Thesis on Topological Insulator

8/15/2019 Thesis on Topological Insulator

1/250

Discoveries of New Topological States

of Matter Beyond Topological

Insulators

Su-Yang Xu

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Advisor: M. Zahid Hasan

January 2015


2/250

c Copyright by Su-Yang Xu, 2014.All Rights Reserved


3/250

Abstract

The discoveries of new forms of matter have been so definitive that they are used to

name periods of the history of mankind, such as the stone age, the bronze age, and

the iron age. Although all matter is composed of component particles, particles can

organize in various ways leading to different states or phases of matter. Finding all

possible distinct states that matter can form and understanding the physics behind

each of them are fundamentally important goals in physics research. Meanwhile, the

discoveries of new phases, such as superconductivity, ferromagnetism, also usually

lead to new technologies, which benefit our society. It had been long believed in

physics that all phases of matter (solid, liquid, gas, superconductivity, etc) are due

to the breaking of certain symmetries and can be described by local order parame-

ters. However, the discovery of the topological phase of matter provided a striking

exception that cannot be described by a local order parameter. A topological phase

fundamentally differs from a conventional material in that a topological phase of mat-

ter is described by a nontrivial topological number, which is a global quantity based

on the bulk electronic wavefunction throughout the momentum-space Brillouin zoneof a crystal. The nontrivial topological number guarantees the existence of protected

surface states, which are of interest because these surface states are insensitive to

smooth changes in material parameters and cannot be removed unless the system

passes through a topological quantum phase transition. In 2007, the 3D Z 2 topo-

logical insulator (TI) state was identified in bismuth-based materials. A 3D TI is a

band insulator in the bulk but features spin-polarized Dirac electronic states on its

surfaces. This discovery has tremendously accelerated the research field of topologi-

cal matter. Not only the 3D Z 2 TI phase itself has attracted huge research interest,

but also, more importantly, it has inspired theoretical predictions for a range of new

topological phases of matter. The primary examples include a topological crystalline

insulator, a topological Dirac/Weyl semimetal, a topological Kondo insulator, a topo-


4/250

logical superconductor, etc. Each of them is predicted to exhibit protected surface

states with a unique set of topologically protected properties. Experimentally identi-

fying these new phases in real materials can tremendously deepen our knowledge of

the physics of topological matter. Moreover, utilization of the protected properties in

topological surface states can lead to future applications such as fault-tolerant topo-

logical quantum computers and low-power spintronic devices, which will revolutionize

our electronic and energy industry.

In this thesis, we present the experimental identifications of several new topolog-

ical phases of matter beyond the Z 2 TI. In Chapter 1, we present the key concepts

for topological phases and the relevant theoretical formulations. In Chapter 2, we

provide a description of the experimental technique of spin-resolved angle-resolved

photoemission spectroscopy (spin-ARPES) and the methodology of using it to probe

the topological number of a 3D topological phase. In Chapter 3, we present our exper-

imental realization of a topological quantum phase transition in the BiTl(S1−δSeδ)2

system, which demonstrates the way a topological phase of matter can arise from an

unremarkable band insulator. In Chapter 4, we demonstrate the observation of the

topological crystalline insulator phase in the Pb1−xSnxTe(Se) systems, where novel

topological surface states protected by the crystalline mirror symmetries and the as-

sociated mirror topological number are experimentally observed. In Chapter 5, we

describe the realization of a topological (Dirac) metal phase in Na3Bi and Cd3As2,

where certain topological state can be stabilized even if the bulk energy gap closes.

We report the observation of Fermi arc surface states, - a new type of 2D electron

gas, revealing the novel topological nature in a topological metal. In Chapter 6,

we present our studies on the topological Kondo insulator candidate SmB6, where

the bulk Kondo gap replaces a simple Bloch gap in a Z2 TI, potentially introducing

electronic correlation to the surface states. Finally, In Chapter 7, we present our

demonstration of superconductivity and magnetism in the topological surface states

iv


5/250

in a prototype Z 2 TI Bi2Se3, which are the keys to realizing a topological supercon-

ductor or a Chern insulator. The works in this thesis constitute the realization of

a number of new topological phases of matter beyond a Z 2 TI state that are being

studied by many groups worldwide. With these discoveries, the world of topologi-

cal matter gradually unveil itself. And we have many reasons to believe that this is

just the tip of the iceberg and there are many more of them with yet more exciting

properties that await their discovery.


6/250


7/250

Acknowledgements

First and foremost, I would like to thank my advisor Prof. M. Zahid Hasan for

his mentorship and support. He has been a constant source of encouragement and

advice throughout my graduate studies. I always remember his words: “The idea

is to keep trying. Everyone will face failure. But if we keep trying, on average we

will do fine.” Every time I run into difficulties in my experiments or in fighting with

the reviewers to get a paper published, these words encouraged me. Indeed, now I

understand that there is no sure path to success, and the key is to “keep trying”. I

also benefited tremendously from his ability to understand and explain the essences

of physics without bringing many formations, and his insights in condensed matter

physics, especially when it came to looking out for the next exciting topics in our

field. This is one the main driving forces to maintain our group in the front line in

the fierce competition. Moreover, I am grateful for the freedom he gave me and also

every other member in our group to pursue our individual interests in our research

work. He allowed us to come up with our own research projects, design our own

experiments and consult experts in relevant areas such as ARPES technique, material

growth and condensed matter theory. I sincerely thank Zahid for every single aspect

of my graduate research.

I am also indebted to my colleagues and friends in my group. I wish to thank David

Hsieh, Dong Qian, and Matthew Xia for introducing me into the group and helping

me settle down. Andrew Wray taught me how to use ARPES and was extremely

generous as a teacher when I started to work on ARPES on topological insulators.

Madhab Neupane, Chang Liu and Nasser Alidoust have been wonderful teammates

in experiments, and Madhab and Chang have also been great friends in my personal

life. I thank Ilya Belopolski for his infinite curiosity when it came to new physics and

for always being unsatisfied with the answers that I provided to his questions. This

drove me crazy but lead to many new and exciting ideas. I thank Guang for being


8/250


9/250

were always ready to help and sent the calculation results on short notices. The

timely response from our sample growth and theoretical calculation collaborators

was crucially important in the competitive research.

I have also been very fortunate to be able to learn from world-renowned scientists

at Princeton Physics. I thank Chen Fang for all the discussions we had in his office, at

lunch or dinner, while we were walking from the parking lot to the department or at

so many random places and random times where we suddenly had some inspiration.

Chen almost single-handedly taught me the physics of topological phases beyond Z 2

topological insulators. The discussions with him led to many fruitful ideas and new

results. I thank Prof. Phuan Ong for agreeing to be second reader on my thesis and

for his kind advice. And I thank Prof. Jason Petta and members of his group from

whom I learned a great deal during my experimental project.

Only people who have experienced the “synchrotron life” would understand the

meaning of “constant night shifts”. I cannot imagine myself going through all the

physical and mental hard times without my friends. I thank my Princeton friends

Jingke Xu, Bo Yang, Jun Xiong, Ke Wang, Zhizhen Zhao, and many others. I also

thank my friends back in China. Especially I want to thank Hei (Liang Zhang) for

being an honest, generous, reliable friend, for bearing my complaints, for comforting

my pains, and for sharing my happiness.

I thank my dear sister Chujun for being connected heart to heart. Wherever she

is and however the world changes, I wish her happiness in her life. I thank Cindy

for her wonderful cooking during “the SF time”. I thank Jingjing and Tian for their

accompany and education. I thank my angel girl momo for her warm care, for her

positive thinking and encouragement, and for always sticking with me to share my

thoughts and feelings.

