scattering module¶

This module, scattering.py contains the setup and calculation of the scattering properties.

Contains routines to set up the scattering of the charge carriers.

scattering.check_scattering(tr)¶

Checks the scattering arrays.

Also that they are dimensionalized to the energy values stored in the current Bandstructure() object.

Parameters:	tr : object A Transport() object.
Returns:	None

scattering.combined_scattering(tr, energy, tau0, energy_trans)¶

Calculates the total relaxation time.

Parameters:

Parameters:	tr : object A Transport() object energy : float The energy of the charge carrier in eV. tau0 : ndarray Dimension: (12) Contains the relaxation time prefactors for the different scattering mechanisms in units of fs. energy_trans : ndarray Dimension: (12) Contains the energy transitions in eV (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms. effmass : float The effective mass in units of the electron mass
Returns:	float The combined relaxation time in fs.

tr : object: A Transport() object
energy : float: The energy of the charge carrier in eV.
tau0 : ndarray: Dimension: (12)

Contains the relaxation time prefactors for the different scattering mechanisms in units of fs.
energy_trans : ndarray: Dimension: (12)

Contains the energy transitions in eV (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms.
effmass : float: The effective mass in units of the electron mass

Returns:

float: The combined relaxation time in fs.

Notes

Calculates the total relaxation time

System Message: WARNING/2 (\\frac{1}{\\tau}=\\sum_i \\frac{1}{\\tau_i},)

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where $\\tau=\\tau_0E^{r-1/2}$ . The array scattering_tau0_select determines which scattering to include in the sum. Consult scattering_parabolic() for additional details. The scattering prefactors $\\tau_0$ are ordered in a sequence described there. The scattering_tau0_select follows this sequence and is set in the bandstructure configuration file.

scattering.find_r_for_closed(tr, band)¶

Analyze the input tau0 and find the associated scattering values r.

Parameters:	tr : object A Transport() object band : integer The band index.
Returns:	integer Two times the r value to avoid half integer values.

Notes

These are necessary for the analytic Fermi integrals.

scattering.interpolate(tr, method='linear')¶

Interpolates the scattering array on all available energies.

Parameters:	tr : object A Transport() object containing the scattering arrays and the energies etc. method : string, optional The interpolation method to use. Uses the `interp1d()` function of Scipy and this sets the parameter kind. Defaults to “linear”.
Returns:	None

See also

scipy.interpolate.interp1d

Notes

Here we only perform an interpolation on the array containing the total relaxation time since this is used during the transport calculations.

scattering.pad_scattering_values(tr)¶

Pad the scattering values.

Parameters:	tr : object A Transport() object.
Returns:	None

Notes

The padded values are stored in the tr object.

We need to pad the energies where the dos is calculated with a larger number of samples such that we cover the whole energy range in the stored bandstructure due to later interpolation routines etc. not going out of bounds when such energies are passed to the interpolator etc.

We can set it to a large number because 1 eV outside the chemical potential. We already know no states contribute at temperatures below 2000 K.

scattering.scattering_dos(tr, dos, energies, select_scattering)¶

Setup scattering mechnisms.

Store values in the scattering arrays using the density of states data as the energy dependency.

Parameters:

Parameters:	tr : object A Transport() object dos : ndarray Dimension: (N,M) Array containing the partial density of states (1/eV/AA^3), where N is the band index and M is the energy index. energies : ndarray Dimension: (M) Array containing the energy in eV at M samplings where the density of states is calculated. select_scattering : ndarray Dimension: (12) Array containing integers. Set to 1 to select the scattering, 0 to exclude. The variables in select_scattering are set in the bandstructure configuration file, one value for each scattering and band. See notes below for the currrently available scattering mechnisms.
Returns:	scattering_inv : ndarray Dimension: (T,N,M,12) The scattering array in fs units in the current Transport() object for T temperature steps, N number of bands, M number of energy steps and 12 number of scattering mechanisms scattering_total_inv : ndarray Dimension: (T, N, M) The total (all mechanisms summed) scattering array in fs units) in the current Transport() object for T temperature steps, N number of bands and M number of energy steps scattering_tau0 : ndarray Dimension: (T, N, 12) The scattering prefactor array, tau0 in units of fs, in the current Transport() object for T temperature steps, N number of bands and 12 number of scattering mechanisms.

tr : object: A Transport() object
dos : ndarray: Dimension: (N,M)

Array containing the partial density of states (1/eV/AA^3), where N is the band index and M is the energy index.
energies : ndarray: Dimension: (M)

Array containing the energy in eV at M samplings where the density of states is calculated.
select_scattering : ndarray: Dimension: (12)

Array containing integers. Set to 1 to select the scattering, 0 to exclude. The variables in select_scattering are set in the bandstructure configuration file, one value for each scattering and band. See notes below for the currrently available scattering mechnisms.

