Angular Spin Current and Persistent Spin Current

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Angular Spin Current and Persistent
Spin Current

• Q.F. Sun
• Institute of Physics
• Chinese Academy of Sciences
• X.C. Xie
• Oklahoma State University

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• Topics
• Angular spin current and
• its physical consequences
• 2. Persistent spin current

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Spin-Orbit Interaction
The interaction describes the effect of an
electrons orbital motion on the orientation of
its spin. The B field due to the relative orbital
motion of nuclear charge is given by
The potential energy of the spin momentum
The relativistic correction factor of ½, the
Thomas precession, has been included.
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Spin-orbit coupling in 2D quantum wells

• Rashba coupling (1960) from structure inversion
  asymmetry

• Dresselhaus coupling (1955) from bulk inversion
  asymmetry.

2DEG
Spin-dependent vector potential
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Spin current density and the spin continuity
equation
2. Is there a spin continuity equation? How
to define the spin current density?
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We have
We have
This is the spin continuity equation
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Notice, due to
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Both linear and angular spin currents can induce
electric fields
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An example
Model Consider a quasi 1D quantum wire having
the Rashba SO coupling
This wave function represents the spin precession
in the x-y plane while moving along the x axis.
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With the definitions given before, the spin
current densities are easily obtained
The non-zero elements of the angular spin
current
Indeed satisfy
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Persistent spin current in spin-orbit coupling
systems
The persistent charge currents in mesoscopic
rings threaded by a magnetic flux have been
extensively investigated. The persistent current
is a pure quantum effect appeared in the
equilibrium situation.
M. Buttiker, et al. Phys.Lett. 96A,
365(1983) L.P. Levy, et al. Phys.Rev.Lett.
64,2074(1990) V. Ambegaokar and U. Eckern,
Phys.Rev.Lett. 65,381(1990).
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Due to the SO coupling, a moving spin is under an
equivalent magnetic field with the field
strength depending on the momentum of the
particle. Thus, it is nature to ask whether there
exists a persistent current in a SO coupling
system, similar as for a system with an external
magnetic field.
Is there a persistent current in a SO coupling
system without an external electric or magnetic
field?
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General discussion from symmetry point of view.
1. The case of without a magnetic field nor the
SO coupling
Two symmetries

• The time-reversal invariance, i.e. T,H0.
• This symmetry leads to the Kramer degeneracy

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In the equilibrium case, the degenerated energy
eigenstates have the same occupational
probability.
For example
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2. Consider a mesoscopic ring threaded by a
magnetic flux
References M. Buttiker, et al. Phys.Lett.
96A, 365(1983) L.P. Levy, et al.
Phys.Rev.Lett. 64,2074(1990) V. Ambegaokar
and U. Eckern, Phys.Rev.Lett. 65,381(1990).
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3. The case of having SO coupling but without
a magnetic field.
The Hamilton is
Then the time-reversal symmetry still holds. But,
the symmetry (ii) is destroyed, i.e.
Therefore the persistent charge current must be
zero everywhere due to the time-reversal
symmetry. However, the persistent spin current
usually exists because of the absence of the
symmetry (ii).
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Is the persistent spin current indeed nonzero
in equilibrium in a system having a SO coupling
? Consider a concrete system with Hamilton
i) An infinite uniform 2DEGs system with
V(x,y)0 everywhere. ii) Having the Rashba
SO coupling.
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This system can be exactly solved.
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This is a very special system with V(x,y)0
everywhere. It has a rather high spatial symmetry.
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where
and
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The spin current density are easily obtained by
using their definitions
Notice that it needs to sum over all the
occupied states
We emphasize that the persistent spin current
indeed is non-zero.
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In order to understand the origin of the non-zero
, we investigate the spin current
density of a single state.
It is surprising that there is a non-zero spin
current flowing along the y-axis for the state.
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Analyze what is the electron motion in the state
A.
The case of without the Rashba coupling
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With the Rashba coupling
The electron motion with its spin precession as
shown in the figure.
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For the y-direction moving wave , its
spin element is negative, so it induces a
negative . For the -y-direction moving
wave , its spin element is
positive, so it also has a negative .
Therefore, a negative emerges for the
state A.
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The other state (state B) in the 1st sub-band
describes the electronic motion as
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The spin currents by the state A and the state B
partially cancel each other out, but the
remaining net spin currents are still quite
large.
The other two states C and D are the
time-reversal states of the states A and B,
respectively. Therefore, the spin currents of C
and D are exactly the same with those of A and
B.
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Conclusions Angular spin current can also
generate an electric field Persistent spin
current exists in a SO system Spin current may
not necessarily induce a spin accumulation.