Topological Insulators

Topological Insulators

OutlineIntroductionBrief history of topological insulatorsBand theoryQuantum Hall effectSuperconducting proximity effectintroductionClose relation between topological insulators and several kinds of Hall effects.Brief history of topological insulatorsThe History of Topological InsulatorQHEQSHE3D TI2005 Kane & Mele 2006 HgTe / CdTe2007 Molenkamp 2007 FuKane Bi1-xSbx 3D TI2008 Hasan ARPES2009 Bi2Se3Bi2Te3Sb2Te32009 ARPES Hasan Bi2Se3 Bi2Te3 Hasan Sb2Te3

1980 1982 5

2D topological insulator

Shou-Cheng Zhang Group. Science 314, 1757 (2006)

2D topological insulatorMolenkamp Group. Science 318, 766 (2007)3D topological insulator

Liang Fu and C. L. Kane Physical Review B, 2007, 76(4): 045302.3D topological insulatorHasan Group. Nature, 2008, 452(7190): 970-974.

Band theory

Figure 1: the band structures of four kinds of material (a) conductors, (b) ordinary insulators, (c) quantum Hall insulators, (d) T invariant topological insulators

Band structuresThe Chern invariant n

Berry phase

Berry flux

The Chern invariant is the total Berry flux in the Brillouin zone

TKNN showed that xy, computed using the Kubo formula, has the same form, so that N in Eq.(1) is identical to n in Eq.(2).

Chern number n is a topological invariant in the sense that it cannot change when the Hamiltonian varies smoothly.

For topological insulators, n0, while for ordinary ones(such as vacuum), n=0.Haldane model

tight-binding model of hexagonal lattice a quantum Hall state with introduces a mass to the Dirac points

Edge states skipping motion electrons bounce off the edgechiral:propagate in one direction only along the edgeinsensitive to disorder :no states available for backscatteringdeeply related to the topology of the bulk quantum Hall state.

Z2 topological insulatorT symmetry operator:

Sy is the spin operator and K is complex conjugation

for spin 1/2 electrons:

A T invariant Bloch Hamiltonian must satisfy

Z2 topological insulator for this constraint,there is an invariant with two possible values: =0 or 1 two topological classes can be understood,is called Z2 invariant. define a unitary matrix:

There are four special points in the bulk 2D Brillouin zone.define:

Z2 topological insulatorthe Z2 invariant is:

if the 2D system conserves the perpendicular spin SzChern integers n, nare independent,the difference

defines a quantized spin Hall conductivity.The Z2 invariant is then simply

Z2 topological insulator

Surface quantum Hall effect

19Integer Quantized Hall Effect

20The explanation for the integer quantized Hall effect can be found in solid state physics textbooks. Here we will use a video for illustration

21**-422

Fig c A thin magnetic film can induce an energy gap at the surface. d A domain wall in the surface magnetization exhibits a chiral fermion mode.b180dbb//pptfor a single surface23Superconducting proximity effect and majorana fermionsMajorana 1937 Ettore Majorana Majorana

when a superconductor (S) is placed in contact with a "normal" (N) non-superconductor. Typically thecritical temperatureof the superconductor is suppressed and signs of weak superconductivity are observed in the normal material overmesoscopicdistances.

Majorana Majorana

Majorana

Majorana

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