Download Spin diffusion equation for nonuniform driving field

The spin-Hall effect and
inverse spin-Hall effect
in a diffusive system
L. Y. Wang1,
Collaborators: C. S. Chu2 and A. G. Mal’shukov3
1Department
of Physics, Fu-Jen Catholic University, New Taipei City, Taiwan
2Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwann
3Institute of Spectroscopy, Russian Academy of Science, Moscow, Russia
Highlight
Motivation and Highlight
 The Rashba-type spin-orbit interaction (RSOI) provides the needed
tunable capability for spin physics by pure electric means.
biased-gate
vg
InGaAs
2DEG
InAlAs
Nitta. et al., Phys.Rev.B 60, 7736(1999)
Highlight
Difficulty confronting RSOI:
 The effect of RSOI on spin accumulation Sz is quenched in
the diffusive regime (lso>>le).
lso is the spin relaxation length and le is the electron mean
free path.
Linear-k SOIs, Rashba or Dresselhaus, do not contribute to
spin accumulation when there are background scatterers.
 J.I. Inoue et al., Phys. Rev. B 70, 041303 (2004)
 E.I. Rashba, Phys. Rev. B 70, 201309 (2004)
 O. Chalaev et al., Phys. Rev. B 71, 245318 (2005)
 E.G. Mishchenko, et al., Phys. Rev. Lett. 93, 226602 (2004)
 A.A. Burkov, et al, Phys. Rev. B 70, 155308 (2004)
 O.V. Dimitrova, Phys. Rev. B 71, 245327 (2005)
 R. Raimondi et al., Phys. Rev. B 71, 033311 (2005)
Highlight
Efforts to recover the RSOI effects
In the mesoscopic ballistic regime:
 Coherence
quantum interference
accumulation
finite spin
 B. K. Nikolić et al., Phys. Rev. Lett. 95, 046601 (2005)
 J. Li and S. Q. Shen, Phys. Rev. B 76, 153302 (2007)
 P. G. Silvestrov et al., Phys. Rev. Lett. 102, 196802 (2009)
Highlight
Residual RSOI effects in the diffusive regime :
 Only along the sample-electrode interfaces was the spin current
found to be nonzero.
 This is understood from the way the spin current vanishes in the
bulk, when exact cancellation occurs between two terms, one
related to spin polarization and the other related to the driving field.
I yz  2 D

S z  2hkF vF  S y  2
y
N 0 e 2 hk2F
*
m
E
E. G. Mishchenko et al., PRL, 93, 226602 (2004)
Highlight
Residual RSOI effects in the diffusive regime :
 Only along the sample-electrode interfaces was the spin current
found to be nonzero.
 This is understood from the way the spin current vanishes in the
bulk, when exact cancellation occurs between two terms, one
related to spin polarization and the other related to the driving field.
 The exact cancellation no longer holds near the electrode interfaces
when the driving field has reached its bulk value but the spin
polarization has not.
E. G. Mishchenko et al., PRL, 93, 226602 (2004)
Highlight
This work:
It is important to see whether the RSOI alone can
still contribute to spin accumulation Sz in the
diffusive regime.
Question we ask in this work:
• Is it possible to recover the RSOI effect on the spin
accumulation Sz in the diffusive regime ?
• And in any location we specify ?
Highlight
Our finding:
The RSOI effect on spin accumulation can be
restored in a nonuniform driving field.
Spin accumulation around a void
Spin accumulation around a
void at an edge
S ztot (1/  m2 )
3
1
2.5
0.5
2
0
1.5
1
-0.5
0.5
0
-3
-1
-2
-1
0
1
2
3
Outline
1. Introduction: spin-Hall effect (SHE)
2. Spin diffusion equation for nonuniform driving field
3. Nonuniform driving field around a circular void in the
bulk.
4. Nonuniform driving field around a edge semicircular .
5. Inverse spin-Hall effect for Rashba spin-obit
interaction
Introduction: spin-Hall effect
Introduction
A driving field results in spin accumulation at the lateral edges
of the sample.
No magnetic field is needed but SOI is crucial.
spin-up
-e
E
z-polarized
spins
-e
spin-down
Physical origins of the SOI
(1) Extrinsic SOI: arises from SOI impurities.
(2) Intrinsic SOI: arises from the lack of structure inversion
asymmetry (SIA) or bulk inversion asymmetry (BIA).
Introduction
Experimental status: extrinsic SHE
The “Extrinsic SHE” arises from SOI impurities:
Experimental result in n-doped
semiconductors.
E
77  m
Y. K. Kato, R. C. Myers, A. C. Gossard,
and D. D. Awschalom, Science, 306, 1910
(2004).
Intrinsic spin-Hall effect in the ballistic regime
Rashba-type SOI
Green arrows: wavevector
Red arrows:
effective magnetic field
py
py
px
px
J. Sinova, et al PRL 92, 126603 (2004)
However, there is no z-polarized spin polarization induced in the
diffusion limit for Rashba SOI.
Introduction
Intrinsic spin-orbit coupling
p2
H
 h   Vimp.  r 
2m
Effective magnetic field:
1. hk (Rashba  SOI)  k  zˆ  k y xˆ  k x yˆ
2. hk (Dresselhaus  SOI)  k x  k 2y   2  xˆ  k y  k 2x   2  yˆ
for growth direction in [001]
Effective
magnetic field
1.
hk
Electron
momentum
2.
hk
Introduction
Electric generation of spin polarization in RSOI
k  dependent effective magnetic field: hk  ẑ  k
where is averaged over the electron distribution given
-τehk E
by a shifted Fermi sphere f   , k  =
δ  F - 
*
m
τ : momentum scattering time
ky
E
The Edelstein bulk spin density in unit of , given by
e
SB   N 0
zˆ  E
N 0 denoting the density of state per spin.
kx  SBy   N 0
e
yˆ
V. M. Edelstein, Solid State Commun. 73, 233 (1990)
Introduction
Spin accumulations induced by the spin current
z  polarized spin current along yˆ

