Effects of Impurity Scattering on theSpin Hall Conductivity in 2D and 3D

Systems

Based on work done withEugene Mishchenko (now at Univ of Utah)

Andrey Shytov (now at BNL) Phys. Rev. Lett., 2004, and unpublished

andHans-Andreas Engel and E. I. Rashba

Cond-mat/0505535

Topics

Theory of Spin Hall Conductivity in n-Doped 3D GaAs

(Engel, Rashba and Halperin)

Small-Angle Impurity Scattering and the Spin Hall Conductivityin 2D Systems, including 2D Hole Systems in GaAs

(Mishchenko, Shytov and Halperin)

Effective Hamiltonian with impurityscattering

(Non-interacting electrons, 2D or 3D)

H = (p2/2m) - b(p) ⋅ s + V(r) + 2λ s⋅(p×∇V) ,

where λ is another spin-orbit coupling constant, which dependson the material. For n-type GaAs, λ = 0.053nm2. This couplingdoes not require broken inversion symmetry in the host material.

Spin-orbit effects due to λ coupling will be called extrinsic.

Spin-orbit effects due to b term will be called intrinsic.

Definition of spin current jmn(r) = (1/4) ∑i { {vi

n , sim} , δ(r-ri) }

In the absence of spin orbit coupling, the velocity operator isgiven by v = p / m .When spin-orbit is important, the velocityoperator is more complicated, and involves the spin. If anelectric field Ex in the x-direction gives rise to a spin-current jzy ,we write: jz

y = - σsH Ex , where σsH is the “spin-Hallconductivity.”.

We use units where h-bar = 1, and sm = ± (1/2) , and e is thecharge of the carrier.

The spin components sx, sy, and sz obey angular momentumcommutation rules, but they are not really spins. They areactually a mixture of spin and translational degrees of freedom.

Experiments which observe a spin Halleffect

Actually, experiments observe a spin polarization that isattributed to spin-Hall conductance.

Kato et al:, Science (2004): 3D n-type GaAs

Wunderlich et al. PRL (2005), 2D hole gas

Sih et al, (2005) 2D electron gas on (110) surface

Interpretation of Spin Hall Obsrvations

Kato et al:, Science (2004): 3D n-type GaAs (in dirty limit)

We interpret as extrinsic effect, (in agreement with Katoet al.)

Wunderlich et al. PRL (2005), 2D hole gas (clean limit)

May be an intrinsic effect

Sih et al, (2005) 2D electron gas on (110) surface (in dirtylimit) Interpret as extrinsic.

3D n-Type GaAs

We have calculated the effects of extrinsic scattering for the3D electron system studied by Kato et al., and find reasonableagreement with the observations The intrinsic effect, due tothe k3 Dresselhaus coupling, gives a contribution which is anorder of magnitude smaller, because of the large scattering rate,(as found by Bernevig and Zhang) .

In dirty limit, impurity scattering always reduces the intrinsicspin Hall effect, typically by a factor ≈ (Δτ)2 .

Three-dimensional samples are always in the dirty limit.

Skew-scattering and side-jumpcontributions

To compare with experiments of Kato et al., in 3D n-type GaAs, werepresent ionized donors by attractive screened Coulomb potentials, withdensity of impurities = density of electrons. Calculations are first order inspin-orbit coupling λ; assume Boltzmann equation. Use previous resultsfrom analyses of the Anomalous Hall Effect.

Extrinsic scattering gives two contributions to the spin-Hallconductance: “skew scattering” (Mott scattering) and the“side-jump” contribution

Results for extrinsic scattering in 3D n-typeGaAs

(H. Engel, E. I. Rashba, and B. I. Halperin)

Skew scattering contribution to spin Hall conductance= 1.7 Ω -1 m-1 × (-2/e).

Side-jump contribution to spin Hall conductance = - 0.8 Ω -1 m-1 × (-2/e)

Total contribution to spin Hall conductance = 0.9 Ω -1 m-1 × (-2/e)

Experimental results of Kato et al. for spin Hall conductance= - 0.5 Ω -1 m-1 × (-2/e) [question about sign]

Remarks on skew scattering

Skew scattering does not occur in the Born approximation,need terms at least of order V3 in the scattering. We obtainskew scattering from exact relativistic solution of scattering bya Coulomb potential. Side jump contribution was adaptedfrom Nozieres and Lewiner calculation for AHE.

Side jump contribution to σsH is of order eλkF2, indep. of τ

Skew scattering contribution is of order eγEFτ , where

γ∝ λkF2 (Vmax / EF) . ==> σsH -> ∞ for τ -> ∞ .

In a system without inversion symmetry, divergence is cut offat τ = b-1, so σsH -> eγEF / b ; could be larger or smalller thanintrinsic contribution.

Effects of impurities on intrinsic spin Hallconductance in 2D case

H = (p2/2m) - b(p) ⋅ s + V(r) + 2λ s⋅(p×∇V)

Effects of impurities on intrinsic spin Hallconductance in 2D case

H = εp - b(p) ⋅ s + V(r)

For simplicity, we shall assume εp is isotropic, but notnecessarily quadratic. Results will depend on parameter α:

1 +α ≡ d log v / d log p .

( α = 0 for parabolic dispersion)

General circularly symmetric 2D modelAssume bz=0,

bx+iby = b(|p|) eNiθ .Results also depend on parameter

N* = d log b / d log p .For 2D electron gas with pure Rashba coupling: N=N*=1.

