Quantum spin Hall effect Shoucheng Zhang (Stanford University) Collaborators: Andrei Bernevig, Congjun Wu (Stanford) Xiaoliang Qi (Tsinghua), Yongshi Wu (Utah) Murakami, Nagaosa (Tokyo) cond-mat/ cond-mat/ cond-mat/ HK meeting, 2005/12

Generalization of the quantum Hall effect Quantum Hall effect exists in D=2, due to Lorentz force. Natural generalization to D=3, due to spin-orbit force: 3D hole systems (Murakami, Nagaosa and Zhang, Science 2003) 2D electron systems (Sinova et al, PRL 2004)

The intrinsic spin Hall effect Key advantage: electric field manipulation, rather than magnetic field. dissipationless response, since both spin current and the electric field are even under time reversal. Topological origin, due to Berry’s phase in momentum space similar to the QHE. Contrast between the spin current and the Ohm’s law:

Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004) Experiment -- Spin Hall effect in a 3D electron film (i) Unstrained n-GaAs (ii) Strained n-In 0.07 Ga 0.93 As T=30K, Hole density: : measured by Kerr rotation

Circular polarization Clean limit : much smaller than spin splitting vertex correction =0 (Bernevig, Zhang (2004)) Experiment -- Spin Hall effect in a 2D hole gas -- J. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, PRL (2005) LED geometry

What about the quantum anomalous Hall effect and the quantum spin Hall effect?

Key ingredients of the quantum Hall effect: Time reversal symmetry breaking. Bulk gap. Gapless chiral edge states. External magnetic field is not necessary! Quantized anomalous Hall effect: Time reversal symmetry breaking due to ferromagnetic moment. Topologically non-trivial bulk band gap. Gapless chiral edge states ensured by the index theorem.

Topological Quantization of the AHE (cond-mat/ ) Magnetic semiconductor with SO coupling (no Landau levels): General 2×2 Hamiltonian Example Rashbar Spin- orbital Coupling

Topological Quantization of the AHE (cond-mat/ ) Hall Conductivity Insulator Condition Quantization Rule The Example

Origin of Quantization: Skyrmion in momentum space Skyrmion number=1 Skyrmion in lattice momentum space (torus) Edge state due to monopole singularity

band structure on stripe geometry and topological edge state

Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, , 2004) In zero-gap semiconductors, such as HgTe, PbTe and  -Sn, the HH band is fully occupied while the LH band is completely empty. A charge gap can be induced by pressure. In this case, charge conductivity vanishes, but the spin Hall conductivity is maximal.

Luttinger Model for spin Hall insulator Bulk Material zero gap Symmetric Quantum Well, z  -z mirror symmetry Decoupled between (-1/2, 3/2) and (1/2, -3/2)  

Topological Quantization of SHE LH HH SHE is topological quantized to be n/2  Luttinger Hamiltonian rewritten as In the presence of mirror symmetry z->-z, =0  d 1 =d 2 =0! In this case, the H becomes block-diagonal:

Topological Quantization of Spin Hall Physical Understanding: Edge states In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states. Energy spectrum on stripe geometry. Laughlin’s Gauge Argument: When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another

Topological Quantization of Spin Hall Physical Understanding: Edge states When an electric field is applied, n edge states with    transfer from left (right) to right (left).   accumulation  Spin accumulation ConservedNon- conserved +=

Quantum spin Hall effect in graphene (Haldane, Kane&Mele) SO coupling opens up a gap at the Dirac point. One pair of TR edge state on each edge.

Edge state contribution to Charge transport Edge state carry chiral spin current but non-chiral charge current. Quantized residual conductance in topological insulator Schematic Picture of Conductance

Stability at the edge The edge states of the QSHE is the 1D helical liquid. Opposite spins have the opposite chirality at the same edge. It is different from the 1D chiral liquid (T breaking), and the 1D spinless fermions. T 2 =1 for spinless fermions and T 2 =-1 for helical liquids. Single particle backscattering is not possible for helical liquids!

Stability at the edge Kane & Mele Wu, Bernevig and Zhang Xu and Moore Sheng et al

Conclusion & Discussion Quantum AHE. Ferromagnetic insulators with spin-orbit coupling. Topologically non-trivial band gap. Hall conductance=Skyrmion number in momentum space. Number of chiral edge modes=Skyrmion number in momentum space. Quantum SHE: Standard semiconductor with a strain gradient, narrow gap semiconductors, and monolayers of graphene. A new type of 1D metal: the helical liquid. Stability ensured by the time reversal symmetry of the spin current.