TOPLOGICAL INSULATORS REVIEW PAPER

NAME: STEPHEN UDOCHUKWU CHUKWUEMEKA
I.D NO: 14210334
LECTURER: Prof. Patrick McNally
MODULE: EE559 FUNDAMENTALS OF NANOELECTRONICS TECHNOLOGY
PAPER TITLE: Topological Insulators
I hereby declare that the attached submission is all my own work, that it has not previously been
submitted for assessment, and that I have not knowingly allowed it to be used by another student.
I understand that deceiving or attempting to deceive examiners by passing off the work of
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doing so will result in loss of marks and possible disciplinary proceedings.
Signed: STEPHEN
Date: 2nd
December, 2015

Stephen Chukwuemeka [14210334] EE559 Fundamentals of Nanoelectronics Technology Review Paper 1

Abstract — Topological Insulators are electronic
devices which possess bulk band gap like an ordinary
insulator in its interior but whose edge or surface
contains protected conducting states.
These states are conceivable because of the merger of
spin-orbit connections and time-inversion symmetry.
This review paper presents a general account of
Topological Insulators with an emphasis on basic
theory and material properties. Topological
Insulators like 2D – two dimensional is known as a
quantum spin Hall insulator which is related to
integer quantum Hall State, while the 3D – three
dimensional topological insulators encourages
innovative spin-polarized 2D Dirac fermions on its
surface. Adequate examination of the distinctive
properties of topological insulators will be observed
in this paper. Specifically, the Dirac fermion material
science and the subsequent unconventional quantum
oscillation examples are talked about in point of
interest. It is underlined that appropriate
examinations of quantum motions make it
conceivable to unambiguously recognize surface
Dirac fermions through transport estimations.
Keywords – topological insulator, quantum oscillation,
Dirac fermions
1. INTRODUCTION
The advancement in consolidated matter material
science is regularly determined by innovations of new
materials. In such manner, materials introducing one of a
kind quantum-mechanical property are of specific
significance. Topological Insulators (TIs) are a class of
such materials and they are presently making a surge of
survey activities.1–3)
Because TIs concern a subjectively
new part of quantum mechanics, that is, the topology of
the Hilbert space, they opened another window for
understanding the complex workings of nature. TIs are
called "topological" in light of the fact that the wave
capacities depicting their electronic states compass a
Hilbert space that has a nontrivial topology. Keep in
mind, quantum-mechanical wave capacities are depicted
by linear blends of orthonormal vectors shaping a
premise set, and the theoretical space spread over by this
orthonormal premise is called Hilbert space. In
crystalline solids, where the wave vector k turns into a
decent quantum number, the wave capacity can be seen
as a mapping from the k-space to a complex in the
Hilbert space (or in its projection), and subsequently the
topology gets to be important to electronic states in
solids. Subject to the way the Hilbert-space topology
gets to be nontrivial, there can be different various types
of TIs.4)
A vital result of a nontrivial topology related with the
wave elements of an insulator is that a gapless interface
state essentially shows up when the insulator is
physically terminated and meets an ordinary insulator
(including the vacuum). This is on the grounds that the
nontrivial topology is a discrete characteristic for gapped
energy states, and in as much as the energy gap stays
open, the topology can't change; thus, all together for the
topology to change over the interface into a trifling one,
the gap must close at the interface. Along these lines,
three-dimensional (3D) TIs are continuously connected
with gapless surface states, likewise two-dimensional
(2D) TIs with gapless edge states. This rule for the
fundamental event of gapless interface states is called
bulk-boundary correspondence in topological stages. 6]
Topological Insulators
Stephen Chukwuemeka, School of Electronic Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland.
stephen.chukwuemeka2@mail.dcu.ie

Figure 1; Edge and surface states of topological
insulators with Dirac dispersions. (a) Schematic real-
space picture of the 1D helical edge state of a 2D TI. (b)
Energy dispersion of the spin non-degenerate edge state
of a 2D TI forming a 1D Dirac cone. (c) Schematic real-
space picture of the 2D helical surface state of a 3D TI.
(d) Energy dispersion of the spin non-degenerate surface
state of a 3D TI forming a 2D Dirac cone; due to the
helical spin polarization, back scattering from k to -k is
prohibited 6]
(Yiochi Ando, 2013)
A huge section of the distinct quantum-mechanical
properties of TIs originate from the curious attributes of
the edge/ surface states. Presently, the TI exploration is
engaged for the most part on time-reversal (TR)
invariant systems, where the nontrivial topology is
ensured by time-inversion symmetry (TRS).1–3)
In those
systems, the edge/surface states present Dirac scatterings
(Fig. 1), and subsequently the material science of
relativistic Dirac fermions gets to be significant.
Moreover, spin degeneracy is lifted in the Dirac
fermions dwelling in the edge/surface conditions of TR-
invariant TIs and their sin is bolted to the force (Fig. 1).
Such a spin state is said to have ''helical spin
polarization'' and it gives a chance to acknowledge
Majorana fermions 5)
in the vicinity of proximity-
induced superconductivity, 6)
also its undeniable
consequences for spintronics applications.
A vital target of the trial investigations of TIs has been
to explain the presence and the technique of such
helically turn spellbound Dirac fermions in the
topological surface state.
