Spin Hall effect and related issues

Dept of Physics Taiwan Normal Univ. Ming-Che Chang

8/22/2005

past/now goal

• magnetic memory, GMR, TMR • generation, manipulation, and detection of spins in metals, semiconductors…

on-going effort • FM/semiconductor spin injection not easy • magnetic semiconductor not easy wish for

• integration with existing semiconductor technology • control via electric field, instead of magnetic field • more researches on the spin-orbit coupling in semiconductors

Spin-orbit interaction in semiconductor (Kittel, Quantum Theory of Solids)

H SO

r r 1 r = ⋅ ∇ S V ( x )×v 2mc 2

(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

r r r ≈ λ SOσ s' s ⋅ k '× k

transition rate,

Wksr → kr ' s '

For strong SO couplings, choose low-symm, narrow-gap materials formed from heavy elements (g*»-50 in InSb) (Rashba, condmat/0309441)

Generation of spin in semiconductor using SO coupling (Rashba PRB 2004) [1] • Hirsch, PRL 1999

• spin Hall effect (SHE), skew scattering

• Voskoboynikov et al, PRB 1999 and many others

• resonant tunneling related ideas

• Kiselev and Kim, APL 2001

• T-shaped filter

• Ioniciociu and D’Amico, PRB 2003 • Stern-Gerlach device

device design

• Ramaglia et al, Euro Phys J B 2003 • quantum point contact • Watson et al, PRL 2003

• adiabatic pumping (need B field)

• Rokhinson et al, PRL 2004

• electron focusing (need B field)

• Bhat and Sipe, PRL 2000

• all-optical technique

• Mal’shukov et al, PRB 2003

• AC gate

[2] • Murakami et al, Science 2003

• SHE, in bulk p-type semiconductor

[3] • Sinova et al, PRL 2004

• SHE, in n-type heterojunction (2DEG)

Hall effect (E.H. Hall, 1879)

[1] Spin Hall effect (J.E. Hirsch, PRL 1999, S Zhang, PRL 2000, Dyakonov and Perel, JETP 1971.)

skew scattering by spinless impurities: no magnetic field required

From spin accumulation to charge accumulation L< spin coherence length ds ds »130 mm at 36 K for Al (Johnson and Silsbee, PRL 1985)

[2] Intrinsic spin Hall effect in p-type semiconductor (I) (Murakami, Nagaosa and Zhang, Science 2003)

Valence band of GaAs:

Luttinger Hamiltonian (1956) (for j=3/2 valence bands) r r 2 1  5  2 H=  γ 1 + γ 2 k − 2γ 2 k ⋅ S  2m  2   r $ λ = k ⋅ S (helicity)

( )

is a good quantum number

(Non-Abelian) gauge potential r r ∂ r Aλλ ' ( k ) = i k , λ r k , λ ' ∂k

Berry curvature, due to monopole field in k-space

FG H

IJ K

r r 7 k$ 2 Ω λ ( k ) = −2 λ λ − 4 k2

Intrinsic spin Hall effect in p-type semiconductor (II) Jy

Semiclassical EOM r

r dk h = eE dt r r r dx ∂E λ ( k ) dk r r r − = × Ωλ (k ) dt dt h∂ k

Ex

Anomalous velocity due to Berry curvature (Chang and Niu, PRL 1995 Sundaram and Niu, PRB 1999)

Spin current r 1 k FH z & nλ ( k ) = − 2 eE x , HH J = ∑ yS 3 λ =±3/ 2 ,kr 4π r 1 k FL z z & nλ ( k ) = + LH J y = eE x , ∑ yS 3 λ =±1/ 2 ,kr 12π 2 z y

Spin Hall conductivity

c c

h h

e 3k FH − k FL 2 12π e H L − k + k F F 12π 2 e = 2 k FH − k FL 6π

|σ zyx | =

c

J yz = σ zyx E x (semiclassical) (Q correction)

h

No magnetic field required Applies to Si as well

[3] Intrinsic spin Hall effect in 2 dimensional electron gas (2DEG) (Sinova, Culcer, Niu, Sinitsyn, Jungwirth, and MacDonald, PRL 2004

