1.20. InAs / GaSb broken gap quantum well (BGQW) (type-II band alignment)

Author: Stefan Birner

Input files required:

  • 1DInAs_GaSb_BGQW_k_zero_nnp.in

  • 1DInAs_GaSb_BGQW_k_parallel_nnp.in

  • 1DInAs_GaSb_BGQW_k_parallel_nnp_01.in

  • 1DInAs_GaSb_BGQW_k_parallel__nnp_11.in

This tutorial aims to reproduce Figs. 1, 2(a), 2(b) and 3 of Hybridization of electron, light-hole, and heavy-hole states in InAs/GaSb quantum wells

Material parameters used are taken from Optical transitions in broken gap heterostructures.

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 type-II 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 electron-like and hole-like 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 (split-off hole valence band edge) in units of [eV]

../../_images/BGQW_band_profile.jpg

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

The input file used here is 1DInAs_GaSb_BGQW_k_zero_nnp.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 (\(\psi^2\)), calculated within 8-band k.p theory at the zone center, i.e. at \(\mathbf{k}_{||}\) :math:` = 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, h2, hh3) are not mixed and thus confined in the GaSb layer.

We use the data files

  • Schroedinger_kp/kp_8x8psi_squared_qc001_el_kpar0001_1D_dir.dat, which contains \(\psi^2\)

  • Schroedinger_kp/kp_8x8psi_squared_qc001_el_kpar0001_1D_dir_shift.dat, which contains \(\psi^2 + \text{E}_{\text{i}}\)

The latter file contains the square of the wave functions (for par0001, 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.

../../_images/BGQW_psi.jpg

The energies of the eigenstates are in units of [eV] and are contained in the file Schroedinger_kp/kp_8x8eigenvalues_qc001_el_kpar0001_1D_dir.dat

The input file 1DInAs_GaSb_BGQW_k_parallel.in was used for the following results. The following figure shows the \(\text{E}(\mathbf{k}_{||})\) 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 \(\mathbf{k}_{||}\) 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 specified that

dispersion{
    path{
        name = "kpar_01_00_10"
        point{ k = [0.0,  0.0,  1.0] }
        point{ k = [0.0,  0.0,  0.0] }
        point{ k = [0.0,  1.0,  0.0] }
        spacing = 1 / $number_k_parallel
    }
    path{
        name = "kpar_10_00_11"
        point{ k = [0.0, 1.0, 0.0] }
        point{ k = [0.0, 0.0, 0.0] }
        point{ k = [0.0, 1.0, 1.0] }
        spacing = 1 / $number_k_parallel
    }

    output_dispersions{}
    output_masses{}
}

The first column contains the \(k_{||}\) value, the other columns contain the eigenvalues for each \(k_{||}\) value: \(\text{E}_{\text{n}}(k_{||}) = \text{E}_{\text{n}}(k_x, k_y) = \text{E}_{\text{n}}(0, k_y)\). Here, n = 1,…,20. (…ev_min 001**_ev_max **020…) 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 nextnano³ software.

../../_images/Zakharova_PRB2001_InAs15nm_GaSb10nm_k_dispersion.jpg

At an in-plane wave vector of 0.014 1/Å, 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 \(\mathbf{k}_{||}\) because of spin-orbit interaction. This is the reason why we have two different dispersions \(\text{E}(\mathbf{k}_{||})\) for “spin up” and “spin down” states. This also means that the wave functions at finite \(\mathbf{k}_{||}\) 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 \(\mathbf{k}_{||}\) 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|| = (kx,ky) = (0,0) [1/nm]
  ...
  29           0.000000E+000   1.400000E+000     ==>   k|| = (kx,ky) = (0,0.14) [1/nm]
1326           1.00000E+000    1.000000E+000     ==>   k|| = (kx,ky) = (1.0,1.0) [1/nm]

In the following figure, we plot the square of the wave functions for \(\mathbf{k}_{||}\) = (0,0.14) nm-1. The corresponding label of our \(\mathbf{k}_{||}\) numbering is 29. Note that this labeling depends on the \(\mathbf{k}_{||}\) space resolution, i.e. the number of \(\mathbf{k}_{||}\) points that have been specified in the input file: num-kp-parallel = 10000

The wavefunctions (\(\psi^2 + \text{E}_\text{i}\))are contained in the file Schroedinger_kp/kp_8x8psi_squared_qc001_hl_kpar00029_1D_dir_shift.dat

../../_images/BGQW_psi_014nm.jpg

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 \(\mathbf{k}_{||}\) \(=0\) which are two-fold spin degenerate as shown in the figure further above. Our results are similar to Fig. 4 of Zakharova’s paper.

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