correlated electron transport in one-dimensional mesoscopic

CORRELATED ELECTRON TRANSPORT IN

ONE-DIMENSIONAL

MESOSCOPIC CONDUCTORS

a dissertation

submitted to the department of applied physics

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

Na Young Kim

September 2006

c© Copyright by Na Young Kim 2006

All Rights Reserved

ii

I certify that I have read this dissertation and that, in

my opinion, it is fully adequate in scope and quality as a

dissertation for the degree of Doctor of Philosophy.

Professor Yoshihisa Yamamoto(Principal Adviser)




Professor Malcolm Roy Beasley




Professor David Goldhaber-Gordon

Approved for the University Committee on Graduate

Studies.

iii

iv

Abstract

Mesoscopic systems have emerged as a result of advanced microfabrication processing.

These systems provide a new playground where tailor-made structures are available.

This enables the study of quantum phenomena due to dimensional confinement and

manipulation of system geometries. Two specific systems are investigated in this

dissertation: single-walled carbon nanotubes and quantum point contacts in a two-

dimensional electron gas.

Single-walled carbon nanotubes (SWNTs) provide a testbed to explore unique

quantum behaviors of one-dimensional (1D) systems. Unlike two- or three-dimensional

counterparts, in which the Coulomb screening justifies an independent single parti-

cle picture, the ground-state properties and system dynamics of 1D conductors are

deeply rooted in more complicated electron-electron interactions. One manifestation

of the 1D features is in electrical transport properties. Recently, the electrical con-

tacts between tubes and metal electrodes have been improved, allowing us to observe

quantum interference in ballistic SWNTs analogous to intensity fringes in Fabry-Perot

cavities. Electron transport properties of well-contacted SWNTs via measurements

of differential conductance and low-frequency shot noise are focused. Experimental

results exhibit strong correlations among conducting channels. The interpretation

of experimental observations within the Tomonaga-Luttinger liquid (TLL) theory is

discussed, which provides qualitative and quantitative agreements with experiments.

Especially, the characteristic TLL parameter inferred from the differential conduc-

tance and the current noise measurements agrees well with the theoretical values

predicted for SWNTs.

Quantum point contacts (QPCs) in a high-mobility two-dimensional electron gas

v

(2DEG) system have been a prototypical device used to investigate low-dimensional

mesoscopic physics as well as a basic ingredient to explore quantum statistics of

particles. The quantized conductance manifests ballistic transport through QPCs and

it is well understood by the wave nature of quantum particles in terms of transmission

probability. An additional remarkable feature has been identified as the 0.7 structure,

reflecting many-body effects. An attempt to explore unresolved features in a QPC is

made with a control of the electron density in a 2DEG. Non-equilibrium transport

properties of differential conductance and current fluctuations in a backgated QPC is

characterized.

vi

Acknowledgements

“If I have seen so far

it is only because I have stood on the shoulder of giants.”

− Isaac Newton

Ph.D. stands for Doctor of Philosophy, but it has meant to me as a process to

learn Patience and Persistence, Humbleness and Humility through Devotion and

Dedication. Not to mention ruminate metaphysical questions over and over like who

I am, where my ways go, and where the truth resides. I admit that I have been

extremely blessed to walk this long journey with wonderful, supportive, precious

people together. These people have helped in various aspects make this thesis possible.

This is the moment I have been dreaming to express my deep gratitude in words

to cherished individuals from the bottom of my heart. Accumulating knowledge of

physics is an important asset I have had here at Stanford, furthermore I have been

incredibly lucky to build up the memorable relationship and friendship with people I

have encountered here.

First, I am deeply indebted to my research advisor, Professor Yoshihisa Ya-

mamoto. Recently, I realized that to do research is like to plant various seeds. Some-

times, you do know what the fruits or flowers of the seeds you plant are at the very

beginning. But sometimes, you plant them since they are seeds and see what the

fruits or flowers would be after gardening. From time to time, you thought you know

what the seeds are, but they turns out to be different than what you thought. Yoshi

has been a limitless source of research ideas and has been granting me freedom how to

garden seeds of research. His big picture and unique approach of physics have always

amazed me, and his tremendous patience and diligence have been exemplifying.

vii

My sincere thanks go to my reading committee, Professor Macolm R. Beasley and

Professor David Goldhaber-Gordon. Professor Beasley has impressed me, showing

humbleness as a mature image of scientists. I enjoyed his class about condensed

matter physics seminar series , which made the seminar approachable as well as made

me to grasp his systematic way to interpret physical concepts displayed in literatures.

Most of all, he is born-nature humorous so that I do now believe that physicists can be

funny and witty. Professor David Goldhaber-Gordon has been very supportive with a

great interest on my research, providing practical advices. His acute assessment and

sharp understanding always challenge me a lot. I learned the breadth and depth of

mesoscopic condensed matter field in his class and I will not forget the thrill of the

random draw to lead the mystery presentation each time.

I am grateful to have Professor Steve Kivelson who accepted the request to be

the chair of the Ph.D. oral committee without hesitation. In addition, his lectures

on superconductivity are one of the best lectures I have had at Stanford due to his

enthusiastic delivery and thorough knowledge in his field. I am very fortunate to

have Professor Katheryn Moler in my Ph.D. oral committe, who is always supportive

and encouraging. Her outgoing and open-minded personality have been unforgettable

since the first day I met her as a perspective student in 1999. One of the regret I

have had at Stanford is that I missed the opportunity to work for her as a rotation

student since I accepted the other offer one day earlier. As her teaching assistant, I

learned that how much she cares for students.

Many thanks are to the collaborators, without them anything in this thesis is

possible. Professor Hongjie Dai initiated the shot noise properties of single-walled

carbon nanotubes with great interest and passion. He has been very patient to accept

my slow pace of research progress. Most samples were prepared by Dr. Jing Kong who

seems to have no give-up and continuously to move forward. I miss a lot the moments

we started our days with prayers together. Other samples I could test were from Dr.

Jien Cao, Dr. Woong Kim and Dr. Ali Javey who are wonderful people whom I share

friendship. High-quality GaAs quantum point contacts have been provided by Dr.

Yoshiro Hirayama at the NTT basic laboratory, who also accepted me as a visiting

student in his lab to experience the fabrication processes and to interact with experts

viii

in his group.

I have been a true beneficiary of the rotation program in the department of applied

physics, who have had no real research experience during undergraduate. I am deeply

thankful for all professors who were willing to accept me as a rotation students,

involving in active research projects to taste what is the research and how to pursue it

: Professor Martin Greven who encouraged me to come one quarter early so that I got

accustomed to a new life at Stanford in advance and from whom I grasped the powder

study of high Tc superconductors with SQUID measurements; Professor Douglas

Osheroff who shared his graduate life to cheer me up when I was in trouble and

who welcomed me with a big smile whenever I bumped into him anywhere; Professor

James Harris who strongly recommended me to work on semiconductors over medical

physics to grasp the basics of material characteristics of GaAs; Professor Martin Fejer

who provided numerous ways and suggestions when I faced difficulties to quantify

material characteristics using proton exchange bath. In addition, I would love to

thank Professor Calvin Quate who always shows sincere and faithful enthusiasm in

our study, allowing me to use his atomic force microscopy in his lab and serving as

one of the qualifying examination committee. I also thank Professor Vahe Petrosian

and Professor Hari Manoharan for being the qualifying examination committee, who

endowed me to have a chance to swallow the fundamentals of physics.

I also take special time to express my gratitude to professors in my undergraduate

institute who have put faith in me that I can finish the degree: Professor Kwun, Sook-

Il who showed innocent passion on physics and loved me as his daughter; Professor

Kim, Sun Kee who was the first professor I can approach without any hesitation;

Professor Jhe, Wonho who prayed for me to fly with ambition and passion under

God’s will; Professor Char, Kookrin who advised me to enjoy and survive graduate

life whenever he visited Stanford over nice dinner; Professor Park, Yungwoo who

invited me as his summer students to have a chance to work on projects; Professor

Park, Byungwoo who supported me very much to receive the admission to Harvard

University with a fellowship which I ended up turned down and trusted me so much

that I could teach his daughter before I left abroad.

I am in debt to all members of Yamamoto group who have been always beside

ix

me whenever I need them. Especially, Dr. William D. Oliver is my first mentor who

has taught me everything with great patience, including all experimental engineering

techniques, theoretical models, and English. In addition he has advised me and

has supported me with his maturity. I cannot really express my deep gratitude in

words when he flew to Stanford for me on the Ph.D. oral defense in the mist of

his busy schedule. Dr. Recher is also my mentor from whom I learn the rigorous

theoretical approaches. Without him, the single-walled carbon nanotube project may

not be further explored and the experimental outcome may be in dark; however, he

persistently tries and manages to acquire quantitative and qualitative explanations.

His patience and persistence are what I would like to have in my personality. I am

grateful to Dr. Jungsang Kim who hosted me without hesitation when I requested the

lab tour of Stanford in 1996 even though I had never met him but since he is my senior

in the same undergraduate. He showed me the very room, Ginzton 25A to explain

his single photon devices very hard to me who had not taken Quantum Mechanics

class back then. I felt quite strange but comfortable when Ginzton 25A came to be

my destiny to commit my real Ph.D. life in 2000. I like to thank Dr. Xavier Maitre

and Dr. Robert Liu who were senior members of mesoscopic transport subgroup, who

influenced my research in various aspects. Professor Matsumoto and Professor Barry

Sanders who visited Y-group had listened to my circuit questions and had discussions

on the role of female physicists with introducing great books. Early generations of

Y-group nurtured me: Dr. Fumiko Yamaguchi, Dr. Matthew Pelton, Dr. Aykutlu

Dana, Dr. Charles Santori, Dr. Edo Waks, Jocylin Plant. Dr. Cyrus Master, Dr.

Thaddues Ladd, Dr. Hui Deng, Dr. Eleni Diamanti, Dr. Jonathan Goldman, Dr.

David Fattal have been friends as well as colleagues who make me feel lucky. Shinichi

has been a great office mate, teaching me the basic knowledge of optical setup and

helping me to overcome hurdles. Current members of y-group are just great as old

folks as my second family in the states: Dr. Bingyang Zhang, Dr. Kaoru Sanaka,

Katsuya Nozawa, Kai-mei Fu, Susan Clark, David Press, Y. C. Neil Na, Young Chul

Yun, Georgious Roumpos, and Kristiaan. In addition, Ms. Yurika Peterman, Ms.

Rieko Sasaki and Ms. Mayumi Hakkaku are the world-best administrators who really

take care of us as a family.

x

Spending two quarters in Fejer group, I have developed wonderful friendship with

Byer/Fejer group folks. Krishnan and Jonathan Kurz were great seniors I worked

with. Social outings with Loren, Frederick, Supriyo, Carlsten, David Hum and Yin-

wen have been joyful. I got to know Yin-Wen as a laser lab partner for two quarters

and thank her a lot for her innocent heart and affections. Moreover, we have had

great girly Christmas eve gathering and she always listens to my distress and concern,

standing in my side. Supriyo, Carlsten and David Hum are great people who make

me laugh and smile and who are always there for me when I need second hands in

the lab. My classmates, Tim, Jason, Adam, Brad, Yu-ju, Sylvia are in my heart and

memories. I am very lucky to have many great student advisors: Patrick Mang, Doug

King, Anu Tewary, Jonathan Goldman and Todd Sulchek, who really wanted me to

get well. Also, Seokchan, Jongkwan, Sanghyun, Yun and Harold have been extremely

helping me a lot to adjust here at the beginning in Korean styles. It was a real fun

to play basketball at 3 in the morning.

I am very grateful to many individuals for their wisdom and kindness with amaz-

ing technical support: Larry Randall for computer and machine shop techniques, Tom

Cover for clean room instruction, and Darla Le-Grand-Sawyer for smooth adminis-

tration.

Hana and Jungsuk have been my sister and brother, sources of encouragement,

who have cared me and my brother with love and prayers. NCBC church families

have been my another family in the states: Patty Kim, Linda Choi, Anthony Song,

Jane Lee, and Soohyun Cho.

I am grateful to my only brother, Sang Hoon who has inspired me with creative

new ideas and sharp questions, who has been my another hands in the lab whenever

I need while performing experiments, and who has challenged me to be professional.

I really do not know how much I depend on my lovely parents who have been patient,

crying with me when I am sad and laughing with me when I am happy and most

of all praying for me all the time at nights and at days. Their love and trust in me

push me to be at this moment and continue to push me to be further. My late grand

mother is still my source of strength and happiness, who prayed for me at 5 o’clock

every morning. Her prayers and love stand me up today. I DO love my family with

xi

my life.

Lastly and mostly, I cannot find the right words to thank our sincere Lord who

is always beside me, wakes me up every time I am in dark, heals me when I mess up

and stands me up when I am down. His soft and tender voice strengthen and guide

me. He has never been disappointed by me, loving me as who I am and teaching me

to be better. I would love to walk through the rest of my life with Him, to be the one

who He wants to be.

xii

Contents

Abstract v

Acknowledgements vii

1 Introduction 1

1.1 Mesoscopic Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Quantum Point Contact . . . . . . . . . . . . . . . . . . . . . 10

1.2 Electrical Transport Properties . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Classical Transport . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Many-Body Physics 18

2.1 Charge Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 The Tomonaga-Luttinger Liquid Theory . . . . . . . . . . . . . . . . 28

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Mesoscopic Electron Transport 34

3.1 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Ballistic Transport . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Fundamentals of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Fluctuation and Dissipation Theorem . . . . . . . . . . . . . . 40

xiii

3.2.2 General Formulation of Noise . . . . . . . . . . . . . . . . . . 42

3.2.3 Classification of Intrinsic Noise . . . . . . . . . . . . . . . . . 43

3.2.4 Crossover of Noise Sources in Frequency Domain . . . . . . . . 51

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Experiment Methodology 54

4.1 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Low-frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Single-Walled Carbon Nanotubes 71

5.1 Single-Walled Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . 72

5.1.1 Electronic Band Structure . . . . . . . . . . . . . . . . . . . . 73

5.1.2 Synthesis and Fabrication . . . . . . . . . . . . . . . . . . . . 86

5.2 Differential Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 Quantum Interference . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2 Spin-Charge Separation . . . . . . . . . . . . . . . . . . . . . 105

5.3 Low-Frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.2 Shot Noise and Fano factor versus the Drain-Source Voltage . 115

5.3.3 Fano Factor versus Transmission Probability . . . . . . . . . . 121

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Quantum Point Contact 126

6.1 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . . . . . 126

6.1.1 Energy Band Profile . . . . . . . . . . . . . . . . . . . . . . . 127

6.1.2 Scattering Mechanism . . . . . . . . . . . . . . . . . . . . . . 131

6.1.3 Backgated 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Quantum Point Contact . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.1 Conductance Quantization . . . . . . . . . . . . . . . . . . . . 136

6.2.2 The 0.7 Structure . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3 Differential Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 152

xiv

6.3.1 Non-integer Conductance Plateaus at Finite Bias Voltage . . . 155

6.3.2 Density Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4 Low-frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4.1 Noise Suppression at Non-integer Conductance Plateaus . . . 160

6.4.2 Density Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7 Conclusions 165

A Physical Constants 167

B Conversion Tables 168

B.1 Energy and temperature . . . . . . . . . . . . . . . . . . . . . . . . . 168

B.2 Frequency, temperature, energy, wavelength and time . . . . . . . . . 169

C Statistics of Particles 170

C.1 The Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . . . 170

C.2 The Fermi-Dirac Distribution . . . . . . . . . . . . . . . . . . . . . . 171

C.3 The Bose-Einstein Distribution . . . . . . . . . . . . . . . . . . . . . 174

C.4 Basic Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 174

D The Dirac Delta Function 176

D.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

D.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

E Useful Mathematical Formulas 178

E.1 Even and Odd functions . . . . . . . . . . . . . . . . . . . . . . . . . 178

E.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

E.3 Fourier-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

E.4 Pauli Spin Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

E.5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 180

E.6 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

E.7 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xv

F Recipe of Making Printed Circuit Boards 184

Bibliography 187

xvi

List of Tables

1.1 Classified dimension of mesoscopic conductors systems. . . . . . . . . 5

1.2 Classified quantum electron transport . . . . . . . . . . . . . . . . . 15

5.1 The g values from the power-law scaling analysis of four samples. . . 121

5.2 The relation between extreme values of F and the overall transmission

probability T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

xvii

List of Figures

1.1 Zone of macroscopic, mesoscopic and microscopic systems. . . . . . . 3

1.2 (a) Semiclassical cartoon of 6C atomic shell structure. (b) The diagram

of s - and p - orbital hybridization: tetragonal sp3 (left) and planer

sp2 (right). (c) Structures of carbon based material: zero-dimensional

C60 (left), three-dimensional diamond (middle) and three-dimensional

graphite (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 History of carbon nanotube research from the discovery to the present

research area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 (a) The current and voltage measurements of a 23.5-kΩ (manufac-

turer’s rating) surface mount chip resistor with HP 4145B semiconduc-

tor parameter analyzer. A linear regression analysis gives conductance

G ∼ 41.316 µS or R ∼ 24.204 kΩ. (b) Measured current in time at 5

mV bias to the surface mount chip resistor. . . . . . . . . . . . . . . 13

2.1 (a) The Fermi sphere of non-interacting electron systems. (b) The

momentum distribution function of the Fermi gas systems. (c) The

spectrum of particle-hole excitations in the Fermi gas systems. . . . 25

2.2 The momentum distribution of the Fermi liquid system. . . . . . . . 27

2.3 The TLL parameter g as a function of Rs/R. Two red lines are marked

at g = 0.2 and g = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 (a) An one-dimensional ballistic conductor in a two-terminal configu-

ration. (b) The energy dispersion of free electrons in a reservoir (left)

and a conductor (right). . . . . . . . . . . . . . . . . . . . . . . . . . 36

xviii

3.2 A two-port system represented by second quantized operators. . . . 38

3.3 Spectrum of dominant noise sources in frequency domain. . . . . . . 43

3.4 A parallel resistor-inductor circuit. . . . . . . . . . . . . . . . . . . . 44

4.1 (a) Two-terminal and (b) four-terminal measurement schematics with

a constant current bias. . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 (a) Johnson-Nyquist noise vs. temperature for different resistance.

(b) Johnson-Nyquist noise of resistors at various temperatures: room

temperature (293 K), liquid nitrogen temperature (77 K), liquid helium

temperature (4 K) and 1 K. . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 (a) The 4 K home-made dipper and (b) Oxford Helium 3 sorption

cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 (a) Estimated dc (dot) and ac (square) voltages at a 1 nA current bias

to a variable resistor together with Johnson-Nyquist noise measured by

a 100 kHz equipment at room temperature and 4 K. Assume ac signal

is hundred times smaller than the dc value. (b) Johnson-Nyquist noise

with a 1 Hz bandwidth at 4 K. . . . . . . . . . . . . . . . . . . . . . 62

4.5 (a) Shot Noise and Thermal Noise crossover. (b) The threshhold R

and I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.6 (a) The circuit diagram of AC modulation scheme. (b) The square-

wave voltage in time. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7 (a) SONY and (b) Fujitsu FSU01LG MESFET bias response at room

temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8 The photograph of a cryogenic amplifier and resonant tank circuit used

in the single-walled carbon nanotube show noise experiments. . . . . 70

5.1 (a) Graphite lattice structure. (b) The direct and (c) the reciprocal lat-

tice space of graphene with unit vectors ~ai, ~bi and translational vectors

~ri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Energy band structure of Graphene. . . . . . . . . . . . . . . . . . . 78

5.3 SWNT geometry on the graphene lattice structure. Chiral vector ~Ch

is drawn for a specific case, a (4,2)-SWNT. . . . . . . . . . . . . . . 79

xix

5.4 Armchair SWNT real (a) and reciprocal (b) lattice space. (c) (10,10)

energy band structure. The first BZ of SWNT is indicated by two thick

vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.5 (a) The µ = 0 band for a (10,10) SWNT along symmetry points Γ, K,

and K ′. The π, π∗ wavefunctions are clearly denoted based on the

energy band coefficients. The real and imaginary coefficients of upper

energy band (b) and lower energy band (c) for µ = 0. The dotted line

is the zone boundary of graphene. . . . . . . . . . . . . . . . . . . . 83

5.6 A zigzag SWNT unit cell in a real (a) and a reciprocal (b) lattice space.

(c) Semiconducting zigzag (10,0) energy band structure. The first BZ

of SWNT is indicated by two thick vertical lines. (d) µ = 6 band for

(10,0). The real and imaginary coefficients of upper energy band (e)

and lower energy band (f) for µ = 6. . . . . . . . . . . . . . . . . . . 84

5.7 (a) The µ = 6 band for a (10,0) zigzag SWNT. The real and imaginary

coefficients of upper energy band (b) and lower energy band (c) for the

µ = 6 band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.8 (a) Schematics of CVD chamber. (b) The mechanism of SWNT growth

from catalysts in CVD chamber adapted from K. J. Cho group. . . . 88

5.9 The schematics of SWNT-device fabrication processes. . . . . . . . . 90

5.10 (a) Optical microscopy picture of portion of a chip containing wire-

bonded devices, (b) a zoom-in view of an individual device, and (c)

atomic force microscopy image of an individual SWNT with patterned

Ti/Au electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.11 (a) The schematics of three-terminal SWNT device. (b) Experimental

two-dimensional image plot of differential conductance versus a drain-

source voltage in y-axis and a backgate voltage in x-axis. . . . . . . . 95

5.12 The differential conductance from a 360 nm-long SWNT device at Vg

= - 5 V. Experimental data are blue circles and the theoretical fitting

of the single-channel double-barrier structure model is in red. . . . . . 97

xx

5.13 (a) Diagram of two-channel double barrier system. (b) Two-dimensional

image plot of differential conductance versus drain-source voltage in y-

axis and backgate voltage in x-axis. . . . . . . . . . . . . . . . . . . . 98

5.14 Illustration of the TLL model on a SWNT device. . . . . . . . . . . 101

5.15 (a) The Vds-dependent f1(Vds, g, T ) at T = 4 K for g = 0.25 (red),

g = 0.75 (green) and g = 1 (blue). (b) The Vds-dependent f2(Vds, g, T )

at T = 4 K for g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). . . 104

5.16 Differential conductance versus Vds at a certain Vg. (a) Experiment

and (b) Theory based on TLL for g = 0.25 (red) and g = 1 (blue). . . 106

5.17 Differential conductance versus Vds at different backgate voltages for

experiment results (a), the theoretical plots from non-interacting Fermi-

liquid theory (b) and from the Tomonaga-Luttinger liquid theory (c). 107

5.18 Schematics of SWNT shot noise measurement setup. . . . . . . . . . 110

5.19 The 1/f noise crossover of SWNT devices for two temperatures 293 K

and 77 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.20 The coupling efficiency α between ILED and IPD at (a) T = 293 K and

(b) T = 4 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.21 Full Shot Noise from the LED/PD pair (a) T = 293 K and (b) T = 4 K 114

5.22 A representative log-log plot of low-frequency shot noise from the

LED/PD pair (upside-down triangle) and the SWNT (diamond) as

a function of Vds. The straight line is the outcome of linear regression

analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.23 (a) The integration in Vds of the Vds-dependent f1(Vds, g, T ) at T = 4

K for g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). (b) The

integration in Vds of the Vds-dependent f2(Vds, g, T ) at T = 4 K for

g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). . . . . . . . . . . . 118

5.24 The experiment data (blue square) of the SWNT noise with the theo-

retical fitting plot(red). . . . . . . . . . . . . . . . . . . . . . . . . . 119

xxi

5.25 A representative log-log plot of Fano factor (open diamond) obtained

from experiments against Vds together with theoretical theoretical fit-

ting of Tomonaga-Luttinger liquid theory for g = 1 (straight line) and

g = 0.25 (dotted line). The broken line on the experimental data

represents the power-law scaling analysis. . . . . . . . . . . . . . . . 120

5.26 Fano factor versus transmission probability taken at Vds = 40 mV

from five SWNT-devices (filled symbols) at varying Vg values. (a)

Ballistic phase-coherent transport theory for one-(dark blue straight)

and two-channel (dotted area) models. (b) Phase-incoherent picture

theory for distributed elastic (square) and inelastic (diamond), inco-

herent double-barrier (light green straight) and many-barrier model

(dark green straight) . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1 (a) Band profile of a GaAs/AlGaAs heterostructure. (b) Band profile

of a 2DEG embedded in a doped GaAs/AlGaAs heterostructure. . . 128

6.2 (a) Growth structure of backgated 2DEG. d1 and d2 are the thickness

of GaAs and the two layers of AlGaAs and superlattice barriers respec-

tively. (b) Band diagram of backgated 2DEG operation. A thick solid

line represents the case of Vb,th, a dotted line is for the above Vb,th and

a thin solid line is for the below Vbg,th. . . . . . . . . . . . . . . . . . 134

6.3 (a) Measured Hall voltage as a function of external magnetic field per-

pendicular to 2DEG by changing the backgate voltages VBG. (b) The

calculated electron density of 2DEG versus VBG. . . . . . . . . . . . 135

6.4 (a) Schottky-split techniques to form a QPC in a 2DEG sysgem. (b)

Sketch of energy dispersion of electron reservoirs and QPC . . . . . . 137

6.5 (a) The saddle-point potential for |ωy/ωx| = 2. (b) computed differen-

tial conductance for the saddle-point potential (a). . . . . . . . . . . 140

6.6 (a) The actual voltage drop effect in left and right moving channels.

(b) Computed differential conductance for non-zero Vds cases. . . . . 141

6.7 The 0.7 structure from differential conductance measurements at 1.5 K. 143

xxii

6.8 (a) The band structure without spin-orbit interaction under zero mag-

netic field ~B = 0 for the first five n. (b) The band structure with

non-zero spin-orbit interaction at ~B = 0. (c) The band structure with

non-zero spin-orbit interaction at finite magnetic field. In all three

cases magnetic field is perfectly aligned in x-direction, i.e. θ = 0. P

and ξ are defined in the context. . . . . . . . . . . . . . . . . . . . . 148

6.9 Computed differential conductance as spin-orbit coupling in a simple

harmonic oscillator potential is on while the magnetic field is kept zero

in (a) and the magnetic field is 0.05 in the unit of g∗µBB/2~ω in (b). 150

6.10 Computed differential conductance as spin-orbit coupling is on while

magnetic field is kept zero. . . . . . . . . . . . . . . . . . . . . . . . 151

6.11 (a) Scanning electron microscope (SEM) image of a quantum point

contact in a AlGaAs/GaAs 2DEG. (b) The Hall-bar structure of the

wirebonded device taken by SEM. . . . . . . . . . . . . . . . . . . . 153

6.12 (a) Experimental differential conductance dG by a sweep of Vg at fixed

Vbg = 2.3 V at finite Vds (b) Transconductance dG/dVg (c)Vds depen-

dence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.13 (a) Observed differential conductance traces at each Schottky gate volt-

age as a function of Vds. All date were taken at Vbg = 2.3V and 1.5 K.

(b) Computed differential conductance with the Vds-dependent saddle-

point potential up to the linear term, i.e. γ = 0. Including second-order

corrections in Vds with two opposite signs of the coefficient, (c) γ > 0

and (d) γ < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.14 The tuning variables are Vg and Vbg and dG is measured at four-probe

techniques with ac signal Vac ∼ 50 µV and the dc bias (a) Vds = 0 mV

and (b) Vds = 2 mV. The Vbg varies from 3.0 V (rightmost) to 2.58 V

(leftmost) by 0.01V interval. . . . . . . . . . . . . . . . . . . . . . . 157

6.15 Bias dependence at Vbg = 2.3 V. (a) Vds = 0.7 mV. (b) Vds = 2 mV.

(c) Vds = 2.5 mV. (d) Vds dependent conductance. . . . . . . . . . . 159

xxiii

6.16 (a) Two-dimensional plot of shot noise raw data at Vbg = 2.3 V with

lines at the conductance values. (b) The contour plot of conductance

values in the unit of GQ simultaneously taken with shot noise measure-

ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.17 Two-dimensional image plot of shot noise versus Vg and Vbg at (a)

Vds = 1 mV. (b) Vds = 2 mV. The straight lines with number are

corresponding to the conductance values normalized by GQ. . . . . . 163

xxiv

Chapter 1

Introduction

Chance favors the prepared mind.

− Louis Pasteur

“Meso-” means middle in Greek, µǫσoς. The term, “mesoscopic system”, was first

coined in 1976 by N. G. van Kampen in the context of statistical physics [1], and

similar usages have come to appear more often in diverse fields since early 1990s

with the advent of microfabrication processing techniques, referring to one whose

dimension lies in between microscopic and macroscopic counterparts. This middle one

has attracted more attention, providing ample room to investigate physical inquiries.

Physicists have been seeking the origin of nature on earth even in a far-off galaxy

with human beings’ imagination and empirical justification. The scope of physics is

very broad, ranging from bulk systems (macroscopic world) to exotic materials re-

sulting from diverse electron configuration (microscopic world). In modern days, the

subject of physics becomes rather subdivided into diverse areas according to primary

interests: for example, astrophysics for universe, particle physics for ultimate con-

stituent objects in atoms, biophysics for biological entities, condensed matter physics

for solid systems and more. Condensed matter physics is particularly concerned

about states and phases of not only naturally existent but also engineered material,

searching for the microscopic level understanding.

1

2 CHAPTER 1. INTRODUCTION

The naturally accessible systems are found in both macroscopic and microscopic

level, but the governing principles of phenomena occurring in two realms are quite

distinct. The statics and kinetics in macroscopic world has been completely assessed

by the deterministic picture in classical Newtonian physics. For example, given an

initial position of a particle with a velocity at a certain time, any succeeding states

at later times can be readily computed along a distinguishable particle trajectory.

This picture, however, turned out to be an inadequate eye to examine properties and

behaviors of individual and aggregates of constituents at the atomic and subatomic

levels. To perceive correct understanding of phenomena in such environment is beyond

the realm of classical physics, requiring alternative paradigm to replace deterministic

reduction of systems.

A tremendous insight has been gained in early twentieth century by brilliant sci-

entists who successfully and beautifully established theoretical framework quantum

physics, game of chance or probability. Quantum physics describes statics and kinet-

ics in microscopic world by introducing the description of wavefunction as a solution

to Schrodinger equation to incorporate the intrinsic particle-like and wave-like nature

of quantum entities. Furthermore, quantum world exhibits substantial character-

istics: duality of wave and particle, indistinguishability, quantization, Heisenberg

uncertainty, finite zero-point energy, and quantum entanglement. The duality fea-

tures coherence in the dynamics of quantum particles in quantum world, linking to

correlation effects in systems. Quantum particles in the coherent sate have a well-

defined energy and a well-defined phase configuration. In the beginning of founding

quantum theory, only Gedanken-experiments with coherent particles had been pos-

sible for validate postulates and predictions of quantum theory due to hindrance to

approach microscopic world. As closing the gap, the new playground to perform

previous gedanken experiments in the lab, mesoscopic world has recently emerged.

What enables us to enlarge our focus beyond preformed area is engineering improve-

ment of fabrication technology and material synthesis. Sub-millimeter structures are

repeatedly patterned by a photolithography, and even smaller features down to tens

of nanometers are constructed by electron-beam lithography with a skillful care in

a reproducible rate. The combination of two lithography methods yields simple and

3

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Figure 1.1: Zone of macroscopic, mesoscopic and microscopic systems.


complex device structures. Of numerous methods in the latter synthesis area to

produce new material, molecular beam expitaxy and chemical vapor deposition are

widely used at present. One outcome of such efforts are mesoscopic conductors.

1.1 Mesoscopic Conductors

Mesoscopic conductors have exhibited subtle and sophisticated phenomena which

are deeply rooted in quantum mechanics and are tightly associated with many-body

interactions. An easy means to classify macro-, micro- and meso-scopic conductors

is physical length. Figure 1.1 presents three regions of systems along the length scale

bar with familiar objects corresponding the size of physical length. Macroscopic ones

are often what our bare eyes can see and what our hands can fold, typically bigger

than millimeters, whereas microscopic systems are too tiny to visualize how behaviors

of atoms are without supplementary equipments, roughly around 0.5 - 10 A close to

the atomic sizes, where A is 1 over 10 billion of millimeter. Mesoscopic world closes

the gap between two regions, spanning from millimeters to nanometers (a billionth of

a millimeter).

The manipulation of system size enables us to obtain all dimensional mesoscopic

conductors. The classification of system dimension is achieved by the comparison

between the physical system length and Fermi wavelength. Fermi wavelength λF is

defined as a wavelength of carriers at the Fermi level, λF = h/pF = 2π/kF where h is

the Planck’s constant and pF, kF are the momentum and wavenumber at the Fermi

surface. Since conduction in mesoscopic systems is dominated by electrons at the

Fermi level, λF means the wavelength of major carriers. λF in metals is typically

atomic dimension, around A, whereas λF in semiconductors is about 10 - 100 nm.

1.1. MESOSCOPIC CONDUCTORS 5

Suppose Lx, Ly and Lz represent physical length of system, satisfying Lx < Ly <

Lz without any loss of generality. Table 1.1 exhibits the condition of each dimension.

Dimension ConditionOne Lx, Ly < λF < Lz

Two λF ∼ Lx ≪ Ly, Lz

Three λF ≪ Lx, Ly, Lz

Table 1.1: Classified dimension of mesoscopic conductors systems.

Two one-dimensional (1D) mesoscopic conductors are studied in this dissertation:

carbon nanotubes and quantum point contact in GaAs/AlGaAs semiconductor het-

erostructures.

1.1.1 Carbon Nanotubes

Carbon (6C), the sixth element in the periodic table, is one of the simple and familiar

atoms. It ubiquitously appears either homogeneous or heterogeneous compounds in

diverse forms from in the atmosphere, on earth, in the sea to inside living bodies.

The abundance of carbon-based chemicals results from three hybridization arrange-

ments of four valence electrons in 2s and 2p orbitals: sp, sp2, and sp3. Even with

carbon element only, different materials are found and any dimensional structures

exist: graphite and diamond in three dimension, graphene in two dimension and the

bulkyball fullernene C60 in zero shown in Fig. 1.2. Carbon bondings are mysterious

that they can be as hard as diamond to scribe other crystals or they can be as weak

as graphite to scribble on paper in a certain circumstance.

In 1991, carbon strands in a furnace for the fullerene production were fortuitously

observed [2]. Since strands which looked like concentric cylinders were unprecedent,

they were denoted as ‘microtubules’ [2], ‘fullerene tubules’ [3] or ‘graphene tubules’ [4]

at first in literatures. These titles reflected different perspectives as to how they were

formed, cylindrical tubes are either elongated fullerene mutations or roll-up wires

with several layers of graphene. Both views envisioned them as 1D objects regardless


(c)

(b)

(a)

(c)

(b)

(a)

Figure 1.2: (a) Semiclassical cartoon of 6C atomic shell structure. (b) The diagramof s - and p - orbital hybridization: tetragonal sp3 (left) and planer sp2 (right). (c)Structures of carbon based material: zero-dimensional C60 (left), three-dimensionaldiamond (middle) and three-dimensional graphite (right).


of whether they start from zero or two-dimensional mother material. Geometric ratio

of diameters and lengths supports the 1D point of view since diameters are around

tens of nm and lengths are around microns even mm. Possibility to access 1D systems

from tiny carbon tubules with ease was of great interest both to scientists who have

longed for a medium to explore unique 1D properties and to engineers who have

searched alternative material to attain further miniaturization of electronic circuits.

