Empirical tight-binding sp³s^* band structure of GaAs, GaP, AlAs, InAs, C (diamond) and Si

The input files to be used are:

1D_TightBinding_bulk_GaAs.in
1D_TightBinding_bulk_GaAs_so.in
1D_TightBinding_bulk_Al0.3Ga0.7As.in
1D_TightBinding_bulk_GaP.in
1D_TightBinding_bulk_GaP_so.in
1D_TightBinding_bulk_AlAs.in
1D_TightBinding_bulk_AlAs_so.in
1D_TightBinding_bulk_C.in
1D_TightBinding_bulk_Si.in
1D_TightBinding_bulk_Ge.in
1D_TightBinding_bulk_InAs_so.in
1D_TightBinding_bulk_AlSb_so.in
1D_TightBinding_bulk_InSb_so.in
1D_TightBinding_bulk_Al0.5In0.5Sb.in

Empirical tight-binding sp³s^* band structure of GaAs and GaP

The empirical tight-binding model that is used here is based on the sp³s^* Hamiltonian, i.e. the 10 x 10 matrix given in Table (A) of [VoglJPCS1983].

In addition, we include spin-orbit coupling leading to a 20 x 20 matrix. The additional terms arising due to spin-orbit coupling are given for instance on p. R5 of [CarloSST2003].

We note that nowadays much better theoretical methods are available for calculating the band structure of bulk materials. However, for educational purposes, the chosen sp³s^* method should be sufficient.

In this tutorial, we calculate the bulk band structure of

GaAs, GaP and AlAs without spin-orbit coupling using the parameters of [VoglJPCS1983] at T = 0 K
GaAs, GaP and AlAs including spin-orbit coupling using the parameters of [KlimeckSM2000] at T = 300 K

Input

The values for the tight binding parametrization have to be specified in the input file:

$numeric-control
...
!------------------------------------------------------------------------------
! Tight-binding parameters for GaAs (values of [Klimeck]). The units are [eV].
!------------------------------------------------------------------------------
!tight-binding-parameters = -3.53284d0             ! Esa (GaAs)
                             0.27772d0             ! Epa
                            -8.11499d0             ! Esc
                             4.57341d0             ! Epc
                            12.33930d0             ! Es_a
                             4.31241d0             ! Es_c
                            -6.87653d0             ! Vss
                             1.33572d0             ! Vxx
                             5.07596d0             ! Vxy
                             0d0                   ! Vs_s_
                             2.85929d0             ! Vsa_pc
                            11.09774d0             ! Vsc_pa
                             6.31619d0             ! Vs_a_pc
                             5.02335d0             ! Vs_c_pa
                             0.32703d0  0.12000d0  ! Delta_so_a Delta_so_c
! Note: a = anion, c = cation
!       s_ = s*

For more information about the meaning of these parameters, refer to the above cited references.

Output

The output of the calculated tight-binding band structure can be found in the following file: TightBinding/BandStructure.dat

The first column contains the number of the grid point in the Brillouin zone. These grid points run

from L point to Gamma point (along Lambda)
from Gamma point to X point (along Delta)
from X point to the U, K points
from U,K points to Gamma point (along Sigma)

The next columns are the eigenvalues of the tight-binding Hamiltonian in units of [eV] for each grid point in k = ( $k_{x}$ , $k_{y}$ , $k_{z}$ ) space.

The file TightBinding/BandStructure_without_so.dat contains the tight-binding band structure without spin-orbit coupling.

The file TightBinding/k_vectors.dat contains for each point the information to which k point it belongs to.

no.	kx	ky	kz	\|k\|	kx [2pi/a]	ky [2pi/a]	kz [2pi/a]	\|k\| [2pi/a]
1	0.314159E+01	0.314159E+01	0.314159E+01	0.544140E+01	0.500000E+00	0.500000E+00	0.500000E+00	0.866025E+00

Note: Currently the units of $k_{x}$ , $k_{y}$ and $k_{z}$ do not take into account the lattice constant a. This should be modified. The values for $k_{x}$ , $k_{y}$ and $k_{z}$ in units of [2pi/a] are correct, however. Another improvement would be to calculate and output the three-dimensional energy dispersion E( $k_{x}$ , $k_{z}$ , $k_{z}$ ) and two-dimensional slices E( $k_{x}$ , $k_{z}$ , 0) through the three-dimensional energy dispersion E( $k_{x}$ , $k_{z}$ , $k_{z}$ ) for a constant value of $k_{z}$ , e.g. $k_{z}$ =0.