Finally but most importantly, I am extremely grateful to my family, my grand-

parents Ruizhi and Ruilan, my mom Xiaobei, my dad Yuping, my aunt Hong, and


10/250

all family members for all their selfless love and care. Family is my forever source of

happiness.

x


11/250

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction 1

1.1 The band insulator state and the band inversion . . . . . . . . . . . . 1

1.2 Previously discovered topological phases. . . . . . . . . . . . . . . . . 5

1.2.1 The integer quantum Hall state and the Chern insulator state 5

1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator 7

1.2.3 The 3D Z 2 topological insulator . . . . . . . . . . . . . . . . . 11

1.3 Theoretical formulations . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.1 The Chern number . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.2 The Z 2 topological invariant in two-dimensions . . . . . . . . 22

1.3.3 The Z 2 topological invariants in three-dimensions . . . . . . . 23

1.4 Inspiration for discovering new topological phases . . . . . . . . . . . 24

1.4.1 Band inversions and spin-orbit interactions . . . . . . . . . . . 24

1.4.2 Constructing new topological phases . . . . . . . . . . . . . . 27

2 Experimental techniques 31

2.1 Spin-integrated Angle-resolved photoemission spectroscopy . . . . . . 31

2.2 Spin-Resolved Angle-resolved photoemission spectroscopy . . . . . . . 34

2.2.1 Mott polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . 36

xi


12/250

CONTENTS

2.2.2 VLEED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Sample preparation for ARPES . . . . . . . . . . . . . . . . . . . . . 42

2.3.1 Single crystal bulk samples . . . . . . . . . . . . . . . . . . . . 42

2.3.2 MBE film samples . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Probing the topological number in 3D bulk materials . . . . . . . . . 45

2.4.1 Separation of insulating bulk from metallic surface states using

incident photon energy modulated ARPES . . . . . . . . . . . 45

2.4.2 Surface Dirac crossing, surface-bulk connectivity, and spin-momentum

locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Topological Quantum Phase Transition in BiTl(S1−δSeδ)2 52

3.1 Evolution of the electronic groudstate across a topological quantum

phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Topological quantum phase transition in BiTl(S1−δSeδ)2 . . . . . . . . 56

3.2.1 Identifying a suitable topological material system . . . . . . . 56

3.2.2 Topological quantum phase transition in BiTl(S1−δSeδ)2 . . . . 58

3.2.3 Spin-orbit coupling vs. lattice constant . . . . . . . . . . . . . 64

3.2.4 3D Dirac semimetal state at the critical point . . . . . . . . . 66

3.3 Topological-critical-point and the “preformed” surface states . . . . . 68

3.4 Topological phase transition and “preformed” critical behavior in (Bi1−δInδ)2Se3 82

4 Topological Crystalline Insulator Phase in Pb1−δSnδTe(Se) 84

4.1 Key theoretical concepts for a topological crystalline insulator state . 85

4.2 Discovery of mirror symmetry protected TCI state in Pb1−δSnδTe . . 89

4.2.1 Band inversions and mirror symmetries in Pb1−δSnδTe . . . . 89

4.2.2 Topological surface states in Pb0.6Sn0.4Te . . . . . . . . . . . . 91

4.2.3 Mirror Chern number and mirror symmetry protection . . . . 98

4.3 Topological surface states in PbδSn1−δSe . . . . . . . . . . . . . . . . 103

xii


13/250

CONTENTS

4.3.1 Lifshitz transition and saddle point singularities . . . . . . . . 103

4.3.2 Temperature-driven topological phase transition . . . . . . . . 108

4.3.3 Topological phase diagram in Pb1−δSnδSe . . . . . . . . . . . . 111

5 Topological Dirac semimetal state in Cd3As2 and Na3Bi 114

5.1 Theoretical concepts for a topological Dirac semimetal . . . . . . . . 116

5.2 3D Dirac semimetal state in high mobility Cd3As2 . . . . . . . . . . . 121

5.3 Fermi arc surface states in topological Dirac semimetal Na3Bi . . . . 132

5.3.1 Choice of the surface termination to observe FASS . . . . . . . 132

5.3.2 Observation of Fermi arc surface states in Na3Bi . . . . . . . . 135

5.3.3 Topological invariant for the Dirac semimetal Na3Bi . . . . . . 141

5.4 Topological semimetals: Dirac, Weyl, and nodal-line . . . . . . . . . . 150

6 Topological states in 4f Kondo systems SmB6 and YbB6 156

6.1 Observation of surface states in topological Kondo insulator candidate

SmB6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Topological surface states in YbB6 . . . . . . . . . . . . . . . . . . . 166

7 Superconductivity and magnetism in topological insulators 174

7.1 Hedgehog spin texture in a magnetic topological insulator . . . . . . 175

7.2 Helical Cooper pairing in topological insulator/superconductor het-

erostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Appendices 231


14/250


15/250


16/250

Figure 1.1: The band insulator state and the band inversion . (a) The crystalstructure of a solid argon lattice. (b) The energy levels of a solid argon in the atomiclimit where the lattice constant goes to infinity resulting in a number of unbound argonatoms. (c) As the lattice constant goes to the experimental value, the Ar3 p and Ar4sorbitals bceome Bloch bands where they gain energy dispersion as a function of the Blochmomentum k. (d) First-principles calculation of the electronic band structure of a solidargon, where the Ar3 p valence band, Ar4s conduction band, and the normal band-gap areseen. (e-h) The same as Panels (a-d) but for a topologically nontrivial band insulatorBi2Se3. In this case, the relative energy positions of the conduction and valence bands are

reversed with respect to their atomic limit (indicated by the blue and red color code inPanels (f,g). This results in a topologically nontrivial state in Bi2Se3. In Panel (h), asingle Dirac cone surface state (red line) enclosing the Γ̄ within the bulk energy gap is seen.


17/250

1.1 THE BAND INSULATOR STATE AND THE BAND INVERSION

the electronic structure where the relative energy positions between the conduction

and valence bands in a solid are reversed with respect to those of in its atomic limit.

Therefore, within band theory, any topologically nontrivial state should have a finite

number of band inversions, because the ones that do not can be adiabatically tuned

into its atomic limit without closing the band-gap. Now we can understand both a

conventional band insulator and a topologically nontrivial insulator under the same

band inversion picture: Let us take an insulator and increase the lattice constant to

infinity. This will result in a number of unbound atoms or ions. We compare the

energy levels for all the filled and unfilled orbitals, and label the last filled orbital as

“V” and the first unfilled orbital as “C”, since these two will form the conduction

and valence bands in the actual compound. Now we gradually change the lattice

constant from infinity to the experimental value, each atomic level will form a Bloch

band and gain dispersion as a function of Bloch momentum k. For a conventional

band insulator, the energy of the valence band “V” will always remain lower than

that of the conduction band “C” at all k points throughout the Brillouin zone (BZ).

Therefore, a conventional band insulator is topologically trivial. In contrast, for a

topological insulator, the energy of “V” becomes higher than that of “C” at some

k points. In this case, in order to restore the relative energy order between “V”

and “C” as in the atomic limit, one has to close the bulk band-gap, which involves

a quantum phase transition that demonstrates the topological inequality between a

topological insulator and the vacuum. We note that band inversion is a necessary

but not a sufficient condition for topological phases. After all, a topological phase is

defined by a topological number - a global quantity calculated using the electronic

wavefunction throughout the BZ. Thus even if a system has nonzero band inversions,

one still needs to evaluate the system’s topological number to confirm its topological

nontriviality. Although band inversion cannot rigorously define any topological phase,

we found that it is in fact very suggestive in terms of identifying real compounds for

4


18/250

new topological states and understanding the topological nature in various topological

materials, which are the main goals of the experiments in this thesis. For example in

numerical band structure calculations, which serves as a useful guide for experiments,

it is usually much easier to identify the band inversion in a material (by computing

the bulk band structure at different lattice constant) than calculating the actual

topological number (e.g. the Chern number involves an integral of the bulk electronic

wavefunction throughout the BZ, which is usually impractical to compute). Similarly,

experimentally driving a band inversion (by applying pressure, varying temperature

or chemical substitution) can drive a conventional material into a topological state

or transform an understood topological phase to an unknown and more exotic one by

going through a topological quantum phase transition. Therefore, it is in this context

that we highlight the importance of band inversion here.

1.2 Previously discovered topological phases.

1.2.1 The integer quantum Hall state and the Chern insula-

tor state

The experimental discovery of the 2D integer quantum Hall (IQH) state [19] in 1980

marks the first realization of a topological phase of matter. The IQH effect is a

quantum version of the Hall effect, which is achieved by applying strong magnetic

field in two-dimensional electron systems at low temperatures. In band theory, the

quantization of the electrons’ circular orbits leads to quantized Landau levels. The

IQH state is an insulator in the bulk because the Fermi level is located in the middle

of two Landau levels. On the other hand, the edges of the IQH state feature chiral

1D metallic states, leading to remarkable quantized charge transport phenomena

(σxy = ne2/h) [Fig. 1.2(a)]. It turns out that the key difference between an IQH state

and a 2D conventional band insulator is a matter of topology [20–22]. The quantized


19/250

1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.

