Spin transport in spin-orbit Spin transport in spin-orbit coupledcoupled bandsbands

Intrinsic contributionsIntrinsic contributions

Qian Niu Qian Niu University of Texas at AustinUniversity of Texas at Austin

AcknowledgementsAcknowledgementsCollaborators: Collaborators:

D. Culcer, A. Dudarev,Yugui Yao, N. Sinitsyn, D. Culcer, A. Dudarev,Yugui Yao, N. Sinitsyn, J. Sinova, T. Jungwirth,J. Sinova, T. Jungwirth, A. H. MacDonald A. H. MacDonald

References by our team:

Culcer et al (PRL,93,046602,2004) Sinova et al (PRL,92,126603,2004)Dudarev et al (PRL92,153005,2004)

Other references:

Murakami et al Science 301, 1248 (2003) Murakami et al cond-mat/0310005 Hu et al cond-mat/0310093 Schliemann et al cond-mat/0310108 Shen cond-mat/0310368 Rashba cond-mat/0311110 Bernevig et al cond-mat/0311024 Inoue et al cond-mat/0402442 Xiong et al cond-mat/0403083

OutlineOutline

MotivationMotivationBoltzmann-wavepacket transportBoltzmann-wavepacket transport

Spin dipole, torque dipoleSpin dipole, torque dipoleEquation of continuity: current & sourceEquation of continuity: current & sourceIntrinsic & extrinsic partsIntrinsic & extrinsic partsSpin current: several attributionsSpin current: several attributions

Spin Hall EffectSpin Hall EffectRashba model, Rashba model, Four band modelFour band modelHgSe, HgTeHgSe, HgTeGaAs, Si, GeGaAs, Si, Ge

Spin accumulation: the role of torque dipole Spin accumulation: the role of torque dipole

ConclusionsConclusions

Magneto-ElectronicsMagneto-Electronics

Magnetic tunneling junction (MTJ) or “spin valve” Nonvolatile MRAM: “Instant on ”

S. Parkin (1990)

Compatibility with Si and GaAs next phase: semiconductor spintronics

GMR read-out heads in hard drives

1st generation spintronic devices based on ferromagnetic metals – already in commercial use

A brighter future with A brighter future with semiconductor spintronicssemiconductor spintronics

•Can do what metals do:GMR, spin transfer, ..., using ferromagnetic semiconductors

•Readily integrated with semiconductor devices: possible way around impedance mismatch in spin injection.

• Tunable: transport, magnetic and optical properties can be readily controlled by doping, gating, and pumping.

• Spin-orbit: strong in semiconductors, may lead to novel effects such as electric generation and manipulation of spins

Boltzmann-wavepacket transportBoltzmann-wavepacket transport

Spin-orbit built into the bandstructure:Spin-orbit built into the bandstructure:– not a perturbation.not a perturbation.

Carrier of charge and spin:Carrier of charge and spin:– represented by wave packets.represented by wave packets.

Effects of external fields: Effects of external fields: – mixing of bands and drifting.mixing of bands and drifting.

Impurity effects: Impurity effects: – scattering and relaxation.scattering and relaxation.

' '

'

'

/

n nn

nn

nnn

Eekuiuuuu

Effect of external fieldsEffect of external fields

Mixing Mixing

DriftingDrifting

nc

c

c

cc

c

kk

r

Brer

k

k

uH

k

uieM

k

u

k

ui

BMre

nn

nn

nnn

ncn

)(

)(

where

Observable and wavepacketObservable and wavepacket

ChargeCharge SpinSpin

(rc, kc) (rc, kc) (rs, ks)

),,( tkrf cc

ccccc rrstkrfkdrdtrS )ˆ(ˆ),,(),( 33

Macroscopic densitiesMacroscopic densities

Spin densitySpin density

Torque densityTorque density

Spin current densitySpin current density

spfkdsfkdtrS 33 ˆ),(

srrp s ˆ)ˆ(

pfkdfkdtrT 33 ˆ),(

ˆ)ˆ( rrp

srrrfkdsrfkdtrJ s ˆˆ)ˆ(ˆˆ),( 33

]ˆ,ˆ[ˆ sHi

Equation of continuityEquation of continuity

sdt

dfkdTJ

t

S s ˆ3

intrinsic extrinsic

Torque density:

sk

tkrfkdEe

ptkrfkdtkrfkdtrT

c

cc

ˆ),,(

),,(ˆ),,(),(

3

33

Electric field induced sourceElectric field induced source

In the Rashba model:In the Rashba model:

Generally nonzero in inversion asymmetric crystalsGenerally nonzero in inversion asymmetric crystals

zEkk

eks

kEe y

y ˆ)(ˆ3

L. S. Levitov et al., Sov. Phys. JETP 61, 133 (1985)

P. R. Hammar and M. Johnson, Phys. Rev. Lett. 88, 066806 (2002)

Y. Kato et al, Cond-mat/0403407 (2004).

Spin current: contributionsSpin current: contributions

Homogeneous systems Homogeneous systems → ignore gradient terms. → ignore gradient terms.

