Topological Aspects of the Spin Hall Effect

1. Topological Aspects of the Spin Hall Effect Yong-Shi Wu Dept. of Physics, University of Utah Collaborators: Xiao-Liang Qi and Shou-Cheng Zhang (XXIII International Conference on Differential Geometric Methods in Theoretical Physics Nankai Institute of Mathematics; August 21, 2005)

2. Motivations • Electrons carry both charge and spin • Charge transport has been exploited in Electric and Electronic Engineering: Numerous applications in modern technology • Spin Transport of Electrons Theory:Spin-orbit coupling and spin transport Experiment:Induce and manipulate spin currents Spintronics and Quantum Information processing • Intrinsic Spin Hall Effect: Impurity-Independent Dissipation-less Current

3. The Spin Hall Effect p-GaAs Electric field induces transverse spin current due to spin-orbit coupling • Key advantages: • Electric field manipulation, rather than magnetic field • Dissipation-less response, since both spin current • and electric field are even under time reversal • Intrinsic SHE of topological origin, due to Berry’s phase inmomentum space, similar to the QHE • Very different from Ohmic current:

4. Family of Hall Effects • Classical Hall Effect Lorentz forcedeflecting like-charge carriers • Quantum Hall Effect Lorentz forcedeflecting like-charge carriers (Quantum regime: Landau levels) • Anomalous (Charge) Hall Effect Spin-orbit couplingdeflecting like-spin carriers (measuring magnetization in ferromagnetic materials) • Spin Hall Effect Spin-orbit couplingdeflecting like-spin carriers (inducing and manipulatingdissipation-lessspin currents without magnetic fields or ferromagnetic elements)

5. Time Reversal Symmetry and Dissipative Transport • Microscopic laws in solid state physics are T invariant • Most known transport processes break T invariance • due to dissipative coupling to the environment • Damped harmonic oscillator • Ohmic conductivity is dissipative: • under T, electric field is even • charge current is odd • Charge supercurrent and Hall current are non-dissipative: (only states close to the Fermi energy contribute!) under T vector potential is odd, while magnetic field is odd

6. Spin-Orbit Coupling • Origin: • ``Relativistic’’ effect in atomic, crystal, impurity or gate electric field • = Momentum-dependent magnetic field • Strength tunable in certain situations • Theoretical Issues: • Consequences of SOC in various situations? • Interplay between SOC and other interactions? • Practical challenge: • Exploit SOC to generate,manipulate and transport spins

7. Spin-orbit couping down-spin up-spin impurity Cf. Mott scattering The Extrinsic Spin Hall effect (due to impurity scattering with spin-orbit coupling) D’yakonov and Perel’ (1971) Hirsch (1999), Zhang (2000) impurity scattering = spin dependent (skew) Mott scattering plus side-jump effect • The Intrinsic Spin Hall Effect • Berry phase in momentum space • Independent of impurities

8. ( : periodic part of the Bloch wf. ) : Magnetic field in momentum space : Band index Berry Phase (Vector Potential) in Momentum Space from Band Structure

9. Wave-Packet Trajectoryin Real Space Anomalousvelocity (perpendicular to and ) Hole spin Spin current(spin//x,velocity//y) Chang and Niu (1995); P. Horvarth et al. (2000)

10. : field strength; : band index (Degeneracy point Magnetic monopole) Intrinsic Hall conductivity (Kubo Formula) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985)

11. Field Theory Approach • Electron propagator in momentum space • Ishikawa’s formula (1986): • Hall Conductance in terms of momentum space topology

12. In p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field. i: spin direction j: current direction k: electric field (Murakami, Nagaosa, Zhang, Science (2003)) :even under time reversal = reactive response (dissipationless) • Nonzero in nonmagnetic materials. Cf. Ohm’s law: p-GaAs Intrinsic spin Hall effect in p-type semiconductors : odd under time reversal = dissipative response

13. Valence band of GaAs S S P3/2 P P1/2 Luttinger Hamiltonian ( : spin-3/2 matrix, describing the P3/2 band)

14. Luttinger model Expressed in terms of the Dirac Gamma matrices:

15. Spin Hall Current (Generalizing TKNN) • Of topological origin(Berry phase • in momentum space) • Dissipation-less • All occupied state contribute Spin Analog of the Quantum Hall Effect At Room Temperature

