Topological Insulators: Theory and Electronic Transport...

Topological Insulators:

Theory and Electronic TransportCalculations

Vadim V. NemytovCenter for the Physics of Materials

Department of Physics

McGill University

Montreal, Quebec

2012

A Thesis submitted to the

Faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of

Master of Science

Contents

Abstract v

Resume vi

Statement of Originality viii

Acknowledgments ix

1 Introduction 1

2 Theory of Topological Insulators 42.1 Topological Insulators in the context of Condensed Matter Theory 42.2 Topological Insulators – Preliminary discussion . . . . . . . . . 92.3 Quantum Spin Hall effect . . . . . . . . . . . . . . . . . . . . . 102.4 Integer Quantum Hall Effect in Graphene . . . . . . . . . . . . . 162.5 Quantum Spin Hall Effect in Perfect Graphene . . . . . . . . . . 27

3 Berry’s phase and the Topological Invariants 343.1 Berry’s Phase and Related Observables . . . . . . . . . . . . . . 353.2 Topological Insulators and the Z2 Topological Invariant . . . . . 413.3 Z2 Invariants and the Spin-resolved Berry’s phase . . . . . . . . 433.4 Summary of the Theory of Topological Insulators . . . . . . . . 45

4 Quantum transport – atomistic point of view 484.1 Tight-Binding Method . . . . . . . . . . . . . . . . . . . . . . . 484.2 Self-Consistency and the Tight-Binding method . . . . . . . . . 514.3 Tight-Binding method and the Density Functional Theory . . . 524.4 Quantum transport . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Quantum transport in Bi2Se3 nanostructures 625.1 Tight-Binding Model for Bi2Se3 . . . . . . . . . . . . . . . . . . 635.2 Transport in Bi2Se3 film with a trench . . . . . . . . . . . . . . 69

6 Discussion and Conclusions 786.1 Cd3As2 - candidate for a new Topological Insulator . . . . . . . 796.2 Berry’s phase and chirality in photonics . . . . . . . . . . . . . . 856.3 Outlook for TB-based numerical study of Bi2Se3 . . . . . . . . . 87

References 89

ii

List of Figures

2.1 SOI-induced spin-momentum locking in 2-D electron gas . . . . 152.2 Unit cell and the Brillouin Zone of Graphene . . . . . . . . . . . 162.3 Energy bands of graphene with Dirac cones at K and K ′ . . . . 172.4 Graphene in Haldane’s model. Two systems with different pa-

rameters are separated by a domain wall in the form of an edge 222.5 Haldane’s model and the equivalence of physics at different edges

and physics on the same edge but with different magnetic fields. 232.6 Edge states of Graphene in Haldane’s model. . . . . . . . . . . . 242.7 Energy bands of Graphene in Haldane’s model in a strip geometry

with two edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Time-reversal invariant momenta is identified for graphene in thebulk and on the “zig-zag” edge. . . . . . . . . . . . . . . . . . . 41

4.1 Diagram of the system in which the central region of interest isconnected to two external leads . . . . . . . . . . . . . . . . . . 59

5.1 Crystal structure of a 6 quintuple layer Bi2Se3 . . . . . . . . . . 625.2 Coordination of neighbouring atoms in Bi2Se3 . . . . . . . . . . 635.3 Energy bands of the Tight-Binding 6 quituple layer Bi2Se3 with

and without the spin-orbit interaction . . . . . . . . . . . . . . . 665.4 Two nearly degenerate Dirac cones in the energy dispersion of

the 6 quintuple layer Bi2Se3 . . . . . . . . . . . . . . . . . . . . 665.5 Helical states associated with different Dirac cones are confined

to opposite surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 675.6 Momentum-Spin Locking in the helical states of Bi2Se3 . . . . . 685.7 Conductance at different energies in 6 quituple layer Bi2Se3, in-

dicating the presence of helical states . . . . . . . . . . . . . . . 695.8 Energy bands of a 9QL slab of Bi2Se3 . . . . . . . . . . . . . . . 715.9 Different set-ups studied are shown schematically and notation

used is explained . . . . . . . . . . . . . . . . . . . . . . . . . . 715.10 Conductance vs. Energy in 9/6/9 quintuple layer type Bi2Se3

systems along primitve vectors ~a1 and ~a2 . . . . . . . . . . . . . 725.11 Conductance vs. Energy in the 9/6/9/6/9 type Bi2Se3 system . 735.12 Conductance vs Energy in 6/9/6 quintuple layer type Bi2Se3 sys-

tems at different extents of the central region . . . . . . . . . . . 745.13 Schematic view of the helical states going around the trenches . 755.14 Energy band diagram of Bi2Se3 in a slab geometry with 2, 3 and

4 quintuple layers . . . . . . . . . . . . . . . . . . . . . . . . . . 76

iii

List of Figures iv

6.1 Photonic bands with Dirac-cone-like dispersion . . . . . . . . . . 866.2 Schemtic diagram of electron crossing between different surfaces

at different incidence angles . . . . . . . . . . . . . . . . . . . . 89

Abstract

In this thesis we investigate quantum transport properties of topological in-

sulator (TI) Bi2Se3 from atomistic point of view. TI is a material having an

energy gap in its bulk but supporting gapless helical states on its boundary.

The helical states have Dirac-like linear energy dispersion continuously cross-

ing the bulk band gap with a spin texture in which the electron spin is locked

perpendicular to the electron momentum. The peculiar electronic structure of

TI material Bi2Se3 is due to a strong spin-orbit interaction and is protected by

the time reversal symmetry. The thesis consists of two main parts. The first

reviews the theory of TI and the second presents our atomistic calculations of

electron transport in the Bi2Se3 material.

In the theoretical review of the physics of TI, I follow the literature and

attempt to present it in a reasonably accessible manner. The theory of TI is

explained in terms of well known physical phenomena including classical and

quantum Hall effects, spin-orbit coupling, spin current, and spin-Hall effect. The

concept of Berry’s phase is then introduced to link with the formal conventional

classification of TI by the topological Z2 invariants. The entire discussion is

within the well known Bloch band theory.

In the second part of this thesis, numerical studies of transport properties

of Bi2Se3 are presented. After a brief discussion of the relevant quantum trans-

port theory and the tight binding atomistic model, we present our calculated

quantum transport results of Bi2Se3 films having a trench in the middle. Such

a large defect, if on normal conductors, would cause significant back scatter-

ing of the carriers. Here, by topological protection of the helical states, back

scattering is forbidden due to the spin-momentum locking. Nevertheless, large

trenches in the film may cause the helical states on the surface to mix inside

the trench, thereby affecting the transmission.

v

Resume

Dans cette these, nous etudions le transport quantique dans l’isolant topologique

(TI) Bi2Se3 a partir d’un modele d’echelle atomique. Un TI est un materiau

ayant une structure de bande de type isolant bien qu’on y retrouve des etats

helicodaux en surface. Ces etats helicoıdaux ont une relation de dispersion

lineaire, dite dispersion de Dirac, qui traverse la bande interdite du cristal. Ces

electrons voyageant selon les relations de Dirac sont contraints a se mouvoir per-

pendiculairement a leur spin. La structure electronique particuliere de l’isolant

topologique Bi2Se3 est due a une forte interaction spin-orbite et est protegee

par une symetrie par renversement du temps. Cette thse comporte deux grands

segments. Dans un premier temps, nous presentons une synthese de la theorie

generale des isolants topologiques. Nous presentons ensuite les resultats de nos

simulation de transport quantique dans le materiau Bi2Se3.

Dans notre resume de la theorie des TI, nous presentons une revue de

litterature et decrivons conceptuellement, dans la mesure du possible, le com-

portement des TI de sorte a rendre notre texte intelligible au non-expert. La

theorie des TI est expliquee a partir de phenomenes classiques et quantiques

connus tels que l’effet Hall, l’interaction spin-orbite, le courant de spin, l’effet

Hall de spin, etc. Le concept de la phase de Berry est ensuite introduit pour

faire le pont avec la classification traditionnelle des TI, laquelle se base sur les

invariants topologiques de Z2. Le tout est presente avec la theorie des bandes

en filigrane.

Dans le second segment de cette these, nous etudions les proprietes physiques

du Bi2Se3 a partir de simulations numeriques. Apres une breve discussion de

certains elements pertinents empruntes de la theorie du transport quantique

et du modele des liens etroits d’echelle atomique, nous presentons les resultats

d’une simulation dans laquelle des electrons voyagent a travers un film de Bi2Se3

ayant une depression en son milieu. Un tel defaut provoquerait une forte diffu-

sion des porteurs de charge dans un conducteur standard. Dans le cas qui nous

concerne, la diffusion des etats helicoıdaux est endiguee par la contrainte qui

force ces etats a voyager perpendiculairement a leur spin. Neanmoins, de larges

vi

Resume vii

depressions dans le film peuvent provoquer le melange des etats helicoıdaux de

surface et des etats localises a l’interieur du cristal, ce qui affecte le transport

des porteurs de charge.

Statement of Originality

This work was produced according to the international standards of conducting

scientific research. Throughout the thesis, due credits were given by citing the

relevant literature to the best of my knowledge.

This thesis contains the following original work and results of my own:

� A comprehensive analysis of the theory of TI is presented in chapters 2 and

3. TI was only recently discovered and no textbooks exist on this topic.

While several reviews can be found in the literature[1],[2], they focus

mostly on the phenomenological aspects of TI. The theoretical review

presented in this thesis is my own account and understanding on the

most important aspects of TI. This account is summarized from many

published literatures as cited in the thesis.

� We carried out numerical studies of quantum transport in Bi2Se3 films

having a central trench. The results were obtained with a model which I

built specifically for this task. To the best of my knowledge, the results of

Chapter 5.2 are original. A manuscript based on the data obtained with

my model is currently in preparation to be published.

� In section 6.1, we argue that Cd3As2 is a possible new TI. Our argument

is based on general considerations of the TI physics. Even though this

has not been confirmed experimentally or by ab initio calculations, due

to the strong argument we present, the new idea is sound.

viii

Acknowledgments

First of all, I would like to thank prof. Hong Guo who granted me a unique

opportunity to engage in an exciting and original research under his direct

supervision and within the environment of prof. Guo’s team. Prof. Guo was

always positive, encouraging and optimistic which helped me overcome my own

limitations and proceed on my path to a completed thesis.

I would like to thank prof. Guo’s entire group for support and the fruitful

discussions. I would like to especially thank Dr. Jingzhe Chen and Dr. Yibin

Hu who had spent a lot of their valuable time answering my trivial questions.

I would like to thank my friends and family for their never-ending support.

Because of them and for them it is merely very difficult to accomplish what

sometimes seems impossible.

ix

1

Introduction

The discovery of topological insulator (TI) has set a fire in the world’s scientific

community and attracted tremendous excitements in physics[3],[4], chemistry[5],

[6],[7],[8], materials science[9],[10] and electrical engineering[11],[12]. TI has an

energy gap in its bulk band structure but supports metallic helical states on

its surface[1],[13]. While all semiconductors have a bulk band gap and some

support surface conducting state, important properties of TI are qualitatively

distinct and new. This is because in TI, the surface helical states are gapless

Dirac fermion-like linear bands crossing the gap with a well defined spin texture

protected by time-reversal symmetry. This spin texture is such that the direc-

tion of the electron spin is locked to perpendicular to electron momentum ~k:

electrons moving in positive ~k have their spins pointing to one direction while

those moving in negative ~k have spins pointing to exactly the opposite direction.

If there is no time-reversal symmetry breaking mechanism in the material such

as spin flipping scattering centers or magnetic fields, the spins cannot be flipped

hence ~k cannot be turned into −~k due to spin-momentum locking, i.e. electron

back-scattering cannot occur. It is thus expected that conduction mediated by

helical states of TI will not suffer disorder scattering. From the point of view of

practical electronic device physics, this property is extremely useful and distinct

from all other known electronic device materials.

TI is also extremely interesting at the fundamental physics level and offers

a real material system for solving long standing scientific puzzles. For instance,

TIs are a manifestation of topological order. This topological order is induced by

the spin-orbit interaction (SOI) and unlike other examples of topological order

such as the fractional quantum Hall effect or chiral p-wave superconductor, the

topologically insulating behavior of TI is a one-particle phenomenon. Thus

many properties can be analyzed exactly or very precisely.

Why the TI physics was not experimentally discovered before? From the

1

1: Introduction 2

materials point of view, this is because to observe TI properties, the bulk band

gap should not be too large – otherwise SOI will not be able to produce Dirac

bands crossing the gap so as to establish topologically distinct TI electronic

structure from that of other “trivial” insulators. The bulk band gap should

not be too small either – otherwise any perturbation (defects etc.) may destroy

the gap. The delicate balance of electronic and material parameters of TI sug-

gests that the topologically protected surface helical states cannot be simply

destroyed by perturbations that respect time-reversal symmetry. However, the

disorder in the bulk can disrupt the band structure and induce charge trans-

port inside the bulk to overwhelm any surface conduction. This is one way in

which electron transport of the helical states may be effectively screened in an

experiment. On the other hand, the helical states themselves may exhibit a

reduced conductance. This is because the helical states require the system to

have suitable dimensions and shape in order for the helical states to establish

and conduct perfectly without any back-scattering. If a helical state is not

isolated from all the other helical states, they will interact and loose some of

the properties associated with a perfect isolated helical state[14],[15]. As such

there is an ongoing research into electron transport properties of TI using both

experimental and numerical techniques.

The work presented in this thesis focuses on Topological Insulators (TI) and

can be divided into two main parts. The first part is comprised of chapters 2 and

3. It investigates the physics that govern TI. Chapter 2 is devoted to a general

discussion of the relevant theory. It firstly introduces the theoretical context and

the necessary background for the latter discussion of TI. Following the initial

introduction, subsequent sections discuss the theory directly related to TI. The

aim of chapter 2 is to build a comprehensive understanding of TI by examining

it from several different angles. Firstly, TI are explained in terms of fairly

intuitive semi-classical well-known phenomena in sections 2.2-2.3. It builds

an intuition but many details are excluded. Then in sections 2.4 and 2.5 the

simplest model of a TI, that occurring in a perfect graphene at zero temperature

is thoroughly studied. The relatively easy mathematics of the model allows us

to understand the key concepts responsible for the formation of the non-trivial

topological phase. Chapter 3 is aimed at building an intuition for TI in terms

of more mathematical and purely quantum mechanical phenomena, at heart of

1: Introduction 3

which is the Berry’s phase. In section 3.1 Berry’s phase is introduced and some

relevant consequences are presented. In section 3.2 the so called “Z2” topological

invariants are formally defined which distinguish regular band insulators and

TI. The formal definition of the invariants by itself offers little insight to most

readers, and so in section 3.3 we relate the Z2 invariants to Berry’s phase.

However it is the hope of the author that the the discussion leading up to the

formal definition gives enough insight to really understand what a TI is.

The second part is devoted to electronic transport theory and its application

to TI. Firstly, in chapter 4 a discussion of the relevant transport theory is

presented. The concepts of a Tight-Binding method and a Landauer-Buttiker

formalism are introduced. This introduces the necessary background for chapter

5.

In chapter 5 we present some theoretical results concerning transport prop-

erties of a well-established TI Bi2Se3. We start by introducing a Tight-Binding

model we used in section 5.1. The model was used to study transport properties

of Bi2Se3 across atomic steps. The chapter is concluded in section 5.2 with the

presentation and discussion of our results.

The thesis is concluded by chapter 6, where we draw some conclusions and

outline possible directions for future research. In section 6.1 we speculate about

Cd3As2 being a new TI. In section 6.2 we bring attention to exciting research

in photonics, which is directly related to the theory discussed in this thesis. We

conclude in section 6.3 by outlining possibilities for future research into electron

transport of Bi2Se3.

2

Theory of Topological Insulators

In this thesis, we are studying TI and in this Chapter we discuss the relevant

theory. We aim to explain the physics behind TI in terms of more familiar

and intuitive concepts such as Hall effect and spin-orbit interaction, in both the

classical and quantum mechanical regimes. First, however, we start in section

2.1 by placing TI in the context of condensed matter physics and introducing

the necessary theoretical framework that shall be used throughout the thesis.

2.1 Topological Insulators in the context of Condensed

Matter Theory

In condensed matter physics one deals with systems consisting of a very large

number of atoms forming a gas, liquid or a solid. The number of atoms can

range from a few hundred to the scale of Avogadro number.

The most general Hamiltonian for protons and electrons interacting in a con-

densed phase can always be written but exact solutions are generally not avail-

able. Therefore, one typically uses some sort of model with its own inherent ap-

proximations. In this thesis we are interested in studying the condensed-matter

systems called Topological Insulators (TI). The most common and straight-

forward way to discuss TIs theoretically is within the framework of the Band

Theory. Thus we proceed the discussion introducing a set of approximations

which lead up to the Band Theory[16]. A common practise is to assume the

Born-Oppenheimer (BO) approximation. It assumes that protons, neutrons

and a certain number of inner-most electrons from a given parent atom can be

taken as a single ion – the rest of the electrons are treated as separate particles.

This approximation is justified because the forces acting on the core electrons

are dominated by the strong attraction from the nucleus of the parent atom.

This allows one to write the total wavefunction as a product of ion wavefunc-

4

2: Theory of Topological Insulators 5

tion and electronic wavefunction, solving for each separately. When solving for

the electronic wavefunction, ionic positions can be taken as essentially static

due to ionic mass being several orders of magnitude higher than that of the

electron, resulting in a much lower ionic velocities on the relevant time-scales.

This approximation is adopted in this thesis, and so the general Hamiltonian

of such a condensed matter system becomes:

H =

Nion∑i=1

P 2i

2Mi

+Ne∑j=1

p2j

2me

+ Vion−ion( ~Ri; ~Ri′) + Ve−e(~rj; ~rj′) + Vion−e( ~Ri; ~rj) + Vext

(2.1)

where the first two sums are kinetic energy terms; ~Pi and ~pj are the momentum

of the ith ion and jth electron respectively; Mi and mj are their masses. The

third term is the Coulomb interaction between all the ions. It depends, in

principle, on all the ions’ positions but the interaction is short ranged and so it

is customary to assume it can be written in terms of few-body integrals, simplest

being a two-body term Vion−ion( ~Ri − ~Ri′). The strength of this term generally

determines whether a gas, liquid or a lattice is formed. The fourth term is the

long range electron-electron (e-e) Coulomb interactions. Due to e-e interactions

being long ranged, generally this term brings the biggest complication to the

attempt at a solution. The fifth term is the ion-electron interactions. The last

term represents some external potential.

In order to solve for the electronic wavefunction the Hamiltonian in equation

(2.1) is rewritten treating ions as static - in line with the BO approximation:

H =Ne∑j=1

p2j

2me

+ Vion−e( ~Ri − ~rj) + Ve−e(~rj − ~rj′) + Vext + Eself−cons. (2.2)

The first and second terms in (2.2) are just the second and fourth terms in (2.1)

respectively and remain unchanged. The third term in (2.2) is the fifth term

in (2.1) together with the additional assumption that ion-electron interactions

are short-ranged and can be written in terms of ~Ri − ~rj. Eself−cons. represents

the energy due to the first and third terms in (2.1); here it is just a constant

calculated in some self-consistent manner. It represents a total shift in energy,

it is not relevant for obtaining the electronic structure and therefore will be

dropped. One can make a more realistic assumption about the motion of ions

which still leaves the problem tractable, namely assume that the ions are lo-

calized and vibrate. Vibrational modes can then be treated as quasi-particles


called phonons, and energy exchange between vibrational degrees of freedom

and electronic motion can be written as phonon-electron interactions. We do

not include phonons nor phonon-electron interactions for the simplicity of the

work.

The final approximation is regarding e-e interaction term in equation (2.2).

One can either drop it entirely assuming e-e interactions are very weak, or treat

them by some other approximation scheme; popular schemes are Hartree-Fock

self-energy representation of e-e interaction or perturbation theory. We simply

drop the e-e interactions. External potential is set to zero. The resultant

Hamiltonian is now:

H =Ne∑j=1

[p2j

2me

+ Vion−e(~rj)

](2.3)

The term Vion−e in equation (2.3) is the background potential due to all the ions

acting on a electron. The above-discussed approximations allow to solve for the

electronic wavefunction relatively easily. The approximations are clearly not

reasonable for all types of condensed matter systems and consequently describe

well only a subset of all the systems. An important property of the Hamiltonian

in (2.3) is that it decouples into single-electron Hamiltonians hj.

H =Ne∑j=1

hj;

hj =p2j

2me

+ Vion−e(~rj)

(2.4)

One can now solve each single-electron Hamiltonian and write the many-electron

wavefunction as a Slater determinant.

We now introduce Band Theory which shall provide the necessary theoret-

ical framework for most of the discussion in the rest of this thesis. Band theory

describes solid state systems which condense into periodic crystal structures.

The key approximations are those discussed above, namely the BO approxima-

tion of static ions and the assumption that e-e interactions are very weak. The

fact that the ions form a periodic crystal means that the single-electron po-

tential Vion−e has the same discrete translational symmetry as the lattice. The

kinetic term is clearly invariant under translations in space and so the Hamil-

tonian in (2.4) as a whole possesses the discrete translational symmetries of the

lattice. As such, Bloch theorem applies. Bloch theorem is well known and so


the detailed discussion is omitted. According to Bloch theorem any solution to

hj in (2.4) must itself be periodic up to a phase and can be written as:

Ψj(~r) =ei~k·~ruj,~k(~r) (2.5)

Here uj,~k is a function respecting full periodicity of the lattice. Momentum

is a good quantum number, and there is a solution for each wave-vector ~k.

