Spin Hall effect and related issues. Dept of Physics Taiwan Normal Univ. Ming-Che Chang

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1 Spin Hall effect and related issues Dept of Physics Taiwan Normal Univ. Ming-Che Chang

2 Basic spin interactions in semiconductors (Dyakonov, cond-mat/ ) spin-orbit interaction required for optical spin orientation major cause of spin relaxation exchange interaction for ferromagnetic semiconductors between FM/Semiconductor junction hyper-fine interaction for materials with non-zero nuclei spin (e.g. GaAs ) dipole-dipole interaction (usually very weak)

3 Spin-orbit interaction in an atom (Eisberg and Resnick, Quantum Physics) An electron moving in a static E field feels an effective B field B E v eff = c v E This B field couples with the electron spin HSO = µ B F eff HG e mc S I = K J F HG E v I E r d c K J φ, = for central force dr F HG 1 dv I S L V e m c rdr K J =, = φ (x 1/ for Thomas precession) fine structure in atomic spectra

4 Spin-orbit interaction in semiconductor (Kittel, Quantum Theory of Solids) H = 1 mc S V x v ( ) SO (V(x) is the lattice potential energy) splitting of valence bands (GaAs, D=0.34 ev) change of g-factor (GaAs, g * =-0.44) for materials without inversion symmetry, lift the spin degeneracy of energy bands (Dresselhaus, Rashba) skew scattering from impurities For strong SO couplings, choose low-symm, narrow-gap materials formed from heavy elements (g * ª-50 in InSb) (Rashba, condmat/ )

5 Generation of spin in semiconductor using SO coupling (Rashba PRB 004) [1] [] [3] Hirsch, PRL 1999 Voskoboynikov et al, PRB 1999 and many others Kiselev and Kim, APL 001 Ioniciociu and D Amico, PRB 003 Ramaglia et al, Euro Phys J B 003 Watson et al, PRL 003 Rokhinson et al, PRL 004 Bhat and Sipe, PRL 000 Mal shukov et al, PRB 003 Murakami et al, Science 003 Sinova et al, PRL 004 spin Hall effect (SHE), skew scattering resonant tunneling related ideas T-shaped filter Stern-Gerlach device quantum point contact adiabatic pumping (need B field) electron focusing (need B field) all-optical technique AC gate device design SHE, in bulk p-type semiconductor SHE, in n-type heterojunction (DEG)

6 Related issue Hall effect in metal/semiconductor: ρ xy R B = 0 Anomalous Hall effect in ferromagnet: ρ xy = R0B + R S M proposed mechanisms: Karplus-Luttinger s mechanism (1954), revived by Jungwirth et al (PRL, 00) Smit s skew scattering mechanism (1955) Berger s side jump mechanism (1970) The Hall effect and its applications, by Chien and Westgate 1979 Intrinsic, related to Berry curvature extrinsic

7 Karplus-Luttinger mechanism (PRB, 1954) Semiclassical EOM (Chang and Niu, PRL 1995) dk ħ dt dx dt = ee E ħ k = λ ( ) k Berry curvature (1983) Ω nz dk Ω dt ( k ) = i F HG λ ( k ) Anomalous velocity due to Berry curvature u k n x u k n y one-band formulation Berry curvature as an effective B field in k-space for multi-band generalization, see Shindou and Imura, cond-mat/ , Culcer et al, Nov, 004 u k n y u k Could be nonzero when (one of the following) time-reversal symmetry is broken (eg. by a B field) n x I KJ lattice inversion asymmetry presence of SO interaction Jungwirth et al, PRL 00 Lee et al, Science 004

8 Impurity (spinless) scattering with SO interaction (I) Skew scattering (Takahashi and Maekawa, PRL, 00, Landau and Lifshitz, QM) Transition rate: (for d impurities, up to nd order Born approx.) H' = V( x) + λs V ( x) v, λ = L F = + H G I M K J N π W = k' s' T ks δcε ε h, ks k ' s' k k ' ħ 1 T = H' + H' H H ' + ε 0 AS ( ) 3 W δ n λv σ k ' k δ ε ε ks k ' s' 1 mc ħ k ' s' H' ks δ iλ σ k ' k V m O P Q s' s s' s k ' k d i c h s' s i i s' s k k ' Would modify the distribution function f ks, through, eg., Boltzmann eq.

