1. Spin-dependent transport in the presence of spin-orbit interaction L.Y. Wang a ( 王律堯 ), C.S. Tang b and C.S. Chu a a Department of Electrophysics, NCTU b Physics 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. InGaAs InAlAs 2DEG Asymmetric heterostructure E V B eff I I : Rashba constant E Rashba Effect (Spin-orbit interaction)

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

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

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

8. The Hamiltonian of an ac-biased finger-gate in the Rashba-type quantum channel 2DEG ac-biased 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-resolved driving 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 There are both of spin-independent and spin-dependent oscillating terms to pump spin. Spin-up Spin-down

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

13. Fig: spin-resolved current transmissions (solid) and (dashed) versus the incident energy. Parameters N=1,,, l=20 (80nm), and with (a) 0.03, (b) 0.04, and (c) 0.05. The corresponding dc spin current is plotted in (d). 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. Numerical Results:

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

15. 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. 2DEG ac-biased

16. Two ac finger-gate with a phase difference InGaAs InAlAs 2DEG 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 current by tuning.

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

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

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

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 zero due to the symmetry configuration.

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

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

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

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

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

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

27. (a)V 0 =0.4 (b)V 0 =0.8 (c)V 0 =1.2 The satellite peaks of the 2nd resonance peak can be resolved by increasing V 0 The spin-resolved peaks become more narrow via increasing V 0. Fig. 8:

28. Fig. 9: The switching point of the spin-polarized direction would be shifted toward the higher energy with increasing V 0.

29. Conclusion: 1.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. 2.The two ac-biased FGs with a fixed phase difference can generate the charge current with spin current. 3.We propose a mechanism to switch the polarized direction of a spin current in the 1FG-1QD-1FG structure.

30. Bound state: