## Charge transport in semiconductors with degeneracy effects.(English)Zbl 0733.76062

This paper is concerned with the following Boltzmann equation, arising in the kinetic theory of semiconductors: $\partial_ tf+\nu \cdot \nabla_ xf-(q/m)E\cdot \nabla_{\nu}f=Q(f).$ Here x and $$\nu$$ are the space and velocity variables, both in $${\mathbb{R}}^ 3$$, q the charge and m the effective mass of an electron, $$E=E(t,x)$$ the electric field. Q (the collision operator) is a nonlinear integral operator, vanishing at $$f=[1+\exp ((m\nu^ 2/2-\mu)/kT)]^{-1}$$, where k is the Boltzmann constant, T a temperature, $$\mu =\mu (t,x)$$ is called the Fermi energy.
After a scaling this equation assumes the following dimensionless form: $$\alpha^ 2\partial_ tf+\alpha (\nu \cdot \nabla_ xf-E\cdot \nabla_{\nu}f)=Q(f),$$ where the same symbols for scaled and unscaled quantities are used; $$\alpha$$ (the scaled mean free path) is a parameter. It is shown by a contraction argument that the Cauchy problem for this equation has an unique weak solution for suitable integrable Cauchy data and smooth electric fields. Moreover, for $$\alpha\to 0$$, the solutions of the scaled equations converge in $$L^ p_{loc}$$ to a Fermi-Dirac function $$F(\mu,\nu)=[1+\exp (\nu^ 2/2-\mu)]^{-1},$$ where $$\mu$$ is a weak solution of a conservation law, called fluid equation. The derivation of this equation is one of the aims of the paper. A study of (a rescaled version of) the fluid equation is also given.
Reviewer: A.Corli (Ferrara)

### MSC:

 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C70 Transport processes in time-dependent statistical mechanics 35D05 Existence of generalized solutions of PDE (MSC2000)
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### References:

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