Bands and Statistics¶

Conduction bands¶

classical{ Gamma{} }¶

By calling this group, a conduction band with a minimum at $$\Gamma$$ point becomes available in the model. This band is referred to as Gamma in output files.

output_bandedge{}

Output minimum (band edge) of this band as energy profile in a single file [eV].

averaged
value:

yes or no

default:

no

If averaged = yes then, for each grid point, the energy profile will be averaged between neighboring material grid points. If averaged = no then abrupt discontinuities at interfaces (in 1D two points, in 2D four points, in 3D eight points for each grid point).

Note

The averaged = yes is similar to boxes = no. Note that boxes is related to output of material grid points while averaged is related to output of simulation grid points.

Warning

2D and 3D simulations can produce a lot of output data (order of GB). It is strongly recommended to use averaged = yes for 2D and 3D simulations to avoid excessive consumption of your hard disk.

classical{ L{} }¶

By calling this group, four conduction bands with minimums at $$L$$ points become available in the model. The bands are referred to as L_1, L_2, L_3, and L_4 for the $$L$$ valleys located at [1 1 1], [1 -1 1], [1 -1 -1], and [1 1 -1] directions, respectively, in output files.

Note

This group does not apply to materials with wurtzite symmetry.

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }.

classical{ X{} }¶

By calling this group, three conduction bands with minimums at $$X$$ points become available in the model. The bands are referred to as X_1, X_2, and X_3 for the $$X$$ valleys located at [1 0 0], [0 1 0], and [0 0 1] directions, respectively, in output files.

Attention

This group does not apply to Si, Ge, GaP, and to materials with wurtzite symmetry

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }.

classical{ Delta{} }¶

By calling this group, three conduction bands with minimums along the $$\Delta$$ lines become available in the model. The bands are referred to as Delta_1, Delta_2, and Delta_3 for the $$\Delta$$ valleys located at [1 0 0], [0 1 0], and [0 0 1] directions, respectively, in output files.

Attention

This group applies to Si, Ge, GaP

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }.

Valence bands¶

classical{ HH{} }¶

By calling this group, a heavy-hole valence band with maximum at $$\Gamma$$ point becomes available in the model. This band is referred to as HH in output files.

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }

classical{ LH{} }¶

By calling this group, a light-hole valence band with maximum at $$\Gamma$$ point becomes available in the model. This band is referred to as LH in output files.

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }

classical{ SO{} }¶

By calling this group, a split-off valence (or crystal-field split-off in wurtzite) band with maximum at $$\Gamma$$ point becomes available in the model. This band is referred to as SO in output files.

output_bandedge{}

This group behaves analogously as for Gamma{ output_bandedge{} }

Carrier statistics for classical densities¶

classical{ carrier_statistics }
options:

fermi_dirac maxwell_boltzmann

default:

fermi_dirac

Optionally, one can use Maxwell-Boltzmann statistics for the classical densities (not recommended as this is only an approximation which is only applicable in certain cases).

In order to maintain consistency, also the (integrated) energy distribution (density_vs_energy) and the classical emission spectra and densities are computed using the same statistics. Use together with quantum regions is possible but not recommended, and convergence of the current-Poisson or quantum-current-Poisson equation may become worse (please readjust convergence parameters accordingly).

Note

• $$n=N_c\ \mathcal{F}_{1/2}\left(\frac{E_F-E_c}{k_BT}\right)$$ (electron density for fermi_dirac)

• $$p=N_c\ \mathcal{F}_{1/2}\left(\frac{E_v-E_F}{k_BT}\right)$$ (hole density for fermi_dirac)

• $$n=N_c\exp\left(\frac{E_F-E_c}{k_BT}\right)$$ (electron density for maxwell_boltzmann)

• $$p=N_c\exp\left(\frac{E_v-E_F}{k_BT}\right)$$ (hole density for maxwell_boltzmann)

• where $$\mathcal{F}_n(E)$$ is a Fermi-Dirac integral of the order $$n$$.