ABSTRACT
Title of Dissertation: NON-LOCAL TRANSPORT SIGNATURES AND
QUALITY FACTORS IN THE
REALISTIC MAJORANA NANOWIRE
Yi-Hua Lai
Doctor of Philosophy, 2022
Dissertation Directed by: Professor Jay Deep Sau
Department of Physics
Majorana zero modes (MZMs) can be fault-tolerant topological qubits due to their topolog-
ical protection property and non-Abelian statistics. Over the last two decades, a deluge of theo-
retical predictions and experimental observations has been actively ongoing in the hope of imple-
menting topological quantum computation upon MZMs. Among several solid-state systems, the
most promising platform to realize MZMs is the one-dimensional semiconductor-superconductor
nanowire (called ?Majorana nanowire? in short), which is the focused system in this thesis. It is
fundamental to identify MZMs as qubits to construct topological quantum computers. Therefore,
the signatures of MZMs become highly crucial for verification. However, the earlier theoret-
ical works demonstrate that topologically-trivial Andreev bound states (ABSs) can mimic the
hallmarks of MZMs in various aspects but do not carry topological properties. As a result, dis-
tinguishing MZMs from ABSs becomes significantly pivotal in the study of Majorana nanowire.
One of the signatures the author studied is the robustness of the quantized zero-bias con-
ductance peak (ZBCP) in the realistic Majorana nanowire. The importance of this signature
becomes further enhanced, particularly after the 2018 Nature paper, which displayed the quan-
tized Majorana conductance, got retracted. In Chapter 2, the proposed quality factors quantify
the robustness of quantized ZBCPs. By comparing the numerical results between different sce-
narios, this study shows that the quality factor F can help distinguish topological MZMs from
trivial subgap bound states in the low-temperature limit.
Another necessary signature of MZMs is the non-local correlation. In Chapter 3, the con-
ductance correlation is demonstrated by modeling the comparing quantum-point-contact (QPC)
conductance from each end. Both the pristine nanowire and the quantum-dot-hybrid-nanowire
system are modeled and compared, which shows the significance of non-local end-to-end corre-
lation for the existence of MZMs. The other approach to simultaneously examining the localiza-
tion of states at both ends of the nanowire is through the Coulomb blockade (CB) measurement.
The lack of sensitivity to the localized state at only one end makes the CB spectroscopy able
to capture the non-local correlation feature of MZMs. However, CB transport in the Majorana
nanowire is much more complicated to analyze than QPC transport because (a) Coulomb inter-
action is treated as equal to MZM physics without perturbation, and (b) there are many energy
levels in the nanowire, which gives rise to an exponential complexity to solve the rate equations.
In Chapter 4, a generalized version of Meir-Wingreen formula for the tunneling conductance of
a two-terminal system is derived. This formula reduces the exponential complexity of the rate
equations to as low as the linear complexity of QPC tunneling, thus allowing multiple energy
levels to be included in the calculation. With dominant realistic effects in the model, the experi-
mental features, such as the bright-dark-bright CB conductance pattern and decreasing oscillation
conductance peak spacings (OCPSs) with the Zeeman field, will be simulated and explained the-
oretically.
In short, the theoretical methods proposed in this thesis, including the quality factors, non-
local correlation ZBCPs, and CB spectroscopy, are intended to distinguish MZMs from other
topologically-trivial bound states. Further investigations on the robustness of quantized conduc-
tance and non-local correlation analysis can clarify the ambiguous signals in the experiments and
push the realization of topological quantum computation to the frontier.
NON-LOCAL TRANSPORT SIGNATURES AND QUALITY FACTORS
IN THE REALISTIC MAJORANA NANOWIRE
by
Yi-Hua Lai
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2022
Advisory Committee:
Professor Jay D. Sau, Chair/Advisor
Professor Sankar Das Sarma, Co-Advisor
Professor Maissam Barkeshli
Professor Johnpierre Paglione
Professor John Cumings
?c Copyright by
Yi-Hua Lai
2022
Acknowledgments
The completion of my Ph.D. does not come easy. Therefore, I would like to express my
gratitude to the people who helped me during my Ph.D. journey. First and foremost, I would like
to thank my advisor, Professor Jay Deep Sau, for his guidance on my research and support on
my career path. As a theoretical physicist who is very familiar with experiments, Prof. Sau can
always reduce a complicated physics problem to a simple formalism through his physics intu-
ition. I especially gained a lot of knowledge from his fruitful perspectives on condensed matter
experiments and his magic numerical tricks that can always help me avoid potential numerical
loopholes. Most importantly, I am indebted to Prof. Sau?s patience with me. He is always ami-
able in answering my questions of different levels. I cannot accomplish my Ph.D. without his
down-to-earth support.
I also owe gratitude to my co-advisor, Professor Sankar Das Sarma, who always gives
me clever suggestions for critical problems. Prof. Das Sarma has a unique and sharp taste for
selecting important problems in condensed matter physics. I always get some physics insights
through working with him on joint projects or watching him ask crucial questions to seminar
speakers. My learning progress benefited a lot from Prof. Das Sarma?s input.
I want to thank Professors Maissam Barkeshli, Johnpierre Paglione, and John Cumings for
making the time and effort to serve as my dissertation committee members.
I feel grateful to have plenty of physics discussions and career suggestions from many
ii
CMTC postdoctoral researchers, including but not limited to Ching-Kai Chiu, Junhyun Lee, Yi-
Ting Hsu, Yang-Zhi Chou, Yunxiang Liao, Sheng-Jie Huang, and Yu-An Chen. They are always
more than willing to share their physics knowledge and valuable experiences. I also feel thankful
for the support I get from many past UMD senior students, including but not limited to Setiawan,
Hoi-Yin Hui, Chun-Xiao Liu, Amit Nag, Yingyi Huang, Yidan Wang, Chiao-Hsuan Wang, Wan-
Ting Liao, and I-Lin Liu. They are always more than happy to follow up with my status in my
Ph.D. and give me practical advice accordingly.
My time at the University of Maryland would not be well-rounded without the accompany
of my Ph.D. peers, including but not limited to Haining Pan, Huan-Kuang Wu, Tomoghna Barik,
DinhDuy Vu, Donovan Buterakos, Subhayan Sahu, Naren Manjunath, Ali Lavasani, Shuyang
Wang, Yong-Chan Yoo, Stuart Thomas, and Navya Gupta from CMTC. I especially thank Hain-
ing Pan and Huan-Kuang Wu for their extra non-physics interactions to help me conquer various
obstacles in my life. I also cherish those fun days with Wen-Chen Lin, Peiyin Lee, Sun-Ting
Tsai, Su-Kuan Chu, Tung-An Wei, Tamin Tai, Jwo-Sy Chen, Chien-Ming Huang, Chih-Yu Lee,
Lauren Lan, Yue Tsao, Chen-Yu Chen, Eugene Yang, and many others. They all give me some
wisdom in my life in different ways.
I am particularly indebted to my lifelong mentor, Professor Ite A. Yu, from my alma mater,
National Tsing Hua University. He has supported my Ph.D. studies virtually since I obtained my
master?s degree under his guidance. Specifically, I want to thank him for writing recommenda-
tion letters for my postdoctoral applications and accomodating the deadlines for me without any
hesitation. Whenever I visited him in Taiwan, I could still feel his lasting passion for physics,
smartness in tackling problems, wisdom towards life, and kindness to all the people. The model
he has set always motivates me to become a better person. I feel lucky and grateful that Prof. Yu
iii
gave me many opportunities, encouraged me, and supported my career path.
Last but not least, I want to thank my parents, my sister, and my husband, who have been
supporting me throughout my Ph.D. years. Particularly, I owe most to my husband, Farhan
Augustine, who always supported me and encouraged me to stay positive when I was in the
valleys of my Ph.D. years. I cannot grow so much and feel fulfilled during my Ph.D. journey
without him accompanying me to pass through every delightful and stressful moment.
iv
Table of Contents
Acknowledgements ii
Table of Contents v
List of Figures vii
List of Abbreviations ix
Chapter 1: Introduction 1
1.1 Majorana zero mode (MZM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Fermion Parity of the Ground State . . . . . . . . . . . . . . . . . . . . . 5
1.2 1D Spinless p-wave Superconductor . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 1D Superconductor-Semiconductor Nanowire . . . . . . . . . . . . . . . . . . . 12
1.4 MZM Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Resonant Andreev Reflection . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Experiment Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.1 The Earliest observed ZBCP . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.2 The Quantized Conductance Experiment . . . . . . . . . . . . . . . . . . 24
1.5.3 Coulomb Blockade Experiment . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 2: Quality Factors for Zero-bias Conductance Peaks 32
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Potential of Good/Bad/Ugly ZBCPs . . . . . . . . . . . . . . . . . . . . 38
2.1.2 Differential Conductance Formalism . . . . . . . . . . . . . . . . . . . . 40
2.1.3 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1 ?Good? ZBCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 ?Bad? ZBCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 ?Ugly? ZBCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4.1 Topological Characteristics from the quality factor F . . . . . . . . . . . 62
2.4.2 Decoherence rate based on the quality factor F . . . . . . . . . . . . . . 69
2.4.3 Multi-channel effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.4 Characterizing MZMs based on J . . . . . . . . . . . . . . . . . . . . . 72
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
v
Chapter 3: Presence versus Absence of End-to-end Non-local Conductance Corre-
lation 75
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Numerical Results: Local Conductance from each end . . . . . . . . . . . . . . . 79
3.3 Numerical Results: Cross Conductance . . . . . . . . . . . . . . . . . . . . . . . 85
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 4: Coulomb-blockade Transport In Realistic Majorana Nanowires 88
4.1 Ideal Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Two-terminal Generalized Meir-Wingreen Formalism . . . . . . . . . . . . . . . 93
4.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2.2 Steady-state Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.3 Linear-Response Conductance . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.4 Generalized Meir-Wingreen Formula . . . . . . . . . . . . . . . . . . . . 99
4.3 Conductance for Few-level Systems . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.1 Rates for Few-electron Process . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.2 Single-bound-state Induced Electron Transport . . . . . . . . . . . . . . 106
4.3.3 One Subgap State At Each End . . . . . . . . . . . . . . . . . . . . . . . 107
4.3.4 Conductance Near N and (N + 2) Degeneracy . . . . . . . . . . . . . . 109
4.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.2 Tunneling Rate From The Density Matrix . . . . . . . . . . . . . . . . . 111
4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5.1 Soft-gap Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5.2 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5.3 Length Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5.4 SC Collapsing Field Dependence . . . . . . . . . . . . . . . . . . . . . . 129
4.5.5 Chemical Potential Dependence . . . . . . . . . . . . . . . . . . . . . . 131
4.5.6 Quantum Dot Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.5.7 Self-energy Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6 Key Features of Coulomb Blockaded Conductance . . . . . . . . . . . . . . . . 140
4.6.1 Bright-dark-bright Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.6.2 Suppression of Normal Coulomb Blockade Peak Relative to ABS and MBS143
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Chapter 5: Conclusion 147
Appendix A: Detailed Derivations for Coulomb Blockade Transport 150
A.1 Microscopic Tunneling Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.2 Steady-state Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.3 Current and Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.4 Meir-Wingreen Formula for One-terminal CB Majorana nanowire . . . . . . . . 155
Bibliography 164
vi
List of Figures
1.1 Kitaev Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Scheme of Superconductor-semiconductor Nanowire . . . . . . . . . . . . . . . 12
1.3 Spectrum of the SOC Semiconducting Nanowire . . . . . . . . . . . . . . . . . . 14
1.4 Scheme of NS Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Schematic plots of the scattering processes in an NS junction . . . . . . . . . . . 18
1.6 Schematic plot of Andreev reflection process . . . . . . . . . . . . . . . . . . . . 19
1.7 Majorana nanowire experiment by Mourik et al. . . . . . . . . . . . . . . . . . . 22
1.8 Quantized Conductance experiment by Zhang et al. . . . . . . . . . . . . . . . . 25
1.9 Majorana Coulomb Blockaded island device by Albretcht et al. . . . . . . . . . . 27
1.10 Coulomb Blockade spectrum with single energy level . . . . . . . . . . . . . . . 29
1.11 Coulomb Blockade Peak splitting in magnetic field . . . . . . . . . . . . . . . . 31
2.1 Schematic plots of the nanowire with different potentials . . . . . . . . . . . . . 37
2.2 Schematic plots to show quality factors . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Numerical results for the ?good? ZBCP . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Numerical results for the ?bad? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Numerical results for the ?bad? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Numerical results for the ?ugly? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 58
2.7 Numerical results for the ?ugly? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 Numerical results for the ?ugly? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 60
2.9 Numerical results for the ?ugly? ZBCP. . . . . . . . . . . . . . . . . . . . . . . . 61
2.10 Statistical histogram for ?bad? and ?ugly? ZBCPs at T = 10 mK and  = 0.05. . 64
2.11 Statistical histogram for ?bad? and ?ugly? ZBCPs at T = 20 mK and  = 0.1. . . 66
3.1 Schematic plot of the hybrid structure composed of leads on both sides . . . . . . 77
3.2 Numerical results with chemical potential ? = 1.0 meV . . . . . . . . . . . . . . 82
3.3 Numerical results with chemical potential ? = 5.0 meV . . . . . . . . . . . . . . 83
3.4 Cross conductance GLR as a function of voltage VR from the right end. . . . . . . 86
4.1 Schematic plot of the Coulomb-blockade Majorana nanowire . . . . . . . . . . . 89
4.2 Ideal Coulomb-blockade results: most similar to experimental data . . . . . . . . 91
4.3 Reference Coulomb-blockade results: for comparison . . . . . . . . . . . . . . . 118
4.4 Temperature dependence of Coulomb-blockade numerical results . . . . . . . . . 122
4.5 Length dependence of Coulomb-blockade numerical results: L = 1.0 ?m . . . . 124
4.6 Length dependence of Coulomb-blockade numerical results: L = 0.6 ?m . . . . 125
4.7 Length dependence of OCPS numerical results . . . . . . . . . . . . . . . . . . . 127
4.8 Vc dependence of Coulomb-blockade numerical results: Vc = 3.6 meV . . . . . . 129
vii
4.9 Vc dependence of Coulomb-blockade numerical results: Vc =? . . . . . . . . . 130
4.10 Chemical potential dependence of Coulomb-blockade numerical results: ? = 2.0
meV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.11 Chemical potential dependence of Coulomb-blockade numerical results: ? = 3.0
meV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.12 Quantum dot dependence of Coulomb-blockade numerical results: one QD . . . 136
4.13 Quantum dot dependence of Coulomb-blockade numerical results: no QD . . . . 137
4.14 Self-energy dependence of Coulomb-blockade numerical results: without self-
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
viii
List of Abbreviations
ABS Andreev Bound state
BdG Bogoliubov-de Gennes
BTK Blonder Tinkham-Klapwijk
CB Coulomb Blockade
DOS Density of States
H.c. Hermitian Conjugation
MF Majorana Fermion
MBS Majorana Bound State
MZM Majorana Zero Mode
NS Normal Metal-Superconductor
OCPS Oscillation Conductance Peak Spacing
QD Quantum Dot
QPC Quantum Point Contact
SC Superconducting / Superconductor / Superconductivity
SM Semiconducting / Semiconductor
SOC Spin-Orbit Coupling / Spin-Orbit Coupled
TQC Topological Quantum Computing
TQPT Topological Quantum Phase Transition
TSC Topological Superconductor
ZBP Zero-Bias Peak
ZBCP Zero-Bias Conductance Peak
ix
Chapter 1: Introduction
In 1937, the charge-neutral Majorana fermion, which is its own anti-particle, was proposed
by Ettore Majorana as an elementary particle that stands as a real solution to the Dirac equa-
tion [1]. This notion has made an impact on several subfields of physics, such as high-energy
physics and condensed matter physics. In high-energy physics, neutrino is the only candidate
that could be a Majorana fermion. Unfortunately, Majorana fermions have not been detected
through the conventional high-energy approach over the past 80 years. On the other hand, in
condensed matter physics, several theoretical models that can host Majorana fermions have been
attracting physicists? attention over the last 20 years. It is not only because of the likelihood to
discover this elementary particle but also because of the potential to construct the non-Abelian
topological quantum computer based on Majorana fermions? unique properties [2?15]. Since
Majorana fermions in solid-state systems appear as zero-energy quasi-particle excitations that
are bound to defects [16] and are not fermions, they are often dubbed as ?Majorana zero modes?
(MZMs) or ?Majorana bound states? (MBSs).
Earlier works for MZMs in condensed matter physics focus on fractional quantum Hall
states [17?19], and p-wave superconductors [2, 19?24] that can support the existence of MZMs
either at the ends of one-dimensional topological superconductors [2] or at the vortices of two-
dimensional topological superconductors [19, 23]. An esssential component of these proposed
1
systems is the exotic p-wave superconducting pairing potential, which is quite scarce in the na-
ture. Therefore, the experimental progress of these exotic systems is slow-paced.
Around 2010, the idea of heterastructure enabled scientists to adapt devices proximitized
by the conventional s-wave superconductor to topological superconductors [25?30]. These ex-
perimentally applicable proposals brought a new phase to the field of Majorana fermions in
condensed matter physics. A significant amount of groundbreaking experiments have been car-
ried out in the past 10 years [31?52]. Among these proposals, the most promising candidate is
the spin-orbit-coupled semiconducting nanowire proximitized by an s-wave superconductor with
an external magnetic field parallel to the axis of the nanowire [26?29]. We call it ?Majorana
nanowire? in short. This system can be tuned into the topological regime, where the MZMs are
supposed to appear at both ends of the nanowire, simply by increasing the magnetic field strength
to be above the topological quantum phase transition (TQPT) field. One of the most essential
features of the MZMs is the zero-bias conductance peak (ZBCP) quantized at 2e2/h at zero tem-
perature, which can be explained by the perfect Andreev reflection facilitated by MZMs [53?57].
While several quantized ZBCPs were observed in several experiments [38, 44, 52], the existence
of MZMs remains elusive because topologically-trivial Andreev bound states (ABSs) or disorder-
induced bound states can exhibit quantized ZBCPs similar to the ones from MZMs [58?69]. In
fact, topologically-trivial subgap bound states can mimic the signatures of MZMs in various way,
making MZMs hard to be distinguished.
In this thesis, we aim to distinguish between the topological MZMs and trivial subgap
bound states by using various signatures of MZMs. The approaches we use to identify MZMs
include: (a) the quality factors to gauge the robustness of quantized ZBCPs [Chapter 2], (b)
the non-local end-to-end conductance correlation [Chapter 3], and (c) the Coulomb blockade
2
spectroscopy [Chapter 4]. In each of the theoretical study, we analyze and simulate the one-
dimensional Majorana nanowire with all the important realistic effects. The first two approaches
provide experiments new directions to search for the elusive MZMs, while the last one explains
the experimental results of the Coulomb blockaded nanowire. The conclusion of the above works
will be provided in Chapter 5.
Before the reader jumps to our core contributed works [Chapter 2- Chapter 4], the author
wants to give the reader a review in this chapter on the theoretical development and understanding
of MZMs, the famous Kitaev chain model, the Majorana nanowire model, several characteristic
signatures of MZMs, and the pioneering experiments related to our theoretical studies. This
chapter should offer the research background for the reader to stay engaged with the problems
the author tries to solve in the following chapters.
1.1 Majorana zero mode (MZM)
Majorana zero mode (MZM), the condensed-matter version of Majorana fermion, has been
vigorously discussed recently mainly because of its special exchange statistics: They are non-
Abelian anyons [70] that do not commute when particles exchanges, unlike bosons and fermions.
MZMs are mixtures of electrons and holes in equal weight because MZMs are their own anti-
particles. Since the Bogoliubov quasi-particles in superconductors can be represented as superpo-
sitions of electrons and holes, MZMs should be able to exist as zero-energy subgap bound states
in superconductors. From the particle-hole symmetry of the Bogoliubov quasi-particle operators,
which satisfy
??() = ?(?), (1.1)
3
the MZMs should appear at the Fermi energy (i.e.,  = 0) such that the self-conjugate relation
?? = ? is satisfied. Note that ?? and ? are Majorana creation and annihilation operators that
satisfy the following anti-commutation relation
{?n, ?m} = 2?nm. (1.2)
Moreover, Majorana operators are required to commute with the Hamiltonian, i.e.,
[H, ?n] = 0. (1.3)
This relation suggests that there is ground-state degeneracy due to the presence of MZMs. It is
because the ground state |GS? and the excited states ?i|GS? correspond to the same energy. In
general, a system that host 2N MZMs ?1, ?2, . . . .?2N has 2N degenerate ground states. We can
understand this further by writing the fermion operators ci in terms of Majorana operators,
1
ci = (?2i?1 + i?2i) , (1.4)
2
for n = 1, . . . , N . Note that the fermion operators ci satisfy the conventional fermionic anti-
commutation relation:
{c , c?i j} = ?ij, (1.5)
{ci, cj} = 0. (1.6)
4
The fermionic number operator
1
n ?i ? cici = (1 + i?2i?1?2i) (1.7)2
can take either 0 or 1, which means the degeneracy of the ground state is 2N -fold for a system of
N fermions. However, since the fermion parity is conserved for an isolated system, this constraint
puts the physical degeneracy to be 2N?1.
Eq. (1.4) also indicates one fermionic operator is composed of two MZMs. One can simply
separate a fermionic operator into a real part and an imaginary part, and claim each part a MZM.
However, it is just a mathemetical concept rather than a physical operation because the two MZMs
close to each other spatially cannot be addressed individually. The MZMs we pay attention
to here are from a delocalized fermionic operator that is composed of two spatially-separated
MZMs. The fermionic state associated with this delocalized fermionic operator can be protected
from most of decoherence because local perturbations that affect only one MZM component
cannot change the whole state. On the other hand, the state can be manipulated by exchanging
MZMs due to their non-Abelian statistics, which leads to the idea of fault-tolerant topological
quantum computer [6].
1.1.1 Fermion Parity of the Ground State
As illustrated as above, there is two-fold degeneracy for the topological ground state, where
the two MZMs are localized separately. One is unoccupied state
|0? ? |GS?, (1.8)
5
and the other is occupied state
|1? ? ?i|GS?. (1.9)
These two states can be characterized by the fermion parity. The fermion parity of site i with
fermion number ni is
Pi = (?1)ni . (1.10)
Since ni = c
?
ici in the language of operators,
(?1)ni = 1? 2ni (1.11)
= ?i?2i?1?2i. (1.12)
The fermion parity operator for the whole system can be thus defined as
?N ( )
P = 1? 2c?F ici (1.13)
?i=1N
= (?i?2i?1?2i) , (1.14)
i=1
with (PF )2 = 1. So its eigenvalues can only be ?1. For the trivial phase, PF is always +1
because ci|GS? = 0. For the topological phase, PF can be either +1 or ?1, as explained below.
In the ideal case of the Kitaev chain model [see Sec. 1.2], a pair of well-separated edge
MZMs (?1 and ?2N ) can be defined as a non-local fermion,
1
f = (?1 + i?2N) , (1.15)
2
f ?
1
= (?1 ? i?2N) . (1.16)
2
6
Similar to the protocol of Eqs. (1.4)-(1.7), the eigenvalues of the fermion number operator can
be 0 when the eigenstate is the unoccupied ground state |0+?, or 1 when the eigenstate is the
occupied ground state |0??. That is to say,
f ?f |0+? = 0, (1.17)
f ?f |0?? = |0??. (1.18)
Moreover,
? i?1? ?2N = 1? 2f f. (1.19)
On the other hand, the fermionic bulk states (in the ideal Kitaev chain model), which are com-
posed of two MZMs next to each other, can be expressed as
1
di = (?2i + i?2i+1) , (1.20)
2
1
d?i = (?2i ? i?2i+1) . (1.21)2
So similarly, ?i? ?2i?2i+1 = 1? 2didi. The fermion parity operator becomes
N??1
PF = ?i?1 (?i?2i?2i+1)?2N (1.22)
i=1
N??1( )
= ?i?1?2N 1? 2d?idi . (1.23)
i=1
Since the bulk states are in the trivial phase, di|GS? = 0 and thus we have
PF |GS? = (?i?1?2N)|GS?. (1.24)
7
By combining Eqs. (1.17)-(1.19) into Eq. (1.24), we can obtain
PF |0?? = (?i?1?2N)|0?? (1.25)
= (1? 2f ?f)|0?? (1.26)
= ?|0??. (1.27)
Therefore, the ground states of the topological phase in the ideal Kitaev chain model [Fig. 1.1(b)]
are 2-fold degenerate that can be categorized by the fermion parity. One can call the ground state
|0+? even-parity state, and the other state |0?? odd-parity state.
1.2 1D Spinless p-wave Superconductor
We want to introduce the first condensed-matter model that can host unpaired MZMs in a
one-dimensional system. In 2001, a toy model of a one-dimensional spinless p-wave supercon-
ductor was proposed by Alexei Y. Kitaev [2]. It is also typically named as ?Kitaev Chain? model.
The Hamiltonian is given by
?N ( ) N1 ??1( )
H = ?? c?c ? ? tc?c i?j j j j+1 + ?pe cjcj+1 + H.c. , (1.28)2
j=1 j=1
where ? is the chemical potential, t ? 0 is the nearest-neighbor hopping, ?pei? is the p-wave
superconducting gap, c?j and cj are the electron creation and annihilation operators at site j. Note
that H.c. means Hermitian conjugation.
The Hamiltonian in Eq. (1.28) can be expressed in terms of Majorana operators by canoni-
8
Figure 1.1: Schematic plots of the Kitaev chain with two distinct phases: The black
dot represents the Majorana fermion, while the blue square represents the electron
site. (a) Trivial phase: ?p = t = 0 and ? < 0. Two Majorana fermions from
the same site are paired up. (b) Topological phase: ?p = t > 0 and ? = 0. Two
Majorana fermions from different sites next to each other are paired up, leaving two
isolated Majorana fermions (denoted by the star sign) at the ends.
cally transforming the electron creation and annihilation operators as follows:
ei?/2
c?j = (?2j?1 ? i?2j) , (1.29)2
e?i?/2
cj = (?2j?1 + i?2j) , (1.30)
2
where ?2j?1 and ?2j are the Majorana operators at the j-th site. In this basis, the Hamiltonian
becomes
N??1i
H = [???2j?1?2j + (?p + t) ?2j?2j+1 + (?p ? t) ?2j?1?2j+2] (1.31)
2
j=1
There are two interesting cases worth of discussion:
(a) The trivial phase: ?p = t = 0 and ? < 0. Then
?N ( )
? ? ? 1H = ? cjcj , (1.32)2
j=1
where the Majorana operators are paired up at the same lattice site [see Fig. 1.1(a)]. The
9
system is gapped by 2|?|with no zero-energy state. So this phase is an insulator.
(b) ?p = t > 0 and ? = 0. In this case,
N??1
H = it ?2j?2j+1. (1.33)
j=1
Now the Majorana operators ?2j and ?2j+1 from different adjacent sites are paired up except
for the Majorana operators at the ends [see Fig. 1.1(b)]. By writing the Majorana operators
into new-defined fermion operator in the bulk, i.e., c?j = (?2j + i?2j+1), the Hamiltonian
becomes
N??1( )
? ? 1H = t c?j c?j . (1.34)2
j=1
The ground states satisfy the condition c?j|GS? = 0 for j = 1, . . . , (N ? 1). On the other
hand, since the two unpaired Majorana modes at each end can form non-local fermionic
states, the phase in this parameter regime is topological. The two degenerate ground states
|0?? composed of the two unpaired Majorana end modes also satisfy the fermion parity
relations Eqs. (1.17)-(1.27).
As above, we analyzed the two distinct phases in two special parameter limits. Each phase
has its own finite parameter space demarcated by SC gap-closure. Generally, the system rep-
resented by the two Hamiltonians are said to be in two distinct topological phases if these two
Hamiltonians cannot be continuously deformed into each other without crossing a gap closure.
Interestingly, both phases have the same bulk properties?the site index of Majorana operators
just need to be shifted by 1 than the other. However, the edge properties of both phases are differ-
ent. Clearly, phase (b) has unpaired Majoranas at both ends. So it is important for us to analyze
10
the parameter space where the bulk quasi-particle energy spectrum has bulk gap closing.
One can set a periodic boundary condition cN+1 = c1 to the Kitaev Hamiltonian as in
Eq. (1.28) to dig into the phase boundary further. Through the Fourier transformation, the Hamil-
tonian in momentum space can be expressed as
?
H = [?2t cos(k)? ?] c?kck + 2?p [i sin(k)ckc?k + H.c.] . (1.35)
k
In the form of BdG Hamiltonian HBdG defined as
1 ?
H = C?kHBdG(k)Ck, (1.36)2
k?BZ
[ ]
where C?k = c
?
k, c?k , BdG Hamltonian is
? ?
??? ?2t cos(k)? ? 2i??p sin(k) ?HBdG(k) = ?? . (1.37)
?2i?p sin(k) 2t cos(k) + ?
By diagonalizing HBdG, the bulk energy spectrum is therefore
?
E(k) = ? [2t cos(k) + ?]2 + 4|? |2 sin2p (k). (1.38)
We can see the bulk gap closes at ? = ?2t when k = 0, and at ? = 2t when k = ?. For the
regime where |?| > 2t, there will always be a bulk gap, which will never host MZMs. So for
|?| > 2t, it is the topologically-trivial phase. In constrast, the regime |?| < 2t is topologically-
non-trivial such that MZMs can exist.
11
Figure 1.2: Schematic illustration of the 1D SOC semiconducting nanowire proxim-
itized by the s-wave SC. The external magnetic field is applied parallel to the axis of
the nanowire.
While this Kitaev chain model is theoretically simple, spinless p-wave SC actually does
not exist in nature intrinsically due to the fact that electrons are spinful. Nevertheless, one can
artificially remove the spin degree of freedom by lifting the Kramer?s degeneracy of the elec-
trons. With this idea, many theoretical models that can realize topological superconductor with
s-wave SC in different types of heterostructures are proposed [25?30, 69, 71?79]. The most
promising proposal to realize topological superconductor is the one-dimensional superconductor-
semiconductor (SC-SM) hybrid nanowire, which will be discussed in the next section.
1.3 1D Superconductor-Semiconductor Nanowire
In 2010, a two-dimensional hybrid structure that can generate MZMs with only conven-
tional SCs and SMs was propose by Sau et al. [26, 27] Later on, Lutchyn et al. [28] and Oreg et
al. [29] proposed a one-dimensional spin-orbit-coupled (SOC) semiconducting nanowire prox-
imitized by the s-wave SC in the presence of a magnetic field parallel to the axis of the nanowire
[see Fig. 1.2]. Majority of the experimental groups working on solid-state systems to realize
MZMs are using this heterostructure recently because the materials are easily accessible in the
12
lab. The Hamiltonian of this SC-SM nanowire is
?
