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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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**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)**Introduction: experimental motivation**Experiments: level splittings of 1-electron quantum dots in B|| Hanson et al, PRL 91,196802 (2003)**| g | = 0.44**splitting ( meV ) | g | = 0.37 B|| (T) Potok et al, PRL 91, 016802 (2003)**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?**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**The Zeeman scenario**sp energy levels eigenstates: Laguerre polynomials eigenspinors in direction of B spin splitting**The SO coupling terms**conduction band (3D) in 2D quantum wells [001]: * linear Dresselhaus term (bulk asymmetry) coupling constant ( z0 vertical width )*** 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**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**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**Typical level spectra with SO**Parameters:**Anisotropy of first two shells at large B**Isotropic when only one source Symmetry! Position of gap minima depend on**Systematics of first-shell gap**anisotropy + zero field splitting + position of minima QD energy levels could determine the lambda’s (need high accuracy!)**In physical units:**below Zeeman |g*|mB B (level repulsion) w0 dependence |g*|mB B**Second shell:**two gaps (inner, outer) zero field value w0 dependence**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?**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)**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**energy levels (without SO)**at large field SO coupling redefines magnetic field weak SO (unitary tranformation)**Spin precession without SO: The Larmor theorem**The Larmor frequency equals the spin-flip gap Spin precession with SO**Real time simulations**No interaction**Real time simulations:**time-dependent LSDA**M. Valín-Rodríguez et al.**Phys. Rev. B 66, 235322 (2002)**Deformation allows the transition between Kramers conjugates**at B=0**M. Valín-Rodríguez et al.**Phys. Rev. B 69, 085306 (2004)**Far Infrared Absorption (without Coulomb interaction):**splitting of the Kohn mode at B=0**Far Infrared Absorption with Coulomb interaction:**restores Kohn mode (fragmented) characteristic spin and density oscillation patterns at B=0**Confinement induced by SO modulation:**Rashba term bulk bands localized states**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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