TRANSPORT MEASUREMENTS ON GRAPHENE P-N A …gn245vw3601/... · Klein tunneling is a relativistic...

TRANSPORT MEASUREMENTS ON GRAPHENE P-N

JUNCTIONS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Nimrod Stander

August 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/gn245vw3601

© 2010 by Nimrod Stander. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/gn245vw3601

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

David Goldhaber-Gordon, Primary Adviser


Malcolm Beasley


Aharon Kapitulnik

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

iv

Abstract

Klein tunneling is a relativistic quantum effect standing at the base of the Quantum

Electrodynamics (QED) theory. It describes a relativistic process where a particle

obeying the Dirac equation - called a Dirac Fermion - propagates through a strong

repulsive potential step without the exponential attenuation typical to quantum tun-

neling. Counter-intuitively, the stronger the potential step, the more transparent it

becomes for the particle. Although the QED theory had proven to be a successful

theory describing reality accurately, Klein tunneling has not yet been observed ex-

perimentally. The extreme conditions required to observe such an effect may occur

in collisions of super-heavy nuclei or pair production at the horizon of a black hole,

difficult or impossible to realize in a laboratory. However, a recently discovered ma-

terial called graphene, which is a one atom-thick lattice of carbons, can be used as a

testbed for such an effect. Since graphene carriers have been proven to be Dirac-like

Fermions, the analog of Klein tunneling can be investigated in a commonly used con-

densed matter setup via electrical transport measurements. In this work we present

experimental evidence for graphene carriers obeying Klein tunneling transport using

a set of metallic gates on graphene to create p-n junctions. These p-n junctions in-

corporate a potential step for graphene Dirac-like Fermions allowing us to investigate

Klein tunneling in graphene. Finally, we also discuss the existence of p-n junction-

like structures between metal contacts and graphene, a topic that has an impact on

graphene’s electronic device applications.

v

Epigraph

vi

Acknowledgements

My work could not have been accomplished without the help and support I have been

given from the following people:

David Goldhaber-Gordon, my advisor, who guided me through all stages of the

PhD. I met David while applying to graduate schools. That brief meeting and later

my visit at Stanford in his lab left no doubt in my mind that David’s lab is where I

would want to pursue my PhD. I was fortunate to be admitted to Stanford and even

more fortunate to start working in David’s lab already the summer before school

started. David not only guided me in research but also served as a mentor to me in

every aspect of the Stanford graduate life.

Benjamin Huard, the best experimentalist I have ever worked with and a true

friend. Working with Benjamin during my first two years, when he was a post-doc in

our lab had a deep impact on me in many aspects.

Joseph Sulpizio, who was involved in many of the projects I worked on during my

studies and spent countless hours in discussing physics with me and helping me with

his great knowledge and expertise on the different lab tools.

Malcolm Beasley, whom I encountered at different points of my PhD career as a

guide and as a collaborator. He gave me sage advice as well as support to keep on

going.

Alexander Fetter, for whom I had the pleasure to work for as a teaching assistant,

and to learn from a great deal about physics as well as ways to approach physics

teaching.

All DGG lab members in the past and present, I am grateful and proud for being

part of the group.

vii

Special thank you to the William R. and Sara Hart Kimball Stanford Graduate

Fellowship which allowed me to be fully dedicated to research in my first three years

at David’s lab.

Finally to my dearest family who supported me with unconditional love. To my

parents, Amnon and Tova Stander, who were with us all the time although living

physically abroad in Israel. To My wife, Shirlee, who supported me throughout the

ups and downs of my studies with infinite love and constant care. And to our beloved

daughter, Lior, who brought unimaginable joy to our lives in the past year.

viii

Contents

Abstract v

Epigraph vi

Acknowledgements vii

1 Why is graphene special? 1

1.1 The missing link - The emergence of a new 2D material . . . . . . . . 1

1.1.1 Seeing a single sheet optically and layers characterization by

Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Band Structure - Massless-Dirac Fermions . . . . . . . . . . . . . . . 6

1.2.1 How the bands bend - Survey of graphene band theory . . . . 6

1.2.2 New degree of freedom - Pseudo-Spin and the absence of backscat-

tering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Integer Quantum Hall Effect in 2DEG - The formation of Lan-

dau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Half Integer Quantum Hall Effect in graphene - Indeed Massless

Dirac Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Disorder in graphene - mobility limiting mechanisms . . . . . . . . . 13

1.4.1 Charge impurities . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.2 Short-range disorder . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.3 Corrugations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

ix

2 Klein tunneling, Klein Paradox and their analog in graphene 16

2.1 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 1D problem with Dirac equation - Transmission and reflection coefficients 17

2.2.1 Particle-antparticle production . . . . . . . . . . . . . . . . . . 17

2.2.2 The Klein Paradox . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Klein tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 The analogs of Klein tunneling and Klein Paradox in graphene . . . . 20

2.3.1 Creating potential steps with graphene p-n junctions . . . . . 20

2.3.2 Reducing the 2D problem into 1D problem . . . . . . . . . . . 21

2.3.3 Relativistic fermions vs. graphene carriers . . . . . . . . . . . 22

2.3.4 Realistic p-n junctions - including non-linear screening and dis-

order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.5 Sharp p-n junctions . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Transport measurements across a tunable potential barrier in graphene 26

3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Creating p-n junctions in graphene . . . . . . . . . . . . . . . . . . . 27

3.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1 Quantifying the p-n junction resistance . . . . . . . . . . . . . 30

3.5 Comparison to theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Magnetic field results . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9.1 Imaging, Raman spectroscopy, and identification of single-layer

graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9.2 Determination of the densities in regions 1 and 2 . . . . . . . 41

3.9.3 Estimation of the mobility . . . . . . . . . . . . . . . . . . . . 42

3.9.4 Measurements at 77 K . . . . . . . . . . . . . . . . . . . . . . 42

3.9.5 Magneto-resistance measurements . . . . . . . . . . . . . . . . 43

x

4 Evidence for Klein tunneling in graphene p-n junctions 47

4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47



4.5 Magnetic field results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


4.8 Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.8.1 Graphene characterization . . . . . . . . . . . . . . . . . . . . 58

4.8.2 Extracting the odd part of the resistance . . . . . . . . . . . . 58

4.8.3 Comparing the experimental value Rodd to theoretical models

of the junction interface resistance . . . . . . . . . . . . . . . . 61

4.8.4 Multiple reflections between interfaces of a potential barrier . 63

4.8.5 Without phase coherence . . . . . . . . . . . . . . . . . . . . . 63

4.8.6 Including phase coherence . . . . . . . . . . . . . . . . . . . . 64

4.8.7 p-n junctions in finite magnetic field . . . . . . . . . . . . . . 65

4.8.8 Fabrication details . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Sharp p-n junctions 68

5.1 Fabrication process of thin dielectrics . . . . . . . . . . . . . . . . . . 68

5.2 Determining the thickness . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Measurements on thin dielectrics . . . . . . . . . . . . . . . . . . . . 72

5.4 Results discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Evidence of the role of contacts on the observed electron-hole asym-

metry in graphene 78

6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79



6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xi


6.7 Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.7.1 Graphene characterization . . . . . . . . . . . . . . . . . . . . 89

6.7.2 Resistance of a p-n junction in graphene . . . . . . . . . . . . 90

6.7.3 Sharp barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.7.4 Smooth barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.7.5 Case of partly invasive probes . . . . . . . . . . . . . . . . . . 92

6.7.6 Wide probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.7.7 Narrow probes . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7.8 Single metal/graphene interface . . . . . . . . . . . . . . . . . 95

6.7.9 Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . 95

6.7.10 Fabrication details . . . . . . . . . . . . . . . . . . . . . . . . 95

A Electronics 98

A.1 Lead connection protocol . . . . . . . . . . . . . . . . . . . . . . . . . 98

A.2 Sample measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B Fabrication of fine features using our lab-shared SEM 101

C 3-axe magnet Dewar procedures 103

C.1 Cooling the Dewar from 300 K to 77 K . . . . . . . . . . . . . . . . . 103

C.2 Cooling the Dewar from 77 K to 4 K . . . . . . . . . . . . . . . . . . 103

C.3 Transfer while at 4K . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C.4 Level monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

C.5 Magnet control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

D Thermal oxidation of Si wafers 108

xii

List of Tables

4.1 Geometrical properties of the samples: top gate length L, graphene

strip width (interface length) w, and top gate dielectric thickness d.

Same letter for two device labels indicates same graphene sheet. All

dimensions were taken by both Scanning Electron Microscopy and

Atomic Force Microscopy. The transition parameter β between clean

and diffusive transport in a single p-n junction is also shown (see text),

averaged across the whole measured voltage range such that nbg < 0

and ntg > 0. Counter-intuitively, despite devices’ low mobility, β ≫ 1

so that Klein tunneling is expected rather than diffusion across the

interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Geometrical properties of the samples: L- top gate length, w- interface

width, and d- top gate dielectric thickness. Same letter for two devices

indicates same graphene sheet. All dimensions were taken by Scanning

Electron Microscope (SEM) and Atomic Force Microscope (AFM) im-

ages. The gate voltage offsets V 0bg and V 0

tg and the capacitance of the

top gate determined from the procedure described in the text are re-

ported here. The mobility µ is estimated from the slope at the origin

of the conductance measured as a function of back gate voltage. These

low values are due to the PMMA cross-linking step. . . . . . . . . . 60

6.1 Geometrical properties of the samples corresponding to Fig. 6.4. The

measurements shown on Figs. 6.1,6.2,6.3 were performed on TiAu1.

The type of metal used as a probe and its thickness is given here

together with the length w of the graphene/metal interface. . . . . . 85

xiii

List of Figures

1.1 Color plot of the contrast as a function of wavelength and SiO2 thick-

ness. Adapted from Blake et al. . . . . . . . . . . . . . . . . . . . . 3

1.2 Diagram of typical Si/SiO2 substrate with exfoliated graphene on top

(in black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Optical image of one of our graphene samples before doing any fabrica-

tion process. The image shows graphene flake and next to it a thicker

graphitic flake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 G and 2D peaks of a single layer graphene and graphite. Adapted from

Ferrari et al.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Left: The honeycomb lattice in real space blue and red dots repre-

sent ‘A’ and ‘B’ lattice sites respectively. One choice of unit cell is

demarcated with a dashed line. Coordinate axes oriented to give the

Pauli and Dirac forms of the electronic Hamiltonian are labeled P and

D. Adapted from Mecklenburg and Regan. Right: Energy contours in

the first Birlioun zone. The thin arrows point to the K+valley and b1

and b2 are the basis vectors for the reciprocal space. Adapted from

Mecklenburg and Regan. . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Pseudo-Spin in graphene: Left: Bloch sphere representation of the

pseudo-spin. θ is the same as the one showing up in Eq. 1.4. Right:

2D Momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Predictions for Quantum Hall Effect: Left: Graphene. Half inte-

ger plateaus and zeroth Landau level Right: 2DEG. Adapted from

Novoselov et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

xiv

2.1 1D problem, sharp case schematic. See condition in the text. . . . . 18

2.2 1D problem, smooth case schematic. See condition in the text. . . . 20

2.3 Left: Bipolar junction in graphene Right: Unipolar junction in graphene

21

2.4 Symmetric p-n junction. F is the constant electric field for linear slope

potential with characteristics length of d. . . . . . . . . . . . . . . . 23

2.5 T as a function of angle of incident with respect to the p-n interface. 23

3.1 a) cross-section view of the top gate device. b) simplified model for the

electrochemical potential U of electrons in graphene along the cross-

section of a). The potential is shifted in region 2 by the top gate

voltage and shifted in both regions 1 and 2 by the back gate voltage.

c) Optical image of the device. The barely visible graphene is outlined

with a dashed line and the PMMA layer appears as a blue shadow.

A schematic of the four-terminal measurement setup used throughout

the paper is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 a) Resistance across the graphene sample at 4 K as a function of the

top gate voltage for several back gate voltages, each denoted by a dif-

ferent color. b) Two-dimensional greyscale plot of the same resistance

as a function of both gate voltages. Traces in a) are cuts along the

correspondingly-colored lines. . . . . . . . . . . . . . . . . . . . . . . 29

3.3 a) Odd component of the resistance: the part which depends on the sign

of the density n2 in region 2. b) Greyscale plot of the odd component of

the resistance for many values of n1. Colored lines are the cuts shown

in a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xv

3.4 a) Using the strongly diffusive model described in the text, one can

predict the resistance as a function of the top gate voltage Vt for sev-

eral values of the density n1 (each represented by the same color as in

Fig. 6.3). Here we plot the odd part of this calculated resistance (cf.

Fig. 6.3) for a barrier smoothness d = 40 nm and a width w = 1.7 µm.

b) Similar curves for several densities n1 using the ballistic model de-

scribed in the text. The curves are plotted only for densities n2 not

too close to zero: n2 > 1011 cm−2 (the model diverges at n2 = 0) and

dashed lines link the curves at opposite sides of n2 = 0. . . . . . . . 33

3.5 Odd part of the resistance as a function of the back gate voltage Vb,

with the constraint n2 = −n1. Measurements were taken at 4 K for

several values of the perpendicular magnetic field B. . . . . . . . . . . 35

3.6 a) Optical image of nearby graphitic flake with labeled thin region ➊

and thick region ➋. b) Optical image of bare graphene sheet (labeled

region ➌) used to create the top-gated device described in the letter.

Note the strong color similarity between the barely visible regions ➊

and ➌, suggesting the graphene thickness in these regions is close. c)

Spatial map of the width of the 2D peak for regions ➊ and ➋. Region

➊ appears less bright than region ➋, indicating a narrower 2D peak

and thinner graphene. d) Spatial map of 2D peak width for region ➌.

The color scale has been chosen differently from c) in order to see the

Au contacts appear as dark lines. . . . . . . . . . . . . . . . . . . . . 37

3.7 Dots: spatially averaged Raman spectra of region ➌ (see Fig. 3.6)

subtracted from a nearly flat background due to the presence of gold

leads. Left panel–line: best fit of the dots using a Lorentzian function

Eq. (3.3). The number gives the integrated intensity of each peak.

Right panel–lines: best fit of the dots using the sum of four Lorentzian

functions Eq. (3.3) using a fixed value for the spacing between the

peaks following Table I of Ref. [15]. The position of the peaks relative

to the 2D1B peak are 34 cm−1 for 2D1A, 54 cm−1 for 2D2A and 69 cm−1

for 2D2B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

xvi

3.8 Spatially averaged Raman spectra of regions ➊, ➋ and ➌ (see Fig. 3.6)

for the G (left) and 2D (right) peaks. The intensity of the spectrum

of region ➌ is scaled differently than the two other spectra for clarity. 39

3.9 Scanning electron microscope image of the fabricated top-gate device.

Note the broken Au contacts due to the final destructive measurement

on this sample. The graphene sheet appears as the dark region in the

center of the contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.10 Dots: Voltage V maxt at which the resistance is maximal as a function

of the back gate voltage. Line: best fit with a straight line having the

same slope at high Vb as the experimental curve. . . . . . . . . . . . 41

3.11 Left: conductance measured at 4 K for n1 = n2 with a line showing a

curve of constant mobility µ = 1800 cm2V−1s−1. Right: conductance

measured at 77 K for n1 = n2 with a line showing a curve of constant

mobility µ = 3000 cm2V−1s−1. . . . . . . . . . . . . . . . . . . . . . . 42

3.12 a) Resistance across the graphene sample at 77 K as a function of the

top gate voltage, for several back gate voltages, each denoted by a

different color. b) Greyscale plot of the same resistance as a function

of both gate voltages. Traces in a) are cuts along the correspondingly-

colored lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.13 a) Odd component of the resistance: the part which depends on the

sign of the density n2 in region 2. Colors correspond to the same values

of n1 as in Fig. 6.2. b) Greyscale plot of the odd component of the

resistance for many values of n1. Colored lines are the cuts shown in a). 44

3.14 Magnetoresistance curves for various densities n1 = n2 measured at 4K. 44

xvii

3.15 Dots: Using the magneto-resistance curves measured at 17 values of

the density n1 = n2 between −3.0 × 1012 cm−2 to −2.3 × 1012 cm−2

and measured at 4 K, we plot the average value of the ratio of the

resistance R to the square of the mean value of R as a function of B√n.

This procedure allows one to average out the universal conductance

fluctuations so the weak localization peak appears more clearly (see

text). Line: theoretical prediction for the same quantity using Lϕ =

4 µm, Li−v = 150 nm and L⋆ = 50 nm in Eq. (3.4). . . . . . . . . . . 45

3.16 Measured magnetoresistance at Vb = −31.8 V and Vt = −1.5 V, thus

close to n1 = n2 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1 Schematic diagram of a top-gated graphene device with a 4-probe mea-

surement setup. Graphene sheet is black, metal contacts and gates

dark grey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 a) 4-probe resistance measured on device C540 (see Table 4.2), as a

function of Vbg and Vtg. The color scale can be inferred from the cuts

shown in b. The densities nbg and ∆ntg are estimated using V 0tg =

2.42 V, V 0bg = 18.65 V and Ctg = 107 nF.cm−2. b) Resistance as a

function of Vtg at several values of Vbg. The two bold curves show a

clear asymmetry with respect to the peak (ntg = 0) for both Vbg < V 0bg

(red) and Vbg > V 0bg (yellow). . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 a) The series resistance 2Rodd of the barrier interfaces as a function of

Vtg, for several values of Vbg for device A60 (corresponding densities nbg

are labeled). The measured resistance 2Rodd (dots) is compared to the

predicted value 2(Rpn − Rpp) using either a diffusive model, Eq. (4.2)

(dashed lines) or a ballistic model Eq. (4.3) with the value c1 = 1.35

chosen to best fit all six devices. (solid lines). b) Same as a) for device

C540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xviii

4.4 Symbols: ratio η = Rodd/ (Rpn − Rpp) as a function of top gate length

L for the devices of Table 1. Rpn is calculated with c1 = 1.35. The

vertical lines show the width of the histogram of η for densities such

that |nbg|,|ntg| > 1012cm−2. The dashed line at η = 1 corresponds to

perfect agreement between theory and experiment, in the case where

the total resistance is the sum of the resistances of two p-n interfaces

in series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 2-probe conductance corrected using the estimated contact resistance

for sample C540 at a magnetic field of 8 T and a temperature 4 K. . 58

4.6 Left: 3-dimensional schematic of representative device. Middle: Atomic

Force Microscope topograph of devices C540 and C1700. Right: Scan-

ning Electron Microscope image with 4-probe measurement scheme for

C540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 2Rodd as a function of Vtg for several nbg shown at the upper left corner

for sample C540. For each chosen nbg, we plot the corresponding curves

with Ctg/Cbg between 7.6 and 7.8 in steps of 0.02. This reflects a spread

of ±1% from the value of 7.7, which we use in the paper. . . . . . . 60

4.8 a) Histograms of η using a diffusive model (red) or a ballistic model

(blue) for device A60. The peak value ηpeak and peak width (2Γ) shown

in the figure were taken from a lorentzian fit Eq. (4.7) to each theory.

The histogram bins are 0.01 wide. b) Same as a) for device C540.

Using the diffusive theory for C540, we could not fit properly η thus

we report the standard deviation in eta as the value of Γ. . . . . . . . 62

4.9 The vertical lines show the spread of η when using diffusive model

(Eq. (2) in the main paper) for densities such that |nbg|,|ntg| > 1012cm−2.

the lines are centered on the average value of the histogram. . . . . . 62

4.10 a) (Rodd)−1 for device C540 as a function of magnetic field B for several

density profiles with nbg = −ntg (nbg is labeled). The theoretical

curves using Eq. (4.18) (solid lines) are fitted with l = 65 nm to the

experimental curves (dots).b) Same as a) for device C1700. The fitting

parameter used was l = 55 nm. . . . . . . . . . . . . . . . . . . . . . 66

xix

4.11 a) Gpnp = R−1pnp as a function of magnetic field for several nbg, with

ntg = −nbg, for device C540. nbg densities are presented on the right

hand side of the figure. b) R−1ppp = Gppp as a function of magnetic field

for several nbg, with ntg = nbg, for device C540. c)-d) Same as a) and

b) for device C1700. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 Left: AFM image of the graphene device with thin Alumina dielectric.