Figure 1.2: The interface between a quantum Hall state and an insulator haschiral edge mode. (a) The skipping cyclotron orbits. (b) The electronic structure of asemi-infinite edge described in a Chern insulator with a Chern number of n = +1. A singleedge state connects the valence band to the conduction band.

number n in the transverse magneto-conductivity σxy = ne2/h is a topological number

(the Chern number) that characterizes the nontrivial topology of the IQH state.

Two years after the discovery of the IQH effect, the fractional quantum Hall (FQH)

effect was observed [23]. The fractional quantum Hall effect (FQHE) is a physical

phenomenon in which the Hall conductance of 2D electrons shows precisely quantized

plateaus at fractional values of e2/h. Unlike the IQH effect, the FQH effect relies

fundamentally on electron-electron interactions and therefore cannot be understood

in the frame of band theory. It is interesting to note that the FQH effect is not only

the second topological state realized in experiments, but also it is believed to have

topological order [24]. We will discuss the difference between the topological order

and the symmetry-protected topological state later in this section.

Let us get back to the IQH state. Since the generators of translations do not

commute with one another in a magnetic field, electronic states cannot be labeled

with momentum k. Can we construct a quantum Hall like state with a nonzero

Chern number but without an external magnetic field so that the system can be

understood by the Bloch band theory? The answer is the Chern insulator. The first

theoretical model example is the Haldane model [25], where a magnetic field that is

zero on the average, but has all of the spatial symmetries of the honeycomb lattice, is

6


20/250

1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator

applied to graphene. And Haldane showed that although the average magnetic field is

zero, it opens up a gap at the Dirac nodes of graphene, and furthermore this gapped

state is not an ordinary insulator, but rather has a quantized Hall conductivity (a

nonzero Chern number). More recently, the Chern insulator state is also theoretically

predicted in the magnetically doped thin film of a 3D Z 2 topological insulator [9],

where no external magnetic field is required and the time-reversal symmetry breaking

is provided by the magnetization of the sample itself. We note that such type of Chern

insulator is also referred as a quantum anomalous Hall insulator, and it was recently

experimentally realized in Cr-doped (Bi1−xSbx)2Te3 thin films [9]. One important

consequence of the Chern insulator is that now we can understand it in a band

theory picture again. As shown in Fig. 1.2(b), a Chern insulator can be understood

as a single edge state that connects the valence band to the conduction band.

1.2.2 The 2D Z 2 topological insulator - the quantum spin

Hall insulator

Although a Chern insulator does not require external magnetic field, its internal

magnetization still breaks time-reversal symmetry. Can we construct a topological

phase that respects time-reversal symmetry? This is particularly motivated by the

fact that there are many more time-reversal invariant materials than the time-reversal

breaking ones in nature. Such motivation leads to the theoretical prediction of the

third type of topological insulator - the quantum spin Hall (QSH) insulator [26,27].

In a QSH insulator, the bulk is insulating and features a pair of counter-propagating

edgestates, which are related by time-reversal symmetry. And the whole system,

unlike a Chern insulator, respects time-reversal symmetry. We provide a physical

picture to build a quantum spin Hall insulator from two Chern insulators: Let us take

a Chern insulator with a Chern number of n = +1, which is achieved by an out-of-

plane magnetization +M [Fig. 1.3(a,b)]. Under time-reversal operation T , quantities


21/250


Figure 1.3: A quantum spin Hall insulator can be constructed by two Cherninsulators that are related by the time-reversal symmetry. (a) A Chern insulatorwith a Chern number of n = +1 has a chiral edge mode as a result of an out-of-planemagnetization. (b) The bottom edge projection of the electronic structure the n = +1Chern insulator. (c,d,) Same as Panels (a,b) but with a time-reversal operation onto thesystem. (e,f,) A quantum spin Hall state can be obtained by combining the above-two

Chern insulator states. The resulting electronic structure on the edges is described by twocouterpropagating edge modes that cross each other at k = 0 (or k = π). The crossing isprotected by the time-reversal symmetry as a result of the Kramers theorem.

8


22/250

1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator

including momentum k, angular momentum (including spin σ), magnitization M will

change sign. The resulting system after time-reversal transformation is shown in

Fig. 1.3(c,d), where a Chern insulator with a Chern number of n = −1 is obtained.

Now if one imagines combining these two Chern insulators into a single system, the

total magnetization goes to zero, and the system is invariant under time-reversal

operation. On the edges of this new system, there are two edge states that are

counter-propagating. We draw the electronic structure that is projected to the bottom

edge. Since the two Chern insulators are symmetric over time-reversal operation, in

the combined system, the two edge modes must “meet” each other at k = 0. In

general, as two states cross, they will hybridize and open up a gap. However, in this

case, these two edge states that have opposite quantum numbers (Left moving: +k

and ↑; Right moving: −k and ↓;) are directly linked by the time-reversal operationbecause they cross each other at k = 0, a time-reversal invariant momentum (TRIM)

(also referred as a Kramers point) [Fig. 1.3(e,f)]. A Kramers point kKramers is defined

as kKramers = G − kKramers, where G is any reciprocal lattice vector. The Kramerstheorem states that the electronic states have to remain doubly degenerate at the

Kramers points in a time-reversal invariant system. And therefore the edge state

crossing at k = 0 is protected by the time-reversal symmetry, giving rise to the

quantum spin Hall phase. From the above pictures, it is evident that the quantum

spin Hall phase is topologically distinct from the vacuum. This is because it is not

possible to remove the metallic edge states from the band-gap as long as the edge

band-crossing remains intact. The topological nontriviality indicates that there exist

a finite number of bulk band inversions at the Kramers points of the BZ. In a real

QSH system, no magnetization is present unlike in our conceptual picture. Thus

one has to identify a physical interaction that can lead to an inverted band-gap in a

time-reversal invariant condition. It turns out that the spin-orbit interaction in heavy

elements plays such a critical role. On the other hand, since spin-orbit interaction


23/250


mixes the spin and orbital degrees of freedom, the physical spin is not a quantum

number. Thus it is more correct to think of the “spin” in the quantum spin Hall state

in strongly spin-orbit coupled systems as the pseudo-spin rather than the real spin.

In 2005, theoretical advances realized that the quantum spin Hall phase is indeed

topologically nontrivial [26]. In fact, it is characterized by a new topological number,

the Z 2 invariant (ν ) [14, 26, 29]. The Z 2 invariant can only take two values, 0 or

1, where ν = 0 (1) is topologically trivial (nontrivial). In 2007, the QSH phase was

experimentally demonstrated in the (Hg,Cd)Te quantum wells using charge transport

by measuring a longitudinal conductance of about 2e2/h (two copies of quantum Hall

currents) at mK temperatures [28]. However, no spin polarization was measured in

this experiment thus spin momentum locking, which is essential for the Z 2 topological

physics, was not experimentally observed [28].

It is important to note that the 2D topological (IQH, FQH, and QSH) insulators

have only been realized at buried interfaces of ultraclean semiconductor heterostruc-

tures at very low temperatures to date. Furthermore, their metallic edge states can

only be probed by the charge transport method. These facts hinder the systematic

studies of many of their important properties, such as their electronic structure, spin

polarization texture, tunneling properties, optical properties, as well as their responses

under heterostructuring or interfacing with broken symmetry states. For example,

the two counter-propagating edge modes in a QSH insulator are predicted to feature

a 1D (usually Dirac) band crossing in energy and momentum space. And edge mode

moving along the +k direction is expected to carry the opposite spin polarization

as compared to that of one moving in the −k direction. However, neither the Diracband crossing nor the spin-momentum locking of the edge modes in a QSH insulator

are experimentally observed, due to the lack of experimental probe that can measure

these key properties for a 1D edge mode at a buried interface at mK temperatures.

10


24/250

1.2.3 The 3D Z 2 topological insulator


In 2007, it was theoretically realized that the Z 2 topological number in a QSH insula-

tor can be generalized to three-dimensions, thus realizing the first three-dimensional

topological phase of matter [14–18]. This is again a critical breakthrough in experi-

mental and material physics because there are more 3D bulk materials than 2D films

in nature and the preparation of bulk materials is usually easier. How do we con-

struct a 3D Z 2 TI? The most straightforward approach is to simply stack a number

of uncoupled 2D quantum spin Hall states to form a 3D bulk. It turns out that such

simple stacking of the 2D QSH states does lead to a type of 3D Z 2 TI, which is the so

called 3D weak TI. A weak TI is not the most topologically protected Z 2 nontrivial

states in bulk materials. But let us start from a weak TI.

