tidy3d.PalankovskiQuayApproxCarrierLifetime

tidy3d.PalankovskiQuayApproxCarrierLifetime#

class PalankovskiQuayApproxCarrierLifetime[source]#

Bases: Tidy3dBaseModel

Doping- and temperature-dependent SRH carrier lifetime, Palankovski–Quay empirical (Scharfetter-style) approximation.

Parameters:

  • tau_max (PositiveFloat) – [units = sec]. Reference lifetime \(\tau_{max}\) from the Palankovski-Quay empirical form (book Table 3.38).

  • N_ref (PositiveFloat) – [units = 1/cm^3]. Reference doping concentration \(N_{ref}\) in the Scharfetter doping-dependence factor.

  • gamma (float = 1.0) – Dimensionless exponent \(\gamma\) of the doping-dependence factor.

  • alpha_T (float = -1.5) – Dimensionless temperature exponent \(\alpha_T\). The Palankovski-Quay model fixes this at \(-3/2\); users may override it.

Notes

This model expresses the Shockley-Read-Hall carrier lifetime as a function of absolute temperature \(T\) and total ionized dopant concentration \(N = N_D + N_A\):

\[\tau(N, T) = \tau_{max}\, \left(\frac{N}{N_{ref}}\right)^{-\gamma}\, \left(\frac{300}{T}\right)^{-\alpha_T}\]

This is the empirical Scharfetter-style form (Palankovski & Quay [1], eqs. 3.157/3.158) combined with the temperature factor of eqs. 3.160/3.161. The book fixes the temperature exponent at \((300/T)^{3/2}\) (i.e. \(\alpha_T = -3/2\)); the parameter is exposed here so users can override it. Material-specific values for \(\tau_{max}\), \(N_{ref}\), and \(\gamma\) are tabulated for Si, SiGe, GaAs, InGaAs, and InAlAs in Table 3.38 of the reference.

The trap-assisted band-to-band tunneling enhancement \(1/(1+r_\nu)\) and the surface-recombination term \(s_\nu/y\) from the book’s full physics-based form are not included in this approximation.

For numerical stability — the unclamped form diverges as \(N \to 0\), which is unphysical (\(\tau_{max}\) is the intrinsic-region upper bound) and would produce NaN in the SRH Jacobian — the backend evaluator floors \(N\) at \(N_{ref}\) before applying the doping factor. The formula above therefore applies verbatim for \(N \ge N_{ref}\); for \(N < N_{ref}\) the lifetime saturates at \(\tau_{max}\,(300/T)^{-\alpha_T}\).

Example

>>> import tidy3d as td
>>> default_Si = td.PalankovskiQuayApproxCarrierLifetime(
...   tau_max=1e-5,
...   N_ref=1e16,
...   gamma=1.0,
...   alpha_T=-1.5,
... )

References

Attributes

tau_max

N_ref

gamma

alpha_T

tau_max#
N_ref#
gamma#
alpha_T#