16. Intrinsic spin Hall effectfor2D n-type semiconductors in heterostructure 2D heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003)) Rashba Hamiltonian Effective magnetic field Kubo formula : independent of SHE: Spin precession by “k-dependent Zeeman field” Note: is not small even when the spin splitting is small. due to an interband effect

17. Spin Hall insulator • Motivation: Truly dissipationless transport • Gapful band insulator • (to get rid of Ohmic currents) • Nonzero spin Hall effect in band insulators: • - Murakami, Nagaosa, Zhang,PRL (2004) • Topological quantization of spin Hall conductance: • - Qi, YSW, Zhang, cond-mat/0505308 (PRL) • Spin current and accumulation: • - Onoda, Nagaosa, cond-mat/0505436 (PRL)

18. Theoretical Approaches • Kubo Formula (Berry phase in Brillouin Zone) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985) • Kubo Formula (Twisted Phases at Boundaries) Niu, Thouless, Wu (1985) (No analog in SHE yet!) • Cylindrical Geometry and Edge States Laughlin (1981) Hatsugai (1993) (convenient for numerical study)

19. Cylindrical Geometry and Edge States Laughlin Gauge Argument (1981): • Adiabatically changing flux • Transport through edge states Bulk-Edge Relation: (Hatsugai,1993) (Spectral Flow of Edge States)

20. Magnetic semiconductor with SO coupling in 2d (no Landau levels) Topological Quantization of the AHE (I) Model Hamilatonian:

21. (c>0) Topological Quantization of the AHE (II) Two bands: Band Insulator: a band gap, if V is large enough, and only the lower band is filled Charge Hall effect of a filled band: Charge Hall conductance is quantized to be n/2p

22. Topological Quantization of the AHE (III) Open boundary condition in x-direction Two arrows: gapless edge states The inset: density of (chiral) edge states at Fermi surface

23. Topological Quantization of Spin Hall Effect I Paramagnetic semiconductors such as HgTe and a-Sn: are Dirac 4x4 matrices (a=1,..,5) With symmetry z->-z, d1=d2=0. Then, H becomes block-diagonal: SHE is topologically quantized to be n/2p

24. LH HH Topological Quantization of Spin Hall Effect II For t/V small: A gap develops between LH and HH bands. Conserved spin quantum number is

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

26. Conserved Non-conserved = + Topological Quantization of Spin Hall Effect IV • Physical Understanding: Edge statesII Apply an electric field n edge states withG12=+1(-1)transfer from left (right) to right (left). G12 accumulation Spin accumulation

27. Effect due to disorder (Green’s function method) Rashba model: Intrinsic spin Hall conductivity (Sinova et al.,2004) + spinless impurities ( -function pot.) + Vertex correction in the clean limit (Inoue et al (2003), Mishchenko et al, Sheng et al (2005)) + spinless impurities ( -function pot.) Luttinger model: Intrinsic spin Hall conductivity (Murakami et al,2003) Vertex correction vanishes identically! (Murakami (2004), Bernevig+Zhang (2004)

28. Topological Orders in Insulators • Simple band insulators: trivial • Superconductors: Helium 3 (vector order-parameter) • Hall Insulators: Non-zero (charge) Hall conductance 2d electrons in magnetic field: TKNN (1982) 3d electrons in magnetic field: Kohmoto, Halperin, Wu (1991) • Spin Hall Insulators: Non-zero spin Hall conductance 2d semiconductors: Qi, Wu, Zhang (2005) 2d graphite film: Kane and Mele (2005) • Discrete Topological Numbers: in 2d systems Z_2: Kane and Mele (2005); Z_n: Hatsigai, Kohmoto , Wu (1990) • 2d Spin Systems and Mott Insulators: Topological Dependent Degeneracy of the ground states Fisher, Sachdev, Sethil, Wen etc (1991-2004)

29. Conclusion & Discussion • Spin HallEffect: A new typeof dissipationless quantum • spin transport, realizable at room temperature • Natural generalization of the quantum Hall effect • Lorentz force vsspin-orbit forces: both velocity dependent • U(1) to SU(2), 2D to 3D • Instrinsic spin injection in spintronics devices • Spin injection without magnetic field or ferromagnet • Spins created inside the semiconductor, • no interface problem • Room temperature injection • Source of polarized LED • Reversible quantum computation? • Many Theoretical Issues: • Effects of Impurities • Effects of Contacts • Random Ensemble with SOC • Topological Order of Quantized Spin Hall Systems