As shown in Fig. 1.4, let us assume that we start from a 2D QSH state that has an

inverted band-gap at the Γ point. The blue and green colors in Fig. 1.4(a) show the

orbital nature of the bulk bands near the Γ point, where a band inversion is evident.

Now let us have N copies of such QSH state and stack them along the out-of-plane

ẑ direction. This will form a bulk material. And in momentum space, it means that

the 2D BZ will gain periodicity in the kz direction and becomes a cube representing

the 3D BZ [Fig. 1.4(b)]. In the 3D BZ, the Kramers points can be sorted by their

origins from the 2D BZ before stacking. For example, both the Γ and the Z points

in the 3D BZ results from the Γ point in the 2D BZ. We assume that the physical

coupling between these 2D QSH slices are extremely weak (e.g. in real materials,

imagine they are coupled by a very weak van der Waals interaction). This means

that the energy dispersion along the kz direction is sufficiently weak, so that it does

not change the orbital nature of the conduction and valence bands by going from the

Γ(3D) point to the Z point. Therefore, the one band inversion in the 2D Γ point in

the QSH slice becomes two band inversions at both the Γ and the Z points in the 3D

BZ after stacking (here only the band inversions at the Kramers points are counted).


25/250


Figure 1.4: A 3D weak topological insulator can be obtained by stacking N copiesof identical 2D quantum spin Hall states along the out-of-plane direction. (a) A2D quantum spin Hall state where there is a band inversion at the 2D BZ center Γ point.(b) stacking N copies of such quantum spin Hall slices along the out-of-plane direction form

a bulk material. In momentum space, it means that the 2D BZ will gain periodicity in the kzdirection and becomes a cube representing the 3D BZ. The time-reversal invariant momentain the 3D BZ are noted. (c) Before turning on the inter-slice coupling, different QSH slicesare completely identical and idenpendent (e.g. Cut2 and Cut3). This also means that thereis no energy dispersion along the out-of-plane Γ − Z direction (Cut1) (d) A finite (butweak) inter-slice coupling means that the states in the QSH system gain energy dispersionalong the out-of-plane kz direction. The inter-slice coupling is weak enough, so that it doesnot change the orbital nature of the conduction and valence bands by going from the Γ(3D)point to the Z point (Cut1). Therefore, there are two band inversions at both the Γ andthe Z points in the 3D BZ (Cuts 2 and 3).

12


26/250


Figure 1.5: The surface of a weak TI shows even number of protected surfacestates or no surface states depending on the surface termination. (a) A bulk BZof the weak TI. The green slices are chosen for detailed studies. (b) The edge projectedelectronic structure for the green slices shown in Panel (a). Without any inter-slice cou-pling, these slices are identical and independent. The surface states for the kx − kz surfaceprojection is a straight line. (c) A bulk BZ showing the two band inversions at the Γ andthe Z points. (d) After turning on the inter-slice coupling, the edge states can hybridize andopen out gaps, except at the Γ and the Z points due to the protection of the time-reversalsymmetry. Therefore, the surface state Fermi surface is described by two dots at the Γ and

the Z points.


27/250


What about the edge states of each QSH slice? Let us look at a few representative

QSH slices at different kz values as shown by the green 2D slices in Fig. 1.5(a). As

seen in Fig. 1.5(b), before we turn on the coupling between these slices, each slice

is a QSH that features two counterpropagating edge modes at its edge. We fix the

chemical potential at the crossing point of the edge modes and investigate the kx −kzplane where all the QSH edges are stacked to form a surface. Thus all the 1D edge

states from the QSH slices are stacked and become a 2D surface state. The surface

state Fermi surface is a straight line that goes from Z − Γ − Z . Now we turn on theinter-slice coupling (hybridization). As we have mentioned above, in general if two

edge states cross, they will open up an energy gap. However, at and only at the Γ and

the Z points, the edge state crossings are preserved because these two points are the

Kramers’ points. Consequently, the surface state Fermi surface of the kx − kz planebecomes two dots at the Γ and the Z points. And if the E F is shifted slightly away

from the crossings, then the two Fermi dots will evolve into two circles that enclose

the Γ and the Z points, respectively. Therefore, the surface states at the kx −kz planeare described by two surface states that enclose the Γ and the Z points, respectively.

In contrast, at the top kx − ky surface, no protected surface state is expected. Thisis because both Γ and Z points project onto the Γ̄ point of the top kx − ky surface.The two surface states that originate from the band inversions at Γ and Z points

can hybridize and open up a gap. Therefore, it can be seen that a weak TI features

protected 2D surface states only at certain crystalline surface terminations.

In fact, from the topology point of view, the 3D weak TI state is essentially

equivalent to a number of independent 2D quantum spin Hall slices, because there

involves no band inversion by decomposing a weak TI into uncoupled 2D QSH slices.

Can we construct a 3D Z 2 TI state that cannot be adiabatically reduced into a bunch

of stacked QSH states? Also, from the surface state point of view, a 3D weak TI

only has protected surface states at certain surface terminations. Is there a 3D Z 2 TI

14


28/250


state that shows protected surface states at all surfaces irrespective of the choice of

the termination? The answers for both questions lead to the 3D strong TI phase.

Let us again start by stacking QSH slices (Fig. 1.6). It is obvious that a weak

inter-slice coupling only leads to the weak TI phase. Now we examine the strong

coupling scenario. We know that at the 2D Γ point the conduction and valence

bands are inverted with respect to the atomic limit. Without interlayer coupling, it

means that there exist two band inversions at both the Γ and the Z points of the

3D BZ. Now we turn on the inter-slice coupling. And in this case, the inter-slice

coupling is strong enough and affects the electronic structure in a way so that the

band inversion at the Z point is in fact removed [Cut1 in Fig. 1.6(d)]. As shown

in the Cuts 2 and 3 in Fig. 1.6(d), while at the Γ point (Cut2) the conduction and

valence bands are still inverted with respect to the atomic limit, at the Z point (Cut3)

these two bands restore the relative energy positions as in the atomic limit (no band

inversion). Therefore, in this case, there is only one band inversion at the Γ point

throughout the 3D BZ. This is an example of a 3D strong TI state. It is important to

note that the 3D strong TI state is topologically distinct from both the weak TI and

the conventional band insulator (the atomic limit) states. This is because (1) it is not

possible to reduce the strong TI to uncoupled 2D QSH slices without going through

a band inversion at the Z point. Therefore a strong TI is topologically inequivalent

to a weak TI. (2) it is also not possible to change a strong TI to a conventional band

insulator without going through a band inversion at the Γ point.

We now investigate the possible existence of protected surface states in the strong

TI state (Fig. 1.7). We note that there is only one band inversion at the Γ point for

the strong TI state, and that at any surface termination the Γ point projects onto

the surface BZ center Γ̄ point. Therefore, one expects a single surface state enclosing

the surface BZ center Γ̄ point at any surface termination. We also note that the case

presented here in Figs. 1.6 and 1.7 is only one of the possible ways to generate a strong


29/250


Figure 1.6: A 3D strong topological insulator can be obtained by coupling N copies of 2D quantum spin Hall states along the out-of-plane direction in a non-trivial way. (a) A 2D quantum spin Hall state where there is a band inversion at the2D BZ center Γ point. (b) The 3D BZ. The time-reversal invariant momenta in the 3DBZ are noted. (c) Before turning on the inter-slice coupling, there is no energy dispersionalong the out-of-plane Γ − Z direction (Cut1). (d) A strong inter-slice coupling can drivea band inversion along the out-of-plane Γ − Z direction (Cut1). In this case, while at the Γpoint (Cut2) the conduction and valence bands are still inverted with respect to the atomiclimit, at the Z point (Cut3) these two bands restore the atomic limit (no band inversion).Therefore, there is only one band inversion at the Γ point of the 3D BZ.

16


30/250


TI. In general, a strong TI should always possess an odd number of band inversions

at the Kramers’ points in a 3D BZ, but the exact number of band inversions (1, 3, 5,

or other odd numbers) depends on the exact form of the inter-slice coupling as well

as the properties of the 2D QSH state to start with (e.g. one can also imagine to have

a QSH state with a band inversion at the 2D BZ corner M point not the Γ point).

