1. Departament de Física, Universitat de les Illes Balears,
Spin and charge oscillation properties of semiconductor quantum dots from real time simulations
Llorenç Serra
Departament de Física, Universitat de les Illes Balears,
IMEDEA (CSIC-UIB)
Outline:
* Model of 2d quantum dot
* Theoretical framework
* Some results
* Spin-orbit coupling effects on spin precession
Collaborators: A. Puente (Mallorca)
M. Valín-Rodríguez (Mallorca)
E. Lipparini (Trento)
V. Gudmundsson (Reykjavik)

2. Semiconductor quantum dots
Vertical quantum dots
GaAs
L  nm z
AlxAsGa1-x
 1 nm
* System of confined electrons in 2D
* Possibilities: Control of
Geometry (circular, elliptic, rectang..)
Size
Number of electrons
ARTIFICIAL ATOMS

3. A SIMPLE MODEL: N electrons in 2D confined by Vext(r)
conduction electrons in GaAs
effective mass m = me
dielectric constant k = 12.4
interaction e2/kr
confinement potential: jellium disk
square well
harmonic potential
effective atomic units

4. Mean field Ground State
Set of orbitals and single-particle energies
* Kohn-Sham version of density-functional-theory:
exchange-correlation in LSDA (or CDFT)
Spin densities:
*Hartree-Fock theory:
exact exchange but no correlation
Numerical methods:
Discretization of the xy plane in a grid
Iterative solution

5. Time evolution td-LSDA td-HF After an initial perturbation on the GS
keep track of observables in ‘real’ time
oscillation frequencies
eigenmodes:
FIR (charge dipole)
spin modes
Fourier analysis
Alternative to perturbation theory
not restricted to small amplitudes or by symmetry

6. Example: Spin-density oscillation

7. Collective excitations in deformed systems
Dipole Excitation
Collective excitations in deformed systems
Free
N=20 electrons in a deformed parabola
wy= 0.75 wx = H*
p-h transitions
Spin
Landau damping atract. residual interaction
Density
Generalized Kohn theorem x,y

8. * td-(mean field) includes correlation effects
* Defines a new correlated ground state (RPA)
Applications:
* Orbital modes (Lz)
* Quadrupole modes (xy)
* Absorption patterns in triangular (square) dots
* Large amplitude motions
* ...

9. FIR Absorption in polygonal dots

10. Local absorption patterns:
* amplitude of oscillating
density
* corner and side modes
B=1T

11. Large amplitude motion in tdHF:
* non parabolic confinement
* CM trajectory
* initial rigid displacement
* 3 intervals of 9000 steps (12 ps each)
* Amplitude shrinks

12. Energy goes to internal modes

13. Spin-orbit coupling and spin precession in quantum dots
Two sources :
*Dresselhaus (bulk asymmetry)
*Rashba (nanostructure asymmetry)
Coupling constants for 2D bulk:
( E vertical electric field )
( z0 vertical width )
* g and a0 known from calculations for the bulk (k.p, tight binding)
* lR and lD uncertain in nanostructures (sample dependent)
in GaAs 2DEG’s: 5 meVA - 50meVA
* Tunability of the Rashba strength

14. *assume given l’s
*sp hamiltonians

15. * spin textures on the ground state
* spinorial orbitals:
* noncollinear SDFT:
* spin textures on the ground state
* time evolution

16. *Analytical solution:
neglect interactions
vertical magnetic field (Bz)
Aleiner-Falk’o transformation
* Spin precession: dl = 0 quasi spin-flip
* Larmor precession:

17. * Dreselhaus SO
* strong (z0=50A, blue symbols)
weak (z0 =85A, green symbols)
* zero field offset
* rearrangements jumps
* results within LSDA

18. time simulation of spin precession in LSDA:

19. In a horizontal B:
*numerical calculation
*first and second levels
*circular parabolic confinement

20. Elliptical dots:
transition between Kramers conjugates