The red arrows point the two top gate metals. The dashed red lines

mark the graphene edges. Alumina is covering the entire area and

extends further out a few microns in each direction. Right: Cut taken

close to the Alumina edge (not shown in the left image). The peak due

to lift-off of a few nm’s is seen close to the step in the middle. . . . . 70

5.2 Step of 4.3nm between 4 layers of Alumina covered with gold and bare

substrate covered with gold. The profile is smoother because of the

additional layer of gold. . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Leakage current as a function of Vtg for the 5nm thin dielectric. . . . 73

5.4 Left: Resistance as a function of Vbg and Vtg. Right: Cuts at fixed Vbg

taken from the left plot . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Density as a function of Vbg (Dirac point chosen at Vbg = 0). Red:

Without the quantum capacitance correction. Green: Taking into ac-

count quantum capacitance correction . . . . . . . . . . . . . . . . . 74

5.6 Mean free path as a function of density. Electron-hole asymmetry

stems from the extended 4-probe geometry of the device. . . . . . . 75

5.7 Experiment vs. theory of Rodd as function of density in a symmetric

p-n junction. Dots: experiment. Solid blue line: Sharp theory. Solid

red line: Smooth theory. . . . . . . . . . . . . . . . . . . . . . . . . . 76

xx

5.8 Filled circles: Transmission coefficient as a function of angle at different

densities for a symmetric smooth p-n junction. Empty squares: The

dimensionless parameter - π~vFk2ye

−1F−1 - as a function of angle at

different densities for a symmetric p-n junction (F electric field at the

interface, e electron charge). Black line: Transmission coefficient as a

function of angle for a sharp p-n junction. . . . . . . . . . . . . . . . 77

6.1 a) 4-probe resistance calculated from voltages measured between in-

vasive probes as a function of gate voltage while a steady oscillating

current of 100 nA runs along the whole graphene sheet. Inset: Scanning

Electron Microscope image of the graphene sample TiAu1 connected to

Hall probes (A-D) and invasive probes (a-e). For clarity, graphene has

been colorized according according to topography measured by atomic

force microscopy. b) Given the charge neutrality gate voltage V 0g iden-

tified from Quantum Hall Effect measurements (see text), we plot here

the asymmetry between electrons and holes by showing the odd part

of the resistance defined in Eq. 6.1. . . . . . . . . . . . . . . . . . . . 80

6.2 a) Odd part of resistance normalized by the extent w of the metal/graphene

interface for four pairs of invasive probes shown in the inset of Fig. 6.1

(same colors). The fluctuating region at densities smaller than 1.2 1012cm−2

has been grayed. The charge density n is measured using the classi-

cal Hall voltage between external probes, implying a capacitance of

13.6 nF.cm−2, consistent with the measured oxide thickness. b) Same

odd part of the resistance scaled by the ratio of the length w on the

distance L between contacts. . . . . . . . . . . . . . . . . . . . . . . 82

xxi

6.3 Conductivity as a function of gate voltage. Each color corresponds to

a pair of probes identified by two letters on Fig. 6.1. The slope of

these curves corresponds to a mobility of µ ≈ 4600 cm2V −1s−1. Inset:

given the charge neutrality gate voltage V 0g identified from Quantum

Hall Effect measurements (see text), we plot here the ratio between

resistivities for electrons and holes as a function of carrier density. In

contrast to the case of invasive probes, the average asymmetry is invis-

ible to the precision of our measurement (note: the observed fluctua-

tions are reproducible). The line corresponds to the ratio 1.20 observed

in Ref. [22] in presence of chemical dopants. . . . . . . . . . . . . . . 83

6.4 Odd part Rodd of the resistance scaled by the inverse width w−1 for

various samples and metals described in Table 6.1. . . . . . . . . . . . 85

6.5 a) From the resistance curves plotted in Fig. 6.1, we show the con-

ductance scaled by the ratio w/L. The non-invasive measurement

between probes B and C from Fig. 6.3 is plotted as a thin line for

reference. b) subtracting λ = 0.135 kΩ.µm divided by the length w of

the metal/graphene interface, each curve from a) is linearized for the

p-type carriers (Vg < V 0g ). Main panel: for each four-probe measure-

ment on the samples from Table 6.1, we plot here the specific resistance

λ which best linearizes the conductance as a function of gate voltage

(see text). The best fit is obtained at the dot and the vertical size of

the corresponding ellipse represents the uncertainty on λ. . . . . . . 87

6.6 Conductivities σxx and σxy measured in sample TiAu1 using external

probes at a magnetic field of 8 T and a temperature 4 K. . . . . . . 89

6.7 Raman spectrum for sample TiAu1 and the spectrum of a few layer

graphene piece next to it. . . . . . . . . . . . . . . . . . . . . . . . . 90

6.8 Cartoon of partly invasive probes on a graphene strip. . . . . . . . . 93

xxii

6.9 Odd part of resistance normalized by the extent w of the metal/graphene

interface for four pairs of invasive probes shown in the inset of Fig. 1 in

the paper (same colors). In a measurement between the invasive probe

b and the external probe A, the odd part is half as large as with two

invasive probes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.10 Image of the external probe region of the sample TiAu1 taken using the

phase-mode of an atomic force microscope. The side graphene arms

are clearly visible here and correspond to the colorization of Fig. 1 in

the letter. Notice the discontinuities in the graphene side arms cor-

responding to an electrical shock after the experiment (no conduction

through the arms after this event). . . . . . . . . . . . . . . . . . . . 97

A.1 Connection check protocol . . . . . . . . . . . . . . . . . . . . . . . . 99

A.2 Measurement setup. 4-probe, 3-probe and 2-probe configurations can

be used here depending on the number of contact leads available to the

graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.1 Tylan1 computer sceen-shot with the time variable. . . . . . . . . . 109

xxiii

xxiv

Chapter 1

Why is graphene special?

1.1 The missing link - The emergence of a new 2D

material

Graphite is a carbon allotrope commonly used as “lead” in pencils. It consists of

many one-atom thick sheets of carbon lattice stacked on top of each other, coupled

via weak Van Der Waals forces. A single sheet of graphite where the carbons are

arranged in a honeycomb lattice is called graphene. Its band structure was already

calculated in 1947 by Wallace, using a tight binding approximation [1]. For many

years graphene theory was used only as a tool to investigate properties of various

carbon-based material such as graphite [2], since as a 2D material it was thought

to be unstable at any finite temperature due to thermal fluctuations [3]. The next

theoretical milestone was achieved by Semenoff in 1984 who showed the equivalence

between graphene carriers’ equation of motion and the 2+1 Dirac equation, which

describes relativistic particles [4]. This discovery made graphene a possible toy model

for bridging between relativistic quantum mechanics and condensed matter physics.

In the 20 years following Semenoff’s discovery, graphene was considered only as an

academic material. In that time frame, devices of other carbon allotropes were pre-

pared and measured such as the 1d carbon nanotube [5, 6] or the 0d fullerene (buck-

yball) [7]. Although graphene can be viewed as the building block for all of these

1

2 CHAPTER 1. WHY IS GRAPHENE SPECIAL?

carbon allotropes, the realization of a graphene device came only in 2004 [8]. Since

then many groups had undertaken the task of investigating experimentally and the-

oretically graphene’s electronic, mechanic and thermal properties [9, 10]. In 2005 it

was experimentally verified that indeed graphene carriers behave like massless Dirac

fermions, as suggested by Semenoff. The experiments probed the unique behavior of

graphene carriers in a high magnetic field, observing the Half Integer Quantum Hall

Effect, a signature of massless Dirac Fermions [11, 12]. These experimental results

opened a bridge between Quantum Electrodynamics and condensed matter physics

allowing the use of graphene as a testbed to investigate relativistic effects, which oth-

erwise requires a setup of great complexity such as a particle accelerator. In this work

we aim to investigate the analog of a fundamental relativistic quantum effect called

Klein tunneling, using electrical transport measurements on graphene p-n junctions.

The first chapter includes a brief review of essential topics of graphene theory. The

second chapter will discuss Klein tunneling and a related effect Klein paradox in the

context of relativistic fermions and their analog in graphene p-n junctions. The ex-

perimental results of the first p-n junction graphene device will be presented in the

third chapter. The fourth chapter will be dedicated to our main experimental results

of the analog of Klein tunneling in smooth p-n junctions in graphene, together with

the verification of graphene’s unique properties such as non-linear screening. Chap-

ter 5 will introduce our efforts in making sharp graphene p-n junctions to probe the

analog of Klein Paradox. Since Graphene is a candidate to replace Silicon in the semi-

conductor industry, the subject of graphene to metal contacts in graphene electronic

devices is of high importance, and will be discussed in the last chapter. The effect

of a metal contact on graphene is similar to a Schottky barrier usually formed in a

metal-semiconductor junction, and as such is close to the concept of p-n junction in

graphene.

1.1. THE MISSING LINK - THE EMERGENCE OF A NEW 2D MATERIAL 3

1.1.1 Seeing a single sheet optically and layers characteriza-

tion by Raman spectroscopy

Laying down graphene on a substrate was first accomplished by mechanical exfoliation

from graphite as described in [8]. The term mechanical exfoliation refers to spreading

a piece of graphite into many flakes using a scotch tape, and then apply it to a pre-

prepared Si/SiO2 substrate (with proper thickness as explained below). Although

graphene is a one-atom thick sheet of carbons, it absorbs a significant fraction of

incident light [13]. This property makes graphene visible under optical microscope

when on top of a substrate. A straight forward calculation of light intensity coming

from a graphene-substrate structure yields high contrast with the substrate at the

visible wavelength (see Fig. 1.1). These results taken from Blake et al. [14] assume the

Figure 1.1: Color plot of the contrast as a function of wavelength and SiO2 thickness.Adapted from Blake et al.

common structure of graphene on Si/SiO2 (See Fig. 1.2) and use graphene thickness

of 0.34nm, and dielectric constant of bulk graphite, matching to theory without any

fitting parameter. Apparently, the thickness of SiO2 used in the first graphene device

(300nm) is at the peak of the contrast between the substrate and graphene, at the

wavelength of visible light. Figure 1.3 shows an optical image of one of our mechanical

exfoliated graphene samples before any fabrication process has taken place. The

almost transparent flake close in contrast to the substrate is a good indication that


this is a single sheet graphene. In order to verify that a flake is indeed a single

layer rather then a multilayer two unequivocal tests can be made. The easier one

which does not need further fabrication steps is by Raman spectroscopy and will be

discussed below. The second one done by measuring the Quantum Hall Effect requires

fabrication of a graphene device and measurements at low temperatures with high

magnetic fields. This method will be discussed in more detail in Section 1.3.2. Both

methods can differentiate between a single layer, a bilayer and many layers.

Figure 1.2: Diagram of typical Si/SiO2 substrate with exfoliated graphene on top (inblack).

Raman spectroscopy probes the electron-phonon scattering processes in graphene

by laser excitations. It was first used on graphene by Ferrari et al. in Ref. [15], where

they show the distinctive signature of the G and 2D peaks in the spectra of a single

layer, a bilayer and a multilayer graphite (see Fig. 1.4). As seen from the figures, a

single layer is characterized by a single 2D peak and a ratio of ≈ 0.3 between the

G and 2D peaks. For a bilayer graphene the 2D peak can be fitted to 4 Lorenzians

corresponding to the additional scattering process between the layers, and also the G

to 2D peak ratio is ≈ 0.5. For higher number of graphene layers the 2D peak consists

of many peaks and also the G to 2D peak ratio is substantially higher.

1.1. THE MISSING LINK - THE EMERGENCE OF A NEW 2D MATERIAL 5

Figure 1.3: Optical image of one of our graphene samples before doing any fabricationprocess. The image shows graphene flake and next to it a thicker graphitic flake.

Figure 1.4: G and 2D peaks of a single layer graphene and graphite. Adapted fromFerrari et al..


1.2 Band Structure - Massless-Dirac Fermions

We follow the tight binding calculation initially described by Wallace [1], and how it

leads to Semenoff’s result of 2+1 Dirac equation in graphene. The band structure

calculated using these methods will show us how graphene carriers can be viewed as

Massless Dirac Fermions. It will also give us insights on a new degree of freedom

for graphene carriers and its implications on scattering processes. To conclude this

part we qoute the experimental evidence that graphene carriers are indeed Massless

Dirac-like Fermions in the context of the Integer Quantum Hall Effect.

1.2.1 How the bands bend - Survey of graphene band theory

Graphene has a honeycomb lattice with two carbon atoms per unit cell , thus it can

be viewed as comprised out of two sub-lattices (blue/red or A/B in Fig. 1.5). In

order to describe the band structure we apply the tight-binding calculation, following

Wallace [1]. As each carbon atom contributes one πz electron, the total wavefunction

on the lattice is given by:

ΨQ(r) =

N∑

j

eQ·Rj

√N

[cAQφ(r − RA

j ) + cBQφ(r − RBj )]

(1.1)

Where j indexes the N sites of the hexagonal lattice Rj = ma1 + na2 and RAj (RB

j )

point to the corresponding ‘A’ (‘B’) sub-lattices in a unit cell j (see Fig. 1.5. The

two terms in the sum correspond to the wavefunction on each sub-lattice ‘A’ or ‘B’

derived from the πz orbitals, with the proper coefficients cAQ and cBQ that solve the

crystal Hamiltonian. The spin degree of freedom is implicit and will not be shown

for convenience (it only adds degeneracy to the problem). It is straightforward to

show that ΨQ(r) fulfill the Bloch condition ΨQ(r) = eiQ·RΨQ(r). The electronic

Hamiltonian is as followed:

H =∑

j

(ǫAA†RjARj

+ ǫBB†RjBRj

) + t∑

(i,j)

(A†RjBRj

+ h.c.) (1.2)

1.2. BAND STRUCTURE - MASSLESS-DIRAC FERMIONS 7

Figure 1.5: Left: The honeycomb lattice in real space blue and red dots represent‘A’ and ‘B’ lattice sites respectively. One choice of unit cell is demarcated witha dashed line. Coordinate axes oriented to give the Pauli and Dirac forms of theelectronic Hamiltonian are labeled P and D. Adapted from Mecklenburg and Regan.Right: Energy contours in the first Birlioun zone. The thin arrows point to theK+valley and b1 and b2 are the basis vectors for the reciprocal space. Adapted fromMecklenburg and Regan.


Where ARj(BRj

) is the operator that creates an electron in the state φ(r − RAj )

(φ(r − RBj )) from the vacuum. The first sum associated with adding an electron

from the vacuum on ‘A’ (‘B’) site with corresponding energy ǫA (ǫB). For graphene

ǫA = ǫB but for now let’s assume ǫA 6= ǫB and take the graphene limit later. The

second sum associated with nearest neighbor hopping (i, j) from one sub-lattice to the

other with t as hopping energy. In reciprocal lattice space the operators transforms

as ARi= 1√

N

∑

j AQjexp(iRi · Qj), where Qj = m

N1b1 + n

N2b2, b1 and b2 are the basis

vectors for the reciprocal space (see Fig. 1.5), and N = N1N2 are the wavevectors in

the first Brillioun Zone. Using a similar transformation for the B operators puts the

Hamiltonian in the following form:

H =∑

j

(

A†Rj

B†Rj

)

H(

ARj

BRj

)

Where we used the relation∑

j exp(iRj · (Q−Q′)) = NδQQ′ and H is the single-

particle Hamiltonian defined by:

H = t

(

−∆/t 1 + e−iQ·a1 + e−iQ·a2

1 + eiQ·a1 + eiQ·a2 ∆/t

)

∆ is defined by (ǫA−ǫB)/2. Since graphene has two electrons per unit cell, its valance

band is full setting the Fermi energy at zero, and we can chose the energy origin as

(ǫA + ǫB)/2 = 0. At the corners of the Brillioun zone Q = K± and the off-diagonal

elements are zero (see Fig. 1.5). This leads to the valley degeneracy of K+ and K−

at the Dirac point. For low energy excitations we denote k = Q − K±, thus the

single-particle Hamiltonian can be approximated as

H = t

(

∆√

(3)tak(±ad + ias)/2√

(3)tak(±ad − ias)/2 −∆

)

Where ± in the off-diagonal terms comes from the two valleys K+ and K− ,

ad = (a1− a2)/a and as = (a1 + a2)/a. In this representation the Hamiltonian at low

energy excitation is isotropic and is entirely in terms of the basis vectors of the direct

1.2. BAND STRUCTURE - MASSLESS-DIRAC FERMIONS 9

and reciprocal lattice. Obviously, the choice of a coordinate system should not affect

the predictions of the theory.

If we choose the coordinate system labeled ‘D’ in Fig. 1 1.5, defining the Fermi

velocity vF =√

2at/2~ and using p = ~k, H becomes: HD = vF (±σxpy−σypx)+σz∆,

where the ± is associated with the different valleys and we used the standard 2x2

Pauli matrices

σx =

(

0 1

1 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 0

0 −1

)

This form shows the connection between the 2+1 problem in graphene and the 3+1

Dirac Hamiltonian. In 3+1 the Dirac Hamiltonian is H = cγ0(γ · p + mc) with

the 4x4 γ matrices as a 4-vector γµ = (γ0, γ) obeying the anti-commutation relation

γµ, γν = 2gµν. In 2+1 the following 2x2 representation obeys the above anti-

commutation relation (adjusted to only 2+1 dimension): γµ = (γ0, ~γ) = (σz , iσx, iσy)

With this representation H becomes H = vFγ0(~γ · ~p+m′vF )

Where the effective mass is defined by m′ = ∆/v2F which cancels out for graphene

as ∆ = 0 and vF plays the role of the speed of light c. This analogy is the base for

treating the graphene carriers as Massless Dirac Fermions.

1.2.2 New degree of freedom - Pseudo-Spin and the absence

of backscattering

Using a different coordinate system donated by ‘P ’ in Fig. 1.5, H becomes HP =

vF (σxpy +σypx)+σz∆. For the K+ valley we have the compact notation ofHP = σ ·p.

Thus, the full 4x4 Hamiltonian including both valleys is

H =

(

vFσ · p 0

0 vFσ · p

)

The above Hamiltonian with 3+1 dimensions and a minus sign in one of valleys is

identical to the Dirac-Weyl equation of massless neutrinons. In this case the additional

minus sign corresponds to the neutrinos chirality since it creates two different kinds

of neutrinos: left-handed and right-handed. In contrast, in 2+1 dimension the valley


degree of freedom does not exist since there is a unitarian transformation that can

change the relative sign between the two blocks. However, in each sub-block the σ

matrices can viewed as pseudo-spin operators creating a new degree of freedom inside

each block. Mathematically, the pseudo-spin is derived from the eigenstates of the

Hamiltonian. As Ando et al. showed [16], the general form of the eigenstates solving

HD is

Fsk = R−1(θ(k))|s 〉 (1.3)

Where θ is defined by

kx + iky = i|k|eiθ(k) (1.4)

and R is spin-rotation matrix given by:

R(θ) =

(

exp(+iθ/2) 0

0 exp(−iθ/2)

)

From the above it can be deduced that the pseudo-spin is a consequence of the

two sub-lattices and is a superposition of the wavefunctions on both sub-lattices. On

the edges, one component of the pseudo-spin is strictly zero and on the other is 1,

while in the bulk both sub-lattices contribute with a phase in between them. This

also rises another interesting property of the pseudo-spin named ‘Chirality’ where

the angle of the pseudo-spin in the Bloch sphere is the same (opposite) as the angle

of momentum in momentum space for electrons (holes) (see Fig. 1.6). Due to this

property, back-scattering is suppressed in all orders since it requires a pseudo-spin

flip, which can only occur if the external potential changes on the scale of the lattice

constant [16].

1.3 Quantum Hall Effect

Quantum Hall Effect (QHE) discovered by Klitizing [17] is one of the richest phe-

nomena in condensed matter physics. Its most pronounce features show up when

measuring longitudinal and transverse conductivity of a 2D Hall Bar shaped sample

1.3. QUANTUM HALL EFFECT 11

Figure 1.6: Pseudo-Spin in graphene: Left: Bloch sphere representation of the pseudo-spin. θ is the same as the one showing up in Eq. 1.4. Right: 2D Momentum space

at high magnetic fields and low temperature. The effect and its origin will be ex-

plained briefly in the following sections, as well as the unique version of the Integer

Quantum Hall Effect that was predicted and observed in graphene.

1.3.1 Integer Quantum Hall Effect in 2DEG - The formation

of Landau levels

At high magnetic fields, the density of states of a 2DEG is separated into discrete

levels with high degeneracy called Landau levels. The formation of Landau levels in

the case of a 2DEG is derived by solving a 2D Schrodinger equation with an external

field. The discrete spectrum is given by the energy eigenvalues En = ~ωc(n + 1/2)

(n=0,1,2...), where the distance between any two levels in energy is proportional to

the cyclotron frequency ωc that is defined by ~ωc = eB/m (h is the Planck’s constant,

e is the electron charge and m is the electron mass). Notice that the zero-mode has

a finite energy above the zero, E0 = ~ωc/2.


As the total energy states is conserved the degeneracy of each level is given by

2eB/~. However, in realistic samples where disorder exists the levels broaden, which

allows the Fermi energy to be set in between Landau levels in the bulk but have con-

tinuous states available on the edges. This has an important role in experimentally

observing the QHE, since the conductance is only mediated via the edge states that

cannot be scattered with states on the opposite edge and can only propagate forward

on the same edge. In this scenario each channel on the edge state will contribute e2/h,

the quantum of conductance that will result in quantized steps of transverse conduc-

tivity, as channels are added, and zero longitudinal resistivity, when the number of

channels is fixed. Figure 1.7 shows the predictions for the transverse conductivity as

function of carrier density for a standard 2DEG in high enough magnetic field.

Figure 1.7: Predictions for Quantum Hall Effect: Left: Graphene. Half integerplateaus and zeroth Landau level Right: 2DEG. Adapted from Novoselov et al.

1.4. DISORDER IN GRAPHENE - MOBILITY LIMITING MECHANISMS 13

1.3.2 Half Integer Quantum Hall Effect in graphene - Indeed

Massless Dirac Fermions

Since graphene carriers obey a Dirac-like equation, the solutions to the external field

problem are rather different. In this case the eigenvalues areEn = sgn(n)√

2e~v2F |n|B

(n=0,1,2...). This spectrum has a zeroth Landau level in contrast to the 2DEG case,

which manifests as a different step-like structure of the transverse conductivity as

function of carrier density. As the zertoth Landau level is unpaired it causes the

conductivity to be quantized at 4e2/h× (n+ 1/2) (n=0,1,2...) rather then 4e2/h×n,

and the first plateau is absent at the energy origin (for comparison see Fig. 1.7). These

two features constitute the relativistic version of the QHE called the Half Integer

Quantum Hall Effect. Experimentally, this effect was observed directly by gating

Hall bar shape graphene devices at high magnetic fields and low temperatures [11,

12], and even in room temperature due to a large gap between the zeroth and first

Landau levels [18]. The experimental results not only prove that graphene carriers

are indeed massless Dirac-like Fermions but also allow to determine unambiguously

whether a graphite flake is truly a single layer. A different version of the QHE that

is distinct from both single layer and standard 2DEG, exists in bilayer graphene, and

was investigated experimentally in Ref. [19]. Thus the Quantum Hall Effect is used

to distinguish between single layer, bi-layer and many layer graphene, which behave

like a standard 2DEG.

