, as illustrated in Fig. 4-21a. Differential

conductance map dI/dV shown in Fig. 4-21b reveals non-linear I-V at low-conductance

points with energy scales of a few meV. At the first glance, they resemble coherent

conductance oscillations of one-dimensional states in carbon nanotubes with semi-

transparent contacts [150], however the details of the oscillations , such as their magnitude

and period, vary among different kink regimes and sometimes between cool-downs for the

same regime. This variation makes it unlikely that the oscillations originate from the

confinement of the kink states themselves. Instead, they can be explained naturally by

considering the parallel conduction of localized charge puddles, i.e. quantum dots of the

non-chiral states formed due to Coulomb potential fluctuations. In Fig. 4-21c, we plot the

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temperature dependence of the junction resistance Rj at selected VSi’s as marked in Fig. 4-

21b. States with more non-linear IV characteristics also exhibit stronger temperature

dependence. At T < 10 K, the T-dependence of the nearly linear kink states is very weak.

Above 10 K, the T-dependence of the kink states is difficult to assess due to the onset of

parallel conduction through the gapped bulk. More studies are required to understand the

role of electron-electron interaction [151] and other predicted temperature dependences

[146, 148, 149].

We also perform numerical studies of the junction conductance using the Landauer-

Büttiker formula and the Green’s function method with on-site Anderson disorder in the

energy range of [-W/2, W/2], where W measures the disorder strength. Although the

Anderson disorder may not reflect all potential valley-mixing mechanisms in experimental

samples, it provides an efficient means to model the conductance loss from inter-valley

scattering and allows us to examine the effect of the controlling factors of the experiment

such as the junction width w, the bulk gap size Δ, and the Fermi level EF of the junction. A

few key findings are highlighted in Fig. 4-20. Figure 4-20a plots the calculated band

structure of device 1 with a bulk band gap Δ = 30 meV. Δ is effectively reduced to ~21

meV due to the presence of the non-chiral states. Figure 4-20b plots three length-dependent

junction conductance σ (L) corresponding to different Fermi levels E1 = 0, E2 = 5 and E3 =

14 meV (dashed lines indicated in Fig. 4-20a). Fitting to Eq. 4-1 yields the MFP L0 of the

kink states, which is 266, 210, and 141 nm respectively for EF = 0, 5, and 14 meV. We

attribute the trend of decreasing conductance σ with increasing EF to the availability of

more inter-valley scattering paths that involve the non-chiral states. The scattering

probability increases as EF is close to or resonates with the energy of an extended or

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localized non-chiral state. This situation is illustrated in Fig. 4-20c. Numerically, we find

that L0 increases with increasing band gap size Δ and decreases with increasing junction

width w. These findings point to effective ways of further improving the conductance of

the kink states.

4.2.7 Towards ballistic kink states in a magnetic field

In device 1, the junction area is connected to three working electrodes as shown in the

inset of Fig. 4-14a. We perform two-terminal resistance measurements R39,39, three-

terminal measurements R39,310 and attribute the difference of the two to the resistance of

electrode R9. This value is used to approximate the resistance of electrode R3, which has

similar aspect ratio. R3 is subtracted from the three-terminal data R39,310 to obtain a four-

terminal reading. As can be seen in Fig. 4-22, R9 is only ~3 kΩ at B = 0, which is a small

Figure 4-22. Magnetoresistance of the kink state in device 1. DL = +0.2 V/nm and DR = -0.3 V/nm,

VSi = -55.5 V. The inset shows the device schematic. R39,310 - R9 is shown in Fig. 4-24a.

fraction of the measured Rj (several tens of kΩ). It however increases significantly with

increasing B-field, reaching ~19 kΩ at B = 7 T. Consequently, the uncertainty in the

approximation of R3 becomes increasingly important. We suspect that a slight increase of

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

Rj (

k

)

B (T)

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Rkink in device 1 above 7 T may be due to this subtraction procedure. This behavior is not

observed in device 2, where the junction resistance is measured using all four electrodes.

The resistance of the kink states in device 2 Rkink is obtained through a two-channel

model as described below. Figures 4-23a and 4-23b plot the COMSOL-simulated potential

profile of the even and odd configurations. Due to the finite width of the junction (w = 110

nm), the bulk gap is reduced to ʹ ~ 1/3 in the junction area in the even configuration.

As Fig. 4-23c shows, at B = 0, non-chiral states exist in both configurations and their band

structures are nearly identical. Near EF = 0, the kink states conduct in parallel with hopping

conduction through charge puddles induced by Coulomb potential fluctuations of the non-

chiral states, i.e.

σj, total = σkink + σpara. (4-2)

Here, σpara is well approximated by the junction conductance in the even configuration σeven.

As Fig. 4-24a shows, at B = 0 σeven is significant in device 2 and as a result, σj, total is below

h/4e2 in the kink regime. As the magnetic field increases, the energies of the non-chiral

states continue to move up in the odd configuration, reaching above the bulk gap at 4 T.

This evolution should reduce the number of puddles co-exiting with the kink states. In

addition, a magnetic field suppresses hopping conduction by localizing carriers. Thus we

expect σpara due to the non-chiral states to rapidly decrease with increasing B, making the

kink state conduction increasingly dominant.

The magnetic field dependence of the band structure in the even configuration is quite

interesting. As Figs. 4-23c – 4-23f show, all non-chiral states in the even configuration

behave similarly to those of the odd configuration except for one pair of states, the energies

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of which remain near ±1/2 ʹ (The state resides at 1/2 ʹ in the K valley and -1/2 ʹ in the

Kʹ valley. Kʹ valley is not shown here). These states roughly correspond to the N = 0 and 1

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Figure 4-23. The kink state in a magnetic field. (a) and (b) COMSOL simulated potential profile of

the even (a) and odd (b) field configurations in device 2. w = 110 nm. The light blue shade mark

the bulk gap Δ (30 meV in (c)-(f)) and the light green shade mark the reduced gap Δ’ in all figures.

(c) to (f) The band structure of the junction at B = 0, 2, 4 and 8 T respectively for the even and odd

field configurations as labeled. Only K valley is shown. The non-chiral states below the bulk gap

are colored blue. The kink states magenta. The N = 0 and 1 LLs of the even configurations are

colored green. The quantum Hall edge states and the zigzag edge states of the system are colored

gray. (g) and (h) Measured σj of the even (blue) and odd (black) bias configurations. The red curves

are fits of σj in the odd configurations using σj, total = σkink + σeven. Rkink = 1 / σkink obtained from each

fit is indicated in the plot.

Landau levels of the junction area with reduced band gap ʹ. Experimentally we see Rj of

the even configuration also rapidly increases with increasing B (Inset of Fig. 4-24a). But

because of these low-lying LLs, we expect the non-chiral states in the even configuration

to have higher conductance than the non-chiral states in the odd configuration, i.e. σeven >

σpara. Thus, approximating σpara with σeven in Eq. 4-2 yields a lower bound of σkink, or higher

bound of Rkink. This approximation is expected to become more accurate as B approaches

zero. Figures 4-23g and 4-23h give two examples of how we use Eq. 4-2 to obtain σkink.

The two-channel model works remarkably well. We are able to fit σj, total of the odd

configuration in the entire VSi range using a single value of σkink and the measured σeven.

The resulting Rkink = 1 / σkink are plotted in Fig. 4-24a as open squares. They represent Rkink

accurately near B = 0 and provide an upper bound of Rkink at B > 0 whereas the raw data

provide a lower bound of Rkink.

We note that for B > 8 T, both the calculated Rkink and the raw Rj merge. This is the

regime where the parallel conduction of the non-chiral states becomes negligible compares

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to h/4e2 (see inset of Fig. 4-24a) and the kink states dominate the junction conductance. In

this regime, we show that Rkink is close to h/4e2 over a range of VSi and is independent of

the bulk gap size (Fig. 4-24b). This observation strongly supports the nearly ballistic

behavior of the kink states.

Figure 4-24. (a) Two representative magnetoresistance traces from device 1. DL = +0.2 V/nm and

DR = -0.3 V/nm, VSi = -55.5 V and -39.3 V for the blue and light blue curve respectively). Symbols

are from Device 2. The solid triangles are raw magnetoresistance data of Rj in the (+ -) field

configuration. Rj can be smaller than h/4e2 due to parallel conduction of the non-chiral states. We

use a two-channel model to estimate the resistance of the kink states, using Rj of the (+ +) field

configuration shown in the inset to approximate the resistance of the parallel channel. The open

squares in (a) plot the estimated resistance of the kink states. At large magnetic field, the non-chiral

states become sufficiently insulating that the raw Rj measures directly the kink state resistance.

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Black symbols: T = 1.6 K. Red symbols: T = 310 mK in a separate cool-down. |DL| = |DR| = 0.3

V/nm. (b) Rj versus VSi in device 2 at B = 8 T for all four field configurations as marked in the plot.

From dark to light colors: |DL| = |DR| = 0.5, 0.4 and 0.3 V/nm. Inset: Potential profile for even (red)

and odd (black) field configurations. (c) The band structure of device 1 shown in Fig. 4-20a

recalculated at B = 6 T. The olive dashed lines mark the edges of the bulk conduction and valance

bands in Fig. 4-20a. Non-chiral states residing below the band edges are now lifted to higher

energies. The kink states are shown in magenta and the quantum valley Hall edge states are in

gray (not relevant in realistic samples). (d) The calculated magneto-conductance for device 1. See

Fig. 4-20b for parameters used in the calculation. EF = 5 meV. Inset: Wave functions of the K (blue)

and K’ (red) valley kink states at B = 6 T showing a spatial separation of 14 nm due to the Lorentz

force. The wave function separation is zero at EF = 0 and increases with increasing EF. It also

increases with increasing B.

Next we demonstrate that the kink states can approach the ballistic conductance limit

of 4 e2/h in the presence of a perpendicular magnetic field B. As Fig. 4-24a shows, the

resistance of the kink states decreases rapidly in the presence of a perpendicular magnetic

field in both devices. Rj reaches about 15 kΩ in device 1 at B ~ 7 T, suggesting a MFP of

approximately L0 ~ 0.8 μm. This is a factor of 4 - 10 increase compared to L0 at B = 0.

More remarkably, in device 2 Rj reaches close to the ballistic limit of h/4e2 = 6.5 kΩ in the

field range of 8 to 14 T, suggesting an L0 that is much longer than 0.4 μm. As an example,

Fig. 4-24b plots Rj of all four field polarities at B = 8 T. Rj is approximately 6.8 kΩ and 8.0

kΩ respectively for the (+ –) and (– +) field configurations. Rj is also insensitive to the

magnitude of the displacement field D, which varies from 0.3 to 0.5 V/nm in the

measurement, suggesting the attainment of steady states. Rj decreases slightly but remains

close to the ballistic limit when B increases to 14 T. In stark contrast, resistances of

hundreds to thousands of kΩ are observed in the even field configurations, as shown in

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Fig. 4-24b. These observations clearly attest to the presence of nearly ballistic conducting

channels in the odd electric field configurations.

The observed magneto-resistance of the kink states is satisfactorily reproduced by our

numerical studies, as shown in Figs. 4-24c and 4-24d. Two potential effects of the applied

magnetic field may play important roles. As Fig. 4-24c displays, the formation of Landau

levels lifts the non-chiral states away from the energy range of the bulk gap/kink states,

which should also reduce the number of co-existing quantum dots. These changes will

reduce the probability of non-chiral state mediated backscattering. In addition, the Lorentz

forces exerted on the counter-propagating kink states by the magnetic field cause their

wave functions to spatially separate towards opposite sides of the junction, as illustrated in

the Inset of Fig. 4-24d. This separation is zero at EF = 0 but widens with increasing EF and

is more pronounced in a wide junction [133, 152]. It also increases with increasing B and

can reach a size comparable to the spread of the kink state wave functions at moderate field

(see Fig. 4-24d). The physical separation of the counter-propagating kink states resembles

what occurs in the conventional quantum Hall effect and strongly diminishes any type of

inter-valley scatterings or potential valley coherence at B = 0 [146-149], thus resulting in a

robust topological protection to the kink states.

4.2.8 Section conclusion

What we have demonstrated in this section – the creation of one-dimensional

topological conducting channels in bilayer graphene using electrical control – opens up

exciting new avenues of implementing valley-controlled valves, beam splitters, and

waveguides to control electron flow in high-quality atomically thin materials [132, 135].

Narrower junctions combined with the use of a larger band gap can potentially enable the

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kink valleytronic devices to operate at non-cryogenic temperatures. The dual-split-gate

structure demonstrated here will enable the realization of potential-controlled few-electron

quantum dots [153] and open the door to the exploration of the fascinating edge and domain

wall physics of the bilayer graphene quantum Hall regime [152, 154].

