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Spin Hall Effect in 1. Rashba Electron Systems in Quantum Hall Regime 2. p-type GaAs Quantum Well with Rashba Coupling Fu-Chun Zhang, The Univ. of Hong Kong Collaborators: Topic 1: Shun-Qing Shen , and Y. Bao (Univ. of Hong Kong), Mike Ma (Cincinnati), X.C. Xie (Oklahoma and IOP, Beijing) Topic 2: Xi Dai (HKU), Zhong Fang , Yu-Gui Yao (IOP, Beijing) PRL 92, 2004, PRB 71, 155316 (2005), cond-mat/0503592; cond-mat/0507603 1. Spin Hall effect in 2D Rashba electron systems in quantum Hall regime Motivation: Why magnetic field? • Rich physics of quantum Hall effect, dissipation-less; Key points • Relation between Spin polarization and spin Hall current • Zeeman and Rashba terms compete to induce level crossings • Resonant spin Hall effect if Fermi energy at level crossing • Experimentally measurable In collaboration with S. Q. Shen, M. Ma, X. C. Xie, Y. J. Bao System and model hamiltonian 2D electron gas with spin-orbit coupling in a magnetic field e → 2 ( p+ A) → → → → r r e c A) × σ ] + g S µ BS • B + λ z • [( p + H = 2m c Kinetic Rashba Zeeman → Spin Hall current and spin polarization λ j z s,x hg sµ B B = − 4m σ y At g_s =0, or Zeeman energy = 0, we have spin Hall current =0. j z s,x = 1 / 2{s z , v x } dsx / dt = (1 / ih)[s x , H ], < dsx / dt >= 0, average − in − an − eigenstate → Single electron ( E = 0 case) with Rashba coupling Rashba, and also Loss et al. → → → → 1 → e→ 2 e→ → H = ( p + A) + g s µ B S • B + λ Z • [( p + A) × σ ] * 2 me c c Kinetic Zeeman Rashba coupling (spin-orbit) Single electron energy ~ 1 2 E ~ = hω (n ± (1 − g ) + 8 n η 2 ) n,± 2 g sm λ2m 2c 2 η = ,g = 3 eB h 2me ~ B Energy levels ~ n = 2,+ 2D electron with Rashba coupling in B-field ~ n = 2,− ~ n =1,− ~ n = 0,+ Mixed spin up and spin down in different Landau level (state entangled: spin and orbit w.f. not decoupled) λ =η = 0 η2 ≠ 0 Spectrum of Rashba system in B-field Landau levels of an electron as functions ηof = λ ml b / h for g = g s m / 2 m e = 0 . 1 (In0.52Al0.48As/In0.53Ga0.47As). Arrows indicate those level crossings giving rise to resonant spin Hall conductance. 2 Magnetization Level crossing Average spin σ Z (unit h / 2 ) per electron as − 11 a function of 1/B. The parameters used λ = 0 .9 × 10 eVm 16 2 n = 1 . 9 × 10 / m are e , g s = 4 , m = 0.05 me taken for the inversion heterostructure In0.53Ga0.47As/In0.52Al0.48As. j xz h = S z vx 2 Fermi level at the level crossing, spin Hall current resonant (peaked) a + λ = λc λ λ =0 λc For any λ ≠ 0 , there is one (B,ne) for resonance Spin Hall conductance v.s. 1/B λ = 0.9 ×10 −11 eVm ne = 1.9 × 1016 / m 2 g s = 4.0; m = 0.05me Resonant Spin Hall Current Density Formalism for charge and spin currents, perturbation H = H 0 (E ) + H ′ H ′ = − (η el bσ y + p x c / B ) E cos θ ns n , p x ~ n , p x , s = i sin θ ns n − 1, p x σ y = Pauli matrix p x = const . ( jc , s ) n , p x , s = ( jc(,0s) ) n , p x , s + ( jc(1, s) ) n , p x , s ( jc(,0s) ) n , p x , s =< n, p x , s | jc , s | n, p x , s > (1) c,s n, px ,s (j ) =∑ n ', s ' < n′p x s′ | H ′ | np x s >< np x s | jc , s | n′p x s′ > (ε ns − ε n′s′ ) + h.c. The effect of E-field 1, ↑ 1, ↑ 0, ↓ 0, ↓ 0, ↑ 0, ↑ E = 0 Carries no current 0, ↑ E=0 + 1 [ 0, ↑ + (iβ 1, ↑ + α 0, ↓ )] 2 State carries a final spin current Discussions on the resonance About anti level crossing. For non-magnetic impurity, u^2/\hbar omega, small. E-field must be larger than all energy scales to see the resonance: temperature, level separation, deviation of B-field from the resonant field Edge spin current and spin polarization in quantum Hall regime Edge state: Rashba coupling=0 MacDonald and Streda (1980’s) Energy spectrum of edge state in quantum Hall system with Rashba coupling (at a distance 4 times of magnetic length) Edge spin Current and spin polarization E=0 V ( y ) = 0 if y ∈ (− L / 2, L / 2); otherwise +∞ Spin polarization and spin current at a small E-field =0. 01 V/m blue: energy separation in bulk > eE l_b, black: similar; red: < eB l_b In calculation, we Assume voltage drops only at the edges, and bulk states contributes no spin current Summary of SHE in quantum Hall region Resonance condition: – Rashba 2DEG with g >0, – Dresselhaus with g<0 Expected to see the resonance near the critical magnetic field at low T At the resonant field, a small E-field changes spin polarization from z to y to -z. Expt.: measure spin polarization Resonant intrinsic spin Hall effect in p-type GaAs quantum well structure Luttinger Hamiltonian with a Rashba spin-orbit coupling arising from the structural inversion symmetry breaking. Rashba term induces an energy level crossing in the lowest heavy hole sub-band, which gives rise to a resonant spin Hall conductance. The resonance may be used to identify the intrinsic spin Hall effect. In collaboration with X. Dai, Z. Fang, Y. G. Yao Hamiltonian (Luttinger + Rushba) H = HL - λ(z × p) · S + V (z) For a given k, in plane wave vector, V(z): infinity potential walls. Limiting cases of the model \lambda =0 , Luttinger hamiltonian. Band structure studied by Yu Cardona and others with a more realistic potential. SHE studied by Bernevig and S. C. Zhang. kL << 1 limit, heavy hole sub-bands studied by Schielmann and Loss by using perturbation theory to the lowest order. Rashba effect: k^3 Here we consider interplay of Rashba and Luttinger terms. Method: using basis of k=0 wavefunctions and truncated numerical method Sub-band dispersion: level crossings Left: No Rashba, similar to Yu & Cardona, Huang et al. \lambda=hbar^2/mL=1; Right: \lambda= 3. K: in-plane momentum, 2L =well width. \gamma_1=7, \gamma_2 =1.9, m=m_e. Mid: Basis function: k=0 eigen states. Spin Hall conductance in GaAs (Kubo formula) hole density = 5x 10^{11} /cm^2, well thickness = 87 A, lifetime = 2 x10^{-11} s, or mobility = 10000 cm^2/sV. Heavy hole sub-band level crossing \lambda_0 = \hbra^2/mL Resonant Rashba coupling Spin Hall conductance for small \lambda (level crossing only in light hole subbands) well thickness= 83A, \lambda = \hbar^2/mL, lifetime= 2x10^{-11}s. Red curve: \lambda=0 Vertex correlations Have not done calculations. Resonance related to heavy hole subband expected to survive with vertex correction. Resonance related to light subband crossing needs more cautious.