The index j here actually also refers to the momentum quantum number, but

by convention wave-vector ~k0 + j · π/2 is identified with the numbers (j, ~k0).

Consequently one can make a plot, called energy bands where for each j, there

is a function Ej(~k) traversing a continuous line. The energy intervals for which

there exist solutions at some ~k and j are called energy bands; energy regions for

which there are no solutions are called energy gaps. Such energy band diagrams

are common in solid state physics.

One invokes the results of Quantum Statistical Mechanics, namely the

Fermi-Dirac distribution, as in (2.6), together with such a band diagram to

understand the behaviour of our many-electron wavefunction. Since the Hamil-

tonian in (2.4) is for a set of decoupled electrons, such an approach is appropri-

ate.

n(E) =1

e(E−EF )/kBT + 1(2.6)

where n(E) is the electron density at energy E, EF is the Fermi energy. One can

easily see that at zero temperature the distribution in (2.6) is a step function

with occupation equal to 1 below EF and 0 above. Once the Fermi energy, EF ,

has been calculated self-consistently or obtained experimentally, it may happen

to fall in a band gap. All the bands below are then called valence bands while

those above – conduction bands. It is customary to call materials with EF

in the band gap band insulators if the gap is on the order of several eV and

semiconductors if it is less. However, semi-conductors and insulators can be

seen as the same class of materials, characterized by a gap.

The above discussed Band Theory predicts correctly a wide range of prop-

erties of insulators, semi-conductors and metals. In this thesis we shall dis-

cuss Topological Insulators (TIs) and the Band Theory provides the theoretical

framework sufficient for most of the discussion that is to follow. Topologi-

cal Insulators are systems possessing time-reversal (T ) symmetry, which have

an energy gap in the bulk but have localized edge/surface states with energy


dispersion continuously crossing the gap and some additional properties to be

discussed. It is because in the their bulk TIs are so similar to ordinary band

insulators that Band Theory turns out to be suitable for the theoretical inves-

tigations of TIs. It happens to be that a particularly suitable balance between

a relatively small band gap and a relatively strong spin-orbit interaction (SOI)

term in TIs induces a unique spin-resolved topology of its bands, distinct from

the trivial band insulators which ultimately leads to the peculiar edge/surface

states. As such in the following sections we shall talk about spin-resolved energy

band topology – something that is not present in the ordinary Band Theory

discussed in textbooks. Also we shall focus on the physics induced by the pres-

ence of a boundary. For this reason we shall use two additional theoretical tools

which can be defined within Band Theory, namely the concepts of a manifold

and of topology. Studying a system on different manifolds simply means solving

the same Hamiltonian with different boundary conditions. In this way one can

include or artificially exclude the surface in our system to explicitly study its

effect. Another concept – that of topology, is intuitively easy to grasp. In this

thesis we shall not calculate topological order, but we shall often refer to it.

The fact that TIs are topologically distinct from trivial band insulators simply

means that the phase transition between the two is necessarily accompanied by

closing and reopening the gap. It is important to know, however, that within

Band Theory one can in principle define topological invariants and calculate

them explicitly. As was mentioned above, in Band Theory one obtains energy

curves Ej(~k) as functions of ~k for any band labelled by j. To each point Ej(~k)

there is a Bloch function uj,~k. One can then define topological invariants in

terms of gradients of uj,~k similar to the way topological invariants are defined

in differential geometry in terms of Gaussian curvature[17]. When we talk of

topological order of the bands we imply its topological uniqueness which can

be captured by topological invariants within Bands theory.

Topological Insulator is an insulator in the bulk and the main physics in

the bulk are typically captured by the Hamiltonian as in equation (2.4) within

the Band Theory. Studying it on different manifolds one can see explicitly

the peculiar surface states arising due to the presence of a surface. Finally, if

necessary, one can in principle define topological invariants within the Band

Theory to identify a given material as either a TI or a trivial band insulator.


The theory presented in this subsection gives sufficient theoretical tools for the

rest of the discussion in this thesis.

2.2 Topological Insulators – Preliminary discussion

Topological Insulators (TIs) are materials with the following characteristic prop-

erties [18],[3]. Firstly, they have a band gap in the interior of the system (also

called bulk) with the Fermi energy inside the gap. As such they cannot conduct

electric current through the bulk at low voltage bias. On the boundaries of

the system – edges and/or surfaces – TIs have states with energy dispersion

continuously crossing the bulk band gap. Topological Insulator can be a 2-

dimensional system or a 3-dimensional system and the boundaries correspond

to 1-dimensional edges or 2 dimensional surfaces respectively[19]. Herefords we

shall just call the surface states as “edge” states. These edge states exponen-

tially decay into the bulk of the system, but along the edge/surface they are not

localized. Furthermore, the edge states are helical. This means firstly that the

spin expectation value of a given edge state is 90° to its momentum expectation

value[20],[18],[19]. Therefore on a given edge/surface two states having opposite

momentum also have opposite spin. Secondly, two states with the same mo-

mentum but residing on opposite edges/surfaces have opposite spins. Here and

throughout the thesis when we speak of momentum or the spin of the state, we

mean expectation value or exact value if momentum and/or spin happen to be

good quantum numbers. The consequence of having such helical states is that

TIs exhibit Quantum Spin Hall Effect (QSHE), which means that in response to

an electric field, there is a quantized spin current in a transverse direction to the

field. Furthermore, Topological Insulators can only occur in systems possessing

time-reversal (T ) symmetry. Lastly, TIs are topologically distinct in a sense

that it is impossible to undergo a phase transition from a TI to an ordinary

insulator without closing energy gap in the bulk of the system[21],[22],[23].

The above paragraph describes the properties of a TI. However, it does not

illuminate what the TI “really is”. More precisely the following questions must

be clarified. What kind of physics make it possible to have a material with

the above described properties? What kind of materials may potentially be

Topological Insulators, i.e. how do we look for new TIs? From the mathematical

physics point of view what are the necessary and sufficient conditions that a


system must posses to be classified as a TI on a firm footing; that is, how do

we classify a TI without listing all of its properties? All of these questions are

addressed in chapters 2 and 3.

Firstly we will talk in general terms about physics such as Hall Effect and

Spin-Orbit coupling which are relevant for understanding TIs. On one hand this

will introduce the concepts necessary for further discussion. On the other hand,

it will build an intuition in terms of simple concepts as to why it is reasonable

to anticipate a material with TI properties at all.

Then in sections 2.4 and 2.5 we study thoroughly two related models which

introduce the simplest possible TI – a two dimensional Tight-Binding Hamil-

tonian with the z-component of spin conserved. The system is described by

relatively simple mathematics which is easy to understand. This allows to un-

derstand thoroughly how the non-trivial topological phase can arise. Other

more complicated TIs can then be understood in terms of the concepts learned

from this section.

While chapter 2 is meant to build an intuitive physical understanding of

TIs, the more mathematical aspects of the theory are left for chapter 3. In

particular it introduces topological invariants as a way to clasify TIs. At the

end of chapter 3, a summary of the entire theory discussion from chapters 2

and 3 is presented, concluding the first part of this thesis.

2.3 Quantum Spin Hall effect

Topological Insulators can be 2D or 3D systems. Both cases exhibit Quantum

Spin Hall Effect (QSHE) but 2D TIs are similar to some previously known 2D

Hall effect systems, while 3D TIs do not have a close relative in the “Hall effect

family”. As such it is best to build understanding about 2D TIs in terms of

well known Hall effect systems first. Having understood the 2D case, one can

then understand a 3D TI by extending the theory behind the 2D TI.

In order to understand QSHE one needs to understand Hall Effect in gen-

eral and then focus on more relevant Hall effect systems. QSHE has two “rel-

atives” in the Hall effect family; they are the Integer Quantum Hall Effect

(IQHE)[24],[18] and intrinsic Spin Hall Effect (iSHE)[20],[18],[19],[25]. The lat-

ter is close to QSHE, first of all, phenomenologically since both produce trans-

verse spin current in response to an electric field. Secondly, both exhibit the


spin Hall effect intrinsicly – i.e. without external agents such as externally ap-

plied fields or doping by magnetic impurities. The similarity between IQHE

and QSHE is a more fundamental one. They both exhibit Hall effect due to

the presence of topologically induced surface states. As such we first present a

preliminary discussion of the Hall effect in general in section 2.3. It is followed

in section 2.3 by a discussion of physics which make intrinsic Spin Hall effects

possible. Integer Quantum Hall system is left for an in-depth analysis in section

2.4. Finally, QSHE in a perfect graphene sheet will be shown to be essentially

due to identical mechanism as that of IQHE.

Thus we proceed with a brief discussion of the Hall Effect.

Hall Effect - Classically and Quantum mechanically

To understand the concept of the Hall Effect it’s worth keeping in mind the

following piece of physics. In classical Electrodynamics (of continuous media,

see for instance ref. [26]) when examining relation of the steady current due to

a constant electric field it is demonstrated that as a consequence of Maxwell’s

laws the most general expression, that of an anisotropic body, is:

ji =σikEk (2.7)

where ~j is the current density, σik – the conductance tensor and ~E – the external

electric field (eqn. 21.8 in ref. [26]). Furthermore, it is shown that while

in the absence of the external magnetic field the conductance is necessarily a

symmetric tensor, once the B-field is turned on – the conductance must acquire

a non-symmetric part as well. Thus in the presence of a B-field:

σik =σIik + σIIik (2.8)

where σIik is the symmetric part and σIIik – the antisymmetric part (eqn. 22.2 in

ref. [26]). The total current can then be written as:

ji =σIikEk + [ ~E × σII ]i (2.9)

The second term is identified as the Hall effect. This formula has an interesting

development in quantum mechanics. P. Streda had showed in 1981 [27] that for

a 2D system together with the external magnetic field (plus a few reasonable

assumptions) the conductance tensor can be written just like in equation (2.8),

with an interesting origin of each term. Equation (12) in ref [27] shows that


the first term in (2.8) is somewhat familiar; it is proportional to the trace of

the Green’s function at Fermi energy multiplied by the density of states at

Fermi energy. This roughly means that conductance is equal to the number of

charge carriers available at a given energy times the probability of getting from

point A to point B. And so in particular if Fermi-energy is in the band gap

then there will be no conductance due to this term. This is a common way to

see/anticipate conductance when looking at E − ~k graph of a given material

within the band theory. The second term of equation 2.8 is more interesting.

It is proportional to the rate of change of charge carriers’ density with respect

to the external magnetic field, evaluated at Fermi energy. This term is said to

have no classical analogies by P. Streda because classically charge carrier density

is independent of the B-field. However, another physicist A. Widom in his

(extremely short) paper [28] related this conductance to the second term in (2.8)

derived in classical Electrodynamics. This illuminates the fact that in Quantum

Mechanics you can have different mechanisms to the antisymmetric tensor in

the classical equation (2.8). The study of different Hall Effects in condensed

matter, such as QSHE, IQHE, etc is the study of different ways you can get

this antisymmetric conductance from first principles. Another important aspect

pointed out by Streda, is that while the first term depends on Green’s function

and thus all the possible impurities, symmetries of the system, etc – the second

term is quite universal, independent of system parameters and thus robust. An

example directly relevant for us is in reference [29], where by treating a periodic

Hamiltonian semi-classically it was shown that σII is proportional to Berry’s

curvature which shall be discussed in greater detail in later chapter.

Intrinsic Spin-Orbital Interaction and the intrinsic Spin Hall Effect

In some materials magnetic impurities may couple to the spin degree of freedom

of the charge carriers and consequently lead to the Spin Hall Effect (SHE) [30].

For the physics of TIs only the intrinsic mechanisms of spin-coupling are relevant

so we proceed with the discussion of the intrinsic SHE (iSHE).

Due to the relativistic effects the Hamiltonian has the term with the Spin

operator S coupled to the electric field. This can either be obtained by using

the Dirac equation and then taking the non-relativistic limit or one can use

classical Electromagnetic theory together with the Special Theory of Relativity

to derive the exact same terms and then second-quantize them in the end. The


correct term for SOI is well known from the rigorous treatment using the former

method (for example see ref. [31] p51 onwards). The correct equation is:

SOI =−ieh8m2

ec2σ · ∇ × ~E − eh

4m2ec

2σ · ( ~E × ~p) (2.10)

where σ is the vector of Pauli matrices, i.e. S = (h/2)σ. Typically electric field

can be written as the gradient of the potential and then the first term vanishes.

For our purposes ~E can be written as the gradient of a potential V (~r) so (2.10)

becomes:

SOI =− h

4m2ec

2σ · (∇V × ~p) (2.11)

We can thus use the the latter method and double check that the final quantized

term is indeed correct. This is an exercise in relativistic EM theory and can be

done; in fact there is a good pedagogical derivation in ref. [32], and we do get

the correct term. What is valuable about the second method is that it makes the

origin of (2.11) more transparent. We can now understand/anticipate 2D QSHE

at least partially using heuristic semi-classical (relativistic) arguments combined

with the results from Quantum mechanics. For details one is referenced to [32]

for an excellent discussion on this subject. In brief, a relativistically-fast moving

spin in an electric field feels a magnetic field in the frame of reference of the

spin and hence feels the force perpendicular to its direction of motion. Opposite

spins feel this force in opposite directions. This force can be expressed in terms

of the original electric field (i.e. electric field in the “lab” frame of reference)

and thus one can get the extra term in the Hamiltonian, namely as in (2.11).

The result of the relativistic EM treatment is that the SOI has the following

form[32]:

SOI =eh

4mec2σ · (~v × ~E) (2.12)

where electron spin is set to be S = (h/2)σ, ~v is the velocity and ~E is the

electric field. To put it into the form appropriate for quantization we write it

with ~v = ~p/me and ~E as a gradient of V (~r) divided by charge and change the

order for cross product, obtaining:

SOI =− h

4m2ec

2σ · (∇V × ~p) (2.13)


which is equivalent to (2.11) which we know to be correct. Now the origin of

the SOI is clear. From this classical term we can learn that moving opposite

spins feel force in opposite directions. We see that the spin is coupled to the

potential gradient which is always present in the condensed matter systems

and so with some additional constraints one can anticipate a sort of intrinsic

Spin Hall effect. That it is linear in momentum will later also prove important.

What is more, (2.13) can be rearranged in a suggestive form, using vector and

del identities (since we are still in a classical regime):

SOI =h

8mec2(σ · ~p×∇V − σ · ∇V × ~p)

=h

8mec2[σ · (~p×∇V ) +∇V · (σ × ~p)]

(2.14)

Now we see some more features of the SOI. For V (~r) being spherically sym-

metric we obtain (working with the form of the first term) the familiar SOI

applicable to an atomic Hamiltonian and in some condensed matter systems.

However, under certain conditions on V (~r), namely V (~r) giving a 2D system

in an antisymmetric well potential, working with the second term in equation

(2.14) and (dV (r)/dz)z, it turns into a Rashba SOI[32]. Thus we also see in a

simple way the heuristic argument for the Rashba-type SOI. What’s more, SOI

interaction in (2.14) appears as a sum of two terms. Indeed, for certain materi-

als such as graphene on a substrate the SOI is a sum of an atomistic term and

a Rashba term. The atomistic SOI and the Rashba SOI considered above are

perhaps the most commonly used in theoretical models. Atomistic SOI comes

from the assumption that the equation (2.14) is dominated by the spherically

symmetric potential in the vicinity of each atom. Rashba SOI is used when

the potential profile of a system is dominated by the global confining potential

which is anti-symmetric. Equation (2.14) is suggestive of these two scenarios

although it is more general. For instance, QSHE has also been realized in a

system where the SOI cames from a globally spherical potential induced by

strain gradient[33]. It is important to understand that the SOI term in (2.14)

is very general. It can sometimes appear in a system intrinsically (atomistic

SOI), sometimes as a side-effect of the experimental set-up (Rashba SOI) and

sometimes induced externally on purpose (Rashba SOI, strain gradient).

For further insight it is inevitable to proceed with more complex quantum

mechanics and unavoidable math that comes with it. SHE prior to the discovery


of the TIs was predicted in several qualitatively different systems. Focusing on

the intrinsic SHE, we look at a simple system producing SHE – a 2D electron

gas together with the SOI [21], as in equation (2.13).

SOI =~p2

2me

− a

hσ · (~z × ~p) (2.15)

This is the system more relevant to our discussion. Overall physics of the model

in ref. [20] are quite different from those responsible for the 2D QSH phase (i.e.

2D TI) but the main importance of that model with respect to ours is two-fold.

First of all, it was shown that a SHE can exist in the absence of a magnetic field

intrinsically due to SOI alone, and secondly they had demonstrated the so-called

spin-momentum “locking”. Spin-momentum locking is when the spin is at 90°

in the E-~p space, as you can see in Figure 2.1. That is, it was demonstrated

that if one only takes free electrons (kinetic terms) and SOI (strong Rashba-

type in their case) some very elegant physics come out leading to the intrinsic

SHE. In addition, in 2D and 3D low energy condensed matter systems one gets

Fermi gas (quasi-particles with ~k as a good quantum number) which in many

ways resembles free electrons and has similar Hamiltonian to the one in (2.15).

Therefore the model in ref. [20] has relevance for QSHE which occurs for a

condensed matter system with a Hamiltonian more complex than that in (2.15).

As the authors of ref. [20] themselves state in the introduction: “In this Letter

we explain a new effect that might suggest a new direction for semiconductor

spintronics research.” We omit the math from the model, because the details

are quite different from the model for the 2D TI. It is, however, worth looking

at the properties that the electrons have in the model of ref. [20] in E-~p space.

Figure 2.1: 2D electron gas with Rashba SOI exhibits “momentum-spin locking”

– electron’s spin at given ~k is perpendicular to it. Figure courtecy of Ref.[20]


We now turn our focus to graphene in the low temperature limit (T ∼ 0).

In references [18] and [24] graphene was shown to exhibit a QSHE and IQHE,

respectively, subject to some constraints. We shall study it in great detail

to understand both the IQHE and QSHE. From two models we shall get a

general insight into a) topological bulk-boundary correspondence which causes

conducting boundary states and b) the effect SOI has on a).

2.4 Integer Quantum Hall Effect in Graphene

The physics of graphene is dominated by nearest neighbour π-type interactions

of the atomic pz-orbitals (~z is normal to the plane of graphene) and graphene’s

honeycomb geometry. The lattice of the graphene is shown in figure 2.2.

Figure 2.2: Graphene unit cell in a) and reciprocal unit cell in b). v1 and v2 are

primitive crystal vectors while r1 and r2 and primitive reciprocal lattice vectors

The main properties of graphene can be studied within the Band theory

together with the Tight-binding approximation which includes arbitrary number

of nearest neighbours. In order to study low energy physics near the Fermi level

it is sufficient to use 1 orbital per atom in the unit cell, i.e. two orbitals per

unit cell, one from atom A and one from inequivalent atom B. A and B are

inequivalent because they are not connected by the primitive lattice vector.

Indeed this is what is done in references [18] and [24]. Constructing thus a

Bloch 2x2 Hamiltonian and only considering nearest neighbours one obtains

(with respect to EF=0):

H(~k)AA =H(~k)BB = 0 (2.16)


H(~k)AB =t1(ei~k·~a′1 + ei

~k·~a′2 + ei~k·~a′3) = t1

∑~a′i

cos(~k · ~a′i) + isin(~k · ~a′i) (2.17)

H(~k)BA =t1∑~a′′i

cos(~k · ~a′′i ) + isin(~k · ~a′′i ) (2.18)

where {~a′i} and {~a′′i } are vectors from atom A to its three nearest neighbour

atoms B and from B to A respectively. They are defined counter-clockwise in a

sense that ~z · ~a1 × ~a2 is positive. To stick to the convention used in ref. [24] we

let {~ai} = {~a′′i }. Then, by symmetry (−~a′2 = ~a′′3,−~a′3 = ~a′′1,−~a′1 = ~a′′2) one gets

(as expected from HBA = H ′AB):

H(~k) =t1∑~ai

cos(~k · ~ai)σx + isin(~k · ~ai)σy (2.19)

where σi are the Pauli matrices and here they act on the orbital space {|A〉, |B〉}.The energy bands are as in figure 2.3.

Figure 2.3: Graphene’s energy bands with vanishing gap at K and K ′ points

giving rise to the so-called Dirac Cones. Figure courtecy of Ref.[34]

The conductance and valence bands touch (gap closes) at E = EF = 0 at

points K and K ′. These points have the property that K = −K ′ which will be

used later on. If we are now to add the six next-nearest neighbours and write

the sum of exponents as cosines and sines, we find that sines cancel out while

cosines double.

H(~k)AA →2t2∑~b′i

cos(~k ·~b′i) (2.20)

H(~k)BB →2t2∑~b′′i

cos(~k ·~b′′i ) (2.21)


where again we define {±~bi} = {±~b′′i } and {±~b′′i } = ± { (~a′′2−~a′′1), (~a′′1−~a′′3), (~a′′3−~a′′2)}. The 2x2 matrix is now:

H(~k) =t1∑~ai

cos(~k · ~ai)σx + isin(~k · ~ai)σy + 2t2∑~bi

cos(~k ·~bi) (2.22)

where I is a 2x2 identity. This is the next-nearest neighbour TB Bloch Hamil-

tonian of graphene in the two orbital approximation. By the symmetry of the

lattice we again get the zero gap at K and K ′. This is the standard way to

study main properties of graphene and can be found in many textbooks (e.g.

ref. [35]). Written in this form the Hamiltonian can be recast into an effective

2-dimensional Dirac equation if expanded about the K and K ′ points using the

~k · ~p approximation (e.g. ref. [36]). It is possible because the Pauli matrices

acting on the orbital space respect the same commutation relations as the Pauli

matrices representing spin (in the original Dirac equation); also because near

K and K ′ one can do linear expansion in ~k (and some other assumptions). For

our purposes, namely to establish the physics behind 2D and 3D Topological

Insulators, this is not desirable. We find the underlying principle not in the

Dirac equation. Instead we shall use another convenient aspect of having our

H(~k) written in term of Pauli matrices as in (2.22). The symmetries of the

system become transparent. This will be especially important in derivations of

ref. [18], discussed in section 2.5.

