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Current induced domain wall motion in Rashba quantum wires Capri Spring School 2013 Martin Stier*, Reinhold Egger** and Michael Thorwart* * ** Universität Hamburg Universität Düsseldorf Outline • domain walls 1D wires (Luttinger liquid) Spin-orbit (Rashba) interaction • numerical results DW velocity current pulses Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 2 11.04.2013 Domain walls • application: racetrack memory • reliable and fast motion of domain walls • motion at low current densities Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 3 11.04.2013 Current induced Domain Wall motion spin polarized current domain wall interaction spin torque on domain wall additional effects may change s Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 4 11.04.2013 Our set-up ● Landau-Lifshitz-Gilbert equation describes DW dynamics ● electronic sub-system includes Rashba term ● describe electrons in 1D wire as Luttinger liquid ● calculate s and with it T by an equation of motion technique M Stier, R Egger, M Thorwart - arXiv:1301.0389, 2013 Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 5 11.04.2013 Luttinger liquid • left and right movers • easy to define currents • 1D system • excitations only around Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 6 11.04.2013 Solution adiabatic non-adiabatic Rashba field Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 7 11.04.2013 Numerical calculations ● solve LLG explicitly ● choose Bloch(y) wall ● Rashba field HR~ey can directly accelerate / decelerate DW R standard parameters: Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 8 11.04.2013 Steady current ● momentary motion determined by both: current and (Rashba) field ● oscillation frequency dependent on Rashba coupling and current density Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 9 11.04.2013 Current pulse DW still in motion after current pulse ('drifting') ● direction of velocity dependent on amplitude at the end of pulse ● drifted distance can be remarkably long ● Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 10 11.04.2013 Controlling effective velocity by pulses ● contains traveled distance while and after current pulse ● depending on pulse length → resulting velocity strongly increased or decreased ● Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 11 11.04.2013 Non-linear dependence on current density oscillation period controlled by current density ● the stronger the current the smaller the period ● this determines if the pulse end at positive or negative amplitude ● Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 12 11.04.2013 Conclusions • derived current induced spin torque under influence of Rashba interaction • Rashba field arises • oscillatory behavior of DW velocity • rich behavior for current pulses strongly in-/decrease of effective velocity possible negative velocities Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 13 11.04.2013 Thank you for your attention! Support by Current induced DW motion in Rashba nanowires is gratefully acknowledged Martin Stier [email protected] Seite 14 11.04.2013 Hamiltonian spin density of left or right (p=L/R) moving particles magnetization of domain wall direction defined by Rashba interaction Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 15 11.04.2013 Equation of motion additional relaxation gradient expansion in derivatives of n: Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 16 11.04.2013 Equation of motion relaxation rate extrinsic relaxation stationary state Current induced DW motion in Rashba nanowires Martin Stier [email protected] Seite 17 11.04.2013