Condensed matter theorists immediately approached and investigated them af-

ter the discover of carbon nanotubes, taking the second view of tubules from two-

dimensional graphene. They simplified and attacked a model problem, one single

tube from one graphene layer in 1992 [3, 4]. Attempts to compute electronic band

structures were readily pursued by applying additional boundary condition resulting

from a spatial confinement to the well-established graphite band structures [5]. The

band structure calculation predicted a surprising insight that the tubules would be

either metallic or semiconducting according to the size of tubes and a roll-up di-

rection [3, 4]. These insightful theoretical prediction launched active research field

with carbon nanotubes (CNTs) since early 1990s. Many terminologies were coined

including single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs)

in order to identify one or more than one tubes, armchair and zigzag tubes inspired by

shapes of carbon hexagon on surfaces. Another landmark in the history of CNT re-

search area emerged in the year 1993, the discovery of theoretically imagined SWNTs.

Real material to test theoretical investigation became alive. This event blossomed the

field furthermore to embrace more and more communities from physicists, material

scientists, chemists to electrical, mechanical, chemical engineers.

The first phase of carbon nanotube experiments was rather limited into material

level investigation such synthesis [2,6–8], chemical treatment [9–11], and surface imag-

ing by scanning electron microscopy [2,8], atomic force microscopy and scanning tun-

nelling microscopy [12,13]. The upturn to obtain transport properties were achieved

once experimentalists found a way to couple grown-nanotubes to metal electrods in

late 1990s [?,14]. In the electron transport context, two tasks were targeted for better

understandings: one was to synthesize nanotubes at designated place and the other

was to improve the coupling between the tube and the electrodes. The first task


Discovery of MWNT

Material characterization

First electrical transport: Coulomb blockade

2002

1991

1993

1992

1998

1997

1999

2000

2001

Discovery of SWNT

Chemical Vapor Deposition SWNT growth

Optical luminescence and florescence

Room temperature single-electron transistor

Orbital Kondo effect

Integrated SWNT device on Si-wafers

Ballistic SWNT devices

Tomonaga-Luttinger behavior in tunnelling regime

Kondo effect

Bandstructure calculation of CNT

2005

2004Ahranove-Bohm effect

Figure 1.3: History of carbon nanotube research from the discovery to the presentresearch area.


was overcome by the chemical vapor deposition technique to produce high-yield and

high quality of SWNTs nearby catalyst islands [15], and the second task was continu-

ously attempted with different metal electrodes and annealing, making nearly Ohmic

contact recently [16–18]. Furthermore, the investigation of unique electronic proper-

ties was expedited as integrated nanotube circuits was accomplished [19]. Numerous

quantum transport phenomena induced from phase-coherent electrons and strong in-

teractions among electrons have been continuously revealed in SWNTs coupled to

electron reservoirs summarized in Fig. 1.3: Coulomb blockade [20–22], Tomonaga-

Luttinger liquid behavior [23,24], quantum ballistic interference [16,17], Kondo [25,26]

and orbital Kondo [27], Aharonov-Bohm interference [28] and magnetic orbital mo-

ment determination [29]. Ballistic transport here means electron conduction does not

experience any types of scattering and the detailed discussion on transport regime is

given in the following subsection.

Along attaining fundamental physical knowledge, advancement of nanotube tran-

sistors has been achieved by making field effect transistors with semiconducting

tubes [18, 20], and interplay of electrons and phonons in transistor-devices was ex-

amined [30–32]. In addition, CNTs showed superior sensitivity to chemical sensor

applications [33]. The extensive study of electrical transport properties leads to ex-

ploring novel quantum Hall effect in graphene very recently [34,35]. Recent interests

span to optical properties based on electronic band structures of semiconducting

carbon nanotubes from exciting florescence spectroscopy [36] and light emission in

field-effect transistor structures [37]. Further photoluminescence [38, 39] and elec-

troluminescence [40–43] results are in question, studying strong exciton effects in

one-dimensional system and illuminating possibilities of optoelectronic components

in future. Growing interests in CNTs since the beginning of discovery is based on

aforementioned novel properties and fabrication advantages of low cost and defect-free

crystalline structures. A big challenge in this mature field is to synthesize particular

types of nanotubes in a controllable manner.


1.1.2 Quantum Point Contact

There are not so many semiconductor elements in nature, and they are neither good

conductors and nor good insulators. But what makes semiconductor materials preva-

lent in everyday life is that they can be tuned and manipulated with one parameter,

their inherent energy gap between conduction and valence bands. They are insulat-

ing but become conducting by low energy excitations. Its adjustability has greatly

revolutionized contemporary life patterns in terms of computational power, telecom-

munication, and electronic gadgets under the motto, faster, smaller, and quieter. Out

of amazing technology breakthroughs, a band engineering drives new material using

various combinations of different semiconductor elements and compounds. Semicon-

ductor Heterostructures composed of more than one element is one of such outcome

by the band engineering.

Semiconductor heterostructures are in general made by epitaxy methods, whose

advantages are high-purity crystalline layer production, thickness control, and easy

material switching. Epitaxy methods are further differentiated according to phase

of sources into liquid phase epitaxy, vapor phase epitaxy and molecular beam epi-

taxy [44]. Fabrication methods have endowed the semiconductor field to control fun-

damental parameters like bandgap energy, effective mass of carriers and mobilities.

There is no fundamental hindrance to mix any combination of semiconductor ele-

ments to form heterostructures; however, there is practical reason not to produce any

random combination of materials. Of importance is to select materials whose lattice

constants are close to each other since strain and roughness are induced at interfaces

of two different materials, encountering strain and roughness scattering. The quality

of interfaces directly affects mobilities of carriers, thus it would be a crucial factor to

shape heterostructures. At present, Group III-V semiconductor heterostructures are

a central system of physics and engineering research.

Semiconductor heterostructures are diversified as two-dimensional quantum well,

two dimensional electron gas systems, one dimensional quantum wires and zero di-

mensional quantum dots. Quantum point contact (QPC) is one of the simplest device

structures on top of two-dimensional electron gas systems. Early interests on QPC

1.2. ELECTRICAL TRANSPORT PROPERTIES 11

were closely linked to the empirical test of quantum conductance behavior in a bal-

listic regime based on theories in 1960s [45]. Unlike metallic point contact formed

in a crude way by adjoining two sharply-edged metals [46], QPC in semiconductor

are fabricated in complicated but reproducible steps of microfabrication processing.

A split-gate technique are a common method to make QPCs based on the idea that

adjusting negative voltage to electrodes on top of the two-dimensional gas systems

induces controllabe electrostatic potential profile underneath. Indeed, nice quantized

plateaus were first observed with a good agreement with theory as a representation

of the ballistic transport in GaAs/AlGaAs heterostructures [47, 48]. A single QPC,

however, still has unresolved feature in conductance traces, so-called “0.7 structure”,

which is on an active research target at present.

QPCs are known as quasi-one dimensional waveguides of degenerate electrons at

low temperatures and sources of monochromatic electrons. Thus, they are elemen-

tary components to realize electro-optics [45] including focusing capability of coherent

electrons under magnetic field. In future, a single and series of QPCs will be expected

to form a solid state qubit for quantum information processing and quantum compu-

tations [49].

1.2 Electrical Transport Properties

Characterizing materials and matter in nature is driven by a yearn for underlying

physical principles which govern nature’s fundamental entities. The principles may be

revealed in ground state properties, excitation spectra and order parameters in phase

transitions. There exist various probes to extract such information which is expected

to be converged to simple governing principles in nature via scattering processes

caused by photons, x-rays and neutrons or transport processes.

Transport properties are obtained from observing a response to an external stim-

ulation according to material characteristics. Thermal transport probes heat con-

duction and electrical transport does charge conduction across materials. In most

cases, the relation between the stimulus and the response is assumed to be linear and

the ratio of the two quantities does carry system information. Similar to conductor


divisions in the previous section, various transport regimes are identified. In order

to appreciate quantum transport as a theme of this dissertation, the discussion of

transport begins with classical transport.

1.2.1 Classical Transport

According to the trend of electrical conductivity with respect to temperature, bulk

elements and compounds are classified into metals or insulators. Metallic material

displays the linear response between current and voltage, so-called Ohm’s law. The

famous Ohm’s law is a good measure of classical electron transport, in which the coeffi-

cient of linear relation between measured current(voltage) and biased voltage(current)

is called resistance. Resistance depends on intrinsic material conductivity(resistivity)

and the geometry of conductors. The origin of resistance in bulk systems is explained

by scattering processes. Electrons or charge carriers are greatly scattered by defects,

impurities or other carriers in materials. The overall scattering processes resulted

from diverse sources become obstacles, reducing conductivity.

Resistance is a macroscopic statistical observable since many electrons are involved

in electrical conduction process. More accurately in this statistical point of view,

resistance is the average value or mean of the ratio of measured current and biased

voltage. There are fluctuations about mean values in the time domain, often called

as noise in electronic circuits. Figure 1.4 presents an example of electrical transport

measurements with a macroscopic resistor. As expected, the linear relation of current

and voltage is shown in Fig. 1.4 (a). The time-trace of current through the same

resistor at a fixed bias voltage V = 5 mV is displayed in Fig. 1.4 (b), exhibiting time

fluctuations. This fluctuation or noise corresponds to variance about the mean in

statistics. The significance of noise lies in two-fold: (1) noise limits the accuracy of

measurement outcomes and (2) intrinsic noise regarding dynamics of charge carriers

delivers transport information.


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20151050

Time (s)

Iavg = 203.995 nA

σ (std) = 9.041*10-11

(b)

Figure 1.4: (a) The current and voltage measurements of a 23.5-kΩ (manufacturer’srating) surface mount chip resistor with HP 4145B semiconductor parameter analyzer.A linear regression analysis gives conductance G ∼ 41.316 µS or R ∼ 24.204 kΩ. (b)Measured current in time at 5 mV bias to the surface mount chip resistor.


1.2.2 Quantum Transport

Discerned from classical transport, features of quantum transport are originated from

aforementioned unique traits such as coherence and quantization well described by

quantum physics. Quantum transport is further resolved into several regimes mainly

by lengthscale comparison. These transport divisions are crucial to understand meso-

scopic conductor transport properties. Besides Fermi wavelength, other relevant char-

acteristic lengths are defined for classified transport regimes: (1) mean free path, lmfp,

(2) thermal diffusion length, lT, and (3) phase coherence length, lφ.

Mean Free Path lmfp

Mean free path, as the name indicates, is an average distance in which particles can

move freely. The hindrance to free motion is due to scattering by defects, impurities

or grain boundaries. Elastic scattering does not conserve momentum but energy,

while inelastic scattering changes both momentum and energy of incident particles.

Thus, mean free paths due to elastic and inelastic scattering should be differentiated

although generally lmfp refers to the elastic mean free path. In semiconductors lmfp

is closely related to the mobility of carriers, and in metals lmfp is much longer than

λF. As lmfp becomes comparable to λF, systems with such lmfp are called in the dirty

limit.

Thermal Diffusion Length lT

At non-zero temperatures, electron wavepackets have energy width about kBT where

kB is the Boltzmann’s constant and T is the temperature. This energy uncertainty

induces diffusion in time. lT is a characteristic length of diffusion process due to

thermal energy.

Phase Coherence Length lφ

Within lφ, particles preserve their phase. Dynamical interactions including mutual

Coulomb interactions among electrons and electron-phonon interactions disturb phase


Regime ConditionBallistic Lx, Ly, Lz < lmfp, lT, lφDiffusive lmfp, lT ≪ Lx, Ly, Lz

Dissipative lφ < Lx, Ly, Lz

Table 1.2: Classified quantum electron transport

coherence. Therefore, this length is important to determine whether quantum inter-

ference effects from phase coherent sources can be detectable or not in systems.

Comparisons of such scales define three distinct transport regimes in Table 1.2.

Varying the physical length of mesoscopic conductors, all enlisted transport regimes

are indeed within practical reach. Condition states that both in dissipative and in

diffusive regime, transport quantities are dominated by scattering process similar to

classical case. In detail, dissipative conductors suffer from inelastic as well as elastic

scattering losing previous information of momentum and energy, whereas diffusive

conductors have elastic scatterers, preserving momentum but not energy. For the

ballistic regime, on the other hand, all dimensions of ballistic conductors are much

smaller than all length scales, namely electrons participating in conduction process

do not encounter any kinds of scattering sources without modifying momentum and

energy.

As an extension from classical argument between resistance and scattering, bal-

listic conductors are not resistive at all in principle. It is true and indeed confirmed

empirically with a special care to eliminate the contact resistance between electron

reservoirs and a ballistic conductor [50]. It implies that ballistic conductors in mea-

surements have non-zero resistance, but it comes not from scattering processes but

from electron modes selected at the interface of a reservoir and a conductor. There

needs to be an alternative way to express resistance beyond Ohm’s law. Landauer

captured the significance of the wave nature of charge carriers in mesoscopic conduc-

tors, and he developed the theory to estimate resistance or conductance in terms of

transmission probabilities of propagating electron modes analogous to electromagnetic


photon modes. He predicted a finite resistance of mesoscopic conductors connected

to electron reservoirs at both ends without introducing scattering [51]. His prediction

which was back then at the heart of controversy against the classical perspective of

resistance had driven intensive experimental efforts on ballistic transport by design-

ing appropriate device structures and geometries QPC in previous section closed the

controversy with observation of conductance plateaus in the ballistic regime [47,48].

Previous perspectives to envision resistance properties are based on single and

independent particle picture. The final quantity of resistance is computed by multi-

plying the one electron value with the total number of electrons. This single-particle

picture works very well in conductivity of bulk systems and Landauer’s theory since

interactions between electrons and nucleus and electrons and electrons are negligible

in high dimension by efficient screening. However, interactions affects electron trans-

port rather significantly in lower dimensions partly because of low electron density and

partly because of insufficient screening among particles. Therefore, single-particle pic-

ture breaks down in lower dimensional conductors and it should take into account of

interactions. It is not a simple task to handle various forms of interactions with many

electrons, especially Coulomb interactions between electrons are notoriously difficult

to be solved in an analytical manner. Such conductors where particle-interactions

cannot be ignored are particularly called ‘strongly correlated systems’.

Besides conductance, noise has also been actively studied in mesoscopic conduc-

tors. The knowledge of noise properties of mesoscopic conductors is invaluable as

quantum limited measurements are performed. Since these empirical results become

the lower bound of performance of quantum information process and quantum compu-

tations [52,53]. There exist different origins to generate fluctuations. Internal micro-

scopic random processes caused by thermal fluctuations, scattering and tunnelling,

quantum effects on noise have been pronounced in mesoscopic conductors [54, 55].

Theories to estimate and experiments to measure such noise in systems have been re-

cently more appreciated since noise is very sensitive to correlations of charge carriers

and scattering mechanisms. In addition, theorists have considered electron trans-

port of mesoscopic systems within quantum coherence transport theory as quantum

1.3. SCOPE OF THE WORK 17

scattering problems. Recent endeavors in this theory are put into gaining a com-

plete statistical analysis of charge transport under the name of “full counting statis-

tics” [56–59]. Conductance and noise from discrete charge carriers correspond to

the first and second moment of characteristic functions in the full counting statistics

respectively. What stimulates the advent of full counting statistics is the fact that

higher moments reveal additional information of systems beyond the first moment.

1.3 Scope of the Work

This dissertation is devoted to investigate correlated electron transport at low tem-

peratures in two one-dimensional ballistic conductors, carbon nanotubes and quan-

tum point contacts. The transport properties exhibit salient features of many-body

electron effects. Chapter 2 first examines two theoretical frameworks in condensed

matter physics to describe how to treat many-body interactions: Fermi liquid theory

and Tomonaga-Luttinger liquid theory. In Chapter 3, in-depth quantum transport to

probe electrical properties of mesoscopic conductors is presented including Landauer-

Buttiker formalism which connects the concepts of conductance and transmission.

Due to increasing attention to intrinsic noise in mesoscopic conductor charge trans-

fer, it discusses the basic knowledge about noise types regarding its origin and roles as

well. Practical strategies and technical description to implement electrical transport

measurements are given in the methodology section, Chapter 4.

Chapter 5 and Chapter 6 are the main context to apply previously described the-

ory and experimental techniques into one-dimensional mesoscopic conductors: single-

walled carbon nanotubes and quantum point contact in two-dimensional electron

gas systems. Two chapters are organized in parallel structures, introducing systems

followed by experiment results and physics interpretations. Interesting many-body

effects disclosed in measurement data are analyzed by various attempts, and the inter-

pretation of results are shown. Summary and outlook of this work concludes Chapter

7.

Chapter 2

Many-Body Physics

The fool collects facts; the wise man selects them.

− John Wesley Powell

The ultimate goal of theoretical physics is to seek a simple way to understand

observations in nature. Beauty of it has lied in the success to model complicated

phenomena elegantly by means of mathematics, a language of physics. The efforts

are driven by a search for microscopic level understandings on natural phenomena in

all physics fields. In particular, condensed matter systems are composed of immense

numbers of particles. To track down all degrees of freedom of all constituents in sys-

tems for microscopic understandings is not only daunting but also impossible. Not to

mention including all sorts of interactions amongst particles. At first, theorists have

attempted to explain, with a few simple and phenomenological parameters, common

classical macroscopic phenomena including transport properties such as phase tran-

sition and resistivity of metals. Statistical mechanics is a consequence of such efforts.

In addition to classical phenomena, quantum phenomena have been included later as

a subject of quantum statistical mechanics. Unlike the classical case, quantum statis-

tical mechanics handles indistinguishable particles. The strategy to treat them is to

use appropriate statistical distribution functions: Bose-Einstein statistics for bosons

and Fermi-Dirac statistics for fermions. In the case of fermions including electrons

18

2.1. CHARGE SCREENING 19

or holes, Pauli exclusion principle regulates their flow. Moreover, it turns out that in

order to understand real electron dynamics, the effect of particle interactions particles

should be taken into account as well together with Fermi-Dirac distribution and Pauli

exclusion principle. Diverse interactions among particles exists including electrostatic

energies between electron and ionized atoms, between electron and lattice vibrations

(phonon), and between electron and electron. How to treat these interactions become

a subtle issue. These interactions come to play crucial roles to exhibit interesting fea-

tures in some systems which are categorized as “strongly correlated systems”. The

examples of such systems are high temperature superconductors and one-dimensional

quantum wires.

This chapter is devoted to introduce two theories which treat interactions in con-

ductors: first, Fermi-liquid (FL) theory which have been successful to exploit bulk

systems for weakly interacting particles; second, Tomonaga-Luttinger liquid (TLL)

which describes interacting one-dimensional conductors whose unique features FL

fails to explain. Before two theories, an important concept is described, “screening”

for electron-electron interactions to validate theories which are applied to specific

systems.

2.1 Charge Screening

Coulomb interactions between charged particles are long-ranged, decaying as a power

law of 1/r, where r is the distance between them. Noticing that people have success-

fully explained physical phenomena in bulk systems without considering such interac-

tions, people have realized that charge screening among many particles attributes to

mitigate the long-range nature of the interactions. In other words, screening causes a

significant reduction of electric field in space. Of many ways to estimate such effect,

an approach introduced below is pedagogical with a link to the elementary electro-

magnetism theory. A basic idea is that the dielectric function is modified due to an

induced electrostatic potential by the induced charge density. Here the argument of

screening is restricted for the static response to long wavelengths in the Thomas-Fermi

approximation [60,61].

20 CHAPTER 2. MANY-BODY PHYSICS

Screening in three dimension

Suppose an induced charge density ρ(~R) of one electron at the origin in three dimen-

sion (3D), ρ(~R) = −eδ(0). The unscreened Coulomb potential Vuns(r) between two

electros far apart by a distance r is

Vuns(r) =e2

4πǫ0ǫbr, (2.1)

in a MKS unit for dielectric constants of vacuum and the media, ǫ0 and ǫb, respectively.

The induced charge polarizes and induces an electrostatic potential, φ(r) satisfying

the Poisson equation

∇2φ(r) =ρ(~R)

ǫ0ǫb= − e

ǫ0ǫb. (2.2)

The Fourier transform of Eq.(2.2) in the reciprocal space ~Q yields

Q2φ = − e

ǫ0ǫb. (2.3)

Therefore, the induced electrostatic potential in the Fourier space is φind = ρind/ǫ0ǫbQ2.

Note that the induced charge ρind is caused by the change in number density of elec-

trons dn such that ρind = −edn and that the total electrostatic potential φtot is due

to the total charge densities including the induced charge as well as external charge

densities. The total electrostatic potential relates to the energy change in the poten-

tial energy dE = −e ˜φtot or the negative change in the Fermi level dE = −dµ for

ρind. These modify the dielectric function in the Fourier space,

ǫ( ~Q) = 1 −˜φind( ~Q)

˜φtot( ~Q)= 1 +

e2

ǫ0ǫbQ2

dn

dµ≡ 1 +

Q2TF

Q2. (2.4)

The Thomas-Fermi screening wavenumberQTF is defined byQTF =√

(e2/ǫ0ǫb)(dn/dµ).

Plugging a new dielectric function into the unscreened potential in the Fourier space

makes the formula of the screened potential Vscr(Q),

˜Vscr(Q) =e2

ǫ(Q)ǫ0ǫbQ2=

e2

ǫ0ǫb

1

Q2 +Q2TF

, (2.5)

2.1. CHARGE SCREENING 21

which is the Lorentzian shape. Converting Eq. (2.5) back into the real space by the

inverse Fourier transform, it gives

Vscr(r) =e2

4πǫ0ǫb

e−QTFr

r, (2.6)

explicitly telling that the long-range Coulomb interaction becomes negligible with an

exponential decay along distance. Therefore, in bulk systems, macroscopic phenom-

ena can be understood without considering Coulomb interactions. Furthermore, at

low temperatures the ratio of dn and dµ is given by the density of states n3D at the

Fermi level, thus a large density of states help the rapid decay of the Coulomb inter-

action in space. As an example, metal has QTF ∼ 1 - 2 A due to a large density of

states at the Fermi level. It explains well why the Coulomb interactions in metal are

insignificant. This result can be understood as follows: repulsive interactions between

electrons leave an electron away from other electron clouds. The emptiness nearby

the electron can be relatively considered as holes, which screen the negative electrons.

Therefore, electrons far away do not feel the potential of these screened electrons: the

more electrons nearby, the more effective screening happens.

Screening in two dimension

Similar to 3D case is the basic idea to compute the screened potential in two-dimension

(2D) (a x − y-plane); however, the difficulty lies in the fact that the induced elec-

trostatic field is in 3D although the charge density is fixed in a 2D plane. Suppose

an induced charge density in 3D , ρ(~R) which is written as ρ(~R) = σ(~r)δ(z) with a

charge density in 2D, σ(~r), at the z = 0 plane. ~R and ~r denote the spatial vector in

3D and 2D. The discontinuity in the electric field at the plane (z = 0) requires the

charge neutrality. In the Fourier space with Gaussian theorem, the induced potential

φind(~q) in 2D is easily computed φind(~q) = ˜σind/2ǫ0ǫbq. The dielectric function in 2D

is written by

ǫ = 1 +e2

2ǫ0ǫbq

dn

dµ≡ 1 +

qTF

q. (2.7)


The density of states at the Fermi level in 2D at low temperatures is constant at

m/π~2 with a mass m. It means that the value of 2D Thomas-Fermi screening qTF

is independent of the electron density. Once again that the above description is valid

for the static response to the long wavelength.

The unscreened potential in 2D real and momentum spaces with the distance r

are

Vuns(r) =e2

4πǫ0ǫbr

˜Vuns =

∫ ∞

0

Vuns(r)eiqr cos θrdrdθ

=e2

2ǫ0ǫbq.

(2.8)

The screened potential with the dielectric function becomes

˜Vscr =e2

2ǫ0ǫb

1

q + qTF

. (2.9)

Note that the analytical result of the Fourier transform does not exist, but the trend

at large r would be found as [61]

Vscr ∼ e2

4πǫ0ǫb

1

q2TFr

3. (2.10)

The asymptotic behavior in Eq. (2.10) describes that the 2D screened potential

follows the power-law in space. The power-law decay with a exponent of 3 is much

better than the inverse r decay, but it is less sufficient to ignore charge interactions

in 2D compared to the bulk case.

Screening in one dimension

The screening in 1D has even more subtlety since the analytical form of Fourier

transform of V (r) = e2/r does not exist

V (q) =

∫ ∞

−∞dre2

reiqr, (2.11)

2.2. THE FERMI LIQUID THEORY 23

where r is 1D distance between electrons. As a simple attempt, suppose that the 1D

conductor with a uniform line charge density σ is screened by a metal conducting

plane at a distance d from the center. The potential energy between two conductors

is V (r) = (σ/2πǫ0ǫb) ln(Rs/r). In this case Rs is the effective length for screening in

1D. The logarithmic nature of the potential remains in much longer scales, meaning

the screening in 1D is not effective at all in comparison of that in higher counterparts.

Therefore, the understanding of properties in 1D should consider carefully the effect

of interactions among electrons.

In summary, the effectiveness of Coulomb interaction screening according to di-

mensionality can be intuitively understood by geometry. With a negative charge at

the origin, the charge repels electrons lying all possible directions in 3D, thus the

charge is completely screened by relatively positive charges and more electrons are

available at a larger distance. In 1D, on the other hand, only left and right electrons

can be pushed away and this push continuously kicks neighbors, yielding collective

behavior.

2.2 The Fermi Liquid Theory

The Fermi liquid (FL) theory is one of the successful theoretical framework to de-

scribe physical properties of weakly interacting many-body condensed matter systems

such as the liquid state of 3He and conductivity in metals and semiconductors. Before

delving into the details of the FL theory, to remind statistical terminologies among

N-particle systems is of help. There are three classical phases in macroscopic world:

gas, liquid and solid. Of many properties to identify phases, the strength of interac-

tions among particles becomes a good measure. An ideal gas phase appears as mutual

interactions between particles are negligibly smaller than kinetic energy, whereas the

solid state is stable as the interactions between particles are strong. The interme-

diate phase, liquid is between gas and solid. Similarly, in the case of electrons in

transport processes, Fermi gas systems refer to ones in which electrons can be treated

independently because electrons are not interacting each other. Thus, understand-

ing physical properties of such systems can be obtained sufficiently enough by an


independent single particle.

The Fermi Gas

The knowledge of any systems is complete when the eigenstates (ground state and

excited states) and the elementary excitations of the system are identified. There is

a N-particle Fermi gas system. It is the goal that finds the spectra of eigenstates

and the excitations. The eigenstates of a single particle within a volume (V) are

simple plane waves in the absence of the potential energy. They are written in the

momentum (~k) space in 3D

|~k〉 =1√Vei~k·~r (2.12)

with the quadratic energy dispersion

E~k =(~~k)2

2m(2.13)

where m is the electron mass. The ground state of the system is as the energy

states are occupied below the Fermi energy (EF ). The phase volume of the complete

Fermi sphere relates to the total number of electrons N taking into account of spin

degeneracy,

4πk3

F

3(2π)3

V

=N

2

N = Vk3

F

3π2,

(2.14)

where kF is the radius of the sphere shown in Fig. 2.1(a), and the Fermi energy

EF is defined as EF = (~ ~kF)2/2m. The momentum distribution of the system is the

sharp step function at kF (Fig. 2.1(b)). The Fermi gas system can be excited by

three ways: first, add a particle into an energy state above EF since all states below

EF are fully occupied; second, remove a particle from a state below EF, leaving a

hole inside the Fermi sphere; third, bring a particle below EF into a state above EF.


kF

ky

kxkFkF

kFkF

kyky

kxkx

(a)

(c)

2kF

E

q

(c)

2kF2kF

E

q

(b)

n~kn~k

kFkF

1

Figure 2.1: (a) The Fermi sphere of non-interacting electron systems. (b) The momen-tum distribution function of the Fermi gas systems. (c) The spectrum of particle-holeexcitations in the Fermi gas systems.


There are respectively named as particle excitations, hole excitations and particle-

hole excitations. The particle-hole excitations are illustrated in Fig. 2.1(c), noting

that the momentum of q due to scattering between a particle and a hole is continuous.

The Fermi Liquid

In the FL system, interactions between electrons need to be considered unlike in the

Fermi gas. Landau approached the FL system from the Fermi gas with a hypothesis,

an adiabatic switch-on of interactions. Further he considered the long wavelength

limit, i.e. low energy excitations near the Fermi energy. Therefore, the eigenstates

of the Fermi gas and the FL are in one-to-one correspondence [?, 62]. He captured

the idea that the interactions would modify the energy dispersion relation of the

Fermi gas, consequently changing the mass of electrons in the system. Introducing

the effective mass m∗ which reflects the strength of mutual interactions, Landau

established the FL theory within the single particle picture. The ground state of the

FL system is still the Fermi surface. The FL theory again studies the elementary

excitations of the system.

The essence of the FL theory is the existence of quasi-particles. They are low-lying

elementary excitations and they consist of electrons with the density fluctuations aris-

ing from the particle interactions. Due to the fact that quasi-particles are formed from

electrons, they also obey fermionic commutation relations. The formal description of

quasi-particles is the Green’s function of electrons:

G(~k, ω) =1

E0(~k) − ω − Σ(~k, ω), (2.15)

where E0 is the energy dispersion of the Fermi gas and Σ is the self-energy due

to many-body interactions. The pole of the above Green’s function provides the

excitation energy, representing quasi-particles.

With the self-energy term, many quantities are derived including the effective mass

m∗ of quasi-particles, the quasi-particle renormalization factor Z and the spectral

density A which describes the probability of finding a certain state [63]. m∗ and Z


are expressed in terms of the self energy as follows:

m∗ = m

(

1 − ∂Σ

∂ω

)(

1 +m

kF

∂Σ

∂k

)−1

(2.16)

Z =

(

1 − ∂Σ

∂ω

)−1

. (2.17)

n~k

kF

1

Z

n~kn~k

kFkF

1

Z

Figure 2.2: The momentum distribution of the Fermi liquid system.

Physically, Z indicates the amplitude that electrons remain as quasi-particles. In

the momentum distribution, the perfect step function is modified by Z in Fig. 2.2.

As Z closes to 1, the system is less interacting. The stronger interactions, the degree

of smearing is larger. The analytical computation of the self-energy regarding the

particle interactions is based on an assumption that the interactions are short-range.

Charge screening discussed in the previous section provides a key to validate the

FL theory in higher dimensions since the effective screening reduces the long-range

Coulomb interactions among electrons. Therefore, the FL theory works well to de-

scribe transport processes in systems whose interactions are short range and isotropic

such as metals, semiconductors and liquid 3He.


2.3 The Tomonaga-Luttinger Liquid Theory

The successful FL theory fails in 1D. The FL theory breakdown in 1D conductors

can be understood roughly in terms of inefficient charge screening in 1D mentioned

earlier. The previous section describes that the long-range Coulomb interactions

survive, correlating electrons in 1D since any excitation at a particular site spreads

over the whole lattice like the domino effect. The collectiveness is the unique feature of

1D excitations, meaning that Landau’s quasi-particles do not exist [64]. The absence

of the quasi-particles in 1D is from the multiple poles of the Green’s function [64]. A

rigorous attempt to describe 1D electron gas systems is formulated in the Tomonaga-

Luttinger liquid (TLL) theory. Tomonaga and Luttinger came up with an exactly

solvable model in 1D with insights that collective modes are bosonic nature and the

linearization of the dispersion near the Fermi level gives low energy properties [62,64].

Bosonization

The Fermi surface of 1D is two points at ±kF. Particle-hole excitations in 1D are only

possible near q = 0 or q = 2kF nearby the Fermi points, whereas any q values below

2kF are allowed for the particle-hole excitations in higher dimensions by conserving

the energy and momentum. In the limit of q → 0 and ω → 0, the excitation spectrum

is linear, resembling a phonon mode. This resemblance hints that the Hamiltonian

of 1D electron gas system can be derived by phonon displacements as a rather in-

tuitive approach. In the extreme limit where the interactions are stronger to form

Wigner crystal, the ground state of such 1D system can be modelled as an equally

spaced particle chain. The lattice constant between particles are a and it is inversely

proportional to the 1D electron density n0, i.e. a = 1/n0. The phonon-like lowest

excitations are written by the displacement ri

ri = r0i +

a

πθi, (2.18)

r0i is the initial displacement and θi is the displacement variable. As one electron

propagates to a nearest neighbor site, the variable advances by π. By definition, the

2.3. THE TOMONAGA-LUTTINGER LIQUID THEORY 29

kinetic energy (K) is derived from the displacement definition:

K =∑

i

1

2mr2

i =

∫

dxma

2π2θ(x)2. (2.19)

If we assume the 1D Coulomb interaction is screened by a ground plane at Rs where

Rs ≫ a, the effective potential V0 is V0 =∫

dx e2/x = 2 e2 log(Rs/a). With this

effective potential, the short-range interaction becomes feasible. If one additional

electron sits inside the chain, the overall change in θ is −π. Thus, the electron

density change δn(x) relates to the change of θ in x such that δn(x) = - ∂xθ(x)/π.

The potential energy (U) is for a short-range interaction

U =

∫

dxdx′1

2V (x− x′)δn(x)δn(x′)

=

∫

dxV0(δn(x))2

=

∫

dxV0

2π2(∂xθ(x))

2.

(2.20)

Therefore, the Hamiltonian density H is given by

H =ma

2π2(∂tθ(x))

2 +V0

2π2(∂xθ(x))

2. (2.21)

Substituting g =√

π~vF/V0 and vρ =√

V0/ma = vF/g, Eq.(2.20) becomes

H =~

2π

[

1

vF

(∂tθ)2 + vρ(∂xθ)

2

]

. (2.22)

Consider a spinless 1D system. The low energy properties can be studied by

linearized dispersion relation near the Fermi level. The positive (negative) slope at

kF(-kF) corresponds to right (left) moving channels. The Hamiltonian H is computed

by integrating the previous Hamiltonian density H =∫

dxH. It consists of the kinetic

energy part and a repulsive Coulomb potential energy with a strength constant λ for


low energy excitations

H = −ivF

∫

dx[

ψ†R∂xψR − ψ†

L∂xψL

]

+ λ

∫

dx(ψ†RψR + ψ†

LψL)2. (2.23)

ψi are field operators for left(L) and right(R) moving electrons. These fields are

expressed by the bosonic fields φ and θ with the cut-off constant Λ such as

ψR(L) =1√2πΛ

ei(φ±θ). (2.24)

Thus, Eq. (2.22) is rewritten by the bosonic fields [65,66]

H =~vF

2π

∫

dx

[

(∂xφ)2 + (∂xθ)2) +

λ

π~vF

(∂xθ)2

]

≡ ~vF

2π

∫ [

(∂xφ)2 +1

g2(∂xθ)

2

]

,

(2.25)

with the renormalization factor g = (1 + λ/π~vF )−1

2 .