In ref. [24] Haldane adds to the equation (2.22) certain extra terms and

studies the resulting effect to demonstrate the possibility of the integer quan-

tized Hall effect without the Landau levels, i.e. no magnetic flux. In ref. [18]

a spin-orbit term as in (2.13) is added and the the spin Hall conductance is

derived. We first focus on the effects of the terms added in ref. [24] to then

better understand ref. [18].

In ref. [24] two terms are added to equation (2.22). The first is added by

hand representing antisymmetric local potential at sites A and B of magnitude

|M | and is written simply as Mσz. The second term arises from adding a

magnetic field which respects full periodicity of the system (with Mσz added)

and so results no net flux through the unit cell. Such B-field can be represented

as a curl of the periodic vector potential ~A(~r) having the effect on the original

system of modifying t1 → t1 and t2 → t2ei(e/h)

∫~A·d~r = t2e

±iφ where φ is a

constant phase and the sign in front depends on the relative orientation of the


two next nearest neighbours (details are in [24]). When one includes the effect

of this periodic ~A(~r) and carefully computes H(~k) again in the {|A〉, |B〉} basis,

one finds that the terms due to next nearest neighbours proportional to sine do

not cancel out anymore. The result is:

H(~k) =t1∑~ai

cos(~k · ~ai)σx + isin(~k · ~ai)σy + 2t2∑~bi

cos(~k ·~bi)

+(M − 2t2sin(φ)

∑~bi

sin(~k ·~bi))σz

(2.23)

Now the first two terms still vanish due to the symmetry of the lattice at points

K and K ′. The term in the second line of equation (2.23) contains the constant

M which clearly doesn’t vanish everywhere unless set to zero; it also contains

the sum over sines – this also doesn’t vanish at K and K ′ unlike the sum of

cosines. In general, the Hamiltonian in (2.23) doesn’t have a vanishing gap

anywhere (provided |t2/t1| < 1/3); the gap-closing can only occur at K or K ′

similar to before provided that the third term has cancellation at K or K ′[24].

This can only occur if M = ±3√

3t2sin(φ) where the sign depends on whether

it is at K or K ′.

We have arrived at the most important feature of the Haldane’s model. The

energy spectrum of the system has, in general, a band gap. This gap varies in

E − ~k space and has local minima at K and K ′ points; in fact, at K or K ′ it

can even vanish provided the above mentioned condition on the third term is

satisfied. We then focus at our system at K and K ′ points.

H( ~K ′) =(M − 2t2sin(φ)∑~bi

sin( ~K ′ · ~bi))σz

H( ~K) =(M − 2t2sin(φ)∑~bi

sin( ~K · ~bi))σz(2.24)

we now defineK ′ as the point where the sum over sines evaluates to (−1/2)(3√

3)

and so at K = −K ′ we have (+1/2)(3√

3). Thus, equations (2.24) evaluate to

H( ~K ′) =(M − 3√

3t2sin(φ))σz

H( ~K) =(M + 3√

3t2sin(φ))σz(2.25)

Note that in both cases we get constant(KorK ′) · σz so that the constant

determines the eigenvalues, but the eigenstates never change; they are [1, 0]T

and [0, 1]T . Label these eigen-states as |1〉 and |2〉 respectively. Also note


that if |1〉 has eigen-value λ1 = constant(KorK ′) then λ2 = −λ1 = −const,because σz = [1 0; 0 − 1]. In terms of the notation of ref.[24], the constant

at K ′ and K is proportional to m− and m+ respectively. Now, note that e-

values depend on three parameters −λ2 = λ1 = λ1(M, t2, φ) and so there are

separate cases to consider depending on the values of the parameters. We define

∆E(K) = E|1〉 −E|2〉 for the energy gap at K and we similarly define ∆E(K ′).

We shall examine several cases depending on the parameters (M, t2, φ) and pay

special attention to ∆E(K) and ∆E(K ′) – these values will prove to be of

significance.

Case 1: M = 3√

3t2sinφ, or else M = −3√

3t2sinφ. We now get a single

vanishing gap in the reduced Brillouin Zone (BZ) (or 3 in the entire BZ). They

occur at, respectively, K or else K ′. At the same time at the opposite point,

i.e. K ′ and K respectively, we get a gap equal to 6√

3t2sinφ. Note, these cases

are equivalent to m− or else m+ being zero in ref. [24].

Notice that M = |M | > 0 and M = −|M | < 0 assign an identical eigenvalue

but of opposite sign to the eigenstate |1〉 (and so |2〉) in ~k-space where there is

a gap, i.e. ∆E( ~K ′) and ∆E( ~K) respectively. The sign of ∆E( ~K ′) and ∆E( ~K)

is opposite but it doesn’t matter since this relative sign difference occurs not

in the same system. We shall see later that the sign of this eigenvalue matters

and has consequences in case 3.

Case 2: M = ∆ε such that |∆ε| > 3√

3t2sinφ. Then no matter whether

∆ε < 0 or ∆ε > 0 you get an energy gap at both ~K ′ and ~K. Moreover the

e-state |1〉 has the same sign of the corresponding e-value at both points ~K ′ and

~K. Therefore ∆E( ~K ′) and ∆E( ~K) have the same sign. This case is equivalent

to m− and m+ having the same sign in ref. [24]. Note that the absolute sign of

∆E is not important since we can always redefine it to get a positive gap. The

important thing is the relative sign of ∆E( ~K) and ∆E( ~K ′), as we shall show.

Case 3: M = ∆δ such that |∆δ| < 3√

3t2sinφ. The sign doesn’t matter so

without loss of generality let ∆δ > 0. We now get a band gap at both ~K ′ and ~K

but ∆E( ~K ′) = 2(∆δ + 3√

3t2sinφ) > 0 while ∆E( ~K) = 2(∆δ − 3√

3t2sinφ) =

−|2(∆δ−3√

3t2sinφ)| < 0. This band structure can be called inverted, although

it is not a convention. What is important to point out is that one cannot go

from case 3 to any other case continuously without closing the gap. This signals

that case 3 is a topologically distinct phase from the rest. Also note that when


the Hamiltonian in equation (2.25) is recast in an effective Dirac equation, like

in ref. [24], case 3 corresponds to quasiparticles associated with wave-vectors

~K ′ and ~K as having opposite mass.

Case 3 puts the system into an Integer Quantum Hall Phase. As we shall

now argue, Case 3 exhibits states confined to the boundary (i.e. edge) which

continuously cross the bulk-gap and carry current with the quantized transverse

Hall conductance σxy = ±e2/h. The sign depends on the sign of ∆δ and is

equivalent to just defining where is “up” in our 2D system; whether current will

flow clockwise or counterclockwise. The author in ref. [24] establishes the case

of quantized Hall conductance for the case 3 by applying small magnetic field,

examining how the resultant Landau levels are filled and then taking the limit of

vanishing field. That is a perfectly valid way to demonstrate the result, however

with the hindsight of the theory of Topological Insulators, we shall argue for

the conductance in terms of topological bulk-boundary correspondence[37].

Let H(~k)i be the Hamiltonian from equation (2.23) with the parameters as

in Case i, where i = 1, 2 or 3. We take case 3, i.e. H(k)3, set ∆δ > 0, define

∆E( ~K) = E|1〉−E|2〉 at ~K and ∆E( ~K ′) = E|1〉−E|2〉 at ~K ′, as before. We thus

have a system which in the bulk has a band gap and an additional property that

in the vicinity of the points ~K and ~K ′ we have ∆E( ~K) < 0 and ∆E( ~K ′) > 0.

Relative sign of ∆E( ~K) and ∆E( ~K ′) matters, and we have an inverted band

structure. We now perform the following thought experiment to demonstrate

the existence of edge states.

Take the system defined by H(k)3 and put it on a 2D half-infinite manifold.

We define x’-y’-z’ axis with z’-axis same as in ref. [24] (normal to the plane,

pointing up, i.e. z’=z), x’-axis is parallel to v2 from figure 2.2 and positive

y’ is at 90° normal to x’ such that x′ × y′ = z′. We shall make our manifold

infinite in negative y’.The positive y’ direction we shall call the front (and -y’

– back). The edge in this way is the “zig-zag” edge. In x’-axis it is infinite

(periodic) so we only focus on the front edge effects. On manifold with edges

one can simply get a Bloch Hamiltonian with periodicity only in one dimension

and obtain bands. The purpose of this thought experiment, however, is to learn

to anticipate the results prior to actually computing such 1D-periodic Bloch

Hamiltonian. Let’s define the system of H(~k)3 together with the half-infinite

manifold as H(~k)3 −M f3 . M stands for manifold and the superscript f means


that it is semi-infinite with the edge at the front.

Now, at the front boundary of H(~k)3 −M f3 add system H(~k)2 on another

half-infinite manifold with the edge at the negative y’ i.e. at the back. This

added system we call H(~k)2−M b2 and the total system that results from fusing

the two we call H(~k)3 −M f3 /H(~k)2 −M b

2 . The total system is schematically

shown in figure 2.4. H(~k)2 has bulk band gap with ∆E( ~K) and ∆E( ~K ′) both

positive, while H(~k)3 has an inverted band structure with alternating signs of

∆E( ~K) and ∆E( ~K ′).

Figure 2.4: The system H(~k)3 −M f3 /H(~k)2 −M b

2 . It is infinite in all extents,

but has a “zig-zag” boundary betweed the systems H(~k)2 (yellow and light blue

atoms) and H(~k)3 (orange and dark blue atoms).

If we separately look at 2D-periodic H(~k)2 and H(~k)3 bulk-bands we re-

alize that one set of bands cannot be continuously transformed into the other

set without closing the energy gap. Formally one can say that they cannot be

transformed one into the other adiabatically without closing the gap due to

different topology of the bands (like a coffee cup cannot be transformed into a

sphere without closing the hole). At the same time, the energy of the above de-

fined “H(k)3−M f3 /H(k)2−M b

2” system must have energy continuously defined

throughout the entire manifold. We conclude that far away beyond and before


the boundary the energy must be well described by the 2D-periodic bulk-bands

of H(~k)2 or H(~k)3 respectively. Thus in the neighbourhood of the boundary

we must have energy states which continuously transform the bands of H(~k)3

into H(~k)2 and therefore cross the gap due to the above-discussed topology

considerations.

We thus have concluded that in the energy region corresponding to bulk-

gap of H(~k)2 and H(~k)3 there exists a continuous energy band. We do not

know the exact shape of it on E − ~k diagram, but we know it either “starts”

on the valence band and “ends” at the conduction band (as we read E − ~kgraph from left to right) or the other way around. We also know that the states

corresponding to these energies are localized along the edge. These edge state

exist due to topological distinction of H(~k)2 and H(~k)3 bulk-bands.

Note it is necessary to have an energy gap in the bulk in order to get edge

states. This is because otherwise bulk states with the same energy and the same

edge-parallel momentum component will couple to our “edge-states-hopefuls”

and result in states that are delocalized throughout the entire system – contrary

to the definition of a boundary state[38].

Figure 2.5: The system H(~k)3 −M b3/H(~k)2 −M f

2 with magnetic field B = B0

on the left is equivalent to the system H(~k)3−M f3 /H(~k)2−M b

2 with magnetic

field B = −B0 on the right.


We now repeat the entire argument but for the edge being at the back of

the half-infinite manifold on which H(~k)3 is defined. Using the above notation

we now have “H(~k)3 − M b3/H(~k)2 − M f

2 ” system. Recall that H(~k)i has a

chirality due to the magnetic field; this will now manifest itself. The situation

is visualized in figure 2.5, studying physics at opposite edges is like studying

physics at the same edge with the opposite magnetic fields. Now the back of

the system corresponds to -y’. More precisely it corresponds to -y’/z’/x’ which

is equivalent to +y’/-z’/x’ which in turn is equivalent to switching B(z′) to

B′(z′) = B(−z′) = −B(z′). Transforming B → −B is equivalent to A → −A,

which in turn is equivalent to φ→ −φ. This interchanges the sign of the gap of

the system at ~K and ~K ′, i.e. the values of ∆E( ~K) and ∆E( ~K ′) get interchanged.

Therefore if we have a positive slope of the edge-band in the above discussed

example, then here we get the negative slope. The BZ of graphene with a single

edge is shown as a projection of the bulk BZ in figure 2.6. In the figure the

energy dispersions of the edge states at the front edge and at the back edge

are shown separately, in green and burgundy respectively. The bulk energy

dispersion is in blue.

a) b)

Figure 2.6: BZ of the system H(~k)3 possessing only a) the front or b) the back

edge. Shown as a projection of the Bulk BZ. Bulk bands are in blue; green and

burgandy are the edge states. a = |v2|


The energy dispersion of graphene in a strip geometry with both the front

and the back edges is shown in figure 2.7.

Figure 2.7: BZ of H(~k)3 which has both the front and the back edges. Green

and burgundy bands correspond to states localized along the front and back

edge respectively. Bulk bands are in blue. a = |v2|

We have found that at each front and back edges of H(~k)3 we have localized

boundary states which are chiral – i.e. positive kx′ states are on one edge and

negative kx′ states are on the opposite edge. This phenomenon is called chirality.

Chirality in our system has as its ultimate origin the fact that the third term

in eq. 2.23 which is responsible for producing an inverted bulk band structure

distinguishes between different directions in real-space. The system is inherently

chiral as can be seen in figure 1 in ref. [24].

An important feature of the chiral edge states follows. Suppose we now add

e-e interactions and try to treat boundary states with an effective 1-dimensional

Hamiltonian. Then we may wish to proceed with the approximations of the Lut-

tinger model. However, in this case we will be forced to set the amplitude for

the back-scattering event to zero. The probability of one electron scattering

another one backward is essentially zero since the forward moving and back-

ward moving electrons are separated by a macroscopic distance. Thus, the

Luttinger liquid (with back-scattering=zero) reduces to the model of the free

quasi-particle rather than charge and spin waves. In other words Luttinger

model is not a suitable approximation for the effective 1-D Hamiltonian of our

edge subsystem. Instead, our edge-states are best described as quasi-particles

with crystal momentum (even in 1-D). Such a 1-D system can only exist as

a part of a bigger 2-D system, otherwise we do not get the spatial separation

of forward-movers and backward-movers. An important feature of such a 1-D


subsystem is that it’s robust against impurities or electron-electron interac-

tions because back-scattering is forbidden. This is contrary to the typical 1-D

fermionic system which is very sensitive to impurities resulting in Anderson

localization[39].

From the fact that we have chiral states it is easy to see how the transverse

Hall conductance arises. Near Fermi energy which is in the gap, the conduction

of current can only take place at the edges. If we establish a small voltage in

x’-direction – a current will flow from, say, left to right. This is equivalent to

right-moving states (+kx′) being more populated than the left-moving states

(−kx′) on our band diagram in figure 2.7. However, since our ±kx′ states reside

on front/back edges this means we have an imbalance of net charge accumulated

on each edge. The front edge with forward-moving electrons has more electrons

than the back edge. This establishes a voltage between the back and front edges

which is nothing but the Hall voltage. This gives a transverse conductance

(charge transfers from one edge to the other). In our earlier general discussion

of the Hall effect in section 2.3, we have seen that the Hall effect is precisely this.

This Hall conductance is quantized because it is proportional to the density of

states at the Fermi energy which gives 1 per edge. It is protected against back-

scattering due to impurities or e-e interaction due to chirality. Thus you get

integer quantized/quantum Hall effect with conductance ±1 e2

h.

This Hall conductance can be shown to be equivalent to the so called topo-

logical Chern number (see ref. [40],[41] for instance); it is a topological invariant

defined for energy bands. Chern number is induced by the so called Berry’s

curvature in momentum space. For now, these concepts over-complicate the

heuristic discussion of the topological boundary states but later on we shall

return to the concept of Berry’s curvature and its consequence for the TIs.

A word of caution is necessary. From the above discussion it follows that the

origin of the boundary states is in the fact that topology of bulk-bands of our

system of interest, H(~k)3, is distinct from that of the system on the other side

of the boundary, H(~k)2. It is then also intuitive that small perturbations can

change the shape of the bands but not the topology and thus the conducting

boundary states persist. It is intuitive but not conclusive and in fact it is a

subject of recent and ongoing research. A careful analysis into the origin and

stability of surface states of IQHE and QSHE had shown[42] that the existence


of gapless (and so conducting) boundary states is not just dependant on the

topology of the bulk but also on the conservation of a certain observable related

to the physics of the boundary states. For IQHE it’s the conservation of charge

and for QSHE it is the conservation of spin (or pseudo-spin in general). We

shall return to the question of the stability of these topological edge states later.

We finish this section by stressing the importance of topology of the bands.

It has been demonstrated above that the gap must close and reopen in order

to go from H(~k)2 to H(~k)3. Thus the edge states can be seen as the system

undergoing a topological phase transition across a boundary. In going from a

trivial band insulator like H(~k)2 to vacuum across a boundary the gap does

not need to close and reopen and so trivial insulators do not have gap-crossing

edge states. One can say that trivial insulators are insulators which can be

adiabatically taken to the atomistic limit without closing the gap. Vacuum and

trivial insulators are topologically equivalent in this sense. It follows that at

the edge between H(~k)3 and vacuum we shall also have the chiral edge states

described above.

2.5 Quantum Spin Hall Effect in Perfect Graphene

We now proceed to the model in ref. [18] which was one of the first to predict

and define the topological QSH phase in 2D. Initially they identified it as a

new type of iSHE system, because the spin current was due to the edge states.

Shortly after the same authors have published a paper where they defined a

Z2 topological invariant for a general class of 2D T -invariant bulk insulators to

uniquely define such a condensed matter state [3]. It deals with the 2D graphene

as defined above in (2.22) with the focus on the effect of adding a SOI term.

In ref. [18] they work on a model of a 2D graphene sheet with a SOI added

to study the effect of SOI at T=0 limit. An important feature of graphene

is that its bands form Dirac cones with Fermi energy falling precisely at the

Dirac point. Given that the gap is closed only at the Dirac points, adding SOI

(a relatively small effect normally) dominates the existence of the energy gap.

This is, in general, the requirement for potentially having a TI.

As has been already discussed in the previous section, the energy eigen-

states near the Fermi energy are near the nonequivalent points ~K and ~K ′. As

such, to study the low energy physics, in ref. [18] they focus on H( ~K) and


H( ~K ′) and model the effective Hamiltonian in terms of the Bloch functions at

~K and ~K ′. In terms of the Tight-Binding Bloch Hamiltonian that we already

introduced in equation (2.22) and thoroughly discussed, what is done in ref.

[18] can be explained as follows. The ~k-space can be represented as an infinite

enumerable set {~ki}, and then we can write H(~k) in a matrix form 〈~k|H|~k′〉 as

an infinite dimensional diagonal matrix:

〈~k|H|~k′〉 =

. . . 0 0

0 H(~ki) 0

0 0. . .

. (2.26)

If we now ask which eigen-vectors correspond to eigenvalue equal to Fermi

energy we shall find that it is a linear combination of those vectors with non-

zero entries only at positions i corresponding to ~K or ~K ′. Therefore we can

write an approximation as in (2.27) leading to the form studied in ref. [18] in

equations (13)-(15).

〈~k|H|~k′〉 =

. . . 0 0

0 H(~ki) 0

0 0. . .

≈(H( ~K) 0

0 H( ~K ′)

). (2.27)

Where each element of the final matrix is a 2× 2 block matrix in {A,B} orbital

space (or equivalently A,B sublattice space). Finally, the authors in ref. [18]

use the result of the k · p expansion [36] around ~K and ~K ′ to get an effective

Hamiltonian in equation (2.29) in terms of slowly varying envelope functions

Ψ(~r):

Ψ(~r) = [(uA, ~K , uB, ~K), (uA, ~K′ , uB, ~K′)]ψ(~r) (2.28)

The effective Hamiltonian in equation (2.29) is written in a matrix form with

respect to the basis { |A, ~K〉,|A, ~K ′〉, |B, ~K〉, |B, ~K ′〉 }.

H0 = hvF ψ†(~r)

(

0 0

0 0

) (kx + iky 0

0 −kx + iky

)(kx − iky 0

0 −kx − iky

) (0 0

0 0

) ψ(~r).

(2.29)


Notice in particular that points ~K and ~K ′ are decoupled. This Hamiltonian

gives gapless energy spectrum E(~q) = ±vF |~q|, where ~q is the wave-vector with

respect to ~K or ~K ′. We now rewrite it in a notation originally used in ref. [18]:

H0 = hvF ψ†(~r)(σxτz ~kx − σyI ~ky)ψ(~r) (2.30)

where I is the 2× 2 identity on {| ~K〉,| ~K ′〉}, σj and τi are Pauli matrices acting

on {|A〉,|B〉} and {| ~K〉,| ~K ′〉} subspaces respectively. In this representation it’s

easier to see which additional term can induce an energy gap and at the same

time respect the existing symmetries.

