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