Drift-Diffusion currents of holes in semi-conductors are usually written as : \begin{equation} \mathbf J_p = -D\nabla p- \mu p \nabla \phi \end{equation}

Where p is the hole density, $\phi$ the electric potential, D the diffusion coefficient, and $\mu$ the mobility.

Sometimes, this Drift-Diffusion term is gathered into :

\begin{equation} \mathbf J_p = p\mu\nabla\Psi_p \end{equation}

Where $\Psi_p$ is the quasi-Fermi energy level.

I can show that assuming Maxwell-Boltzmann statistics but I really wonder if that works as well when using Fermi-Dirac statistics?

This is indeed assumed in the following paper "A Self-Consistent Static Model of the Double-Heterostructure Laser " which uses Fermi-Dirac statistics in a regime where Maxwell-Boltzmann is not valid.

  • $\begingroup$ A link or citation for the paper might help. $\endgroup$
    – Jon Custer
    Apr 8, 2017 at 17:20

1 Answer 1


Firstly, in the industry, people do use the 2nd equation widely even in modeling degenerate semiconductors (2DEG in GaAs or inversion layer in FinFET). I wonder if there is any publication studying the validity of such approach.

Then regarding Einstein equation, it actually is only valid for non-degenerate cases. E.g. "Generalized Einstein Relation for Degenerate Semiconductors", by Lindholm, Proceedings of the IEEE, March 1968.

And of course, besides Einstein Equation being wrong, it is also wrong in the process when deriving the 2nd equation from the 1st one as you pointed out.

However, it works "well" so far in the industry when modeling degenerate semiconductor, so it will be interesting to study how much error this has and why it works.

  • $\begingroup$ Great answer! I even forgot about the Einstein relationship which are indeed not necessarily valid. They are a kind of wiedemann-Franz law for semi-conductor hence not surprising they ve got a limited validity. I checked the difference numerically of 1) and 2). As long as $|\Psi_p-E_v| \approx k_BT$ it is still fairly accurate but beyond $3k_BT$ the error becomes significant. $\endgroup$ Apr 10, 2017 at 8:09

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