Electron transport in quantum wires
subjected to the Rashba SOI
Almas Sadreev, Lab. of Theory of Nonlinear Processes,
Institute of Physics, Siberian Branch of Russian Academy of Sciences
Krasnoyarsk, Russia
• My thanks to
• Karl-Fredrik Berggren and Ivan Shelykh for invitation
And to whom I collaborate and collaborated :
Karl-Fredrik Berggren (Linkoping University, Sweden)
Evgeny Bulgakov and Konstantin Pichugin (Institite of
Physics, Krasnoyarsk, Russia)
Pavel Exner, Pavel Streda and Petr Seba (Chech Rep.)
Stockholm, Nordita
Evgeny
Sherman
workshop 2012,
September(University of Basque Country, Bilbao, Spain)
Spintronics (nickname for spin-based electronics):
•
•
•
•
We want to use spins of single electrons for
storage,
transfer,
and manipulation of information and not only ...
Stockholm, Nordita
workshop 2012, September
Spin-orbit interaction
From Nitta’s lecture
Stockholm, Nordita
workshop 2012, September
U
E
2DEG
z
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
H R   dz  (z) H R (x,y,z)
2
=  z (E x p y -E y p x ) +  E z ( x p y - y p x )
>
Stockholm, Nordita
workshop 2012, September
In infinite interface electric field is uniform and directed perpendicular to
the interface.
HR 
a ( x p y   y px )
H R  ia ( y  x   x  y )
where a ~ Ez
The Razhba spin-orbit interaction
E.I. Razhba, Sov. Phys. Solid State 2, 1224 (1960)
Stockholm, Nordita
workshop 2012, September
Emmanuil Rashba
Lusakowski et al, PRB68,081201R (2003)
Stockholm, Nordita
workshop 2012, September
SOI in 2DEG
J. Phys. C (2007)
1
Hˆ 
( pˆ x2  pˆ y2 )  Hˆ R ,
*
2m
k
Hˆ R  SO
(ˆ x pˆ y  ˆ y pˆ x ). [ pˆ a , Hˆ ]  0
*
m
k
1 2
Hˆ 
p  SO
(ˆ x p y  ˆ y px )
*
2m
m*
px  ip y   a 
0
a 
HR     : 
 



 p  ip 0    
 
y
 x

   p, px  p cos  , p y  p sin 
   p, px  p cos  , p y  p sin 
0
ei   a   a 
  p :  -i
      ;
e
0

     
a  ei / 2 ,   e-i / 2 .
ˆ x   e
-i / 2
e
i / 2
i / 2

 0 1  e
 1 0   e-i / 2   cos  , ˆ y =sin



 e i / 2 
 k  ( x, y )  exp(ik x x  ik y y )  -i / 2 
e

 e-i / 2 
 k  ( x, y )  exp(ik x x  ik y y )  i / 2 
e

Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
HR 
a ( x p y   y px )
Infinite 1d wire
2 2
pˆ x2
kx
2
ˆ
ˆ
ˆ
H

a
p
.
[
p
,
H
]

0.



1,
E


a kx .
y x
x
y
*
*
2m
2m
ik1 x x


e
1
a  0 :   ( x) 
 ik2 x x 
2 e 
 z  const
 x  cos k x x,
 y  sin kikx xx x
a  0 :  ( x)  e
k x  k1x  k2 x
Supriyo Datta, Purdue Univ.
Datta-Das spin transistor
• S.Datta, Electronic Transport in Mesoscopic Systems (Cambridge
University Press, Cambridge, England, 1995).
• S. Datta, Quantum transport: Atom to transistor
Stockholm, Nordita
workshop 2012, September
Datta-Das spin transistor
Stockholm, Nordita
workshop 2012, September
• In 2007 Albert Fert (France) and Peter Grunberg (Germany) have got
Nobel prize for discovery of giant magnetoresistance in multilayers of
Fe/Cr, one after the other. Chromium of order of 1nm of thickness. The
resistance increase between the parallel and anti-parallel
configurations of the layers of iron went up to 80%!
Albert Fert
Stockholm, Nordita
workshop 2012, September
Peter Grunberg
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
H  H 0  VR
2
k SO
( pˆ xˆ z  pˆ zˆ x ).
*
2m
m
H 0 m, k x ,   Em (k x ) m, k x ,  ,
H0 
( pˆ x2  pˆ z2 ), VR 
*
  2 m2
2
Em (k x ) 

