Interplay of intrinsic and extrinsic spin-orbit coupling in atwo-dimensional electron gas

Advanced Workshop: Spin and Charge Properties of Low DimensionalSystems

Roberto Raimondi

Dipartimento di Fisica, Università degli Studi Roma Trehttp://www.fis.uniroma3.it/raimondi

June 29-July 4, 2009, Sibiu, Romania

In collaboration with Peter Schwab (Augsburg University)arXiv:0905.1269

http://www.fis.uniroma3.it/raimondi

Motivations and background

Why spin-orbit (SO) is attractive: spin manipulation by electric fieldsDescribe spin-charge phenomena in diffusive systems

spin Hall effect (SHE)Dyakonov and Perel, JETP Lett. 13, 467 (1971); Hirsch, PRL 83, 1834 (1999); Zhang, PRL 85, 393 (2000)

Murakami et al. Science 301, 1348 (2003); Sinova et al. PRL 92, 126603 (2004)

current-induced spin polarization (CISP)spin injection Hall effect (SIHE) (cf. Yungwirth’s talk)

Most theoretical activity focused on either intrinsic SO or extrinsicInoue et al., PRB 70, 041303 (2004); Mishchenko et al., PRL 93, 226602 (2004); Raimondi and Schwab, PRB 71, 033311(2005)

Chalaev and Loss, PRB 71, 245318 (2005); Dimitrova, PRB 71, 245327 (2005); Khaetskii, PRL 96, 056602 (2006)

Engel et al., PRL 95,166005 (2005); Tse and Das Sarma, PRL 96, 056601 (2006)

Interplay of extrinsic and intrinsic SO coupling important for experimentsSih et at., Nature Phys. 1, 31 (2005) SHE and CISPWunderlich et al., arXiv:0811.3486 SIHE

Contrasting views in the present status of theoryTse and Das Sarma, PRB 74, 245309 (2006)

Huang and Hu, PRB 73, 235314 (2006)

Hankiewicz and Vignale, PRL 100, 026602 (2008)

Outline

Overview of intrinsic and extrinsic SO mechanisms

Symmetry and spin continuity equation

Microscopic formulation

User-friendly spin diffusion equation

Applications to SHE and CISP

Conclusions

The model: 2DEG with SO coupling

Interplay of disorder, intrinsic and extrinsic SO

H =p2

2m− σ ×

„αêz +

λ204∂xV (x)

«· p + V (x)

δpF = 2mα

LSO = !mα L2SO = DτDP

Dyakonov-Perel spin relaxation

xy

z

E

SHE

Elliott-Yafet

spin relaxation

τ−1s ∼ |B|2 ∝ λ40

l = vF τ

mean free path

∆x ∼ λ20s×∆kSide-jump

skew-scattering

S = A + k× k′ · σB

∼ R(AB∗) ∝ λ20

SU(2) formulation

Spin-dependent vector potential

à = m σ ׄαêz +

λ204∂xV (x)

«≡ Ã = 1

2Ãaσa

Jin, Li, Zhang, J. Phys. A 39, 7115 (2006).

Tokatly, PRL 101, 106601 (2008).

Spin density along the a-axis

Spin current

Covariant derivative

sa = 12 Trσa

ja = 14 Trnσa, p−Ãm

o∂̃x · ja = ∂x · ja + εabcÃb · jc

Spin continuity equation

∂tsa + ∂̃x · ja = 0

This operator equation cannot be applied directlyto disorder averaged quantities because

Ãb · jc 6= Ãb·jc

The strategy to deal with disorder in SO coupling

space-independent vector potential

A = αm σ × êz

spin-dependent disorder potential

U(x) = V (x)−λ204

σ × ∂xV (x) · p

normal current

ja0 =14

Trnσa,

p− Am

oanomalous current

jaav =14

Trnσa,

Ã− Am

o

Spin continuity equation with disorder-dependent extra terms

∂tsa + ∂̃x · ja0 =?

What is the form of the RHS?We expect

spin relaxation

anomalous current

The technical tool: the Keldysh Green function

Equation of motion (EOM) Raimondi et al., 74, 035340 (2006)

∂tǦ + ∂̃x ·12

np− Am

, Ǧo

= −ihΣ̌, Ǧ

iTechnical Details

Include electric field

∂̃x(·) = (∂x − eE∂�)(·)− i[A, (·)]

Wigner representation

Ǧ(�,p, x, t) =Z ∞−∞

dηei�ηZ

dre−ip·rǦ(x +r2, t +

η

2; x−

r2, t −

η

2)

Spin and Keldysh matrix structure

Ǧ =„

GR G0 GA

«G = G0σ0 + Gaσa, a = x , y , z G< =

12

(G − GR + GA)

Physical observables

sa(x, t) = − i2R d�

2π

Pp Tr(σ

aG

The self-energy

Standard white-noise disorder model

V (x1)V (x2) = niv2o δ(x1 − x2)