4.3 Ballistic kink states and a valleytronic device in bilayer graphene

4.3.1 Motivation for building a valleytronic device

Overcoming the limit of the Moore’s law necessitates the exploration of new paradigms

of electronics using alternative state variables other than charge, and new material

platforms beyond traditional semiconductors. The advent of two-dimensional layered

materials such as graphene and transition metal dichalcogenides has inspired the concept

of a new type of devices, the operation of which exploits the valley degrees of freedom in

materials with hexagonal symmetry [38, 120, 125, 138]. Experiments have shown that a

net valley polarization can be created by current [39, 126, 139] or optical excitation [127].

However the realization of valleytronic devices remains challenging. In the previous

section, we have shown that valley-momentum locked topological conduction channels can

be created in bilayer graphene using a scalable, gate-defined structure [112]. This

construction enables the possibility of an all-electric valley valve [132], which controls the

transmission of ballistic current flow in the topological channels by controlling their valley

indices using gates, in analogy to the action of a Datta-Das spin valve [155]. In this section

we experimentally demonstrate the realization of the valley valve with a transmission

on/off ratio of 8 at zero magnetic field and more than 100 at several Teslas. We further

demonstrate the operation of the device as a full-ranged tunable beam splitter and a

reconfigurable waveguide for coherent electron flow. These successful operations serve as

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building blocks of an electron optical network. The ability to route ballistic current flow in

situ is a significant step forward towards employing topological wires as interconnects for

ultra-low power electronics.


In bilayer graphene, a perpendicular electric field applied through a pair of top and

bottom gates breaks the symmetry of the two constituent layers and opens a gap in its

band structure [43, 47]. Previously we have demonstrated that this gap can be inverted

using two opposite electric fields and along the zero gap line emerges metallic, helical,

quantum valley Hall kink states (kink states for short) [112]. Topological in origin, the kink

states are chiral in each valley with opposite chiralities, i.e. group velocities, in K and Kʹ (-

K), as illustrated in Fig. 4-25a. They are immune from backscattering in the absence of

valley-mixing scatterings and are expected to carry current ballistically over a long distance

Figure 4-25. (a) illustrates the creation of the valley-momentum locked kink states at the junction

of two oppositely gapped bilayer graphene regions. D > 0 corresponds to a positive voltage on the

bottom gate. States from different valleys have opposite chiralities. Including spin and layer isospin,

there are four modes in each valley. The magenta curve is the calculated wave function distribution

of the kink states in a 70-nm wide junction9. States in K and Kʹ overlap in space. (b) Schematics of

our quad-split-gated valley router device. The four graphite bottom gates are gray. The four top

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gates are gold. The blue sheet and dashed lines represent the bilayer graphene sheet. The global

Si backgate is light green. The four gold arcs are Cr/Au side contacts.

without dissipation [42, 99, 128, 129, 131]. Harnessing the power of ballistic edge

transport, such as that realized in quantum Hall, quantum spin Hall [156, 157] and quantum

anomalous Hall [158] systems, could be a promising pathway towards the realization of

electronics with ultra-low power consumptions. The ability to manipulate the current path

in situ will be a key asset to the development of such circuits. So far, this is only possible

for the quantum Hall edge states, the creation of which requires a strong magnetic field.

The valley-dependent chirality of the kink states in bilayer graphene offers an elegant

alternative that can be magnetic field free. Figure 4-25a illustrates the creation of kink

states in bilayer graphene through asymmetric gapping [99, 112]. The shown chiralities

correspond to the (- +) gating configuration. A (+ -) configuration simultaneously flips the

chirality of the kink states in both valleys. The existence of two flavors (helicities) is a

unique attribute of the kink states that can be exploited to for device functionalities. Indeed,

the potential to control the transmission of the kink states in a “valley router” setup shown

in Fig. 4-25b has been discussed theoretically [132]. In this work, we report on the

experimental realization of the valley router and demonstrate its operations as a

reconfigurable waveguide, a valley valve and an electron beam splitter. Both the

waveguiding and the valley valve effects function well without a magnetic field. Our work

represents the first proof-of-concept demonstration of a valleytronic device.

4.3.3 Device fabrication and characterization

We first exfoliate multi-layer graphite (Kish graphite) sheets to heavily doped Si/SiO2

substrate then use standard e-beam lithography and oxygen plasma etching to pattern a

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Figure 4-26. (a) A scanning electron microscopy (SEM) image of a bottom graphite split gate

structure. (b) An optical micrograph of device 1 before the top gates are made. The dashed lines

outline the edges of the bilayer graphene sheet. Scale bar is 5 μm in both images.

series of bottom split gates as shown in Fig. 4-26a. The width of the splits is 70 nm in both

directions. We then use a van der Waals dry transfer technique [26] to make a h-BN/bilayer

graphene/h-BN stack. The stack is then transferred to the bottom split gate structure. We

intentionally misalign the crystallographic edges of the bilayer sheet with the directions of

the splits to ensure no channel is along an armchair direction. Then, the sample is annealed

in an Ar/ H2 (90%/10%) atmosphere at 450 °C for 3 hours to reduce bubbles introduced by

the transfer process. We then examine the covered intersections in an atomic force

microscope (AFM) to identify clean and bubble-free candidate for device fabrication.

Following Ref. [66], we use a negative-tone resist hydrogen silsesquioxane (HSQ) as the

etching mask, and use reactive ion etching (CHF3/O2 at 10:1 ratio) to define the device area

shown in Fig. 4-26b. We then pattern and deposit the Cr/Au side contacts and electrodes

that will connect to the top gates. Lastly, we pattern and deposit the four Ti/Au top gates.

Using an alignment procedure detailed in Ref. [112], we can match the width and position

of the top and bottom splits to a few nm in precision.

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Figure 4-27. (a) Schematics of our quad-split-gated valley router device. The magenta cross

represents the four kink channels. Each is 70 nm wide and 300 nm long. (b) An optical image of

device 1. (c) A false colored scanning electron micrograph of the central region taken on another

device. Scale bars in (b) and (c) are 3 μm and 100 nm respectively.

Figures 4-27a - c show schematics, optical and scanning electron micrographs of our

four-terminal valley router device, which consists of four pairs of split top and bottom gates

and a global Si backgate. The aligned edges of the eight gates define the four kink channels

shown in magenta in Fig. 4-27a. The dual-gated region (orange areas in Fig. 4-27a) in each

quadrant is gapped and placed at the charge neutrality point (CNP) determined using

methods detailed in the following paragraphs.

We employ the following steps to determine the gating efficiencies of the top and

bottom gates and the gate voltages that correspond to the charge neutrality points (CNPs)

of the dual-gated area in each quadrant. We first obtain an approximate Vtg - Vbg relation

for the CNPs of all four quadrants by shorting all top gates together and all bottom gates

together and perform two-terminal resistance measurements as a function of sweeping Vtg

at fixed Vbg’s, similar to measurements shown in the Fig. S5c in Ref. [112]. The resistance

maxima are identified as the CNPs. We then repeat the same measurement for each

quadrant by tuning its top and bottom gates (e.g. measure R23 while changing VNWt and

VNWb) while setting the other three quadrants at their CNPs so that they are insulating. This

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procedure allows us to find the precise gate voltages of the CNPs for each quadrant. They

differ slightly from each other due to slightly different environmental doping in different

parts of the device while the gating efficiency ratio is found to be 1:1.8 for all our top and

bottom gates. The voltage on the doped Si gate is fixed at VSi = 50 V in the measurements

above so that the channels are conducting in parallel but do not affect the determination of

the gate voltages of the CNPs. Figure 4-28 plots the Vtg - Vbg relation for the CNP of the

northwest quadrant as an example.

Figure 4-28. Vtg - Vbg relation for the charge neutrality point of the northwest quadrant. Red line is

a linear fit with a slope of -1.8.

To determine the gating efficiencies, we measure the magneto-resistance of the

northwest quadrant R23 as a function of VNWt while fixing the magnetic field B = 2 T and

VNWb = 1 V. The other three quadrants are set at their CNPs so that they are insulating. We

calculate the gating efficiency from the quantum oscillations and obtain 6.13 × 1011 cm-2V-

1 for the top gates in device 1. This value is in good agreement with the measured top h-

BN thickness of 27nm and the dielectric constant ε = 3 for our h-BN. The gate efficiency

of the bottom gates is calculated to be 1.1 × 1012 cm-2V-1. In our measurements, gating

configurations are labeled by voltages applied to the bottom gates while voltages on the

top gates are set accordingly to keep the dual-gated areas at their respective CNPs. In

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5

1

2

3

4

5

6

V

tg (

V)

Vbg

(V)

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addition, we set the magnitudes of the bottom gate voltages to be the same in all four

quadrants, e.g. ± 3V.

4.3.4 Ballistic kink states

Figure 4-29. (a) A band diagram of the channel in magnetic field. See Ref.9 for calculations. (b)

Resistance of the east channel as a function of the Si backgate voltage RE (VSi) at different magnetic

fields as labeled in the graph. As the inset shows, RE is obtained by measuring R24 while doping

the left quadrants heavily so that R24 = RE + Rc. Rc ~ 800 W. (c) Resistance of the kink state as a

function of the magnetic field Rkink (B) in the east (black) and south (blue) channels of device 1 and

south (red) channel of device 2. The right axis labels the corresponding transmission coefficient

τkink. Throughout our experiments, the bottom gates are set to ± 3 V with the polarity given in

diagrams unless otherwise specified. The top gates are set accordingly to place the dual-gated

regions at the CNP.

We first measure the resistance of each kink channel Rkink separately. As an example,

Fig. 4-29b plots the resistance of the east channel RE as a function of VSi, which controls

the Fermi level EF in the channel, at fixed magnetic fields B = 0 to 8 T. RE exhibits a broad

peak at B = 0, which evolves into a wide plateau as B increases and saturates at about 7.3

kΩ. This plateaued region corresponds to the gapped regime of the channel, where the kink

states reside. Its resistance value of 7.3 kΩ is a sum of the ballistic resistance of the kink

129

states, i.e. h/4e2 = 6.5 kΩ and contact resistance Rc ~ 800 Ω (discussed in the following

paragraphs). Additional plateaus outside the kink regime correspond to the sequential

addition of 4-fold degenerate quantum Hall edge states in the channel. The devices studied

here are of considerably higher quality than those reported in our previous work [112]. We

attribute the improvement to cleaner interfaces achieved by the van der Waals dry transfer

method [26].

Figure 4-30a shows a schematic of how the kink states are contacted. Each channel is

connected to the Cr/Au side contact through an access region, which is gated by two bottom

gates and the Si backgate. The contact resistance Rc combines the resistance of the

metal/graphene interface and the resistance of the bilayer graphene access region.

Figure 4-30b shows measurements of the south channel Rs at magnetic fields B = 0 to

8T. The resistance plateaus correspond to an integer number of edge states (kink + quantum

Hall) contributing to the conduction, as labeled in Fig. 4-30b. Away from VSi = 0 V, their

resistances are well described by R = h/ne2 + Rc, where the index n is the number of edge

sates. Fig. 4-30c plots the resistance of the four plateaus from the hole side of the B = 4 T

curve, as a function of 1/n. A linear fit through the data indicates Rc is roughly a constant

for all four plateaus and Rc = 1140 Ω. This corresponds to ~ 600 Ω per contact. We attribute

the low contact resistance to the large width of the metal/graphene interface and the heavy

doping of the access region. We also see that the well-developed plateaus remain at the

same value as B increases. This shows that Rc is field independent.

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Figure 4-30. (a) A schematic of the contacting scheme. (b) R13 (VSi) at selected B-fields from 0 T to

8 T measuring the resistance of the south channel. The measurement scheme is shown in the

inset, where the north quadrants are heavily doped and the south quadrants are gapped with the

bottom gate voltage (VSWb, VSEb) set at (-2 V, +2 V). (c) R13 vs the plateau index 1/n. Points are

taken from the B = 4 T trace on the hole side. A linear fit (red line) to the data yields an intercept Rc

= 1140 Ω. (d) Bottom: Resistance of the kink states Rkink + Rc for the south (magenta) and east

(black) channels in device 1 and the south channel (blue) in device 2. Black and blue curves are

shifted in x and y to overlap with the magenta curve. B = 6 T. Top: R13 measured with the top and

bottom gates set to 0 V. The change reflects the VSi -dependence of the bilayer graphene channel,

which includes the access region. Its profile strongly resembles that of the magenta curve in the

131

kink regime. This led us to conclude that the VSi – dependence of the magenta curve in this regime

is due to Rc.