k

z  , m  1, 2,3,... channel number
2m*  L2x

2
Quasi 1d wire
x, z m, k x ,    m ,k x ( x, z ) 
2
 mx
1
sin
exp(ik z z ) 
Lx
Lx
2 kz
Datta& Das, Appl. Phys. Lett.(1989);
Perroni et al J. Phys. C (2007);
SJeong&Lee PRB (2006);
Zhang, Brusheim and Xu, PRB (2005)
Mireles& Kirczenow, PRB (2001);
Knobbe&Schapers PRB (2005);
Moroz & Barnes, PRB (1999) ;
Pramanik et al PRB (2007);
Debald&Kramer, PRB (2005);
Governale&Zulicke PRB (2002);
Rodriguez, Puente and Serra (2003)
Erlingsson et al PRB (2010);
VR  a ( pˆ xˆ y  pˆ yˆ x ).
0 1
 0 i 

0
ˆ x  
 ; ˆ y  
1
0


i
m ,  | VR | m ',   0, m ,  | VR | m ',   0
2i a
m ,  | VR | m ',   a k x m ,m ' 
Ly
Ly
 dy sin
0
m 
Ly y
sin
m'
Ly
 a k x m ,m '  vmm '
Approximate by two channels
m=1
Stockholm, Nordita
workshop 2012, September
m=3
m=2
Two-band (two-channel) model
We take into account the exact solution of the quantum wire in the absence of SOI,
then we study its effect on the sub-bands. Because of SOI, a coupling between subbands with opposite spins occurs. In order to study the effects of this coupling,
we will discuss the results within the first- and second-order perturbation theory
approach with respect to the SOI.
This assumption is valid if the wire is very narrow. Moreover, this simple system is
studied since it provides a simple understanding of the transport properties.
l n
ln
Stockholm, Nordita
workshop 2012, September
Two-band model
Energy is measured in terms of E0 
  2  k z2  2k SO k z

0

Hˆ  
0


J12


2
2m* L2x


 2  k z2  2k SO k z J12
0


J12
4 2  k z2  2k SO k z 0
 J12  16ikSO / 3
0
0
4 2  k z2  2k SO k z 


0
0
J12
E   2 n 2  k z2  2kSO k z ,   1
Stockholm, Nordita
workshop 2012, September
two band model
  2  k z2  2kSO k z

0

Hˆ  
0


J12


Stockholm, Nordita
workshop 2012, September


 2  k z2  2kSO k z J12
0


J12
4 2  k z2  2kSO k z 0

2
2
0
0
4  k z  2kSO k z 


0
0
J12
n=1,1
n=1,1
n=2,1
n=2,1
Two-band model (wave functions)
 sin  x

;
 if (k z ) sin 2 x 
 1 ( x, z )  C (k z )eik z 
z
 1 ( x, z )  C (k z )e
Sa ( x, z ) 
 if (k z ) sin 2 x 

;
sin

x


 1 ( x, z ) 
1 *
*
ˆ

(
x
,
z
)

(
x
,
z
)

 1
 a  ( x, z ) 
1
2
 1

3
f (k x ) 
16kSO
C (k x ) 
ik z z
2
Lx
 3 2



2
k
k

g
(
k
)
SO x
1
x ;
 2


1
1  f 2 (k x )
Stockholm, Nordita
workshop 2012, September
•
Current induced spin polarization –spin Hall effect
Reynoso, Usaj, and Balseiro, PRB70, 235344 (2004); PRB73, 115342 (2006)
SSxx((xx,, y)
Ly  5
Ly  2
S y ( x, y)
Ly  1
S z ( x, y )
Stockholm, Nordita
workshop 2012, September
Ly  1/ 2
PHYSICAL REVIEW B 72, 045353 2005
Charge- and spin-density modulations in semiconductor quantum wires
Minchul Lee and Christoph Bruder
kSO Ly  1
kSO Ly  2
Stockholm, Nordita
workshop 2012, September
Gate Control of Spin-Orbit Interaction in an Inverted
In0.53Ga0.47AsIn0.52Al0.48As Heterostructure
• J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, PRL (1999)
Stockholm, Nordita
workshop 2012, September
Quantum wire Hamiltonian
2
ˆ
p
Hˆ 0 
 Vc ( z )  Vc ( y )
*
2m
Finger gate potential (Davies, Larkin, and Sukhorukov, J. Appl. Phys.
77, 4504 (1995).
The simplest model of ballistic transport
ˆ2
kSO
p
ˆ ).
ˆ
H

e
(


p
z
2m* m*
Magnetic field 
ˆ  e A) 2
(
p
kSO
c
ˆ  e A)]
ˆ
H

e
[


(
p
z
c
2m*
m*
Current density
ˆ
k
1  |H |
e
j 
 * Re{  | pˆ |   SO
 | (ez   ) | 
*
c
m
m
A
P
| t |2  | t |2
| t |2  | t |2
Parabolic wire
kSO  0
Stockholm, Nordita
workshop 2012, September
Strong
weak
SOI SOI
A. V. Moroz and C. H. W. Barnes
Phys. Rev.B60,14272 (1999); B61, R2464 (2000)
px  k x
gates
1d wire
x
Is the only integral of
motion
From parabolic potential
Asymmetry of heterostructure
Landauer two-probe conductance with spin splitting
e2
e2

G  Sp(T T ) 
h
h
 | T 
n
nn ',
e2
e2

G  Sp(T T ) 
h
h
2
|
, n ' '
'
2
|
T
|
 n ,n ' '
nn ', '
Curvilinear wire
Cylindrical coordinates
    1

,

sin

,

cos






 x y  R

H R  i  ( x sin    y cos  ) ,   2m*a R

Bulgakov &Sadreev PRB 66 (2002)
2

H   2   ( x sin    y cos  )


1
1

Jˆ z   z  lˆz   z  i ,
2
2

1
[ z , H ]   [i y sin   i x cos  ]
2

[  i , H ]  i  [ y sin   i x cos  ]

[ Jˆ , H ]  0
i 


Ae
1
 ( ) 
 i (  1) 
2 
1    Be

z
Stockholm, Nordita
workshop 2012, September
2
H  H0  H R. H0   2 .

0
e i  

H R  i  ( x sin    y cos  )
   -i
 .

 e 0  
  A 2 ei

H 0   

2 i( 1) 
  B(   1) e

0
 iB(   1)ei 
ei    Aei 
H R     -i
  i( 1)    
.
i( 1) 
 e 0    Be

 iA e

H    ,
2 
i  (  1)
i 
(  1)  
2
0
Stockholm, Nordita
workshop 2012, September
 1 ( ) 
 i ei1

2  i ( 1 1)
1   e
 2 ( ) 
 ei2

i (  1)
2 
1    i e 2

1



1

1 1 
2



;
General solution is
Two channels of transmission
1 
 (0)   
0
Let electron is injecting with
spin directed along the z-axis
Stockholm, Nordita
workshop 2012, September
Spin polarization by magnetic field or ferromagnetic layer
No spin flip
1
 ikx
[eikx   r e ]
2 k

 ( x)   (a eik x  b e ik x )
  L ( x) 

1
ikx
t
e
 
2 k 
  L (0, L)   (0, L); ' L (0, L)  ' (0, L).
  L ( x) 
| t |2  | t |2
P
| t |2  | t |2
ik x
1  e 1x 
a  0 :   ( x) 
 ik2 x x 
2 e 
Stockholm, Nordita
workshop 2012, September
Spin polarization by Rashba SOI
Kisilev&Kim J. Appl. Phys. (2001); (2003)
Bulgakov &Sadreev PRB 66. 075331 (2002)
Zhai &Xu, PRL 94, 246601 (2005).
Time reversal
 0 1 ˆ
Tˆ  i y Kˆ  
 K.
1 0 
Tˆ n    ( n )*    n .
Tˆ   (  an* n    bn* n )
n
Stockholm, Nordita
workshop 2012, September
Spin filtering
'
L
 bLL   r L L' t L R'  a L' 

;
 R 
 a   t ' r ' '  b R' 
 R    R L  R R   R 
'
L
  L aL* L   r L L' t L R'   L a L' 

;


  RbR*   t ' r ' '  b R'


R 
  R L  R R   R 
*
*
L
 aLL   r L'  L t R'  L  b L' 

;
S  S  1   R    '*
'*
R
 b   t ' r '  a ' 
 R    L R  R R   R 
bRR  t'*'  bL'  r'*'  aR' ;
L
r
  R R ' r
L
R
R
R
L
R
R
R
bR*R  t' '  bL'*  r' '  aR'* ;
L
'
R L
R
L
R
 t
Spin polarization
| t |2  | t |2 (| t |2  | t |2 )
Pz 
. Px  iPy 
2
|
t
|
  '
R
 R bR*   R t'
R
 '
R
bRR*  t' '  bL'*  r' '  aR'* ;
L
'
 R  R'
R
 t
 t L t* L
L
2
|
t
|
  '
 '
'
R L
.
R
'
L  R
 L' bL*  r'
'
L
'
R  L
'
L
'
L  R
L R
L
'
R R
 L' bL*  r'
 R t'
 t'
bL'*   R r' '  aR'*
R
 R' aR*
'
R  R
 t
R
R
'
R
 R' aR* ;
'
R  L
'
R
 L' ;
  L R t
R  L
Therefore, for the single channel transmission through the two-terminal QD with the
Rashba spin-orbit interaction there is NO spin polarization (filtering).
• Unitarity of S-matrix:

S S 1
 r * t *   r t '  1 0 
 * *  


 t ' r '   t r '   0 1
2
2
| r |  | t | 1
r t ' r ' t  0
*
*
rt '  r ' t  0
*
*
| r '|  | t '| 1
2
2
| t ' |2 | r ' |2 1 | t ' |2
2
2



|
t
|

|
t
'
|
2
2
2
|t |
|r|
1 | t |
Kisilev&Kim J. Appl. Phys. (2001); (2003)
Bulgakov &Sadreev PRB 66. 075331 (2002)
Liu et al PRB (2007)
Spin transistor
Perroni et al J. Phys. (2007)
The Hall like effect, induced by the Rashba spin-orbit.
Landau&Liphshitz, Quantum Mechanics.
d
d
 const  b(l  )
n
n
s
s
Stockholm, Nordita
workshop 2012, September
Four-terminal structure
Bulgakov et al, PRL (1999)
Two effects
output
1) Spin polarization
Stockholm, Nordita
workshop 2012, September
SOI
output
output
2) Hall-like effect
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
Nitta et al, Gate Control of Spin-Orbit Interaction in an Inverted
In0.53Ga0.47AsIn0.52Al0.48As Heterostructure , PRL (1997,1999)
Three-terminal
Yokoyama&Eto PRB (2009)
Four-terminal
Stockholm, Nordita
workshop 2012, September
Theory and experiments by Nitta group
Physica E(2005)
Stockholm, Nordita
workshop 2012, September
Bulgakov&Sadreev, JETP Lett. (2001):
1) Exact solution for circular QD with Rashba SOI
2) Spin polarization 90% by circular polarized laser field
PRB (2004): Exact solution with Rashba SOI and magnetic field

R
 0 1 ˆ
ˆ
ˆ
T  i y K  
 K.
1
0


1
1

Jˆ z   z  lˆz   z  i ,
2
2

[Tˆ , Hˆ ]  0, [ Jˆ , H ]  0
No Rashba SOI
 (r ,  )  CJ m ( n|m| r )eim , r  r / R
 2 d2
d
2
r

r



m
J m ( r )  0
 dr 2

dr


J m ( n|m| )  1,
z
  2m*a R
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
Eigen energies
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workshop 2012, September
Laser field with circular polarization
 

0

z 
z  x  iy, VR  2  




 * 0 
 z

1 ˆ e 2
Hˆ 
( p  A)  V (r )  VR ,
2m
c

i
 Hˆ  ,   exp(itJˆ z ) , 
t



i
 H , H  H   Jˆ z  2iA(  * );
t
z z
Stockholm, Nordita
workshop 2012, September
Stockholm, Nordita
workshop 2012, September
1d packet:
t=0
t=1.5
t=7
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workshop 2012, September
2d case
Landauer conductance