 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Parabolic Quantum Well (GaAs / AlAs)
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DGaAs_ParabolicQW.in
> 1DGaAs_ParabolicQW_infinite.in
> 1DGaAs_ParabolicQW_infinite_half.in
Parabolic Quantum Well (GaAs / AlAs)
This tutorial aims to reproduce figures 3.11 and 3.12 (pp. 8384) of
Paul Harrison's
excellent book "Quantum
Wells, Wires and Dots" (1^{st} edition, Section 3.5 "The
parabolic quantum well"), thus the following description is based on the
explanations made therein.
We are grateful that the book comes along with a CD so that we were able to
look up the relevant material parameters and to check the results for
consistency.
General comments on the solutions of a parabolic potential
An ideal parabolic potential represents a "harmonic oscillator" which is
described in nearly every beginner's textbook on quantum mechanics.
The eigenstates can be calculated analytically and are given by the following
relationship:
E_{n} = ( n  1/2 ) h_{bar}w_{0}
where n = 1, 2, 3, ...
One feature of a particle that is confined in such a well is that the energy
levels are equally spaced by h_{bar}w_{0} above the zero point
energy of 1/2 h_{bar}w_{0}.
The eigenfunctions show an evenodd alternation which is also the case in
symmetric, square quantum wells.
The eigenenergies can be measured experimentally by analyzing the optical
transitions between the conduction and the valence band states, taking into
account the selection rules (both states must have the same parity, see tutorial
on interband transitions). For
intersubband transitions, different selection rules apply (see tutorial on
intersubband transitions). Such
an experiment can be used to measure the conduction and valence band offsets
because the curvature of the conduction and valence band edges (and thus the
eigenstates) depends on the offsets.
(More information on this can be found in The Physics of LowDimensional
Semiconductors  An Introduction, John H. Davies, Cambridge University Press
(1998).)
Parabolic quantum well: 10 nm AlAs / 10 nm AlGaAs / 10 nm AlAs
> 1DGaAs_ParabolicQW.in
 It is possible to grow parabolic quantum wells by continuously varying the
composition of an alloy.
 Our structure consists of a 10 nm Al_{x}Ga_{1x}As
parabolic quantum well (the x alloy content varies parabolically) that is surrounded by
10 nm AlAs barriers on each side.
We thus have the following layer sequence: 10 nm AlAs
/ 10 nm Al_{x}Ga_{1x}As / 10 nm AlAs.
The barriers are printed in bold.
 This figure shows the conduction band edge and the three lowest electron
wave functions (psi) that are confined inside the parabolic quantum well.
All other states are not confined any more.
(Note that the energies were shifted so that the conduction band edge of GaAs
equals 0 eV.)
The figure is in perfect agreement with Fig. 3.11 (p. 83) of
Paul Harrison's
book "Quantum
Wells, Wires and Dots" (1^{st} edition).

Technical details:
The parabolic potential is specified by using a parabolic alloy
profile.
$material
...
materialnumber = 2
materialname =
Al(x)Ga(1x)As ! Al_{x}Ga_{1x}As
(parabolic quantum well)
clusternumbers = 2
alloyfunction =
parabolic
$end_material
$alloyfunction
...
materialnumber = 2
functionname =
parabolic ! Al_{x}Ga_{1x}As
(parabolic quantum well)
orientation =
0 0 1
! along z direction
varyfrompostopos = 10d0 20d0
! from 10 nm to 20 nm
xalloyfromto =
0.0d0 1.0d0 ! from
Al_{0.0}Ga_{1.0}As = AlAs to Al_{1.0}Ga_{0.0}As
= GaAs
$end_alloyfunction
In agreement with Paul Harrison,
 we assumed a constant effective mass of 0.067 m_{0} throughout the
whole sample and
 assumed the conduction band offset between GaAs and AlAs to be 0.83549 eV.
 Output
a) The conduction band edge of the Gamma conduction band can be
found here:
band_structure / cb1D_001.dat The 1^{st}
column contains the position in units of [nm] .
The 2^{nd} column contains the conduction band edge in
units of [eV] .
b) This file contains the
eigenenergies and the squared wave functions (Psi²):
Schroedinger_1band / cb001_qc001_sg001_deg001_dir_psi_squared_shift.dat
The 1^{st}
column contains the position in units of [nm] .