However, regardless of these details, a strong TI will always have an odd number of

band inversions in the bulk and feature an odd number of surface states enclosing the

Kramers points irrespective of the surface termination. It is also important to note

that the surface state of 3D TIs are usually referred as “Dirac cone” surface states

in many research works. “Dirac cone” means that the surface states cross at the

Kramers point with a linear dispersion. This is, however, not always true. In fact,

the Kramers theorem only requires the surface states to remain doubly degenerate at

a Kramers point, or, in other words, the surface states have to cross each other at the

Kramers point. But it does not require a specific fashion of the crossing in terms of

the energy dispersion of the surface states. It is actually also possible to have surface

states with cubic or even higher order crossings in principle. But a cubic or high

order crossing means that the linear term is somehow forbidden due to the presence

of certain additional symmetry. This is quite hard and it has not been experimentally

achieved yet. Therefore, “Dirac” surface state is usually used to describe the metallic

boundary mode of 3D strong or weak TIs.

The above physical pictures can be mathematically formulated, which results in

the Z 2 topological numbers for a 3D bulk system [14–16,29]. In 2007, it was theoreti-

cally realized that in three-dimensions, there exist four Z2 topological invariants that

define the topological property of a 3D bulk material, namely (ν 0; ν 1ν 2ν 3), where ν 0

is the strong topological invariant, and ν 1 − ν 3 are the weak topological invariants,respectively [14]. If all four invariants are zero, then the system is a conventional

band insulator. If the strong invariant ν 0 = 0 but one or more than one of the weak


31/250


Figure 1.7: The surface of a strong TI features an odd number of protected surfacestates at all surfaces irrespective of the surface termination. (a) A bulk BZ of thestrong TI. The green slices are chosen for detailed studies. (b) The edge projected electronicstructure for the green slices shown in Panel (a). Without any inter-slice coupling, theseslices are identical and independent. The surface states for the kx − kz surface projectionis a straight line. (c) A bulk BZ showing the one band inversion at the Γ point. (d) Afterturning on the inter-slice coupling, in the case of a strong TI, only the band inversion atthe Γ point remains whereas the band inversion at the Z point is removed. Furthermore,for k−space region near the Γ point where the conduction and valence bands are inverted,the edge states can hybridize and open out gaps, except at the Γ point itself due to the

protection of the time-reversal symmetry. Therefore, the surface state Fermi surface isdescribed by one dots at the Γ point.

18


32/250


invariants is nonzero (ν 1 + ν 2 + ν 3 = 0), then the system is a weak TI. Finally, if the strong topological invariant is nonzero (ν 0 = 1), the system is a 3D strong Z 2

topological insulator.

Experimentally, in 2007, the 3D strong TI phase was experimentally identified in

the Bi1−xSbx semiconducting alloy system, marking the first realization of a topologi-

cally nontrivial phase of matter in 3D bulk materials [30,31]. Shortly after Bi1−xSbx,

another class of strong TI - the Bi2Se3 class - was experimentally discovered [32–36].

The Bi2Se3 class was found to feature a single Dirac cone surface state due to a sin-

gle bulk band inversion at the Γ point (same as the example given in Figs. 1.6 and

1.7) and therefore it serves as the prototype 3D TI, which has been the most widely

researched topological insulator even to date.

More importantly, it turns out that the experimental discovery of the 3D topologi-

cal insulator phase in 2007 opened a new experimental era in fundamental topological

physics [30–127]. In contrast to its 2D analogs, (1) a 3D topological insulator can be

realized at room temperatures without magnetic fields. Their metallic surface states

exist at bare surfaces rather than only at buried interfaces. (2) The electronic and

spin groundstate of the topological surface states can be systematically studied by

spin- and angle-resolved photoemission spectroscopy (spin-ARPES), which provides a

unique and powerful method for probing the topological number in three-dimensional

topological phases. (3) Due to the accessible conditions (room temperature, no mag-

netic field, bare surface), it is also possible to study the tunneling [87–94], electrical

transport [95–106], optical [107–114], and many other key properties of the topo-

logical surface states. (4) The 3D topological insulator materials can be doped or

interfaced to realize superconductivity or magnetism [52–60]. (5) Since its discovery

in 2007, there have been more than a hundred compounds identified as 3D Z 2 topo-

logical insulators. These conditions not only make the 3D Z 2 TI state a topic of huge

research interest worldwide, but also lay the experimental foundation for discovering


33/250


new topological phases of matter [37–51], which is the main theme of this thesis.

We note that time-reversal symmetry is required for a Z 2 topological insulator

phase. This can be visualized in the surface state point of view. If time-reversal

symmetry is broken, the surface state degeneracy at the Kramers point can be lifted.

And in that case, the distinction between a Z 2 topological insulator and a conventional

band insulator no longer exists. In the topological theory, a state that is topologically

nontrivial only if a symmetry is respected is called a symmetry protected topological

(SPT) state. A Z 2 TI is a SPT state [128]. On the other hand, a state whose

topological nontriviality does not rely on any additional symmetry is defined as a

state that features topological order [24]. In general, the realization of a topological

order inherently requires strong electron-electron interaction, which is beyond band

theory [24]. Strong interaction leads to fractionalized non-electron-like quasi-particle

boundary excitations, which is believed to be one of the key properties for a system

with a topological order. A fractional quantum Hall (FQH) state, whose edges feature

gapless excitations with fractional charges, is an example of a state with a nontrivial

topological order. This is in contrast to the Z 2 TI phase (a SPT state) where the

metallic surface states are electron quasi-particles. In theory, it is believed that the

boundary modes in a state with topological order can be robust against any local

perturbation, whereas the gapless boundary modes in a SPT state are robust against

local perturbations that do not break the corresponding symmetry. It is in this

sense that a state with topological order is more robust than a SPT state. However,

let us also view this issue based on experimental and material facts. Due to the

requirement of very strong electron-electron interaction, topological order is more

difficult to realize and its existence is rarer in real solid-state materials as compared

to a SPT state. As a matter of fact, the FQH state is the only state with topological

order that has been experimental realized so far. Other theoretical proposals, such as

a fractionalized Chern insulator [129], or a fractionalized topological insulator [130],

20


34/250

etc., remain elusive due to the difficulty of identifying a real system that has the

necessary material parameters. Moreover, the experimental study of a topological

phase relies on measuring its gapless boundary states. In a topologically ordered state,

the boundary modes are fractionalized non-electron like quasi-particles. Although in

the case of FQH, their existence can be demonstrated by charge transport. But it is in

general extremely challenging to systematically study the energy dispersion, angular

momentum (spin polarization), and other key properties of fractionalized non-eletron-

like quasiparticles. On the other hand, these difficulties do not exist in a SPT state

such as a Z 2 TI state in Bi2Se3. Furthermore, the requirement of a symmetry in

a SPT state is usually a “given” in a solid state system. For example, in a solid-

state crystal, the translational symmetry or sometimes also space group symmetries

(mirror, C n) is a given. Furthermore, time-reversal symmetry is also a very common

property that exists in most non-magnetic materials. In fact, in experiments, it is

quite often that one has to make considerable effort to break a certain symmetry (e.g.

to apply strong magnetic field to break time-reversal symmetry for a FQH state)

in order to realize certain topological order. Therefore, by these (experimental or

material) considerations, the realization, the systematic study and the utilization of

SPT states are of importance and interest. In this thesis, we focus on the experimental

discoveries of new symmetry-protected topological states (mostly weakly interacting

systems within the band theory) beyond a Z 2 TI.

1.3 Theoretical formulations

In this section, we provide the theoretical formulations for the Chern number and the

Z 2 invariants.


35/250


36/250

1.3.3 The Z 2 topological invariants in three-dimensions

Λa in the bulk 2D BZ, where k and −k coincide. In the square 2D BZ as shown inFig. 1.4(a), these four points are the Kramers points, namely 1 Γ, 2 X , and 1 M .