Results

GaAs without spin-orbit coupling from 1D_TightBinding_bulk_GaAs.in

../../../_images/BandStructureGaAs_Vogl.jpg

The calculated band structure is in excellent agreement with Fig. 11(d) of [VoglJPCS1983]. The conduction band minimum is at the Gamma point (direct band gap). Because spin-orbit coupling is not included in the Hamiltonian, the sp³s^* empirical tight-binding parameters were taken from [VoglJPCS1983] at T = 0 K.

GaAs including spin-orbit coupling from 1D_TightBinding_bulk_GaAs_so.in

../../../_images/BandStructureGaAs_so_Klimeck.jpg

The calculated band structure is in excellent agreement with Fig. 1 of [KlimeckSM2000]. The conduction band minimum is at the Gamma point (irect band gap). Spin-orbit coupling lifts the degeneracy of heavy/light hole and split-off hole at the Gamma point. Heavy and light hole are still degerate at the Gamma point. The sp³s^* empirical tight-binding parameters were taken from [KlimeckSM2000] at T = 300 K.

GaP without spin-orbit coupling from 1D_TightBinding_bulk_GaP.in

../../../_images/BandStructureGaP_Vogl.jpg

The calculated band structure is in excellent agreement with Fig. 2 of [VoglJPCS1983]. The conduction band minimum is calculated to be at the X point (indirect band gap). Because spin-orbit coupling is not included in the Hamiltonian, heavy, light and the split-off hole are degenerate at the Gamma point, i.e at k = ( $k_{x}$ , $k_{y}$ , $k_{z}$ ) = 0. The sp³s^* empirical tight-binding parameters were taken from [VoglJPCS1983] at T = 0 K.

GaP including spin-orbit coupling from 1D_TightBinding_bulk_GaP_so.in

../../../_images/BandStructureGaP_so_Klimeck.jpg

The calculated band structure is in excellent agreement with Fig. 1 of [KlimeckSM2000]. The conduction band minimum is in the vincinity of the X point at the Delta line (indirect band gap), so-called camel’s back. Spin-orbit coupling lifts the degeneracy of heavy/light hole and split-off hole at the Gamma point. Heavy and light hole are still degenerate at the Gamma point. The sp³s^* empirical tight-binding parameters were taken from [KlimeckSM2000] at T = 300 K.

AlAs without spin-orbit coupling from 1D_TightBinding_bulk_AlAs.in

../../../_images/BandStructureAlAs_Vogl.jpg

InAs including spin-orbit coupling from 1D_TightBinding_bulk_InAs_so.in

../../../_images/BandStructureInAs_so_Klimeck.jpg

C (diamond) without spin-orbit coupling from 1D_TightBinding_bulk_C.in

../../../_images/BandStructureC_Vogl.jpg

Si (silicon) without spin-orbit coupling from 1D_TightBinding_bulk_Si.in

../../../_images/BandStructureSi_Vogl.jpg

The k space resolution, i.e. the number of grid points on the axis of these plots can be adjusted. This can be done with:

$tighten
calculate-tight-binding-tighten = no
destination-directory           = TightBinding/
number-of-k-points              = 50             ! This number corresponds to 50 k points between the Gamma point and the X point
                                                 ! The number of k points along the other directions are scaled accordingly.

Empirical tight-binding sp3s* band structure of GaAs, GaP, AlAs, InAs, C (diamond) and Si

Empirical tight-binding sp3s* band structure of GaAs and GaP

Input

Output

Results

Empirical tight-binding sp³s^* band structure of GaAs, GaP, AlAs, InAs, C (diamond) and Si

Empirical tight-binding sp³s^* band structure of GaAs and GaP