 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
InAs / GaSb broken gap quantum well (BGQW) (typeII band alignment)
Author:
Stefan Birner
> 1DInAs_GaSb_BGQW_k_zero_nn3.in /
*_nnp.in
 input file for the nextnano^{3} and nextnano++ software
> 1DInAs_GaSb_BGQW_k_parallel_nn3.in / *_nnp_01.in / *_nnp_11.in
 input file for the nextnano^{3} and nextnano++ software
These input files are included in the latest version.
InAs / GaSb broken gap quantum well (BGQW) (typeII band alignment)
This tutorial aims to reproduce Figs. 1, 2(a), 2(b) and 3 of
Hybridization of electron, lighthole, and heavyhole states in
InAs/GaSb quantum wells
A. Zakharova, S.T. Yen, K.A. Chao
Physical Review B 64, 235332 (2001)
The material parameters were taken from
Optical transitions in broken gap heterostructures E. Halvorsen, Y. Galperin, K.A. Chao Physical Review B
61 (24), 16743 (2000)
The heterostructure is a broken gap quantum well (BGQW) with 15 nm InAs and 10
nm GaSb, sandwiched between two 10 nm AlSb layers.
Note that this heterostructure is asymmetric.
To be consistent with the above cited papers, strain is not included into the
calculations although this would be possible.
The structure has a typeII band alignment, i.e. the electrons are
confined in the InAs layer,
whereas the holes are confined in the GaSb layer.
Depending on the width of the InAs and/or GaSb layers, things can be even more
complicated because the hole states can hybridize with the electron states,
making it difficult to distinguish between electronlike and holelike states.
Another difficulty arises because the lowest electron states might be located
below the highest hole states. This requires a new algorithm to occupy the
states according to a suitable Fermi level.
The following figure shows the electron and hole band edges of the BGQW
structure.
 band_structure/cb1D_001.dat (Gamma conduction
band edge) in units of [eV]
 band_structure/vb1D_001.dat (heavy hole
valence band edge) in units of [eV]
 band_structure/vb1D_002.dat (light
hole valence band edge) in units of [eV]
 band_structure/vb1D_003.dat (splitoff hole valence
band edge) in units of [eV]
The origin of the energy scale is set to the InAs conduction band edge energy.
The heavy hole and light hole band edges are degenerate because we neglect the
effects of strain to be consistent with the above cited papers.
Results

> 1DInAs_GaSb_BGQW_k_zero.in
The following figure shows the conduction band edge and the heavy/light hole valence band edges in this
BGQW structure together with the electron (e1, e2), heavy hole (hh1,
hh2, hh3)
and light hole (lh1) energies and
wave functions (psi²), calculated within 8band k.p theory at the zone
center, i.e. at k_{} = 0.
One can clearly see that the electron state (e1,
e2) are confined in the
InAs layer (left part of the figure), whereas the heavy (hh1,
hh2, hh3) and light hole (lh1)
states are confined in the GaSb layer (right part
of the figure).
One can see a slight hybridization of the e1 and lh1
states, i.e. these states are mixed states whereas the heavy hole states (hh1,
hh2, hh3) are not mixed and
thus confined in the GaSb layer.
 Schroedinger_kp/kp_8x8psi_squared_qc001_el _kpar0001 _1D_dir.dat
 contains Psi_{i}^{2}
 Schroedinger_kp/kp_8x8psi_squared_qc001_el_kpar0001_1D_dir_shift.dat  contains
Psi_{i}^{2} + E_{i}
The latter file contains the square of the wave functions (for kpar0001 ,
i.e k_{} =
0, i.e. k_{x} = k_{y} = 0), shifted by their energies, so that
one can nicely plot the conduction and valence band edges together with the
square of the wave functions.
The energies of the eigenstates are in units of [eV] and are contained in
this file:
 Schroedinger_kp/kp_8x8eigenvalues_qc001_el_kpar0001_1D_dir.dat.dat
> 1DInAs_GaSb_BGQW_k_parallel.in
The following figure shows the E(k_{}) dispersion of the
electron and hole states along the [10] direction and along the [11] direction in (k_{x},k_{y}) space.
The [01] direction has the same dispersion due to symmetry arguments.
In this input file, the energy levels and wave functions for 24 k_{}
points along a line from (k_{x},k_{y}) = (0,0) to (k_{x},k_{y})
= (0,k_{y}) have been calculated.
 Schroedinger_kp/kpar1D_disp_01_00el_8x8kp_ev_min001_ev_max020.dat
contains the k_{} dispersion from [00] to [01] because
in the input file it is specifed:
$outputkpdata
!kpardispersion = 0100 ! plot k_{} dispersion from [00] to [01]
kpardispersion = 010011 ! plot k_{} dispersion from
[01] to [00] to [11]
The first column contains the k_{} value, the other
columns contain the eigenvalues for each k_{} value: E_{n}(k_{})
= E_{n}(k_{x},k_{y}) = E_{n}(0,k_{y})
Here, n = 1,...,20. (...ev_min001_ev_max020... )
Note that for this particular example, the eigenvalues have to be
sorted manually if you want to connect the energy values,
i.e. to include lines ("lines are a guide to the eye").
The black lines are the results of
the nextnano++ software, the red dots
are the results of the nextnano^{3} software.
At an inplane wave vector of 0.014 1/Angstrom, strong intermixing between the
e1 and the lh1 states occurs.
In contrast to the wave functions at k_{} = 0, where the e1
and lh1 wave functions are nearly purely electron or hole like, the
wave functions at k_{} = (0,0.014) = (0.014,0) are a
mixture of electron and light hole wave functions. Compare with Fig. 4 of the
A. Zakharova et al.
In asymmetric quantum wells, the double spin degeneracy is lifted at finite
values of k_{} because of spinorbit interaction. This is
the reason why we have two different dispersions E(k_{})
for "spin up" and "spin down" states.
This also means that the wave functions at finite k_{} are
different for "spin up" and "spin down" states.
 The file
Schroedinger_kp/kp_8x8k_parallel_qc001_el1D_dir.dat
tells us which number of k_{} vector corresponds to (k_{x},k_{y}).
k_par_number k_x [1/nm]
k_y [1/nm]
1
0.000000E+000 0.000000E+000 ==>
k_{} = (k_{x},k_{y}) = (0,0)
[1/nm]
...
29
0.000000E+000 1.400000E+000
==> k_{} = (k_{x},k_{y})
= (0,0.14) [1/nm]
1326
1.00000E+000 1.000000E+000 ==>
k_{} = (k_{x},k_{y}) = (1.0,1.0)
[1/nm]
 In the following figure, we plot the square of the wave functions for k_{} = (0,0.14) nm^{1}.
The corresponding label of our k_{} numbering is
29. Note that this labeling depends on
the k_{} space resolution, i.e. the number of k_{}
points that have been specified in the input file: numkpparallel =
10000
The wave functions are contained in this file:
 Schroedinger_kp/kp_8x8psi_squared_qc001_hl_kpar00029_1D_dir_shift.dat  contains
Psi_{i}^{2} + E_{i}
The electron states (e1)
couple strongly with the light hole states (lh1).
This is expected from the energy dispersion plot because at 0.14 nm^{1}
a strong anticrossing is present for these states.
One can also clearly see that for spin up and spin down states, different
energy levels and different probability densities exist.
This is in contrast to the states at k_{} = 0 which are
twofold spin degenerate as shown in the figure further above.
Our results are similar to Fig. 4 of Zakharova's paper.
