1. Andres Reynoso, Gonzalo Usaj and C. A. Balseiro
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, (2006)
G. Usaj and C. A. Balseiro, Europhysics Letters 72, 621 (2005)

2. 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.

3. Motivation
Prediction and observation of the Spin Hall effect

4. also

5. 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

6. 2DEGs and Rashba spin-orbit coupling
Relativistic correction
Rashba E. I., Sov. Phys. Solid State, (1960).
Can be controlled
Nitta et al. PRL 78 (1997)
Miller et al PRL (2003)
GaAs/AlGaAs
InSb/InAlSb
0.5-1 meV nm
5-10 meV nm a

7. Bulk solution of the Rashba Hamiltonian
Fermi surface
Spin  k
Two bands

8. evanescent modes  interesting properties
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

9. 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

10. Spin polarization Linear response:
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

11. There is spin polarization at the edge!
EF = 0
EF =5meV
There is spin polarization at the edge!
Few remarks:
This is a geometric effect
The polarization decays with EF
is zero y
Characteristic lenght

12. 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

13. 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

14. Two terminal spin polarization
250 nm x 1500 nm
500 nm x 2500 nm x y z
z,conv

15. Effect of the sample-lead 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

16. 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.

17. 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

18. Shape effects Since the effect is originated in the surface:x1
Since the effect is originated in the surface:
What happens if we modify it?
EF~5.1meV

19. Shape effects EF=5.1meV EF=4.9meV
Non-uniform patterns of spin accumulation.
Shape effects
Spin polarization can be enhanced  10 to 100T
EF=5.1meV
EF=4.9meV

20. 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.

21. 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.

22. 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 1 2 B A C D
(a) y x

23. TEF - 2DEG with Rashba coupling. Review
Usaj Gonzalo y Balseiro C.A., Phys. Rev. B 70, (R) (2004).
Reynosoa A., G. Usaj , Sánchez M.J. y Balseiro C.A., Phys. Rev. B 70, (2004).

24. P and D definition T+=Tu,u+Td,u T-=Tu,d+Td,d Tu=Tu,d+Tu,u Td=Td,d+Td,u
Spin up*
Unpolarized
Incident electrons
DEVICE
Spin down*
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
Incident electrons
Total output
Tu=Tu,d+Tu,u
DEVICE
DOWN polarized
Incident electrons
Total output
Td=Td,d+Td,u
DEVICE
D goes from 1 (only spin up produces output) to -1 (only spin down produces output)

25. 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.

26. QPCs en presencia de interacción Rashba: Polarización [4]
Reescribimos el Hamiltoniano
Debido al confinamiento lateral ky esta cuantizado, las bandas quedan: y x 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

27. Effect of Polarizing QPCs in the TEF1 2 B A C D
(a) y x
Detector QPC:
VG is changed
Injector QPC:
VG fixed
The polarizing
QPCs
umbalances the
amplitude of the
peaks

28. 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:

29. 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.

30. 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

31. Can we induce a charge current with a magnetic field?
Meaning?
less clear since charge is conserved
It might be observable in transport measurements

32. Nanowire: enhanced effects