Scattering time Side-jump Spin relaxation

∼ 1τ ∼1τs

skew-scattering

First correction to Born

scattering amplitude

Goal: evaluate the self-energy in Wigner coordinates

Σ̌(p, (x1 + x2)/2) =Z

d(x1 − x2)e−ip·(x1−x2)Σ̌(x1, x2)

An important example

The scattering time contribution

Σ̌0(x1, x2) = V (x1)Ǧ(x1, x2)V (x2)= v2o niδ(x1 − x2)Ǧ(x1, x1)

By Fourier transform

Σ̌0(p, x) = v2o niX

p′Ǧ(p′, x)

= v2o niN0Z

dp̂′Z

d ξǦ(p′, x)

=i

2τ(−i)π〈Z

d ξǦ(p′, x)〉

The microscopic expression of the elastic scattering time

τ−1 = 2πN0niv2o

The quasiclassical approximation

The quasiclassical Green function describes length scales L� λF

ǧ(�, p̂, x, t) =iπ

Zd ξǦ(�,p, x, t) ξ =

p2

2m− µ

The Eilenberger EOM for λ0 = 0

∂t ǧ + ∂̃x ·12

np− Am

, ǧo

= − 12τ

[〈ǧ〉, ǧ]

How to go

1 Keldysh ”12” component⇒ kinetic equation2 Project on σ0 ⇒ charge equation ρ = eN02

Rd�〈g0〉

3 Project on σa ⇒ spin equation sa = −N04R

d�〈ga〉

The kinetic equation (including λ0)

∂tg + ∂̃x ·12

np− Am

, go

=1τ

„12

n1 +

A · p̂2pF

, 〈g〉o− g

«+

λ20pF8τ

εabc b∂an(〈p̂cg〉 − p̂c〈g〉), σbo+

(λ0pF )2

8τ2πN0voεabc p̂a

n〈p̂bg〉, σc

o− 1

τs

12`〈g〉 − σz〈g〉σz

´

momentumrelaxation and D-Pspin relaxation

side-jump

skew-scattering

E-Y spin relaxation

A first consequence: the anomalous current

jaav,i = εiajN04

Zd �λ20pF4τ〈p̂jg0〉 = εiaj

12

eλ204

nEj ≡ εiaj12σsHSJ Ej

anomalous velocity contribution as in AHE(Nozieres and Lewiner, J. Phys. (Paris) 34, 901 (1973))

Similar contribution in SHE in the absence of Rashba(Engel, Halperin, Rashba, PRL 95, 166605 (2005); Tse and Das Sarma, PRL 96, 056601 (2006))

This term is unaffected by the presence of the Rashba term

More consequences

The second bit of the side-jump

Notice the covariant derivative acting on the second term of the side-jumpcontribution b∂a〈g0〉 ≡ (∂a − eEa∂�)〈g0〉

In the absence of Rashba this leads to a contribution identical to the previous one

With Rashba this term is modified

The skew scattering

In the absence of Rashba

σsHSS =14

(pF l)(2πN0v0)σsHSJ

In the presence of Rashba this term is modified

E-Y spin-relaxation time

This term was not considered in previous analyses

τ−1s = τ−1(λ0pF/2))4

User friendly spin-density diffusion equation

Physical assumptions

τ � T vF τ � L vF τ � Lso ⇒ αpF τ � 1

Spin current and continuity equation

jai = µsaEi − D∂̃isa + (σsHSJ + σsHSS)εiabEb +

e8π

(τD)εikjBak Ej

∂tsa + ∂̃x · ja −12εabcAb · jcSJ +

1τs

sa = 0

Drift-Diffusion terms and extrinsic spin-charge coupling Dyakonov and Perel, Physics Letters A35, 459 (1971)

Covariant derivative in the diffusion term Kalevich, Korenev, Merkulov, Solid State Commun. 73, 559(1994); PRB 74, 041308 (2006)

New source of spin current in the last term due to SU(2) magnetic field

Ba = ∂̃x × Aa ⇒ σsHint =e

8πτDBzz

The SHE

Main formula

σsH =1/τs

1/τs + 1/τDP

„σsHint + σ

sHSS +

12σsHSJ

«+

12σsHSJ

1 Is analytic in both limits λ0 → 0 and α→ 0Inoue et al. PRB 70, 041308 (2004); Raimondi and Schwab, 71, 033311 (2005)

Mishchenko et al. PRL 93, 226602 (2004)

Engel, Halperin, Rashba, PRL 95, 166605 (2005); Tse and Das Sarma, PRL 96, 056601 (2006)

2 σsHint is the diffusive limit of the bubble contribution of the diagrammaticlanguage

3 The ratio τs/τDP acts as a tuner of the spin Hall effect4 Since 1/τs ∝ λ40 and 1/τDP ∝ α2, it would be unphysical to take the limitα→ 0, with 1/τs = 0