The Tomonaga-Luttinger Liquid Parameter g

The TLL parameter, g is a measure of the interaction strength, defined as a dimen-

sionless quantity,

g =

(

1 +V0

π~vF

)− 1

2

, (2.26)

for a interaction potential V0. The second term in Eq. (2.26) indicates the competition

of the potential and kinetic energy in a system. In the absence of V0, g becomes 1,

recovering the non-interacting Fermi gas system, whereas V0 > ~vF > 0 for a repulsive

Coulomb interaction leads to g < 1. The stronger interactions V0, the smaller g. Note

that g can be also greater than 1 if attractive Coulomb interactions are dominant

among particles. The TLL parameter g emerges in various 1D properties such as

the fractional charge ge, the charge mode velocity vF/g, and the power-exponents of

correlation functions.

2.3. THE TOMONAGA-LUTTINGER LIQUID THEORY 31

Single-Walled Carbon Nanotubes

Single-walled carbon nanotubes (SWNTs) are one specific example of one-dimensional

conductors. Metallic SWNTs have been predicted as the TLL system [65, 66]. The

transport properties in the tunnelling regime, where tubes are isolated from metal

reservoirs, exhibited the TLL features as the power-scaling conductance by means of

the bias voltage and the temperatures [23]. Recently, angle-integrated photoemission

measurements obtained the spectral function from SWNT mats, claiming the direct

observation of the TLL features in SWNTs [67]. The search of the TLL behavior in

SWNTs is active since the strongly correlated SWNTs serve as a basic ingredient of

quantum electron entanglers [68–71].

The lowest bands of metallic SWNTs in the Brillouin zone (BZ) are linear. Since

the hexagonal BZ of SWNTs contains two inequivalent K and K ′, four bands are

at the same energy reflecting the orbital and spin degeneracies. The Hamiltonian

HSWNT,0 of an infinite SWNT without interactions can be derived in a similar way

with bosonic fields with a band index i= 1,2 and a spin index σ =↑, ↓

HSWNT,0 =∑

i,σ

∫

dx[

i~vF (ψ†Riσ∂xψRiσ) − ψ†

Liσ∂xψLiσ

]

. (2.27)

Similar to the above the electron fields are expressed by bosonic fields, φRiσ(Liσ) =

(1/√

2πΛ)ei(φiσ+θiσ). The bosonic fields obey the commutation relation [φiσ(x), θjσ′(x′)] =

−iπδijδσσ′Θ(x − x′). Known the fact that the interaction term relates to the total

charge density, new bosonic fields are defined for convenience such that the total

charge and spin fields are denoted as θic(s) = (θi↑±θi↓)/√

2 and θc(s)± = (θ1µ±θ2µ)/√

2.

φ fields are defined similarly. The new fields denoting as a = (c(s),±) obey the same

commutation relations [φa(x), θb(x′)] = −iπδabΘ(x − x′). These fields convert the

free Hamiltonian into the bosonized form of H with one charged excitation and three

neutral excitations [65, 66]. In an infinite SWNT, the possible scattering process

is repulsive forward scattering which requires a small momentum transfer q ∼ 0,

whereas backscattering processes are allowed between two branches with a big mo-

mentum transfer q ∼ 2kF. The forward scattering arises from a long-ranged repulsive


Coulomb interaction. The Hamiltonian of this part is written as Hint =∫

dxV0ρ2tot.

Assume that the Coulomb interaction is screened by the ground plane at Rs, the

effective screened potential is V0 = e2 ln(Rs/R) for a SWNT radius R. Note that the

interactions only change the total charge part, making the total Hamiltonian HSWNT

to be

HSWNT =vc

2π

∫

dx

[

g(∂xφc+)2 +1

g(∂xθc+)2

]

+∑

a=(c−,s+,s−)

vF

2π

∫

dx[(∂xφa)2 +(∂xθa)

2].

(2.28)

The propagating velocity of the total charge mode vc is faster than the Fermi velocity

by 1/g.

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rs/R

g

Figure 2.3: The TLL parameter g as a function of Rs/R. Two red lines are markedat g = 0.2 and g = 0.3.

The TLL parameter g for the SWNT is written in the CGS unit as

g =

[

1 +8e2

π~vF

ln

(

Rs

R

)]− 1

2

. (2.29)

2.4. SUMMARY 33

Figure 2.3 plots g vs Rs/R with vF = 8 × 107 cm/s. It shows that the g value falls

quickly between 0.2 and 0.3 for Rs/R > 5. The logarithmic dependence on Rs/R

explains that the value of g is rather insensitive to the actual value of Rs [65]. The

small value of g indicates that strong Coulomb interaction is present in SWNTs.

2.4 Summary

Chapter two has presented the background level of theoretical aspects regarding elec-

tron transport. The important concept “screening” has been examined in terms of di-

mensions. In higher dimensional systems, weakly interacting (or non-interacting) FL

theory has been a successful description for electrical properties, whereas in the lower

dimensional conductors, especially one-dimensional cases, there has been observed

that many quantities and phenomena are beyond the FL theory, where interactions

among charged particles are not negligible due to insufficient screening. The TLL

theory particularly focuses on the 1D system and as an example, the application of

the TLL on single-walled carbon nanotubes has been introduced. Such systems are

in the correlated transport regime, which is the main theme of this thesis. The next

chapter will discuss much narrower topics but more practical observable quantities

:conductance and shot noise.

Chapter 3

Mesoscopic Electron Transport

I know of no other advice than this:

Go within and scale the depths

of your being from which your very life

springs forth.

− Rainer Maria Rilke

The preceding chapter discussed theoretical aspects of many-body physics and in-

troduced two backbone descriptions for weakly and strongly interacting many-electron

condensed matter systems: Fermi liquid theory and Tomonaga-Luttinger liquid the-

ory. Both theories have nicely established a mathematical framework to study the

ground state properties and elementary excitations in terms of second quantization

techniques. Furthermore, they link correlations between system observables in equi-

librium and in non-equilibrium situations to scattering and transport processes.

Transport processes is related to the response of a system to external stimulation.

For example, firing up a part of a system generates a temperature gradient, inducing

a net heat flow across it. Closely connected to the thermal conduction, the movement

of electrons due to a non-zero electrical potential along a system yields electrical con-

ductivity, one of the material characteristics. Conductivity measurements by probing

a current change according to a bias voltage across a system, have provided valuable

34

3.1. LINEAR RESPONSE THEORY 35

information to identify the states of matter, metal or insulator. This chapter pays

a particular attention to the electrical transport properties, discussing fundamentals

and the implications in mesoscopic conductors.

3.1 Linear Response Theory

Linear response theory (LRT) raises a practical question: how a system in equilibrium

responds as its equilibrium state is disturbed. It formulates the response function of

a many-particle system which is stimulated by an external source. LRT assumes that

the external stimulation is weak enough that it can be treated as a perturbation,

justifying the Taylor series expansion. Plus, the perturbation expansion series are

converging rapidly after the first linear term; thus, considering the first non-trivial

linear term would be sufficient to describe the response of systems. This response

function is a measurable quantity, therefore it is real-valued. In transport, the re-

sponse function is a macroscopic transport coefficient. Since it is shown that the

response function relates to the correlation functions in the system, LRT describes a

nonequilibrium system in terms of fluctuations about its equilibrium state. There-

fore, understanding the dynamics of a system in equilibrium is essential to predict

nonequilibrium situations.

Suppose a system whose isolated Hamiltonian is denoted as H0. If a weak time-

dependent disturbing field A ·F (t) is applied to the system at time t0, the perturbed

Hamiltonian H at later time t becomes H = H0 − A · F (t) where A is the inter-

nal quantity conjugate to the field F (t). LRT says that the average of A in the

nonequilibrium 〈A(t)〉 can be written as

〈A(t)〉 = 〈A(t)〉0 +

∫ t

−∞dt′R(t, t′)F (t′) +O(F (t)2), (3.1)

where 〈...〉0 is the average over equilibrium ensembles. R(t, t′) is the linear response

function, which relates two times t′ and t. t′ is the time at which the external

field acts on the system and t is the time of measurement. Thus t > t0, it is the

causality property. A simple example of the response function is the conductivity in

36 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT

equilibrium, which in connection with Eq.(3.1) is known as the Green-Kubo formula.

It states that the equilibrium conductivity σ of a one-dimensional system subjected

to a constant voltage V at time t = 0 is given in terms of the current density jx(t)

along the x-direction,

σ =V

kBT

∫ ∞

0

dt〈jx(0)jx(t)〉0.

Conductance in mesoscopic conductors can be computed as the response function

described above. Note that the mathematical strategies are different depending on

which regime (either ballistic or diffusive) the actual transport occurs.

3.1.1 Ballistic Transport

Ballistic transport refers to the transport of electrons without encountering any types

of scattering sources. In other words, the system size is smaller than the mean free

path and the inelastic scattering length. Based on the point of view that conductance

arises from scattering, conductance is predicted to be infinite in the ballistic regime;

however, finite conductance has been measured in the ballistic conductors. This ob-

servation boosted theoretical interests to understand the origin of finite conductance.

(a) (b)c

E

k

E

k

(a) (b)c

E

k

E

k

(b)cc

E

k

E

k

Figure 3.1: (a) An one-dimensional ballistic conductor in a two-terminal configura-tion. (b) The energy dispersion of free electrons in a reservoir (left) and a conductor(right).


Laudauer-Buttiker Formalism

Landauer brilliantly captured the wave nature of electrons in mesoscopoic conductors,

and he interpreted the conductance as the transmission probabilities of propagating

modes analogous to electromagnetic fields in optical waveguides.

Suppose a simple one-dimensional (1D) ballistic conductor with two leads coupled

to bulk electron reservoirs, as illustrated in Fig. 3.1(a). Adiabatic transition from bulk

reservoirs to the device and zero temperature are assumed. In case of free electrons,

Fig. 3.1 (b) presents the energy dispersion relations in bulk reservoirs (left) and the

conductor (right). The horizontal axis represents the longitudinal wavenumber k. Due

to the spatial confinement, the allowed modes in the conductor are discrete, while the

modes in the bulk are relatively dense. Therefore, not all modes below the Fermi

energy can propagate into the conductor due to energy and momentum conservation,

yielding that only certain modes can be matched in both regions. Mode reflection

at the interface of two dissimilar materials causes finite conductance even with a

ballistic conductor. Sometimes this finite resistance is called ‘contact resistance’. In

the simplest case, one channel in the conductor exists. The current I across the

conductor with the applied bias voltage V is given as I =∫ EF +eV

EFeρ(E)vg(E) with

energy-dependent density of states and group velocity. The density of states ρ in

1D is given ρ = 1/2π~vg(E). Note that in 1D, there is a magic cancellation of the

velocity component, yielding the product of ρ(E) and vg(E) is constant 1/h. Thus,

the current including spin degeneracy is

I =

∫ EF +eV

EF

e2

hdE =

2e2

hV,

reducing the conductance G to G = I/V = 2e2/h ≡ GQ denoted as the spin-

degenerate quantum unit of conductance. GQ is measured when the mode is com-

pletely transmitting into the opposite reservoir. For a mode which is transmitting

with a probability T , the conductance G is G = GQT . Moreover, if there are more

than one channel involved in the transport process and each mode has an individual

transmission probability Ti, then the conductance G is obtained as a sum over all


modes

G = GQ

∑

i

Ti,

known as the ‘Landauer formula’ [72, 73]. Then, Buttiker further extended the Lan-

dauer’s formula into multi-lead and multi-mode systems even in presence of a mag-

netic field. He established coherent scattering formalism. Conductance measured in

two leads α and β is ,

Gα→β = GQ

Nα∑

n=1

Nβ∑

m=1

|tβα,mn|2,

with channel modes m,n [74].

Scattering Matrix and Transmission Matrix

a1

b1

(a) (b)

±°

® ¯a2

b2

a1a1

b1b1

(a) (b)

±°

® ¯

(b)

±±°°

®® ¯a2a2

b2b2

Figure 3.2: A two-port system represented by second quantized operators.

Second quantization representation is an elegant way to describe mesoscopic con-

ductors. This is powerful in many aspects: first, it deals straightforwardly with

indistinguishable many particles; second, it automatically satisfies exchange rules of

bosons or fermions. The previous 1D, one-channel conductor is regarded as a two-port

system drawn in Fig. 3.2(a). The operators ai annihilate particles in the incoming

channels into the scattering site, and the operators bi do particles in the outgoing


channels. The index i is either 1 or 2. How incoming and outgoing operators are

related is written in a compact matrix form. There are two different ways to connect

those operators: (1) scattering matrix S and (2) transfer matrix T . A certain form

is more efficient than the other, appropriate for situations. The components of these

matrices are transmission and reflection coefficients between corresponding modes.

Consider an example to view how to form the matrices with a two-port system

with one channel. First, the S-matrix gives an obvious connection of the incoming

channels versus the outgoing channels such that

(

b1

b2

)

=

(

r11 t12

t21 r22

)(

a1

a2

)

≡ S

(

a1

a2

)

. (3.2)

Since S is unitary, i.e. SS† = S†S = 1, two conditions among components should be

met:

SS† =

(

r11 t12

t21 r22

)(

r∗11 t∗21

t∗12 r∗22

)

=

(

1 0

0 1

)

, (3.3)

reading that

|r11|2 + |t12|2 = |r22|2 + |t21|2 = 1

r11t∗21 + t12r

∗22 = r∗11t21 + t∗12r22 = 0.

(3.4)

Second, a T-matrix describes how the left operators propagate to the right side:

(

b2

a2

)

=

(

T11 T12

T21 T22

)(

a1

b1

)

≡ T

(

a1

b1

)

. (3.5)

The benefit of the T-matrix representation is to readily compute the overall T-matrix

Tall as a particle propagates several T-matrices until it reaches the final location.

Explicitly, it means

(

bN

aN

)

= T (N)T (N−1) · · · T (1)

(

a1

b1

)

≡ Tall

(

a1

b1

)

. (3.6)


Thus, Tall expresses Tall = T (N)T (N−1) · · ·T (1). The components of T are rewritten in

terms of rij and tij,

T11 = t21 −r11r22t12

,

T12 =r22t12,

T21 = −r11t12,

T22 =1

t12.

(3.7)

3.2 Fundamentals of Noise

The term ‘Noise’ in electronic circuits has dominantly negative meanings since it of-

ten refers to spurious, unwanted uncontrollable contamination to a signal. Therefore,

noise, in other words signal fluctuations, is a limiting factor of the sensitivity of equip-

ments and apparatus. Electronic systems are vulnerable to external noise sources via

magnetic, capacitive, radio frequency couplings [75]. These couplings ruin main sig-

nals unless they are well taken care of in right manners. Grounding and shielding are

primary solutions to fight against external noise sources [76]. In addition to cumber-

some degradations, there exist other types of noise, so called intrinsic noise of systems.

These noise sources cannot be eliminated by grounding and shielding techniques, and

they reflect statistical features arising from enormous numbers of electrons in trans-

port processes. This section is devoted to introduce some background knowledge of

these intrinsic noises.

3.2.1 Fluctuation and Dissipation Theorem

Statistical mechanics is a study of many-body systems, asking the effective way to

treat countless degrees of freedom [77]. It has been successful to describe macro-

scopic thermodynamic phenomena in equilibrium. It has been investigating the non-

equilibrium and irreversible processes as well. Of numerous approaches, the fluctua-

tion and dissipation theorem (FDT) is of importance in that it basically tells that the

3.2. FUNDAMENTALS OF NOISE 41

non-equilibrium properties are closely related to the equilibrium quantities [78]. In

detail, it provides a general relationship between the response of a system disturbed

by an external source and the internal fluctuations of the system without the distur-

bance. The validity of FDT lies in the linear response regime, indicating that the

external disturbance is weak and the dominant term is the linear one. The response

of the system is often characterized by a response function, for example admittance

or impedance in the electronic circuits. On the other hand, the internal fluctuations

reflect correlation functions of physical quantities in thermal equilibrium. Therefore,

the roles of the FDT can be summarized into two: first, the FDT predicts the fluctua-

tion characteristics or intrinsic noise of the system from the known properties; second,

the FDT provides a basic formula to derive the known properties such as resistance

from the analysis of fluctuations in the system.

Brownian motion is a prototypical phenomenon to provide insight into the FDT,

describing random behavior of objects. And it results in statistical fluctuations in

thermal equilibrium systems. For convenience, consider a one-dimensional system

that moves in one direction x with velocity v(t) at time t. The system is not com-

pletely isolated from the external world, but only couples with the outside environ-

ment weakly. The slowly varying external coupling is given as F(t), whereas F (t), the

interaction of the system with other degrees of freedom, is rapidly fluctuating. The

latter sets one time scale, the ‘correlation time’ τ ∗, which measures roughly the mean

time between maxima of F (t). For macroscopic times τ , i.e. τ ≫ τ ∗, the equation of

motion is

mdv

dt= F(t) + F (t). (3.8)

After the integration over τ followed by the ensemble average, Eq.(3.8) is rewritten

as

m〈v(t+ τ) − v(t)〉 = F(t)τ +

∫ t+τ

t

〈F (t′)〉dt′. (3.9)

The integrand 〈F (t)〉 in the second term on the right side is associated with an energy

change ∆E in the external world at temperature T such that

〈F (t)〉 =1

kBT〈F (t)∆E〉0, (3.10)


with the ensemble average 〈....〉0 in equilibrium state [77]. Note that the energy

change in the external world is equivalent to the negative work done by the force

F (t),

∆E = −∫ t′

t

dt′′v(t′′)F (t′′) ≈ −v(t)∫ t′

t

dt′′F (t′′). (3.11)

The approximation in the last equivalence is valid since the system velocity vary

negligibly over τ . Plugging the last equation into Eq. (3.10), the mean of F (t) is

written with β = (kBT )−1 as

〈F (t′)〉 = −β〈F (t)v(t)

∫ t′

t

dt′′F (t′′)〉0

= −βv(t)∫ t′

t

dt′′〈F (t′)F (t′′)〉0.(3.12)

Since what physically matters is the time change, dummy variables can be changed

accordingly s ≡ t′′ − t′. Equation (3.9) reads

m〈v(t+ τ) − v(t)〉 = F(t)τ − βv(t)

∫ t+τ

t

dt′∫ 0

t−t′ds〈F (t′)F (t′ + s)〉0. (3.13)

In statistical mechanics, 〈F (t′)F (t′ + s)〉0 is referred to as “ correlation function ” of

F (t). Note that the second term on the right side leads to ‘dissipation’ in the system.

3.2.2 General Formulation of Noise

The internal fluctuations in the time domain look random, thus it is quite challenging

to extract quantitative information. Therefore, often the time-information of the

fluctuations is converted into its conjugate parameter, frequency. The frequency

response gives the spectral content of the fluctuations. The quantity in the frequency

domain for correlation functions is called “spectral density”, J(ω). And the relation

between two quantities is via Fourier transformation,

J(ω) =1

2π

∫ ∞

−∞〈F (t′)F (t′ + s)〉0eiωsds,


this relation is especially called as “Wiener-Khintchine Theorm”.

Particularly in electronic circuits, the variables which fluctuate according to ex-

ternal stimulations are current or voltage. The Fourier components of fluctuating

currents or voltages are named “power spectral density” because the unit is either

[A2/Hz] or [V2/Hz]. Explicitly, the current noise spectral density SI and the voltage

noise spectral density are expressed in terms of current and voltage operators, Î(t)

and ˆV (t),

SI(ω) =

∫ ∞

−∞〈 Î(t), Î(0)〉eiωtdt

SV (ω) =

∫ ∞

−∞〈 ˆV (t), ˆV (0)〉eiωtdt.

(3.14)

3.2.3 Classification of Intrinsic Noise

10-25

2

4

6

10-24

2

4

6

10-23

2

4

6

10-22

SI

(A2/

Hz

)

100 10

3 106 10

9 1012 10

15

Frequency (Hz)

1/f noise

Shot noise

Thermal noise

Quantum noise

10-25

2

4

6

10-24

2

4

6

10-23

2

4

6

10-22

SI

(A2/

Hz

)

100 10

3 106 10

9 1012 10

15

Frequency (Hz)

1/f noise

Shot noise

Thermal noise

Quantum noise

Figure 3.3: Spectrum of dominant noise sources in frequency domain.


There are several sources causing fluctuations in time-varying currents: thermal

noise, shot noise, 1/f noise, and quantum noise. In terms of frequency behavior,

two distinct trends are identified. Some noise is constant over some frequency range,

whereas other noise diverges as frequency is decreased. These are often named by

colors such that the former case is white nose and the latter one is pink noise. The

examples of white noise are thermal and shot noise, and the 1/f noise belongs to

the pink noise category. In addition, the noise grows as frequency increases, and

quantum noise falls into this type. Figure 3.3 exhibits the spectra of intrinsic noise

sources. This subsection describes the origin of the intrinsic noise sources and their

mathematical formulations.

Johnson-Nyquist Noise

At finite temperature, the electron collisions with the lattice vibrations and impuri-

ties induce fluctuations in the electrons’ velocity and position. The electronic motion

is well explained by a Brownian particle model. The net microscopic electronic ther-

mal agitation drives a fluctuating voltage across the resistor. Since it exists without

voltage applied to system, it is equilibrium quantity. The microscopic thermal fluc-

tuations provide correlations physical quantities either current or voltage, and they

are associated with dissipative term in the circuit, conductance or resistance. Such

outcome is called the Johnson-Nyquist noise at non-zero temperature of statistical

objects. It is one of prototypical example to which FDT applies explicitly.

R L

I (t)

V0 (t)+

-R L

I (t)

V0 (t)+

-

Figure 3.4: A parallel resistor-inductor circuit.


Consider a parallel resistor-inductor circuit drawn Fig 3.4. R is the resistance and

L is the inductance. Given the voltage across the resistor V0(t), the Kirchhoff voltage

law tells that LdI(t)/dt = V0(t), where I(t) is the current of the circuit. Suppose

V0(t) decomposes into a slowly-varying part V (t) and a rapidly-varying part v(t).

Note that in this circuit,there are three relevant time constants: τ ,τm,τ ∗. τ is the

time constant for the macroscopic quantities (e.g.I(t)) of the circuit to change, τm is

the time constant for a change in the electron’s velocity (momentum), and τ ∗ is the

mean-free time between successive collisions of an electron with the lattice. These

three are assumed to obey the following inequality: τ ≫ τm ≫ τ ∗. The slowly-varying

part V (t) changes on the time scale τ and the rapidly-varying part v(t) changes on the

time scale τ ∗. Here, V (t) acts to keep I(t) around I(t) = 0,in other words, V (t) is a

relaxation term for the departure of I(t) from its steady-state value. Upon the above

time scales, the equation of motion, the ‘Langevin equation’ can be derived from the

Kirchhoff voltage law. Now both I(t) and V0(t) can be decomposed into the slowly-

varying part and the rapidly-fluctuating one: I(t) = I(t) + i(t); V0(t) = V (t) + v(t).

Over τm, rapidly-fluctuating components can be averaged out, you can write the

Kirchhoff voltage law as

LdI(t)

dt= V (t).

Considering that V (t) is a restoring force for the steady-state value of I(t) with a

coefficient, R, then

V (t) = −RI(t).

If we approximate I(t) = I(t)+ i(t) ≈ I(t), neglecting the small modulating signal in

current, then the equation becomes

LdI(t)

dt= V (t) + v(t)

≈ −RI(t) + v(t).

(3.15)

Putting γ = R/L, the Langevin equation is re-written as

dI(t)

dt+ γI(t) =

1

Lv(t).


As a mathematical trick, multiplying eγt and rearranging the terms yield

d

dt[eγtI(t)] =

1

Leγtv(t). (3.16)

Integrating Eq.(3.16) from −∞ to 0,

I(0) =1

L

∫ 0

−∞eγtv(t)dt. (3.17)

Suppose the voltage fluctuation v(t) is stationary. Its covariance function depends on

the time difference so that v(t′)v(t′′) = v(t′ − t′′)v(0) for a stationary process.

I(0)2 =1

L2

∫ 0

−∞

∫ 0

−∞eγ(t+t′)v(t)v(t′)dtdt′,

=1

L2

∫ 0

−∞

∫ 0

−∞eγ(t+t′)v(0)v(t− t′)dtdt′.

(3.18)

Then, changing the integration variables to s = t′ + t′′ and s′ = t′ − t′′, the integral

in Eq. (3.18) becomes,

I(0)2 =1

L2

∫ 0

−∞eγsds

∫ 0

−∞v(0)v(s′)ds′

=1

L2

1

γ

1

2

∫ ∞

−∞v(0)v(s′)ds′

(3.19)

Inserting I(0)2 = kBT/L from the equipartition theorem at temperature T , Eq. (3.19)

iskBT

L=

1

L2

L

R

1

2

∫ ∞

−∞v(0)v(s′)ds′. (3.20)

After replacing a dummy variable s′ by t,the FDT for thermal noise can be written

as

R =1

2kBT

∫ ∞

−∞v(0)v(t)dt. (3.21)

Basically, Eq. (3.20) indicates that the microscopic voltage correlations due to ther-

mal random motions of electrons are related to equilibrium resistance, dissipative


term in the circuit. The Wiener-Khintchine theorem is expressed such that

Sv(ω) = 4

∫ ∞

0

v(0)v(t) cos(ωt)dt. (3.22)

Since v(0) and v(t) are correlated only over τ ∗, it would be fair to approximate

ωτ ∗ ≪ 1, leading cos(ωt) ≈ 1. Combining Eq.(3.21) and Eq.(3.22), the Johnson-

Nyquist thermal noise is expressed as

Sv(ω) = 4

∫ ∞

0

v(0)v(t) cos(ωt)dt

≈ 2

∫ ∞

−∞v(0)v(t)dt

= 4kBTR.

(3.23)

Note that the expression does not have frequency dependence, indicating it is white

up to a cut-off frequency. Similarly, the Johnson-Nyquist thermal noise in currents

can be derived such that SI(ω) = 4kBTG, with conductance G. The way to remove

thermal noise is to reach the absolute zero temperature, thus cooling the system would

reduce the amount of thermal noise in systems in a linear manner.

1/f Noise

The 1/f noise has several alternative name, for example, flicker noise, pink noise

or telegraph noise. However, often the 1/f noise or low-frequency noise are rather

conventionally used due to the frequency trend. As name indicates, it is the dominant

noise sources in low frequency. In the circuit, the following equality holds valid with

C1/f , a measure of the relative noise of the sample,

SI(f)

I2=SV (f)

V 2=SR(f)

R2=SG(f)

G2=C1/f

f. (3.24)

Experimentally the above quantities Si(f) are proportional to fα where the power

exponent is roughly α = −1.0±0.1 [79]. The microscopic origin of this noise is unclear

and under investigation [80]. Of many theoretical speculations, a random telegraph


signal would lead to 1/f noise in that trapping centers in system capture and release

electrons or holes in a random fashion. The question remains what are candidates

of trapping centers. Defects on surface or interfaces act as such centers. Consider a

generation-recombination noise whose physical quantity is X its fluctuation is ∆X.

Suppose ∆X decays with time scale τ [79]. The decay differential equation is

−d∆X(t)

dt=

∆X(t)

τ. (3.25)

The integration of the above differential equation gives solution that

∆X(t) = ∆X(t0)e− 1

τ(t−t0).

The correlation function of X is

ϕX(t) = 〈∆X(t0)∆X(t0 + t)〉 = 〈∆X(t0)∆X(t0)e− t

τ 〉= 〈(∆X)2〉e− t

τ .(3.26)

Plugging the expression of the correlation function into the Wiener-Khintchine theo-

rem, the power spectral density SX(f) is given

Sx(f) = 4

∫ ∞

0

ϕX(t) cos(2πft)dt

= 4

∫ ∞

0

〈(∆X)2〉e− tτ cos(2πft)dt

= 4〈(∆X)2〉[

τ

1 + (2πft)2

]

.

(3.27)

This independent-electron process yields Lorentzian spectrum, exhibiting that it is

white for fτ ≪ 1 while it shows 1f2 for fτ ≫ 1. If there are a large number of

Lorentzian spectra, the overall noise is from a summation of independent process.

Mathematically, it can be computed with a weighting factor g(τ) , which is inversely

proportional to a time τ . Meaning that the Lorentzian spectra have relaxation time

τ satisfying the inequality τ1 < τ < τ2, g(τ) can be written as g(τ)dτ = 1ln(

τ2τ1

)1τdτ .


With this g(τ), the spectral density SX(f) becomes

SX(f) =

∫ τ2

τ1

g(τ)〈(∆X)2〉 4τ

1 + (2πfτ)2dτ

=4

ln( τ2τ1

)〈(∆X)2〉

∫ τ2

τ1

1

1 + (2πfτ)2dτ

=2

π ln( τ2τ1

)

(

1

f

)

〈(∆X)2〉[

tan−1(2πfτ2) − tan−1(2πfτ1)]

.

(3.28)

There are three regimes: (1) For f < 1/2πτ2, Eq.(3.28) becomes SX(f) = 4τ2〈(∆X)2〉/ ln(τ2/τ1), which is constant in frequency; (2) For 1/2πτ2 < f < 1/2πτ1, Eq. (3.28)

becomes SX(f) = 〈(∆X)2〉/ ln(τ2/τ1)f , which is 1/f spectrum; (3) For For f >

1/2πτ1, Eq. (3.28) becomes SX(f) = 〈(∆X)2〉/π2τ1 ln(τ2/τ1)f2, which is rather rapid

decay to the power 2 in f . This attempt is successful in that it produces the spectra

of 1/f which is common in experiments. Equation (3.25) tells that the 1/f noise is

related to current flowing. However, experiments showed that even in equilibrium

situation without driving current or voltage, 1/f noise existed although the strict

thermal equilibrium may not be confirmed in experiments [81,82]

Hooge proposed an empirical expression of the 1/f noise for homogeneous samples

as

SI(f) = γI

2+β

Ncfα

SV (f) = γV

2+β

Ncfα,

(3.29)

with constants α, β, γ and the total number of charge carriers in the system, Nc [79].

He found that the value of γ is, surprisingly, constant about 2×10−3 for independent

electrons in homogeneous samples.

Shot Noise

The electrical conduction in circuit components is typically by majorities of electrons

or holes. Due to the nature of discreteness of electrons or holes, current becomes


statistically fluctuating around the mean value. The associated noise is called shot

noise. It results from the emission of discrete electrons. When the distribution of

electron emission follows the Poisson statistics, in other words, emitted electrons are

independent of each other, the shot noise is proportional to the average current value

I, i.e.

SI(ω) = 2eI, (3.30)

where e is the elementary charge. This noise is strong Particularly, this formula

is denoted as ‘full shot noise’. Unlike thermal noise, it emerges in non-equilibrium

situation, meaning that either voltage or current applies to the system. Shot noise

is also flat in range of frequency, satisfying fτ1 ≪ 1 where τ1 is the transit time of

electrons, supposedly very short. It is present in vacuum tubes and semiconductors,

where electrons leaves a cathode or passes a potential barriers.

Partition Noise

Ballistic conductors have no fluctuations in principle due to definite conductance

values. However, scatterers along the path of electrons or charge carriers contribute

a new source of noise. It is called partition noise. The incoming wave of particles is

partially reflected and partially transmitted after facing a scatterer. The probability of

the transmission by the scatter is denoted as T . The process is binomial distribution,

yielding the (1-T) fluctuations. The current fluctuation spectral density becomes

reduced from the full shot noise by this factor: SI = 2eI(1 − T ). When T is much

smaller than 1, this form goes back to the full shot noise value. The characteristic

of such scattering is elastic by conserving energy and momentum so that phase of

electrons is preserved, i.e. phase-coherent picture.

Quantum Noise

Quantum noise is the frequency-dependent excess noise. It is proportional to the fre-

quency, becoming the dominant fluctuations at high frequencies. To describe quan-

tum noise, correlations among electrons originated from Coulomb interactions and the

Pauli exclusion principle should be considered similar to the shot noise. Moreover,


quantum noise includes vacuum fluctuations. The finite frequency current spectral

density is written with energy-independent transmission probabilities Tn

SI(ν) =N∑

n

Tn(1 − Tn)2e2

h

[

(eV + hν) coth

(

eV + hν

2kBT

)

+ (eV − hν) coth

(

eV − hν

2kBT

)]

+N∑

n

T 2n

2e2

h

[

2hν coth

(

hν

2kBT

)]

.

(3.31)

It is reduced to a simple form SI = 2hνG in the limit where hν ≫ eV, kBT with

G = 2e2/h∑

Tn. Equation (3.31) is a complete and general form of the current

spectral density including shot noise, thermal noise and quantum noise.

3.2.4 Crossover of Noise Sources in Frequency Domain

Figure 3.3 presents the frequency dependent current fluctuations SI versus frequency

f . Three distinct regimes are identified: (1) Region I: low frequency where 1/f noise

is a dominant noise source; (2) Region II: intermediate frequency where white noise

sources both thermal and shot noise are the biggest; (3) Region III: high frequency

where quantum noise exceeds other sources.

Pink Noise versus White Noise

Consider a system whose current power spectral density is composed of three main

components under assumption that the frequency is not high enough where quantum

noise does not play a dominant role: 1/f noise, thermal noise and shot noise. Assume

that all three are uncorrelated, then the overall noise power spectral density is a linear

combination such that

SI(ω = 0) = S1/f + SThermal + SShot

= AI2DC

f+ 4kBTG+ 2eIDC .

(3.32)


This equation produces the crossover from pink noise to white noise, which corre-

sponds to the left part of the plot in Fig. 3.3. In mesoscopic phase-coherent conduc-

tors, the excess noise is partition noise, so that the total noise power spectral density

at zero frequency is expressed with the transmission probability Tn

SI(ω = 0) = AI2DC

f+ 4kBTG

∑

n

Tn

+ 2kBTG∑

n

Tn(1 − Tn)

[

eV

2kBTcoth

eV

2kBT− 1

]

.

(3.33)

Note that there exists another crossover between thermal and shot noise. From

the second and third term in Eq.(3.33), thermal noise is bigger than the partition

noise as kBT ≫ eV . On the other hand, taking the limit of eV ≫ kBT , the last two

terms in Eq.(3.33) are reduced to∑

n(4kBTGQTn) + 2eI∑

Tn(1 − Tn)/∑

Tn. As T

goes to 0, the term equals to the shot noise value.

White Noise versus Quantum Noise

The second crossover occurs between white noise and quantum noise. The appropriate

formula is given by Eq.(3.31). There are several limiting cases in Eq.(3.31). First,

when eV ≫ kBT , Eq.(3.31) becomes SI = 2eI∑

Tn(1 − Tn)/∑

Tn, partition noise.

Second, when kBT ≫ eV , Eq.(3.31) recovers the Nyquist-Johnson noise such that

SI = 4kBTG. Last, when hν ≫ eV, kBT , the current spectral density is SI = 2hνG.

Therefore, the right part of Fig. 3.3 exhibits this crossover from shot and thermal

noise to quantum noise.

3.3 Summary

In this chapter, the basic theoretical descriptions of transport processes and transport

properties. The LRT as a fundamental theory to treat transport processes has been

reviewed and the Landauer-Buttiker formalism within the LRT has been introduced.

Based on this formalism, conductance and the current fluctuations are introduced.

The fluctuations and dissipation theorem connects non-equilibrium quantity with

3.3. SUMMARY 53

equilibrium quantity such that the shot noise is computed by the average value of

current. Various sources of current fluctuations have been discussed: thermal noise,

1/f noise, shot noise and quantum noise. The crossovers between different noise

sources in the frequency range have been examined.

