Electron transport in quantum wires subjected to the Rashba SOI

(1)

Electron transport in quantum wires subjected to the Rashba SOI

• My thanks to

• Karl-Fredrik Berggren and Ivan Shelykh for invitation

Stockholm, Nordita

workshop 2012, September

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.)

Evgeny Sherman (University of Basque Country, Bilbao, Spain)

Almas Sadreev, Lab. of Theory of Nonlinear Processes,

Institute of Physics, Siberian Branch of Russian Academy of Sciences

Krasnoyarsk, Russia

(2)

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

(3)

Spin-orbit interaction

(4)

From Nitta’s lecture

Stockholm, Nordita

(5)

U

z 2DEG

Stockholm, Nordita

E

(6)

Stockholm, Nordita

(7)

Stockholm, Nordita

(8)

>

2 R R

z x y y x z x y y x

(z) (x,y,z)

= (E p -E p ) + E ( p - p )

H dz  H

   

 

Stockholm, Nordita

(9)

where  ~ E

_z

The Razhba spin-orbit interaction

E.I. Razhba, Sov. Phys. Solid State 2, 1224 (1960)

In infinite interface electric field is uniform and directed perpendicular to the interface.

( )

R x y y x

H      p   p

Emmanuil Rashba

Stockholm, Nordita

( )

R y x x y

H  i      

(10)

Lusakowski et al, PRB 68,081201R (2003)

Stockholm, Nordita

(11)

SOI in 2DEG

2 2

*

2

* *

ˆ 1 ( ˆ ˆ ) ˆ , 2

ˆ ( ˆ ˆ ˆ ˆ ). [ ˆ , ] 0 ˆ

ˆ 1 ( ˆ ˆ )

2 0 : 0

, cos , sin

x y R

SO

R x y y x

SO

x y y x

x y

R

x y

H p p H

m

H k p p p H

m

H p k p p

m m

p ip

H p ip

p p p p p

 



 

 

   

 

  

  

  

  

     

               

   



J. Phys. C (2007)

 

i -i

i / 2 -i / 2

i / 2 -i / 2 i / 2

-i / 2

i / 2 -i / 2

, cos , sin

0 e

: ;

e 0 e , e .

0 1 e

ˆ e e cos , ˆ =sin

1 0 e ( , ) exp( ) e

e

x y

k x y

p p p p p

p

x y ik x ik y



 

  



  

 

  

 

   



_

   

    

             

 

 

         

  



-i / 2 i / 2

( , ) exp( ) e

k

x y ik x ik y

x y

e





_ 

  

  

 

     

 

Stockholm, Nordita

(12)

Infinite 1d wire



²



_y ^{2 2} ²

* *

ˆ ˆ ˆ . [ , ] 0. ˆ 1, .

2 2

x x

y x x x

p k

H p p H E k

m   m 

       

 

0 : ( ) x e ^{ik x}

^x

   

1

2

0 : ( ) 1

2

x

ik x ik x

x e

 e

    ^    ^   

1 2

cos , sin

z

x x y x

x x x

const

k x k x

k k k



 



  

  



(

_x _y _y _x

) H

R

    p   p

Stockholm, Nordita

(13)

• S.Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, England, 1995).

• S. Datta, Quantum transport: Atom to transistor

Supriyo Datta, Purdue Univ.

Datta-Das spin transistor

Stockholm, Nordita

(14)

Datta-Das spin transistor

Stockholm, Nordita

(15)

• 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 Peter Grunberg

Stockholm, Nordita

(16)

Stockholm, Nordita

(17)

Quasi 1d wire

0 2

2 2

0 * *

0

2 2 2

2

* 2

,

ˆ ˆ ˆ ˆ ˆ ˆ

( ), ( ).

2 , , ( ) , , ,

( ) , 1, 2,3,...

2 2 1

, , , ( , ) si

channel numb

n exp( )

r

2 e

x

R

SO

x z R x z z x

x m x x

m x z

x

x m k z

x x

H H V

H p p V k p p

m m

H m k E k m k

E k m k m

m L

x z m k x z mx ik z

L L 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);

Stockholm, Nordita

(18)

Approximate by two channels

, '

0

, ' '

ˆ ˆ ˆ ˆ

( ).

0 1 0

ˆ ; ˆ

1 0 0

, | | ', 0, , | | ', 0

2 '

, | | ', sin sin

y

R x y y x

x y

R R

L

R x m m

y y y

x m m mm

V p p

i i

m V m m V m

i m m

m V m k dy

L L y L

k v

  

 

  

 

 

    

     

   

     

    



 





 



Stockholm, Nordita

m=1 m=2 m=3

(19)

Two-band (two-channel) model

l n  l n 

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 sub- bands 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.

