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How to calculate the mobility?

Could you please share with a simple example (or simpler equation) to do this?

Or in other words, if we have $E_g$, $E_F$, $P$, $V_{cell}$, $E_{vb,max}$, $E_{vb,min}$, $\epsilon_1$, $\epsilon_2$ and $\omega$ (eV) parameters from our calculation, is there any way to calculate the mobility?

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A simple equation for the electron mobility is $\mu=e\tau/m^*$.

Here, $e$ is the electron charge, and the effective mass $m^*$ can be obtained by fitting a parabola to a band minimum/maximum as obtained, e.g., with Quantum ESPRESSO. There are packages that can help with this, see this question.

Computing the relaxation time $\tau$ between scattering events is orders of magnitude more expensive and, to my knowledge, rarely done ab initio - depending on the material, you may need to consider scattering between electrons (e.g. via a many-body perturbation theory code like yambo), between electrons and lattice vibrations (e.g. via EPW), and scattering between electrons and defects/impurities.

I'm not familiar with computing relaxation times though, perhaps others can comment.

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