Next chapter will discuss how to design practical experimental setup to measure

conductance and the current fluctuations. It will also present various techniques for

an improved signal to noise ratio.

Chapter 4

Experiment Methodology

Creative scientists have faith that

well-thought-out hypotheses, good experimental design, and persistence

will lead to truth through research.

− Robert V. Smith

Physics is a subject to explore natural and engineering phenomena with seeking

for fundamental understandings and to stimulate a search to discover intrinsic prop-

erties predicted by theoretical models. In all cases, both experiments and theories are

so tightly interconnected that theories without being confirmed by experiments and

experiments without being understood by theory are just incomplete. As an experi-

mentalist, I believe that the knowledge of physics is strongly rooted in empirical facts

and I emphasize the crucial roles of experiments not only as verification of existent the-

oretical speculations and hypotheses but also as a driving force to investigate further

unresolved phenomena beyond established models. Therefore, experiments should

be pursued for accurate measurements with a careful implementation of appropriate

apparatus. This chapter begins with a general guideline for electron transport mea-

surements with mesoscopic conductors whose specific outcome ensues in subsequent

chapters, and the guideline applies the hardware preparation to the specific case of

carbon nanotubes with specific numbers.

54

4.1. CONDUCTANCE 55

Of interest is to investigate electrical properties via conductance and shot noise

measurements in this thesis. For any experiments before actual implementation, the

initial step is to check the plausibility by rough estimate. It includes the recognition

of system limiting factors. As an example, conventional resistors cannot hold above

a certain voltage indicated by a power-rating: a 1/2-Watt 50 Ω carbon-composition

resistor can be biased up to 5 V before frying it. Similarly, mesoscopic conductors have

even tighter limit since they are too tiny to hold high voltage or current through them.

External excitation range would be constrained by the comparison of relevant energy

scalings. In order to observe the intrinsic properties in non-equilibrium states, the

bias energy to a system is greater than the thermal energy from non-zero temperature

of the ambience environment.

Next, the signal-to-noise ratio (SNR) of a system, one of the essential number

estimate, should be concerned. The SNR tells whether the signal is big enough to

be measured in the presence of possible noise sources. Plus, it provides how much

the signal should be increased for the bigger SNR with available techniques including

amplification or phase-sensitive detections. The SNR is defined in decibels (dB),

SNR = 10 log10

(

v2s

v2n

)

, (4.1)

where vs and vn denote a root-mean-square (rms) signal and a rms noise respectively.

It is unavoidable to handle weak signals from mesoscopic conductors, thus the main

efforts is devoted to improve the SNR. Here, two-terminal transport measurements

are focused.

4.1 Conductance

Mesoscopic field tends to report conductance rather than resistance from electron

transport measurements although conductance and resistance are reciprocal in gen-

eral. The conventional tradition is adopted here. The preceding chapter mentions

the distinction between linear conductance G and differential conductance dG. For a

linear current-voltage (I − V ) relation, two quantities are exactly same; however, the

56 CHAPTER 4. EXPERIMENT METHODOLOGY

nonlinear I − V trend occurs in most cases of mesoscopic systems, in which G and

dG become distinct. Since G is the integral of dG or dG is the differentiation of G

in mathematical perspective, one quantity can be computed from the other by post-

analysis. Instead of relying on numerical process, both are intended to be measured

experimentally. The method to extract a G value is straightforward such as applying

a dc voltage and measuring a dc current, whereas the method to find a dG value

requires some thoughts. The dG value corresponds to the tangential component at

a specific dc voltage or current, the bias components consist of both dc and ac such

that a tiny ac wiggle on top of a dc value. The ac change represents the slope at the

dc value in a I − V characteristic.

rIN

r1

r2

Rd

rINr1

r2

Rd

rIN

(a) (b)rIN

r1

r2

Rd

rIN

r1

r2

Rd

rINr1

r2

Rd

rIN

rINr1

r2

Rd

rIN

(a) (b)

Figure 4.1: (a) Two-terminal and (b) four-terminal measurement schematics with aconstant current bias.

Conductance is the response to the external stimulations acting on a system. A

couple of excitation schemes exist : current-bias and voltage-bias. It is not an ab-

solute rule to choose one scheme from the other, but a common sense for the best

results has been formed according to the size of conductance, such as a system whose

G is less than a quantum unit of conductance GQ (i.e. a high resistive conductor)

had better be biased by constant voltage source, whereas a highly conducting sys-

tem is biased by constant current source. A rationale behind the practical setup is

based on the stability of the bias source and the high SNR. In any bias schemes,

series resistance is unavoidable in a practical setup, originated from various sources

4.1. CONDUCTANCE 57

like wirings between equipments and contact interface between electrodes and de-

vice materials. Note that this issue is discussed readily in terms of resistance, thus

the following argument goes on with resistance not conductance. A typical two-

terminal measurement shown in Fig. 4.1 includes all spurious series lead resistances

ri together with a device resistance Rd. On the other hand, a four-terminal configu-

ration can eliminate series resistances, enabling us to access selectively an interesting

portion of circuits. Quantitatively, V = I(Rd + r1 + r2) in the former case and

V = V1 − V2 = I(Rd + r2)− Ir2 = IRd in the latter configuration under the assump-

tion that the output impedance of current sources and the input impedance rIN of

voltmeters are infinite.

Based on the actual implementation of transport measurements with single-walled

carbon nanotubes and quantum point contacts in the upcoming chapters, the setups

have several wiring options: coax cables, low-resistive wires and high-resistive wires.

We use a BeCu center conductor coax cables, whose resistance per a meter is 0.4 - 1

Ω, and a Cu wire has a low resistance around 3 Ω per meter and a Mn wire is used

as a high resistive wire, which has resistance around 200 Ω per meter. Coax cables

are necessary to protect signals from noise due to grounding of outer conductors,

low-resistive wires are preferable to bias electronic components, and high-resistive

wires are used to apply gate voltages where the fluctuations may be filtered out by

effective low-pass filter formed by resistance and capacitance. Since the resistance of

mesoscopic device is typically a range of 10 - 100 kΩ, the lead resistance of wirings

contributes negligibly.

The contact resistance, other source of series resistance, becomes an issue since

it may not be controllable easily. For a two-terminal single-walled carbon nanotube

(SWNT) device, the ideal resistance with a perfect Ohmic contact is 6.5 kΩ. The

resistance bigger than the ideal value indicates that the contact between the metal

electrode and the tube is just imperfect. Depending on how good the contact is,

the electrical transport regimes of the nanotube devices are classified as a tunnelling

regime with a strong barrier, namely poor contacts or a ballistic regime with a weak

barrier, namely good contacts. For SWNT devices, the physical contact cannot be

be separated from the intrinsic tube device since it is determined during fabrication;


thus, a four- terminal configuration does not win over a two-terminal counterpart. On

the other hand, the resistance of quantum point contact (QPC) can be extracted well

from all spurious resistance factors by the four-probe measurement in the Hall bar

geometry. In this reasoning, we performed a two-terminal measurement for SWNTs

and a four-terminal measurement for QPCs.

Suppose an experiment to measure a resistance around (10 kΩ)−1 with a 1 nA

current-bias. The expected voltage value according to Ohm’s law is around 1 µV,

which is too small to be detected in regular voltmeters. Due to engineering ad-

vancement, measuring 10 µV is not even a challenge any more by using commer-

cially available high-sensitivity multimeters or placing a low noise preamplifier af-

ter a device. The limiting factors to ruin an accuracy are noise sources. Whereas

prevalent extrinsic noise sources are eliminated by proper grounding and shielding of

noise reduction techniques [76], intrinsic noise sources cannot be completely excluded

so that strategic approach should be made. Amongst unavoidable intrinsic noise,

Johnson-Nyquist thermal noise is concerned, as an example, in a finite resistance R,

vthermal =√

4kBTRB where vthermal is a rms thermal noise in a unit of Volts, kB

Boltzmann constant, T temperature, and B the equivalent noise bandwidth. With a

100 kHz-bandwidth voltmeter at room temperature, vthermal ∼ 4µV, corresponding

to 40 % error and SNR = 20 log10 (10µV/4µV) ∼ 8 dB. The situation for the dG

case is even worse since an ac signal is supposed to be at least ten or hundred times

smaller than the dc value, thus the thermal noise in a wide frequency range exceeds

completely over the interesting signal, yielding SNR = 20 log10 (0.1µV/4µV) ∼ −32

dB. Given a R value, SNR enhancement can be achieved by decreasing T (cooling)

and B (narrow-band detection).

4.1. CONDUCTANCE 59

(a)

(b)

10-18

10-17

10-16

10-15

10-14

10-13

en

(V2/

Hz)

3 4 5 6 7 8

102 3 4 5 6 7 8

1002 3

T (K)

R = 10 kΩ

R = 100 kΩ

R = 1 MΩ

R = 10 MΩ

10-18

10-17

10-16

10-15

10-14

en

(V2/

Hz)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

T = 293 K

T = 77 K

T = 4 K

T = 1 K

(a)

(b)

10-18

10-17

10-16

10-15

10-14

10-13

en

(V2/

Hz)

3 4 5 6 7 8

102 3 4 5 6 7 8

1002 3

T (K)

R = 10 kΩ

R = 100 kΩ

R = 1 MΩ

R = 10 MΩ

10-18

10-17

10-16

10-15

10-14

en

(V2/

Hz)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

T = 293 K

T = 77 K

T = 4 K

T = 1 K

Figure 4.2: (a) Johnson-Nyquist noise vs. temperature for different resistance. (b)Johnson-Nyquist noise of resistors at various temperatures: room temperature (293K), liquid nitrogen temperature (77 K), liquid helium temperature (4 K) and 1 K.


(a)

(b)

Figure 4.3: (a) The 4 K home-made dipper and (b) Oxford Helium 3 sorption cryostat

4.1. CONDUCTANCE 61

Cooling Technique

Cooling is essential not only for a sake of thermal noise reduction but also to attain

degenerate electrons in mesoscopic conductors which is deeply related to the Fermi-

Dirac distribution at a given temperature. Figure 4.2 demonstrates the former role

of cooling that Johnson-Nyquist noise en = 4kBTR (V2/Hz) is dramatically reduced

at low temperatures. It is straightforward to cool down at 77 K and 4 K by simply

putting the system into appropriate cryogen, liquid nitrogen and liquid helium 4

(4He). Further temperature drop requires a well-thought design of cryostat. Pumping4He in a finite volume V reduces temperature down to ∼ 1.5 K, which is expected from

the ideal gas law PV = nRgT , the relation of pressure P and V and T with universal

gas constant Rg for n moles of ideal gas. Below 1.5 K, the isotope of helium 4, 3He

with a lighter mass due to one neutron deficiency is introduced. Condensation of3He gas and succeeding vapor pressure reduction bring the system temperature down

to 300 mK. As even lower temperature is required in transport measurements, the

dilution refrigerator is utilized, whose operation principle is based on natural phase

separation of the mixture of 3He and 4He below 700 mK [83]. The base temperature in

the dilution refrigerator remains continuously around 10 - 50 mK. The aforementioned

cryostats are commercially available even with magnet inside. The measurements

presented in succeeding chapters were done in a home-made 4 K dipper and a Helium

3 cryostat (Fig. 4.3).

Lock-in Detection

Lock-in amplifier often prevails over low-noise voltage or current preamplifier for a

weak signal. Its benefit lies on the concept of phase-sensitive detection [75]. Basically,

a signal is modulated at low frequency less than 100 Hz and a output signal at this

particular frequency is only captured by comparison to the same frequency reference.

In this way, the bandwidth as narrow as 10 mHz can be realized. All experimental

data in this dissertation were taken by 22 Hz modulation with SR830 lock-in amplifiers

and SR554 transformer preamplifier. The transformer preamplifier helps to reduce

a lock-in amplifier’s input noise with a gain of 100 or 500. This method applies


10-8

10-7

10-6

10-5

10-4

10-3

V

(V

)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

dc Voltage

ac Voltage

BW = 100 kHz BW = 1 Hz

(a)

(b)

10-7

10-6

10-5

10-4

10-3

V

(V

)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

dc Voltage

ac Voltage T = 293 K T = 4 K

10-8

10-7

10-6

10-5

10-4

10-3

V

(V

)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

dc Voltage

ac Voltage

BW = 100 kHz BW = 1 Hz

(a)

(b)

10-7

10-6

10-5

10-4

10-3

V

(V

)

104

2 3 4 5 6 7 8

105

2 3 4 5 6 7 8

106

R (Ω)

dc Voltage

ac Voltage T = 293 K T = 4 K

Figure 4.4: (a) Estimated dc (dot) and ac (square) voltages at a 1 nA current biasto a variable resistor together with Johnson-Nyquist noise measured by a 100 kHzequipment at room temperature and 4 K. Assume ac signal is hundred times smallerthan the dc value. (b) Johnson-Nyquist noise with a 1 Hz bandwidth at 4 K.

4.2. LOW-FREQUENCY SHOT NOISE 63

to any intrinsic noise reduction since the noise contribution is proportional to the

measurement bandwidth.

Combination of two methods boosts the ac signal up above the noise by two orders

of magnitude shown in Fig. 4.4 , enabling us to obtain the signal reliably. SNR for

both conductance and differential conductance is greatly improved by ∼ 80 dB and

∼ 40 dB respectively for a 10 kΩ resistor with a 1 Hz bandwidth detection at 4 K.

4.2 Low-frequency Shot Noise

The rationale described in conductance section applies to shot noise measurement

setup as well. As the first step, the signal and the noise are identified and SNR is

estimated accordingly whether present strategies are sufficient for accurate measure-

ment outcome. The signal carrying out useful information in this case is shot noise

and the remaining noise sources arise from thermal fluctuations and the amplifier

noise, assuming the extraneous noise sources are already well taken care of by the

noise reduction techniques.

For a maximum value of possible SNR, suppose the noise portion is small and the

signal is large such that the amplifier noise is zero and the value of current fluctuations

is the full shot noise since the mesoscopic conductor’s shot noise is suppressed due to

correlations among charge carriers. Then Eq. (4.1) in this extreme case is

SNR = 20 log10

( √2eIBR√

4kBTRB

)

.

Putting into some real values, consider that R equals to 20 kΩ and an average current

I is around 1 nA at 4 K. The estimated SNR is 20 log10(3.57 · 10−10/2.1 · 10−9) ∼ −15

dB. Since two noise sources are sharing the same bandwidth, there is no significant

gain by reducing the bandwidth. Certainly, cooling helps to enhance SNR; however,

the above value is at the best case. In other words, if we include the amplifier noise

and the suppressed shot noise, SNR is even lower than -15 dB. Therefore, other

strategic movements should be incorporated, which are AC modulation, a cryogenic

amplifier, resonant tank circuit design described below.


10-10

10-9

10-8

10-7

10-6

I (A)

103

104

105

106

107

R (

Ω)

Full shot noise dominant

Thermal noise dominant

10-21

10-20

10-19

10-18

10-17

10-16

10-15

SV

(V

2 /H

z)

10-8

2 3 4 5 6 7

10-7

2 3 4 5 6 7

10-6

I (A)

SI = 2eI, S

V= S

I(R)

2

SI(10k)2 SI(100k)

2

SI(1M)2 S

I(10M)

2

Sthermal = 10 kΩ

Sthermal = 100 kΩ

Sthermal = 1 MΩ

Sthermal = 10 MΩ @ 4 K(a)

(b)

10-10

10-9

10-8

10-7

10-6

I (A)

103

104

105

106

107

R (

Ω)

Full shot noise dominant

Thermal noise dominant

10-21

10-20

10-19

10-18

10-17

10-16

10-15

SV

(V

2 /H

z)

10-8

2 3 4 5 6 7

10-7

2 3 4 5 6 7

10-6

I (A)

SI = 2eI, S

V= S

I(R)

2

SI(10k)2 SI(100k)

2

SI(1M)2 S

I(10M)

2

Sthermal = 10 kΩ

Sthermal = 100 kΩ

Sthermal = 1 MΩ

Sthermal = 10 MΩ @ 4 K(a)

(b)

Figure 4.5: (a) Shot Noise and Thermal Noise crossover. (b) The threshhold R andI.


AC Modulation

in

V1

Rd

en

Lock

In( ) 2

V2 V3 V4

V

(a)

10

8

6

4

2

0

V (

mV

)

0.140.120.100.080.060.040.020.00

time (s)

(b)

in

V1

Rd

en

Lock

In( ) 2Lock

In( ) 2

V2 V3 V4

V

(a)

10

8

6

4

2

0

V (

mV

)

0.140.120.100.080.060.040.020.00

time (s)

(b)

Figure 4.6: (a) The circuit diagram of AC modulation scheme. (b) The square-wavevoltage in time.

Since the shot noise exists only when the average current flows (non-equilibrium

condition), AC full modulation of voltage bias, in principle, gets rid of unmodulated

noise, for example, thermal noise or amplifier noise together with the lock-in tech-

nique. Consider a simple conductor which has current fluctuations in (shot noise)

and voltage fluctuations en from thermal noise and amplifier noise shown in Fig. 4.6


(a). The voltage applying to the conductor to generate in is a square wave in time

domain at ωm (Fig. 4.6 (b)), and 100% modulation brings the offset to V = 0. Thus,

the bias voltage pattern indicates the conductor is biased on and off at frequency ωm,

resulting in a on-off in at the same frequency. Mathematically, a square wave consists

of harmonics of sine waves, thus time-dependent AC full modulation voltage bias and

in are written as,

V (t) =Vpp

2+Vpp

2

4

π

( ∞∑

n=1

1

(2n− 1)sin((2n− 1)ωmt)

)

=Vpp

2

(

1 +4

πsin(ωmt) +

4

3πsin(3ωmt) + ...

)

,

(4.2)

in(t) = in(0)

(

1 +4

πsin(ωmt) +

4

3πsin(3ωmt) + ...

)

, (4.3)

where Vpp is a peak-to-peak voltage value and in(0) is the device shot noise; for

example, in(0) =√

2eI for the full shot noise.

Now we can compute the values of Vi at each node i in Fig. 4.6(a):

V1 = en + in(t) ·Rd,

V2 = Av(V1) = Av(en + in(t) ·Rd),

V3 = V2 ·√B,

V4 = V 23 = A2

v(en + in(t) ·Rd))2B. (4.4)

Note that the dimension of V1 and V2 are [V/√

Hz] since en and in(t) are the square

of the noise power spectral density, but V3 after the band pass filter becomes [V] with

noise-equivalent bandwidth B. Av is the voltage gain of the amplifier. V4 is the value

proportional to the square of V3 in the unit of [V2]. The final readout of V at the

lock-in amplifier is from the demodulation of lock-in reference signal sin(ωmt). What

the demodulation at the lock-in does is to bring the component at sin(ωmt) to the

DC. Therefore, the surviving term in the end is the sin(ωmt) part. Plugging Eq. (4.3)


into Eq. (4.4) and assuming en and in are not correlated, V4 becomes

V4 = A2vB(e2n + (in(t) ·Rd)

2)

= A2vB[(e2

n + (in(0) ·Rd)2(1 +

(

4

πsin(ωmt)

)2

+

(

4

3πsin(3ωmt)

)2

+ 2

(

4

πsin(ωmt)

)

+ 2

(

4

3πsin(3ωmt)

)

+ 2

(

4

πsin(ωmt)

)(

4

3πsin(3ωmt)

)

+ ...)].

(4.5)

The relevant term for the demodulation is

A2vB(in(0) ·Rd)

22

(

4

πsin(ωmt)

)

,

which contains only the shot noise current fluctuations. Recall that sin2(ωmt) =1−cos(2ωmt)

2, the voltage at the lock-in is

V = A2vB(in(0) ·Rd)

22

(

4

πsin(ωmt)

)

sin(ωmt)

= A2vB(in(0) ·Rd)

22

(

4

π

1 − cos(2ωmt)

2

)

,

(4.6)

and the value of V measured is the dc component, Vout = A2vB(in(0) ·Rd)

2( 4π).

Cryogenic Amplifier

The benefits of placing amplifier close to the device are mainly two-fold: (1) to reduce

contamination to the signal from the device along the wirings between the device and

the amplifier. (2)to reduce the thermal noise of a transistor in the amplifier. We

used metal-semiconductor field effect transistors (MESFET) constructed in GaAs,

whose charge carriers are not frozen at low temperatures unlike Si. Plus, MESFETs

are advantageous due to a couple of features: high mobility of the carriers and high

transit frequency. The performance of a transistor depends on the bias points of

drain-source voltage VDS and gate-source voltage VGS as the source is referenced as a

ground. Figure 4.7 presents the IDS versus VDS at different VGS. The IDS−VDS trend


30x10-3

20

10

0

I DS (A

)

543210 VDS (V)

SONY MESFET

40x10-3

30

20

10

0

ID

S (A

)

2.52.01.51.00.50.0

VDS (V)

Fujitsu MESFET

(a)

(b)

30x10-3

20

10

0

I DS (A

)

543210 VDS (V)

SONY MESFET

40x10-3

30

20

10

0

ID

S (A

)

2.52.01.51.00.50.0

VDS (V)

Fujitsu MESFET

(a)

(b)

Figure 4.7: (a) SONY and (b) Fujitsu FSU01LG MESFET bias response at roomtemperature.


can be divided into three sections: Section I is where IDS increases as VDS increases

below 1 V; Section II is a upper right part in which IDS becomes constant for lower

absolute values of VGS; Section III is where IDS saturates around larger negative values

of VGS. In both Section II and III, the transistors have high gain, but they have high

power in Section II and low power in Section III. Meanwhile, when the transistors are

biased in Section I, they have reasonable gain but low power and low noise. Although

we cannot take advantage of high gain in Section I, achieved low power and low noise

are a good trade-off for a stable transistor performance.

Resonant Tank Circuit

In the shot noise measurements, there are several energy scales: thermal energy kBT ,

bias energy eV and measurement energy ~ω, which are relevant to thermal noise, shot

noise and quantum noise. In order to measure the shot noise, the following condition

should be satisfied, ~ω < kBT < eV . The lower bound of the measurement frequency

choice is limited by a 1/f noise corner frequency. A tank circuit is inserted after a

device to choose the measurement frequency at the resonant frequency of the circuit

ω = 1√LC

with a inductance L and a capacitance C. A simple tank circuit has two

degrees of freedom, resonant frequency ω and a quality factor (Q). We used the tapped

inductor tank circuit in order to introduce additional degree of freedom, impedance

matching because the optimal input impedance of a transistor is around 1 - 2 kΩ

lower than the device impedance ∼ 10 - 100 kΩ. A nice thing of this tank circuit

design is that it absorbs parasitic capacitance from coax cables, chip sockets and pc

boards and capacitance value change at different temperatures, shifting a resonant

frequency accordingly. We selected the adequate values of passive components for an

aimed frequency at low temperature around 10 - 20 MHz, which is high enough where

1/f noise is highly suppressed and ~ω ∼ 40 - 80 µ eV much less than kBT ∼ 0.3 meV

at 4 K and eV ∼ 1 meV.


Figure 4.8: The photograph of a cryogenic amplifier and resonant tank circuit usedin the single-walled carbon nanotube show noise experiments.

4.3 Summary

This chapter has discussed experimental schemes and techniques : measurement con-

figurations, bias schemes, cooling techniques, lock-in detection, AC modulation, cryo-

genic amplifiers, and a resonant tank circuit. These are practically used in actual

electrical measurements for conductance and shot noise properties towards the reli-

able and accurate measurement data. The outcomes and interpretations of applying

the techniques to specific conductors, single-walled carbon nanotubes and quantum

point contacts are discussed in two subsequent chapters.

Chapter 5


We shall not cease from exploration.

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time.

− T. S. Eliot

The following two chapters present the direct application of the previously ac-

quired knowledge to actual systems : one class is single-walled carbon nanotubes and

the other is quantum point contacts in semiconductors. Both systems share common

features such that they are regarded as (quasi-) one-dimensional electron waveguides.

Chapter 5 is devoted to experimental and theoretical study of low-temperature trans-

port measurements through carbon nanotubes. The content starts off from an in-

troduction to single-walled carbon nanotubes, including band structure and device

fabrication and synthesis followed by experimental data of nanotube devices and the-

oretical interpretations of them.

71

72 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES

5.1 Single-Walled Carbon Nanotubes

The discovery of single-walled carbon nanotubes (SWNTs) dating back to the be-

ginning of the 1990s has set out a new research trend widely spread over numerous

scientific and engineering fields. SWNT research themes evolve during the last two

decades, giving a clear hint that there is plenty of room to explore SWNTs in future.

The incessant interests on SWNTs arise from their distinct characteristics - in chem-

ical, mechanical, electrical, and optical aspects - and their potential contributions to

molecular electronics and optoelectronics. Almost defect-free surface, relatively easy

growth condition, and high geometric ratio of transverse dimension to longitudinal

length make SWNTs an ideal one-dimensional (1D) system. Therefore, SWNTs be-

come attractive study materials for low-dimensional physics in which strong electron-

electron interactions are unavoidable; and the innate tininess may allow SWNTs to

be a prospective replacement of silicon (Si)-based electronics.

It is a breakthrough in the SWNT research area that isolated individual SWNTs

were synthesized on Si-wafer with a reasonable yield by chemical vapor deposition

(CVD) in 1998 [15], enabling scientists and engineers to imagine limitless dream and

administer diverse tests. In addition, A part of reasons to achieve dense knowledge of

SWNTs in a concentrated period is the adaptation of Si-technology to SWNT-devices,

which has been well-established and optimized in a great extent. Indeed many en-

gineering components have been developed including sensitive chemical sensors [33],

intrajunction diodes [24], p-n junction [84], field-effect transistors [21], and single elec-

tron transistors [20, 22, 23]. Practically, various essential logic gates - AND/NAND,

OR/NOR, SRAM, inverter, ring oscillator - for memory storage and bit manipulation

for computation with superior performance over existent CMOS -electronics in terms

of low threshold and power consumption, implying a bright future [85,86].

Out of manifold approaches to assess characteristics, transport experiments have

been a straightforward and initial probe to extract thermal and electronic properties

arising from electrical and heat conduction by valence electrons and/or phonons in

device level once fabrication steps to place metal electrodes either on top of or under

5.1. SINGLE-WALLED CARBON NANOTUBES 73

carbon nanotubes were developed [19]. In particular, the electron transport measure-

ments have actively been performed all over the world last decade and have revealed

fascinating phenomena both in physics and in engineering aspects. In the contri-

butions to the fundamental physical knowledge, novel electronic properties through

SWNTs have been revealed such as Coulomb blockade oscillation [20], the Kondo

effect [25], ballistic quantum interference [16,17], and Tomonaga-Luttinger liquid be-

havior [23].

In this section, the basic knowledge about SWNTs nanotubes is introduced from

the bandstructure calculation and the different types of SWNTs are presented. And

the synthesis of SWNTs and the fabrication of SWNT devices are explained at the

end.

5.1.1 Electronic Band Structure

The carbon nanotube field was first initiated by identifying cylindrical objects ( now

called multi-walled carbon nanotubes ) in entangled carbon soots [2]. Immediately

theorists picked up the subject and moved rapidly to model this new system by sim-

plifying a one-shell tubule even before SWNTs were discovered in the lab yet. The

swift theoretical understanding of new materials was possibly obtained because their

mother material, the two-dimensional (2D) graphite or graphene, had been a long-

time subject in condensed matter physics. Regardless of having no idea how to form

such tubule things at lab benches, theorists envisioned that they would be shaped by

rolling one- or more than one- sheet of graphene seamlessly. Extra confinement in two

dimensions by this roll-up requires to satisfy a periodic boundary condition around

the circumferential direction and leaves room to have only one freely propagating di-

rection. The task became to apply the new boundary condition to the band theory of

graphene based on a tight-binding calculation of unpaired π-orbital [5]. This yielded

an appropriate band structure of SWNTs in 1992 by several groups [3, 4]. Following

the historical path of obtaining SWNT bandstructure, the beginning step is to study

graphene band structure.


a1

a2

O A B

a1

a2

O A B

x

y

x

y

r1

r2

r3

r1

r2

r3

kx

ky

kx

ky

K

K′

b1

b2

M

Γ

(a)

(b) (c)

Figure 5.1: (a) Graphite lattice structure. (b) The direct and (c) the reciprocal lattice

space of graphene with unit vectors ~ai, ~bi and translational vectors ~ri.


Graphene

Graphene is a name to indicate only one sheet of three dimensional graphite which

contains many layers of sp2 carbon hexagons shown in Figure 5.1(a). Considering

the fact that the spacing between layers (3.35 A) is longer than the neighbor atom

distance aC−C (1.42 A) within the same plane, thus two dimensional graphene can

exist in nature. Unlike the previous statement, it has been a challenging to isolate one

layer from bulk graphite. Recently several groups succeeded to construct graphene

reproducibly by mechanical exfoliation, which fires the second boom of graphene

research [34,35].

Figure 5.1 presents the direct and reciprocal lattice structures of graphene, show-

ing a hexagonal Brillouin Zone (BZ). For consistency, a horizontal axis in figures

always represents ~x and a vertical one does ~y. Two unorthogonal unit vectors ~a1 and

~a2 form a unit cell. The unit cell contains two carbon atoms A and B in the direct

lattice space. A single carbon atom couples to three nearest neighbor carbon atoms

and their corresponding translational vectors are denoted as ~ri where i goes 1, 2, and

3. Similarly, two unit vectors ~b1 and ~b2 are readily found from the direct lattice unit

vectors in 2D, satisfying ~bi · ~aj = 2πσij by definition. The (x, y) coordinates of these

unit vectors from the origin O = (0, 0) are specified with the size of the unit vectors

a = |~a1| = |~a2| =√

3ac−c,

~a1 = (a

√3

2, a

1

2),

~a2 = (a

√3

2,−a1

2),

~b1 = (2π√3a,2π

a),

~b2 = (2π√3a,−2π

a),


~r1 = (−a√

3

6, a

1

2),

~r2 = (−a√

3

6,−a1

2),

~r3 = (a1√3, 0).

In momentum k-space (Fig. 5.1 (c) ), the high symmetry points Γ (the center of the

first BZ), K, K ′ (vertices of a hexagon) and M (a middle point in the side of the

hexagon) are indicated. Note here that all vertices of the first BZ are inequivalent.

They are distinguished by either K or K ′. This inequivalence can be easily checked

by rotating each vertex by 2π/3. And the alternate order of K and K ′ are easily

assigned. The unit cell in the BZ contains two vertices K and K ′.

Let us restrict our focus on the unit cell and the first BZ. A Hamiltonian H is writ-

ten using the tight-binding calculation. Tight-binding theory is a good approximation

of atoms in well-defined lattice sites. It describes the effect on a localized electron in

an atom due to the existence of nearest neighbor entities perturbatively [60]. It is a

well-known fact that three valence electrons per each carbon atom form sp2 hybridiza-

tion. The chemical properties are described by the remaining unpaired π electron in

a 2 pz orbital. Thus, the Hamiltonian H can be expressed by a simple 2-by-2 matrix

for the unpaired electron in two carbon atoms in the unit cell. Obviously, the bases

are pz atomic orbitals of carbon atom A and B denoted as Φi(~k, ~r) where i = A and

B. These wavefunctions should satisfy Bloch’s theorem due to periodicity or transla-

tional symmetry i.e. Φi(~k, ~r) = Φi(~k, ~r+~a) where ~a is a translation vector. The form

of H is

H =

(

HAA HAB

HBA HBB

)

,

where HAA = HBB = ǫ2p and HAB = t∑

~riei~k·~ri . ǫ2p is the pz orbital energy and

it is often set to 0 for convenience. t is the transfer integral between two atoms A

and B. Conventionally, t is negative and ~ri is a nearest neighbor translation vector.


Therefore, the matrix elements are explicitly rewritten as

H =

(

ǫ2p tf(~k)

tf∗(~k) ǫ2p

)

where f(~k) = ei~k·~r1 + ei~k·~r2 + ei~k·~r3 .

The remaining tasks are to obtain eigenvectors and eigenvalues by solving a secular

equation, det(H−EI) = 0 where I is a 2-by-2 identity matrix. The two eigenenergies,

E1 and E2 are ~k-dependent. The corresponding eigenfunctions Ψ1 and Ψ2 are a linear

combination of the two bases vectors:

Ψ1 ≡(

CA1

CB1

)

= CA1ΦA(~k) + CB1ΦB(~k),

Ψ2 ≡(

CA2

CB2

)

= CA2ΦA(~k) + CB2ΦB(~k).

Chemists have special terminologies for orbital bonding. When CA/CB = +1,

two pz orbitals face each other in the same direction, technically called ‘π-bonding’

whereas the negative -1 state forms ‘π∗-bonding (anti-π bonding)’

Generally the overlap matrix S of orbitals is not zero and needed to be considered

as well since the two wavefunctions are not completely isolated from each other,

S =

(

1 sf(~k)

sf ∗(~k) 1

)

where s is the overlap integral between two Bloch wavefunctions. In the Slater-Koster

scheme, s is set to 0 for a simple approximation of the band structure calculation in

graphite.

When both ǫ2p and s are assumed to be zero, eigenvalues are

Ei(~k) = ±t√

∣

∣

∣f(~k)

∣

∣

∣

2


where i = 1 and 2. The interesting feature in Ei(~k) versus ~k is that a gapless disper-

sion at vertices of the first BZ occurs. That is why graphene is semi-metals shown in

Fig. 5.2. Also note that near EF , the dispersion is linear so that it becomes massless

Dirac dispersion. The related phenomena due to massless Dirac dispersion in the

quantum Hall regime have been reported with graphene recently [34,35].

Figure 5.2: Energy band structure of Graphene.


Figure 5.3 describes an imaginative way to form SWNTs by placing a line OB over

a line AB′. There are infinite ways to roll up the graphene sheet in principle. A

specific SWNT among many possible configuration is identified by two orthogonal

vectors, chiral vector ~Ch and the translation vector ~T . Two vectors are linked to the

graphene unit vectors as a linear combination with positive integers n and m such as

~Ch = n~a1 +m~a2. Since hexagonal symmetry sets that in the regime of 0 < |m| < n,

all possible SWNTs configurations appear. Therefore, a group of coefficients (n,m)

classifies SWNTs. This notation is very convenient: once (n,m) is determined, all the

relevant parameters can be readily computed, for example, ~T , the chiral angle θ, the


number N of hexagons per unit cell , the diameter of a tube dt, and corresponding

reciprocal vectors ~K1, ~K2.

Figure 5.3: SWNT geometry on the graphene lattice structure. Chiral vector ~Ch isdrawn for a specific case, a (4,2)-SWNT.

A periodic boundary condition along the chiral vector or ~K1 quantizes reciprocal

vectors in infinitely long SWNTs so that the true reciprocal lattice vectors are in the

direction of ~K2. The reciprocal vectors are, therefore, represented as k ~K2/∣

∣

∣

~K2

∣

∣

∣+µ ~K1

where k is a continuous-variable wavenumber in the longitudinal direction. Since the

number of reciprocal vectors is N , the reciprocal vector index µ takes an integer value

from 0 to N − 1 in a unit cell.

