Transcript PowerPoint 簡報 - National Tsing Hua University

NCTU
Electrical Means of Manipulating
Electron Spins in Semiconductors
C. S. Chu
Dept. of Electrophysics
National Chiao Tung University
Collaborators:
A.G. Mal’shukov (RAS)
L. Y. Wang (NCTU)
NTHU Colloquium
11.22.2006
NCTU
NTHU Colloquium
11.22.2006
NCTU
Quoted from the abstract of
“Spintronics: Fundamentals and applications”
Spintronics, or spin electronics, involves the
study of active control and manipulation of
spin degrees of freedom in solid-state
systems.
in Reviews of Modern Physics,
vol. 76, p.323-410, 2004,
by I. Žutić, J. Fabian, and S. Das Sarma.
NTHU Colloquium
11.22.2006
NCTU
How about transport of spins in non-magnetic
semiconductor using only electrical control?
Schemes making use of spin Hall effect
Schemes other than spin Hall effect
NTHU Colloquium
11.22.2006
NCTU
A simplest version of a spin Hall effect:
An electrical current passes through a sample
with spin-orbit interaction, and induces a spin
polarization near the lateral edges, with opposite
polarization at opposing edges (M.I. D’yakonov
and V.I. Perel’, JEPT Lett., 13, 467 (1971)).
This effect does not require an external magnetic
field or magnetic order in the equilibrium state
before the current is applied.
M.I. D’yakonov and V.I. Perel’ (1971) proposed
an extrinsic mechanism for the spin Hall effect in
the paper: “Possibility of orienting electron spins
with current”.
cond-mat/0603306
H. Engel, E.I. Rashba, B.I. Halperin
NTHU Colloquium
11.22.2006
NCTU
V.M. Edelstein, Solid State Commun. 73, 233 (1990)
“Spin polarization of conduction electrons induced by electric
current in two-dimensional asymmetric electron systems”
S. Murakami, N. Nagaosa, S.C. Zhang,
Science 301, 1348 (2003)
“Dissipationless quantum spin current at room temperature”
J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T.
Jungwirth, and A.H. MacDonald,
Physical Review Letters 92, 126603 (2004)
“Universal Intrinsic Spin Hall Effect”
NTHU Colloquium
11.22.2006
NCTU
A simple picture for the extrinsic spin Hall effect
J.E. Hirsch, PRL 83, 1834 (1999)
NTHU Colloquium
11.22.2006
NCTU
More on the simple picture for the extrinsic spin Hall effect
p2
H
 V (r )    ( p  V )
2m
p2
H
 V (r )    ( p  E)
2m
Potential energy
mr  V  ( E   )     ( p  E)

E
For electron incident upon the
LHS of the attractive scatterer,
again the spin up particle is
deflected more to
the left and the spin down particle
is deflected more to the right.
NTHU Colloquium
11.22.2006

NCTU
A “simple picture” for the intrinsic spin Hall effect
PRL 92, 126603 (2004)
J. Sinova, D. Culcer,
Q. Niu et al.
NTHU Colloquium
11.22.2006
Asymmetric heterostructure that has
Spin-orbit interaction
Schematic layer structure of an inverted
In0.53Ga0.47As / In0.52Al0.48As
heterostructure.
(Nitta et al. Phys. Rev. Lett.78, 1355(1997))
Calculated conduction band diagram (solid
line) and electron distribution (dash line).
(Nitta et al. Physica E, 2, 527(1998))
NTHU Colloquium
11.22.2006
Rashba effect (spin-orbit interaction )
Heterostructure:
InGaAs
2DEG
InAlAs
Structure inversion asymmetry:
E
E
analog
NTHU Colloquium
11.22.2006
An electron moves between
two charged plane
V
Effective magnetic field
induced by the effective
current I.
I
E
Beff
I
In Lab. frame
The SOI hamiltonian is given by
H     Beff
In the rest frame of an
electron

