## An asymptotic analysis of one-dimensional models of semiconductur devices.(English)Zbl 0639.34016

One dimensional, macroscopic models for several types of semiconductor devices are analyzed by means of singular perturbation techniques. In the steady state drift diffusion approximation, the distribution of charges and the electric field in a semiconductor is governed by a two point boundary value problem for the ODE system $\lambda^ 2\frac{d^ 2}{dx^ 2}\psi =n-p-C(x);\quad \frac{d^ 2}{dx^ 2}J_ n=R(n,p),\quad \frac{d^ 2}{dx^ 2}J_ p=-R(n,p)$
$J_ n=\mu_ n(\frac{d}{dx}n- n\frac{d}{dx}\psi),\quad J_ p=-\mu_ p(\frac{d}{dx}p+p\frac{d}{dx}\psi).$ $$\psi$$ is the electric potential, n and p are the densities of negative and positive carriers and $$J_ n$$ and $$J_ p$$ are the current densities. $$\lambda$$ 2 is the scaled, minimal Debye length and acts as the perturbation parameter. Different types of devices are modelled by different sets of boundary conditions and different forms of the doping concentration C(x). In this paper the cases of a p-n junction diode, a MOS diode and a Schottky diode are treated. Asymptotic expansions of the solutions in powers of $$\lambda$$ are derived. Internal layers occur in the p-n diode where, due to abrupt chances in the doping concentration, C(x) is discontinuous. Boundary layers occur in the MOS and the Schottky diode at contacts. In all three cases proofs for the existence of isolated outer solutions of the corresponding reduced problems are given under certain restrictions on the size of the applied bias. Using this fact, the asymptotic validity of the expansions, and with it the existence of isolated solutions of the full problem, is shown.
Reviewer: C.Ringhofer

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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