2DEGs with Rashba spinorbit coupling: Current induced spin polarization

1
2DEGs with Rashba spin-orbit coupling Current
induced spin polarization and QPCs polarization
measurement via transverse electron focusing
Andres Reynoso, Gonzalo Usaj and C. A.
Balseiro Instituto Balseiro and Centro Atómico
Bariloche, Argentina
A.Reynoso, G. Usaj and C. A. Balseiro, PRB
73, 115342 (2006) G. Usaj and C. A.
Balseiro, Europhysics Letters 72, 621 (2005)
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Abstract
In clean two dimensional electron gases with
Rashba spin-orbit coupling a current flow induces
a spin polarization. This geometric effect
originates from special properties of the
electron's scattering at the edges of the sample.
In wide samples, the spin polarization has its
largest value at low energies (close to the
bottom of the band) and goes to zero at higher
energies. In this case, the spin polarization is
dominated by the presence of evanescent modes
which have an explicit spin component outside the
plane. In quantum wires, on the other hand, the
spin polarization is dominated by interference
effects induced by multiple scattering at the
edges. Here, the spin polarization is quite
sensitive to the value of the Fermi energy,
especially close to the point where a new channel
opens up. I will present results for different
geometries showing that the spin polarization can
be strongly enhanced. If time permits I will
mention how the transverse electron focusing in
the presence of Rashba spin-orbit coupling can be
used to measure the polarization induced by the
injector and detector QPCs.
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Motivation

• Prediction and observation of the Spin Hall
  effect

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also
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Motivation

• Observation (and Prediction) of the Spin Hall
  effect
• Is it intrinsic or extrinsic? Still open
  question.
• Despite the effect is intrinsically associated to
  finite systems, most (not all) of the theoretical
  approaches deal with infinite systems
• Let us see what the presence of a boundary does
  to the simplest escenario ? quantum transport in
  the ballistic regime

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2DEGs and Rashba spin-orbit coupling
Relativistic correction
Rashba E. I., Sov. Phys. Solid State, 2 1109
(1960).
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Bulk solution of the Rashba Hamiltonian
Spin ? k
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Reflection at a hard-wall potential

• Because of the translational invariance in the
  x-direction, the ky component of the momentum is
  conserved.
• Two reflected waves are required by the boundary
  condition.

This leads to a oscillating spin density
evanescent modes ? interesting properties
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evanescent mode

• Spin component outside the plane, a? b
• It does not depend on the sign of ?!
• It depends on the sign of kx and the boundary

• For any incident angle, the z-component of the
  spin density depends on kx

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eV/2
Spin polarization
-eV/2
Linear response
Analytic solutions are complicated (boundary
conditions) ? we use a tight binding hamiltonian
Physical quantities are calculated using Green
functions
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There is spin polarization at the edge!

• Few remarks
• This is a geometric effect
• The polarization decays with EF

y
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Narrow systems
y
y
y
y

• The sign of the spin accumulation depends on the
  relation between Ly and ?SO
• The sign of the accumulation can change close to
  the entrance of a new channel

Different widths
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Spin accumulation in small system

• Symmetries
• Sx (x,y) -Sx (-x,y) -Sx (x,-y)
• Sy (x,y) Sy (-x,y) Sy (x,-y)
• Sz (x,y) Sz (-x,y) -Sz (x,-y)

This symmetries are valid in linear response only
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Two terminal spin polarization
250 nm x 1500 nm
500 nm x 2500 nm
?x
?y
?z
?z,conv
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Effect of the sample-lead interface

• When the spin orbit coupling is turned on
  abruptly, lt?xgt can becomes non-zero
• Additional structure appears due to multiple
  reflection at the interfaces

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Fermi Energy dependence of the effect

• When EF coincides with the energy of a transverse
  mode the spin accumulation grows and can change
  its sign.

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Edge roughness effect
p Probability of modyfing a site ½
Probability of adding or substracting sites
p0
EF5.1meV
p
EF5.1meV
p0
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Shape effects
Since the effect is originated in the
surface What happens if we modify it?
EF5.1meV
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Non-uniform patterns of spin accumulation.
Shape effects
Spin polarization can be enhanced ? 10 to 100T
EF5.1meV
EF4.9meV
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L-shaped 2DEG

• The non-uniform patterns of spin accumulation
  also show that
• The inplane spin component tends to be
  perpendicular to the electron impulse
• The accumulated normal spin component is mostly
  positive in one edge and negative in the other
  edge of the sample.

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Summary

• Geometric effects in ballistic systems with
  spin-orbit coupling are important.
• When the system is biased, there is a spin
  polarization at the edges of the sample.
• It is important to take this effects into account
  when analyzing numerical data in confined
  systems.
• Although this theory (as it is) can explain some
  of the features observed in recent experiments,
  it cannot account for the magnitude of the
  observed SHE.

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Transverse electron focusing (TEF). 2DEG with
Rashba coupling.
Bulk states
Beenakker C.W. and van Houten H., in Solid State
Physics vol. 44, Academic Press, Boston, (1991).
Edge states
Experimental Setup
Due to spin-orbit coupling there are two states
with different cyclotron radius for that Fermi
energy
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TEF - 2DEG with Rashba coupling. Review
Usaj Gonzalo y Balseiro C.A., Phys. Rev. B 70,
041301(R) (2004). Reynosoa A., G. Usaj , Sánchez
M.J. y Balseiro C.A., Phys. Rev. B 70, 235344
(2004).
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P and D definition
TTu,uTd,u
Spin up
Unpolarized Incident electrons
DEVICE
Spin down
T-Tu,dTd,d
P (polarization) goes from 1 (only spin up goes
out at the output) to -1 (only spin down at the
output of the device)
Total output
UP polarized Incident electrons
TuTu,dTu,u
DEVICE
DOWN polarized Incident electrons
Total output
TdTd,dTd,u
DEVICE
D goes from 1 (only spin up produces output) to
-1 (only spin down produces output)
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QPC - 2DEG with Rashba coupling.
QPC in ballistic regime with Rashba coupling
POLARIZES!
Eto et. al. J. Phys. Soc. Jpn. 74, 1934 (2005)
Of course spin hall effect is also present! It
changes with the gate voltage.
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QPCs en presencia de interacción Rashba
Polarización 4
Reescribimos el Hamiltoniano
Debido al confinamiento lateral ky esta
cuantizado, las bandas quedan
x
Autoestados de H0 (también autoestados de sy) de
distinta banda y distinto espín son mezclados
por H ? cruce evitado
Un flujo de electrones no polarizado que
atraviesa el QPC debido a estos cruces evitados
sale con una polarización de espín no nula en
dirección y.
x
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Effect of Polarizing QPCs in the TEF
Detector QPC VG is changed
Injector QPC VG fixed
The polarizing QPCs umbalances the amplitude
of the peaks
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Effect of Polarizing QPCs in the TEF
A measure of the peak umbalance is given by
We show that FP is related to the characteristics
of the QPCs as follows
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TEF with QPCs

• In the transverse electron focusing QPCs
  introduces a umbalance in ballistic systems with
  spin-orbit coupling are important.
• This conductance umbalance in the first peak of
  focusing can be correlated with the
  characteristics of the QPCs P and D.

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Spin and charge currents
In large systems it is localized near the
boundaries It is non-zero even at equilibrium ?
meaning? No much ? it does not leave the sample
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Can we induce a charge current with a magnetic
field?
Meaning? less clear since charge is conserved It
might be observable in transport measurements
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Nanowire enhanced effects