)

Semiconductor heterojunction

z z

» triangular quantum well

QW with structure inversion asymmetry (SIA): Rashba coupling (Sov. Phys. Solid State, 1960) p2 α r r H= + σ × p ⋅ z$ 2m h

• 1974 Ohkawa and Uemura, due to gradient of the confinement potential ∂V / ∂z • 1976 Darr, ∂V / ∂z for a bound state is actually zero • 1985 Lassnig, interface/valence band are crucial No easy way to calculate a Zawadzki’s, Semi Sci Tech 2004) VG-dependence of the Bychkov-Rashba parameter • Can be determined from the beating of dHvA oscillation • tunable by gate voltage

Engels et al 1997 PRB, InP/In 0.77 Ga 0.23 As/InP

Intrinsic spin Hall effect in 2DEG Rashba Hamiltonian (1960)

p2 α r r H= + σ × p ⋅ z$ 2m h r λ = (σ × p$ ) ⋅ z$ (helicity)

r r r r ασ × k ⋅ z$ = − µ Bσ ⋅ Beff r r r Beff ( k ) ≈ λz$ × k

is a good quantum number Eigen-energies

r h2 k 2 Eλ ( k ) = + λαk , λ = ±1 2m E

l=-1

El(k)=El(-k) Kramer degeneracy

• no space inversion symmetry k

• invariant under time reversal

Dynamics of spin under electric perturbation

(l=-1)

dk = -eEt // -x dBeff » lz´dk // -ly Landau-Lifshitz eq. r r r r dS r r r = S × Beff ( k ) + γS × S × Beff dt

d

i

damping When both bands are filled, spin Hall conductivity: e |σ zyx | = independent of α 8π • not so for non-parabolic bands

Jy Ex

• only for clean system • not related to Berry curvature No magnetic field required

Effect of disorder on the intrinsic spin Hall effect (I) • Rashba system with short-range impurities • Inoue et al (2003) • Sheng et al, cond-mat/0504218 • Dimitrova (2004)

• Nomura et al cond-mat/0506189

• Khaetskii (2004)

• Raimonde and Schwab (2004) σ SH = σ

clean SH

+σ

vertex SH

FG H

IJ K

e e = 0! = + − 8π 8π

• Perturbative calculations for other systems • If H(k)=H(-k), eg. Luttinger model then vertex correction is zero (Murakami, PRB 2004)

r r r r • For systems with H ( k ) = E 0 ( k ) + σ x d y ( k ) − σ y d x ( k )

r r If ∂E0 / ∂k ∝ d , then perfect cancelation (eg. Rashba) otherwise σ s remains finite. (quoted from Murakami' s talk)

Spin Hall effect is finite in general

Effect of disorder on the spin Hall effect in Rashba system (II) • sSH robust against weak disorder in finite systems • Nikolic et al, cond-mat/0408693 • Hankiewicz et al, PRB 2004 • Sheng et al, PRL 2005

Stronger SO coupling

Spin Hall effect observed (I) (Kato et al, Science 2004) • Local Kerr effect in strained n-type bulk InGaAs, 0.03% polarization

Mostly likely extrinsic.

