spin transverse force on spin current in an...
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SPIN TRANSVERSE FORCEon spin current in an electric field
Dr. Shun-Qing Shen沈顺清博士
Department of PhysicsThe University of Hong Kong
August 9, 2005
Reference: S. Q. Shen, cond-mat/0506676
This work was supported by the Research Grant Council of Hong Kong.
Lorentz Force:on a moving charge in a magnetic field
BcvqF ×=
BcjF c ×=
What type of forceexerting on a moving spin in an electric field?
Spin-orbit coupling
( ) ⋅⋅⋅+∇×⋅
++
⋅+++=
VAcep
cm
BVAcep
mH B
σ
σµ
42
2
4
)(21
h
1. From Dirac equation of an electron in an electromagnetic field;2. From the k.p theory: k.p coupling between conduction band and valence band.
SU(2) gauge field
[ ] ( )
Λ++== A
cep
mHr
iv 1,1
h
VSmce
Vmce
∇×=∇×=Λ21
4σh
( )[ ]VVmce
∇∇⋅−∇=Λ×∇ σσ 2
4h
Magnetic field induced by the gauge field.
Spin Current and Vector Potential
.
;),(
),(
↓
↓↑
↑
↓↑
∂∂
↓
∂∂
↑
=
=
AAAE
AAAE
J
J
↑↓↓↑ −=⇒−= JJAA
In many body system if the vector potential for electrons is spin dependent
Special Case:
VAceP
mH
ii +
−=∑
2
, 21
σσ
Spin-dependent current
Spin current: 0≠− ↓↑ JJ
[ ] ( ) 22 )4/(2,21
4mceVVi
VSmce
Vmce
γαβγβα εσ
σ
∂∇⋅=ΛΛ
∇×=∇×=Λ
h
h
[ ] 0,212
=ΛΛ
∇×−=Λ
classical
classical Ve
βα
µ
In quantum case:
In a classical case, σµµ B−→
0:2:
=×=×
µµσσσ
ClassicaliQuantum
Quantum and classical
Non-canonical commutator of velocity
],[],[ 2
2
βαγαβγβα ε ΛΛ+−=ceB
ceivv h
Quantum mechanical version of Newton’s law
[ ]Hvim
dtdvmF ,
h==
fgh FFFF ++=
⋅⋅⋅+⋅+=Λ×∇+=
∇−×=
BVVBB
VBvceF
Beffeff
effeffh
σµ
Lorentz force Spin force in Stern-Gerlach
Ehrenfest’s Theorem
0→h
Spin electromagnetic force
( ) ( )[ ]VBVBmc
F Bg ∇⋅−∇⋅= σσµ
22
In a special case, eEVVBB zz −=∂=∇=
[ ]yxEBmceF yxz
Bg ˆˆ
2 2
σσµ+−=
This term will play an essential role in generating spin current in the Rashba system.
Spin transverse force
EJcmEe
F ESf ×= 42
2
4
{ }EEvJ ES /,4
⋅= σh
{ } VvVcm
Ff ∇×∇⋅= ,16 42 σh
eEV −=∇
Application of spin transverse force
• A single spin solution in an electric field;• Spin force balance: a relation between
spin current and spin polarization• Charge Hall effect driven by spin current
A 2D electron in a perpendicular field
)(2
2
xyyx ppmpH σσλ −+=
zjmF zsf ˆ4
2
22
×=h
λ
Li, Hu, and Shen, PRB71, 241305 (R) (2005)/cond-mat
Spin precession in spin-orbit coupling
Spin precession:
)ˆ()(2)( zptdttd
××= σλσh
The initial condition: xpp =
tttt
ttt
cxczz
yy
czcxx
ωσωσσ
σσωσωσσ
sincos)(
)(sincos)(
+=
=−=
Shen, PRB (2004)
Spin current and acceleration
( )ttmpj cxczxz
x ωσωσ sincos2
+=h
( )ttvmpv
czcxy
yx
x
ωσωσλ
λσ
sincos −−=
+=
dtdvmzjmF z
f =×= 2
224h
λ
−−=
+=
2sin
2cos
2sin ttt
py
tmpx
cz
cx
c
xt
yx
t
ωσωσω
λσ
h
For a typical system: 21211 /1010 cmne −=
Amplitude of oscillation: nmpx 0.55.12 −≈/h
Frequency of oscillation: spxc /100.13.0/2 14−×−== hλω
Relation: λλ=×
h
h x
x
pp
22
c410−=λ
Wave length X frequency = velocity?
Zitterbewegung (wriggle) of electron wave packet
Absence of transverse spin current
( )ttmp
tmp
tv
tmptv
czcxy
xy
y
yx
x
ωσωσλλσ
λσ
sincos)()(
);()(
−−=−=
+=
{ } 0, == zy
zy mp
v σσ
A moving spin in an electric field generates spin dependent transverse oscillation which does not contribute to transverse spin current.