General expressions of the aforementioned parameters for a (n,m) tube are sum-

marized as follows:

• ~Ch = n~a1 +m~a2 ≡ (n,m) where 0 < |m| < n,

• dR = gcd(2n+m, 2m+ n) where ‘gcd’ means the greatest common divisor,

• ~T = t1 ~a1 + t2 ~a2 where t1 = n+2mdR

and t2 = −2n+mdR

,

• cos θ =~Ch· ~a1

| ~Ch|| ~a1|= 2n+m

2√

n2+nm+m2

• N =| ~Ch×~T || ~a1× ~a2| = 2(n2+nm+m2)

dR


• dt =| ~Ch|

π= a

π

√n2 + nm+m2

• ~K1 = 1N

(−t2~b1 + t1~b2),

• ~K2 = 1N

(m~b1 − n~b2)

•∣

∣

∣

~K2

∣

∣

∣= 2π

|~T |

Inserting the relations of ~K1 and ~K2 into the E(~k) obtained in the above, the band

structure of SWNTs are analytically calculated. Notice that according to where the

one-dimensional BZ of SWNTs meets the graphene BZ, SWNTs exhibit metallic or

semiconducting electronic properties. Two extreme cases are considered as examples:

Armchair (n,n) SWNTs for a truly metallic case and zigzag (n,0) SWNTs for either

metallic or semiconducting cases. The denotation of armchair and zigzag comes from

the shape of carbon hexagon array along the SWNT surface.

Armchair((n,n)) Single-Walled Carbon Nanotubes

Armchair SWNTs are special in a sense that ~Ch is in x direction from the same

coefficients of ~a1 and ~a2 due to n = m. Consequently, allowed reciprocal vectors are

parallel to y direction (Fig. 5.4). The 1st BZ has a length of 2π/a, independent of

index n. Note that the energy band of µ = n is the lowest state in which ~K2 passes

through inequivalent vertices of the hexagon. This µ value comes from the ratio of

two length scales |ΓM | and∣

∣

∣

~K1

∣

∣

∣.

The above parameters are reduced to have simple expressions:

• Ch = n~a1 + n~a2 ≡ (n, n),

• dR = 3n,

• T = t1 ~a1 + t2 ~a2 where t1 = 1 and t2 = −1,

• cos θ =√

32,

• N = 2n


a1

a2

O A

x

y

Ch

T K

K’

K2

K1kx

ky

b1

b2

(a) (b)

(c)

¡B

a1

a2

O A

x

y

x

y

Ch

T K

K’

K2

K1kx

ky

kx

ky

b1

b2

(a) (b)

(c)

¡¡B

Figure 5.4: Armchair SWNT real (a) and reciprocal (b) lattice space. (c) (10,10)energy band structure. The first BZ of SWNT is indicated by two thick vertical lines.


• dt = aπ

√3n

• ~K1 = 12n

(~b1 + ~b2)

• ~K2 = 12(~b1 − ~b2)

•∣

∣

∣

~K2

∣

∣

∣ = 2πa

The case n = 10 is computed as a specific example. SWNTs with the (10,10)

configuration are obtained dominantly in laser ablation synthesis according to the lit-

erature. The bandstructure clearly shows that there is no gap at the Fermi level since

the lowest energy band always crosses at the vertices within the 1st BZ. Therefore,

the finite density of states at the Fermi level confirm that armchair tubes are metallic.

In addition, the real and imaginary coefficients of eigenfunctions assign the π and π∗

bands without confusion. The crossing bands at the Fermi energy are orthogonal,

therefore, the interband scattering is blocked by symmetry. This explains well that

metallic tubes do not suffer from backscattering processes near kF and they have long

mean free paths [87].

Zigzag((n,0)) Single-Walled Carbon Nanotubes

Zigzag SWNTs have quantized reciprocal vectors placed with a spacing of∣

∣

∣

~K1

∣

∣

∣. They

are perpendicular to a line between Γ and K shown in Fig. 5.6. These vectors are

along a ~b2. Let us compare two length scales.

|ΓK|∣

∣

∣

~K1

∣

∣

∣

=

(

4π3a

)

(

2πan

) =2n

3. (5.1)

Equation (5.1) differentiates two groups of (n,0) SWNTs. As n is a multiple of 3,

a reciprocal vector overlaps with one of vertices in a hexagonal BZ. While another

group exists when the ratio is not an integer because n is not a multiple of 3. Then,

a reciprocal vector misses any of vertices K or K′, creating the energy band gap.

Depending on n values, zigzag tubes are either metallic (n = 3q where q is a positive

integer) or semiconducting. Semiconducting tubes have a direct band gap at a certain


K’K

Γ

π

*π

*π

π*π

π

(a)

(b)

(c)(c)

(c)

Figure 5.5: (a) The µ = 0 band for a (10,10) SWNT along symmetry points Γ, K, andK ′. The π, π∗ wavefunctions are clearly denoted based on the energy band coefficients.The real and imaginary coefficients of upper energy band (b) and lower energy band(c) for µ = 0. The dotted line is the zone boundary of graphene.


a1

a2

O A

x

y

T Ch K

K’

K2

K1

kx

ky

b1

b2

(a) (b)

(c)

B ¡

a1

a2

O A

x

y

T Ch K

K’

K2

K1

kx

ky

kx

ky

b1

b2

(a) (b)

(c)

B ¡¡

Figure 5.6: A zigzag SWNT unit cell in a real (a) and a reciprocal (b) lattice space.(c) Semiconducting zigzag (10,0) energy band structure. The first BZ of SWNTis indicated by two thick vertical lines. (d) µ = 6 band for (10,0). The real andimaginary coefficients of upper energy band (e) and lower energy band (f) for µ = 6.


(a)

(b)

(c)

(a)

(b)

(c)

Figure 5.7: (a) The µ = 6 band for a (10,0) zigzag SWNT. The real and imaginarycoefficients of upper energy band (b) and lower energy band (c) for the µ = 6 band.


wavenumber k. The gap energy is typically about 1 eV or so and its size is inversely

proportional to a diameter (d), Eg ∝ eV/d. The fact that they are direct bandgap

materials and the bandgap 1 eV is close to the communication optical wavelength

is advantageous for optoelectronic and photonic devices as engineering applications

in future. Figure 5.7 shows a particular case for n = 10, which corresponds to a

semiconducting zigzag tube. In this case, the assignment of π and π∗ is not clear

since the bands are actually mixed and expressed as a superposition of two bands

plotted in Fig. 5.8. This reason makes semiconducting SWNTS to have shorter mean

free paths due to backscattering processes.

This rather simple theory with a (n,m) notation has enabled us to gain remarkable

insights into material structures and these knowledge have been confirmed by scanning

tunnelling microscopy experiments [13,88].

5.1.2 Synthesis and Fabrication

Discovering multi-walled carbon nanotubes (MWNTs) seems fortuitous in a carbon

arc-discharge chamber which was designed to the production of fullerenes [2]. Two

years later, SWNTs were found by the arc-discharge method similar to MWNTs this

time except for adding catalytic components in the chamber [8]. Although these

works sparkled scientists and engineers’s interest significantly enough to establish

a huge community of theoretical and experimental research on nanotubes, efficient

synthesis methods have been on demand in order to isolate nanotubes, to grow specific

nanotubes, and to build up refined devices to investigate novel quantum phenomena

in one dimension for quantitative assessment. Out of three major approaches to

synthesize SWNTs using catalytic nanoparticles, electric arc-discharge, laser ablation

and chemical vapor deposition (CVD), the CVD method has been superior to produce

high-quality SWNTs.

Synthesis: Chemical Vapor Deposition

CVD refers to a chemical process to deposit a thin-film or dense structures like pow-

ders or fibers on substrates using gaseous reactants. The basic principle of operation is


to flow a gas phase of elements or compounds into a substrate, on which the supplied

gas will undergo thermal reaction and will be decomposed while reaction byprod-

ucts are flushed out. Its versatility of any element or compound, high-purity, high

density, low cost, simplicity and flexibility to variations explicate the wide-usage and

numerous forms of CVD in semiconductor industries. Diverse CVD systems share

four common structures: a reaction vessel, a source of reactants, a substrate, and an

exhaust system for byproduct removal.

The CVD chamber which was used to grow SWNT for devices presented in this

dissertation also consists of a similar structure ( Fig. 5.8 (a)): a 1-inch diameter tube

vessel inserted into a furnace, sources of gas CH4, H2 and Ar, a Si-substrate con-

taining catalyst islands and an exhaust system. Using iron based alumina-supported

catalyst, SWNTs were grown with a carbon feedstock, 99.999 % CH4 and H2 at right

concentration for 5 - 7 minutes at 900 - 1000 oC, followed by a Ar flush and a cool-

down to room temperature. The success to synthesize high-yield of SWNTs near

the catalyst islands with low resistance has advanced the SWNT research field into

ballistic transport studies and prototype of nanotube-electronics [15].

The synthesis mechanism in the catalytic CVD method is associated with the

details of nanoparticles since the catalytic nanoparticles are essential to form the

SWNTs unlike then MWNTs. Recently, Li et al. have attempted to assess the role

of catalysts. The authors have showed that the diameter of SWNTs indeed closely

linked to the nanoparticle size in terms of statistical analysis [89]. According to

this report, the synthesis can be understood in three stages: first, nanoparticles as

catalysts absorb decomposed carbon atoms from CH4 or other carbon feedstock in the

CVD process; second, the absorption of carbon atoms to nanoparticles would continue

until the saturation. Once it reaches the saturation, carbon atoms become to grow out

from the catalysts with a closed-end; third, an excess carbon supply adds to carbon

precipitation on surface and it yields finite-length nanotubes in the end. Figure 5.8

(b) illustrates such speculative synthesis mechanism by computer simulations from

Professor K. J. Cho group. It is reasonable, therefore, that the SWNT diameter

would be determined by the nanoparticle size as the initial basis. Although Li et al.

[89] provided valuable information as to the microscopic level understanding of the


CH4

H2

Ar

valve

Furnace

Exhaust

(a)

(b)

CH4

H2

Ar

valve

Furnace

Exhaust

CH4

H2

Ar

valve

Furnace

Exhaust

(a)

(b)

Figure 5.8: (a) Schematics of CVD chamber. (b) The mechanism of SWNT growthfrom catalysts in CVD chamber adapted from K. J. Cho group.


synthesis in the catalytic CVD process, the complete controllability to produce tailor-

made SWNTs with an expected diameter, chirality, length, position and orientation

at will is yet to be far from reality. This is the SWNT fabrication challenge at

present. Once this goal is achieved, it is not difficult to imagine that SWNTs would

become ubiquitous in various applications as electrical, chemical, mechanical and

optical components.

Fabrication Processes

The configuration of SWNT devices this thesis studies resembles conventional semi-

conductor field-effect transistors, which have three terminals: source, drain and gate.

The fabrication goal is to produce three-terminal isolated SWNT nanotube devices

on top of a Si-wafer.

Figure 5.9 shows the steps of processes. The procedure has been optimized to have

a high yield of SWNTs devices appropriate for experimental purpose. The starting

material is a four-inch heavily doped p-type Si-wafer. It is substrate which serves

as a backgate. Due to multi-layer lithographic steps, first, global wafer marks and

chip marks should be placed in the blank wafer. These marks are very useful firstly

that the overlapping processes can be performed within the lithographic resolution

limit and secondly that they draw boundaries of chips in the whole wafer, where total

64 chips are located. Later the wafer is broken into each chip along the alignment

marks for further steps. In general the sizes of marks are 2 µm - features which are

large enough to be recognized easily. Those marks are patterned by a standard 1

µm-photolithography recipe developed in the Stanford Nanofabrication Facility. The

wafer is coated by a 3612 positive photoresist for 1 µm-thick resist layer in a svgcoat

machine, and it is exposed for 30-40 seconds using a EV aligner or Karl Suss exposers.

After developing at a svgdev track, the wafer is loaded in a Drytek for etching. It

etches the Si-substrate by 1 µm in depth. Descum process before etching to get rid

of residual photoresist around trenches would be of use in order to produce sharp

edges. The photoresist layer is removed at wet-bench non-metal. The marks are

basically engraved in the Si-wafer, they are robust and chemically inactive for any

further steps.


Thermal oxidation

Si

SiO2

PMMA

Si

SiO2

E-beam for catalyst islands

SWNT synthesis

Si

SiO2

PMMA

E-beam for electrodes

Si

SiO2

Metal deposition and Lift-off

Alignment marks Thermal oxidation

Si

SiO2

PMMA

Si

SiO2

Si

SiO2

E-beam for catalyst islands

SWNT synthesis

Si

SiO2

PMMA

Si

SiO2

Si

SiO2

PMMA

E-beam for electrodes

Si

SiO2

Si

SiO2

Metal deposition and Lift-off

Alignment marks

Figure 5.9: The schematics of SWNT-device fabrication processes.


The second step is the thermal oxidation on top of the marked Si-wafer. It is

a very critical step to avoid any possible impurities on wafers, since any dirts on

the wafers will lead to a leakage when devices are characterized. Therefore, before

inserting the wafer to a diffusion furnace named Tylan for thermal oxidation, the

wafer should be cleaned thoroughly and properly through the diffusion wet bench

process. The 0.5 -µm oxide layer is grown on the wafer. Again up to this point,

an extreme care to prohibit contaminations should be taken. With the oxide-layer,

the four-inch wafer is now broken into pieces for further delicate processes as a chip.

A rough dimension of chip is about 4 mm × 4 mm. The next task is to pattern

catalyst islands at specific locations for growing SWNTs. The size of catalyst isalnds

are about 5 µm × 5 µm and they are defined relative to chip marks by electron-beam

lithography (EBL) using Hitachi H-700 F11 Electron Beam System. The Hitachi

system has a minimum feature size around 70 nm reproducibly. One of common EBL

resist is polymethylmethacrylate (PMMA). For this step, a 5 % 495 K PMMA is

coated. Alumina and iron based catalyst material solution is dropped on the surface

and the chip is inserted into an oven for 5 minute to dry out the catalyst solution.

The PMMA layer is removed in acetone.

Nanotubes are synthesized by the aforementioned CVD method with methane

and hydrogen gas at 900 ∼ 100 o C at the furnace for 5 ∼ 7 minutes. Once the

furnace is cooled down to room temperatures, the growth yield is roughly checked

by atomic force microscopy (AFM) in order to know the distribution of tube density

near the catalysts. Since an isolated SWNT is in question, the low number of SWNTs

grown near the catalyst island is perferrable. The temperature and the duration of

the growth are slightly modified each time according to the tube density analysis.

The second EBL is proceeded for patterning metal electrods after PMMA coating,

exposure and development. This is followed by the metal deposition in the Innotec or

Dai-group metal evaporator. The choices of metals have been Titanium (Ti) only, Ti

and gold (Au), and palladium (Pd) only. The typical thickness of metals are around

50 ∼ 70 nm. The liftoff in acetone to remove PMMA is executed as the final step.

The portion of some devices in one chip out of the fabrication steps is imaged by

optical microscopy shown in Fig. 5.10(a). The total 98 individual devices are made


in one chip and figure 5.10 presents the zoom-in image of an individual device. The

two big squares are the top and bottom 100 µm × 100 µm pads for wire-bondings

and the two black middle squares are the catalyst islands. Two narrow thin lines are

electrodes on top of the tube to be source and drain terminals.

Room Temperature Characterization

As a chip containing 98 devices is ready to be tested from the fabrication processes

described above, it undergoes a couple of characterization steps at room temperature

(RT), which facilitate to filter out well-contacted devices with an individual metallic

SWNT before a cool-down. First, the I − Vds at a few Vg is measured of each device

at a probe station. According to the resistance (R) and its dependence on Vg, devices

whose R is between 10 ∼ 30 kΩ and which have weak or no dependence on Vg (metallic

behavior) are selected for the atomic force microscopy (AFM) imaging. The AFM

images determines the number of nanotubes across the source and drain electrodes

and the diameter of SWNTS (Fig. 5.10(c)). Nanotubes, whose diameters are 1.5 ∼3.5 nm from AFM images, are presumably regarded as SWNTs based on statistics.

AFM images cannot identify a SWNT, a double-walled nanotube or multi-walled

nanotube at all unlike transmission electron microscopy (TEM) which requires a

conducting substrate. The current device configurations for transport measurements

are inadequate to TEM, thus the choice of SWNTs relies on statistics of TEM results

with synthesized nanotubes on conducting substrates by the same recipe.

5.2 Differential Conductance

According to the RT characterization results, chosen SWNT-devices were wirebonded

at a chip socket. They were loaded in the dipper for a cool-down at 4 K. Figure 5.10(a)

shows the optical microscopy picture of several wirebonded devices in a chip. The

SWNT devices have a three-terminal geometry: source, drain and backgate whose

diagram is simplified in Fig. 5.11 (a). The Si substrate was used as the backgate. The

dimension of metal electrodes was aimed as around 200 nm, and the spacing between

two electrodes was determined by the EBL pattern, between 200 and 600 nm. The

5.2. DIFFERENTIAL CONDUCTANCE 93

(a)

(b)

(c)

500nm

100¹m

100¹m

(a)

(b)

(c)

500nm500nm

100¹m100¹m100¹m

100¹m100¹m

Figure 5.10: (a) Optical microscopy picture of portion of a chip containing wire-bonded devices, (b) a zoom-in view of an individual device, and (c) atomic forcemicroscopy image of an individual SWNT with patterned Ti/Au electrodes.


Ti/Au, Ti-only, and Pd metal electrodes were used, which featured low-resistance

contacts. The resistance of the selected metallic SWNT devices was typically 12 ∼50 kΩ at room temperatures and about 9 ∼ 25 kΩ at 4 K. Note that the resistance

of tube devices decreases as the temperature decreases. The trend of resistance and

temperature can be explained by phonons, which are frozen at low temperatures. It

is distinct from devices which are well isolated from the electrodes, in that resistance

becomes higher at low temperatures.

5.2.1 Quantum Interference

SWNTs well-contacted to two electrodes with finite reflection coefficients produce

quantum interference pattern in differential conductance ( dI/dVds ) at 4 K. Predic-

tions [87, 90] and experiments [16, 17, 20, 21] have shown that both the elastic and

the inelastic mean free path are at least on the order of microns in metallic nan-

otubes at low temperatures. Therefore, the electron transport within 200 ∼ 600

nm-long SWNTs is believed to be ballistic [16, 17]. Figure 5.11 (b) is a represen-

tative two-dimensional image plot of dI/dVds as a function of a drain-source voltage

Vds in y-axis and a backgate voltage Vg in x-axis. The color bar scale is renormalized

by 2GQ = 4e2/h. The interference pattern arises from the fact that SWNT has a

finite length determined by the electrode spacing. The diamond structures at low

Vds are an additional confinement along the longitudinal direction due to the poten-

tial barriers at the interfaces with two metal electrodes. The confinement quantizes

energy levels and the energy spacing between maxima corresponds to ∆E = ~vF/L.

The Fermi-velocity of SWNT is adopted from the value of graphite Fermi-velocity,

8 × 105 m/s. The oscillations are measured from a 360 nm-long SWNT. The cor-

responding energy spacing is ∆E ∼ 10 meV. The size of the pattern is consistent

with the energy spacing in a good agreement with the experiment results. Liang et

al. reported similar interference features up to 5 mV Vds values with 200 nm and

500 nm-long Ohmic contacted SWNT devices at 4 K [16]. They modelled the system

as an electronic analog to the Fabry-Perot (FP) cavities, and they claimed that the

observation of quantum interference is an evidence of the ballistic transport. More


Gate

SiO2

g = 1 g = 1g < 1

DrainSource

Vg

Vds

-10 -9 -8 -7 -6 Vg (V)

-20

-10

0

10

20

Vds

(mV

)0.40

0.38

0.36

0.34

0.32

0.30

0.28

0.26

-9.0 -8.5 -8.0 -7.5 -7.0

-40

40

Vg (V)

(a)

(b)

Gate

SiO2

g = 1 g = 1g < 1

DrainSource

Vg

Vds

Gate

SiO2

g = 1 g = 1g < 1

DrainSource

Vg

Vds

-10 -9 -8 -7 -6 Vg (V)

-20

-10

0

10

20

Vds

(mV

)0.40

0.38

0.36

0.34

0.32

0.30

0.28

0.26

-9.0 -8.5 -8.0 -7.5 -7.0

-40

40

Vg (V)

-10 -9 -8 -7 -6 Vg (V)

-20

-10

0

10

20

Vds

(mV

)0.40

0.38

0.36

0.34

0.32

0.30

0.28

0.26

-9.0 -8.5 -8.0 -7.5 -7.0

-40

40

Vg (V)

(a)

(b)

Figure 5.11: (a) The schematics of three-terminal SWNT device. (b) Experimentaltwo-dimensional image plot of differential conductance versus a drain-source voltagein y-axis and a backgate voltage in x-axis.


specifically, the phase information preserves along the nanotube. Contrary to the

usual FP oscillations, we found in all devices that the interference pattern fringe con-

trast is reduced in magnitude at high Vds shown in Fig. 5.11 (b). In order to explain

experimental observations in dI/dVds, three theoretical modesl are investigated in

this section: A single-channel double-barrier structure model, two-channel double-

barrier model in the Landauer-Buttiker formalism and a double-barrier cavity in the

Tomonaga-Luttinger liquid theory.

Double-Barrier Structure

A simple theoretical attempt to study Vds-dependent differential conductance (dI/dVds)

for a finite-length device is to compute dI/dVds in a double-barrier structure with two

transmission coefficients TL and TR of left and right barriers. Assume that all four

channels from spin and orbital degeneracy are acting identically, thus a factor 4 ap-

pears in the expression of dI/dVds. Considering the multiple reflections between two

barriers, the overall transmission coefficient T is energy (E) dependent,

T (E) =TLTR

1 + (1 − TL)(1 − TR) − 2√

(1 − TL)(1 − TR) cos(φ(E)), (5.2)

where the accumulated phase from the reflection paths is φ(E) and it is a function

of Vds, Vg and the SWNT length L. From an empirical trace, the maximum and

minimum value of T can be extracted because the maximum of Eq.(5.1) occurs at

cos(φ) = 1 and the minimum does at cos(φ) = −1, respectively. The explicit expres-

sion of φ is φ = EL~vF +αVg = eVdsL/~vF +αVg. The constant α is the effectiveness

of the gate voltage to the device through the oxide layer. In our case, α is around

0.01 from the quantum dot devices with the same thickness of a SiO2 oxide layer.

From the Landauer-Buttiker formula, dI/dVds is given by

dI

dVds

=4e2

hT (E) =

4e2

hT (Vds, Vg). (5.3)

The theoretical fitting to the experimental data is done in Fig. 5.12. This simple

model fits well the experimental dI/dVds data at low Vds, but it deviates significantly


as Vds increases. Quantitatively, two regions are divided around Vds ∼ 10 mV, which

corresponds to the energy spacing. In this model, one possible interpretation can

be as follows: When Vds is smaller than ∆E/e, the coherence length of the electron

wavepackets is longer than the tube length, L. In this region, the SWNT operates

as an isolated zero-dimensional quantum dot between two leads. For Vds > ∆E/e,

the coherence length of electron wavepackets is shorter than L. In this limit, each

wavepacket propagates through the one-dimensional conductor, and the oscillating

period increases. This phenomenon may indicate the correlated electrons since iso-

lated wavepackets are likely to experience unscreened Coulomb interactions. [91].

0.52

0.50

0.48

0.46

0.44

0.42

0.40

0.38

dI/

dV

(/

2GQ

)

403020100 Vds (mV)

Figure 5.12: The differential conductance from a 360 nm-long SWNT device at Vg = -5 V. Experimental data are blue circles and the theoretical fitting of the single-channeldouble-barrier structure model is in red.

Fabry-Perot Interferometer

Liang et. al. captured the wave nature of electrons through a isolated nanotube as an

electron waveguide. Two interfaces at metal and tubes are one-to-one correspondence


(b)

(a)

a1r c1l

d1rb1l

c1r a1l

b1rd1l

a2r c2l

d2rb2l

c2r a2l

b2rd2l

a1r c1l

d1rb1l

c1r a1l

b1rd1l

a2r c2l

d2rb2l

c2r a2l

b2rd2l

Figure 5.13: (a) Diagram of two-channel double barrier system. (b) Two-dimensionalimage plot of differential conductance versus drain-source voltage in y-axis and back-gate voltage in x-axis.


to partially reflecting mirrors in a FP interferometer [16]. Similar to photons in

the FP cavity, electrons would experience multiple reflections between two barriers

separating the metal reservoirs from the SWNT before escaping. The theoretical

model is based on the Landauer-Buttiker formalism with four conducting channels

including spin and orbital degeneracies. The authors in Ref [16] dealed with two

spin-degenereate transverse channels by lifting the orbital degeneracy although they

did not address what causes the orbital degeneray lifted. The approach is to establish

three scattering (S) matrices at left and right interfaces and inside the tube denoted

as SL, SR, and SN . The sketchy of the model is drawn in Fig. 5.13(a). The numbers

1 and 2 mean the spin-degenerate channels, yielding 4×4 scattering matrices. As

mentioned above, metallic infinite SWNTs have dominantly forward scatterings while

backscattering and interbranch scatterings are prohibited due to a big momentum

transfer 2kF between two K points and the symmetry between two orthogonal π and

π∗ orbitals. Therefore, the backscattering which leads to resistance increase occurs

possibly only at the interfaces between metal electrodes and the tube. Explicitly,

the incoming and outgoing operators for two channels in a system have the following

notation in Fig. 5.13(a). Two scattering matrices SL and SR contain this assumption

such that

d1r

d2r

b1l

b2l

= exp

i

0 0 r2 r1eiδ1

0 0 r1eiδ1 r2e

iδ2

r2 r1e−iδ1 0 0

r1e−iδ1 r2e

−iδ2 0 0

a1r

a2r

c1l

c2l

, (5.4)

and

b1r

b2r

d1l

d2l

= exp

i

0 0 r2 r1e−iδ1

0 0 r1e−iδ1 r2e

−iδ2

r2 r1eiδ1 0 0

r1eiδ1 r2e

iδ2 0 0

c1r

c2r

a1l

a2l

. (5.5)

All scattering matrices should satisfy the Unitarity, so they are taken in exponential

forms under Born approximation. r1 and r2 are intra-mode and inter-mode reflection

coefficients. δ1 and δ2 are phase shift for intra-mode and inter-mode scatterings.


These represent weak backscattering and inter-mode mixing. It is fair to assume

symmetric barriers so that the four parameters are commonly shared in both matrices.

In addition, the multiple reflection part in the middle comes in SN is

c1r

c2r

c1l

c2l

=

eiφ1 0 0 0

0 eiφ2 0 0

0 0 eiφ1 0

0 0 0 eiφ2

d1r

d2r

d1l

d2l

. (5.6)

In the section of the tube, since inter-mode mixings are not allowed. φ1 and φ2 are

accumulated phase of individual channels. The total scattering matrix ST is a matrix

product of all three matrices, ST = SL

⊗

SN

⊗

SR. The electronic FP interferometer

is written with the ST matrix,

b1r

b2r

b1l

b2l

=

t1r,1r t1r,2r r1r,1l r1r,2l

t2r,1r t2r,2r r2r,1l r2r,2l

r1l,1r r1l,2r t1l,1l t1l,2l

r2l,1r r2l,2r t2l,1l t2l,2l

a1r

a2r

a1l

a2l

. (5.7)

The differential conductance is

dI

dVds

=2e2

h

2∑

i=1

|ti|2 =2e2

h

2∑

i=1

Tr(S†TST ). (5.8)

Similar to the double-barrier case, the energy dependence is included in the phase

parameters. Figure 5.13(b) presents the two-dimensional image plot from theoretical

modelling with r1 = 0.5, r5 = 0.25, δ = 0.4, δ = 0.95 given in Ref. [16] and for L = 360

nm. Since the model is valid for weak backscattering, the values of r1 and r2 should

not be big, whereas two other fitting parameters are randomly chosen until the fit is

close to the data. It would be the drawback of the model in that there are too many

free parameters which are not experimentally accessible.


¡L2

L2

glead = 1 glead = 1g < 1

¡1 1

1 2

¡L2

¡L2

L2

L2

glead = 1glead = 1 glead = 1glead = 1g < 1g < 1

¡1¡1 11

1 2

Figure 5.14: Illustration of the TLL model on a SWNT device.

Tomonaga-Luttinger Liquid

The third attempt to understand the empirical behaviors in dI/dVds is motivated by

the fact that SWNTs have exhibited the features of strong correlations among charge

carriers in experiments [23] and in theories [65, 66, 91]. SWNTs are one of the ideal

systems to investigate one-dimensional physics since their geometric ratio between ra-

dius and length ranges from 100 to 106. Plus, a long-range Coulomb interaction makes

metallic SWNTs to be a Tomonaga-Luttinger liquid (TLL), a non-Fermi liquid. The

TLL theory is a framework to describe the low-energy properties of one-dimensional

conductors [62, 63, 92, 93]. The interaction strength in SWNTs is estimated theoreti-

cally with the a TLL interaction parameter g ∼ 0.2 − 0.3. In comparison with g = 1

for the non-interacting Fermi gas, SWNTs are strongly correlated systems, in which

Coulomb interactions play a crucial role in the current flow.

The ballistic SWNT device including metal electrodes are theoretically modelled

as an infinite one-dimensional conductors with inhomogeneous the TLL parameter

g(x). The interaction is assumed to be strong in the SWNT (0 < g < 1) and weak

in the higher dimensional metal reservoirs (g = 1) for metals [91, 94]. The schematic

of the model is illustrated in Fig. 5.14. The TLL without the barriers is described by


the bosonized Hamiltonian [65]

Hswnt = (vF/2π)∑

a

∫

dx[(∂xφa)2 + g−2

a (x)(∂xθa)2]

, where θa(x) and πa(x) = −∂xφa/π are conjugated bosonic variables, i.e. [θa(x), πb(x′)] =

iδabδ(x− x′). The four conducting transverse channels of the SWNTs in the FL the-

ory are transformed to four collective excitations in the TLL theory: one interacting

collective mode (a = 1, ga ≡ g) of the total charge and three neutral non-interacting

collective modes (a = 2−4, ga = 1) including spin. There are two distinct propagating

velocities vc = vF/g and vF . The inter-channel and intra-channel scattering occurs at

the barriers. These modes are partially reflected at the two barriers. The backscatter-

ing is supposed to be weak enough that the backscattering Hamiltonian term can be

treated as a perturbation. Non-equilibrium situation is treated within the Keldysh

formalism. The transport properties are computed from correlation and retarded

Green’s functions [94]. Three non-interacting modes encounter backscattering at the

physical barrier, whereas the interacting mode encounters the momentum-conserving

backscattering due to g mismatch at the interfaces as well as the physical barrier

backscatterings. The theoretical differential conductance I ≡ e(2/π)θ1 = 2GQVds−IBwhere IB is given to leading order in the backscattering amplitudes as [91,94]

IB =2e

πt2F

∑

b=1,2

Ub

∫ ∞

0

dteCb(t) sin

(

Rb(t)

2

)

sin

(

eVdst

~

)

, (5.9)

where tF = L/vF is the travelling time for a non-interacting mode along the SWNT

length L. The backscattered current IB consists of two contributions: the term pro-

portional to U1 represents the incoherent sum of backscattering events at the two

barriers and the term associated with U2 results in the FP oscillations due to the

coherent interference between backscattering events from different barriers. At high

Vds, the U1-term in Eq. (5.9) dominates and the oscillation amplitude decreases. U1

and U2 are independent of Vds, but U2 depends periodically on Vg. The interaction

parameter g is involved in the time integral through Cb(t) and Rb(t), which are cor-

relation and retarded functions, respectively. These Green’s functions contain a sum


over all four collective modes, and their forms are obtained at zero [91,94] and finite

temperatures [94]. Mathematically, the terms are proportional to the backscattering

amplitudes uαi of each mode (i = 1,2,3,4) at left and right barriers (α = l, r),

U1 ∝4∑

i=1

(uli)

2 + (uri )

2

U2 ∝4∑

i=1

2uliu

rj cos(

VgCL

e+ 2∆ij)

(5.10)

where C is the capacitance from the backgate and ∆ij the phases for inter- and

intra-channel mixing.

A simplified form of the differential conductance including Eq. (5.9) in the unit

of 4GQ is written with two Vds-dependent f1 and f2 functions,

dI

dVds

= 4e2

h[1 + U1 · f1(Vds, g) + U2 · f2(Vds, g)] . (5.11)

Both two backscattering contributions have the sine function of the drain-source volt-

age and contain the TLL parameter through Green’s functions.

In Fig. 5.15, the Vds dependent functions f1 and f2 are displayed for different g

values at T = 4 K and the tube length L is 360 nm. The function f1 exhibits the

power-law behavior for g less than 1 in Vds, and the function f2 exhibits oscillation

amplitudes due to the Fabry-Perot interference. The backscattering amplitudes are

renormalized by the interaction parameter g, reducing the oscillation amplitude in Vds.

Figure 5.16 (b) shows overall theoretical fitting plots combining two terms for g = 0.25

(red) and g = 1 (blue) with U1 = 0.14 and U2 = 0.1. The plot with g = 1 represents

the case that all four modes are non-interacting, thus the interference amplitudes

are constant regardless of the bias voltage size. Clearly, the theoretical differential

conductance trace for g = 0.25 has qualitative agreement with the experimental data:

the amplitude of the FP oscillation is damped at high Vds compared to that at low

Vds. The differences between the TLL theory and the experimental data are found

in that the overall conductance level in the data is much lower than the theory, and

the oscillation period in Vds increases in experiments. In addition, the asymmetry


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

403020100 Vds (mV)

(b)

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

403020100 Vds (mV)

g = 0.25 g = 0.75 g = 1

(a)

Figure 5.15: (a) The Vds-dependent f1(Vds, g, T ) at T = 4 K for g = 0.25 (red),g = 0.75 (green) and g = 1 (blue). (b) The Vds-dependent f2(Vds, g, T ) at T = 4 Kfor g = 0.25 (red), g = 0.75 (green) and g = 1 (blue).

in positive and negative Vds is vivid in experiments, but the cause is not definitely

understood yet.

To identify the TLL feature uniquely in experiments requires to increase Vds above


the levelspacing ~/2gtF. Note that the tendency of amplitude reduction in experimen-

tal data cannot be reproduced by the reservoir heating model [95] which asserts that

the dissipated power V 2ds(dI/dVds) leads to a bias-voltage dependent electron temper-

ature. We have tested this effect for the non-interacting case (g=1) in our theory

and have found that it causes a slight damping of the FP-oscillations (U2-term) but

the incoherent part (U1-term) is independent of temperature. The temperature effect

fails to account for the experimentally observed enhanced backscattering amplitude

at low Vds [94]. In addition, the conductance is relatively small (on the order of GQ)

so that heating effects should not be pronounced in the bias window considered.

5.2.2 Spin-Charge Separation

The TLL theory provided a clue to interpret the Vds-dependent differential conduc-

tance traces, a qualitative feature of interactions. Figure 5.17(a) explicitly shows

several traces in Vds at different values of Vg with the following pronounced features:

the oscillations in low Vds have different periods at different Vg and the period of

oscillations become elongated at high Vds.