1
H = dx??NW (x)HBdG(x)?(x), (1.39)
2
( )
2
HBdG = ?
~
??
2
x ? i?R?x?y ? ? ?z + Vz?x + ??x, (1.40)2m
where m? is the effective electron mass, ?R is the SOC constant, ? is the chemical potential,
Vz = g?BB is the spin-splitting Zeeman field due to magnetic field B applied perpendicular to
SOC, and ? is the proximity-induced s-wave SC gap. HBdG is expressed in the basis of Nambu
spinor ?(x) = [c?(x), c?, c
?
?(x),?c
?(x)]T? . ?x,y,z are spin Pauli matrices, and ?x,y,z are particle-
hole Pauli matrices.
One can understand the physics of this SOC semiconducting nanowire further by analyti-
cally solve the energy spectrum firstly in the limit where the proximitized SC gap is absent, i.e.,
setting ? = 0 in Eq. (1.40). The analytical energy solution will be
~2k2 ?
E x?(kx) = ? ? ?? V
2
z + (?Rk
2
x) , (1.41)
2m
where E+(kx) and E?(kx) correspond to the upper band and lower band, respectively [see
Fig. 1.3 (a)-(c)]. Firstly, at Vz = 0, the SOC term shifts the parabolic bands towards the pos-
itive and negative momentum directions depending on their spin polarization, as in Fig. 1.3(a).
As soon as the Zeeman field is turned on, the band crossing at kx = 0 turns into band anti-
crossing, resulting a band gap opening, as in Fig. 1.3(b). If ? is set inside the gap, there is only
one effective spin direction even though this direction depends on momentum. In this case, spin-
less SC can be induced by the proximity effect. Stronger magnetic field can increase the gap
13
Figure 1.3: Energy spectrum E(kx) of the SOC semiconducting nanowire. The spin
polarizations are denoted by the red arrows. (a) ? = 0 and Vz = 0. (b) ? = 0 and
small Vz > 0. (c) ? = 0 and large Vz > 0. (d) ? > 0 and large Vz > 0. This figure
is adapted from Ref. [80].
14
and make the spins aligned to the same direction such that the spin polarization is independent
of momentum, as in Fig. 1.3(c). However, it will be hard for the SC-proximity effect to kick in
because the SC pairing term only couples electrons with anti-parallel spins. Finally, if the s-wave
SC gap is switched on, i.e., ? > 0, as in Fig. 1.3(d) the nanowire becomes topological. Focusing
on the lower band and writing the Hamiltonian in the basis of the lower band, i.e., ???(kx) and
??(kx), we can get the Hamiltonian describing the occupied band as [30]
? [ ]
Hoccupied = dkx E?(k )?
?
x ?(k )?
? ?
x ?(kx) + ??(kx)??(kx)??(?kx) + H.c. , (1.42)
where
? i?Rkx?? = ? (1.43)
2 V 2 + (? k )2z R x
is p-wave symmetry. Here the inter-band pairing terms are ignored because ? is assumed to
be small. Therefore, this one-dimensional SC-SM heterostructure, in the limit of small ?, can
effectively turn into a spinless p-wave superconductor if ? is placed within the band gap, i.e., only
the lower band is occupied. Using an analysis of the spectrum of the superconductor at k = 0,
one can show the topological regime satisfies [28, 29]
?
V > ?2z + ?2. (1.44)
A pair of MZMs should appe?ar at each end of the nanowire when the Zeeman field Vz is larger
than the TQPT field V = ?2TQPT + ?2.
15
1.4 MZM Signatures
In this section, we want to understand the physics behind the experimental signatures of
Majorana zero modes (MZMs) in this one-dimensional SC-SM nanowire. There are several
signatures of MZMs theoretically predicted, including
(a) Zero-bias Conductance Peak (ZBCP) [54]
Since Majorana operator is its own anti-particle, it should be a zero-energy mode in the SC
system. The differential conductance should show a peak at zero-bias voltage.
(b) Quantized Conductance [54?56]
The ZBCP probed by a single-channel lead should be quantized at 2e2/h at zero tempera-
ture.
(c) Non-local Correlation [81?83]
The ZBCPs for a pair of MZMs should show up at each end at once.
(d) Gap Closing and Reopening [84, 85]
The bulk gap closes as the system approaches the TQPT field from below, and the bulk gap
reopens as the system goes away from the TQPT field to further above.
(e) Majorana Oscillations [81]
For a finite length of the nanowire, the overlap of wave functions result in zero-energy
mode splitting, whose amplitude increases as the magnetic field increases.
However, topologically-trivial Andreev bound states (ABSs) can mimic several signatures of
MZMs, such as the above (a)-(c) [62, 83, 86]. Also, the experimental signal from gap closing
16
Figure 1.4: Schematic plot of the normal metal-superconductor (NS) junction. A
bias voltage is applied to the normal metal, with the superconductor grounded.
and reopening is usually too weak to identify TQPT field [87]. Thus, the path to verify the
existence of MZMs becomes harder in the presence of these experimental complexities. Since
these hallmarks that MZMs exhibit as above are mostly based on conductance measurements
through a normal metal-superconductor (NS) junction [see Fig. 1.4], it is essential to understand
the quantum transport across the NS junction, which will be discuss in Sec. 1.4.1.
1.4.1 Resonant Andreev Reflection
To understand the differential conductance measured across a NS junction, we need to an-
alyze the electron/hole scattering problem so as to count the net flowing charge per unit time.
Realistically, a tunnel barrier is usually produced at the interface of normal metal and supercon-
ductor. This tunnel barrier can therefore affect the scattering results as well. There can be three
situations when an electron is incident from the normal metal to the SC:
(a) Normal reflection: reflected back as a regular electron, as in Fig. 1.5(a)
(b) Andreev reflection: reflected back as a hole, generating one Cooper pair in the SC, as in
Fig. 1.5(b)
17
Figure 1.5: Schematic plots of the scattering processes in an NS junction: (a) Normal
reflection. (b) Andreev reflection. (c) Quasi-particle transmission.
(c) Transmission: transmitted as a propagating quasi-particle in the SC, as in Fig. 1.5(c)
When the incident electron holds the energy smaller than the SC gap (i.e., E < ?), only
normal reflection [case (a)] and Andreev reflection [case (b)] can happen. Only when the energy
of the incident electron is larger than the SC gap (i.e., E > ?) can the transmission [case (c)]
happen and quasi-particles are allowed to propogate in the SC. Normal reflection does not con-
tribute any charged particle to the current in the SC [see Fig. 1.5(a)]. On the other hand, for each
electron going through Andreev reflection, it contributes a Cooper pair of charge 2e to the SC [see
Fig. 1.5(b)]. We can understand that Andreev reflection is a transmission process for an incoming
particle from the electron channel to the hole channel in the normal-metal lead [see Fig. 1.6]. In
generic case, the transmission is less than one due to the tunnel barrier. However, in a special
case where the zero-energy Majorana bound state (i.e., MZM) is produced in the SC, the resonant
transmission happens such that an electron is completely transmitted into a hole in the normal
lead. Therefore, Majorana zero modes can induce the perfect resonant Andreev reflection. More
rigorous theoretical arguments regarding this can be found from Ref. [54?57].
Here we provide a simple way via S-matrix to show the resonant Andreev reflection. Af-
18
Figure 1.6: Schematic plot of Andreev reflection viewd as a transmission phe-
nomenon from an electron channel to a hold channel of the normal-metal lead.
ter all, it is efficient and standard to introduce the S-matrix method for the quantum transport
problem. Suppose the incident wave function with energy within the SC gap (i.e., E < ?)
is ? ?in = [?(0 ), ??(0?)]T and the reflected output wave function is ?out = [?(0+), ??(0+)]T ,
where x = 0 represents the position where the scattering occurs. In our case, NS junction is
the scattering region. By matching the wave functions from both sides of the scattering region
with the boundary conditions, the particle-hole reflection coefficients can be summarized into the
reflection matrix ? ?
??? ree reh ?R(E) = ?? (1.45)
rhe rhh
such that
?out = R(E)?in. (1.46)
ree and rhe are the normal and Andreev reflection coefficients based on an incoming electron,
while rhh and reh are the normal and Andreev reflection coefficients based on an incoming hole.
Since our system is SC, particle-hole symmetry constrains the reflection matrices by the relation
? R?x (?E)?x = R(E). (1.47)
19
Let?s take E = 0 in Eq. (1.47) because we want to analyze the case of Majorana zero modes.
Then Eq. (1.47) becomes
? R?x 0?x = R0, (1.48)
where R0 = R(E = 0). By taking the determinant of Eq. (1.47), we get
det(R0) = det(? R
? ? ?
x 0?x) = det(R0) = [det(R0)] , (1.49)
which implies det(R0) to be real.
With the energy conservation condition (i.e., the sum of the normal reflected energy and the
Andreev reflected energy is constant), the reflection matrix requires to be unitary R?(E)R(E) =
1, or
|r 2ee| + |r |2he = |reh|2 + |r 2hh| = 1 (1.50)
in the single-channel lead case. The unitarity condition of reflected matrices indicates
1 = det(R?0R0) = | det(R0)|2. (1.51)
So the quantity Q ? det(R0) has two solutions
Q = |r |2ee ? |r 2he| = ?1. (1.52)
Together with Eq. (1.50), one can find out there are only two situations in the case of a single-
channel lead when the energy is zero:
(a) Perfect normal reflection: |ree| = 1, |rhe| = 0, and Q = 1.
20
(b) Perfect Andreev reflection: |ree| = 0, |rhe| = 1, and Q = ?1.
The differential conductance measured across a NS junction is defined as
?I
G = , (1.53)
?V
where I is the current passing through the normal lead and V is the bias voltage of the normal
lead relative to the grounded SC. Since the perfect resonant Andreev reflection corresponds to
the existence of MZMs in the topological SC, the ZBCP introduced by the MZM is
e2 ( ) 2
G = 1? |r |2ee + |reh|2
2e
= (1.54)
h h
based on BTK formalism [88]. Therefore, the presence of MZMs should exhibit a quantized
ZBCP at 2e2/h at zero temperature in the case of a single-channel lead.
1.5 Experiment Review
1.5.1 The Earliest observed ZBCP
There has been a deluge of experimental works [31?38, 41?46, 48?50, 52, 89] trying to
detect MZMs in the 1D SC-SM nanowire following the theoretical proposals in 2010 [26?29].
Here we intend to review one of the first experimental transport studies done by Mourik et al. in
2012 [31].
In this experiment, the indium-antimonide (InSb) semiconducting nanowire of strong SOC
is deposited on the niobium-titanium-nitride (NbTiN) superconductor [see Fig. 1.7(a)]. A normal
21
Figure 1.7: 2012 Delft experiment on the 1D superconductor-semiconductor
nanowire. (a) The scanning electron microscope image of the setup. A semiconduct-
ing nanowire made of InSb is in contact with a normal lead (N) and a superconductor
(S). Four gates (marked by numbers 1-4) underneath the nanowire are for tuning the
chemical potential of the nanowire. A gate (marked in green) is used to tune the
tunnel barrier across the tunnel junction. (b) Differential conductance dI/dV as a
function of bias voltage V for different strengths of magnetic field ranging from 0 to
490 mT (in steps of 10 mT). The traces are offset for clarity. (c) Differential conduc-
tance dI/dV as a function of bias voltage V and angle of the magnetic field. When
the angle is 0 or ?, the magnetic fields are perpendicular to the SOC direction. When
the angle is ?/2 or 3?/2, the magnetic fields are parallel to the SOC direction. (d)
Differential conductance dI/dV as a function of bias voltage V and angle of the
magnetic field. The magnetic field is always perpendicular to the SOC direction.
This figure is adapted from Ref. [31].
22
lead, which is attached to the end of the nanowire, is used to measure the current across the
tunnel junction (N-SM-SC). A tunnel barrier between the normal lead and the nanowire can be
adjusted by a gate at the interface. An external magnetic field, whose strength and direction can
be controlled, is applied parallel to the surface of the superconductor.
The key results that show the ZBCP emerges as the magnetic field increases are in Fig. 1.7(b).
Since the theoretical proposal suggests the ZBCP induced from the topological MZM only ap-
pears above the TQPT field, this experimentally observed ZBCP was seen as a breakthrough in
search of MZMs in the nanowire system. Another theoretically-consistent evidence is shown in
Fig. 1.7 (c)(d), where the conductance is measured with a variation of the magnetic field angle.
The ZBCP is the most pronounced when the magnetic field is perpendicular to the SOC direc-
tion, which is when the angle is 0 and ? in Fig. 1.7(c) and for all angles in Fig. 1.7(d). The ZBCP
vanishes as the magnetic field approaches parallel to the SOC direction, which is when the angle
is ?/2 and 3?/2 in Fig. 1.7(c). These experimental features are in line with the theory, which
strongly suggests this ZBCP is a clear sign of the existence of MZMs.
However, there are still several defects in these results compared to the ideal theoretical
prediction. First of all, after calibrated by the offset, the observed ZBCP is at most 0.1 times
of the predicted quantized value (2e2/h). Secondly, the induced SC gaps (marked by the green
arrows in Fig. 1.7(b)) are not as obvious as required to facilitate the appearance of topological
MZMs. This is the so-called ?soft gap? issue. Recent experiments made a lot of efforts to
consolidate a hard gap in the nanowire, mostly by fabricating a better interfacial contact between
the superconductor and the nanowire [36, 39].
23
1.5.2 The Quantized Conductance Experiment
While the fabrication of the Majorana nanowire device has improved a lot over the past
10 years and the quantized ZBCP has finally been observed [38, 44, 52], several theoretical
works [58?69] show that trivial QD-induced ABSs, inhomogeneous potential, or disorder-induced
random potential can produce quantized ZBCP very similar to the MZM-induced ZBCP. So the
quantized ZBCP is not a sufficient indication for the existence of MZMs. Nonetheless, MZM
should exhibit a robust quantized plateau near zero temperature. This idea becomes more em-
phasized especially after the 2018 Nature paper [44], which showed the quantized conductance,
got retracted in part because of a calibration error. Fig. 1.8 demonstrates the recalibrated version
of the quantized conductance experiment conducted by Zhang et al. [52] Fig. 1.8(a)(b) show the
schemes of the 1D nanowire device with the tunnel gate controlling the tunnel barrier height and
the corresponding ZBCP [Fig. 1.8(e)] and normal-state conductance [Fig. 1.8(f)]. The ZBCP
does not show robust quantization when the charge jump occurs near the white arrow. Moreover,
the ZBCP in Fig. 1.8(c) can be tuned to be above or below the quantization value by the tunnel
gate, which does not exhibit the robustness of the quantized ZBCP, either.
One may therefore be motivated to study the robustness of quantized ZBCPs quantitatively
in both topological (pristine nanowire) and trivial scenarios (with QD, with smooth-varying po-
tential, or with disorder-induced random potential). In Chapter 2, we propose the quality factors
to quantify the robustness of ZBCPs. We also calculate the ZBCP as a function of the normal-
state conductance in different scenarios, and use the quality factors to analyze these scenarios,
aiming at distinguishing the topological MZMs from other trivial bound states. This project leads
to the work of Ref. [90].
24
Figure 1.8: 2018 Delft experiment [44] showing the conductance on the order of
2e2/h on the 1D nanowire device. The data here is the calibrated version [52]. (a)
False color scanning electron micrograph of the nanowire device. (b) Schematic plot
of the nanowire device. A InSb nanowire is covered by a thin superconducting Al
shell on two sides. Side gates and contacts are Cr/Au. A bias voltage V is applied
to the contact (yellow) on the left and a current I is drained from the contact on the
right. (c) Differential conductance dI/dV as a function of the magentic field B for
three different tunnel gate voltages. (d) Differential conductance dI/dV as a function
of tunnel gate voltage VTG and bias voltage V . White arrow indicates a charge jump
with a large effect on the conductance. (e) Horizontal linecut from (d) at V = 0. (f)
Horizontal linecuts at finite bias V = ?0.2 mV (black, blue), showing the out-of-
gap conductance increasing over the same range in tunnel gate voltage. The normal
conductance GN (green) is taken as the average value of the other two out-of-gap
linecuts (black, blue). This figure is adapted from Ref. [52].
25
Since a bare ZBCP probed as usual from one end does not prove the appearance of MZMs,
detecting the non-local property of MZMs is highly encouraged. One may think if there is a
way to measure the MZM wave function to analyze the its locality. However, there is no way
to measure the wave functions in experiments though it is simple to calculate the wave function
distribution in real space. The only way to check the non-local correlation of MZMs at the ends
of the wire is through the QPC conductance measurement. Therefore, it is necessary to detect
ZBCPs at both ends of the nanowire simultaneously. In the scenario where the unintentional QD
exists only at one end of the wire, the QD-induced ZBCP should not show at both ends. In Chap-
ter 3, we will provide detailed simulation results for these two scenarios to study the presence
and absence of end-to-end non-local conductance correlations, which leads to the publication of
Ref. [83].
1.5.3 Coulomb Blockade Experiment
The other way to probe the localization of states at both ends of the nanowire is through
the Coulomb blockade spectroscopy because the CB conductance does not reflect the localized
state only at one end. An experimental work [91] in 2016 shows the exponential decay of spin-
splitting amplitude with the nanowire length increasing [see Fig. 1.11(d)], which seems to be an
indication for the existence of MZMs. No matter if it is a correct interpretation or not, it is worth
reviewing these experimental results [91] before the readers dig into our theoretical study on the
CB Majorana nanowire.
The CB tunneling occurs when the scale of the device is small enough such that each
tunneling electron needs to overcome the electrostatic charging potential carried by the device.
26
Figure 1.9: Majorana Coulomb Blockaded island device by Albretcht et al. (a) False
color scanning electron micrograph of the InAs nanowire device. The source-to-
drain voltage VSD is applied on the left side of Ti/Au contact (yellow), whilte the gate
voltage VG is applied to change the chemical potential of the nanowire. Two-facet
Al shell (light blue) of length L is covering the nanowire. (b) Cross-section of a
hexagonal InAs nanowire showing the orientation of the Al shell and magnetic field
directions B? and B?. (c) Profile of the electrostatic potential energy (solid curve)
through the tunnel barriers. Ther Fermi levels in the left and right reservoirs, and the
discrete energy levels in the quantum dot, are indicated (dashed lines). Panel (a)(b)
is adapted from Ref. [91] and Panel (c) is adapted from Ref. [92].
The Majorana nanowire can easily fall into the CB regime when the wire is relatively short
(? 1 ?m) and the transmission to the end of the wire is lowered by the tunnel barriers. Since
the Majorana nanowire is isolated from normal-metal leads in this regime, we also called the
nanowire ?Majorana island? when the CB effect kicks in. A source-to-drain bias voltage VSD is
applied to one end of the nanowire relative to the other grounded end [Fig. 1.9(a)], which can
change the Fermi levels of the source and drain terminals. The gate voltage VG is applied to the
nanowire to change its chemical potential. Only when the chemical potential of the nanowire is
within the voltage window formed by the Fermi energies of source and drain terminals can the
transport occurs [see Fig. 1.9(c)].
27
The simplest way to understand the CB conductance peaks is from the ground-state energy
spectrum, as in Fig. 1.10. The spectrum can be described by
EN(NG) = EC(NG ?N)2 + pNE0, (1.55)
where NG = CVG/e is the gate-induced charge (with electron charge e and gate capacitance C)
and N is the electron occupancy of the nanowire. It is easy to see that this charging energy is pe-
riodic inNG asN is shifted equally from the formula when the ground-state quasi-particle energy
E0 = 0. Since the nanowire is proximitized by the superconductor, the ground-state energy is
zero when the electron occupancy N is even, and equal to the lowest quasi-particle energy when
N is odd. The CB transport happens at the intersection of the two parabolas. Therefore, for a
SC system, the CB peaks are intrinsically 2e-periodic due to the Cooper-pair transport [Fig. 1.10
(blue curve)]. The Majorana nanowire can contribute to 1e-periodic transport when MZM ap-
pears and the lowest quasi-particle energy becomes zero [Fig. 1.10 (red curve)]. There can be
a transitioning regime from 2e-periodic transport to 1e-periodic transport, which is expressed
in green in Fig. 1.10. In Majorana nanowires, this transition is typically done by adjusting the
magnetic field, as one can see in the experimental CB conductance in Fig. 1.11.
Fig. 1.11 (a)(e)(f)(g) show the CB conductance as a function of gate voltage VG and mag-
netic field B. It is clear that the CB peaks change from 2e periodicity (below 0.1 T) to 1e period-
icity (? 0.2 T) as the magnetic field is increasing. One can see this CB conductance shows the
?bright-dark-bright? pattern, especially in the long nanowire case [Fig. 1.11(g)]. Another way to
analyze the CB conductance results is by tracking the CB peak positions in the parameter space
and calculating the spacing difference between peaks, as in Fig. 1.11(b)(c). We call this quantity
28
Figure 1.10: Coulomb blockade spectrum with single quasi-particle energy level.
(a) The CB energy of the superconducting island, with or without sub-gap states:
EN(NG) = EC(NG ?N)2 + pNE0, where NG = CVG/e is the gate-induced charge
(with electron charge e and gate capacitance C) and N is the electron occupancy. E0
is the energy of the lowest quasiparticle state, which is filled for odd parity (pN = 1
for odd N ) and empty for even parity (pN = 0, for even N ). Ground-state ener-
gies for even (odd) N are shown in black (color). Transport occurs at the intersec-
tions of the parabolas, indicated by the filled circles. (b) Differential conductance
g = dI/dVSD versus gate voltage VG at zero bias (VSD = 0) for magnetic fields
B? = {0, 80, 220} mT. The splitting of the 2e-periodic peak (light blue light) re-
flects a transition from Cooper-pair tunneling to single-quasi-particle charging of the
Coulomb island. Evenly spaced 1e-periodic Coulomb peaks are characteristic of a
zero-energy state. This figure is adapted from Ref. [91].
29
?oscillation conductance peak spacing? (OCPS). In Fig. 1.11(b)(c), OCPSs are decreasing as the
magnefic field is increasing. Finally, Albrecht et al. also provides the first oscillation amplitude
of OCPSs for five devices of different length, as in Fig. 1.11(d). The data in Fig. 1.11(d) is
fitted well with an exponential function A = A0 exp(?L/?) that characterizes the topological
protection of MZMs. We would like to explain if these experimental features are tenable for the
existence of MZMs, with solid theories and realistic simulations, as in Chapter 4. One can find
the corresponding published work in Ref. [93].
30
Figure 1.11: Coulomb blockade Peak splitting in magnetic field. (a) Zero-bias con-
ductance g as a function of gate voltage VG and parallel magnetic field B? for the
L = 0.9 ?m device, showing a series of 2e-periodic Coulomb peaks below about
100 mT and nearly 1e-periodic peaks above about 100 mT. (b) ?Se,o?: Average peak
spacing as a function of magnetic field B? for even and odd parities measured in a
L = 0.9 ?m device. The Coulomb peaks become evenly spaced at B? = 110 mT; at
higher fields, their spacing oscillates around ?Se? = ?S0? (c) Similar to (b), but for a
longer wire, L = 1.5 ?m. (d) Oscillatory amplitudeA plotted against the wire length
for 5 devices (L ranging from 330 nm to 1.5 ?m; black dots). The green line is an
exponential fit to the data: A = A0 exp(?L/?) with A0 = 300 ?eV and ? = 260
nm. (e) Conductance g as a function of magnetic field B? and VG for device length
L = 400 nm. (f) Similar to (a), but for device length L = 790 nm. (g) Similar to (a),
but for device length L = 1.5 ?m. This figure is adapted from Ref. [91].
31
Chapter 2: Quality Factors for Zero-bias Conductance Peaks
As of the introduction in Chapter 1, particularly the sub-section 1.5.2 on Zhang et al. ex-
periment [44, 52], the observation of ZBCPs quantized near ? 2e2/h is insufficient to be the
evidence of MZMs. Instead, the robustness of the quantized zero-bias conductance becomes a
crucial feature of MZMs that could possibly help the ZBCPs associated with topological MZMs
(which are called ?good? ZBCPs [58]) stand out from other non-topological ZBCPs. In this chap-
ter, we would like to introduce new quality factors that not only can quantify the robustness of
quantization, but also can characterize the topological property of the ZBCP. These new quanti-
ties should be able to directly assess the quantized ZBCPs measured in the experiment, giving the
experiments practical use. Since the reported ZBCPs so far, which are claimed to be from MZMs,
do not exhibit the robustness of quantization, it is most likely that these experimentally-observed
ZBCPs are either associated with ABSs induced from QDs or inhomogeneous potential (which
are called ?bad? ZBCPs [58]), or associated with disorder-induced fermionic bound states (which
are called ?ugly? ZBCPs [58]). While the MZM-based ZBCPs intrinsically show robust quanti-
zation of 2e2/h at zero temperature when probed from a single-channel tunnel contact, ZBCPs
coming from other non-topological sources can also possibly reach somewhat robust quantiza-
tion with enough parameter fine-tuning [86, 94?97]. The main question we try to answer in this
chapter is whether we can take advantages of the new quality factors, based on the robustness
32
of quantization, to distinguish topological ?good? ZBCPs from trivial ?bad? and ?ugly? ZBCPs.
In this sense, we would like to compare results of different models by calculating the conduc-
tance scanning through Zeeman field and tunnel barrier height, and even presenting the zero-bias
conductance in terms of normal-state conductance GN to simulate the experimentally-accessible
data.
The quality factor F proposed here will be to gauge the robust quantized ZBCP to the
changes of the tunnel barrier height because the Majorana quantization should be independent
of the tunnel barrier height in the topological phase. To match the measurable quantity GN ,
we will convert the tunnel barrier height to the corresponding GN , and make the quality factor F
dimensionless. The other proposed quality factor J will be to gauge the robust quantized ZBCP to
the changes of Zeeman field. It is known that the parameter, Zeeman field, can change the system
from topological phase to trivial phase, or vice versa. So the robustness over Zeeman field will be
restricted by the boundaries of topological phase. However, we still want to introduce the quality
factor J because the plot of conductance versus Zeeman field is the first step for searching MZMs.
Using these two quality factors to analyze the simulated conductance of ?good?, ?bad?, and
?ugly? scenarios, we can see if the quality factors work out to distinguish MZMs by verification
of the calculated wave function.
2.1 Model
In this section, we will describe the 1D SC-SM nanowire [26?29] as in Fig. 2.1, which
gives rise to ?good?, ?bad?, and ?ugly? ZBCPs depending on the form of the potential in the real
space [58].
33
We use the minimal single-band model to describe the 1D superconductor-proximitized
semiconductor nanowire with intrinsic Rashba spin-orbit coupling and external Zeeman-field-
induced spin splitting, in the form of a Bogoliubov-de Gennes (BdG) Hamiltonian
?
1 L
H? = dx???(x)HNW??(x) (2.1)
2 0
with
( )
~2
HNW(?) = ? ??
2
x ? i?R?x?y ? ?+ V (x) ?z + Vz?x + ?(?, Vz)? i?, (2.2)2m
where m? = 0.015me is the effective mass of an electron (me is the rest mass of an electron),
? is the chemical potential, and ? is an infinitesimal dissipation parameter included to avoid
singularities in G0(E) from resonant transmission. The Rashba spin-orbit coupling with strength
?R is perpendicular to the wire [98] and the magnetic field B is applied along the nanowire
longitudinally such that the Zeeman te(rm Vz = 1g?BB, where ?B)is Bohr magneton. The2 T
wave function in Eq. (2.1) is ??(x) = ???(x), ???(x), ??
?
?(x),???
?
?(x) in Nambu space, with
?x,y,z(?x,y,z) being Pauli matrices in spin (particle-hole) space. ?(?, Vz) is a superconducting
self-energy
??0 + ?(Vz)?x
?(?, Vz) = ??? ?0, (2.3)
?2(V )? ?2z
where ? is the self-energy coupling strength with the parent SC. The self-energy is the effective
renormalized energy introduced into the system when proximitized by the SC in the intermediate
34
regime [99, 100]. The Zeeman-field-varying SC gap is
?
?(V 2z) = ?0 1? (Vz/Vc) ? ?(Vc ? Vz), (2.4)
which hosts a bulk parent SC gap ?0 without Zeeman field and vanishes above the SC collapsing
field V 1c .The Heaviside-step function ?(Vc ? Vz) indicates that the proximitized SC effect no
longer exists in the nanowire when Vz > Vc, which is not the interest of this study. We will only
numerically show the calculated conductance, energy spectrum, and wave functions below Vc.
To implement the numerical calculation, we have to discretize the continuum Hamiltonian as in
Eq. (2.2) into a lattice chain of tight-binding model [101] with the lattice constant a = 10 nm.
Then the effective tight-binding tunneling strength is t = ~2/(2m?a2) ? 25 meV, the effective
spin-orbit coupling strength is ? = ?R/(2a), and the wire length is given by L = Na, where N
is the total number of the atoms constituting the nanowire.
The Hamiltonian in Eq. (2.2) becomes energy-dependent when the self-energy ?(?, Vz) is
included. Thus, to get the energy spectrum, instead of diagonalizing HNW (?) directly, we can
specify the peaks of the density of states (DOS) located at energies ?0 from the Green?s function,
i.e.,
1 { [ ]}
?tot(?) = ? Im Tr G?(?) , (2.5)
?
and the Green?s function is
1
G?(?) = , (2.6)
? ?HNW(?)
1The reason for this gap collapse is unclear. One possiblity is a strong magnetic-field-induced orbital effect in the
parent SC. It could also simply result from the parent SC reaching the Clogston limit with the Zeeman spin splitting
becoming equal to the bulk SC gap.
35
where ? = ?0 + i? and ? is an infinitesimal real number for DOS brodening. The trace function
Tr(. . . ) in Eq. (2.5) is over the spatial space and sub-space (i.e., particle-hole and spins) of the
Green?s function matrix G?(?). Note that all the numerical results in this study include the self-
energy because this is close to the real experimental situations.
We use the minimal model to simulate the nanowire instead of a more complex 3D model
because it is found that the minimal model gives results very similar to complex models [86,102].
In addition, a more complex model will have many more unknown parameters, making it less
useful.
