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2D Tutorial
FockDarwin states of a 2D parabolic potential in a magnetic field
Note: This tutorial's copyright is owned by Stefan Birner,
www.nextnano.com.
Author:
Stefan Birner.
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 2DGaAs_BiParabolicQW_4meV_GovernalePRB1998.in
> 2DGaAs_BiParabolicQW_3meV_FockDarwin.in
FockDarwin states of a 2D parabolic potential in a magnetic field
In this tutorial we study the electron energy levels of a twodimensional
parabolic 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,y) plane.
 The motion in the z direction is not influenced by the magnetic field and
is thus that of a free particle with energies and wave functions given by:
E_{z} = h_{bar}^{2} k_{z}^{2}
/ (2 m*) psi(z) = exp (+ i k_{z} z)
For that reason, we do not include the z direction into our simulation
domain, and thus only simulate in the (x,y) plane (twodimensional
simulation).
2D parabolic confinement with h_{bar}w_{0} = 4 meV
> 2DGaAs_BiParabolicQW_4meV_GovernalePRB1998.in
First, we want to benchmark the nextnano³ code to some other numerical
calculation:
This input file aims to reproduce the figures 1, 2, 3 and 4 of
M. Governale, C. Ungarelli
Gaugeinvariant grid discretization of the Schrödinger
equation
Phys. Rev. B 58 (12), 7816 (1998).
 The GaAs sample extends in the x and y directions (i.e. this is a
twodimensional simulation) and has the size of 240 nm x 240 nm.
At the domain boundaries we employ Dirichlet boundary conditions to the
Schrödinger equation, i.e. infinite barriers.
The grid is chosen to be rectangular with a grid spacing of 2.4 nm, in
agreement with Governale's paper.
 The magnetic field is oriented along the z direction, i.e. it it
perpendicular to the simulation plane which is oriented in the (x,y) plane).
We calculate the eigenstates for different magnetic field strengths (1 T, 2 T,
..., 20 T), i.e. we make use of the magnetic field sweep.
$magneticfield magneticfieldon
= yes magneticfieldstrength
= 0.0d0 ! 1 Tesla = 1 Vs/m^{2} magneticfielddirection
= 0 0 1 ! [001] direction magneticfieldsweepactive
= yes ! magneticfieldsweepstepsize
= 0.5d0 ! 0.5 Tesla = 0.5 Vs/m^{2} magneticfieldsweepnumberofsteps =
40
! 40 magnetic field sweep steps
$end_magneticfield
 A useful quantitiy is the magnetic length (or Landau magnetic length)
which is defined as:
l_{B} = [h_{bar} /
(m_{e}* w_{c})]^{1/2}
= [h_{bar} / (e B)]^{1/2} It is independent of the mass of the particle and depends only on the magnetic
field strength:
 1 T: 25.6556 nm
 2 T:
18.1413 nm
 3 T: 14.8123 nm
 ...
 20 T: 5.7368 nm
 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).
Another useful quantity is the cyclotron frequency: w_{c}
= e B / m_{e}* Thus for the electrons in GaAs, it holds for the different magnetic field
strengths:
 1 T: h_{bar}w_{c} = 1.7279 meV
 2 T: h_{bar}w_{c} = 3.4558 meV
 3 T: h_{bar}w_{c} = 5.1836 meV
 ...
 20 T: h_{bar}w_{c} = 34.5575 meV
 The twodimensional parabolic confinement (conduction band edge
confinement) is chosen so that the electron ground state has the following
energy: E_{1} = h_{bar}w_{0} = 4 meV
(without magnetic field)
 The following figure shows the lowest fifteen eigenvalues for a magnetic
field magnitude of B = 10 T. It is in perfect agreement with Fig. 1 of
Governale's paper. The ground state has the energy E_{1} = 9.44 meV
(at B = 10 T).
 The following figure shows the ground state probability density (psi^{2})
for a magnetic field magnitude of B = 10 T. It is in perfect agreement with
Fig. 2(a) of Governale's paper.
The ground state has the energy E_{1} = 9.44 meV (at B = 10 T).
The left, vertical axis shows psi^{2} in units of nm^{2} (the
peak value is 0.00267 nm^{2}).
The parabolic conduction band edge confinement potential is also shown.
The horizontal axis shows the colormap of the conduction band edge values. In
the middle of the sample the conduction band edge is 0 eV, and at the boundary
region, the conduction band edge has the value 1.0092 eV.