Lorentz force & spin transverse force
Zitterbewegung of electron in 1D quantum wire
Schliemann et al, PRL (2005)
In the Lunttinger model there exists also the zitterbewegungof electronic wave packet.
Jiang et al, PRB (2005)
Resonant Spin Hall Effectin the presence of magnetic field and Rashba spin-orbit coupling
Shen, Ma, Xie, and Zhang, PRL 92, 256603 (2004)Shen, Bao, Ma, Xie, and Zhang, PRB 71, 155316 (2005)
The spin-orbit coupling competes with Zeeman splitting to introduce additional degeneracy between different Landau levels at certain magnetic fields. This degeneracy, if occurring at the Fermi level, gives rise a resonant spin Hall conductance. The spin current as well as the charge current is non-dissipative in the quantum regime.
Spin current and spin polarizationIn the presence of Zeeman splitting,
yBszx SBgjm µλ −=
Bao, Zhuang, Shen and Zhang, Cond-mat/0503592
Current and spin polarization distributionnear the edge of the sample
Spin Force in a magnetic field
],[ HcieFF fg Λ=+h[ ]
zjmF
yxBgmF
zsf
yxBs
g
ˆ4
ˆˆ2
2
22
×=
+=
h
h
λ
σσµλ
In Rashba system Fg becomes a driven force to form a spin current.
Linear Responses
[ ] ( ) [ ]( )
[ ]
[ ][ ] [ ] 0,,
)(,
)(1,1
,Im,
22
=−=
−=
−==
+−−−=
∑
∑≠
nrnfeH
mnEEmHn
nrmEEi
nHrmi
nvm
EEnvmmHn
ffeH
nn
nm
mn
mn mnmn
αα
αα
αα
σσ
σσ
ησ
σ
hh
Rashba’s work (2004) in the absence of Zeeman splitting
Balance of two spin forces
[ ] ( ) ( )
[ ] ( ) ( )xBzyxBzyy
yBzxyBzxx
BSgmjiBgpiH
BSgmjiBgpiH
µλσµσλσ
µλσµσλσ
+=+=
−=−=
h
h
42,
42,
yBzx
xBzy
Bgjm
Bgjm
σµλ
σµλ
+=
−=
fFF −=g
Non-zero Px will generatean effective magnetic field along the y direction in the Rashba coupling. As a result it induces a non-zero polarization.
Spin transverse force & charge Hall effect
τβλ
βλ
vzjmFdtdv
zjmF
zsdriven
zsf
−×−+=
×−=
)ˆ)((4
)ˆ)((4
222
222
2
h
h
In a system with the Rashba and Dresselhaus coupling,
Assume the spin current goes along the x direction, and the drift velocity orthogonal to the spin current is
τβλ zxsy jmv ,
222 )(4
−−=h
Generating spin current
By the spin force:zzBzzB BgBg σµσµ )()( ∇−=∇−
Ping Zhang et al proposed the spin force will generate a reciprocal spin Hall effect by introducing a conserved spin current.
By circularly polarized light injection
Sherman, Najmaie, and Sipe, APL (2005)Najmaie, Sherman, and Sipe, PRL (2005)
Charge Hall effect driven by spin current
−15
−10
−5
0
5
10
15
20
25
30
35
−15
−10
−5
0
5
10
15
20
25
30
35
46
810
1214
020
4060
80
−20
−10
0
10
20
30
40
n2D
[1011] λ [meVnm]
I [n
A]
a) b) laser
beam splitter
sample
µ = µ
= −µµ1
1
µ4= µ4=0
µ3= µ3=0
µ2= µ
µ2= −µ
a) b)
µ3= µ3=−µ
µ4= µ4=µ
µ1= µ1=0 µ2= µ2=0
x
y
1 Linear Rashba system2 Cubic Rashba system
Hankiewicz, Li, Jungwirth, Niu, Shen, and Sinova, cond-mat/0505597
A system with R & D coupling
)ˆ)((4 222
2
zjmf zs ×−= βλ
hLi, Hu, and Shen, PRB 71, 241305(R) (2004)
zz
xy
yx
σσ
σσ
σσ
−→
→
→
Conclusions
• An electric field will exert a transverse force on a moving spin
• Physical origin of zitterbewegung: spin precession produces an oscillating spin force
• Zeeman splitting and spin current• Charge Hall effect driven by the spin current
Berry phase
πγ +=
πγ −=
πβλβλγ22
22
−−
=
βλγ == when0riki
ei
ek ⋅
−
×
±=±
θ
θ21,,
yx
xy
kkkk
βλβλ
θ−
−=tan
0=λ
βλ ≠
Spin Hall Conductance and Berry Phase in 2DEG
γπ 28eGs =
for Rashba couplingSinova et al, PRL (2004)π8
eGs =
for Rashba and Dresselhaus couplingShen, PRB 70, 081311 (R) (2004)
Berry phase Condition: Two bands are filled. The conductance is not a constant if only one band is filled.
xzys EjG /=
In a clean limit, the linear response theory gives
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