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
2: Theory of Topological Insulators 6
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
2: Theory of Topological Insulators 7
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
2: Theory of Topological Insulators 8
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.
2: Theory of Topological Insulators 9
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
2: Theory of Topological Insulators 10
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
2: Theory of Topological Insulators 11
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
2: Theory of Topological Insulators 12
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
2: Theory of Topological Insulators 13
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)
2: Theory of Topological Insulators 14
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
2: Theory of Topological Insulators 15
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]
2: Theory of Topological Insulators 16
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)
2: Theory of Topological Insulators 17
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)
2: Theory of Topological Insulators 18
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
2: Theory of Topological Insulators 19
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
2: Theory of Topological Insulators 20
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
2: Theory of Topological Insulators 21
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
2: Theory of Topological Insulators 22
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
2: Theory of Topological Insulators 23
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.
2: Theory of Topological Insulators 24
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|
2: Theory of Topological Insulators 25
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
2: Theory of Topological Insulators 26
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
2: Theory of Topological Insulators 27
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
2: Theory of Topological Insulators 28
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)
2: Theory of Topological Insulators 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)
2: Theory of Topological Insulators 30
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)
2: Theory of Topological Insulators 31
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
2: Theory of Topological Insulators 32
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
2: Theory of Topological Insulators 33
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)
3: Berry’s phase and the Topological Invariants 36
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.
3: Berry’s phase and the Topological Invariants 37
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
3: Berry’s phase and the Topological Invariants 38
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-
3: Berry’s phase and the Topological Invariants 39
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
3: Berry’s phase and the Topological Invariants 40
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: Berry’s phase and the Topological Invariants 41
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
3: Berry’s phase and the Topological Invariants 42
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
3: Berry’s phase and the Topological Invariants 43
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
3: Berry’s phase and the Topological Invariants 44
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)
3: Berry’s phase and the Topological Invariants 45
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
3: Berry’s phase and the Topological Invariants 46
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
3: Berry’s phase and the Topological Invariants 47
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.
4: Quantum transport – atomistic point of view 50
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)
4: Quantum transport – atomistic point of view 51
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
4: Quantum transport – atomistic point of view 52
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.
4: Quantum transport – atomistic point of view 53
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
4: Quantum transport – atomistic point of view 54
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)
4: Quantum transport – atomistic point of view 55
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)
4: Quantum transport – atomistic point of view 56
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)
4: Quantum transport – atomistic point of view 57
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)
4: Quantum transport – atomistic point of view 58
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
4: Quantum transport – atomistic point of view 59
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
4: Quantum transport – atomistic point of view 60
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 ,
4: Quantum transport – atomistic point of view 61
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-
5: Quantum transport in Bi2Se3 nanostructures 64
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)
5: Quantum transport in Bi2Se3 nanostructures 65
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.
5: Quantum transport in Bi2Se3 nanostructures 66
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.
5: Quantum transport in Bi2Se3 nanostructures 67
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
5: Quantum transport in Bi2Se3 nanostructures 68
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
5: Quantum transport in Bi2Se3 nanostructures 69
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.
5: Quantum transport in Bi2Se3 nanostructures 70
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.
5: Quantum transport in Bi2Se3 nanostructures 71
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”
5: Quantum transport in Bi2Se3 nanostructures 72
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
5: Quantum transport in Bi2Se3 nanostructures 73
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.
5: Quantum transport in Bi2Se3 nanostructures 74
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
5: Quantum transport in Bi2Se3 nanostructures 75
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-
5: Quantum transport in Bi2Se3 nanostructures 76
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
5: Quantum transport in Bi2Se3 nanostructures 77
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
6: Discussion and Conclusions 80
“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.
6: Discussion and Conclusions 81
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
6: Discussion and Conclusions 82
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
6: Discussion and Conclusions 83
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
6: Discussion and Conclusions 84
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
6: Discussion and Conclusions 85
(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.
6: Discussion and Conclusions 86
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].
6: Discussion and Conclusions 87
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
6: Discussion and Conclusions 88
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.
6: Discussion and Conclusions 89
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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