   V  E
HRashba  0  p  zˆ   
NTHU Colloquium
11.22.2006

where  0 is called the
Rashba constant.
E2 D  kx2  k y2+ 0 kx2  k y2
E
ky
kx
Fig.3.Dispersion relation for a 2D
Rashba-type system and the Rashba
constant 0  0.13.
NTHU Colloquium
11.22.2006
Rashba spin-orbit interaction (SOI)
• SOI is significant in narrow gap
semiconductor heterostructures.
• Large variation (up to 50%) of
the SOI coupling constant ,
tuned by metal gates, has been
observed experimentally.
Rashba term:
H so    p  vˆ   
vˆ : normal to interface
 : the Pauli spin operator
[ Nitta et. al. PRL 78 (1997)
Engels et. al. PRB 55 (1997)
Grundler, PRL 84 (2000) ]
• Static gate control of  has been
the focus of previous proposals
on spin polarized transistors.
[ Datta et. al. APL 56 (1990), ……]
NTHU Colloquium
11.22.2006
Tuning of the coupling constant  0 by a metal gate
Vg
InGaAs
InAlAs
2DEG
Spin-orbit coupling parameter  of the
first (circle) and second (square) subband
as a function of the gate voltage: including
(solid) and not including (open) band
nonparabolicity correlation.
(Nitta. et al. Phys.Rev.B 60,7736(1999))
NTHU Colloquium
11.22.2006
NCTU
A “simple picture” for the intrinsic spin Hall effect

E
J. Sinova, et al PRL 92, 126603 (2004)
This picture is
subject to change
since it has not
incorporated the
scattering picture
as well as the
form of the spinorbit interaction
NTHU Colloquium
11.22.2006
Experimental observation of extrinsic spin Hall Effect
in thin 3D layers
Y.K. Kato, R.C.Myers, A.C. Gossard, D.D. Awschalom, Science 306,
1910 (2004)
NTHU Colloquium
11.22.2006
Experimental confirmation of spin Hall Effect in a 2D
hole gas
J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys.
Rev. Lett. 94, 047204 (2005)
NTHU Colloquium
11.22.2006
NTHU Colloquium
11.22.2006
SHE in n-type Rashba spin-orbit systems
vanishes in the presence of weak disorder
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)
A.G. Mal’shukov et al, Phys. Rev. B 71, 121308(R) (2005)
B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95, 016801
(2005)
Vertex correction is important !
NTHU Colloquium
11.22.2006
Would the entire intrinsic spin Hall story collapse
due to the presence of impurities?
SHE is found in Dresselhaus-type spin-orbit
systems
A.G. Mal’shukov et al, Phys. Rev. B 71, 121308(R) (2005)
SHE is found in cubic Rashba-type hole systems
B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95, 016801
(2005)
NTHU Colloquium
11.22.2006
NCTU
Thus far, the research on spin Hall focused on
physical quantities such as:
spin current; spin Hall conductivity.
The system of interest was mostly Rashba-type
rather than Dresselhaus-type.
Spin accumulation at the edges was essentially
obtained from the bulk spin current plus some
plausible arguments.
Explicit calculation of the spin accumulation at the
edges in a diffusive sample was in order at the time.
NTHU Colloquium
11.22.2006
Question: What sort of spin accumulation
could instrinsic Rashba SOI or
Dresselhaus SOI induce near a
diffuse sample boundaries?
Outline: Derivation of a diffusion equation for the
spin and charge densities in a 2D strip
Spin accumulation at the strip edges
and its symmetry properties
Connection between the spin flux and
the spin densities
Summary
NTHU Colloquium
11.22.2006
Derivation of a diffusion equation for the
spin and charge densities in a 2D strip
2
p
H0 =
+ hp  
2m
Rashba SOI:
hk =  (k  zˆ)
InGaAs
InAlAs
Asymmetric heterostructure
Dresselhaus SOI:
hkx =  k x (k y2   2 );
Symmetric quantum well
hky =   k y (k x2   2 )
NTHU Colloquium
11.22.2006
2DEG
2
p
H0 =
+ hp  
2m
 