This review paper is expected to give an informative
prologue to the field of topological insulators, laying
emphasis on the essential hypothesis and materials
properties. It likewise expounds on the fundamentals of
the portrayals of TI materials utilizing transport
estimations.
2. HISTORY
With the intention of comprehending the historical
background of TIs, we have to appreciate the
significance of TIs in condensed matter physics. In this
segment, I will shortly discuss the important topics that
paved way to the discovery of TIs as well as a
description on the advancement of the researches that
were carried out based on TIs in its earlier days.
2.1 Integer Quantum Hall Effect
In the year 1980, von Klitzing et al. found the quantum
Hall effect in a high-versatility 2D semiconductor under
high attractive fields.7)
The event of this impact is more
often constrained to low temperatures, where
localization of electrons and Landau quantization of their
energy range lead to vanishing longitudinal conductivity
𝜎xx together with quantization of the Hall conductivity
𝜎xy to integer number products of 𝑒2
/h when the
synthetic potential is situated in the middle Landau
levels. Such a quantization of transport coefficients
clearly indicated a perceptible quantum phenomenon, as
was clarified by Laughlin's gauge argument.8)
It is
reasonable to specify that this quantization phenomenon
was hypothetically foreseen already in 1974.9)
In 1982,
it was perceived by Thouless, Kohmoto, Nightingale,
and den Nijs (TKNN)10)
that this phenomenon not only
is quantum mechanical but it is also topological; to be
specific, TKNN showed10)
that in the quantum Hall
system the k-space is mapped to a topologically-
nontrivial Hilbert space, whose topology can be
determined by a whole number topological invariant
called TKNN invariant 𝑣, and that 𝜎xy becomes equal to
𝑣 times 𝑒2
/h. The TKNN invariant is also called the first
Chern number or the winding number, and it is

equivalent to the Berry phase of the Bloch wave function
calculated around the Brillouin zone (BZ) limit separated
by 2𝜋 (genuine computations are presented in the
subsequent section). 6]
In knowledge of the past, the quantum Hall system could
be accepted as the first topological insulator that became
known to physicists, in light of the fact that when the
quantization is occurring, the energy range is gapped
because of the Landau quantization furthermore, the
chemical potential is situated inside of the crevice, which
is a situation akin to an insulator. For this situation, the
nontrivial topology indicated by the TKNN invariant is
trademark of a 2D system with broken TRS. Likewise,
as it was indicated by Halperin,11)
the integer quantum
Hall effect is dependably joined by chiral edge states,
and those gapless states living at the interface to the
vacuum can be comprehended to be an aftereffect of the
bulk-boundary correspondence because of the
topological 2D "bulk" state.
It is judicious to say that the integer quantum Hall effect
was just the beginning. The fractional quantum Hall
(FQH) effect found in 1982 by Tsui, Stormer, and
Gossard12)
ended up containing wealthier material
science, on the grounds that electron relationships
assume fundamental parts in the FQH effect and also,
they prompt the presence of partially charged
quasiparticles.13)
As regards topology, nevertheless,
FQH states try not to have much pertinence to
topological insulators, since the previous present
ground-state decline and their topological character is
portrayed by a significant conceptual idea of topological
order.14)
2.2 Quantum spin Hall effect and Z2 topology
On an alternate front in consolidated matter material
science, generation and control of spin currents have
been drawing in a great deal of enthusiasm, since they
will have a significant sway on future spintronics.15)
In
such manner, the spin Hall effect, the presence of
transverse spin current accordingly to longitudinal
electric field, has been talked about hypothetically since
1970s, 16–20)
however its trial affirmation by Kato
et al. 21)
in 2004 gave a major help to the examination of
this phenomenon. It was soon perceived that the spin
Hall effect in nonmagnetic systems is on a very basic
level identified with the anomalous Hall effect in
ferromagnets,22)
and likewise to the last effect, there are
both intrinsic and extrinsic starting points of the spin
Hall effect. The intrinsic system of the spin Hall effect
originates from the Berry bend of the valence band
Bloch wave capacities incorporated over the Brillouin
zone.19, 20)
Since such an integral can get to be limited
even in an insulator, Murakami, Nagaosa, and Zhang
went ahead to propose the thought of spin Hall insulator,
23)
which is a gapped insulator with zero charge
conductivity yet has a limited spin Hall conductivity
because of a limited Berry phase of the involved states.
Later it was shown 24)
that the proposed spin Hall
insulators can't actually produce spin currents without
any electrons at the Fermi level, however this thought
activated consequent proposition of its quantized
version, the quantum spin Hall (QSH) insulator, by Kane
and Mele, 25,26)
trailed by an autonomous proposition by
Bernevig and Zhang. 27)
The QSH insulators are
basically two duplicates of the quantum Hall system, in
which the chiral edge state is spin polarized,
furthermore, the two states form a time-switched pair to
regain the general TRS. At the point when current
streams utilizing the edge conditions of a QSH protector,
a quantized adaptation of the twist Hall impact, the QSH
impact, is anticipated to be watched. Since the
anticipated marvel depends on the quantum Hall impact,
it just exists in 2D. While it is not clear from the onset
on how one can accomplish such a state with quantized
edge states in zero magnetic field, the clever proposition
by Kane and Mele gave a solid model to understand the
QSH insulator.25)
Their model is basically a graphene
model with spin–orbit coupling (SOC).
In graphene, the band structure close to the Fermi level
comprises of two linearly scattering cones situated at 𝐾
and 𝐾’ focuses in the BZ; 28)
since the low-energy
material science on these cones is portrayed by utilizing
the Dirac equation with the rest mass set to zero, 29)
this
scattering is called Dirac cone and the electrons are said

to carry on as massless Dirac fermions. Kane and Mele
showed 25)
that a limited SOC prompts an opening of a
hole at the intersection purpose of the cone (called Dirac
point) and, moreover, that a time-switched pair of spin-
polarized one-dimensional (1D) states in reality appear
at the edge in some parameter range; in this model, the
preferred spin polarization of the edge state is
accomplished because of the SOC which has a
characteristic inclination to adjust spins in connection to
the momentum path. This curious spin- non-degenerate
state [Figs. 1(a) and 1(b)] is regularly said to have
helical spin polarization or spin-momentum locking.
Intriguingly, those electrons in the gapless edge state act
as 1D massless Dirac fermions inside of the hole opened
in the 2D Dirac cone. For this situation, the 2D "bulk"
electrons can be seen as huge Dirac fermions in view of
the limited energy gap at the Dirac point.
In particular, Kane and Mele perceived that the
electronic conditions of their QSH insulator is described
by a novel topology indicated by a Z2 index, 26)
which
communicates whether the amount of times the 1D edge
state crosses the Fermi level somewhere around 0 and
𝜋/𝑎 is even or odd (𝑎 is the lattice constant). Keep in
mind, in arithmetic the group of integer numbers is
called Z and its quotient set sorting even and odd
numbers is called Z2; henceforth, a Z2 list for the most
part gives a topological arrangement in view of equality.
(A definite portrayal of the Z2 record for TR-invariant
TIs is given in the following segment.) The hypothetical
innovation of the Z2 topology in insulators was a major
stride in our comprehension of topological stages of
matter, on the grounds that it demonstrated that
nontrivial topology can be inserted in the band structure
of a normal insulator and that breaking of TRS by use of
magnetic fields is not obligatory for figuring it out a
topological stage.
Shockingly, the SOC in graphene is exceptionally weak,
and consequently it is hard to experimentally detect the
QSH effect expected in the Kane–Mele model. Be that
as it may, another hypothetical increase was soon made
by Bernevig, Hughes, and Zhang (BHZ),30)
who
developed a 2D model to deliver a Z2 topological stage
taking into account the band structure of HgTe; taking
into account their model, BHZ anticipated that a
CdTe/HgTe/CdTe quantum well ought to offer rise to
the QSH effect. This forecast was confirmed in 2007 by
Konig et al., 31)
who watched 𝜎xx to be quantized to
2𝑒2
/h in zero magnetic field when the chemical potential
is tuned into the bulk band gap(Fig. 2), giving proof for
the gapless edge states in the offending administration.
This was the first test affirmation of the TR-invariant TI
described by the Z2 topology.
Figure 2; Longitudinal four-terminal resistance of
various CdTe/HgTe/CdTe quantum-well structures as a
function of the gate voltage measured in zero magnetic
field at 30 mK. Devices with the size of 1 × 1 µ𝑚2 or
less in the band-inverted regime (III and IV) show
quantized conductance of 2𝑒2
/h, giving evidence for the
2D TR-invariant TI phase. (Yiochi Ando, 2013)
Figure 3; Spin-resolved surface band structure of
Bi1-x Sbx (x=0.12- 0.13) on the (111) cleaved surface. Its
surface BZ and the momentum direction of the data are
shown in the inset. Dispersions shown by symbols are

determined from the peak positions of the spin-resolved
energy dispersion curves and they are plotted on top of
the spin-integrated ARPES data shown in grey scale.
There are three Fermi-level crossings of the surface
states, which signify the Z2-nontrivial nature of this
system. (Yiochi Ando, 2013)
Without sitting tight for test check of the Z2 topology in
2D, scholars saw that this topological grouping of
insulators can be reached out to 3D systems, where there
are four Z2 invariants to completely describe the
topology.32–34)
actually, the term ''topological insulator''
was created by Moore and Balents in their paper to
propose the presence of TIs in 3D systems.32)
For 3D
TIs, Fu and Kane made a solid forecast in 2006 that the
Bi1-xSbx alloy in the insulating structure ought to be a
TI, and they further recommended that the nontrivial
topology can be checked by taking a glance at the
surface states utilizing the angle-resolved photoemission
spectroscopy (ARPES) and calculating the amount of
times the surface states cross the Fermi energy between
two TR-invariant momenta.35)
The proposed test was
directed by Hsieh et al. who reported in 2008 that
Bi1-xSbx is without a doubt a 3D TI.36)
The experimental
distinguishing proof of Bi1-x Sbx as a TR-invariant TI
opened a great deal of new experimental opportunities to
address a topological stage of matter. For instance, the
first transport investigation of Bi1-xSbx to recognize
topological 2D transport channels was accounted for by
Taskin and Ando,37)
and the first scanning tunneling
spectroscopy (STS) study that tended to the exceptional
spin polarization was accounted for by Roushan et al.,38)
both in 2009. Direct awareness of the helical spin
polarization of the surface states in Bi1-x Sbx utilizing
spin resolved ARPES was first mostly done by Hsieh et
al.39)
and after that completely refined by Nishide et al.
