Electron transport in quantum wires subjected to the
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. ) Stockholm, Nordita Evgeny Sherman workshop 2012, September(University of Basque Country, Bilbao, Spain)
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
U E 2 DEG z Stockholm, Nordita workshop 2012, September
Stockholm, Nordita workshop 2012, September
Stockholm, Nordita workshop 2012, September
> Stockholm, Nordita workshop 2012, September
In infinite interface electric field is uniform and directed perpendicular to the interface. where a ~ Ez The Razhba spin-orbit interaction E. I. Razhba, Sov. Phys. Solid State 2, 1224 (1960) Stockholm, Nordita workshop 2012, September Emmanuil Rashba
Lusakowski et al, PRB 68, 081201 R (2003) Stockholm, Nordita workshop 2012, September
SOI in 2 DEG J. Phys. C (2007) Stockholm, Nordita workshop 2012, September
Stockholm, Nordita workshop 2012, September Infinite 1 d wire
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 1 nm of thickness. The resistance increase between the parallel and anti-parallel configurations of the layers of iron went up to 80%! Albert Fert Stockholm, Nordita workshop 2012, September Peter Grunberg
Stockholm, Nordita workshop 2012, September
Stockholm, Nordita workshop 2012, September Quasi 1 d 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 m=1 Stockholm, Nordita workshop 2012, September m=2 m=3
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 subbands 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 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, PRB 70, 235344 (2004); PRB 73, 115342 (2006) Stockholm, Nordita workshop 2012, September
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
Gate Control of Spin-Orbit Interaction in an Inverted In 0. 53 Ga 0. 47 As. In 0. 52 Al 0. 48 As 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 weak SOI Stockholm, Nordita workshop 2012, September
A. V. Moroz and C. H. W. Barnes Phys. Rev. B 60, 14272 (1999); B 61, R 2464 (2000) Is the only integral of motion gates 1 d 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:
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. n n s s Stockholm, Nordita workshop 2012, September
Four-terminal structure Bulgakov et al, PRL (1999) Two effects output 1) Spin polarization Stockholm, Nordita workshop 2012, September SOI output 2) Hall-like effect
Stockholm, Nordita workshop 2012, September
Stockholm, Nordita workshop 2012, September
Nitta et al, Gate Control of Spin-Orbit Interaction in an Inverted In 0. 53 Ga 0. 47 As. In 0. 52 Al 0. 48 As 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): 1) Exact solution for circular QD with Rashba SOI 2) 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
1 d packet: t=0 t=1. 5 t=7 Stockholm, Nordita workshop 2012, September
2 d case
Landauer conductance
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