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)

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

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 in momentum space, similar to the QHE Very different from Ohmic current: Electric field induces transverse spin current due to spin-orbit coupling The Spin Hall Effect p-GaAs

Family of Hall Effects Classical Hall Effect Lorentz force deflecting like-charge carriers Quantum Hall Effect Lorentz force deflecting like-charge carriers (Quantum regime: Landau levels) Anomalous (Charge) Hall Effect Spin-orbit coupling deflecting like-spin carriers (measuring magnetization in ferromagnetic materials) Spin Hall Effect Spin-orbit coupling deflecting like-spin carriers (inducing and manipulating dissipation-less spin currents without magnetic fields or ferromagnetic elements)

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 ( only states close to the Fermi energy contribute!) Ohmic conductivity is dissipative: under T, electric field is even charge current is odd Charge supercurrent and Hall current are non-dissipative: under T vector potential is odd, while magnetic field is odd

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

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

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

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

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

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

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

Valence band of GaAs Luttinger Hamiltonian ( : spin-3/2 matrix, describing the P 3/2 band) S P S P 3/2 P 1/2

Luttinger model Expressed in terms of the Dirac Gamma matrices:

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

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

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/ (PRL) Spin current and accumulation: - Onoda, Nagaosa, cond-mat/ (PRL)

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)

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

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

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

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

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

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

Topological Quantization of Spin Hall Effect III Physical Understanding: Edge states I In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states Energy spectrum for cylindrical 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 Effect IV Physical Understanding: Edge states II Apply an electric field n edge states with    transfer from left (right) to right (left).   accumulation Spin accumulation ConservedNon-conserved +=

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

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 ( )

Conclusion & Discussion Spin Hall Effect: A new type of dissipationless quantum spin transport, realizable at room temperature Natural generalization of the quantum Hall effect Lorentz force vs spin-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