Spinorbit 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 dEstudis 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)
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Introduction experimental motivation
Experiments level splittings of 1-electron
quantum dots in B
Hanson et al, PRL 91,196802 (2003)
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Potok et al, PRL 91, 016802 (2003)
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Origin of the deviations ? Extension of the
wfs in AlGaAs region (g0.4) Nuclear
polarization effects (hyperfine) Non
parabolicity of the bands What is the role of
typical spin-orbit couplings of semiconductors?
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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
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The Zeeman scenario
sp energy levels
eigenstates Laguerre polynomials
eigenspinors in direction of B
spin splitting
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Natural units
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The SO coupling terms
conduction band (3D)
in 2D quantum wells 001
linear Dresselhaus term (bulk asymmetry)
coupling constant
( z0 vertical width )
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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
2DEGs 5 meVÅ - 50 meVÅ tunability of the
Rashba strength with external fields (basis of
spintronic devices)
We shall treat lR and lD as parameters
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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
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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
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Typical level spectra with SO
Parameters
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Anisotropy of first two shells at large B
Isotropic when only one source Symmetry!
Position of gap minima depend on
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Systematics of first-shell gap
anisotropy zero field splitting position of
minima QD energy levels could determine the
lambdas (need high accuracy!)
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In physical units
below Zeeman gmB B (level repulsion) w0
dependence
gmB B
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Second shell
two gaps (inner, outer) zero field value w0
dependence
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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?
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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)
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II. QD levels in a vertical B
As before, the Zeeman term
BUT now, B also in spatial parts
Symmetric gauge
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energy levels (without SO)
at large field
SO coupling redefines magnetic field weak SO
(unitary tranformation)
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Spin precession without SO The Larmor theorem
The Larmor frequency
equals the spin-flip gap
Spin precession with SO
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spin-flip (precessional) transition (N 7, 9, 11)
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Real time simulations
No interaction
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Real time simulations time-dependent LSDA
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M. Valín-Rodríguez et al. Phys. Rev. B 66,
235322 (2002)
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Deformation allows the transition between Kramers
conjugates at B0
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M. Valín-Rodríguez et al. Phys. Rev. B 69,
085306 (2004)
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Strong variation with tilting angle
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Far Infrared Absorption (without Coulomb
interaction)
splitting of the Kohn mode
at B0
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Far Infrared Absorption with Coulomb interaction
restores Kohn mode (fragmented) characteristic
spin and density oscillation patterns
at B0
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Confinement induced by SO modulation Rashba
term
bulk bands
localized states
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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 ?