Spin current can be decomposed into:Spin current can be decomposed into:

Spin Hall agrees with the Kubo formula.Spin Hall agrees with the Kubo formula.

]ˆ[),( 3 pdt

pdsrfkdtrJ

s

cs

Convective termd/dt (Spin dipole)

Torque dipole

Spin Current: intrinsic & extrinsicSpin Current: intrinsic & extrinsic

Distribution: equilibrium part + shiftDistribution: equilibrium part + shift

Extrinsic spin currentExtrinsic spin current

Intrinsic spin currentIntrinsic spin current

fff 0

0

3 ˆ1

sk

fkdJ exts

]ˆˆ1

[010

3int

p

dt

pdsE

qs

kfkdJ

ss

Spin Current: Rashba modelSpin Current: Rashba model

Rashba Hamiltonian:Rashba Hamiltonian:

Spin current per carrier:Spin current per carrier:

Spin-Hall conductivity:Spin-Hall conductivity:

zEpm

ej s ˆ

8

2

8

esH

zpm

pH ˆ

2

2

Optical LatticeOptical Lattice

FrBrr )()()( VV

6Li

(PRL 70, 2249 (1993)).

E

q

2/1F JIF

iiiiii

ii

zVVzVB

VV

)cosˆsinsinˆ()(

cos)(

321

0

rkkrkkrkr

rkr

Bands & Spin HallBands & Spin Hall

like Rashba coupling!

Spin Current: four-band modelSpin Current: four-band model

Luttinger Hamiltonian: Luttinger Hamiltonian:

The intrinsic spin current:The intrinsic spin current:

Spin-Hall conductivity:Spin-Hall conductivity:

hhs

lhs

lhh

hhs

jj

zEekkkm

j

ˆ]6

1

12

1

4

1

)(6[

222

2

h

l

hsH

mm

ke

13 2

Energy

k

Heavy holes

Light holes

])(2)2

5[(

22

22

21

2

Jkkm

H

Spin accumulationSpin accumulation

areacellspinsEJSdx sHs /10 4

equation of continuity:equation of continuity:

spin accumulation: spin accumulation: x

sample

ls=spin diffusion length

Js

S

S

Jt

S s

30 ps 20000 V/cm

2.5x1017 spins x [v]/[E]

Zero-gapZero-gap semiconductorssemiconductors

HgSeHgSe

HgTe HgTe

= 0.0023 e / a

= 0.0031 e / a

InsulatorsInsulators

1% strained HgSe and HgTe1% strained HgSe and HgTeenergy gap: ~ 40 meVenergy gap: ~ 40 meV

Spin Hall conductivity: unchanged.Spin Hall conductivity: unchanged.

GaAs GaAs

GeGe

SiSi

= 0.001 e / a

= 0.0015 e / a

= 0.00017 e / a

Symmetry:

js = E

Covariant under time reversal and spatial inversion

The torque dipole and spin accumulationThe torque dipole and spin accumulation

Source term (no bulk spin generation):Source term (no bulk spin generation):

The equation of continuity: The equation of continuity:

Justifies the definition of the spin transport current:Justifies the definition of the spin transport current:

In insulators, JIn insulators, Jstst believed to be zero (in progress). believed to be zero (in progress). ]ˆ[3

dt

pdsrfkdJ

s

c

st

S

P

S

PJt

S s

)(

Spin transport currentSpin transport current

Without the torque dipole, the spin-Hall Without the torque dipole, the spin-Hall conductivity for the four band model is:conductivity for the four band model is:

Similar magnitude to the original, but differs Similar magnitude to the original, but differs by a sign.by a sign.

ml

m

kekk

e

h

llh

sH

13

)(6 22

ConclusionsConclusions

Boltzmann-wavepacket transport Boltzmann-wavepacket transport – Intuitive, rigorous, localIntuitive, rigorous, local

intrinsicintrinsic contributions to spin source & current contributions to spin source & current– Berry phase and much more.Berry phase and much more.

Intrinsic spin current: Intrinsic spin current: – convective termconvective term, a , a spin dipolespin dipole and a and a torque dipoletorque dipole..

Intrinsic spin Hall agrees with Kubo formula.Intrinsic spin Hall agrees with Kubo formula.– Rashba, four-band, zero-gap, insulatorsRashba, four-band, zero-gap, insulators

Spin Accumulation: Spin Accumulation: – cancellation of torque dipole termscancellation of torque dipole terms