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Spin-orbit effects in semiconductor quantum dots

Spin-orbit effects in semiconductor quantum dots. Llorenç Serra. Departament de Física, Universitat de les Illes Balears Institut Mediterrani d’Estudis Avançats IMEDEA (CSIC-UIB) Palma de Mallorca (SPAIN). Outline: Introduction: experimental motivation

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Spin-orbit effects in semiconductor quantum dots

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  1. Spin-orbit effects in semiconductor quantum dots Llorenç Serra Departament de Física, Universitat de les Illes Balears Institut Mediterrani d’Estudis Avançats IMEDEA (CSIC-UIB) Palma de Mallorca (SPAIN) Outline: Introduction: experimental motivation Level structure in horizontal B Vertical B: spin precession Far Infrared absorption Confinement induced by SO Collaborators: Manuel Valín-Rodríguez (Mallorca) Antonio Puente (Mallorca) Enrico Lipparini (Trento)

  2. Introduction: experimental motivation Experiments: level splittings of 1-electron quantum dots in B|| Hanson et al, PRL 91,196802 (2003)

  3. | g | = 0.44 splitting ( meV ) | g | = 0.37 B|| (T) Potok et al, PRL 91, 016802 (2003)

  4. Origin of the deviations ? * Extension of the wf’s in AlGaAs region (g=+0.4) * Nuclear polarization effects (hyperfine) * Non parabolicity of the bands What is the role of typical spin-orbit couplings of semiconductors?

  5. z y B q x I. QD levels in a horizontal B Model of spatial confinement: 2D representation (strong z confinement) effective mass model (GaAs conduction band) parabolic potential in xy plane The Zeeman term: bulk GaAs gyromagnetic factor Bohr magneton Pauli matrices

  6. The Zeeman scenario sp energy levels eigenstates: Laguerre polynomials eigenspinors in direction of B spin splitting

  7. Natural units:

  8. The SO coupling terms conduction band (3D) in 2D quantum wells [001]: * linear Dresselhaus term (bulk asymmetry) coupling constant ( z0 vertical width )

  9. * Rashba term (nanostructure z asymmetry) ( E vertical electric field ) Rashba and Dresselhaus terms: * used to analyze the conductance of quantum wells and large (chaotic) dots * lR and lD uncertain in nanostructures (sample dependent!) in GaAs 2DEG’s: 5 meVÅ - 50 meVÅ * tunability of the Rashba strength with external fields (basis of spintronic devices) We shall treat lR and lD as parameters

  10. No exact solution with SO, but analytical approximations in limits: a) Weak SO in zero field 2nd order degenerate pert. theory fine structure: zero-field up-down splitting ! Kramers degeneracy an alternative method: unitary transformation

  11. b) Weak SO in large field definitions - new fine structure of the major shell - (q dependence) anisotropy! Intermediate cases only numerically, - xy grid - Fock-Darwin basis

  12. Typical level spectra with SO Parameters:

  13. Anisotropy of first two shells at large B Isotropic when only one source Symmetry! Position of gap minima depend on

  14. Systematics of first-shell gap anisotropy + zero field splitting + position of minima QD energy levels could determine the lambda’s (need high accuracy!)

  15. In physical units: below Zeeman |g*|mB B (level repulsion) w0 dependence |g*|mB B

  16. Second shell: two gaps (inner, outer) zero field value w0 dependence

  17. Experimental results from QD conductance: 1 electron occupancy Potok et al., Phys. Rev Lett. 91, 018802 (2003) Hanson et al., Phys. Rev Lett. 91, 196802 (2003) | g | = 0.44 splitting ( meV ) | g | = 0.37 B|| (T) BUT: zero field splitting of 2nd shell? q - anisotropies?

  18. SO effects in GaAs are close to the observations BUT only for a given B orientation. Determination of the angular anisotropy and zero field splittings are important to check the relevance of SO in these experiments. M. Valín-Rodríguez et al. Eur. Phys. J. B 39, 87 (2004)

  19. z y B x II. QD levels in a vertical B As before, the Zeeman term: BUT now, B also in spatial parts: Symmetric gauge

  20. energy levels (without SO) at large field SO coupling redefines magnetic field weak SO (unitary tranformation)

  21. Spin precession without SO: The Larmor theorem The Larmor frequency equals the spin-flip gap Spin precession with SO

  22. spin-flip (precessional) transition (N = 7, 9, 11)

  23. Real time simulations No interaction

  24. Real time simulations: time-dependent LSDA

  25. M. Valín-Rodríguez et al. Phys. Rev. B 66, 235322 (2002)

  26. Deformation allows the transition between Kramers conjugates at B=0

  27. M. Valín-Rodríguez et al. Phys. Rev. B 69, 085306 (2004)

  28. Strong variation with tilting angle:

  29. Far Infrared Absorption (without Coulomb interaction): splitting of the Kohn mode at B=0

  30. Far Infrared Absorption with Coulomb interaction: restores Kohn mode (fragmented) characteristic spin and density oscillation patterns at B=0

  31. Confinement induced by SO modulation: Rashba term bulk bands localized states

  32. Conclusions: * In horizontal fields SO effects are small, but they are close to recent observations. Zero field splittings and anisotropies are also predicted. * In vertical fields the SO-induced modifications of the g-factors are quite important. * Possibility of confinement induced by SO ?

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