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)


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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