I  2 D S z  2hkF vF S y  2
y
z
y
(1) Spin diffusion
N 0eE 2 hk2F
(2) Spin precession
vF2
diffusion constant : D 
2
m*
(3) Induced by spincharge coupling
via a driving field
B
S
For RSOI, the bulk spin polarization y results in exact
cancellation between (2) and (3).
Outline
1. Introduction: spin-Hall effect (SHE)
2. Spin diffusion equation for nonuniform driving field
3. Nonuniform driving field around a circular void in the
bulk.
4. Nonuniform driving field around a edge semicircular .
5. Inverse spin-Hall effect for Rashba spin-obit
interaction
The diffusive regime
k F l  1:
Fermi wave length F
impurity
l  F
Mean free path l
spin relaxation length Lso  l
impurity
Mean free path l
Lso
Spin diffusion regime
The four densities
t  

i
Di  r , t   Tloop [ Dˆ i  r , t  Sloop  down , up ]  i 
G  r , r , t , t 
By using Keldysh Green’s function and consider the linear
response approximation to treat the external electric field:

G


†
ˆ
ˆ
 r , r , t , t   i Tl    r , t    r , t  i   H '   d 


loop
ˆ   r'' ,    r'' ,  
ˆ  r'' , 
H '     d 2 r 

j

imp
Keldysh Green’s function is represented by retarded and advanced
Green’s functions:
dN F r
d '
 j  r'' ,  
G  r , r'' ,    '  Ga  r'' , r ,  ' 
2
d
dN F r
d '
j
2
r


    d r 
 j r , 
[G  r , r'' ,    '  G
 r'', r ,  '  
2
d
a
G
 r , r'' ,    ' Ga  r'' , r ,  ']
G  r , r ,     j  d 2 r 
The densities are obtained by
Di  r ,    Di0  r ,     d 2 r ' ij  r , r ',    j  r ',  
j
Response function is given by
d ' dN F
 ij  r , r' ,    i 
Tr G a  r' , r ,   i G r  r , r' ,    ' j 
2 d '
Di0  q ,    2 N 0 ( EF ) i ( q ,  )
D00  2 N 0 ( EF ) 0 (q ,  )
Electric potential energy
Di0 x , y , z  0
Retarded Green's function: G  , k  
r
   p  i  hp  
  
Advanced Green's function: G  , k  
a
1
=
with  is the scattering time
2
 i   h
2
p
2
p
;
   p  i  hp  
  