(Fig. 3).40)
2.3 Topological Field Theory
While the idea of topological insulators got to be
prevalent when the disclosure of the Z2 topology by
Kane and Mele, 26)
there had been hypothetical
endeavors to imagine topological conditions of matter
past the extent of the quantum Hall system. With regards
to this, an imperative advancement was made in 2001 by
Zhang and Hu, who summed up the 2D quantum Hall
state to a four-dimensional (4D) TR-invariant state
having a whole number topological invariant.41)
The
compelling field theory for this 4D topological system
was developed by Bernevig et al.42)
After the Z2
topology was found for TR-invariant systems in 2D and
3D, 26, 32–34)
it was portrayed by Qi, Hughes, and Zhang
43)
that the framework of topological field hypothesis is
valuable for portraying those systems too, and they
further showed that the Z2 TIs in 2D and 3D can really
be reasoned from the 4D compelling field theory by
utilizing the dimensional reduction. From a functional
perspective, the topological field hypothesis is suitable
for depicting the electromagnetic reaction of TIs what's
more, has been utilized for anticipating novel topological
magneto electric effects.43
2.4 Dirac Materials
It is intriguing to take note of that the initial 3D TI
material Bi1-x Sbx whose topological surface state
comprises of 2D massless Dirac fermions, has long been
known not impossible to miss band structure to offer
ascent to 3D monstrous Dirac fermions in the bulk.44]
This circumstance is like the Kane– Mele model where
1D massless Dirac fermions rise out of 2D enormous
Dirac fermions. Since Dirac fermions play essential parts
in TIs, it is helpful to specify the history of Dirac
material science in consolidated matter.
The semimetal Bi has been an essential testing ground of
quantum material science since the time that the
Shubnikov–de Haas (SdH) and de Haas–van Alphen
(dHvA) motions were found in Bi in 1930.45]
This is
basically on the grounds that the great degree low carrier
density ~10−5
per atom) and the precise long mean free
path (reaching ~1 mm) effortlessly put the framework in
''quantum limit” at generally low magnetic fields.46]
In
the mid-twentieth century, one of the long-standing
riddles in Bi was its extensive diamagnetism, which
opposes the regular intelligence for attraction in metals
including Pauli para-magnetism what's more, Landau
diamagnetism.47]
Intriguingly, in Bi1-x Sbx at low Sb
concentration, the carrier density turns out to be even
lower than in Bi, while the diamagnetic susceptibility

rises, which is likewise inverse to the desire from
Landau diamagnetism.47]
To address the bizarre electronic properties of Bi, a
compelling two-band model was developed by Cohen
and Blount in 1960.48]
In 1964, Wolff perceived that this
two-band model can be changed into the four-part
enormous Dirac Hamiltonian, and he introduced a rich
investigation of the choice principles utilizing the Dirac
theory.44]
This was the start of the idea of Dirac fermions
in strong states, albeit a portion of the impossible to miss
material science of massless Dirac fermions were
perceived in as right on time as 1956 by McClure in the
setting of graphite.
Talking about graphite, the mapping of the k ∙ p
Hamiltonian of its 2D sheet (i.e., graphene) 50]
to the
massless Dirac Hamiltonian was initially utilized by
Semenoff in 1984. 51]
With the test acknowledgment of
graphene, 28]
this framework has turned into a
prototypical Dirac material. One of the recognizing
properties of massless Dirac fermions is the Berry period
of 𝜋; an imperative outcome of the 𝜋 Berry stage in the
dense matter setting is the truancy of backscattering,
which was pointed out first by Ando, Nakanishi, and
Saito in 1998. 52]
An imperative part of the Dirac material science is that
magnetic fields essentially cause collaborations in the
middle of upper and lower Dirac cones. Indeed, the
