Electron transport in quantum wires subjected to the Rashba SOI Almas Sadreev, Lab. of Theory of Nonlinear Processes, Institute of Physics, Siberian Branch of Russian Academy of Sciences Krasnoyarsk, Russia My thanks to Karl-Fredrik Berggren and Ivan Shelykh for invitation And to whom I collaborate and collaborated : Karl-Fredrik Berggren (Linkoping University, Sweden) Evgeny Bulgakov and Konstantin Pichugin (Institite of Physics, Krasnoyarsk, Russia) Pavel Exner, Pavel Streda and Petr Seba (Chech Rep.) Evgeny Sherman (University of Basque Country, Bilbao, Spain) Stockholm, Nordita workshop 2012, September

Spintronics (nickname for spin-based electronics): We want to use spins of single electrons for storage, transfer, and manipulation of information and not only ... Stockholm, Nordita workshop 2012, September

Spin-orbit interaction

From Nitta’s lecture Stockholm, Nordita workshop 2012, September

E U 2DEG z GaAs AlGaAs n - AlGaAs GaAs Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

> Stockholm, Nordita workshop 2012, September

The Razhba spin-orbit interaction In infinite interface electric field is uniform and directed perpendicular to the interface. Emmanuil Rashba where a ~ Ez The Razhba spin-orbit interaction E.I. Razhba, Sov. Phys. Solid State 2, 1224 (1960) Stockholm, Nordita workshop 2012, September

Lusakowski et al, PRB68,081201R (2003) Stockholm, Nordita workshop 2012, September

SOI in 2DEG J. Phys. C (2007) Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September Infinite 1d wire

S. Datta, Quantum transport: Atom to transistor Supriyo Datta, Purdue Univ. Datta-Das spin transistor S.Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, England, 1995). S. Datta, Quantum transport: Atom to transistor Stockholm, Nordita workshop 2012, September

Datta-Das spin transistor Stockholm, Nordita workshop 2012, September

In 2007 Albert Fert (France) and Peter Grunberg (Germany) have got Nobel prize for discovery of giant magnetoresistance in multilayers of Fe/Cr, one after the other. Chromium of order of 1nm of thickness. The resistance increase between the parallel and anti-parallel configurations of the layers of iron went up to 80%! Albert Fert Peter Grunberg Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

Quasi 1d wire Datta& Das, Appl. Phys. Lett.(1989); Stockholm, Nordita workshop 2012, September Quasi 1d wire Datta& Das, Appl. Phys. Lett.(1989); Perroni et al J. Phys. C (2007); SJeong&Lee PRB (2006); Zhang, Brusheim and Xu, PRB (2005) Mireles& Kirczenow, PRB (2001); Knobbe&Schapers PRB (2005); Moroz & Barnes, PRB (1999) ; Pramanik et al PRB (2007); Debald&Kramer, PRB (2005); Governale&Zulicke PRB (2002); Rodriguez, Puente and Serra (2003) Erlingsson et al PRB (2010);

Approximate by two channels Stockholm, Nordita workshop 2012, September

Two-band (two-channel) model We take into account the exact solution of the quantum wire in the absence of SOI, then we study its effect on the sub-bands. Because of SOI, a coupling between sub-bands with opposite spins occurs. In order to study the effects of this coupling, we will discuss the results within the first- and second-order perturbation theory approach with respect to the SOI. This assumption is valid if the wire is very narrow. Moreover, this simple system is studied since it provides a simple understanding of the transport properties. Stockholm, Nordita workshop 2012, September

Two-band model Energy is measured in terms of Stockholm, Nordita workshop 2012, September

two band model n=1,s=1 n=1,s=-1 n=2,s=-1 n=2,s=1 Stockholm, Nordita workshop 2012, September

Two-band model (wave functions) Stockholm, Nordita workshop 2012, September

Current induced spin polarization –spin Hall effect Reynoso, Usaj, and Balseiro, PRB70, 235344 (2004); PRB73, 115342 (2006) Stockholm, Nordita workshop 2012, September

Charge- and spin-density modulations in semiconductor quantum wires PHYSICAL REVIEW B 72, 045353 2005 Charge- and spin-density modulations in semiconductor quantum wires Minchul Lee and Christoph Bruder Stockholm, Nordita workshop 2012, September

J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, PRL (1999) Gate Control of Spin-Orbit Interaction in an Inverted In0.53Ga0.47AsIn0.52Al0.48As Heterostructure J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, PRL (1999) Stockholm, Nordita workshop 2012, September

Quantum wire Hamiltonian Finger gate potential (Davies, Larkin, and Sukhorukov, J. Appl. Phys. 77, 4504 (1995).

The simplest model of ballistic transport

Parabolic wire Strong SOI weak SOI Stockholm, Nordita workshop 2012, September

Is the only integral of motion A. V. Moroz and C. H. W. Barnes Phys. Rev.B60,14272 (1999); B61, R2464 (2000) Is the only integral of motion GaAs AlGaAs n-AlGaAs 2 D E G gates 1d wire x

From parabolic potential Asymmetry of heterostructure

Landauer two-probe conductance with spin splitting

Curvilinear wire Bulgakov &Sadreev PRB 66 (2002) Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

General solution is Two channels of transmission Let electron is injecting with spin directed along the z-axis

Stockholm, Nordita workshop 2012, September

Spin polarization by magnetic field or ferromagnetic layer No spin flip Stockholm, Nordita workshop 2012, September

Spin polarization by Rashba SOI Kisilev&Kim J. Appl. Phys. (2001); (2003) Bulgakov &Sadreev PRB 66. 075331 (2002) Zhai &Xu, PRL 94, 246601 (2005). Time reversal Stockholm, Nordita workshop 2012, September

Spin filtering Spin polarization Therefore, for the single channel transmission through the two-terminal QD with the Rashba spin-orbit interaction there is NO spin polarization (filtering).

Unitarity of S-matrix:

Spin transistor Kisilev&Kim J. Appl. Phys. (2001); (2003) Bulgakov &Sadreev PRB 66. 075331 (2002) Liu et al PRB (2007) Spin transistor Perroni et al J. Phys. (2007)

The Hall like effect, induced by the Rashba spin-orbit. Landau&Liphshitz, Quantum Mechanics. s n n s Stockholm, Nordita workshop 2012, September

Four-terminal structure Bulgakov et al, PRL (1999) Two effects output 1) Spin polarization 2) Hall-like effect SOI output output Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

Nitta et al, Gate Control of Spin-Orbit Interaction in an Inverted In0.53Ga0.47AsIn0.52Al0.48As Heterostructure , PRL (1997,1999) Three-terminal Yokoyama&Eto PRB (2009) Four-terminal Stockholm, Nordita workshop 2012, September

Theory and experiments by Nitta group Physica E(2005) Stockholm, Nordita workshop 2012, September

Bulgakov&Sadreev, JETP Lett. (2001): Exact solution for circular QD with Rashba SOI Spin polarization 90% by circular polarized laser field PRB (2004): Exact solution with Rashba SOI and magnetic field f R Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

Eigen energies Stockholm, Nordita workshop 2012, September

Laser field with circular polarization Stockholm, Nordita workshop 2012, September

Stockholm, Nordita workshop 2012, September

1d packet: t=0 t=1.5 t=7 Stockholm, Nordita workshop 2012, September

2d case

Landauer conductance