2

i


h

p
p
2
imp .
SHE accumulation
Impurity averaging by ladder series
p




+

pq

p'
p ' p

p ' q 
p



,  ', q  
Building
block
p


pq


+
+
+…..
pq

ci
 U
v

2
r
a
G
p
,



'
G


 
  p  q,  ' 
p
ci impurity concentration
v system volume
U impurity strength
Diffusion equation in stationary state
Di  r ,    Di0  r ,     d 2 r ' ij  r , r ',    j  r ',  
j
[1   ]il  Dl  Dl0   i il Dl0
we know that:
 il  1  i  Dq 2   il   (...)il
[1   ]il  Dl  Dl0   i  il Dl0  i 1 Dl0
[1   ]il

 D  D   i D
0
l
l
stationary state:  =0:
0
l
 D il  Dl  Dl0 
D il  Dl  Dl0   0
It is convenient to introduce the diffusion propagator
D 
il
 1   

il
 F    hk in the diffusion regime
il ,  ', q  
1 ci
U
2V
2
i
a
l
r

Tr

G
p

q
,

'

G


 p,    '
 
p
Finally, we obtain that:
 il  ,  ', q    il 1  i  D q 2  ...........h 0
 iq 4 2 ilm [v F hPmF ]................qh
 4 2 hP2F  il  nki nkl ..............h 2
 i  4 3  q h3p
nip
p
 i 0 .........qh3
(Spin-charge coupling)
Diffusion equations:
density of state at EF
for stationary state ( =0):
Dil  Dl  Dl0   0,
with Dl0  2 N 0 ( EF ) 0 l 0
homogeneous electric field E along xˆ
 0  eEx, (e  0)
Dil  r    il D 2  R ilm m  il  M i 0
1 2
D  vF : diffusion constant
2
R ilm  4 i ln [hFn vFm ] : spin precession term due to SOI field
il  4 hF2  il  nki nkl  : D'yakonov Perel' relaxation
M i 0  4 2 hF3
nip  hpi / | hp |
nip
p
 : spin-charge coupling without the magnetic field
Spin precession due to either the SOI field or
the external magnetic field:
BSOI
D’yakonov Perel’ (DP) spin relaxation:
Spin relaxation due to spin precession between collisions with
no spin-flipping impurities. A change in momentum changes
the precession axis of the spin.
B’
B
spin
p
p’
Scattering from a
normal scatterer :
no spin-flipping
 D xx Dx  D xy Dy  D xz Dz  D x 0 D00  0,
 yx
yy
yz
y0 0
 D Dx  D Dy  D Dz  D D0  0,
 D zx D  D zy D  D zz D  D z 0 D 0  0,
x
y
z
0

Di  2 Si , D00  2 N 0 e  x, y 
Rashba SOI case: hk   k  zˆ   k y xˆ   k x yˆ
D xx  D 2   xx /
2
 D 2  2hF2 /
2
,
D yy  D 2   yy /
2
 D 2  2hF2 /
2
,
D zz  D 2   zz /
2
 D 2  4hF2 /
2
,
D xy  D yx  0,
R xzx  2hF vF 
R yzy  2hF vF 
yz
D 

,D 

,
x
x
y
y
xz
zxx
zyy
2
h
v

2h v  
R


R

D zx 
 F F
, D zy 
 F F
,
x
x
y
y
D
x0
2 hF2 2 
2 hF2 2 
y0
z0

,
D


,
D
0
3
3
y
x
Spin diffusion equations for Rashba spin-orbit interaction:
2
2 2

2
h

2
h
v

2

h


2
F
F F
F
Sz 
2 N 0 e  x, y   0,
 D S x  2 S x 
3
x
y

2
2 2

2
h

2
h
v

2

h


2
F
F F
F
Sz 
2 N 0 e  x, y   0,
 D S y  2 S y 
3
y
x

2

4
h
2hF vF 
2hF vF 
2
F
D S z  2 S z 
Sx 
S y  0,

x
y

In the bulk, applying a homogeneous electric field Exˆ
All spin densities becomes position-independent in the bulk for stationary state.
  x, y    Ex
2
2 2

