def _get_conductivity(self, n_idx, t_idx):
velocities = self._get_group_velocity()
lifetimes = 1 / self._get_scattering_rates(n_idx, t_idx)
energies = self._get_energies()
_, weights = np.unique(self.ir_to_full_kpoint_mapping,
return_counts=True)
ef = self.fermi_levels[n_idx, t_idx]
temp = self.temperatures[t_idx]
dfde = -dFDde(energies, ef, temp * units.BOLTZMANN)
nkpoints = len(self.kpoints)
integrand = velocities**2 * lifetimes * dfde * weights[
None, :] / nkpoints
conductivity = np.sum(integrand)
return integrand / conductivity
def calculate_fd_cutoffs(
self, fd_tolerance: Optional[float] = 0.01, cutoff_pad: float = 0.0
):
energies = self.dos.energies
# three fermi integrals govern transport properties:
# 1. df/de controls conductivity and mobility
# 2. (e-u) * df/de controls Seebeck
# 3. (e-u)^2 df/de controls electronic thermal conductivity
# take the absolute sum of the integrals across all doping and
# temperatures. this gives us the energies that are important for
# transport
weights = np.zeros(energies.shape)
for n, t in np.ndindex(self.fermi_levels.shape):
ef = self.fermi_levels[n, t]
temp = self.temperatures[t]
dfde = -dFDde(energies, ef, temp * units.BOLTZMANN)
sigma_int = np.abs(dfde)
seeb_int = np.abs((energies - ef) * dfde)
ke_int = np.abs((energies - ef) ** 2 * dfde)
# normalize the transport integrals and sum
nt_weights = sigma_int / sigma_int.max()
nt_weights += seeb_int / seeb_int.max()
nt_weights += ke_int / ke_int.max()
weights = np.maximum(weights, nt_weights)
if not self.is_metal:
# weights should be zero in the band gap as there will be no density
vb_bands = [
np.max(self.energies[s][: self.vb_idx[s] + 1]) for s in self.spins
]
cb_bands = [
np.min(self.energies[s][self.vb_idx[s] + 1 :]) for s in self.spins
]
vbm_e = np.max(vb_bands)
cbm_e = np.min(cb_bands)
weights[(energies > vbm_e) & (energies < cbm_e)] = 0
weights /= np.max(weights)
cumsum = np.cumsum(weights)
cumsum /= np.max(cumsum)
if fd_tolerance:
min_cutoff = energies[cumsum < fd_tolerance / 2].max()
max_cutoff = energies[cumsum > 1 - fd_tolerance / 2].min()
else:
min_cutoff = energies.min()
max_cutoff = energies.max()
min_cutoff -= cutoff_pad
max_cutoff += cutoff_pad
logger.info("Calculated Fermi–Dirac cut-offs:")
log_list(
[
"min: {:.3f} eV".format(min_cutoff / units.eV),
"max: {:.3f} eV".format(max_cutoff / units.eV),
]
)
self.fd_cutoffs = (min_cutoff, max_cutoff)