Dirac formalism permits one to normally incorporate
such ''interband effects'' of magnetic fields into
computations. 6]
In 1970, Fukuyama and Kubo showed,
by expressly performing the assessments, that the
extensive diamagnetism in Bi and Bi1-x Sbx is actually a
result of their Dirac nature. 47]
Intriguingly, because of
such interband effects of magnetic fields, the Hall
coefficient does not deviate but rather gets to be zero
when the carrier density vanishes at the Dirac point, 53]
which is surely seen in a topological insulator. 54]
3 FUNDAMENTALS OF THE TOPOLOGY IN
TOPOLOGICAL INSULATORS
3.1 TKNN Invariant
The topological invariant characterized for the whole
integer quantum Hall system, the TKNN invariant, 10)
is
firmly identified with the Berry stage. To see this, we
determine the TKNN invariant by computing the Hall
conductivity of a 2D electron arrangement of size L × L
in opposite magnetic fields, where the electric field E
and the magnetic field B are connected along the y-and
z-axes, separately. By treating the effect of the electric
field E as an irritation potential V = -eEy, one may
utilize the irritation theory 6]
to estimate the distressed
eigen state |𝑛> E as
|𝑛> E = |𝑛> + ∑
⟨ 𝑚|(−𝑒𝐸𝑦)|𝑛⟩
𝐸𝑛−𝐸𝑚
|𝑚𝑚(≠𝑛) > + ... (1)
In order to use this distressed eigenstate, the expectation
value of the current density along the x-axis is
obtainable, jx, in the existence of the E field along the y-
axis as
(𝑗𝑥)E = ∑ 𝑓(𝐸𝑛 n)⟨𝑛|𝐸(
𝑒𝑣 𝑥
𝐿2 |𝑛⟩ 𝐸
= (𝑗𝑥)E= 0 +
1
𝐿2
∑ 𝑓(𝐸𝑛 𝑛)
× ∑ (
⟨𝑛|(𝑒𝑣 𝑥)|𝑚⟩ ⟨𝑚|(−𝑒𝐸𝑦|𝑛⟩
𝐸𝑛 − 𝐸𝑚
𝑚(≠𝑛)
+
⟨ 𝑛|(−𝑒𝐸𝑦| 𝑚⟩⟨ 𝑚|−𝑒𝐸𝑦| 𝑛⟩
𝐸𝑛−𝐸𝑚
), (2)
where vx is the electron velocity along the x-direction
and 𝑓(𝐸𝑛) is the Fermi distribution function. The
Heisenberg equation of motion
𝑑
𝑑𝑡
𝑦 = 𝑣 𝑦 =
1
𝑖ħ
[y, H]
leads to
⟨𝑚|𝑣 𝑦|𝑛⟩ =
1
𝑖ħ
(𝐸𝑛 − 𝐸𝑚)⟨𝑚|𝑦|𝑛⟩, (3)
which allows one to evaluate
𝜎 𝑥𝑦 =
〈𝑗𝑥〉𝐸
𝐸
= -
𝑖ħ𝑒2
𝐿2
∑ 𝑓(𝐸 𝑛)𝑛≠𝑚

⟨ 𝑛| 𝑣 𝑥| 𝑚⟩⟨ 𝑚| 𝑣 𝑦| 𝑛⟩−⟨ 𝑛| 𝑣 𝑦| 𝑚⟩⟨ 𝑚| 𝑣 𝑥| 𝑛⟩
(𝐸𝑛−𝐸𝑚)2 (4)
The identity of the system considered in a periodic
potential and its blocks are as follows:
⟨𝑢 𝑚𝑘′|𝑣𝜇|𝜇 𝑛𝑘⟩ =
1
ħ
(𝐸 𝑛𝑘 − 𝐸 𝑚𝑘′) ⟨𝑢 𝑚𝑘′|
𝜕
𝜕 𝑘𝜇
|𝑢 𝑛𝑘⟩ (5)
Rewrite equation (4) into the form
𝜎 𝑥𝑦 = −
𝑖𝑒2
ħ𝐿2
∑ ∑ 𝑓(𝐸 𝑛𝑘)
𝑛≠𝑚𝑘
× (
𝜕
𝜕𝑘 𝑥
⟨𝑢 𝑛𝑘|
𝜕
𝜕𝑘 𝑦
𝑢 𝑛𝑘⟩ −
𝜕
𝜕𝑘 𝑦
⟨𝑢 𝑛𝑘|
𝜕
𝜕𝑘 𝑥
𝑢 𝑛𝑘⟩) (6)
Berry connection equation written for Bloch states:
𝑎 𝑛(𝑘) = −𝑖⟨𝑢 𝑛𝑘|∇ 𝑘|𝑢 𝑛𝑘⟩ = −𝑖 ⟨𝑢 𝑛𝑘|
𝜕
𝜕𝑘 𝑥
|𝑢 𝑛𝑘⟩ (7)
The Hall conductivity reduces to
𝜎 𝑥𝑦 = 𝑣
𝑒2
ℎ
(8)
With
𝑣 = ∑ ∫
𝑑2 𝑘
2𝜋𝐵𝑍𝑛 (
𝜕𝑎 𝑛,𝑦
𝜕𝑘 𝑥
−
𝜕𝑎 𝑛,𝑥
𝜕𝑘 𝑦
) (9)
This 𝑣 can be expressed as 𝑣 = ∑ 𝑣 𝑛𝑛 with 𝑣 𝑛 the
contribution from the nth band, and one can easily see
that 𝑣 𝑛 is related to the Berry period, namely
𝑣 𝑛 = ∫
𝑑2 𝑘
2𝜋𝐵𝑍
(
𝜕𝑎 𝑛,𝑦
𝜕𝑘 𝑥
−
𝜕𝑎 𝑛,𝑥
𝜕𝑘 𝑦
)
=
1
2𝜋
∮ 𝑑𝒌. 𝑎 𝑛(𝒌)𝜕𝐵𝑍
= -
1
2𝜋
𝑦𝑛 [𝜕𝐵𝑍]. (10)
Due to the nature of the single valued wave function,
changes in the period factor as soon as it’s done with the
encircling the Brillouin zone boundary [𝜕𝐵𝑍] 𝑐𝑎𝑛 only
be an integer multiple of 2𝜋, which means
𝑦𝑛 [𝜕𝐵𝑍] = 2𝜋𝑚 (𝑚 𝜖 𝑍). (11)
Therefore, 𝑣 𝑛 can only take an integer value, and hence
𝜎 𝑥𝑦 is quantized to integer multiples of
𝑒2
ℎ
. The integer 𝑣
is called TKNN invariant, and it plays the role of the
topological invariant of the quantum Hall system, which
is a TRS breaking TI. It is clear from Equation (10) that
the TKNN invariant becomes nonzero (i.e., the system
becomes topological) when the Berry connection
𝒂 𝑛(𝒌) is not a single-valued function 6]
3.2 Extension to 3D systems
Taking into account a general homotopy contention,
Moore and Balents showed 32)
that there are four Z2
invariants for 3D systems. While the development of the
homotopy was included, the physical birthplace of the
four invariants can be effectively caught on. For
effortlessness, consider a cubic system furthermore, take
the cross section steady a = 1. In the 3D BZ of this
system, there are eight TRIMs meant as ʌ0,0,0, ʌ 𝜋,0,0,
ʌ0,𝜋,0, ʌ0,0,𝜋, ʌ 𝜋,0,𝜋, ʌ0,𝜋,𝜋, ʌ 𝜋,𝜋,0, and ʌ 𝜋,𝜋,𝜋, [see Fig.
