 
nextnano^{3}  Tutorial
2D Tutorial
FockDarwin states of a 2D parabolic, anisotropic (elliptical) potential in a magnetic field
Author:
Stefan Birner
==> This is the old website:
A new version of this tutorial can be found
here.
> 1DGaAs_ParabolicQW_infinite_4_6meV.in
> 1DGaAs_ParabolicQW_infinite_6_1meV.in
> 2DGaAs_BiParabolicEllipticQD_Austing.in
FockDarwin states of a 2D parabolic, anisotropic (elliptical) potential in a magnetic field
In this tutorial we study the electron energy levels of a twodimensional
parabolic, anisotropic (elliptical) confinement potential
that is subject to a magnetic field.
Such a potential can be constructed by surrounding GaAs with an Al_{x}Ga_{1x}As
alloy that has a parabolic alloy profile in the x and y directions.
First, it is necessary to study the energy states of a 1D parabolic
confinement.
1D parabolic confinement along the x direction with h_{bar}w_{0} = 4.6 meV
(1D simulation)
> 1DGaAs_ParabolicQW_infinite_4_6meV.in
For similar results and a discussion, we refer to this tutorial:
Parabolic Quantum Well (GaAs / AlAs)
1D parabolic confinement along the y direction with h_{bar}w_{0}
= 6.1 meV (1D simulation)
> 1DGaAs_ParabolicQW_infinite_6_1meV.in
For similar results and a discussion, we refer to this tutorial:
Parabolic Quantum Well (GaAs / AlAs)
First, it is a good idea to get familiar with the results of a 2D parabolic
and isotropic confinement: FockDarwin states of a 2D parabolic potential in a magnetic field
Now we turn to the anisotropic confinement...
2D parabolic, anisotropic (elliptical) confinement with h_{bar}w_{x} =
4.6 meV and and h_{bar}w_{y} = 6.1 meV 
FockDarwinlike spectrum (2D simulation)
> 2DGaAs_BiParabolicEllipticQD_Austing.in
 The electron effective mass in GaAs is m_{e}* = 0.067 m_{0}.
We assume this value for the effective mass in the whole region (i.e. also
inside the AlGaAs alloy).
 The following figure shows the parabolic, anisotropic (elliptical) conduction band edge confinement potential,
as well as the ground state wave function (psi^{2}) at B = 0 T.
In
the middle of the sample the conduction band edge is at
0 eV and at the boundary
region the conduction band edge has the value 0.84 eV.
The radii of the ellipse are 300 nm along the x axis and 226 nm along the y
axis.
The parabolic confinement along the x direction is: h_{bar}w_{x} =
4.6 meV
The parabolic confinement along the y direction is: h_{bar}w_{y} =
6.1 meV
Thus the ellipticity is roughly 4/3.
At zero magnetic field, the eigenvalues for such a system are given by:
E_{nx,ny} = (n_{x} + 1/2) h_{bar}w_{x} + (n_{y} +
1/2) h_{bar }w_{y}
n_{x} = n + 1/2 l  1/2 l
n_{y} = n + 1/2 l + 1/2 l
_{ } for n = 0,1,2,3,... and l =
0,+ 1,+ 2,...
(n = radial quantum number, l = angular momentum quantum number, w_{x}
and w_{y} =
oscillator frequencies)
For more details, see A.V. Madhav, T. Chakraborty, Physical Review B 49,
8163 (1994).
The eigenvalue spectrum of a 2D parabolic and isotropic
potential shows a shelllike structure:
Energy levels of an "artificial atom"  2D harmonic potential
For the anisotropic elliptical potential, this degeneracy at B
= 0 T is lifted.
The following figure shows the calculated FockDarwinlike spectrum, i.e. the eigenstates as a function of magnetic field magnitude.
Here, each of these states is twofold spindegenerate. However, a magnetic
field lifts this degeneracy (Zeeman splitting) but this effect is not taking into account
in this tutorial.
 Such a spectrum can be related to experimental transport measurements
which
give insight into the singleparticle energy spectrum of a quantum dot.
The rectangles in the above figure are related to the figures of the
following publications:
cyan rectangle: Fig. 2 of
Twolevel anticrossings high up in the
singleparticle energy spectrum of a quantum dot
C. Payette, D.G. Austing, G. Yu, J.A. Gupta,
S.V. Nair, B. Partoens, S. Amaha, S. Tarucha
arXiv:0710.1035v1
[condmat.meshall] (2007)
green rectangle: Fig. 2(b) of
red rectangle: Fig. 3(a) of
Probing by transport the singleparticle energy
spectrum up to high energy of
one quantum dot with the ground state of an adjacent
weakly coupled quantum dot
D.G. Austing, G. Yu, C. Payette, J.A. Gupta, M.
Korkusinski, G.C. Aers
physica status solidi (a), 508 (2007)
(Comments red rectangle: In Fig. 3(a) of
the publication by Austing et al., the ground state energy has been
subtracted from the excited states. Thus the slope of the energy spectrum
look slightly different.)
It is interesting to note that there are exact crossings in the calculated spectrum
whereas the experiment reveals anticrossings.
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