APCTP Workshop on Semiconductor Nano-Spintronics:
Spin-Hall Effect and Related Issues
Aug.8-11,2005
Spin Hall Effect in 2D disordered systems
with spin-orbit coupling
Shuichi Murakami
(Department of Applied Physics, University of Tokyo)
Collaborators：Naoto Nagaosa (Tokyo)
Shoucheng Zhang (Stanford)
Naoyuki Sugimoto (Tokyo)
Shigeki Onoda (Tokyo)
Spin Hall effect (SHE)
r
E
Electric field induces a transverse spin current.
• Extrinsic spin Hall effect
D’yakonov and Perel’ (1971)
Hirsch (1999), Zhang (2000)
impurity scattering = spin dependent (skew-scattering)
Spin-orbit couping
up-spin
• Intrinsic spin Hall effect
No impurities required !
down-spin
impurity
Berry phase in momentum space
Intrinsic spin Hall effect in semiconductors
y
x
z
r
E
j ij = σ sε ijk Ek
p-GaAs
i: spin direction
j: current direction
k: electric field
σ s : even under time reversal = reactive response
(dissipationless)
• Nonzero in nonmagnetic materials.
Cf. Ohm’s law: j = σE
σ : odd under time reversal
= dissipative response
y
Intrinsic spin Hall effect
• p-type semiconductors
(SM, Nagaosa, Zhang (2003))
x
z
r
E
p-GaAs
Luttinger model
( )
r r 2
5  2
h 2 
H=
 γ 1 + γ 2 k − 2γ 2 k ⋅ S 
2m 
2 