Because a portion of the access region (the 70 nm-wide channel) is controlled by the

Si backgate, Rc can exhibit a noticeable dependence on VSi and this is included in the

measurement of the channel resistance. Away from VSi = 0, this dependence manifests as

an electron-hole asymmetry, as shown by the systematically lower R13 values in Fig. 4-30b

on the electron side. Near VSi = 0, Rc may increase by a few hundred Ω. This also leads to

an increase in the measured R13, as shown in Fig. 4-30b. To examine the impact of Rc, Fig.

4-30d plots the measured resistance of several kink channels from two devices together. Rc

is nearly a constant in the entire range of VSi for the blue curve (south channel of device 2).

This leads to well quantized resistance plateaus for the kink states and the quantum Hall

states, with the difference between the two very close to h/4e2. The magenta curve (south

channel in device 1), on the other hand, exhibits a maximum near VSi = 0 V. By measuring

the resistance of the bilayer graphene channel from contacts 3 to 1 independently and

comparing its shape to that of the magenta curve, we conclude that the south channel most

likely remains quantized in the entire kink regime while Rc (VSi) accounts for the excess

over h/4e2. As a third example, we show that the black curve (east channel in device 1)

exhibits a slowly decreasing Rc, resulting in a tilted plateau in the kink regime.

However, the large width of the side contacts (golden arcs in Fig. 4-27b) and the finger-

like top gates enhance parallel conduction through the dual-gated regions. Their resistance

Rpara is non-negligible at small magnetic fields and this is why RE is less than h/4e2 at B =

0 and 2 T in Fig. 4-29b. Using a two-channel model developed in Ref. [112], we measure

Rpara independently and extract RE using 𝑅𝐸 =(𝑅24−𝑅𝐶)𝑅𝑝𝑎𝑟𝑎

𝑅𝑝𝑎𝑟𝑎−(𝑅24−𝑅𝐶)(discussed in the following

132

paragraphs). Rpara increases with B rapidly, the effect of which becomes negligible at 4 T.

This leads to the observed saturation of R24 = h/4e2 + Rc in Fig. 4-29b.

Figure 4-31. (a) A schematic of the two-channel model. (b) and (c), R13 (VSi) at selected B-fields

from 0 T to 8 T. The color scheme follows that of Fig. S3b. Insets show the measurement schemes

and illustrate the parallel conduction paths through the gapped quadrants. Rpara remains the same

in (b) and (c) while the kink states are absent in (c). We use Rpara measured in (c), R13 measured

in (b), together with Rc = (1440 ± 70) Ω to calculate Rkink of the south channel at VSi = -11 V (red

arrow). (VSWb, VSEb) = (+3 V, -3 V) in (b) and (-3 V, -3 V) in (c). From device 1.

In data shown in Fig. 4-29b, the kink state resistance drops below h/4e2 at B = 0 and

2T. This is due to finite parallel conduction. We use a two-channel model first established

in Ref. [112] and illustrated in Fig. 4-31a to calculate the kink state resistance Rkink, and

the transmission coefficient 𝜏𝑘𝑖𝑛𝑘 = ℎ/4𝑒2

𝑅𝑘𝑖𝑛𝑘. Rpapa represents the resistance of the parallel

conduction paths. Rpara can be simulated by an “even” field configuration as shown in the

inset of Fig. 4-31c. Using data shown in Figs. 4-31b and 4-31c, we can extract the resistance

of the kink states at VSi = -11 V (red arrow in 4b) using 𝑅𝑘𝑖𝑛𝑘 =(𝑅13−𝑅𝐶)𝑅𝑝𝑎𝑟𝑎

𝑅𝑝𝑎𝑟𝑎−(𝑅13−𝑅𝐶). Here Rc =

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R13 - h/4e2 = (1440 ± 70) Ω is the contact resistance. Similar analyses are done for two

other channels and the results are plotted in Fig. 4-29c.

Similar measurements and analyses were performed in two devices. Fig. 4-29c plots

several extracted Rkink (B). At B = 0, Rkink is approximately 7000 Ω for our 300 nm-long

channels, which corresponds to a transmission coefficient τkink = h/4e2 / Rkink = 0.92 and a

mean free path of 3.5 μm [112]. This is on par with that of the quantum spin Hall edge

states [156, 157] and affirms the topological protection provided by the valley-momentum

locking of the kink states. Since spin plays no role in the origin of the kink states, the

magnetic field becomes a versatile control knob in our experiments. It is effective in

suppressing both parallel bulk conduction and weak backscattering of the kink states at B

= 0. The former is due to increased difficulty of hopping conduction in a magnetic field

while the latter is attributed to reduced coupling to non-chiral states as they are lifted away

from the kink states in a magnetic field [112].

4.3.5 Waveguide operations

Figure 4-32 demonstrates the operation of the valley router as a reconfigurable

waveguide for the kink states. Figures 4-32a - 4-32c show three configurations of the

waveguide, which we label as “through”, “right turn” and “left turn”, respectively. In all

three, the kink states only exist in two of the four channels and the chirality in each valley

is preserved along the paths. In measurements described in Fig. 4-32, we keep the same

gate voltages on the bottom gates when measuring the north channel, the south channel and

the “through” configuration and use the top gates to control doping. This strategy ensures

the contact resistance included in R13 is roughly the same for RN, RS and Rthrough. This is

indeed the case as the data in Fig. 4-32d shows where the measured “through” resistance

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R13, together with the resistance of each individual kink channel RN and RS. It is

immediately clear that the transmission through the intersection region is ballistic since all

three curves overlap in the kink regime. As shown in Fig. 4-32e, similar ballistic

transmission is also observed in the two 90 bends, consistent with the results of numerical

simulations [132, 135]. This four-terminal device thus serves as an in situ reconfigurable

electronic waveguide of the kink states.

Figure 4-32. (a) – (c) illustrate the “through”, “right turn” and “left turn” configurations of the

waveguide respectively. (d) Two-terminal resistance R13 (VSi) corresponding to the resistance of

the north channel RN (blue), the south channel RS (red) and the “through” configuration (black),

respectively. The overlap of all three indicates ballistic transmission through the intersection region

in (a), i.e. τi = 1. B = 6 T. (e) R34 (orange) and R14 (olive) as a function of VSi in the configurations

shown in (b) and (c) respectively. B = 5 T. The kink states are also transmitted through the bends

ballistically. The variation of R in the kink regime is likely due to the change of Rc. (f) The

transmission coefficient τi of the intersection region in (a) as a function of the magnetic field. τi is

determined using the model shown in the inset.

135

Deviation from perfect transmission starts to occur as the magnetic field B is lowered

to B < 6 T. Figure 4-32f plots the transmission coefficient of the intersection region τi (B)

determined using the model shown as the inset.

Figure 4-33. (a) A schematic of the model used to determine the transmission coefficient τi of the

intersection region. (b) The measured R13 in the “through” configuration and the calculated τi as a

function of the magnetic field. Rc = 1075 W. (VSWb, VSEb) = (-3 V, +3 V). (VNWb, VNEb) = (-3 V, +3 V).

From device 1.

We use a modified two-channel model shown in Fig. 4-33a to extract the transmission

coefficient of the intersection τi in the “through” configuration. 𝑅13 = 𝑅𝐶 + (∑ 𝑅𝑝𝑎𝑟𝑎 //

ℎ/4𝑒2

𝜏𝑘𝑖𝑛𝑘𝑁 𝜏𝑖𝜏𝑘𝑖𝑛𝑘

𝑁 ), where RNpara, R

Spara, τ

Nkink and τS

kink are obtained from measurements described

above. Fig. 4-33b plots the measured R13 and the corresponding τi. τi increases from 0.63 at

B = 0 to 1 at B = 6 T. τi is smaller than τkink of individual channels shown in Fig. 4-29c,

which points to additional backscattering sources at the intersection. This behavior is

consistent with our understanding of the backscattering mechanisms of the kink states

[112] and its dependence on the bulk gap size Δ. The increase of Δ should enable further

136

increase of τi to achieve fully ballistic guiding of the kink states at B = 0, however this is

not allowed in our device due to gate leakage.

4.3.6 Valley valve and electron beam splitter

Figure 4-34. (a) illustrates the chiralities of the kink states in this gating configuration and the

blocking of straight transmission, i.e., the valley valve effect, in the absence of inter-valley

scattering. The wave functions of the K and Kʹ states remain overlapped in space in the absence

of a magnetic field or when EF is at the CNP (e = 0). (b) and (c) illustrate the shifts of the wave

function centers in a positive magnetic field for positive (b) and negative (c) energies.

A more powerful operation of the valley router enables it to function as a valley valve

and a coherent electron beam splitter simultaneously. In this operation, the polarity of the

electric field changes sign between adjacent quadrants. The resulting chiralities of the kink

states in all four channels are shown in Fig. 4-34a. Kink states in opposing channels have

opposite helicities, that is, states with the same chirality carry opposite valley indices K

and Kʹ and vice versa. In stark contrast to the “through” configuration shown in Fig. 4-32a,

here straight current transmission is forbidden in the absence of inter-valley scattering. This

valley valve effect occurs regardless of the presence of a magnetic field, as we will show.

The application of a magnetic field, however, offers an additional control. Calculations

137

have shown that away from the CNP, the wave functions of the kink states from the two

valleys shift in opposite directions in a magnetic field [112, 133, 159]. Figures 4-34b and

4-34c illustrate the situations of positive and negative EF, respectively. As the kink states

shift away from the mid lines of the channels, the wave function of a state coming from a

particular channel has unequal overlap with states of adjacent channels, thus leading to

unequal current partition. The wave function separation is tunable as a function of EF and

B and can become comparable or greater than the width of the wave function themselves

at several Teslas [112, 159]. Consequently, a current partition from 0 to 1 is possible.

Figure 4-35. (a) Measurements of the normalized percentage current received at terminals 2 - 4

respectively as labeled in the graph while using terminal 1 as the current source. A small I3 in the

entire kink regime demonstrates the valley valve effect. The inset illustrates the six independent

current transmission coefficients used in our S-matrix model. The diagram reflects an

experimentally observed C2 rotational symmetry. B = 8 T. (b) I3 (VSi) at fixed magnetic fields from 0

to 11 T demonstrating the robustness of the valley valve effect.

To test these predictions experimentally, we source a constant current from one

terminal and measure the normalized percentage current Ii = (current to drain i/ total

138

current) received at the other three terminals simultaneously. Figure 4-35a plots I2, I3, I4 at

B = 8 T using S1 as the current source. Remarkably, I3 remains low over the entire range

of VSi when all four channels are in the kink regime (-20 V < VSi < 20 V). Similar behavior

is also observed in measurements using other source terminals further discussed in the

following paragraphs. Figure 4-35e examines the impact of a magnetic field on I3 (VSi).

Near the CNP, the valley valve effect is already very strong at B = 0 with a small I3 of 8%.

I3 further decreases to less than 1% at B = 11 T, as the magnetic field works to suppress

residual inter-valley scattering in the intersection region. The suppression of current flow

between opposing terminals provides compelling evidence of the valley valve effect. The

transmission on/off ratio between the “through” configuration and the blocking

configuration is approximately 8 at B = 0 and more than 100 in a magnetic field. Although

far behind that of conventional Si transistors, the performance of the valley valve already

compares favorably with a recent all-electric spin valve [160].

Instead of propagating forward, the kink state wave functions from S1 split at the

intersection and propagate towards D2 and D4. Both I2 and I4 vary co-linearly with VSi,

forming a prominent X-like feature in Fig. 4-35d. The tunable range of I2 (I4) increases

with B until it saturates close to the full range of 0 to 1 at ~ 5 T. This current partition

behavior is reproducible using different source terminals and in different devices. It is in

excellent agreement with the wave function separation scenarios depicted in Figs. 4-34a –

4-34c and represents an electron analog of an optical beam splitter. Here the application of

a magnetic field is essential to its operation. However, calculations have suggested

alternative means of controlling the current partition by adjusting the size of Δ in different

quadrants [159]. This possibility will be explored in future experiments.

139

Next we show more data and analysis of the beam splitting operation discussed earlier.

Expanding the data shown in Fig. 4-35a. Figure 4-36a plots the normalized percentage

current I2 at selected magnetic fields from 1 T to 8 T. I2 is linear with VSi near the

equipartition point with a positive slope. Both the slope and the tunable range expanded by

I2 increase with increasing B at small B but quickly saturate. This behavior is in good

agreement with the numerical calculations of Ren et al [159]. Figure 4-36b plots the tunable

range of I2 vs. B. It increases from 28% at B = 1 T to 96% at B = 5 T.

Figure 4-36. Properties of the beam splitter. (a) Normalized percentage current I2 at selected B -

fields from 1 to 8 T. Dashed lines mark the tunable range of I2 for the 3 T data. (b) The tunable

range of I2 as a function of the magnetic field. (c) Transmission coefficients T0, T2 and T1’ measured

using terminal 2 (solid curves) or 4 (dashed curves) as the current source. The color scheme follows

that of the inset. The excellent agreement between the solid and dashed curves indicates an

approximate C2 rotational symmetry. (VNWb, VNEb, VSWb, VSEb) = (-3 V, +3 V, +3 V, -3 V). B = 8 T.

