Spin-dependent transport in the presence of spin-orbit interaction

1. Spin-dependent transport in the presence of spin-orbit interaction L.Y. Wanga(王律堯), C.S. Tangb and C.S. Chua aDepartment of Electrophysics, NCTU bPhysics Division, NCTS 2005.08.05

2. Outline: 1. Introduction of Rashba-type spin-orbit interaction 2. Spin-pumping via ac-biased finger- gates 3. Spin current generation involving a quantum dot 4. Conclusions

3. E V E I Beff I : Rashba constant Rashba Effect (Spin-orbit interaction) InGaAs 2DEG InAlAs Asymmetric heterostructure

4. No SOI Energy dispersion relation The Hamiltonian of an electron in a quantum channel Fig: Dispersion relation for

5. VG改變介電常數 L x Rashba-SOI VG改變SOI-coupling constant (APL. Vol.56 p665, Datta and Das.)

6. AC gate & spin current generation:Schematic distribution of spin currents induced by a time dependent circular gate Under the gate, electrons with opposite spins move in opposite direction (dashed-line arrows). Arrows outside the gate area show the accumulated spin polarization during a half period /. Mal’shukov, C.S. Tang, C.S. Chu and K. A. Chao (PRB 68, 233307)

7. Tuning Rashba constant by a metal gate Experimental data Gate Rashba constant InGaAs 2DEG InAlAs Gate voltage Nitta. et al. Phys.Rev.B 60, 7736(1999)

8. ac-biased 2DEG 2DEG The Hamiltonian of an ac-biased finger-gate in the Rashba-type quantum channel FG

9. The mechanism of spin pumping under the ac-biased finger-gate The longitudinal Hamiltonian: Spin-dependent vector potential: The spin-dependent vector potential gives rise to a spin-resolveddriving electric field.

10. It turns out that the linear term of vector potential is given by , manages to give rise to nontrivial spin-resolved transmission. The spin-up transition amplitude is stronger than spin-down one.

11. After unitary transformation Spin-indep. Oscillating term Spin-dep. Oscillating term Spin-up Spin-down There are both of spin-independent and spin-dependent oscillating terms to pump spin.

12. What is spin current? Charge current Spin current (up)

13. Numerical Results: The main dip structures are corresponding to the process due to the spin-resolved inelastic mechanism. The brown circle is related to the process due to the stronger pumping strength. Fig: spin-resolved current transmissions (solid) and (dashed) versus the incident energy . Parameters N=1, , , l=20 (80nm), andwith(a) 0.03, (b) 0.04, and (c) 0.05. The corresponding dc spin current is plotted in (d).

14. Transmission Time-dependent perturbation approach Here, only is right (left)-going spin-dependent wave vector. (1) For : (2) For :

15. ac-biased 2DEG 2DEG Fig: Current transmission versus for FG number N= (a) 1, and (b) 2. Pumping spin per cycle is plotted in (c) for N=1 (think curve) and N=2 (thick curve) with driving frequency . Other parameters are , , and the edge-to-edge separation .

16. Two ac finger-gate with a phase difference InGaAs 2DEG InAlAs The phase difference is maintained between the ac biases of the two finger gates. In such case, the charge current is nonzero due to the breaking the symmetry. This configuration would yield spin-polarized charge currentby tuning.

17. Spin current Charge current Negative charge current Spin (charge) Current (nA) Positive charge current Fig: Spin (charge) Current vs. phase difference . , , , , .

18. 2DEG QD 2DEG FD FD DD 1FG-1QD-1Fg System configuration Fig : A quantum dot locates between two ac-biased finger-gate in a Rashba-type quantum channel.

19. p=3.93q=4.93r=5.93 Spin-dependent transport with varying 1 Fig : q 1st resonance peak 2nd resonance peak p r

20. Spin-current with varying 1 Fig: The polarized direction of a spin current can be changed in opposite direction when an electron is incident in the front and in the back of resonance energies (switching point) within a quantum dot. But the net charge current is zerodue to the symmetry configuration.

21. Spin-up Spin-down For the main-peak (resonance energy) q=4.93 ( ) y=5 FG Dot FG |(x,y)|2 |(x,y)|2 y x x Fig : The magnitude of the spatial wave function is plotted when the incident energy is approached to the resonance energy.

22. Spin-up Spin-down For the satellite-peak p=3.93 ( ) y=5 Dot FG FG |(x,y)|2 |(x,y)|2 y x x Fig : The magnitude of the spatial wave function is plotted when the incident energy is equal to .

23. For the satellite-peak r=5.93 ( ) Spin-up Spin-down y=5 Dot FG FG |(x,y)|2 |(x,y)|2 y x x Fig : The magnitude of the spatial wave function is plotted when the incident energy is equal to .

24. One-sideband approximation in an ac-FG 1st-order 0th-order 1st-order L

25. One-sideband contribution and approximation: Fig: The transmission of spin-up electron is larger than spin-down one in T1 and T-1 processes.

26. The transition rate of a spin-up electron is larger than spin-down one

27. (a)V0=0.4 (b)V0=0.8 (c)V0=1.2 The satellite peaks of the 2nd resonance peak can be resolved by increasing V0 Fig. 8: The spin-resolved peaks become more narrow via increasing V0.

28. Fig. 9: The switching point of the spin-polarized direction would be shifted toward the higher energy with increasing V0.

29. Conclusion: • We have proposed a generation of dc spin current without charge current via ac-biased FGs in a Rashba-type quantum channel in the absence of magnetic field. • The two ac-biased FGs with a fixed phase difference can generate the charge current with spin current. • We propose a mechanism to switch the polarized direction of a spin current in the 1FG-1QD-1FG structure.

30. Bound state: