Spin-orbit effects in semiconductor quantum dots Departament de Física, Universitat de les Illes Balears Institut Mediterrani d’Estudis Avançats IMEDEA (CSIC-UIB) Palma de Mallorca (SPAIN) Llorenç Serra 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, (2003)

Potok et al, PRL 91, (2003) splitting (  eV ) B || (T) | g | = 0.37 | g | = 0.44

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?

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 B x y  z

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

Natural units:

The SO coupling terms conduction band (3D) * linear Dresselhaus term (bulk asymmetry) in 2D quantum wells [001]: ( z 0 vertical width ) coupling constant

* 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  R and D uncertain in nanostructures (sample dependent!) in GaAs 2DEG’s: 5 meV Å - 50 meV Å * tunability of the Rashba strength with external fields (basis of spintronic devices) We shall treat R and D as parameters

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

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

Parameters : Typical level spectra with SO

Anisotropy of first two shells at large B Isotropic when only one source S ymmetry! Position of gap minima depend on

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

In physical units: below Zeeman | g *|  B B (level repulsion)  0 dependence | g *|  B B

Second shell: two gaps (inner, outer) zero field value  0 dependence

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

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)

II. QD levels in a vertical B As before, the Zeeman term: B x y z 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

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

Real time simulations No interaction

Real time simulations: time-dependent LSDA

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

Deformation allows the transition between Kramers conjugates at B=0

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

Strong variation with tilting angle:

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 ?