Quantum Spin Hall Effect and Topological Insulator Weisong Tu Department of Physics and Astronomy University of Tennessee Instructor: Dr. George Siopsis

Introduction Quantum Hall Effect 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. In the quantum hall effect, and the conductivity can be represented as 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 effect do not need a strong magnetic field, which is different from the quantum Hall effect.

Introduction Topological Insulator 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

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:

From quantum Hall effect to quantum spin Hall effect he most important aspect of quantum Hall effect and quantum spin Hall effect is spatial separation of electron movement. 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". 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. 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.

From quantum Hall effect to quantum spin Hall effect 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 2 π rotation of electron spin. Thus the two backscattering paths always interfere destructively which leads to perfect transmission.

The theory and models of 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. Inside an insulator, the electric field E and the magnetic field B are both well defined.

Topological classification of insulators In a Lagrangian-based field theory, the insulator’s electromagnetic response can be described by the effective action where 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.

Outlook and Application 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. Beside teaching us about the quantum world, the exotic particles discovered 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.

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