SMR 1646 - 11 ___________________________________________________________ Conference on Higher Dimensional Quantum Hall Effect, Chern-Simons Theory and Non-Commutative Geometry in Condensed Matter Physics and Field Theory 1 - 4 March 2005 ___________________________________________________________ Intrinsic Spin Hall Effect Shuichi MURAKAMI Department of Applied Physics, University of Tokyo Tokyo, Japan These are preliminary lecture notes, intended only for distribution to participants. March 3,2005, ICTP, Trieste, Italy Intrinsic Spin Hall Effect Shuichi Murakami (Department of Applied Physics, University of Tokyo) Collaborators：Naoto Nagaosa (U.Tokyo) Shoucheng Zhang (Stanford) Masaru Onoda (AIST, Japan) Spin Hall effect (SHE) r E Electric field induces a transverse spin current. • Extrinsic spin Hall effect D’yakonov and Perel’ (1971) Hirsch (1999), Zhang (2000) impurity scattering = spin dependent (skew-scattering) Spin-orbit couping up-spin down-spin impurity Cf. Mott scattering • Intrinsic spin Hall effect Berry phase in momentum space Independent of impurities ! y Intrinsic spin Hall effect x z • p-type semiconductors (SM, Nagaosa, Zhang, Science (2003)) r E p-GaAs Luttinger model 2 ⎡ 5 r r 2 ⎤ r = h ⎛γ + γ ⎞ 2 − γ ()⋅ ( S : spin-3/2 matrix) H ⎢⎜ 1 2 ⎟k 2 2 k S ⎥ 2m ⎣⎝ 2 ⎠ ⎦ • 2D n-type semiconductors in heterostructure r (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL (2003)) E z Rashba model k 2 r y H = + λ()σr × k 2m z x Berry phase in momentum space ( U(1) gauge field) ∂ ∂ r r r r * unk d A (k ) = −i nk nk = −i u r d x : Gauge field ni ∂ ∫ nk ∂ ki unit cell ki r antimonopole ( u n k : periodic part of the Bloch wf.) r ik⋅xr ψ r r = r r nk (x) unk (x)e r r r r = ∇ r × Bn (k ) k An (k ) : Field strength ( n : band index) monopole Intrinsic Hall conductivity 2 σ = − e ()r r xy nF En (k ) Bnz (k ) ∑r h n,k Hall conductivity due to Berry phase (intrinsic contribution) 2 σ = − e ()r r xy nF En (k ) Bnz (k ) ∑r h n,k Berry phase Kubo formula (Thouless et al. (1982)) Semiclassical eq. of motion (Sundaram,Niu (1999)) r 1 ∂E (k ) r r r xr& = n + k& × B (k ) ∂ r n Anomalous velocity due to Berry phase h k • Quantum Hall effect r r r k& = −eE − exr& × B(xr) • Anomalous Hall effect • Spin Hall effect r r = Bn (k ) “magnetic field in k-space” Valence band of semiconductors (diamond (Si, Ge) or zincblende (GaAs)) p-orbit (x,y,z)×（↑,↓） + spin-orbit coupling split-off band (SO) heavy-hole band (HH) doubly degenerate light-hole band (LH) (Kramers) Luttinger Hamiltonian (Luttinger(1956)) 2 = h ⎡⎛γ + 5 γ ⎞ 2 − γ ()r ⋅ r 2 ⎤ H ⎢⎜ 1 2 ⎟k 2 2 k S ⎥ 2m ⎣⎝ 2 ⎠ ⎦ r ( S : spin-3/2 matrix) r Helicityλ = kˆ ⋅ S is a good quantum number. r 3 γ − 2γ λ = kˆ⋅ S = ± ⇒ E = 1 2 2k 2 : heavy hole (HH) 2 2m h Helicity r 1 γ + 2γ λ = kˆ ⋅ S = ± ⇒ E = 1 2 2k 2 : light hole (LH) 2 2m h Semiclassical Equation of motion r r& r r hk e r r (λ) r hk = eE , x& = + E × B (k ) ( λ : helicity= band index) mλ h ∂ = E Anomalous velocity Drift velocity ∂ ki (due to Berry phase) Two bands touch at k=0 r k = 0 Æ monopole at k=0 r r λ r ⎛ 7 ⎞ k B( ) (k ) = λ⎜2λ2 − ⎟ ⎝ 2 ⎠ k 3 3 1 λ = ± : HH, λ = ± : LH 2 2 A hole obtains a velocity perpendicular to both k and E. r r λ r ⎛ 7 ⎞ k Real-space trajectory B( ) (k ) = λ⎜2λ2 − ⎟ ⎝ 2 ⎠ k 3 r λ = ± 3 λ = ± 1 r& = r r = hk + e r × r (λ) r : HH, : LH hk eE , x& E B (k ) 2 2 m r λ h λ = kˆ ⋅ S r r Anomalous velocity (perpendicular to andS )E λ < 0 r // k λ > 0 r E // z Hole spin Spin current (spin//x, velocity//y) H H = h λ r = Ez kF jyx y&Sxn (k ) 2 , ∑ r π 3 λ=± 3,k 4 2 σ = e H − L L s (3kF kF ) λ r E k π 2 j L = h yS n (k ) = − z F , 12 yx ∑ & x 2 3 λ=± 1 r 12π 2 ,k Intrinsic spin Hall effect in p-type semiconductors y x In p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field. z (SM, Nagaosa, Zhang, Science (2003)) r E i: spin direction p-GaAs i = σ ε j: current direction j j s ijk Ek k: electric field σ s : even under time reversal = reactive response (dissipationless) • Nonzero in nonmagnetic materials. Cf. Ohm’s law: j = σE σ : odd under time reversal = dissipative response • topological origin (Berry phase in momentum