1.4 Disorder in graphene - mobility limiting mech-

anisms

There are three general mechanisms for scattering in graphene: Charge impurities,

Short-range disorder and corrugations, each contributes differently to the transport

properties of graphene such as it’s conductivity and mobility. We briefly discuss each

one separately.


1.4.1 Charge impurities

Charge impurities originate either from the substrate surface underneath the graphene,

from trapped molecules between the substrate and graphene (either below or above

the graphene), and from dopants on top of the graphene (if no dielectric capping

it). The presence of charge impurities lead to a model showing the following linear

relation between conductivity and density [20, 21]:

σ(n) = Ce

∣∣∣∣

n

nimp

∣∣∣∣

+ σres (1.5)

Where C ≈ 5×1015V −1s−1,nimp is the charge impurity density, and σres is the residual

conductivity at the Dirac point which is impurity density dependent. This results

implies a constant mobility far from the Dirac point which is confirmed experimentally

in many of the graphene experiments [12, 11]. In addition two more experimental

evidences corroborate with charge impurities as a dominant mechanism for scattering

in graphene. The first one is the work by Chen et al. [22], where they dope pristine

graphene in vacuum with Potassium atoms, which serve as charge impurities on the

surface of graphene. Their results show substantial degrading in mobility and also

a better fit to linear conductivity when adding more charge impurities. The second

evidence originates in the work of Bolotin et al. [23] showing ultra-high mobilities on

graphene, after suspending a graphene sheet by etching the substrate underneath it,

and preforming current-driven annealing, which is assumed to heat the graphene and

evaporate water from its surface.

1.4.2 Short-range disorder

Short-range scattering can occur from either defects in the graphene lattice or its

edges. Far from the Dirac point the effect due to these scatters is a constant contri-

bution to conductivity which manifests as sub-linear behavior at high carrier density

in many experiments on graphene. Also, STM studies of exfoliated graphene [24] show

low defects density in the lattice indicating that the mobility-limiting mechanism in

graphene is probably dominated by mechanism other than short-range disorder.

1.4. DISORDER IN GRAPHENE - MOBILITY LIMITING MECHANISMS 15

1.4.3 Corrugations

Graphene is not completely flat and ripples can be created over the sheet. The effect

of the ripples was studied in Ref. [25] where they show the ripples with height-height

correlation function given by 〈[h(r)− h(0)]2〉 give rise to an exponent dependent

conductivity:

σ(n) = Ce(

nnimp

)2H−1

However, the experiment shows [26] that 2H = 1.1 thus ripples like short-range

disorder contribute constant conductivity rather than linear (which were the case if

2H = 2).

Chapter 2

Klein tunneling, Klein Paradox

and their analog in graphene

2.1 The Dirac equation

The Dirac equation is the first-order relativistic version of the Schrodinger equation,

which describes elementary spin-1/2 particles. Using the more general, relativis-

tic energy-momentum relation E =√

c2p2 + (mc2)2 in the form of operators, Dirac

showed in an ingenuous way how to take the derivative of theses operators [27],

rewriting the equation as

(

βmc2 +

3∑

j=1

αjpj

)

ψ(x, t) = i~∂ψ

∂t(2.1)

where c is the speed of light, and αi’s and β are 4x4 matrices that obey the following

anti-commutation relation: αi, αj = αi, β = 0 (i, j = 1, 2, 3).

16

2.2. 1D PROBLEM WITH DIRAC EQUATION 17

2.2 1D problem with Dirac equation - Transmis-

sion and reflection coefficients

Quantum tunneling in the context of non-relativistic problems, which has no classic

analog, is usually described by exponential decay of the wavefunction underneath a

potential barrier. As the barrier becomes infinitely wide, the particle is expected to be

fully reflected back. In contrast, for a relativistic particle obeying the Dirac equation,

the theory predicts three kinds of possible counter-intuitive tunneling mechanism:

Particle-antiparticle production, Klein Paradox and Klein tunneling [28]. We will

discuss each individually after solving a simpler 1D version of the Dirac equation,

which can be generalized in a straightforward manner to 3D.

In 1D we have the following Hamiltonian (choosing natural units c = ~ = 1 to

simplify the results):

(

σx∂

∂x− (E − V (x))σZ +m

)

ψ = 0 (2.2)

where V (x) = V for x > 0 and V (x) = 0 for x < 0 (see Fig. 2.1). In this case the

reflection and transmission coefficients are as follow:

R =

(1− κ1 + κ

)2

(2.3)

and

T =4κ

(1 + κ)2(2.4)

where κ = pk

E+mE+m−V

, E and m are the total energy of the particle and its mass

respectively, k is the momentum of the incoming particle, and p is the momentum of

the transmitted particle.

2.2.1 Particle-antparticle production

Although the following does not show up in Klein’s original paper, many authors

treat the following as The Klein Paradox. When E < V − m, κ is negative and

18 KLEIN TUNNELING, KLEIN PARADOX AND THEIR ANALOG

Figure 2.1: 1D problem, sharp case schematic. See condition in the text.

the reflection and transmission coefficients are R > 1 and T < 0, which predicts

more particles being reflected from the potential barrier, and the unphysical negative

transmission. This result of particle-antiparticle production out of the vacuum is still

controversial [29]. It is related to the pathological case of infinite potential step, and

does not occur when the problem is solved first with a finite-width barrier and later

taking the limit of finfinite width. Also the extreme conditions required to for this

phenomena, like in the horizon of a black hole or in a collapse of superheavy nucleus,

render it hard to impossible to experiment.

2.2.2 The Klein Paradox

The result that Klein did discuss in his original paper is still counter-intuitive but

has no unphysical consequences. Klein showed that for x > 0 for the particle to

propagate to the right, the group velocity defined by vg = dEdp

= pE−V

implies that

the momentum p must be negative. Using the relativistic energy-momentum relation

p = −√

(V − E)2 −m2 gives κ =√

(V −E+m)(E+m)(V −E−m)(E−m)

. But now, when m < E < V −mκ ≥ 1 and both R and T are between 0 and 1, and satisfy R + T = 1. It is still a

counter-intuitive effect, since for fixed E, at the limit V →∞, where the potential is

becoming strongly repulsive to the particle, T is non-zero. Still, the Klein Paradox is

2.2. 1D PROBLEM WITH DIRAC EQUATION 19

problematic from the experimental point of view, as it requires the potential step to

be sharp (will be called from now on the ’sharp case’) on the order of the particle’s

Compton wavelength λcompton = hmc

, which requires electric fields of the order of

1016MV/cm. Explicitly, the sharp case means

∆V

L>

mc2

λcompton(2.5)

(see Fig. 2.1 for definition of ∆V and L). This condition arises naturally from the

discussion in Klein tunneling effect, which will be considered next.

2.2.3 Klein tunneling

Although not unphysical, the Klein paradox in the sharp case has an unrealistic

potential step that creates enormous electric fields. A more realistic potential (see

Fig. 2.2) in the same relativistic context, was considered by Sauter in Ref. [30]. The

potential in this case can be described by V (x) = vx, where the slope v determines

a constant electric field. Sauter showed that for high electric fields meaning large v,

his results are reduced back to the Klein Paradox, but for weaker fields there is a

forbidden region for both particle and anti-particle, where its extent along the energy

axis is proportional to the rest energy. Thus, the particle has to tunnel through the

potential step in order to become an anti-particle, with the following T :

T (m) = exp

(

−πc3

~m2

(∆V

L

)−1)

(2.6)

In order for the exponent to be of order one condition [2.5] must be satisfied. Oth-

erwise, the exponent will be negatively large and T will go to zero. Thus, for the

realistic smooth case T will be practically zero and the quantum tunneling behavior

with exponential attenuation is attained.


Figure 2.2: 1D problem, smooth case schematic. See condition in the text.

2.3 The analogs of Klein tunneling and Klein Para-

dox in graphene

As massless Dirac-like Fermions graphene can be used as a testbed to probe both Klein

tunneling and Klein Paradox, in a simpler condensed matter experiment setup. In

the next sections we show the equivalence between potential steps and p-n junctions

in graphene, and how the later can be used to probe experimentally the analogs of

Klein tunneling and Klein Paradox in graphene.

2.3.1 Creating potential steps with graphene p-n junctions

The carriers’ density in graphene can be tuned by an external gates, such as a back

gate, common in many graphene devices. The gate is capacitively coupled to the

graphene sheet and tunes its density in the following way ne = Cbg(Vbg − V 0bg), where

n is the carriers density in graphene, Cbg is the gate capacitance, Vbg is the voltage

applied on the gate, and V 0bg is the Dirac point shift in gate voltage probably due

to doping (in ideal graphene V 0bg = 0). Adding another local gate, such as a top

gate, will create two regions where the density can be tuned independently. As seen

in Fig. 2.3, a potential step can be created with either same polarity or opposite

2.3. ANALOGS IN GRAPHENE 21

polarity of the density in the two regions. Electrons and holes in graphene can be

viewed as the analog of relativistic particles and anti-particles, respectively. In the

same manner that anti-particles are particles with anti-parallel group velocity and

momentum, holes are electrons in the valence band with anti-parallel group velocity

and momentum. Thus in a p-n junction in graphene electrons propagate on the n-

region while in the p-region holes propagate. The fabrication of such devices and

their operation will be discussed in further detail in chapter 3.

Figure 2.3: Left: Bipolar junction in graphene Right: Unipolar junction in graphene

2.3.2 Reducing the 2D problem into 1D problem

In order to solve the 2D problem in graphene we set the total energy at zero where

the chemical potential µ is zero as well. Also, we start from a simpler case described

in Ref. [31] where the potential step arising from the p-n junction is symmetric,

and approximated as linear V (x) = Fx (F = vFkF/d, see Fig. 2.4 for relevant

parameters). We also assume that the potential step is smooth, which in this context

means π~vFk2y < F , and will be verified in the next section.


Since the 2D problem is translation invariant in the y direction, we can choose

a fixed ky and solve the problem for kx. From conservation of total energy, an im-

pinging electron will satisfy ~kx(x) =√

V 2(x)/v2F − ~2k2

y by adding kinetic energy

and potential energy terms. As ky is fixed and V(x) approaches zero, as long as ky

is non-zero, there will be a forbidden region very similar to the one in the relativistic

case, where an electron tunnel through the p-n junction in order to become a hole.

Effectively, in this situation a gap opens up in the size of ~vFkF . In contrast, for

ky = 0 as the electron motion is perpendicular to the p-n interface, the forbidden

region shrinks to nothing. Thus, as long as the potential changes slowly on the scale

of the lattice constant, the electron will be fully transmitted due to the absence of

back-scattering [16]. Quantitatively, the dependence of T in ky was calculated in

Ref. [31], assuming the transport is ballistic along the p-n junction:

T (ky) = exp

(

−π~vFk2y

(edV

dx

)−1)

(2.7)

using the relation ky = kF sin(θ) we plot T (θ) in Fig. 2.5, which shows a selective

transmission around the perpendicular to interface trajectory. Note that this result

although derived for a symmetric potential is actually presented in a more general

way and is valid for non-symmetric potentials as well. This result is also known in the

context of inter-band tunneling in semiconducting p-n junctions as the Landua-Zener

tunneling (for more details see Ref. [32]).

2.3.3 Relativistic fermions vs. graphene carriers

Equations 2.6 and 2.7 are equivalent, if one makes the following substituation:

m = ~

vFky and c = vF The ‘Compton’ wavelength in graphene is then ~

mc→ 2π

ky,

thus for each ky there is a ‘Compton’ wavelength associated with. Condition 2.5 in

the case of graphene translates into π~vFk2y > F . For small angles perpendicularly

to the interface, this condition will never be satisfied. Theses angles are associated

with ‘Compton’ wavelengths diverging to infinity or ky = 0, where the sharp condition

applies, but as we will see in the last section of this chapter, T (ky) can be concatenated


Figure 2.4: Symmetric p-n junction. F is the constant electric field for linear slopepotential with characteristics length of d.

Figure 2.5: T as a function of angle of incident with respect to the p-n interface.


over the two regimes (smooth and sharp), where at small angles both sharp and

smooth allow full or close to 1 transmission.

2.3.4 Realistic p-n junctions - including non-linear screening

and disorder

Equation 2.7 is still valid even in the case of disorder. Zhang and Fogler [33] showed

that as long as the interface is ballistic close to the Dirac point where the junction

changes polarity, equation 2.7 holds and the resistance is

Rpn =π

2

h

e2

√

~vF (e|Fpn|)−1 (2.8)

where Fpn is the electric field at the point where the junction changes polarity.

However, due to poor screening properties of graphene at this point (as density of

carriers is close to zero), the electric field is enhanced and is different from the field

derived in the approximated linear potential case. Including the effects non-linear

screening, Fogler et al. [34] derived the resistance of a p-n junction

Rpn = c1h

e2wα−1/6|n′|−1/3 (2.9)

where c1 is a numerical factor that depends on the specific problem, w is the width

of the graphene p-n junction, α is a dimensionless parameter that represents the

Coloumb interaction for specific gate dielectrics, and n′ is the density slope at the

point where the junction changes polarity. The density is calculated assuming graphene

is a good metal and by solving the Poisson equation for a specific geometry and gate

voltages. Note that approximating the potential as linear will have an infinite density

slope since V is proportional to k, and k is proportional to√n.

2.3.5 Sharp p-n junctions

As described in previous section, the condition for having a sharp p-n junction is

π~vFk2y < F , which requires a gate that is strongly coupled to the graphene sheet.


Since the characteristic length of the gate is proportional to the dielectric thickness,

a thin dielectric is favorable in order to reach the sharp junction regime within a

practical voltage applied on the gates. Cayssol et al. calculated in Ref. [35] a general

formula for the transmission coefficient as a function of angle, with any type of junc-

tion (unipolar or bipolar) and with arbitrary sharpness (by varying the characteristic

length of the interface continuously).

Chapter 3

Transport measurements across a

tunable potential barrier in

graphene

B. Huard, J.A. Sulpizio, N. Stander, K. Todd, B. Yang and D. Goldhaber-Gordon

Stanford University, Department of Physics, Stanford, California, USA 1

3.1 Abstract

The peculiar nature of electron scattering in graphene is among many exciting theo-

retical predictions for the physical properties of this material. To investigate electron

scattering properties in a graphene plane, we have created a gate-tunable potential

barrier within a single-layer graphene sheet. We report measurements of electrical

transport across this structure as the tunable barrier potential is swept through a

range of heights. When the barrier is sufficiently strong to form a bipolar junctions

(npn or pnp) within the graphene sheet, the resistance across the barrier sharply

increases. We compare these results to predictions for both diffusive and ballistic

transport, as the barrier rises on a length scale comparable to the mean free path.

1This chapter is adapted with permission from Phys. Rev. Lett. 98, 236803 (2007). c©(2007) bythe American Physical Society.

26

3.2. INTRODUCTION 27

Finally, we show how a magnetic field modifies transport across the barrier.

3.2 Introduction

The recent discovery by Novoselov et al. [8] of a new two-dimensional carbon material

– graphene – has triggered an intense research effort [11, 36, 9]. Carriers in graphene

have two unusual characteristics: they exist on two equivalent but separate sublat-

tices, and they have a nearly linear dispersion relation. Together these ingredients

should give rise to remarkable transport properties including an unusual suppression

of backscattering [16]. Unlike in conventional metals and semiconductors, electrons

in graphene normally incident on a potential barrier should be perfectly transmit-

ted, by analogy to the Klein paradox of relativistic quantum mechanics [37, 38, 31].

Backscattering would require either breaking of the “pseudospin” symmetry between

electrons living on the two atomic sublattices of the graphene sheet [31] or a mo-

mentum transfer of order the Fermi wavevector kF , which can only be produced by

a sharp, atomic-scale step in the potential. In addition to the obvious relevance to

future graphene-based electronics, understanding transport across potential barriers

in graphene is essential for explaining transport in graphene close to zero average den-

sity, where local potential fluctuations produce puddles of n- and p-type carriers [39].

We report an experiment in which a tunable potential barrier has been fabricated

in graphene, and we present measurements of the resistance across the barrier as a

function of the barrier height and the bulk carrier density.

3.3 Creating p-n junctions in graphene

To create a tunable potential barrier in graphene, we implemented a design with two

electrostatic gates, a global back gate and a local top gate (Fig. 6.1a.) A voltage Vb

applied to the back gate tunes the carrier density in the bulk of the graphene sheet,

whereas a voltage Vt applied to the top gate tunes the density only in the narrow strip

below the gate. These gates define two areas in the graphene sheet whose densities

28 TRANSPORT MEASUREMENTS OF A BARRIER IN GRAPHENE

n2 – underneath the top gate – and n1 – everywhere else – can be controlled indepen-

dently (Fig. 6.1b.) The graphene was deposited by successive mechanical exfoliation

of natural graphite crystals using an adhesive tape (Nitto Denko Corp.) [8]. After

Figure 3.1: a) cross-section view of the top gate device. b) simplified model forthe electrochemical potential U of electrons in graphene along the cross-section ofa). The potential is shifted in region 2 by the top gate voltage and shifted in bothregions 1 and 2 by the back gate voltage. c) Optical image of the device. The barelyvisible graphene is outlined with a dashed line and the PMMA layer appears as ablue shadow. A schematic of the four-terminal measurement setup used throughoutthe paper is shown.

exfoliation, thin graphite flakes were transferred onto a chip with 280 nm thermal

oxide on top of an n++ Si substrate, used as the back gate. A single-layer graphene

sheet was identified using an optical microscope, and 30 nm thick Ti/Au leads were

evaporated onto the sheet using standard e-beam lithography. To form the dielectric

layer for the top gate, a thin layer of PMMA (molecular mass 950K, 2% in anisole)

was then spun onto the Si chip at 4000 rpm for 35 s and baked at 180C for 2 minutes.

On one section of the graphene sheet, the PMMA was cross-linked by exposure to

30 keV electrons at a dose of 21, 000 µC cm−2 [40]. The unexposed PMMA was re-

moved by soaking the chip for 10 minutes in acetone. Finally, a 50 nm thick, 300 nm

wide Ti/Au strip was deposited on top of the cross-linked PMMA using standard e-

beam lithography to form the top gate (Fig. 6.1c.) Raman spectroscopy of the sheet

indicates that it is in fact composed of a single layer [15, 41] (see also Section 3.9).

3.4. EXPERIMENTAL RESULTS 29

-7.5 -5 -2.5 0 2.5 5 7.5

Vt HVL

0

1

2

3

4

5

6

7

RHk

WL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

aL

-8 -4 0 4 8

Vt HVL

0

10

20

30

40

50

60

VbHVL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

-3

-2

-1

0

1

2

n1H1

012

cm

-2L

bL

Figure 3.2: a) Resistance across the graphene sample at 4 K as a function of thetop gate voltage for several back gate voltages, each denoted by a different color.b) Two-dimensional greyscale plot of the same resistance as a function of both gatevoltages. Traces in a) are cuts along the correspondingly-colored lines.

3.4 Experimental results

We performed conventional four-terminal current-biased lockin measurements (100 nA

at 12.5 Hz) of the resistance of a PMMA-covered L = 1.3 µm by w = 1.7 µm sec-

tion of a graphene flake, in which a potential barrier can be induced by the top

gate (Fig. 6.1c). All the measurements were performed in liquid Helium at 4 K. In

Fig. 6.2b, this resistance is plotted as a function of both the back gate voltage Vb and

the top gate voltage Vt2. Two clear white lines appear, indicating local maxima in

the resistance in the different regions of the graphene sheet. In the bulk of the sheet

(region 1), the average carrier density is given by n1 = Cb(Vb − V 0b )/e, where the

back gate capacitance per area Cb ≈ 14 nF cm−2 is inferred from Hall measurements

of other graphene flakes on the same oxide layer. The horizontal white line marks

the neutrality point (n1 = 0) in this region, allowing us to estimate V 0b ≈ 31.5 V.

In region 2, the average density n2 = n1 + Ct(Vt − V 0t )/e is modulated not only by

the back gate, but also by the top gate, with capacitive coupling Ct. The voltage

2The leakage currents of back and top gates are linear in voltage over the full range studied, withleakage resistances Rb ≈ 3.7 GΩ and Rt ≈ 1.7 GΩ


V 0t is not necessarily zero, since the chemical doping in regions 1 and 2 can differ,

and this difference can vary widely from sample to sample. The diagonal white line

marks the neutrality point underneath the top gate (n2 = 0). The slope of this line

provides the relative coupling of the graphene sheet to the two gates: Ct/Cg ≈ 6.8,

so Ct ≈ 1.0 × 102 nF.cm−2. Using the PMMA thickness of 40 nm as measured by

AFM, this value leads to a dielectric constant ǫPMMA = 4.5 (close to the accepted

room temperature value for non-cross-linked PMMA). The crossing point of these

white lines yields V 0t = −1.4 V (see Section 3.9).

