nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Empirical tight-binding (sp³s*) band structure of GaAs, GaP and AlAs

Note: This tutorial's copyright is owned by Stefan Birner, www.nextnano.de.

Author: Stefan Birner
Please send comments to nextnano3 (-at-) wsi.tum.de.

If you are interested in the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
1) -> 1D_TightBinding_bulk_GaAs.in
2) -> 1D_TightBinding_bulk_GaAs_so.in
3) -> 1D_TightBinding_bulk_GaP.in
4) -> 1D_TightBinding_bulk_GaP_so.in
5) -> 1D_TightBinding_bulk_AlAs.in
6) -> 1D_TightBinding_bulk_AlAs_so.in
7) -> 1D_TightBinding_bulk_C.in

Note: The nextnano³ version of 2004_08_24 does not include the tight-binding feature. If you are interested in the executable that includes the electric field, please contact stefan.birner@nextnano.de.

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

[Vogl]
A semi-empirical tight-binding theory of the electronic structure of semiconductors
P. Vogl, H.P. Hjalmarson, J.D. Dow
J. Phys. Chem. Solids 44 (5), 365 (1983)

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

Microscopic theory of nanostructured semiconductor devices: beyond the envelope-function approximation
A. Di Carlo
Semiconductor Science and Technology 18, R1 (2003)

We note that nowadays much better 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 [Vogl]    at T =    0 K
• GaAs, GaP and AlAs including spin-orbit coupling using the parameters of [Klimeck] at T = 300 K

[Klimeck]
sp3s* Tight-Binding parameters for transport simulations in compound semiconductors
G. Klimeck, R.C. Bowen, T.B. Boykin, T.A. Cwik
Superlattices and Microstructures 27 (5), 519 (2000)

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, we 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_values.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 modfied.
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_y,k_z),
and two-dimensional slices E(k_x,k_y,0) through the three-dimensional energy dispersion E(k_x,k_y,k_z) for a constant value of k_z, e.g. k_z = 0.

Results

GaAs without spin-orbit coupling

-> 1D_TightBinding_bulk_GaAs.in

The calculated band structure is in excellent agreement with Fig. 11(d) of [Vogl].
The conduction band minimum is at the Gamma point (direct band gap).
Because spin-orbit coupling is not included in the Hamiltonian,
heavy, light and 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 [Vogl] at T = 0 K.

GaAs including spin-orbit coupling

-> 1D_TightBinding_bulk_GaAs_so.in

The calculated band structure is in excellent agreement with Fig. 1 of [Klimeck].
The conduction band minimum is at the Gamma point (direct 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 degenerate at the Gamma point.
The sp³s* empirical tight-binding parameters were taken from [Klimeck] at T = 300 K.

GaP without spin-orbit coupling

-> 1D_TightBinding_bulk_GaP.in

The calculated band structure is in excellent agreement with Fig. 2 of [Vogl].
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 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 [Vogl] at T = 0 K.

GaP including spin-orbit coupling

-> 1D_TightBinding_bulk_GaP_so.in

The calculated band structure is in excellent agreement with Fig. 1 of [Klimeck].
The conduction band minimum is in the vicinity 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 [Klimeck] at T = 300 K.

AlAs without spin-orbit coupling

-> 1D_TightBinding_bulk_AlAs.in

AlAs including spin-orbit coupling

-> 1D_TightBinding_bulk_AlAs_so.in

C (diamond) without spin-orbit coupling

-> 1D_TightBinding_bulk_C.in

• Please help us to improve our tutorial. Send comments to nextnano3 (-at-) wsi.tum.de.