 The following figure shows the probability density (psi^{2}) of
the 14th excited state (i.e. E_{15}) for a magnetic field magnitude of
B = 10 T. It is in perfect agreement with Fig. 3(a) of Governale's paper.
Its energy is E_{15} = 21.72 meV (at B = 10 T).
The left, vertical axis shows psi^{2} in units of nm^{2} (the
peak value is 0.000284 nm^{2}).
The parabolic conduction band edge confinement potential is also shown.
The horizontal axis shows the colormap of the conduction band edge values. In
the middle of the sample the conduction band edge is 0 eV, and at the boundary
region, the conduction band edge has the value 1.0092 eV.
 The following figure shows the ground state energy as a function of
magnetic field magnitude. It is in perfect agreement with Fig. 4 of
Governale's paper. The ground state has the energy E_{1} = 4.01 meV at
B = 0 T.
2D parabolic confinement with h_{bar}w_{0} = 3 meV 
FockDarwin spectrum
> 2DGaAs_BiParabolicQW_3meV_FockDarwin.in
Here, we calculate the singleparticle states of a twodimensional harmonic
oscillator.
The eigenvalues for such a system are given by
E_{n,l} = (2n + l  1) h_{bar}w_{0
} for n = 1,2,3,... and l =
0,+ 1,+ 2,...
(n = radial quantum number, l = angular momentum quantum number, w_{0} =
oscillator frequency)
The degeneracy of the eigenvalues is as follows (neglecting spin, for zero
magnetic field):
 the ground state is not
degenerate
 the second state is twofold degenerate
 the third state is threefold degenerate
 the forth state is fourfold
degenerate
 ...
Magnetic field
This input file aims to reproduce the Figs. 5(a) and 6(a) (which are analytical
results) of
L.P. Kouwenhoven, D.G. Austing, S. Tarucha
Fewelectron quantum dots
Rep. Prog. Phys. 64, 701 (2001).
 We chose the parabolic confinement such that h_{bar}w_{0}
= 3 meV in agreement to this paper.
The electron effective mass is taken to be m_{e}* = 0.067 m_{0}
(GaAs).
 The eigenvalues of a twodimensional parabolic potential that is subject
to a magnetic field can be solved analytically. The spectrum of the resulting
eigenstates is known as the FockDarwin states (1928):
E_{n,l} = (2n + l + 1) h_{bar }
[w_{0}^{2} + 1/4 w_{c}^{2}]^{1/2}
 1/2 l h_{bar}w_{c} for
n = 0,1,2,3,... and l = 0,+ 1,+ 2,...
(Note that the last term is w_{c} and not w_{0}
as in Kouwenhoven's paper.)
(w_{c} = e B / m_{e}* = cyclotron
frequency, for GaAs: h_{bar}w_{c} = 1.728 meV at 1 T)
Each of these states is twofold spindegenerate. A magnetic field lifts this
degeneracy (Zeeman splitting). However, this effect is not taking into account
in this tutorial.
 The following figure shows the calculated FockDarwin spectrum, i.e. the
eigenstates as a function of magnetic field magnitude:
The figure is in excellent agreement with Fig. 5(a) of Kouwenhoven's paper.
 The following figure show the probability densities (psi^{2}) of
some of these eigenstates for a magnetic field of B = 0.05 T.
The figures are in excellent agreement with Fig. 6(a) of Kouwenhoven's paper.
The parabolic conduction band edges are also shown.



(n,l) = (0,0) (1^{st}) 
(n,l) = (0,1)
(2^{nd}) 
(n,l) = (0,2)
(4^{th}) 






(n,l) = (1,0)
(5^{th}) 
(n,l) = (2,0) (13^{th}) 
(n,l) = (2,2)
(18^{th}) 
'
 For very high magnetic fields, eventually all states become degenerate
Landau levels as can be seen in this figure. The reason is that the electrons
are confined only by the magnetic field and not any longer by the parabolic
conduction band edge.
The red line shows the fan of the lowest
Landau level at 1/2 h_{bar}w_{c}. The higher lying states (not
shown) will collect in the second, third, ..., and higher Landau fans (not
shown).
The left part of the figure (black region) contains exactly the same
FockDarwin spectrum that has been shown in the figure further above (from 0 T
to 3.5 T).
 Please help us to improve our tutorial. Send comments to
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[at] nextnano.com .