Transport fluctuations seen in Fig. 6.2 are a reproducible function of gate voltages

and magnetic field (universal conductance fluctuations [42]) with amplitude δG ≈0.2e2/h 3. The magnetoresistance is almost perfectly symmetric in magnetic field, as

expected(see Section 3.9). We extract the phase coherence length Lϕ ≈ 4 µm and

the inter-valley scattering length Li−v ≈ 0.15 µm from the weak localization peak.

Lϕ > Li−v indicates that the sample is lying flat on the substrate [43, 44] (see also

Section 3.9).

3.4.1 Quantifying the p-n junction resistance

Fig. 6.2a shows selected cuts through Fig. 6.2b: device resistance across the potential

barrier as a function of the top gate voltage Vt, for several values of the bulk density

n1. Each curve has a maximum close to n2 = 0, arising from the enhanced resistivity

of the graphene in region 2. However, each curve is noticeably asymmetric with

respect to this maximum: the resistance depends on whether or not the carriers in

region 2 (electrons or holes) are the same type as those in region 1. Specifically,

for given absolute values of the densities n1 and n2, the resistance is always higher

if the carrier types in the two regions are opposite (n1n2 < 0) than if the carriers

are the same throughout. In order to highlight the effects of pn junctions between

regions 1 and 2, we extract for each value of n1 the part of the resistance Rodd which

depends on the sign of the carriers in region 2: Rn1(n2) = Revenn1

(n2) + Roddn1

(n2),

3This value is measured by sweeping magnetic field at 17 values of the density n1 = n2 between−3.0× 1012 and −2.3× 1012cm−2.


Revenn1

(−n2) = Revenn1

(n2) and Roddn1

(−n2) = −Roddn1

(n2) (Fig. 6.3) 4. The resistance of

the device away from the junctions (region 1 and possibly the interior of region 2)

does not depend on the sign of n2 and hence is entirely contained in Reven, which we

do not examine further.

-7.5 -5 -2.5 0 2.5 5 7.5

Vt HVL

-0.4

-0.2

0

0.2

0.4

Rn

1

od

dHk

WL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

aL

-5 0 5

Vt HVL

0

10

20

30

40

50

VbHVL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

-3

-2

-1

0

1

2

n1H1

012

cm

-2L

bL

nnn

ppp

npn

pnp

Figure 3.3: a) Odd component of the resistance: the part which depends on the signof the density n2 in region 2. b) Greyscale plot of the odd component of the resistancefor many values of n1. Colored lines are the cuts shown in a).

The presence of pn junctions between regions 1 and 2 is associated with a sub-

stantial increase of the overall resistance (white areas compared to black areas of

Fig. 6.3b). We label each section of Fig. 6.3b with the carrier types in region 1

and 2, “p” for holes and “n” for electrons, to emphasize that enhanced resistance

is associated with an npn or pnp junction. The junction-sensitive resistance curves

are almost symmetric upon simultaneous sign change of both densities n1 and n2:

left-right reflection and color swap in Fig. 6.3a, or 180 degree rotation about the

center in Fig. 6.3b. Deviations from this symmetry are presumably associated with

uncontrolled spatial fluctuations in the density. Strikingly, a sharp step appears in

the resistance as the boundary between nnn and npn or pnp configurations is crossed.

We can explore the underlying physics of such a resistance increase in two opposite

4The voltage V(n2=0)t at which the density n2 is zero is determined using the above equation

n2 = n1 + Ct(Vt − V 0t )/e(see Section 3.9).


regimes: strongly diffusive or ballistic. Which applies to experiments depends on

whether the elastic mean free path is greater than or less than the length over which

the potential barrier rises.

3.5 Comparison to theory

In a simple model, the electrostatic potential U(x) induced by the set of two gates

in the graphene sheet changes linearly between regions 1 and 2 over a width 2d

(Fig. 6.1b). Assuming perfect screening in the graphene sheet (a good approximation

for our relevant density range), we estimate d ≈ 40 nm independent of carrier density,

by solving the Laplace potential in the region between the top gate and the graphene.

If the elastic mean free path le is much shorter than d, the total resistance can be

estimated by integrating the local resistivity across the device. The conductivity

σ(x) at any position x depends only on the density of charge carriers n(x) and can be

inferred from resistance measurements where the density is uniform (see Section 3.9).

It can be approximated by an interpolation between low and high density behavior

σ(x) ≈[(eµn(x))2 + σ2

min

]1/2[9] where the mobility is µ ≈ 2 × 103cm2V−1s−1 and

σmin ≈ 4e2/h. Using d = 40 nm and a width w = 1.7 µm, we plot the predicted odd

part of R(Vt) with respect to the voltage V(n2=0)t for each value of n1 on Fig. 6.4a.

At high enough density, where the conductivity σ of graphene is proportional to

the density of charges, one can define a mean free path le in an isotropic diffusive

model by σ = 2kF lee2/h, where kF =

√πn is the effective Fermi wavevector defined

relative to the K-point of the Brillouin zone 5. For n1 = n2 ≈ −1012 cm−2, one can

see in Fig. 6.2a that σ ≈ 62e2

h6. Therefore, le ≈ 0.03 µm for n ≈ −1×1012 cm−2. The

mean free path is then already of the order of d for a few volts applied to the back

gate. Hence, the diffusive model is a poor approximation and is unable to reproduce

the data shown in Fig. 6.3.

The opposite regime of ballistic transport has been considered in the limits of sharp

5Notice that the prefactor 2, which is related to the inverse of the dimension of the diffusivematerial in this model, might be a slight underestimate because of the angle-dependent scatteringin graphene.

6The conductivity can be estimated for n2 = n1 as σ = L

Rwwhere R is the resistance.

3.5. COMPARISON TO THEORY 33

-7.5 -5 -2.5 0 2.5 5 7.5

Vt HVL

-0.4

-0.2

0

0.2

0.4

Rn

1

oddHk

WL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

aL

-7.5 -5 -2.5 0 2.5 5 7.5

Vt HVL

-0.4

-0.2

0

0.2

0.4

Rn

1

oddHk

WL

-4 -2 0 2 4

n2-n1 H1012 cm-2L

bL

Figure 3.4: a) Using the strongly diffusive model described in the text, one can predictthe resistance as a function of the top gate voltage Vt for several values of the densityn1 (each represented by the same color as in Fig. 6.3). Here we plot the odd partof this calculated resistance (cf. Fig. 6.3) for a barrier smoothness d = 40 nm and awidth w = 1.7 µm. b) Similar curves for several densities n1 using the ballistic modeldescribed in the text. The curves are plotted only for densities n2 not too close tozero: n2 > 1011 cm−2 (the model diverges at n2 = 0) and dashed lines link the curvesat opposite sides of n2 = 0.

and smooth potential steps. We use here the results of Cheianov et al. [31], which

are valid in the limit |n1|, |n2| ≫ d−2. Note that the results for a sharp barrier [37]

lead to the same qualitative results for Roddn1

(n2) but are more than 10 times smaller

than what follows. The transmission probability τ(θ1) for electrons impinging from

region 1 on the interface between regions 1 and 2 depends strongly on the angle of

incidence θ1 with respect to normal incidence, and is given by

τ(θ1) = e−2π3/2d

|n1|

|n1|1/2+|n2|

1/2sin2 θ1

if n1n2 < 0. (3.1)

We incorporate both interfaces of the potential barrier into the resistance calcu-

lation in the following way. In the fully ballistic regime (le ≫ L2, d, where L2 is the

distance between the two interfaces), all possible reflections at the interfaces, and their

interferences, must be taken into account. However, in the regime of this experiment,

where le ∼ d ≪ L2, the two interfaces can be considered as independent resistors in

series. Using a Landauer picture, the (antisymmetric) conductance through the full


barrier is then

Gnpn =1

2

4e2

h

mmaxy∑

my=−mmaxy

e−2π1/2d

(2πmy/w)2

|n1|1/2+|n2|

1/2 , (3.2)

where we sum the transmission coefficients of incident modes having all possible values

of the quantized wavevector component perpendicular to the interface ky = 2πmy/w,

with my integer and mmaxy =

⌊√

πmin(|n1|, |n2|)w/2π⌋

. Each such mode carries a

conductance 4e2/h, where the degeneracy factor 4 is characteristic for graphene. The

presence of two interfaces in series gives an overall prefactor of 1/2.

Using d = 40 nm and w = 1.7 µm, we plot the calculated odd part of the resistance

in our ballistic model as a function of Vt in Fig. 6.4b for the same densities n1 as in

the strongly diffusive case, assuming that the resistance in the case where the sign of

n is uniform is much smaller than 1/Gnpn. Although the results of the ballistic model

are of the same order of magnitude as the measured values of the odd part of the

resistance, the model is unable to reproduce some aspects of the experimental data.

In particular, the model does not explain why Roddn1

does not only jump but continues

to increase as |n2| passes beyond zero density. This behavior is particularly surprising

as, for fixed n2, Roddn1

decreases with increasing |n1|. Furthermore, by construction,

the model does not apply close to zero density n1 or n2.

3.6 Magnetic field results

A complementary test of the unusual transmission through a potential barrier in

graphene as a function of the angle of incidence was proposed by Cheianov et al. [31].

In the fully ballistic regime, the resistance across the barrier is predicted to increase as

soon as a magnetic field applied perpendicular to the graphene sheet gets higher than

a typical value B⋆ ≈ ~/e(√

π|n2|/πL22d)1/2

, where L2 is the barrier length (0.3 µm

via SEM). Using the values d = 40 nm, L2 = 0.3 µm, and n2 = 2 × 1012cm−2, one

gets B⋆ ≈ 0.1 T.

Fig. 6.5 shows measurements of the antisymmetrized resistance across the device

for several values of a magnetic field B applied perpendicular to the graphene sheet,

with the top gate voltage Vt adjusted to maintain n2 ≈ −n1. This odd part of the

3.7. CONCLUSIONS 35

-15-20-25-30

Vb-Vb0 HVL

0

0.1

0.2

0.3

0.4

0.5

0.6

Rn

1

oddHk

WL

-1.5-2.-2.5-3.

n1»-n2 H1012cm-2L

B=0B=1 TB=2 TB=3 TB=4 T

Figure 3.5: Odd part of the resistance as a function of the back gate voltage Vb, withthe constraint n2 = −n1. Measurements were taken at 4 K for several values of theperpendicular magnetic field B.

resistance shows an enhancement as the magnetic field increases above 2.5 T, an

order of magnitude higher than the predicted transition field B⋆. The fully ballistic

model is therefore unable to explain the results of our measurements in magnetic

field. In fact, the diffusive model also predicts an increase of Roddn1

as B increases, due

to magnetoresistance of the strip near zero density between p and n regions. If the

entire graphene sheet is set to zero density, its resistance increases with magnetic field

similarly to Roddn1

at n2 ≈ −n1 ≈ 2×10−12cm−2 (Fig. 6.5). This qualitative agreement

indicates that while ballistic physics plays a role in transport across the individual

pn junctions, transport through the full npn barrier is far from ballistic.

3.7 Conclusions

In conclusion, we have fabricated a gate-tunable barrier device from a single-layer

graphene sheet, and have hence created bipolar junctions within graphene. We study

transport across the potential barrier as a function of Fermi level and barrier height,

as well as in the presence of an external magnetic field, and demonstrate that a sharp

increase in the resistance occurs as the potential crosses the Fermi level. This increase


is better described by a ballistic than a diffusive model, but the dependence of the

additional resistance as a function of the barrier height is not yet understood. A

clear explanation of this behavior is essential for realizing many exciting proposed

graphene electronics applications, such as electron focusing [31].

3.8 Acknowledgements

We thank V.I. Fal’ko, E.A. Kim, D.P. Arovas and A. Yacoby for enlightening discus-

sions, A. Geim for supplying the natural graphite and M. Brongersma, A. Guichard

and E. Barnard for the Raman spectrometry. This work was supported by MARCO/FENA

program and the Office of Naval Research. K.T. was supported by a Hertz Founda-

tion graduate fellowship, N.S was supported by a William R. and Sara Hart Kimball

Stanford Graduate Fellowship, J.A.S. by a National Science Foundation graduate fel-

lowship. B.H. was supported in part by the Lavoisier fellowship. Work was performed

in part at the Stanford Nanofabrication Facility of NNIN supported by the National

Science Foundation under Grant ECS-9731293.

3.9. SUPPLEMENTARY MATERIAL 37

3.9 Supplementary Material

3.9.1 Imaging, Raman spectroscopy, and identification of single-

layer graphene

After micromechanical cleavage and deposition of graphitic flakes onto a 280 nm

thermally-grown oxide on Si, optical microscopy was used to identify the thinnest

flakes [9]. Microscope settings (brightness, contrast, color balance) were tuned so

that these thinnest flakes, presumed to be few- or single-layer graphene, could be

consistently identified as verified by measurements with atomic force microscope and

3D white-light interferometry (using a profiler from Zygo corp.).

Figure 3.6: a) Optical image of nearby graphitic flake with labeled thin region ➊ andthick region ➋. b) Optical image of bare graphene sheet (labeled region ➌) used tocreate the top-gated device described in the letter. Note the strong color similaritybetween the barely visible regions ➊ and ➌, suggesting the graphene thickness inthese regions is close. c) Spatial map of the width of the 2D peak for regions ➊ and➋. Region ➊ appears less bright than region ➋, indicating a narrower 2D peak andthinner graphene. d) Spatial map of 2D peak width for region ➌. The color scale hasbeen chosen differently from c) in order to see the Au contacts appear as dark lines.

An optical image of the bare graphene sheet studied in the letter after deposition

on the oxide and before any further fabrication is shown in region ➌ of Fig. 3.6b. As

a comparison, Fig 3.6a shows a similar optical image of a nearby graphitic flake which


consists of a thin region and a thick region, labeled as regions ➊ and ➋, respectively.

The color matching between the optical images of regions ➊ and ➌ indicates that the

thickness in these regions is close.

2500 2600 2700 2800 2900 3000

DkRaman Hcm-1L

2

4

6

8

10

12

14

Inte

nsityHa

.u.L

2D #932

2500 2600 2700 2800 2900 3000

DkRaman Hcm-1L

2

4

6

8

10

12

14

Inte

nsityHa

.u.L

2D1 B#11

2D1 A#509 2D2 A#370

2D2 B#0.03

Figure 3.7: Dots: spatially averaged Raman spectra of region ➌ (see Fig. 3.6) sub-tracted from a nearly flat background due to the presence of gold leads. Left panel–line: best fit of the dots using a Lorentzian function Eq. (3.3). The number gives theintegrated intensity of each peak. Right panel–lines: best fit of the dots using thesum of four Lorentzian functions Eq. (3.3) using a fixed value for the spacing betweenthe peaks following Table I of Ref. [15]. The position of the peaks relative to the2D1B peak are 34 cm−1 for 2D1A, 54 cm−1 for 2D2A and 69 cm−1 for 2D2B.

Raman spectroscopy was performed to determine layer thickness in the three

regions using a scanning microscope with 514 nm laser excitation. Figs. 3.6c and

3.6d show the width of the 2D peak (near 2700 cm−1) as a function of position. The

graphitic regions appear clearly on these maps and are perfectly correlated with the

optical images. The spectra within the three labeled regions in Figs. 3.6c and 3.6d

were spatially averaged to obtain a characteristic spectrum for each region. Raman

spectroscopy was performed on region ➌ after device fabrication and measurement

were completed, and after a voltage bias was applied to the device that was sufficiently

large to break the contacts (Fig. 3.9). The presence of the Au leads and contacts on

the device, which can be seen as dark lines in Fig. 3.6d, significantly modifies the

Raman spectrum of bare graphene (region ➌). Due to this modification, only the 2D

peak (Fig. 3.7) could be extracted from the spectrum in this region without significant

distortion by subtracting an almost linear background. On Fig. 3.7, we plot the best

fit of this peak using a single Lorentzian function

I(∆kRaman) ∝ 1

Γ2/4 + (∆kRaman −∆k0Raman)2

. (3.3)


The best fit is obtained for a width Γ = 43 cm−1 and a center at ∆k0Raman = 2693 cm−1.

The fact that a single Lorentzian fits the 2D peak of the device so well indicates that

it is made of a single layer of graphene [15, 41]. For consistency, we also check whether

this peak can correspond to bilayer graphene. In Fig. 3.7, we plot the best fit of the

2D peak assuming it is made of the sum of four Lorentzian peaks following Ref. [15].

The spacing between these 4 peaks is fixed for bilayers and only the peak widths,

peak intensities, and a global offset are fitted. In Fig. 3.7, it appears that the two

components 2D1B and 2D2B are negligible, which is not the case for bilayer graphene.

Figure 3.8: Spatially averaged Raman spectra of regions ➊, ➋ and ➌ (see Fig. 3.6)for the G (left) and 2D (right) peaks. The intensity of the spectrum of region ➌ isscaled differently than the two other spectra for clarity.

Furthermore, Graf et al. have shown that the ratio of the integrated intensities

of the G peak (near 1600 cm−1) to the 2D peak gives another measurement of the

number of layers [41]. Since the Raman spectrum of the graphene in region ➌ cannot

safely be extracted from the measurement near the G peak due to strong contribution

of the Au leads to the Raman signal, it is meaningless to try this analysis on the actual

device. However, we performed such a measurement on the nearby graphitic flakes in

regions ➊ and ➋ (Fig. 3.8), which is informative since optical measurement and the

2D Raman peak on regions ➊ and ➌ are similar. For region ➊, the G/2D ratio is 0.33,

which is close to the value obtained in [41] for monolayer graphene (0.25± 0.04). A

single Lorentzian also fits the 2D peak in this region well, providing further evidence


that it consists of a single layer. By contrast, for region ➋, the G/2D ratio is 1.05 and

the 2D peak is centered7 at 2718 cm−1 and appears quite broad and non-Lorentzian,

suggesting the flake in this region is many layers thick. As a final clue of the flake

thickness, one can compare the positions of the G peaks in regions ➊ and ➋ on

Fig. 3.8. The peak in region ➊ is displaced to the right by about 4 cm−1 from the

peak in region ➋ which can only be true if region ➊ is a monolayer according to Graf

et al. [41]. All these measurements show that the device investigated in the letter is

made of a single layer of graphene.

Figure 3.9: Scanning electron microscope image of the fabricated top-gate device.Note the broken Au contacts due to the final destructive measurement on this sample.The graphene sheet appears as the dark region in the center of the contacts.

7The position of the peaks may depend on the uncontrolled doping of graphene, thus our mea-surement, the one from Ref. [41] and the one from Ref. [15] give slightly different peak positions.However, the fact that the peaks of regions ➊ and ➌ sit at the same position which is different fromthe one of ➋ is informative.


3.9.2 Determination of the densities in regions 1 and 2

As shown in Fig. 2(a) in the letter, the resistance is the sum of a symmetric curve

and of a non-zero antisymmetric curve with respect to a sign change of the density

n2. Therefore, at a fixed value of Vb (or n1), the voltage V(n2=0)t corresponding to

n2 = 0 can be slightly different from the voltage V maxt maximizing the resistance.

On Fig. 3.10, we plot the fitted voltage V maxt as a function of the back gate voltage

Vb. Far from V minb , we expect ∂V max

t /∂Vb to be constant and equal to the ratio

0 10 20 30 40 50 60

Vb HVL

-6

-4

-2

0

2

4

Vtm

axHVL

Figure 3.10: Dots: Voltage V maxt at which the resistance is maximal as a function of

the back gate voltage. Line: best fit with a straight line having the same slope athigh Vb as the experimental curve.

Ct/Cg between the capacitance per area of the top and back gates (see the letter).

This leads to a good estimate of this ratio Ct/Cg ≈ 6.8. Besides, the shift of the

resistance peak position due to the antisymmetric part should be exactly opposite

for opposite values of n1. One can see this on Fig. 3.10 as a global shift towards

the positive (resp. negative) Vt when Vb < V minb (resp. Vb > V min

b ). Therefore, since

n2 = n1+Ct(Vt−V mint )/e, the voltage V

(n2=0)t can be plotted on Fig. 3.10 as a straight

line whose slope is 6.8 and whose position is such that its distance from V maxt is the

same at positive and negative doping n1. This gives an estimate for V mint = −1.4 V.


-30 -20 -10 0 10 20 30

Vb-Vbmin HVL

0

0.2

0.4

0.6

0.8

1

GHm

SL

-3 -2 -1 0 1 2

n1=n2 H1012 cm-2L

-30 -20 -10 0 10 20

Vb-Vbmin HVL

0

0.25

0.5

0.75

1

1.25

1.5

1.75

GHm

SL

-3 -2 -1 0 1 2

n1=n2 H1012 cm-2L

Figure 3.11: Left: conductance measured at 4 K for n1 = n2 with a line showing acurve of constant mobility µ = 1800 cm2V−1s−1. Right: conductance measured at77 K for n1 = n2 with a line showing a curve of constant mobility µ = 3000 cm2V−1s−1.

3.9.3 Estimation of the mobility

In order to estimate the mobility of the graphene sheet, we have measured the 4-

terminal conductance of the graphene sheet as function of the back gate voltage Vb

for a fixed value Vt = V mint (see Fig. 3.11). On the first cool-down of the sample, down

to 4 K, the conductance is almost linear in Vb, which indicates that the mobility of the

graphene sheet µ ≡ dσ/dn is almost constant at high density, as already observed

by several groups [9]. Its slope, together with the known capacitance of the SiO2

dielectric, allows one to estimate the mobility µ ≈ 1800 cm2V−1s−1. After having

exposed the sample to air, on the second cool-down of the sample down to 77 K,

the conductance as a function of back gate voltage seems to deviate from the linear

behavior (Fig. 3.11). This behavior remains unexplained but could be due to deposits

that might have appeared between the two cool-downs.

