nextnano^{3}  Tutorial
next generation 3D nano device simulator
Empirical tightbinding (sp^{3}s*) band structure of GaAs, GaP,
AlAs, InAs, C (diamond) and Si
If you are interested in the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 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 tightbinding (sp^{3}s*) band structure of GaAs and GaP
The empirical tightbinding model that is used here is based on the sp^{3}s*
Hamiltonian, i.e. the 10 x 10 matrix given in Table (A) of
[Vogl]
A semiempirical tightbinding theory of the electronic structure of
semiconductors
P. Vogl, H.P. Hjalmarson, J.D. Dow
J. Phys. Chem. Solids 44 (5), 365 (1983)
Download the paper including corrections.
In addition, we include spinorbit coupling leading to a 20 x 20 matrix.
The additional terms arising due to spinorbit coupling are given for instance
on p. R5 of
Microscopic theory of nanostructured semiconductor devices: beyond the
envelopefunction approximation
A. Di Carlo
Semiconductor Science and Technology 18, R1 (2003)
We note that nowadays much better theoretical methods are available for calculating the
band structure of bulk materials.
However, for educational purposes, the chosen sp^{3}s* method should be
sufficient.
In this tutorial, we calculate the bulk band structure of
 GaAs, GaP and AlAs without spinorbit coupling using
the parameters of
[Vogl] at T =
0 K
 GaAs, GaP and AlAs including spinorbit coupling using the parameters
of
[Klimeck] at T = 300 K
[Klimeck]
sp3s* TightBinding 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 tightbinding parametrization have to be specified in the
input file:
$numericcontrol
...
!
! Tightbinding parameters for GaAs (values of [Klimeck]). The units are
[eV].
!
!tightbindingparameters = 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 tightbinding 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 tightbinding 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 tightbinding band structure without spinorbit 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 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 threedimensional
energy dispersion E(k_{x},k_{y},k_{z}),
and twodimensional slices E(k_{x},k_{y},0) through the
threedimensional energy dispersion E(k_{x},k_{y},k_{z})
for a constant value of k_{z}, e.g. k_{z} = 0.
Results
GaAs without spinorbit 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 spinorbit coupling is not included in the Hamiltonian,
heavy, light and splitoff hole are degenerate at the Gamma point, i.e. at k
= (k_{x},k_{y},k_{z}) = 0.
The sp^{3}s* empirical tightbinding parameters were taken from
[Vogl] at T = 0 K.
GaAs including spinorbit 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).
Spinorbit coupling lifts the degeneracy of heavy/light hole and splitoff hole
at the Gamma point.
Heavy and light hole are still degenerate at the Gamma point.
The sp^{3}s* empirical tightbinding parameters were taken from
[Klimeck] at T = 300 K.
GaP without spinorbit 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 spinorbit coupling is not included in the Hamiltonian,
heavy, light and splitoff hole are degenerate at the Gamma point, i.e. at k
= (k_{x},k_{y},k_{z}) = 0.
The sp^{3}s* empirical tightbinding parameters were taken from
[Vogl] at T = 0 K.
GaP including spinorbit 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), socalled camel's back.
Spinorbit coupling lifts the degeneracy of heavy/light hole and splitoff hole
at the Gamma point.
Heavy and light hole are still degenerate at the Gamma point.
The sp^{3}s* empirical tightbinding parameters were taken from
[Klimeck] at T = 300 K.
AlAs without spinorbit coupling
> 1D_TightBinding_ bulk_ AlAs.in
AlAs including spinorbit coupling
> 1D_TightBinding_bulk_AlAs_so.in
InAs including spinorbit coupling
> 1D_TightBinding_bulk_InAs_so.in
C (diamond) without spinorbit coupling
> 1D_TightBinding_ bulk_ C.in
Si (silicon) without spinorbit coupling
> 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.
$tighten
calculatetightbindingtighten = no
!
destinationdirectory
= TightBinding/
numberofkpoints
= 50
! This corresponds to 50 k
points between the Gamma point and the X point.
! The number of k points along the other directions are scaled
correspondingly.
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
