Download Quantum Spin Hall Effect and Topological Insulator

Quantum Spin Hall Effect and Topological Insulator
Abstract:
In the 1980s, Dr. Klaus von Klitzing's group discovered the quantum Hall effect as a result of
systematic measurement on silicon field transistors. The discovery changed the condensed
matter field and he won the Nobel prize on the year of 1985. The quantum Hall effect is really
important in the study of microelectronic device. One of the most important condition of
quantum Hall effect is the external magnetic field. However, the researchers discovered a
phenomena which is quite like quantum Hall effect, but it is time reversal invariant and do not
require an applied field. The researchers called it quantum spin Hall effect. From the study of
quantum spin Hall effect, they found some material behaved only depend on its topology and not
on its specific geometry, and it was topologically distinct from all previously known states of
matter, so they call this kind of material as topological insulator. In my project, I will introduce the
theory of quantum spin Hall effect and how can we find topological insulator though quantum
spin Hall effect.
Introduction
The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed
in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. It
is discovered by Dr. Klaus von Klitzing's group. In the quantum Hall effect, and the conductivity
can be represented as σ=
=v
. The quantum Hall effect is referred to as the integer or
fractional quantum Hall effect depending on the ν, which is known as filling factor, is an integer or
fraction respectively. In 2000s, researchers found a new class of topological states, which is
topologically distinct from all other known states of matter including quantum Hall effect. They
call it quantum spin Hall effect. It is a state of matter proposed to exist in special two-dimensional
semiconductors, which have a quantized spin-Hall conductance and a vanishing charge-Hall
conductance. The quantum spin Hall state of matter is the cousin of the integer quantum
Hall state, and both states can be realized on lattice that does not require the application of a
large magnetic field[1]. From the study of quantum spin Hall effect, scientists found a material
which behaves as an insulator in its interior but contains conducting states on its surface. In the
bulk of a non-interacting topological insulator, the electronic band structure resembles an
ordinary band insulator, with the Fermi level falling between the conduction and valence bands.
On the surface of a topological insulator there are special states that fall within the bulk energy
gap and allow surface metallic conduction[2]. In my project, starting from brief introduction of
quantum Hall effect, I will explain how we get the theory of quantum spin Hall effect and how
can we find topological insulator though the quantum spin Hall effect.
From quantum Hall effect to quantum spin Hall effect
In the surface of most material, the gas of electrons go both forward and backward if we see
in one-dimension, and the direction of motion are mixed. If we want to study properties of one
kind of material, it is better make electrons move in separate lines with same direction, for we
can avoid rand collisions. Quantum Hall effect is one most important way to make this condition
happen in surface of material. However, the quantum Hall effect occurs only when a strong
magnetic field is applied to a 2-dimensional gas of electrons in a semiconductor. Electrons will
only travel along at the edge of the semiconductor at low temperature in the condition of low
temperature and strong magnetic field. The electrons will only go in two lanes at the top and
bottom of sample's edges with opposite direction. We can compare the edge of quantum Hall
effect and quantum spin Hall effect by the figure below:
Figure 1. The spatially separated movement of one dimensional spinless chain in quantum Hall bar
and one dimensional spinful chain in quantum spin Hall bar.[3]
The most important aspect of quantum Hall effect and quantum spin Hall effect is spatial
separation of electron movement. In the left part of figure 1, we can see the one dimensional
electron chain moves forward and backward separately on the two edges. On the upper edge,
the electron only move forward and the electron on the lower edge moves only backward. Those
two basic degrees of freedom are spatially in a quantum Hall bar as labeled with "2=1+1". This is
the key reason why the quantum Hall effect is topologically robust, for the electron will go
around an impurity without scattering. This transportation is extremely useful but it requires a
strong magnetic field, which severely limits the application potential of the quantum Hall effect.
In the real one dimensional system, the forward and backward moving channels will split
into four channels with spin-up and spin-down electrons in both direction, which is shown in the
right part of figure 1. We can leave the spin-up forward electrons and spin-down backward
electrons at the top edge and the other two on the bottom edge. A model with such edge states
distribution is called quantum spin Hall effect, for the transportation is just like quantum Hall
effect. In the quantum spin Hall edge, the backscattering impurities are forbidden on both the
top and bottom edges. This is a result of antireflection coating of electrons with different spin
direction. In a quantum spin Hall edge stat, electrons can be scattered in two direction by a
nonmagnetic impurity. The directions can be clockwise whose spin is rotate by π, and the other
one is counterclockwise with spin rotation by -π, so the two path related by time reversal
symmetry, differ by a full 2π rotation of electron spin. Thus the two backscattering paths always
interfere destructively which leads to perfect transmission. If the impurity carries a magnetic
moment, the time reversal symmetry is broken and the two reflected waves on longer interfere
destructively. In this case, the robustness of quantum spin Hall effect is protected by the time
reversal symmetry. Those edge state impurity is the phenomena of quantum spin Hall effect.[3]
The theory and models of topological insulator
From the 2000s, researchers has already theoretically predicted several kinds of materials
can behave like quantum spin Hall effect, which turn out to be a quantum spin Hall insulator or
topological insulator. The essence of the quantum spin Hall effect in real materials can be
captured in explicit models that are particularly simple to solve. In my project, I will use
HgTe/CdTe quantum wells, which is already predicted to be a topological insulator and used very
often, as a example to explain the theory of topological insulator.