3.9.4 Measurements at 77 K

We performed the same measurements as the one described in the letter during a later

cool-down at 77 K. We report these measurements in Figs. 6.2,6.3. A clear difference

between the measurements at the two temperatures comes from the suppression of the


universal conductance fluctuations due to an enhanced dephasing rate at 77 K. How-

ever, due to the unexplained behavior of the conductance (see Fig. 3.11), we cannot

rule out the contribution of unknown channels of conductance in these measurements.

-10 -5 0 5 10

Vt HVL

0

1

2

3

4

RHk

WL

-6 -4 -2 0 2 4 6

n2-n1 H1012 cm-2L

aL

-10 -5 0 5 10

Vt HVL

-30

-20

-10

0

10

20

30

40

VbHVL

-6 -4 -2 0 2 4 6

n2-n1 H1012 cm-2L

-3

-2

-1

0

1

2

3

n1H1

012

cm

-2L

bL

Figure 3.12: a) Resistance across the graphene sample at 77 K as a function of thetop gate voltage, for several back gate voltages, each denoted by a different color. b)Greyscale plot of the same resistance as a function of both gate voltages. Traces ina) are cuts along the correspondingly-colored lines.

3.9.5 Magneto-resistance measurements

We performed magneto-resistance measurements on the embedded graphene at 4 K

in order to estimate the phase coherence length τϕ and the intervalley scattering

time τi. We have thus measured the resistance of embedded graphene as a function of

magnetic field for 17 values of the charge carrier density ranging from−3.0×1012 cm−2

to−2.3×1012 cm−2 between -1 and 1 T (Fig 3.14). The magneto-resistance fluctuates

strongly but is almost perfectly symmetric upon magnetic field reversal, indicating

that the fluctuations results from quantum interferences in the electron trajectories.

Moreover, a weak localization peak is present at zero magnetic field, but is of the

same order of magnitude as the fluctuations. Far from the peak at zero field, one can

estimate the average resistance 〈R〉 for each density and the standard deviation of

the magneto-conductance which is δG ≡ (〈(R− 〈R〉)2〉)1/2/〈R〉2 ≈ 0.2e2/h indicating


-10 -5 0 5 10

Vt HVL

-0.2

-0.1

0

0.1

0.2

Rn

1

od

dHk

WL

-6 -4 -2 0 2 4 6

n2-n1 H1012 cm-2L

aL

-10 -5 0 5 10

Vt HVL

-30

-20

-10

0

10

20

30

40

VbHVL

-6 -4 -2 0 2 4 6

n2-n1 H1012 cm-2L

-3

-2

-1

0

1

2

3

n1H1

012

cm

-2L

bL

nnn

ppp

npn

pnp

Figure 3.13: a) Odd component of the resistance: the part which depends on thesign of the density n2 in region 2. Colors correspond to the same values of n1 as inFig. 6.2. b) Greyscale plot of the odd component of the resistance for many values ofn1. Colored lines are the cuts shown in a).

-1 -0.5 0 0.5 1

B HTL

0.85

0.9

0.95

1

1.05

1.1

RHk

WL

n1=-3.0 1012cm-2

n1=-2.3 1012cm-2

Figure 3.14: Magnetoresistance curves for various densities n1 = n2 measured at 4K.


-1.5 -1 -0.5 0 0.5 1 1.5!!!!

nB HT 106cm-1L

26.4

26.6

26.8

27

27.2

27.4

27.6

Me

an

of

RXR\2He

2hL

Figure 3.15: Dots: Using the magneto-resistance curves measured at 17 values of thedensity n1 = n2 between −3.0 × 1012 cm−2 to −2.3 × 1012 cm−2 and measured at4 K, we plot the average value of the ratio of the resistance R to the square of themean value of R as a function of B

√n. This procedure allows one to average out

the universal conductance fluctuations so the weak localization peak appears moreclearly (see text). Line: theoretical prediction for the same quantity using Lϕ = 4 µm,Li−v = 150 nm and L⋆ = 50 nm in Eq. (3.4).

that the phase coherence length is of the order of or longer than the size of the sample,

L = 1.3 µm in this case. Since the universal conductance fluctuations are different

at each value of the density, it is tempting to average them out by averaging the

magneto-resistance on the density. It is predicted that, without universal conductance

fluctuations, the magneto-resistance reads [43]

∆R

〈R〉2 = − e2

πh

w

L

[

F (B

Bϕ)− F (

B

Bϕ + 2Bi−v)− 2F (

B

Bϕ +B⋆)

]

(3.4)

where

F (z) = ln z + ψ(1

2+

1

z) (3.5)

and, for a mobility µ,

Bα =1

2√πnµLα

. (3.6)

These equations explicit the dependence of the magneto-resistance on the density

n: the main contribution comes from the term 〈R〉2, then the characteristic fields

Bα grow as√n and finally, the critical length Lα themselves can depend on n (due

to trigonal warping for instance). We can take into account the first two of these

effects in order to average out the conductance fluctuations. Indeed, assuming that


the lengths Lα are constant of n, one can write the magnetoresistance as

∆R

〈R〉2 (B, n) = f(B√n) (3.7)

therefore averaging R(B√n)/〈R〉2 at each value of B

√n is the good way to proceed.

Plotted in Fig. 3.15, the resulting curve is in good agreement with the theoretical

expression (3.4) of McCann et al. assuming that the phase coherence length is Lϕ =

4 µm, the inter-valley scattering length is Li−v = 150 nm and L⋆ = 50 nm.

The big value of Lϕ indicates that the inelastic scattering properties in the inves-

tigated sample are similar if not weaker than in the samples investigated in Ref. [44].

Besides, the fact that Li−v < Lϕ is consistent with a tight bond between graphene

and the substrate [43, 44]. Finally, we show in Fig. 3.16 the magnetoresistance mea-

sured close to zero density to support the discussion on the measurements at finite

magnetic field in the letter. Notice how the increase in field seems similar to Fig. 5

of the letter.

-4 -2 0 2 4

B HTL

5

6

7

8

RHk

WL

Figure 3.16: Measured magnetoresistance at Vb = −31.8 V and Vt = −1.5 V, thusclose to n1 = n2 = 0.

Chapter 4

Evidence for Klein tunneling in

graphene p-n junctions

N. Stander, B. Huard and D. Goldhaber-Gordon Stanford University, Department of

Physics, Stanford, California, USA 1

4.1 Abstract

Transport through potential barriers in graphene is investigated using a set of metallic

gates capacitively coupled to graphene to modulate the potential landscape. When a

gate-induced potential step is steep enough, disorder becomes less important and the

resistance across the step is in quantitative agreement with predictions of Klein tun-

neling of Dirac fermions up to a small correction. We also perform magnetoresistance

measurements at low magnetic fields and compare them to recent predictions.

4.2 Introduction

Graphene is promising for novel applications and fundamental physics due to its re-

markable electronic, optical and mechanical properties [10]. At energies relevant to

1This chapter is adapted with permission from Phys. Rev. Lett. 102, 026807 (2009). c©(2009)by the American Physical Society

47

48CHAPTER 4. EVIDENCE FOR KLEIN TUNNELING IN GRAPHENE P-N JUNCTIONS

electrical transport, quasi-particles are believed to behave like Dirac fermions with

a constant velocity vF ≃ 1.1 × 106 m.s−1 characterizing their dispersion relation

E = ~vFk. The Klein paradox for massless Dirac fermions predicts that carriers in

graphene hitting a potential step at normal incidence transmit with probability one

regardless of the height and width of the step [37]. At non-normal incidence, this tun-

neling problem for 2D massless fermions can be represented as a 1D problem for mas-

sive Dirac fermions, with the effective mass proportional to the conserved transverse

momentum. The Klein tunneling probability should then depend on the profile of

the potential step [37, 31, 45]. Recent experiments have investigated transport across

potential steps imposed by a set of electrostatic gates [46, 47, 48, 49, 50, 51] and

results of Ref. [50] support an interpretation of Klein tunneling. We present measure-

ments on six devices which allow a quantitative comparison with Klein tunneling in

graphene when the potential profile created by the gates is evaluated realistically [33].

Disorder is sufficiently strong in all our devices to mask effects of multiple reflections

between the two steps of a potential barrier, so that all data can be accounted for

by considering two independent steps adding ohmically in series. Finally, we probe

the transition from clean to disordered transport across a single potential step, and

we refine the accuracy of the transition parameter introduced by Fogler et al. [34].

In a complementary measurement, we show that the effect of a low magnetic field on

the Klein tunneling across a potential step in graphene is not explained by existing

predictions in the clean limit [52].

Figure 4.1: Schematic diagram of a top-gated graphene device with a 4-probe mea-surement setup. Graphene sheet is black, metal contacts and gates dark grey.


Sample L (nm) w (µm) d (nm) 〈β〉 µ (cm2V−1s−1)A60 60 4.3 34 7.6 1800B100 100 2.1 42 3.8 1700B220 220 2.1 42 3.5 1700C540 540 1.74 25 7.9 1400A860 860 3.6 34 7.9 1800C1700 1700 1.74 47 1.9 1300

Table 4.1: Geometrical properties of the samples: top gate length L, graphene stripwidth (interface length) w, and top gate dielectric thickness d. Same letter for twodevice labels indicates same graphene sheet. All dimensions were taken by both Scan-ning Electron Microscopy and Atomic Force Microscopy. The transition parameter βbetween clean and diffusive transport in a single p-n junction is also shown (see text),averaged across the whole measured voltage range such that nbg < 0 and ntg > 0.Counter-intuitively, despite devices’ low mobility, β ≫ 1 so that Klein tunneling isexpected rather than diffusion across the interface.

aL

-10 -5 0 5 10-80

-40

0

40

80-4 0 4 8

-8

-4

0

4

Vtg HVL

VbgHVL

Dntg H1012cm-2L

n bgH1

012cm

-2 L

bL

-10 -5 0 5 10

1

2

3

4

-4 0 4 8

Vtg HVL

RHk

WL

Dntg H1012cm-2L

Figure 4.2: a) 4-probe resistance measured on device C540 (see Table 4.2), as afunction of Vbg and Vtg. The color scale can be inferred from the cuts shown in b.The densities nbg and ∆ntg are estimated using V 0

tg = 2.42 V, V 0bg = 18.65 V and

Ctg = 107 nF.cm−2. b) Resistance as a function of Vtg at several values of Vbg. Thetwo bold curves show a clear asymmetry with respect to the peak (ntg = 0) for bothVbg < V 0

bg (red) and Vbg > V 0bg (yellow).



We measure six top-gated graphene devices (typical schematic shown in Fig. 4.1),

whose essential parameters are listed in Table 4.2. The density nbg far from the

top-gated region is set by the back gate according to nbg =Cbg(Vbg−V 0

bg)

ewhere Cbg =

13.6 nF.cm−2 is the back gate capacitance per area (from Hall effect measurements

on a similar wafer oxidized in the same furnace run), e is the electron charge, and

V 0bg is the gate voltage required to attain zero average density [22]. The density

ntg well inside the top gated region is set by both back gate and top gate voltages

according to ntg = nbg +Ctg(Vtg−V 0

tg)

e, where Ctg and V 0

tg are the top gate counterparts

of Cbg and V 0bg. Throughout this letter we use the notation ∆ntg = ntg − nbg to

identify the contribution of the top gate voltage only, which tunes the potential step

height. As described in previous work [46], an asymmetry with respect to ntg = 0

appears in the 4-probe resistance measured across a top-gated region as a function

of Vtg for fixed back gate voltages Vbg (Fig. 6.1b). This asymmetry quantifies the

resistance across the potential step in graphene created by the gates. All graphene

top-gated devices were fabricated in the same way, which is described in detail in

Section 4.8. For electrical characterization, samples are immersed in liquid Helium at

4 K and four-terminal measurements are made using a lock-in amplifier at a frequency

32 Hz with a bias current of 100 nA. All samples show typical monolayer graphene

spectra measured by Raman spectroscopy and exhibit the quantum Hall plateaus

characteristic of graphene when measured in perpendicular magnetic fields up to 8 T

at 4 K (see Section 4.8).


In order to extract the resistance of the p-n interfaces only, we measure the odd part

of resistance Rodd about ntg = 0 [46]:

2Rodd(nbg, ntg) ≡ R(nbg, ntg)−R(nbg,−ntg), (4.1)


where R is the four-terminal resistance as a function of the densities far from the top

gated region and well inside that region. Extracting the odd part Rodd from the mea-

sured resistance requires an accurate determination of the densities nbg and ntg. This

is made by the measurement of three independent quantities V 0bg, V 0

tg, and Ctg/Cbg.

We carefully measure these quantities by using the quantum Hall measurements at

8 T and electron-hole symmetry (see Section 4.8). There are two physical interpre-

tations for Rodd depending on the relative magnitude of two length scales: the mean

free path le = he2

σ2√

πn(well defined for kF le ≫ 1 or equivalently for a conductivity

σ ≫ 2e2/h) and the top gate length L. For L ≫ le, after crossing the first interface

of the barrier carriers lose all momentum information before impinging on the second

interface. In this case, the total barrier resistance can be modeled by two junctions

in series. The expression 2(Rpn−Rpp) where Rpn (Rpp) denotes the theoretical value

of the resistance of a single p-n ( p-p) interface, can then be compared directly to

the experimental quantity 2Rodd [46]. For L≪ le, multiple reflections occur between

the two interfaces of the barrier, which is predicted to reduce the total barrier re-

sistance(see Section 4.8). As all devices have modest mobility, we start by using a

diffusive model to calculate Rpn and Rpp. In this model, due to disorder the resistance

depends on the local resistivity ρ(n) (measured for a uniform density at Vtg = V 0tg) at

each position x:

R(dif)pn − R(dif)

pp =1

w

∫

ρ(n(nbg, ntg, x))− ρ(n(nbg,−ntg, x))dx (4.2)

Figure 6.3 compares the experimental curves for 2Rodd as a function of Vtg at several

Vbg for samples A60 and C540 to the corresponding predictions. Clearly, the diffusive

model represented by the dashed lines predicts resistance values considerably below

the experimental curves, hinting that transport through the device cannot be viewed

as entirely diffusive. Following the calculation by Fogler et al. [34], we retain the

diffusive model for the region away from the interface, but replace it by a ballistic

interface model for a region extending one mean free path in either direction from


nbg

H1012cm-2L

-4.7

-3.5

-2.3

-1.2

aL

0 2 4 6 8 100.

0.1

0.2

0.3

0.40 2 4 6

Vtg HVL

2Rod

dHk

WL

Dntg H1012cm-2L

nbg

H1012cm-2L

-4.7

-3.5

-2.3

-1.2

bL

-2 0 2 4 6 8 100.

0.2

0.4

0.6

0.80 2 4 6

Vtg HVL

2Rod

dHk

WL

Dntg H1012cm-2L

Figure 4.3: a) The series resistance 2Rodd of the barrier interfaces as a function of Vtg,for several values of Vbg for device A60 (corresponding densities nbg are labeled). Themeasured resistance 2Rodd (dots) is compared to the predicted value 2(Rpn − Rpp)using either a diffusive model, Eq. (4.2) (dashed lines) or a ballistic model Eq. (4.3)with the value c1 = 1.35 chosen to best fit all six devices. (solid lines). b) Same asa) for device C540.


the location where density changes polarity 2. Thus,

Rpn − Rpp = R(bal)pn − R(bal)

pp +R(dif)pn |x≥|le| − R(dif)

pp |x≥|le| (4.3)

where the two last terms are taken from Eq. 4.2, but with the integral excluding x ∈[−le, le]. The first two terms are the ballistic contributions to the interface resistance

for bipolar and monopolar configurations, and can be calculated individually as fol-

lows. All conduction channels on the low-density side of a monopolar junction should

have transmission nearly 1 through the junction [35], so R(bal)pp = 4e2

h

w√

πmin(|nbg|,|ntg|))2π

.

The bipolar case was addressed by Zhang and Fogler [33]:

R(bal)pn = c1

h

e2wα−1/6|n′|−1/3, (4.4)

where h is Planck’s constant, α = e2

ǫr~vF∼ 0.56 is the dimensionless strength of

Coulomb interactions (ǫr ≈ 3.9 is the average dielectric constant of SiO2 and cross-

linked PMMA measured at 4K), and n′ is the slope of the density profile at the

position where the density crosses zero (density profile calculated from the classical

Poisson equation with realistic gate geometry, temporarily treating graphene as a per-

fect conductor). Expression 4.4 refines this calculation to take into account non-linear

screening of graphene close to zero density, going beyond the linear model used in

Ref. [31]. The prefactor c1 in Eq. (4.4) is determined numerically [33]. In our case,

α = 0.56 and the prefactor is predicted to be c1 = 1.10± 0.03 [53]. In order to test

this prediction c1 will be used as a single fit parameter across all samples and densi-

ties. The solid lines in Fig 6.3 were generated by Eq. 4.3, choosing c1 = 1.35 to best

account for all experimental curves in all devices (voltages Vbg > V 0bg give a similar

agreement, not shown for clarity). The slight discrepancy between theoretical and

experimental values of c1 might be due in part to exchange and correlation effects.

2The mean free path is a function of carrier density, and hence it varies as one approaches theinterface. Which value of mean free path is appropriate to characterize the region near the interfaceover which transport is ballistic? Following Fogler et al. [34] we self-consistently define a distancexbal over which transport near the interface is ballistic: le(n(xbal)) = xbal, where the interface is atx = 0, n(x) is derived from solution of the classical Laplace equation for the actual device geometry,

and le = h

2eµ√

n(x)π

.


Trying to fit the data using a naive linear potential model requires an independent

fitting parameter for each device, and even with the best fit to the data, some qual-

itative trends of the experimental data cannot be accounted for by this model, as

described in detail in the Section 4.8. This mismatch between the linear model and

the data indicates the importance of accounting for non-linear screening close to zero

average density. We continue by calculating the ratio η = Rodd/ (Rpn −Rpp), for all

devices, for all measured Vbg and Vtg, using Eq. 4.3. The histogram of η is sharply

peaked at a certain value ηpeak with a small peak width (see Section 4.8). For all

devices except C1700, regardless of their length L, η is close or slightly higher than

1 when using c1 = 1.35 (Fig. 6.4), which indicates that the resistances of both inter-

faces of the potential barrier simply add in series, and a single p-n junction is less

sensitive to disorder than transport between the two interfaces of a potential barrier.

Fogler et al. introduced the parameter β = n′n− 3

2i to describe the clean/disordered

transition in a single p-n junction, where ni is related to the mobility by ni = eµh

[34].

According to Ref. [34], when β ≫ 1 the ballistic contribution in Eq. (4.3) dominates

and the junction is in the clean limit, whereas for β ≪ 1, the diffusive contribution in

Eq. (4.3) dominates and the junction is in the disordered limit. The threshold β = 1

marks the transition where ballistic contribution must be taken into account since it

is comparable to the diffusive contribution. In the following, we refine this transition

threshold experimentally. From Fig. 6.4 and Table 1, it seems that transport is indeed

well described by Eq. (4.3) when β > 3.5 but more poorly for C1700 where β . 2,

where we find that η is further than 1 and has a large spread of values. In addition,

Fogler et al. predict that the diffusive contribution to the the interface resistance

will be negligible for β > 10, which is reached in several of our devices for densities

ntg > 3 · 1012cm−2. At these densities, in spite of our devices’ modest mobility, the

junction can be considered as disorder-free since the calculated ballistic contribution

to Rodd is 10 times higher than the diffusive one, which allows us to make a rather ac-

curate measurement of the ballistic contribution alone in this clean limit, and match

it well with the ballistic terms in Eq. (4.3). In a recent experiment where suspended

top gates were used, for one sample the agreement with Eq. 4.2 – the disordered

limit – was very good (sample S3 in Ref. [9]). This is due to a much larger distance


àà

à

à

à à

200 200050 500100 10000

0.5

1

1.5

2

L HnmL

Η

Figure 4.4: Symbols: ratio η = Rodd/ (Rpn − Rpp) as a function of top gate length Lfor the devices of Table 1. Rpn is calculated with c1 = 1.35. The vertical lines show thewidth of the histogram of η for densities such that |nbg|,|ntg| > 1012cm−2. The dashedline at η = 1 corresponds to perfect agreement between theory and experiment, inthe case where the total resistance is the sum of the resistances of two p-n interfacesin series.


between the top gate and the graphene sheet, and much smaller density range than

in the present work, likely due to lower dielectric constant combined with mechanical

instability of the top gate when applying higher voltages. These two factors consider-

ably reduce n′ (around 80 times), which is not fully balanced by the cleaner graphene

of Ref. [50] (ni 2-5 times smaller). We estimate 〈β〉 ≈ 0.7 for device S3 reported in

Ref. [50]. Note that two other devices on substantially cleaner graphene (S1 and S2

in Ref. [50]) support an interpretation of Klein tunneling with β = 2.5 and β = 4

respectively. From the present work and from the result of Ref. [50], one can see that

the transition between clean and disordered transport in p-n junctions seems to be

sharp: for β > 2.5 the clean limit applies, for β < 0.7 the disordered limit applies

and in between neither limit is valid. 3.

