Transcript Document

2DEG’s with Rashba spin-orbit coupling:
Current induced spin polarization# and
QPC’s 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)
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 spinorbit coupling can be used to measure the polarization induced by
the injector and detector QPCs.
Motivation
 Prediction and observation of the Spin Hall effect
also
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
2DEGs and Rashba spin-orbit coupling
H SO 
2 2
2m c
 .[V (r )  p]
Relativistic correction
Rashba E. I., Sov. Phys.
Solid State, 2 1109
(1960).
H SO 

( p y x  px y )

GaAs/AlGaAs
InSb/InAlSb
0.51 meV nm
510 meV nm
Can be controlled
Nitta et al. PRL 78 (1997)
Miller et al PRL 90 076807 (2003)
Bulk solution of the Rashba Hamiltonian
Spin  k
Two bands
Fermi surface
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
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
Spin polarization
eV/2
-eV/2
Linear response:
Analytic solutions are complicated (boundary conditions)
 we use a ‘tight binding’ hamiltonian
Physical quantities are calculated
using Green functions
EF = 0
There is spin polarization
at the edge!
Few remarks:
This is a geometric effect
 The polarization decays
with EF
EF =5meV
y
Characteristic lenght

is zero
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
Spin accumulation in small system
In linear response
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
Two terminal spin polarization
250 nm x 1500 nm
x
y
z
z,conv
500 nm x 2500 nm
Effect of the samplelead interface
 When the spin orbit
coupling is turned on
abruptly, <x> can
becomes non-zero
Additional structure
appears due to multiple
reflection at the
interfaces
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.
Edge roughness effect
p: Probability of modyfing a site
½ : Probability of adding or
substracting sites
p=0
EF~5.1meV
p=
EF~5.1meV
p=0
Shape effects
Since the effect is originated in
the surface:
x1
Transmitance
2
DOS[1/( t a )]
What happens if we modify it?
EF~5.1meV Resonant
30
state
15
14
12
10
4,0
4,5
5,0
E [meV]
5,5
6,0
Shape effects
Non-uniform patterns of spin accumulation.
Spin polarization can be enhanced  10 to 100T
EF=4.9meV
EF=5.1meV
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.
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.
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
C
B
x
(a)
y
2
1
D
A
Due to spin-orbit
coupling
there are two
states with
different
cyclotron radius
for that Fermi
energy
TEF - 2DEG with Rashba coupling. Review
O2
(a)
(b)
O1
x
y
2
1
O2
(c)
O1
2
1
O1
2
O2
(d)
1
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).
P and D definition
Spin up*
Unpolarized
DEVICE
Incident electrons
T+=Tu,u+Td,u
Spin down*
P*  (T  T ) / T
T-=Tu,d+Td,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)
UP polarized
DEVICE
Incident electrons
DOWN polarized
Incident electrons
DEVICE
Total output
Tu=Tu,d+Tu,u
Total output
Td=Td,d+Td,u
D*  (Tu  Td ) / T
D goes from 1 (only spin up produces output) to -1 (only spin
down produces output)
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.
QPCs en presencia de interacción Rashba: Polarización [4]
Reescribimos el Hamiltoniano H
p2 
H 
 px y
*
2m 
0
 H0  H'
H '


p y x
Debido al confinamiento lateral ky esta
cuantizado, las bandas quedan:
y
2
x
x
2

kx
0
E   E y ,n 
  kx
*
2m
Autoestados de H0 (también
autoestados de sy) de distinta
banda y distinto espín son
mezclados por H’  cruce evitado
x
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.
Effect of Polarizing QPCs in the TEF
C
B
x
(a)
Detector QPC:
VG is changed
y
2
1
D
A
Injector QPC:
VG fixed
The polarizing
QPCs
umbalances the
amplitude of the
peaks
Effect of Polarizing QPCs in the TEF
A measure of the peak umbalance is given by
FP
0,5
AB1,1  AB1, 2
=10
0,0
-0,5
0,8
FP
FP 
AB1,1  AB1, 2
este es uno
By=0
By=10T
By=-10T
0,4
0,0 =20
0,6
0,8
1,0
Vg/EF
We show that FP is related to the
characteristics of the QPCs as follows:
~ ~
P1  D2
P1  D2
FP 
~~ 
1  P1D2 1  P1D2
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.
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
Can we induce a charge current with a magnetic field?
Meaning?
less clear since charge is
conserved
It might be observable in
transport measurements
Nanowire: enhanced effects