Stockholm, Nordita

(20)

Two-band model

2 0

2

* 2_x

E   m L

Energy is measured in terms of

2 2

12

2 2

12

2 2

12

2 0 0 J 0 2 J 0 ˆ 0 J 4 2 0

z SO z

k k k

H k k k



 

 

  

2 2

J 0 0 4

12

 k

_z

2 k k

_{SO z}

 

 

   

 

 

2 2 _z2

2

_{SO z}

, 1 E    n  k  k k    

12

16

_SO

/ 3 J   ik

Stockholm, Nordita

(21)

two band model

2 2

12

2 2

12

2 2

12

2 0 0 J 0 2 J 0 ˆ 0 J 4 2 0

z SO z

k k k

H k k k



 

 

  

2 2

J 0 0 4

12

 k

_z

2 k k

_{SO z}

 

 

   

 

 

n=1

n=1

n=2

n=2

Stockholm, Nordita

(22)

Two-band model (wave functions)

 

1

* * 1

1 1

1

( , ) ( ) sin ;

( )sin 2 ( )sin 2

( , ) ( ) ;

sin

( , )

1 ˆ

( , ) ( , ) ( , )

( , ) 2

z

ik z z

z

ik z z z

x z C k e x

if k x

x z C k e

x

S x z x z x z x z

 

x z

 



 



   









 



 

  

 

  

  

 

 

  

 

2

1

2

3 3

( ) 2 ( ) ;

16 2

2 1

( ) 1 ( )

x SO x x

SO

x

x x

f k k k g k

k

C k L f k



 

     

 

 

Stockholm, Nordita

(23)

• Current induced spin polarization –spin Hall effect

( , ) S x y

x

Reynoso, Usaj, and Balseiro, PRB70, 235344 (2004); PRB73, 115342 (2006)

( , ) S x y

y

( , ) S x y x

( , ) S x y

z

y

5 L  L

_y

 2

y

1 L  L

_y

 1/ 2

Stockholm, Nordita

(24)

PHYSICAL REVIEW B 72, 045353 2005

Charge- and spin-density modulations in semiconductor quantum wires Minchul Lee and Christoph Bruder

SO y 1 k L 

SO y

2 k L 

Stockholm, Nordita

(25)

• J . Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, PRL (1999)

Gate Control of Spin-Orbit Interaction in an Inverted In0.53Ga0.47AsIn0.52Al0.48As Heterostructure

Stockholm, Nordita

(26)

2

0 *

ˆ ˆ ( ) ( )

2

^c ^c

H p V z V y

 m  

Quantum wire Hamiltonian

Finger gate potential (Davies, Larkin, and Sukhorukov, J. Appl.

Phys.

77, 4504 (1995).

(27)

The simplest model of ballistic transport

2 2

| | | |

t t

P t t

 

 



2

* *

2

* *

ˆ ˆ

ˆ ( ).

2 ( ˆ ) ˆ

ˆ [ (

2 )]

| ˆ |

1 Re{ | | ˆ | ( ) |

SO z

ec SO e

z c

SO z

Magnetic field

C

k

H p e p

m m

p A k

H e p A

m m

H e k

j p e

c

urrent densit

m y

A m



  

    

  

        





    

      

     

(28)

Parabolic wire

SO

0 k  weak SOI Strong SOI

Stockholm, Nordita

(29)

A. V. Moroz and C. H. W. Barnes

Phys. Rev.B60,14272 (1999); B61, R2464 (2000)

gates

1d wire ^x

Is the only integral of motion

x x

p   k

(30)

Asymmetry of heterostructure

From parabolic potential

(31)

Landauer two-probe conductance with spin splitting

2 2

2 , ' ' ', '

2 2

2 , ' ' ', '

( ) | |

n n nn

e e

G Sp T T T

h h

e e

G Sp T T T

h h

 



  





 



(32)

Curvilinear wire

 

*

, 1 sin , cos

( sin cos ) , 2

R x y

x y R

H i m

Cylindrical coordin

R ates

 



      



      

    

 

   



2

( sin cos )

1 ˆ 1

ˆ ,

2 2

1 [ , ] [ sin cos ]

2 [ , ] [ sin cos ]

[ , ] 0 ˆ

x y

z z z z

z y x

y x

z

H

J l i

H i i

i H i i

J H

    

 

 



     

    



 

   

 

    



 

    







Bulgakov &Sadreev PRB 66 (2002)

( 1) 2

( ) 1

1 i i

Ae Be



 

   ^

 

       

Stockholm, Nordita

(33)

2

0 0 2

i -i

2 i

0 2 i( 1)

i i i

-i i( 1) i( 1)

. .

