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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). NCTU. NCTU. - PowerPoint PPT Presentation
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NTHU Colloquium 11.22.2006
Electrical Means of ManipulatingElectron Spins in Semiconductors
C. S. ChuDept. of Electrophysics
National Chiao Tung University
NCTU
Collaborators: A.G. Mal’shukov (RAS) L. Y. Wang (NCTU)
NTHU Colloquium 11.22.2006
NCTU
NTHU Colloquium 11.22.2006
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.
Quoted from the abstract of “Spintronics: Fundamentals and applications”
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NTHU Colloquium 11.22.2006
How about transport of spins in non-magnetic semiconductor using only electrical control?
Schemes making use of spin Hall effect
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Schemes other than spin Hall effect
NTHU Colloquium 11.22.2006
NCTUA 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
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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
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A simple picture for the extrinsic spin Hall effect
J.E. Hirsch, PRL 83, 1834 (1999)
NTHU Colloquium 11.22.2006
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2( ) ( )
2pH V r p Vm
2( ) ( )
2pH V r p Em
( ) ( )mr V E p E
E
Potential energy
For electron incident upon theLHS of the attractive scatterer, again the spin up particle is deflected more tothe left and the spin down particle is deflected more to the right.
More on the simple picture for the extrinsic spin Hall effect
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 )
analogE
InGaAs
InAlAs
2DEG
Heterostructure:
Structure inversion asymmetry:
E
NTHU Colloquium 11.22.2006
EV
In Lab. frame
Beff
In the rest frame of an electron
I
I
Effective magnetic field induced by the effective current I.
Effective magnetic field induced by the effective current I.
An electron moves between
two charged plane
An electron moves between
two charged plane
The SOI hamiltonian is given by
B H V Eeff
Rashba 0 ˆH p z
where is called the Rashba constant.
0
NTHU Colloquium 11.22.2006
Fig.3.Dispersion relation for a 2D Rashba-type system and the Rashba constant .
ky
kx
E
0 0.13
2 2 2 22 0 D x y x yE k k k k +
NTHU Colloquium 11.22.2006
Rashba spin-orbit interaction (SOI)
: normal to interface: the Pauli spin operato
Rashba term:
r
so ˆ
ˆ
H
v
p v
• 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.
[ 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 by a metal gateTuning of the coupling constant by a metal gate0
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))
InGaAs
InAlAs
2DEG2DEG
Gat
e
gV
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 spin-orbit interaction
NTHU Colloquium 11.22.2006
Experimental observation of extrinsic spin Hall Effect in thin 3D layersY.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 gasJ. 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
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)
Would the entire intrinsic spin Hall story collapse
due to the presence of impurities?
NTHU Colloquium 11.22.2006
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Thus far, the research on spin Hall focused onphysical quantities such as:
spin current; spin Hall conductivity.