We determine the transmission coefficient Ti from measurements conducted using

different terminals as the current source. Ideally a C4 rotational symmetry is expected.

However our data show an approximate C2 rotational symmetry, as illustrated in Fig. 4-

36c. T0 and T2 shown in Fig. 4-36c are different from T0ʹ and T2ʹ, which are measured while

140

sourcing the current from terminal 1 (Fig. 4-35a). We suspect that the C2 symmetry arises

from the unequal gap size Δ on the four quadrants. Our device characterization lack the

resolution to precisely locate the Δ = 0 condition for each quadrant. If the Δ = 0 condition

corresponds to positive offsets on the bottom gate voltages (The offsets are usually within

± 0.2 V from our experiences), (VNWb, VNEb, VSWb, VSEb) = (-3 V, +3 V, +3 V, -3V) would

lead to larger Δ on the NW and SE quadrants and smaller Δ on the NE and SW quadrants.

This gating configuration would shift the zero potential lines toward the NE and SW

quadrants. Indeed, measurements of the two-terminal resistances show R12 R34 < R23

R14 (See later discussion), in agreement with this hypothesis.

Lastly we explain the origin of the abrupt rise of I3 and concurrent drop of I2, both to

approximately 50% at VSi > 30V in Fig. 4-35a. The VSi range of the kink regime in each

channel can be identified in individual measurements. VSi ~ 30V corresponds to the onset

of the first Landau level (LL) in the conduction band of the south channel. At this voltage,

the first LL is also present in the north channel, but not in the east and west channels.

Therefore, all current carried by the quantum Hall edge states flows to D3 while all current

carried by the kink states flow to D2. Since both have conductance σ = 4 e2/h an

equipartition between I2 and I3 is expected. This is indeed what’s observed. In Fig. 4-35a,

I4 also shows a small dip at about VSi = -20 V then recovers. The beginning of the dip

corresponds to the onset of the first LL in the valence band of the south channel. At this

voltage, the first LL is also present in the west and east channels. The Landauer-Büttiker

formula dictates, however, that the quantum Hall edge current flows entirely to D4 in an

ideal scenario; thus no dramatic redistribution of current is expected. Our data is consistent

141

with this interpretation. These features enable us to confirm our understandings of the band

structure of the channels in a magnetic field.

4.3.7 Comparisons between experiment and theoretic modeling

We developed an S-matrix model to describe the transmissions of the kink states

between different channels with six independent coefficients schematically shown in the

inset of Fig. 4-35a. These coefficients are obtained directly from the normalized percentage

current measurement. Figure 4-37a plots the four non-zero Ti (VSi) together. Using the

experimental input and the Landauer-Büttiker formula, we have calculated the resistance

for various two-terminal and non-local measurement geometries. Figure 4-37b compares

the calculated and measured two-terminal resistance R13, while other cases are discussed

later. The excellent agreement between theory and experiment affirms the ballistic edge

transport nature of the kink states.

Figure 4-37. (a) Transmission coefficients T0 (orange), T0ʹ (blue), T2 (red), T2ʹ (olive) as a function

of VSi. (b) The measured (magenta) and calculated (black) two-terminal resistance R13. A contact

resistance of Rc = 1 kΩ is added to the theoretical curve to match data. B = 8 T. Small discrepancy

between theory and experiment is attributed to microscopic imperfections of the device, which

142

cannot be fully captured by the simple model. Gray dashed lines indicate the corresponding

plateaus in T and R13, with the value of R13 labeled for each line.

S-matrix model is employed to correlate the outgoing modes Ψout =

(𝜓1,𝑜𝑢𝑡, 𝜓2,𝑜𝑢𝑡, 𝜓3,𝑜𝑢𝑡, 𝜓4,𝑜𝑢𝑡)𝑇 with the incoming modesΨin = (𝜓1,𝑖𝑛, 𝜓2,𝑖𝑛, 𝜓3,𝑖𝑛, 𝜓4,𝑖𝑛)

𝑇

through Ψout = 𝑆 Ψin where 𝜓i,in/out denotes the in/out mode for the i-th terminal and we

express

𝑆 = (

𝑟0 𝑡0 𝑡1 𝑡2

𝑡2′ 𝑟0′ 𝑡0′ 𝑡1′𝑡1 𝑡2 𝑟0 𝑡0

𝑡0′ 𝑡1′ 𝑡2′ 𝑟0′

).

Figure 4-38. (a) and (b) The S-matrix model describing the transmission of kink states between

different channels. Two sets of parameters are used to describe the scattering amplitude as

illustrated in the diagrams. The diagrams are drawn for e > 0 (See Fig. 3b of the text). Blue (red)

arrows denote the kink states at K (Kʹ).

Figures 4-38a and 4-38b illustrate the possible scattering processes between all

terminals. The transmission and reflection parameters are labelled constrained by the

experimentally observed C2 symmetry. The conductance matrix 𝐺 is related to the S-matrix

by

𝐺i←j =4𝑒2

ℎ|𝑆𝑖𝑗|

2 (4-3)

143

where the coefficient 4 e2/h accounts for the four modes of the kink states in each channel

(two due to spin, two due to layer isospin). Using Eq. 4-3 and the Landauer-Büttiker

formula shown in Eq. 4-4

𝐼i = ∑ (𝑉𝑖 − 𝑉𝑗)𝐺𝑖←𝑗𝑗 (4-4)

we obtain

(

𝐼1

𝐼2

𝐼3

𝐼4

) =4𝑒2

ℎ× (

𝜏 −𝑇0 −𝑇1 −𝑇2

−𝑇2′ 𝜏′ −𝑇0′ −𝑇1′−𝑇1 −𝑇2 𝜏 −𝑇0

−𝑇0′ −𝑇1′ −𝑇2′ 𝜏′

) (

𝑉1

𝑉2

𝑉3

𝑉4

), (4-5)

where 𝑇i = |𝑡𝑖|2 and 𝑇i

′ = |𝑡𝑖′|2 for 𝑖 ∈ 0,1,2 , and 𝜏 = 𝑇0 + 𝑇1 + 𝑇2 = 1 − |𝑟0|2 and

𝜏′ = 𝑇0′ + 𝑇1

′ + 𝑇2′ = 1 − |𝑟0

′|2. The current transmission coefficients Ti’s are shown in the

inset of Fig. 4-35a. By setting 𝑉1 = 𝑉 and 𝑉2 = 𝑉3 = 𝑉4 = 0 , we find that 𝜏 =

ℎ

4𝑒2

𝐼1

𝑉, 𝑇0

′ = −𝐼4

𝐼1𝜏, 𝑇1 = −

𝐼3

𝐼1𝜏, 𝑇2

′ = −𝐼2

𝐼1𝜏.

Figure 4-39. Comparison between theory (black curves) and experiment (magenta and green

curves) on two-terminal resistances R43, R14 and the partner non-local resistances R12,43 and R23,14.

In R43 and R14, a constant Rc is added to the theoretical curves to match data. Diagrams illustrate

how the non-local resistances are measured.

144

Our measurements are in the ballistic regime with 𝜏 ≈ 𝜏′ ≈ 1 . Therefore, the

transmission coefficients can be directly obtained from the measured percentage current as

described in Fig. 4-35. Equation 4-5 allows us to predict the outcomes of various

measurement geometries using the experimentally obtained coefficients as input, and

compare to the corresponding measurements. The results on the two-terminal resistance

R13 are given in Fig. 4-37b. Figures 4-39a and 4-39b show additional examples. In all cases,

theory and experiment reach excellent agreement. As Figs. 4-39a and 4-39b show, both

experiment and theory show R43 < R14, as a consequence of the C2 symmetry (See Section

S5 for more discussions). It is also worthwhile pointing out that R43 and the corresponding

non-local resistance R12,43 (and similar R14 and R23,14) follow a simple relation in the entire

kink regime, which reads

𝑅43 − 𝑅12,43 =ℎ

4𝑒2 + 𝑅𝑐. (4-6)

This relation is successfully reproduced in our calculated curves (without Rc) as well.

In the next section, we rewrite the Landauer-Büttiker formula in terms of quantized

transmission coefficients. We can show that the two-terminal conductance 𝐺43 is 𝐺43 =

8+𝑀𝑁−2(𝑀+𝑁)

16+𝑀𝑁−4𝑁

8𝑒2

ℎ, while its partner non-local conductance G12,43 is given by 𝐺12,43 =

(16

𝑀+

16

4−𝑁− 8)

𝑒2

ℎ. It is then straightforward to show that Eq. 4-6 is satisfied analytically.

4.3.8 Quantized transport in high magnetic field

When the magnetic field increases to B > 6 T, quantization at fractional values of ~ 0,

¼, ½, ¾, and 1 start to appear in Ti (Fig. 4-37a) and corresponding plateaus are seen in R13

(Fig. 4-37b). Both become more precise at higher field as discussed later. Their presence

145

suggests that the four-fold degeneracy of the kink states is lifted in a strong magnetic field

[152] and the transmission coefficients of each mode take on different and discrete values

of 0 or 1.

In the following, we provide a detailed discussion of the origin of the fractional

quantization and its manifestation in transport, which reflect the coexisting helical and

chiral nature of the kink states in a magnetic field.

Figure 4-40. (a) Quantized transmission coefficients T0 and T2 at fractional values marked in the

plot at selected magnetic fields from 6 to 18 T. The centers of the curves are aligned to facilitate

comparison. (b) The measurement setup.

Figure 4-40 plots transmission coefficients T0 and T2 at fixed magnetic fields from 6 T

to 18 T, where quantizations at fractional values of ¼, ½ and ¾ are clearly seen. These

observations can be understood by taking into account the Landau level quantization of the

bulk bands of bilayer graphene [161] and the resulting evolution of the kink states to valley

kink states in a strong magnetic field [152, 162]. Figure 4-41a shows a likely energy

spectrum of the valley kink states in a guiding center description for the north channel. The

energy splitting between the different modes of the valley kink states results in four distinct

146

crossing points of the states from K and Kʹ valleys. As EF sweeps through the crossing

points, the physical separation between the K and Kʹ states is different in size and direction

for each mode; thus each mode can take on different values of transmission coefficients.

As the crossing points grow farther apart in stronger magnetic fields, the attainment of the

0 or 1 limit for each of the four modes manifests as plateaus of fractional coefficients at 0,

¼, ½, ¾, and 1. This process is illustrated in Fig. 4-41b - 4-41f.

Figure 4-41. (a) An energy spectrum of the valley kink states in strong magnetic and electric fields

in a guiding center description. (b) - (f) The evolution and quantization of T0 as a function of EF. T0

represents current transmission from terminal 4 to 3, or equivalently K valley states flowing from

terminal 3 to 4. The connections are made based on the physical distance between modes. A

connecting line represents a transmission probability of 1. (g) An idealized plot of T0 and T2 as a

147

function of EF. Similar analysis applies to T0ʹ and T2

ʹ. Here we have neglected the effect of intervalley

scattering and assumed that transmissions preserve the spin and isospin indices of the valley kink

states.

We can rewrite the two-terminal conductance G13 = 4e2/h (1 - T0T2ʹ - T2T0ʹ) obtained in

the Landauer –Büttiker theory in terms of quantized coefficients:

𝑇0 =𝑀

4, 𝑇2 =

4 − 𝑀

4, 𝑇0

′ =𝑁

4, 𝑇2

′ =4 − 𝑁

4, 𝑀, 𝑁 ∈ 0,1,2,3,4,

where M is the number of modes flowing from terminal 3 (1) to terminal 4 (2) and N is the

number of modes flowing from terminal 2 (4) to terminal 3 (1). We obtain

𝐺13 = [4 +𝑀𝑁

4− (𝑀 + 𝑁)]

𝑒2

ℎ. (4-7)

Table 4-3 lists all possible values of 𝐺13 for given values of 𝑇0(𝑀) and 𝑇0′(𝑁). Equation

(5) makes it clear that the quantization of G13 directly results from the quantized

transmission coefficients. Figure 4-42a plots the measured coefficients and G13 at B = 16

T. As Figure 4-42b illustrates, the agreement between experiment and Eq. 4-7 is excellent.

Table 4-3. Possible values of G13

as a combination of M and N.

148

Next, we use the microscopic guiding center picture to provide a physical and intuitive

interpretation of the quantization in G13 (R13). We examine two scenarios, which

correspond to M = 4, N = 4 (T0 = T0ʹ = 1) and M = 3, N = 2 (T0 = 3/4, T0ʹ = 1/2) respectively,

to illustrate the essential points of the picture.

