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2D Tutorial
Vertically coupled quantum wires in a longitudinal magnetic field
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DAlGaAs_GaAs_DQW.in
> 1DGaAs_ParabolicQW_10meV.in
> 2DGaAs_CoupledQWRs_parabolic_10meV_APL2007.in
Vertically coupled quantum wires in a longitudinal magnetic field
In this tutorial we study the electron energy levels of two coupled quantum
wires as a function of a longitudinal (i.e. perpendicular) magnetic field.
This input file aims to reproduce Fig. 2 of
Vertically coupled quantum wires in a
longitudinal magnetic field
L.G. Mourokh, A.Y. Smirnov, S.F. Fischer
Applied Physics Letters 90, 132108 (2007).
Thus the following description is based on the explanations made therein.
The idea is to model experimental results which are shown in Fig. 8 (b) of this
paper:
Tunnelcoupled onedimensional electron systems
with large subband separations
S.F. Fischer, G. Apetrii, U. Kunze, D. Schuh, G.
Abstreiter
Physical Review B 74, 115324 (2006)
The following figure shows the layout of the structure in the (y,z) plane.
The blue regions are the barrier materials (Al_{0.32}Ga_{0.68}As)
and
the red regions are 14.5 nm GaAs quantum wells
that are separated by a 1 nm thin Al_{0.32}Ga_{0.68}As
tunnel barrier.
1D simulations
The following figure shows the confined eigenstates E_{z} of the
coupled, symmetric QW system (1D simulation along the z direction).
Note that the states have bonding and
antibonding character.
> 1DAlGaAs_GaAs_DQW.in
The following material parameters were used:
 conduction band offset GaAs/Al_{0.32}Ga_{0.68}As:
CBO = 0.27882 eV
 electron effective mass GaAs:
m_{e }= 0.067 m_{0}
 electron effective mass Al_{0.32}Ga_{0.68}As:
m_{e }= 0.09356 m_{0}
The ground state (bonding state) has the
energy E_{z,1} = 13.86 meV,
the antibonding state has the energy
E_{z,2} = 19.78 meV,
thus the bondingantibonding level separation is Delta_{SAS} = 5.92 meV.
To model quantum wires, instead of quantum wells, we follow the methodology
of the above cited paper and apply a harmonic oscillator potential in the
quantum well plane.
We construct such a potential by surrounding GaAs with an In_{x}Ga_{1x}As
alloy that has a parabolic alloy profile in the y direction.
(Our "InAs" has the same material parameters than GaAs apart from the conduction
band edge energy which is used to model the parabolic confinement.)
We illustrate this by a 1D simulation along the y direction: This
figure shows the eigenstates of the parabolic confinement potential.
The harmonic oscillator potential was chosen such that the energy separation of
the eigenstates is 10 meV, consequently the ground state is at 5 meV.
> 1DGaAs_ParabolicQW_10meV.in
 2D parabolic confinement with h_{bar}w_{0} = 10 meV
Making use of the equation
E_{n} = ( n  1/2 ) h_{bar}w_{0}
where n = 1, 2, 3, ... and w_{0} = (C/m*)^{1/2}
(m* = effective mass, C = constant which is related to the parabolic potential
V(y) = 1/2 K y^{2} )
one can calculate h_{bar}w_{0}:
h_{bar}w_{0} = 2 E_{y,1 }
 0 eV = 10.06 meV
h_{bar}w_{0} = E_{y,2
}
 E_{y,1} = 10.05 meV
h_{bar}w_{0} = E_{y,3
}
 E_{y,2} = 10.05 meV
h_{bar}w_{0} = E_{y,4
}
 E_{y,3} = 10.06 meV
...
E_{y,1} =
0.00503 eV
E_{y,2} =
0.01508 eV
E_{y,3} =
0.02513 eV
E_{y,4} =
0.03519 eV
The following figure shows the conduction band edge profile of the two
coupled quantum wires. The confinement is as follows:
 along the y direction: parabolic confinement
 along the z direction: AlGaAs / GaAs / AlGaAs / GaAs / AlGaAs
heterostructure confinement
Along the x direction (perpendicular to the simulation plane), we will apply a
magnetic field to study the energy spectrum of the coupled quantum wires as a
function of longitudinal magnetic field.
> 2DGaAs_CoupledQWRs_parabolic_10meV_APL2007.in
Magnetic field
 The magnetic field is oriented along the x direction, i.e. it is
perpendicular to the simulation plane which is oriented in the (y,z) plane.
We calculate the eigenstates for different magnetic field strengths (0
T, 0.5 T, 1.0 T,
..., 16 T), i.e. we make use of the magnetic field sweep.
$magneticfield magneticfieldon
= yes magneticfieldstrength
= 0.0d0 ! 1 Tesla = 1 Vs/m^{2} magneticfielddirection
= 1 0 0 ! [100] direction magneticfieldsweepactive
= yes ! magneticfieldsweepstepsize
= 0.50d0 ! 0.50 Tesla = 0.50 Vs/m^{2} magneticfieldsweepnumberofsteps =
32
! 32 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}.
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 onedimensional parabolic confinement (conduction band edge
confinement) was chosen so that the electron ground state has the energy of E_{1} = h_{bar}w_{0}
= 5 meV in the 1D simulation.
In the 2D simulation, the ground state has the energy: E_{1} = 18.64 meV (without magnetic field)
which corresponds approximately to
E_{1} ~= E_{y,1} +
E_{z,1} = 5.03 eV +
13.86 meV = 18.89 meV. (In 2D, we
use a different grid resolution compared to 1D simulations.)
Magnetic field
 The following figure shows the calculated energy spectrum, i.e. the eigenstates as a function of magnetic field magnitude:
The figure is in excellent agreement with Fig. 2 of the above cited paper and
reproduces Fig. 8 (b) of the experimental paper very well.
For the physical significance of such a spectrum, we refer to the discussion
in the above cited papers.
 Note that each of the electron 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 figures show wave functions (psi^{2}) of the lowest four eigenstates
 at zero magnetic field
 at 5.5 T (where E_{3} and E_{4} have similar
energies)
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
