Transcript Spin transport in spin

Spin transport in spin-orbit
coupled bands
Intrinsic contributions
Qian Niu
University of Texas at Austin
Acknowledgements
Collaborators:
D. Culcer, A. Dudarev,Yugui Yao, N. Sinitsyn,
J. Sinova, T. Jungwirth, 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)
Hu et al cond-mat/0310093
Shen cond-mat/0310368
Bernevig et al cond-mat/0311024
Xiong et al cond-mat/0403083
Murakami et al cond-mat/0310005
Schliemann et al cond-mat/0310108
Rashba cond-mat/0311110
Inoue et al cond-mat/0402442
Outline
Motivation
Boltzmann-wavepacket transport
Spin dipole, torque dipole
Equation of continuity: current & source
Intrinsic & extrinsic parts
Spin current: several attributions
Spin Hall Effect
Rashba model,
Four band model
HgSe, HgTe
GaAs, Si, Ge
Spin accumulation: the role of torque dipole
Conclusions
Magneto-Electronics
1st generation spintronic devices based on
ferromagnetic metals – already in commercial use
GMR  read-out heads in hard drives
Magnetic tunneling junction
(MTJ) or “spin valve” 
Nonvolatile MRAM:
“Instant on ”
S. Parkin (1990)
Compatibility with Si and GaAs
 next phase:
semiconductor spintronics
A brighter future with
semiconductor 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 transport
Spin-orbit built into the bandstructure:
– not a perturbation.
Carrier of charge and spin:
– represented by wave packets.
Effects of external fields:
– mixing of bands and drifting.
Impurity effects:
– scattering and relaxation.
Effect of external fields
Mixing
un  un   un '
n'
un '


iun / k  eE
 n   n'
Drifting

 

kc     erc  B
rc
   
rc    kc   n
kc
where



   n  e (rc )  M n  B

u
u
 n  i n  n
k
k

u
u
M n  ie n  ( n  H ) n
k
k
Observable and wavepacket
Charge
Spin
(rc, kc)
 
f (rc , kc , t )
(rc, kc)
(rs, ks)

 

3
3
S (r , t )   d rc d kc f (rc , kc , t ) sˆ (r  rˆ)
c
Macroscopic densities
Spin density

s
3
3
S (r , t )  d k f sˆ     d k f p
s

p  (rˆ  r ) sˆ
Torque density


3
3
T (r , t )   d k f ˆ     d k f p
i

ˆ  [ Hˆ , sˆ]
Spin current density


ˆ
p  (r  r )ˆ

 
3
3

J (r , t )   d k f rˆsˆ     d k f (rˆ  r )rˆsˆ
s
Equation of continuity
s
S
df
3
 J T  d k
sˆ
t
dt
intrinsic
extrinsic
Torque density:

 
  
3
3
T (r , t )   d k f (rc , k , t ) ˆ     d k f (rc , k , t ) p

 

3
 eE   d k f (rc , k , t )  sˆ
k
Electric field induced source
In the Rashba model:
 
 ek y  
 eE   sˆ y  3 (k  E )  zˆ
k
k
Generally nonzero in inversion asymmetric crystals
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: contributions
Homogeneous systems → ignore gradient terms.
Spin current can be decomposed into:
s


dp
s 
3
J (r , t )   d k f [rc sˆ 
p ]
dt
Convective term
Torque dipole
d/dt (Spin dipole)
Spin Hall agrees with the Kubo formula.
Spin Current: intrinsic & extrinsic
Distribution: equilibrium part + shift
f  f 0  f
Extrinsic spin current
J
s
ext
1 
 sˆ
  d k f
 k
3
0
Intrinsic spin current
J s int
1 
q  
3
 sˆ 1  E   sˆ
  d k f0 [
 k

s

dp

p ]
0
dt
Spin Current: Rashba model
Rashba Hamiltonian:
2
 
p
H
   p  zˆ
2m
Spin current per carrier:
s
e 2 
j  
E  zˆ
8mp
Spin-Hall conductivity:

sH
e

8
Optical Lattice
E
q
6Li
F  1/ 2
F  I  J 
V (r)  V (r)  B(r)  F
V (r )  V0 cosk ir
i
B(r )   (V1 zˆ sin k ir  V2k i sin k ir  V3 zˆ  k i cosk ir )
i
(PRL 70, 2249 (1993)).
Bands & Spin Hall
like Rashba coupling!
Spin Current: four-band model
Energy
Luttinger Hamiltonian:
  2
2
5
2
H 
[( 1   2 )k  2 2 (k  J ) ]
2m
2
k
The intrinsic spin current:
Heavy holes
2

1
1
1 
s
j hh  [
 2
 2 ]eE  zˆ
2
6mh ( h   l ) 4k 12k 6k
j s lh   j s hh
Light holes
Spin-Hall conductivity:

sH

kh
e
3 2
1
ml
mh
Spin accumulation
equation of continuity:
sample
s
S
S
 J  
t

Js
S
x
spin accumulation:
ls=spin diffusion length
 Sdx  J
s
30 ps
  E  10 spins / cell area
4
sH
20000 V/cm
2.5x1017 spins x [v]/[E]
Zero-gap semiconductors
HgSe
= 0.0031 e / a
HgTe
= 0.0023 e / a
Insulators
1% strained HgSe and HgTe
energy gap: ~ 40 meV
Spin Hall conductivity: unchanged.
GaAs
Ge
Si
= 0.001 e / a
Symmetry:
js =  E
= 0.0015 e / a
= 0.00017 e / a
Covariant under time reversal
and spatial inversion
The torque dipole and spin accumulation
Source term (no bulk spin generation):
 S
 P 

The equation of continuity:
 s 
S
S
   (J  P )  
t

Justifies the definition of the spin transport current:
s
 st

dp
3
J   d k f [ r sˆ c 
]
dt
In insulators, Jst believed to be zero (in progress).
Spin transport current
Without the torque dipole, the spin-Hall
conductivity for the four band model is:

sH

e
6
2
( k h  kl ) 
kl
e
3
2
mh
1
ml
Similar magnitude to the original, but differs
by a sign.
Conclusions
Boltzmann-wavepacket transport
– Intuitive, rigorous, local
intrinsic contributions to spin source & current
– Berry phase and much more.
Intrinsic spin current:
– convective term, a spin dipole and a torque dipole.
Intrinsic spin Hall agrees with Kubo formula.
– Rashba, four-band, zero-gap, insulators
Spin Accumulation:
– cancellation of torque dipole terms