4.5 Magnetic field results

Being sharply dependent on angle of incidence, transport through potential steps in

graphene should be sensitive to the presence of a magnetic field, which bends electron

trajectories. For nbg = −ntg the predicted interface conductance in the clean limit is

Gpn(B) = Gpn(0)(1− (B/B⋆)2)3/4, (4.5)

where Gpn(0) is the conductance at zero field, B⋆ = ~(el)−1√π∆ntg and l is the

distance over which the potential rises, which is proportional to the thickness d of the

oxide [52]. We measure R−1odd as a function of magnetic field B in two devices C540

and C1700 on the same graphene sheet but with different top gate dielectric thickness

d (Table 1). We use the experimental Gpn(0) and the best parameter l to fit all curves

within the same device (see Section 4.8). The parameters l for C540 and C1700 are

found to be 65 nm and 55 nm respectively, whereas C1700 has the thicker dielectric

(see Table 4.2). Further theoretical work is needed to explain this discrepancy.

3Near ntg ≈ 0, where density fluctuations are bigger than ntg, disorder should dominate [34].

4.6. CONCLUSIONS 57

4.6 Conclusions

In conclusion, we show evidence for Klein tunneling across potential steps in graphene

with a quantitative agreement to a model with one free parameter describing screening

properties in graphene. The crossover between clean and disordered regimes occurs

as a function of the parameter β around 1 as predicted by Fogler et al. [34]. More

work is needed to go into the fully ballistic regime, and also to measure directly the

angle dependence of Klein tunneling [31].


We thank J. A. Sulpizio for help with fabrication and characterization, and M. Fogler,

D. Novikov, L. Levitov, and A. Young for enlightening discussions. We also thank A.

Savchenko for pointing out the need to take into account the resistance of the monopo-

lar junction in the predictions for Rodd. While this work was under review, we became

aware of related work by A. Young et al., in which evidence is seen for ballistic trans-

port across a full npn junction [54]. This work was supported by the MARCO/FENA

program and the Office of Naval Research contract N00014-02-1-0986. N. Stander was

supported by a William R. and Sara Hart Kimball Stanford Graduate Grant. Work

was performed in part at the Stanford Nanofabrication Facility of NNIN supported

by the National Science Foundation under Grant ECS-9731293. Critical equipment

(SEM,AFM) was obtained partly on Air Force Grants FA9550-04-1-0384 and F49620-

03-1-0256.


4.8 Supplementary material

4.8.1 Graphene characterization

We measure 2-probe conductance in each sample at high magnetic field (8 T), in

order to verify it has the unique behavior of a single sheet. For example Fig. 6.6

shows conductance G⋆ measured in sample C540 at 8 T (note that an estimated

contact resistance Rcon = 1830 Ω has been taken into account G⋆ = (G−1−Rcon)−1).

The plateaus in G⋆ are at values 2e2/h, 6e2/h . . . , characteristic of a single layer.

Appearance of peaks between plateaus was predicted by Abanin and Levitov, for a

2-probe measurement. [55].

Figure 4.5: 2-probe conductance corrected using the estimated contact resistance forsample C540 at a magnetic field of 8 T and a temperature 4 K.

4.8.2 Extracting the odd part of the resistance

Extracting the odd part of the resistance requires the determination of three quanti-

ties: The ratio between the top gate capacitance Ctg and the back gate capacitance

Cbg, V 0bg, which corresponds to zero average density far from the top gated region,and

V 0tg which corresponds to zero average density below the top gated region, when

Vbg = V 0bg. A good approximation to these parameters can be extracted from Fig. 1a

of the paper, since the voltage offsets are the coordinates of the global maximum in


Figure 4.6: Left: 3-dimensional schematic of representative device. Middle: AtomicForce Microscope topograph of devices C540 and C1700. Right: Scanning ElectronMicroscope image with 4-probe measurement scheme for C540.

resistance and the slope of the diagonal peaked line gives the ratio Ctg/Cbg. How-

ever, the odd part turns out to be particularly sensitive to Ctg/Cbg, so that a mere

estimation of the peak position is not enough.

Instead, we measure the resistance for each device as a function of voltages Vbg

and Vtg at 8 T in the Quantum Hall regime. The position of the transition between

the first and second conductance plateaus in Vtg for each value of Vbg leads to a

determination of the ratio Ctg/Cbg within 1%. The determination of the voltages V 0bg

and V 0tg can be done accurately by symmetrizing the resistance in Fig. 1 of the paper

with respect to the point (V 0bg, V

0tg):

R(V 0bg + ∆Vbg, V

0tg + ∆Vtg)← R(V 0

bg −∆Vbg, V0tg −∆Vtg) (4.6)

and choosing the point (V 0bg, V

0tg) which leaves this resistance the most unchanged.

Still, the uncertainty of 1% on the capacitance ratio Ctg/Cbg leads to some un-

certainty on the odd part Rodd of the resistance. However, this uncertainty remains

negligible except at low densities ntg (see Fig. 4.7).


Sample L (nm) w (µm) d (nm) V 0bg (V) V 0

tg (V) Ctg (nF.cm−2) µ (cm2V−1s−1) 〈β〉A60 60 4.3 34 25.65 -1.36 92 1800 7.6B100 100 2.1 42 9.35 -0.49 69 1700 3.8B220 220 2.1 42 10.95 -0.73 69 1700 3.5C540 540 1.74 25 18.65 -2.42 107 1400 7.9A860 860 3.6 34 25.5 -2.35 92 1800 7.9C1700 1700 1.74 47 13.4 -1.35 52 1300 1.9

Table 4.2: Geometrical properties of the samples: L- top gate length, w- interfacewidth, and d- top gate dielectric thickness. Same letter for two devices indicates samegraphene sheet. All dimensions were taken by Scanning Electron Microscope (SEM)and Atomic Force Microscope (AFM) images. The gate voltage offsets V 0

bg and V 0tg

and the capacitance of the top gate determined from the procedure described in thetext are reported here. The mobility µ is estimated from the slope at the origin ofthe conductance measured as a function of back gate voltage. These low values aredue to the PMMA cross-linking step.

Figure 4.7: 2Rodd as a function of Vtg for several nbg shown at the upper left cornerfor sample C540. For each chosen nbg, we plot the corresponding curves with Ctg/Cbg

between 7.6 and 7.8 in steps of 0.02. This reflects a spread of ±1% from the value of7.7, which we use in the paper.


4.8.3 Comparing the experimental value Rodd to theoretical

models of the junction interface resistance

Figures 3a and 3b in the paper show the experimental Rodd in comparison to the

theoretical Rnp − Rpp, for the two devices A60 and C540 at several nbg within both

clean and disordered models. In order to quantify the compatibility of the theory to

the experiment, we define η(Vbg, Vtg) as the ratio Rodd/(Rnp − Rpp). We determine

the ratio η for all measured densities |nbg|, ntg > 1012cm−2 and calculate the corre-

sponding histogram for η using two models of transmission across a single potential

step in graphene: diffusive and ballistic. η = 1 corresponds to perfect agreement be-

tween theory and experiment in the limit where L≫ le (see paper). Figures 4.8a and

Figs. 4.8b show two histograms of η each, for devices A60 and C540, respectively. The

red histogram is using a diffusive model while the blue one is using a ballistic model.

We follow the same procedure for all devices, and extract the value η associated with

the the diffusive theory and ballistic theory, at the center of the peaked histogram

together with its width (2Γ) , by fitting the data to the following Lorentzian:

Frequency =a

(η − ηpeak)2 + (Γ)2(4.7)

In the paper, Fig. 4 presents ηpeak and small error bars for η when using the ballistic

model with a fitting parameter c1 = 1.35. This is complemented here by Fig. 4.9

showing the wide spread of η for some devices, along with a much lower predicted

value of Rnp − Rpp when using the diffusive model. Note that B100 and B220 have

a relatively smaller spread in η when using the diffusive model, which is due to a

smaller range of densities. In contrast the relatively small spread of η when using the

diffusive theory for A60 is not due to a smaller range of densities nor due to the short

dimension of the top gate. We currently do not understand this feature, although the

spread is still larger than the spread in η when using the ballistic model.


G=0.059Ηpeak=1.1A60

G=0.11Ηpeak=1.9

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

50

100

150

Η

Freq

uenc

y

G=0.075Ηpeak=1.3

C540

G=0.995

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

20

40

60

80

ΗFr

eque

ncy

Figure 4.8: a) Histograms of η using a diffusive model (red) or a ballistic model (blue)for device A60. The peak value ηpeak and peak width (2Γ) shown in the figure weretaken from a lorentzian fit Eq. (4.7) to each theory. The histogram bins are 0.01 wide.b) Same as a) for device C540. Using the diffusive theory for C540, we could not fitproperly η thus we report the standard deviation in eta as the value of Γ.

à

à

à

à

à à

200 200050 500100 10000

1

2

3

4

5

L HnmL

Rodd

RpnHdisL

- RppHdisL

Figure 4.9: The vertical lines show the spread of η when using diffusive model (Eq. (2)in the main paper) for densities such that |nbg|,|ntg| > 1012cm−2. the lines are centeredon the average value of the histogram.


4.8.4 Multiple reflections between interfaces of a potential

barrier

One of the goals of the main paper was to investigate the transition from diffusive

to ballistic transport through the potential barrier by making the top gate length

smaller than the mean free path of the carriers. In this limit the transport across the

whole potential barrier is expected to be ballistic (no disorder), and charge carriers

are subject to multiple reflections on the two interfaces of the barrier.

4.8.5 Without phase coherence

The transmission probability TΣ across the whole potential barrier is related to the

transmission probability T across a single interface by:

TΣ = T (T + (1− T )2(T + (1− T )2(T . . . (4.8)

Hence,

TΣ =1

2T−1 − 1. (4.9)

Therefore the total conductance for a width w is given by

GΣ =4e2

h

w

2π

∫ +∞

−∞TΣ(ky)dky (4.10)

where ky is the component of the wavevector ~k along the potential interface. One can

compare this to the conductance across a single interface

G =4e2

h

w

2π

∫ +∞

−∞T (ky)dky (4.11)

so that

GΣ/G =

∫ +∞−∞ (2T (ky)−1 − 1)−1dky

∫ +∞−∞ T (ky)dky

(4.12)


According to Ref. [31], T (ky) = e−γk2y with γ > 0 therefore,

GΣ/G =

∞∑

n=1

1

2k√k≈ 0.81 (4.13)

Using the notations of our paper, this translates into a resistance

RΣ ≈ 1.24R(bal)np . (4.14)

Therefore, for a length L small enough (L≪ le), the odd part of the resistance should

be such that

2Rodd ≈ 1.24R(bal)np . (4.15)

As seen from Fig. 4 in the paper, this regime is never achieved fully in the exper-

iments but may be the cause of the smaller η for the shortest top gate.

4.8.6 Including phase coherence

In the phase coherent regime, the above derivation remains valid up to a phase term

in the transmission:

TΣ = |T (T + (1− T )2eiE∆t/~(T + (1− T )2eiE∆t/~(T . . . | (4.16)

where ∆t = 2L cos θ is the time spent between back and forth bounces and θ is the

angle of incidence. This simplifies into

TΣ = T 2∣∣1− (1− T )2 exp

[i2πnL(πn− k2

y)−1/2]∣∣−1

(4.17)

with n the density below the top gate. Phase coherent length in our devices is of the

order of a few microns, extracted from a similar device in Ref. [46]


4.8.7 p-n junctions in finite magnetic field

As explained in the paper transport through potential steps in graphene should be

sensitive to the presence of a magnetic field, which bends electron trajectories. For

instance, in the clean limit the angle at which carriers are transmitted perfectly should

be given by arcsin(B/B⋆) where B⋆ = ~(el)−1√π∆ntg and l is the distance over

which the potential rises, which is proportional to the thickness d of the oxide [52].

For nbg = −ntg the predicted interface conductance is

Gpn(B) = Gpn(0)(1− (B/B⋆)2)3/4, (4.18)

where Gpn(0) is the conductance at zero field. Since Eq. (4.18) is a prediction for the

conductance of a single p-n interface and le ≪ L in both devices, R−1odd can be inter-

preted as the conductance of a single p-n interface (canceling out the monopolar bulk

magnetoresistance, whose source in not well understood). For several gate voltages

such that nbg = −ntg, we measure R−1odd as a function of magnetic field B (Fig. 4.10)

in two devices C540 and C1700 on the same graphene sheet but with different top

gate dielectric thickness d (Table 1). We use the experimental Gpn(0) and the best

parameter l to fit all curves within the same device. The parameters l for C540 and

C1700 are found to be 65 nm and 55 nm respectively, whereas C1700 has the thicker

dielectric (see Table 4.2).

We also show here how to extract the p-n interface conductance in the presence

of magnetic field. Both C540 and C1700 satisfy the condition le ≪ L (Fig. 4 of our

paper), thus the barrier resistance can be viewed as that of two p-n interfaces in

series. In this case (2Rodd)−1 = (Rpnp−Rppp)−1, where Rpnp = G−1pnp (Rppp = G−1

ppp) is

the resistance of the barrier when nbg = −ntg (nbg = ntg). Figures 4.11a-d show Gpnp

and Gppp for C540 and C1700, as a function of magnetic field, at several nbg. The

flatness of the nbg = 0 curve is a measure of how well V 0bg and V 0

tg were determined.

Also, Gppp > Gpnp at all measured densities, in both devices, which is consistent with

the zero magnetic field case. Finally, we note a weak localization dip in both devices

C540 and C1700 in the conductance near B = 0 for all densities.


-4.0×1012cm-2

-3.0×1012cm-2

-2.0×1012cm-2

-1.0×1012cm-2

aL

0 1 2 3 4 5 60.

1.

2.

3.

B HTL

H2R

oddL

-1Hm

SL -2.0×1012cm-2

-1.5×1012cm-2

-1.0×1012cm-2

-0.5×1012cm-2

bL

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

B HTL

H2R

oddL-

1Hm

SL

Figure 4.10: a) (Rodd)−1 for device C540 as a function of magnetic field B for sev-eral density profiles with nbg = −ntg (nbg is labeled). The theoretical curves usingEq. (4.18) (solid lines) are fitted with l = 65 nm to the experimental curves (dots).b)Same as a) for device C1700. The fitting parameter used was l = 55 nm.

-4.0×1012cm-2

-3.0×1012cm-2

-2.0×1012cm-2

-1.0×1012cm-2

0 cm-2

aL

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

B HTL

Gpn

pHm

SL

-4.0×1012cm-2

-3.0×1012cm-2

-2.0×1012cm-2

-1.0×1012cm-2

0 cm-2

bL

0 1 2 3 4 5 60.00.51.01.52.02.53.03.5

B HTL

Gpp

pHm

SL

-2.0×1012cm-2

-1.5×1012cm-2

-1.0×1012cm-2

-0.5×1012cm-2

0 cm-2

cL

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

B HTL

Gpn

pHm

SL

-2.0×1012cm-2

-1.5×1012cm-2

-1.0×1012cm-2

-0.5×1012cm-2

0 cm-2

dL

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

B HTL

Gpp

pHm

SL

Figure 4.11: a) Gpnp = R−1pnp as a function of magnetic field for several nbg, with

ntg = −nbg, for device C540. nbg densities are presented on the right hand side of thefigure. b) R−1

ppp = Gppp as a function of magnetic field for several nbg, with ntg = nbg,for device C540. c)-d) Same as a) and b) for device C1700.


4.8.8 Fabrication details

The substrate used in these experiments is a highly n-doped Si wafer with a nominal

resistivity of less than 0.005 Ω · cm. Standard 1− 10 Ω · cm wafers experience carrier

freeze-out and hence hysteretic response to applied gate voltage at temperatures below

4K.

All graphene sheets were produced by successive mechanical exfoliation of Highly

Oriented Pyrolytic Graphite grade ZYA from General Electric (distributed by SPI)

using an adhesive tape (3M Scotch Multitask tape with gloss finish), then deposited

onto a layer of SiO2 297 nm thick grown by dry oxidation at 1500 C on a highly n-

doped Si substrate, which serves as a global back gate. Before deposition of graphene,

the substrate was cleaned by Piranha etch. After suitable sheets were located with

respect to alignment marks by optical microscopy, metallic probes were patterned

using standard electron beam lithography followed by electron beam evaporation

of Ti/Au (5 nm/25 nm thick). Afterward, the graphene sheets were etched in dry

oxygen plasma (1:9 O2:Ar) into the desired shape, and one or two layers of Polymethyl

Methacrylate (PMMA, molecular mass 950K or 495K at 2% in anisole) were spun on

top of it, then cross-linked using 30 keV electron beam with a dose of 2×104 µC.cm−2.

In a final e-beam lithography step, the top gates were patterned on top of the cross-

linked layer, followed by electron beam evaporation of Ti/Au (5 nm/45 nm-55 nm

thick).

Chapter 5

Sharp p-n junctions

As discussed in Section 2.3.5, a sharp potential step requires a gate strongly coupled

to the graphene sheet. Below we describe our fabrication method for creating a thin

dielectric on top of a graphene sheet, and then present the measurements on such a

device.

5.1 Fabrication process of thin dielectrics

For the gates we need the dielectric to satisfy the following:

1. It needs to have a low leakage current to the graphene sheet. The current should

be much smaller than the typical current bias used in our setup.

2. Be thin enough. The characteristic length of the potential step is proportional

to the thickness of the dielectric. However, it should not be too thin, otherwise

tunneling could occur from the metal gate to the graphene sheet (Alumina is a

common tunneling barrier in break-junction at thicknesses of 1− 2nm [56].

In our case the dielectric is made of thin Aluminum films deposited by e-beam evapo-

ration, and oxidized in ambient conditions. We first follow the procedure of exfoliating

graphene on a substrate, and depositing metal contacts to it, as described in the sup-

plementary materials section of Chapter [46]. There are three choices in patterning

the dielectric:

68

5.1. FABRICATION PROCESS OF THIN DIELECTRICS 69

1. Evaporating Aluminum everywhere without the use of PMMA mask. The main

advantage is avoiding an ebeam lithography step as no mask is needed, which

might lead to a higher mobility device. However, since the metal leads already

deposited, they will be covered with Alumina as well, which might cause inter-

mittent electrical contact between the wire bond and the pads. Such problems

occasionally happened with some of the faulty devices. Also, thickness measure-

ment by AFM cannot be taken since Alumina is everywhere. However, since

the dielectric is covering a large chip, elipsometry can be done to characterize

the thickness and dielectric in-tandem. Another option is evaporation of a test

chip with well defined pattern that can be used in the AFM.

2. Patterning the Aluminum using a PMMA mask, to cover the graphene sheet

only. This way the entire flake is protected under the Alumina, and its edges

are well defined. In this case, AFM spectroscopy can be done directly on the

actual device but the extra ebeam lithography step concentrated on the device

and its vicinity might degrade the device quality leading to lower mobilities.

3. Patterning the Aluminum using a PMMA mask, to cover part of the graphene

sheet. If the top gate dielectric is strictly under the top gate metal, then the

discontinuity between the interfaces -Alumina and Air - creates a very sharp

slope at the interface, which generates a large electric field.

Option 3 is the best one for getting high electric fields at the interface of a p-n

junction. It is also the only option for making devices with multiple steps of dielectrics.

However, it requires oxidizing the Aluminum in-situ, and evaporating the top gate

metal immediately after oxidization takes place in order to self-aligned the gate metal

and the gate dielectric. It also requires a special care with the e-beam lithography.

In this work, every evaporation of Aluminum of a few nm thickness created edges

as high as 10nm with width of a few 100’s of nm. Such peaks close to the gated

region makes it nonuniform as well as difficult to overcome with the gate metal. This

findings drove us to chose option 2 where the dielectric pattern is big enough, and the

graphene itself is covered with uniform 5nm dielectric (see Fig. 5.1). It is also easier

70 CHAPTER 5. SHARP P-N JUNCTIONS

this way to characterize the dielectric thickness with AFM in comparison to option

1.

After patterning the PMMA mask, we initiate 4 steps of Alumina deposition in

an ebeam evaporator. In each step we evaporate 1nm of 99.999% Aluminum at a rate

of 1A× sec−1 inside an e-beam evaporator at a base pressure of around 1e − 7torr.

The low evaporation rate makes the surface smoother. After each evaporation the

chamber is vented, and the sample stays in ambient conditions for a period of at

least one hour, in order to be fully oxidized (although according to Ref. [57], full

oxidation takes place as soon as the sample is taken out of the evaporator’s chamber).

After 4 such evaporations the thickness of the Alumina is expected to be higher than

the actual thickness of the evaporated Aluminum because of the oxidizing process.

Figure 5.1 shows the AFM image of the device and the measured thickness of the

dielectric.

Figure 5.1: Left: AFM image of the graphene device with thin Alumina dielectric.The red arrows point the two top gate metals. The dashed red lines mark the grapheneedges. Alumina is covering the entire area and extends further out a few microns ineach direction. Right: Cut taken close to the Alumina edge (not shown in the leftimage). The peak due to lift-off of a few nm’s is seen close to the step in the middle.

Another approach for the Alumina deposition is evaporating only 1 thin layer of

Alumina (1-2nm) and then using the ALD chamber to deposit an arbitrary-thickness

high quality Alumina. The evaporated layer is used as a functionalizing layer which

5.2. DETERMINING THE THICKNESS 71

allows the growth of Alumina using ALD [58]. This technique can be used as long as

no lift-off is needed (as in option 1 above), since the growth of ALD Alumina occurs

on the walls of the PMMA as well, which leads to very difficult lift-offs that creates

rough edges. Such edges are highly non-uniform and will be hard to overcome in the

next step of top gate metal deposition.