At these four points, ω(Λ) is antisymmetric. The determinant of an antisymmetric

matrix is the square of its pfaffian (P f ), which allows us to define

δ a = P f [ω(Λa)]/

Det[ω(Λa)] = ±1 (1.4)

Provided the Bloch wavefunction |um(k) > is chosen continuously throughoutthe Brillouin zone (which is always possible), the branch of the square root can be

specified globally, and the Z 2 invariant is

(−1)ν =4

a=1

δ a (1.5)

If the crystal has inversion symmetry, there exists a shortcut to obtain ν [29]. At

the Kramers points Λa the Bloch states um(Λa) are also the eigenstates of the parity

operator with the eigenvalue ξ m(Λa) = ±1. The Z 2 invariant then simply followsfrom with

δ a =

m

ξ m(Λa) (1.6)

where the product is over the Kramers pairs of occupied bands. This has proven

quite useful for identifying Z 2 topological insulators from numerical band structure

calculations.

1.3.3 The Z 2 topological invariants in three-dimensions

In three dimensions there are eight Kramers points, which are expressed in terms of

primitive reciprocal lattice vectors as Λa = (n1b1 + n2b2 + n3b3)/2, with n = 0, 1.

This leads to four independent Z 2 topological invariants to define the Z 2 topological


37/250

1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES

properties of a 3D band insulator, namely [ν 0; ν 1ν 2ν 3], where ν 0 is the strong invariant

and (ν 1, ν 2, ν 3) are the weak invariants.

The strong invariant can be calculated by the pfaffian over all eight Kramers points

(−1)ν 0 =8

a=1

δ a (1.7)

where again we have δ a = P f [ω(Λa)]/

Det[ω(Λa)] = ±1.Besides the strong invariant, it has been shown in theory that one can define

another 3 independent weak invariants (ν 1, ν 2, ν 3) as

(−1)ν i=1,2,3 = 8ni=1;nj=i=0,1

δ a. (1.8)

Again, the conclusion from the above topological theory is that if all four invariants

are zero (ν 0 = ν 1 = ν 2 = ν 3 = 0), then the system is a conventional band insulator; if

ν 0 = 1, then the system is a strong (Z 2) topological insulator; if ν 0 = 1 but at least

one of the weak invariants is nonzero, then the system is a weak (Z 2) topological

insulator.

1.4 Inspiration for discovering new topological phases

Now that we have gone through all previous topological phases, we discuss the in-

spirations that are brought by these previous works for discovering new topological

matter.

1.4.1 Band inversions and spin-orbit interactions

First of all, it is evident that the band inversion is a key (a critical necessary condition)

that gives rise to the topological insulator state. More specifically, by band inversion,

we mean two phenomena in the bulk electronic structure: (1) At certain k points, the

24


38/250

1.4.1 Band inversions and spin-orbit interactions

relative energy positions between the conduction and the valence bands are reversed

with respect to those of in the atomic limit. (2) An insulating gap is opened once the

conduction and valence bands are inverted.

How do we realize a band inversion with these two phenomena in real materials?

Is there any clue or hint regarding how to search for materials with band inversions?

Let us again start from a very simple case, which is NaCl (an ionic insulator). We

write it as Na+Cl− to highlight the distinct ionic states of sodium and chlorine in

this insulator. As shown in Fig. 1.8, the lowest valence and conduction bands near

the Fermi level are the Cl−3 p and the Na+3s orbitals, respectively. As the system

goes from the atomic limit [Fig. 1.8(a)] to the actual lattice constant [Fig. 1.8(b)],

these bands gain energy dispersion as a function of Bloch momentum k. In the case

of Na+Cl−, the relative energy positions between the conduction and valence bands

remain the same as the atomic limit at all k points throughout the BZ. Thus a large

full band-gap is formed. The physical meaning of the band-gap can be understood

in the picture that it costs a finite amount of energy (∼ the band-gap) to excite anelectron from the valence band to the conduction band, or more loosely speaking,

to give the electron that is transferred to Cl− back to the Na+ [Fig. 1.8(c)]. Such a

process is highly energetically unfavorable because the system gains much energy as

Na atom transfers its 3s1 electron to the Cl atom to form Na+Cl−.

Now let us imagine another compound A+B−, where we label the lowest valence

and conduction bands as B− and A+, respectively as shown in Fig. 1.8(d). Now as we

tune the lattice constant to the actual value, the valence and conduction bands gain

energy dispersion in a way so that these two bands actually cross at certain Kramers

point, as shown in Fig. 1.8(e). Now in this scenario, at the k−space vicinity to theKramers point where band-crossing occurs, the system actually gains energy to give

the electron that is transferred to the B atom back to the A atom. Furthermore, it

is important to note that as the conduction and valence bands cross, they become


39/250


Figure 1.8: Band Inversions and the spin-orbit interaction. (a-c) In the case of sodium chloride Na+Cl−, the system does not have any band inversion. The fact that thereis a large band-gap between the Cl−3 p and the Na+3s bands is consistent with that it isenergetically costly to give the electron that is taken by the Cl− back to the Na+. (d-f)

In a material A+

B−

, the valence and conduction bands gain energy dispersion in a wayso that these two bands actually cross at certain Kramers point. Further consideration of the spin-orbit interaction opens a full insulating gap. SnTe is a real material example forthis case. (g-i) In this case, without spin-orbit coupling, there is no band crossing, and thespin-orbit interaction is responsible for both inverting the conduction and valence bandsand for opening up the full insulating gap. We note that Bi2Se3 belongs to this case.

26


40/250

1.4.2 Constructing new topological phases

degenerate at certain k points in the BZ [see Fig. 1.8(e)]. Thus there does not exist

a full insulating gap as required by the Z 2 topological insulator state. However, as

we further consider the effect of the spin-orbit interaction, it causes the bands that

cross to hybridize and open up a full band-gap. In terms of real materials, SnTe is

a famous example of this kind [131], where a band inversion between the Sn5 p and

Te5 p orbitals is found even without the including the spin-orbit interaction at each

L point of its BZ. And the spin-orbit interaction is responsible for opening up a full

band-gap. There is another slightly different case, as shown in Figs. 1.8(g-i), where

without spin-orbit coupling, there is no band crossing, and the spin-orbit interaction

is responsible for both inverting the conduction and valence bands and for opening up

the full insulating gap. We note that Bi2Se3 belongs to this case [33]. From the above

physical picture, it is quite clear that we need to work with materials with heavy

(high Z ) elements for a strong spin-orbit coupling, which is necessary to open up a

full band-gap. Furthermore, it can be seen that we cannot have A+B− materials like

NaCl, where Na strongly prefers to loose an electron and Cl strongly prefers to take

an electron. In this case, it is hard to imagine that one could get a band crossing at

certain k− points because a band crossing creates a local k− space regime, where itis energetically favorable to return the electron taken by the B− back to the A+. This

means that we should look for materials that are composed of heavy “semimetal”

elements. Therefore, bismuth (Bi) is a good candidate because it can form both

a positive ion as Bi3+ or Bi5+, and a negative ion as Bi3−, whose energies are not

significantly different. In fact, Bi2Se3 (Bi3+) and Na3Bi (Bi

3−) are two good examples

for the above argument in searching for materials with band inversions.


How do we construct new topological phases? From the discussions in the previous

sections, we know that the band inversion is a necessary but not a sufficient condition


41/250


to realize a topological state. In the case of the Z 2 TI state, the argument is that

the bulk band inversions occur at the Kramers points of the BZ. Therefore, a surface

(edge) state band crossing can be protected by the Kramers theorem.

(1) Inspired by this logic, can we find other types of symmetries that can also

protect a surface (edge) state band crossing (a doubly degeneracy of the surface

bands)? If a symmetry can protect a surface state band crossing at certain momentum

space locations in the surface BZ, then we just need to construct a material that

has a number of bulk band inversions that locate at these special momentum space

locations. In a solid state crystal system, the most common symmetries besides the

time-reversal symmetry are the various space group symmetries. In Chapter 4, we

will systematically discuss such possibilities, leading to our experimental identification

(one of the three concurrent ARPES works) of the first topological crystalline insulator

(TCI) phase in the Pb1−xSnxTe(Se) system.

(2) In general, in a 3D bulk crystal, if two bands cross each other as shown in

Fig. 1.8(e), spin-orbit interaction can cause hybridization between the two bands that

cross and open a full energy gap. As two 3D bulk bands cross, they become degenerateat multiple k points in the BZ. Thus a full energy gap means spin-orbit interaction

lifts up the degeneracy at all k points where they cross. This is actually quite intuitive

because without consideration of extra symmetries, there is no reason a certain k point

is more special than other k points where the two bands are degenerate. Therefore,

if spin-orbit interaction breaks the degeneracy, it should do that at all bulk band

crossing k points.

