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# nextnano3 - Tutorial

## k.p dispersion in bulk GaAs (strained / unstrained)

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 6-band and 8-band k.p theory results.
• We calculate E(k) for bulk GaAs at a temperature of 300 K.

## Bulk dispersion along [001] and along [011]

• \$output-kp-data
destination-directory    = kp/

bulk-kp-dispersion       = yes
grid-position            = 5d0                      !
in units of [nm]
!----------------------------------------
! Dispersion along [011] direction
! Dispersion along [001] direction
! maximum |k| vector = 1.0 [1/nm]
!----------------------------------------
k-direction-from-k-point = 0d0  0.7071d0  0.7071d0  !
k-direction and range for dispersion plot [1/nm]
k-direction-to-k-point   = 0d0  0d0       1.0d0     !
k-direction and range for dispersion plot [1/nm]

!
The dispersion is calculated from the k point 'k-direction-from-k-point' to Gamma, and then from the Gamma point to 'k-direction-to-k-point'.

number-of-k-points       = 100                      !
number of k points to be calculated (resolution)
shift-holes-to-zero      = yes                      ! 'yes' or 'no'
\$end_output-kp-data
• We calculate the pure bulk dispersion at grid-position=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 Bir-Pikus strain Hamiltonian will be diagonalized.
The grid point at grid-position must be located inside a quantum cluster.
shift-holes-to-zero = 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 number-of-k-points. 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 split-off energy of 0.341 eV is identical to the split-off energy as defined in the database:
6x6-kp-parameters = ... 0.341d0 !  [eV]

• If one zooms into the holes and compares 6-band vs. 8-band k.p, one can see that 6-band and 8-band coincide for |k| < 1.0 nm-1 for the heavy and light hole but differ for the split-off hole at larger |k| values.

To switch between 6-band and 8-band k.p one only has to change this entry in the input file:
\$quantum-model-holes
...
model-name = 8x8kp !
for 8-band k.p
= 6x6kp !
for 6-band k.p

## 8-band k.p vs. effective-mass approximation

• Now we want to compare the 8-band k.p dispersion with the effective-mass 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 In0.2Ga0.8As. 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:

\$simulation-flow-control
...
strain-calculation  = homogeneous-strain
\$end_simulation-flow-control

\$domain-coordinates
...
pseudomorphic-on    = In(x)Ga(1-x)As
alloy-concentration = 0.20d0
\$end_domain-coordinates

As substrate material we take In0.2Ga0.8As and assume that GaAs is strained pseudomorphically (homogeneous-strain) 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 anti-crossing along [001]. (Actually, the anti-crossing 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 6-band vs. 8-band k.p, one can see that the agreement between heavy and light holes is not as good as in the unstrained case where 6-band and 8-band k.p lead to almost identical dispersions.

Note that in the strained case, the effective-mass approximation is very poor.

• Please help us to improve our tutorial! Send comments to support [at] nextnano.com.