(Note that Psi_{n}² is shifted with respect to its
energy E_{n} so that they can be nicely plotted into the conduction
band profile.)
This file contains the eigenenergies and the wave functions (Psi): Schroedinger_1band / cb001_qc001_sg001_deg001_dir_psi_shift.dat
The 1^{st}
column contains the position in units of [nm] .
(Note that Psi_{n} is shifted with respect to its
energy E_{n} so that they can be nicely plotted into the conduction
band profile.)
a) and b) can be used to plot the data as shown in the figure above.
c)
This file contains the eigenenergies of the electron states. The units are [eV] . Schroedinger_1band / ev1D_cb001_qc001_sg001_deg001_dir.dat
Paul Harrison uses a 0.01 nm grid whereas we use the
0.01 nm grid only in the middle of the device (or 0.02 nm)
but at the boundaries (i.e. from 0 nm to 5 nm and from 25
nm to 30 nm) we use a 0.1 nm grid to avoid long CPU times:
The eigenvalues read:
nextnano³:
num_ev: eigenvalue [eV]:
1 0.1377775566
(0.10 / 0.05 / 0.02 / 0.05 / 0.10 nm grid)
2 0.4121053675
(0.10 / 0.05 / 0.02 / 0.05 / 0.10 nm grid)
3 0.6754933822
(0.10 / 0.05 / 0.02 / 0.05 / 0.10 nm grid)
1 0.1377754485
(0.10 / 0.01 / 0.01 / 0.01 / 0.10 nm grid)
2 0.4121049460
(0.10 / 0.01 / 0.01 / 0.01 / 0.10 nm grid)
3 0.6755000401
(0.10 / 0.01 / 0.01 / 0.01 / 0.10 nm grid)
Paul Harrison's book: 1
0.1377751623 (0.01
nm grid)
2 0.4121058503
(0.01 nm grid)
3 0.6755025905
(0.01 nm grid)
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(z) = 1/2 K z^{2} )
one can calculate h_{bar}w_{0}:
h_{bar}w_{0} = 2 E_{1 }
 0 eV = 0.276 eV
h_{bar}w_{0} = E_{2
} E_{1} = 0.274 eV
h_{bar}w_{0} = E_{3
} E_{2} = 0.263 eV
Obviously, due to the finite AlAs barrier that we have employed,
the higher lying states deviate slightly from the analytical results where
infinite barriers have been assumed.
This figure shows the eigenenergies for the confined states E_{1},
E_{2} and E_{3}. As expected they are lying on a straight line
because they are separated by h_{bar}w_{0}.
The figure is in perfect agreement with Fig. 3.12 (p. 84) of
Paul Harrison's
book "Quantum
Wells, Wires and Dots" (1^{st} edition).
 Matrix elements
The following matrix elements have been calculated:
intrabandmatrixelements = o !
matrix element < psi_{f}*  psi_{i} >
This spatial overlap matrix elements simply returns the Kronecker
delta as expected because the wave functions are orthogonal.
==> Schroedinger_1band/intraband_o1D_cb001_qc001_sg001_deg001_dir.txt
intrabandmatrixelements = p !
matrix element < psi_{f}*  p  psi_{i} >
==> Schroedinger_1band/intraband_pz1D_cb001_qc001_sg001_deg001_dir.txt
More details...
intrabandmatrixelements = z ! dipole matrix
element < psi_{f}*  z  psi_{i} >
==> Schroedinger_1band/intraband_z1D_cb001_qc001_sg001_deg001_dir.txt
More details...
"Infinite" (30 eV) parabolic QW confinement for GaAs
> 1DGaAs_ParabolicQW_infinite.in
 The following figure shows the eigenstates of a parabolic quantum well
(GaAs) where the confinement is assumed to be 30 eV.
Now up to 37 eigenstates are confined in the quantum well (grid resolution:
0.025 nm inside the well, 0.05 nm inside the barrier). The figure shows the
conduction band profile and the square of the wave functions (psi_{n}^{2})
for eigenstate n (n = 1, 2, ..., 37).
 This next figure shows the energies of the 37 confined electron states as
a funtion of eigenstate n.