5.2 Determining the thickness

To determine the thickness, we measure the dielectric step by AFM 1 near the edge of

the Alumina to the substrate (shown on Fig. 5.1), which is a few microns away from

the graphene sheet. As the step contains Alumina on one side and SiO2 on the other

side, the AFM tip to substrate interaction might be different on each side, giving

wrong thickness value. In order to overcome this problem, we made similar devices

without the graphene sheet and evaporated Alumina with the same conditions as the

actual device. We then evaporate Au everywhere and measure the step height using

AFM between the gold coated Alumina and the gold coated SiO2. Figure 5.2 show the

step height which is indeed close to 5nm. During the same Alumina evaporation steps

we also deposit Alumina everywhere on bare SiO2 chips using tape to roughly mask

part of the chip. We use three different such chips, where after the first and second

evaporations we take out the first and second chips respectively out and do not proceed

with further evaporations for them. This way one chip has only one layer of Alumina,

one has two layers of Alumina and the third, which undergone all four evaporations

has four layers of Alumina. After all evaporations we remove the tape from each chip

and wash the chips in Acetone and IPA to remove the tape residue. Then we measure

thickness using a Woollam M2000 Spectroscopic Ellipsometer, which gives thicknesses

of 1.8nm, 2.7nm and 5.5nm for one, two and four evaporations respectively (Same

heights were also measured with AFM). The dielectric constant extracted from the

ellipsometer is close to 8 as expected from Alumina layers.

1MFP-3D Asylum AFM


Figure 5.2: Step of 4.3nm between 4 layers of Alumina covered with gold and baresubstrate covered with gold. The profile is smoother because of the additional layerof gold.

5.3 Measurements on thin dielectrics

Measurements were done at T = 4.2K using current bias of 100nA. Fig. 5.1 show the

4-probe configuration that was used between the two voltage probes (marked as ’P’),

and the source and drain (marked with ’S’ and ’D’ respectively). Typical leakage

currents are shown in Fig. 5.3 as a function of the top gate voltage. The top gate

dielectric resistance is of a fewGΩ, and the leakage is smaller than 1nA. In comparison

to the current bias of 100nA the leakage current has a negligible effect on the graphene

sheet conductivity. The low comparable leakage current on the top gate indicates high

quality dielectric. We measure 4-probe resistance on two separate regions each has

one top gate and is covered fully with Alumina (see Fig. 5.1). The results shown in

Fig. 5.4 has similar characteristics to the ones achieved in Refs. [46, 59]. However, the

ratio between the top gate capacitance and back gate capacitance is almost an order

of magnitude higher than previous results. Since the geometric capacitance becomes

5.4. RESULTS DISCUSSION 73

very large in thin dielectric structures, the quantum capacitance correction is non-

negligible, and the slope marked in dashed line in Fig. 5.4 represents the ratio between

the back gate capacitance and the total top gate capacitance, which is comprised out

of two capacitors in series, the geometric capacitance and the quantum capacitance.

The quantum capacitance in graphene is proportional to D, the density of states

Cq = e2D, where e is the electron charge. Extracting a fixed geometric capacitance

from the total capacitance value along the slope (at zero density below the top gated

region) yield top gate capacitance which corresponds to a thickness of 5nm with a

dielectric close to 8. Also, the relation n ∝ Vg no longer holds and we calculate it self

consistently, using the experimental parameters (See Fig. 5.5).

Figure 5.3: Leakage current as a function of Vtg for the 5nm thin dielectric.

5.4 Results discussion

Since the top-gated region is much longer (300nm and 500nm) than the mean free

path (Fig. 5.6), the odd part of the resistance is a measure of a single p-n junction

interface resistance. In order to eliminate the mono-polar resistance contribution to


Figure 5.4: Left: Resistance as a function of Vbg and Vtg. Right: Cuts at fixed Vbg

taken from the left plot

Figure 5.5: Density as a function of Vbg (Dirac point chosen at Vbg = 0). Red:Without the quantum capacitance correction. Green: Taking into account quantumcapacitance correction


the odd part of the resistance, we look only at a symmetric p-n junction where the

densities p and n under the two regions satisfy p = −n . We compare these results of

the region gated with 300nm long top gate with the theory of sharp and smooth p-n

junction (similar results are extracted from the longer gate). The smooth p-n junction

resistance is calculated using the transmission coefficient taken from Ref. [31] with the

correction for electric field at the interface due to non-linear screening [34]. We use

the same c1 used for fitting in Ref [59]. The sharp p-n junction resistance is calculated

using equation 10 in Ref [35], which gives the proper transmission coefficients for the

sharp case by taking the limit d → 0. Figure 5.7 show the experimental results for

different symmetric densities compared with the two theories.

Figure 5.6: Mean free path as a function of density. Electron-hole asymmetry stemsfrom the extended 4-probe geometry of the device.

The theory of smooth potential step supports the experimental results, although

there is a strong evidence that the dielectric is only 5nm thick, and the top gate

is strongly coupled to the graphene sheet. Figure 5.8 shows that for a symmetric

p-n junction the range of angles in the sharp limit, satisfying π~vFk2ye

−1F−1 < 1 is

substantial (about 70 degrees at the lowest density of 0.5× 1012cm−2). At this range


Figure 5.7: Experiment vs. theory of Rodd as function of density in a symmetric p-n

junction. Dots: experiment. Solid blue line: Sharp theory. Solid red line: Smooththeory.

the sharp transmission coefficient and smooth transmission coefficient differ enough

to expect the experimental p-n junction resistance to be between the smooth and

sharp predictions and not following directly the smooth theory.


Figure 5.8: Filled circles: Transmission coefficient as a function of angle at differentdensities for a symmetric smooth p-n junction. Empty squares: The dimensionlessparameter - π~vFk

2ye

−1F−1 - as a function of angle at different densities for a sym-metric p-n junction (F electric field at the interface, e electron charge). Black line:Transmission coefficient as a function of angle for a sharp p-n junction.

Chapter 6

Evidence of the role of contacts on

the observed electron-hole

asymmetry in graphene

B. Huard, N. Stander, J. A. Sulpizio and D. Goldhaber-Gordon Stanford University,

Department of Physics, Stanford, California, USA 1

6.1 Abstract

We perform electrical transport measurements in graphene with several sample ge-

ometries. In particular, we design “invasive” probes crossing the whole graphene sheet

as well as “external” probes connected through graphene side arms. The four-probe

conductance measured between external probes varies linearly with charge density

and is symmetric between electron and hole types of carriers. In contrast measure-

ments with invasive probes give a strong electron-hole asymmetry and a sub-linear

conductance as a function of density. By comparing various geometries and types

of contact metal, we show that these two observations are due to transport prop-

erties of the metal/graphene interface. The asymmetry originates from the pinning

1This chapter is adapted with permission from Phys. Rev. B. 12, 78 (2008). c©(2008) by theAmerican Physical Society

78


of the charge density below the metal, which thereby forms a p-n or p-p junction

depending on the polarity of the carriers in the bulk graphene sheet. Our results also

explain part of the sub-linearity observed in conductance as a function of density in

a large number of experiments on graphene, which has generally been attributed to

short-range scattering only.

6.2 Introduction

Graphene, a crystalline monolayer of carbon, has a remarkable band structure in

which low energy charge carriers behave similarly to relativistic fermions, making

graphene a promising material for both fundamental physics and potential applica-

tions [10]. Most interesting predicted transport properties require that charge carriers

propagate with minimal scattering. Recently experimentalists have succeeded in re-

ducing disorder [23, 60] and have shown the important role of nearby impurities on

the mobility of charge carriers [61, 62]. In contrast, the effect of metallic contacts

on transport has received little attention in experiments. For instance, most experi-

ments show a clear difference between the conductances at exactly opposite densities,

a phenomenon previously attributed to different scattering cross-sections off charged

impurities for opposite carrier polarities[63, 22]. In this letter, we show that trans-

port properties of the interface between graphene and metal contacts can also lead to

such an asymmetry. This effect is due to charge transfer from the metal to graphene

leading to a p-p or p-n junction in graphene depending on the polarity of carriers in

the bulk of the sheet. We also show that this leads to sub-linear conductance as a

function of gate voltage, which is traditionally attributed to short-range scattering

[64, 65, 66, 25]. With a proper measurement geometry, we find conductivity linear in

density up to at least n = 7× 1012 cm−2 showing that short range scattering plays a

negligible role.

80 ROLE OF CONTACTS IN ELECTRON-HOLE ASYMMETRY IN GRAPHENE

aL

bL

a-bc-dd-eb-cI=100 nA

-60 -40 -20 0 20 40 60

0

10

20

0-0.2

0.2

-0.4

0.4

Vg HVL

RHk

WL

Rod

dHk

WL

Figure 6.1: a) 4-probe resistance calculated from voltages measured between invasiveprobes as a function of gate voltage while a steady oscillating current of 100 nAruns along the whole graphene sheet. Inset: Scanning Electron Microscope imageof the graphene sample TiAu1 connected to Hall probes (A-D) and invasive probes(a-e). For clarity, graphene has been colorized according according to topographymeasured by atomic force microscopy. b) Given the charge neutrality gate voltageV 0

g identified from Quantum Hall Effect measurements (see text), we plot here theasymmetry between electrons and holes by showing the odd part of the resistancedefined in Eq. 6.1.



In order to investigate the properties of the graphene/metal interface, we used two

types of metallic voltage probes (see inset Fig. 6.1). “Invasive” probes (like a-e in

Fig. 6.1) extending across the full graphene strip width are sensitive to contact and

sheet properties while “external” probes (like A-D in Fig. 6.1) connected to narrow

graphene arms on the side of the strip are sensitive to sheet properties only. All

the graphene samples described in this letter are prepared by successive mechanical

exfoliation of Highly Oriented Pyrolytic Graphite (HOPG) grade ZYA from General

Electric (distributed by SPI) using an adhesive tape (3M Scotch Multitask tape with

gloss finish). The substrate is a highly n-doped Si wafer, used as a gate (capacitance

13.6 nF.cm−2 from Hall effect measurements), on which a layer of SiO2 297 nm thick is

grown by dry oxidation. Metallic probes are patterned using standard electron beam

lithography followed by electron beam evaporation of metal (see Table 6.1). Finally,

the graphene sheets are etched in dry oxygen plasma (1:9 O2:Ar) into the desired

shape. The voltage measurements between probes are performed in liquid Helium at

4 K using a lock-in amplifier at a frequency between 10 and 150 Hz with a bias current

of 100 nA. All samples were also measured in perpendicular magnetic fields up to 8T,

and show the quantum Hall plateaus characteristic of monolayer graphene. Most

samples were additionally characterized by Raman scattering, in each case showing

the typical graphene spectrum(see Supplementary material in Chapter 6.7).

Fig. 6.1 shows the 4-probe resistances measured between four pairs of invasive

probes in the sample TiAu1, as a function of gate voltage Vg. The resistance is

maximal close to the value V 0g of the gate voltage where the average charge density

is zero. In order to quantify the asymmetry between electron and hole transport, we

plot in Fig. 6.1b the odd part of the resistance defined as

Rodd(∆Vg) =1

2[R(V 0

g + ∆Vg)−R(V 0g −∆Vg)]. (6.1)

We determined the voltage V 0g with good precision using the sharp features of resis-

tance as a function of density in the Quantum Hall regime at 8 T. Two regimes of

density can be distinguished. For low densities n . 1.2 1012 cm−2, Rodd fluctuates


aL0 1 2 3 4

-0.4

-0.2

0.0

0.2

0.4

n H1012cm-2L

Rod

dwHk

W.Μ

mL

bL0 1 2 3 4

-0.6-0.4-0.2

0.00.20.40.6

n H1012cm-2L

Rod

dwLHk

WL

Figure 6.2: a) Odd part of resistance normalized by the extent w of themetal/graphene interface for four pairs of invasive probes shown in the inset of Fig. 6.1(same colors). The fluctuating region at densities smaller than 1.2 1012cm−2 has beengrayed. The charge density n is measured using the classical Hall voltage between ex-ternal probes, implying a capacitance of 13.6 nF.cm−2, consistent with the measuredoxide thickness. b) Same odd part of the resistance scaled by the ratio of the lengthw on the distance L between contacts.

widely. The extent of this fluctuating regime is consistent with the density of charged

impurities ni = e(hc2µ)−1 ≈ 0.5 1012 cm−2 one would calculate from the assumption

that the mobility µ ≈ 4600 cm2V −1s−1 (see Fig. 6.3) is dominated by scattering off

charged impurities, where c2 ≈ 0.1 for graphene on SiO2 [61, 62]. For larger densi-

ties n & 1.2 1012 cm−2, Rodd saturates to a finite value, corresponding to a higher

resistance for electrons (Vg > V 0g ) than for holes (Vg < V 0

g ). Such an asymmetry

was previously predicted and observed in the presence of charge impurity scattering

in graphene [63, 61, 22, 62]. In that case, the asymmetry comes from a difference

between the scattering cross-sections of positive and negative charge carriers on a

charged impurity. Let us define two different resistivity functions ρe(|n|) for electrons

and ρh(|n|) for holes as a function of density n. If this is the source of the asymmetry

in resistance, the odd part Rodd should be given by 2Rodd(n) = (ρe(|n|)−ρh(|n|))L/wfor electrons (n > 0), where L is the distance between voltage probes and w is the

width of the graphene strip. However, as can be seen in Fig. 6.2b, the asymmetry

of resistivity inferred in this way from our four measurements from Fig. 6.1 varies

widely with changing L. On the contrary, if we associate Rodd with a specific inter-

face resistance r(n), all curves for different geometries collapse together (Fig. 6.2a).


Therefore, we propose a more general expression for Rodd:

sheet property ≪ interface

2Rodd(n) =

︷︸︸︷

[ρe(|n|)− ρh(|n|)]Lw

+

︷︸︸︷

r(n)1

w.

(6.2)

Repeating the resistance measurements using external probes instead of invasive

probes, we can get rid of the interface term r(n)/w in Eq. (6.2) and measure the

sheet asymmetry only. To the precision of our measurements, ρe/ρh = 1± 0.03 when

averaged on all densities (Fig. 6.3 inset). The absence of asymmetry between ρe and

A-B

B-C

C-D

1.20

0 1 2 3 40.7

0.8

0.9

1.0

1.1

1.2

1.3

n H1012 cm-2L

Ρe

Ρh

-60 -40 -20 0 20 40 600

1

2

3

4

5-6 -4 -2 0 2 4

Vg HVL

Σ=HR

wLL-

1Hm

SL

nC-D H1012cm-2L

Figure 6.3: Conductivity as a function of gate voltage. Each color corresponds to apair of probes identified by two letters on Fig. 6.1. The slope of these curves cor-responds to a mobility of µ ≈ 4600 cm2V −1s−1. Inset: given the charge neutralitygate voltage V 0

g identified from Quantum Hall Effect measurements (see text), we plothere the ratio between resistivities for electrons and holes as a function of carrier den-sity. In contrast to the case of invasive probes, the average asymmetry is invisible tothe precision of our measurement (note: the observed fluctuations are reproducible).The line corresponds to the ratio 1.20 observed in Ref. [22] in presence of chemicaldopants.

ρh is in contrast with the ratio of about 1.20 Chen et al. observed [22] when graphene

was exposed to chemical dopants. To understand this apparent discrepancy, let us

consider the three proposed sources of scattering in graphene: short-range scatterers,

charged impurities and corrugation in the graphene sheet. First, short-range scatter-

ers add a term ρs almost independent of n to the resistivity. From Fig. 6.3, we can set


an upper bound ρs < 15 Ω/ surprisingly small compared to other reported values

[67]. Charged impurities naturally lead to the observed linear dependence of con-

ductivity on n, whereas corrugation requires a particular height correlation function

to give the same behavior, which is thus less likely [26]. As has been predicted and

shown experimentally, scattering off charged impurities of a given polarity occurs at

a different rate for electrons and holes [63, 22], and it also shifts the voltage V 0g and

decreases the mobility. However, both in our measurements and in those of Ref. [22]

prior to doping, there is no asymmetry in the resistivity. This could be due to some

equilibration between impurities of opposite polarities, but in this case, the difference

in V 0g between the experiments is somewhat surprising and would be worthy of further

investigation.


As we have seen, for invasive probes, Rodd scales inversely with the extent w of the

metal/graphene interface. Metallic probes in contact with graphene are expected

to pin the charge density nc in the graphene below the metal thereby creating a

density step along the graphene strip [68, 69, 70]. The height of this step and the

sign of nc depends on the mismatch between the work functions of the metal and the

graphene sheet. As we will see, for our choices of contact metal the charge density in

graphene is pinned to a negative value nc (p-type) below the metal. Thus depending

on the polarity of the carriers in bulk graphene sheet a p-n junction or a p-p junction

develops close to the metal/graphene interface. We have shown elsewhere [46, 59]

that the resistances associated with these junctions for opposite values of the charge

density n in the sheet differ by an amount rnc(n)/w where rnc depends only on nc and

on the length over which the density varies across the junction [31, 34, 37, 33]. This

is consistent with the observed positive Rodd; with n-type graphene below the contact

one should observe a negative Rodd. If we further assume that the density changes

from nc to n on a very short scale compared to (|nc|+ |n|)−1/2, we obtain an analytical

expression for rnc (see Supplementary material in Chapter 6.7). If this is the origin of

the observed asymmetry, Rodd should counter-intuitively decrease when the mismatch


between metal and graphene work functions increases. The limit where nc goes to

infinity gives the lowest possible value of Rodd in this sharp-junction approximation

(see Supplementary material in Chapter 6.7)

rnc(n) > 2.06h

4e2n−1/2. (6.3)

TiAu2TiAu1

Pd2Pd1

theoretical min

1 2 3 4 5-0.4

-0.2

0.0

0.2

0.4

n H1012 cm-2L

rHk

W.Μ

mL

Figure 6.4: Odd part Rodd of the resistance scaled by the inverse width w−1 for varioussamples and metals described in Table 6.1.

Sample Metal thickness w (µm)

Pd1 Pd(30 nm) 0.4Pd2 Pd(30 nm) 0.9

TiAu1 Ti(5 nm)/Au(25 nm) 0.8TiAu2 Ti(3 nm)/Au(15 nm) 2.4

Table 6.1: Geometrical properties of the samples corresponding to Fig. 6.4. Themeasurements shown on Figs. 6.1,6.2,6.3 were performed on TiAu1. The type ofmetal used as a probe and its thickness is given here together with the length w ofthe graphene/metal interface.

On Fig. 6.4, we show the function rnc measured in several graphene sheets con-

tacted with two types of metal (see Table 6.1). For Pd, which is expected to have

a high work function (ΦPd ≈ 5.1 eV < Φgraphene = 4.5 eV with the prediction


for graphene, see e.g. Ref. [70]), the function rnc is very close to the lower bound

Eq. (6.3). In contrast, for Ti covered with a layer of Au, where the work function

mismatch should be smaller (ΦTi ≈ 4.3 eV and ΦAu ≈ 5.1 eV), the function rnc was

larger at high densities n, suggesting that the densities nc and n are of the same

order of magnitude. We notice that for Pd, rnc decreases with n whereas for Ti/Au

it increases, but it is hard to explain this increase since it would require knowing the

potential profile close to the lead. Finally, as expected[70] (ΦAl ≈ 4.2 eV < Φgraphene),

Ti/Al probes lead to the opposite doping: Rodd that we estimate from other works

[71, 72] is negative.

Charge transfer from the metallic probes has yet another observable effect on

transport. On Fig. 6.5a, we show the conductance measured using invasive probes

scaled by the geometrical aspect ratio of each section. Even on the hole side (Vg < V 0g )

where there is no p-n junction, a sub-linearity is striking when compared to the

external probe measurement shown in the same figure. We find that there is a constant

specific contact resistance λ such that (R−λ/w)−1 is linear in density in the hole region

(see Fig. 6.5b). This contact resistance independent of density n can be attributed

to a higher concentration of short-range scatterers near the contact (perhaps due to

e-beam exposure during lithography) and/or to the region of constant density nc near

the contacts. In order to determine which effect is dominant, we compare the value

of λ for the two different metals of Table 6.1. We find that this resistance is small for

Pd probes compared to Ti/Au probes, consistent with sub-linearity coming from a

region of larger constant density nc for Pd than for Ti/Au, and not from short-range

scatterers.

6.5 Conclusions

In conclusion, we have shown that all measurements using invasive metallic probes

should exhibit an asymmetry between hole and electron conductances due to charge

transfer at the graphene/metal interface. Similarly, invasive probes lead to a sub-

linearity in the conductance as a function of density, even in a 4-probe geometry.

In every experiment using invasive probes, one should consider these effects in the

6.5. CONCLUSIONS 87

aLB-C

-50 0 500

2

4

Vg HVL

HRwLL-

1Hm

SL bL

-50 0 500

2

4

Vg HVL

HR-

ΛwL-

1 LwHm

SL

Pd1 Pd2 TiAu1 TiAu20

0.1

0.2

0.3

0.4

sample

ΛHk

W.Μ

mL

Figure 6.5: a) From the resistance curves plotted in Fig. 6.1, we show the conductancescaled by the ratio w/L. The non-invasive measurement between probes B and C fromFig. 6.3 is plotted as a thin line for reference. b) subtracting λ = 0.135 kΩ.µm dividedby the length w of the metal/graphene interface, each curve from a) is linearized forthe p-type carriers (Vg < V 0

g ). Main panel: for each four-probe measurement on thesamples from Table 6.1, we plot here the specific resistance λ which best linearizesthe conductance as a function of gate voltage (see text). The best fit is obtained atthe dot and the vertical size of the corresponding ellipse represents the uncertaintyon λ.


calculation of the conductivity from the resistance measurement and sample geometry.