First, consider the former feature: Vg-dependent oscillation periods. The theory

predicted the existence of two propagating velocities by means of the backgate volt-

ages. This feature is captured from the fact that the U2 backscattering term of FP

interference contains a cosine function of Vg. The oscillation patterns from the U2

term are governed by three non-interacting modes. Although the interacting charge

mode behaves differently from the other three modes, the modification of the inter-

acting charge mode is too slight to be recognized. On the other hand, the U1 term

shows the enhanced backscattering amplitude at low Vds with the power-law scalings

and contains the interference of only the charge mode due to the mismatch of g at the

interface (Fig. 5.15(a)). If the interaction-induced interference can be observed, the

quantitative information of the TLL behavior in ballistic SWNTs is obtained. The

key to observe the charge mode interference is to remove the FP contribution which

masks the interaction effect.

Since Vg determines the location of the Fermi level in the bandstructure, the choice


0.38

0.36

0.34

0.32

0.30

0.28

dI/d

V

-40 -20 0 20 40 V

ds(mV)

(a)

(b) 1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

dI/

dV

ds

-40 -20 0 20 40

Vds (mV)

Figure 5.16: Differential conductance versus Vds at a certain Vg. (a) Experiment and(b) Theory based on TLL for g = 0.25 (red) and g = 1 (blue).


0.38

0.36

0.34

0.32

0.30

dI/

dV

ds

-40 -20 0 20 40 V

ds (mV)

-9V

-8.3V

-7.7V

1.00

0.95

0.90

0.85

0.80

0.75

0.70

d

I/d

Vd

s

-20 -10 0 10 20 V

ds (mV)

(a)

(b)

(c)

Figure 5.17: Differential conductance versus Vds at different backgate voltages forexperiment results (a), the theoretical plots from non-interacting Fermi-liquid theory(b) and from the Tomonaga-Luttinger liquid theory (c).


of U2 sets up whether the maximum or minimum dI/dV starts at the zero Vds shown

in Fig. 5.17(c). Furthermore, note that the FP oscillations are absent at a certain Vg

according to the cosine function of Vg. It means that the backscattering contributions

are π/2 out of phase, leading to destructive interference. It is speculated that the

contribution from ac signals on top of Vds/2 and −Vds/2 from the Fermi level add

up constructively at the node of cosine function in Vg, whereas they cancel out at

the antinode of cosine function in Vg since the k values within the ac signal window

involve in transport processes. In the non-interacting two-channel model, the effect of

Vg shows a simple oscillating amplitude behavior in all regions of Vds in Fig. 5.17(b).

On the other hand, the outcome of the TLL shows the dramatic change such that

the weak oscillation emerges at U2 = 0. The position of the peak sits in higher Vds

compared to the non-zero U2. From the period of the interference patterns, the values

of the mode velocities are extracted due to the fact that the periods in Vds relates to

the travelling distance (l) and velocity (v) such that e∆Vds = hv/l. With vc = vF/g

and the tube length L, the TLL parameter g is written as

g =∆Vds(U2 6= 0)

2∆Vds(U2 = 0). (5.12)

Sorting the dI/dV traces into two groups and measuring the periods, an indication of

this effect is found by comparing the primary periods (2gtF if U2 ∼ 0 and tF when U2

is maximal) of these traces, which gives g ∼ 0.22. This fact hints that the total charge

mode has its own velocity vF/g in the SWNT. In the literature, spin-charge separation

in semiconducting wires has been recently observed by mapping the distinct charge

and spin mode velocities [96].

The latter signature of the longer period at high Vds is beyond our theory, but

is likely to be caused by a strong barrier asymmetry at high Vds which would also

suppress the U2-term. The ratio of primary and elongated periods along Vds relates

to g ∼ 0.22. To reach a conclusive claim, further experiments focusing on this aspect

should be performed.


5.3 Low-Frequency Shot Noise

5.3.1 Experiment Setup

Two-terminal shot noise measurements were implemented using all technical strate-

gies presented in Chapter 4. Figure 5.11 illustrates the simplified diagram of the

SWNT shoe noise measurement setup. The design of the experiments has required

to consider several technical issues: the shot noise measurement frequency and cali-

bration.

1/f noise crossover

To determine the shot noise measurement frequency f is one of the first things to

be considered before implementation. The upper bound of f is, in principle, limited

by the energy scaling comparison among eVds, kBT, and ~ω, where ω is the angular

frequency, equal to 2πf . For SWNTs, Vds can be applied up to 40 ∼ 50 mV without

saturating the current. Thus the biggest energy scale is eV ∼ 40 meV. The thermal

energy at 4 K is 0.33 meV. Therefore, any frequency as ~ω is lower than kBT works

well, which corresponds to f ∼ 400 GHz. 400 GHz is in the microwave regime. To ex-

ecute any measurements at such high frequency in the microwave regime is extremely

difficult because there are very delicate technical treatments involved. Therefore, the

practical upper bound is taken by the wavelength of chosen frequency. As long as

the wavelength is much larger than the system size (rigorously, the system size is

smaller than a fourth of the wavelength as a rule of thumb), subtle microwave issues

can be ignored. Since the cryostat including wirings is around 2 m, the wavelength 8

m corresponds to around 37.5 MHz. Roughly speaking, tens of MHz range is a good

upper limit. The lower bound of the measurement frequency is set by the 1/f noise

crossover mentioned earlier in Chapter 3.

The 1/f noise of carbon nanotubes has been measured in various forms: SWNT

mats and isolated SWNTs [97], two-crossing MWNTs [98], SWNTs at room tempera-

tures (RT) [99] and MWNTs at low temperatures [100]. The 1/f noise measurements

were executed with a current bias scheme at three temperatures: RT, 77 K(liquid

110C

HA

PT

ER

5.

SIN

GLE

-WA

LLE

DC

AR

BO

NN

AN

OT

UB

ES

Gv

Resonant

Circuit

RPD>>RSWNT

Signal

DCVg

RSWNT

-20V

LED

Vdc

Vac

SWNTVdc

Vac

+

+Cparasitic

*

#

( ) 2 Lock-InGv

Resonant

Circuit

Resonant

Circuit

Resonant

Circuit

RPD>>RSWNT

Signal

DCVg

RSWNT

-20V

LED

Vdc

Vac

Vdc

Vac

SWNTVdc

Vac

Vdc

Vac

+

+Cparasitic

*

#

( ) 2 Lock-In( ) 2 Lock-In

Figu

re5.18:

Sch

ematics

ofSW

NT

shot

noise

measu

remen

tsetu

p.


10-16

10-15

10-14

10-13

10-12

10-11

SV (

V2 /

Hz

)

103

104

105

106

Frequency (Hz)

I = 1125 nA @ T = 77 K

I = 1209 nA @ T = 293 K

Figure 5.19: The 1/f noise crossover of SWNT devices for two temperatures 293 Kand 77 K.

nitrogen) and 4 K. The representative 1/f voltage noise spectral densities are shown

in Fig. 5.19. The calibration of 1/f noise measurements can be achieved by comparing

with the theoretical thermal noise because without the bias, the only possible noise

source is the thermal noise. The red straight lines are thermal noise voltage 4kBTR

at T = 283 K and T = 77 K. with measuring linear resistance R at given tempera-

tures. Indeed, both noise traces without any bias at RT and 77 K sit exactly on top

of theoretical values. As the non-zero current is applied to the SWNT, the excess 1/f

noise signals emerge. The total noise spectral density is

SV,Tot = AV 2

fα+ 2eIR2 + 4kBTR. (5.13)


The corner frequency fc is found as the pink noise is equal to the white noise level,

fc =[

AV 2(2eIR2 + 4kBTR)−1]1/α

. (5.14)

Fitting the 1/f noise trace with the Hooge’s formula gives constant A and the power

exponent α of frequency. Experimentally, the conservative value of fc is estimated

from setting the 1/f noise equal to the thermal noise since the white noise level is

higher with adding the shot noise. The values of fc is around 5 MHz at RT and

becomes lower to 1 MHz at 77 K. The origin of the 1/f noise would relate to the

trapping impurities in oxides. They may be frozen at lower temperature, reducing

the 1/f noise and fc. From the lower bound of the fc and the upper bound of 40

MHz, the range of the shot noise frequency is tens of MHz. The actual measurement

implementation of the frequency choice is done by a tank circuit with passive com-

ponents, inductors and capacitors. The resonant frequency at RT in the experiment

is aimed at 20 MHz and it is reduced down to 14 ∼ 15 MHz at 4 K due to the

parasitic capacitance. The RT bandpass filter is placed with the low and high cutoff

frequencies of 12 and 21.4 MHz to carry the noise signals to the outside world.

A Full Shot Noise Source

The shot noise measurements were performed by placing two current noise sources

in parallel: a SWNT device and a full shot noise generator. The role of the full

shot noise is to calibrate the circuit. Using a cryogenic amplifier boosts the signal-

to-noise ratio but adds the complexity of the calibration due to transfer function of

the amplifier. The ideal full shot noise source has the particle statistics governed by

the Poisson distribution. A pair of a light emitting diode (LED) and a photodiode

(PD) is selected as a full shot noise candidate. PD converts the photon energy by

accepting photons emitted from the LED into electrical current. In particular, as

long as the lower coupling efficiency between the LED current and the PD current is,

the shot noise of the PD current is closer to the ideal full shot noise. The coupling

efficiency between two currents is measured at RT and 4 K. Photons from the LED is

proportional to the LED current. At 4K, the ratio of LED and PD current is about 0.1


1400

1200

1000

800

600

400

200

0

IP

D (

nA

)

400x103

3002001000 ILED (nA)

T = 4.2 Kα ~ 0.38 %

2000

1500

1000

500

0

IP

D (

nA

)

1.6x106

1.41.21.00.80.60.40.20.0 ILED (nA)

T = 293 Kα ~ 0.13 %

(a)

(b)1400

1200

1000

800

600

400

200

0

IP

D (

nA

)

400x103

3002001000 ILED (nA)

T = 4.2 Kα ~ 0.38 %

2000

1500

1000

500

0

IP

D (

nA

)

1.6x106

1.41.21.00.80.60.40.20.0 ILED (nA)

T = 293 Kα ~ 0.13 %

(a)

(b)

Figure 5.20: The coupling efficiency α between ILED and IPD at (a) T = 293 K and(b) T = 4 K.


(a)

(b)

250

200

150

100

50

0

SP

D (a

.u.)

2000150010005000 IPD (nA)

T = 293 KRinput = 10 kΩ

400

300

200

100

0

SP

D (a

.u.)

160012008004000 IPD (nA)

T = 4.2 KRinput = 10 kΩ

(a)

(b)

250

200

150

100

50

0

SP

D (a

.u.)

2000150010005000 IPD (nA)

T = 293 KRinput = 10 kΩ

400

300

200

100

0

SP

D (a

.u.)

160012008004000 IPD (nA)

T = 4.2 KRinput = 10 kΩ

Figure 5.21: Full Shot Noise from the LED/PD pair (a) T = 293 K and (b) T = 4 K


- 0.4 %, eliminating completely the shot noise squeezing effect due to constant current

operation [101]. The full shot noise is confirmed with a constant input resistor R =

10 kΩ. The shot noise signal after the tank circuit is fed into the cryogenic amplifier

followed by the bandpass filter, a square-law detector and a lock-in amplifier. The

signals of the lock-in amplifier at both RT and 4 K exhibit the linear relation of the

PD current as expected (Fig. 5.21). The slope of the noise and the current is related

to the following quantities: the elementary charge e, resistance, the equivalent noise

bandwidth and the gain of the amplifier. The output spectrum before the square

law detector is captured with a HP8561E spectrum analyzer, from which the voltage

gain and the area of the spectrum are measured. Then, the unknown value from the

slope is the size of the elementary charge. With a R = 10 kΩ at RT and 4 K shown

in Fig. 5.21, the linear regression analysis tells the slopes of the full shot noise are

0.2544 ± 0.000543 (0.11853±0.0003) at 4 K (RT). Together with the voltage gain Av

= 1111.6037 (676.005) and the equivalent noise bandwidth BW = 1.330 MHz (1.584

MHz) are obtained from the spectrum, the elementary charge is 1.52 × 10 −19 (1.61

× 10 −19) with a 5 % (0.6 %) error to the ideal value e = 1.6× 10 −19.

5.3.2 Shot Noise and Fano factor versus the Drain-Source

Voltage

Two-terminal shot noise measurements are performed at a particular Vg value. Fig-

ure 5.22 shows a typical shot noise SSWNT versus Vds on a log-log scale for a particular

Vg. SSWNT (dot) is clearly suppressed to value below the full shot noise SPD (triangle).

It means the charge flow is regulated further beyond Poisson statistics and it suggests

that the relevant backscattering for shot noise is indeed weak. It can be understood

as the partition noise due to the partial transmission between the metal reservoir and

the tube. Another noticeable feature between two noise graphs is clearly different

scaling slopes versus Vds.

The Green’s function theory of the TLL model is extended to calculate the shot


2.5

2.0

1.5

1.0

0.5

lo

g (

SP

D, S

SW

NT )

1.61.41.21.00.80.60.40.20.0 log ( V

ds )

Figure 5.22: A representative log-log plot of low-frequency shot noise from theLED/PD pair (upside-down triangle) and the SWNT (diamond) as a function ofVds. The straight line is the outcome of linear regression analysis.

noise spectral density in the zero-frequency limit,

Sswnt(ω) =

∫

dteiωt〈δI(t), δI(0)〉

with δI(t) = I(t)− I the current fluctuation operator and · · · the anticommutator

[94]. In this model, the SWNT noise is expressed as Sswnt = 2e coth(eVds/2kBT )IB +

4kBT (dI/dVds − dIB/dVds), becoming SSWNT = 2eIB for eVds > kBT . This simple

result indicates that the charge carriers in low frequency transport processes are

electrons not fractional charges in the TLL. It is argued rather intuitively that the

carriers in the metal electrodes are recovered to electrons for a long average over the

transit time. IB is computed by integrating backscattering contribution in dI/dVds

with respect to Vds.


IB ≡∫ ∞

0

dVds

(

4e2

h− dI

dV

)

=

∫ ∞

0

dVds(U1f1(Vds, g, T ) + U2f2(Vds, g, T )).

(5.15)

Figure 5.23 displays the g-value effect on two backscattering contributions. Again

the U1 term shows the asymptotic behavior in Vds. Without interaction (g = 1),

the backscattered current is in a perfect linear relation with Vds. As soon as the

interaction turns on, the correction due to interactions deviates from the linearity.

The deviation becomes larger for the smaller g. The interference in the U2 term

remains in IB. The values of the fitting parameters U1 and U2 at 4 K for a 360

nm SWNT device produces theoretical shot noise in red together with experimental

data in blue. The TLL theory explains the interference pattern in the noise very well

(Fig. 5.24).

Moreover, the asymptotic behavior of IB from the dominant U1-term when eVds >

~/2gtF yields the power-law relation IB ∼ V 1+αds with α = −(1/2)(1−g)/(1+g). Note

that the power-law scaling exponent α is uniquely determined by the TLL parameter

g. The power exponent in this sample (Fig. 5.22) at Vg = −7.9 V is - 0.36 for the

SWNT, whereas α for the PD is zero. The average values of power-law exponent over

seven different gate voltages are estimated to be α ∼ - 0.31± 0.027, corresponding

to g ∼ 0.25 ± 0.049, a value which closely matches the experimental value g ∼ 0.22

obtained by the differential conductance oscillation mentioned above and also the

theoretical g-value for SWNTs.

The Fano factor F (I) was obtained at each current value by taking the ratio of

the SWNT noise and the PD noise, and it is presented on a log-log scale in Fig. 5.25.

The theoretical Fano factor F (I) ≡ §SWNT/2eI also manifests an asymptotic power

behavior in the high bias regime (eVds > ~/2gtF ), F ∼ V αds, assuming the backscat-

tered current is smaller than its ideal value 2GQVds. A linear regression analysis of the

Fano factor F with Vds, therefore, is another means to obtain the g value. The Fano


40

30

20

10

0403020100

Vds

(mV)

g = 0.25 g = 0.75 g = 1

(a)

(b)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

403020100 V

ds (mV)

Figure 5.23: (a) The integration in Vds of the Vds-dependent f1(Vds, g, T ) at T = 4 Kfor g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). (b) The integration in Vds ofthe Vds-dependent f2(Vds, g, T ) at T = 4 K for g = 0.25 (red), g = 0.75 (green) andg = 1 (blue).


25

20

15

10

5

SS

WN

T (

a.u

)

403020100V

ds (mV)

3.5

3.0

2.5

2.0

1.5

1.0

0.5

§ EXP

Theory

14.01 =U

1.02 =U

KT 4=

nm360=L

Fitting Parameters

25

20

15

10

5

SS

WN

T (

a.u

)

403020100V

ds (mV)

3.5

3.0

2.5

2.0

1.5

1.0

0.5

§ EXP

Theory

14.01 =U

1.02 =U

KT 4=

nm360=L

14.01 =U

1.02 =U

KT 4=

nm360=L

Fitting Parameters

Figure 5.24: The experiment data (blue square) of the SWNT noise with the theo-retical fitting plot(red).

factor F for g = 0.25 (red) and g = 1 (yellow) is displayed n Fig. 5.25 (a). The log-

log scale presentation of the Fano factor is easy to appreciate the power-law scalings

and the linear scale presentation of F shows well that theoretical model for g = 0.25

gives the oscillatory behavior which matches well with experimental data (diamonds).

The stiffer slope (α) corresponds to stronger electron-electron interaction. The mean

value of the exponent α and the inferred g derived over seven Vg values are α = - 0.33

± 0.029 and g = 0.22 ± 0.046 respectively for this particular sample. We find that

the measured exponents α and inferred g values from the spectral density Sswnt and

the Fano factor F from four different devices with various metal electrodes (Ti/Au,

Ti-only, Pd) show similar statistics α ∼ −0.31±0.047 and g ∼ 0.26±0.071 as derived

from several Vg values for each sample. We stress that the non-linear decay of the

experimental F along Vds indeed starts at a voltage scale log(~/2getF) ∼ 0.61 for

g ∼ 0.18 in Fig. 4 as a manifestation of a collective electron effect.

The inferred g values from the exponents of the spectral density SSWNT and the


(a)

(b)

0.30

0.25

0.20

0.15

0.10

Fan

o f

ac

tor

403020100Vds (mV)

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4 lo

g (

Fan

o f

acto

r)

1.51.00.50.0

log ( Vds )

Figure 5.25: A representative log-log plot of Fano factor (open diamond) ob-tained from experiments against Vds together with theoretical theoretical fitting ofTomonaga-Luttinger liquid theory for g = 1 (straight line) and g = 0.25 (dottedline). The broken line on the experimental data represents the power-law scalinganalysis.


Sample Electrodes g(Noise) g(Fano factor)1 Ti/Au 0.25 ± 0.13 0.19 ± 0.072 Ti/Au 0.31 ± 0.086 0.31 ± 0.203 Pd 0.36 ± 0.18 0.11 ± 0.0784 Pd 0.28 ± 0.12 0.23 ± 0.057

Table 5.1: The g values from the power-law scaling analysis of four samples.

Fano factor F from four different devices with various metal electrodes (Ti/Au, Ti-

only, Pd) are listed in the Table 1. The overall statistical result is summarized as

α ∼ - 0.31 ± 0.047 and g ∼ 0.26 ± 0.071 as derived from several Vg values for each

sample. The experiments from the shot noise and the Fano factor are consistent with

each other as well as theoretical prediction.

5.3.3 Fano Factor versus Transmission Probability

Ballistic Phase-Coherent Picture

Previous subsection has focused on the trend of the Fano factor as a function of

Vds. In this section, the absolute values of the Fano factor are considered in terms

of the transmission probabilities. The definition of the transmission probability T

is not very obvious from experimental data since two-terminal resistance in SWNTs

contains not only the intrinsic resistance but also the physical contact resistance

and the lead resistance. Since the contributions of different resistance sources are

not distinguished, experimentally the possible way to estimate T is the ratio of the

measured resistance and the quantum unit of resistance.

Model I : One Transmission Probability T Mesoscopic partition noise with a

lumped elastic scatterer with a transmission probability T is a form of SI = 2eI(1−T ).

In this case, the Fano factor is simply reduced to (1-T) because F = SI/2eI = 1−T .

Model II : Two-Channel Model Suppose two differential transmission probabilities

T1 and T2 exist in the system. It can be either two channels in the system or one

channel in the double barrier system. In this case, the total transmission probability

T is related to T1 and T2 such that T = (T1 + T2)/2. The shot noise SI with two


values is written as SI = eGQV (T1(1 − T1) + T2(1 − T2)) where the current I is

I = GQV T = GQV (T1 + T2)/2. This can be expanded into a n-channel model:

T =

∑ni=1 Ti∑n

i=1

, (5.16)

F =

∑ni=1 Ti(1 − Ti)∑n

i=1 Ti

. (5.17)

In this model for a fixed T , F is not single-valued but it has the range bounded

by the maximum and minimum values from differentiating F with respect to T and

setting to zero. Table 5.2 summarizes the range of the allowed Fano factor values as

a function of T in the two-channel model:

F expressionsFmax = 1 − T for 0 ≦ T ≦ 1

Fmin = 1 − 2T for 0 ≦ T ≦ 1/2

= (2T−1)(1−T )T

for 1/2 < T < 1

Table 5.2: The relation between extreme values of F and the overall transmissionprobability T

Figure 5.26 (a) shows the experimental data as well as the above Fano factor

relations for one and two-channel cases. Experimental data are the Fano factor at

40 mV from five SWNT-devices denoted as different symbols. Given a device, many

points were taken by sweeping gate voltage. Some devices show the resistance change

over the gate voltage change. Apparently, experimental data are completely off from

any models in the ballistic phase-coherent conductors.

Phase-Incoherent Picture

Figure 5.26 clearly states that the Fano factor of experimental data is further sup-

pressed below the value of F with a lumped elastic scatterer in the ballistic regime.

Further suppression can be caused by correlations. In the literature, the effects on


1.0

0.8

0.6

0.4

0.2

0.0

F

ano

Facto

r

1.00.80.60.40.20.0

T (=G/(2GQ) )

1-T two-mode model

1.0

0.8

0.6

0.4

0.2

0.0

F

an

o F

acto

r

1.00.80.60.40.20.0

T (=G/(2GQ) )

distributed elastic scattering

distributed inelastic scattering

incoherent double-barrier model incoherent many-barriers model

(a)

(b)

Figure 5.26: Fano factor versus transmission probability taken at Vds = 40 mV fromfive SWNT-devices (filled symbols) at varying Vg values. (a) Ballistic phase-coherenttransport theory for one-(dark blue straight) and two-channel (dotted area) models.(b) Phase-incoherent picture theory for distributed elastic (square) and inelastic (dia-mond), incoherent double-barrier (light green straight) and many-barrier model (darkgreen straight)


the suppressed shot noise of disorder has been studied both in experiments and in

theory [95, 102–110]. Considering the involvement of distributed elastic and/or dis-

tributed inelastic scatterers in transport processes, such systems are no longer in the

ballistic regime but rather in the diffusive regime. Theorists have showed that a dis-

ordered phase-coherent conductor exhibits one-third of the Poisson shot noise value

using the random-matrix theory [103]. The degree of suppression in the Fano factor

value (1/3) is also predicted in a semi-classical picture without phase coherence but

allowing to have the fluctuations of the distribution function [106]. Liu studied several

models based on the previous pictures by quantum Monte Carlo simulation in order

to produce the relation of the Fano factor and the transmission probabilities plot-

ted in Fig. 5.26 (b) [111, 112]. He tracked down the electron phase according to the

sources of scattering: elastic scatterers reverse the momentum of electrons whereas

inelastic scatterers lose the energy of electrons. Similar to the literature, he found

that distributed elastic scatterers reduced the shot noise by 1/3 from the Poisson

value as the transmission probability approaches to 0. However, distributed inelastic

scatters further suppress the shot noise close to 0 due to the complete heat removal

by phonons such as macroscopic resistor.

n-barrier models in the phase-incoherent picture were simulated [112]. For n

identical barriers with the same transmission probability t, the total transmission

probability T is expressed as T = t/(t + n(1 − t)). For this T , the Fano factor F is

estimated as

F =1

3

(

1 +n(1 − T )2(2 + (3n− 2)T ) − (nT )3

n3

)

.

In the case of a double barrier n = 2, F is reduced to F = (1/2)(1 − T )(1 + T ).

In Fig. 5.26(b), the light green line says F approaches to 0.5 as T decreases, while

the case of many-barrier structure yields F to be 1/3.

Unfortunately, all above models do not match any point of experimental data in

terms of the degree and the trend of the suppressed Fano factor along the transmission

probability. Therefore, the strongly suppressed Fano factor in SWNTs is due to some

other correlations among charge carriers. To understand the absolute value of the

5.4. SUMMARY 125

Fano factor in non-chiral Luttinger liquid conductors such as SWNTs is missing at

present and this is an interesting problem to study further.

5.4 Summary

Chapter 5 has discussed the low temperature electrical properties of metallic SWNTs.

The discussion started from the bandstructure of SWNTs in terms of the tight-binding

approximation to understand electronic properties. Focusing on metallic SWNTs,

the conductance and the current fluctuations have been measured. The experimental

outcome has exhibited unique features originated from correlations among charge car-

riers. The experimental data have been examined by both non-interacting and inter-

acting theories. A non-interacting picture may describe conductance data in the low

bias regime; however, it fails to explain high-energy data. The Tomonaga-Luttinger

liquid theory has explained the qualitative trend of conductance as a function of the

drain-source voltage. In addition, it has captured the quantitative information of the

strong electron-electron interactions both in conductance and the shot noise quanti-

ties. The strength of the interactions is parameterized by g, which has been obtained

from the conductance period at various Vg and power-law scaling exponents from

both shot noise and the Fano factor. The g values are consistent from two indepen-

dent measurements. This work contributes the first quantitative investigation of TLL

interaction effects in the shot noise of well-contacted SWNTs.

Chapter 6

Quantum Point Contact

The modern world,

despite a surfeit of obfuscation, complication, and downright deceit,

is not impenetrable, is not unknowable,

and – if the right questions are asked –

is even more intriguing than we think.

All it takes is a new way of looking.

− Steven D. Levitt

6.1 Two-Dimensional Electron Gas

A two-dimensional electron gas (2DEG) has emerged along the route to improve the

functional capacities of semiconductor transistor. It is a material where electrons’

free motion is only possible in a two-dimensional plane. It was found early on inside

silicon-based MOSFETs (metal-oxide field effect transistor), where electrons beneath

the gate oxide were trapped at the semiconductor-oxide interface. A 2DEG is also

formed in other elemental and compounds semiconductors, germanium, gallium ar-

senide (GaAs) and aluminum arsenide (AlAs). For compound semiconductors, the

2DEG is embedded in a heterojunction where two different materials face. The good

126

6.1. TWO-DIMENSIONAL ELECTRON GAS 127

example is a doped GaAs/AlGaAs heterostructure and the 2DEG is placed in the

GaAs layer according to the bandgap arrangement of GaAs and AlGaAs.

This new structure has endowed physicists with many opportunities to explore

the low dimensional physics. Further spatial confinements allow us to create one-

dimensional and zero-dimensional structures with advanced microfabrication tech-

nologies. The confinement is directly responsible for quantum-size effect. The 2DEG

has marked a new era in condensed matter physics since it reveals novel quantum

phenomena due to low dimensionality (example: quantum Hall effect) and also serves

a test bed of quantum mechanics postulates. The field has been very productive to

understand the following basic features: universal conductance fluctuations [?, 113],

localization [114], quantized conductance [45,115], Aharonov-Bohm interference [116],

Coulomb Blockade [117,118], Kondo effect [119,120], integer quantum Hall effect [121],

fractional quantum Hall effect [122] and more. In addition, a dream to establish solid-

state qubits for quantum computation in a 2DEG is actively pursued at present [49].

In addition, the quality of 2DEGs determines the transport regimes among dissipative,

diffusive and ballistic ones. This section discusses the principle of a 2DEG formation

in a GaAs/AlGaAs heterostructure, the realization of a GaAs/AlGaAs 2DEG and

the scattering mechanisms to explain the high quality of 2DEG traits.

6.1.1 Energy Band Profile

The understanding of how to form a 2DEG system in doped heterostructures starts

from energy band profiles. Let’s first consider undoped heterostructures composed of

GaAs and AlGaAs as common choices since their lattice constants are very close to

each other. Due to the almost perfect lattice match, the GaAs/AlGaAs heterostruc-

ture yields almost no strain at the interface. Often the strain becomes a possible scat-

tering source. The lattice constants of GaAs and AlAs are 0.56533 nm and 0.56611

nm respectively. In actual growth, instead of using AlAs, the alloy of GaAs and AlAs,

AlxGa1−xAs, is used for a larger bandgap layer. Lattice constant and bandgap energy

Eg are depending on the concentration (x) of Al in the alloy, and typically x is about

128C

HA

PT

ER

6.

QU

AN

TU

MP

OIN

TC

ON

TA

CT

Vacuum levels

i - GaAs

Eg = 1.414 eV

i – Al0.3Ga0.7As

Eg = 1.798 eV

Gro

wth

direct

ion

EcEv

E

(a)

i - GaAsEg = 1.414 eV

n – Alc.3Ga0.7As

Eg = 1.798 eV

2DEG

(b)Ev Ec

Vacuum levels

i - GaAs

Eg = 1.414 eV

i – Al0.3Ga0.7As

Eg = 1.798 eV

Gro

wth

direct

ion

EcEv

E

(a)

i - GaAsEg = 1.414 eV

n – Alc.3Ga0.7As

Eg = 1.798 eV

2DEG

(b)Ev Ec

Figu

re6.1:

(a)B

and

profi

leof

aG

aAs/A

lGaA

sheterostru

cture.

(b)

Ban

dprofi

leof

a2D

EG

embed

ded

ina

dop

edG

aAs/A

lGaA

sheterostru

cture.


0.30. The lattice constant and bandgap energy for x < 0.45 have the following rela-

tions; lattice constant = 0.56533+0.00078x and Eg = 1.424+1.247x at the high sym-

metric Γ point from the band structure at room temperature. The bandgap energy

difference between two layers of GaAs and Al0.3Ga0.7As is ∆Eg = Eg(Al0.3Ga0.7As)

-Eg(GaAs) = 1.798 − 1.424 = 0.374 eV with lattice mismatch ∼ 0.000234nm by 0.04

% difference, which is ever better than the mismatch between GaAs and AlAs. The

question remains how to construct energy bands of two materials at the interface

where two layers meet. Anderson’s rule is a simple theory to answer the energy band

alignment question based on electron affinity from the vacuum levels [?, 61, 123]. It

says that the electron affinity of two materials should be in the same line. Following

Anderson’s rule, the GaAs/AlGaAs heterostructure energy band profile is shown in

Fig. 6.1 (a) by taking the affinity values of GaAs and Al0.3Ga0.7As, 4.07 eV and

3.74 eV. The affinity line match fixed the conduction band difference by ∆Ec ∼ 0.33

eV. Then, the remaining portion 0.04 eV from the bandgap energy 0.37 eV should

drop across the valence band, ∆Ev ∼ 0.04 eV. This heterojunction is a basic compo-

nent to grow quantum wells by sandwiching a narrow band GaAs with broad band

AlxGa1−xAs materials for exploring optical and electrical properties.

In the above heterostructure or quantum wells, all valence bands are occupied

at zero temperature while conduction bands are empty, which means systems are

insulating. In order to probe transport characteristics, charge carriers should be

introduced in the conduction band. There are several ways to do it: one is to use

optical pumping which excites electrons from valence bands to conduction bands

by light sources like lasers; another is to place electrically electrons or holes in the

bands, namely doping. A straightforward way to dope materials is to insert donors

or acceptors in designated locations; however, it is not really gainful since ionized

donors or acceptors become scatterers which ruin the quality , thus this way should

be avoided for the high mobility sample. The solution to this issue is modulation

doping. Its advantage is to separate ionized atoms from the interface, the location of

a 2DEG. In this case, the concentration of scatters is significantly reduced, yielding

the improved mobility of electrons in a 2DEG. The presence of ionized donors or

acceptors and migrating electrons or holes break neutrality in the growth direction.

130 CHAPTER 6. QUANTUM POINT CONTACT

This creates a built-in potential near the interface. As a result, the Fermi energy sits

in the conduction band, leaving the non-zero population of electrons in the potential

whose shape is more like a triangular well in a simple picture. Figure 6.1 (b) presents

a 2DEG within the GaAs conduction band, accordingly.

A Confinement Potential

A potential to trap a 2DEG near the interface is simply modelled as a triangular

well potential created by a linear electric field along the growth direction in z- axis,

Vz = eEz where e is the electron charge and E is the electric field magnitude. Note

that without losing a generality and with following the conventional choice, the growth

direction is designated in z-axis. The triangular well potential profile is valid only

for small z since the potential in the real heterostructure becomes flat far away from

the interface. Suppose z = 0 denotes the interface between GaAs and AlGaAs. The

corresponding Schrodinger equation with wavefunction in z-axis φ(z) is written as

[

− ~2

2m∗d2

dz2+ eEz

]

φ(z) = ǫφ(z). (6.1)

A strategy to solve Eq. (6.1) is to reduce it with dimensionless parameters which

replace z and ǫ. As a mathematical point of view, how to define such parameters

relies on pure dimensional analysis. Consider the following rearranged equation,

−d2φ(z)

dz2+

2m

~2eEzφ(z) =

2m

~2ǫφ(z). (6.2)

Since the wavefunction φ(z) has dimension in [length−1/2] but it is present in all

terms, its dimension can be neglected. The first term has the unit of [length−2] by

second derivative in z whereas the second is that of [length] from z. In order to

match dimension for both terms, the second term should be multiplied by something

of [length−3]. Thus, we can define the length scale z0 from the coefficients in front of

z to make a dimensionless parameter, z = z/z0, yielding z0 = (~2/2meE)1/3. A next

task is to express Eq. (6.2) in terms of z. Because of z = z/z0, the relations of the first

and second derivative between two parameters z and z are d/dz = (d/dz)(dz/dz) =


(1/z0)(d/dz) and d2/dz2 = (1/z20)(d

2/dz2). Therefore, Eq. (6.2) is restated as

−d2φ(z)

dz2+ zφ(z) =

2mǫ

~2z20φ(z). (6.3)

Now, we are able to define the energy scale by ǫ = ǫ/ǫ0 where ǫ0 = ((eE~)2/2m)1/3.

Finally, the Schrodinger equation simplifies into

d2φ(z)

dz2+ (ǫ− z)φ(z) = 0. (6.4)

Note that indeed z0 and ǫ0 have the unit of [length] and [energy] automatically since

expression of z0 is in MKS[

kg·m2/s2·sJskgJ/m

]1/3

= [m3]1/3 = [m] and ǫ0 = eEz0 = [J ]. The

choice of parameters is just good. Equation (6.4) resembles Airy or Stokes equation,

d2y/dx2 = xy, which has two independent solutions called Airy functions Ai(x) and

Bi(x). The solutions are not simply expressed in an analytical form, but in modified

Bessel functions of order 1/3, whose numerical tables are available. Although both Ai

and Bi oscillate for a negative x, the asymptotic behaviors of Ai and Bi are distinct

in that Ai converges but Bi diverges as x grows in positive end. Thus, the solution to

Eq. (6.4) should be Ai not Bi due to convergence. Satisfying the boundary condition

φ(z = 0) = 0 at z = 0 yields quantized bound state energies ǫn = cn((eE~)2/2m)1/3.

