SPIN TRANSVERSE FORCE on spin current in an electric field

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SPIN TRANSVERSE FORCEon spin current in an
electric field

• Dr. Shun-Qing Shen
• ?????
• Department of Physics
• The University of Hong Kong
• August 19, 2005

Reference S. Q. Shen, cond-mat/0506676 This
work was supported by the Research Grant Council
of Hong Kong.
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Lorentz Forceon a moving charge in a magnetic
field
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What type of forceexerts on a moving spin in an
electric field?
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Spin-orbit coupling
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.
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SU(2) gauge field
Magnetic field induced by the gauge field.
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Spin Current and Vector Potential
In many body system if the vector potential for
electrons is spin dependent
Spin-dependent current
Special Case
Spin current
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Quantum and classical
In quantum case
In a classical case,
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Non-canonical commutator of velocity
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Quantum mechanical version of Newtons law
Relativistic quantum correction
Ehrenfests Theorem
Lorentz force
Spin force in Stern-Gerlach
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Spin electromagnetic force
In a special case,
This term will play an essential role in
generating spin current in the Rashba system.
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Spin transverse force
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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

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A 2D electron in a perpendicular field
Li, Hu, and Shen, PRB71, 241305 (R)
(2005) /cond-mat0502102
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Spin precession in spin-orbit coupling
Spin precession
Spin operator at t0.
The initial condition
Shen, PRB (2004)
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Spin current and acceleration
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Zitterbewegung (wriggle) of electron wave packet
For a typical system
Amplitude of oscillation
Frequency of oscillation
?
Relation
Wave length X frequency velocity
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Absence of transverse spin current
A moving spin in an electric field generates spin
dependent transverse oscillation which does not
contribute to transverse spin current.
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Lorentz force spin transverse force
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Zitterbewegung of electron in 1D quantum wire
In the Lunttinger model there exists also the
zitterbewegung of electronic wave packet. Jiang
et al, PRB (2005)
Schliemann et al, PRL (2005)
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Resonant Spin Hall Effectin the presence of
magnetic field and Rashba spin-orbit coupling
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.
Shen, Ma, Xie, and Zhang, PRL 92, 256603
(2004) Shen, Bao, Ma, Xie, and Zhang, PRB 71,
155316 (2005)
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Spin current and spin polarization
In the presence of Zeeman splitting,
Current and spin polarization distribution near
the edge of the sample
Bao, Zhuang, Shen and Zhang, Cond-mat/0503592
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Spin Force in a magnetic field
In Rashba system Fg becomes a driven force to
form a spin current.
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Linear Responses
Rashbas work (2004) in the absence of Zeeman
splitting
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Balance of two spin forces
Non-zero Px will generate an effective magnetic
field along the y direction in the Rashba
coupling. As a result it induces a non-zero
polarization.
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Spin transverse force charge Hall effect
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
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Generation of spin current
By the spin force
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)
By spin Hall effect?
2DEG? 2DHG,
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Charge Hall effect driven by spin current
1 Linear Rashba system 2 Cubic Rashba system
Hankiewicz, Li, Jungwirth, Niu, Shen, and Sinova,
cond-mat/0505597
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A system with R D coupling
Li, Hu, and Shen, PRB 71, 241305(R) (2004)
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Conclusions

• An electric field will exert a transverse force
  on a moving spin
• Physical origin of zitterbewegung spin
  precession produces an oscillating spin force
• Balance of spin force a relation between the
  Zeeman splitting and spin current, a practicable
  way to extract the distribution of spin current
  by measuring spin polarization.
• Charge Hall effect driven by the spin current

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Berry phase
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Spin Hall Conductance and Berry Phase in 2DEG
In a clean limit, the linear response theory gives
for Rashba coupling Sinova et al, PRL
(2004)
for Rashba and Dresselhaus coupling Shen,
PRB 70, 081311 (R) (2004)
Berry phase
Condition Two bands are filled. The conductance
is not a constant if only one band is filled.