Besides the SC-SM nanowire itself, the normal lead attached to the end of the nanowire (as
the left green block in Fig. 2.1) is where the tunneling conductance is measured. The Hamiltonian
of the normal lead is
( )
~2
H 2lead(?) = ? ??x ? i?R?x?y ? ?+ Elead ?z + Vz?x (2.7)2m
with the on-site energy Elead ? ?25 meV in the lead controlled by the gate voltage. A tunnel
barrier induced at the interface of the normal-superconductor (NS) junction can be described by
replacing V (x) in Eq. (2.2) with a boxlike potential
Vbarrier(x) = Ebarrier?lbarrier(x), (2.8)
along with ?(x) = 0, i.e., uncovered by the SC, as the blue barrier potential at the left end of the
nanowire in Fig. 2.1.
36
Figure 2.1: Schematic plots of the hybrid structures of a 1D superconductor-
semiconductor nanowire and an attached lead, with different potentials. (a) Pris-
tine nanowire: V (x) = 0. (b) Nanowire with a SC-uncovered quantum dot
V (x) = VD exp(?x2/?2D) with length ?D. (c) Nanowire with an inhomogeneous
potential V (x) = VD exp(?x2/(2?2D)). (d) Nanowire with random disorder V (x).
37
2.1.1 Potential of Good/Bad/Ugly ZBCPs
The potential V (x) in Eq. (2.2) determines whether the ZBCP belongs to ?good?, ?bad? or
?ugly? type [58]. There are four kinds of potentials we set up for numerical simulations to discuss
the quality factor of quantization for different types of ZBCPs. First of all, Eq. (2.2) displays as
a pristine nanowire that can produce ?good? ZBCPs above TQPT field when V (x) = 0, as in
Fig. 2.1(a). This Hamiltonian can definitely generates the genuine MZMs in the topological
regime [26?29]. However, in the realistic experimental system, the unintentional potentials can
produce ?bad? and ?ugly? ZBCPs.
The ?bad? ZBCPs are defined as those ZBCPs appearing in the topologically-trivial regime,
resulting from the deterministic spatially-varying potential. These ?bad? ZBCPs are asscociated
with so-called ABSs [59?69]. Such ABSs result from unintended quantum dots created when
the lead is attached to the end of the nanowire. Such a quantum dot can arise either from mis-
match of Fermi energies between the normal lead, semiconductor and the superconductor or from
screened charged impurities or their combination [62,64]. In our model, the quantum dot hosts a
Gaussian potential at the end of the nanowire, part of which is not covered by the parent SC, as
in Fig. 2.1(b). That is to say, the quantum dot potential is
( )
2
V (x) = VD exp ?
x
?(?D ? x), (2.9)
?2D
where VD is the dot barrier height and ?D is the dot length. Also, the parent SC is mathematically
expressed as
?SC(x, Vz) = ?(Vz) ? ?(x? ?D), (2.10)
38
where ?(Vz) is defined as Eq. (2.4). The Heaviside step function ?(x) describes that the SC
only covers the nanowire outside of the quantum dot. A variation of the quantum dot model
?
with ??D = 2?D to replace ?D in Eq. (2.9) can occur where the quantum dot potential extends
into the superconductor as shown in Fig.2.1(c). This can be accomplished by dropping the factor
?(x ? ?D) in both Eqs. (2.9) and (2.10). Results for the case of a ?bad? potential that do not
specify an ?D should be understood as being obtained from calculations which use this variant of
the ?bad? potential.
The ?ugly? ZBCPs are defined as those ZBCPs showing up in the trivial regime, induced
by random strong disorder. The typical potential that accounts for ?ugly? ZBCPs is the onsite
disorder-induced random potential which follows an uncorrelated Gaussian distribution statisti-
cally with zero mean and standard deviation ??, i.e.,
V (x) = Vimp(x), (2.11)
?Vimp(x)? = 0, (2.12)
?Vimp(x)Vimp(x?)? = ?2??(x? x?), (2.13)
as in Fig. 2.1(d). Each set of ?ugly? results in Sec. 2.3.3 is based on just one particular config-
uration of Vimp(x), which is unpredictable contrary to the deterministic quantum dot potential in
Eq. (2.9). This random impurity potential can induce trivial ?ugly? ZBCPs which mimic ?good?
ZBCPs.
39
2.1.2 Differential Conductance Formalism
We numerically compute the local differential tunneling conductance G0 = dIL/dVL from
the normal lead at the left end through the NS junction by the scattering matrix (S-matrix) method
[Eq. (2.14)]. Specifically, we use Python package KWANT [103] to compute the differential
conductance with the in-built scattering matrix derived from the known Hamiltonian. The con-
ductance at zero temperature can be expressed and computed by the scattering matrix elements
as follows:
G0 = N ? Tr(r ? ?eeree ? rehreh) (2.14)
in the unit of e2/h, where N is the number of the conducting channels in the lead, ree is the
normal reflection matrix, and reh is the Andreev reflection matrix. In our system with only one-
subband lead, N = 2 counts the two spin modes. The conductance at finite temperature G(V )
can be further calculated by the convolution of the zero-temperature conductance G0(E) and the
derivative of the Fermi-Dirac distribution ?f(E, T )/?E, i.e.,
? ? ?f(E ? V, T )
G(V ) = ?? G0(E) [ ( dE )] (2.15)?? ?E? 1 V ? E
= ? G0(E) sech2 dE. (2.16)
?? 4T 2T
The above is the formalism for simulating the differential tunneling conductance in our results.
40
2.1.3 Wave Functions
To determine the topological superconducting characteristic of a one dimensional system,
it is necessary to look at the structure of the Majorana modes. Specifically, a topological super-
conductor is characterized by spatially well-separated Majorana wave functions [65,85,104]. On
the other hand, the ABSs display two overlapping wave functions at one end of the nanowire [83],
even we construct their wave functions in the Majorana mode. The 4N component Nambu wave
function ?(x) for an N -site system, corresponding to any eigenenergy ?0 obtained from peaks
of ?(?) in Eq. (2.5) can be obtained as the eigenstate of HNW(?0) with eigenvalue ?0. In a clean
or weakly disordered system, one could determine the topological characteristic of the system by
a straightforward calculation of the topological invariant [2, 101, 105]. However, this approach
is not well-defined for systems without a spectral or transmission gap, e.g. short strongly disor-
dered nanowires. The systems of interest here, shown in Fig. 2.1, are systems with a few sub-gap
states. In this case, it is simpler to directly determine the topological character of the system by
analyzing the low-energy wave functions directly as we describe below.
Majorana mode wave functions, even for a topological superconducting system of finite
size, split into non-zero energy (?0 = ?) eigenstates ?(x) and ??(x) that are not themselves
Majorana (i.e., particle-hole symmetric). For a low-energy eigen-wave function ?(x) corre-
sponding to a positive energy , one can use the particle-hole symmetry to define an orthogonal
wave function ??(x) = ?y?y??(x), which can be checked to be an eigenstate of HNW(?) with
41
eigenvalue ?. Then we can reconstruct the particle-hole symmetric Majorana wavefunctions as
?A(x) = ?
1
[?(x) + ??(x)] , (2.17)
2
?B(x) = ??
i
[?(x)???(x)] , (2.18)
2
( )
where T?A,B(x) = ? (x), ? (x), ??? ? ?(x),????(x) are manifestly particle-hole symmetric. Note
that this recipe suffers from a phase ambiguity of the eigenstate ?(x) in the case of a general
class D Hamiltonian, where it needs to be refined. This is not a problem for our case where
the BdG Hamiltonian is real, which allows ?(x) to be real. In general, ?A,B(x) are not the
eigenfunctions of the BdG Hamiltonian, except when  = 0, they represent the MZMs. However,
when  is much smaller than the SC gap, the off-diagonal matrix element of the Hamiltonian
HNW(? = 0) between ?A,B(x) is suppressed by a factor proportional to . The matrix elements
of HNW(? = 0) between these states and any other excited states vanish as well. A system would
be characterized as topological if the densities |??(x)|2 corresponding to the state ??(x) are
spatially separated.
If |? 2 2A(x)| and |?B(x)| are localized on the opposite ends of the nanowire without over-
lapping, then we have a pair of MZMs. On the contrary, if |?A(x)|2 and |?B(x)|2 are clearly
overlapped with each other, then the system hosts ABSs. When one of |? 2A,B(x)| is localized on
one end of the wire, while the other one is localized in the middle of the wire, then this pair can be
quasi-Majorana bound states when they are partially-overlapped with each other [65, 104, 106].
42
2.2 Quality Factor
The highlights of this work is the newly proposed quality factors to quantify the robust-
ness of quantized conductance plateau. One of the most characteristic features of an MZM is
quantized tunneling conductance [53?56]. To be precise, the ZBCP associated with an MZM is
predicted to be exactly quantized at T = 0 for a sufficiently long topological wire, even when the
transmission of the tunnel contact becomes vanishingly small [53?56]. One may feel this phe-
nomenon counter-intuitive because the ?normal?-state conductance GN (for Vbias  ?) diverges
as the transmission of the tunnel contact is reduced. We use quotation marks over ?normal? here
to emphasize that this quantity is close to the actual normal-state conductance only in the limit
when the SC gap is the smallest energy scale in the problem. However, we will refer to this
quantity as the normal-state conductance in the rest of the thesis because this is the most con-
venient quantity to measure. The large-series resistance that suppresses the transmission should
decrease the conductance of the MZM, which contradicts the theoretical quantization. One can
have insight into this counter-intuitive behavior by considering the analytic form of conductance
2e2 (2V ?)2
G0(V ) = (2.19)
h (V 2 ? 4t2)2 + (2V ?)2
that probes into a Majorana ?, which is weakly coupled to the Majorana ?? at the other end [55].
Note that t is the splitting of the Majorana modes and ? is the tunnel broadening [55], which
vanishes with the normal-state conductance, i.e., G 2N ? (2e /h)?/?. As the Majorana splitting
t vanishes, the above conductance takes the form of a Lorentzian with quantized height (2e2/h),
but with a width ?. However, following the finite-temperature modification as Eq. (2.15), the
43
quantized height is reduced as T ? ? such that it approaches G(V = 0) ? (2e2/h)?/T . The
latter form is more consistent with an expectation of a conductance limited by the normal-state
conductance GN .
The conductance Gz versus GN plot for an ideal Majorana wire shown in Fig. 2.2(a) can
further confirm the theoretically expected behavior for the ZBCP into an MZM described in the
last paragraph. Ideally, the so-called normal-state conductance GN is defined as the conductance
without SC. Since it is often non-trivial to remove SC from a device without changing other
factors, GN is practically taken as an average of the conductance at Vbias = ?Vabove where Vabove
is a large bias voltage above the SC gap. The variation ofGN in Fig. 2.2(a) is obtained by varying
the barrier height, which can be implemented by tuning a tunnel gate voltage [44,52] or changing
the tip-sample distance in STM [107] in the experiment. As a result of finite-temperature effect,
the zero-bias conductance Gz in Fig. 2.2(a) shows a nearly quantized plateau when the normal-
state conductance GN is larger, but then decreases to zero linearly as GN is reduced to zero,
which is consistent with the theoretical analysis discussed in the last paragraph.
The quantized plateau independent of GN is a phenomenal signatures of MZMs. While
one can tune GN and other parameters sufficiently to make the conductance into a SC quantized
without the appearance of an MZM, this quantization is not expected to be robust. Nonetheless,
the conductance into an MZM cannot be precisely quantized either due to the finite-temperature
and finite-size effects. In this chapter, we aim to distinguish MZMs from other SC bound states
by quantifying the robustness of the conductance plateau as shown in Fig. 2.2(a). To do this, we
first identify the largest continuous interval [GN,1, GN,2] on the x-axis over which the zero-bias
conductance Gz is within a tolerance  of quantization, i.e., |Gz ? 1| <  (in units of 2e2/h).
Then, a quality factor F = GN,2/GN,1 that gauges the quantized degree of the conductance into
44
Figure 2.2: Schematic plots to show quality factors. The tolerance factor for
quantized conductance is  = 0.05. (a) Black curve is the zero-bias conductance
peak Gz versus normal-metal conductance (above SC gap) GN . The quality factor
F ? GN,2/GN,1, where GN,1 and GN,2 are defined by the consecutive normal-metal
range for which the corresponding zero-bias peak is quantized within [1 ? , 1 + ]
in the unit of 2e2/h. (b) Black curve is the zero-bias conductance peak Gz versus
Zeeman field Vz. The quality factor J ? VZ,2/VZ,1, where VZ,1 and VZ,2 are defined
by the consecutive Zeeman field range for which the corresponding zero-bias peak is
quantized within [1? , 1 + ] in the unit of 2e2/h.
45
the MZM can be assigned. In Fig. 2.2(a), we set  = 0.05 as an example, meaning as long as the
ZBCP is above 95% of 2e2/h and below 105% of 2e2/h [within pink region in Fig. 2.2(a)], then
we take the ZBCP as a ?well-quantized? ZBCP. In Sec. 2.3, we also demonstrate the numerical
results of  = 0.10 and  = 0.20 for comparison. In some special cases, where the ZBCP over
the visible range is sectioned into several parts [e.g. Fig. 2.4(f)], we take the largest consecutive
normal-metal range to define GN,1 and GN,2.
The robustness of the topological phase should facilitate the quality factor to be applied to
other parameters as well. The Zeeman splitting Vz controlled by tuning the applied magnetic field
is one such parameter. As long as the topological gap is not destroyed, an MZM system associated
with a reasonably large topological gap is expected to be robust to changes of Vz. Here, we will
quantify the robustness of the MZM to the Zeeman field variations using a quality factor J defined
in a similar way to the tunnel-gate quality factor F proposed in the last paragraph. Fig. 2.2(b) is
the plot of zero-bias conductance peak Gz versus Zeeman field. This is the conductance linecut
as in Fig. 2.3(e), which can be extracted from Fig. 2.3(a) with only Vbias = 0. The definition of
quality factor J is similar to F as illustrated in the previous paragraph, except that we change
the normal-metal conductance GN part to Zeeman field Vz for the quality factor J . Formally,
it is defined as J ? VZ,2/VZ,1, where [VZ,1, VZ,2] is the range over which |Gz ? 1| <  in the
unit of 2e2/h and the tolerance factor for quantized conductance  is a small number. Same as
Fig. 2.2(a) as an example, we set  = 0.05, meaning 5% of the difference from the quantized
value 2e2/h as highlighted in the pink region in Fig. 2.2(b) are considered ?well-quantized?. All
the explanations for F can also be applied to J as long as we replace GN,1 and GN,2 by VZ,1 and
VZ,2, respectively.
So far, the definitions of F and J are not well-defined in cases where the zero-bias con-
46
ductance Gz does not lie in the quantized range (or as the pink regions in Fig. 2.2). In this case,
we formally define F and J to be zero. In non-topological cases, where the zero-bias conduc-
tance Gz touches the quantized range slightly, F and J are just a bit higher but close to 1. In
contrast, we will see from our numerical results in Sec. 2.3.1 that F and J can be much larger
than 1 in the ideal MZM case at low temperature. The main goal of our work is to study F and J
for various models to determine if they can be used to distinguish topological MZMs from other
non-topological ZBCPs.
2.3 Numerical Results
In this section, we demonstrate simulated conductance plots for various tunnel gate strengths
and magnetic field strengths to determine the robustness of quantized ZBCPs to the variety of pa-
rameter tuning. We will see that the quality factors F and J for the ideal MZM will be limited by
temperature, topological gap and length. This section is mainly to compare the quality factors for
the ?good? ZBCP (i.e., ideal MZM) with other non-topological models for the ZBCP (i.e., ?bad?
and ?ugly? ZBCPs). We believe that the quality factors F and J obtained from these numerical
results can be directly applied to experiments because the quantities shown in this section are
typically measured in most MZM experiments [38, 43, 44, 52, 107].
2.3.1 ?Good? ZBCP
MZM-induced ZBCPs at the ends of topological superconducting nanowires are referred
to as ?good? ZBCPs [58]. Theoretically, they can be produced above the TQPT field in Majorana
nanowire model as Fig. 2.1(a), which is the setup for the numerical results in Fig. 2.3. While this
47
Figure 2.3: Numerical results for the ?good? ZBCP. Parameters: ? = 3.0 meV,
?0 = 0.2 meV, Vc = 2.5 meV, L = 3.0 ??m, ? = 0.5 meV, ? = 1.0 meV, and ? =
10?4 meV. The TQPT field is at V = ?2 + ?2TQPT = 1.118 meV. (a) Conductance
false-color plot as a function of bias voltage V and Zeeman field Vz, with the fixed
tunneling barrier height at Ebarrier = 10 meV. (b) Conductance false-color plot as a
function of bias voltage V and tunneling barrier heightEbarrier, with the fixed Zeeman
field at Vz = 1.5 meV. (c) Lowest-lying wave-function probability density |?|2 as a
function of nanowire position x, with fixed Vz = 1.5 meV and Ebarrier = 10 meV. (d)
Quality factor F as a function of temperature T for three different tolerance factors
, with the fixed Vz = 1.5 meV. The inset figure gives an overall trend starting from
T = 0. At T = 10 mK, F = 3.25 for  = 0.05. At T = 20 mK, F = 2.74 for
 = 0.1. (e) Conductance linecuts as a function of Zeeman field Vz for three different
bias voltages Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks
Vz = 1.5 meV. (f) Conductance linecuts as a function of tunneling barrier height
Ebarrier for three different bias voltages Vbias, with the fixed Vz = 1.5 meV. (g) Zero-
bias conductanceGz as a function of normal-metal conductanceGN for five different
temperatures T , with the fixed Vz = 1.5 meV. (h) Quality factor J as a function of
temperature T for three different tolerance factors , with the fixed Ebarrier = 10 meV.
48
ideal model is possibly not realizable at current experimental stage, the results in this sub-section
will serve as a reference for the signatures of a topological superconductor.
As expected from the literature [53?56], we observe a nearly quantized ZBCP appearing
above the TQPT field Vz > VTQPT = 1.118 meV in Fig. 2.3(a), which shows the conductance
at zero temperature as a false-color plot versus the bias voltage V and Zeeman field Vz. The
topological origin of this ZBCP can be seen in Fig. 2.3(c), which shows the lowest-lying wave-
function probabilities |? (x)|2A,B [see Sec. 2.1.3] at Vz = 1.5 meV> VTQPT [yellow dashed line
in Fig. 2.3(a)]. The localized wave functions |? 2A,B(x)| at opposite ends of the wire verify that
the ZBCP at Vz = 1.5 meV comes from topological MZMs. However, the wave function is not
accessible in a transport experiment in a nanowire. On the other hand, as discussed in the intro-
duction and Sec. 2.2, the quantization of the ZBCP associated with a topological MZM should
be robust to changes in the barrier height. This is consistent with Fig. 2.3(b), which shows the
conductance false-color plot as a function of bias voltage V and tunneling barrier height Ebarrier at
the fixed Zeeman field Vz = 1.5 meV. The plot shows the strength of the MZM-based ZBCP re-
mains widely constant as the tunnel barrier height Ebarrier increases, while the width of the ZBCP
decreases. This can be further confirmed by the linecuts obtained from Fig. 2.3(b) at Vbias = 0
that are shown in Fig. 2.3(f), where the zero-bias conductance Gz is found to drop by less than
5% from quantization. Fig. 2.3(f) also includes linecuts from Fig. 2.3(b) at Vbias = ?0.2 meV.
In contrast to the robust Gz, these finite-bias linecuts, which may be interpreted as the normal-
state conductance, change significantly with the tunnel barrier height Ebarrier. In fact, the tunnel
barrier height Ebarrier cannot be obtained directly through experimental measurements because
tunnel gates have lever arms that rely on complicated structures of capacitance. Meanwhile, the
normal-state conductance GN determined by Fig. 2.3(f) at Vbias = ?0.2 meV provides a way to
49
quantify the tunnel barrier height that is directly comparable to experiments. One subtlety that
arises in SM systems is that the conductance at the non-zero bias voltages in Fig. 2.3(f) can be
different in SM systems. This can be compensated by defining GN to be the average between the
conductances at Vbias = ?0.2 meV.
Therefore, to demonstrate a result directly compared to experiments, we plot Gz versus
GN in Fig. 2.3(g), while keeping Vz = 1.5 meV [yellow dashed line in Fig. 2.3(a)] to ensure the
wire is in the topological SC phase. Consistent with our discussion in the previous paragraph
about Fig. 2.3(f), we find that the ZBCP height does not change significantly as GN is reduced
at zero temperature T = 0. On the contrary, the results at finite temperature T > 0 obtained
from Eq. (2.15) show that Gz goes to zero as GN decreases, as expected from the discussion in
Sec. 2.2. The results in Fig. 2.3(g) clearly show that the robustness of the quantized ZBCP to the
tunnel barrier changes would be limited by the experimental temperature, even in the case of ideal
MZMs. The degree of robustness of the ZBCP quantization over GN can be characterized by the
quality factor F defined in Sec. 2.2, where the ?well-quantized? zero-bias conductance Gz lies
within a tolerance factor . The plot of the quality factor F versus temperature as in Fig. 2.3(d)
shows that while the quality factor F associated with MZMs can be quite large (i.e., more than 80
at T = 0), F decreases quite rapidly as the temperature T becomes comparable to the topological
SC gap. The main figure in Fig. 2.3(d) is to show the detailed variations of F in the experimental
temperature range from T = 2 mK to T = 30 mK. The inset figure starting from T = 0 gives an
overall trend to show how sharply F decreases from zero temperature to finite temperature. Since
we only want to demonstrate how sharp the change of F is by the temperature variations, the tick
numbers in the inset figure are not crucial here. While the quantitative value of F increases as
the tolerance factor  increases, the qualitative behavior does not seem to be remarkably affected
50
by the value of .
ZBCPs associated with the ideal MZMs are expected to be robust to variations in the Zee-
man field because MZMs arise in a topological SC phase. This is consistent with the result in
Fig. 2.3(e), where we find that the zero-bias conductance (i.e., Vbias = 0) starts out low in the
non-topological regime (i.e., Vz < VTQPT), but then sharply grows into a quantized plateau for
Vz > VTQPT. Same as the robustness with respect to the tunnel barrier height seen in Fig. 2.3(f),
the result in Fig. 2.3(e) shows that the ZBCP associated with a topological MZM are also quite
robust to variations in the Zeeman potential. Based on the discussion in Sec. 2.2, this robustness
can be quantified with the definition of parameter J , which is plotted as a function of temperature
in Fig. 2.3(h). Similar to the quality factor F associated with tunnel gate robustness, the Zeeman-
field quality factor J also decreases with temperature in a way that is qualitatively independent
of the tolerance  used in the calculations of J .
2.3.2 ?Bad? ZBCP
Now we consider the quality factor of the?bad? ZBCPs, which arise from an inhomoge-
neous potential near the end of the nanowire as shown in Fig. 2.1(b,c). Such an inhomogeneous
potential can lead to ZBCPs [62], which can even exhibit nearly quantized conductance [65].
Thus, it is interesting to analyze the robustness of the quantizated ZBCPs and see if the quality
factors can distinguish between such ABSs that give rise to ?bad? ZBCPs and the topological
MZMs discussed in sub-section 2.3.1.
The conductance false-color plot in Fig. 2.4(a) based on the setup in Fig. 2.1(b) clearly
shows that a pair of finite-energy conductance peaks merge together into a ZBCP, quite similar
51
to the topological result as in Fig. 2.3(a). Based on the ZBCP linecut shown in Fig. 2.4(e), we
observe that unlike the ideal MZM case, the ZBCP below the TQPT at Vz = 0.87 meV [marked
by the black dashed line in Fig. 2.4(e), or the yellow dashed line in Fig. 2.4(a)] is only near
the quantized value 2e2/h. We can find that the ZBCP splitting for Vz > 0.87 meV as seen in
Fig. 2.4(a) is very likely to cause the reduction of the ZBCP seen in Fig. 2.4(e). To study if the
tunnel-gate robustness of MZMs seen in the last sub-section 2.3.1 applies to this nearly quantized
ZBCP, we examine the ZBCP height as the tunnel barrier height varies. As in Fig. 2.4(b), the
ZBCP height appears to remain nearly constant while the width of the ZBCP changes, which is
quite similar to the case of ideal MZMs. However, a closer checkup on the quantization using the
linecut of Vbias = 0 shown in Fig. 2.4(f) demonstrates that the ZBCP height at Vz = 0.87 meV
varies by a substantial amount as the tunnel barrier is changed. This huge variation can also be
observed by plotting the ZBCP height versus the normal-state conductance shown in Fig. 2.4(g).
Contrary to the case of the ideal MZM, the ZBCP height overshoots the quantized value at low
temperatures when the normal-state conductance GN small. This explains why the value of the
quality factor F shown in Fig. 2.4(d) is relatively small compared to the topological case. The
quality factor F for  = 0.05 at 10 mK in Fig. 2.4(d) is below 1.5. Interestingly, the quality
factor F turns out to be quite large when  = 0.2, which makes sense based on the fact that the
conductance Gz in Fig. 2.4(g) remains close to quantized to some extent. These observations
along with the small ZBCP splitting seen in Fig. 2.4(a) can be understood by the low-energy
wave functions decomposed in terms of Majorana basis shown in Fig. 2.4(c). We can confirm
that the system is non-topological because both Majorana components of the wave functions are
strongly overlapping at the same end. However, one of the components has a stronger spatial
modulation, which means it has a strong weight at a different Fermi level. This suppresses the
52
overlap between the two states and ensures that only one of the states can couple strongly to
the lead, explaining the observations made about Fig. 2.4(a) and Fig. 2.4(g). Given the ZBCP
splitting observed in Fig. 2.4(a), the ZBCP quantization in Fig. 2.4(e) can only survive over a
rather narrow range of Zeeman field Vz, which consequently leads to the relatively smaller values
of the quality factor J seen in Fig. 2.4(h) compared to the topological case.
Fig. 2.5 that shows a ?bad? ZBCP comes from another inhomogeneous potential configura-
tion as in Fig. 2.1(c). While the ZBCPs shown in Fig. 2.5(a) are qualitatively similar to the ?bad?
ZBCP in Fig. 2.4(a), the quantization of the ZBCP at Vz = 0.9 meV, based on Figs. 2.5(b,d,e,f,g),
appears to be more robust compared to Fig. 2.4. In fact, relative to the previous ?bad? ZBCP
[Fig. 2.4(d)], the quality factor F in Fig. 2.5(d) at temperature of 20 mK and  = 0.1 is 1.79,
closer to the value of 2.74 in the ?good? case [see Fig. 2.3(d)]. This can be understood from the
Majorana decomposition in Fig. 2.5(c) that shows a pair of partially-separated modes, which have
been described as quasi-Majoranas [65]. Such separated segments between spatially-separated
MZMs exhibit its own topological characteristics, which explains the high value of F in this
?bad? ZBCP. The quality factor J shown in Fig. 2.5(h) also turns out to be intermediate between
the ?good? ZBCP shown in Fig.2.3(h) and the previous ?bad? ZBCP shown in Fig. 2.4(h).
2.3.3 ?Ugly? ZBCP
The last sets of results [Figs. 2.6-2.9] belong to the ?ugly? scenario, where the disorder-
induced random potential, as shown in Fig. 2.1(d), generates a ZBCP by chance. Such ZBCPs
only show up in only a small proportion of disorder configurations with the same parameters.
Figs. 2.6-2.9 show results that span the set of possibilities, but their occurrence is somewhat
53
Figure 2.4: Numerical results for the ?bad? ZBCP. Parameters: ? = 2.51 meV, ?0 =
0.2 meV, Vc = 2.1 meV, L = 3.0 ??m, ? = 1.0 meV, ? = 1.0 meV, and ? = 10?4
meV. The TQPT field is at V 2 2TQPT = ? + ? = 1.414 meV. The parameters for the
quantum dot on the left end are: VD = 0.3 meV and ?D = 0.15 ?m. (a) Conductance
false-color plot as a function of bias voltage V and Zeeman field Vz, with the fixed
tunneling barrier height at Ebarrier = 75 meV. (b) Conductance false-color plot as a
function of bias voltage V and tunneling barrier heightEbarrier, with the fixed Zeeman
field at Vz = 0.87 meV. (c) Lowest-lying wave-function probability density |?|2 as a
function of nanowire position x, with fixed Vz = 0.87 meV andEbarrier = 75 meV. (d)
Quality factor F as a function of temperature T for three different tolerance factors
, with the fixed Vz = 0.87 meV. The inset figure gives an overall trend starting from
T = 0. At T = 10 mK, F = 1.331 for  = 0.05. At T = 20 mK, F = 1.789 for
 = 0.1. (e) Conductance linecuts as a function of Zeeman field Vz for three different
bias voltage Vbias, with the fixed Ebarrier = 75 meV. The black dashed line marks
Vz = 0.87 meV. (f) Conductance linecuts as a function of tunneling barrier height
Ebarrier for three different bias voltage Vbias, with the fixed Vz = 0.87 meV. (g) Zero-
bias conductanceGz as a function of normal-metal conductanceGN for five different
temperatures T , with the fixed Vz = 0.87 meV. (h) Quality factor J as a function of
temperature T for three different tolerance factor , with the fixed Ebarrier = 75 meV.
54
Figure 2.5: Numerical results for the ?bad? ZBCP. Parameters: ? = 2.5 meV, ?0 =
0.2 meV, Vc = 1.2 meV, L = 3.0 ?m?, ? = 1.0 meV, ? = 0.2 meV, and ? = 10?4
meV. The TQPT field is at VTQPT = ?2 + ?2 = 1.020 meV. The parameters for
the inhomog?eneous potential on the left end are: VD = 1.2 meV and ?D = 0.4
?m (??D = 2 ? 0.4 ?m). (a) Conductance false-color plot as a function of bias
voltage V and Zeeman field Vz, with the fixed tunneling barrier height atEbarrier = 10
meV. (b) Conductance false-color plot as a function of bias voltage V and tunneling
barrier height Ebarrier, with the fixed Zeeman field at Vz = 0.9 meV. (c) Lowest-
lying wave-function probability density |?|2 as a function of nanowire position x,
with fixed Vz = 0.9 meV and Ebarrier = 10 meV. (d) Quality factor F as a function
of temperature T for three different tolerance factors , with the fixed Vz = 0.9
meV. The inset figure gives an overall trend starting from T = 0. At T = 10
mK, F = 2.148 for  = 0.05. At T = 20 mK, F = 1.785 for  = 0.1. (e)
Conductance linecuts as a function of Zeeman field Vz for three different bias voltage
Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks Vz = 0.9 meV.