The Hamiltonian in (2.30) has inversion symmetry with the centre of inver-

sion being the point between (any) atom A and B, it also has time-reversal (T )

symmetry. The time-reversal symmetry is easy to see since the Hamiltonian of

pure graphene has no magnetic terms. The inversion symmetry of the Hamilto-

nian in equation (2.30) can be demonstrated in a few steps. If a Hamiltonian H

has an inversion symmetry, it means that it commutes with the parity operator

π. Therefore, it suffices to show that Hπ|Ψ〉 = πH|Ψ〉 for any state |Ψ〉. We

show this is true for the basis ket |A, ~K〉; it can be shown for the rest of the

basis kets similarly.

Hπ|A, ~K〉 =H|B,− ~K〉

=H|B, ~K ′〉

=(−kx + iky)|A, ~K ′〉

(2.31a)

πH|A, ~K〉 =π(kx − iky)|B, ~K〉

=(−kx + iky)π|B, ~K〉

=(−kx + iky)|A,− ~K〉

=(−kx + iky)|A, ~K ′〉

(2.31b)

The last line of equation (2.31a) equals the last line of equation (2.31b) and so

the inversion symmetry of H in equation (2.30) follows.

We now rewrite equation (2.25) of Haldane model to make easy compar-

isons:

H( ~K ′) =(M − 3√

3t2sin(φ))σz

H( ~K) =(M + 3√

3t2sin(φ))σz(2.32)


Recall, that here σz acts on the { |A〉,|B〉 } subspace. For further convenience,

we rewrite it in { |A, ~K〉,|A, ~K ′〉, |B, ~K〉, |B, ~K ′〉 }, and add a subscript “Hal-

dane”:

HHaldane = MσzI − 3√

3t2sin(φ)σzτz (2.33)

As before, here σz acts on {|A〉,|B〉} subspace, while I and τz act on {| ~K〉,| ~K ′〉}.We can now easily see that the term MσzI which was added in Haldane’s

model opens the gap but also breaks the inversion symmetry:

MσzIπ|A, ~K〉 =MσzI|B,− ~K〉

=MσzI|B, ~K ′〉

=Mσz|B, ~K ′〉

=M |A, ~K ′〉

(2.34a)

πMσzI|A, ~K〉 =πMσz|A, ~K〉

=πM |A, ~K〉

=M |B,− ~K〉

=M |B, ~K ′〉

(2.34b)

The last lines of equations (2.34a) and (2.34b) are not equal. The term MσzI

is extrinsic and naturally it is not present in a perfect graphene. We can also

see that the other term, the extrinsic term due to periodic magnetic field,

3√

3t2sinφ · σzτz, opens the gap but breaks another symmetry of (2.30) – the

T -symetry. The T operator is T = exp(−iπσy/h)K, where K is the complex

conjugation. We check the commutation [σzτz, T ]; note that T | ~K〉 = |− ~K〉 in

~k-space:

σzτzT |A, ~K〉 =σzτz|A,− ~K〉

=σzτz|A, ~K ′〉

=σz|A,− ~K ′〉

=σz|A, ~K〉

=|A, ~K〉

(2.35a)


T σzτz|A, ~K〉 =T σz|A, ~K〉

=T |A, ~K〉

=|A,− ~K〉

=|A, ~K ′〉

(2.35b)

One can see that [σzτz, T ] 6= 0 since the last lines in equations (2.35a) and

(2.35b) are not equal. T -symmetry is broken by the term 3√

3t2sinφ ·σzτz, and

so this term is not present in a naturally occurring graphene either.

Now, as was discussed in the section 2.3 the SOI term is always present in

a condensed matter system. What was realized in ref. [18] is that if we add the

spin degree of freedom in (2.30) and consider the naturally occurring SOI then

it will be of the form ∆SOσzτzsz, where sz is a Pauli matrix acting on spin. An

additional term of this form gives a system of two decoupled Hamiltonians –

one for spin up and one for spin down. Each of these subsystems corresponds to

Haldane’s model in equation (2.33) with M = 0, ∆SO = 3√

3t2sinφ and gives a

non-trivial topology, i.e. a system we reffered to as H(~k)3 in section 2.4. The

spin degree of freedom made it possible to have an effective magnetic field for

each spin without breaking the T -symmetry. To see that the SOI is indeed

of the form ∆SOσzτzsz consider the following three arguments. Firstly, rewrite

SOI and observe that SOI( ~K) = −SOI( ~K ′).

SOI(~k) = − h2

4m2ec

2~σ · (~∇V (~r)× ~k) (2.36)

Now, using the fact that ~K ′ = − ~K, plugging K and K ′ for k in (2.36) we see

that SOI( ~K) = −SOI( ~K ′). Therefore SOI is proportional to τz in {| ~K〉,| ~K ′〉}subspace. Secondly, ~σ can be re-written sz because the z-component of spin is

conserved. Lastly to see that it is proportional to σz use the fact that |A〉 = π|B〉where π is the parity operator and where we define ~r = 0 to be the inversion

centre.

〈A|SOI(~k)|A〉 =(〈B|π)SOI(~k)(π|B〉)

=〈B|(πSOI(~k)π)|B〉

=〈A|SOI( ~−k)|A〉

(2.37)

where we used π = π†, and that parity effect on the operator is to take ~k to

−~k. Now it follows from SOI(−~k) = −SOI(~k) that SOI is also proportional


to σz in {|A〉,|B〉 } subspace. Therefore the full Hamiltonian of graphene is

H0 = hvF ψ†(~r)(σxτzIs ~kx + σyIkIs ~ky + ∆SOσzτzsz)ψ(~r) (2.38)

where Ik and Is are identity operators on space {|K〉,|K ′〉} and {|↑〉, |↓〉} re-

spectively.

As was already mentioned, the Hamilotnian in (2.38) has the z-component

of the electron spin as a good quantum number. Therefore it decouples into

two Hamiltonians, one for spin down and one for spin up. Each one is a copy

of Haldane’s model for case 3 of section 2.4 (case 3: |M | < 3√

3t2sinφ ) with

M = 0, 3√

3t2sinφ = ∆SO for spin up, and 3√

3t2sinφ = −∆SO for spin down.

Therefore one gets one pair of chiral edge states for spin up and one pair of

chiral states for spin down, but with the opposite chirality. Note we again use

the notation of section 2.4. The net effect is that on the front edge you have

spin up states with +kx′ momentum and spin down with −kx′ ; at the back edge

you have the reverse situation – spin up with −kx′ and spin down with +kx′ .

Such edge states are often called “spin filtered” or helical. They are not chiral

because each edge has both +kx′ and −kx′ states. We can see that under a small

bias favouring current from, say, left to right – the front edge will have spin up

states (+kx′) more populated while the back edge will have spin down states

(also +kx′) more populated. Thus the back edge has the expectation value of

the z-component of the spin skewed towards net spin down polarization. The

front edge has net spin up. Therefore there is a net “spin voltage” between

the two edges, i.e. there is a spin Hall conductance – transverse transport of

spin in response to an electric field. The value of the spin conductance can be

induced from the Hall conductance for each spin, using the standard formula

Js = (h/2e)(J↑−J↓)[18]. From section 2.4 we know that a quantized charge Hall

conductance of magnitude e2/h is associated with a set of chiral states. For each

spin the current is in opposite direction and so the charge Hall conductance is

of opposite sign. Therefore the spin Hall conductance is quantized and is given

by:

σxyspin =h

2e

[e2

h−(−e

2

h

)]=

e

2π(2.39)

A general discussion of robustness of these helical states will take place

further down in greater detail. It is important to note, however, that in gen-

eral z-component of the spin may not be conserved. For instance there may be


a small Rashba-type potential present, for example due to graphene being de-

posited onto a substrate. In that case you still get a QSHE and topological edge

states but the spin Hall conductance is not quantized anymore. It is important

to understand that if the energy of SOI term in equation (2.38) is bigger than

the added Rashba term, then the gap at ~K and ~K ′ is still dominated by the

∆SOσzτzsz term. Therefore edge states will exist due to non-trivial topology of

the bulk-bands. It is still the case that for spin up and spin down the effect is

opposite and so you still expect spin filtered edge states. This has been demon-

strated rigorously by Murakami et al. [43] who defined a quantity s(c)z which is a

sort of conserved component of sz; it also represents a type of spin polarization.

They showed that you still get QSHE. It has also been demonstrated numeri-

cally that for 2-D Topological Insulator the QSHE persists in the presence of

Rashba SOI for a certain range of Rashba SOI strengths[44].

We have thus seen that in 2D graphene with T -symmetry and intrinsic

SOI you get a QSHE. The system in (2.38) described above is a 2D TI. In fact

in the literature sometimes Topological Insulator is called a QSH insulator or

a QSH phase. What creates the topological edge states is the fact that you

have topologically-nontrivial band structure in the bulk with alternating sign

at ~K and ~K ′. What gives this system helicity and transverse spin conductance

is the fact that the term creating inverted gaps in the bulk is the SOI which

distinguishes between “up” and “down” in real space. This was discussed in

section 2.4.

The robustness to the back-scattering for these helical states is not the

same as for the chiral states of Haldane’s model in ref. [24]. For instance a 4

electron interaction term can back-scatter a pair of electrons on the same edge

in the QSHE whereas this is not possible for IQHE. More generally, it has been

shown that since spin is not a conserved quantity, helical states of the TI are

less robust than the chiral states of the IQHE system from Haldane’s model

[42]. The existence of the Helical states depends on the assumption that the

edge states do not undergo spontaneous time-reversal symmetry breaking. The

stability of the edge states is an ongoing research.

3

Berry’s phase and the Topological Invariants

In this chapter we shall talk about the phenomenon called Berry’s phase. First,

Berry’s phase can be used to define topological invariants distinguishing topo-

logically non-equivalent phases of matter. Second, Berry’s phase is related to

the motion of the centre of charge inside of the material, i.e. it can induce

a non-trivial electronic transport. More generally Berry’s phase has a related

observable and can be measured in experiment. Lastly, Berry’s phase strongly

depends on the energy bands of the material. As such Berry’s phase is a pow-

erful concept, which gives an alternative explanation unifying the topology of

energy bands and the QSHE.

Section 3.1 goes into the discussion of Berry’s phase and related phenomena.

Whereas earlier discussion was focused on introducing TI in terms of intuitive

semi-classical concepts, this section lays out purely quantum mechanical con-

cepts which ultimately lead to QSH effect and hence a TI. On one hand, this

section gives an alternative way to build an intuition for why a TI may exist, in

terms of purely quantum mechanical concepts. It introduces concepts that are

more purely quantum mechanical and mathematical in nature and yet are still

relatively easy to grasp; these concepts can then be used to explain the formal

non-intuitive mathematical definition of a TI.

Finally we introduce the full formal definition of a Topological Insulator

in section 3.2. A formal definition for a topological Z2 invariant uniquely dis-

tinguishing TIs is presented. The formal definition is not transparent and to

people who are not in mathematical physics or are field theorists it offers little

insight. The concepts of Berry’s phase and related concepts are used to explain

the formal definitions in more transparent terms in section 3.3.

34

3: Berry’s phase and the Topological Invariants 35

3.1 Berry’s Phase and Related Observables

We proceed to the discussion of the Berry’s phase. The concept of Berry phase

arises from studying the evolution of the state of the system under adiabatic

conditions [45]. Suppose we have a system which is described by the Hamil-

tonian H and suppose that H depends on some parameters ~R = [R1, R2, ...],

H(~R), which we can be varied externally, and adiabatically slowly. An example

of ~R could be an external electromagnetic field varying adiabatically slowly. At

each fixed value of ~R, there is a natural basis of energy eigenstates satisfying:

H(~R)|n(~R)〉 = En(~R)|n(~R)〉 (3.1)

Here |n(~R)〉 is the nth eigenstate with energy En(~R). The adiabatic evolution

of the state is well known [45], it is:

|Ψ(t)〉 = e−1h

∫ t0 dt′En(~R(t′))eiγn|n(~R(t))〉 (3.2)

The first exponential in equation (3.2) is the familiar adiabatic dynamical phase

factor. The second exponential is added by hand and has to be solved for. One

then plugs equation (3.2) into time-dependent Schrodinger’s equation and solves

it under the adiabatic assumption. The derivation is straight-forward [45] and

leads to the expression for the phase γ:

γn(t) = i

∫ ~R(t)

~R(0)

〈n(~R(t))|~∇~R n( ~R(t))〉 · d~R (3.3)

The phase defined by equation (3.3) is called Berry’s phase. We see it is not,

in general, equal to zero. We are especially interested what happens after an

adiabatic cycle. Let ~R(0) = ~R(T ) after having traversed some closed path C in

the parameter space in time T:

γn(C) ≡∮〈n(~R)|i~∇~R|n(~R)〉 · d~R (3.4)

Now, it is particularly useful to rewrite Berry’s phase in several different forms

to get a better insight into its significance. First we rewrite (3.4) using the

Stoke’s theorem which states that for a vector field ~F the integral over a closed

contour can be transformed into an integral over an area enclosed:∮∂S

~F · d~s =

∫S

(~∇× ~F ) · d~a (3.5)


Here ∂S is a closed contour, d~s is the infinitesimal vector tangential to this

contour; S is the area enclosed by ∂S and d~a is an infinitesimal area element,

pointing as a vector normal to the surface. Now setting ∂S = C and d~s = d~R,

~F = ~∇~Rn(~R) we get:

γn(C) =− Im(∫ ∫

AC

∇× 〈n|~∇n〉 · d ~A)

=− Im(∫ ∫

AC

〈~∇n| × |~∇n〉 · d ~A) (3.6)

The quantity 〈~∇n| × |~∇n〉 is called the Berry’s curvature. We can further

modify equation (3.6) to obtain more useful information. We insert identity in

a form of projections onto eigenstates:

γn(C) = −Im

(∫ ∫AC

d ~A ·

(∑m6=n

〈∇n|m〉 × 〈m|∇n〉

))(3.7)

Now we use 〈m|∇n〉 = 〈m|∇H|n〉/(En − Em);n 6= m and obtain:

γn(C) = −∫ ∫

AC

d ~A · ~Vn(~R) (3.8a)

~Vn(~R) = Im

(∑m 6=n

〈n(~R)|∇~RH(~R)|m(~R)〉 × 〈m(~R)|∇~RH(~R)|n(~R)〉(En − Em)2

)(3.8b)

From the above equations we see several important properties of the Berry’s

phase. First of all, it does not only depend on the Hamiltonian and correspond-

ing eigenstates defined on the path ~R ⊆ C. Instead from equations (3.6)-(3.8)

we see that Berry’s phase is sensitive to the system’s ~R-dependence throughout

the entire area enclosed by the closed path C. What is especially relevant for

the discussion of the Topological Insulators, Berry’s phase is sensitive to energy

dispersion of H(~R) inside AC . In particular, due to the denominator in (3.8b),

for a given n the value of γn is dominated by those states |m(~R)〉 that share a

degeneracy or near-degeneracy with the states |n(~R)〉.Berry-phase most naturally occurs in the solid state systems described by

Bloch Hamiltonians where external adiabatic perturbation usually comes from

electromagnetic fields and the phase space is provided by the Brillouin Zone

itself[46]. To see this explicitly we now review the main features of the systems

with Bloch symmetry and connect it to the Berry-phase formalism.


First of all, the many-electron wavefunction for the Bloch system is the

Slater determinant made of single quasi-particle wavefunctions. Therefore the

many-electron wavefunction is a product of independent quasi-particles labelled

by ~k and n (nth band) with energy dispersion of each quasi-particle dictated

by the shape of the band. These electrons are decoupled and therefore under

external single-particle perturbation the evolution of the many-electron state

is the product of independently evolving single-electron wavefunctions. Each

single electron wavefunction is subject to (i.e. “feels”) a single particle Hamil-

tonian labelled by k, i.e. H(~k)Ψn,~k(~r) = En~kΨn,~k(~r). Now, if we introduce

an electromagnetic vector potential ~A, the single electron Hamiltonian is now

modified to be:

H =1

2m

(~p− e

c~A)2

+ V (~r) (3.9)

Here V (~r) possesses periodicity of the crystal. Now assume that ~A varies in time

adiabatically slowly, ~A = ~A(t). Let’s stick to 1D for simplicity, so V (x + a) =

V (x) where a is the periodicity of the crystal. The time-dependent Schrodinger’s

equation in the adiabatic approximation is then:

H(t)Ψn(t;x) = En(t)Ψn(t;x) (3.10)

The solution to this is well known and can be written in the form [46]:

Ψn(t;x) = eikxun,k(t)(x) (3.11)

where the exponent is time independent but the function un,k(t)(x) corresponds

to the solution of the original unperturbed Hamiltonian for the quasi-momentum

quantum number k(t) = k − (e/ch)A(t). Recall that u(x) is the periodic part

of the Bloch eigen-function. The above equations can be rewritten just for the

u(x) to get:

H(t)un,k(t) =H(k(t))un,k(t)

=

[1

2m(p+ k(t))2 + V (x)

]un,k(t)

=En(k(t))un,k(t)

(3.12)

Thus we see that what we have achieved is an adiabatically changing Hamlto-

nian where the parameter is simply the wave-vector k(t). The corresponding


periodic eigen-function is just the periodic function at the quantum number

k(t) = k − (e/ch)A(t). In terms of the notation used in equations (3.1) - (3.8),

we identify ~R = k(t) and |n〉 = un so that H(~R)|n(~R)〉 = H(k(t))un(k(t)). As

an example for the external field, take A(t) = −c∫ t

0E(t′)dt′ and set E(t′) =

constant = −E0. This gives h(d/dt)k(t) = −eE0, so that k(t) = (e/h)E0t. For

small enough E0 this is an adiabatically changing parameter k(t).

From the above discussion we learn that if such A(t) is turned on, our

many-body wavefunction will “explore” the Brilouin Zone. The single-electron

wavefunctions constituting the overall function sort of “slide” across k so that,

phase aside, the many-electron wavefunction remains the same throughout. It

is important to keep in mind that if we have an insulator, a weak electric field

will not induce a conventional current since there are no states near Fermi

energy. The above adiabatic phase acquisition will still take place. One more

thing to note, is that in Brillouin Zone the points connected by a reciprocal

lattice vector are identified. As such, in our 1D case discussed above, once k(t)

traverses from −π/a to π/a it completes a closed adiabatic loop. This is a

special feature of Brilouin Zone; more generally one typically needs at least two

adiabatic parameters to be able to traverse a non-trivial adiabatic closed path.

Above we had briefly introduced the concept of Berry’s phase and have then

shown that in crystalline solids the many-body wavefunction naturally “picks

up” Berry’s phase in the ~k-space if subjected to weak external electric field.

Thus one might rightfully anticipate to see some observable effects due to the

Berry’s phase acquired under adiabatic external fields. We now show that this

Berry’s phase is directly related to an important observable – the net motion of

the centre of charge. This link was originally made in the study of change in the

charge polarization of a material. The following closely follows the derivation

in the original paper [47].

Experimentally, changes in charge polarization (CP) of solids can be in-

duced by various external means such as application of a strain (piezoelectric-

ity), changes in temperature (pyroelectricity) and electric field (ferroelectricity).

Thus the physically unambiguous observable is the change in CP. If a crystalline

solid with non-interacting electrons is assumed then the CP can be calculated

within Bloch Band formalism as a function of the externally controlled adia-


batically changing parameter λ. For a band insulator it is [47]:

~P (λ)α =

fqe8π3

M∑n=1

∫BZ

d~k〈u(λ)

n,~k|i∇~kα|u

(λ)

n,~k〉 (3.13)

where α = x,y or z ; f is the occupation number for states in the valence bands (2

for spin-degenerate systems); M is the number of filled valence bands. We have

skipped the full derivation here but to give a very simple intuitive justification

for the formula, consider the commutation relations between momentum and

position.

[~x, ~p] = ih (3.14a)

~p = −ih dd~x

(3.14b)

This is the familiar relation between momentum and position operators. Po-

sition operator in momentum space, however, can also be written in a similar

way:

~x =ihd

d~p

=i∇~k(3.15)

Thus, in (3.13) the polarization is simply proportional to the average centre

of the many-body wavecuntion. It is written as an integral over each band in

terms of functions un,~k summed over all the filled valence bands. By comparing

equation (3.13) with (3.4) one can notice that aside from the proportionality

constant the CP is just the the sum of Berry’s phases for each band. To see

this, we identify |n〉 with u(λ)n and ~R with ~k; the closed path C we identify with

Brillouin Zone boundary. We rewrite (3.13) to get:

~P (λ) =fqe8π3

M∑n=1

γ(λ)n (C) (3.16)

Notice, that in equation (3.4) the integral was over ~R-space and ~R was the

adiabatic parameter. Here, equations (3.13) and (3.16) are in terms of the

integrals over ~k-space but it is not the case that ~k is the adiabatic parameter.

The adiabatic parameter is λ and has not been specified so far. Thus the

equation (3.13) is mathematically equivalent to Berry’s phase – a phase which

can also be induced physically by adiabatic fields as has been shown above in

equation (3.12). What we are trying to demonstrate is that Berry’s phase which


can be induced in Bloch periodic systems tracks the motion of the net centre

of charge. One can evolve the system adiabatically from λ = 0 to λ = T over

a closed path such that the Hamiltonian at λ = 0 and λ = T is the same. The

difference ∆~P ≡ ~P λ=T − ~P λ=0 is the change of the net centre of charge that

takes place over a closed adiabatic cycle. It can be shown that the difference

between ~P λ=0 and ~P λ=T is equal to the derivatives of Berry’s phases [47]:

∆~P = −fqe8π3

M∑n+1

∫BZ

d~k∇~kγn,~k (3.17)

Thus, we see that Berry’s phase associated with each filled band of the material

is directly related to the property of this material to change the net centre of

charge under an external adiabatic field. This relation is captured by equations

(3.13) and (3.16), where the centre of charge is shown to be proportional to the

sum over Berry phases; equation (3.17) shows that the change in the centre of

charge is proportional to the derivatives of the Berry’s phases.