H '  V (r , t )  B(r , t )   

i
i (r , t ) 
i
where  0  1, and  i   i for i  x, y, z.
H = H0 +
Vimp
+H'
Four vector density Di (r , t ) :
D0 (r , t ) = n(r , t )
Di (r , t ) = 2 Si (r , t )
NTHU Colloquium
11.22.2006
Random distribution of
Isotropic scatterers
ˆ †  r , t  i 
ˆ  r , t  S   ,  
Di  r , t   Tl 



l
down
up 
Di  r , t   i Tr  i G   r , r , t , t  
Linear response:
G

 r , r , t, t 


†
ˆ
ˆ
 i Tl    r , t     r , t  i   H '   d 


loop

j
ˆ
ˆ  r , 



H '     d r   r ,   j  r ,    

2
< ….. > denotes averaging over impurity configuration
NTHU Colloquium
11.22.2006

 r , r ,
G


d '
j
2




d



  r  2
 
 j  r ,  


G
r
,
r
,



'
G


   r, r , '
G   r , r ,    nF   G r  r , r ,    G a  r , r ,   
G   G r  G 
G

G

G
a
   k  i  hk  
G k , 
2
2




i


h

 k
k
r


   k  i  hk  
G k , 
2
2




i


h

 k
k
a


NTHU Colloquium
11.22.2006
0

Di  r ,     d r ' ij  r , r ,    j  r ',    Di  r ,  
2
j
 ij  r , r ,  
d  ' dnF
 i 
Tr G a  r , r ,  '   i G r  r , r ,    '   j 
2 d  '
Di0  r ,  

 i d 2r '
  j  r ', 
j
d '
nF  '  Tr[G r  r ', r ,  ' iG r  r , r ',  ' j
2
 G a  r ', r , ' iG a  r , r ',  ' j ] 
NTHU Colloquium
11.22.2006
Evaluation of Di0 (r , )

 
r
G
p1 , k1  q,  ' Gr k1 , p1  q,  '
 , k1  q
 , k1


' '
Q
k1
 , p1
q
q
k1  q

' '
 , p1  q
p1


Ladder diagrams do not contribute
p1  q
NTHU Colloquium
11.22.2006
NCTU
p1
 
i
q
p1




q
p1  q
j
 
p1  q
D  q,    2N0 (EF )i (q, )
0
i
N0 ( EF ) is the density of states per spin
at the Fermi energy
NTHU Colloquium
11.22.2006
Derivation of the diffusion equation of Di (r , )
Di  r ,    D  r ,   
0
i
 d r '  r , r,     r ',  
2
ij
j
j
Fourier transform of  d r ' 
2
dnF i j
i
d

'
   

2
d '
p

ij
 r , r ',    j  r ',  
:
r
G
 p, p ' q,    ' Ga  p ', p  q,  '  j  q,  
p'
NTHU Colloquium
11.22.2006
Treating the disorder within the
ladder diagram approximation
r
G
 p, p ',    ' Ga  p ' q , p  q ,  '
im .
r
 G
 p,    ' Ga  p  q ,  '  pp '
 ' r
a
Gr  p ',    ' Ga  p ' q ,  ' K 
G
p
,



'
G


'  '
 '   p  q,  '
p



+

pq


p'
p
p ' p

p ' q 

K ,  ', q  

+

+…..
+
pq


+


+….
+



NTHU Colloquium
11.22.2006


NCTU


K ,  ', q  

+



,  ', q  





pq

+….
+

p

ci
 VS
V


2

r
a
G
p
,



'
G


 
  p  q ,  ' 
p
1
ci
2
K  ,  ', q   VS 1    ,  ', q  
V
NTHU Colloquium
11.22.2006
Treating the disorder within the
ladder diagram approximation
r
G
 p, p ',    ' Ga  p ' q , p  q ,  '
im .
r
 G
 p,    ' Ga  p  q ,  '  pp '
 ' r
a
Gr  p ',    ' Ga  p ' q ,  ' K 
G
p
,



'
G


'  '
 '   p  q,  '
1
ci
2
K  ,  ', q   VS 1    ,  ', q  
V
Di  r ,    Di0  r ,   
2
d
 r 'ij  r , r,    j  r ',  
j
dnF i j
i
d '
   