space) • dissipationless Spin analog of the quantum Hall effect • All occupied state contribute. Order estimate (at room temperature) : GaAs eE x = z ()H − L ≡ h σ −1 -1 j y 2 3k F k F s E z σ (Ω cm ) 12π 2e s : Unit of conductivity carrier mobility Charge Spin (Hall) density conductivity conductivity −3 µ 2 σ Ω−1 -1 σ Ω−1 -1 n (cm ) (cm /Vs) ( cm ) s ( cm ) 1019 50 80 73 1018 150 24 34 1017 350 5.6 16 1016 400 0.64 7.3 σ = enµ σ ∝ ∝ 1/3 S kF n σ σ As the hole density decreases, both and decrease.S σ σ decreases faster than .S Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure (Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003)) Rashba Hamiltonian 2 z ⎛ k λ + ⎞ 2 ⎜ (k y ik x )⎟ = k + λ()σr × r = ⎜ 2m ⎟ y H k z 2 2m ⎜λ − k ⎟ ⎜ (k y ik x ) ⎟ x ⎝ 2m ⎠ Effective electric field along z e Kubo formula : J J Sz σ = x y s 8π 1 J S z = {}J , S independent of λ 2D heterostructure y 2 y z σ Note: Sis not small even when the spin splitting is λsmall. interband effect spin Hall effect in the Rashba model ≈ Spin precession by “k-dependent Zeeman field” k 2 r H = + λ()σr × k 2m z r = λ × r Bint (zˆ k ) • Semiclassical theory Culcer et al., PRL(2004) • Rashba + Dresselhaus • Sinitsyn et al., PRB(2004) • Shen, PRB(2004) Disorder effect, edge effect Green’s function method k 2 δ Rashba model: H = + λ()σ k −σ k + spinless impurities ( -function pot.) 2m x y y x σ = e z Intrinsic spin Hall conductivity (Sinova et al.(2003)) J y J x S 8π + Vertex correction in the clean limit σ vertex = − e + S +⋅⋅⋅ (Inoue, Bauer, Molenkamp(2003)) 8π z J y J x σ = S 0 • Inoue, Bauer, Molenkamp (2004) • Rashba (2004) • Raimondi, Schwab (2004) • Dimitrova (2004) • Calculation by Keldysh formalism (Mishchenko, Shytov, Halperin (2004)) σ = Spin Hall current does not flow at the bulk – consistent with S 0 Spin current only flows near the electrodes Spin accumulation SHE No SHE SHE σ ω = = •S ( 0) 0 in Rashba model in the clean limit σ ω = = • (several papers incorrectly claiming S ( 0 ) in0 general.) Question Intrinsic spin Hall effect is so fragile to impurities that it vanishes at the bulk? Answer NO! Not in general. • Rashba model is an exception. σ ω = there are models with nonzero S ( . 0) • two experimental reports • 2D electron gas, spin accumulation at the edges Y.K.Kato et al., Science (2004) • 2D hole gas, spin LED J. Wunderlich et al., cond-mat (2004), to appear in PRL. Intrinsic spin Hall effect is nonzero in general. r r •If Hˆ (k ) = Hˆ (−k ) e.g. Luttinger model (p-type semicond.) Æ in the clean limit the vertex correction is ZERO. (SM (2004)) SHE is finite. r = r + σ r − σ r • For models with H (k ) E0 (k ) x d y (k ) y d x (k ) : n-type semicond. in heterostructure r k 2 r r e.g. Rashba model: E ( k ) = , d = λk 0 2m ∂E r (a) If r 0 = A d for constant ,A SHE vanishes. ∂k (b) Otherwise, SHE is finite in general. λ σr × r Rashba model ( k ) z Æ (a) β σ −σ Dresselhaus model ( x k x y k yÆ) (b) In real materials SHE is finite in general. Tight-binding model on a square lattice − V3 −V + iV σ 1 2 x − − σ V1 iV2 y without vertex correction with vertex correction SHE -- nonzero in general EF Definition of spin current is not unique in the presence of spin-orbit coupling • Spin-orbit coupling Æ spin is not conserved Æ no unique def. of spin current • Noether’s theorem cannot be applied. ∂S ∂ • i + ∇⋅ J (spin) = 0 ⇐ 0 = S d d r = i[]H, S d d r ∂t i ∂t ∫ i ∫ i Eq. of continuity requires conservation of spin, but the spin is not conserved in these models • In some models, spin current is covariantly conserved. (Zhang) = 1 ()r + r 2 r = k k : gauge field associated with the spin-orbit coupling H p A , A Aj t 2m (e.g.) Rashba model = ∂ + : covariant derivative D j j iAj i + + J i = []()D ψ t iψ −ψ t i ()D ψ : spin current j 2m j j ∂ i + ()i = t S D j J j 0 Criterion for nonzero spin Hall conductivity r r r • Different filling for bands in the same multiplet of J = L . + S ≠ c HS.O. v 0 (Example) : GaAs Valence band: J=3/2 Conduction band: J=1/2 Hole-doping gives a Electron-doping does not give different filling for HH and rise to different filling for two LH bands .