The system of interest was mostly Rashba-typerather than Dresselhaus-type.
Spin accumulation at the edges was essentially obtained from the bulk spin current plus someplausible 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
0 = + 2 p
pH h
m
Rashba SOI:
ˆ = ( )k
h k z
InGaAs
InAlAs
2DEG
Asymmetric heterostructureDresselhaus SOI:
2 2
2 2
= ( );
= ( )
xk x y
yk y x
h k k
h k k
Symmetric quantum well
NTHU Colloquium 11.22.2006
2
0 = + 2 p
pH h
m
Four vector dens (ity , ) : iD r t
0 ( , ) = ( , )D r t n r t
( , ) = 2 ( , )i iD r t S r t
0 imp = + + 'H H HV
0where 1, and for , , .i i i x y z
i
ii trtrBtrVH ),( ),(),( '
Random distribution of Isotropic scatterers
NTHU Colloquium 11.22.2006
†ˆ ˆ, , , ,ii l l down upD r t T r t r t S
, ,, ,ii G rD r t i T rr t t
†ˆ ˆ, , ', , , l
loop
i T r t r t i H dG r r t t
2 ˆ ˆ' , , ,jjH d r r r r
Linear response:
< ….. > denotes averaging over impurity configuration
NTHU Colloquium 11.22.2006
2 '
2
, , , ' , , '
, ,
jj r
dd r G r r G r r
G r r
, , , ,, , r aFn G r r G r rG r r
r
a
G G G
G G G
2 2
2 2
,
,
kr
k k
a
k k
k
k k
G ki h
G
h
i h
i
hik
NTHU Colloquium 11.22.2006
2 0, ' ,',, ,i j ij
ij rD r d r r rr D
'
, , ' , , ' 2
'
, ,
a rF ji
ij
dndi Tr G r r G r r
d
r r
0
2 [ ', , ' , ', '
',
', ,
,
'' '
' , ', '
2
]
jr rj
ja a
i
iF
j
i
r Tr G r r G r r
G r r G r r
D r
di d r n
NTHU Colloquium 11.22.2006
1 1 1 1, , ' , , 'r rG p k q G k p q
0Evaluation of ( , )iD r
q q
1,k q
1, p
1, p q
1, k
Q
1k q ''
' '
1p
1k
1p q
Ladder diagrams do not contribute
NTHU Colloquium 11.22.2006
00, 2 ( ) ( , )i F iD q N E q
1p
1p
1p q
1p q
ji
q q
0 ( ) is the density of states per spin
at the Fermi energyFN E
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NTHU Colloquium 11.22.2006
2Fourier transform of ' , ', ' :,ij jd r rr r
0 2, , ' , , ',i i ij jj
D r D r d r r r r
Derivation of the diffusion equation of ( , )iD r
'
, ' ,'2
, ' ', , ''
r a
pj
iF j
p
G p p qq G p p qdni
dd
NTHU Colloquium 11.22.2006
+
p
p q
'p p
p
'p
p q
'p q
+ + +…..
Treating the disorder within the ladder diagram approximation
, ',K q
+
+
+….
'
' '
.
''
, ', ' '
, ' , '
', ' ' , ' , ' '
, '
,
,
r app
r a r
m
a
r a
i
G p G p
G p p G p q
q
G p G p q G p G p
q
K q
p
NTHU Colloquium 11.22.2006
, ',K q
+
+
+….
12, ', 1 , ',i
S
cK q V q
V
, ', q
2, ' , 'r ai
Sp
cV G p G p q
V
p
p q
NCTU
NTHU Colloquium 11.22.2006
Treating the disorder within the ladder diagram approximation
'
' '
.
''
, ', ' '
, ' , '
', ' ' , ' , ' '
, '
,
,
r app
r a r
m
a
r a
i
G p G p
G p p G p q
q
G p G p q G p G p
q
K q
p
12, ', 1 , ',i
S
cK q V q
V
'
, ' ,'2
, ' ', , ''
r a
pj
iF j
p
G p p qq G p p qdni
dd
0 2, , ' , , ',i i ij jj
D r D r d r r r r
NTHU Colloquium 11.22.2006
'' ' '
' '
1
2n m
nmO O
'' ' '
1
2i j
ijij
O O
21, ', , ' , '
2is i a s ri
Sp
cq V Tr G p q G p
V
To get some feeling, let’s consider the case h0:
2
0, ', 1is is
hq i Dq
2
0
2
1
2
/ 2
i S F
F
c V N E
D v
F
F
p
E
qv
h
, ,
:limitDirty
NTHU Colloquium 11.22.2006
First order in
, ', ?is
hq
2linear in h and =0, ',
F
ismis m
F p
iq q v h
Need to evaluate up to first order in . q
kkhh
Angular averagePrecession of the inhomogeneous spin polarization about the effective SOI field.
NTHU Colloquium 11.22.2006
2 2
22
3
2
3
2 0, =0 term
, ',
' '
1 1
2 ''
1
' '
is
p p qi s
p p q
pi siS
p p qp
p qi s
p p q
qh
q
h hTr
i i
hcV Tr
V ii
hTr
i i
It is diagonal, and is nonzero for i, s limited to x, y, and z.