2
d '
p

r
G
 p, p ' q,    ' Ga  p ', p  q,  '  j  q,  
p'
NTHU Colloquium
11.22.2006
 '
 '
O
1
i
  
Oij j'  '
2 ij
1 ci
 ,  ', q  
VS
2V
is
2
Onm
1
 ' m
  n O
'  ' '
2 
 ' '
i a
s r

Tr

G
p

q
,

'

G  p,    '


 
p
To get some feeling, let’s consider the case h0:
is ,  ', q 

h0

1
2
   ci VS N 0  EF 
2
D  vF2 / 2
 is 1  i  Dq2 
Dirtylimit : hp  
 
   , vF  q   ,
  EF
NTHU Colloquium
11.22.2006
 is  ,  ', q 


hk   hk
First order in h

?

Need to evaluate up to first order in q .
 is  ,  ', q 
linear in h and  =0
i ism
m

q

v
h
 F  pF
2

Precession of the inhomogeneous spin
polarization about the effective SOI field.
NTHU Colloquium
11.22.2006
Angular average
 is  ,  ', q  q 0, =0
h2 term




 



h


h
q

p
p
s
Tr  i


   '   i 2    '   i 2  
pq
p

 


2


hp
1
1 ci
2


s
i



VS  Tr 


3


2V
   '  p  q  i  
p 


i



'



p




2



h
1
pq

Tr  i
s
3
  '  p  i     '   i   

pq



It is diagonal, and is nonzero for i, s limited to
x, y, and z.
NTHU Colloquium
11.22.2006
 ij  ,  ', q  q 0, =0 ,
i , j  x, y , z
h2 term
 ij  ,  ', q  q 0, =0  4 2 hp2F  ij  nki nkj 
h2 term

nˆk  hk hk
D’akonov-Perel
spin relaxation
Charge-spin coupling
 l 0  ,  ', q  

2
hp3F
nlpF
p
  iq    l 0  ,  ', q 
NTHU Colloquium
11.22.2006
Diffusion equation of Di (r , )
Di  r ,    Di0  r ,   
2
d
 r 'ij  r , r,    j  r ',  
j
D ij  D  D 0   i D j
j
D ij   ij D 2  4 ijm hpmF vFm m  4 hp2F  ij  nip n pj  
Rijmm
 0  eEx
 ij
h3pF nipF

2
p
M i0
(e > 0)
0
0
0
Colloquium
D00  2 N 0eEx ; NTHU
D11.22.2006

0
;
D

D

D
0
x
y
z 0

Rashba-type strip
Equation for the spin densities for Rashba-type
semiconductor strip
y
2S z

S
D 2   zz S z   R zyy
y
y
z
y0
0
2S y

S
M

D
0
D 2   yy S y   R yzy
 x
y
y
2 x
x
E
y
2S x
D 2   xxS x  0
y
Bulk spin density : Sx = Sz = 0

Sby   M xy 0 2 yy

D00
  N0eE
NTHU Colloquium
x 11.22.2006
V.M. Edelstein
Solid State Comm. 1990
k
J.I. Inoue
et al, PRB 2003
h =  (k  zˆ)
What boundary conditions do we have for the solving
of the spin densities ?
Answer: Connecting spin flux and spin densities
Iil  r , t    d 2r ' dt '  ijl  r , r ', t  t '  j  r ', t '
j
lij  r , r ',  
d ' nF  '
 i 
Tr[G a  r , r ',  ' J il G r  r , r ',  '   j ]
2
 '


Jil   ivl  vl i 4
kl

vl 

hk  
m * kl
NTHU Colloquium
11.22.2006


Connecting spin flux and spin densities
S i 1 ijy j
I  r   D
 R  S  Sbj    iz I SH
y 2
y
i
I SH

N 0 y  hk
1 zjy j
  R Sb  eE 2 vF 
 hk 
2
2
 k x
z
NTHU Colloquium
11.22.2006
Rashba-type strip
S 1 zyy y
I  D
 R (S  Sby )
y 2
z
y
z
y

