# Drift Diffusion currents in semi-conductors

Drift-Diffusion currents of holes in semi-conductors are usually written as : $$\mathbf J_p = -D\nabla p- \mu p \nabla \phi$$

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 :

$$\mathbf J_p = p\mu\nabla\Psi_p$$

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.

• A link or citation for the paper might help. Apr 8, 2017 at 17:20

• 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. Apr 10, 2017 at 8:09