Figure 4-42. (a) Measured transmission coefficients T0, T2, T0ʹ and T2

ʹ (top) and G13 (bottom) as a

function of VSi at B = 16 T. G13=1/(R13-Rc). RC = 1174 Ω. “Glitches” in the middle of a plateau are

likely due to microscopic potential irregularities. (b) Idealized data in (a) demonstrating the working

of Table 4-3. The color scheme follows that of Table 4-3. Experimentally observed plateaus in G13

are shown in bold in Table 4-3.

Figure 4-43b depicts the flow of the kink states in the case of 𝑇0 = 𝑇0′ = 1 and 𝑇2 =

𝑇2′ = 0. Although the incoming and outgoing modes carry different valley indices, this

situation resembles that of a chiral-edge-state system in transport and it is straightforward

to obtain G13 = 4 e2/h since four edge modes are present. G13 of the scenario illustrated in

Fig. 4-43c for T0 = 3/4, T2 = 1/4, T0ʹ = 1/2 and T2

ʹ = 1/2 is less obvious. In this case, it is

helpful to view it as the sum of three subsystems as shown in Fig. 4-43d - 4-43f. A single

chiral mode contributes e2/h conductance. The resistance of the helical system is (h/e2 +

149

h/e2) // (h/e2 + h/e2) = h/e2, with each pair of counter-propagating modes contributing to a

quantum resistance of h/e2. The third subsystem does not contribute to G13 since no mode

connects terminals 1 and 3. Together, we recover G13 = 2 e2/h. The same method can be

used to understand other plateaus in G13.

Figure 4-43. (a) The energy diagram of the valley kink states shown in Fig. S11a. The blue lines

mark EF in (b) and (c). (b) The transmission scenario for T0 = T0ʹ = 1. This situation mimics that of

a chiral quantum Hall system with G13 = 4 e2/h. (c) T0 = 3/4. T0ʹ = 1/2. The black dashed lines

illustrate the shift of the crossing points towards the NE and SW quadrants, as discussed in Section

5. This shift can account for the difference between T0 and T0’, as the diagram illustrates. (a) - (f)

Decomposition of (c) into the sum of three subsystems, the conductance of which add up to the

observed 2 e2/h.

150

4.3.9 Section conclusion

In this section we have demonstrated ballistic kink states and near ballistic transmission

of kink states with the same helicity. We also showed the capability of our device as a valve

and electron beam splitter by utilizing the valley DOF. And this is the first all electrically

controlled valleytronic operations since the advent of graphene. We envision that an

electron optical network built with ballistic transport channels, waveguides, valves, and

beam splitters demonstrated here, together with other graphene-based components such as

lenses [110, 163] and filters [164], could achieve rather complex functionalities. The high

quality and high controllability of the kink states also enable exciting new avenues to

explore topological superconductivity [165].

151

Chapter 5

Valley, spin and orbital competing zero energy Landau levels and gate-

controlled transmission of quantum Hall edges in bilayer graphene

5.1 Introduction

5.1.1 Brief introduction to quantum Hall effect

In a magnetic field, charged particles pick up circular motions due to Lorentz force,

and classical Hall effect was discovered by Edwin Hall in 1879. At low temperature and

stronger magnetic field, more striking phenomena, e.g. De Hass-van Alphen effect, was

observed due to the quantization of the cyclotron orbit. In 1980 von Klitzing

experimentally demonstrated quantum Hall effect in a GaAs 2DEG [166]. In the quantum

Hall experiments, both the Hall resistance ρxy and the longitudinal resistance ρxx exhibit

very interesting behavior, i.e. ρxx vanishes while ρxy plateaus at value 2e

hxy

with ultra-

high precision, when the magnetic field satisfies the condition 0

nB , where n is the

carrier density, ν is an integer which is known as filling factor and eh /0 is the

magnetic flux quantum.

In the rest of this subsection, we use a semiclassical approach to demonstrate these key

features in a quantum Hall system by solving a single electron Schrödinger equation [167].

In a magnetic field B along the z direction, let us consider a 2DEG on the x – y plane with

finite dimensions of Lx and Ly along the x and y direction respectively. The Hamiltonian of

an electron in a magnetic field can be written as 22

1Ap q

mH , q = -e is the charge of

152

an electron. We can choose a vector potential 0 , ,0 BxA such that the translational

symmetry is preserved in the y direction, and the Hamiltonian becomes

22

2

1eBxpp

mH yx . Since the translational symmetry is preserved on the y

direction, we can expect a wavefunction which has a plane wave component in the y

direction such as xeyx k

iky

k , , and kpy . Then the Hamiltonian can be written

as 222

2

22

1B

Bx klx

mp

mH

, which is the Hamiltonian for a harmonic oscillator in the

x direction centered at 2

Bklx . meBB / is the cyclotron frequency and eB

lB

is

the magnetic length. Hence the discrete energy eigenvalues are

2

1jE Bj , where j

is an integer. And the associated wavefunction for the jth energy state is

222 2/)(2

, ,, BB lklx

Bj

iky

jj

iky

kj eklxHeAkxeyx

(5-1)

where Aj is a normalization prefactor and Hj are the Hermite polynomials. In a magnetic

field, the continuous conduction and valence band in 2DEGs become these discrete energy

levels called Landau levels (LLs). The wavefunction for the these LLs exponentially decay

around the center of the harmonic oscillator on the length scale of Bl , therefore we have

0/ 2 klL Bx for a finite sample constrained within xLx 0 . Hence the number of

states allowed for a certain LL is h

eBA

l

LLdk

LN

B

yx

lL

y

Bx

2

0

/ 222

, where A is the area of

the 2DEG. The degeneracy is a very large number, therefore each LL is independent of k

153

as illustrated in Fig. 5-1a. If the νth LL is fully occupied then the carrier density

h

eB

A

Nn

, which also satisfies

0

nB .

Figure 5-1. (a) Landau level band diagram. (b) Potential profile along x axis shows the potential

steeply increases towards sample edges. (c) Schematic shows EF is in the middle of two Landau

levels.

The potential rises sharply towards the physical edge of the sample as shown, e.g. on

the x direction, in Fig. 5-1b. We assume the sample geometry is only finite in the x

direction, and the potential is smoothly varied and described by V (x). The Hamiltonian is

then modified as xVeBxppm

H yx 22

2

1. With a first order approximation we

can expand the potential to be 00 / xxxVxVxV , and the consequence is

creating a drift velocity on the y direction x

V

eBvy

1. Wavefunction associated with

momentum k is localized within a length scale of Bl at the center of harmonic oscillator

2

Bklx and carries a drift velocity x

V

eBvy

1. V (x) is flat in the bulk and carries a

negative/ positive derivative on the left/ right sample edge, therefore vy = 0 in the bulk and

the states at the sample edges are chiral, i.e. states on the same edge travel towards the

154

same direction, and travel oppositely on two edges: vy > 0 on the left and vy < 0 on the

right. Sample width is usually much larger than Bl , thus backscatter edge modes from one

edge to the other is eliminated and ρxx vanishes. This reveals the chirality nature of quantum

Hall edge modes. Furthermore if a potential difference of Δμ is introduced on the two

sample edges as shown in Fig. 5-1b, then more states are occupied on the right edge, i.e.

Fermi level of the left edge EFL is lower than that of the right edge EER. The net current is

contributed by all occupied states and can be calculated as

Δh

edx

x

V

eBl

ekv

dkeI

B

yy

1

22 2. Thus the Hall conductance contributed by

edge states from one LL is h

e

eΔ

I y

xy

2

/

. If multiple LLs are occupied as displayed in

Fig. 5-1c, e.g. the νth LL is fully occupied, then h

exy

2 . This captures the key

observations in quantum Hall experiments.

5.1.2 Quantum Hall effect in bilayer graphene

With a low energy approximation, we only consider the nearest neighbor hopping in

bilayer graphene, then the Hamiltonian can be reduced into a two-fold format on an A1 –

B2 base, where A1 and B2 are the non-stacked low energy atom sites. Now the wavefunction

is TBA 21, , and the Hamiltonian can be expressed as

0

0

2

12

2

*

yx

yx

ipp

ipp

mH

(5-2)

155

where m* is the bilayer graphene effective mass and 1 labels the K and Kʹ valley

[43]. In a magnetic field the momentum operator gains an extra term of –qA. Let us choose

a Landau gauge defined by 0 , ,0 BxA such that we have a magnetic field of strength B

on the z direction meanwhile we can also preserve the translational symmetry in the y

direction. We define a new set of operators ieBipp yx and ieBipp yx ,

then the Hamiltonian described in Eq. 5-2 becomes

0

0

2

12

2

*

mH K , and

0

0

2

12

2

*

mH K (5-3)

for the K and Kʹ valley respectively in a magnetic field. Since the translational invariance

is preserved on the y direction, we can express the eigenstates as plane waves in the y

direction and harmonic oscillators in the x direction. Each component in can be

expressed as Eq. 5-1.

In the K valley, and in the Hamiltonian acts as lowering and raising operator

respectively for the harmonic oscillator states such that

1

2 j

B

j jl

i

and

112

j

B

j jl

i

. (5-4)

Then the eigenvalues (Landau level energy) can be obtained as

)1(, jjE Bj , (5-5)

where ± denotes the electron and hole energy states, and the corresponding wavefunction

2

,2

1

j

j

j

for integers 2j . (5-6)

156

Both j = 0 and j = 1 gives 010 EE with corresponding wavefunctions

0

0

0

and

0

1

1

. (5-7)

In the Kʹ valley, the function of and reverses, and they now become raising and

lowering operators for the harmonic oscillator states such that

11

2 j

B

j jl

i

and

1

2

j

B

j jl

i

. (5-8)

The eigenvalues are also )1(, jjE Bj with corresponding wavefunctions

j

j

j

2

,2

1 for integers 2j . (5-9)

Both j = 0 and j = 1 again gives 010 EE , however the wavefunctions now only have

the B2 component such that

0

0

0

and

1

1

0

. (5-10)

The discussion above shows there is a 4-fold degeneracy for all the non-zero energy

LLs due to K and Kʹ valley degeneracy and 2-fold spin degeneracy. Take the extra 2-fold

orbital degeneracy into account, the zero energy LL has an 8-fold degeneracy. Interestingly,

the wavefunctions for the 8-fold degenerate LL has nonzero component for only one

graphene layer for each valley, i.e. the electrons/ holes in the K valley only occupy the A1

sites which are on the top graphene layer, and the Kʹ valley carriers only have non-zero

wavefunction distribution on the B2 sites located on the bottom graphene layer as suggested

157

in Eqs. 5-7 and 5-10. Hence in the low energy approximation, the valley degeneracy is

equivalent to the layer degeneracy in bilayer graphene for the 8-fold zero energy LL. The

valley, spin and orbital orders compete in a clean sample at low temperature and generate

various ordering states [161], and this makes the 8-fold zero energy LL in bilayer graphene

a fascinating element for engineering exotic phase condensed matter.

5.1.3 Tunneling of quantum Hall edge modes

When two sets of quantum Hall edge modes are brought close, the edge modes from

one set interact with the ones the other set. The interaction strength depends on the barrier

height between the two sets of quantum Hall edge modes. With a weak barrier, e.g. a

smooth potential barrier, all the edge modes from the two sets can fully mix with each other

[168], while with a stronger barrier only the closest edge modes, i.e. the outmost edge mode

from each set can mix [169]. The two quantum Hall systems become completely isolated

in the strong barrier limit. In this subsection, we discuss a few examples of quantum Hall

edge modes tunneling with a barrier of tunable strength regardless of the detailed barrier

format.

Figure 5-2. A diagram shows quantum Hall edge modes tunneling in a two zone uni-polar junction.

Black dashed line indicates the junction center.

158

Two-zone uni-polar junction The diagram in Fig. 5-2 shows two sets of quantum Hall

edge modes with filling factor ν1 and ν2 on the left and right side of the sample respectively.

The filling factors have the same polarity, i.e. 021 and both set of edge modes travel

clockwise, while21 . There is a tunable barrier between the two sets of edge modes,

and the barrier strength is characterized by parameter α such that only 21 portion of

edge modes from the right side can tunnel through the barrier and reach the left side of the

sample. In the limit of no barrier 0 , and 1 when the two sets of edge modes are

isolated. Let’s assume current I is sent from pin 1, and pin 4 is grounded. Since quantum

Hall edge modes are ballistic edge modes, we adopt Landauer Büttiker theory to calculate

the current transport in this system. We can write down the Ii-Vi relation for each pin as

0)1()1(

0

0)1(

0

2215261

2

6

452

2

5

342

2

4

2532

2

3

121

2

2

611

2

1

VVVh

eI

VVh

eI

IVVh

eI

VVVh

eI

VVh

eI

IVVh

eI

(5-11)

Then we can calculate the resistance between the following pairs of pins,

2

2

4141

2

12

3256

2

2

3232

1

1

1

1

1

1

1

e

h

I

VVR

e

h

I

VVR

e

h

I

VVR

. (5-12)

159

We define the tunnel resistance RT in the uni-polar tunnel junction as the resistance across

the barrier on the side where more edge modes run into the less, so 32 RRT in this

example. The two terminal source-drain conductance h

e

RG

2

2

41

41 11

becomes

h

eG

2

2141 ,min when 0 in the fully mixing situation as measured in Ref.

