1. Denis Bulaev Department of Physics University of Basel, Switzerland Spectral Properties of a 2D Spin-Orbit Hamiltonian

2. Outline Motivation k.p method 2DEG Quantum Dots Summary

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4. k.p method Pauli HamiltonianThomas term (s-o coupling)

5. Inversion asymmetric strs. (T d ) TdTd E8 C 3 3 C 2 6  6 S 4  11111  111   2200   30 1   301 E k EgEg  15 11 CB l=0 (s) j=l+s=1/2 VB l=1 (p) j=3/2 & 1/2 Bir and Pikus. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).

6. Inversion asymmetric strs. (T d ) E k EgEg  j=3/2  8  j=1/2  6 E k  15 l=1  1 l=0 Single group Double group  j=1/2  7  D x  1 =  6 D x  15 =  7 +  8 Optical Orientation, ed. by Zakharchenya and F. Meier (North - Holland, Amsterdam, 1984) Bir and Pikus. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).

7. Kane Hamiltonian Folding down

8. Electron effective Hamiltonian Dresselhaus SO (DSO) coupling (GaAs, InAs, InSb, etc - inversion asymmetry) For Ge, Si - inversion symmetric strs (point group O h = T d x C i ) DSO = 0! Remark No. 1 DSO is due to bulk inversion asymmetry (BIA) Dresselhaus, Phys. Rev. 100, 580 (1955).

9. 2DEG GaAs Al x Ga 1-x As GaAs z V(z) z D 2d (E; C 2 ; 2C 2 ; 2  d ; 2S 4 ) C 2v (E; C 2 ; 2  v ) Al x Ga 1-x As Al y Ga 1-y As

10. Dresselhaus SO interaction D'yakonov & Kocharovskii, Sov. Phys. Semicond. 20, 110 (1986)

11. Rashba SO interaction After folding down Bychkov & Rashba, JETP Lett. 39, 78 (1984). Remark No. 2 RSO is due to structure inversion asymmetry (SIA)

12. Spin degeneracy & splitting without SO couplingwith SO coupling time inversion symmetry (Krames degeneracy) space inversion symmetryspace inversion asymmetry spin degeneracyspin splitting

13. Energy spectrum of 2DEG Ganichev, et al., PRL 92, 256601 (2004).

14. Spin decoherence anisotropy Averkiev & Golub PRB 60, 15582 (1999). Remark No. 3 SO coupling leads to anisotropy in dispersion and spin decoherence

15. Effective Hamiltonian for a QD

16. Canonical transformation Geyler, Margulis, Shorokhov, PRB 63, 245316 (2001).

17. Dresselhaus SO coupling Rashba SO coupling Anti-crossing (crossing) of the levels E 2 and E 3 at Three lowest electron energy levels

18. 0 0.05 0.10 0.15 0.20 0.25 2 46 8 10 Energy [meV] B [T] E 2 – E 1 E 3 – E 1 E 1 – E 1 orbital Zeeman Anticrossing due to Rashba coupling Bulaev, Loss, PRB 71, 205324 (2005).

19. Summary SO coupling is due to space inversion asymmetry Dispersion anisotropy in a 2DEG Anticrossing due to RSO in a QD