The ground state energy has c1 ∼ 0.23381 from the exact Airy function evaluation.

Although this simple triangular-well model provides a qualitatively correct picture.

More realistic potentials would shape close to flat for large z including many-electron

effects. Variational Hartree approximations with Fang-Howard wavefunctions or nu-

merical Hartree self-consistent calculations have been pursued in these efforts [61].

6.1.2 Scattering Mechanism

Since the advent of 2DEGs in semiconductors, one of the main focus in material

preparation is to produce good quality 2DEGs. A good quality 2DEG is crucial

to study intrinsic features according to dynamics of electrons and their governing

principles. A figure of merit to identify the quality of a 2DEG is the mobility µ,

which provides information as to scattering processes in 2DEGs. The mobility µ is


a transport quantity and is given by the ratio of the drift velocity and an applied

electric field. Several lengthscales in mesoscopic systems introduced in Chapter 1 are

closely linked to this quantity µ. First of all, mean free path is lmfp = vF τ , where the

relaxation time τ is proportional to µ through µ = eτ/m∗. Phase-coherence length lφ

is also a function of µ in a slightly more complicated way via the relation lφ =√

Dτφ.

The mobility µ is related to the diffusion constant D via D = v2F τ/2 = v2

Fm∗µ/2e. In

this case, there is another time scale τφ, which represents the phase relaxation time.

Typically this τφ would be longer than τ . These two examples of µ involvement in

dynamics of electrons clearly state that µ is determined by scattering sources in the

2DEG.

There exist several sources to scatter electrons in semiconductor heterostructures:

defects of crystal structures, impurities in structure layers, and lattice vibrations

(phonons). At low temperatures, the last option, phonons would be negligible; how-

ever, other scattering centers will remain to limit the quality of systems. The previous

subsection briefly mentioned that the way to introduce doping contents greatly af-

fects scattering mechanisms. The standard reported number of the high mobility in a

2DEG is around µ = 107 cm2/Vs with a carrier density n2D = 2×1011 cm−2 achieved in

a delta-doped GaAs/AlGaAs heterostructure which has an inserting undoped spacer.

The spacer spatially separates ionized donors from a 2DEG. The values are truly

related to the structure design of a doping layer as well as the purity of molecular

beam epitaxy machine and target sources, which unintentionally introduce impurities

during growth processes.

Several research groups scrutinized scattering mechanisms as a function of electron

sheet density in a 2DEG [124–126]. The existent scattering sources in a delta-doped

heterostructure at low temperatures are divided into three categories: (1) remote

ionized donors, (2) unintentional background impurities in GaAs and the undoped

AlGaAs spacers and (3) interface roughness. The dominant scattering mechanisms

are different depending on electron density. At low n2D less than 5×1010 cm−2, the

major disorder comes from remote ionized impurites [124,125], whereas as n2D is more

than 5×1010 cm−2, homogenous background doping impurities are dominant with an

empirical trend µ ∝ nδ2D where δ ≈ 0.6 ∼ 1.1. The density-dependent trends in µ


would be understood in a handwaving way that in the case of high n2D, more elec-

trons involve to screen effectively the potential induced by remote ionized impurities,

while background doping impurities remain unaffected by screening. According to

the literature, the 2DEGs with n2D ∼ 1012 cm−2 would have µ ∼ 106−7 cm2/Vs, so

that the corresponding mean free path is extended into 10 - 100 µm at submilli K

and clear features such as quantized conductance plateaus and quantum Hall effects

in a ballistic transport regime are exhibited.

6.1.3 Backgated 2DEG

From the previous discussion regarding scattering mechanisms, delta-doped 2DEG

structures at low temperature are superb to produce clean experimental results. Here

the devices we study have different methods to introduce conduction electrons in an

heterostructure not by doping but by backgating to n-type GaAs layer at the bot-

tom [127]. Such devices whose growth structure is sketched in Fig. 6.2 (a) are

designed for probing electron-electron interactions as a function of electron density

in the 2DEG because the strength of Coulomb interaction is directly influenced by

the number of electrons in a system. Figure 6.2 (b) shows a band diagram of induc-

ing carriers in a 2DEG by positive backgate voltages. The operation of a backgated

2DEG starts off the assumption that a surface charge density and its correspond-

ing electric field at the 2DEG surface are constant. As voltages get larger than the

threshold value Vbg,th applied to the n-type GaAs with respect to ground, the con-

duction band shifts downward, making the levels in a 2DEG align with the chemical

potential of the source. Consequently, electrons from the source electrode travel into

the 2DEG. The presence of electrons in the 2DEG screens the electric field induced

by the constant surface charge, thus the electron density is linearly increasing with

Vbg,th to values between 0.5 × 1011 cm−2 and 3 × 1011 cm−2 [128]. Irrespective of

undoping, the mobility after illumination at 1.6 K reaches a remarkably high value

of 5 × 106 cm2/Vs for n2D ∼ 3.5 × 1011 cm−2 , which is a record mobility value for

inverted GaAs/AlGaAs heterostructures [127]. Besides density controllability, these

devices enjoy the benefit of greatly reduced background impurity scattering. One of


(a)

(b)

Vbth

d2d1

2DEG

Φs0Φ i

Vbth

d2d1

2DEG

Φs0Φ i

Vb,th

2 DEG

Figure 6.2: (a) Growth structure of backgated 2DEG. d1 and d2 are the thickness ofGaAs and the two layers of AlGaAs and superlattice barriers respectively. (b) Banddiagram of backgated 2DEG operation. A thick solid line represents the case of Vb,th,a dotted line is for the above Vb,th and a thin solid line is for the below Vbg,th.

6.2. QUANTUM POINT CONTACT 135

120

100

80

60

40

20

0

V

H (

µV

)

1.00.80.60.40.20.0

B (T)

VB=2.2V

VB=2.3V

VB=2.4V

VB=2.5V

VB=2.7V

,BV n

3.0

2.8

2.6

2.4

2.2

D

en

sit

y (

10

15 m

-2)

2.72.62.52.42.32.2

VBG (V)

(a) (b)

Figure 6.3: (a) Measured Hall voltage as a function of external magnetic field perpen-dicular to 2DEG by changing the backgate voltages VBG. (b) The calculated electrondensity of 2DEG versus VBG.

these devices show density variations in measurements by applying a magnetic field

perpendicular to the 2DEG in Fig. 6.3.

6.2 Quantum Point Contact

Early attempts to form point contacts were tried with metals in a crude manner by

pressing two sharp ends [46,129]. It was extremely challenging by this method to fab-

ricate small confinements comparable to the Fermi wavelength λF around 1 A, not to

mention the lack of reproducibility. In these aspects, quantum point contacts (QPCs)

in semiconductors have superior benefits over metallic ones: relatively straightforward

width control around a few λF ∼ 40 nm in a high-mobility 2DEG and reproducible

fabrication recipe development. A QPC in a two-dimensional electron gas (2DEG)

system has been a prototypical device used to investigate low-dimensional mesoscopic

physics. Several methods to produce such constrictions are imprint lithography [130],


in-plane gate by focused ion etching [131], and split-gate technique [45]. The last

method is the most commonly used one based on the fact that negatively applied

voltages to lithographically patterned Schottky gates on top of a 2DEG creates an

electrostatic potential deep in a 2DEG. The electrostatic potential induces additional

spatial confinement to leave only one freely propagating direction (Fig. 6.4). It also

depletes electrons underneath to separate the electron reservoirs into two regions,

so-called source and drain. To adjust the effect of QPC potential in a controllable

manner with a Schottky gate voltage governs the width of the constriction (W ),

the route between source and drain, ultimately conductance. The QPC acts as an

electron waveguide, allowing certain modes to propagate through as W equals to mul-

tiples of half wavelength, W = nλ/2 with integer n. Note that a QPC is not truly

one-dimensional since W and length (L) of such confinements, in general, are rougly

λF ∼ W < L. It is rigorously quasi-one dimensional.

6.2.1 Conductance Quantization

As a manifestation of coherent electron propagation in a ballistic one-dimensional

electron waveguide, there are quantized conductance plateaus in integer-multiples

of the spin-degenerate quantum unit of conductance GQ = 2e2/h as the width of

a QPC increases. Recently, the quantum modes of coherent electrons under QPCs

were imaged by atomic force microscopy [132]. The simplest expression of current

in terms of GQ and the energy-independent transmission probabilities of transverse

channels Tn under finite drain-source voltages Vds is I =∑

nGQVdsTn. This simple

integer-plateau picture is true as long as Vds is kept small. A quantitative model to

produce such features and the non-zero Vds effects is presented and discussed in this

subsection.

A Saddle-Point Constriction

An electrostatic potential in a 2DEG associated with the negative gate voltage to the

Schottky electrodes is nicely modelled as a saddle-point potential, a linear combina-

tion of two harmonic oscillator potentials in a x−y plane. The saddle-point potential


y

xZ

Vg < 0

y

xZ

y

x

y

xZ

Vg < 0

(a)

(b)

k

E

Figure 6.4: (a) Schottky-split techniques to form a QPC in a 2DEG sysgem. (b)Sketch of energy dispersion of electron reservoirs and QPC


under a zero magnetic field is written as

U(x, y) = U0 +1

2m∗ω2

yy2 − 1

2m∗ω2

xx2, (6.5)

assuming x is the longitudinal coordinate. U0, the potential at the saddle point has a

bias-voltage dependence, m∗ is the effective electron mass, and ωx, ωy are frequencies

of harmonic potentials. Transmission coefficients in the saddle-point constriction were

calculated analytically [133,134],

Tn(E) =1

1 + e−πǫn(E),

and ǫn, the lowest value of each transverse channel due to confinement potentials, is

found

ǫn(E) =2

~ωx

[

E − ~ωy(n+1

2− U0)

]

,

normalized by ~ωx. By definition, the current is computed from Tn(E) for EF −eVds >

0

I =2e

h

∫ EF

EF−eVds

∑

n

Tn(E).

Since Tn(E) is not a function of Vds, the differential conductance is dG = dI/dVds =

GQ

∑

n Tn(E). Note that ~ωx is taken as a energy scale throughout the theoretical

consideration. Figure 6.5 (a) depicts a saddle-point potential expressed in Eq. (6.5)

with the ratio of two frequencies ωy/ωx ≡ a = 2 and m∗ ∼ 0.067me for GaAs where

me is the electron mass. At first, the term U0 is assumed to be constant in Vds. In

the case of such a simple potential, the quantized energy level are readily expressed

ǫn(E) = 2

[

E − U0

~ωx

− ~ωy

~ωx

(n+1

2)

]

. (6.6)

In Eq. (6.6), the first term in the bracket is a variable corresponding to the voltage

applied to the Schottky gates on top of the 2DEG, eVg/~ωx. Defining z ≡ (E −


U0)/~ωx, transmission coefficients are

Tn(z) =e2π[z−a(n+1/2)]

1 + e2π[z−a(n+1/2)].

Quantized plateaus in differential conductance are drawn in Fig. 6.5 (b) as Vg in-

creases, in other words, the constriction width becomes wider.

Non-Integer Conductance Plateaus

The previous saddle-point potential model is successful to describe quantized conduc-

tance plateaus at zero drain-source voltage. Natural questions arise that what would

happen as the drain-source voltage increases. Since transmission coefficients in this

model do not depend on the drain-source voltage, it is expected to have no drain-

source voltage effect on conductance trace. However, Kouwenhoven et al. reported

that nonlinear current-voltage characteristics were observed for high drain-source volt-

age opposed to the theoretical model prediction [135]. Based on experimental obser-

vation, the authors considered carefully the position of the Fermi level (EF ), chemical

potentials of left and right reservoirs, µL, µR in the energy dispersion diagram. As

long as the source-drain voltage is small enough, the number of transverse channels

is always the same for both forward and backward electron flow direction. However,

the story becomes different as soon as the difference of drain and source chemical

potentials is larger than the energy spacing of allowed bands. Figure 6.6 visualizes

the latter case in that µL lies in even below the lowest energy E0; thus, there is no

left moving channel. They claimed that the difference between forward and backward

propagating transverse channels within the bias energy window (µR −µL) would gen-

erate non-integer conductance plateaus for large Vds. A mathematical formulation

of this argument is quite straightforward. Suppose that the actual voltage drop at

left and right electrodes would be βeVds and (1 − β)eVds where 0 ≦ β ≦ 1. Then,

µR = EF + βeVds and µL = EF − (1 − β)eVds, which satisfies the initial construction

µR − µL = eVds.

A similar approach is taken to Eq. (6.5), introducing a drain-source voltage (Vds)

effect. Phenomenologically, the term U0 can be replaced by U0(Vds) = U0 − βeVds to


(b) 4

3

2

1

0

dG

(/

GQ

)

1086420 eVg (2π/hωx)

Vds = 0 V

xy

U

xy

U

xy

U

(a)

Figure 6.5: (a) The saddle-point potential for |ωy/ωx| = 2. (b) computed differentialconductance for the saddle-point potential (a).


(1¡ ¯)eVds

¯eVds¹R

¹L

EF

E1

E0(1¡ ¯)eVds(1¡ ¯)eVds

¯eVds¯eVds¹R¹R

¹L¹L

EFEF

E1E1

E0E0

2.0

1.5

1.0

0.5

0.0

dG

(/

GQ

)

76543210eVg(2π/hωx)

Vds = 0 mV

Vds = 1 mV

Vds = 2 mV

(a)

(b)

Figure 6.6: (a) The actual voltage drop effect in left and right moving channels. (b)Computed differential conductance for non-zero Vds cases.


first order in Vds. β indicates how much portion of Vds drops across the source-device

interface and across the drain side, thus the range of β is [0,1]. U(Vds) yields Tn such

as

Tn(ǫn(E, Vds)) =eπǫn

1 + eπǫn, (6.7)

where ǫn(E, Vds) = 2~ωx

[

E − ~ωy(n+ 12) − U0(Vds)

]

.

The corresponding current for EF − eVds > 0 is

I(Vds) =2e

h

∫ µR

µL

Tn(ǫn(E, Vds))dE

=2e

h

∑

n

∫ µR

µL

eπǫn

1 + eπǫndE.

(6.8)

Substituting y = 1+ eπǫn = 1+ e2π

~ωx[E−~ωy(n+ 1

2)−U0(Vds)], the integral in E converts

into dy with dy = 2π/~ωxeπǫndE. This integral can be easily computed by simple

calculus:∫ y2

y1(~ωx/2π)(dy/y) = (~ωx/2π) ln(y2/y1). Equation (6.8) becomes

I(Vds) =2e

h

∑

n

~ωx

2πln

[

1 + e2π

~ωx[EF−~ωy(n+ 1

2)−U0(Vds)]

1 + e2π

~ωx[EF−eVds−~ωy(n+ 1

2)−U0(Vds)]

]

. (6.9)

Since transmission coefficients are a function of Vds as well as E, computing the

differential conductance from current requires extra care since both integrand and

integral ranges contain Vds. Suppose a case to differentiate with respect to x a integral

in y of a function f(x, y). If F (x, y) =∫

f(x, y), the remaining task is the following:

d

dx

∫ b(x)

a(x)

f(x, y)dy =d

dx[F (x, b(x)) − F (x, a(x))]

= [f(x, b(x)) − f(x, a(x)) + b′(x)f(x, b(x)) − a′(x)f(x, a(x))]

= b′(x)f(x, b(x)) − a′(x)f(x, a(x)) +

∫ b(x)

a(x)

d

dxf(x, y)dy.

Taking into account of the above derivative result, the differential conductance dG is


derived as [136],

dG =2e2

h

∑

n

[

(

1 +dU0(Vds)

d(eVds)

)

e2π


2)−U0(Vds)]

1 + e2π


2)−U0(Vds)]

− dU0(Vds)

d(eVds)

e2π

~ωx[EF−~ωy(n+ 1

2)−U0(Vds)]

1 + e2π

~ωx[EF−~ωy(n+ 1

2)−U0(Vds)]

].

(6.10)

A natural choice of β is 0.5, which means that Vds drops symmetrically across drain

and source. Figure 6.6 (b) shows the differential conductance traces at three Vds

values, the evolution trend of integer plateaus to half-integer plateaus. In principle,

any non-integer conductance plateaus would emerge according to the value of β,

revealing the degree of symmetry between two reservoirs and a 2DEG.

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.30 -0.25 -0.20 -0.15 -0.10 VSchottky (V)

Date: 07-07-03

Vbg=2.5V400 uVpp -200uVoffset

Figure 6.7: The 0.7 structure from differential conductance measurements at 1.5 K.


6.2.2 The 0.7 Structure

In addition to conductance plateaus at integer multiples of quantum unit of conduc-

tance for low bias voltages across QPC, a remarkable feature around 0.6 - 0.8 GQ

has been identified [137]. It becomes denoted as the ‘0.7 structure’ or ‘0.7 anomaly’

since this feature is beyond simple non-interacting single particle picture. It is quite

robust and universal in a sense that any group observes this shoulder structure below

the first conductance plateau regardless of QPC shapes, 2DEG structures and het-

erostructure material. Not only QPC structures, quantum wires formed by cleaved

edge growth techniques preserve the feature as well [138]. Figure 6.7 plots a differen-

tial conductance of backgated quantum point contact at 1.5 K by sweeping Schottky

gate voltages, containing that deviating feature from the smooth steps starts near 0.7

GQ. Although its physical origin is under investigation and a microscopic-level un-

derstanding has yet been established, it is believed to be related to electron-electron

interactions unlike conductance plateaus, which are explained very well in terms of

non-interacting single electron approximation.

This speculation on correlated electrons was initiated from an experimental fact

that the 0.7 structure evolves to 0.5 when the spin degeneracy is lifted by in-plane

magnetic field [137] and enhanced g-factor under in-plane magnetic field [139], im-

plying that two-electron effect is reminiscent in a zero-field. In order to ferret out

such conjecture, many groups have been performing conductance measurement by

choosing practical tuning variables which may elucidate the behavior of interacting

electrons behavior in systems since late 1990s: channel length [140, 141], tempera-

ture [141–143], in-plane B-field [137, 143–146] bias Vds [142, 144, 145] and electron

density in 2DEG [140, 144, 147, 148]. Recently, more attentions move to shot noise

measurements near such structures [149,150].

There have been numerous theoretical models to explain the 0.7 structure in terms

of experimental variables such as temperature, in-plane magnetic field , bias Vds and

electron density: thermal activation field [151], Kondo effect [152], phenomenological

N-electron bound state model [153], spontaneous spin-polarization [154, 155] spin-

orbit coupling [156, 157], and phenomenological model [148]. In this chapter, the

spin-orbit coupling in the non-interacting single electron picture is discussed as a


candidate to reproduce the 0.7 structure. This consideration lacks of many-body

effects, thus the further thought should be included in this approach; however, it

certainly has a pedagogical purpose.

Spin-Orbit Coupling

The spin-orbit (SO) coupling model takes a position of spontaneous spin polariza-

tion to a large extent. The formation of the 0.7 structure is caused by lifting spin

degeneracy due to spin-orbit interactions.

Spin-orbit coupling is pure quantum mechanical effect. It occurs when an electron

moves with a momentum ~p along the electric field ~E. Due to the relativistic motion,

the electron does feel the effective magnetic field ~B induced as ~B = −(~p × ~E)/mc

with an electron mass m and the speed of light c. Its Hamiltonian operator has a

form of HSO = ~S · ~L where ~S and ~L are spin angular momentum and linear angular

momentum operators. The spin-orbit (SO) Hamiltonian then is written in terms of

the momentum operator ~p, Pauli spin matrices ~σ and the gradient of a electrostatic

potential ~∇V ,

HSO =~

4m2c2(~∇V × ~p) · ~σ. (6.11)

In semiconductors, there are two types of SO interactions. These are related to

broken symmetries: Dresselhaus SO interaction is one due to bulk inversion asymme-

try described by the Hamiltonian HSO,D = β(σxkx−σyky) and Rashba SO interaction

is caused by structure inversion asymmetry described by the Rashba Hamiltonian

HSO,R = α(σxky −σykx) where α and β are coupling constants depending on material

characteristics. Between two SO interactions, for a QPC in a 2D plane, the Rashba

SO has bigger effect on transport properties. Therefore, here the Rashba SO effect is

focused on.

Case 1: Simple Harmonic Confinement Potential

Suppose the confinement potential in y-direction is U(y) = (1/2)~ω2, a simple har-

monic oscillator with an oscillating frequency ω. The Hamiltonian operator including


SO interaction becomes [156]

H =p2

2m∗ + U(y) − iασy∂

∂x+g∗µB

2~σ · ~B

=p2

2m∗ +1

2~ω2 − iασy

∂

∂x+g∗µB

2~σ · ~B,

(6.12)

under the influence of the magnetic field ~B. Herem∗, g∗ are effective electron mass and

effective g-factor and µB is the Bohr magneton. An ansatz solution of the Schrodinger

equation with Hamiltonian Eq. (6.12) has an easy form since x and y are separable,

ψ(x, y) = eikxφ(y)

(

ϕ↑

ϕ↓

)

, (6.13)

where k is the wavevector in x-direction, φ(y) is the wavefunction for the transverse

direction y and ϕ↑, ϕ↓ denote spinors with up and down spin configuration respec-

tively. Writing the Hamitonian operator in matrix using the basis of spinors is very

conveninent in order to calculate eigenenergies and eigenfunctions. Using the matri-

ces of spinors, ϕ↑ =

(

1

0

)

≡ | ↑〉 and ϕ↓ =

(

0

1

)

≡ | ↓〉, action of Pauli matrices

on the spinors yields:

σx| ↑〉 =

(

0 1

1 0

)(

1

0

)

=

(

0

1

)

= | ↓〉

σx| ↓〉 = | ↑〉

σy| ↑〉 =

(

0 −ii 0

)(

1

0

)

=

(

0

i

)

= i| ↓〉

σy| ↓〉 = −i| ↑〉〈↑ |σx| ↑〉 = 〈↓ |σx| ↓〉 = 〈↑ |σy| ↑〉 = 〈↓ |σy| ↓〉 = 0

〈↑ |σy| ↓〉 = −〈↓ |σy| ↑〉 = −i.

(6.14)

In the third term of Eq.(6.12) one should consider the eikx term of the wavefunction


ψ due to the partial derivative with respect to x,

〈ψ(↑)| − iασy∂

∂x|ψ(↓)〉 = e−ikxφ∗(y)〈↑ | − iασy

∂

∂xeikxφ∗(y)| ↓〉

= −iα(ik)〈↑ |σy| ↓〉 = iαk

〈ψ(↓)| − iασy∂

∂x|ψ(↑)〉 = −iαk.

(6.15)

Together with Eqs.(6.14) and (6.15), the Hamiltonian matrix is

H =

(

~2k2

2m∗+ Etr

n iαk + g∗µB

2(Bx + iBy)

−iαk + g∗µB

2(Bx − iBy)

~2k2

2m∗+ Etr

n

)

, (6.16)

where Etrn is the quantized energy level from y-confinement equal to ~ω(n + 1/2).

Diagonalizing the Hamiltonian, the eigenenergies are

E±n =

~2k2

2m∗ + ~ω(n+1

2) ∓

[

(

g∗µBB

2

)2

+ 2µBαkB sin θ + (αk)2

] 1

2

. (6.17)

Since B2 = B2x + B2

y and By = B sin θ where θ is the angle along the y-direction,

Eq.(6.17) is expressed by the magnitude of total magnetic field, B and the angle θ.

SO - Effect in the Band Structure

Figure 6.8 compares two dispersion relations for zero (a) and non-zero (b) spin-

orbit coupling α by plotting Eq.(6.17) with dimensionless variables ξ(n) ≡ E±n /~ω,

P ≡ ~k/√

2m∗~ω and α′ ≡ α√

2m∗/~3ω. When magnetic field is absent, Eq.(6.17)

is much simplified to ξ(n) = P 2 ± α′P + (n + 1/2), which is used to generate plots

with α′ = 0 for (a) and α′ = 1 for (b) in Fig. 6.8. Note that non-zero magnetic field

induces anticrossings of channels which produce gaps in the spectra (Fig. 6.8 (c)).

The general formula for both non-zero B and θ is

ξ(n) = P 2 ∓√

B′2 + (α′P )2 + α′B′P sin θ + (n+1

2), (6.18)

where B′ is the normalized magnetic field such that B′ = g∗µB/2~ω.


P

(a)

(b)

P(c)

P

»

»

»

P

(a)

(b)

P(c)

P

»»»

»»»

»»»

Figure 6.8: (a) The band structure without spin-orbit interaction under zero magnetic

field ~B = 0 for the first five n. (b) The band structure with non-zero spin-orbit

interaction at ~B = 0. (c) The band structure with non-zero spin-orbit interactionat finite magnetic field. In all three cases magnetic field is perfectly aligned in x-direction, i.e. θ = 0. P and ξ are defined in the context.


SO - Effect in Differential Conductance

The current from such band structures is written as

I =e

2π

∑

n,s

∫ ∞

−∞vns[Θ(vns)f(E, µL) + Θ(vns)f(E, µR)]dk, (6.19)

with the electron velocity vns for channel with band index n and spin s, step function

Θ and Fermi function of left and right reservoirs whose chemical potentials are µL

and µR respectively [156]. The velocity is the slope of the dispersion relation, vns =

(1/~)(∂En±/∂k), where

∂En±

∂k=

~2k2

m∗ ± 1

2

2α2k + αg∗µBB sin θ√

g∗µBB2

+ αg∗µBB sin θk + (αk)2

. (6.20)

Care should be taken to the integral range according to the positive and negative

velocities for forward and backward propagating modes in the band structures. The

interesting quantity now is the differential conductance obtained from taking a deriva-

tive of Eq.(6.18) with respect to the bias voltage eV = µL − µR. The explicit V

dependence appears in the Fermi functions . If µL = µR + eV , the surviving term is

only f(E, µL) = f(E, µR + eV ) after differentiation.

Assume that the magnetic field is aligned along x-direction. The differential con-

ductance is computed considering the velocity direction as well as temperature. The

results are plotted in Fig. 6.9 without and with the magnetic field for various tem-

peratures. All energy scales are renormalized by the confinement potential energy ~ω.

Note that the unit of differential conductance in this numerical analysis is taken to

be e2/h, the quantum unit of conductance without spin-degeneracy. When tempera-

ture is zero, the conductance trace for zero-magnetic field (Fig. 6.9 (a)) shows ideal

step-like plateaus, indicating that the Fermi functions are exactly step functions as

expected. In addition, the number of propagating channels in both directions remains

the same. However, as soon as the magnetic field is turned on, due to the presence

of anticrossings in the band structure, there exist dips which correspond to energy


¹¹

dG(=(e2 h))

dG(=(e2 h))

dG(=(e2 h))

dG(=(e2 h))

¹¹

(a)

(b)

Figure 6.9: Computed differential conductance as spin-orbit coupling in a simpleharmonic oscillator potential is on while the magnetic field is kept zero in (a) and themagnetic field is 0.05 in the unit of g∗µBB/2~ω in (b).


gaps among bands, making the participant channel numbers different. The trace at

zero temperature evolves smoothly to the ones at finite temperature, resembling the

observed 0.7 structure before conductance plateaus are completely washed out due to

thermal broadening. The key to validate this model by experiments is to choose the

appropriate condition to satisfy the numerical energy scaling ranges. Up to now, the

electrostatic potential formed in the 2DEG to confine systems is not well controllable

practicallyt; however, other parameters would be rather flexible to be selected for

tests.

dG(=(e

2 h))

dG(=(e

2 h))

¹¹

Figure 6.10: Computed differential conductance as spin-orbit coupling is on whilemagnetic field is kept zero.

Case 2: Saddle-Point Potential

Although the simple harmonic potential produces a promising perspective towards

the 0.7 structure, it is a natural extension to apply the spin-orbit concept to the

saddle-point potential, which is rather physical model as a confined potential in real

2DEGs. The simplest attempt is done for the potential keeping only the linear Vds


term and for spin-orbit coupling without a magnetic field applied in the system. The

absence of the magnetic field does not induce anticrossings, consequently no bumps

below plateaus for the saddle-point potential shown in Fig. 6.10. Therefore, in this

story there should be non-zero magnetic field to cause anticrossings between bands.

The tasks become quite complicated to compute the differential conductance with

nontrivial band structures arising from a nonzero magnetic field and its orientation

in the system. However, these remaining tasks may be worthy of being pursued

in future, examining the effect of combining the potential shape and the spin-orbit

interaction as one of the approaches regarding the microscopic understanding of the

0.7 structure.

The initial attempt to incorporate the spin-orbit interactions in the single-electron

picture with a harmonic oscillator seems promising to produce a similar shape of the

0.7 structure near the 0.7 GQ value. This interaction differentiates spin-up and spin-

down electrons which propagate through the narrow channel. This concept of this

model is reasonable and plausible in the real system, thus the further rigorous calcu-

lation including the saddle-point potential and careful experimental implementation

would be highly on demand.

6.3 Differential Conductance

Implementing experimental methodology described in Chapter 4, nonequilibrium

transport of a single QPC through differential conductance and low-frequency shot

noise were studied at 1.5 K. Figure 6.11 shows the shape of a Schottky split-gate on top

of a 2DEG taken by a scanning electron microscope and the image of a wire-bonded

whole chip which has the Hall-bar structure with Ohmic contacts and voltage probes.

The Hall-bar structure has a capability to characterize the quality of the 2DEG as

well as to access quantum Hall regimes by applying a perpendicular magnetic field

to the 2DEG plane. Backgated QPCs in question have several ways to regulate the

electron density. Three experimental variables are used:(1) backgate voltage (Vbg) to

vary the electron sheet density in the 2DEG, (2) Schottky split-gate voltage (Vg) to

control the entrance of electrons from reservoirs, and (3) drain-source voltage (Vds)


(a)

(b)

(a)

(b)

Figure 6.11: (a) Scanning electron microscope (SEM) image of a quantum pointcontact in a AlGaAs/GaAs 2DEG. (b) The Hall-bar structure of the wirebondeddevice taken by SEM.


-0.30

-0.25

-0.20

-0.15

-0.10

Vg (

V)

3210-1-2-3

Vds (mV)

(b)

1.00.9 0.9

0.50.5

2.5

2.0

1.5

1.0

0.5

dG (

/GQ

)

-3 -2 -1 0 1 2 3

Vds (mV)

(a)

1.5

1.0

0.5

dG

(/G

Q)

-0.32 -0.28 -0.24 -0.20

Vg (V)

Vds = 0.0 mV = 0.7 mV

= 1.0 mV

= 1.5 mV

= 2.0 mV = 2.5 mV

(c)

Figure 6.12: (a) Experimental differential conductance dG by a sweep of Vg at fixedVbg = 2.3 V at finite Vds (b) Transconductance dG/dVg (c)Vds dependence


to determine number of electrons to flow through the potential constriction. This

section discusses experimental data in terms of three knobs, Vds, Vg and Vbg.

6.3.1 Non-integer Conductance Plateaus at Finite Bias Volt-

age

The Hall-bar structure allowed us to measure four-probe measurements, taking the

actual voltage drop across the QPC. At first, we fixed a 2DEG electron density by

applying a certain backgate voltage Vbg. The Vbg was chosed at 2.3 V for Fig. 6.12.

The measured dG with an ac bias voltage Vac ∼ 100 µV is plotted as a function

of the drain-source voltage Vds and the split-gate voltage Vg. The values of dG are

normalized by the spin-degenerate quantum unit of conductance GQ. Dark regions in

Fig. 6.12(a) emerge as dG traces converge according to Vg change. They correspond to

the conductance plateaus. As Patel and his colleagues pointed out, dG flattens around

integer-multiples of GQ along Vds ∼ 0, whereas away from Vds ∼ 0 dG approaches

plateaus but at different positions [139]. The non-integer conductance plateaus at

different Vds values are clearly illustrated by plotting individual graphs in Fig. 6.12

(c), where the first step emerges below 0.5 GQ when Vds = - 2.5 mV.

We compute the transconductance d(dG)/dVg by differentiating dG in terms of

Vg as a post-analysis. The quantity is presented in a two-dimensional image graph

(Fig. 6.12(b)). Here, black areas correspond the plateaus due to the small difference

between traces along Vg axis. Inside the first big diamond black area, a V-shape red

structure exists. It recognizes the 0.9 structure from the GQ plateau. Furthermore,

we notice that the transition behavior is not identical over the whole conductance

values for finite Vds. Below GQ, an additional shoulder structure around 0.7 GQ is

manifest and it moves to 0.9 GQ. Then the plateau clearly forms below 0.5 GQ at

a large Vds. In contrast, above GQ, as Vds increases, no structure similar to the 0.7

anomaly is apparent and the plateau shows an increasing manner. The appearance of

the non-integer conductance plateaus in terms of Vds is understood quantitatively by a

Vds-dependent saddle-point potential model in x and y directions described previously

upto second-order Vds terms U0(Vds) = U0 − βeVds + γeV 2ds/2 [136]. A higher-order


2.5

2.0

1.5

1.0

0.5

dG

(/

GQ

)

-3 -2 -1 0 1 2 3

Vds (mV)

(a) (b)

(d)(c)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

2.5

2.0

1.5

1.0

0.5

dG

(/

GQ

)

-3 -2 -1 0 1 2 3

Vds (mV)

(a) (b)

(d)(c)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

dG( =GQ)

Figure 6.13: (a) Observed differential conductance traces at each Schottky gate volt-age as a function of Vds. All date were taken at Vbg = 2.3V and 1.5 K. (b) Computeddifferential conductance with the Vds-dependent saddle-point potential up to the lin-ear term, i.e. γ = 0. Including second-order corrections in Vds with two oppositesigns of the coefficient, (c) γ > 0 and (d) γ < 0.


term would exhibit some aspects of the experiment. It indicates that γ relates to the

trend of plateau movements for finite Vds values: A negative quadratic term yields a

decreasing pattern while a positive term yields an increasing pattern shown in Fig.

6.12(c) and (d). The contribution of the theoretical model is to provide qualitative

picture of plateau evolution along with Vds; however, the model fails to replicate the

abnormal plateaus, the 0.7 GQ and the 0.9 GQ, which is beyond the present theoret-

ical model. As suggested in the previous spin-orbit coupling section, one immediate

action is to incorporate the spin-orbit coupling and the second-order saddle-point

potential in order to see whether this action provides promising perspectives or not.

Otherwise, the next candidate is to approach the whole system in the consideration

of many-body interactions.

6.3.2 Density Effect

5

4

3

2

1

G (

/G

Q)

-0.6 -0.5 -0.4 -0.3 -0.2 Vg (V)

5

4

3

2

1

G

(/

GQ

)

-0.6 -0.5 -0.4 -0.3 -0.2 Vg (V)

(a) (b)

Figure 6.14: The tuning variables are Vg and Vbg and dG is measured at four-probetechniques with ac signal Vac ∼ 50 µV and the dc bias (a) Vds = 0 mV and (b) Vds =2 mV. The Vbg varies from 3.0 V (rightmost) to 2.58 V (leftmost) by 0.01V interval.

One of the aim to fabricate backgated QPCs is to study the electron density effect

on the 0.7 anomaly. This time the constant variable throughout measurements is the


drain-source voltages and the sweeping variables are Vg and Vbg.