However, this inspires us to ask a question whether we can identify additional

symmetries that can protect the bulk band crossings at certain k points even with

the consideration of the spin-orbit coupling? If that scenario can be realized, then an

important consequence is that the system becomes a semimetal, where the conduc-

tion and valence bands have finite overlap and there does not exist a full band-gap

28


42/250


irrespective of the choice of the chemical potential. Then an interesting question is

whether the system can still be topological if the bulk band gap closes (meaning that

there is no full energy gap)? If so, how do we define its topological number and what

would the surface state be like? In Chapter 5, we will deal with these intriguing ques-

tions, which lead to our experimental realization of the topological Dirac semimetal

phase in Cd3As2 and Na3Bi.

(3) In the case of Bi2Se3 as shown in Figs. 1.8(g-i), the inverted band gap is solely

opened due to the spin-orbit coupling. The effect of electron-electron correlation is

negligible. Thus the low energy physics within the inverted band-gap is described

by the single-particle picture. Can we find a material where the electron-electron

interaction plays a non-negligible role in opening up the inverted bulk band-gap? In

Chapter 6, we present our experimental studies on a well-known f -electron system

SmB6. In SmB6, a narrow (∼ 15 meV) band-gap opens at low temperatures ( 30 K)due to the coherent hybridization between the conduction d electrons and the localized

f electrons near the the Fermi level. We show experimental evidence consistent with

the existence of topological surface states within the Kondo gap, supporting the

existence of a topological Kondo insulator phase in SmB6.

(4) None of the known TI materials (represented by Bi2Se3) are naturally super-

conductors or magnets. It is of interest to study the interplay between a topological

state and a broken-symmetry (superconducting or ferromagnetic) state. In Chapter 7,

we present systematic studies of the electronic and spin groundstate of the prototype

TI Bi2Se3 as it is doped into or in proximity to a symmetry-broken (superconducting

or magnetic) state. We identify the key experimental signatures in our experiments

that are considered as the keys to realizing a topological superconductor or a Chern

insulator.

(5) Finallly, all the states mentioned above are new (symmetry-protected) topo-

logical phases beyond a Z 2 TI, which means that they cannot be smoothly deformed


43/250


into each other without going through a band inversion (a topological quantum phase

transition). Can we experimentally realize a band inversion and a topological quan-

tum phase transition? This will bring valuable insights in understanding the way

how two states with distinct topological numbers can be turned into each other. In

Chapter 3, we present our our experimental realization of a topological quantum

phase transition in the BiTl(S1−δSeδ)2 system, which demonstrates the one of the

most basic topological quantum phase transitions - a topological transition between

a conventional band insulator and a Z 2 topological insulator.

30


44/250

Chapter 2

Experimental techniques

In this chapter, we introduce the basic principles of the experimental technique used

in this thesis (the spin-resolved angle-resolved photoemission spectroscopy) and the

review how such technique is used to reveal the 3D Z 2 topological insulator state in

Bi1−xSbx and Bi2Se3 bulk materials.

2.1 Spin-integrated Angle-resolved photoemission

spectroscopy

Angle-resolved photoemission spectroscopy (ARPES) is a direct experimental tech-

nique to observe the distribution of the electrons (more precisely, the density of single-

particle electronic excitations) in the reciprocal space of solids [132,133]. ARPES is

one of the most direct methods of studying the electronic structure of the surface of

solids. The photoelectric effect was first observed by Hertz. When photons of energy

hν , strike a sample surface, electrons absorb the energy and escape into the vacuum

with certain kinetic energies. A good approximation of the photoemission process

can be described by the three-step model, as shown in Fig. 2.1:

• An electron is first excited by the photon inside the bulk.31


45/250

2.1 SPIN-INTEGRATED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY

Figure 2.1: The three-step model used to describe the photoemission process inside thesample.

• The excited electron then travels to the sample surface.

• The electron escapes from the sample into the vacuum.

After a photoelectron comes out of the sample, it is being collected by an electron

analyzer, as a function of its kinetic energy as well as the surface emission angles θ

and φ (Fig. 2.2). The excited photoelectron then follows the kinematics below which

describes the momentum p = k

kx = 1

2mE kin sin θ cos θ (2.1)

ky

= 1

2mE kin sin θ sin θ (2.2)kz =

1

2mE kin cos θ (2.3)

The conservation of energy gives:

E kin = hν − φ − |E B| (2.4)32


46/250

Figure 2.2: (a) The Scienta analyzer setup used to measure emitted photoelectrons fromthe sample surface. The two hemispheres probe the electrons to produce a 2D image of energy (E ) vs. momentum (k) in one measurements. (b) The geometry of the detectorrelative to the sample surface. The momentum of the electron inside the sample can b eextracted from the measured values of E kin, θ , φ.


47/250

2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY

where φ is the work function of the sample and E B is the binding energy of the

electrons. Due to the existence of a surface potential experienced by the escaping

electron, the component of the momentum perpendicular to the sample surface is not

conserved. To capture the effect of the surface potential, we assume that the energy

of the “real” final state (free from the influence of the surface potential) E f has an

energy offset of V 0 with respect to the kinetic energy measured by the analyzer E kin,

namely

E f = E kin + V 0 (2.5)

where V 0 is usually referred as the inner potential.

Since the in-plane momentum is conserved, thus we know that k =√

2mE kin sin θ.

Moreover, we know that 2k

2

2m +

2k⊥2

2m = E f . Therefore, from these relationships, we

get:

k⊥ = 1

2m(E kincos2θ + V 0) (2.6)

where V 0 is usually referred as the inner potential.

The value of V 0 can be determined experimentally by measuring the periodicity

of the energy dispersion along the out-of-plane momentum k⊥ direction at the nor-

mal emission angle. However, for samples where the dispersion perpendicular to the

sample surface is small, such a measurement can be difficult.

2.2 Spin-Resolved Angle-resolved photoemission spec-

troscopy

The Spin-Resolved Angle-resolved photoemission spectroscopy not only measures the

energy E and the Bloch momentum k but also the spin polarization of an electronic

34


48/250

2.2.1 Mott polarimetry

state. In this section, we review two currently used techniques for spin-resolved

photoemission experiments. To perform spin-resolved electron studies, an electron

spectrometer has to be combined with a suitable spin polarimeter. To determine

the spin of an ensemble of electrons, one needs to utilize a spin-dependent scattering

process.

The traditional one is the Mott polarimetry. Such a technique arises from spin-

orbit interaction of heavy elements (such as gold) that are used as the detector target

where a spin polarized electron beam is injected, causing spin-dependent scattering.

However, since the spin-orbit interaction is usually a small effect compared to the

Coulomb interaction between electrons, the spin dependence for most scattering tar-

gets and a wide range of parameters is small. The relative influence of the electron

spin on the scattering cross section only becomes significant under conditions, where

the mean scattering cross section is small. Thus in general, the efficiency of the Mott

polarimetry method is not so high. For example, measuring the spin polarization of

the surface state of TI Bi2Se3 at a fixed energy E along a fixed momentum space di-

rection cut takes about 6 - 8 hours to gain good statistics using the Mott polarimetry

with synchrotron radiation light source.

A newer technology, the very low energy electron diffraction (VLEED) spin de-

tector was recently applied. A VLEED uses a magnetic scattering target. In electron

diffraction on ferromagnets, the intensities of diffracted beams in general depend on

the spin polarization of the primary electron beam relative to the sample magnetiza-

tion. If the incident electron beam has a very low energy (usually < 10 eV), then the

efficiency of measuring spin polarization is dramatically improved as compared to a

Mott polarimetry.


49/250


Figure 2.3: Mott polarimetry. Schematic of a Mott scattering geometry. Incidentelectrons (red dot) with a polarization P on a high Z nucleus (yellow dot) are backscatteredto the left and right with a probability that is dependent on P . The figure is adapted fromRef. [135].


The interaction Hamiltonian between a photon and spin 12

electron can be described

by the Dirac equation:

H int = 1

2m( p −e

c A)2

+ eΦ − e

2mcσ · (▽ × A) + ie

4m2c2 E · p − e

4m2c2σ · ( E × p) (2.7)

where p is the electron momentum, A is the photon vector potential, Φ is the scalar

potential, E is the electric field and σ is the electron spin. However by using linearly

polarized photons in the UV to soft x-ray regime, it has been shown ( [134]) that

the spin dependent terms are greatly suppressed, and the photon electron interaction

Hamiltonian can be well approximated by the Schrodinger model H int =

− emc

A

· p,

which conserves spin.