9 Anomalous Hall effect in ferromagnet spin-polarized incident current charge-polarized outgoing current Ø Spin Hall effect in semiconductor unpolarized incident current charge-unpolarized outgoing current but spin-polarized outgoing current

10 Hall effect (E.H. Hall, 1879) [1] Spin Hall effect (Dyakonov and Perel, JETP 1971, J.E. Hirsch, PRL 1999) skew scattering (due to SO coupling) by spinless impurities: transition rate, W λ k k ks k s SOσ s s ' ' ' ' no magnetic field required From spin accumulation to charge accumulation L< spin coherence length d s d s ª130 mm at 36 K for Al (Johnson and Silsbee, PRL 1985)

11 [] Intrinsic spin Hall effect in p-type semiconductor (I) (Murakami, Nagaosa and Zhang, Science 003) Valence band of GaAs: Luttinger Hamiltonian (1956) (for j=3/ valence bands) H = 1 5 γ + γ k m λ = k S (helicity) 1 is a good quantum number γ ( k S ) Semiclassical EOM (Chang and Niu, PRL 1995) dk ħ dt dx dt = ee E ħ k = λ ( ) k dk Ω dt λ ( k ) Anomalous velocity due to Berry curvature

12 Intrinsic spin Hall effect in p-type semiconductor (II) (Non-Abelian) gauge potential A k i k i k k λλ'( ) =, λ, λ' Berry curvature, due to monopole field in k-space (Neglecting off-diagonal elements) F k Ω λ ( k ) = λhg I λ K J 7 4 k Spin current H z ħ z k F HH J y = ys n k = ee λ ( ) x, 3 4π LH λ = ± 3/, k L z ħ z k F J y = ys n k = + ee λ ( ) x, 3 1π λ =± 1/, k J y Spin Hall conductivity z e H σ yx = c3k F k 1π e H ck F + k 1π 1 H L = ck F k h F 6π E x No magnetic field required L F L F h h J z y = σ z yx E (semiclassical) (Q correction) x Applies to Si as well

13 [3] Intrinsic spin Hall effect in dimensional electron gas (DEG) (I) (Sinova, Culcer, Niu, Sinitsyn, Jungwirth, and MacDonald, PRL 004) Semiconductor heterojunction z z ª triangular quantum well

14 Intrinsic spin Hall effect in DEG (II) Structure Inversion Asymm (SIA) Rashba Hamiltonian (1960) E l=-1 k Eigen-energies ħ k Eλ ( k ) = + λαk, λ = ± 1 m ασ k z = µ Bσ B B ( k) λz k eff eff H p α = + σ p z m ħ λ = ( σ p ) z (helicity) is a good quantum number no space inversion symmetry invariant under time reversal E s (k)=e -s (-k) Kramer degeneracy

15 Dynamics of spin under electric perturbation Landau-Lifshitz eq. ds = S Beff k + S S B dt ( ) γ d i eff (l=-1) dk = -eet // -x db eff ª lz dk // -ly damping When both bands are filled, spin Hall conductivity: z e σ yx = independent of α! 8 π not so for non-parabolic bands only for clean system J y E x not related to Berry curvature (?) No magnetic field required

16 Effect of disorder on the intrinsic spin Hall effect (I) Rashba system with short-range impurities Inoue et al (003) Dimitrova (004) Khaetskii (004) Raimonde and Schwab (004) F HG I K J = clean vertex e e σ SH = σ SH + σ SH = + 8π 8π 0! Perturbative calculations for other systems If H(k)=H(-k), eg. Luttinger model then vertex correction is zero (Murakami, PRB 004) For systems with H( k ) = E0( k ) + σ xd y ( k ) σ yd x( k ) If E / k d, then perfect cancelation (eg. Rashba) 0 otherwise σ remains finite. (quoted from Murakami' s talk) s Spin Hall effect is finite in general

17 Effect of disorder on the spin Hall effect in Rashba system (II) Numerical calculations: s SH robust against weak disorder Nikolic et al, cond-mat/ Hankiewicz et al, PRB 004 Sheng et al, PRL 005 Nomura et al, PRB 005 Stronger SO coupling Stronger SO coupling Weaker disorder

18 Spin Hall effect observed (I) (Kato et al, Science 004) Local Kerr effect in strained n-type bulk GaAs/InGaAs, 0.03% polarization The effect is independent of the direction of strain. Mostly likely extrinsic.