5(c)]. At these focuses in the BZ, the Bloch Hamiltonian
gets to be TR symmetric, i.e., 𝛩𝐻(ʌ𝑖)𝛩−1
= 𝐻(ʌ𝑖), and
every Kramers pair of groups get to be deteriorate.
Notice that the six planes in the 3D BZ, x = 0, x = ±𝜋,
y = 0, y =±𝜋, z = 0, and z = ±𝜋 have the symmetries
of the 2D BZ, and subsequently they each have a Z2
invariant. The six invariants may be meant as x0, x1, y0,
y1, z0, and z1, be that as it may, those six invariants are
not all independent. 32)
This is since the items x0x1, y0y1,
and z0z1 are excess, which comes from the way that
those three items include Pf[w(ʌ𝑖)]/√det[𝑤(ʌ𝑖)] from
each of the eight TRIMs and henceforth are the same.
This implies there are two compelling relations x0x1 =
y0y1 = z0z1, which directs that just four invariants can be
autonomously decided in a 3D system.
The solid development of the four Z2 invariants was
given by Fu and Kane. 33)
For each TRIM ʌ𝑖 we define
𝛿(ʌ𝑖) ≡
Pf[𝑤(ʌ 𝑖)]
√det[𝑤(ʌ 𝑖)]
(12)
Using this 𝛿(ʌ𝑖), the four Z2 invariants V0, V1, V2, V3
are defined as

(−1) 𝑣0
= ∏ 𝛿(ʌ 𝑛1,𝑛2,𝑛3),
𝑛𝑗=0,𝜋
(13)
(−1) 𝑣1
= ∏ 𝛿(Ʌ 𝑛1,𝑛2,𝑛3),
𝑛≠𝑖=0,𝜋;𝑛𝑖=𝜋
(𝑖 = 1,2,3) (14)
The invariant V0 is given by a product of all eight
𝛿(ʌ𝑖)′s, so it is unique to a 3D system. Alternatively,
other Vi’s are a product of four 𝛿(ʌ𝑖)′s and is similar to
the Z2 invariant in the 2D case. For example,
(−1) 𝑣3
= 𝛿(ʌ0,0,𝜋) 𝛿(ʌ 𝜋,0,𝜋)𝛿(ʌ0,𝜋,𝜋)𝛿(ʌ 𝜋𝜋,,𝜋) (15)
corresponds to the Z2 invariant on the Z = 𝜋 plane, which
can be seen by considering the TR polarization defined
on this plane, 𝑃𝜃(𝑦 = 0) 𝑍=𝜋 =
1
𝑖𝜋
log[𝛿(ʌ0,0,𝜋)𝛿(ʌ 𝜋,0,𝜋)]
and
𝑃𝜃(𝑦 = 0) 𝑍=𝜋 =
1
𝑖𝜋
log[𝛿(ʌ0,𝜋,𝜋)𝛿(ʌ 𝜋,𝜋,𝜋)]. When the
TR polarization changes between y = 0 and y = 𝜋 [i.e.,
Figure 6. Schematic pictures of the surface states
between two surface TRIMs, ʌ 𝑎
𝑠
and ʌ 𝑏
𝑠
, for (a)
topologically trivial and (b) topologically nontrivial
cases. In the latter, the ‘‘switch-partner’’ characteristic is
a reflection of the change in the TR polarization (see
text). The shaded regions represent the bulk continuum
states. To enhance the visibility of the Kramers
degeneracy, surface state dispersions are shown
from −ʌ 𝑎
𝑠
to −ʌ 𝑏
𝑠
through ʌ 𝑎
𝑠
.
𝑃𝜃(𝑦 = 0) 𝑍=𝜋 and 𝑃𝜃(𝑦 = 𝜋) 𝑍=𝜋 are different, then the
Z2 topology is nontrivial and V3 becomes 1.
The physical outcome of a nontrivial Z2 invariant is the
presence of topologically-ensured surface states. This is
graphically appeared in Fig. 6, in which topologically
trifling and nontrivial surface states are analysed. In the
nontrivial case, the Kramers sets in the surface state
''switch accomplices'', and subsequently, the surface state
is ensured to cross any Fermi energy inside the bulk gap.
This switch-accomplice trademark is an impression of
the change in the TR polarization talked about above. It
is standard to compose the blend of the four invariants in
the structure (V0; V1, V2, V3), in light of the fact that (V1,
V2, V3), can be translated as Miller files to indicate the
bearing of vector ʌi in the proportional space.
