nextnano³ - Tutorial

next generation 3D nano device simulator

1D 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 check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
-> bulk_kp_dispersion_GaAs_nn3.in / *_nnp.in -input file for nextnano³ / nextnano++ software-> bulk_kp_dispersion_GaAs_nn3_3D.in -input file for nextnano³-> bulk_kp_dispersion_GaAs_nn3_strained.in / *_nnp_strained.in -input file for nextnano³ / nextnano++ software

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 In_0.2Ga_0.8As. 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-coordinatesAs substrate material we take In_0.2Ga_0.8As 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.

Analysis of eigenvectors

(preliminary)

Usingschroedinger-kp-basis = Voon-Willatzen--Bastard--Foreman(box-integration) one obtains the following output for the eigenvectors at the Gamma point, k = (k_x,k_y,k_z) = 0.

The square of the spinor is contained in this file:bulk_8x8kp_dispersion_eigenvectors_squared_000.datExample: The X_up component contains a complex number. Here, we output the square of X_up. This file then gives us information on the strength of the coupling of the mixed states.
This file is easier to analyze then the file containing the complex numbers (bulk_8x8kp_dispersion_eigenvectors_000.dat).

eigenvalue S+ S- HH LH LH LH SO SO 1 0 1.0 0 0 0 0 0 0 2 1.0 0 0 0 0 0 0 0 3 0 0 0 1.0 0 0 0 0 4 0 0 0 0 1.0 0 0 0 5 0 0 0 0 0 1.0 0 0 6 0 0 1.0 0 0 0 0 0 7 0 0 0 0 0 0 0 1.0 8 0 0 0 0 0 0 1.0 0

eigenvalue S+ S- X+ Y+ Z+ X- Y- Z- 1 1.0 0 0 0 0 0 0 0 2 0 1.0 0 0 0 0 0 0 3 0 0 0 0 0.5 0.5 0 0 4 0 0 0 0 0.166 0.166 0.666 0 5 0 0 0.5 0 0 0 0 0.5 6 0 0 0.166 0.666 0 0 0 0.166 7 0 0 0 0 0.333 0.333 0.333 0 8 0 0 0.333 0.333 0 0 0 0.333

+: spin up, -: spin down

The electron eigenstates are 2-fold degenerate, i.e. have the same energy, and are decoupled from the holes.

1: S_down 2: S_up

The hole eigenstates are 4-fold (heavy and light holes) and 2-fold degenerate (split-off holes).

3: |3/2, 3/2> hh spin up 1/SQRT(2) | ( X + iY ) up > 4: |3/2, 1/2> lh 1/SQRT(6) | ( X + iY ) down > - SQRT(2/3) | Z up > 5: |3/2,-1/2> lh -1/SQRT(6) | ( X - iY ) up > - SQRT(2/3) | Z down > 6: |3/2,-3/2> hh spin down 1/SQRT(2) | ( X - iY ) down > 7: |1/2, 1/2> s/o split 1/SQRT(3) | ( X + iY ) down > + 1/SQRT(3) | Z up > 8: |1/2,-1/2> s/o split -1/SQRT(3) | ( X - iY ) up > + 1/SQRT(3) | Z down >
up: spin up, down: spin down1/SQRT(2) = 0.707 ==> 1/SQRT(2)^2 = 0.5 1/SQRT(3) = 0.577 ==> 1/SQRT(3)^2 = 0.333 1/SQRT(6) = 0.408 ==> 1/SQRT(6)^2 = 0.166

nextnano3 - Tutorial