[168]. In the device geometry displayed in Fig. 5-2 RT vanishes when the barrier is very

weak, and RT becomes very large when the two quantum Hall systems are isolated.

Interestingly, the difference of the resistance measured across the junction from the two

sides, i.e. 3256 RR , is independent of α, which is always 2

12

11

e

h

as long as there

are well established quantum Hall edge states on the two sides of the barrier.

Figure 5-3. A diagram shows quantum Hall edge modes tunneling in a two zone bi-polar junction.

Black dashed line indicates the junction center.

Two-zone bi-polar junction The diagram in Fig. 5-2 is similar to the one shown in Fig.

5-2 except for that, in this case the filling factors on the two sides of the sample carries

opposite polarities, i.e. 021 and the edge modes travel clockwise and counter

160

clockwise on the left and right side respectively. Similarly we can write the Ii-Vi relation

for each pin as

0)1()1(

)1(

0)1()1(

)1()1(

0

0

2

21

13

21

261

2

6

2

21

13

21

223252

2

5

542

2

4

432

2

3

121

2

2

611

2

1

VVVh

eI

VVVVh

eI

IVVh

eI

VVh

eI

VVh

eI

IVVh

eI

. (5-13)

Then we get the resistance between the following pairs of pins,

2

21

2141

2

2

56

2

12

32

1

1

1

1

1

1

e

hR

e

hR

e

hR

. (5-14)

In the limit of fully mixing and 0 , the two terminal source-drain conductance

h

eG

2

21

2141

1

1

becomes

h

eG

2

21

2141

as measured in Ref. [168]. In the two-

zone bi-polar tunnel junction, we define the tunnel resistance RT as the resistance across

the barrier on the side where edge states from the two sides travel away from each other,

so 56 RRT in this example. RT vanishes when the barrier is very weak, and RT becomes

161

very large when the two quantum Hall systems are isolated. 2

12

5632

11

e

hRR

is

again independent of α, and checks the quality of the quantum Hall on the two sides of the

sample.

Figure 5-4. A diagram shows quantum Hall edge modes tunneling in a three zone uni-polar junction

where edge modes fully mixed with each other. Black dashed lines indicate the junction centers.

Three-zone uni-polar junction Finally we discuss a situation that three sets of quantum

Hall edge states fully mix across two barriers, which correspond to 0 described in the

previous two examples. We assume ,max and all the filling factors carry the

same sign. In the two junction areas illustrated by black dashed lines in Fig. 5-4, the

proceeding edge modes are dominated and only contributed from the middle set of quantum

Hall edge modes. We can then easily obtain

256

232

e

hR

e

hR

. (5-15)

162

5.2 An empirical diagram of the valley, spin and orbital ordering in the 8-fold zero

energy LL in bilayer graphene

5.2.1 Motivation of this study

Bilayer graphene provides a fascinating platform to explore potentially new

phenomena in the quantum Hall regime of a two-dimensional electron gas (2DEG). The

existence of two spins, two valley indices K and Kʹ, and two isospins corresponding to the

n = 0 and 1 orbital wave functions result in an eight-fold degeneracy of the single-particle

E = 0 Landau level (LL) in a perpendicular magnetic field B [43, 170, 171]. This SU (8)

phase space provides ample opportunities for the emergence of broken-symmetry many-

body ground states [60, 78, 80, 172-181]. The application of a transverse electric field E

drives valley polarization through their respective occupancy of the two constituent layers

[43, 170, 171]. Coulomb exchange interactions, on the other hand, enhance spin ordering

and promote isospin doublets [172, 174, 182]. As a result of these intricate competitions,

the E = 0 octet of bilayer graphene (filling factor range -4 < < 4) exhibits a far richer

phase diagram than their semiconductor counterparts. Experiments have uncovered 4, 3, 2,

1 coincidence points for filing factors = 0, ±1, ±2 and ±3 respectively, where the crossing

of two LLs leads to the closing of the gap and signals the phase transition of the ground

state from one order to another [78, 172, 182-184]. Their appearance provides key

information to the energetics of the LLs and the nature of the ground states involved. Indeed,

coincidence studies on semiconducting 2DEGs are used to probe the magnetization of

quantum Hall states [185] and measure the many-body enhanced spin susceptibility [186].

In bilayer graphene, the valley and isospin degrees of freedom increase the number of

potential many-body coherent ground states. Furthermore, the impact of actively

163

controlling these degrees of freedom became evident in the recent observations of

fractional and even-denominator fractional quantum Hall effects in bilayer graphene [114,

183, 187-190].

A good starting point of exploring this rich landscape would be a single-particle, or

single-particle-like LL diagram, upon which interaction effects can be elucidated

perturbatively. Indeed, even in the inherently strongly interacting fractional quantum Hall

effect, effective single-particle models, e.g. the composite fermion model [191], can

capture the bulk of the interaction effects and provide conceptually simple and elegant

ways to understand complex many-body phenomena. In bilayer graphene, a LL diagram

that provides a basis to interpret and reconcile the large amount of experimental findings

to date has yet to emerge. Predictions of tight-binding models involving two or four bands

[171, 173] are both at odds with experiments [182]. Exchange interactions are shown to

lead to large corrections, which completely alter the single-particle spectrum [182].

5.2.2 Building an empirical LL diagram based on our measurements

Given the non-perturbative nature of this problem, here we take an empirical approach

to construct an effective single-particle-like LL diagram of bilayer graphene in the presence

of a perpendicular magnetic field and a transverse electric field. This diagram is built upon

three energy scales, namely the interlayer potential difference (E), the exchange-

enhanced spin gap s (B) and the exchange-reduced splitting between the n = 0 and 1

orbitals E01. All three are obtained from our own measurements and corroborated by others

in the literature. The effective LL diagram can quantitatively reproduce the observed

coincidence conditions of = 0, ±1, ±2 and ±3 and the large gap energies of = ±2 and

reconcile the seemingly disparate results on the gap energies of = ±1. It also captures the

164

five filling sequences of the LLs from =-4 to +4 reported in Ref. [182]. Because of its

single-particle nature, it does not produce states with spin or valley coherence, such as

those observed at = 0 and ±2 at small electric fields. We hope that it provides a basis to

guide the design of future experiments and a starting point upon which more sophisticated

theoretical tools can be applied to address its deficiencies.

Figure 5-5. RCNP (D) for five devices measured at T = 1.6 K. Device numbers are labelled with

corresponding color. Figure 5-5. RCNP (D) for five devices measured at T = 1.6 K. Device numbers

are labelled with corresponding color.

We begin by measuring the interlay potential difference (E), which gives rise to the

band gap of bilayer graphene at B = 0. All 8 devices reported in this section are dual-gated,

with the bilayer sheet sandwiched between two h-BN dielectric layers. These devices are

fabricated with two methods. Devices 1, 3 and 24 are fabricated using van der Waals dry

transfer plus one-dimensional side contacts techniques first introduced by Wang et al [26],

while devices 6, 23L, 23R, 34 and 43 are fabricated using PMMA assisted layer by layer

transfer techniques first introduced by Dean et al [66]. Fabrication process is detailed in

Chapter 4. The mobility of these bilayer graphene devices ranges from 20, 000 to 100, 000

-400 -200 0 200 400

10k

100k

1M

10M

RC

NP (

)

D (mV/nm)

T = 1.6 K

device #

06

23

43

34

24

165

cm2V-1s-1. RCNP (D) for five devices measured at T = 1.6 K are plotted in Fig. 5-5. RCNP (D)

increases near exponentially with D field, and this suggest the high quality of our devices.

As described in Chapter 2, we measure as a function of the applied displacement

electric field D using thermally activated transport at the charge neutrality point (CNP). At

D < 800 mV/nm, which covers majority of the quantum Hall studies in the literature, (D)

is well approximated by (meV) = 0.13D (mV/nm). This implies an effective dielectric

constant of = 2.6 for the interlayer screening of bilayer.

In a perpendicular magnetic field, the gapped bands of bilayer graphene evolve into

discrete LLs. In the large-D limit, the dominant energy splitting is that of the K and Kʹ

valleys. In the two-band tight-binding model, the energies of the n = 0 orbitals in K (

and Kʹ ( valleys split by the interlayer potential difference , since |+,0> and |-,0>

occupies opposite layers. The splitting between |+,1> and |-,1> is smaller, as they occupy

both layers [171]. A much larger correction, however, arises from including the effect of

the electron-hole asymmetry represented by the hopping parameter 4 in a four-band model

[173], where BEEE )/( 0410110 is positive in both valleys and lifts the |,1> states

above the |,0> states. This leads to a sequence of |+,1>, |+,0>, |-,1>, |-,0> with E10 (meV)

= 0.48B (T) obtained using measurements of 1eV4 =0.22 eV and 0 =3.43 eV

[100]. Since E10 >> the bare Zeeman splitting z (meV) = 0.11B (T), the four-band model

would favor orbital isospin polarization and produce spin doublets, e.g. |0, > followed by

|0, >. Contrary to this prediction, Hunt et al observed a LL filling sequence that prefers

orbital doublets, i.e, |0, > closely followed by |1, >, which was interpreted as due to large

166

exchange correction to E10, which counters the effect of 4 and reduces the magnitude of

E10 significantly [182].

Figure 5-6. A qualitative LLs for the E = 0 octet of bilayer graphene at a fixed B field as an evolution

of D field. K and Kʹ valley states are in red-like and blue-like colors respectively.

Motivated by the experimental results, we sketch a qualitative LL diagram of bilayer

graphene at a fixed magnetic field B and as an evolution of the D-field in Fig. 5-6. In the

large-D limit, the sequence of the LLs follows that of Ref. [182], where measurements

show explicit breaking of the electron-hole symmetry. E10 (D, B) > 0 is the same for the K

and Kʹ valleys and is to be determined by experiments. The spin gap s (B) is made to be

larger than E10 to favor orbital doublets and adds energy of ½ s to spin up and down

levels respectively. We postulate that the splitting between E+, 0 and E-, 1 is the interlayer

potential difference , with 1021

21

0, EΔE and 1021

21

1, EΔE The offset ½ E10

allows us to maintain the e-h symmetry of the n = 0 orbitals, i.e, E+,0 = ½ ½ E10 = - E-,0.

As D decreases, this single-particle LL diagram predicts two coincidence D fields, at

which the = 0 gap closes. As illustrated in Fig. 5-6, a larger D*h corresponds to the

coincidence of the (+, 0, ) and the (-, 1, ) levels, i.e. D*h = s (B), while a smaller

D*l corresponds to the coincidence of the (+, 1, ) and (-, 0, ) levels, i.e. D*

l+ 2E10

(D*l, B) = s (B). They are distinguishable so long as E10 can be resolved experimentally.

167

Figure 5-7. (a) Color map of Rxx (Vtg, Vbg) in device 6 at B = 8.9 T. Dashed lines show constant

filling factors ν = 0, 1 and D = 0. The positive D*0, ±1 is also marked in figure. White out region

shows where contacts (also bilayer graphene) reach CNP. (b) and (c), Rxx (D) at selected B-fields

from 10 - 16 T in device 24 (b), and at B = 25 T and 31 T in device 6 (c). (d) An example of ν = -2

gap measurement with method described in Ref. [72]. Arrows mark the gap size. (e) Δ (B) for ν = -

2 (symbols), and a linear fit passes through origin with 0.94 meV/T slope.

The measurements of D*h and D*

l for = 0 are presented in Figs. 5-7a – 5-7c. Figure

5-7a shows a colored map of magnetoresistance Rxx as a function of the top and bottom

gate voltages Vtg and Vbg in device 06 at B = 8.9 T. Lines corresponding to constant filling

factors = 0, and D = 0 are marked in the plot. We sweep the top and bottom gates in

a synchronized fashion to follow a line of constant and measure Rxx (D). A dip (peak) in

Rxx (D) is identified as the coincidence field D*0 (D

*±1) for = 0 (. D* is symmetric

about 0 and the positive D*0, ±1 is marked in Fig. 5-7a. Figure 5-7b plots a few examples of

168

Rxx (D) at fixed B-fields from 10 - 16 T in device 24. A double-dip structure starts to appear

at B ~ 12 T and the difference between D*h and D*

l rapidly increases with B. Higher field

data obtained on device 06 are shown in Fig. 5-7c. The cumulated results of D*, D*h and

D*l obtained from 4 devices are plotted together in Fig. 5-8a. Above B ~ 7 T, D*

h (D* at

low field) exhibits a remarkably linear dependence on B, with a slope of 8.3 mV/nm/T

Both the linear trend and the slope are in good agreement with measurements on other h-

BN encapsulated bilayers [78, 182-184]. The linear dependence of D*h (B), together with

(D) = 0.13D, leads to s (B) = B with 1.08 meV/T. The large magnitude of s

validates the LL sequence drawn in Fig. 5-6 and suggests an exchange-enhanced spin gap

2 at = 2, which should be insensitive to the D-field as long as E10 is small. This is

indeed consistent with our observations as shown in Fig. 5-9a.