5 In appropriate limits agrees with previous analysesTse and Das Sarma, PRB 74, 245309 (2006)

Huang and Hu, PRB 73, 235314 (2006)

Hankiewicz and Vignale, PRL 100, 026602 (2008)

Numbers

GaAs: input parameters Sih et al., Nature Phys. 1, 31 (2005)

ns = 1012cm−2, µ = 103cm2Vs, λ0 = 4.7× 10−8cm, α = 10−12eVm, N0vo = − 12

Estimates

eσsHSJ = 1.3× 10−7Ω−1, eσsHSS = −4.3× 10−7Ω−1, eσsHint = 8.2× 10−9Ω−1

Spin relaxation times

τs = 8.4× 103ps� τDP = 90ps

By having α smaller one may reach theinteresting region τs ≈ τDP

0.00 0.01 0.02 0.03 0.04 0.05

!0.2

0.0

0.2

0.4

0.6

0.8

1.0

2αpF τ

α = 10−12 eVmσsH(α)σsH(0)

CISP: in-plane vs out-of-plane

The classical Edelstein result (Solid State Commun. 73, 233 (1990))

sy = −mµαN0E , λ0 = 0

The effect of extrinsic SO

sE = −2mα

1/τs + 1/τDP

„σsHint + σ

sHSS +

12σsHSJ

«E .

Origin of out-of-plane CISP with in-plane magnetic field

∂tsy = −„

1τDP

+1τs

«(sy − sE ) + bx sz

∂tsz = −2τDP

sz − bx sy + mαµN0bx E

With λ0 = 0 no out-of-plane CISP unless with angle-dependent scattering (Milletarı̀ etal. EPL 82, 67005 (2008)) or energy-dependent scattering and non parabolic dispersion(Engel et al. PRL 98, 036602 (2007))

In general sz ∼ bx (mµαN0E − sE )

Out-of-plane CISP without magnetic field

Linear Dresselhaus SO for (110) QW (Hankiewicz, Vignale, Flatté, PRL 97, 266601 (2006))

AD = −mβσz êz BaD = ∂̃x × AaD = 0

For electric-field induced spin polarization both Rashba and Dresselhaus arenecessary!

A→ A + AD

Electric-field orientation dependence

sz = Ey 2mβσsHSS +

12σ

sHSJ +

τe4π (τ

−1s + 4m2D(2α2 + β2))

4m2D(2α2 + β2) + 2τ−1s

interplay of Rashba and linear Dresselhaus provides a mechanism forout-of-plane CISP

CISP sz ∝ Ey shows a strong dependence on the orientation of the appliedelectric field

Comparison with measured CISP

x

y

z

2DEG

[1, 1, 0]

[1, 1, 2

]

[1, 1, 0

]

[0, 0, 1]

growth direction

Electric field

[1, 1, 1

] (110) GaAs quantum well (Sih et al., Nature Phys. 1, 31 (2005))

Electric field along various crystal directions

Out-of-plane CISP measured via Kerrspectroscopy

Experimental findings1 For E‖ [0, 0, 1] CISP only at edges and no in the bulk⇒ SHE2 CISP in the bulk without magnetic field for E‖

ˆ1, 1, 0

˜,ˆ1, 1, 1

˜andˆ

1, 1, 2˜

in agreement with theory3 Surprisingly CISP has the same sign for E‖

ˆ1, 1, 0

˜and

ˆ1, 1, 2

˜, which

is opposite forˆ1, 1, 1

˜, not explained by the theory

4 cubic Dresselhaus ∼ ky (k2y /2− k2x ) and non-linear effects may beimportant

Conclusions

Theory for both, intrinsic and extrinsic, SO mechanisms

User friendly spin diffusion equation

SHE: non-analytic puzzle settled

SHE: intrinsic mechanism as tuner of the extrinsic-driven effect

CISP: due to interplay of intrinsic and extrinsic, even without magneticfield

Why the electric field does not contribute to the side-jump?

Consider the Hamiltonian

H =k2

2m− eE · x + λeE · k× s

and the associated velocity operator and spin current

v =km

+ λes× E ja = 12

nv, sa

oTo first order in E, the first SJ term is due to the anomalous velocity

δja = TrX

k

f (k)λe12

ns× E, sa

o=

14λen êa × E

The second SJ is due to a change of the distribution function

ja0 = TrX

k

km

sa∂f∂�λeE · k× s = −1

4λen êa × E

The cancellation is similar to that for the diamagnetic term and only exists atzero frequency

What is the connections with diagrams? SJ terms

Γx = −α ττsσy

1τtτDP

+ ττs

Γyz =vF4

2αpF τ1 + (2αpF τ)2

„1− τ

τs

«σy

1τtτDP

+ ττs

For τs →∞, Γx = 0 so that C + D = 0. Also Γyz = (vF/4αpF τ)σy whichleads to the cancellation E + F + G + H = 0. Only A + B survives which ishalf the SJ and is not changed by Rashba.