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1D Tutorial
Dispersion in infinite superlattices: Minibands (KronigPenney model)
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1Dsuperlattice_dispersion_4nm.in
1Dsuperlattice_dispersion_6nm.in
1Dsuperlattice_dispersion_bulk_GaAs.in
Dispersion in infinite superlattices: Minibands (KronigPenney model)
This tutorial aims to reproduce two figures (Figs. 2.27, 2.28, p. 56f) of
Paul Harrison's
excellent book "Quantum
Wells, Wires and Dots", 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.
Superlattice 1: 4 nm AlGaAs / 4 nm GaAs
 Our infinite superlattice consists of a 4 nm GaAs quantum well
surrounded by 2 nm Al_{0.4}Ga_{0.6}As barriers on each side.
The choice of periodic boundary conditions leads to the following sequence of
identical quantum wells: 4 nm AlGaAs / 4 nm GaAs / 4 nm AlGaAs / 4 nm GaAs /
... . So our superlattice period has the length L=8 nm.
(Actually it has the length L = 8.25 due to the grid point resolution of 0.25
nm.)
This figure shows the conduction band edge and the first eigenstate that is
confined inside the well and its corresponding charge density (psi²) for
the superlattice vector k_{z }= 0. Note
that periodic boundary conditions are employed for solving the Schrödinger
equation. The second eigenstate is not confined inside the well and is
therefore not shown here.
(Note that the energies were shifted so that the conduction band edge of GaAs
equals 0 eV.)
 In a superlattice the electrons (and holes) see a periodic potential which
is similar to the periodic potential in bulk crystals. This means that the
particle wave functions are no longer localized in one quantum well. They
extend to infinity and they are equally likely to be found in any of
the quantum wells. The eigenstates are called Bloch states (as in bulk)
and the wave functions are periodic:
Psi (z) = Psi (z + L)
For a travelling wave of the form exp(ik_{z}z) it holds:
Psi (z + L) = exp(ik_{z}(z + L)) = exp(ik_{z}z) exp(ik_{z}L)
i.e. Psi (z + L) = Psi (z)
exp(ik_{z}L)
k_{z} is the momentum of the electron (or hole) along the growth
direction of the infinite superlattice.
 Here we plot the superlattice dispersion curve, i.e. the energy of the
electron as a function of its superlattice wave vector k_{z} for the
lowest eigenstate. As the energy is a periodic function of k_{z} with
period 2pi/L, we plot only the interval [  pi / L , + pi / L].
The plot is in excellent agreement with Fig. 2.27 (page 56) of
Paul Harrison's
book "Quantum
Wells, Wires and Dots".
When the electron is at rest (k_{z}=0), the dispersion curve shows a
minimum. As the electron momentum k_{z} increases, its energy also
increases and reaches a maximum at k =  pi/L and k = + pi/L. Thus the
electron within the superlattice occupies a continuum of energies. This
continuum that is bound by a maximum and a minimum of energy is called
miniband. Due to the similarity with the energy bands of a bulk crystal,
the point in the superlattice Brillouin zone for k_{z}=0 is
called Gamma and for k_{z}=pi/L it is called X.
Superlattice 2: 6 nm AlGaAs / 6 nm GaAs
 Our second infinite superlattice consists of a 6 nm GaAs quantum
well surrounded by 3 nm Al_{0.4}Ga_{0.6}As barriers on each
side. The choice of periodic boundary conditions leads to the following
sequence of identical quantum wells: 6 nm AlGaAs / 6 nm GaAs / 6 nm AlGaAs / 6
nm GaAs / ... . So our superlattice period has the length L=12 nm.
(Actually it has the length L = 12.25 due to the grid point resolution of 0.25
nm.)
This figure shows the conduction band edge and the two lowest eigenstates that
are confined inside the well and their corresponding charge density (psi²)
for the superlattice vector k_{z }= 0.
Note that periodic boundary conditions are employed for solving the
Schrödinger equation. The third eigenstate is not confined inside the well and
is therefore not shown here.