Berry’s phase has direct relevance for Topological Insulators. It can be

shown that the nontrivial topology of the bulk-bands results in a non-trivial

Berry’s phase. Under external electric field this gives rise to the net motion of

charge. For an Integer Quantum Hall phase, this net motion of charge causes

the current along the edges, giving the quantized Hall coefficient. For the Quan-

tum Spin Hall phase, it is more subtle. One has to project periodic functions

un,~k onto spin up and spin down subspaces first. The net result is that spin gets

transferred from one edge to the opposite one given spin Hall effect. The full

mathematical treatment necessary to rigorously demonstrate the claim of this

paragraph is very involved and is not presented here. Hopefully the reader has

been presented with enough preliminary theoretical and mathematical back-

ground to complete this one more step on their own if necessary.

In the next section we first present the full mathematical definition of a

Topological Insulator in 3 and 2 dimensions. That definition is not unique but

is generally accepted as the convention, since it is formulated within the widely

used Bloch Band theory. However, the formal definition offers little insight to

a lot of readers who are not themselves mathematical physicists. We therefore

shall link the formal definitions to the concepts related to Berry’s phase which

were presented in this section.


3.2 Topological Insulators and the Z2 Topological In-

variant

In this section a formal definition of the topological invariants is presented which

distinguish between the TI and the trivial band insulator. In the next section

these definitions are revisited in terms of more intuitive phenomena related to

Berry’s phase.

In three dimensions there are 8 distinct time-reversal invariant momenta

(TRIM), which are expressed in terms of primitive reciprocal lattice vectors ~bi

as Γi=(n1,n2,n3) = (n1~b1 + n2

~b2 + n3~b3)/2 with nj = 0, 1 [19]. The operation of

the time-reversal operator T in ~k-space takes ~k to −~k, and in general the Bloch

Hamiltonian H(~k) 6= H(−~k). However the above defined Γi are special points.

They are at the Brillouin Zone boundaries or at the origin and are connected

by a reciprocal lattice vector. Hence TRIM are equivalent and we indeed have

H(~k) = H(−~k) if ~k = Γi. In two dimensions there are 4 such points. The points

are identified for the familiar Brillouin Zone of graphene in figure 3.1a.

a) b)

Figure 3.1: BZ of graphene in a TI state is shown a) in the bulk, b) on a

single “zig-zag” edge; energy bands in dark and light green are included for

spin up and down helical states respectively. Time-reversal-invariant momenta

are identified, using the notation defined in text. a = |v2| as before


A 3D system has 2D surfaces and corresponding 2D Brillouin Zones. Simi-

larly a 2D system, such as graphene, has a 1D edge and a corresponding 1D BZ.

Brillouin zones of these subsystems also have T -invariant momenta. A given

crystal does not have a unique surface or an edge, it can be cut in many ways.

A surface or a boundary can then be defined as perpendicular to the vector

~G = n1~b1 + n2

~b2 + n3~b3. For such an edge/surface, its edge/surface T -invariant

momenta (sTRIM) are then the projections of the bulk TRIM defined by the

relation ±~G/2 = Γi1 − Γi2. The two edge T -invariant momenta Λ0 and Λ1

are shown in figure 3.1b for the graphene with the “zig-zag” edge. On the 2D

surface of a 3D system there are 4 sTRIM.

The topological invariant distinguishing Topological Insulators from regular

band insulators can be expressed in terms of the function defined on TRIM. To

arrive at the expression for the invariant we start by defining a matrix w(~k):

wij(~k) ≡ 〈un=i( ~−k)|T |un=j(~k)〉 (3.18)

Here T is the time-reversal operator. Next we define a number δi as

δi =

√det[w(Γi)]

Pf [w(Γi)](3.19)

Pf in equation (3.19) is the Pfaffian of the matrix. Pfaffian is similar to deter-

minant and can be defined for skew-symmetric matrices of even dimension; it

has the property that for any matrix A, (Pf [A])2 = det[A]. Recall that each

sTRIM Λj is a projection of two bulk TRIM Γj1 and Γj2. We define a number

ηj associated with sTRIM Λj in terms of δj1 and δj2

ηj = δj1δj2 (3.20)

For any two points Λj and Λj′ in the surface BZ, they will be connected by

energy bands intersecting EF an odd number of times if Λa and Λb have op-

posite sign [19], i.e. ηaηb = −1, and an even (or zero) number of times if the

sign is the same. The former case gives the so called Strong TI (STI), whereas

the latter gives the Weak TI (WTI). If we now look at a differently terminated

surface we shall find that for a STI you are guaranteed to have these surface

excitations in the bulk-gap, but in a WTI you may or may not. More generally,

WTI is adiabatically connected to a trivial band insulator, it is not topologi-

cally distinct. It is common to call WTI just a regular insulator, while STI a


Topological Insulator. Thus the presence or absence of the non-trivial topolog-

ical phase depends on the value of ηaηb which in turn depends on δi, defined

in the bulk BZ. Therefore one can express the topological Z2 invariant which

captures the physics on the surface in terms of the bulk BZ. Here Z2 means

that the invariant, call it v0 is unique only modulo 2. The invariant is defined

by the equation:

(−1)v0 =∏

nj=0,1

δi=(n1,n2,n3) (3.21)

The value of v0 = 0 or 1 mod 2 distinguishes the trivial insulating and the QSH

phases respectively.

The above thus gives a concise definition of a Topological Insulator within

the formalism of Bloch Band theory. Next we make a connection of the above

definitions to the more intuitive notions of the Berry’s phase and related phe-

nomena.

3.3 Z2 Invariants and the Spin-resolved Berry’s phase

What is not clear from the above definition of a Z2 topological invariant is

that it tracks the centre of “spin” in a material. We have already seen that

Berry’s phase is directly related to the net centre of charge in a material. Since

Topological Insulator exhibit a Spin Hall Effect, what needs to be tracked is,

loosely speaking, the “centre of spin”. In a case when the z-component of spin

is conserved this is easy to do. Spin is a good quantum number in such a case

and the natural way to define a spin current is

Is =h

2e(I↑ − I↓) (3.22)

This spin current can be seen as the motion of the “centre of spin”. Generally

spin is not conserved. However, one can still resolve the spin degree of freedom

with the aid of the Kramer’s theorem. Kramer’s theorem says that for system

of spin-1/2 particles with time-reversal symmetry, each state must be at least

doubly degenerate. Within Bloch formalism, this degeneracy occurs at states

with quantum numbers ~k and −~k. We also know that in Topological Insulators

time-reversal symmetry is respected. For a state with spin and wave-vector as

good quantum numbers, the operation of the time-reversal operator takes ~k to

−~k as well as the spin gets flipped. These relations between momentum and


spin of the states can be used to derive a useful relation within the Bloch Band

formalism [48]:

|uIn,−~k〉 = −eiχn,~k T |uII

n,~k〉 (3.23a)

|uIIn,−~k〉 = eiχn,−~k T |uI

n,~k〉 (3.23b)

In this equation labelling I an II is arbitrary and is meant to capture the spin

degree of freedom; χn,~k is some phase, generally different at different n and

~k. If the spin is conserved, one can simply set I =↑ and II =↓. The above

equations in (3.23) capture a simple idea in a rigorous way. The idea is that the

states come with spin up and down and therefore it should be possible to track

their relative motion. Kramer’s theorem together with time-reversal symmetry

allows us to define a pair of states at ~k and −~k equivalent to states “up” and

“down”.

In the same work, ref. [48], it was demonstrated that the Berry’s phase

associated with the Brillouin zone can be decomposed into contribution due to

the states labelled I and II. In section 3.1 we have shown in equations (3.13)

and (3.16) that Berry’s phase associated with the Brillouin zone is precisely the

net centre of charge. We rewrite those equations here in terms of the integrand

of the Berry’s phase, called Berry’s connection; call it ~A(~k):

~P (λ) =fqe8π3

M∑n=1

∫BZ

d~k〈u(λ)

n,~k|i∇~k|u

(λ)

n,~k〉

=fqe8π3

M∑n=1

∫BZ

d~k ~A(~k)

(3.24)

Relation (3.23) can be used to decompose Berry’s connection ~A(~k) as[48]:

~A(~k) = ~AI(~k) + ~AII(~k) (3.25)

Now we write the centre of charge decomposed into spin degree of freedom:

~P (λ) = ~P I,(λ) + ~P II,(λ) (3.26)

Finally, one can then difine a new quantity which would track the “centre of

spin” analogously to equation (3.22):

~P(λ)spin = ~P I,(λ) − ~P II,(λ) (3.27)


This last quantity has been rigorously demonstrated [48] to serve as an equiv-

alent definition to the topological invariant in the following sense; in 2D:

(−1)v0 = (−1)∆Pspin (3.28a)

∆Pspin = P(λ=Λ0)spin − P (λ=Λ1)

spin (3.28b)

Basically the invariant tracks whether the spin current takes place along the

path defined by the direction Λ1 − Λ0.

3.4 Summary of the Theory of Topological Insulators

Any finite system can be naturally divided into its interior and a subsystem

corresponding to its boundaries – surfaces and edges. If in the interior the

system has an energy band gap, with Fermi energy EF lying inside, then it can,

in principle, support boundary states with energy inside the band gap. The

band gap is needed to support boundary states for the following reasons. By

definition the wavefunction of a boundary state decays exponentially into the

interior of the system. Within Bloch Band formalism it is easy to see the need

for a band gap; the boundary states and bulk states must have different energies.

A given boundary state has fixed momentum and energy quantum numbers ~k0,⊥

and E0. If there is also a state from the bulk with the same energy E0 and a

component of the wavevector perpendicular to the surface being ~k0,⊥ – the two

states will couple. If the two states couple, the state assumed to be a boundary

state gets delocalized throughout the system and will not correspond to the

definition of a boundary state.

Thus a system with a bulk gap may support boundary states. The next

question is whether a system has boundary states which continuously cross

the band-gap energy range, crossing the Fermi energy. In section 2.4 we have

studied Haldane’s model of graphene under the external magnetic field with

a certain chirality property. We learned how the band structure of the bulk

may be topologically non-trivial. Topologically non-trivial energy bands must

necessarily close and re-open the band gap in order to undergo a phase transition

to the trivial band insulator. A system with distinct topology of its energy bands

in the bulk can be seen as undergoing a phase transition across its boundary,

if the boundary separates it from a regular insulator or vacuum. We also saw


that the chirality of the magnetic field manifests itself in the chirality of the

boundary states.

As was already mentioned, however, Haldane’s model induced band-gap

crossing boundary states by the means of external magnetic field. Topologically

Insulator is a material which has such band-gap crossing states intrinsically.

By definition, Topological Insulators are systems which posses time-reversal

symmetry, which is to say no external magnetic field is present. In section

2.3 we have seen how in condensed matter systems, a spin-orbit interaction is

generally present. Spin-orbit interaction means that an electric charge with a

spin in the presence of an electric field can “feel” the effective magnetic field; the

direction of the field is opposite for opposite spin. In section 2.5 we have seen

how the naturally occurring intrinsic spin-orbit interaction can induce band-

gap crossing boundary states. Spin-orbit interaction can be seen as an effective

magnetic field acting on each spin separately and in opposite directions, thus the

effect is similar to the one in Haldane’s model. In 2-D the boundary states are

helical. This means that on the same edge states moving in opposite directions

have opposite spin and states travelling in the same direction but opposite edges

also have opposite spin. This can be seen as two chiral states superimposed –

one for spin up and one for spin down. Such a physical picture works well for 2-D

system when the z-component of spin is a good quantum number. In a general

2-D and 3-D system without spin conservation, the mathematical treatment of

the problem becomes more complicated but the basic idea remains the same.

From the rigorous study of the Haldane’s model in 2.4 we have learned that

a peculiar magnetic field can induce chiral edge states. However, we also learned

that the relative strength of the term containing the effect of the magnetic field

matters. If it is too weak the system remains in the trivial insulating phase; if

it is strong enough then distinct topology of the energy bands is induced. The

situation with a Topological Insulator is analogous. It must have a spin-orbit

interaction term in its Hamiltonian, and this term must be sufficiently strong

to induce a non-trivial topology of the bands. In a “toy model” where a SOI

term can be added by hand, the above can be re-stated in the following way.

Without the SOI a system is a regular insulator; when the SOI is added and

strength is increased gradually up to its full value – the energy gap in the bulk

must close and reopen. At the same time, energy bands corresponding to the


boundary will have the gap closing, forming continuous energy dispersion. In

the literature this sort of material is often referred to as having a SOI-induced

band gap.

In section 3.1 we introduced the concept of Berry’s phase and its conse-

quences for condensed matter systems. We have shown that under external

electromagnetic field, the many-body wavefunction of the system picks up a

Berry’s phase. This phase is not equal to zero in general. It’s precise value is

a function of the filled energy bands in the Brillouin Zone. More so, Berry’s

phase is large at those points in the Brillouin Zone where two or more bands

become degenerate or, more generally, have a very small energy difference. On

the other hand, we have shown that when a wavefunction picks up a Berry’s

phase in an adiabatic cycle, the net centre of charge may be shifting. This gave

an alternative way to see why topology of the bands may give rise to charge Hall

Effect or spin Hall Effect. Under a small external electric field, the many-body

wavefunction of the system picks up Berry’s phase. This Berry’s phase, on one

hand, is a function of energy bands and on the other hand, manifests itself as

a motion of the centre of charge, i.e. current.

In section 3.2 we have presented the definition of a Topological Insulator

which serves as the standard non-phenomenological definition in mathematical

physics. A real material can be a Topological Insulator depending on its prop-

erties. However, within Bloch Band formalism one can rigorously define a Z2

topological invariant v0, which indicates a trivial insulator if it’s zero, or a TI if

it’s equal to 1. After the formal definition, we have linked it to the concept of

Berry’s phase. It can be shown that the Z2 topological invariant is equivalent

to whether spin current is supported on the boundary or not.

4

Quantum transport – atomistic point of view

Having rather thoroughly reviewed the theory of TI, we now move on to the

second part of this thesis in which we investigate quantum transport properties

of a known TI, Bi2Se3, using an atomistic model. The idea is to use an atomistic

model which correctly reproduces the main properties of the TI yet does not

require a prohibitively large computation. A natural approach is the tight-

binding (TB) model. In section 4.1, a general discussion of the TB method is

presented. In section 4.2, the TB approach is compared to the density functional

theory (DFT) to establish the validity of the approximations in TB. An optimal

way to parameterize the TB model is also presented.

TB provides the Hamiltonian of the TI material from which quantum

transport properties are obtained by applying the Landauer-Buttiker formal-

ism which is discussed in section 4.4. In particular, the transmission coefficients

are calculated by the Green’s function method which we shall briefly review.

Numerical applications of our TB approach will be the subject of next Chapter.

4.1 Tight-Binding Method

Tight-Binding method is well known and there are numerous reviews available

[49]. The original idea was based on an assumption that the underlying basis set

for the eigenstates of the many-body system is atomic-like, and hence localized

[50]. We use the results of the Bloch’s theorem for periodic crystals and write

down a Bloch wave-function constructed from the atomic-like orbitals localized

at each atom of the system. Let φi,α(~r − ~Rj) be αth atomic-like orbital (e.g

α = px) localized on the ith atom of the unit cell at ~Rj; the crystal is divided

into unit cells, which are enumerated by {~Rj}. The (i, α)th Bloch wave-function

48

4: Quantum transport – atomistic point of view 49

can then be written as:

ψi,α(~r,~k) =1√N

∑~Rj

ei~k·~Rjφi,α(~r − ~Rj) (4.1)

where N is the number of unit cells of the system. We have constructed Bloch

wave-functions in terms of linear combinations of atomic-like localized functions.

However, we also know, due to Bloch, that the eigenstates of the system are

themselves Bloch wave-functions, i.e. they respect the periodicity conditions:

Ψ(~r − ~aj;~k) = ei~k·~ajΨ(~r;~k) (4.2)

where ~aj is one of the primitive lattice vectors. Thus we have established two

facts, namely a) the set of Bloch functions in (4.1) is itself spanned by the

complete (assumed) basis set of atomic-like functions φi,α(~r − ~Rj) – therefore

this set is also complete, b) all the eigenfunctions of the system are Bloch

functions; It follows that any eigenstate can be written as linear combinations

of the set in (4.1).

ΨEn(~r;~k) =∑i,α

C(n)i,α ψi,α(~r;~k) (4.3)

Note that ~k is a good quantum number. Now, one can notice that the set

in (4.1) has the cardinality (size) equal to the (number of atoms in the unit

cell)×(atomic-like orbitals per atom). Therefore the set is finite and the set of

eigenstates is also finite. Clearly this means that we find an incomplete set of

solutions since the true set is infinite. However, the approximations are still

good, since the solutions that we get under the TB approximations correspond

to those eigenstates from the true set of eigenstates which lie in the relevant

energy range. That is, the underlying atomic-like orbitals chosen for the TB

approximation themselves correspond to certain energies as solutions to the

isolated atom problem. This gives physical motivation for why an incomplete

set can give all the relevant solutions – i.e. solution in energy range of interest.

We shall shortly see, that TB model is often used as a parameter-based model in

which case one can “force” the TB eigenstates correspond exactly to an isolated

subset of true solutions.

We can now write down the Hamiltonian and overlap matrices in terms of

the states in equation (4.1) which are themselves written in terms of the atomic-

like functions. We write down eigen-value equation in terms of these matrices.


The different ways to use TB then reduce to different ways of obtaining the

matrix elements:

Hi,α;j,β(~k) =1

N

∑l

∑ll

ei~k·(~Rl−~Rll)

∫d~rφ∗i,α(~r − ~Rl;~k)Hφj,β(~r − ~Rll;~k) (4.4a)

Si,α;j,β(~k) =1

N

∑l

∑ll

ei~k·(~Rl−~Rll)

∫d~rφ∗i,α(~r − ~Rl;~k)φj,β(~r − ~Rll;~k) (4.4b)

The double sum reduces to a single sum and cancels out N.

Hi,α;j,β(~k) =∑l

ei~k·~Rl

∫d~rφ∗i,α(~r − ~Rl;~k)Hφj,β(~r;~k) (4.5a)

Si,α;j,β(~k) =∑l

ei~k·~Rl

∫d~rφ∗i,α(~r − ~Rl;~k)φj,β(~r;~k) (4.5b)

where ~Rll is now fixed and was taken to be the zero vector. The integrals in a TB

method are taken to be parameters; different ways to obtain these parameters

give rise to different versions of a TB model. Typically different ways reduce

to two main approaches. First one is trying to build parameters based on some

heuristic physical arguments – which makes this a phenomenological model

and usually not very accurate. The other approach is treating them as pure

parameters when fitting some observable such as energy bands to a given set of

known data – from experiment or first-principle calculations.

The second aspect of the TB approximation is to reduce the sum over all

the unit cells ~Rl to just a few terms corresponding to nearest neighbors, or up to

next nearest neighbors, etc. – depending on the level of accuracy one wishes to

achieve. We now write down the generalized eigen-value Shrodinger’s equation

in terms of equations (4.5).

HΨEn(~r;~k) =EnΨEn(~r;~k)

=En∑i,α

C(n)i,α ψi,α~r;

~k(4.6)

In this set of equalities apply the Bra 〈ΨEn(~k)| on the first and the last expres-

sion.∫d~rΨ∗En(~r;~k)HΨEn(~r;~k) = En

∑j,β

∑i,α

C∗(n)j,β C

(n)i,α

∫d~rψ∗En(~r;~k)ψEn(~r;~k)

(4.7)


This can be rewritten as many simultaneous linear equations for each j, β = n,

solving for the vector of coefficients [C(n)i,α ] = ~C(n).

Hi,α;j,β(~k)~C(n) = EnSi,α;j,β(~k)~C(n) (4.8)

This generalized eigenvalue equation can be solved if we know the TB parame-

ters in equations (4.5). This completes the general discussion of the TB model.

The next step is to discuss different approaches to obtaining TB parameters in

(4.5). Once parameters are known, the full set of vectors ~C(n) and eigen values

En can be obtained.

The most general TB model is less restrictive than presented in the above

discussion. Namely, we assumed Bloch periodic system. In general the TB

method only means that localized orbitals are assumed to make a complete

basis and that the interactions beyond few nearest neighbours are negligible.

This way we can write down relations such as in (4.5). It is common to assume

Bloch theorem in conjunction with TB.

4.2 Self-Consistency and the Tight-Binding method

One may have noticed that in the above discussion there were no equations of

self-consistency derived. On the other hand, having solved a set of equations

(4.8), one may then project onto each atom in the system the net charge,

thus obtaining electron density and hence charge density. This would give

us information about ionicity of the atoms in our many-body system. We

have no energy term which would depend on the charge density to account

for these Coulombic interactions so the results of (4.8) are not self-consistent.

Often when TB model is used, the parametrization of (4.5) is assumed to give

results which are reasonable enough and the lack of self-consistency is ignored.