Figure 6.14 compares two cases of different dc drain-source voltages. The common

feature regardless of Vds sizes is that the threshold value of Vg at which electrons start

flow through the constriction moves leftward as larger Vbg is applied. Vbg indeed

controlls the electron density in a 2DEG. The leftward motion of Vg means that the

more electrons reside in the 2DEG at large Vbg, the bigger pinch-off voltage is required

to deplete all 2DEG electrons. There is rather weak structure near the 0.7 GQ, and no

significant evolving trend of the 0.7 structure is captured as a function of the electron

density or Vbg. However, as the dc bias voltage increases in (b), the robust plateaus

below GQ emerge near 0.5 GQ as well as integer-GQ plateaus. Nuttinck and NTT

colleagues reported a vivid trend about the 0.7 structure evolution into the 0.5 GQ

with the same device structures in the limit of low electron density, claiming that two

degenerate channels are splitting and a two-channel model in the Landauer - Buttiker

formalism may explain it in a phenomenological level [147]. The failure to reproduce

their results would be conjectured that the density is not significantly varying from

the initial and the final Vbg. Thus, the future work should be scrutinizing the role

of electron density in an improved setup, resolving the anomaly structures and the

emergence of integer plateaus together with 0.5 GQ steps.

6.4 Low-frequency Shot Noise

The early shot noise measurements of a single QPC have revealed important in-

formation of ballistic quantum transport. The ballistic transport behaviors related

the conductance plateaus obtained from the shot noise measurements have been

well interpreted in terms of noise suppression within the Landauer-Buttiker formal-

ism [158]. Recent efforts of shot noise measurements have focused the 0.7 anomaly

structures [149, 150, 159]. These work have reported complete shot noise suppres-

sion [159] and a partial suppression in terms of Fano factor whose number is in the

range predicted by a two-channel model [149]. New shot noise measurements of the

0.7 structure provide further evidence that the evolution of noise behavior from the

point of 0.7 GQ to the symmetric point 0.5 GQ occurs as the in-plane magnetic field is


3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20Vg (V)

25x10-6

20

15

10

5

Un

no

rma

lize N

ois

e (a

.u.)

Vds = 0.7 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G

(/

GQ

)

-0.35 -0.30 -0.25 -0.20 -0.15 V g (V)

Vds = - 0.7 mV

= - 1.0 mV

= - 1.5 mV

= - 2.0 mV

= - 2.5 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)

100x10-6

80

60

40

20

0

Un

no

rm

aliz

e N

ois

e (a

.u.)

Vds = 2 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)

80x10-6

60

40

20

0

Un

no

rm

aliz

e N

ois

e (a

.u.)

Vds = 2.5 mV

(a) (b)

(c) (d)

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20Vg (V)

25x10-6

20

15

10

5

Un

no

rma

lize N

ois

e (a

.u.)

Vds = 0.7 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G

(/

GQ

)

-0.35 -0.30 -0.25 -0.20 -0.15 V g (V)

Vds = - 0.7 mV

= - 1.0 mV

= - 1.5 mV

= - 2.0 mV

= - 2.5 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)

100x10-6

80

60

40

20

0

Un

no

rm

aliz

e N

ois

e (a

.u.)

Vds = 2 mV

3.0

2.5

2.0

1.5

1.0

0.5

0.0

G (

/G

Q)

-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)

80x10-6

60

40

20

0

Un

no

rm

aliz

e N

ois

e (a

.u.)

Vds = 2.5 mV

(a) (b)

(c) (d)

Figure 6.15: Bias dependence at Vbg = 2.3 V. (a) Vds = 0.7 mV. (b) Vds = 2 mV. (c)Vds = 2.5 mV. (d) Vds dependent conductance.


applied although the paper does not explicitly state the value of the Fano factor [150].

Using the quantitative information in the paper [150], the Fano factor is managed to

be estimated. The estimate says that the Fano factor is around 0.18 near the 0.7

structure. Thus it supports again that repulsive correlation among electrons near

the structure may be involved; however, these experimental data are yet to solve the

puzzle of the 0.7 structure origin. This section discusses rather conceptually trans-

parent aspects of noise properties based on experimental results in conjunction with

differential conductance.

6.4.1 Noise Suppression at Non-integer Conductance Plateaus

Together with the differential conductance, two-terminal shot noise measurements

were executed at 1.5 K with all necessary technical strategies described in the Chapter

4. The tank circuit resonance occurs around 12 MHz at 1.5 K and a bandpass filter

on the output signal line from the device after the cryogenic amplifier comprise of a

21.4 MHz low-pass filter and a 5.6 MHz high-pass filter. One difference in the setup

from the experimental setup of carbon nanotubes presented in Chapter 5 is that there

is no full-shot noise source in parallel with a QPC device. Therefore, we have not

yet succeeded in absolute calibration of measurement apparatus; however, we are still

able to extract interesting and reproducible trend. For the shot noise signal, there

is the upper bound of Vds to beat the background noise related to the signal-to-the

noise ratio. This is is the limit of the present circuit and the drain-source voltage

should be more than 500 µV.

Figure 6.15 (a)-(c) show three representative plots of shot noise along with linear

conductance versus Vg at three different Vds in Fig. 6.15 (d). Similar behaviors are

observed in other devices as well. No matter what values of Vds are applied, the shot

noise level is clearly minimal when conductance G = I/Vds reached about GQ and

2 GQ regardless of calibration. The degree of the suppression at 3 GQ becomes less

significant for a large Vds. In the transient zones between the multiples ofGQ, the noise

characteristic is rather complex. Below the first plateau, the noise suppression appears

around 0.6 GQ and 0.9 GQ until Vds ∼ 1.5 mV (Fig. 6.15(a)). As Vds further increases,


-0.35

-0.30

-0.25

-0.20

-0.15

V

g (

V)

-3.0 -2.0 -1.0

Vds (mV)

(a) (b)

-0.35

-0.30

-0.25

-0.20

-0.15

V

g (

V)

-3.0 -2.0 -1.0

Vds (mV)

2.5

2

1.5

1

0.9

0.7 0.5

2 GQ

1 GQ

1.5 GQ

0.5 GQ

-0.35

-0.30

-0.25

-0.20

-0.15

V

g (

V)

-3.0 -2.0 -1.0

Vds (mV)

(a) (b)

-0.35

-0.30

-0.25

-0.20

-0.15

V

g (

V)

-3.0 -2.0 -1.0

Vds (mV)

2.5

2

1.5

1

0.9

0.7 0.5

2 GQ

1 GQ

1.5 GQ

0.5 GQ

2 GQ

1 GQ

1.5 GQ

0.5 GQ

Figure 6.16: (a) Two-dimensional plot of shot noise raw data at Vbg = 2.3 V withlines at the conductance values. (b) The contour plot of conductance values in theunit of GQ simultaneously taken with shot noise measurements.

these locations move down to 0.5 GQ and 0.8 GQ (Fig. 6.15(b)). And eventually the

suppressed noise is found only at 0.4 GQ for Vds > 2.5 mV (Fig. 6.15(c)). In contrast,

when G is higher than GQ, only one additional noise reduction is found about 1.6 GQ

or 1.7 GQ regardless of the magnitude of Vds. The plateau structures in G gradually

washes out as Vds increases as shown in Fig. 6.15(d). Although the conductance loses

its step-behavior in high Vds, the noise suppression appears in a clear and robust

manner.

The two-dimensional image plot of shot noise as a function of Vg and Vds is of

ease to perceive the continuous shot noise behavior. The black color depicts the base

shot noise level. The occurrence of the suppressed shot noise can be easily seen in

units of GQ. Furthermore, the actual plot contains other noticeable features. The left


figure has eye-guide lines for conductance values taken simultaneously along the dc

line. The colored contour plot of conductance G = I/Vds(Fig. 6.16(b)) helps to see

the relation of G and the shot noise. Again under GQ, several black strips are visible:

The upper strip relates to the shot noise suppression around GQ, and the lower two

ones start at the conductance values 0.7 GQ and 0.9 GQ. For a large Vds, the shot

noise suppression occurs at less than 0.5 GQ. The shot noise signal in higher G has

a rather simple pattern: the reduced noises are observed around 1.6 or 1.7 GQ and 2

GQ as previously stated.

We notice that the shot noise behavior in the transient zone between the integer

multiples of GQ shares some features with the transconductance two-dimensional

image plot (Fig. 6.12(b)). The peaks in the transconductance correspond to the

larger shot noise signals and the dark areas in the transconductance match to the

black strips in the shot noise image. Moreover, both the transconductance and the

shot noise share common features for G < GQ; the 0.7 structure can be distinctive

and the location of the noise suppression and the new plateaus in d(dG)/dVg occur

around 0.4 GQ as Vds > 2 mV. Within the saddle-point potential model, d(dG)/dVg

is expressed in terms of Ti(1 − Ti) where Ti is the i-th one-dimensional (1D) channel

transmission probability. Since the shot noise has a term of Ti(1−Ti) for a small energy

window, two quantities are closely related. It is not, however, obvious to predict the

response of the shot noise for a large Vds because the shot noise is obtained from the

integral of the energy dependent transmission probability. Qualitatively, the noise

suppression around the plateaus can be expected based on the fact that the current

fluctuations can be zero or low when the current remains constant.

The different characteristics in both the transconductance and the shot noise

are observed in the region of G < GQ and G > GQ. This observation is certainly

beyond the simple saddle-point potential model in a single-particle approximation. In

particular, it is surprising to have the strongly suppressed shot noise at 0.7 GQ. This

empirical fact means that electrons are regulated at 0.7 GQ by a certain governing

physical mechanism. The possible factor relating to the mechanism of the 0.7 anomaly

would be the density of electrons. The shot noise study in terms of the electron density

would provide more information to explore this question in the future.


3.0

2.8

2.6

2.4

V

bg (

V)

-0.6 -0.5 -0.4 -0.3 -0.2

Vg (V)

4

3.6

3

2.6

2

1.5

1

0.9

500

400

300

200

100

0

x1

0-6

3.0

2.9

2.8

2.7

Vb

g (V

)

-0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30

Vg (V)

3.6

3

.5

3

3

2.6

2.5

2

2

1.5

1

1 0

.5

100

80

60

40

20

0

x1

0-6

(a)

(b)

3.0

2.8

2.6

2.4

V

bg (

V)

-0.6 -0.5 -0.4 -0.3 -0.2

Vg (V)

4

3.6

3

2.6

2

1.5

1

0.9

500

400

300

200

100

0

x1

0-6

3.0

2.9

2.8

2.7

Vb

g (V

)

-0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30

Vg (V)

3.6

3

.5

3

3

2.6

2.5

2

2

1.5

1

1 0

.5

100

80

60

40

20

0

x1

0-6

(a)

(b)

Figure 6.17: Two-dimensional image plot of shot noise versus Vg and Vbg at (a)Vds = 1 mV. (b) Vds = 2 mV. The straight lines with number are corresponding tothe conductance values normalized by GQ.


6.4.2 Density Effect

Similar to the differential conductance, density effect on the shot noise has been

studied in the preliminary level. Again, the sweeping variables are Vg and Vbg while

Vds is fixed at a certain value. Two-dimensional image plots of the shot noise signals in

Fig. ?? exhibit the consistent features of the differential conductance: the threshold Vg

appears at higher values for a high 2DEG density; and the suppressed noise at 0.9GQ

evolves to 0.5GQ as Vds increases. Unfortunately, the 0.7 structure is not recognizable

in these plots although individual noise trace contains a partial or full suppression

discussed in the previous subsection. The better resolution of the experiment is

required by improving technical side.

6.5 Summary

The main theme of Chapter 6 is to investigate quantum ballistic transport properties

of a quantum point contact in a GaAs/AlGaAs heterostructure. The general char-

acteristics of two dimensional electron gas systems as a mother material of a QPC

have been discussed in terms of energy band diagrams followed by the formation of

a QPC, introducing the spatial confinement potential models. The transport quanti-

ties, current and conductance are computed in the non-interacting Landauer-Buttiker

formalism. Chapter 6 has also discussed the unresolved feature so-called the 0.7 struc-

ture, and as a theoretical attempt to understand the origin of the 0.7 structure, the

effect of the spin-orbit interaction has been studied. In the last two sections, exper-

imental results of differential conductance and shot noise measurements performed

at 1.5 K are presented and the observations are analyzed. Two independent mea-

surements are closely tightened such as the suppressed noise values are founded near

the conductance plateaus. Therefore, further measurements on both quantities are

essential to explore transport properties including the 0.7 structure in future.

Chapter 7

Conclusions

If I can stop one heart from breaking,

I shall not live in vain;

If I can ease one life the aching,

or cool one pain,

or help one fainting robin

Onto his nest

I shall not live in vain.

− Emily Dickinson

Mesoscopic systems become essential to explore the transition between classical

physics and quantum physics for last several decades. The continuing fabrication

advancement has expanded the coverage of regime and the diversity of structures.

Indeed, unique physics phenomena have been revealed in designed systems, validating

the postulates of quantum physics and providing the new insights of perspective

prototypical devices in the areas of information storage, information transportation,

and information manipulation.

Among numerous specific systems, this thesis has put particular attention to two

one-dimensional mesoscopic structures, investigating the electron motions in non-

equilibrium condition at cryogenic temperatures. It has inquired governing principles

165

166 CHAPTER 7. CONCLUSIONS

to control the behavior of charge carriers given situation. It has described theoreti-

cal aspects of many-electron systems and applied such knowledge into experimental

observations, attempting to understand empirical facts as much as possible and pos-

tulating hypotheses for further investigation.

The study of single-walled carbon nanotubes (SWNTs) has raised interesting is-

sues to probe intrinsic electron-electron interactions which are unavoidable in one-

dimensional systems. The electrical measurements of differential conductance and

shot noise exhibited the signatures of strong interaction among electrons, which

are qualitatively and quantitatively explained in Tomonaga-Luttinger liquid (TLL)

model. Still some features appearing in data are beyond the TLL model, requiring

modified assumptions and better modelling of systems in question. The study can

claim that our experimental techniques provide a way to quantify the contributions

of transmitted and backscattered currents in a three-terminal device which couples

well to the electron reservoirs. The natural extension is to investigate semiconducting

SWNTs, multi-walled carbon nanotubes, and single metallic quantum wire in terms

of electron-electron interactions. It can apply to understand spintronic devices with

ferromagnetic reservoirs with considerations.

The study of quantum point contact has also explored low-dimensional physics,

mainly focusing on the effect of physical variables to regulate electron density. The

new contribution is the detailed shot noise measurement complimentary to differential

conductance. They will be of importance to seek the origin of the universal system

‘0.7 structure’. At present, there are many attempts to explain the feature, but the

complete picture has yet to come. The thesis gives a try to analyze the single-particle

picture with spin-orbit coupling for the 0.7 structure although it is mostly considered

as the symptom of many-body effect. In order to identify the microscopic level un-

derstanding of such feature, careful experiments should be designed and performed

together with advanced theoretical models.

This thesis is along the journey to explore novel physical phenomena occurring

in mesoscopic structures. It is based on previous understanding on them and it is

hoped to add even a slight knowledge in mesoscopic field in order to attain the truth

of science and nature.

Appendix A

Physical Constants

Symbols Physical Constants MKS unit CGS unit eV unit

h Planck's constant 6.62618 £10¡34 J-s 6.62618 £10¡27 erg-s~ h/2¼ 1.05459 £10¡34 J-s 1.05459 £10¡27 erg-s 6.58217 £10¡16 eV-se elementary charge 1.6 £10¡19 C 4.803 £10¡10 esuc speed of light 2.99792 £108 m/s 2.99792 £1010 cm/s® ¯ne structure constant = e2/hc 1/137.036me electron mass 9.10953 £10¡31 kg 9.10953 £10¡28 g 0.511 MeV/c2

mh proton mass 1.67265 £10¡27 kg 1.67265 £10¡24 g 938.279 MeV/c2

aB Bohr radius 0.52918 ºA 0.52918 £10¡8 cm¹B Bohr magneton = e~/2mec 0.9273 £10¡27 J/gauss 0.9273 £10¡34 erg/gauss 5.78838 £10¡9 eV/gausskB Boltzmann's constant 1.3806 £10¡23 J/K 1.3806 £10¡30 erg/KR Gas constant 8.3145 J-K/molNA Avogadro's number 6.02205 £1023 / mol

Symbols Physical Constants MKS unit CGS unit eV unit

h Planck's constant 6.62618 £10¡34 J-s 6.62618 £10¡27 erg-s~ h/2¼ 1.05459 £10¡34 J-s 1.05459 £10¡27 erg-s 6.58217 £10¡16 eV-se elementary charge 1.6 £10¡19 C 4.803 £10¡10 esuc speed of light 2.99792 £108 m/s 2.99792 £1010 cm/s® ¯ne structure constant = e2/hc 1/137.036me electron mass 9.10953 £10¡31 kg 9.10953 £10¡28 g 0.511 MeV/c2

mh proton mass 1.67265 £10¡27 kg 1.67265 £10¡24 g 938.279 MeV/c2

aB Bohr radius 0.52918 ºA 0.52918 £10¡8 cm¹B Bohr magneton = e~/2mec 0.9273 £10¡27 J/gauss 0.9273 £10¡34 erg/gauss 5.78838 £10¡9 eV/gausskB Boltzmann's constant 1.3806 £10¡23 J/K 1.3806 £10¡30 erg/KR Gas constant 8.3145 J-K/molNA Avogadro's number 6.02205 £1023 / mol

167

Appendix B

Conversion Tables

hν = kBT,

ν =c

λ,

ν =1

time.

(B.1)

B.1 Energy and temperature

temperature energy

1 K 86 µeV

4 K 0.344 meV

300 K 25 meV

168

B.2. FREQUENCY, TEMPERATURE, ENERGY, WAVELENGTH AND TIME169

B.2 Frequency, temperature, energy, wavelength

and time

frequency temperature energy wavelength time

10 MHz 500 µK 40 neV 30 m 100 ns

1 GHz 50 mK 4 µeV 30 cm 1 ns

21 GHz 1 K 90 µeV 14 mm 50 ps

240 GHz 12 K 1 meV 1.24 mm 4 ps

360 THz 17400 K 1.5 eV 824 nm 3 fs

Rule of Thumbs

• Temperature vs. Energy : 1 K ∼ 80µ eV

• Wavelength vs. Energy : 1 nm ∼ 2 meV at λ ∼ 780 nm

Wavelength vs. Frequency : 1 nm ∼ 500 GHz at λ ∼ 780 nm

• LC circuit resonant frequeny: 1 nH + 1 pF ∼ 5 GHz, 1 µ H + 1 nF = 5 MHz

Appendix C

Statistics of Particles

The distribution function is a statistical concept, which provides the information

about the probability of a particle in energy state E.

C.1 The Maxwell-Boltzmann Distribution

Identical but distinguishable (classical) particles such obey the Maxwell-Bolzmann

(MB) distribution at given temperature T and at energy E

fMB = e−(E−µ)/kBT .

As an example, for a particle whose mass is m with a velocity ~v, such molecular

distribution is specified as

f~v = 4π

(

m

2πkBT

)3/2

~v2exp

[−m(~v)2

2kBT

]

,

from which many fundamental gas properties can be explained.

170

C.2. THE FERMI-DIRAC DISTRIBUTION 171

C.2 The Fermi-Dirac Distribution

Identical indistinguishable particles with half-integer spin obey the Fermi-Dirac (FD)

distribution, fulfilling the Pauli’s exclusion principle in the quantum world. It is

essential to study electrons in matels and conduction processes in semiconductors.

fFD(E, µ, T ) =1

e(E−µ)/kBT + 1,

where µ is the chemical potential.

The FD distribution approaches to the MB distribution in the limit of high tem-

perature and low density.

Integrals

(i)∫∞0fi(E)dE

Substituting x = e(E−µi)/kBΘ, kBΘdx = xdE and integrands come from x =

e−µi/kBΘ ≪ 1 ( µ1 ≫ kBΘ) to x→ ∞.

∫ ∞

0

1

e(E−µi)/kBΘ + 1dE ≈ kBΘ

∫ ∞

0

(

1

x+ 1

)(

1

x

)

dx

= kBΘ

∫ ∞

0

(

1

x− 1

x+ 1

)

dx

≈ kBΘ

[

ln

(

x

1 + x

)]∞

e−µi/kBΘ

≈ kBΘ(0 − ln(e−µi/kBΘ)) ∼ µi.

(ii)∫∞0fi(E)(1 − fi(E))dE

172 APPENDIX C. STATISTICS OF PARTICLES

Similarly, put x = e(E−µi)/kBΘ.

∫ ∞

0

fi(E)(1 − fi(E))dE =

∫ ∞

0

(

1

e(E−µi)/kBΘ + 1

)(

e(E−µi)/kBΘ

e(E−µi)/kBΘ + 1

)

dE

=

∫ ∞

e−µ/kBΘ

kBΘ1

(x+ 1)2dx

= kBΘ

[

− 1

x+ 1

]∞

e−µi/kBΘ

≈ kBΘ

(

1

1 + e−µi/kBΘ

)

≈ kBΘ.

(iii)∫∞0fi(E)(1 − fj(E))dE

Again, set x = e(E−µi)/kBΘ and a = e(µi−µj)/kBΘ

∫ ∞

0

fi(E)(1 − fj(E))dE =

∫ ∞

0

(

1

e(E−µi)/kBΘ + 1

)(

e(E−µj)/kBΘ

e(E−µj)/kBΘ + 1

)

dE

=

∫ ∞

e−µ/kBΘ

kBΘ

(

1

x+ 1

)(

a

ax+ 1

)

dx

= kBΘa

1 − a

∫ ∞

e−µ/kBΘ

(

1

x+ 1− a

ax+ 1

)

dx

= kBΘa

1 − a

[

ln

(

x+ 1

ax+ 1

)]∞

e−µi/kBΘ

≈ kBΘa

1 − aln

∣

∣

∣

∣

1

a

∣

∣

∣

∣

≈ (µi − µj)

(

e(µi−µj)/kBΘ

e(µi−µj)/kBΘ − 1

)

.

Interchanging i and j and doing some algebra,

∫ ∞

0

fj(E)(1 − fi(E))dE ≈ (µi − µj)

(

1


)

.

C.2. THE FERMI-DIRAC DISTRIBUTION 173

Thus,

∫ ∞

0

fi(E) [1 − fj(E)] + fj(E) [1 − fi(E)]dE

≈ (µi − µj)

(

e(µi−µj)/kBΘ + 1


)

= (µi − µj) coth

(

µi − µj

2kBΘ

)

.

(iv)∫∞0fi(E)(1 − fj(E + C))dE where C is a constant

Since C is a constant, modifying the above ‘a’ provides the answer. Let b =

e(C+µi−µj)/kBΘ with the same ‘x’,

∫ ∞

0

fi(E) [1 − fj(E + C)] dE

=

∫ ∞

0

(

1

e(E−µi)/kBΘ + 1

)(

e(E+C−µj)/kBΘ

e(E+C−µj)/kBΘ + 1

)

dE

=

∫ ∞

e−µ/kBΘ

kBΘ

(

1

x+ 1

)(

b

bx+ 1

)

dx

≈ (C + µi − µj)

(

e(C+µi−µj)/kBΘ

e(C+µi−µj)/kBΘ − 1

)

.

(v)∫∞0fj(E + C) [1 − fi(E)] dE where C is a constant

With b = e(C+µi−µj)/kBΘ,

∫ ∞

0

fj(E + C) [1 − fi(E))] dE

=

∫ ∞

0

(

1

be(E−µi)/kBΘ + 1

)(

e(E+C−µi)/kBΘ

e(E+C−µi)/kBΘ + 1

)

dE

≈ (C + µi − µj)

(

1


)

.

174 APPENDIX C. STATISTICS OF PARTICLES

Combining (iv) and (v),

∫ ∞

0

fi(E) [1 − fj(E + C)] + fj(E + C) [1 − fi(E)]dE

≈ (C + µi − µj)

(

e(C+µi−µj)/kBΘ + 1


)

= (C + µi − µj) coth

(

C + µi − µj

2kBΘ

)

.

C.3 The Bose-Einstein Distribution

Identical indistinguishable particles with integer spin obey the Bose-Einstein (BE)

statistics, as a governing principle in the quantum world. A example of BE distribu-

tion is the Planck radiation formula.

fBE(E, µ, T ) =1

e(E−µ)/kBT − 1

The BEredu to the MB distribution in the limit of high temperature and low

density.

C.4 Basic Distribution Functions

A. Binomial Distribution Function

It is the probability of an event occuring at x times out of n trials with a success

probability p in a single trial.

fb(x) =n!px(1 − p)n−x

x!(n− x)!

MEAN = np

STANDARD DEVIATION =√

np(1 − p)

B. Gaussian Distribution Function

The limit of the binomial distribution function at a large n is known as Gaussian

C.4. BASIC DISTRIBUTION FUNCTIONS 175

or normal distribution function. With a mean a and a standard deviation σ,

fg(x) =1√

2πσ2exp

[

−(x− a)2

2σ2

]

.

C. Poisson Distribution Function

Poisson distribution function is another limit of the binomial distribution func-

tion for a small probability p, expressed as a mean a

fp(x) =e−aax

x!

Appendix D

The Dirac Delta Function

D.1 Representations

•δ(x− x0) =

1

2π

∫ ∞

−∞dkeik(x−x0)

•δ(x) = lim

g→∞

sin(gx)

πx

•

δ(x) =1

2πlimg→∞

∫ g

−g

dkeikx

=1

2πlimg→∞

eigx − e−igx

ix

= limg→∞

sin(gx)

πx

•δ(x) = lim

a→0

1

π

a

x2 + a2

•δ(x) = lim

a→∞

a√πe−a2x2

176

D.2. PROPERTIES 177

D.2 Properties

•∫

dxδ(x− x0) = 1

•∫

dxδ(x)f(x) = f(0)

• δ∗(x) = δ(x)

• δ(−x) = δ(x)

• δ(ax) = 1|a|δ(x)

• f(x)δ(x− a) = f(a)δ(x− a)

• δ(x2 − a2) = 12|a| [δ(x− a) + δ(x+ a)]

•∫

δ(x− b)δ(a− x)dx = δ(a− b)

•∫

dxδ′(x)f(x) = −f ′(0)

• δ(x− a) = dΘ(x−a)dx

Appendix E

Useful Mathematical Formulas

E.1 Even and Odd functions

∫ l

−l

f(x)dx =

0 if f(x) is odd,

2∫ l

0f(x)dx if f(x) is even.

E.2 Taylor Series

Definition

f(x) = f(a) + f ′(a)(x− a) + f′′

(a)2!

(x− a)2 + ...fn(a)n!

(x− a)n + ...

Examples

• ex =∑∞

n=0xn

n!for all x

• ln(1 + x) =∑∞

n=0(−1)n

n+1xn+1 for |x| < 1

• xm

1−x=∑∞

n=m xn for |x| < 1

• (1 + x)α =∑∞

n=0

(

α

n

)

xn for all |x| < 1 and all α ∈ C

• sin x =∑∞

n=0(−1)n

(2n+1)!x2n+1 for all x

178

E.3. FOURIER-TRANSFORM 179

E.3 Fourier-Transform

Fourier transforms are commonly used in various fields. They are very useful to

convert information easily between two conjugate variables such as position x and

momentum p, energy E and time t or frequency ω and time t.

Continuous Fourier Transform

Suppose f(t) is square-integrable function. It can be decomposed by complex expo-

nentials with frequency components F(ω)

f(t) =1√2π

∫ ∞

−∞F(ω)eiωtdω.

The inverse Fourier transform is also defined as

F(ω) =1√2π

∫ ∞

−∞f(t)e−iωtdt.

The coefficient 1/√

2π is chosen symmetrically for both t and ω, which is known as

unitary. A common non-unitary transform is

f(t) =1

2π

∫ ∞

−∞F(ω)eiωtdω.

F(ω) =

∫ ∞

−∞f(t)e−iωtdt.

Finite Fourier Transform

Instead of continuous variable, finite Fourier transform is often employed handling

given number of data. It is called the discrete Fourier transform as well. Suppose N

complex numbers x0, ..., xN−1 which are converted into the sequency of N complex

numbers X0, ..., XN−1. The Fourier and inverse Fourier transforms are as follows:

Xk =N−1∑

n=0

xne− 2πi

Nkn k = 0, ...., N − 1,

180 APPENDIX E. USEFUL MATHEMATICAL FORMULAS

xn =1

N

N−1∑

k=0

Xke2πiN

kn n = 0, ..., N − 1

E.4 Pauli Spin Matrices

Representation

σx =

(

0 1

1 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 0

0 −1

)

.

Properties

• σi · σj = 2δij, for i 6= j.

• [σi, σj] = 2iǫijkσk.

• det(σi) = −1.

• Tr(σi) = 0.

• σ†i = σi.

• (~σ · ~a)(~σ ·~b) = ~a ·~b+ i~σ · (~a×~b).

• (~σ · ~a)2 = |~a|2.

E.5 Trigonometric Functions

A,B are the angle in radian.

• sin2A+ cos2A = 1.

• sec2A− tan2A = 1.

• sin(A±B) = sinA cosB ± cosA sinB.

• cos(A±B) = cosA cosB ∓ sinA sinB.

E.6. SPECIAL FUNCTIONS 181

• tan(A±B) = tan A±tan B1∓tan A tan B

.

• sinA± sinB = 2 sin (A±B)2

cos (A∓B)2

.

• cosA+ cosB = 2 cos (A+B)2

cos (A−B)2

.

• cosA− cosB = 2 sin (A+B)2

sin (B−A)2

.

• sin 2A = 2 sinA cosA = 2 tan A1+tan2 A

.

• cos 2A = cos2A− sin2A = 2 cos2A− 1 = 1 − 2 sin2A

= 1−tan2 A1+tan2 A

= cot A−tan Acot A+tan A

.

• tan 2A = 2 tan A1−tan2 A

= 2 cot Acot2 A−1

= 2cot A−tan A

.

• sin2A = 1−cos 2A2

.

• cos2A = 1+cos 2A2

.

E.6 Special Functions

• Gamma function

Γ(p) =

∫ ∞

0

xp−1e−xdx, p > 0.

• Error function

erf(x) =2√π

∫ x

0

e−t2dt.

• Lorentzian function

L(x) =2α

x2 + α2.

• Gaussian function

G(x) = ae−(x−b)2/c2 .

182 APPENDIX E. USEFUL MATHEMATICAL FORMULAS

E.7 Vector Operators

Some vector operators frequently used in various occasions are summarized in three

different coordinates: Cartesian, cylindrical and spherical. Note that the angle θ

in the spherical coordinate is defined in the x − y plane such as in the cylindrical

coordinates ad φ is defined the angle from the z-axis.

Gradient

A. Cartesian Coordinates

~∇ = i∂

∂x+ j

∂

∂y+ k

∂

∂z

B. Cylindrical Coordinates

~∇ = r∂

∂r+ θ

1

r

∂

∂θ+ z

∂

∂z

C. Spherical Coordinates

~∇ = r∂

∂r+ φ

1

r

∂

∂φ+ θ

1

r sinφ

∂

∂θ

Divergence


~∇ · ~F =∂Fx

∂x+∂Fy

∂y+∂Fz

∂z


~∇ · ~F =1

r

∂(rFr)

∂r+

1

r

∂Fθ

∂θ+∂Fz

∂z


~∇ · ~F =1

r2

∂(r2Fr)

∂r+

1

r

∂Fθ

∂θ+

1

r sinφ

∂Fφ sinφ

∂φ

E.7. VECTOR OPERATORS 183

Curl


~∇× ~F =

∣

∣

∣

∣

∣

∣

∣

∣

i j k∂∂x

∂∂y

∂∂z

Fx Fy Fz

∣

∣

∣

∣

∣

∣

∣

∣


~∇× ~F =1

r

∣

∣

∣

∣

∣

∣

∣

∣

r θ z∂∂r

∂∂θ

∂∂z

Fr rFθ Fz

∣

∣

∣

∣

∣

∣

∣

∣


~∇× ~F =

∣

∣

∣

∣

∣

∣

∣

∣

1r2 sin φ

r 1r sin φ

φ 1rθ

∂∂r

∂∂φ

∂∂θ

Fr rFφ r sinφFθ

∣

∣

∣

∣

∣

∣

∣

∣

Laplacian


∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2


∇2 =1

r

∂

∂r

(

r∂

∂r

)

+1

r2

∂2

∂θ2+

∂2

∂z2


∇2 =1

r2

∂

∂r

(

r2 ∂

∂r

)

+1

r2 sin2 φ

∂2

∂θ2+

1

r2 sinφ

∂

∂φ

(

sinφ∂

∂φ

)

Appendix F

Recipe of Making Printed Circuit

Boards

This appendix introduces a simple recipe to build a prototype of printed circuit

boards (PCB) with a resolution around 10 µm in a quick and dirty method. The

PCBs generated by this recipe works quite well using photolithography and etching.

Step 1: Photomask Design

1. Generate a pattern of PCBs using softwares such as power point, illustrator or

CAD.

2. Print the pattern in the overhead projector transparencies with a laser printer.

3. Make sure all structures and patterns come out as black to block the light. If not,

paint thoroughly the incomplete structures and/or patterns with black marker pens.

Step 2: Substrate Preparation

1. Clean the surface of substrates. For Cu PCBs, a sand paper works well.

2. Resolve the grease by soaking substrates into the following sequential solvents:

Acetone → Methanol → Isopronanol

3. If critical, ultrasonic the substrates in Leksol, the detergent about 20 minutes.

184

185

Step 3: Resist Coating

1. Spin one side of the substrate with Shipley 1813 for 40 - 60 seconds at 2000 -

3000 rpm. Typically, 2000 rpm gives 2 µm-thick resist coatig and 3000 rpm coats 1

µm-thick resist on the top. (NOTE: Any thickness between 1 and 2 µm works well.)

2. Bake the coated substrate for 20 minutes in the oven at 80 oC in order to remove

remaining solvents in the resist.

3. Do the other side of the substrate, following the above for protection from etching

if necessary.

Step 4: Exposure

In the case of the EV aligner in Ginzton Clean room,

1. Turn on the lamp for more than 10 minutes before actual exposure. (NOTE: Do

not push the button too long, around 10 seconds are enough to enlighten the lamp.)

2. Power on the EV aligner.

3. Press “Load” button to place the substrate and mask.

4. Flip the “Shutter open” to expose the light for 48 seconds.

5. Flip the “Shutter open” back to the original position to stop exposure.

Step 5: Develop

1. Prepare the developing solution by mixing the Develop and water as a 1:1 ratio.

2. Develop the substrate around 45 seconds. (NOTE: around 10 seconds, the pattern

starts to appear.)

3. Clean the substrate in water.

Step 6: Etching

1. Prepare two beakers: One is for etchant and the other for clean water.

2. Soak the substrate into the etchants.

(a) Cu substrate: Ferric Chloride for more than 40 minutes. (NOTE: Heating up

the substrate may facilitate the etching process.)

186 APPENDIX F. RECIPE OF MAKING PRINTED CIRCUIT BOARDS

(b) Au (Cr) substrate: Au (Cr) etchant for a few minutes

3. Alternate two beakers of the etchant and water for speeding up the etching.

Step 7: Cleaning

1. Clean the substrate in the running distilled water.

2. Blow up the substrate with a nitrogen gun.

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