(f) Conductance linecuts as a function of tunneling barrier height Ebarrier for three
different bias voltage Vbias, with the fixed Vz = 0.9 meV. (g) Zero-bias conductance
Gz as a function of normal-metal conductance GN for five different temperatures T ,
with the fixed Vz = 0.9 meV. (h) Quality factor J as a function of temperature T for
three different tolerance factor , with the fixed Ebarrier = 10 meV.
55
rare. More generic results that do not show any quantization can be found in the Supplementary
Material [90]. Figs. 2.6-2.9(a) show conductance peaks that may be interpreted as bound states
or gap closure [94]. Specifically, even though ZBCPs appear in all these plots are below the
nominal TQPT field (i.e.,Vz/VTQPT = 1), Figs. 2.6-2.7(a) exhibit the gap closing feature merging
into a ZBCP, similar to the way ?good? and ?bad? ZBCPs [i.e., Figs. 2.3-2.5(a)] does, while
Figs. 2.8-2.9(a) show a separation between the ZBCP and the gap closing peaks. What makes
the presented results in Figs. 2.6-2.9 so special is that all the ZBCPs in Figs. 2.6-2.9(a) appear
to approach close to the quantized value 2e2/h in various degrees, as one can see them clearly
through the Vbias = 0 linecuts in Figs. 2.6-2.9(e). Fig. 2.6(e) shows a comparatively wide range
of ZBCP plateau, but it deviates significantly from the quantized value, which gives rise to a
vanishing quality factor J at  = 0.05 plotted in Fig. 2.6(h). Fig. 2.7(e) shows quantized ZBCPs
limited to small ranges, with a strong variation of the zero-bias conductance value between the
quantized peaks that give rise to a very low quality factor J ' 1.2 even when the tolerance factor
is  = 0.2. Fig. 2.8(e) shows a nearly quantized plateau starting just slightly below the TQPT
field (i.e., Vz ? VTQPT) and maintaining the quantization through the topological regime (i.e.,
Vz > VTQPT) till the SC collapses. This can be interpreted as a result of a reduction of the actual
TQPT field due to the disorder potential. While disorder are typically inclined to suppress the
topological phase and increase VTQPT, it could possibly reduce VTQPT in rare fluctuations [108].
Finally, Fig. 2.9(e) exhibits a nearly quantized ZBCP, which is constrained to a small range of
Zeeman field that cannot be categorized as a plateau.
Figs. 2.6-2.9(b) show the ZBCP height as the tunnel barrier height varies, with the Zeeman
field fixed near the peak of the ZBCP [i.e., dashed line in Figs. 2.6-2.9(a)]. Figs. 2.6-2.7(b)
show that the ZBCP splits as the peak is reduced, revealing that the ZBCP was indeed non-
56
topological, as can be confirmed from the Majorana decomposition shown in Figs. 2.6-2.7(c).
However, these ZBCP splittings in Figs. 2.6-2.7(b) can merge and mimic zero-bias peaks when
the finite-temperature effect kicks in, preventing ZBCPs to vary sharply as temperature changes,
as in Figs. 2.6-2.7(g), which are opposite to what we have seen in Figs. 2.3-2.5(g). This is why
we do not observe a sharp drop from the zero-temperature value of F in Figs. 2.6-2.7(d) to a
finite-temperature one, and instead observe almost flat profiles of F values as the temperature
changes for  = 0.1 and  = 0.2. Due to their non-topological nature of the ZBCPs, the quality
factors F at T = 0 are much smaller (below 3) in Figs. 2.6-2.7(d) compared to other topological
cases [Figs. 2.3, 2.5, 2.8, 2.9(d)]. With the apparent overlapping wave functions localized at one
end in Fig. 2.6(c), the robustness of the ZBCP to changes in the tunnel barrier height cannot
sustain at a practically measurable temperature of 20 mK, which is quantified to be F = 0 for
 = 0.05, indicating the complete lack of quantization for this parameter. On the other hand,
the partially-overlapping Majorana modes in Fig. 2.7(c), which exhibit a bit more topological
character compared to the previous case, reflect this part on the quality factor F? the value is
F = 1.36 at T = 20 mK for  = 0.05.
On the contrary, Fig. 2.8(b) shows the results of another disorder configuration, where a
ZBCP remains unsplit even under changes of the tunnel barrier height. This can be verified with
the Majorana decomposition plotted in Fig. 2.8(c), which shows a pair of separated Majoranas on
opposite ends that indicate a topological state. The ZBCP plotted as a function of normal-state
conductance GN in Fig. 2.8(g) turns out to be quantized only at temperatures below 10 mK, as
one can also check from the quality factor F in Fig. 2.8(d). This can be explained by the fact that
the nearly quantized ZBCP in Fig. 2.8 occurs only slightly below the nominal TQPT field.
Finally, the ZBCP shown in Fig. 2.9(g) produced by another disorder configuration demon-
57
Figure 2.6: Numerical results for the ?ugly? ZBCP. Parameters: ? = 2.5 meV, ?0 =
0.2 meV, Vc = 1.2 meV, L = 3.0 ?m?, ? = 1.0 meV, ? = 0.2 meV, and ? = 10?4
meV. The TQPT field is at VTQPT = ?2 + ?2 = 1.020 meV. The parameters for
the onsite random potential is: ?? = 2.0 meV. (a) Conductance false-color plot as
a function of bias voltage V and Zeeman field Vz, with the fixed tunneling barrier
height at Ebarrier = 10 meV. (b) Conductance false-color plot as a function of bias
voltage V and tunneling barrier height Ebarrier, with the fixed Zeeman field at Vz =
0.8 meV. (c) Lowest-lying wave-function probability density |?|2 as a function of
nanowire position x, with fixed Vz = 0.8 meV and Ebarrier = 10 meV. (d) Quality
factor F as a function of temperature T for three different tolerance factors , with
the fixed Vz = 0.8 meV. The inset figure gives an overall trend starting from T = 0.
At T = 10 mK, F = 1.126 for  = 0.05. At T = 20 mK, F = 1.352 for  = 0.1. (e)
Conductance linecuts as a function of Zeeman field Vz for three different bias voltage
Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks Vz = 0.8 meV.
(f) Conductance linecuts as a function of tunneling barrier height Ebarrier for three
different bias voltage Vbias, with the fixed Vz = 0.8 meV. (g) Zero-bias conductance
Gz as a function of normal-metal conductance GN for five different temperatures T ,
with the fixed Vz = 0.8 meV. (h) Quality factor J as a function of temperature T for
three different tolerance factor , with the fixed Ebarrier = 10 meV.
58
Figure 2.7: Numerical results for the ?ugly? ZBCP. Parameters: ? = 2.5 meV, ?0 =
0.2 meV, Vc = 1.2 meV, L = 3.0 ?m?, ? = 1.0 meV, ? = 0.2 meV, and ? = 10?4
meV. The TQPT field is at V 2TQPT = ? + ?2 = 1.020 meV. The parameters for
the onsite random potential is: ?? = 2.0 meV. (a) Conductance false-color plot as
a function of bias voltage V and Zeeman field Vz, with the fixed tunneling barrier
height at Ebarrier = 10 meV. (b) Conductance false-color plot as a function of bias
voltage V and tunneling barrier height Ebarrier, with the fixed Zeeman field at Vz =
0.66 meV. (c) Lowest-lying wave-function probability density |?|2 as a function of
nanowire position x, with fixed Vz = 0.66 meV and Ebarrier = 10 meV. (d) Quality
factor F as a function of temperature T for three different tolerance factors , with
the fixed Vz = 0.66 meV. The inset figure gives an overall trend starting from T = 0.
At T = 10 mK, F = 1.490 for  = 0.05. At T = 20 mK, F = 2.638 for  = 0.1. (e)
Conductance linecuts as a function of Zeeman field Vz for three different bias voltage
Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks Vz = 0.66 meV.
(f) Conductance linecuts as a function of tunneling barrier height Ebarrier for three
different bias voltage Vbias, with the fixed Vz = 0.66 meV. (g) Zero-bias conductance
Gz as a function of normal-metal conductance GN for five different temperatures T ,
with the fixed Vz = 0.66 meV. (h) Quality factor J as a function of temperature T
for three different tolerance factor , with the fixed Ebarrier = 10 meV.
59
Figure 2.8: Numerical results for the ?ugly? ZBCP. Parameters: ? = 2.5 meV, ?0 =
0.2 meV, Vc = 1.2 meV, L = 5.0 ?m?, ? = 1.0 meV, ? = 0.2 meV, and ? = 10?4
meV. The TQPT field is at VTQPT = ?2 + ?2 = 1.020 meV. The parameters for
the onsite random potential is: ?? = 1.0 meV. (a) Conductance false-color plot as
a function of bias voltage V and Zeeman field Vz, with the fixed tunneling barrier
height at Ebarrier = 10 meV. (b) Conductance false-color plot as a function of bias
voltage V and tunneling barrier height Ebarrier, with the fixed Zeeman field at Vz =
0.95 meV. (c) Lowest-lying wave-function probability density |?|2 as a function of
nanowire position x, with fixed Vz = 0.95 meV and Ebarrier = 10 meV. (d) Quality
factor F as a function of temperature T for three different tolerance factors , with
the fixed Vz = 0.95 meV. The inset figure gives an overall trend starting from T = 0.
At T = 10 mK, F = 1.084 for  = 0.05. At T = 20 mK, F = 0 for  = 0.1. (e)
Conductance linecuts as a function of Zeeman field Vz for three different bias voltage
Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks Vz = 0.95 meV.
(f) Conductance linecuts as a function of tunneling barrier height Ebarrier for three
different bias voltage Vbias, with the fixed Vz = 0.95 meV. (g) Zero-bias conductance
Gz as a function of normal-metal conductance GN for five different temperatures T ,
with the fixed Vz = 0.95 meV. (h) Quality factor J as a function of temperature T
for three different tolerance factor , with the fixed Ebarrier = 10 meV.
60
Figure 2.9: Numerical results for the ?ugly? ZBCP. Parameters: ? = 2.5 meV, ?0 =
0.2 meV, Vc = 1.2 meV, L = 5.0 ?m?, ? = 1.0 meV, ? = 0.2 meV, and ? = 10?4
meV. The TQPT field is at VTQPT = ?2 + ?2 = 1.020 meV. The parameters for
the onsite random potential is: ?? = 2.0 meV. (a) Conductance false-color plot as
a function of bias voltage V and Zeeman field Vz, with the fixed tunneling barrier
height at Ebarrier = 10 meV. (b) Conductance false-color plot as a function of bias
voltage V and tunneling barrier height Ebarrier, with the fixed Zeeman field at Vz =
0.8 meV. (c) Lowest-lying wave-function probability density |?|2 as a function of
nanowire position x, with fixed Vz = 0.8 meV and Ebarrier = 10 meV. (d) Quality
factor F as a function of temperature T for three different tolerance factors , with
the fixed Vz = 0.8 meV. The inset figure gives an overall trend starting from T = 0.
At T = 10 mK, F = 1.624 for  = 0.05. At T = 20 mK, F = 1.419 for  = 0.1. (e)
Conductance linecuts as a function of Zeeman field Vz for three different bias voltage
Vbias, with the fixed Ebarrier = 10 meV. The black dashed line marks Vz = 0.8 meV.
(f) Conductance linecuts as a function of tunneling barrier height Ebarrier for three
different bias voltage Vbias, with the fixed Vz = 0.8 meV. (g) Zero-bias conductance
Gz as a function of normal-metal conductance GN for five different temperatures T ,
with the fixed Vz = 0.8 meV. (h) Quality factor J as a function of temperature T for
three different tolerance factor , with the fixed Ebarrier = 10 meV.
61
strates a plateau that appears nearly as robust as the ideal topological ZBCP plateau shown in
Fig. 2.3. This can be quantified by the quality factor F plotted in Fig. 2.9(d) at T = 20 mK,
 = 0.05, which is close to that of the ?good? ZBCP shown in Fig. 2.3. We can cross-check the
nearly topological behavior seen from ZBCP height by the Majorana wave functions shown in
Fig. 2.9(c), which show that the left component is spatially-separated from the right component.
Therefore, the ZBCP in this parameter should be identified as topological. However, Fig. 2.9(c)
also shows a disorder configuration where the right Majorana would not be accessible to tunnel-
ing, which will fail the test of end-to-end conductance correlation [81, 83, 87] between MZMs.
2.4 Discussion
2.4.1 Topological Characteristics from the quality factor F
Now we are going to analyze the quality factor F to see if it can allow us to distinguish
the case of topological (i.e., ?good?) MZMs from the ?bad? and ?ugly? ZBCPs that can arise
from quantum dots and disorder. If we focus on the quality factor F with the smallest value
 = 0.05 and the lowest temperature T ? 0 [i.e., insets in panel (d) of Figs. 2.3-2.9], the ideal
MZM is predicted to reach an F in excess of 80, while this ideal value of the quality factors in
the ?bad? and the ?ugly? cases remain below 20. Unfortunately, this ideal value for the quality
factor F is not measurable in a finite-temperature experiment. Panel (d) of Figs. 2.3-2.9 show
the quality factor over a more realistic range of temperatures. We see that even measurements
performed on an ideal MZM at a low temperature of 10 mK, as one can see in Fig. 2.3(d), can
only contribute to a quality factor of 3.3 for  = 0.05, which is much below the value of 84 at
T ? 0 . Similarly, the quality factors F for the ?bad? and the ?ugly? cases are reduced to be below
62
2 at T ? 10 mK for  = 0.05. The significant reduction of the quality factor at experimentally-
accessible temperatures potentially could make it difficult to distinguish between topological
and non-topological MZMs. However, the good-bad-ugly paradigm is more of a classification
of ZBCPs based on microscopic models rather than their ultimate topological characteristics.
More specifically, one can find that the wave functions plotted in Panel (c) reveal that many
ZBCPs classified as ?bad? and ?ugly? based on models show spatially-separated MZM wave
functions that allow them to be considered as topological. While this occurs in more than one
result shown in Sec. 2.3, these are rare for the ?bad? and ?ugly? model, unless one tunes the
parameters specifically to obtain a ZBCP stable with the change of magnetic field. One can find
more generic examples of ?bad? and ?ugly? models with lower F values that do not demonstrate
separated wave functions in the Supplementary Material [90]. In fact, among all the results
presented in Sec. 2.3, only Figs. 2.4, 2.6 and 2.8 display truly non-topological results, which
show a quality factor F below 1.5 for  = 0.05 at 10 mK. While the ideal topological ?good?
ZBCP in Fig. 2.3 gives F = 3.25 at 10 mK for  = 0.05, which is way above 1.5, a threshold of F
to distinguish topological ZBCPs can be carefully determined by the statistical distribution of the
F values for the non-topological ?bad? and ?ugly? cases (see the extended data in Supplementary
Material [90]).
In Fig. 2.10(a), the blue dots are occurrence times for a case with a quality factor Fn larger
than a given F value. The complementary cumulative distribution function (CCDF), as the red
dashed curve in Fig. 2.10(a), is determined by fitting the blue dots with spline interpolation. The
probability density function (PDF), as the blue curve in Fig. 2.10(b), can be numerically obtained
by taking the derivative of the CCDF shown in Fig. 2.10(a). Fig. 2.10(a) shows that there is no
non-topological case (?bad? and ?ugly?) that exhibits a quality factor F larger than 3, while the
63
Figure 2.10: Statistical histogram for ?bad? and ?ugly? ZBCPs at T = 10 mK and
 = 0.05. (a) Blue dots: number of cases whose quality values Fn (n denotes each
case) are larger than a given F value on the x-axis. The data can be found in the
Supplementary Material [90]. Red dashed curve: complementary cumulative distri-
bution function (CCDF) is determined by fitting the blue dot data with spline inter-
polation. (b) Probability density function (PDF) is the (negative) derivative of the
CCDF shown in (a).
64
occurence times for F > 2 is already rare. With the extended data of ?good? ZBCPs, which all
demonstrate F > 3 at T = 10 mK for  = 0.05 (summarized on the first page of the pdf file of
the Supplementary Material [90]), the cumulative histogram for the trivial ZBCPs in Fig. 2.10(a)
suggests a threshold of approximately F = 3 would be sufficient to separate topological and
non-topological ZBCPs. In fact, while the data of F < 1.5 (at T = 10 mK for  = 0.05)
demonstrate overlapping wave functions clearly, the data that linger above F = 2.0 in Fig. 2.10(a)
demonstrate partially-separated quasi-Majoranas, which can be classified as between topological
and non-topological. In short nanowires, which are most of the current experimental setups,
MZMs appear as partially-separated quasi-Majoranas, which have been proposed to be able to
implement braiding [65]. Nonetheless, quasi-Majoranas are non-topological in the sense that the
two Majorana modes would not localize ideally at both ends, which will fail the examination of
the end-to-end conductance correlation [83]. An example can be visualized in Fig. 2.5, which is
labelled as a model of a ?bad? ZBCP with a F value of 2.15 for  = 0.05 at T = 10 mK. Since
Fig. 2.5(c) shows that the Majorana wave functions in this system are quite well-separated but
inaccessible from the right end, this case would correspond to the quasi-Majorana scenario [65].
Because of the separated MZMs, one can interpret this scenario as having a short segment of a
topological superconductor at one end of the wire.
The temperature criterion 10 mK discussed in the last paragraph may be relaxed to 20 mK if
the tolerance factor  associated with F is increased to  = 0.1. Similar to the way we determine a
threshold of F at T = 10 mK for  = 0.05 to separate topological and non-topological ZBCPs, a
threshold of F = 2.5 for  = 0.1 at T = 20 mK can be scrupulously determined by a combination
of the quality factors of the ideal ?good? ZBCPs from the extended data [90], and the statistical
histogram in Fig. 2.11 from the extended non-topological ?bad? and ?ugly? data [90]. To get a
65
Figure 2.11: Statistical histogram for ?bad? and ?ugly? ZBCPs at T = 20 mK and
 = 0.1. (a) Blue dots: number of cases whose quality values Fn (n denotes each
case) are larger than a given F value on the x-axis. The data can be found in the
Supplementary Material [90]. Red dashed curve: complementary cumulative distri-
bution function (CCDF) is determined by fitting the blue dot data with spline inter-
polation. (b) Probability density function (PDF) is the (negative) derivative of the
CCDF shown in (a).
66
sense of this distinction, one can find that the ideal MZM shown in Fig. 2.3 is associated with
an F of 2.74 for  = 0.1 at T = 20 mK, while all the non-topological cases are below 1.8,
except for Fig. 2.7. In fact, we particularly present the ?ugly? ZBCP results in Fig. 2.7, which is
associated with a value of F = 2.64 for  = 0.1 at T = 20 mK, to show an example that an ?ugly?
ZBCP could produce a quality factor F larger than the threshold to be mistakenly distinguished
as topological, but does not possess topological characters based on its Majorana-decomposed
wave functions (as we discussed in Sec. 2.3.3). Fig. 2.7 is also the only one case among the 30
non-topological ZBCPs that shows up with F > 2.5 in Fig. 2.11(a). While this false-positive case
only accounts 3.3% of our simulated non-topological samples [90], giving rise to a precision rate
of 96.7% to distinguish the non-topological ZBCPs from topological ZBCPs, the value of 96.7%
is actually a conservative estimation in the sense that we dropped out hundreds of ?ugly? cases
that do not reach quantization (say F = 0 even for  = 0.2) at all and purposely kept as many
as we can the ?bad? and ?ugly? cases that show quantization similar to the ones from MZMs.
Therefore, we believe the precision rate to identify non-topological ZBCPs can be higher than
96.7%, which is already a high proportion. Similar to Fig. 2.10(a), one can find those cases [90]
associated with F ' 2 in Fig. 2.11(a) would exhibit partially-separated wave functions, known
as quasi-Majoranas, while those cases associated with F < 1.5 mostly show overlapping wave
functions, which are clearly non-topological. So the quality factor F indeed demonstrate the
quality of MZMs of different levels gradually. A high value of F is seen as being associated with
a MZM of good quality with sufficient confidence statistically.
This distinction between MZMs and quasi-Majoranas (or ABSs) is further blurred at  =
0.2 where Figs. 2.3-2.5 show an F value of approximately 5 at T = 20 mK. However, raising
this tolerance  seems to increase the separation between these MZMs/quasi-Majoranas and the
67
?ugly? ZBCPs. In fact, the threshold for the quality factor F depends on several factors, including
the topological superconducting gap Eg (which can be enhanced by increasing the spin-orbit
coupling constant ?), nanowire length L, temperature T , and tolerance factor . For a system with
largeEg and long L, the MZMs produced in the nanowires are expected to be quite robust against
changes in the tunnel barrier height, so we can pick a high value of threshold, below which would
be the non-topological cases. As the temperature T is tuned lower, the MZM-induced ZBCPs will
become closer to the ideal quantized plateau, giving a high value of F [as we can see in Fig. 2.3],
while the ZBCPs induced from trivial subgap states can barely do the same [as we can see in
Fig. 2.4-2.9]. The difference of F values between topological and trivial ZBCPs can even become
two orders of magnitude as the temperature approaches absolute zero. In this case, it is also easy
to choose a higher value of threshold to separate the topological and non-topological ZBCPs. The
tolerance factor  can be viewed as the sensitivity to the quantization plateau. A smaller value of
 is only applicable when the ZBCP is closer to quantization, which can be realized by increasing
the topological superconducting gap Eg, increasing the nanowire length L, and/or lowering the
temperature T . Under these conditions, the gap of F values between the topological and non-
topological scenarios can be large, which makes us easy to choose the threshold of F . In contrast,
when the experimental conditions do not facilitate the production of MZMs, the distinction of the
F values between topological case and trivial case can be comparatively minor, making it hard
for choosing the threshold. This issue can also be seen from the false-positive case popping up in
Fig. 2.11 under the condition of T = 20 mK with  = 0.1 with a threshold of 2.5, while this issue
is completely diminished in Fig. 2.10 under the condition of T = 10 mK with  = 0.05 with a
threshold of 3.0.
Among the ?bad? and ?ugly? results we present, the ones that demonstrate some kind of
68
topological character (i.e., separated Majorana decomposed wave functions), such as Figs. 2.5, 2.8,
and 2.9, in fact do not pass the thresholds determined by the statistical histograms in Figs. 2.10-
2.11. We can interpret these as the topological ZBCPs that cannot maintain their topological
character in the finite-temperature environment when quantum dots or disorders interfere with
the system. Since the threshold we set is to rigorously distinguish the ZBCP induced by MZMs,
a measured value of F higher than the threshold indicates a high probability (> 95%) of getting a
topological ZBCP. On the other hand, a value of F below the threshold does not tell the scenario
? it could be topological or non-topological. Therefore, our proposed quality factor F can help
filter out those ZBCPs with F values larger than the threshold as potential MZM candidates to
proceed with other Majorana examinations.
To apply this new metric F to the experiments, we estimate the quality factor F values
through the recalibrated quantized conductance data from Zhang et al. [52], which is our moti-
vation to study the robustness of quantized ZBCPs. Under the experimental temperature 20 mK,
the quantized plateau in Fig. 4(b) of Zhang et al?s paper [52] gives rise to F ' 1.16 for  = 0.1.
Their extra data as shown in Fig. G2(b) also exhibits a quality factor value of F ? 1 (for  = 0.1
at 20 mK), which are way below F = 2.5 as the suggested threshold in our simulation. These
low values of F demonstrating fragile quantization indicate the ZBCPs shown in Zhang et al?s
paper [44, 52] are very likely arising from non-topological subgap states.
2.4.2 Decoherence rate based on the quality factor F
Based on the analysis and estimations in the previous sub-section, it might be experimen-
tally challenging to distinguish MZMs from other non-topological ?bad? and ?ugly? ZBCPs. This
69
leads to doubts about our motivation of using transport to make this distinction, as opposed to
time-domain techniques that work directly with nanowires in qubit configuration [109]. There-
fore, it is critical for us to argue here that the quality factor F can provide a preliminary estimate
of the decoherence rate for the MZM or the quasi-Majorana as a qubit. Alternatively, the best
quality factor we can obtain from a class of Majorana device provides an estimated bound of the
coherence time for a Majorana qubit based on such devices. To proceed this quote, firstly, we
assume the nanowire is long enough such that the overlap of Majoranas is smaller than the mea-
sured temperature. In this limit, the estimated bit-flip rate of a topological qubit depends on the
topological SC gap Eg and the temperature T according to the relation Te?Eg/2T [5]. Note that
the precision of quantization for the MZM-induced ZBCP is limited by the same ratio Eg/T . In
detail, the width of ZBCP at T = 0, denoted by ? in Eq. (2.19), is proportional to the normal-state
conductance GN ? ? [110]. Thus, the quality factor F can be approximated as
GN,2 ? ?maxF = . (2.20)
GN,1 ?min
For lower normal-state conductance, GN , the conductance can be quantized only if ?min ? T .
The largest ?, for which the ZBCP can be approximated as Lorentzian shape based on Eq. (2.19),
has the same order of the topological gap, i.e., ?max ? Eg. Based on the above arguments, the
quality factor for a ZBCP in a long Majorana wire with a topological gap Eg at temperature T is
estimated to be F ? Eg/T .
Given the analysis in the previous sub-section, we conclude that among the types of models
studied here, the quality factor F needs to be greater than 2 to be confidently identified as a
topological MZM. Assuming a qubit is build from an MZM of this quality, using the estimated
70
decoherence rate quoted earlier, we would conclude a rate of 56 MHz at 20 mK. Compared
to most non-topological qubits at current stage, this decoherence rate is already higher, which
means the threshold F needs to be higher than the one that can already convincingly distinguish
a MZM-based quantized ZBCP.
2.4.3 Multi-channel effects
For the simplicity of presentation, we apply the single-channel semiconductor nanowire
model for simulation. Nonetheless, based on the analysis of TSC conductance probed through a
QPC [56], the quantization of the measured conductance entirely depends on the number of chan-
nels in the tunnel contact, rather than the number of channels in the nanowire. In fact, even if the
number of channels in the TSC nanowire is unknown, the quantization of the conductance is still
measurable as long as the tunnel contact is gated by a tunnel barrier. Typically, the tunnel barrier
is tuned to be high, which leads the normal-state conductance to be below 2e2/h accordingly [52].
This implies that the QPC is likely to be single-channel. It is because the nanowire mobilities are
large enough (i.e., ballistic transport) while the tunnel barrier is adiabatic potential (i.e., smooth
Gaussian potential with the width larger than Landau Fermi wavelength) so as to show a few
quantized steps as a function of gate voltage [111]. Thus, even if the contacts of the tunnel bar-
rier are multi-channel, the transmission eigenvalues through the tunnel barrier are probably quite
small for the higher-order channels. Still, the small transmission in the higher-order channels
may cause the topological conductance a small deviation from quantization, which could be cru-
cial to explain observed conductances above 2e2/h [38, 52]. In this sense, our analysis on the
relation between the quality factor and the topological quantum computing (TQC), as discussed
71
in the last seb-section, seems to have a limitation. Nevertheless, this can be avoided by making
sure the MZM conductance is measured through a clean segment of semiconductor nanowire,
which can be controlled to be single-channel. This should be feasible because clear conductance
steps through non-SC segments of nanowire can be demonstrated [111, 112].
2.4.4 Characterizing MZMs based on J
Contrary to the quality factor F , which can be extremely large, the quality factor J is lim-
ited. From the numerical results in Sec. 2.3, J values are all below 2.1, which are comparatively
lower when F values can reach almost 50 in the zero-temperature limit. In general cases of
?good? ZBCPs, J values could even be lower than those from ?bad? or ?ugly? ZBCPs when the
Majorana splitting oscillations dominate. The best value of J we can get for the ?good? ZBCP
is Vc/VTQPT when the nanowire is longer than the SC coherence length and therefore the Majo-
rana splitting is suppressed. The quality factor J in this case turns to be contrained by how soon
the SC gap collapses and how late the system enters the topological regime. There is not much
significant difference for the J values between the ?good? ZBCPs and ?bad? or ?ugly? ZBCPs.
Therefore, based on the models studied here, the robustness to the Zeeman field is difficult to use
for characterizing MZMs. However, a low value of the quality factor J suggests a topological
gap that may be too small for practical use.
2.5 Summary
To sum up, we have studied the robustness of the ZBCP quantization to the variations of the
tunnel barrier height and Zeeman field, which can be a new way to separate topological MZMs
72
from trivial ?bad? and ?ugly? ZBCPs associated with subgap ABSs and random disorder, respec-
tively. This is motivated by the robustness nature of ?good? ZBCPs associated with MZMs. In
addition, contrary to the experimentally-inaccessible wave functions, the quality factors obtained
from the tunneling experimental results can show the topological character of the ZBCP. While
the quality factor J , which quantifies the quantized plateau over Zeeman field, is not a particu-
larly strong indicator to show the topological character of the system, it can be a useful diagnostic
to avoid a small topological gap. On the other hand, the dimensionless quality factor F , which
takes on values  80 only in the topological phase at zero temperature, can sharply quantify
the stability of the ZBCP to changes in normal-state conductance. In contrast, non-topological
systems with strongly overlapping MZMs only show quantized plateau over a very narrow range
of normal-state conductance GN , which typically leads to quality factors F less than 10 at zero
temperature. Therefore, the value of quality factor F at low temperature is a rather strong indica-
tor of the topological character of the system. Unfortunately, the values of F drop very quickly
as soon as the finite-temperature effect kicks in such that it becomes smaller even for the ?good?
ZBCP. So the use of F to distinguish MZMs from trivial ZBCPs is restricted by the current
experimental temperature limit ? 20 mK. Though the results of finite temperature are kind of
disappointing, they do not relegate the quality factor F to limbo. In fact, the quality factor F
could still play a single diagnostic for identifying emergent MZMs if the topological gap can be
enhanced with the improved material fabrication. Another challenge that threatens the current
experimentally-observed quantization is the need to design the tunnel barrier contact carefully
so the contact is ensured to only hold a single channel. According to Wimmer et al. [56], only
when the tunnel contact is single-channel can the measured conductance from TSC be guaran-
teed to reach quantization 2e2/h at zero temperature. This is very likely the reason why robust
73
quantized ZBCPs of MZMs are still elusive. Once these design, gap, and temperature constraints
are resolved, a ZBCP with a quality factor F greater than the threshold should tell a ZBCP as a
topological MZM.