19 Spin Hall effect observed (II) (Wunderlich et al, to appear PRL 005) spin LED in GaAs D hole gas, 1% polarization p n might be intrinsic? (Bernevig and Zhang, cond-mat/ )

20 Beyond Rashba (clean): In a III-V quantum well (SIA) J. Schlieman and D. Loss, PRB 003 H Rashba p α = + d σ m ħ p σ β + d σ ħ p σ p x y y x p x x y y Dresselhaus (1955) [001] QW Effective magnetic field: i i (BIA) BIA SIA BIA=SIA BIAπSIA Ganichev and Prettl, cond-mat/030466

21 Rashba-Dresselhaus system in an in-plane magnetic field p α H = + d σ p σ p i γ + d σ p σ p i + β σ + β σ m * ħ ħ x y y x x x y y x x y y Eigen-energies: Eλ ( k ) = E0( k ) + λ γk x + αk y + β x + αk x + γk y β y, λ = ± d i d i Distorted Fermi surfaces (generic cases): (c) (b) (a) Parameters: α γ Point of degeneracy 1 ev A (tunable by gate voltage) β = g */ µ B, µ mev / T 11 k = πn 10 / A for n 10 / cm F of the same order b g B B k F HG γβ + αβ x y x y 0 =, α γ αβ α + γβ γ I KJ

22 Effect of in-plane magnetic field on spin Hall conductivity J y E x B σ = η µν 1 i ħ k, λ, λ ' λ λ ' b g f ω Kubo formula k, λ k, λ ' λλ' f k d i η ħ η η jµ = dvµ σ + σ v i; j = ev 4 η k, λ j k, λ' k, λ' j k, λ, µ µ ν ν ν For g = 0 (pure Rashba) E +, min E(k 0 ) E(k 0 ) E +, min E -, min E -, min z σ xy ( B) could be changed by 100% simply by rotating the magnetic field

23 Spin Hall conductivity (electron density fixed) σ x xy = σ = 0 y xy Boundary of plateau E(k 0 ) = m β αγβ β β ζ α γ x + 4 x y + y = α + γ c h () (1) (3) () M.C. Chang, PRB 005 Acknowledgement: M.F. Yang

24 Existence of charge Hall effect? Thouless formula (PRL 198) e σ λ xy = Ω λ ( k ), ħ k filled Berry curvature Ω λ k, λ vx k, λ' k, λ' vy k, λ k, λ v y k, λ' k, λ' vx k, λ ( k ) = i = 0 ω ( k ) λ' λ Berry phase z λπ α > γ S Γλ = dk k λ i λ = α = γ k k for,, 0 for T + λπ for α < γ Ω ( k ) = sgn α γ λπδ k k λ λλ' R c h d i 0 iθ F ie k, + = iθ, k, =, ie KJ 1 tanθ F HG I γk + αk + β = αk + γk β (c) (a) x y x x y y at every k, except at degenerate point k 0 HG I KJ Hall conductivity is zero wherever the chemical potential is 0+0=0 (-p)+p =0

25 Issues on the use of SO coupling for spin injection: (Rashba, cond-mat/ ) spin current is not well defined (total spin not conserved) no experimental procedure to measure it directly (accumulation? Induced electric field?) existence of background spin current (in noncentrosymmetric materials) Rashba, PRB 003 the role of spin current in Maxwell electrodynamics? (c M is part of the charge current) Rashba, cond-mat/040473

26 Definition of spin current α α + b g α σ + J = Re ψ λ 0 s p V ψ, t α + α where σ ψ s ψ α 1 c + α h α J Reψ s v + vs ψ Spin torque Spin flux α α α d α J s r + s r = s r dt Advantage: source term (RHS) vanishes under certain conditions c h P.Zhang et al, cond-mat/ Onsager relations rescued would remove most of the spin Hall insulators most of the results on spin Hall conductivity need to be re-calculated (?)