topological insulators - the budker groupbudker.berkeley.edu/physics141_2013/yize jin and lu...

IntroductionQuantum Hall Effect

Topological InsulatorsApplication

Topological Insulators

Yize Jin, Lu Zheng

Department of Physics, Fudan University, China

Yize Jin,Lu Zheng Topological Insulators



Outline

1 Introduction

2 Quantum Hall Effect

3 Topological InsulatorsQuantum Spin Hall EffectBand StructureThe First Found Topological Insulators

4 Application




Topology




Genus




Topological Insulators

Topological insulator is insulator in bulk but conductor only onedge.




Quantum Hall Effect

|u(~k)〉 is an eigenstate of the Hamiltonian, the Berry’sphase of this is Am = i〈um|∇k |um〉 this may be expressedas a surface integral of the Berry flux

Fm = ∇× Am

The first chern number of each state of an electron

nm =1

2π

∫d2~kFm

The first chern number of an electron

n =∑

m

nm

Hall conductivityσxy = Ne2/~




Quantum Spin Hall EffectBand StructureThe First Found Topological Insulators

Outline

1 Introduction



4 Application





Quantum Spin Hall Effect

The conditions of QHE(Quantum Hall Effect) are strongmagnetic field and low temperature ,which are hard toorealize.QHE =⇒ QSHE(Quantum Spin Hall Effect):no magneticfield






Electronic current =⇒ Spin current






Edge-states electrons in QSHE are immune to impurityscattering






In figure (a), reflected light from upper edge and bottomedge interfere with each other destructively.In figure (b),the upper electron is scattered clockwise(π)while the lower counterclockwise(−π).Since an electron is a spin-1/2 particle, a 2π(= π − (−π))rotation difference will cause a phase difference of -1,resulting in destructive interference.





Outline

1 Introduction



4 Application





Band Structure

Topologically Inequivalent





Band Structure

Why are they topologically inequivalent?=⇒ Theintersections of trivial insulators can be seperated, but nottopological insulators.Why can’t the intersections of topological insulators beseperated?=⇒ Kramers theoremWhat is Kramers theorem?=⇒The energy levels ofsystems with an odd total number of fermions remain atleast doubly degenerate in the presence of purely electricfields.In topological insulators, the red line(edge states) doesn’tcome back to valence band like trivial insulators. =⇒Bandinversion.





Outline

1 Introduction



4 Application





Forecast





HgTe: Eg < 0 ⇐⇒ p orbital band is above the s orbitalband.CdTe: Eg > 0 ⇐⇒ s orbital band is above the p orbitalband.Make a sandwich=⇒band inversion=⇒topologicalinsulator?





The First Found Topological Insulator

E1 is the s-likeconduction subbandand H1 is p-likevalence subband.dc = 6.5nmThick quantum wellhas a quantizedresistance plateau at

R =h

2e2

due to the conductingedge states




Possible Application

SuperconductorTopological Quantum ComputationAnomalous Quantum Hall EffectMajorana Fermion,,




Spintronic Devices

Impurity scattering =⇒ Heat dissipationSolution:QSHE




References

X.-L Qi,S.-C Zhang, Phys. Today 63(1), 33(2010).B.A. Bernevig, T. L. Hughes, S.-C. Zhang, Science 314,1757(2006).M.König et al., Science 318, 766 (2007).M.Z.Hasan, and C.L.Kane (2010), e-print arXiv:1002.3895.




THANK YOU!


topological insulators - the budker groupbudker.berkeley.edu/physics141_2013/yize jin and lu...

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