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Spectral Properties of a 2D Spin-Orbit Hamiltonian

Spectral Properties of a 2D Spin-Orbit Hamiltonian. Denis Bulaev Department of Physics University of Basel, Switzerland. Outline. Motivation k.p method 2DEG Quantum Dots Summary. Motivation. ….. Quantum Computing. Supercoducting

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Spectral Properties of a 2D Spin-Orbit Hamiltonian

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

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

  3. Motivation • ….. • Quantum Computing Supercoducting [A.Shnirman, G.Shön, Z.Herman, PRL 79, 2371 (1997)] Quantum-Dot-based [D.Loss and D.P.DiVincenzo, PRA 57, 120 (1998)] Nano’ll make $1T/yr by 2015

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

  5. E G1 Eg G15 k Inversion asymmetric strs. (Td) 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. (Td) Single group Double group E E G1 l=0 j=1/2 G6 DxG1= G6 DxG15 = G7+ G8 Eg D j=3/2 G8 G15 l=1 j=1/2 G7 k k 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 Dresselhaus, Phys. Rev. 100, 580 (1955). (GaAs, InAs, InSb, etc - inversion asymmetry) For Ge, Si - inversion symmetric strs (point group Oh = Td x Ci ) DSO = 0! Remark No. 1 DSO is due to bulk inversion asymmetry (BIA)

  9. 2DEG GaAs GaAs AlyGa1-yAs AlxGa1-xAs AlxGa1-xAs AlxGa1-xAs V(z) V(z) z z D2d (E; C2; 2C2; 2sd; 2S4) C2v (E; C2; 2sv)

  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

  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. Three lowest electron energy levels DresselhausSO couplingRashba SO coupling Anti-crossing (crossing) of the levels E2 and E3 at

  18. Anticrossing due to Rashba coupling E3 – E1 0.25 0.20 orbital Energy [meV] 0.15 0.10 Zeeman E2 – E1 0.05 E1 – E1 8 0 2 4 6 10 B [T] 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

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