2
h

2
h
v

2

h


2
F
F F
F
D

S

S

S

2 N 0 e  x, y   0,
x
x
z

2
3
x
y

2
2 2

2
h

2
h
v

2

h


2
F
F F
F
D

S

S

S

2 N 0 e  x, y   0,

y
y
z
2
3
y
x

2

4
h

2h v  
2h v  
D 2 S z  F2 S z  F F
Sx  F F
S y  0,

x
y

The bulk spin densities are:
S x  0;
S z  0;
Edlestein spin polarization
S yED  
2 N 0 eE
.
RSOI
field
k
SDE for nonuniform field
Spin diffusion equation for nonuniform driving field
RSOI Hamiltonian : hk  and hk  zˆ  k

 xx
R xzx
2
 D S x  2 S x 


 yy
R yzy
2
 D S y  2 S y 


 zz
R zxx
2
 D S z  2 S z 


M x0  0
Sz 
D0  0,
x
2 3

M y0  0
Sz 
D0  0,
3
y
2

R zyy 
Sx 
S y  0,
x
y
spin precession arising from diffusive flow: R ilm  4 iln hkn vFm
D'yakonov-Perel' spin-relaxation rates: il  4 hk2  il  nki nkl 
spin-charge coupling terms:  M
i0
nki
  4 h

k
2
3
k
nk  hk / | hk |
D00  2 N 0 e    : effective local equilibrium density
(The overlines denote the angular averaging)
 hk
 hk2
 hk3 q
SDE for nonuniform field
For uniform driving field, the spin-charge coupling term, given by
 i0 D 00 , has  i0 kept up to the order h 3F q . This is appropriate
because  is position independent.
 il 

2 N 0
i r  0
l a 0

Tr

G
p',

+

'


 G  p' - q,'  ,
 
p'
G r / a 0 : retarded (advanced) Green's functions averaged over
impurity configuration.
  1/ 2 ,  0  1,  x,y,z   x , y , z
However, for the case of nonuniform driving field, we need to
include terms in  i0 D 00 that are higher order in q.
SDE for nonuniform field

 xx
R xzx
2
 D S x  2 S x 


 yy
R yzy
2
 D S y  2 S y 

zz
zxx


R
 D 2 S z  2 S z 


M x0  0
Sz 
D0  0,
3
x
2

M y0  0
Sz 
D0  0,
3
y
2

R zyy 
Sx 
S y  0,
x
y
To identify additional expansion terms in  i0 for
nonuniform driving field, we note first of all that SEd is of
order hF q .
If SEd is to satisfy the spin diffusion equations, all the terms
involving Si will be replace by Si  SEd .
For spin diffusion equations, the terms of order hF q 3 and
hF2 q 2 will be needed.
Outline
1. Introduction: spin-Hall effect (SHE)
2. Spin diffusion equation for nonuniform driving field
3. Nonuniform driving field around a circular void in the
bulk.
4. Nonuniform driving field around a edge semicircular .
5. Inverse spin-Hall effect for Rashba spin-obit
interaction
Nonuniform field with a void
Nonuniform driving field around a circular void in the
bulk
We study the spin accumulation in the vicinity of a circular void,
with radius R0~lso, where the driving field becomes nonuniform.
Here lso is the spin relaxation length, and lso >> le, the electron
mean free path.
E           0 j has to satisfy
the steady state condition  j = 0 and
the boundary condition j = 0.
E

 0 : electric conductivity
E  E0 xˆ  E0  R0 /    cos 2 xˆ  sin 2 yˆ 
2

R0
y
x
I
Nonuniform field with a void
In-plane spin currents can recover the RSOI-induced
spin accumulation
The conventional restoration of RSOI-induced spin
accumulation is associated with the nonvanishing of out-ofz
plane spin current I n .
The key process we find is associated with the in-plane
spin currents, I nx or I ny , and with the way they vanishes at
the void boundary. (n denotes the flow direction normal to
the void boundary)
Spin current
Spin current operator:
1
1
J il  Vl i   iVl  please note that spin factor have not included
2
2

H
  p2
velocity operator Vl  i  i 
 hp  
p p  2m

1
1
1
Vl i   iVl   vl i  hpi

2
2
2
we consider the spin-Hall current in z-direction:(hpz =0)
J il 
J il  vl i
Next, we only consider this term contribution in spin-Hall current.
I il  r , t   Tl  J il  r , t  Sl   down , up  
1 '