Examples would be fitting parameters to experimental data such as energy

bands, elasticity, etc., or extracting those parameters from DFT first-principles

calculations.

Historically, TB model was widely used when the computer power available

was very low, starting in 1950s. Therefore, in addition to non-self-consistent

use, there were many different approaches developed to introduce a level of self-

consistency to the TB model as presented above. The most common extension


is to write the total energy of a system as:

Etotal =N∑i=1

εi +1

2

∑α

∑β 6=α

U(|~Rα − ~Rβ|) (4.9)

where the first sum is over the single particle eigenvalues obtained in equations

(4.8), while the second term is supposed to capture all the other energy contri-

butions and is assumed to only depend on inter-atomic distance. The function

U in the second term depends on the electron density and thus introduces a

level of self-consistency.

However, in the opinion of the author, today the most common TB model is

used in conjunction with the more accurate DFT calculations. There are several

ways to extract information from DFT to build a reliable TB model. This allows

easy calculations for which DFT would be unsuitable, such as dealing with very

large systems. Thus in the next section we present the TB theory in the context

of DFT. This will illuminate rigorously just how reasonable TB approximations

are. It will also give a better overall physical picture and a better understanding

of the degree of error introduced in any one parametrization scheme.

4.3 Tight-Binding method and the Density Functional

Theory

Let’s remind ourselves the key equation expressing total energy in DFT and in

the TB model. For TB it is equation (4.9):

E =N∑i=1

εi +1

2

∑α

∑β 6=α

U(|~Rα − ~Rβ|)

where the second term is added to introduce some semi-empirical self-consistency.

The first term is always present and is the only term in the common non-self-

consistent TB approaches. The energies εi come, in principle, from a single-

particle Shrodinger’s equation:

HΨi(~r) =

[−1

2∇2 + V (~r)

]Ψi(~r) = εiΨi(~r) (4.10)

As was already discussed, for the non-self-consistent TB model this equation is

often not solved but rather the integrals involving H, as in equation (4.5), are

parameterized in some fashion.


In DFT, the underlying energy functional is:

E[n] = Ts[n] + F [n] (4.11)

where n(~r) is the electron density, Ts is the kinetic term representing non-

interacting quasi-particles and F is the functional including all other terms.

The functional F consists of electron-nuclei electrostatic interaction, electron-

electron interactions and exchange-correlation energy which are written, respec-

tively, as

F [n] =

∫Vnucl(~r)n(~r)d3r +

1

2

∫ ∫n(~r)n(~r′)

|~r − ~r′|d3rd3r′ + Exc[n(~r)] (4.12)

The exchange-correlation energy Exc is defined by the above equation [51]. More

precisely, Exc is the only term without an explicit expression. All the remaining

terms do not give the true energy. The missing energy is ascribed to exchange-

correlation effect and Exc is meant to balance out the total expression giving

the true total energy. Often reliable estimate of Exc can be represented in terms

of a functional εxc, available in the literature, so that

Exc[n(~r)] =

∫εxc[n(~r)]n(~r)d3r (4.13)

The kinetic term is that of non-interacting quasiparticles and therefore comes

from a one-electron Shrodinger’s equation just like in (4.10). We rewrite it to

reiterate that this is a different formalism and the potential used is, in general,

different. [−1

2∇2 + V (~r)

]Ψi(~r) = εiΨi(~r) (4.14a)

Ts[n(~r)] =N∑i

∫Ψ∗i (~r)

(−1

2∇2

)Ψi(~r)d

3r

=N∑i

εi −∫V (~r)n(~r)d3r

(4.14b)

Comparing equation (4.9) with the set of equations (4.11), (4.12) and (4.14b)

one can draw the first important conclusion. While the last expression of equa-

tion (4.14b) may be attempted to be set in correspondence with the first term

of (4.9), the second term of (4.9) clearly requires a set of approximations if

one hopes to reach a full correspondence. More importantly, (4.9) is not self-

consistent in a rigorous sense since its one particle potential is inherently non-

self-consistent unlike in DFT. We now demonstrate this below. According to


DFT, there exists a ground state density function n0(~r) at which the energy

functional in (4.11) is a global minimum. Therefore we have:

δE[n0(~r)] = E[n0(~r) + δn0(~r)]− E[n0(~r)] = O((δn()~r)2) (4.15a)∫δn0(~r)d3r = 0 (4.15b)

This leads to a self-consistency equation by putting a restriction on the one

particle potential V (~r).

δF

δn[n0(~r)] = V (~r) + constant (4.16)

where the arbitrary constant can be set to zero. Thus in DFT we have, on one

hand, that equation (4.14a) is an independent-particle equation which generates

electron density for a given potential V (~r); on the other hand equation (4.16)

is an independent equation where the left hand side is a functional written in

terms of the density and the right hand side is V (~r). Equations (4.14a) and

(4.16) together give the self-consistency condition similar to that of the Hartree

theory.

To illuminate the connection between DFT and TB, we proceed with de-

scribing a step-by-step process of evaluating equation (4.11) at a guessed elec-

tron density which we assume to be very close to the true ground state density.

First, take a guess for the ground state density, call it n1(~r). Then define V1(~r)

in terms of n1(~r) based on equation (4.16). Next, we solve the separable equa-

tion (4.14a) and construct the resultant density, call it n2(~r). Since we made

a guess at the ground state energy, in general we have that n1(~r) 6= n2(~r). We

can evaluate the total energy using equations (4.11) and (4.14b)

E[n2] =N∑i=1

εi −∫V1(~r)n2(~r)d3r + F [n2] (4.17)

Thus, comparing it to the TB energy expression which is in terms of a single

density function it is not clear how to relate the two. One of the keys to proceed

is to work with the assumption that n1(~r) is a very good guess at the ground

state. This way even though DFT self-consistency is not met, i.e. n1(~r) 6= n2(~r),

it is almost met, i.e. n1(~r)− n2(~r) = ∆n(~r) is very small. Then we can rewrite

equation (4.17) as an expansion about the function n1(~r):

E[n2] =N∑i=1

εi+F [n1]−∫δF

δn[n1]n1(~r)d3r+

1

2

∫ ∫δ2F

δn2[n1]∆n(~r)∆n(~r′)d3rd3r′

(4.18)


Notice that the dependence on n2(~r) is only through the last term via the

function ∆n(~r). The next step is the following realization. On physical grounds

and as is evident from equations (4.17) and (4.18), the energy functional is equal

to or greater than ground state energy. This is captured by the last term in

(4.18) which is strictly positive. However, one cannot expect this from the semi-

empirical TB energy expression. Instead one can hope that for a correct ground

state density TB gives correct ground state energy, but expanding about ground

state density may give higher as well as lower energy. Therefore in ref. [51], it

was suggested to define a different energy functional which coincides with E[n]

at the ground state density and, when expanded about it, the results only differ

at the second order in ∆n.

ETB[n1] ≡ E[n2]− 1

2

∫ ∫δ2F

δn2[n1]∆n(~r)∆n(~r′)d3rd3r′ (4.19)

This functional is now entirely in terms of a single density function, here n1. To

show more clearly the difference and similarity between the two functionals, we

expand both about the ground state density n0. Recall that neither n1 nor n2 in

general are equal to n0. For n1(~r)−n0(~r) = ∆n1(~r) and n2(~r)−n0(~r) = ∆n2(~r)

one can obtain the following expressions [51]:

E[n1] = E[n0] +1

2

∫ ∫δ2E

δn2[n0]∆n1(~r)∆n1(~r′)d3rd3r′ + ... (4.20a)

E[n2] = E[n0] +1

2

∫ ∫δ2E

δn2[n0]∆n2(~r)∆n2(~r′)d3rd3r′ + ... (4.20b)

ETB[n1] = E[n0] +1

2

∫ ∫δ2E

δn2[n0]∆n1(~r)∆n2(~r′)d3rd3r′ + ... (4.21)

where one can clearly see that the new functional ETB[n] varies about the

ground state density in a similar way to E[n], but unlike E[n] it can both

increase and decrease due to the product ∆n1∆n2.

The relation between DFT and semi-empirical TB can now be explicitly

demonstrated by relating the expression in equation (4.9) to ETB[n]. First let’s

rewrite equation (4.21) again:

ETB[n] =N∑i=1

εi + F [n]−∫δF

δn[n]n(~r)d3r (4.22a)

ETB[n] =N∑i=1

εi − EH [n]−∫µxc[n]n(~r)d3r + Exc[n] (4.22b)


Here µxc = δExc/δn, the eigen-values of the first term come from equation

(4.14a) and potential V (~r) = V [n] comes from equation (4.16). This form is

particularly suitable for isolating core electrons and valence electrons. It can be

shown [51] that writing n = nc + nv, i.e. separating the total electron density

into core electrons and valence electrons respectively, equation (4.22b) can be

rewritten in a frozen core pseudopotential approximation as

ETB[n] =N(v)∑i=1

ε(v)i −EH [nv]−

∫µxc[nv]nv(~r)d

3r+Exc[nv]+1

2

∑α

∑β 6=α

ZαZβ

|~Rα − ~Rβ|(4.23)

where Z(v)α is the number of valence electrons on atom α. This approximation

is well established[51].

In TB it is assumed that the electron density is a sum of atomic-like localized

electron wave-functions, so let’s write our electron density nv as just such a sum.

Note that at this point it is not important to what extent the wave functions

are localized.

nv(~r) =∑α

nv,α(~r) (4.24)

Plugging this into (4.23), some terms separate into the form similar to the

second term of TB equation (4.9),

EH;ZαZβ ≡1

2

∑α

∑β 6=α

ZαZβ

|~Rα − ~Rβ|− EH [nv]

EH;ZαZβ =1

2

∑α

∑β 6=α

(ZαZβ

|~Rα − ~Rβ|−∫ ∫

nv,α(~r)nv,β(~r′)

|~r − ~r′|d3rd3r′

) (4.25)

The exchange-correlation terms require further approximations, because the

dependence on density of these terms is not linear. To proceed we assume that

regions in a solid with the overlap of densities from three or more atoms is small.

In this case a cluster expansion can be used and many-body terms are dropped

beyond two-body interactions.

Dxc[n] =

∫(εxc[nv(~r)]− µxc[nv(~r)])nv(~r)d3r (4.26a)


Dxc

[∑α

nv,α(~r)

]≈∑α

Dxc[nv,α]+

+1

2

∑α

∑β 6=α

(Dxc[nv,α + nv,β]−Dxc[nv,α]−Dxc[nv,β]) + ...

(4.26b)

The three dots mean that the terms involving many-body interactions were

dropped. Estimates suggest that truncating many-body terms in the above

expression introduces only very small errors[51].

We can now see what has been accomplished. Under the assumption that

exchange-correlation energy depends on many-body interactions only negligibly,

the TB and the DFT energy expressions can be put into correspondence. We

rewrite them again in a form which makes it easy to compare the two; frozen

core pseudopotential is assumed although it is not necessary:

Etotal =N(v)∑i=1

ε(v)i +

1

2

∑α

∑β 6=α

U(|~Rα − ~Rβ|) (4.27)

ETB =N(v)∑i=1

ε(v)i +

∑α

Dxc[nv,α]+

+1

2

∑α

∑β 6=α

(Dxc[nv,α + nv,β]−Dxc[nv,α]−Dxc[nv,β] +

+ZαZβ

|~Rα − ~Rβ|−∫ ∫

nv,α(~r)nv,β(~r′)

|~r − ~r′|d3rd3r′

) (4.28)

Now it is important to remind ourselves that ETB in the above equation sat-

isfies two conditions. Firstly, taking ground state density n0 as an input, i.e.

the density which minimizes the DFT functional E[n], gives ETB[n0] = E[n0].

Secondly, if the input density n1 6= n0, but is still a good guess, then the func-

tionals E[n1] and ETB[n1] are the same up to the second order in ∆n = n1−n2.

This means that ETB is essentially a good functional, meaning it gives correct

total energy provided that correct or nearly correct input density is used. How-

ever, unlike in DFT, ETB does not have this inherit property that minimizing

it gives the ground state energy. That is, the ground state density n0 is not

necessarily the global minimum of ETB. Also we see that ETB can be put into

a direct correspondence with the TB equation (4.9). The first term of (4.28)


corresponds to the first term of (4.9). The second and third line of (4.28) can

be put in correspondence with the second term of (4.9). The second term of

(4.28) can also be considered as in correspondence with U(|~Rα − ~Rβ|) if it is

modified slightly to include single-body energy terms.

What we can conclude from this exercise is the following. Firstly, we showed

that the TB equation of the total energy given in (4.9) can, in principle, capture

the true total energy only requiring minimal approximations. In other words,

the approximation is taking the exchange-correlation terms depending on three

or more bodies as negligible. It is quite remarkable that the total energy of a

condensed matter system can be written only in terms of single and two-body

energy terms.

We can also clearly see the shortcomings of the TB model. Firstly, in

practice the functional form of U(~Rα − ~Rβ) is taken to be something simple,

e.g. inverse exponential of the distance or inverse distance squared. It cannot

be taken to reliably capture the true energy. Secondly, we have learned that

the single electron energies εi (or ε(v)i ) in a TB method are based on a potential

which is not self-consistent. That is, even with some self-consistency due to the

term U(~Rα − ~Rβ), we showed that the corresponding energy functional ETB

does not have the property that the ground energy of the system is its global

minimum.

It seems that the TB model is most useful when applied in conjunction with

DFT calculations. If a TB model is built by itself – even with some level of self-

consistency, one cannot know to what extent the results obtained are accurate.

On the other hand, one can use the TB method in conjunction with an exist-

ing DFT calculation in order to go beyond the DFT limitations on large sizes.

There is more than one way to base a TB model on DFT. A few examples can

be found in references [52], [53] and [54] among many. The simplest thing to do

is to take the matrix and overlap elements directly from DFT calculations. This

guarantees that the energies εi and the corresponding electron eigenfunctions

are based on a self-consistent potential V (~r)[n0]. There are a few shortcomings

of such an approach. Firstly, the typical basis set used in DFT is very large,

between several hundred and thousands. Secondly, for a given basis, the interac-

tion well beyond nearest-neighbors may be non-negligible. These shortcomings

do not pose a stumbling block in general, and a TB model obtained this way


will always produce calculations much faster than the DFT.

4.4 Quantum transport

The aim of this section is to introduce the theoretical means to mimic an exper-

iment in which a mesoscopic system is connected to two external leads. A tiny

bias voltage is applied to the leads and we wish to measure the conductance

of the system. The Landauer-Buttiker formalism for electron transport will be

introduced with the aim to combine it with our TB model. In the next Chap-

ter we shall use the TB method together with Landauer-Buttiker formalism to

study electron transport in Bi2Se3 films.

The general system of interest is shown in figure 4.1. The central region is

connected to two leads to the left and right which we treat as semi-infinite.

Figure 4.1: General set-up of the problem. We study the transport properties

of the central region labelled C by sending current through it from the left wire

L into the right wire R.

This set up mimics typical experiments in which the transport properties

of the central region are investigated. The picture in Fig.4.1 is one dimensional

(1D), but our actual calculations are done on 2D and 3D systems[55]. The

length of the L and R leads is on the scale much larger than the size of the

central region. Consequently they are treated as semi-infinite, connected to

electron reservoirs far away. The fundamental result of the theory of electronic

transport is that the conductance through region C can be expressed in terms

scattering properties of that same region via the Landauer formula[56],[55]:

C =2e2

hT (4.29a)

T = Trace(ΓLG

RCΓRG

AC

)(4.29b)

In equation (4.29a) C and T stand for conductance and the transmission co-

efficient respectively; h is the Plack’s constant and e the electron charge. One

can see that conductance is proportional to transmission which is an intuitive


result. In equation (4.29b), the transmission coefficient is expressed in terms of

the retarded and advanced Green’s functions GRC and GA

C respectively as well

as two functions related to the left and right leads, – ΓL and ΓR, respectively.

ΓL,R is called line-width function and they describe how the central region is

coupled to the two leads. We now discuss in some detail how equation (4.29b)

is obtained and what it means.

To derive equation (4.29b) we start by writing the Hamiltonian for the

entire system shown in figure 4.1. This system is infinite and can be divided

into two semi-infinite subsystems L and R, and a finite system C. We write

down the Hamiltonian in matrix form:

H =

HL hLC hLR

hCL HC hCR

hRL hRC HR

(4.30)

Here HL, HC and HR are the Hamiltonians of the left, central and right sub-

systems respectively; hAB is the coupling between subsystems A and B. The

left and right leads are separated by the central region, they typically do not

directly couple and so we set hLR = hRL = 0.

H =

HL hLC 0

hCL HC hCR

0 hRC HR

(4.31)

Now we can write down the equation for Green’s function, writing the Green’s

function in terms of subsystems as well.

(ε−H)G = I (4.32a)GL GLC GLCR

GCL GC GCR

GRCL GRC GR

=

ε−HL hLC 0

hCL ε−HC hCR

0 hRC ε−HR

−1

(4.32b)

Identity on the right hand side of equation (4.32a) assumes an orthogonal basis.

One can rewrite it in a more general form by replacing I with the overlap matrix

S, the derivation does not change from this substitution. So far, the system is

still infinite and so it is not clear how to obtain the solutions since the matrix

is infinite in dimensions. However, if one writes out the expression for GC ,


by formally solving equation (4.32), one can re-express the original infinite

dimensional problem in terms of the effective finite system,

GC = (ε−HC − ΣL − ΣR)−1 (4.33a)

ΣL = h†LCGLhLC (4.33b)

ΣR = hRCGRh†RC (4.33c)

HereGL andGR are the Green’s functions of the semi-infinite left and right leads

respectively. The problem of an isolated semi-infinite and infinite leads are well

studied and Green’s functions are relatively easy to obtain in most cases[57],[58].

If one checks carefully, one will find that the newly defined functions ΣL and ΣR

have finite dimensions. It is customary to think of the finite system defined by

HC + ΣL + ΣR as the central system, namely HC plus the self-energies imposed

by the leads. Note that the expression in (4.33a) is exact, no approximation

has been made.

What has been achieved so far is the following. An effective Hamiltonian

for the central region with the effect of the leads is Heff = HC + ΣL + ΣR.

The Hamiltonian Heff is exact, and it has finite dimensions. For this effective

Hamiltonian there is an effective Green’s function, in equation (4.33a); it also

has finite dimensions. One can now obtain transmission coefficient, treating

this problem as a scattering problem. Electron enters from the left lead, gets

scattered by the central region and has a certain probability to transmit to

the right lead. This scattering problem has been solved, and the solution is

the well-known Fisher-Lee relation[55]. The result is the equation (4.29b). We

rewrite the relevant formulas again:

T = Trace(ΓLGRCΓRG

AC) (4.34a)

ΓL = i(ΣRL − ΣA

L) (4.34b)

ΓR = i(ΣRR − ΣA

R) (4.34c)

These equations are known as the “Landauer-Buttiker” formulation which are

widely used in quantum transport theory. By the TB Hamiltonian, we can cal-

culate the Green’s function and thus obtain the transmission coefficient. In the

next Chapter, quantum transport properties of Bi2Se3 films will be investigated

this way.

5

Quantum transport in Bi2Se3 nanostructures

In this Chapter we present our calculations of transport properties of the known

TI Bi2Se3[4]. Crystal structure of Bi2Se3 consists of quintuple layers (QL) held

together by van der Waals forces. Each QL consists of five layers of atoms, with

inter-layer bonding primarily of the covalent type[59]. This inter-layer covalent

bonding is dominated by the σ-type bonding between the p-orbitals of Bi and

Se atoms. For each atom its nearest-neighbour atoms are nearly octahedrally

coordinated, three atoms in the layer above and three atoms in the layer below.

We have studied Bi2Se3 in a slab geometry, meaning there are a finite num-

ber of atomic layers in one direction (call it z-direction) while Bloch periodicity

is assumed in the other two directions (a film). This geometry explicitly in-

cludes two surfaces, call them top and bottom, where the helical states manifest

themselves. The unit cell of Bi2Se3 is shown in a 6QL film geometry in figures

5.1a and 5.1b.

a) b)

Figure 5.1: Unit cell of 6 QL Bi2Se3 slab in a) 3D and b) side view with 6 QL

identified. Blue are the Bi atoms, red and green are non-equivalent Se atoms.

Figures 5.2a and 5.2b show how nearest and next-nearest neighbours are

coordinated. The diagram is for the neighbours of a Se atom in the 4th atomic

62

5: Quantum transport in Bi2Se3 nanostructures 63

layer, but the neighbours are coordinated in the same way for all atoms.

a) b)

Figure 5.2: Coordination of neighbouring atoms in Bi2Se3 for Se atom in 4th

atomic layer. Black – reference unit cell, red – unit cells at ±~a2, blue at ±~a1,

cyan at ±(~a1 + ~a2). a) top view, b) relevant atoms are shown in 3-D. Black

dashed lines indicate next-nearest neighbours, blue and red solid lines – nearest

neighbours in the atomic layer above and below respectively

The main physics of Bi2Se3 is captured by the nearest neighbour (NN) and

next-nearest neighbour (NNN) interactions. These interactions are dominated

by the atomic p-type orbitals. Electrons corresponding to the s and d orbitals

also contribute to the overall physics since they have similar energies. The

following section describes how our TB model was constructed to capture the

main physics of Bi2Se3.

