5.9.1. k.p dispersion in bulk GaAs (strained / unstrained)
- Input files:
bulk_kp_dispersion_GaAs_nnp.nnp
bulk_kp_dispersion_GaAs_nnp_strained.nnp
- Scope:
We calculate \(E(k)\) of strained and unstrained \(\rm GaAs\).
Band structure of bulk \(\rm GaAs\)
Input file: bulk_kp_dispersion_GaAs_nnp.nnp
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 [110]
[000] to [100]
We compare 6-band and 8-band k.p theory results. We calculate \(E(k)\) for bulk \(\rm GaAs\) at a temperature of 300 K.
Bulk dispersion along [100] and along [110]
quantum{
region{
...
bulk_dispersion{
lines{ # set of dispersion lines along crystal directions of high symmetry
name = "lines"
position{ x = 5.0 }
k_max = 1.0
spacing = 0.01
shift_holes_to_zero = yes
}
path{ # dispersion along arbitrary path in k-space
name = "user_defined_path"
position{ x = 5.0 }
point{ k = [0.7071, 0.7071, 0.0] }
point{ k = [0.0, 0.0, 0.0] }
point{ k = [1.0, 0.0, 0.0] }
spacing = 0.01
shift_holes_to_zero = yes
}
}
}
}
We calculate the pure bulk dispersion at position x = 5 nm. In our case this is \(\rm GaAs\), 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 position{ x = 5.0 } must be located inside a quantum region.
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 spacing. To increase the resolution, just increase this number.
We use two direction in k space, i.e. from [000] to [110] and from [000] to [100]. In the latter case the maximum value of \(|k|\) is
Note that for values of \(|k|\) larger than 1.0 nm-1, k.p theory might not be a good approximation anymore.
The results of the calculation can be found in the folder bias_00000\Quantum\Bulk_dispersions. Figure 5.9.1.1 visualizes the results.
Figure 5.9.1.1 Bulk k.p dispersion in \(\rm GaAs\): \(E(k)\) along [100] and [110].
The split-off energy of 0.341 eV is identical to the split-off energy as defined in the database:
...
valence_bands{ delta_SO = 0.341 } # [eV] Vurgaftman1
...
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, see Figure 5.9.1.2.
Figure 5.9.1.2 Bulk k.p dispersion in GaAs: \(E(k)\) along [100] and [110] - Comparision between 6x6 and 8x8 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, see Figure 5.9.1.3.
Figure 5.9.1.3 Bulk k.p dispersion in \(\rm GaAs\): \(E(k)\) along [100] and [110] - Comparision between 8x8 k.p and effective-mass approximation
Band structure of strained \(\rm GaAs\)
Input file: bulk_kp_dispersion_GaAs_nnp_strained.nnp
Now we perform these calculations again for \(\rm GaAs\) that is strained with respect to \(In_{0.2}Ga_{0.8}As\). The \(InGaAs\) lattice constant is larger than the \(\rm GaAs\) one, thus \(\rm GaAs\) is strained tensely. The changes that we have to make in the input file are the following:
strain{
pseudomorphic_strain{ }
}
run{
strain{ }
}
As substrate material we take \(In_{0.2}Ga_{0.8}As\) and assume that \(\rm GaAs\) is strained pseudomorphically (pseudomorphic_strain{ }) with respect to this substrate, i.e. \(\rm GaAs\) 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 \(\rm GaAs\).
Furthermore, the degeneracy of the heavy and light hole at \(k`= 0 is lifted, see :numref:`fig-1D-kp-dispersion-bulk-GaAs-kp-bandedges-strained\).
Now, the anisotropy of the holes along the different directions [100] and [110] is very pronounced. There is even a band anti-crossing along [100]. (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 anymore. This means that the dispersion is also different in these directions ([100], [010], [001]).
Figure 5.9.1.4 Bulk k.p dispersion in \(\rm GaAs\) strained with respect to \(In_{0.2}Ga_{0.8}As\) : \(E(k)\) along [100] and [110].
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, compare Figure 5.9.1.5.
Figure 5.9.1.5 Bulk valence band k.p dispersion in \(\rm GaAs\) strained with respect to \(In_{0.2}Ga_{0.8}As\) : \(E(k)\) along [100] and [110] - Comparision between 6x6 and 8x8 k.p approximation.
Note that in the strained case, the effective-mass approximation is very poor.
Analysis of eigenvectors
(preliminary)
Using the Voon-Willatzen-Bastard-Foreman k.p basis one obtains the following output for the eigenvectors at the Gamma point, \(k\) = (\(k_x\), \(k_y\), \(k_z\)) = 0.
Example: The x_up component contains a complex number. Here, we show the square of X_up. This gives us information on the strength of the coupling of the mixed states.
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
\(| S \downarrow{}\rangle\) \(\;\)
2
\(| S \uparrow{}\rangle\) \(\;\)
The hole eigenstates are 4-fold (heavy and light holes) and 2-fold degenerate (split-off holes).
3
\(\left|\frac{3}{2}, \frac{3}{2}\right\rangle\) \(\;\) hh spin up
\(\frac{1}{\sqrt{2}}\left| (X +iY) \uparrow{}\right\rangle\)
4
\(\left| \frac{3}{2}, \frac{1}{2}\right\rangle\) \(\;\) lh
\(\frac{1}{\sqrt{6}}\left| (X + iY) \downarrow{} \right\rangle - \sqrt{\frac{2}{3}} \left| Z \uparrow{}\right\rangle\)
5
\(\left|\frac{3}{2}, -\frac{1}{2}\right\rangle\) \(\;\) lh
\(\frac{1}{\sqrt{6}}\left| (X - iY) \uparrow{} \right\rangle - \sqrt{\frac{2}{3}} \left| Z \downarrow{}\right\rangle\)
6
\(\left|\frac{3}{2}, -\frac{3}{2}\right\rangle\) \(\;\) hh spin down
\(\frac{1}{\sqrt{2}}\left| (X - iY) \downarrow{} \right\rangle\)
7
\(\left|\frac{1}{2}, \frac{1}{2}\right\rangle\) \(\;\) s/o split
\(\frac{1}{\sqrt{3}}\left| (X + iY) \downarrow{} \right\rangle - \frac{1}{\sqrt{3}} \left| Z \uparrow{}\right\rangle\)
8
\(\left|\frac{1}{2}, -\frac{1}{2}\right\rangle\) \(\;\) s/o split
\(\frac{1}{\sqrt{3}}\left| (X - iY) \downarrow{} \right\rangle - \frac{1}{\sqrt{3}} \left| Z \downarrow{}\right\rangle\)
\(\frac{1}{\sqrt{2}}\) = 0.707 \(\rightarrow{}\) \(\left(\frac{1}{2}\right)^2\) = 0.5\(\frac{1}{\sqrt{3}}\) = 0.577 \(\rightarrow{}\) \(\left(\frac{1}{3}\right)^2\) = 0.333\(\frac{1}{\sqrt{6}}\) = 0.408 \(\rightarrow{}\) \(\left(\frac{1}{6}\right)^2\) = 0.166
Last update: 2025-10-16