{lTr  i G   r ', r , t , t    l' Tr  i G   r , r ', t , t  }r 'r
2m
G  r , r ,        j  d 2 r 
 
d '

 j  r ,   G
 r , r ,    ' G  r , r ,  '
2
Useful formula:
G   r , r ,     N F   G r  r , r ,    G a  r , r ,   
G   G r  G 
G   G   G a
d ' j
   j  r ,    N F  '   N F    '   
2
r
G
 r , r ,    ' Ga  r , r ',  '
G  r , r ,     d 2 r 
d ' j
a
   j  r ,   {N F    ' Ga  r , r ,    ' G
 r , r ',  '
2
r
 N F  '  G
 r , r ,    ' Gr  r , r ',  '}
  d 2 r 
Connecting the spin densities to the
spin current
Si
ijy
b
I  y   2 D
 R  S j  S j   2 I sH  iz
y
where spin-Hall current:
i
y
I sH
 hk

  R S  4 eEN v 
 hk 
 k x
z
zjy
b
j
2
y
0 F
Nonuniform field with a void
In-plane spin currents can recover the RSOI-induced
spin accumulation
i  x or y
I

 2 D
Si  Rizi S z ˆ iˆ

diffusion term
Related to spin
accumulation
In-plane spin current: no driving field term
Rizi denotes the precession of Si into S z when it flows along iˆ.
ˆ iˆ denotes the projection of the flow.
Rashba SOI governs the symmetry of Rijk such that Rizj =0 for i  j.
Nonuniform field with a void
In-plane spin currents can recover the RSOI-induced
spin accumulation
In-plane spin polarization is given by
S  S  S Ed
Edelstein-like spin polarization for the case of nonuniform
driving field
S Ed  r    N0 e / zˆ  E   
N0 : density of state,  : electron scattering time, and e>0
In-plane spin currents can recover the RSOI-induced
spin accumulation
Boundary conditions:
E



I     R0    2 D
Si  Rizi S z ˆ


i
ˆi 
0

   R0

R0
y
x
If Sz=0, spin current indicates S|| is position-independent.(Wrong!)
S  Si  x or y alone fails to satisfy the boundary condition,
because of its radial dependence results in generation of
spin accumulation S z and Si .
Spin diffusion equation for nonuniform driving field
SEd alone cannot satisfy the boundary condition, thus the spin
density have an additional S so that S  SEd  S .
Spin diffusion equation for S :

 2
  S x  4S x  4 x S z  0


 2
  S y  4S y  4 S z  0
y

 2 S z  8S z  4S x  4S y  0


2 2
Dimensionless length in unit of spin relaxation length lso  D so , so  2
hF
S j q  =  a jq  eim   H m1   q  
m
H m1   q   is the first kind Hankel function.
Spin diffusion equation for nonuniform driving field
The asymptotic behavior required of S jq  leads to Im  >0.
The eigen-modes  q :
 1  2i


 2  2  i 2 7
    *
3
2

In terms of these modes S j is expanded in the form
2
3
S j   d  a j
0
q 1 m
The coefficient a j
q
q
im  
1

H


e
.
  m q 
  is determined by the boundary condition!
Boundary conditions related to spin currents
Spin current
hp 

I  2 D j Si  R S x  R S y  R S z   4  v  hp 
 eN 0l  r 
kl  z
l  x, y

i
j
Spin diffusion
ixj
iyj
izj
Spin precession
2
xyi
j
F
It is contributed from driving
z
I
field and is nonzero for j only.
The boundary conditions are
   S x  2cos S z  2 E /  sin 2 
0
  0
   S y  2sin S z  2 E /  cos 2 
0
  0
I i  0  0
=0
   S z  2cos S x  2sin S y 
0
  0
    /  , 0  R0 / lso , and E  eE0 N 0 /
E
The analytic approach is assumed
 ax1    it x sin 2
  2
 az    t z cos 
  3
*
 2
az    az  
t x is real and t z is complex
tx 
tz 
Substituting into
boundary conditions
tx , tz
Im YZ * 
4 E
 8H11  z1  Im  XY *    1 z1 H11  z1  Im  ZX *   2 1 z1 H 21  z1  Im  ZY * 