r
( S : spin-3/2 matrix)
• 2D n-type semiconductors in heterostructure
(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald (2003))
r
E
Rashba model
(
r r
k2
H=
+λ σ ×k
2m
)
z
z
y
x
Experiments on spin Hall effect
• 3D n-type, spin accumulation at the edges
Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science (2004)
• 2D p-type, spin LED
J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth,
PRL (2005)
Criterion for nonzero spin Hall conductivity
r
r
r
• Different filling for bands in the same multiplet of J = L + S .
c H S.O. v ≠ 0
(Example) : GaAs
Valence band: J=3/2
Conduction band: J=1/2
Hole-doping gives a
different filling for HH
and LH bands.
Æ Spin Hall effect
Electron-doping does not
give rise to different filling
for two conduction bands
Æ NO spin Hall effect
In 2D heterostructure,
Rashba coupling lifts the
degeneracy
Æ Spin Hall effect
Disorder effect, edge effect
Green’s function method
Rashba model:
k2
H=
+ λ (σ x k y − σ y k x )
2m
+ spinless impurities (short-range pot.)
Intrinsic spin Hall conductivity (Sinova et al.(2003)) σ S =
+ Vertex correction in the clean limit
(Inoue, Bauer, Molenkamp(2003))
σS = 0
Inoue, Bauer, Molenkamp (2004)
• clean limit
• δ -fn. impurity
σ S vertex = −
e
8π
e
8π
+
+ ⋅⋅⋅
• Calculation by Keldysh formalism (Mishchenko, Shytov, Halperin (2004))
Spin Hall current does not flow at the bulk – consistent withσ S = 0
Spin current only flows near the electrodes
SHE
Spin accumulation
No SHE
SHE
SHE in disordered Rashba model -- Green’s function-Inoue, Bauer, Molenkamp
(2004)
Mishchenko, Shytov,Halperin
(2004)
Raimondi,Schwab (2004)
Dimitrova (2004)
Liu, Lei (2004,2005)
Chalaev Loss (2004)
Dimitrova; Chalaev,Loss : J s ≡
E Fτ
∆τ
Potential
∞
∞
δ-fn.
any
any
δ-fn.
∞
∞
large
large
∞
large
∞
large
1
{v y , S z }∝ dS y
2
dt
any
δ-fn
any
δ -fn
•diffusion equation
•Might be nonzero for finiteEFτ
•Arbitrary dispersion (?)
•Arbitrary form of Rashba coupling (?)
•Zero even with Dresselhaus (’05)
•Might be nonzero for T>0
•Zero even with Dresselhaus.
•Weak localization correction=0
J s = 0 for steady state
Definition of spin current is not unique
in the presence of spin-orbit coupling
• Spin-orbit coupling Æ spin is not conserved Æ no unique def. of spin current
• Noether’s theorem cannot be applied.
•
[
∂Si
∂
+ ∇ ⋅ J i(spin) = 0 ⇐ 0 = ∫ S i d d r = i H , ∫ S i d d r
∂t
∂t
]
Eq. of continuity requires conservation of spin, but the
spin is not conserved in these models
“Conventional” definition of the spin current:
Js ≡
1
{v y , S z }
2
SHE in disordered Rashba model -- Green’s function + numerics --
Nomura, Sinova, Sinitsyn, MacDonald,
cond-mat/0506189
DC spin Hall conductivity is likely to vanish.
Sheng,Sheng, Weng, Haldane
cond-mat/0504218
σs ≈ 0
Outline
Disorder effect in Rashba model
• Convensitonal spin current J s ≡
1
{v y , S z }
2
• Keldysh formalism
σs = 0
in the bulk
under general conditions
• Conserved effective spin current
(Zhang et al.(’05))
Depends on impurity potential : Extrinsic
Nonzero but much smaller than
e
8π
For general system:
σ s is nonzero.
SHE in disordered Rashba model -- Green’s function-Inoue, Bauer, Molenkamp
(2004)
Mishchenko, Shytov,Halperin
(2004)
Raimondi,Schwab (2004)
Dimitrova (2004)
Liu, Lei (2004,2005)
Chalaev Loss (2004)
This talk
Dimitrova; Chalaev,Loss : J s ≡
E Fτ
∆τ
Potential
∞
∞
δ-fn.
any
any
δ-fn.
∞
∞
large
large
∞
large
∞
large
any
1
{v y , S z }∝ dS y
2
dt
any
δ-fn
any
δ -fn
•diffusion equation
•Might be nonzero for finiteEFτ
•Arbitrary dispersion (?)
•Arbitrary form of Rashba coupling (?)
•Zero even with Dresselhaus (’05)
•Might be nonzero for T>0
•Zero even with Dresselhaus.
•Weak localization correction=0
•Even wiith higher-order Born approx.
•Weak localization correction=0
J s = 0 for steady state
Js
1 dω d 2 p
K
= ∫
tr
J
G
s
2i 2π ∫ (2π )2
(
)
(
r r
k2
(e.g.) Rashba model : H =
+λ σ ×k
2m
When J s is proportional to S& y ...
Js ≡
Js
1 1
dω d 2 p
=
tr σ y H , G K
2
∫
∫
2i 4imλ 2π (2π )
( [
[H , G ]
K
[
1
{v y , S z } = 1 H , σ y
2
4imλ
)
z
]
])
No contribution for
• 1st Born
• higher Born
• weak localization corr.
r
i r
K
= −ieE ⋅ ∂ pr G − eE ⋅ ∂ pr H , ∂ ω G K
2
i r
− eE ⋅ (∂ pr Σ∂ ω G − ∂ ω Σ∂ pr G )K
2
i r
+ eE ⋅ (∂ pr G∂ ω Σ − ∂ ω G∂ pr Σ )K − ([Σ, G ])K
2
{
No contribution to J s
}
Js = 0
If Ô
r
is independent of r
and
t
,
dOˆ (t )
=0
dt
r
in the steady state under uniform E
• Rashba model
dS
1
(conventional) spin current operator J s ≡ {v y , S z }∝ y
2
dt
r
r
dr
• Charge current in diffusive metal J ≡ e
dt
r
J ≠0
Js = 0
Conserved spin current
Js ≡
d
( yS z )
dt
Calculation of
(Zhang, Shi, Xiao, Niu, cond-mat/0503505)
Eq. of continuity for spin
r
d
∇ ⋅ Js + S z = 0
dt
Js
in response to electric field ?
r
r
E
1) Spatial modulation of
: E = ( Ee iqy ,0,0)
2) Calculate
1 dω d 2 p
&
Sz = ∫
tr S& z G K
2
∫
2i 2π (2π )
3) Calculate J s
(
)
by taking ∂ q
1
dω d 2 p
& GK
J s = lim (− i∂ q )∫
tr
S
z
2i q →0
2π ∫ (2π )2
(
)
1
1 dω d 2 p
J s = lim ∫
tr S& z G K
2
∫
2i q →0 iq 2π (2π )
(
)
No contribution
[H , G ] = −ieEe
K
iqY
(
)
( [
i tr [H , S z ]G K = −i tr S z H , G K
])
Conventional
spin current
{
} {
i
q
∂ p x G K − eEeiqy ∂ p x H , ∂ ω G K − ∂ p x H , GEK
2
2
{
}
Js
}
1
− eEtqeiqyε ijz ∂ pi H , ∂ p j G K
2
q
− eE ∂ pi Σ∂ p j G − ∂ pi G∂ p j Σ
K
2
i ~
~
~
~
r
~
− ∂t Σ∂ ω G − ∂ ω Σ ∂t G − ∂t G∂ ω Σ + ∂ ω G∂t Σ K
(
∂t = ∂ t − eE ⋅ ∂ pr )
2
A
A
q  ∂Σ R K
∂G R K 
K ∂Σ
K ∂G
GE + GE
Σ E − ([Σ, G ])K
−
− ΣE
−


2  ∂p y
∂p y
∂p y
∂p y

No contribution for
(
)
(
)
Torque dipole density
Ps
• 1st Born
• higher Born
• weak localization corr.
∂Σ
1 dω d 2 p   ∂ Σ
Ps = ∫
S
G
G
+
tr
z
∂p y
4i 2π ∫ (2π )2   ∂p y