74
Chapter 3: Presence versus Absence of End-to-end Non-local Conductance
Correlation
Based on the fundamental theory of the MZM, the appearance of ZBCP in the topologi-
cal regime is an important basic hallmark for the MZM. Unfortunately, it is later found that the
topologically-trivial ABSs could also produce ZBCPs, making the signature of ZBCP not unique
for the topological MZMs. On the other hand, as we illustrated in Chapter. 2, non-topological sub-
gap states can also mimic the quantization of ZBCPs, a signature of MZMs at zero temperature,
with appropriate fine-tuning. All these complications brought by the unintentional QD-induced
ABSs have obscured the signatures for the existence of MZMs. One may propose to focus on
the signatures of MZMs only in the topological regime. However, the TQPT field which sepa-
rates the topological regime from the trivial regime is typically too weak to be measured. This
is why making a distinction between MZMs and ABSs is tremendously important in search of
Majoranas.
In this chapter, we utilize the fundamental non-local nature of MZMs [2, 81] to establish a
protocol that could decisively distinguish between ZBCPs arising from MZMs and ABSs. Moti-
vated by the three-terminal setup Deng et al. [36] carried out tunnel conductance measurements,
we propose that by carrying out the tunnel conductance measurements from both ends of the wire
and comparing the two sets of tunneling data, the topological character of ZBCPs can be identi-
75
fied. The ZBCPs arising from MZMs (or ABSs) will be correlated (or uncorrelated) between the
two ends, meaning if the ZBCPs show up from both ends, it is likely to be associated with MZMs,
whereas if the ZBCP only exists at one end but not the other, then it is likely to come from ABS.
We will demonstrate extensive numerical simulations to stress that the presence of this end-to-
end conductance correlation is associated with the presence of MZMs, while the absence of the
end-to-end conductance correlation is most likely to be come from the ABS localized at one end.
For completeness of the study on the same three-terminal setup, we also calculate the cross
conductance in the nanowire with and without the embedded QD. Such a measurement has been
proposed a way to detect the TQPT [82]. We suggest that by combining the conductance correla-
tion test and cross conductance measurement, one should be able to distinguish the MZM-induced
ZBCPs from ABS-induced ZBCPs.
3.1 Model
In this chapter, we consider a hybrid superconductor-semiconductor nanowire, with an
external quantum dot (QD) embedded near the left end of the wire, as shown in Fig. 3.1. One can
find the same protocol to numerically calculate the QPC differential conductance G = dI/dV ,
wave functions, and energy spectrum, as we already illustrated in detail in Sec. 2.1, except that
now the spatial arrangement of the Hamiltonian is a bit modified. As in Fig. 3.1, from the leftmost
end, it starts with Hlead (described by Eq. (2.7)), and then Hbarrier (the tunnel barrier is described
by Eq. (2.8)), HQD (described by Eq. (2.2), with V (x) described as Eq. (3.1)), HNW (described
76
Figure 3.1: Schematic plot of the hybrid structures composed of leads on both sides.
This setup is basically motivated by Deng et al. experiment [36]. The semiconductor
nanowire is covered by a parent s-wave superconductor, except for the part embedded
by the quantum dot (shown in the red dashed line). Note that the quantum dot is
strongly coupled to the nanowire, which may not exhibit Coulomb blockade effect.
The leads used to measure the conductance are set on both sides, which also induce
the tunnel barriers (green parts). The differential conductances calculated in this
paper are probed either from the left lead or from the right lead.
by Eq. (2.2)), Hbarrier, and Hlead to the right. The QD potential in our numerics is of the form
( )
3?x
VQD(x) = VD cos , (3.1)
2lD
where lD only counts a fraction of the total nanowire length L. In fact, the precise form of the QD
potential is irrelevant for our consideration because no qualitative conclusion depends on these
details.
Similar to Chapter 2, we calculate the Majorana-decomposed wave function probabilities
|?A(x)|2 and |? (x)|2B (which follows by Eq. (2.17) and Eq. (2.18)) over the spatial space in
the nanowire, by first summing over the inner degrees of freedom (particle-hole space and spin
space). If |? 2A(x)| and |?B(x)| are localized separately at each end of the wire, then they are
77
characterized as an MZM pair. In this case, the ZBCPs we observe in the corresponding con-
ductance plot come from MZMs. In contrast, if |?A(x)| and |?B(x)|2 overlap with each other,
possibly only localized on one end of the wire, then the observed ZBCP arises from ABS. It is
also possible that the nanowire is not long enough such that the tails of the two Majorana bound
states (MBSs) at each end overlap with each other pretty much. In this case, we cannot have the
topological MZMs. Only the well-separated MBSs without overlapping can facilitate the exis-
tence of MZMs. This also means one zero-energy ABS can be considered to be composed of
double MBS modes which overlap considerably in the real space. Therefore, one can theoret-
ically distinguish the sources of ZBCPs by just comparing the behaviors of wave functions on
both ends of the wire. However, as mentioned in Chapter 2, while the wave functions are easy
to theoretically calculated, there is no way to access the wave functions in the experiments. The
wave function we present in our numerical results in this chapter are mainly for verifying the
topological character of the corresponding calculated ZBCP, which is typical measured in the
experiments to search for the MZM.
The parameters in all of our numerical results (Figs. 3.2-3.4) are chosen as follows: the
effective mass is m? = 0.015me, nanowire length L = 5 ?m, induced SC gap ?0 = 0.9 meV,
spin-orbit coupling ?R = 0.5 eVA?. The gate voltage in the lead is Elead = ?25 meV, with the
induced tunnel barrier height Ebarrier = 10 meV, and the barrier length lbarrier = 20 nm. The
strength of the confinement potential in the quantum dot is VD = 4 meV, with length lD = 0.3
?m. The temperature, which smears the conductance profile by thermally broadening all sharp
features, is set at 0.02 meV. The phenomenological dissipation parameter is ? = 0.01 meV. The
above parameters will be fixed throughout all our results, and other tuning parameters, including
the chemical potential ?, the Zeeman energy Vz, the SC gap collapsing point Vc, along with the
78
?
TQPT field V 2 2TQPT = ? + ?0, will be provided in the captions of the figures. In our presented
results of this chapter, we do not include the self-energy, i.e., ? = 0 in Eq. (2.3). Nevertheless,
the presence of non-local correlation that is associated with MZM-induced ZBCPs should not
qualitatively depend on the strength of self-energy coupling, as one can find the extensive self-
energy results in the appendix of this work [83]. Our strategy to distinguish MZMs from ABSs
is valid no matter self-energy effect is included or not. Similar for the parameters ? and Vc ? they
are nonessential aspects for our theory with respect to the MZM/ABS distinction. We include
them in our simulation to make the results realistic, not because they are necessary for the main
point of end-to-end ZBCP corralations being stressed in this work.
3.2 Numerical Results: Local Conductance from each end
The main idea to distinguish between MZMs and ABSs in this work is quite fundamental:
ZBCP associated with the topological MZM should exhibit non-local correlations because topo-
logical MZMs always appear in pairs at both ends of the wire, which contribute to ZBCPs at each
end. On the contrary, an ABS is a non-topological fermionic bound state arbitrarily localized
near one or the other end of the wire. Moreover, the ZBCP associated with the ABS only shows
up when the tunneling end connects to the relevant ABS localized at the same end. So the ABS-
induced ZBCPs do not show up at both ends of the wire simultaneously and thus should have no
non-local correlatioins. Since the regime where the ABS sticks to zero-energy depends on a lot
of different parameters and therefore kind of random, we present extensive numerical results to
touch various scenarios for complete discussion. Particularly, we will vary the chemical potential
for each simulated set since the chemical potential can determine the nominal TQPT field, which
79
divide the trivial regime and the topological regime.
In Fig. 3.2, we show an example (chemical potential ? = 1.0 meV) where there is a pair of
ABSs that approach and stick to near zero energy near VTQPT = 1.35 meV, but the Zeeman field
range where the ABSs are zero-energy is very limited, as one can see from the energy spectrum
in Fig. 3.2(a). Above VTQPT, the expected zero-energy MZMs appear. The corresponding conduc-
tance spectra probed by the left lead and the right lead are shown in Fig. 3.2(b) and Fig. 3.2(c),
respectively. The dominant ZBCPs only show up above the TQPT field (black dashed line) from
both ends, which are consistent with the spectrum in Fig. 3.2(a). One thing to pay attention
to in these conductance spectra [Figs. 3.2(b)(c)] is that the conductance pattern that appears to
follow the energy pattern induced from ABSs in Fig. 3.2(a) below VTQPT only shows up in the
conductance from the left end, as in Fig. 3.2(b). It is clearly because the QD is embedded near
the left end of the wire (as shown in Fig. 3.1), which leads to the ABS localized on the left end.
On the contrary, the conductance probed from the right end does not demonstrate this subgap
feature. Even though the conductance profiles in Fig. 3.2(b) and Fig. 3.2(c) appear quite unlike,
the ZBCPs probed from both the left and the right ends show up at the same Zeeman field. This
is because both ZBCPs arise from MZMs, while the ABSs exist in this parameter regime do not
stick to zero energy [see the subgap states below VTQPT in Fig. 3.2(a)]. One can examine the con-
ductance further by taking the linecuts of conductance from both ends at the same Zeeman field
in Figs. 3.2(b)(c) and presenting them in the same plot, such as Fig. 3.2(d) for Vz = 2.0 meV.
Obviously, despite that the heights of the ZBCPs at the left and the right ends have a big differ-
ence, the ZBCPs appear to be correlated. As the Zeeman field where the conductance linecuts
are checked increases (such as Fig. 3.2(e) at Vz = 3.0 meV), the ZBCP heights from both ends
become comparable. In this case where the chemical potential is small, the zero-energy states
80
come from MZMs appearing at each end of the wire. This can be confirmed from an examina-
tion of the lowerst-lying wave functions, which should display the near zero-energy states, both
when Vz is close to VTQPT [as Fig. 3.2(f)] and when Vz is deeper into the topological regime [as
Fig. 3.2(g)]. Both sets of linecuts show the Majorana-decomposed wave functions localized at
each end with finite weight.
The situation changes qualitatively when the chemical potential is raised to ? = 5.0 meV.
As shown in the energy spectrum [Fig. 3.3(a)], now even ABSs appearing below VTQPT (black
dashed line) have several chances to stick to zero energy. In this case, the ZBCP on the left end
[Fig. 3.3(b)] also shows up in the trivial regime (Vz < VTQPT), while the ZBCP on the right end
[Fig. 3.3(c)] does not show up until the Zeeman field is in the vicinity of VTQPT. The linecuts of
the conductance in the trivial regime at Vz = 1.9 meV [yellow dashed lines in Figs. 3.3(b)(c)]
exhibit a dramatic difference between the left end and the right end, as one can see in Fig. 3.3(d).
The linecut from the left end shows an obvious ZBCP, while the one from the right end only
shows strong gaps located near V = ?0.5 meV. However, as the Zeeman field is increased to
Vz = 6.0 meV [purple dashed lines in Figs. 3.3(b)(c)], ZBCPs with comparable height appear on
both ends [Fig. 3.3(e)], which is expected since Vz = 6.0 meV is in the topological regime. The
origin of the ZBCP on the left end in Fig. 3.3(d) can be cross-checked with the corresponding
wave functions as shown in Fig. 3.3(f). We can see the Majorana-decomposed wave functions are
entirely localized ar the left end, which is the same way as ABSs exhibit. In contrast, the wave
functions are separated on each end of the wire at a higher Zeeman field, as in Fig. 3.3(g), which
is consistent with the behavior of the topological MZMs. Interestingly, the ABS-induced ZBCP
on the left end as in Fig. 3.3(d) is very similar to the MZM-induced ZBCPs in Fig. 3.3(e) though
the former one does not possess any topological character.
81
Figure 3.2: Numerical results with chemical potential ? = 1.0 meV. (a) Energy
spectrum. (b) Conductance G(V ) measured from the left lead. (c) Conductance
G(V ) measured from the right lead. (d) Conductance linecuts from both leads at
Vz = 2.0 meV. (e) Conductance linecuts from both leads at Vz = 3.0 meV. (f)
Majorana components of lowest-lying wave functions in the nanowire at Vz = 2.0
meV, above VTQPT. (g) Majorana components of lowest-lying wave functions in the
nanowire at Vz = 3.0 meV, above VTQPT. The SC collapsing field is at Vc = 3.2 meV.
The black, yellow and purple vertical dashed lines in the panels (a,b,c) represent the
TQPT field VTQPT = 1.35 meV, Vz = 2.0 meV and 3.0 meV respectively.
82
Figure 3.3: Numerical results with chemical potential ? = 5.0 meV. (a) Energy
spectrum. (b) Conductance G(V ) measured from the left lead. (c) Conductance
G(V ) measured from the right lead. (d) Conductance linecuts from both leads at
Vz = 1.9 meV. (e) Conductance linecuts from both leads at Vz = 6.0 meV. (f)
Majorana components of lowest-lying wave functions in the nanowire at Vz = 1.9
meV, below VTQPT. (g) Majorana components of lowest-lying wave functions in the
nanowire at Vz = 6.0 meV, above VTQPT. The SC collapsing field is at Vc = 10.4 meV.
The black, yellow and purple vertical dashed lines in the panels (a,b,c) represent the
TQPT field VTQPT = 5.08 meV, Vz = 1.9 meV and 6.0 meV respectively.
83
Here we only show results for two chemical potentials as in Fig. 3.2 and Fig. 3.3 to demon-
strate the difference of the end-to-end conductance comparisons between MZMs and ABSs, re-
spectively. One can find the extensive results in the appendix of Ref. [83] for many other values
of chemical potential and magnetic field to confirm further that the presence (or absence) of end-
to-end conductance correlations implies the presence (or absence) of MZMs (or ABSs) in the
nanowire. There are several subtleties for our proposed non-local correlation method to work in
the experiments.
One subtlety that we would like to address for our method is that all the results we demon-
strate in this work satisfy the relation VTQPT < Vc, meaning there is always a topological regime
that exists before the SC collapses. If VTQPT > Vc, then there will never be any MZM pair show-
ing up before the SC gap crashes and destroys the TSC nanowire. In this case, all the measured
ZBCPs are unfortunately induced from the trivial ABSs and thus manifest no correlations in the
tunneling signals from both ends of the wire. Therefore, making sure VTQPT < Vc is an important
ingredient in experiments to realize our proposed non-local correlation method.
Another subtlety we want to emphasize is that the nanowire needs to be long enough (i.e.,
longer than the SC coherence length) such that the topological physics can manifest itself. If the
wire is short, the tail of ABS wave function could extend to the other end, making the ZBCP on
the other end also show up. In this case, there will also be end-to-end correlation even for trivial
ABSs. The presence of non-local correlation will not be exclusive for the well-separated MZMs
at this point. On the other hand, in short wires, it will be easier to discern MZMs by the Majorana
splitting oscillations as studied in Ref. [81] already. Our work applies to long wires, where there
is a fundamental difference between MZMs and ABSs.
The other false-positive scenario that could accidentally deduce the ABS-induced ZBCPs
84
to come from MZM pairs is when two different QDs fortuitously locate respectively at each end
of the wire. In this case, each QD can exhibit ABS-induced ZBCP on both ends, leading to the
presence of non-local correlations in the tunneling measurements from the two ends. This issue is
statistically unlike to happen, but cannot be ruled out. Therefore, one should pay more attention
to the presence of non-local correlation before jumping to the conclusion about the existence
of MZMs. This issue can be taken care better if one can also show the bulk gap closing and
reopening near the TQPT using the same setup [Fig. 3.1] as a cross-check, which we will discuss
in the next sub-section.
3.3 Numerical Results: Cross Conductance
Other than the conductance correlation between the two ends of the wire, our three-terminal
setup as in Fig. 3.1 allows one to measure the cross conductances, GLR = dIL/dVR or GRL =
dIR/dVL. This non-local conductance is useful because it can distinguish the bulk states from
non-topological subgap states induced by potential inhomogeneity [82]. It is important in the
sense that bulk states appearing at the gap closure is a hallmark of the TQPT, which should
tell us the boundary between the trivial regime and the topological regime. One can find the
cross conductance GLR as a function of voltage for a pristine nanowire model showing bulk gap
closure [Fig. 3.4(a)], which is a signature of TQPT. Specifically, with the Zeeman field varying,
the cross conductance area that is suppressed to zero becomes narrower from Vz = 2.8 meV
to Vz = VTQPT = 3.13 meV, but then becomes wider again as the Zeeman field is increased to
Vz = 3.3 meV, which demonstrates bulk gap closing and reopening near the TQPT. On the other
hand, in Fig. 3.4(b), we show the cross conductance for the nanowire with an embedded QD on
85
Figure 3.4: Cross conductance GLR as a function of voltage VR from the right end
applied to the nanowire, for different Zeeman fields Vz. Note that the induced tunnel
barrier height is Ebarrier = 1.0 meV. (a) GLR for a pristine Majorana nanowire. The
bulk gap appears to close and reopen as Vz crosses the TQPT field VTQPT. (b) GLR for
a Majorana nanowire with an embedded QD on the left end. The bulk gap closing
and reopening signature is difficult to see in (b).
the left end. In this case, the cross conductance is almost zero in an apparently wide voltage
range, and the signature of bulk gap closing and reopening near TQPT is clearly suppressed with
this extra QD embedded. One may therefore need to implement extra tuning to compensate this
suppressing effect brought by the QD. Nevertheless, the cross conductance one can check upon
the same setup can be used as a cross-check method for the existence of MZMs.
3.4 Summary
Through our extensive numerical simulations, the ABS-induced ZBCP can be distinguished
from the topological MZM-induced ZBCP by comparing independent tunneling measurements
carried out from both ends of the wire at the same time. The basic idea behind our work is
simple: Topological MZMs exhibit non-local nature intrisically, whereas the topologically-trivial
86
ABSs can only exist near one or the other end depending on the extrinsic details. Therefore, a
tunneling measurement from a particular end can only detect an ABS-induced ZBCP if the ABS
locates at that end, whereas the MZM-induced ZBCPs must be equally accessible from both wire
ends. Note that the signature of non-local correlated ZBCPs for MZMs has been well-studied
in Ref. [81]; nevertheless, the work presented in this chapter incorporates detailed studied in the
scenario of trivial ABSs existing in the wire, which is outside the domain of Ref. [81]. As long
as the subtle conditions, including VTQPT < Vc, L  lc (nanowire length is much longer than
the SC coherence length), and two unintended QDs do not accidentally locate at both ends of the
wire, one should be able to claim that the presence (absence) of correlated ZBCPs from both ends
indicate the presence (absence) of the topological MZMs (trivial ABSs) in the nanowire.
For completeness, we also studied the cross conductance that can be measured in the same
three-terminal setup as the conductance correlation. We have shown that the bulk gap closing
and reopening feature near TQPT can be clearly identified in the pristine nanowire, where only
MZMs exist. However, the QD-induced ABSs could obscure this feature, making the TQPT
field hard to identify. MZMs can exist in the nanowire even when the cross-conductance fails
to demonstrate a TQPT. Therefore, we suggest using a combination of end-to-end conductance
correlation together with cross conductance to confirm the existence of MZMs so as to avoid
false-positive conclusions.
87
Chapter 4: Coulomb-blockade Transport In Realistic Majorana Nanowires
Motivated by the Coulomb blockade experiments introduced in Chapter 1 (Sub-section 1.5.3),
we want to theoretically study the Majorana nanowire in the CB regime with the realistic mod-
eling in this chapter. Particularly, we intend to explain the features, the ?bright-dark-bright?
pattern, and the decreasing OCPS with the increasing Zeeman field that appear in the exper-
iments. [91, 113, 114]. In addition, we will justify if the experimental results of the length-
dependent OCPS [91] are an indication of MZMs.
The relatively short experimental device length for epitaxially grown SC-SM nanowires
allows for a measurement of transport in the CB regime [91, 113, 114] simply by lowering the
transmission to either end of the wire. The schematic setup for CB transport measurements is
shown in Fig. 4.1, which describes a CB SC-SM island between two leads. Transport in the CB
regime, while somewhat more complicated to theoretically interpret relative to QPC tunneling,
provides information about the states at both ends and possibly also bulk transport [91,115,116].
The theoretical complication arises from the fact that one must take into account both MBS and
CB physics on an equal and non-perturbative footing while at the same time addressing the non-
equilibrium physics of tunneling transport. Direct measurement of tunnel conductance at both
ends requires a three-terminal configuration [49], which risks generating additional spurious sub-
gap states at the third contact [117]. The CB measurement dispenses with the need for such a
88
Figure 4.1: Schematic plot of the Coulomb blockaded Majorana nanowire. This
semiconducting nanowire (e.g. InAs or InSb) is proximitized by the parent s-wave
superconductor (e.g. Al) which only covers part of the nanowire. The gate with
voltage VG controls charge on the nanowire. The Coulomb blockaded transport is
measured between the leads shown at the two ends of the Majorana nanowire. The
coupling to the leads is pinched off to the required degree by the tunnel gates shown.
third contact, leading to a more straightforward measurement. Additionally, as we will discuss
later in this work, the CB conductance is less sensitive to subgap states that are localized only at
one end or another, with the exception of MBSs, making it a particularly attractive experimental
approach. This possibility of exploring both wire ends and the bulk so that the bulk-boundary
topological correspondence could be investigated in a single experiment in a single sample com-
pels us to carry out the extensive theoretical analysis presented in this work.
Despite the advantages of CB measurements, they are more complicated to interpret relative
to QPC both because of the involvement of tunneling at both ends and Coulomb interaction. As
a result, there is no standard formalism to model such transport analogous to the BTK formalism
used to model QPC tunneling [59?69]. In fact, the interplay of Coulomb interactions and low-
temperature Fermi-liquid correlations can lead to intricate many-body physics, such as the Kondo
effect [118?122], which can further complicate the interpretation of data. Such complications can
be avoided for temperatures above the Kondo temperature, where transport can be modeled by
perturbation theory in tunneling [115]. However, CB transport through a complex system, such
89
as an SC-SM nanowire which has many low-energy levels, is challenging to treat numerically
and is characterized by an exponential complexity of the perturbative rate equations [116].
In this work, we show that the CB transport can be modeled at the same level of com-
plexity as QPC tunneling, provided we consider the regime where the tunneling rate is below
the equilibration rate of the nanowire. This assumption leads us to a generalized Meir-Wingreen
formula for tunnel conductance, which we use to study various effects in the nanowire, such as
ABSs, self-energy, and soft gap. We anticipate that our theory will provide a route to interpret
CB transport in hybrid Majorana systems resulting from these important realistic effects.
4.1 Ideal Simulation Result
One of the main motivations of our work is to provide a qualitative understanding of mea-
sured CB transport experiments [91,113,114]. Therefore, before presenting the details of the for-
malism in Secs. 4.2, 4.3 and 4.4, we present a preliminary description of our main results to moti-
vate our work. In Fig. 4.2(a), we show a representative result that qualitatively resembles some of
the recent experimental data [91, 113, 114] for the CB conductance as a function of gate-induced
charge number ng and Zeeman field Vz. As discussed in more detail in Sec. 4.5, Figs. 4.2(b,c,d)
allow us to compare the information from Fig. 4.2(a) to other characteristics of the wire, such as
end conductance [Fig. 4.2(b)] that is measured more typically (i.e., QPC) [31?35,37,43,123] and
the nanowire spectrum [Fig. 4.2(c)], which is the information desired from tunneling transport.
The OCPS shown in Fig. 4.2(d) is directly obtained from the peak spacing along ng in Fig. 4.2(a)
and can be compared to parts of the spectrum in Fig. 4.2(c). The model used to obtain the re-
sults in Fig. 4.2(a), which most closely resembles experimental data [91,113,114], is a Majorana
90
Figure 4.2: The ideal case results, which show the most similar features as experi-
mental data by fine-tuning. The parameters are: the temperature T = 0.01 meV, the
wire length L = 1.5 ?m, the SC gap at zero Zeeman field ?0 = 0.9 meV, the SC
collapsing field Vc = 4.2 meV. Other relevant parameters are given in Sec. 4.5. (a)
Coulomb-blockade conductance G as a function of the gate-induced charge number
ng and Zeeman field Vz at Ec = 0.13 meV. We only show two periods in the range
of ng ? [17, 21]. (b) Non-Coulomb-blockade conductance G from the left lead as a
function of bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function
of Zeeman field Vz. (d) OCPS as a function of Zeeman field Vz that is extracted from
the vertical peak spacing in panel (a).
91
nanowire model that includes quantum dots, self-energy effects, suppression of SC, and soft SC
gap, in addition to CB (see Sec. 4.4 for details).
Figure 4.2(a) does not show the 2e-periodic (in ng) part of the CB conductance that appears
as the brightest features in the experimental data [91,113,114] at the lowest part of Zeeman field
range. This is because the quasi-particle gap in this range of Zeeman field, which is below that
shown in Fig. 4.2(a), is larger than the charging energy Ec, so the transport is dominated by
Cooper-pair transport [115]. The charging energy Ec is defined by the electrostatic energy
U(N) = Ec(N ? ng)2, (4.1)
where N is the total electron number in the Majorana nanowire. The range of Zeeman energy
plotted in Fig. 4.2(a) is where the quasi-particle gap is below Ec, so the conductance is domi-
nated by single-electron transport. As will be discussed in more detail in Sec. 4.2, the calculated
conductance G is 2e-periodic in the gate charge ng. As a result, the conductance plot shown in
Fig. 4.2(a) is representative of the conductance over the entire range of gate charge ng. Note
that the CB conductance [e.g. Fig. 4.2(a)] in this work is shown in units where the normal-state
conductance peak (i.e., at high Vz) is equal to one. This is a natural unit to use in CB situations,
and further details of this choice are discussed following Eq. (4.36).
The energy spectrum in Fig. 4.2(c) shows (as elaborated in Sec. 4.5) that the oscillations
seen in the OCPS in Fig. 4.2(d) arise from ABSs [59, 61?65] (and not MBSs) at the ends of
the wire. The dark region of the conductance in Fig. 4.2(a) arises from the part of the ABS
where the energy difference between the two ABSs at the two ends exceeds the temperature.
The structure of this energy spectrum contains multiple subgap states as in Shen et al. [113] As
92
illustrated in Sec. 4.5, we find such oscillations in OCPS with decreasing amplitude only in the
case of QD-generated ABSs before the TQPT shown by the first dashed line in Fig. 4.2(a). One
of the puzzling features of the CB conductance data [91, 113, 114] in Majorana nanowires is the
intensity of the 2e-periodic peaks being higher than the 1e-periodic part, which is the opposite
of what the naive expectation is. This occurs despite the 2e conductance arising from Cooper-
pair transport, which is higher-order in the tunneling, while the 1e-periodic peaks being from
electron transport. As a result, as discussed in Sec. 4.3, the conductance of the 2e-periodic part is
theoretically expected to be smaller [115]. However, our results in Fig. 4.2(a) show a significant
suppression of the 1e-periodic conductance both from the self-energy corrections as well as from
the soft-gap, compared with systems where these effects are not included. This suppression can
explain the discrepancy between the experiments [91,113,114] and the simple model of Majorana
tunneling for 1e-periodic conductance [115]. A more detailed understanding of the features in
Fig. 4.2 as well as the role of the various contributing factors, such as soft-gap, self-energy, and
quantum dots, are provided in Sec. 4.5.
4.2 Two-terminal Generalized Meir-Wingreen Formalism
We will start by combining the rate equation formalism with an equilibration assumption
to derive a two-terminal generalization of the Meir-Wingreen mesoscopic theoretic formalism
for the interacting CB system. We will then apply this formalism to a semi-realistic model for a
semiconductor nanowire that has been used to study QPC tunneling [27,59?69,83,104,110,124]
in Majorana nanowires. We will then study various limits of the model to develop a generic
correspondence between features seen in CB transport experiments [91, 113, 114] manifesting
93
characteristics of the semiconductor wire model such as self-energy, soft gap etc.
4.2.1 Setup
Let us consider transport through a thermalizing dot Q, which has N units of charge on
it. Note that this thermalizing dot is our main system, i.e., the nanowire in our case, which
has nothing to do with the unintentional quantum dots induced by disorder in our context. It is
described by the Hamiltonian ?
HQ = Ej|?j???j|, (4.2)
j
where {Ej} is the set of energies of the system. The dot Q is coupled to two leads L and R, with
chemical potential ?L and ?R, respectively, through a tunneling Hamiltonian
? [ ]
H ?t = t? ak?c? + H.c. , (4.3)
k,?=L,R
where c?=L,R is an electron annihilation operator on the left and right ends of Q. The operators
a?k,? create electrons in the leads ? = L,R. t? is the tunneling matrix element at the ? end.
We treat the tunneling to the lead perturbatively as in Ref. [125]. The tunnel coupling is
assumed to be weak enough so that the quantum dot Q can reach thermal equilibrium between
successive tunneling events that change the conserved change N . The assumption here is that the
tunneling is slow compared with the equilibration time, which can always be ensured by tuning
the tunnel barrier. Within this framework, the state of dot Q within a specific charge sector N ,
labeled i, have a conditional probability given by the Boltzmann distribution
PN(Ei) = Z
?1e??EiN , (4.4)
94
where Ei is the energy of state i, ? is the inverse of temperature, and
?
Z = e??EiN (4.5)
i
is the normalized partition function for each charge N , i.e., state i has N electrons. Note that
P (Ei)
PN(Ei) = , (4.6)
P0,N
where P (Ei) is the probability of the state i with energy Ei and P0,N is the probability of having
N electrons. It is assumed that the state i has N electrons.
The probability distribution P0,N of havingN electrons is determined by the balance of two
processes where the system Q either gains or loses the electron from the leads via a tunneling
process. The tunneling rate of electrons from the leads into Q, which is assumed to be in a charge
state N , can be computed using Fermi?s golden rule to be
? ?
??N = ?? PN(Ei) df(? ??)?(Ej ? Ei ? )|??j|c? 2?|?i?| , (4.7)
i,j
where f() = (1 + e?)?1 is the Fermi function, c?? is the electron creation operator at the end
? = L,R of the system, and ? 2? = t??? is the basic tunneling rate into the lead ? with ?? being
the density of states in lead ? = L,R. Considering the current from tunneling at voltages large
compared to the SC gap, the tunneling rate ?? can be written as ?? = g?/2??1D,? [115], where
g? is the normal-state conductance at the end ? and ?1D,? is the density of states at the end of
the Majorana nanowire. Following the derivation in Appendix A.1, one can show that the reverse
95
tunneling rate of electrons ??N , i.e., from dot Q into the leads, is related to the rate ?
?
N as
?? = e???
Z
? N?1 ??N N?1. (4.8)ZN
Such a relation is consistent with the requirement of satisfying the correct number distribution in
Q when the system is decoupled from one of the leads.
