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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 The Spin Hall Effect Electric field induces transverse spin current due to spin-orbit coupling y x z E J j Hs ijk Ek p-GaAs i • 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: e2 2 J j c E j where c k F l h 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 mx x kx • Ohmic conductivity is dissipative: under T, electric field is even e2 2 J j E j where k F l charge current is odd h (only states close to the Fermi energy contribute!) • Charge supercurrent and Hall current are non-dissipative: 1 A j J j S Aj , E j c t J H E , H 1 / B 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) impurity scattering = spin dependent (skew) Mott scattering plus side-jump effect Spin-orbit couping up-spin down-spin impurity Cf. Mott scattering • The Intrinsic Spin Hall Effect Berry phase in momentum space Independent of impurities Berry Phase (Vector Potential) in Momentum Space from Band Structure unk d * Ani (k ) i nk nk i unk d x ki ki unit cell ( u nk : periodic part of the Bloch wf. ik x nk ( x ) u nk ( x ) e ) Bn (k ) k An (k ): Magnetic field in momentum space n : Band index Wave-Packet Trajectory in Real Space Chang and Niu (1995); P. Horvarth et al. (2000) k eE , k e ( ) x E B (k ) m Anomalous velocity (perpendicular to S and E ) 0 // k 0 E // z Hole spin Spin current (spin//x,velocity//y) Intrinsic Hall conductivity (Kubo Formula) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985) xy e2 h nF En (k ) Bnz (k ) n,k Bn (k ) k An (k ): field strength; n : band index (Degeneracy point Magnetic monopole) Field Theory Approach • Electron propagator in momentum space with p (p, p0 ) S F ( p) • Ishikawa’s formula (1986): i e2 σ xy 24π 2 h 1 1 1 d p ε tr[ μ S F (p) ν S F (p) λ S F (p)] 3 μνλ • Hall Conductance in terms of momentum space topology Intrinsic spin Hall effect in p-type semiconductors In p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field. y x (Murakami, Nagaosa, Zhang, Science (2003)) z E j s ijk Ek i j i: spin direction j: current direction k: electric field p-GaAs s : even under time reversal = reactive response (dissipationless) • Nonzero in nonmagnetic materials. Cf. Ohm’s law: j E : odd under time reversal = dissipative response Valence band of GaAs S S P P3/2 P1/2 Luttinger Hamiltonian 2 1 5 2 H 1 2 k 2 2 k S 2m 2 ( S : spin-3/2 matrix, describing the P3/2 band) Sx 0 3i / 2 0 0 3i / 2 0 i 0 0 i 0 3i / 2 0 0 3/2 0 S y 3i / 2 0 0 0 3/2 0 1 0 0 1 0 3/2 3/ 2 0 0 0 Sz 0 3 / 2 0 0 0 0 0 1/ 2 0 0 0 1/ 2 0 0 0 3 / 2 Luttinger model Expressed in terms of the Dirac Gamma matrices: Spin Hall Current (Generalizing TKNN) ji ijk Ek j 4 ijk [nL (k ) nH (k )]Gij k (k ) V k e 2 k FH k FL ijk 6 J yz Jx • Of topological origin (Berry phase in momentum space) Spin Analog of the • Dissipation-less Quantum Hall Effect • All occupied state contribute At Room Temperature Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003)) Rashba Hamiltonian k2 (k y ik x ) k2 2m H k z 2 k 2m ( k ik ) y x 2m Effective magnetic field Bint ( zˆ k ) Kubo formula : J Sz y J x J yS z 1 J y , S z 2 e s 8 independent of z y x 2D heterostructure SHE: Spin precession by “k-dependent Zeeman field” Note: S 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/0505308 (PRL) • Spin current and accumulation: - Onoda, Nagaosa, cond-mat/0505436 (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: (Hatsugai,1993) (Spectral Flow of Edge States) Topological Quantization of the AHE (I) Magnetic semiconductor with SO coupling in 2d (no Landau levels) Model Hamilatonian: d x sin k y , d y sin k x d z c(2 cos k x cos k y es ) Topological Quantization of the AHE (II) E (k ) (k ) Vd (k ) 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/2 n 1, if 0 es 4 1, if 2 es 4 0, if es 0 or es 4 (c>0) Topological Quantization of the AHE (III) Open boundary condition in x-direction (c 1,V / t 3, es 0.5) Two arrows: gapless edge states The inset: density of (chiral) edge states at Fermi surface Topological Quantization of Spin Hall Effect I Paramagnetic semiconductors such as HgTe and a-Sn: a 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/2 Topological Quantization of Spin Hall Effect II For t/V small: A gap develops between LH and HH bands. LH d 3 (k ) 3 sin k x sin k y d 4 (k ) 3 (cos k x cos k y ) d 5 (k ) 2 es cos k x cos k y Conserved spin quantum number 12 1 2 is [ , ] / 2 HH n 1, if 0 es 4 1, if 2 es 4 0, if es 0 or es 4 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 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 Topological Quantization of Spin Hall Effect IV • Physical Understanding: Edge states II Apply an electric field n edge states with 1211 transfer from left (right) to right (left). 12 accumulation Spin accumulation Conserved = Non-conserved + Effect due to disorder (Green’s function method) Rashba model: k2 H x k y y k x 2m + spinless impurities ( -function pot.) S Intrinsic spin Hall conductivity (Sinova et al.,2004) + Vertex correction in the clean limit vertex (Inoue et al (2003), Mishchenko et al, S Sheng et al (2005)) e 8 e 8 J z y J yz Jx Jx S 0 k2 2 H k S 2 k y S x k x S y Luttinger model: 1 2m + spinless impurities ( -function pot. Intrinsic spin Hall conductivity (Murakami et al,2003) S Vertex correction vanishes identically! (Murakami (2004), Bernevig+Zhang (2004) e 6 H L z J ( k k ) y F F 2 S vertex 0 J z y Jx Jx 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) 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