# 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 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 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 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 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 InAs including spin-orbit coupling from 1D_TightBinding_bulk_InAs_so.in C (diamond) without spin-orbit coupling from 1D_TightBinding_bulk_C.in Si (silicon) without spin-orbit coupling from 1D_TightBinding_bulk_Si.in 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.