Spin Hall effect observed (II) (Wunderlich et al, PRL 2005) • spin LED in GaAs 2D hole gas, 1% polarization

p

n

might be intrinsic? (Bernevig and Zhang, PRL July 2005)

Spin Hall effect observed (III) (Sih et al, cond-mat/0506704) • n-type GaAs [110] QW

Dresselhaus coupling (PRB 1955): III-V semiconductor with bulk inversion asymm (BIA) (BIA)

[001] r r QW,2 linear Ω( k ) ≈ k n ( − k x , k y )

r r r r H ( k ) = S ⋅ Ω( k ) r r Ω( k ) ≈ k x ( k y2 − k z2 ), k y ( k z2 − k x2 ), k z ( k x2 − k y2 )

d

[110] QW

r r Ω( k ) ≈ k n2 ( − k x / 2,− k x / 2)

Rashba Rashba and Dresselhaus, p2 α [001] quantum well: H= + σ x p y − σ y px * 2m h β + σ x px − σ y p y h

d d

Dresselhaus

[111] QW

r r Ω( k ) ≈ ( 2 / 3 ) k n2 ( k y , − k x )

(SIA)

i i

i

Effective magnetic field: BIA

SIA

BIA=SIA

BIA¹SIA

Ganichev and Prettl, cond-mat/0304266

σ zxy =

e N 8π

For 2D electron systems, with Rashba and Dresselhaus coupling, N=1 if Rashba > Dresselhuas N=-1 if Dresselhaus > Rashba (Shen, PRB 2004) For 2D hole system with (cubic) Rashba, N=9 (Schliemann and Loss, PRB 2005)

Rashba-Dresselhaus system in an in-plane magnetic field p2 α γ H= + σ p − σ p + σ x px − σ y p y + β xσ x + β yσ y x y y x * 2m h h

d

i d

Eigen-energies:

r r Eλ ( k ) = E0 ( k ) + λ

d

γk x + αk y + β x

i

i d 2

i

2

+ αk x + γk y − β y ,

λ=±

Distorted Fermi surfaces (generic cases): (c) (b) (a)

FG H

r γβ x + αβ y αβ x + γβ y Point of degeneracy k 0 = ,− 2 α2 −γ 2 α −γ 2

Parameters: α ≈ 1 eV ⋅ A (tunable by gate voltage)

γ of the same order

b

g

β = g * /2 µ B B, µ B ≈ 0.06 meV / T k F = 2πn ≈ 102 / A for n ≈ 1011 / cm2

IJ K

Effect of in-plane magnetic field on spin Hall conductivity σ ηµν

B

Kubo formula

r r r f kr ,λ − f kr ,λ ' r 1 η k , λ jµ k , λ ' k , λ ' jν k , λ , = r ∑ 2 ih k ,λ ,λ ' ω λλ ' k

Jy

b λ ≠ λ 'g

Ex

jµη =

di

h vµσ η + σ η v µ ; 4

d

For g = 0 (pure Rashba)

i

E+, min

jν = − evν

E(k0)

E(k0) E-, min

E+, min E-, min

r σ ( B ) could be changed by 100% simply z xy

by rotating the magnetic field

Spin Hall conductivity (electron density fixed)

y σ xxy = σ xy =0

Boundary of plateau E(k0) = m β + 4αγβ x β y + β 2 x

2 y

α c =ζ

2

−γ

(1) (3)

h

2 2

(2)

α2 +γ 2

(2)

M.C. Chang, PRB 2005 Acknowledgement: M.F. Yang

Existence of charge Hall effect?

FG H

Thouless formula (PRL 1982)

σ

λ xy

e2 = h

IJ K

r 1 1 , k ,+ = iθ 2 −ie

r Ω λ ( k ), r∑

tan θ =

k filled

F GH

I JK

iθ r 1 −ie , k ,− = 2 1

γk x + αk y + β x αk x + γk y − β y

Berry curvature r Ωλ (k ) = i ∑

λ '≠ λ

r r r r r r r r k , λ vx k , λ ' k , λ ' v y k , λ − k , λ v y k , λ ' k , λ ' vx k , λ r =0 2 k ω λλ ( ) '

R − λπ for α r r r ∂ | for α Γ = z dk ⋅ k , λ i r k , λ = S0 ∂k |T+λπ for α r r r ⇒ Ω ( k ) = − sgncα − γ hλπδ d k − k i

Berry phase λ

λ

2

2

>γ 2

2

=γ2

2