More generally, let us consider the process of transferring j electrons into the dot as ??,jN .
Equilibrium with the lead ? requires the rate of the reverse process to be
?,j ?j?? ZN?j?N = e
? ??,jN?j. (4.9)ZN
4.2.2 Steady-state Rate Equations
The steady-state probability distribution P0,N can be determined from the rates ??N and ?
?
N
by equating the rate of electrons transitioning from having N electrons to N ? 1 electrons to the
rate of electrons making the reverse transition. This equation, following the definitions of the
rates ??N and ?
?
N , can be written to be:
? ?[ ]
P (??,j + ??,j) = P ??,j ?,j0,N N N 0,N?j N?j + P0,N+j?N+j (4.10)
?,j ?,j
Substituting ??,jN from Eq. (4.9) and rearranging the terms, the above condition is given by
? ?
???,j{P? ?j??? ?,j ?j???N 0,N ? P?0,N+je } = ??N?j{P?0,N?j ? P?0,Ne } (4.11)
?,j ?,j
96
where ???,j = ??,jN N ZN and P?0,N = P0,N/ZN . The above equation is solved by the detailed balance
condition, where both sides of the above equation vanish such that
(? ? )P? ?,j ?j????,j 0,N+j??N eP?0,N = ?,j . (4.12)
?,j ??N
Redefining P ?0,N = P? e
?N??L
0,N to simplify the equilibrium solution and defining ?? = V? + ?L
as the voltages for linear response, the above equation becomes
(? )? ?,j ?j?V?
? ?,j P?0,N+j??N eP0,N = (? ?,j (4.13)?,j ??N? ) ( )P ? ???,j ? jP ? ???,j? ?,j 0,N+j N V?,j ? ?,j 0,N+j N ?? ??? ???,j (4.14)?,j N ?,j N
There is a trivial solution for equilibrium with V? = 0, which is constant with P ?0,N = P0,eq + vN ,
so (?
? ? )?,j?,j vN+j??NvN
???,j
? ?P0,eq?N (4.15)
(??,j N? )j???,j?,j N V??N = ?,j . (4.16)
?,j ??N
This solution is invariant under a constant shift, which in principle is fixed by normalization. In
the case of a two-terminal case, only VR = V is non-zero, so we can expand ?N = ?NV , where
(?? )j j??R,jN?N =
???,j
. (4.17)
?,j N
97
The fluctuations vN can now be expanded vN = ?P0,eqV ?N , which satisfies
(? )
?,j ?
?,j
N+j??N
?N ? ?
???,j
? ?N . (4.18)
?,j N
4.2.3 Linear-Response Conductance
The current at the left lead L is determined by the balance of electrons tunneling in and out
of L. Using the definitions of the tunneling rate ??,j and ??,jN N , this current can be written as
? ( )
I = jP0,N ?
L,j ? ?L,jN N . (4.19)
N,j
Using the rescaled variables in Eq. (4.11), the current is re-written as
?
I = j??L,j ?N?LN e {P
? ?
0,N ? P0,N+j}. (4.20)
N,j
Substituting the current to linear order is
?
I ? j??L,je?N?LN {vN ? vN+j}. (4.21)
N,j
Fortunately, the current is not affected by the constant shift ambiguity.
Divided by V , the conductance is found to be
?
G = ? j?P ??
L,j ?N?L
0,eq N e {?N ? ?N+j} (4.22)
N,j
= j?L,jN {?N ? ?N+j}, (4.23)
N,j
98
where ??,jN = ??P ?
?,j
0,N N . Since the redefinition is a scaling that depends only on N and the
equation for ?N only involves ratios of ??, the equation for ? can be re-written in terms of ?
?,j
N as
(?? )j?R,jj N?N = ?,j . (4.24)
?,j ?N
The fluctuations vN can now be expanded as vN = ?P0,eqV ?N , which satisfies
(? )?,j
?,j ?N+j?N
?N ? ? ?,j ? ?N . (4.25)
?,j ?N
4.2.4 Generalized Meir-Wingreen Formula
In this case, we limit to one-electron processes that should dominate in the strict tunneling
limit. If necessary, the generalization to multi-electron processes is straightforward, albeit quite
cumbersome, but multi-electron transport should be negligible in the tunneling limit of interest
here.
The zero-bias conductance G can be calculated by expanding the current I to the lowest
order in the bias voltage ?R ? ?L = V . Refer to the Appendix A.3,
dI ? R L
G = | ??N ??NV=0 = ?? P?0,N . (4.26)
dV ??RN + ??
L
N N
Restoring the variable change from Eq. (4.11), the conductance can be re-written in terms of a
re-scaled lead conductance ??N = ??P ?0,N?N as
? ?R?L
G = N N . (4.27)
?RN + ?
L
N N
99
Incidentally (as detailed in the Appendix A.4), applying all the variable transformation to Eq. (4.7),
the rescaled transition rate ??N is given by
? ? ?
??
2
N = ?? {P (E ) + P (E )}f ?(E ? E ? ?) ??? |c? |? ??i j j i j ? i , (4.28)
i,j
which is very similar to the effective one-terminal conductance in the Meir-Wingreen?s pa-
per [126].
In the strong CB limit, where only two charge states N and N ? 1 participate in transport,
?N can be assumed to vanish except for one value of N , so
?R?L
G = N N , (4.29)
?R LN + ?N
which is physically the series formula for the conductance at each end. This equation is the same
as Eq.(176) of the Aleiner et al. review [127], except that the matrix elements of Eq.(130) are
replaced by the Meir-Wingreen formula.
To proceed further, we assume the system Hamiltonian [i.e., Eq. (4.2)] to be of the form
?
HQ = 
?
pdpdp + U(N), (4.30)
p
where p is the eigen-energy of the quasi-state p. The electron number variableN is in general dif-
ferent from the total occupation of Bogoliubov quasi-particles d?p and instead is equivalent to the
parity of the number of quasi-particles in this case. Specifically, in the limit of strong Coulomb
blockade where only two consecutive values of electron numberN0, N0?1 are allowed, the elec-
? ?
tron number N ? {N0, N0? 1} can be uniquely fixed by the relation (?1)N = Q0 ? (?1) p dpdp ,
100
where Q0 is the ground-state fermion parity of the first part (i.e., BdG) of the Hamiltonian HQ,
written as
Q0 = Pf{HQ,BdG(E = 0)}, (4.31)
where HQ,BdG is the first part of HQ written in a Majorana basis. Applying this relation to
? ?
Eq. (4.1), we can show that U(N) = (?U/2) ?Q ? (?1) p dpdp0 , where ?U = (?1)N0 [U(N0)?
U(N0?1)] is the electrostatic energy difference between the two transition-allowed charge states
N0, N0 ? 1. Substituting the energy eigenvalues Ei and wave-functions |?i? for HQ, the coeffi-
cients ??N in Eq. (4.28) can be written in a more explicit form (details are in Appendix A.4):
e2 ?? ? [ ]
??N = ? F?p(n,Q) (1? n)A?p + nB?~ p
, (4.32)
p n=0,1 Q=?1
where
F?p(n,Q) = feq [(1? 2n)p ?Q?U ] ? Fp(n,Q) (4.33)
with [ ]
Q?U ? ( )
? e??(? +n ?2 p) 1 +QQ (?[1)n0 s 6=p t?anh s2Fp(n,Q) = ??(Q?U ( )] . (4.34)e +n2 p)Q=?1 n=0,1 1 +QQ tanh ?s0 s 2
Note that f () = (1 + e?)?1eq is the Fermi distribution at equilibrium and, following Eqs. (4.56)
and (4.57), ?
A?p = |up,??|
2 (4.35)
?=?,?
is the tunneling rate for the electron from the lead at ? end to the nanowire (same for the opposite
direction), and ?
B? = |v 2p p,??| (4.36)
?=?,?
101
is the tunneling rate for the hole from the lead at ? end to the nanowire (same for the opposite
direction), where ?? are assumed to be the same at both ends ?. Within the tunneling limit
considered here, the value of the tunneling amplitudes ?? determines the overall scale of the
conductanceG. In our calculation, the value of ?? has been chosen such that the peak height in the
normal metal (i.e., large Vz) regime is equal to 1. This should be considered a choice of units for
our calculation. Comparison to experimental data can be made by scaling the experimental data
in a similar way by the normal state conductance. We note that this choice of unit is the natural
one in the tunneling limit under consideration here. Note that up,?? and vp,?? are coefficients of
electron and hole relation with quasi-particle and quasi-hole that are discussed in more detail in
Sec. 4.4.1.
The equation for G in Eq. (4.27), together with the definitions (4.31)-(4.36), is the central
formalism used in this work to compute the conductance of a system Q coupled to separate
leads L and R. Since the only constraint in equations for the conductance [Eqs. (4.31)-(4.36)]
connecting the number of electrons N to the quasi-particle degrees of freedom is through the
parity, the results are invariant as long as N changes by 2. Using Eq. (4.1), this also implies that
the conductance G is periodic in ng with period 2. Because of this, in this paper, we will only
plot the gate charge ng over two periods i.e., a range of length 4. This formalism reduces, in the
case where leads L and R coincide in space, to the well-known conductance derived by Meir and
Wingreen [126] for interacting systems. Our work generalizes the formalism to the situation with
arbitrarily spatially separated L and R leads as appropriate for Majorana nanowire experiments.
102
4.3 Conductance for Few-level Systems
The evaluation of the conductance G using Eq. (4.27) for a realistic Majorana system,
which has a complicated spectrum, requires a rigorous numerical treatment. In this section,
we analytically evaluate Eq. (4.27) in cases where the system Q has one or two levels in the
low-energy spectrum. We will find that the conductance G can be written analytically in these
cases. The results in these cases will help understand the numerical results for the more complex
Majorana wire system, which in certain parameter regimes contains only a few low-energy levels
relevant for these analytical results.
4.3.1 Rates for Few-electron Process
In addition to electron tunneling processes, transport through the system Q also occurs
through Cooper-pair tunneling because of the proximity-induced superconductivity. In order
to place these two processes on a comparable footing, we rewrite the equation of the scaled
transition rate for tunneling of electrons [Eq. (4.28)] as
? ?
??,1 = ?? (1)
(1)
N ? P?(Ei) df(? ?)?(Ej ? Ei ? )Mij (4.37)i,j
(1)
= ?? (1) ?1 ??(Ei?N?)? Ztot e f(Ej ? Ei ? ?)Mij , (4.38)
i,j
? ?
where (1) 2Mij = ??|c??|?? is the transition matrix element for transferring one electron from the
leads to the dot, and (1)?? is the one-electron tunneling rate, which was referred to as ?? in
103
Eq. (4.7). Similarly, we can write the rate for two-electron (or Cooper-pair) transfer
? ?
??,2
(2)
N = ??
(2)Z?1 e??(Ei?N?)? tot? df(? ?)f(Ej ? Ei ? ? ?)Mij (4.39)i,j
(2) ?1 ??(Ei?N?) (2) (2)= ?? Ztot e f (Ej ? Ei ? 2?)Mij , (4.40)
i,j
?
where f (2) (2)() = (?)/(e? ? 1) and Mij is the matrix element of transferring a Cooper pair
into Q. The charge of the system changes by 2, preserving parity under this tunneling process.
The parameter (2)?? sets the scale of the Cooper-pair tunneling rate analogous to the one-electron
tunneling rate (1)?? .
Using the fact that ? and the gate voltage entering Ej play equivalent roles, we can set ? to
zero. In that case, we can write the rates in a more symmetric form
?
??,1 = ?? (1) ?1
1 (1)
N ? Ztot M , (4.41)e?Ei + e?Ej ij
i,j
?
?,2 (2) ?1 Ej ? Ei (2)?N = ??? Ztot M , (4.42)e?Ej ? e?Ei ij
i,j
where Ei is understood to be replaced by Ei ? Ei ? miniEi. The latter can be done since
only ratios of Ei enter the formula. In this form, it is clear that any ?N rate is significant if both
energies are less than ??1.
The tunneling matrix elements (i=1,2)?? can be estimated by considering the limits of trans-
port without a SC gap and without s?ub-gap states respectively. In the case without SC, we can
rewrite Eq. (4.41) as ?1 ? ?? (1)?Q d??M (1)(?)(1 + e??)?, where ?Q is the normal-state?>0
density of states (DOS) in the system Q. Ignoring the frequency dependence of M (1)(?) on
104
the scale of the temperature T so that ? M (1)Q (?) ? ?1D,?, we can write the end conductance
(1) ? ? (1)? P0,Ng? P0,N?? ?1D,? so that g? ? (1)?? ?1D,?, which is similar to the normal-state
conductance discussed below Eq. (4.7). In the limit of large conductance at the opposite end,
which maintains the equilibrium distribution for the number N , the normal-state conductance is
G ? g?. In the case of Cooper-pair transport with no subgap quasi-particle state and negligible
charging energy, we can assume Ej ? E so that ?(2) ? (2) (2)i ?? M . The parameter in this ap-
proximation is the single-end NS conductance g(SC) calculated from the BTK formalism [88] so
that ? (2)M (2) ? g(SC). The Beenakker formula [128] suggests that the gapped SC conductance
g(SC) ? g2? in the limit where g? is the conductance in units of the quantum of conductance and
is assumed to be much smaller than unity. Therefore, in the tunneling limit ?(2)  ?(1), leading
to the expectation that the conductance from Cooper-pair transport processes should be much
smaller than arising from electron transport [129].
The constraint (2)  (1)?? ?? may be alleviated by enhancement of Cooper-pair transport in
the presence of ABSs. To understand this, we note that the Cooper-pair tunneling amplitude (2)??
is generated by elastic co-tunneling of two electrons into the SC through virtual states
? ?
? (2) 2
unv
? = t
n
? (4.43)En ? ?n
= t2 d?Tr[?(?)?+](? ??U)?1, (4.44)
?>?U
where un, vn and En are the particle and hole components of the wave functions of states with
Bogoliubov-de Gennes (BdG) eigenvalue En > 0. Here ?(?) is the local density of the SC
wire in Nambu space with a particle-hole matrix ?+ = ?x + i?y. Considering a simplified SC
model where we apply a uniform pair potential to the states of a normal metal such that the
105
?
SC densit?y matrix is given by Tr[?( ?
2 + ?2)?+] = ?
?
0(?)? 2 2 . Within this approxmation,? +?
(2)
? = t2 d?? (?)? ??? 0 2 2 2 2? . In the limit of a uniform density of states, we can?>?U ? +? ( ? +? ?U)
scale the integration variable ? ? ?? so that (2)? 2? ? t ?0. The conductance ? (2)2?2 ? 4 2? ? t ?0 ?
?21 
(1)
?? . Alternatively, if we consider a scenario that may be realistic for a SM/SC structure
where the local density of states in the SM is suppressed near the Fermi level but enhanced above
energy (1)? ? ?, ?? ? ?0(? ? 0) may be suppressed without changing (2)?? . This allows a
situation where the 2e conductance peaks with height ?2 may exceed the normal CB conductance
peaks at high magnetic fields.
4.3.2 Single-bound-state Induced Electron Transport
In the case of one ?active? level, i.e., within the range of thermal activation, there are only
two states: one with electron number N and another with N + 1, where the quasi-particle energy
 = EN+1 ? EN is the energy difference between the two states. Substituting the quasi-particle
energy into Eq. (4.41) leads to
(1) ( )
??,1
???? ?
N = sech
2 , (4.45)
4 2
where (1) (1) ?,(1)??? = ?? Mij . In this case, the conductance becomes
(1) (1) ( )
? ??R ??L ?G = sech2 , (4.46)
4 (1) (1)??R + ?? 2L
which is consistent with Chiu et al. [116]. The single-level case is also consistent with Meir-
Wingreen?s original formula [126]. Note here that the conductance of a state that is localized at
106
one of the ends of the system is substantially suppressed since one of (1)?R,L is small. From the
last paragraph, ??? ? g??1D so that G ? ?? gRgL sech21D (?/2). This conductance is enhancedgR+gL
compared to the non-Coulomb blockaded conductance.
One can use Eq. (4.46) to estimate the conductance in the case of a large number of lev-
els with similar transmissions ?? . Assuming that the conductance is split among N levels with
conductance ?? /N spread out over a range ?, the resulting conductance can be approximated by
(1) (1) ? ( )
G ' ? ??R ??L ??1 sech2 ? d (4.47)
4 (1) (1)??R + ?? 2L
(1) (1)
? ? ??R ??L 2T . (4.48)
4 (1) (1)?? ?R + ??L
We note that the conductance G in this case is suppressed relative to Eq. (4.46) by a temperature-
dependent factor of (T/?). This factor cancels the factor ??1D so that the conductance is now
temperature independent and comparable with the conductance of the non-Coulomb blockaded
case [127, 130].
4.3.3 One Subgap State At Each End
Let us consider the case of a long wire with a pair of levels, one at each of the left and right
ends: ?
?,1 1 (1)?N = ??
(1)
? Z
?1
tot Mij (4.49)e?Ei + e?Ej
i,j
107
We assume that there are levels at the two ends of a wire with energy ?. Generalizing Eq. (4.45)
to this case, the left and right conductances would be given by
( )
??
??,1 = (2Z )?1 ??? (1)e???/2N tot ? sech . (4.50)2
Using Eq. (4.29), the conductance can be written as
(1) (1)
? ?? ??
G =
(1) ( )R L ( ) , (4.51)4 ?? cosh2 ?1 (1)0 + ??1 cosh2 ?02 2
where 0 = min?(?) and 1 = max?(?). Note that at the CB resonance, 0 ? 0 while 1 stays
positive. This means that the conductance G is exponentially suppressed if a pair of levels near
the left end and right end have different energies.
If the energy levels of the ABSs on the left and the right are nearly degenerate, i.e.,
L ? R = , the above equation reduces to the result for a single level, i.e., Eq. (4.46) and
the conductance suppression is eliminated. This can be seen in the short nanowire case, consider-
ing that the ABSs on both ends are delocalized such that one bound state occupies both ends. This
also means the exponential suppression in Eq. (4.51) only applies to the long nanowire, where the
bound states are localized enough. The conductance for the two-state system in Eq. (4.51) can
also be suppressed even in the case of nearly degenerate level ? =  in the presence of gapless
states in the bulk of the superconductor that are generated by a magnetic field, as is assumed
for the results in Fig. 4.2. This suppression can be understood as a suppression of the tunneling
matrix elements ?? resulting from hybridization between the bound states and the bulk states.
As will be elaborated in the discussion, this suppression will play a role in understanding the
108
suppression of conductance relative to that from the 2e periodic Cooper-pair transport.
4.3.4 Conductance Near N and (N + 2) Degeneracy
In this case, transport is dominated by Cooper-pair transfer processes and Eq. (4.29) can be
generalized to
?L,2?R,2
G = (4.52)
?L,2 + ?R,2
Using Eq. (4.42), so that ??,2 = ???,2  , where  = EN+2 ? EN , the Cooper-pair conduc-sinh(?)
tance is written as
?L,2?R,2 ?
G = . (4.53)
?L,2 + ?R,2 sinh(?)
We notice that this reaches a maximum value comparable to gSC that is independent of tempera-
ture as  approaches 0. This is different from the suppression factor for the single-level case in
Sec. 4.3.3 and the maximum value in this case is simply the non-CB conductance gSC.
For an ideal superconductor, the number of electrons in the system N is even. However,
sub-gap bound states that are not directly coupled to the leads can play an important role in the
CB transport. For example, applied Zeeman fields can drive a state to cross zero energy changing
the ground-state parity of N . This leads to a shift in the periodicity of the CB conductance of the
system [113]. Another possibility is where the system has states on the order of or lower than the
temperature of the system. In this case, the N and N + 2 degeneracy cannot be reached, because
this gate voltage would correspond to a ground state of N + 1, which has no degeneracy.
109
4.4 Model
4.4.1 Hamiltonian
The 1D superconductor-proximitized semiconductor nanowire with spin-orbit coupling in
the presence of a field-induced Zeeman spin splitting can be described in the following discretized
form
?{ [ ]}
H? ?BdG() = Cx [(2t? ?)?z?0 + Vz?0?x + ?()]C + C
?
x x+a(?t?z?0 + i??z?y)Cx + H.c. ,
x
(4.54)
where Cx = (c , c , c
?
x? x? x?,?c
?
x?) is the electron operator at position x, and ?() is the self-
energy [99] following Eq. (2.3) (replacing ? by ), with the Zeeman-field-varying SC gap fol-
lowing Eq. (2.4). We can also add the quantum dot (QD) into the nanowire. As an example, the
potential confining quantum dots for our numerical results follows the form of Eq. (3.1) at the
left and right ends of the nanowire, but the potential depth VD value is different on both sides. lD
is the QD length. The whole Hamiltonian with the QD is
?{ [ ]}
H? () = C?QD x [(2t? ?+ Vdot(x))?z?0 + Vz?0?x]Cx + C
?
x+a(?t?z?0 + i??z?y)Cx + H.c. .
x
(4.55)
The Hamiltonian of the leads can be described by Eq. (4.55) but replacing Vdot(x) by a
constant Elead, which represents the gate voltage applied on the lead. There is also a NS tunnel
barrier at the junction between the leads and the nanowire. The Hamiltonian in the area with the
tunnel barrier is described by Eq. (4.55) but replacing Vdot(x) by Vbarrier(x) = Ebarrier?lbarrier(x), a
110
square potential with potential strength Ebarrier and width lbarrier.
The electron creation and annihilation operators are written in terms of quasi-particles and
quasi-holes ?
? ( )cx? = u? ?p,x?dp + vp,x?dp , (4.56)
?p ( ? ? )cx? = vp,x?dp + up,x?dp . (4.57)
? p
The normalization leads to x,? (|u 2p,x?| + |vp,x?|2) = 1. The quasi-particle and quasi-hole for
the energy level p are given by
?
? ( )dp = u ?p,x?cx? + vp,x?cx? , (4.58)
x,?=?,?
? ( )
dp = v
? ? ?
p,x?cx? + up,x?cx? . (4.59)
x,?=?,?
4.4.2 Tunneling Rate From The Density Matrix
From Eqs. (4.35) and (4.36), the tunneling rate can be expressed as the square of wave
function coeffi?cients, b?ased on? Eqs. (4?.58) and (4.59). So, the expression in Eq. (4.35) and (4.36)2 2
is technically ????e | ? ?x?? and ???h ?  |x?? , respectively. Note that ?e (?h  ) is the electron (hole)-p p p p
part of wave function with eigen-energy p. We choose to use the density matrix approach to find
the degeneracy and degenerate wave functions for the tunneling rates.
We start with the density matrix obtained by taking the anti-Hermitian part of the Green?s
function, i.e., ( )
G(?)?G?(?)
?wire(?) = (4.60)
2i?
111
or
?wire(?) ? ? ? ? ??wire(x?? ;x ? ? ;?) (4.61)
= ?n,m(x??)?
? ? ? ?
n,m(x ? ? )?(? ? n) (4.62)
n,m
(n is the energy-level index and m is the degeneracy index of an energy level), while we express
the Green?s function in the basis of eigenstates, i.e.,
G(?) ? G( wire(x?? ;x
???? ?;?)) (4.63)?1
= ?H?BdG(?)? ? + i? (4.64)? (x??)??(x???? ?p p )
= (4.65)
? p ? ? + i?p 1 ?
= ? ? (x??)?
?(x???? ?)P ( ) + i? ?(? ?  )? (x??)??(x???? ?p p ?? ? ? p ??p p ?),(4.66)p ? ?p p
Hermitian anti-Hermitian
where the linewidth ? is assumed to be infinitesimal. We can pick out the sub-density matrix at
energy level n by integrating the density matrix over the bound state range [n ? a, n + b], i.e.,
? n+b
(n)
?wire(x?? ;x
???? ?) ? ? ?wire(?)d? (4.67)n?a
= ? ? ? ? ?n,m(x??)?n,m(x ? ? ). (4.68)
m
Note that the integration grid needs to be much finer than the linewidth ?, so the numerical
discrete integration can be close to the continuous integration. The benefit of using density matrix
is that we can find the degenerate wave functions of energy n by diagonalizing the sub-density
112
matrix (n)?wire and get eigenvalues ?m and eigenwave function ?n,m. Then we can express the
sub-density matrix as
?
(n)
?wire(x?? ;x
???? ?) = ?m?n,m(x??)?
?
n,m(x
???? ?). (4.69)
m
Comparing Eqs. (4.67) and (4.69), we can get the effective wave function of degeneracy label m
of energy n:
?n,m(x??) = (?m)
1/2?n,m(x??). (4.70)
By default, we will get (4Ntot) of ?m by diagonalizing
(n)
?wire(x?? ;x
???? ?) with the dimension
(4Ntot)? (4Ntot), if Ntot is the number of lattice site. In order to get the correct degeneracy for the
bound states, we need to technically set some threshold for ?m, i.e., only those degenerate states
with ?m larger than the threshold can be picked as the degenerate states we are going to include
into the calculations. This threshold is also kind of constrained by the linewidth of the Green?s
function ? in Eq. (4.63). When ? is small, the peak of DOS is very sharp and narrow, even a
small threshold can select the eigenvalue ?m precisely. On the other hand, when ? is larger, the
peak of DOS becomes broadened, then we need a higher threshold to select out the positions of
the central peaks. This threshold for ?m should be universally the same over the whole parameter
space, in order to make sure the conductance depends only on the LDOS at both ends of the
nanowire near zero energy.
The above formula Eq. (4.70) applies to the bound states below the gap. For SC states
(above the gap before the gap collapses) and metallic states (after the gap collapses), the integra-
113
tion range in Eq. (4.67) will be a bit different, i.e.,
? n+?n/2
(n)
? ? ? ?wire(x?? ;x ? ? ) ? ?wire(?)d? (4.71)
n??n?(1/2 )
? ?n + ?n?1?wire(n) (4.72)
2
? ??wire(n) ??n (4.73)
= ?m?n,m(x??)?
? ? ? ?
n,m(x ? ? ) ??n (4.74)
m
with the energy spacing picked as
Dn
?n ? n ? n?1 ? , (4.75)
?SC(n)VSC
where the presumed degeneracy Dn is the size of the density matrix, the bulk BCS DOS of the
superconductor ?SC(n) is
2?F 
?SC() = ? ? [??(Vz)] , (4.76)
2 ??(V )2z
where ?F is the DOS at Fermi level. VSC is defined from the total DOS above the SC gap as
?tot() = VSC?SC(). (4.77)
This total DOS is under the assumption that the superconductor is much larger than the nanowire
so that the component of the wave function in the nanowire is negligible. After the superconduct-
114
ing gap collapses, i.e., ?(Vz > Vc) = 0, Eq. (4.76) reduces to
?metal() = 2?F , (4.78)
a constant. Note that the factor 2 here is due to the spins.
To avoid the singularity at ?SC(n = ?) [check out Eq. (4.76)], we define each energy level
(for SC states and metallic states) as follows. First, we define the states? density (states existing
per unit volume) as
?  ?  2? x ?F
F () ? ?SC(x)dx = ? dx = 2?F 2 ??2. (4.79)
2 2
0 ? x ??
At the same time, F (n) can also be written as
1 ?
F (n) = Dm, (4.80)
VSC m?n
where Dm is the degeneracy of m for m ? n. (Note that 1 = ?+0+.) By equating Eqs. (4.79)
and (4.80), we can find the energy level (for SC states and metallic states) is at
? ?
m?nDm
 2n = ? + ( )2. (4.81)
2?FVSC
The benefit from the technical side is that we only need coarse profiles of DOS. The DOS does
not need to have very sharp and precise peaks of each of the energy states. We can just presume
there are a lot of degenerate states occupying one dominant peak. The details, which may not be
precisely known for the experimental system, would not be crucial in such a situation.
115
4.5 Numerical Results
In this section, we will use the formula derived from Sec. 4.2.4 to calculate the CB con-
ductance from single-electron tunneling processes similar to the one shown in Fig. 4.2(a) and
discussed in the beginning of the chapter. In the strong CB limit, transport can only occur near
a degeneracy between the energy of the SC island with N and N + 1 electrons. This occurs for
magnetic fields that are large enough to reduce the SC gap below the charging energy [91, 115].
The range of Zeeman field over which the conductance is shown [similar to Fig. 4.2(a)] is thus
limited to the range where the 2e periodic CB peaks seen at small Zeeman field are split. As men-
tioned in the discussion of the results in Fig. 4.2, the CB conductance depends on many details of
the Hamiltonian such as soft gap, self-energy, quantum dots, etc. To dig into each factor, we will
systematically study the contribution of various mechanisms and ingredients used in Fig. 4.2 by
changing one parameter at one time from the reference case shown in Fig. 4.3.
The reference result shown in Fig. 4.3 is calculated from the Hamiltonian described in
Sec. 4.4.1 with the following parameters: the temperature T = 0.01 meV, the nanowire length
L = 1.5 ?m (150 sites, with the lattice space a = 10 nm), the hopping strength t = 25 meV, the
SC gap at zero Zeeman field ?0 = 0.9 meV, the SC collapsing field Vc = 4.2 meV, the spin-orbit
coupling constant ? = 2.5 meV, the chemical potential ? = 2.5?meV, the self-energy coupling
constant ? = 1.4 meV, the TQPT field is theoretically at VTQPT = ?2 + ?2 = 2.87 meV for this
parameter choice, the QD potential height at the left end VD = 1.0 meV, the QD potential height
at the right end VD = 4.0 meV, and the two QD widths are both lD = 0.26 ?m. The tunneling
barrier induced by the lead occupiesNbarrier = 20 nm (2 sites), with the energy heightEbarrier = 10
meV. DOS at Fermi level ?F = 0.1 (?m3 ? meV)?1, SC bulk volume V 5SC = 10 ?m3, and the
116
upper bound of energy level is 2.5 meV (roughly 3 times larger than ?0). The numerical results
in this work will use variations around these standard parameters as described in the various
sub-sections. Our choice of parameters is generic for the currently used experimental Majorana
nanowire systems.