Figure 5-8. (a) A collection of coincident points D field D* versus B from 5 devices. D*h (D* at low B

field) and D*l for ν = 0 states are shown with squares and circles respectively. Data in devices 24,

6 and 34 are plotted in red, black and orange colors. Stars show D*-1 in device 43 (blue) and data

adapted from Ref. [182] (olive). Top and bottom black dashed lines plot calculated D*h and D*

l for ν

= 0, dark yellow dashed line plots calculated D*1. (b) E10 versus D*

l, corresponding B fields are

marked at the top axis. (c) E10/B versus √D*l and a linear fit. Data in device 24 and 6 with same

color scheme in (a).

169

Measurements of -2 in device 3 are shown in Figs. 5-7d and 5-7e. Here we have

followed methods used in Refs. [60, 172] to extract the value of -2 from two-terminal

differential conductance measurements, an example of which is shown in Fig. 5-7d. The

resulting -2 (B) are plotted in Fig. 5-7e and is well described by a linear line with a slope

of 0.94 meV/T. Similar linear dependence of ±2 on B are obtained in the literature, with a

slope of 1 - 1.3 meV/T [172, 184, 192, 193]. These observations support the association of

the LL gap at = 2 with the spin degree of freedom, although we caution against a literary

interpretation of the gap as simply exchange-enhanced Zeeman splitting as = 2 may be

a many-body coherent ground state.

Figure 5-9. (a) Two-terminal conductance versus filling factors taken at selected D and B field as

labelled in the figure. In Device 1. (b) Calculated LLs for the E = 0 octet of bilayer graphene at D =

0.2 V/nm as an evolution of B field. (c) Calculated LLs at B = 12 T as an evolution of D field. (d)

170

Calculated energy gap for ν = ±1 and ±3 for a singly gated device where Dν=0 = 220 mV/nm as an

evolution of B field.

The appearance of D*l at B > 12 T enables us to determine the magnitude of E10 using

D*l+ 2E10 (D

*l, B) = s (B). Figure 5-8b plots E10 obtained from devices 06 and 24. E10

is 0.2 meV at D = 96 mV/nm (B = 12 T) and rapidly increases to 5.2 meV at D = 167

mV/nm (B = 31 T). Here the electric field and magnetic field vary codependently.

Motivated by the reported linear B-dependence of the = 1 gap in the literature [188, 192-

194], we hypothesize that E10/ B is a pure function of D and obtain its functional form

empirically. Figure 5-8c plots E10/B and a fit to the data. E10 appears to be a non-linear

function of D and only rises sharply after a large threshold D-field is reached. We obtain

E10 (meV)/B = 0.058(√𝐷 − 9.43) in the regime of D 96 mV/nm, with the choice of the

√𝐷 form motivated by the apparent linear dependence.

5.2.3 Using our empirical LL diagram to explain measurements in literature

The magnitude and D-dependence of E10 we obtained immediately explains a few

experimental findings regarding the size of the = 1 and = 3 gaps. 1 is often too

small to be measured experimentally and the reported values vary widely [188, 192-194].

In a single-gated sample without unintentional doping, the D field increases linearly with

doping and for = 1, D = 2.2B (mV/nm) is always small. Consequently, the gap at = 1

is minute and difficult to detect experimentally. However, uncontrolled unintentional

doping in single-gated samples can lead to an offset D-field D0 that serves to enhance 1

while generating an extrinsic asymmetry between +1 and -1, but also produce variations

among samples. Simultaneous knowledge of D and 1 can serve as a test on our empirical

expression of E10. For example, in Ref. [188], Kou et al reported a large constant gap of ~

171

25 meV at = 0. We interpret the = 0 gap as arising from an unintentional D0 ~ 220

mV/nm. Consequently, our E10 would predict asymmetric energy gap for ν = ±1 and ν =

±3 as shown in Fig. 5-9d, i.e. the energy gaps for negative filling factors are larger than

their positive counterpart. Furthermore, our calculated energy gaps can quantitatively

capture their measured gap values.

The D-dependence of E10 suggests that a large D-field can effectively enhance 1. This

is indeed the case. In Fig. 5-9a, we show the appearance of = 1 in magnetic fields as

low as 4 T at D = 0.2 V/nm.

The knowledge of D, s (B) and E10 (D, B) enables us to construct a quantitative

LL diagram of bilayer graphene at any arbitrary electric and magnetic fields. Figure 4(a)

plots an example at B = 31 T and as a function of varying D-field. A few more scenarios

(lower B-field or varying B at fixed D) are shown in Figs. 5-9b and 5-9c. The two

coincidence fields D*h and D*

l for the = 0 state are evident in Fig. 5-10a and their values

match data well at all fields (Black dashed lines in Fig. 5-8a). This is expected since we

have reverse-engineered our diagram based on these observations. In addition, the diagram

predicts the closing of the = 1 gaps at D*1, the value of which is calculated and plot in

Fig. 5-8a as a dark yellow dashed line. Also plotted there are our measurements for D*-1

(blue stars) obtained in device 43 using maps similar to that shown in Fig. 5-7a, and data

points obtained by Hunt et al [182] at B = 31 T (green stars). The calculated D*1 is e-h

symmetric and captures the average of the measured D*+1 and D*

-1 very well. However

both our data and that of Ref. [182] deviate from the calculated curve with D*-1 tending

towards D*l and D*

+1 tending towards D*h. This intrinsic asymmetry between = 1 is

172

illustrated in the inset of Fig. 5-10a, which is also reflected in the gap energies of = 1

discussed earlier [188].

Figure 5-10. (a) Calculated LLs for the E = 0 octet of bilayer graphene at B = 31 T as an evolution

of D field where D 96 mV/nm. Quantum states with corresponding color are labelled on the right.

(b) A sketch of LL diagram at small D field based on gathered experimental results from literature.

Circles show coincident points for ν = 0 and ±1 states (a) and ν = -2 states (b). Inset in (a), a sketch

shows the e-h asymmetry effect on LL diagrams. Color bars above LL diagrams show five different

D-dependent filling sequences also reported in Ref. [182].

Our measurements are not able to resolve small value of E10 at D < 96 mV/nm. It’s

attempting to smoothly extrapolate E10 to zero at D = 0, as the dashed lines in Fig. 5-10a

show. However, coincidence measurements reported for = 2 [172, 182, 183] suggest

that this is not the case. D*-2 is shown to occurs at approximately 27 mV/nm at 31 T [182]

and exhibits a rough slope of ~ 0.9 mV/nm/T at lower field [172, 183]. This information,

173

together with the filling sequence of the LLs observed at small D fields in Ref. [182], and

the closing of the = 1 and 3 gaps at D* = 0 [182, 183], led us to a sketch of the LLs at

small D fields, as shown in Fig. 5-10b. In this diagram, D*-2 occurs at E+,0 = E-,1, where the

= -2 state transitions from an orbital isospin polarized state to a valley polarized state.

We estimate E10 (D = 0) to be ~ 0.1B meV, using = 0.13D*-2 and D*

-2 ~ 0.8 mV/nm/T.

This estimate is roughly 20% of the E10 given by the four-band tight-binding model [173].

A more quantitative understanding of this part of the LL diagram would require careful

measurements of E10, especially at low D-fields. An accurate knowledge of E10 would

greatly benefit the understanding and control of even-denominator fractional quantum Hall

states in bilayer graphene, which so far only occur in the n = 1 orbitals [114, 187, 189].

The LL diagrams shown in Figs. 5-10a and 5-10b together can successfully reproduce

the five D-dependent filling sequences reported in Ref. [182] for the E = 0 octet. They are

illustrated above the diagrams, using the same color scheme as Ref. [182] to facilitate a

direct comparison. The agreement is quite remarkable and attests to the strength of the

effective single-particle approach in capturing many features of the many-body problem.

5.2.4 Limitation of our empirical LL diagram and section summary

The simplicity of the single-particle approach, however, also leads to visible

deficiencies. Apart from the absence of the experimentally observed e-h asymmetry in D*1

and D*2, and the -dependent D* within a LL [182], the simple diagram is likely to fail

near crossing points, where LLs in close proximity to each other can potentially order into

coherent quantum Hall ferromagnets [172, 194]. Another clear failure is the spin

configuration of the = 0 state at low D-field. Our diagram predicts a spin ferromagnet

174

while experiments point to a putative canted antiferromagnet with spin-valley coherence

[78, 174, 175]. Clearly, more sophisticated theoretical tools and measurements are needed

to illuminate the rich quantum Hall physics bilayer graphene has to offer.

In summary, we constructed an empirical LL diagram for the E = 0 octet of bilayer

graphene in the presence of a perpendicular magnetic field and a transverse electric field.

This diagram offers a unified, intuitive framework to interpret a large number of

experimental findings to date, complete with quantitative energy scales. We expect it

serves as a good base to launch future experiments and calculations.

5.3 Gate-controlled transmission of quantum Hall edges in bilayer graphene

5.3.1 Motivation of this study

It has been a long history that people are studying quantum Hall edge tunneling

problems since the discovery of quantum Hall and fractional quantum Hall effect (FQHE)

in high quality 2DEGs in GaAs [166, 195]. As a semiconductor, one can use gate structures

to position the Fermi level into the band gap, therefore deplete part of the sample and

constrain the spatial distribution of carriers. Initially a line junction structure is used to

create a narrow depleted region as a barrier between two adjacent quantum Hall systems

[196]. Later quantum point contact structures and interferometers are used to study the

interaction of quantum Hall edge modes and the exotic properties of quasiparticle in FQHE.

With all these tools, people have probed the 1/3 fractional charge [197] and fractional

statistics for 5/2 states [198]. However, the gating in this system is somewhat weak since

the distance between the 2DEG and gate is on the order of 1 μm, which leads to complex

edge reconstruction therefore make it hard to compare experiments with theories. In this

section we demonstrate quantum Hall edge tunneling experiments in graphene systems,

175

where we can realize much sharper gating using thinner dielectrics. In addition, the flexible

gating in graphene systems enables us to independently control the filling factors on the

two sides of tunnel barriers.

Figure 5-11. (a) Schematic shows a typical global back gate plus local top gate device. Black curve

shows a smoothly potential profile across the junction area. (b) and (c) Schematic diagrams show

quantum Hall edge modes fully mix in a bi-polar (b) and uni-polar junction (c). Adapted and modified

from Ref. [168]

Since the advent of graphene, numerous investigations on the quantum Hall edge

tunneling have been carried out [168, 169, 199-201]. Without a band gap, it’s very

challenging to electrostatically confine Dirac fermions. So far, most experiments adopt one

global gate plus local gates structures, an example is shown in Fig. 5-11a. This gating

structure leaves the junction potential uncontrolled, and usually leads to a smooth junction

potential on the length scale of gate dielectrics. Also many experiments only measure a

two terminal conductance, therefore the results are not conclusive. With a smoothly

connected potential across the junction, quantum Hall edge modes from the two sides of

176

the junction can mix well. Figure 5-11b shows a schematic diagram of such bi-polar

junction, where the edge modes from the left and right flow along the same direction,

completely mix and become indistinguishable in the junction region, therefore leads to a

two terminal conductance of h

eG

2

21

21

as discussed in Section 5.1. In unipolar

junctions as shown in Fig. 5-11c, the number of smaller filling factor edge modes are fully

mixed and get eliminated from the junction, and this leads to the measured two terminal

conductance showing the smaller filling factor. The fully mixing scenario captures the

majority of the current experiments in graphene systems. A better control in the junction

area is needed to pursue more sophisticated experiments discussed in the previous

paragraph. In the rest of this section we will show a controlled tunnel junction in bilayer

graphene nanostructures.