0 e

( sin cos ) .

e 0 e

( 1) e

0 e e ( 1)e

e 0 e e

R

R x y

R

H H H H

H i H A

B

A iB

H B iA





 

  

    



     

 

 



   

 



 

    



 

 

           

  

        

     

                

   



2

. ( 1)

, 0

( 1) H i

i

   

  

  

 

 

 

 

 

  



Stockholm, Nordita

(34)

1

2

1 2 ( 1)

2 2 ( 1)

2

( ) 1

1 ( ) 1

1 1 1 ;

i i

i e e

e i e

 

  



   

 





 

       

 

       

  

Two channels of transmission General solution is

Let electron is injecting with spin directed along the z-axis (0) 1

 _{  } ^{ } 0

 

(35)

Stockholm, Nordita

(36)

(37)

Spin polarization by magnetic field or ferromagnetic layer

' '

2 2

( ) 1 [ ]

2 ( ) ( )

( ) 1

2 (0, ) (0, ); (0, ) (0, ).

| | | |

ikx ikx L

ik x ik x

ikx L

L L

x e r e

k

x a e b e

x t e

k

L L L L

t t

P t t

 

 



  



 



   

 



 

   



 

 



 

 





No spin flip

1

2

0 : ( ) 1

2

x

ik x ik x

x e



e

    ^   ^  

 

Stockholm, Nordita

(38)

Spin polarization by Rashba SOI

Bulgakov &Sadreev PRB 66. 075331 (2002) Zhai &Xu, PRL 94, 246601 (2005).

Kisilev&Kim J. Appl. Phys. (2001); (2003)

*

* *

0 1

ˆ ˆ ˆ .

1 0

ˆ ( ) .

ˆ ( )

y

n n n

n n n n

n

T i K K

T

T a b

  

    

   

     



    

 

 

 

 

  

   

 

 

  



Time reversal

Stockholm, Nordita

(39)

Spin filtering

' ' '

' '

* '

;

L L L L R L

R R L R R R

L L L L R L

R R L R R R

L L

R R

L L L L

R R

R

r t a

b

a t r b

r t a

a

b t r b

     

 





  

 

  

  

    

    

  

 

  

  

    

    

' ' '

* *

'* '*

1

^L ^{L L} ^{R L} ^L

;

R L R R R R

L L

R R

r t b

S S a

b t r a

     



    

  

 

  

    

    

' ' ' '

'* '*

;

R L R L R R R

R L R

b

_

 t

_{ }

b

_

 r

_{ }

a

_

' ' ' '

* ' * ' *

;

R L R L R R R

R L R

b

_

 t

_{ }

b

_

 r

_{ }

a

_

' ' ' '

* ' * ' *

;

R L R L R R R

R L R

b

__

 t

_{ }_

b

_

 r

_ __

a

_

' ' ' '

* ' * ' *

' * ' ' *

;

R L R L R R R

R L L R R R

R L L R R R

R L R

R R R

L R

b t b r a

t b r a

      

     

  

 

  

 

 

  

'

L R L R R L

t _{ }   t _ _ _ _

 

' '

;

L R R L

R

t

_ _

t

_ _ L



_

 

_

 Spin polarization

*

2 2 2 2

2 2

' '

| | | | (| | | | )

. .

| | | |

L L

L

z x y

t t t t t t

P P iP

t t

 



 

 















   

 

'

' '

r '

R R R R

r

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).

(40)

• Unitarity of S-matrix:

* *

2 2

* *

2 2

2 2 2

2 2

2 2 2

1 ' 1 0 ' 0 1 ' '

| | | | 1

' ' 0

| ' | | ' | 1

| ' | | ' | 1 | ' |

| | | ' |

| | | | 1 | | S S

r t r t

t r t r

r t

r t r t rt r t

r t

t r t

t t

t r t

 

    

     

    

 

 

    



(41)

Kisilev&Kim J. Appl. Phys. (2001); (2003)

Bulgakov &Sadreev PRB 66. 075331 (2002)

Liu et al PRB (2007)

Perroni et al J. Phys. (2007)

Spin transistor

(42)

The Hall like effect, induced by the Rashba spin-orbit.



n n



( )

d d ^ _  const b l    

Landau&Liphshitz, Quantum Mechanics.

Stockholm, Nordita

(43)

output

o u t p u t

o u t p u t SOI

Four-terminal structure

Bulgakov et al, PRL (1999)

Two effects

1) Spin polarization 1) Spin polarization

2) Hall-like effect 2) Hall-like effect

Stockholm, Nordita

(44)

Stockholm, Nordita

(45)

Stockholm, Nordita

(46)

Yokoyama&Eto PRB (2009)

Three-terminal

Four-terminal Nitta et al, Gate Control of Spin-Orbit Interaction in an Inverted