As expected, the curve shows a linear dependence because the eigenstates are
equally spaced by
h_{bar}w_{0} = 0.826 eV (where we used E_{n}
= ( n  1/2 ) h_{bar}w_{0}).
h_{bar}w_{0} = 2 E_{1
} 0 eV = 0.8261 eV
E_{1} / (2 E_{1}) = 0.5000
h_{bar}w_{0} = E_{2
} E_{1} = 0.8260 eV
E_{2} / (2 E_{1}) = 1.4999
h_{bar}w_{0} = E_{3
} E_{2} = 0.8260 eV
E_{3} / (2 E_{1}) = 2.4997
h_{bar}w_{0} = E_{4
} E_{3} = 0.8259 eV
E_{4} / (2 E_{1}) = 3.4994
h_{bar}w_{0} = E_{5
} E_{4} = 0.8259 eV
E_{5} / (2 E_{1}) = 4.4991
h_{bar}w_{0} = E_{6
} E_{5} = 0.8258 eV
E_{6} / (2 E_{1}) = 5.4987
h_{bar}w_{0} = E_{7
} E_{6} = 0.8257 eV
E_{7} / (2 E_{1}) = 6.4982
h_{bar}w_{0} = E_{8
} E_{7} = 0.8257 eV
E_{8} / (2 E_{1}) = 7.4978
Still, due to the "infinite" barrier of 30 eV (which is still a
finite barrier) that we have employed, the higher lying states deviate
slightly from the analytical results where infinite barriers have been
assumed. Thus a much higher barrier sho.
 One should bare in mind that the energy level spacing of such parabolic
quantum wells is inversely proportional to both the well width and the square
root of the effective mass.
 It is also interesting to look at the intraband matrix elements, i.e. to
investigate the probability for
intersubband transitions.
The relevant output is contained in these two files:
 Schroedinger_1band / intraband_pz1D_cb001_qc001_sg001_deg001_dir.txt 
p_{z}
 Schroedinger_1band / intraband_z1D_cb001_qc001_sg001_deg001_dir.txt 
z
From the calculated oscillator
strengths it can be seen that only transitions from one level to the
neighboring levels (+1 and 1) are allowed.
Because in the case of a harmonic oscillator the momentum operator is
proportional to the sum of the creation and the annihilation operators, thus
only states can couple that have different occupation numbers with the
difference equal to 1.
"Infinite" (30 eV) halfparabolic QW confinement for GaAs
(Thanks to Michael Povolotskyi who suggested this halfparabolic structure!)
> 1DGaAs_ParabolicQW_infinite_half.in
 The following figure shows the eigenstates when taking only the right half
of the parabolic quantum well
(GaAs) that has been calculated above. The confinement is 30 eV on the right
and infinite confinement on the left (Dirichlet boundary conditions).
Now only 18 eigenstates are confined in the quantum well, i.e. half the number
of the eigenvalues compared with the full parabolic QW (grid resolution:
0.025 nm inside the well, 0.05 nm inside the barrier). The figure shows the
conduction band profile and the square of the wave functions (psi_{n}^{2})
for eigenstate n (n = 1, 2, ..., 18).
 Again, the eigenstates are
equally spaced. However, the separation energy is now twice the one as before:
h_{bar}w_{0} = 2 * 0.826 eV = 1.65.
The ground state energy this time is given by:
E_{1}
= ( 3/2 ) h_{bar}w_{0 }/ 2.
h_{bar}w_{0} = 4/3 E_{1} =
1.639 eV
h_{bar}w_{0} = E_{2
} E_{1} = 1.647 eV
h_{bar}w_{0} = E_{3
} E_{2} = 1.648 eV
h_{bar}w_{0} = E_{4
} E_{3} = 1.648 eV
 It is also interesting to look at the intraband matrix elements, i.e. to
investigate the probability for
intersubband transitions.
The relevant output is contained in these two files:
 Schroedinger_1band / intraband_pz1D_cb001_qc001_sg001_deg001_dir.txt 
p_{z}
 Schroedinger_1band / intraband_z1D_cb001_qc001_sg001_deg001_dir.txt 
z
 We note that also more realistic parabolic quantum wells can be calculated
with nextnano³.
Assuming that the alloy profile is parabolic,
 strain can be included (the strain tensor depends on the alloy
profile),
 as well as effective masses that depend on the alloy profile,
 an 8band k.p model (necessary to get correct
intersubband transition energies)
 and bowing parameters (especially important for AlGaAs).
All these features are automatically included in the nextnano³ code.
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