5.1 Tight-Binding Model for Bi2Se3

Our non-self-consistent TB model is based on DFT calculations of Bi2Se3 as

reported in Ref.[14]. In that work the basis set which produces the correct

ground state density, was assumed to be a finite set of atomistic functions, they

are described as follows. The total set of functions consists of subsets of func-

tions localized on each atom. For each atom, the subset of localized functions

consists of functions with s-, p- or d -type angular part. In order to make sure

the basis set approaches completeness, there are several functions with different

radial parts corresponding to each angular momentum. For example, for each

Se atom there are three localised functions of the type px, which can be labeled

as |p′x〉, |p′′x〉 and |p′′′x 〉. Any two functions localized on the same atom hav-


ing different angular momentum are orthogonal. Any two functions localized

on the same atom corresponding to the same angular momentum are highly

non-orthogonal, for example 〈p′x|p′′x〉 ' 0.99. A set of evaluated overlap and in-

teraction integrals is available between these localized functions from the DFT

analysis.

This basis set together with the information about overlap and interaction

integrals can be naturally converted into a TB model. There is, however, a

difficulty which has to be addressed. Recall the key equations defining the

non-self-consistent TB model, (4.5) and (4.8), which we rewrite here:

Hi,α;j,β(~k) =∑l

ei~k·~Rl

∫d~rφ∗i,α(~r − ~Rl;~k)Hφj,β(~r;~k) (5.1a)

Si,α;j,β(~k) =∑l

ei~k·~Rl

∫d~rφ∗i,α(~r − ~Rl;~k)φj,β(~r;~k) (5.1b)

Hi,α;j,β(~k)~C(n) = EnSi,α;j,β(~k)~C(n) (5.2)

Solving the generalized eigen-value equation (5.2) is equivalent to solving a

secular equation:

det(HS−1 − E

)= 0 (5.3)

When equation (5.2) is put into the form of equation (5.3), the difficulty of

transforming DFT parameters into TB model becomes transparent. Namely,

as the off-diagonal overlap matrix elements of the type 〈p′x|p′′x〉 approach unity,

the elements of S−1 become large. This is simply because the overlap matrix

written in terms of a linearly dependent basis does not have an inverse. The fact

that 〈p′x|p′′x〉 ' 0.99 means that these functions are nearly linearly dependent.

Consequently, the full original basis with all the interactions included produces

correct energy bands. However, if we drop some of the intereactions, say beyond

next-nearest neighbours, the delicate balance of large elements of S−1 is lost and

the matrix becomes divergent.

As such, in our model we took a reduced basis set defined in terms of

the original basis set. The localized functions defined in the original DFT

calculation were converted into the TB basis function by “normalizing out”

the different radial functions. For a given atom, say atom “A”, and for a given

angular type, say px, the original basis set has n(A, px) = no functions {|(A)p(1)x 〉,

... , |(A)p(no)x 〉}. We obtain a TB basis function |(A)pTBx 〉 by converting:

|(A)pTBx 〉 = NA

(|(A)p(1)

x 〉+ ...+ |(A)p(no)x 〉

)(5.4)


where NA is defined by imposing the normalization condition:

〈(A)pTBx |(A)pTBx 〉 =N2A

(〈(A)p(1)

x |+ ...+ 〈(A)p(no)x |

) (|(A)p(1)

x 〉+ ...+ |(A)p(no)x 〉

)=1

(5.5)

Introducing this conversion takes care of the problem of the divergent secular

equation. However, in doing so we also introduced some numerical error: the

energy bands as well as other properties do not exactly agree with those obtained

from the original DFT calculation anymore.

In addition, two more approximations are introduced. First, when convert-

ing parameters into the TB model, only those overlap and interaction integrals

were considered which correspond to NNs or NNNs. This introduces some error

but also makes the matrices of equation (5.1) in block tri-diagonal form. Block

tri-diagonal matrices can be inverted very fast using an algorithm in Ref.[60].

The less off-diagonal elements there are the faster is the inversion, and the aim

of TB model is fast calculations. Second, in the original DFT work[14], the

system was solved in two ways, once with SOI and once without SOI. We took

the parameters from the DFT calculation which excludes SOI. This allows us to

add our own TB SOI explicitly. The atomistic SOI term in the TB form is[61]:

SOI =1

2λSO

0 −i 0 0 0 1

i 0 0 0 0 −i0 0 0 −1 i 0

0 0 −1 0 i 0

0 0 −i −i 0 0

1 i 0 0 0 0

(5.6)

The SOI matrix should be read with respect to the basis {|px ↑〉, |py ↑〉, |pz ↑〉,|px ↓〉, |py ↓〉, |pz ↓〉}; λSO = 0.22eV for Se and 1.25eV for Bi [62].

As a result of the above mentioned approximations, the energy bands ob-

tained had correct overall features but lacked in detailed accuracy. For instance,

the band gap was smaller than it should be. As such, the final step in construct-

ing our TB model was altering some values of the interaction integrals by hand,

increasing or decreasing them slightly from the original DFT values. The net

result is a TB model which reproduces a series of observables well and is very

fast at performing numerical calculations.


First, we present energy bands for Bi2Se3 6QL slab in figure 5.3, with and

without the SOI. One can see explicitly that introducing the SOI closes the gap

with energy dispersion resembling that of the helical states,

a) b)

Figure 5.3: Energy bands of Bi2Se3 in a 6-QL slab geometry. Only 10 bands

above (green) and below (black) EF are shown. a) shows the case without SOI;

exact value of EF is not known. b) is the case with SOI included; EF = 2.88eV

exactly at the Dirac point

From figure 5.3b we can see energy bands continuously crossing the Fermi

level with a dispersion being essentially linear in ~k. This very much resembles

the Dirac cone that we expect for a TI. For Bi2Se3 we expect to have one Dirac

cone for each surface[4]. Our system has two surfaces and indeed, as can be

seen from figure 5.4, we have two Dirac cones essentially superimposed.

Figure 5.4: Energy Bands of Bi2Se3 zoomed in at energies near EF and around

the Γ point. One can clearly see two nearly degenerate Dirac cones.


The eigen-states which lie on these two Dirac cones should be confined to

the two opposite surfaces. To confirm this is indeed the case, in figure 5.5 we

plot two wave functions lying at the same fixed wavecetor ~k0 and with nearly

the same energy corresponding to the two different Dirac cones. The wave

functions from different Dirac cones are confined to opposite surfaces, exactly

as expected. We can also see explicitly the extent of wave function localization.

The helical state does not reside solely in the first QL, it has considerable

amplitude in the second QL away from the surface. In fact, at least 5-6 QLs

are needed to spatially separate the helical states corresponding to the top and

bottom surfaces of Bi2Se3 TI[14]. The two Dirac cones in figure 5.4 are nearly

degenerate instead of fully degenerate precisely because there is some small but

non-zero overlap between the states localized at opposite surfaces.

Figure 5.5: Normalized probability distribution of states on the two Dirac cones.

Red and blue correspond to different Dirac Cones. Enumeration is over TB

basis, there are 5(atoms)*9(orbitals per atom)=45(orbitals) per QL

Finally we project out the spin polarization of the states corresponding to

the Dirac cones. Recall that TI has spin-momentum locking, i.e. the state

on the Dirac cone must have its spin, 〈~S〉, at 90° to its momentum quantum

number. In figure 5.6 we show spin-momentum locking for two sets of helical

states from two different Dirac cones. In order to obtain those plots, eigen-states

were extracted corresponding to wave vector of a fixed small radius, rotating

it 360° around the Γ point. At each such ~k, two eigen-states were extracted in

the energy window corresponding to the nearly degenerate Dirac cones. These


states were separated into two batches of slightly higher and slightly lower

energies; this way the states from different Dirac cones were separated. Our

calculations show that the spin polarization at each ~k point is at 90°±2°, in

excellent consistency to the spin-momentum locking.

a) b)

Figure 5.6: Momentum-Spin Locking at energy range 2.9eV < E < 2.92eV . At

each wavevector ~k a small arrow shows the direction of spin polarization. a)

Red and b) blue correspond to different Dirac cones and show opposite chirality

locked at 90° to ~k. All quantities are normalized for easy visualization.

The data presented so far suffice to convince oneself that our TB model

gives the proper qualitative (and even semi-quantitative) physics of the TI. We

also point out that the energy bands obtained in figure 5.3b correspond to the

bulk band gap of 0.207eV . This can be deduced from the dispersion curves

by looking at the bands away from the Dirac cone. The value of 0.207eV for

a 6QL system underestimates the gap of 0.26eV obtained in the original DFT

calculation by 0.053eV , which is somewhat large but it does not affect the

transport physics we aim to investigate.

This gap is also easily seen in the Conductance vs. Energy plot, presented in

figure 5.7. The conductance was obtained within Landauer-Buttiker formalism,

discussed in section 4.4. The conductance is measured in the direction of the

crystal primitive vector ~a1, at a fixed transverse wave vector which is set to

zero. In this way, the quantized nature of conductance is easy to see. The

conductance of 2Go (where Go ≡ 2e2/h is the conductance quanta) in the energy


range 2.847eV < E < 3.054eV corresponds to the helical surface states. This

is expected from our theoretical discussion, i.e. each helical state contributes a

conductance quanta. Conductance at energies outside of this range corresponds

to electrons in the valence and conduction bands of the bulk.

Figure 5.7: Conductance vs Energy curve in the direction of the primitive lattice

vector ~a1 at transverse wavevector set to zero. Vertical dashed line indicates

Fermi energy, EF = 2.88eV

In the next section we shall examine how “T vs. E” plot changes as we

simulate transport through a TI film having atomic steps. The robustness

against back-scattering has been confirmed experimentally using TI of high-

quality [63],[64],[65], but effects of atomic trenches or bumps have not been

studied experimentally and, as we show, they can lead to the derailment of the

surface helical states.

5.2 Transport in Bi2Se3 film with a trench

TB model is particularly useful for theoretically studying large systems that

are beyond the capabilities of DFT. Also, in a TB model it is easy to change

different parameters and study the response of the system. Even though some

atomic configurations that can be easily modeled by TB are perhaps hard to

achieve experimentally, such investigations are still interesting as they reveal

important features of the TI physics.


It is very interesting to investigate the ability of helical states to traverse

robustly across atomic trenches on the Bi2Se3 film. We are aware of one re-

cent work along this direction[15] where conductance was calculated for a 5-QL

Bi2Se3 slab having a step-like structure of one or two QLs. In that work, the

scattering region was defined by a gradual increase (decrease) of atomic layers

at the top (bottom) surface - one atomic layer at a time, smoothly changing

the overall width of the film by one or two QLs. As we discussed above, DFT

calculations[14] showed that films thicker than 5-QL are necessary to isolate the

two surfaces and, for thinner films, the states on the top surface can back-scatter

into the states on the bottom surface through the film.

Instead of the stair-case like structure of Ref.[15], here we investigate quan-

tum transport through Bi2Se3 films where a trench is cleaved on the surface.

The atomic structures are schematically shown in Fig.5.9. The trenches are

severe defects on the surface which typically provide large scattering and sig-

nificantly affect the value of conductance. For TI, however, it is expected that

defect scattering is drastically suppressed due to the topological nature of the

helical states. By investigating trenches having various different configurations,

we shall examine the robustness of the helical states against defect scattering.

In our calculations, we always keep the minimum thickness of the Bi2Se3

film (inside the trench) to be 6QL so that the surface helical states on the top

and bottom surfaces are spatially separated. Fig.5.9(d) shows the side view of a

two-lead device in the form of 9QL/6QL/9QL, where the trench has a thickness

of 6QL and device leads are 9QL thick perfect films extending to z = ±∞. The

film is periodic in the transverse direction. We are interested in transport along

the z-direction, from left to the right in Fig.5.9(d). The TB model discussed in

section 5.1 is used. Even though the TB parameters were obtained by fitting

to the 6QL DFT results, these parameters can be applied for thicker films such

as 9QL etc..

Before the transport calculation of the two-lead device, we first confirmed

that the 9QL leads are indeed TI – as they should be, and have correct energy

dispersion, surface helical states, spin-momentum locking, etc.. This also pro-

vides a check on the transferability of our TB parameters. The energy bands of

the 9QL lead is plotted in Fig.5.8, showing the relevant energy range and the

Dirac point inside the bulk band gap.


Figure 5.8: Energy bands of a 9QL slab of Bi2Se3. Fermi energy is at EF =

3.123eV

Figure 5.9 clarifies the notation that shall be used to describe each set-up.

a) b)

c) d)

Figure 5.9: Different set-ups are shown schematically: a) top view of the general

system with parallelograms as unit cells; b)“969a1” example and c) – “9669a2”

with 9QL and 6QL cells labelled in red and green. d) is a side view of “966669a1”


Each configuration can be described by a set of numbers plus a vector de-

scribing direction along which the trenches are oriented. For instance “96669a1”

means that electrons are in-coming from a semi-infinite 9QL lead (the first 9),

entering a central region having 6QL thickness and three unit cells in extension

(the three 6), and exit into the right 9QL semi-infinite lead (the last 9); a1

means that the trench is introduced along the direction of a primitive lattice

vector ~a1. Some configurations may be hard to realize experimentally, for in-

stance the “969a1” configuration – a system which has a sudden trench of 15

atoms deep but only about 2 atoms wide (one unit cell wide). However, wider

trenches may well be realized experimentally by advanced fabrication facilities.

For the 9QL/6QL/9QL two-lead device involving a 6QL trench, different

configurations are studied where the central region has a trench represented by

a sudden decrease of the thickness of film. As we shall see, a sudden decrease

and a sudden increase of the thickness of the slab have somewhat different effect

on the conductance.

First we present the calculated conductance versus electron energy for sys-

tems “96669a1” and “96669a2” in figures 5.10a and 5.10b, respectively.

a) b)

Figure 5.10: Conductance vs Energy. Red – 9QL slab, blue – a) “96669a1”

system and b) “96669a2”. Dashed vertical line represents Fermi energy at

EF = 3.123eV . Both systems exhibit conductance of 2 in the energy range

corresponding to the helical states, i.e. no back-scattering

One can see – extremely remarkably, that the helical states exhibit no back-

scattering by the trench! Namely, conductance G = 2Go in the band gap

regardless of the trench. This result is exactly in accordance with the physics


of TI. It is well known that introducing a sudden decrease in the thickness of

the film typically causes considerable back-scattering. This is because in the

vicinity of the atomic trench, the potential rapidly increases due to the presence

of vacuum and electrons incoming to near the vicinity of the trench “feel” a force

in the opposite direction to their motion. However, such a potential barrier

acts only on the charge degree of freedom and cannot flip the spin. For helical

states, flipping spin is a necessary condition for back-scattering due to the spin-

momentum locking. Since this cannot happen, the trench does not back-scatter

the incoming electrons and G = 2Go is therefore obtained. In other words, if the

conducting state were not protected topologically, the trench – which is a large

defect, would significantly scatter the electrons and G would be reduced from

2Go. We have also calculated other systems of the type “96669” but having

various extensions of the 6QL trench region. The effect is always the same,

namely no back-scattering in the helical states and G = 2Go in the bulk gap.

Since the results all look similar to those of Figs. 5.10a and 5.10b, we do not

plot them here.

It is thus obvious that a defect such as a single trench, does not derail the

helical states. Nevertheless, if the scattering region is so irregular as to mix

the helical states together, back-scattering will be possible. We now investigate

such drastically irregular structures. In figure 5.11 we present conductance

for a device in the “96969a1” configuration. This structure has a 6QL thick

double-trench.

Figure 5.11: Conductance vs Energy plot in the direction of ~a1; red – 9QL slab,

blue – “96969a1”. Conductance of the Helical states starts to deviate away

from the value of 2Go near the conduction and valence bands.


Overall, the conductance of the helical states is still fairly robust, G = 2Go

for a wide range of energies inside the bulk gap. However, near the energy of

the conduction band or the valence band, the conductance gradually drops to

G < 2Go. In particular, near the valence bands the conductance drops below

2Go in the region just below EF (vertical dashed line). Since helical states

cannot elastically back-scatter, something must be happening to them. It is

safe to claim that the reduced conductance is due to the helical states on the top

surface, since the bottom surface is perfectly flat so there is no scattering region

for the helical states at the bottom surface. In addition, the bottom surface

is separated from the top by at least 6QLs so that the interaction between

the two surfaces through the film can be excluded. We postpone presenting

possible reasons for the reduced conductance and study a few more relevant

systems before returning to this issue.

The principal difference between systems “96669a1” and “96969a1” is the

presence of the sudden increase in the slab thickness in the latter, that makes

structure more irregular. Let’s examine systems of the type “696a1”, i.e. system

where electrons coming from a 6QL semi-infinite lead and have to overcome an

abrupt increase in the film thickness, as if to go over a sudden mountain. In

figure 5.12 we present a series of results where the extension of the central

9QL region becomes gradually longer, from 1 to 7 unit cells (i.e. from 696 to

699999996).

a) b)

Figure 5.12: Conductance vs. Energy in “696a1”-type systems. Central 9QL

region’s length is varied in a) and in b) only 9QL region of 1 and 7 unit cells

are compared. Arrows indicate the improvement in conductance with length


The last system, having a 9QL bump and 7 unit cells in extension, is fairly

realistic and fabricatable by advanced techniques. It represents an atomic step

of 15 atoms in height and about 15 atoms in lateral extension. We can see that

the net effect of increasing the lateral extension of the 9QL central region is

that the conductance more and more approaches that of the perfect 666 film

(red line). This is especially noticeable near the Fermi energy. Clearly, if the

9QL central region had infinitely long extension, we would recover perfectly

G = 2Go. Here we observe that a seven unit cell extension is long enough

to almost achieve this limit. The result also shows that it is more difficult to

traverse a “mountain” than to go over a trench.

We now want to draw conclusions from the data showing reduced conduc-

tance of the helical states in the mountain-climbing situation. Recall that it

takes 6QL in order to spatially separate the helical states associated with the

top and the bottom surfaces. It is our opinion that the reduced conductance

arises due to the top surface states having overlap with itself in the central

region of the type “696”. To aid our discussion, in figure 5.13 we plot schematic

diagrams of the helical states going through a sudden decrease-and-increase in

thickness (“969” type) or a sudden increase-and-decrease (“696” type).

a) b)

Figure 5.13: Schematic view of the helical states confined to the top and bottom

surfaces, in blue and orange respectively. In a) the system of the type “969a1”

is shown and in b) – “696a1”. The images are not to scale.

The helical states in the presence of an atomic step (going up mountain)

have an amazing ability to simply “go around” – thanks to their topological

nature, as discussed in Ref.[15]. However, sometimes there is “no room” for

the surface subsystem to establish in a way that keeps it isolated from all

other helical states. We have seen in figure 5.5 that the surface states have non-


negligible amplitude in 2QL closest to the surface – this is a considerable spread

over 10 atomic layers (2QL = 10 atomic layer). We thus conclude that helical

states associated with a given surface, e.g. top, can overlap with itself depending

on the exact geometry of the system. More precisely, helical states with opposite

wave vector and same spin may not be separated in space anymore. This

situation is shown schematically in figure 5.13b; the helical states inside the

9QL region overlap and so that they are somewhat derailed leading to G < 2Go.

Furthermore, in figures 5.12a and 5.12b, one can see that elongating the 9QL

region has the main effect on conductance near the Fermi energy. Elongating

the 9QL region leads to reducing the overlap of the helical states inside that

same region. The fact that we see the main effect in the vicinity of the Fermi

level can be understood if we recall the effect on the energy band diagram of

Bi2Se3 of having less than 6QL separation. As an example, we include energy

bands for 2, 3 and 4 QL Bi2Se3 system in figure 5.14, taken from the ref. [15].

a) b) c)

Figure 5.14: Energy band diagram of Bi2Se3 in a slab geometry with a) 2QL b)

3QL and c) 4QL slab thickness. The gap gradually closes reaching a degeneracy

at a Dirac point as the separation between surfaces increases. Figure courtecy

of Ref.[15]

Looking at figure 5.14 one should remember that for our 6 and 9 QL systems

the Fermi energy is exactly at the Dirac point. Therefore, it is plausible that

what we see in figure 5.12 is directly related to the gap-opening due to overlap.

In a “696a1”-type system the helical states are injected into the central region,

where the helical states overlap with themselves. Those states with energies

in the neighborhood of the Fermi energy seem to be affected the most by such

an overlap. Inside the scattering region, instead of having states with opposite


momentum having opposite spin, we get new states which are superpositions of

the original helical states, thus G < 2Go becomes possible.

The above discussion gives a qualitative reasoning behind the reduced G.

Our data predict that whenever a surface has “mountains”, i.e. sudden elevation

followed by a sudden descent, helical states can be derailed to produce a reduced

conductance – equivalent to the occurrence of back-scattering. In order to cause

any noticeable conductance reduction, these mountains should have appropriate

scale. The height should be on the order of 1QL or more, which compares to

the approximate spatial extension of the surface helical states. The extent of

the mountain should be 25 atoms or less, since it takes about 25 atoms (∼ 5QL)

to spatially separate helical states corresponding to different surfaces.

Such mountains or surface bumps can certainly exist in real topological

insulators. Our analysis shows an effective mechanism to induce elastic back-

scattering of the helical states. This is despite a well established understanding

that helical states cannot elastically back-scatter in the absence of electron-

electron interactions or time-reversal symmetry breaking disorder[19],[23]. This

is true for an isolated helical state on a perfect surface but, as we have demon-

strated, not true for a very irregular surface.

Our calculation shows that effective back-scattering is possible without

electron-electron interactions or time-reversal breaking impurities: by mixing

the surface states in the bulk region. To our knowledge this is the first work

that points out this possibility. In a perfect sample this back-scattering does

not occur, but we need to keep this effect in mind when designing TI systems

involving non-trivial topology of the surfaces.