Returns:

scattering_inv : ndarray: Dimension: (T,N,M,12)

The scattering array in fs units in the current Transport() object for T temperature steps, N number of bands, M number of energy steps and 12 number of scattering mechanisms
scattering_total_inv : ndarray: Dimension: (T, N, M)

The total (all mechanisms summed) scattering array in fs units) in the current Transport() object for T temperature steps, N number of bands and M number of energy steps
scattering_tau0 : ndarray: Dimension: (T, N, 12)

The scattering prefactor array, tau0 in units of fs, in the current Transport() object for T temperature steps, N number of bands and 12 number of scattering mechanisms.

Notes

Currently only the following scattering mechanisms are supported:

select_scattering index	scattering mechanism
1	Acoustic phonon scattering from def. pot.
2	Non-polar optical phonon scattering from def. pot. (alpha stage)
3	Intervalley phonon scattering (alpha stage)
4	Polar optical phonon scattering (alpha stage)
5	None
6	None
7	None
8	None
9	None
10	None
11	None
12	Constant (energy and k-point independent)

Only the acoustic phonon scattering has been tested.

Consult the bandstructure configuration file for the respective constants that have to be set besides select_scattering and their units.

Todo

Add more extensive documentation for the different scattering mechanisms.

Warning

The scattering models based on density of states does currently not properly involve the energy shift required for the transfer energies. This is quite serious, but does not influene the acoustic phonon scattering (taken to be zero in the model implemented here). The current approach if not using the analytic parabolic models is that the transfer energies for all scattering mechanisms are summed and the energy of where the relaxation time is evaluated is shifted by this amount during interpolation. The sum approximation is not physically justified and needs additional investigation, also the interpolation are sensitive and can fail close to the van Hove singularities since the relaxation time is propotional to the inverse of the density of states. All known problems pertaining to density of states are also manifested here for the scattering.

scattering.scattering_parabolic(tr, energies, select_scattering, use_eonk=False)¶

Setup scattering mechnisms.

Store values in the scattering arrays using parabolic band dispersions as an approximation.

Parameters:

Parameters:	tr : object A Transport() object energies : ndarray Dimension: (N) Array containing the energy in eV at N samplings where the scattering values are o be calculated. select_scattering : ndarray Dimension: (12) Array containing integers. Set to 1 to select the scattering, 0 to exclude. The variables in select_scattering are set in the bandstructure configuration file, one value for each scattering and band. See notes below for the currrently available scattering mechnisms. use_eonk : boolean If set to True, generate the scattering values on the supplied energy for each band and on its k-points
Returns:	scattering_inv : ndarray Dimension: (T,N,M,12) The scattering array in fs units in the current Transport() object for T temperature steps, N number of bands, M number of energy steps and 12 number of scattering mechanisms scattering_total_inv : ndarray Dimension: (T, N, M) The total (all mechanisms summed) scattering array in fs units) in the current Transport() object for T temperature steps, N number of bands and M number of energy steps scattering_tau0 : ndarray Dimension: (T, N, 12) The scattering prefactor array, tau0 in units of fs, in the current Transport() object for T temperature steps, N number of bands and 12 number of scattering mechanisms.

tr : object: A Transport() object
energies : ndarray: Dimension: (N)

Array containing the energy in eV at N samplings where the scattering values are o be calculated.
select_scattering : ndarray: Dimension: (12)

Array containing integers. Set to 1 to select the scattering, 0 to exclude. The variables in select_scattering are set in the bandstructure configuration file, one value for each scattering and band. See notes below for the currrently available scattering mechnisms.
use_eonk : boolean: If set to True, generate the scattering values on the supplied energy for each band and on its k-points

Returns:

scattering_inv : ndarray: Dimension: (T,N,M,12)

The scattering array in fs units in the current Transport() object for T temperature steps, N number of bands, M number of energy steps and 12 number of scattering mechanisms
scattering_total_inv : ndarray: Dimension: (T, N, M)

The total (all mechanisms summed) scattering array in fs units) in the current Transport() object for T temperature steps, N number of bands and M number of energy steps
scattering_tau0 : ndarray: Dimension: (T, N, 12)

The scattering prefactor array, tau0 in units of fs, in the current Transport() object for T temperature steps, N number of bands and 12 number of scattering mechanisms.

Notes

Currently only the following scattering mechanisms are supported:

select_scattering index	scattering mechanism
1	Acoustic phonon scattering from def. pot.
2	Non-polar optical phonon scattering from def. pot.
3	Intervalley phonon scattering
4	Polar optical phonon scattering
5	Piezoelectric acoustic phonon scattering
6	Ionized impurity scattering, Brooks-Herring
7	Ionized impurity scattering, Conwell-Weisskopf
8	Alloy scattering
9	None
10	None
11	None
12	Constant (energy and k-point independent)

Also consult the bandstructure configuration file for the respective constants that have to be set besides select_scattering and their units.

Todo

Add more extensive documentation for the different scattering mechanisms.