NTHU Colloquium 11.22.2006
2
0, =0 term
, ', , , , ,ijqh
q i j x y z
2 2
2 0, =0 term
, ', 4F
ij ij i jp k kq
hq h n n
Charge-spin coupling
kkkhhn
ˆ
D’akonov-Perelspin relaxation
0 3 02
, ', , ',F
F
lpl l
p
nq h iq q
p
NTHU Colloquium 11.22.2006
Diffusion equation of ( , )iD r
0 2, , ' , , ',i i ij jj
D r D r d r r r r
0
32 2
24 4 F F
F F
ijjj
ip pij ij ijm m m ij i j
p F m p p p
D D D i D
h nD D h v h n n
p
ijmmR ij 0iM
0
0 0 0 00 0 0
(e > 0)
2 ; 0 ; 0x y z
eEx
D N eEx D D D D
NTHU Colloquium 11.22.2006
Equation for the spin densities for Rashba-typesemiconductor strip
y
SRS
y
SD
yzyyzzz
z
2
2
y
x
E
0 2
2
xxx
x
Sy
SD
x
DM
y
SRS
y
SD
yx
zyzyyyy
y
0
00
2
2
2
Bulk spin density : Sx = Sz = 0
eENx
DMS yyy
xy
0
000
b 2
ˆ = ( )k
h k z V.M. Edelstein
Solid State Comm. 1990J.I. Inoue et al, PRB 2003
Rashba-type strip
NTHU Colloquium 11.22.2006
What boundary conditions do we have for the solving of the spin densities ?
Answer: Connecting spin flux and spin densities
2, ' ' , ', ' ', 'l li ij j
j
I r t d r dt r r t t r t
, ',
''[ , ', ' , ', ' ]
2 '
lij
F a li
jr
r r
ndi Tr G r r G rJ r
4ill
ili vvJ
*l
l kl
kv h
m k
NTHU Colloquium 11.22.2006
Connecting spin flux and spin densities
b
1
2
iy ijy j j
i iz SH
SI r D R S S I
y
20
b2 2
1 zjy j y kSH F k
x z
N hI R S eE v h
k
NTHU Colloquium 11.22.2006
Rashba-type strip
)(2
1byyzyy
zyz SSR
y
SDI
zyzyy
yy SR
y
SDI
2
1
y
SDI
xyx
0 2
:conditionBoundary
dI yi
0SHI
NO Spin Accumulation at edges for Rashba-type strip.
NTHU Colloquium 11.22.2006
Equation for the spin densities for Rashba-typesemiconductor strip
y
SRS
y
SD
yzyyzzz
z
2
2
y
x
E
0 2
2
xxx
x
Sy
SD
x
DM
y
SRS
y
SD y
x
zyzyyyy
y
0
002
2
Bulk spin density : Sx = Sz = 0
eENx
DMS yyy
xy
0
000
b 2
ˆ = ( )k
h k z V.M. Edelstein
Solid State Comm. 1990J.I. Inoue et al, PRB 2003
Rashba-type strip
NTHU Colloquium 11.22.2006
Equation for the spin densities for Dresselhaus-typesemiconductor strip
y
SRS
y
SD
xzxyzzz
z
2
2
y
x
E
0 2
2
yyy
y
Sy
SD
x
DM
y
SRS
y
SD x
x
zxzyxxx
x
0
002
2
Bulk spin density : Sy = Sz = 0
)48(
)2(162
2244
2240
000
b
FF
Fxxxx
x
kk
keEN
x
DMS
Dresselhaus-type strip
NTHU Colloquium 11.22.2006
Spin density diffusion equation in a 2D stripdriven by a homogeneous electric field
y
SRS
y
SD
zxzyxxx
x
2
2
0 2
2
yyy
y
Sy
SD
y
SRS
y
SD
xzxyzzz
z
2
2
Edges of the strip are at y = ± d/2
0 yS
parity opposite of bemust and zx SS 0 2
:conditionBoundary
dI yi
Dresselhaus-type strip
NTHU Colloquium 11.22.2006
Spin density diffusion equation in a 2D stripdriven by a homogeneous electric field
y
SRS
y
SD
zxzyxxx
x
2
2
y
SRS
y
SD
xzxyzzz
z
2
2
Edges of the strip are at y = ± d/2
parity opposite of bemust and zx SS
0 2
:conditionBoundary
dI yi
x
yx
SI y D
y
1
2
zy zxy xz SH
SI y D R S I
y
odd
even z
x
S
S
Dresselhaus-type strip
NTHU Colloquium 11.22.2006
Spin densities for i = x, z as a functions of its width d.
The inset shows the dependence of Sz(y) on the transverse coordinate y. Lengths are measured in unit of .
(PRL.95, 146601(2005))
/ 2 iiS d S
2 / 2SO F Fy kyl v v h
NTHU Colloquium 11.22.2006
xx
ZS ZS
yy
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.
(a) (b)
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NTHU Colloquium 11.22.2006
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.
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NTHU Colloquium 11.22.2006
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at/0
6101
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Thank you !
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