6

Discussion and Conclusions

In this thesis we have first presented an in-depth discussion of the theory behind

TIs in chapters 2 and 3. This was followed by chapter 4 about electron transport

theory suitable for fast calculations of large systems. In chapter 5 we presented

our numerical study into electron transport in Bi2Se3 subject to non-trivial

surface topology. In this last chapter we draw some conclusions from the work

presented in chapters 2 – 5 and outline some possible directions for future

research. The aim is to leave the reader with ideas for a new research.

In chapter 2 we have seen how a TI comes to existence. As a researcher

one is interested in discovering a new TI, among other things. Having identified

several different TIs may prove very useful when trying to build a device utilizing

TI in the future. As such, in section 6.1 we present a speculative argument for

a possible new TI. We argue that Cd3As2 is potentially a TI. Primarily, section

6.1 is meant to bring attention to Cd3As2 for possible future research. However,

it also gives some clues as to how to search for new TIs in general. Unlike section

2 which presented a general discussion of the theory of TI, section 6.1 focuses

on the practical difficulties of identifying a new TI.

In section 6.2 we present a brief discussion meant to point out the impor-

tance of the concept of Berry’s phase. We have seen in section 3 that one way

to understand TI is as a consequence of Berry’s phase. We have linked Berry’s

phase, on one hand, to the topology of the energy bands and on the other

hand to the spin current on the system’s boundary. In section 6.2 we point out

how similar physics based on Berry’s phase had paved the way for producing a

system similar to a TI but in the realm of photonics.

In section 6.3 we review again the benefits and limitations of a TB model,

which were originally presented in section 4.1 - 4.4. We then discuss future

work that can be done based on our existing TB model of Bi2Se3.

78

6: Discussion and Conclusions 79

6.1 Cd3As2 - candidate for a new Topological Insula-

tor

In this section we shall present an argument for why Cd3As2 may be 3D Strong

Topological Insulator. The argument is made using the theoretical understand-

ing established in Chapters 2 and 3. Currently there is no such claim in the

scientific literature. By the end of the chapter we may also gain a better un-

derstanding of a TI.

In order not to loose the track of logical cause-and-effect steps which lead

up to the above stated claim, we first write a step-by-step summary, and then

elaborate on each step in more detail. The following provides a summary of how

we deduced that Cd3As2 is potentially a TI, based on the available scientific

literature regarding this material:

1. Between a trivial phase and a TI phase there is a phase transition;

2. This phase transition requires closing and re-opening the gap precisely

because the two phases are topologically distinct;

3. This phase transition between a regular insulator and a Topological Insu-

lator is induced by spin-orbit interaction;

4. This means that for a given TI the Hamiltonian can, in principle, be

written as H0 + λ∆SO with λ = 1. Taking λ → 0 gradually should

produce the phase transition towards the regular insulator;

5. Therefore, if a Hamiltonian of the form H0 +λ∆SO is available for a given

material with a bulk band gap, say “A2B3”, one can deduce whether this

is a TI or not depending on whether the gap closes and reopens upon

λ→ 0 transition or remains open;

6. If for A2B3 an explicit Hamiltonian in the form H0 +λ∆SO is not available

and the transition λ → 0 has not been studied there is an alternative

approach. One can look at different published theoretical works regarding

A2B3, some of which disregarded SOI as unimportant and some of which

included it;

7. It turns out that the system “before” the phase triansition (i.e. λ = 0;

regular insulator) and the system “after” (λ = 1;TI) leave a characteristic


“signature”. This makes it possible to look at the system with λ = 1 and

deduce whether taking λ = 0 → 1 gradually would close and reopen the

gap. The signature of being in a Topologically non-trivial phase is the

presence of the so-called inverted gap;

8. The definition of an inverted gap will be given below. What is important

is that a given material with a band gap in the bulk can give either a

“regular” band gap or an “inverted” one. A transition from the regular

band structure to an inverted one requires closing and reopening the band

gap;

9. This means that if the material without SOI has regular band gap, while

with SOI this same material has an inverted band gap – the material must

have undergone a topological phase transition induced by the SOI;

10. This suffices to propose that Cd3As2 as a candidate for a 3D TI, upon

examining existing published papers on this material. Cd3As2 has an

inverted band gap in the bulk, and it has a strong SOI.

Now we present an expanded discussion on each of the above points. The first

point follows by the definition. The discovery of a TI is precisely the realization

that band insulators with time-reversal symmetry come in two distinct phases.

As was already discussed, these two phases are distinguished by the topol-

ogy of the bands – the bands of a TI cannot be adiabatically transformed into

the bands of a trivial band insulator without closing and reopening the gap.

This is conveyed in the point 2. This mechanism has already been pointed out

as particularly important in the search for TIs by Fu and Kane[66], Murakami

et. al.[21] and others[67].

Point 3 follows from the discussion in section 2.5 and then again in the

summary, section 3.4.

Points 4 and 5 are essentially restatements of the point 3. The topological

phase transition is the hallmark of a Topological Insulator. One should reiterate,

however, that it is impossible to have helical boundary states without SOI. Thus

if in going from λ = 1→ 0 the gap does not close and reopen, we deduce that

we started with a regular insulator and ended up in a regular insulating phase.

At λ = 0 it is impossible to have a Topological Insulator, while at λ = 1 one

may or may not have a Topological Insulator.


Point 6 simply dwells on the issue of testing different materials for such a

topological phase transition in practice. Topological Insulators were discovered

only several years ago. Thus older scientific work was typically not focused on

the SOI effect. In the majority of the materials SOI interaction only produces a

negligible effect. This fact, coupled to the fact that adding SOI into the model

introduces extra complications and requires longer computation times meant

that typically in the past SOI was not included in theory. In addition, it is

essentially a convention to study the bulk of the material whenever using Bloch

Band formalism. As such, even if SOI is included but one is concerned with the

physics in the bulk only – the helical edge states and all the peculiar properties

that follow will remain unnoticed.

Point 7 is the key realization which makes it possible to look through the

scientific literature to find new Topological Insulators. We must say a few

words about what the inverted gap is. For a band insulator, there is a band

gap with Fermi energy EF in the gap. There are valence bands below EF and

conduction bands above. At zero temperature such a material cannot conduct

since energy is required to promote electrons from a valence band into the

conduction bands. If the gap is large enough, then even at room temperature

there is not enough thermal energy to promote a significant number of electrons

into the conduction bands. Either way, as one gradually increases electric field

across such a material, at some critical point there is enough energy for electrons

to be promoted into the conduction band and support current. This process will

be dominated by regions of the band structure where the band gap is minimal.

The valence and the conduction bands in the vicinity of a band-gap minimum

in most cases have negative and positive curvatures. It is a convention to

think of quasi-particles within Bloch formalism in the following way. A particle

with wave-vector ~k has an effective mass proportional to the curvature of the

corresponding band at ~k. Thus, in most cases, when an electron is promoted into

the conduction band with a positive curvature it is viewed as an electron with

positive mass; the hole that it leaves behind corresponds to a negative curvature

band and is treated as a positron with negative mass and a positive charge. An

electron can give off energy returning back to the valence band, mimicking an

electron combining with a positron and giving off energy. The inverted gap is

when this order is reversed, i.e. in the vicinity of the minimum, conduction band


has negative curvature and the valence band has positive curvature. Sometimes

this effect is very noticeable in the band structure, but often the valence band,

conduction band or both are nearly flat having only slightly negative curvature

in the small neighbourhood of the minimum band gap. In the literature people

sometimes refer to quasi-particles in such systems as having negative mass.

From the paragraph above it is easy to see why a transition from a regular

band gap to an inverted one requires closing and reopening of the gap. This is

point 8. The transition can be envisioned as follows. In the materials with

an inverted band gap the conduction band in the vicinity of the band-gap

minimum moves down, while the valence band moves up. The original curvature

is retained however. This often occurs at the Γ point.

Point 9 simply states that if SOI induces an inverted gap then it induces a

topological phase transition. The converse may or may not be true however. It

is possible to have an inverted gap without SOI.

In point 10 we claim that Cd3As2 is a TI candidate. We now present a

literature review of the relevant work in order to demonstrate this claim.

The crystal structure of Cd3As2 had gotten firmly established by 1967 from

the work of Goodyear and Steigmann[68]. It has an unusually large unit cell

consisting of 160 atoms due to the presence of vacancies; the unit cell can be

divided into 16 cubic sub-cells each of which has a different configuration of

vacancies. This is the reason for why it took relatively a lot of time to examine

this material theoretically. The knowledge of the crystal structure paved the

way for theoretical calculations of its band structure. Initially it was not well

established whether Cd3As2 had an inverted band structure or a direct gap

like other materials of the II-V group such as Zn3As2. In 1969 P.J. Lin-Chung

had produced the first attempt at obtaining the band structure of Cd3As2[69].

Among other approximations, the SOI was neglected. This produced a band

insulator with the direct gap of about 0.6eV approximately at the Γ point.

At around the same time, some other authors started arguing for the in-

verted band structure[69],[70],[71]. In ref. [71] for instance, they have studied

the change in the energy gap of alloy Cd3−xZnxAs2, where x is taken from 3

to 0. A combination of measurements at different values of x and interpolation

of the data to x = 0 had shown that the gap goes from positive to negative,

with the final value of about −0.1eV . Importantly, the gap closes and reopens


as x changes from 3 to 0 – which is an indication that Zn3As2 and Cd3As2

are potentially topologically distinct. A convincing solution to this problem

was finally presented in 1976 by M.J. Aubin, L.G. Caron and J.-P. Jay-Gerin

concluding that Cd3As2 has an inverted band gap of about −0.19eV [70]. Also,

as one can learn from ref. [70], other transport measurements performed on

Cd3−xZnxAs2 alloys at room temperature produce an anomaly which is consis-

tent with the crossover of conduction and valence bands – again, an indication

of the possible topological phase transition. M.J. Aubin et. al. then go on

to perform the calculations of band structure of Cd3As2 among other things.

In their calculations, unlike Lin-Chung, SOI was included and it was explicitly

argued that it must be included since its value is comparable in energy to the

size of the gap – this is generally the case for TIs. Recalling that Lin-Chung

had obtained a regular gap rather than inverted, it is thus tempting to conclude

that with SOI you get an inverted gap and without – regular gap. However,

these two authors used different models so it is not completely clear what is

responsible for the inverted gap. If the inverted gap is SOI-induced, then it is

indeed a very strong indication that Cd3As2 is a TI. We need the inverted gap

to be SOI-induced since this is equivalent to saying that the topological phase

transition is SOI-induced and hence we have a QSH phase.

In 1979 B. Dowgiallo-Plenkiewicz and P. Plenkiewicz had returned to this

problem. Theoretically, using pseudo-potential band structure calculations on

the real non-approximated crystal structure, they found that both with and

without SOI they get an inverted gap[72]. This goes contrary to our hopes that

the inverted gap is SOI-generated. However, their atomic pseudo-potential form

factors were determined by extrapolation from those used in other related ma-

terials such as CdTe, GaAs, etc. Also, the secular determinant was solved using

a perturbation technique. That is, it is hard to estimate the accuracy of their

results. Those familiar with calculations of band structures of materials with

a very small gap (∼ 0.1eV − 0.2eV ) know that even with the well established

DFT and the currently available computer power it is not a trivial task.

Later on, in 1983 another team, including participation from McGill, had

again studied the band structure of Cd3As2 theoretically and had reached con-

clusions that neither confirm nor dispute those reached by Plankiewicz et.

al.[73]. In particular in ref. [73] they had focused on how the band structure


changes as we decrease the number of approximations step by step, starting with

Lin Chung’s model. They start by making an independent calculation with the

same approximated crystal structure as used by Lin Chung; SOI is excluded.

In the next step, the calculations are based on the same crystal structure but

with SOI included. Finally in ref. [73] they make a calculation which is based

on the true crystal structure of Cd3As2 taking vacancies into account; SOI is

included. The following results were found. The first and the second steps

which used the fictitious fluorite crystal structure both produced similar band

structure with a regular band gap. In the last step, with vacancies represented

by appropriate pseudo-potentials and SOI included one gets drastic changes to

the band structure and importantly one gets inverted band gap. Unfortunately,

B. Plenkiewicz et. al. did not calculate the band structure with vacancies but

without SOI.

From the above discussion one can conclude, if nothing else, that the origin

of the inverted gap is inconclusive. By now it is well-established that Cd3As2

has an inverted gap of ∼ 0.19eV and a SOI strength of ∼ 0.3eV at Γ. It may

or may not be SOI-induced.

We make an argument in favour of the inverted gap being SOI-induced.

A material from the same family, Zn3As2 has an identical electronic structure

as Cd3As2 except Zn has its principal quantum number n = 4, one less than

the Cadmium’s n = 5. In the periodic table Zn is right above Cd correspond-

ing to full s, p and d orbitals. From the papers by Plenkiewicz et. al. and

from the work conducted at McGill[73], it seems like the presence of vacan-

cies is crucial for having an inverted gap. But the two crystals Zn3As2 and

Cd3As2 have identical positions of vacancies within the crystal structure. Yet,

the first one has a regular band gap while the latter has an inverted band gap.

Furthermore, a phase transition which closes the gap was directly observed in

alloys Cd3−xZnxAs2. We now point to the fact that Cd is much heavier than

Zn and so it has much bigger SOI strength. The atomic number of Cd is 48,

as compared to 30 of Zn. This is 60% increase in going from Zn to Cd. This

is somewhat reminiscent of the case of Se3Sb2 and Te3Sb2. Both have identical

electronic structure, the difference is that Se is directly above Te in the periodic

table with the difference of 1 in their principle quantum numbers n. For Se3Sb2

and Te3Sb2, however, it is well established in an experiment[4] that the former


(lighter) material is not a Topological Insulator while the latter (heavier) is.

For Zn3As2 and Cd3As2 where the former has regular gap and the later has an

inverted gap we suggest that the situation is analogous.

In conclusion, it is the opinion of the author that Cd3As2 is a strong can-

didate for being a Topological Insulator. The modern computer power and

advances in theoretical computation techniques are probably sufficient to reli-

ably answer this question.

6.2 Berry’s phase and chirality in photonics

In chapter 3 we have introduced the concept of Berry’s phase and related phe-

nomena. We saw that Berry’s phase can be seen as a mechanism that converts a

non-trivial band topology into observable surface spin currents. Our discussion

was within the context of TIs so we had avoided venturing into other systems

where Berry’s phase manifests itself. In this section we would like to bring to

attention just one of the recently discovered systems in which Berry’s phase

plays an important role. We briefly mention some other examples as well.

The system is, broadly, a Photonic Crystal with Broken Time-Reversal Sym-

metry [74]. It has direct relevance to our discussion of Haldane’s model in section

2.4 which produces Integer Quantum Hall Effect. IQHE involved electrons un-

der chiral magnetic field exhibiting chiral edge states under suitable conditions.

Here, instead, a somewhat similar effect is produced by photons. There are

many differences between a system of photons and electrons. Electrons are

fermions, have charge and keep a conserved total number of particles. Photons

are bosons, are electrically neutral and can be absorbed or emitted by the crys-

tal. The two systems seem quite different but it turns out that physics related

to Berry’s phase are similar for both and produce similar effects.

By a Photonic crystal one means a periodic “metamaterial” that transmits

electromagnetic waves. The work in ref. [74] outlines the possibility of having

a localised 1-dimensional channel which acts as a directional waveguide. The

unidirectional photonic modes confined to this channel cannot back-scatter (re-

flect) at bends or imperfections[74]. This is analogous to the chiral edge states

of the IQHE. The “ingredients” to have a system producing these unidirec-

tional modes are as follows. One needs a 2D photonic crystal which exhibits

a photonic band gap (PBG). The crystal must be the “Faraday-effect” crystal.


One may recall that magnetic field can interact with light making the plane

of polarization rotare; this is called the Faraday Effect. The photonic crystal

must produce this effect (and hence break the time-reversal symmetry). In such

a material there is a well-defined axis associated with the Faraday’s effect. If

one is to join two such materials with the opposite Faraday’s axis one obtains

the desired unidirectional waveguide with the energy dispersion inside the PBG.

Figure 6.1 shows a band diagram for different transverse electric (TE) and mag-

netic (TM) modes that are allowed in the material. A photonic band gap is

formed of modes that are not supported by the material. The above described

effect produces a 1D channel with energy dispersion resembling that of a Dirac

cone.

Figure 6.1: Photonic Bands with a band gap. Dirac-cone-like dispersion corre-

sponds to the 1D waveguide localised at the junction of two photonic crystals

with opposite Faraday’s axis. Figure courtecy of Ref.[74]

The details are left out[74],[75]. In brief, these photonic crystals have pho-

tonic bands. In analogy with the energy bands of Bloch states there are Berry’s

phases that can be associated with each photonic band. One can then define

the topology of the photonic bands in terms of these Berry’s phases. The two

materials with the opposite Faraday’s axes have distinct topology and so they

are connected by a continuous state, much like the “edge” state in IQHE. This

exciting theoretical proposition has been recently realized in an experiment in

March, 2012[76].


It is quite fascinating that in a system of photons, quite different from an

electronic system exhibiting IQHE, we get a very similar effect. In fact, many

different teams are currently trying to utilize in optics the robustness that the

topologically induced states offer. Recently a paper in Nature Physics had

come out that demonstrated a way to have a photonic system analogous to

the QSHE[77]. It does not require external magnetic field which breaks time-

reversal symmetry. Instead an effective SOI mechanism is described, mimicking

the way SOI in a QSHE produces an effective magnetic field for each spin.

The effective spin is the clockwise and counter-clockwise polarization of the

electromagnetic modes while the SOI comes from coupling of these modes to

resonator optical waveguides[77]. The outcome is the topologically protected

propagating modes, robust against back-scattering.

Another exciting devolpment is in the field of x-ray optics. X-ray beams

interact very little with matter because their refraction scales with the square of

the wavelength[78]. Therefore, x-ray optics often uses diffraction to control the

x-ray beam, and for many optical elements a high degree of crystal perfection

is desirable and necessary[78]. It turns out that due to Berry’s phase effect the

x-ray beam is sensitive to the Bragg reflection points where its refraction scales

as the inverse square of wavelength[78],[79]. The net effect is that crystals with

gradually deformed crystal structures can be utilized as waveguides.

In this section we have seen how an IQH-like effect can take place in photonic

crystals. We also listed few recent develoments in the related fields, all of which

have Berry’s phase effect in common. Many more examples where excluded;

topologically protected edge states have been shown to exist even in phonon

systems[80],[81],[82]. Berry’s phase and the related topological arguments may

not always be the most intuitive way of understanding the physics in a material.

However, the phenomenon of Berry’s phase is a universal effect that transcends a

wide range of different systems. Having understood well topological arguments

for bulk-boundary correspondence giving rise to the edge states in IQHE and

QSHE one can then easily understand similar effect in complitely different fields.

6.3 Outlook for TB-based numerical study of Bi2Se3

In chapter 4 we have presented a the framework for the numerical study of trans-

port properties of Bi2Se3. In section 4.1 we have introduced the TB model, a


model which is very simple, intuitive and does not require a lot of computational

power. In section 4.3 we examined the extent to which TB model is reliable

by formulating it within the DFT formalism. We have reached a conclusion

that DFT-based TB model can give reliable results. The optimal use of a TB

model is then to theoretically study those systems which are too large for a

DFT calculation.

In chapter 5 we have presented our TB model of Bi2Se3 and presented some

preliminary results on transport properties of this material. To the best of

our knowledge, there are currently two groups who have used an atomistic TB

model to study the helical states of Bi2Se3[15],[83]. There still remain questions

regarding Bi2Se3 that can be tackled using our TB model. We list two examples.

First, there is no reliable quantitative investigation into the effect of diag-

onal disorder on the helical states of Bi2Se3. Some preliminary results have

been obtained using DFT[84]. The work in ref. [84] suggests that disorder

can destroy topological states with the value of perturbing potential of 0.3eV .

However, the system studied contained two unit cells and cannot mimic disorder

realistically. This encourages further research into the effect of disorder using

our TB model.

Another possible use of our TB model can be in testing the recent exciting

prediction of refraction effects in Bi2Se3[85]. In ref. [85] a DFT calculation

have been performed to obtain, among other things, energy dispersion of surface

states of Bi2Se3. What is new about this work is that the surface chosen was

(221) in Miller indices unlike the conventional (111) surface. The surface (221)

has a reduced symmetry as compared to (111) which manifests itself in the

energy dispersion. This surface also has a single Dirac cone but it is highly

anisotropic. One can then construct a low energy effective model where the

anisotrpy of the Dirac cone manifests itself as electrons having different speeds

depending on direction of motion. The electrons that travel from a (111) surface

onto a (221) surface or vice versa experience an effect similar to that of light

which crosses the boundary between two mediums of different refractive indeces.

Remarkably, it is predicted that it is possible to have an effect equivalent to a

total internal refraction. The effective model predicts a peculiar dependence of

conductance on the angle of incidence between different surfaces. Two examples

are shown in figures 6.2a and 6.2b.


a) b)

Figure 6.2: Transition between surfaces at an incidence angle φ – a) (111) to

(221) and b) (111) to (221) to (111), in Miller indeces. Figure courtecy of

Ref.[85]

This dependence can be tested numericaly using our TB model.

We have listed but a few ways in which to continue research based on a

TB model of a Bi2Se3. This concludes this section, this chapter and the entire

thesis. It is the hope of the author that this thesis brings useful information to

the reader and gives ideas for future research.

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