Firstly , the two-dimensional topological insulator mercury telluride can be described by an
effective Hamilton that is essentially a Taylor expansion in the wave factor
of the interactions
between the lowest conduction band and the highest valence band:
H(
M( ) = M - B
Where the upper 2 x 2 block describes spin-up electrons in the s-like E1 conduction and the
p-like H1 valence bands, and the lower block describes the spin-down electrons in those bands.
The energy gap between the bands is 2M, which is usually negative. In the case M/B < 0, the
quantum wells, the bands are inverted, M becomes negative, and the solution yield the edge
states of a quantum spin Hall insulator. After we know the structure of mercury telluride, then we
can move to HgTe/CdTe quantum wells. The structure of are HgTe/CdTe quantum well is
zincblende-type semiconductor quantum wells, in which there are four relevant bands close to
the Fermi level. In the E1 band, there are two spin states of the s orbital, whereas the HH1 band
consists of the | +i , ↑> and | −| -i ), ↓> orbitals.[4] The effective Hamiltonian near the G
point, the center of the Brillouin zone, is given by the equation:
)=
,
and H = ε(k) +
where
are the Pauli matrices, and we also have:
+i
Here,
(k)
= A(
)=A
= M - B(
+
),
= C - D(
+
).
are momenta in the plane of the two-dimensional electron gas (2DEG), and A,
B, C, and D are material specific constants. Spin-orbit coupling is naturally built in this
Hamiltonian through the spin-orbit coupled p orbital | +i , ↑> and | −| -i ), ↓>.
Two-dimensional materials can be grouped into three types according to the sign of the Dirac
mass parameter M. In conventional semiconductors such as GaAs and CdTe, the s-like E1 band
lies above the p-like HH1 band, and the mass parameter M is positive. Different from the
semiconductors(HgTe) we mentioned before, the s-like orbital lies below the p-like orbitals;
therefore, the Dirac mass parameter M in the HgTe/(Hg,Cd)Te quantum wells can be continuously
tuned from a positive value M > 0 for thin quantum wells with thickness d < dc to a negative
value M < 0 for thick quantum wells with d > dc. A topological quantum phase transition occurs at
a critical thickness d = dc. The quantum spin Hall phase occurs in the inverted regime where M <
0, which is d<dc. The sample edge can be viewed as a domain wall of the mass parameter M,
separating the topologically nontrivial phase with M < 0 from the topologically trivial phase with
M > 0.
The theory above is mostly about 2-D topological insulator, the 3D topological insulator can
be described by similar model:[3]
H(
M(
=M-
-
The model become more complicated in 3D. The curvature parameters
the same sign. As in the 2D model, the solution for M/
M/
and
have
<0 describe a trivial insulator, but for
>0, the bands are inverted and the system is topological insulator.
Topological classification of insulators
According to the time reversal symmetry by now, we can firstly divide insulators into two
broad classes, presence or absence of time reversal symmetry. The quantum Hall state is a
topological insulator state which breaks the time reversal symmetry. We have two definitions of
time reversal invariant topological insulators, one in terms of non-interaction topological band
theory and one in terms of topological field theory[5]. Inside an insulator, the electric field E and
the magnetic field B are both well defined. In a Lagrangian-based field theory, the insulator’s
electromagnetic response can be described by the effective action
= π
dt(ε
-
)
where ϵ the electric permittivity and μ the magnetic permeability, from which Maxwell’s
equations can be derived. The integrand depends on geometry, though, so it is not topological. To
see that dependence, one can write the action in terms of
tensor:
=
π
dt
, the 4D electromagnetic field
The implied summation over the repeated indices μ and ν depends on the metric tensor—that is,
on geometry.
Such classification is valid for a periodic system. For a real solid with a finite boundary, a
topological insulator is insulating only in the bulk; it has an odd number of gapless Dirac cones on
the surface that describe conducting surface states. If we uniformly cover the surface with a thin
ferromagnetic film, an insulating gap also opens up on the boundary; the TR symmetry is
preserved in the bulk but broken on the surface.
Outlook and applications of topological insulator
From the 2000s, the study of quantum spin Hall effect and topological insulator started and
grow rapidly. A lot of researchers has already predicted some kinds of quantum spin Hall
insulator material. Some experiment through molecular-beam epitaxy and scanning tunneling
microscopy also have done to detect more properties of those kinds of material. In recent study,
the 2D topological insulator is predicted to have fractional charge at the edge and spin–charge
separation in the bulk. In introductory physics classes we learned that a point charge above a
metal or an insulator can be viewed as inducing an image charge below the surface. A point
charge above the surface of a 3D topological insulator is predicted to induce not only an image
electric charge but also an image magnetic monopole below the surface. Such a composite object
of electric and magnetic charges, called a dyon, would obey neither Bose nor Fermi statistics but
would behave like a so-called anyon with any possible statistics. Dislocations inside a 3D
topological insulator contain electronic states that behave similarly to quantum spin Hall edge
states. Beside teaching us about the quantum world, the exotic particles in topological insulators
could find novel uses. For example, image monopoles could be used to write magnetic memory
by purely electric means, and the Majorana fermions could be used for topological quantum
computing.
Reference
[1]. C.L. Kane and E.J. Mele, Physical Review Letters 95, 226801 (2005).
[2]. Kane, C. L.; Mele, E. J. (30. September 2005) 95 (14): 146802
[3]. 2009 American Institute of physics, S-0031-9228-1001-020-3
[4]. X Qi and S Zhang, science vol 318, nov 2 2007.
[5]. J.E, Moore, L.Balent, phys. rev. lett. 75, 121306(2007).