1
H1   z1  Im 8Y *   1 z1Z * 
E
0 8H11  z1  Im  XY *    1 z1 H11  z1  Im  ZX *   2 1 z1 H 21  z1  Im  ZY * 






zi   i 0 , g 2   2 /   22  4  , X  2 H1   z1   2 g 2 2 H 2   z2  ,
1
Y   g 2 2  1 H11  z2  , Z  2   2  4 g 2  H11  z2 
1
Spin density Si
We have obtained a j
q
  and from the expression
S j   d  a jq   H m1  q   eim  
2
0


3
q 1 m


S  2 it H 1      2 Im t g H 1      sin 2
x 2
1
z 2 2 2 
 x

1
1

  S x cot 2

S

2


it
H



2
Im
t
g
H




 y
x 0  1 
z
2
0
2



 S z  4 Im t z H11   2    sin 



ΔSx and ΔSz are even parities in φ, ΔSy is odd parity in φ .
ΔSz  0 confirms that RSOI’s contribution to spin accumulation
can be restored in a nonuniform driving field .
Total spin density Si
We can calculate the total spin density:
Sx  SEd
x  S x
Sy  SEd
y  S y
S z  S z
GaAs material parameters:
electron density: n  11012 cm 2
effective mass: m*  0.067m0
Rashba SOI constant:   0.3  10-12 eVgm
electron mean free path: le  0.43m
spin-relaxation length: lso  3.76m
Total spin density Si=x,y
E
E
Fig.1: Total spin densities Sx, Sy with radius R0=0.5lso.
Total spin density Sz
Fig.2: Total spin densities Sz is shown the dipole form
normal to a driving field E with radius R0=0.5lso. The spin
accumulation is restored due to nonuniform driving field.
Number of electron spin Sz
d
Fig.3: Number of out-of-plane electron spin within a
circular area the same size as void. The center is shifted
by a distance d from the void center along =/2.
probe
beam size
Spin currents
 S

 S 
I x    x  2Sz  xˆ    x  yˆ
 x

 y 
 Sy 
 Sy

I  
 2Sz  yˆ
 xˆ   
 x 
 y

y
 Sz
 Sz

 02

02
I  
 2Sx  2E 2 sin 2  xˆ   
 2Sy  2E 1  2 cos 2   yˆ

 x

 

 y
z
Summary I:
We have demonstrated that nonuniform driving
field can give rise to spin accumulation in a
diffusive Rashba-type 2DEG.
The nonuniform driving field can be realized by
patterning the sample such as with a circular void
in the sample or with a semicircular void at the
sample edge.
The physical process is identified to be associated
with spin current for the in-plane spin at the
boundary.
Outline
1. Introduction: spin-Hall effect (SHE)
2. Spin diffusion equation for nonuniform driving field
3. Nonuniform driving field around a circular void in the
bulk.
4. Nonuniform driving field around a edge semicircular .
5. Inverse spin-Hall effect for Rashba spin-obit
interaction
Spin accumulation for edge-semicircular void (ESV)
by nonuniform driving field
y
R0
x

E  E0 xˆ  E0  R0 /    cos 2 xˆ  sin 2 yˆ 
2
The nonuniform driving field E() for the circular void satisfies
also the additional boundary condition jy=0 imposed by the ESV case
at the sample edge, =(0, ) . Thus E() holds in the two cases.
To satisfy the additional boundary condition for spin current at
sample edge, a further additional spin accumulation SESV is needed,
leading to
Stot  SEd  S  SESV
Spin accumulation for ESV
y
x
-R0
R0
For the case of ESV, the void spin density SEd  S no longer
satisfies the ESV boundary condition.
The ESV spin density becomes SEd  S  SESV .
Ed
S
 S will induces new source terms
The void spin density
Ivoid at y=0 through the spin currents.
Spin diffusion equation for ESV

 ESV
2 ESV
ESV
 S x  4S x  4 S z  0

x


 ESV
2 ESV
ESV
 S y  4S y  4 S z  0

y

 2 ESV
 ESV
 ESV
ESV
 S z  8S z  4 S x  4 S y  0
x
y

y
x
y=0
First we assume that the sample edge is translation invariant at y=0
such that we can transform the spin diffusion equation into the kspace.
The ESV-induced spin density can be expressed in plane wave
form with a decay mode in y direction:
3
SiESV   dk 
j 1

i x, y, z
ai0   k  eikx e  y ,
j
k  real ,   complex, Re[  ]  0
Substituting SESV into the spin diffusion equation, one
can solve the eigen-modes:
 k 2   2  4
  S xESV 
0
4ik

  ESV 
2
2
0

k



4

4


 S y   0
2
2
 4ik
  S zESV 
4


k



8




S xESV   dkeikx ax(1)0  k  e  1 y  az(2)0  k  g 2e  2 y  az(3)0  k    g 2*  e  3 y

a k e
S yESV   dkeikx ax(1)0  k  g1e  1 y  az(2)0  k  g3e  2 y  az(3)0  k  g3*e  3 y
S zESV   dkeikx
(2)
z0
 2 y
 az(3)0  k  e  3 y
1  k 2  4,  2  k 2  2  2 7i ,
g1 =

3   2*
k
 k 2  2  2 7i
, g 2  3i  7 ,g 3
3  7i
2
8
8
k 4
ik






Procedure to calculate the spin density SESV
generate a
additional source
induce SESV(0) but
I ri  x, y , z
r  R0
0
at ESV boundary
Applying axiliary source
to obtain Saux leading to
I ri  x, y , z
0
r R
0
The additional source terms induced by SEd  S at y=0
for the semi-infinite system

  ESV (0) 

Ed 
add

S


S

S

I



x
x
x
x
 y

 y


 y 0 
 y 0


    S ESV (0)  2S ESV (0)      S  S Ed  2S   0
z
y 
z

 y  y
  y y
 y 0 
 y 0


2
 





ESV
(0)
ESV
(0)
Ed
add
0
 Sz
 2S y



S

2

S

S

2

E
1

cos
2


I








z
y
y
z

2
  y

y



  y 0

y 0

S ESV (0) satisfies the boundary condition at y  0
Iadd
x
-R0
Iadd
z
R0
-R0
I yi  x , y , z
y 0
0
y
R0
x
The spin current related to SESV(0) can not satisfy the ESV
boundary condition
I ri  x, y , z
-R0
r  R0
0
R0
I yi  x , y , z
y 0
0
New artificial source within (-R0,R0) is needed to satisfy the
ESV boundary condition
I i  x, y , z
-R0
R0
ESV
0
The new artificial source can generate a spin density Sart to
satisfy the ESV boundary condition
Spin currents normal to ESV:
   art 
art

S

I

x 
x

y
 y 0


    S art  2S art   I art

y
z 
y

y
 y 0


    S zart  2S yart   I zart
  y
 y 0
we can obtain spin density S aux
such that S ESV  S
ESV  0 
I i  x, y , z
  0
0
 S aux can statisfy the ESV boundary condition
Spin density: S z  S
Ed
z
lso  3.76m, R 0 =0.5 lso
S z  S zEd (1/  m 2 )
3
1
2.5
0.5
2
0
1.5
1
-0.5
0.5
0
-3
-1
-2
-1
0
1
2
3
Fig.4: Spin density S z  S zEd are contributed from a circular void.
Spin density: S
ESV (0)
z
lso  3.76m, R 0 =0.5 lso
3
SzESV 0 (1/ m2 )
1
2.5
2
0.5
1.5
0
1
-0.5
0.5
0
-3
-1
-2
-1
ESV (0)
0
1
2
3
Fig.5: Spin density S z
is contributed from the additional
source induced by S z  S zEd .
Spin density: S
art
z
lso  3.76m, R 0 =0.5 lso
S zaux (1/  m 2 )
3
1
2.5
0.5
2
1.5
0
1
-0.5
0.5
0
-3
-1
-2
-1
0
1
2
3
art
Fig.6: Spin density S z is contributed from the artificial source
within (-R0, R0).
Spin density: S
S
tot
z
 S z  S
Ed
z
S
ESV (0)
z
S
tot
z
lso  3.76m, R 0 =0.5 lso
aux
z
S ztot (1/  m2 )
3
1
2.5
0.5
2
0
1.5
1
-0.5
0.5
0
-3
-1
-2
-1
0
1
Fig.7: Total Spin density S ztot .
2
3
Outline
1. Introduction: spin-Hall effect (SHE)
2. Spin diffusion equation for nonuniform driving field
3. Nonuniform driving field around a circular void in the
bulk.
4. Nonuniform driving field around a edge semicircular .
5. Inverse spin-Hall effect for Rashba spin-obit
interaction
Inverse Spin-Hall Effect (ISHE)
Inverse Spin-Hall Effect (ISHE)
Inverse Spin-Hall Effect (ISHE)
Linear response theory
H  H 0  H so  Vi
H 0 :unperturbed Hamiltonian of a 2DEG with random
elastic spin-independent impurities
H so =h k  :spin-orbit interaction
Vi :the interaction of electrons with the auxiliary fields
We consider two kind of spin current sources:
(i)Inhomogeneous spin polarization Sz creates the spin flux due to
spin diffusion.
(ii)Spin current is driving by spatially uniform spin-dependent
“electric” field.
V1   z B
:slowly varying in time nonuniform Zeeman field B
V2   z k A :uniform spin-dependent field  z A
Spin-up driving field
Spin-down driving field
Spin/charge currents for linear response approach
  0:
I
s/c
d
 ,    i  Tr{Gkr ,k '    Gka,k '   js / cGka'q ,k q    V  , q  N F  
k , k ' 2
Gkr ,k '   js / c Gkr ' q ,k  q      Gka' q ,k  q      Vi  , q  N F    }
Vi  , q  

1
  N F  Ek  Tr  s / c ,
,
2 k
k


where  s   z , c  e
1
k hk 
js / c   s / c , v  ,
v *
2
m
k
N F   : Fermi distribution
For very low temperature, so that N F      N F      
(I)For V1  B z case:
1
2

Dij  1 | U |  / 2  

 ij
 ij   Tr  i Gkr  q    j Gka   
k
ki  r
K ij   * Tr Gk  q    j Gka   
k m
For Rashba spin-orbit interaction:
2 N 0
 

I i
B  K yz Dzz  K yx Dxz  
Dxz 
2 


c
y
For Rashba spin-orbit interaction:
2 N 0
K yx  
;

 2 k F2 N 0
K yz  i q
2m*3
2 N 0  
 

c
Iy  i
B  K yz Dzz   K yx  
 Dxz 
2 
  

 2 k F2 N 0 
 
i
B  i q
D
* 3  zz
2 
2m  
Our goal is to express the charge current through the spin
current Ixs.
The charge current induces by V1
For Rashba spin-orbit interaction:
ki 
i
R jk   * Tr  j Gkr  q    k Gka    ;
m
k

kx 
  x
r
a
I i
B  Rzx Dxz  Dzz  *Tr  z Gk  q    z Gk    
2 
k m

s
x
Finally, we have
2 *

e

m s
s
Ix 
Ix

Inverse spin-Hall effect
No ISHE for V2
(II)For V2   z k A case:
For Rashba spin-orbit interaction, the charge current is:

I i
A
2 i
c
y
i 
 | U |2
hk
x
K

D
R

 ii iz  0
yi
k y 
 2
The spin current is:
2
N
v
s
*
I x  im A 0 F
2
For V2 case, there is no inverse spin-Hall effect.
Conclusion
The spin accumulation can be restored by nonuniform
driving field for RSOI in the diffusive regime.
The spin accumulation is shown the dipole form normal
to the driving field for the case of a circular void.
The spin accumulation is corrected in the corner of
edge semicircular void at the sample.
The inverse spin-Hall effect depends on the specific
form of spin current sources for Rashba SOI case.
Thank you!
Spin accumulations with Rashba SOI
(near the electrode interfaces only)
Spin relaxation
length
N 0 eE 2 hk2F

S z  2hkF vF S y  2
y
m*
(1)
(2)
(3)
When the exact cancellation occurs between (2) and (3), spin
current vanishes in the bulk.
I yz  2 D
The exact cancellation between (2) and (3) no longer holds near the
electrode interface when driving field has reached its bulk value but
the spin polarization has not.
E. G. Mishchenko et al., PRL, 93, 226602 (2004).
Summary I:
The intrinsic SHE provides a possible way to
manipulate electron spins by electric means
The RSOI + driving field can generate
a bulk spin polarization due to a shifted Fermi
sphere.
The spin current can be contributed from the spin
diffusion, spin precession, and the driving field.
The SHE is quenched in the RSOI due
to the exact cancellation between the bulk spin
polarization and contribution of a driving filed.
I ij  2 D j Si  Rixj S x  Riyj S y  R izj S z
hp 

  4  v  hp 
 eN 0l  r 
kl  z
l  x, y

2
xyi
j
F