Provided the photoemission process is spin-conserving, the spin of the initial state

of an electron in a solid can be determined by measuring its spin after it has been

photoemitted. Mott electron polarimetry ( [134]) is a method of separating electrons

of different spin from such a photoemitted beam based on the use of spin-orbit (Mott)

36


50/250


scattering of electrons from nuclei.

The physical principle of Mott scattering can be understood from the classical

picture of a moving electron scattering off of a stationary bare nucleus of charge Z e.

At low incident energies, the electron interacts with the nucleus predominantly via its

charge, and scattering is described by the Rutherford cross section σR(θ), where the

scattering angle θ is typically small. At high incident energies and in cases where Z

is large, the velocity v of the electron in the electric field E of the nucleus can result

in a considerable magnetic field B in its rest frame given by:

B = −

1

c v × E (2.8)

which, using E = (Ze/r3)r, can be written as

B = Ze

cr3r × v = Ze

mcr3 L (2.9)

where L = mr × v is the electron orbital angular momentum. The interaction of this magnetic field with the electron spin S creates a spin-orbit ( L

· S ) term in the

scattering potential and introduces a spin dependent correction to the Rutherford

cross section:

σ(θ) = σR(θ)[1 + S (θ) P · n̂] (2.10)

where S (θ) is the asymmetry or Sherman function, P is the polarization 2

(< S x >

, < S y >,< S z >), and n̂ is the unit normal to the scattering plane defined by

n̂ = ki × kf | ki × kf |

(2.11)

Where ki and kf are the initial and final wave vectors of the electron respectively.

The direction n̂ depends on whether scattering to the left or right is being considered.


51/250


Figure 2.4: Spin-resolved ARPES (Mott polarimetry) experimental setup. (a)Schematic of the spin-resolved ARPES spectrometer COPHEE. Photoelectrons are energyand momentum analyzed using a hemispherical electrostatic analyzer and are alternatelydeflected at a frequency of 1 Hz into two orthogonally mounted Mott polarimeters. The dualpolarimeter system is shown rotated by 90◦ for clarity. (b) The relationship between thesample and Mott coordinate systems. When θ and φ are both zero, the sample coordinatescan be transformed into the Mott coordinates via a 45◦ rotation about their common z axis.The Mott axes marked red denote the spin components that the polarimeter is sensitive to.The figure is adapted from Ref. [135].

This spin-orbit scattering relation allows for the measurement of the component of

spin polarization perpendicular to the scattering plane in the following way. Consider

a beam of N incident electrons with N ↑ of them polarized along +z and N ↓ of them

along -z, which leads to a net polarization P z = (N ↑ − N ↓)/(N ↑ + N ↓). When thescattering of this beam from a nucleus takes place in the xy plane, there results a

left-right scattering asymmetry Az(θ) defined as:

Az(θ) = N L − N RN L + N R

(2.12)

where N L and N R are the number of electrons scattered to the left and right

respectively through an angle θ. Substituting the relations N L∝N ↑[1+S (θ)]+N ↓][1−S (θ)] and N R∝N ↑[1 − S (θ)] + N ↓][1 + S (θ)] derived from Equation 5 to Equation 7

38


52/250

2.2.2 VLEED

yields:

P z = Az(θ)

S (θ) (2.13)

which shows that given the Sherman function, measurement of Az(θ) yields P z. In

a single Mott polarimeter, it is therefore possible to measure two orthogonal spin com-

ponents of an electron beam by arranging four detectors in two orthogonal scattering

planes in front of a target.

2.2.2 VLEED

The concept of VLEED spin detector is based on a simple and intuitive phenomenon,

which is that as electrons are injected to a ferromagnetically ordered material, the in-

tensities of diffracted beams in general depend on the spin polarization of the primary

electron beam relative to the sample magnetization.

Obviously, the efficiency of such detector depends on the ratio of electrons that

are being reflected with respect to other processes. In a simplified kinematic picture,

the intensity modulation can be described in terms of constructive or destructiveinterference of reflections from different lattice planes. As the energy of the primary

beam is lowered, the radius of the Ewald sphere decreases until it is smaller than the

smallest reciprocal lattice vector. In this situation there are no higher-order diffracted

beams, and only the specularly reflected beam can be observed. This leads to an

increase of the intensity of the specular beam in LEED at very low energies VLEED,

which sets one of the foundation for using VLEED as an efficient spin detector.

So far in the above-paragraph, the discussion only contains the issue how to en-

hance the the simple reflection of an electron beam, but it has not covered the spin

sensitivity. Let us consider a VLEED process on a ferromagnet, such as an iron

(Fe) crystal. In a ferromagnetic material the spin degeneracy of the electronic states

is lifted by the exchange interaction, such that, e.g., a critical point in the band


53/250


structure occurs at lower energy for the majority states than for the minority states.

Therefore, the features in the energy dependence of the absorption and reflection

probability occur at different energies for majority and minority states. This may be

used to realize a spin-dependent scattering for electron spin analysis. Of the three

elemental 3d ferromagnets, Fe has the largest exchange splitting of about 2.5 eV.

Therefore Fe is a good candidate for this type of spin polarimeter.

Figure 2.5: Spin-resolved ARPES (VLEED) experimental setup. Left: Schematiclayout of hemispherical energy analyzer with VLEED spin p olarimeter. A normal hemi-spherical electron analyzer (HA 50 are operated by the HAC 300 unit) is first used to detectthe energy and the momentum of the electronic states. The spin p olarimeter requires threeadditional voltage supplies E P , lens, and V sc. The long dashed line shows light incidence,the short dashed line shows an electron trajectory. Channeltron 1 (labeled ch. 1) is foracquiring non spin-resolved data when the scattering target is removed. Channeltron 2 isfor counting the electrons scattered back under 10◦ from the magnetized Fe surface. Right:Spin-dependent specular electron scattering off a Fe(100) film at about 10◦ incidence angle.(a) Number of scattered electrons; filled symbols are for parallel magnetization of scat-tering target and photoemission source, empty symbols for antiparallel magnetization; (b)asymmetry; and (c) figure of merit. The figure is adapted from Ref. [136].

Fig. 2.5 (left panel) shows schematically the layout of the Fe VLEED spin detector.

The spectrometer is a commercial hemispherical analyser (HA 50, 50 mm mean radius,

with power supply HAC 300 by VSW). Behind the exit slit of the analyzer, the

40


54/250


55/250

2.3 SAMPLE PREPARATION FOR ARPES

preoxidized Fe(001)-p(1 × 1)-O targets as the spin detector [137].

2.3 Sample preparation for ARPES

2.3.1 Single crystal bulk samples

Figure 2.6: A schematic of the single crystal bulk sample geometry mounted inside theARPES measurement chamber. The ceramic top post was used to cleave the sample surfacein situ, to ensure a clean surface for measurements.

For single crystal bulk samples, they are grown by our collaborators including

Prof. R. J. Cava’s group in the Department of Chemistry in Princeton University,

Prof. F. C. Chou’s group in the Center for Condensed Matter Sciences in National

Taiwan University, Prof. Y. Chen’s group in the Physics Department in Purdue

University, Prof. S. Jia’s group in the International Center for Quantum Materials,

Peking University, Beijing 100871, China.

In order to measure them in ARPES, one needs a very clean and flat surface

because ARPES is a very surface-sensitive probe. To do so, the sample is then

42


56/250

2.3.2 MBE film samples

mounted in a geometry as depicted by Fig. 2.6. The crystals are first cut into pieces

of approximately 2 mm × 2 mm × 0.5 mm. A small piece is then mounted onto acopper post, using the commercial epoxy resin Torr seal. For other samples with poor

electrical conductivity, silver epoxy is used instead to ensure good electrical contact

with the sample holder. A ceramic top post is then attached onto the sample with

another layer of epoxy. After the sample is mounted inside the ARPES measurement

chamber, it is cooled and pumped down for a few hours. The sample was then cleaved

insitu at pressures of less than 1 ×10−10 torr, by knocking down the ceramic top-postwith a cleaver. The exposed surface was typically visually shiny, indicating a flat

surface.

2.3.2 MBE film samples

For MBE film samples, they are grown by our collaborators such as

Thesis on Topological Insulator

Documents

Transcript of Thesis on Topological Insulator