A 3D TI is called "solid" at the point when V0 = 1, while
it is called "frail" when V= 0 and Vi = 1 for some i
=1,2,3.
4. TWO-DIMENSIONAL TI’S
The primary material that was tentatively distinguished
as a TR-invariant TI was CdTe/HgTe/CdTe quantum
well, 31)
in particular, a slim layer of HgTe sandwiched
by CdTe. This is a 2D framework where the level of
opportunity for the opposite bearing is extinguished
because of the quantum imprisonment of the electronic
states in the HgTe unit and the subsequent subband
development. Both HgTe and CdTe solidify in
zincblende structure, and their superlattices have been
effectively concentrated on in light of their application to
infrared indicators.
In this way, the fundamental innovation to blend the
required quantum well was at that point created before
the expectation of its TI nature, despite the fact that it
includes an exceptionally specific sub-atomic shaft
epitaxy (MBE) strategy to manage mercury.31)
As we
have found in the examination of the BHZ model in the
past area, the reversal in the middle of p-and s-orbital
groups is vital for the framework to acquire the TI
nature. Mass HgTe acknowledges such a band reversal,
while CdTe does not.
Along these lines, HgTe is a decent beginning material
for considering a TI stage; on the other hand, there is an
issue in the band structure of mass HgTe, that is, a
precious stone symmetry-ensured decadence at the point
makes the framework to be naturally a zero-crevice
semiconductor,30)
which implies that there is no band
crevice between the p-and s-orbital groups and the
framework is not qualified as a cover. On the other hand,
by sandwiching HgTe by CdTe, which has marginally
bigger cross section steady, the epitaxial strain applied
on HgTe breaks the cubic cross section symmetry and

prompts a hole opening, and thus the framework can turn
into a honest to goodness separator. It was anticipated by
BHZ that over a certain basic thickness the strained
HgTe unit holds the band reversal and ought to be a
TI.30)
This forecast was affirmed by a gathering at the
College of Wu¨rzburg drove by Molenkamp by means of
transport examination of small scale manufactured
samples.31)
They found that, at the point when the
thickness of the HgTe unit is over the basic thickness dc
of 6.3 nm, their examples demonstrate a ''negative
energy gap” (i.e., band is upset) and quantization of the
conductance to 2e2
/h in zero attractive field was watched
at the point when the synthetic potential is tuned into the
hole (Fig. 2). In contrast, when the thickness is beneath
dc, the band reversal is lost and they watched separating
resistivity.
The Wu¨rzburg gather later reported the perception of
nonlocal transport, 55)
which gave further backing to the
presence of an edge state. They additionally
demonstrated that the 1D edge state in charge of the
quantized transport in the 2D TI stage is prone to be
helically turn spellbound, by manufacturing an intricate
gadget structure which depends on the routine twist Hall
impact that happens in metallic (doped) HgTe.56)
The CdTe/HgTe/CdTe quantum well can be made
extremely clean with the transporter versatility of up to
~105
cm2
V-1
s-1
, which makes it conceivable to study
quantum transport properties. Then again, the downside
of this framework is that the mass band hole that opens
because of the epitaxial strain is small (up to ~10 meV,
contingent upon the thickness 57]
, which makes the
recognition of the TI stage to be conceivable just at low
temperature. Additionally, since the combination of
CdTe/HgTe/CdTe quantum wells requires devoted MBE
machines, the wellsprings of tests for fundamental
material science investigations are at present extremely
constrained, which has made the advancement of test
investigations of 2D TIs (QSH encasings) moderately
moderate.
As of late, another 2D TI framework,
AlSb/InAs/GaSb/AlSb quantum well, was hypothetically
predicted58]
and tentatively
confirmed.59–61]
The vital workings of this system are the
accompanying: The valence-band top of GaSb lies above
the conduction-band base of InAs. Consequently, when
InAs what's more, GaSb are in direct contact and they
are both quantum limited (by the external units of AlSb
which has an extensive band hole and acts as a
boundary), the subsequent gap subband in GaSb may lie
over the electron subband in InAs, and in this way the
band request of this quantum well is transformed. The
band hole in this quantum well emerges from hostile to
crossing of the two subbands at limited force and thus is
exceptionally little (~ 4 meV), which mentions clean
objective fact of the helical edge state exceptionally
difficult.59,62)
Reasonably persuading proof for the TI
stage was gotten by means of perception of the 2e2
/h
quantization of the zero-predisposition Andreev
reflection conductance through Nb point contacts,60)
yet
all the more as of late, direct perception of the
conductance quantization to 2e2
/h has been
accomplished by acquainting issue with the InAs/GaSb
interface by Si doping to confine the undesirable mass
carriers.
5. THREE-DIMENSIONAL TI’S
As specified in Sect. 2, the initial 3D TI material that
was tentatively distinguished was Bi1-x Sbx,36)
taking
after the particular expectation by Fu and Kane.35)
This
material is a combination of Bi and Sb and it normally
has the two vital elements, (i) band reversal at an odd
number of TRIMs and (ii) opening of a mass band hole,
in the Sb focus scope of 0.09 to 0.23.63]
The 3D Z2
invariant has been recognized as (1;111).