For 2D electron gas on (001) surface with pure linearDresselhaus coupling N*=1, N= -1 .

2D hole gas in a III-V semiconductor

(Simplified model, Rashba coupling only, neglects tetragonalanisotropy )

H = εp - b(p) ⋅ s ,

bx+iby = β (px+ipy)3 = β |p|3 e3iθ

Angular dependence (N=3) reflects fact that holes are a two-dimensional subspace of a four-dimensional space of states thattransform under rotations like j=3/2. For small holeconcentrations, band is parabolic and constant β is independentof |p|.

Hence N = N* = 3, α = 0

Results for circularly symmetric 2Dmodel (bx+iby) = b(|p|) eNiθ

For N = ± 1 (i e., 2D electron gas with pure Rashbaor linear Dresselhaus coupling):

Spin Hall conductance = 0,

for arbitrary angular-dependence of the impurityscattering, arbitrary values of N* and α

Results for circularly symmetric 2Dmodel (bx+iby) = b(|p|) eNiθ

For |N| > 1 , in the clean limit bτ >> 1 :

For isotropic impurity scattering: Find “universalvalue”: σsH = N e / 8π

For angle-dependent scattering, results depend onenergy dependence of b and on non-parabolicity ofthe energy dispersion, as well as on the form of thescattering. In the limit of small angle scattering wefind:

Result for 2D Hole System

Model with N = N* = 3 , α = 0 , in the clean limit :

In the dirty limit, spin Hall conductance is always stronglysuppressed by a factor (bτ)2<<1.

More Results

.(From Mishchenko, Shytov & Halperin, PRL 2004)

“Universal” spin Hall conductance does occur for ac field, ifτ -1 << ω << Δ .

Although there is no bulk dc spin Hall current in the 2Delectron system with pure Rashba or linear Dresselhauscoupling, dc spin Hall currents do occur near contacts in afinite sample, could lead to z-polarized spin accumulation incorners of the sample.

Bulk spin-polarization does occur, with polarization in x-yplane

Spin-currents and polarization in afinite sample

Notes. Spin-orbit length LS=vF/2Δ is the distance for spin to rotate by oneradian. Ex is the voltage gradient, including both chemical potential andelectrostatic contributions.

Effects at edges

More detailed analyses by various authors have shown thatout of plane polarization can also take place at lateral edgesif spin flip processes occur there or if carriers can crossboundary into a different conductor.

Rashba cond-mat/0507007

Adagideli and Bauer, cond-mat/0506531

Nikolic et al.

Boltzmann Equation approach

Let n(θ)dθ be the excess density of electrons whosemomentum direction makes an angle θ with the x-axis, ina wedge of size dθ, after subtracting the equilibriumdensity. Let the vector function Φ(θ)dθ be the excessspin-density of electrons in this wedge

The equations of motion for these densities havecontributions from the electric field, from impurityscattering, and from precession about the effectivemagnetic field b(p).

:

Impurity scattering contributionImpurity scattering gives contributions to the equations ofmotion of the following form:

∂n(θ)/∂t = ∫dθ′ K(θ-θ′) [n(θ′) - n (θ)]

where, in the Born approx., K(θ-θ′)=<|V(q)|2> m/(2π).

∂Φµ(θ)/∂t = ∫dθ′ K(θ-θ′) [Φµ(θ′) - Φµ(θ)] +

∫dθ′ [Mµ(θ,θ′) n(θ′) - Mµ(θ′,θ ) n(θ)] ,

for µ=x,y,z , where….

Mµ(θ,θ′) = (vF/4pF) K(θ-θ′) ∂/∂ε [p bµ(θ) / v]

+ (1/4pFvF) [bµ(θ)+bµ(θ′)] tan [(θ-θ′)/2] ∂K(θ-θ′)/∂θ

The first term in the kernel Mµ reflects the difference in densityof states of the two spin-orientations in the scattered direction θ.

The second term reflects the difference in the scattering matrixelement because the scattering wavevector q is slightly differentfor the two spin states.

(We have neglected terms higher order in b/EF.)

Two Fermi Surfaces

Green region is doubly occupied.Arrows show spin directions for singly-occupied states.

Electric field and precessioncontributions

Electric field terms arise from the momentum drift dp/dt = eE..This leads to terms in linearized equations of motion that aredetermined by the equilibrium distributions, and are given by

∂n(θ)/∂t = (ekF/2π2) E cos θ

∂Φµ(θ)/∂t = (e/8π2vF) E ∂[bµ(θ) sin θ ] /∂θ .

The precession contribution is given by

∂Φ(θ)/∂t = b(θ) × Φ(θ)

Solution of the equations of motion

Equations can be attacked by Fourier transform in θ. Thereare couplings between different Fourier components, and ingeneral there are an infinite number of non-zero terms.

If bx+iby = b(|p|) eNiθ , bz = 0 ,then the only non-zero Fourier components of Φµ are:

m=± 1, for Φz , m=N± 1, for Φx+iΦy .

Summary of ResultsFor 3D electrons in GaAs, estimates of extrinsic effect seemconsistent with experiments of Kato et al. Intrinsic effect notimportant.

For 2D electrons in GaAs: with pure Rashba or linearDresselhaus coupling, intrinsic spin Hall effect is absent evenin clean limit. (The cubic Dresselhaus term could lead to anon-zero result)

For 2D holes in GaAs, intrinsic effect should be present.May be responsible for observations of Wunderlich et al.However, possible boundary effects and/or extrinsicscattering contribution should also be examined.