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Landau levels of a bulk GaAs sample 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.
> 2DBulkGaAs_LandauLevels.in
Landau levels of a bulk GaAs sample in a magnetic field
In this tutorial we study the electron energy levels of a bulk GaAs sample
that is subject to a magnetic field.
 The GaAs sample extends in the x and y directions (i.e. this is a
twodimensional simulation) and has the size of 300 nm x 300 nm.
At the domain boundaries we employ Dirichlet boundary conditions to the
Schrödinger equation, i.e. infinite barriers.
 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,
3 T), i.e. we make use of the magnetic field sweep.
$magneticfield
magneticfieldon
= yes
magneticfieldstrength
= 1.0d0 ! 1 Tesla = 1 Vs/m^{2}
magneticfielddirection
= 0 0 1 ! [001] direction
magneticfieldsweepactive
= yes !
magneticfieldsweepstepsize
= 1.0d0 ! 1 Tesla = 1 Vs/m^{2}
magneticfieldsweepnumberofsteps = 2
! 2 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
 The electron effective mass in GaAs is m_{e}* = 0.067 m_{0}.
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
 The energy spectra for different magnetic fields (1 T, 2 T, 3 T) look as
follows:
 The Landau levels are given by E_{n} = (n  1/2) h_{bar}w_{c}
where n = 1,2,3,...
 The number of states for each Landau level can be calculated as follows
(see P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, p. 536, 3^{rd}
ed.):
N = L_{x}L_{y} e B / h = 1/(2pi) L_{x}L_{y} / l_{B}^{2}
(ignoring spin)
where L_{x} and L_{y} are the lengths in the x and y
directions (300 nm in this example) and l_{B} is the magnetic length.
N(1 T) = 21.76 ==> ~ 22 states per Landau level (in the
figure above: 22 as it should be)
N(2 T) = 43.52 ==> ~ 44 states per Landau level (in the
figure above: 44 as it should be)
N(3 T) = 65.29 ==> ~ 66 states per Landau level (in the
figure above: 66 as it should be)
Note that N is independent of n.
 For the calculations, we used the symmetric gauge A =

1/2 r x B = 1/2 B x r
leading to the following energies (see J.H. Davies, The Physics of
LowDimensional Semiconductors, p. 222):
E_{n,l} = (n + 1/2 l + 1/2 l  1/2) h_{bar}w_{c}
One can see that all states having a negative value of 'l' are degenerate with
the states with l=0, i.e. the allowed energies are independent of l if l < 0
(for the same n).
The energies increase if l increases (for l > 0 and for the same n).
 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 did not include the z direction into our simulation
domain, and thus only simulate in the (x,y) plane (twodimensional
simulation).
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