In contrast to the 4 nm quantum well superlattice described above, two
confined electron states exist.
(Note that the energies were shifted so that the conduction band edge of GaAs
equals 0 eV.)
 The following figure shows the first two minibands for this superlattice.
They arise from the first and the second eigenstate. Note that due to the
scale of this figure the first miniband looks almost flat. It is also
interesting that for the second miniband the minimum is not at the center
(i.e. at Gamma) but at the edges of the superlattice Brillouin zone at X (and
X).
Again, the plot is in excellent agreement with Fig. 2.28 (page 57) of
Paul Harrison's
book "Quantum
Wells, Wires and Dots". However, the caption of Fig. 2.28 incorrectly
states that this should be a 8 nm GaAs / 8 nm Al_{0.4}Ga_{0.6}As
superlattice. In fact, it must be a 6 nm GaAs / 6 nm Al_{0.4}Ga_{0.6}As
superlattice.
Technical details
 The resolution of the miniband plot has to be specified within the keyword
$quantummodelelectrons :
$quantummodelelectrons
...
boundarycondition001 = periodic !
periodic boundary conditions are necessary for superlattices
numks001
= 21 !
number of superlattice vectors along z direction
For each superlattice vector k_{z}, the Schrödinger equation
has to be solved. The 11^{th} superlattice vector corresponds to k_{z}=0
which is obviously identical to the case when no superlattice is specified at
all.
The miniband dispersion is written to this file:
Schroedinger_1band/sg_dispSL1D_el_qc001_sg001_deg001_evmin001_evmax002.dat
It contains the following data:
k_z [pi/L] k_z [1/AA]
1^{st} eigenvalue 2^{nd}
eigenvalue
1.0
...
...
...
Dispersion in bulk GaAs with periodic boundary conditions
We take the same input file as 1Dsuperlattice_dispersion_6nm.in
but this time we replace the AlGaAs barrier with GaAs so that we have
only pure bulk GaAs with a length of 12 nm. So our superlattice period has the
length L=12 nm.
(Actually it has the length L = 12.25 due to the grid point resolution of 0.25
nm.)
At the boundaries we apply periodic boundary conditions and the same
superlattice options as above:
1Dsuperlattice_dispersion_bulk_GaAs.in:
$quantummodelelectrons
...
boundarycondition001 = periodic !
periodic boundary conditions are necessary for superlattices
numks001
= 21 !
number of superlattice vectors along z direction
This figure shows the conduction band edge and the three lowest eigenstates and their corresponding charge density (psi²)
for the superlattice vector k_{z }= 0.
Note that periodic boundary conditions are employed for solving the
Schrödinger equation.
 The ground state wave function is constant with its energy equal to the
conduction band edge energy.
 The energies of the second and third eigenstate are degenerate.
(Note that the energies were shifted so that the conduction band edge of GaAs
equals 0 eV.)
The following figure shows the first three minibands for this superlattice.
They arise from the first, second and third eigenstate.
The second and third eigenstate are degenerate at k_{z }= 0 as can be
seen also in the figure above. Also at k_{z }= 1 and k_{z }= 1,
the first and second eigenstate are degenerate. This is as expected because the
dispersion should look like the parabolic dispersion E(k) of bulk GaAs.
 If you are interested in simulations of
superlattices in 2D and 3D, please contact stefan.birner@nextnano.de.
 Please help us to improve our tutorial Send comments to
support [at]
nextnano.com .
Comments:
 The original input file now leads to a slightly different superlattice
dispersion (checked for the 4 nm GaAs QW tutorial, and first eigenvalue).
Maybe this has to do with the length of the superlattice region or boundary
conditions.
k.p (boxintegration) with parabolic/isotropic electron mass (E_P =
0, S=1/me) leads to exactly the same eigenvalues as in tutorial.
k.p (finitedifferences1D) with parabolic/isotropic electron mass
(E_P = 0, S=1/me) leads to exactly the same eigenvalues as in tutorial.
