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TOPOLOGICAL INSULATORS

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OUTLINE Introduction Brief history of topological insulators Band theory Quantum Hall effect Superconducting proximity effect

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INTRODUCTION Close relation between topological insulators and several kinds of Hall effects. Hall effect Anomalous Hall effect Spin Hall effect Quantum Hall effect Quantum Anomalous Hall effect Quantum Spin Hall effect

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BRIEF HISTORY OF TOPOLOGICAL INSULATORS

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THE HISTORY OF TOPOLOGICAL INSULATOR QHE QSHE 3D TI 2005 Kane & Mele 2006 HgTe / CdTe 2007 Molenkamp 2007 Fu Kane Bi1-xSbx 3D TI 2008 Hasan ARPES 2009 Bi2Se3 Bi2Te3 Sb2Te3 2009 ARPES Hasan Bi2Se3 Bi2Te3 Hasan Sb2Te3 1980 1982

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2D topological insulator Shou-Cheng Zhang Group. Science 314, 1757 (2006)

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

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3D topological insulator Liang Fu and C. L. Kane Physical Review B, 2007, 76(4): 045302.

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3D topological insulator Hasan Group. Nature, 2008, 452(7190): 970- 974.

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BAND THEORY

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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 structures

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THE 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.

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

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EDGE STATES skipping motion electrons bounce off the edge chiral:propagate in one direction only along the edge insensitive to disorder :no states available for backscattering deeply related to the topology of the bulk quantum Hall state.

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Z 2 TOPOLOGICAL INSULATOR T symmetry operator: Sy is the spin operator and K is complex conjugation for spin 1/2 electrons: A T invariant Bloch Hamiltonian must satisfy

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Z 2 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:

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Z 2 TOPOLOGICAL INSULATOR the Z2 invariant is: if the 2D system conserves the perpendicular spin Sz Chern integers n, nare independent,the difference defines a quantized spin Hall conductivity. The Z2 invariant is then simply

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Z 2 TOPOLOGICAL INSULATOR

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SURFACE QUANTUM HALL EFFECT

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INTEGER QUANTIZED HALL EFFECT

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

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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.

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SUPERCONDUCTING PROXIMITY EFFECT AND MAJORANA FERMIONS

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MAJORANA 1937 Ettore Majorana Majorana

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when a superconductor (S) is placed in contact with a "normal" (N) non- superconductor. Typically the critical temperature of the superconductor is suppressed and signs of weak superconductivity are observed in the normal material over mesoscopic distances.

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Majorana Majorana

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MAJORANA

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MAJORANA

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