Semiconductors pp 159-183 | Cite as

# Derivation of the High Field Semiconductor Equations

## Abstract

Electron and hole densities evolve in *x* — *z* phase space according to Boltzmann equations. When the mean free path of the particles is short *and* the electric force on the particles is weak, a well-known expansion (the Hilbert expansion) can be used to solve the Boltzmann equation. This asymptotic solution shows that the spatial density of electrons and holes evolves according to diffusion-drift equations. In fact, the Hilbert expansion leads directly to the Basic Semiconductor (van Roosbroeck) Equations.

As devices become smaller, electric fields become stronger, which renders the Basic Semiconductor Equations increasingly inaccurate. To remedy this problem, we use singular perturbation techniques to obtain a new asymptotic expansion for the Boltzmann equation. Like the Hilbert expansion, the new expansion requires the mean free path to be short compared to all macroscopic length scales. However, it does *not* require the electric forces to be weak. The new expansion shows that spatial densities obey diffusion-drift equations as before, but the diffusivity *D* and mobility *μ* turn out to be nonlinear functions of the electric field. In particular, our analysis determines the field-dependent mobilities *µ*(*E*) and diffusivities *D*(*E*) directly from the scattering operator. By carrying out this asymptotic expansion to higher order, we obtain the high frequency corrections to the drift velocity and diffusivity, and also the corrections due to gradients in the electric field. Remarkably, we find that Einstein’s relation is still satisfied, even with these corrections.

The new diffusion-drift equations, together with Poisson’s equation for the electric field, form the *high-field semiconductor equations*, which can be expected to be accurate regardless of the strength of the electric fields within the semiconductor. In addition, our analysis determines the entire momentum distribution of the particles, so we derive a very accurate *first moment model* for semiconductors by substituting the asymptotically-correct distribution back into the Boltzmann equation and taking moments; this model is roughly analogous to a hydrodynamic model without an energy equation. Finally, we present the extension of the high field diffusion-drift equations to three dimensions.

### Keywords

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