S
1 yzy z
y
I y  D
 R S
y 2
x

S
I xy   D
y
Boundarycondition:
Iiy  d 2  0
I SH  0
NTHU Colloquium
11.22.2006
NO Spin
Accumulation
at edges for
Rashba-type
strip.
Rashba-type strip
Equation for the spin densities for Rashba-type
semiconductor strip
S
zz z
zyy S
D 2  S   R
y
y
2
z
y
z
0
2S y

S

D
D 2   yy S y   R yzy
 M xy 0 0
y
y
x
x
E
y
S
D 2   xxS x  0
y
2
x
Bulk spin density : Sx = Sz = 0

Sby   M xy 0 2 yy

D00
  N0eE
NTHU Colloquium
x 11.22.2006
V.M. Edelstein
Solid State Comm. 1990
k
J.I. Inoue
et al, PRB 2003
h =  (k  zˆ)
Dresselhaus-type strip
Equation for the spin densities for Dresselhaus-type
semiconductor strip
x
x
2S z

S
zz z
zxy
D 2  S   R
y
y
E
S
xx x
xzy S
x 0 D
D 2  S   R
 Mx
y
y
x
2
x
z
0
0
2S y
D 2   yy S y  0
y
Bulk spin density : Sy = Sz = 0

S  M
x
b
x0
x
2
xx

D00
16N0eE4 ( 2  kF2 2)

NTHU Colloquium
x 11.22.2006 (8 4  kF4  4kF2 2 )
y
Dresselhaus-type strip
Spin density diffusion equation in a 2D strip
driven by a homogeneous electric field
z
 2S x


S
xx
x
xzy
D
  S   R
2
y
y
Edges of the
strip are at
y = ± d/2
 S
yy
y
D



S
0
2
y
2
y
S  0
y
x
 2S z


S
zz
z
zxy
D



S


R
y 2
y
S x and S z must be of opposite parity Boundarycondition:
Iiy  d 2  0
NTHU Colloquium
11.22.2006
Spin density diffusion equation in a 2D strip
driven by a homogeneous electric field
z
 2S x


S
xx
x
xzy
D
  S   R
2
y
y
x
 2S z


S
zz
z
zxy
D



S


R
y 2
y
Dresselhaus-type strip
Boundarycondition:
Iiy  d 2  0
Edges of the S x and S z must be of oppositeparity
strip are at
x
y = ± d/2

S
y
I x  y   D
y
S
1 zxy x
I  y   D
 R S  I SH
y
2
z
y
z
NTHU Colloquium
11.22.2006
S even
x
S odd
z
Spin densities Si  d / 2  S
for i = x, z as a functions of its width
d.
i
The inset shows the dependence of
Sz(y) on the transverse coordinate
y. Lengths are measured in unit of
lSO  vF2 / 2  vFy hky  .
(PRL.95, 146601(2005))
NTHU Colloquium
11.22.2006
NCTU
(a)
SZ
SZ
x
(b)
y
x
y
Spin densities of Sz are of odd parity in a 2D strip with /k=1.3
for the strip width d = (a) 1; (b) 10, respectively.
NTHU Colloquium
11.22.2006
NCTU
Summary
1. A diffusion approximation has been derived for the
spin density.
2. Spin accumulation at the two edges of a Dresselhaus
2D strip associated with the spin Hall effect is
obtained.
3. The spin accumulation exhibits damped oscillations
as a function of the strip width.
4. Our analysis shows that the spin current decreases
as τ2 whereas the strip spin density decrease as τ.
This explains why we still obtain noticeable spin
polarization in our dirty regime examples.
NTHU Colloquium
11.22.2006
NCTU
Latest development
NTHU Colloquium
11.22.2006
NCTU
Latest development
NTHU Colloquium
11.22.2006
NCTU
Latest development
NTHU Colloquium
11.22.2006
NCTU
Latest development
NTHU Colloquium
11.22.2006
NCTU
Thank you !
NTHU Colloquium
11.22.2006