A Nonlinear Eigenvalue Problem Modelling the Avalanche Effect in Semiconductor Diodes.

This paper is concerned with the analysis of the solution set of the two-point boundary value problems modelling the avalanche effect in semiconductor diodes for negative applied voltage. We interpret the avalanche-model as a nonlinear eigenvalue problem with the current as eigen parameter and show using a priori estimates and a well known theorem on the structure of solution sets of nonlinear eigenvalue problems for compact operators that there exists an unbounded continuum of solutions which contains a solution corresponding to every negative voltage. This effect also called avalanche generation is characterized by a sudden increase of the current flowing through the device starting at a certain negative voltage. Physically, the diode breaks down shortly after the onset of avalanche generation. Therefore, it was conjectured that there is a threshold voltage beyond which no solutions of the avalanche model exists. We show that this conjecture is false more precisely a continuous branch of solution along which every negative voltage and every negative bias is assumed at least once exists. Mathematically, the avalanche-effect only becomes apparent through an exponential increase of the absolute value of the current starting at a certain negative voltage.