External probes do not have this issue and reveal a conductance linear in density.


We thank D. Novikov, M. Fogler and J. Cayssol for enlightening discussions. This

work was supported by the MARCO/FENA program and the Office of Naval Research

# N00014-02-1-0986. N. Stander was supported by a William R. and Sara Hart Kim-

ball Stanford Graduate Fellowship, J.A. Sulpizio by a National Science Foundation

graduate fellowship. Work was performed in part at the Stanford Nanofabrication

Facility of NNIN supported by the National Science Foundation under Grant ECS-

9731293. Critical equipments were obtained on Air Force Grant.


6.7 Supplementary Material

6.7.1 Graphene characterization

We checked that the Quantum Hall effect in our samples corresponds to graphene.

For example, Fig. 6.6 shows the conductivities measured in sample TiAu1 using ex-

ternal probes at 8 T. The plateaus in σxy show indeed the values characteristic of

graphene ±2e2/h, ±6e2/h . . . [10] We also performed Raman spectroscopy on most of

Figure 6.6: Conductivities σxx and σxy measured in sample TiAu1 using externalprobes at a magnetic field of 8 T and a temperature 4 K.

the graphene samples. For example, Fig. 6.7 shows the spectrum for sample TiAu1

and a piece of few layer graphene (most likely bilayer) next to it. The spectrum of

graphene is also striking here [10].


Figure 6.7: Raman spectrum for sample TiAu1 and the spectrum of a few layergraphene piece next to it.

6.7.2 Resistance of a p-n junction in graphene

We consider here the transport across a monotoneous potential step eU(x) between

regions of wavevector kL and kR:

eU(−∞) = −~vF |kL|; eU(+∞) = ~vF |kR|.

Two cases must be distinguished depending on the transversal component ky of the

wavevector and the slope deU/dx of the potential at the position of zero energy.

6.7.3 Sharp barrier

In the case where ky ≪ (deU/dx)[~vF (kL + kR)]−1, the barrier is sharp and the

transmission is given by

T (θ) = 1−∣∣∣∣

eiθ − eiϕ

e−iθ + eiϕ

∣∣∣∣

2


where θ is the angle of incidence and ϕ the outgoing angle [37]. Besides, Snell’s law

imposes that

|kL| sin θ = −|kR| sinϕ.

We can express the transmission as a function of the vector ky = kL sin θ by

T (ky) =2√kL

2 − k2y

√kR

2 − k2y

|kL||kR|+ k2y +

√kL

2 − k2y

√kR

2 − k2y

.

The conductance is given by

G =4e2

h

Nc/2∑

m=−Nc/2

T (k(m)y )

where k(m)y = 2πm/w and Nc = min(|kL|, |kR|)w/π. Therefore, assuming that |kL| <

|kR|, we get

G ≈ 8e2w

3πh(|kL|+ |kR|)−2

[2|kL|(|kL|2 + 3|kR|2)

+|kR|(|kL|2 − 6|kL||kR|+ |kR|2)E(|kL|2/|kR|2)

−(|kR| − 3|kL|)(|kR| − |kL|)(|kL|+ |kR|)K(|kL|2/|kR|2)]

where K(x) is the complete elliptic integral of the first kind and E(x) is the complete

elliptic integral of the second kind 2. In the limit of a high barrier where |kL| ≪ |kR|,we get

G ≈ 8e2w

3πh(6− 3π/2)|kL| ≈ 0.86

4e2w

πh|kL|.

2From http://mathworld.wolfram.com,

K(x) =π

2

+∞∑

n=0

[(2n− 1)!!

(2n)!!

]2

x2n

and

E(x) =π

2− π

2

+∞∑

n=1

[(2n− 1)!!

(2n)!!

]2x2n

2n− 1


Therefore

r(n) ≈ 1.16

√πh

4e2

√

|nL|−1

6.7.4 Smooth barrier

In the case where ky ≫ (deU/dx)[~vF (kL + kR)]−1, the barrier is smooth and the

transmission is given by [31, 33]

T (ky) = e−π~vF k2y(deU/dx)−1

.

In order to go further, we make the oversimplifying assumption that the potential

U varies linearly with x and is characterized by a length d such that deU/dx =

~vF (|kL|+ |kR|)/d. This leads to a conductance

G ≈ 2e2w

πh

√

|kL|+ |kR|d

erf

(√

π|kL|2d|kL|+ |kR|

)

which in the limit of a high barrier where |kL| ≪ |kR| gives

G ≈ 4e2w

πh|kL|.

Therefore,

r(n) ≈√πh

4e2

√

|nL|−1.

6.7.5 Case of partly invasive probes

Since the experiment by Chen et al. [22] was performed with partly invasive probes,

one can wonder if the observed asymmetry could be due to transport properties of

the graphene/metallic probe interface instead of a difference between electron and

hole mobilities.

We therefore consider the case of partly invasive probes. These probes extent on

a length w1 and leave a transversal length w2 free of metal. The width of the probes

(along the current lines) is l and the distance between probes is L. In this case, the


Figure 6.8: Cartoon of partly invasive probes on a graphene strip.


measured resistances for electrons or holes read

Rh = ρhL+ l

w1 + w2

Re =ρel/w2(2r/w1 + ρel/w1)

2r/w1 + ρelw1+w2

w1w2

+ ρeL

w1 + w2

6.7.6 Wide probes

In this case where 2r ≫ ρel, these expressions simplify into

Rh = ρhL+ l

w1 + w2

Re ≈ρel

w2+ ρe

L

w1 + w2

This leads to a ratio of conductances for electrons and holes given by

Ge

Gh=ρh

ρe

L + l

l(1 + w1/w2) + L

hence,Ge

Gh=ρh

ρe(1− lw1/w2

l(1 + w1/w2) + L)

and since l ≪ L,Ge

Gh≈ ρh

ρe

(

1− l

L

w1

w2

)

In the experiment of Chen et al. [22], the geometry of the sample is such that

L ≈ 8.2 µm, l ≈ 1.6 µm, w1 ≈ 2.9 µm and w2 ≈ 10.6 µm. This leads to a correction

in the measured mobility ratio given by

Ge

Gh≈ 0.95

µe

µh


6.7.7 Narrow probes

In this case where 2r ≪ ρel, we eventually get

Ge

Gh≈ ρh

ρe

(

1− l

l + L

w1

w1 + w2

2r

ρel

)

which is even closer to the ratio of mobilities than the first case.

Ge

Gh

≈(

1− 0.052r

ρel

)µe

µh

According to these results, it seems that the role of the contacts in the asymmetry

observed in the experiment of Ref. [22] is negligible.

6.7.8 Single metal/graphene interface

The measurements discussed in the letter are performed either on a pair of invasive

probes or on a pair of external probes. In the picture described in the letter, the

asymmetry comes from the crossing of interfaces metal/graphene. In order to illus-

trate this point further, we show a measurement between an external probe (A) and

an invasive probe (b) in Fig. 6.9. The odd part of the resistance is then half as large

as when measured using a pair of invasive probes, consistent with a single interface

being crossed instead of two.

6.7.9 Atomic Force Microscopy

Graphene is difficult to see on the raw Scanning Electron Microscope image shown

in the paper, and we used Atomic Force Microscopy (AFM) to colorize it. An AFM

measurement of the graphene arms is shown in Fig. 6.10.

6.7.10 Fabrication details

The substrate used in these experiments is a highly n-doped Si wafer with a resistivity

of less than 0.005 Ohm-cm. The wafer was dried oxidized.


Figure 6.9: Odd part of resistance normalized by the extent w of the metal/grapheneinterface for four pairs of invasive probes shown in the inset of Fig. 1 in the paper(same colors). In a measurement between the invasive probe b and the external probeA, the odd part is half as large as with two invasive probes.


Figure 6.10: Image of the external probe region of the sample TiAu1 taken usingthe phase-mode of an atomic force microscope. The side graphene arms are clearlyvisible here and correspond to the colorization of Fig. 1 in the letter. Notice thediscontinuities in the graphene side arms corresponding to an electrical shock afterthe experiment (no conduction through the arms after this event).

Appendix A

Electronics

Two electronic setups were used in the projects described in previous chapters. The

first one is designed to verify which leads to the sample work well. it is done on every

sample prior to any electronic measurement. The second one is the actual current

bias measurement of 2-probe, 3-probe or 4-probe measurements. The two setups are

described in the following sections.

A.1 Lead connection protocol

Graphene flakes can sustain very large current densities. However, if not handled

properly, the leads connecting to the graphene can break, especially at the contact

interface, due to high changes of voltages on the device. Thus, it is important to

ground yourself to the dunker-stick or fridge when loading the sample. Also, making

sure that all the pins are grounded before loading the sample and at least one pin

connected from the sample to the one of the measurement instruments (e.g. lock-

in amplifier) via the breakout box. This way the virtual ground of the lock-in is

the common ground for the dunker-stick and the graphene flake. Figure A.1 shows a

schematic of the setup used to check the contacts integrity. The graphene is connected

in one side in series to a current-limiting resistor and a voltage divider, and on the

other side it is grounded. Applying 1V on the lock-in will have 1mV across both

graphene and the limiting- resistor. In case of a short, the limiting-resistor will limit

98

A.2. SAMPLE MEASUREMENTS 99

the current passing through the device. As the graphene resistance expected to be

of the same order of the limiting resistor (of 26kΩ), a certain drop of voltage is

expected on the two lock-in terminals (differential mode). In each iteration, only one

of the graphene leads is grounded, while all the rest are floating (except for the one

connected to the rest of the circuit). This way for each combination of contact it is

possible to check whether there is a connection or not (no voltage drop on the limiting

resistor indicates a very large resistance associated with an open circuit between the

two contacts).

Figure A.1: Connection check protocol

A.2 Sample measurements

Most of the measurements were done by current biasing and measuring 2-probe,3-

probe or 4-probe measurements. The lock-in excitation was 1VAC on a 10MΩ,

which gives a current excitation of 100nA. For a typical device geometry and typical

graphene resistivity, the resistance measured is of the order of a few kΩ at 4K.

Figure A.2 shows a schematic circuit.

100 APPENDIX A. ELECTRONICS

Figure A.2: Measurement setup. 4-probe, 3-probe and 2-probe configurations can beused here depending on the number of contact leads available to the graphene

Appendix B

Fabrication of fine features using

our lab-shared SEM

Fabrication process for contacts and etch patterns is presented in the supplementary

material of Chapters 3-4. Features of single 50nm wide lines were patterned with

a recipe developed by Joseph Sulpizio (See Sulpizio’s thesis for more details). The

shared SEM in our lab is FEI Sirion SEM that is typically used for imaging (although

its UHR mode is not functional) with the additional Nabity Pattern Generation Sys-

tem (NPGS) for patterning. Typical dose for patterning large features (larger than

100nm width) is around 270µCcm−2 in area mode and using 30kV (other voltages

are hard to focus). As a rule of thumb, before any new pattern design and after

every substantial maintenance to the SEM (such as tip replacement), it is beneficial

to conduct a dose area that includes metal evaporation to optimize the dose required.

For fine features less than 100nm we used a single layer of PMMA 950K spun 40

seconds at 4000rpm and then placed on a hot plate for 5 minutes at 180 degrees. The

dose used for fine features is typically higher than the normal area dose. Area dose is

used as it has better control over using the line dose. Doses as high as 400µCcm−2 can

be used to achieve sharp fine lines. Before performing the write, the writing sweep

direction must be changed in the .dc2 file to run on the longer side otherwise features

will come out much thicker than designed due to over exposure (see the NPGS manual

for specific details). In the .run file the point and line spacings should be adjusted to

101

102APPENDIX B. FABRICATION OF FINE FEATURES USING OUR LAB-SHARED SEM

the minimal value otherwise the pattern might be underexposed. Alignment might be

an issue as well when trying to write a top gate in between pre-made metal contacts.

In order to overcome this problem, it is best to pattern four smaller alignment marks

in the first ebeam lithography together with the contacts. Using the large alignment

marks done by photolithography is not recommend as they are typically good only

down to 500nm accuracy.

Appendix C

3-axe magnet Dewar procedures

C.1 Cooling the Dewar from 300 K to 77 K

1. Connect one port of the outer jacket to a LN2 dewar using rubber hoses and

connect the others to disconnected rubber hoses, and transfer LN2 (filled at 50

L)

2. Connect the fill port (F) of the belly to a LN2 dewar and start transferring

LN2. If there is no fridge or dunker stick in place, you can safely remove the

cap of the central port and use it as an exhaust. If there is already something

through this port, transfer more slowly and use the vent port V as an exhaust.

The noise can be painful, but it will be greatly diminished if you put a rubber

hose at the exhausting port.

3. Stop the transfer after around 30 L and wait for 2 hours at least and until the

magnet leads Z have a resistance of about 20 Ohms (19.7 Ohms is 77K)

C.2 Cooling the Dewar from 77 K to 4 K

1. You first need to remove all the LN2 still present in the belly. In order to do

that, you must use the blowout tube through the F port (NOT the V port). It

is tricky to go pass the baffle on top of the magnet, but you have to do it. Try

103

104 APPENDIX C. 3-AXE MAGNET DEWAR PROCEDURES

rotating the blowout tube until it goes through this baffle. It is useless to bend

the tube. In its final position, the tube should be nearly all the way down.

2. Using a rubber hose, connect the blowout tube to the outer jacket using any

port or let the rubber hose go into a LN2 recipient. Put the green rubber plug

into the vinyl hose exiting from the leads. Finally, pressurize the belly using

N2 gas on the V port.

3. Do not stop until all the LN2 has been removed from the belly. You should

check this by looking directly through the center using a flashlight.

4. When all the LN2 has been removed, pressurize the V port using He gas instead

of N2 for 5 minutes and remove the blowout tube. Keeping the flow of He gas

will prevent clogging the transfer line for the next step.

5. Get ready to transfer LHe. For this first transfer you need to use the short

transfer tube WITH the new extension on the 3-axe dewar side. First put the

transfer line with extension all the way through the F port (it has to go all the

way). It will be hard to pass the baffle above the magnet, but you have to do

it to cool the magnet efficiently without loosing too much He. The tube should

be in ALL the way Try not to bend the transfer line, just rotate it until it goes

through. The fact that the extension has a smaller diameter will be helpful

here.

6. Once it is done, put the other end of the transfer line into a LHe dewar and

stop pressurizing the belly through the V port. Open the V port and remove

the plug off the vinyl tube. Do not go too fast as you will use the cooling power

of the He gas and not the liquid Helium to cool the dewar.

7. Check the coil temperature by measuring the resistance of the Z magnet. When

it is below 1 Ohm, wait for a decrease in the exiting gas flow. This is a sign

that liquid He starts to be transferred. You can also check this by turning on

the Level Monitor and pushing the button on “update”. As soon as you see a

C.3. TRANSFER WHILE AT 4K 105

level higher than 0, you can safely increase the pressure in the dewar to increase

the flow of liquid He.

8. Fill the belly with the desired amount of He, knowing that the magnet cannot

be operated if this level goes below 15”.

9. Stop the transfer and remove the transfer line.

C.3 Transfer while at 4K

1. For this transfer you will need the short transfer tube. The short part goes to

the Cryostat, while the longer one goes to the He dewar.

2. Depressurize the dewar by closing the security valve and opening the vent one.

3. The next three steps are done immediately one after the other, so please read

all of them before starting the transfer:

(a) Open the F port by removing the cap and unscrew half way the fitting on

it. Also, prepare the V port by inserting the small tube as exhaust.

(b) Insert the longer part to the He dewar, you can do it fairly quick until

you reach a point when you start feeling a steady flow from the other end.

Once there is a flow, hold the dewar part in the same position (do not go

further yet).

(c) Once you see “white” gas flowing out of the other side (the one which is

going to enter the cryostat), you need to dunk the tube into the F port.

Make sure you pass at least the first baffle. It is important at the beginning

of this step to have the tube very close to the open F port. Once you see

the “white” gas, it takes about 2 seconds for the He to start flowing out,

so be prepared.

4. Change the switch on the upper panel of the Magnet controller to “update”.

106 APPENDIX C. 3-AXE MAGNET DEWAR PROCEDURES

5. Check He level and make sure your dewar is not over pressurized which might

lead to a high loss of He.

6. If the He level increases too slow, you can pressurize the dewar by plugging He

gas to it.

C.4 Level monitor

1. The level monitoring is done by a superconducting wire that goes from 0.5

inches above the bottom of the dewar up to 35 (+0.5) inches above the bottom.

The part of the wire which is covered by He will not have any resistance, while

the part that sticks out will have some resistance. The monitoring device apply

some current to the wire, and measure the voltage. The output is given by

inches/cm above 0 (which is the +0.5 inches from the dewar’s bottom) or by

percentage from the active length of 35 inches.

2. the level monitor is very simple. You turn it on and choose between inches,

cms or percentage of the maximum He height. In OSampleO mode, the level is

measured every 60s, in “update” mode, it is continuous. Of course, in order to

limit he consumption, it should stay in “Sample” mode, except during transfer

where the update mode can be useful.

3. the rotating knob can go from Silence to any other position in which the appa-

ratus produces an unbearable noise as soon as the He level goes outside of the

prescribed boundaries for operating the magnet

C.5 Magnet control

1. the magnet is made of three perpendicular coils. The maximum field of each

coil independently is higher than when operated together. One should be very

careful when pushing the limits of the coils. The records now are

C.5. MAGNET CONTROL 107

(a) X=0.5,Y=0 T and Z=7.5 T at a rate of 0.2 A/s for X and Y and 0.1 A/s

for Z

(b) X=0,Y=0.5 T and Z=7.5 T at a rate of 0.2 A/s for X and Y and 0.1 A/s

for Z

2. there are three options to operate the magnet. First, a completely manual mode

where the user asks for the field to go indefinitely up or down or pause. This

is not recommended. Second, a nice mode consists in setting the ramp rate

and the destination field on each magnet and asking the magnet to reach this

field. This is the safest mode. Third, a Labview program can be used to set the

magnet in a particular configuration. This is a bit dangerous in case Windows

has to reboot or something like we are used to.

3. The magnet has a persistent mode which is great for minimizing He consumption

but which comes with hazards. Always put a note hiding the switch heater

button with the current field in the magnet before switching the heater off.

This should avoid any mistakes leading to a quench of the superconducting

magnet, and a destruction of McCullough building.

Appendix D

Thermal oxidation of Si wafers

Oxidation of n++ 4” < 1 − 0 − 0− > Si wafers 1 is done at SNF using the Ty-

lan1 furnace. Wafers must be cleaned at the wet bench prior to using it in Tylan1

(cleaning procedure can be found at the SNF wiki under the wet bench diffusion, wbd-

iff “https://snf.stanford.edu/SNF/ equipment/wet-benches/wet-bench-diffusion/Wet

Bench Diffusion”). After cleaning, wafers should be dried well in the dryer. While

drying the wafers, you should load the recipe for Tylan1 on its state-of-the-art pc

computer. There are two floppy diskettes that can be used. They are labeled “Dry1

Directory” and “Dry2 Directory”. The recipe “DRY 1100” is available on both (the

difference between the two is when it was last updated. Dry1 was updated on 1982,

while Dry2 was updated on 2008).

The following webpage in the SNF wiki has all the commands needed to load a

receipe and run it, “https://snf.stanford.edu/SNF/ equipment/annealing-oxidation-

doping/furnaces/tylan-oxidation-and-annealling-furnaces-tylan1-4”

When loading the recipe it has either 1 or 2 time variables (see Fig. D.1). The first

variable is the oxidation time and the second one (if there is one) is the annealing

time. To determine the time needed for certain thickness, use the following web-

site “http://www.lelandstanfordjunior.com/thermaloxide.html” (made by Eric Per-

ozziello). The time depends strongly on the dopants density in the wafers. The

1Wafers bought from SILICON QUEST INT’L with the following characteristics: 4” N/AS ¡1-0-0-¿ ¡.005 OHM-CM, 525+-25micron PRIME N/ARSENIC, ETCH BACK, 2 STD. FLATS

108

109

difference between intrinsically doped Si and heavily dopes Si can be a difference of

a couple of hours in the furnace (where doped Si takes less time to oxidize). Also,

the dry process is chosen as the oxidization is more controllable with slight errors in

thickness (up to 1% ) during longer time periods (300nm of SiO2 takes about 10 min-

utes using wet oxidation, while using dry oxidization it can take 5-6 hours. However,

a small mistake in doping concentration or actual time in the furnace can cause more

than a 5% error in thickness in the wet case that is crucial for seeing graphene on a

substrate). Thus a trial wafer must be prepared first with an estimate density, and

then according to the actual measured thickness by the elipsometer, calculate back

the dopants density, and use it to oxidize a full batch of wafers.

After each batch of wafer oxidation, at least one wafer must be tested for leakage

current. Our best breakdown voltage achieved within the projects described in this

thesis was 180V on the back gate of a sample, which is equivalent to 6MV/cm and a

density of 1.5 × 1013cm−2 . In most samples where the breakdown was much lower,

the gate did not break but rather the metal contacts to graphene blew up probably

due to current leakage from the back gate wire to the other wires.

Figure D.1: Tylan1 computer sceen-shot with the time variable.

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