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