 

 
 E ,K 
∂H
1 dω d 2 p   ∂H
S
G
G
= ∫
+
tr
z
∂p y
4i 2π ∫ (2π )2   ∂p y

 
1 dω d 2 p  1  ∂H
= ∫
tr 
, S z GK 
2
∫
2i 2π (2π )  2  ∂p y
 
 

 
 E ,K 
Js
Js ≡
1
{v y , S z }
2
1 dω d p   ∂ Σ
∂Σ
Ps = ∫
S
G
G
tr
+
z
4i 2π ∫ (2π )2   ∂p y
∂p y

2
In general systems
 

 
 E ,K 
• No divergence in the clean limit
• ∝ ∂Vr
∂p
Æ Ps
always extrinsic
zero for
•1st Born approx.
Js
1st
δ
-fn. impurity
=0
• 2nd Born approx
2
Rashba model
2 nd
σsH/α2pF [e/8π]
Js
α : range of impurity pot.
(short but finite)
≠0
0.02
0.02
0.01
0.01
0.00
0.00
0.8
• σ s = 0 in the clean limit
• σ s = 0 in the short-range limit
0.6
1/τεF
0.4
0.4
0.3
0.2
typically
λp F
EF
< 0.2,
1
< 0.05
τE F
σ s << 10 −5 ⋅
e
8π
0.2
0.1
λpF/εF
Spin Hall effect in the Rashba model
Conserved spin current J s = J s + Ps
Impurity
potential
r
δ (r )
r r
V ( p − p′)
r r
V ( p, p′)
Born approx.
Conventional
spin current
Torque dipole
density
Js
Ps
1st
0
0
higher
0
0
0
0
0
finite
0
finite
0
finite
1st
higher
1st
higher
Nonzero for general
spin-orbit-coupled system
Zero in general
-- Ps is extrinsic.
On Rashba-like tight-binding models …
− V1 + iV 2σ x
− V1 − iV 2σ y
without
vertex
correction
“Ando model”
This case also satisfies J s ∝ S& y
Js = 0
No intrinsic SHE
with
vertex
correction
− V3
− V1 + iV 2σ x
− V1 − iV 2σ y
intrinsic SHE
σs ≈
e
⋅ (const.)
8π
EF
Modified Rashba model
(
r r
k2
2
H=
+ λ (a + bk ) σ × k
2m
)
z
-- SHE is nonzero (SM, PRB(2004))
iλ (k −σ + − k +σ − )
(
iλ k − σ + − k + σ −
3
3
)
iλk 2 (k −σ + − k +σ − )
Nomura, Sinova, Sinitsyn, MacDonald,
cond-mat/0506189
In Rashba-like systems
• Intrinsic SHE (if nonzero) is of the order of
e
8π
.
Summary
• Rashba model
conserved effective spin current J s = J s + Ps
Conventional spin current
Js =
1
{v y , S z }
2
J s ∝ S& y yields J s = 0 under general conditions.
• Any impurity potential
• Finite τ
• Higher-order Born approx.
+ weak localization corr.
Torque dipole density Ps
Nonzero but depends
on the impurity pot. -- extrinsic
Zero in the clean limit
Zero for δ -fn. potential
Small for Rashba model : σ s << 10−5 ⋅
e
8π
For generalized models
it gives σ s ≈ O e 
 8π 
Nonzero spin Hall effect in band insulators
1) Zero-gap semiconductors : α-Sn, HgSe, HgTe, β-HgS…
conventional
zero-gap
CB
EF
EF
α-Sn under
9
2
uniaxial stress = 3.2 ×10 dyn/cm
Æ energy gap 44meV
“LH”
0.2
σs
( e/a)
Estrain=0
Estrain=0.002
0.15
Estrain=0.02
Estrain=0.2
0.1
HH
“HH”
0.05
0
“CB”
LH
-0.05
-0.1
k =0
k
k =0
k
-0.15
-2
-1.5
-1
-0.5
0
0.5
γ2 γ 3
/
• Spin Hall effect is nonzero (≈ 0.1e / a ) in band insulators
• Uniaxial strain Æ finite gap at k=0
Spin Hall insulator
1
1.5
2
2) Narrow-gap semiconductors : PbS, PbSe, PbTe
Doubly degenerate
0.05
σs
EF
( e/a)
Mva=0.15
Mva=0.25
Mva=0.35
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-4
k=
π
a
-3
(1,1,1)
Direct gap (0.15eV-0.3eV) at 4 equivalent L-points
r
r
r
H = vk ⋅ pˆ τ 1 + λvk ⋅ ( pˆ × σ )τ 2 + Mv 2τ 3
r
r
σ : spin τ :orbital
• Spin Hall effect is nonzero (≈ 0.04e / a ) in band insulators
Spin Hall insulator
-2
-1
0
1
2
3
4
λ
Mva
λ
PbS
0.26
1.2
PbSe
0.16
1.4
PbTe
0.35
3.3