Sadly, it worked out that this framework is not extremely
suitable for definite investigations of the topological
surface state because of the convoluted surface-state
structure,36,40)
as can be found in Fig. 3. This is on the
grounds that its guardian material, Bi, as of now
harbours noticeable twist non-ruffian surface states
because of the solid Rashba impact on the surface of this
material.64]
In Bi1-x Sbx, such non-topological,
Rashbasplit surface states are in charge of 2 or 4 Fermi-
level intersections of the surface states (contingent upon
the compound potential), 36,40)
and the topological one
contributes only one extra Fermi-level intersection.

To start with guideline counts of the surface conditions
of Bi1-x Sbx have been reported, 65,66]
however the
anticipated surface-state structure does not by any means
concur with the test results. Such a fragmented
comprehension of the way of the surface state is mostly
in charge of disarrays every so often found in
understandings of exploratory information.
For instance, in the STS work which tended to the
security from backscattering in the surface condition of
Bi1-x Sbx,38)
the examination considered the turn
polarizations of just those surface expresses that are
additionally present in topologically insignificant Bi,87)
but then, it was inferred that the outcome gives proof for
topological assurance. By the by, Bi1-x Sbx is interesting
among known 3D TI materials in that it has a
characteristically high 2D transporter versatility of ~104
cm2
V-1
s-1
(regardless of the fact that it is an alloy),
which makes it simple to study novel 2D quantum
transport.37,67]
Also, it is generally simple for this framework to
decrease the mass bearer thickness to ~1016
cm-3
in high
caliber single gems, making it conceivable, for instance,
to perform Landau level spectroscopy of the surface
states through magnetooptics. 68]
The mass band hole of
Bi1-x Sbx is not substantial (up to ~30 meV relying upon
x), 63]
yet it is sufficiently substantial to distinguish the
2D transport properties at 4K. Alongside Bi1-x Sbx, Fu
and Kane predicted 35)
that HgTe, -period of Sn (called
''dim tin''), and Pb-1
xSnxTe would gotten to be 3D TIs
under uniaxial strain to break the cubic grid symmetry.
They additionally proposed that Bi2Te3 would be an
applicant, yet they didn't perform band estimations to
explain the equality eigenvalues. Such computations
were done by Zhang et al., 69]
who concocted a solid
expectation that Bi2Se3, Bi2Te3, and Sb2Te3 ought to be
3D TIs be that as it may, Sb2Se3 is not; moreover, Zhang
et al. proposed a low energy viable model to portray the
mass band structure of this class of materials.
This model, with a few amendments made later, 70]
has
turned into a famous model for hypothetically examining
the properties of a 3D TI. Tentatively, presence of a
solitary Dirac-cone surface state was accounted for in
2009 for Bi2Se3 by Xia et al.71]
and for Bi2Te3 by Chen et
al.72]
furthermore by Hsieh et al., 73]
Sb2Te3 was
measured by Hsieh et al.73]
alongside Bi2Te3, yet the
presence of the topological surface state was left
unsubstantiated because of the vigorously p-sort nature
of the deliberate specimens. The topological nature of
Sb2Te3 was affirmed just as of late in meagre film tests
utilizing STS.74]
Bi2Se3, Bi2Te3, and Sb2Te3 all take shape in tetra-dymite
structure, which comprises of covalently fortified
quintuple layers (e.g., Se–Bi–Se–Bi–Se) that are stacked
in –A–B–C– A–B–C– way and are pitifully
collaborating with van der Waals power (Fig. 7); thusly,
those materials separate effectively between quintuple
layers (QLs). Since each QL is around 1 nm thick, the
cross section consistent along the c-hub is around 3 nm.
The 3D Z2 invariant of these tetra-dymite frameworks is
(1;000), which implies that topological Dirac-cone
surface state is focused at the ¬ purpose of the surface
BZ (Figure 8).
This openness of the topological surface state and the
nonappearance of non-topological surface states make
those materials appropriate for tentatively tending to the
properties of the topological surface state. Additionally,
single crystal development of those materials is
generally straightforward, which made it simple for
some experimentalists to begin dealing with them.
Moreover, the mass band crevice of Bi2Se3 is generally
substantial, 0.3 eV, and along these lines one can see
mechanical pertinence that topological properties of this
material may conceivably be abused at room
temperature. Every one of those components made a
difference start a surge of examination exercises on 3D
TIs.
The surface-state structure of Bi2Se3 is moderately
straightforward what's more, shows a practically
admired Dirac cone with just slight ebb and flow, as
demonstrated schematically in Figure. 8(a).
Interestingly, the surface condition of Bi2Te3 is more
convoluted [see Figure. 8(b)] and the Dirac point is
situated underneath the highest point of the valence
band, which makes it hard to test the surface transport
properties close to the Dirac point without being
bothered by mass transporters in Bi2Te3.

Fig. 7. Crystal structure of tetra-dymite chalcogenides, Taken
(Yiochi Ando, 2013)
Fig, 8. Schematic bulk and surface band structures of
(a) Bi2Se3 and (b) Bi2Te3.
CONCLUSION
In this paper, general accounts of Topological Insulators
have been presented. A holistic idea of the basic theories
related to topological insulators and material properties
were presented. Furthermore, there were discussions on
the 2D-Two dimensional and 3D- three dimensional
topological insulators. The Dirac fermion material
science as well as the examples of subsequent quantum
oscillations was discussed in this paper.
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