We build a dual-split-gated bilayer graphene device with a global silicon back gate as

illustrated in Fig. 5-12a. The global backgate can control the potential barrier in the line

junction, also we can tune the filling factor on the two sides independently. Furthermore,

we can control the energies of Landau levels by controlling the displacement field of DL

and DR. With this suite of control knobs, we study quantum Hall edges tunneling across

this tunable potential barrier and we will focus on the tunneling effects in the 8-fold zeroth

LL. The device fabrication is similar with that described in Section 4.2. First we exfoliate

a graphite flake on to silica, then etch it into a split gate structure. Bottom h-BN flake is

then transferred, followed by transferring bilayer graphene flake. Then we etch the

graphene to define device geometry, afterwards a top h-BN flake is transferred. Then Ti/Au

177

electrodes are made with e-beam lithography, and metal top split gates are separately made

with good alignment precision. An optical image of finished device is shown in Fig. 5-12b.

Figure 5-12. Schematic of a dual-split gate plus global back gate device (a), and an optical image

of such device (b). (c) and (d) shows the dimensions in the device.

As shown in Figs. 5-12c and 5-12d, the h-BN dielectrics thickness are 20 and 28nm,

the Hall bar width is 300nm and our split junction size is 70nm. The dielectric is thin and

the junction width is small, therefore we can obtain sharp gating profile and wide range of

tunability of the junction potential profile. With all these we can obtain good Quantum Hall

States on both sides, and study how they tunnel though the line junction barrier.

Figure 5-13 is a schematics showing a unipolar junction, and let’s assume νR is larger

than νL and they carries the same sign. Source and drain are on the two ends of the device,

we measure the tunnel resistance RT across the junction and it’s defined as the resistance

measured along the side where more edges flows into the less. We can also monitor Rxx

and Rxy in the two sides of the sample. The parameter α shows the fraction of left edges

flow back, and 1- α shows the fraction of left edges go through the barrier. In Section 5.1,

178

Figure 5-13. Schematic shows the measurement setup.

we have calculated RT and the opposite side resistance R34 using Landauer Büttiker

equation and the results are displayed in Eq. 5-12. The subtraction of the two resistance

yields a simple formula independent of α, and this can check whether we have good

quantum Hall on the two sides. In the case α = 0, this means all edge can perfectly go

through the barrier, and RT vanishes. On the other hand α = 1 means all edges are reflected

back, and we call this complete pinch-off, and the two quantum Hall systems are isolated,

meanwhile RT becomes very large. We are going to explore the edges get pinched off one

by one.

As discussed in details in Section 5.2, in bilayer graphene, both n = 0 and 1 orbital

gives E = 0. The competing valley, spin and orbital ordering makes this 8-fold degenerated

E = 0 LLs extremely interesting. A perpendicular D field brings potential energy difference

on the two layers of graphene therefore breaks the layer degeneracy, opens up a band gap

Δ and separates the LLs into two groups. Interaction, large B and D field also introduce a

non-zero E10 term that lifts the orbital degeneracy. Together with interaction enhanced

Zeeman energies, which break the spin degeneracy, all 8 degeneracy can be fully lifted,

and we can observe integer quantum Hall states with filling factor incensement of 1 at large

B and D field.

179

5.3.3 Perfect and sequential transmission of quantum Hall edges

We first replicate the fully mixing of quantum Hall edge states experiments in our

devices. Figure 5-14a presents examples of perfect transmission of quantum Hall edge

modes in the limit of α = 0. We fix νL = -4, and vary νR from -12 to -1, and the silicon gate

voltage VSi is fixed at -40 V, which is a good choice for promoting perfect transmission of

edge states. When the right side has more edge modes, based on our definition RT should

Figure 5-14. (a) RT (VRT) with fixed νL = -4, νR varies from -12 to -4 (black curve) and from -4 to -1

(red curve). Inset shows the definition of RT for the two scenario with corresponding color. (b)

Absolute value of R34 – R12 as an evolution of VRT (solid curve) and calculated curve following (1/νL

- 1/νR)h/e2 (dashed curve). Well developed νR is labelled in the figures. B = 18 T, T = 300 mK.

read as R12, and we indeed observes vanishing RT when quantum Hall is well developed

on the right such as νR = -12, -8 and -4. When right side has less edge modes, we now read

R34 as RT, and again RT = 0 is observed when quantum Hall states are well established on

the right side, e.g. νR = -3 and -1. There is a special case when νL = νR = -4, both the black

and red curves represents RT and they both vanishes. One more thing to check, we subtract

R34 and R12 as plotted in blue curve in Fig. 5-14b. Calculated curve following (1/νL - 1/νR)

h/e2 is also plotted in dashed line, and indeed the measured curve is consistent with the

180

calculated curve when quantum Hall states are well established on the right side of the

device. With the great tunability of device, we can further control the potential in the line

junction and explore the quantum Hall edge modes transmission in a controlled fashion.

Figure 5-15. (a) RT as a function of VSi at B = 8.9 T with fixed νL = -8 and νR = -4. RT undergoes

three stages illustrated with schematics from left to right as VSi varies from -60 V to 0 V, which are

3-zone regime (b), perfect transmission regime (c) and towards pinch-off regime (d).

Now we proceed to operate the control knob in the line junction, which is VSi that

defines the junction potential profile. Figure 5-15 shows an example of measuring RT as a

function of VSi at B = 8.9 T, and filling factors on the two sides are fixed at νL = -8 and νR

= -4. The behavior of RT can be classified into three regimes marked in Fig. 5-15a. When

-46 V < VSi < -22 V, RT stays at zero independent of VSi, suggests a scenario that edge

181

modes from the two sides fully mix in the junction. We call this range perfect transmission

regime, the corresponding schematic diagrams showing the edge modes routing scheme

and band structures across the device are shown in Fig. 5-15c. The orange shade in the

diagrams indicate the junction area, dashed line indicates EF in the device. At relative low

magnetic field, the LLs in bilayer graphene carry 4-fold degeneracy as the valley and spin

degeneracy remains. The lines inside the junction area indicate the potential profile that

connects the two sets of LLs. In the perfect transmission regime, two sets of LLs are

smoothly connected within the junction. EF cuts through one set of band, hence there are

four edge modes traveling out-of-paper direction towards the left boundary of the junction,

while the rest of edge states from both side mix well and get “eliminated”. As VSi further

increases, the potential in the junction deeps below EF as illustrated in Fig. 5-15d. EF cuts

through two bands and one band respectively on the left and right side of the junction,

therefore there are two sets, i.e. eight, edge modes traveling out-of-paper direction towards

the left boundary of the junction, while one set, i.e. four, edge modes traveling into-paper

direction towards the right boundary of the junction. The two quantum Hall systems seem

isolated, and RT increases with VSi. We call this range of VSi towards pinch-off regime

because edge modes are getting reflected by the junction. When VSi is very low, e.g. VSi <

-46 V in this case, the junction potential is pulled up as illustrated in Fig. 5-15b. We can

compare this situation to a three-zone bi-polar junction problem discussed in Section 5.1,

and indeed our measured RT qualitatively follows the trend described in Eq. 5-15. We call

this range of VSi 3-zone regime. We are especially interested in the towards pinch-off

regime, as it demonstrates a possibility of sequentially transmitting edge states. Large B

182

field can lift the LL degeneracy, and enables us to observe edge mode gets pinched off one

by one.

Figure 5-16. RT as a function of VSi in a semi-log scale measured at B = 18 T with fixed νL = -8 and

νR = -4. DL = -0.2 V/nm, DR varies from -0.36 V/nm to -0.58 V/nm marked in figure. Schematics

show the evolution of the potential profile in the junction area, which leads to sequential pinch off

of edge state.

Figure 5-16 plots a set of RT as a function of VSi measured at B = 18 T, we fix the filling

factors on the two sides at νL = -8 and νR = -4. DL is fixed at -0.2 V/nm, DR varies from -

0.36 V/nm to -0.58 V/nm. As VSi increases, RT establishes a sequence of plateau features

at approximately 1/12 h/e2, 1/4 h/e2 and 3/4 h/e2, which correspond to α = 1/4, 2/4 and 3/4

respectively. Schematic diagrams in Fig. 5-16 shows the evolution of transmission rate of

quantum Hall edge modes from perfect transmission to sequential pinch off. When VSi is

183

very negative, the LLs of the two sides are smoothly connected, this corresponds to α = 0.

As VSi increases, the potential in the junction is pulled down with one LL passing through

EF, which leads to 1 out of 4 edge modes gets reflected by the junction. Similarly, as VSi

further increases, more and more LLs pass through EF and leads to more edge modes get

reflected, until all of them reach EF when all edge modes get reflected and RT becomes very

large.

Figure 5-17. (a) and (b) Schematic diagrams show the 2-fold LL degeneracy gets lifted as DR

increases from 0.05 V/nm to more than 0.2 V/nm. (c) RT as a function of VSi measured at B = 18 T

with fixed νL = -4 and νR = -2. DL = -0.2 V/nm, DR is -0.05 V/nm (red), -0.2 V/nm (blue) and -0.3

V/nm (olive). Insets show three different scenarios of edge states mixing in the junction. From left

to right, perfect transmission, one edge mode gets reflected, both edge modes get reflected by the

junction. (d) Schematic shows small misalignment of the top right gate.

Thanks to the great tunability of our device, we can also modify the LL structure by

controlling the D field. Let’s take transmission of edge modes between νL = -4 and νR = -2

184

as an example. The schematic diagrams in Fig. 5-17a and 5-17b illustrate D field increases

E10, therefore removes the 2-fold degeneracy. Figure 5-17c plots a sequence of RT as a

function of VSi with same fixed DL = -0.2 V/nm, and DR varies from -0.05 V/nm to -0.3

V/nm. The LLs in the right side of sample carry a 2-fold degeneracy with DR = -0.05 V/nm,

hence the degenerated 2 edge modes get pinched off together and explains why RT does

not plateau at 1/2 h/e2. Since the 2-fold degeneracy is lifted by larger D field, sequential

pinch off of edge modes are observed in the blue and olive curves in Fig. 5-17c. Due to the

small misalignment of the top split gate illustrated in Fig. 5-17d, the increasing DR also

dopes more negative charges into the junction, which means VSi needs to shift towards

negative to compensate this effect, and this qualitatively explains why the curves in Fig. 5-

17c shifts towards negative VSi direction as D field increases.

In conclusion, we have demonstrated using a gate-controlled potential barrier to

transmit individual quantum Hall edge mode in bilayer graphene devices. We can further

modify the arrangement of gates to build quantum point contact structures in bilayer

graphene to study the non-Abelian properties of the quasiparticle in recently discovered

robust even denominator fractional quantum Hall states [114, 187, 189].

185

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VITA

Jing Li

Education

Ph.D. in Physics, 2017, The Pennsylvania State University, United States of America

B.S. in Physics, 2010, Wuhan University, Wuhan, China

Awards and Fellowships

David C. Duncan Graduate Fellowship in Physics, Then Pennsylvania State University, 2014-

2015

David H. Rank Memorial Physics Award, The Pennsylvania State University, 2011

Duncan D Grad Fellow Scholarship and Braddock Homer Fellow Scholarship, The Pennsylvania

State University, 2010

The National Scholarship, Ministry of Education of China, 2007

The first-class People’s Scholarship, Wuhan University, 2007-2010

Publications

J. Li, K. Wang, K. McFaul, Z. Zern, Y. F. Ren, K. Watanabe, T. Taniguchi, Z. H. Qiao, J. Zhu,

“Gate-controlled topological conducting channels in bilayer graphene”, Nature Nanotech. 11,

1060–1065 (2016).

J. Li, L. Z. Tan, K. Zou, A. A. Stabile, D. J. Seiwell, K. Watanabe, T. Taniguchi, S. G. Louie, J.

Zhu, “Effective Mass in Bilayer Graphene at Low Carrier Densities: the Role of Potential Disorder

and Electron-Electron Interaction”, Phys. Rev. B Rapid Comm. 94, 161406(R) (2016).

A. A. Stabile, A. Ferreira, J. Li, N. M. R. Peres and J. Zhu, “Electrically tunable resonant scattering

in fluorinated bilayer graphene”, Phys. Rev. B Rapid Comm. 92, 121411(R) (2015).

J. Li, R. Zhang, Z. Yin, J. Zhang, K. Watanabe, T. Taniguchi, C. Liu, J. Zhu, “A valley valve and

electron splitter in bilayer graphene”, submitted (2017).

J. Li, Y. Tupikov, K. Watanabe, T. Taniguchi, J. Zhu, “An effective Landau level diagram of bilayer

graphene”, submitted (2017).

J. Li, H. Wen, K. Watanabe, T. Taniguchi, J. Zhu, “Gate-controlled transmission of quantum Hall

edge states in bilayer graphene”, manuscript in preparation.

J. Li, Z. Yin, K. Watanabe, T. Taniguchi, J. Zhu, “Temperature dependent study on the ν = 0 state

in CAF and Ferromagnet phase in bilayer graphene”, manuscript in preparation.

a transport study of emerging phenomena in bilayer

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