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, 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 spin-orbit 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 Relativistic correction Rashba E. I., Sov. Phys. Solid State, 2 1109 (1960). Can be controlled Nitta et al. PRL 78 (1997) Miller et al PRL 90 076807 (2003) GaAs/AlGaAs InSb/InAlSb 0.5-1 meV nm 5-10 meV nm a

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

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

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 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

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

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 500 nm x 2500 nm x y z z,conv

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

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 Since the effect is originated in the surface: What happens if we modify it? EF~5.1meV

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

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 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

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).

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)

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 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

Effect of Polarizing QPCs in the TEF 1 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

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:

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