For all the results in this chapter: Panel (a) shows the Coulomb-blockade conductance
as a function of the gate-induced charge number ng and the Zeeman field Vz. We show the
calculated CB conductance for two periods in ng space, which has identical conductance pattern,
thus explicitly verifying the periodicity of the conductance shown in Sec. 4.2.4. Panel (b) is
the regular QPC tunnel conductance probed locally from the left lead as a function of the bias
voltage V and the Zeeman field Vz without CB. Panel (c) is the energy spectrum as a function of
Zeeman field Vz, which is aligned with the conductance in Panel (a). The spectrum in Fig. 4.3(c)
is identical to the one in Fig. 4.2(c) below the TQPT field [marked by the dashed line at the lower
Zeeman field Vz], where there is a closure of the bulk spectrum. The 1e periodic conductance
features in Fig. 4.3(a) showing up above TQPT are found to be brighter than in the case of the
Fig. 4.2(a), and show an abrupt drop in intensity following the Zeeman field Vc [marked by the
dashed line at the higher Zeeman field Vz], where the superconductivity of the parent Al SC is
destroyed. The drop in intensity of the 1e periodic conductance peak in Fig. 4.3(a) is a result of
a transition from transport through MBSs suggested in earlier works [91, 115] to 1e periodic CB
transport in a normal metal [130]. The enhanced CB peak in Fig. 4.3(a) is a result of resonant
transport through MBS, similar to the case in quantum dots [130], and has been discussed in more
detail in sub-section 4.3.2.
Panel (d) is obtained by tracing the maximum conductance value ng for each Vz value
from Panel (a) for both even and odd N , and then calculating the absolute value of difference of
117
Figure 4.3: The reference case: We will use this set of results as reference for the
later results in this section. The difference from Fig. 4.2 is that this set of result
does not have soft gap. The parameters are: the temperature T = 0.01 meV, the
wire length L = 1.5 ?m, the SC gap at zero Zeeman field ?0 = 0.9 meV, the SC
collapsing field Vc = 4.2 meV. Other relevant parameters are given in Sec. 4.5. (a)
Coulomb-blockade conductance G as a function of the gate-induced charge number
ng and Zeeman field Vz at Ec = 0.13 meV. We only show two periods in the range
of ng ? [17, 21]. (b) Non-Coulomb-blockade conductance G from the left lead as a
function of bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function
of Zeeman field Vz. (d) OCPS as a function of Zeeman field Vz that is extracted from
the vertical peak spacing in panel (a).
118
these two tracking ng for both even and odd N . Comparing the positions of the dashed lines in
Fig. 4.3(d) with the energy spectrum shown in Fig. 4.3(c), we see that the OCPS arises from the
ABS states before the TQPT. The oscillatory splitting of ABSs at each end appears in the local
tunneling conductance spectrum at the left lead as seen in Fig. 4.3(b) though the oscillation in
this case does not show significant suppression with Zeeman potential. On the other hand, the
decreasing OCPS as in Fig. 4.3(d) demonstrates the lobes coming from the two ABSs at both
ends exhibited in Fig. 4.3(c). Therefore, OCPSs, which we obtain from the CB conductance
in Panel (a), give information about the non-local states, rather than just local states. The non-
local transport through a pair of localized levels at each end, which is what is responsible for
the shift of the energy of the resonance between the two ABSs, is described in more detail by
Eq. (4.51) in Sec. 4.3.3. The main conclusion in our model, where occupation of one ABS is
allowed to relax to the other ABS, is that the intensity of the transport peak is suppressed by the
energy difference between the two ABSs relative to temperature. This is in contrast to the case
where such relaxation is forbidden [91] and non-local transport requires both ABSs to be near
zero energy relative to temperature. Both these models would lead to suppression of conductance
from ABS states for Zeeman field in the beginning of the range in Fig. 4.3(a) [and Fig. 4.2(a)]
where the ABS energy difference is much larger than temperature.
Aside from demonstrating the ideal case (the most similar to the experimental data) as in
Fig. 4.2, we will also discuss different effects by changing various parameters relative to the refer-
ence case shown in Fig. 4.3, such as wire length, temperature, chemical potential, SC collapsing
field, with and without self-energy, and with and without QDs, in the following subsections of
Sec. 4.5.
119
4.5.1 Soft-gap Dependence
Most of the CB experiments [91,113,114] in nanowires do not show the abrupt drop in the
1e periodic conductance seen in the ideal case plotted in Fig. 4.3(a). Additionally, as discussed
in Sec. 4.3.4 as well as in previous work [115], the conductance into the ideal MBS seen in
Fig. 4.3(a) is expected to have a significantly higher intensity than the 2e periodic conductance
from Cooper-pair transport. Such an enhanced intensity is not seen in experiments [91, 113,
114], leading to a contradiction between theory and experiment. These issues are resolved in
the calculated conductance in Fig. 4.2, in which the soft SC gap is considered in the topological
regime completely. The CB conductance difference of 1e transition between the topological
regime and the normal-metal regime (i.e., regime above Vc) do not show any visible difference.
Clearly, soft gap plays a key role in the experimental CB phenomenology, and brings agreement
between theory and experiment.
Physically, the soft SC gap arises from impurity-induced bound states in the SC that is
subject to a strong Zeeman field. Experimentally, Majorana nanowires always manifest soft gaps
at finite magnetic fields even if the gap is hard at zero field. We model the soft gap phenomeno-
logically by splitting the superconductivity of the proximity-inducing superconductivity into two
parts with two different gaps ?1(VZ) and ?2(VZ) with different critical Zeeman fields such that
?
?1(Vz) = ?0 1? (V /V )2? z TQPT , (4.82a)
? 22(Vz) = ?0 1? (Vz/Vc) . (4.82b)
The SC gap collapse is thought to arise from the magnetic field entering the parent SC (i.e.,
120
Al in these CB experiments), destroying the parent superconductivity and hence all Majorana
physics. The soft gap regime (VTQPT < Vz < Vc) is characterized by weakened SC, which we
model using a self-energy that is as an average of the self-energy from a clean SC and that from
a normal metal. The soft gap is the generic experimental situation in Majorana nanowires at
finite magnetic field values even if the zero field system has a hard SC gap. With this choice, the
non-vanishing sub-gap density of states is generated in the bulk SC above Vz > VTQPT from the
closing of the first SC gap ?1, which is given by
{? ? }1 2?F ? 2?F?SC() = ?( ?1(Vz)) + ?(??2(Vz)) . (4.83)
2 2 ?? (V )2 2 ?? (V )21 z 2 z
At the same time, the system remains superconducting below Vz < Vc from the second super-
conducting part Eq. (4.82b). We will then follow the same procedure as Eqs. (4.79)-(4.81) to
obtain the energy levels within the soft-gap regime, i.e., ?1(Vz) <  < ?2(Vz), and the hard gap
regime i.e.,  > ?2(Vz), as before. The two SC gaps ?j(Vz) also correspondingly modify the SC
self-energy in Eq. (2.3) to a form which averages between the two SC gaps as in Eq. (4.83).
The effect of the soft SC gap on the CB conductance is a reduction of the peak height
associated with the MBS in the topological regime of the Zeeman field (VTQPT < Vz < Vc) from
Fig. 4.3(a) to Fig. 4.2(a). For the parameters chosen in Fig. 4.2(a), the MBS conductance is found
to be almost identical to the normal state conductance above Vz > Vc, which appears to be the case
in the experimental data [91, 113]. Our results suggest that an MBS peak height comparable to
the normal CB peak at higher Zeeman field would be indicative of a soft gap. While the precise
matching of the MBS and normal state CB peaks might be a result of our parameter choices,
the smooth intensity variations of the CB peak in experiments [91, 113] suggest the emergence
121
Figure 4.4: Line cuts of the conductance as a function of the gate-induced charge
number ng (i.e., Fig. 4.3), for fixed Zeeman fields Vz, with different temperature
values T = 0.005 meV, T = 0.01 meV, and T = 0.02 meV in a single panel. (a)
At Vz = 2.0 meV, the nanowire is in the Andreev-bound-state/trivial regime. (b) At
Vz = 3.5 meV, the nanowire is in the Majorana/topological regime. (c) At Vz = 5.0
meV, the nanowire is in the normal-metal regime.
(possibly gradual) of a soft SC gap at some Zeeman field above the TQPT. This is of course
the experimental phenomenology observed in the usual tunneling spectroscopy of all Majorana
nanowires studied so far, where the SC gap always becomes monotonically softer with increasing
magnetic field.
4.5.2 Temperature Dependence
As discucssed in Sec. 4.3.2, the 1e periodic CB peaks arising from MBS and normal metal
behavior show, respectively, an inverse and vanishing temperature dependence of the CB peak
height. This is a characteristic difference of a CB peak arising from a resonant bound state or
a continuum of states [127, 130]. In Fig. 4.4(a), Vz = 2.0 meV, which is in the topologically-
trivial regime below TQPT, the conductance peak heights for T = 0.005, 0.01, and 0.02 meV
are roughly 10, 5, 2.5 in arbitrary units. As explained in Sec. 4.3.1, the overall scale of the
conductance in our work is determined by the tunneling parameter ? . Our results can be compared
122
to experiment by setting the normal state CB conductance for Vz > Vc to the normal-state tunnel
conductance. Since the CB conductance peaks arise from ABSs, which are isolated bound states,
the peak heights vary inversely with temperature as discussed in Sec. 4.3.2 [127, 130]. A similar
temperature dependence is seen for the resonance in the topological regime (Vz = 3.5 meV)
in Fig. 4.4(b), where the peak heights are approximately 12, 7, and 3.5 at T = 0.005, 0.01,
and 0.02 meV, respectively. This is in contrast to the CB conductance peak in the normal-metal
regime at Vz = 5.0 meV, which is shown in Fig. 4.4(c), where we find the peak heights to be
almost temperature independent as with normal metal CB (see Sec. 4.3.2) [127,130]. The results
in Fig. 4.4 show that the temperature dependence of the CB peaks can be used to distinguish
between 1e periodic CB peaks arising from MBSs or trivial non-superconducting CB effect in
regular metallic grains.
4.5.3 Length Dependence
In Fig. 4.5, we decrease the nanowire length to L = 1.0 ?m, compared to Fig. 4.3 (L =
1.5 ?m). The Majorana oscillations between the two dashed lines in Fig. 4.5 become more
obvious in the shorter wire, which satisfies the trend of the Majorana splitting e?2L/? [81] which
is necessarily enhanced in shorter wires indicating an exponential weakening of the topological
protection. Due to the Majorana splitting, the lobes of the OCPS, which project the combination
of the lowest-energy states on both ends, start to increase as soon as we enter the topological
regime for the shorter wire. On the other hand, Majorana oscillation is suppressed for the longer
wire, as in Fig. 4.3 for the case of L = 1.5 ?m. Therefore, we can see that lobes of the OCPS
for L = 1.5 ?m decrease monotonously as the Zeeman field increases, while the counterparts of
123
Figure 4.5: This set of results has the shorter wire length L = 1.0 ?m, while keeping
all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade conductance G
as a function of the gate-induced charge number ng and Zeeman field Vz atEc = 0.13
meV. (b) Non-Coulomb-blockade conductance G as a function of bias voltage V and
Zeeman field Vz. (c) Energy spectrum as a function of Zeeman field Vz. (d) OCPS as
a function of Zeeman field Vz.
124
Figure 4.6: This set of results has the shorter wire length L = 0.6 ?m, while keeping
all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade conductance G
as a function of the gate-induced charge number ng and Zeeman field Vz atEc = 0.13
meV. (b) Non-Coulomb-blockade conductance G as a function of bias voltage V and
Zeeman field Vz. (c) Energy spectrum as a function of Zeeman field Vz. (d) OCPS as
a function of Zeeman field Vz.
125
L = 1.0 ?m decrease only before the TQPT field. We also expect that the OCPS for the wire
length longer than L = 1.5 ?m will look no different from the one of L = 1.5 ?m, considering
that both ABS and MBS will be even more localized and stable, and thus the oscillations in the
topological regime will be suppressed and negligible as in Fig. 4.3. Based on Figs. 4.3(c) and
4.5(c), both of which show the decreasing lowest-lying energies as a function of Zeeman field
below the TQPT field (first dashed line), we can also observe that the decreasing lobes of OCPS
in Fig. 4.5(d) mainly come from the ABSs induced by the two quantum dots at both ends. The
decreasing OCPS coming from the lowest-lying ABSs on both ends will be destroyed when the
nanowire is too short so that the ABSs on both ends interfere with each other, such as the L = 0.6
?m case in Fig. 4.6.
In Fig. 4.6(a), with shorter nanowire length L = 0.6 ?m, the conductance peak is vis-
ible in the range of Zeeman potential at the lowest end in Fig. 4.6(a), which is in contrast to
Figs. 4.2(a), 4.3(a), and 4.5(a), where the conductance is suppressed based on Eq. (4.51) in
Sec. 4.3.3 for the longer wire case.
The suppression of the conductance based on Eq. (4.51) in the long wire case is eliminated
for shorter wires with length comparable to the coherence (or the localization) length of the bound
states. In this case, electrons from either end can tunnel into each of the ABSs so that electron
tunneling between the ends of the wire through either of the ABSs described by Eq. (4.46) pro-
vides a measurable contribution to the CB conductance in shorter wires such as in Fig. 4.6(a).
These results lead to the conclusion that the observation of the dark region at the transition from
2e periodic conductance near the lowest part of the Zeeman range shown in the CB conductance
plots [e.g. Figs. 4.2(a), 4.3(a), 4.5(a). etc.] can be understood to be a consequence of ABSs at
the ends of the wire.
126
Figure 4.7: The oscillatory amplitude of the first lobe in the OCPS (|So ? Se|/2) for
different wire lengths. All the other parameters are kept the same as Fig. 4.3, except
for the nanowire length. The maximum oscillatory amplitude does not display strong
length dependence as experimental data.
127
The length dependence of the first lobe of the OCPS, which is plotted in Fig. 4.7 shows
a 40% increase in magnitude of the OCPS with increasing length. This is in contrast to the
exponential decrease in the magnitude of OCPS expected from MBS splitting oscillations, which
has been claimed to be observed experimentally [91,114]. Additionally, the change in the OCPS
observed by Albrecht et al. [91, 114] is two orders of magnitude in contrast to the 40% we see
in Fig. 4.7 over the same range of lengths. A similar (? 40%) length dependence of OCPS has
been obtained in earlier theoretical work [116] using a master-equation approach. The origin
of the large discrepancy of the length dependence of our OCPS with experiment is the fact that
the OCPS in our models arise from ABSs rather than MBSs, as expected in the experiments.
Since the ABS energy is dominated by the profile of the confining potential at the end, we do not
expect them to have the exponential length dependence induced by MBS. While this might be a
motivation to restrict to models where OCPS from ABSs are absent, one should note that OCPS
from MBSs are found numerically to increase or show no significant decrease with increasing
applied Zeeman field [116] as seen in Fig. 4.5(d). Thus the OCPS pattern from a single device in
the experiment [91,113,114] is significantly more consistent with those arising from ABSs [such
as Fig. 4.2(d)] than from MBSs. Therefore, while our results show qualitative consistency of the
OCPS with single device, the range of models we study cannot reproduce both the decreasing
OCPS with field as well as with length seen in the experiments [91,114]. Again, this is consistent
with the earlier theoretical conclusion based on the master-equation approach [116], and we
believe that the conclusion of an ?exponential protection? made in Ref. [91] is incorrect and is
an artifact of using very few samples with each sample having its own different sets of ABS, etc.
(i.e., the sample length was not varied in situ, but only by going from sample to sample where
obviously many things, not just the sample length, are changing in an uncontrolled manner).
128
Figure 4.8: This set of results has the lower SC collapsing field Vc = 3.6 meV,
while keeping all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade
conductance G as a function of the gate-induced charge number ng and Zeeman field
Vz at Ec = 0.13 meV. (b) Non-Coulomb-blockade conductance G as a function of
bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function of Zeeman
field Vz. (d) OCPS as a function of Zeeman field Vz.
4.5.4 SC Collapsing Field Dependence
In this subsection, we change the SC collapsing field in Fig. 4.3 from Vc = 4.2 meV to
lower value Vc = 3.6 meV, as in Fig. 4.8, and to higher value Vc = ?, i.e., ?(Vz) = ?0,
as in Fig. 4.9. The value of Vc determines the size of the topological regime relative to the
normal metal regime, where the SC gap is destroyed. Since the CB conductance for a normal
metal is exactly 1e periodic [92, 131?135], a low value of Vc can appear as a suppression of the
Majorana splitting oscillations expected in the topological superconducting regime. The range
129
Figure 4.9: This set of results has the infinite SC collapsing field, i.e., ?(Vz) = ?0,
while keeping all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade
conductance G as a function of the gate-induced charge number ng and Zeeman field
Vz at Ec = 0.13 meV. (b) Non-Coulomb-blockade conductance G as a function of
bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function of Zeeman
field Vz. (d) OCPS as a function of Zeeman field Vz.
130
of the topological regime in Figs. 4.3 and 4.8, which starts at the TQPT field and ends at the SC
collapsing field Vc (i.e., region between the two dashed lines), does not give enough parameter
space for the MBS to be delocalized by the strong external magnetic field. On the contrary, if the
parent SC gap is highly robust to the applied magnetic field (i.e., Vc  VTQPT) as is the case in
Fig. 4.9, the range of topological SC becomes large enough to accommodate an observable range
of Majorana splitting oscillations with an amplitude that increases with increasing Zeeman field.
Thus, the oscillations in the topological superconducting phase are found to have an amplitude
increasing with field, which is in contradiction with the experiments. As discussed in the previous
subsections, the experimental observations of OCPSs [91, 113, 114] are more consistent with
ABSs which dominate when Vc is not too large relative to VTQPT as in Fig. 4.8 or Figs. 4.2 or 4.3.
4.5.5 Chemical Potential Dependence
Unlike the dependencies discussed so far, changing the chemical potential can substantially
change the spectrum of the nanowire even below the TQPT in a way that has been studied in the
context of tunneling transport [58, 59, 62, 64, 65, 124]. Aside from changing the value of the
TQPT field (the first dashed line), changing the chemical potential ? also modifies the spectrum
of subgap ABS energies [59,62,64,65,124]. In addition, the ABSs induced from both ends of the
nanowire do not follow a monotonic trend as the chemical potential is varied, due to the fact that
the potential heights of QDs are not the same on both ends. Since the OCPS roughly projects the
combination of the lowest energies on both ends, it is not guaranteed to generate the decreasing
trend of OCPS by simply tuning the chemical potential. We can compare Fig. 4.3 with Figs. 4.10
and 4.11, where the chemical potential changes from ? = 2.5 meV to ? = 2.0 meV and ? = 3.0
131
meV, respectively. The energy spectrum below TQPT field in Fig. 4.10(c) has no similarity to the
one in Fig. 4.3(c), while the structure in Fig. 4.11(c) is similar to Fig. 4.3(c), even the chemical
potentials in Figs. 4.10 and 4.11 are both just away from ? = 2.5 meV by 0.5 meV. Nevertheless,
the behaviors in the topological regime for these three plot sets do not seem to show a significant
difference when the nanowire is long enough to suppress the Majorana splitting oscillation. Thus,
as seen in Figs. 4.3(d), 4.10(d), and 4.11(d), the first OCPS lobes arise from end ABS. Comparing
the different chemical potential cases in Figs. 4.3, 4.10, and 4.11, we see that while the OCPSs do
not show any strong increase with Zeeman potential, observing a strong decrease with Zeeman
field as in Fig. 4.3 and seen in experiments [91, 113, 114] depends on the choice of chemical
potential.
While the results in Figs. 4.2, 4.3, 4.8, 4.9, and 4.12 suggest the decreasing oscillations of
CB conductance peaks are indicative of conductance in the ABS regime, the results in Figs. 4.10
and 4.11 suggest otherwise. These latter results show small decreasing oscillations across the
Vz = VTQPT, making it difficult to distinguish the ABS regime from the MBS regime. This is
especially so in Fig. 4.11(a), where the CB conductance patterns between the ABS regime and
MBS regime look similar (i.e., both display almost 1e periodic CB transitions) because the ABSs
stay close to zero energy, giving rise to the brightness pattern, which barely has oscillations. This
result means that the CB conductance profiles by themselves cannot provide enough information
about the difference between the ABS regime and the MBS regime. However, it should be noted
that in Fig. 4.11(a), a little patch of darkness near TQPT separates the ABS regime and MBS
regime though it is not so visible in the general case. This will be discussed in more detail in
Sec. 4.6.1. In general, however, our detailed theoretical results indicate that particular caution is
warranted in interpreting experimental CB conductance peaks as arising from MZMs since the
132
Figure 4.10: This set of results has the lower chemical potential ? = 2.0 meV,
while keeping all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade
conductance G as a function of the gate-induced charge number ng and Zeeman field
Vz at Ec = 0.16 meV. (b) Non-Coulomb-blockade conductance G as a function of
bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function of Zeeman
field Vz. (d) OCPS as a function of Zeeman field Vz.
133
Figure 4.11: This set of results has the lower chemical potential ? = 3.0 meV,
while keeping all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade
conductance G as a function of the gate-induced charge number ng and Zeeman field
Vz at Ec = 0.21 meV. (b) Non-Coulomb-blockade conductance G as a function of
bias voltage V and Zeeman field Vz. (c) Energy spectrum as a function of Zeeman
field Vz. (d) OCPS as a function of Zeeman field Vz.
134
distinction between the manifestations of ABS and MBS in CB conductance is rather subtle and
small.
4.5.6 Quantum Dot Dependence
In Fig. 4.3, we show the calculated results with two QDs on both ends of the nanowire.
In this subsection, we show the results of one QD in Fig. 4.12 by removing the QD on the right
end but keeping the left one. We also show the results without any QD in Fig. 4.13. Compared
to Fig. 4.12(c), the non-Coulomb-blockade conductance in Fig. 4.12(b) reflects the ABS located
at the left end as it is measured from the left lead. We also observe that there is no difference
between Figs. 4.3(b) and 4.12(b), even though there are two low-energy ABSs in Fig. 4.3. Both
Fig. 4.3(b) and Fig. 4.12(b) only demonstrate the ABS induced from the QD at the left end. So
the regular tunnel conductance (without Coulomb blockade) can only provide local information.
On the other hand, Fig. 4.3(d) projects the combination of the two ABSs located at both ends, by
comparing it with Fig. 4.3(c). Therefore, the OCPS obtained from the Coulomb blockade con-
ductance gives us non-local information that the regular (non-Coulomb-blockade) conductance
cannot provide.
In Fig. 4.12(a), the CB conductance peak is suppressed below the TQPT field (first dashed
line), relatively lower than the Majorana peak, which lies between the two dashed lines. The
diminished conductance associated with the ABSs in Fig. 4.12 arises from the difference in the
ABS energies (seen in Fig. 4.12(c)) at the two ends. Such a difference in energies leads to
suppression in conductance as seen in Eq. (4.51). The CB conductance intensity increases once
the ABS energies approach zero energy and continues to remain high past the TQPT until the SC
135
Figure 4.12: This set of results has only one QD on the left end of the wire, while
keeping all the other parameters the same as Fig. 4.3. (a) Coulomb-blockade con-
ductance G as a function of the gate-induced charge number ng and Zeeman field Vz
at Ec = 0.13 meV. (b) Non-Coulomb-blockade conductance G as a function of bias
voltage V and Zeeman field Vz. (c) Energy spectrum as a function of Zeeman field
Vz. (d) OCPS as a function of Zeeman field Vz.
136
Figure 4.13: This set of results has none QD on both ends of wire, while keeping all
the other parameters the same as Fig. 4.3. (a) Coulomb-blockade conductance G as
a function of the gate-induced charge number ng and Zeeman field Vz at Ec = 0.40
meV. (b) Non-Coulomb-blockade conductance G as a function of bias voltage V and
Zeeman field Vz. (c) Energy spectrum as a function of Zeeman field Vz. (d) OCPS as
a function of Zeeman field Vz.
gap closes.
Comparing the OCPS of two-QD case [Fig. 4.3(d)] with the case without QDs [Fig. 4.13(d)],
we conclude that the Zeeman-field oscillations of OCPS depend on the presence of QDs. This
is consistent with the absence of oscillations in the ABS spectrum in Fig. 4.13. While the MBS
spectrum typically shows oscillations with Zeeman energy [136], the splitting amplitude is sig-
nificantly below the thermal resolution [62] at the temperatures we consider. Indeed, neither the
non-CB conductance [Fig. 4.13(b)] nor the OCPS from CB conductance [Fig. 4.13(d)] show any
137
Figure 4.14: This set of results does not include the self-energy, while keeping all
the other parameters the same as Fig. 4.3. (a) Coulomb-blockade conductance G as
a function of the gate-induced charge number ng and Zeeman field Vz at Ec = 0.13
meV. (b) Non-Coulomb-blockade conductance G as a function of bias voltage V and
Zeeman field Vz. (c) Energy spectrum as a function of Zeeman field Vz. (d) OCPS as
a function of Zeeman field Vz.
oscillations, which is different from the results in Figs. 4.3 or 4.12, both of which have at least
one QD. Therefore, the oscillatory lobes of OCPSs in the experimental data [91, 113, 114] most
likely come from the ABSs induced by the QDs.
4.5.7 Self-energy Dependence
In Fig. 4.14, we remove the self-energy term induced by the parent SC. The energy split-
tings from both ABSs and MBSs are enlarged without self-energy, compared to Fig. 4.3, which
138
incorporates self-energy. The self-energy suppresses the energy splitting by a factor of 1/(1 +
?/?) [104]. Aside from the effective ?SC gap ? being replaced by the self-energy coupling ?,
?the TQPT field is shifted to VTQPT = ?2 + ?2 for the self-energy case, rather than VTQPT =
?20 + ?
2 for the non-self-energy case. According to weak-coupling BCS theory, coherence
length is ? = ~vF/(??), so the coherence length becomes shorter when ? > ?. Therefore,
the Majorana splitting, which follows e?L/?, becomes smaller with parent SC coupling [79,137].
The subgap states become more localized as we increase the coupling strength ? too. The self-
energy generically reduces the Majorana oscillation amplitude above TQPT, but the quantitative
suppression depends on many details of the parameters.
The inclusion of self-ernergy may be a key to understanding why MBS oscillations in-
creasing in amplitude with Zeeman field are not observed in OCPS experiments or conventional
conductance experiments without CB. For the non-self-energy results in Fig. 4.14, we do not
observe the OCPS decreasing with the Zeeman field monotonously. In fact, the splitting in the
absence of self-energy seen in Fig. 4.14 is large enough to eliminate any 1e periodic features that
are seen in the experiments and in Fig. 4.3 beyond a magnetic field which is not large enough to
kill SC completely. Whether the almost universal experimental absence of Majorana oscillations
with increasing Zeeman field is a consequence of the self-energy effect or simply a manifesta-
tion of the dominance of ABS in the experiments is an important question. Obviously, if the
experiments are manifesting only ABS and no MBS, there will not be any Majorana oscillations
whether self-energy effects are included or not. The current nanowire experiments, including the
CB experiments, are certainly more consistent with the physics of ABS being dominant, which
provides a natural explanation for why the MBS oscillations are never seen.
139
4.6 Key Features of Coulomb Blockaded Conductance
In this section, we summarize how the generic features of the experiment can be understood
from a compilation of our results in Sec. 4.5 together with the analytical arguments in Sec. 4.3.
4.6.1 Bright-dark-bright Pattern
One of the commonly seen features in the CB experiments [91,113,114] is the bright-dark-
bright intensity pattern that is seen as the Zeeman field increases. Specifically, the 2e periodic
CB peaks at small Zeeman field are seen to be bright followed by a dark region, which then be-
comes bright as the CB resonances approach the purely 1e periodic regime. While the numerical
results shown in Fig. 4.3-4.14 do not show the 2e periodic region, they do show a dark region of
suppressed conductance at the lower end of the Zeeman scale corresponding to immediately after
the 2e periodic region, consistent with experiments. The suppressed conductance seen in our nu-
merical plots in this so-called dark region can be understood from Eq. (4.51), which shows that
the conductance intensity will become suppressed when the eigen-energy difference from both
ends is larger than the temperature. From the spectrum in Figs. 4.2(c), 4.3(c) and 4.5(c), there are
two sub-gap ABSs which are located on both wire ends respectively. The higher-energy subgap
ABS suppresses the resonant conductance as the arguments below Eq. (4.51) suggest. There-
fore, we are unable to observe the conductance below Vz = 1.7 meV. The brightness appears
above Vz = 1.7 meV, when the energy difference from L and R ends is within the temperature
range, i.e., |L ? R| < T . These numerical results also prove that Eq. (4.51) applies in the
long nanowire, where the ABSs on both ends are localized. On the other hand, when the wire is
short, the ABSs on both ends are delocalized and one can use Eq. (4.46) instead of Eq. (4.51) to
140
describe conductance through each of these states. The resulting conductance is no longer sup-
pressed by the energy difference between the ABSs, though it is instead suppressed by the short
length. The topological MBS peaks are associated with a single ABS that is therefore described
by Eq. (4.46) and shows a bright CB resonance even when split [115]. This Majorana peak (in
the absence of a soft gap) described by Eq. (4.46) is the brightest in the CB conductance plot.
Thus, the presence of exponential suppression based on Eq. (4.51) can in principle explain the
bright-dark-bright feature seen in experiments on long Majorana wires [91, 113, 114] although
the experimental wires are unlikely to be in the long wire regime given the rather small induced
SC gap.
The mechanism for the dark feature discussed in the previous paragraph is different from
that proposed previously [115], where the dark feature resulted from a continuum of delocalized
states. The suppression in the latter mechanism arises from the delocalization of the continuum
states, as opposed to the localization of the ABSs considered in our work. The numerical results
in Figs. 4.3-4.14(a) also contain contribution from continuum states, but only near the TQPT
when the bulk gap closes. This is seen as a separate small dark region in the vicinity of the
TQPT (dashed line at the lower Vz) in Figs. 4.3-4.14. Since this region is characterized by a
large number of states, the numerical treatment of this region requires the generalized Meir-
Wingreen formalism described in Sec. 4.2 of this work. One can use this small dark region as a
signature to distinguish the ABS regime and the MBS regime, as seen in Fig. 4.11(a). But we can
barely pinpoint this small dark patch generically because this signature may be confused with a
larger range of dark conductance arising from localized ABSs at the two ends as seen in other
CB conductance numerical results, i.e., Panel (a) of Figs. 4.3, 4.5, 4.6, 4.8, 4.9, 4.10, and 4.14.
Furthermore, the soft-gap effect in the MBS regime together with certain ABS wave-function
141
profiles could obscure the darkness associated with the TQPT patch, making the presence of
MZM challenging to distinguish, as in Fig. 4.2(a). This further reinforces our earlier comment
that CB conductance studies may not be a good experimental technique to discern MBS from
ABS.
