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
Top Related