 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
k.p dispersion in bulk GaAs (strained / unstrained)
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.
> 1Dbulk_kp_dispersion_GaAs.in
> 1Dbulk_kp_dispersion_GaAs_3D.in
> 1Dbulk_kp_dispersion_GaAs_strained.in
Band structure of bulk GaAs
 We want to calculate the dispersion E(k) from k=0 nm^{1} to k=1.0
nm^{1} along the
following directions in k space:
 [000] to [001]
 [000] to [011]
We compare 6band and
8band k.p theory results.
 We calculate E(k) for bulk GaAs at a temperature of 300 K.
Bulk dispersion along [001] and along [011]
$outputkpdata
destinationdirectory = kp/
bulkkpdispersion = yes
gridposition = 5d0 ! in units of [nm]
!
! Dispersion along [011] direction
! Dispersion along [001] direction
! maximum k vector = 1.0 [1/nm]
!
kdirectionfromkpoint =
0d0 0.7071d0 0.7071d0 ! kdirection
and range for dispersion plot [1/nm]
kdirectiontokpoint =
0d0 0d0 1.0d0 !
kdirection
and range for dispersion plot [1/nm]
!
The dispersion is calculated from the k point 'kdirectionfromkpoint '
to Gamma, and then from the Gamma point to 'kdirectiontokpoint '.
numberofkpoints = 100 ! number of k points to be calculated (resolution)
shiftholestozero = yes
! 'yes' or 'no'
$end_outputkpdata
 We calculate the pure bulk dispersion at
gridposition=5d0 ,
i.e. for the material located at the grid point at 5 nm. In our case this is
GaAs but it could be any strained alloy. In the latter case, the k.p
BirPikus strain Hamiltonian will be diagonalized.
The grid point at gridposition must be located inside a quantum cluster.
shiftholestozero = yes forces the
top of the valence band to be located at 0 eV.
How often the bulk k.p Hamiltonian should be solved can be specified
via numberofkpoints . To increase the resolution, just increase
this number.
 We use two direction is k space, i.e. from [000] to [001] and from [000] to [011].
In the latter case the maximum value
of k is SQRT(0.7071² + 0.7071²) = 1.0.
Note that for values of k larger than 1.0 nm^{1},
k.p theory might not
be a good
approximation any more.Start the calculation.
The results can be found in kp_bulk/bulk_8x8kp_dispersion_as_in_inputfile_kxkykz_000_kxkykz.dat .
Step 3: Plotting E(k)
 Here we visualize the results. The final figure will look like this:
The splitoff energy of 0.341 eV is identical to the splitoff energy as
defined in the database:
6x6kpparameters = ...
0.341d0 ! [eV]
 If one zooms into the holes and compares 6band vs. 8band k.p, one can
see that 6band and 8band coincide for k < 1.0 nm^{1} for the heavy and light hole but
differ for the splitoff hole at larger k values.
To switch between 6band and 8band k.p one only has to change this entry in
the input file:
$quantummodelholes
...
modelname = 8x8kp ! for 8band k.p
=
6x6kp ! for 6band k.p
8band k.p vs. effectivemass approximation
 Now we want to compare the 8band k.p dispersion with the
effectivemass approximation. The effective mass approximation is a simple
parabolic dispersion which is isotropic (i.e. no dependence on the k
vector direction). For low values of k (k < 0.4 nm^{1}) it is in good agreement
with k.p theory.
The output data can be find here:
kp_bulk/bulk_sg_dispersion.dat .
Band structure of strained GaAs
 Now we perform these calculations again for GaAs that is strained
with respect to In_{0.2}Ga_{0.8}As. The InGaAs lattice
constant is larger than the GaAs one, thus GaAs is strained tensilely.
 The changes that we have to make in the input file are the following:
$simulationflowcontrol
...
straincalculation = homogeneousstrain
$end_simulationflowcontrol
$domaincoordinates
...
pseudomorphicon = In(x)Ga(1x)As
alloyconcentration = 0.20d0
$end_domaincoordinates
As substrate material we take In_{0.2}Ga_{0.8}As and
assume that GaAs is strained pseudomorphically (homogeneousstrain )
with respect to this substrate, i.e. GaAs is subject to a biaxial strain.
 Due to the positive hydrostatic strain (i.e. increase in volume or
negative hydrostatic pressure) we obtain a reduced band gap with respect to
the unstrained GaAs.
Furthermore, the degeneracy of the heavy and light hole at k=0 is
lifted.
Now, the anisotropy of the holes along the different directions [001] and
[011] is very pronounced. There is even a band anticrossing along [001].
(Actually, the anticrossing looks like a "crossing" of the bands but if one
zooms into it (not shown in this tutorial), one can easily see it.)
Note: If biaxial strain is present, the directions along x, y or z are not
equivalent any more. This means that the dispersion is also different in
these directions ([100], [010], [001]).
 If one zooms into the holes and compares 6band vs. 8band k.p, one can
see that the agreement between heavy and light holes is not as good as in the
unstrained case where 6band and 8band k.p lead to almost identical
dispersions.
Note that in the strained case, the effectivemass approximation is very poor.
 Please help us to improve our tutorial! Send comments to
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[at] nextnano.com .