A separate puzzle that is not immediately resolved by our numerical treatment is the relative
intensity between the two bright regions. The first 2e periodic bright region results from elastic
cotunneling of Cooper pairs and the second one arises from tunneling of electrons. Thus, one
expects the second bright region to be brighter than the first region [115], which is quite different
from what is seen in experiments [91,113,114]. A potential resolution of this puzzle is provided in
this work by details of the proximity-induced semiconductor structure. Specifically as described
in Sec. 4.3.1, the normal-state CB conductance at high fields, which is equal to the normal-state
conductance, is suppressed in semiconductor structures if the local density of states is suppressed
away from an ABS resonance in the semiconductor at low densities. Such a suppression does not
affect the virtual elastic cotunneling process since a non-resonant ABS can contribute to Cooper-
pair tunneling. Thus, the inclusion of the transmission resonance associated with an ABS can
explain the enhanced 2e periodic Cooper-pair tunneling. Resolution of this puzzling brightness
paradox in the CB experiments is one of the major conceptual achievements of our theory. Our
explanation, however, further reinforces the dominance of trivial ABS over topological MBS in
the existing Majorana nanowire experiments.
142
4.6.2 Suppression of Normal Coulomb Blockade Peak Relative to ABS and
MBS
The results for the CB conductance in Figs. 4.3, 4.5-4.8 in Sec. 4.5 show 1e periodic con-
ductance both from MBSs in the topological region (i.e., VTQPT < Vz < Vc) as well as normal
metallic CB for Vz > Vc. In all these cases, the MBS conductance peak, being resonant follows
Eq. (4.46) and is higher than the normal-state CB conductance described by Eq. (4.48). In prin-
ciple, this makes the bright 1e periodic conductance from MBS even brighter relative to the 2e
periodic conductance mentioned in the previous subsection. Additionally, such a difference in
brightness between the MBS and normal conductance is not seen in experiments [91, 113]. Both
these issues get resolved in Fig. 4.2, where we have included the effect of a soft-gap. Such a soft-
gap arises from the interplay of disorder and magnetic field on the parent SC [138]. Interestingly,
the presence of large oscillations from the ABS states prior to the TQPT suggests the absence of
disorder-induced subgap states below the TQPT.
4.7 Summary
We have developed a theory for and numerically calculated the two-terminal conductance
of a Majorana nanowire in the CB regime, including all the important realistic effects, such as
the soft-gap, SC proximity effect, temperature, nanowire length, SC collapsing field, chemical
potential, QDs, self-energy, SC states, and metallic states. The realistic model for the wire used
in our work, in certain parameter regimes such as near the TQPT or beyond the critical Zeeman
field Vc, contains a large number of low-energy states. In order to compute CB conductance in
143
this region, we have derived a generalized Meir-Wingreen formula, which is based on assuming
the tunneling rate is lower than the equilibration rate in the nanowire. This assumption reduces
the complexity of the rate equation formalism from the exponential [116] to linear in the number
of low-energy levels. However, the assumption requires an equilibration process that might not be
very efficient in the limit of a few levels. We have discussed the resulting differences in Sec. 4.3.2.
Our calculation also entirely focuses on the 1e tunneling regime and we have only provided
analytic estimates for the Cooper-pair tunneling regime relative to the normal-state conductance.
The normal-state conductance seen in the high Zeeman field regime of our numerical results thus
provides a calibration scale to compare the experimental results.
Our optimal results are summarized by Fig. 4.2(a), which shows an example of the CB
conductance that is closest to the experimental results. While this plot excludes the 2e periodic
Cooper-pair tunneling part of the transport seen at low Zeeman fields as the brightest feature in the
experiments [91,113,114], our CB conductance plot shows a range of Zeeman field, which starts
with darkness followed by a bright region, similar to experiments [91, 113, 114]. By comparing
with the energy spectrum shown in Fig. 4.2(c), the dark region in our simulations arises from
ABSs, in contrast to the mechanisms studied in earlier works [115]. The CB conductance be-
comes visible when the ABSs approach zero energy and remains bright in the topological region,
where the conductance is due to MBSs, except for a dark patch near the TQPT. At large enough
Zeeman field, the SC is driven to normal-state and the CB conductance peak from MBSs crosses
over to the 1e periodic CB peak of a normal metal. While the normal-metal CB peak is higher
than the dark region from ABSs, it is substantially (in all results, except for Fig. 4.2) weaker than
the resonant CB peaks for MBSs. We have verified that resonant CB peaks, such as those from
MBSs and ABSs, can be distinguished from normal-metal and TQPT-related CB peaks by their
144
temperature dependence. The difference in intensities between the MBS and normal CB regime,
which is not seen in experiments [91, 113], is suppressed in Fig. 4.2 by the introduction of a soft
gap, i.e., subgap density of states in the SC that is introduced by semiconductor disorder. The
complete bright-dark-bright feature is not naturally included in our numerical simulations since
the first bright feature in the experiments results from 2e periodic Cooper-pair tunneling. How-
ever, we have argued in Sec. 4.3.1 that resonant transmission features associated with the barrier
potential can suppress the normal CB conductance (at high Zeeman field) relative to the 2e peri-
odic Cooper-pair conductance (at low Zeeman field). Thus, we find that the intensity pattern seen
in experiments can be matched by a specific nanowire model in our generalized Meir-Wingreen
formalism.
Aside from the intensity fluctuations, the positions of the CB peaks that deviate from 1e
periodicity provide interesting spectroscopic information, such as MBS splitting [91, 115]. In
fact, the observation of such breaking of 1e periodicity allows one to verify the absence of quasi-
particle poisoning by low-energy subgap states [139]. Following Albrecht et al. [91], we have
characterized the positions of the peaks through the OCPS [plotted in panel (d) of Figs. 4.2-4.14,
except Figs. 4.4 and 4.7]. We find that in the case of ABSs at both ends of the wire, the first lobes
of the OCPS result from a combination of both ABSs. This is distinct from tunneling conductance
at a single end, which is sensitive to the spectrum only at one end. We find that this model of
ABSs at both ends produces OCPS that decreases with increasing Zeeman field. The OCPS
arising from MBSs [91, 115, 116], which increases in amplitude with increasing Zeeman field, is
suppressed by the inclusion of self-energy from the proximity-inducing SC. By comparing OCPS
from models with and without QDs, we find that QDs are necessary to obtain oscillations that
decrease with increasing Zeeman field.
145
We emphasize that the experimentally claimed ?exponential protection? in Refs. [91, 114]
may be a misleading artifact of the data being taken at very few samples with each sample having
its own set of ABS dominating the CB transport. Our work shows that because of the non-
universal nature of ABS dominating CB tunneling transport in the currently available SC-SM-
QD nanowire samples, there is no universal length dependence in the CB physics. We believe
that the strong length dependence in the claimed experimental data can be an artifact of the few
samples considered in each of the systems [91, 114]. Since each sample has a totally different
parameter set (and not just different length), such experiments can tell us absolutely nothing about
the intrinsic length dependence of Majorana physics since all system parameters are varied along
with the wire length in such experiments.
To sum up, the qualitative features, i.e., bright-dark-bright intensity patterns as well as de-
creasing OCPS of CB oscillations in Majorana nanowires, can be understood in terms of our
semi-realistic model for the superconductor-semiconductor structure. We found that ABSs at
both ends, self-energy and soft gap are all necessary ingredients to explain the features of the
experiment. We find that the CB conductance has certain distinct advantages over direct tunnel-
ing conductance. Specifically, the CB conductance is sensitive to the lowest-energy states, unlike
single-end tunneling conductance which picks up signal from all states. Furthermore, the CB con-
ductance is sensitive to the delocalization of states. Away from zero energy, conductance through
localized states is suppressed. An experimental characterization of the temperature dependence
of the CB peak intensities would be important to verify the characters of the states contributing
to the conductance. An important point to keep in mind in this context is, however, the fact that
our current zero-bias conductance theory indicates that CB conductance measurements may not
be particularly useful in distinguishing topological MBS signatures from trivial ABS signatures.
146
Chapter 5: Conclusion
Over the last decade, the one-dimensional nanowire of superconductor-semiconductor hy-
brid structure has attracted a lot of condensed matter physicists because the fault-tolerant Majo-
rana zero modes (MZMs) existing in this nanowire can facilitate the development of topological
quantum computer. Since the theoretical model was proposed in 2010, a deluge of experiments
has been reported in search of MZMs. While this field is active, and Microsoft Corporation also
invests in the Majorana nanowire research, the current experimental results are not solid enough
to prove the existence of MZMs as feasible topological qubits. This thesis intends to provide sev-
eral methods through theoretical calculations and arguments to help the experiments distinguish
the topological MZMs.
In Chapter 1, we introduced the Majorana zero modes from the famous toy model, Kitaev
chain, to show the properties of MZMs. We also introduced the one-dimensional superconductor-
semiconductor nanowire model, which is the main focus of this thesis. Later, we elaborated on
the MZM signatures one should observe when the Majorana bound states exist in the nanowire.
Finally, three topic-related experiments, i.e., earliest observed ZBCP, quantized conductance,
and Coulomb blockade experiments, that motivate the theoretical works in this thesis are also
presented.
In Chapter 2, we carried out the standard QPC tunneling conductance simulations with
147
various potentials that lead to ?good?, ?bad?, and ?ugly? ZBCPs. Only the ?good? ZBCPs arise
from the genuine MZMs, but the other two kinds of trivial ZBCPs can still resemble the quanti-
zation of MZM-induced ZBCPs. We proposed a new quality factor F to quantify the robustness
of quantized ZBCP over the normal-state conductance GN . This quality factor F is useful to
the conductance data because Gz (zero-bias peak) versus GN is the typical way to show the
experimentally-measured data. Since the quantization of ZBCPs is intrinsic for MZMs at low
temperatures, small values of F should quantitatively reflect those fine-tuned quantization in-
duced by trivial subgap bound states. The conclusion of this work is that we can use this metric
F to distinguish the MZM-induced ZBCPs (F ? 2) from other topologically-trivial ZBCPs, with
large SC gaps at low-temperature limits.
In Chapter 3, we demonstrated the numerical results of local conductance from each end
of the wire in a Majorana nanowire with one QD embedded near the wire end. Comparing the
conductance data obtained from both ends, the one that shows the end-to-end ZBCP correla-
tion, meaning ZBCPs are showing up on both ends, indicates a pair of MZMs is present in the
nanowire. In contrast, the one that does not show end-to-end ZBCP correlation, meaning there
could be a ZBCP from one end but not the other, indicates this ZBCP does not come from MZMs.
In the model we studied, this one-end ZBCP comes from the quantum dot located at the same
end. Apart from comparing local tunneling conductance, the same three-terminal system can
support cross conductance measurements, which should help identify the TQPT field with the
bulk gap closing and reopening signature if the external QD is absent. We suggest the experi-
mentalists carry out the end-to-end conductance correlation measurements and cross conductance
measurements together as cross checks for the appearance of MZMs.
In Chapter 4, we studied the Majorana nanowire in the Coulomb-blockade limit by deriv-
148
ing the generalized version of the Meir-Wingreen formula that applies to a two-terminal system
and numerically calculating the CB conductance with all the important realistic effects such as
soft gap, self-energy, QDs...etc. Assuming the tunneling rate is below the equilibrium rate of the
nanowire, the Meir-Wingreen formula is reduced from exponential complexity to linear complex-
ity, which is at the same level as the QPC tunneling conductance calculation. This computational
advantage allows us to include thousands of states in the normal-metal regime or above the SC
gap, which should contribute to the CB conductance realistically. With this new theoretical tool,
we simulated the experimental features [91, 113, 114], including the ?bright-dark-bright? pattern
and the ?decreasing OCPS as a function of Zeeman field?, and explained the mechanisms behind
these features. At the same time, we argued that the length-dependent OCPS, which indicates
topological protection from MZMs [91], is not consistent with the decreasing OCPS as the Zee-
man field increases. In the model we studied, the decreasing OCPS with increasing Zeeman field
comes from the multiple subgap Andreev bound states, with self-energy, long wire, and early SC
collapsing field suppressing the Majorana oscillations. Since the QD-induced ABSs do not have
length dependence, the first amplitude of OCPS should be wire length-independent.
To sum up, we have investigated the non-local transport signatures (i.e., end-to-end conduc-
tance correlation and CB conductance) and the robustness of ZBCPs (i.e., quality factors) by sim-
ulating the one-dimensional SC-SM model with various realistic effects. Given that topologically-
trivial ABSs and disorder-induced subgap states can mimic several signatures of MZMs, we par-
ticularly compare the numerical results coming from purely MZMs and results from ABSs or
disorder-induced subgap states. We are sure these comparisons can provide realistic references
to the experiments, and our theoretical proposals should help experiments anchor the Majorana
zero modes in the near future.
149
Appendix A: Detailed Derivations for Coulomb Blockade Transport
A.1 Microscopic Tunneling Rates
The rate of absorption of electrons from lead ? = L,R is given by
? ? ? ?
??
2
N = ?? PN(Ei) df(? ??)?(Ej ? Ei ? ) ???j|c? ??|?i?
?i,j ? ? (A.1)2
= ?? PN(Ei)f(Ej ? E ?i ? ??) ??j|c??|?i?? ,
i,j
where f() = (1 + e?)?1 is the Fermi function.
Similarly, the rate of emission of electrons into lead ? is given by
? ?
??N = ?? PN(Ei) d [1? f(? ??)] ?(Ej + ? Ei) |??j|c?|? ?|
2
i
?i,j (A.2)
= ?? PN(Ei) [1? f(Ei ? Ej ? ??)] |??j|c?|?i?|2 .
i,j
This equation can be simplified and related to the coefficient ??N by interchanging the indices i, j
150
and considering the charge state (N + 1) as
? ?
?? = ? P (E ) [1? f(E ? E ? ? )] ??? |c? |? ???2N+1 ? N+1 j j i ? j ? i
?i,j ? ?
= ?? P ?N+1(Ej)f(Ei ? Ej + ??) ??j| 2c??|?i??
i,j
Z ????? N ? ?2= ??e PN(E )e??(Ej?Ei???)i f(Ei ? Ej + ??) ???j|c? |? ??i (A.3)
Z ?N+1 ?i,jZN ? ?2
= ??e
???? PN(Ei)f(?Ei + E ? ? ? ? ?j ?) ??j|c
Z ?
|?i?
N+1 i,j
= e???
Z
? N ??
Z N
.
N+1
A.2 Steady-state Probabilities
The rate of the system leaving the state N must match the rate that is entering into the state
N . This leads to the equation
? ?
P0,N (?
?
N + ?
?
N) = (P
? ?
0,N?1?N?1 + P0,N+1?N+1). (A.4)
? ?
Substituting ? in terms of ?, the equilibrium condition becomes,
?
? Z
?
N?1 ZN
P0,N (?N + e
???? ??N?1) = (P
? ???? ?
0,N?1?
Z N?1
+ P0,N+1e ?N). (A.5)
? N
Z
? N+1
Collecting the rates ??N , the equilibrium condition becomes
? [ ] ? [ ]
??
ZN ZN?1
N P ? P e???? = ?? P ????0,N 0,N+1 N?1 0,N?1 ? P0,Ne . (A.6)ZN+1 Z? ? N
151
Defining ?? ?NZN = ??N and P0,N/ZN = P?0,N , the steady-state condition simplifies to
? [ ] ? [ ]
???N P? ? P? e???? = ??? P? ? P? e????0,N 0,N+1 N?1 0,N?1 0,N . (A.7)
? ?
The above equation is solved by the detailed balance condition, where both sides of the above
equation vanish so that
(? )? ?1 ( ? [ )? ? ] ?1??? e ??? ??? e ?(?? ?) ? 1
P? ? N ?? ? N0,N+1 = P?0,N ? = P?0,Ne 1 + ? . (A.8)
? ??
? ?
N ? ??N
In the limit of a small applied voltage ?? = ?+ V? and expanding to linear order in V?,
( ? )
?
?? ? ??NV?P?0,N+1 = P?0,Ne 1 + ? ? . (A.9)
???? N
Solving the recursion, we get
( ?? )?
P? = Z?
?? V
1 ?N? j ?
0,N tot e 1 + ? ?? . (A.10)
???
j<N ? j
In equilibrium, if V? = 0, the above equations become
(eq)
P? ?1 N??0,N = Ztot e , (A.11)
? ? ?where Z = Z e N??tot N N (noting that (eq)N ZN P?0,N = 1).
152
A.3 Current and Conductance
The current at the left lead is given by
?
I = P L L0,N(?(N
? ?N)
?N )
= P L ???
Z
L N?1 L
0,N ?N ? e ?N?1
?N ( ZN ) (A.12)
= P? ??L ? e???L L0,N N ??N?1
?N ( )
= ??L P? ? e???LN 0,N P?0,N+1 .
N
This current vanished at V? = 0 because the combination of probability factors vanishes. Since
the probability factor vanishes to linear order, the conductance can be extracted by expanding this
factor to linear order in V?:
P? ? e???L0,N P?0,N+1 = P?0,N [? (1? ?VL)e??(?P?0,N+1?? )]???? NV?= P?0,N (1? (1? ?V ) 1 + ?? L ) ? ???N
???? V?P? ? N0,N (VL ?? ? ???? N ) (A.13)
? ? ????N(V? ? VL)?P?0,N ( ? ???N )
??R? N(VR ? VL)= ?P?0,N ?
? ??
?
N
153
Writing V = VR ? VL, the conductance is written as
dI ?| ??R ??LG = N NV=0 = ?? P?0,N
dV ??R? N + ??
L
N N
?R?L
= ?? P N N0,N (A.14)
? ?
R
N + ?
L
N N
?R L
= N
?N ,
?RN + ?
L
N N
where ??N = ??P ?0,N?N .
In the above expression, we use the equilibrium tunneling rate
? ? ?
??N = ?
2
???Z
?1 e?N?tot ZNPN(Ei)f(Ej ? Ei ? ?) ??? |c? ?j ?|?i?
? i,j
= ??? P (E )f(E ? E ? ?) ?? ? (A.15)? i j i ??j| 2c? |? ?? i? .
i,j
Here we have used the identity we showed earlier, i.e.,
e?N? e??(Ei?N?)
P (Ei) = PN(Ei)P0,N = PN(Ei)ZN = . (A.16)
Ztot Ztot
Using the identity
??Ei
e??N?
e
P (Ei)f(Ej ? Ei ? ?) =
1 + e?(Ej?Ei??)
1
=
e?Ei +[e?(Ej??)? ] (A.17)= ?? 1 e??Ei + e??(Ej??) f ?(Ej ? Ei ? ?)
= ???1e??N? [P (Ei) + P (Ej)] f ?(Ej ? Ei ? ?),
154
the above conductance can be written in the Meir-Wingreen form
?
? ?? ? ??? 2?N = ?? [P (Ei) + P (Ej)] f (Ej ? Ei ? ?) ??j|c?|?i? . (A.18)
i,j
A.4 Meir-Wingreen Formula for One-terminal CB Majorana nanowire
The original Meir-Wingreen?s formula is the Landauer formula, when we consider the
transport through an interacting region, with one terminal connecting the system and the en-
vironment [126]. The general Hamiltonian is of this form
? ?
H = k?a
? ? ?
k?ak? +Hint({cn}; {cn}) + (Vk?,nak?cn + H.c.), (A.19)
k,??L,R k,n,??L,R
where a?k? (ak?) creates (destroys) an electron with momentum k in channel ? from either left (L)
or the right (R) lead, and {c?n} and {cn} form a complete and orthonormal set of single-electron
creation and annihilation operators in the interacting region. The channel index includes spin and
all other quantum numbers which, in addition to k, are necessary to define uniquely a state in the
leads.
Through the Keldysh formalism, one can get the linear-response conductanceG in the form
[ ]
e2 ? ?
G = ?n,m(Ej ? Ei) (Pi + Pj) ?
?feq(Ej ? Ei) ?? ?j|cn|?i???i|cm|?j?, (A.20)~ ?
m,n i,j
where the ?i are eigenstates, with energies Ei, of the uncoupled interacting region, and Pi is
the equilibrium probability of state ?i. For non-interacting electrons, one can choose the c?s to
correspond to single-particle eigenstates of the uncoupled system, and the overlap factor in each
155
term, ??j|c?n|?i???i|cm|?j?, is trivially 0 or 1.
Now we apply our system to the Meir-Wingreen?s formula in Eq. (A.20). In our case, we
consider only m = n = {L,R}?{?, ?}, where the transmission only occurs on the left and right
ends of the nanowire, and the tunneling factor is independent of the energy (setting ?n,m = 1),
then Eq. (A.20) becomes
e2 ? ?? [ ]?feq(Ej ? Ei)
G = (Pi + Pj) ? |??j|c? 2~ ? x,?
|?i?| , (A.21)
x=L,R ?=?,? i,j
where |?i? = |{ni}? is the state with some quasi-electron configuration over Nl energy levels,
and ?
Ei = nss + U(N) (A.22)
s
is the energy of configuration {ni} with electrostatic energy U(N) = Ec(N ? n )2g . ns is the
occupation number of s state of the quasiparticle, i.e., ns is the eigenvalue of the operator d?sds
w?hile d
?
s and ds are creation and annihilation operators for the quasi-particles. (Note that N =
s ns + 2Nc is the total electron number and Nc is the additional number of the Cooper pairs
away from the charge-neutral point as the gate voltage of the nanowire is zero, and ng is the
number of electrons corresponding to the gate voltage.) s is the eigen-energy of s state of the
Hamiltonian as Eq.(4.54). The probability for quasi-particle distribution configuration {ni} is
described by the Gibbs distribution
[ (? )]
P = e??Eii /Z = Z
?1 exp ?? nss + U(N) , (A.23)
s
156
where the canonical partition function is given by
? ? [ (? )]
Z = e??Ei = exp ?? nss + U(N) . (A.24)
i i s
Due to fermion parity conservation, we only need to separate the total electron number N
into even and odd groups. The superconductor ground state favors the even number of electrons
due to the condensate. If one more electron adds in the superconductor and overcomes the su-
perconducting gap, then there will be one extra quasiparticle besides the condensate. Otherwise,
the condensate ground state will remain and only allow 2e transport. Therefore, the behaviors of
even parity and odd parity for this superconductor-induced nanowire are different. We only need
to discuss the parity of the total electron number N , i.e., we discuss only the two values from (N
mod 2). To simplify the formula later, we can rewrite Eqs. (A.23) and (A.24) as
[ (? )]
P = Z?1
1
i exp ?? nss + Qi ??U (A.25)
2
s
and ? [ (? )]
Z = exp ? 1? nss + Qi ??U , (A.26)
2
i s
?
where Qi = Q0 ? (?1) s ns is the fermion parity (note that Q0 = ?1 is the ground-state
fermion parity, and the subscript i means the quasi-particle distribution configuration.), and
?U = U(N) ? U(N ? 1). The parity of N and (N ? 1) is opposite. Therefore, we can
shift U(N) and U(N ? 1) by the mean value [U(N ? 1) + U(N)] /2 and express them with the
parity Qi and the electrostatic energy difference ?U . Equation (A.26) can be further simplified
157
as
? [ (? )]1
Z = exp ?? nss[+ Q2?{ni}? s (
i?U
? )]1
= ?(Q?Qi) exp ?? nss + Q?U
2
?Q {n?i} [ s? ] [ (? )]1 1
= 1 +Q ?Q (?1) s ns0 [ exp ?? nss + Q?U ] (A.27)2 2Q?=?1 {ni} ( ) s1 ??Q ??U ?( ) ?( )= exp 1 + e??s +Q ?Q ? 1? e??s0
? 2 2Q=?1 s [ s ( )]1 ? ?
= e??(Q??U/2) ? (1 + e?? ?s s) 1 +Q ?Q0 ? tanh .
2 2
Q=?1 s s
The energy of configuration {ni}, i.e., Ei [the original definition is in Eq. (A.22)], can also be
shifted and redefined as ? Qi?U
Ei = nss + . (A.28)
2
s
Since the electron operator is of the form as Eqs. (4.56) and (4.57), the quasi-particle cre-
ation and annihilation operators can only change the orbital occupation number by one, say in
the pth orbital. Hence, the eligible final transition state can only be the same configuration with
one orbital occupation changed, i.e.,
|?j? = |{nj 6=p, n?p}? = (1? n ?p)(dp|{ni}?) + np(dp|{ni}?), (A.29)
with np = 1 (np = 0) if the pth orbital is occupied (empty) in the configuration {ni}. Then the
158
energy difference between these two states is
Ep = Ej ? Ei = (1? 2np)p ?Qi?U. (A.30)
( The transition matrix eleme)nt is therefore ?? |c
?
j x?|?i? ? ?{nj}|c?x?|{ni}? = ?{n ?i=6 p, n?p}|cx?|{ni}? =
(1? np)?{ni}|dp + np?{ni}|d?p c?x?|{ni}?. With Eqs. (4.56) and (4.57),
??j|c?x?|?i? = (1? n ?p)up,x? + npvp,x?. (A.31)
Hence,
|??i|c? |? ?|2 = (1? n )?x,? + n ?x,?x? j p p p p , (A.32)
where
Ax,?p = |u
2
p,x?| (A.33)
is the tunneling rate for the electron to tunnel from the lead at x to the nanowire (same for the
opposite direction), and
Bx,? = |v |2p p,x? (A.34)
is the tunneling rate for the hole to?tunnel from the lead?at x to the nanowire (same for the
opposite direction). Note thatAx = Ax,? andBx = Bx,?p ?=?,? p p ?=?,? p as defined in Eqs. (4.35)
and (4.36).
159
With the ingredients above, we can rewrite Eq. (A.21) as
( )[ ]
e2 ? ?? ? Pj ?feq ((1? 2np)p ?Qi?U) [ ]
G = P 1 + ? ? (1? n )Ax,? + n Bx,?
~ i p p? Pi ?
p p
x=L,R ??=?,? ie2 ? p
= Pi (1 + exp [?? ((1? 2np)p ?Qi?U)])~
x={L,R i p
? ?f ?
} [ ]
eq ((1? 2np)p??Qi?U) (1? n )Axp p + npBx[ (p
e2 ? ?? ?
?
= ?? ? ? )]?Z? ?1 ? 1exp ?? nss + Q?U ?~ ? 2 ?x=L,R p n=0,1 Q=?1,1 ? i:n=np sQ=Q0?(?1) ns ?? ?
[ Fp(n,Q)? ] [ ]? (1 + exp [?? ((1? 2n)p ?Q?U??)]) ? ?f ?eq ((1? 2n)p ?Q?U)? ? (1? n)Axp + nBxp
e2 ? ?? ? =??feq((1?2n)p?Q?U) [ ]
= ? f?eq((1? 2n)p ???Q?U) ? Fp(n,Q?) ? (1? n)Axp + nBxp .~
x=L,R p n=0,1 Q=?1,1
F?p(n,Q)
(A.35)
Fp(n,Q) sums over all the possible configurations, but with the constriction that n = np and
160
?
Q = Q0(?1) s ns being selected properly, i.e.,
? ( ? ) ( ? )
F (n,Q) = Z?1e??Q?U/2p ?(n? np)? Q?Q0(?1) ns exp ?? n(ss{ns} ? [ s? ] ? )
= Z?1e??Q?U/2e??np ?1 1 +QQ ? ?1?) s=6 p ns0 (?1)np? exp ?? nss[ 2{ns=6 p} ] s 6=p
1 ? ? ?
= e? ?
( ? ) ( )?Q?U/2e ?np 1 + e ?s +QQ0(?1)np 1? e??s
2Z [?s=6 p ?s 6=p? ]1 (1 + e ?s) ??s
= e??(Q?U/2+np) s ? [ +QQ0(?
n s(1? e )1)
2Z ?(1 + e ?p) (1? e???p) ]1
= e??(Q?U/2+n ) ??
1 ? n s tanh (?s/2)p[ (1 + e
s) ? +QQ? 0( 1)2Z 1 + e p 1? e??p
s ? ]
e??(?Q?U/2+np) 1 [ ?? tanh(? /2)??p +QQ0( 1)n s(1+e ) (1(?e??= )p]s)
e??(Q?U/2) ?sQ=?1 1 +QQ0 s tanh 2
(A.36)
Note that the star ? part in Eq. (A.36) is technically a delta function for Q when we sum over
(?1) and 1.
1 [ ? ]
f ? 1 +QQ (?1) s=6 p ns0 (?1)np . (A.37)
2
If Q is the correct parity, i.e.,
? ?
Q = Q0(?1) s ns = Q0(?1) s=6 p ns(?1)np , (A.38)
?
then f = (1 + 1)/2 = 1. On the contrary, if Q = ?Q0(?1) s ns (not correct parity), then
f = (1? 1) = 0. So only Q that represents the correct fermion parity is picked and evaluated.
161
We can actually simplify Eq. (A.36) further.
? ( )? ? ( )
s tanh
s
2 tanh (?p/2) ?s
? = ? ? tanh1? e ?p e ?p/2 (e?p/2 ? e ?p/2? ) 2s 6=p( )
= e? /2
tanh (?p/2) ?p stanh
2 sinh (??p/2) s=6 p( ) 2e?p/2 ?s
= tanh (A.39)
2 cosh (?p/2) 2
s=6 p ( )
e?p/2 ? ?s
= tanh
e?p/2 + e??p/2? s 6=p( ) 21 ?s
= ? tanh1 + e ?p 2
s 6=p
Then, the Fp(n,Q) factor is simplified to be
[
? ? ( )]e ?(Q?U/2+np) 1 +QQ (?1)n 1 tanh ?s
(1+e??p ) 0 (1+e??p ) s=6 p 2
Fp(n,Q) = ? [ [ ? ( )]e??(Q?U/2) ?1 +QQ ( ta)n]h ?sQ=?1 0 s 2
e?
= ??(Q?U/2+np)??p 1 +Q[Q (?1)n ?s(1+e ) 0 ?s 6=p tanh( 2)] (A.40)
Q=?1 e
??(Q?U/[2) 1 +QQ0 s t?anh ?s2 ( )]? e??(Q??U/2+np) 1 +QQ0 ?[1)n ?ss=6 p= ?tanh 2 ( )] .
e??(Q?U/2+np) ?sQ=?1 n=0,1 1 +QQ0 s tanh 2
For numerical evaluation [in MATLAB, tanh(x) gives 1, when x is above?some threshold, and
this causes the conductance computation unable], it is necessary to replace s tanh(?s/2) with
(1? e?y), where ( ? )
y = ? log 1? ?stanh( ) (A.41)
2
s
162
is evaluated by the identity
?2x
tanh(x) = 1? 2e?[2x+log(1+e )]. (A.42)
163
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