A Nonlinear Eigenvalue Problem Modelling the Avalanche Effect in Semiconductor Diodes

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Abstract
This paper is concerned with the analysis of the solution set of the two-point boundary value problem modelling the avalanche effect in semiconductor diodes for negative applied voltage. This effect is represented by a large increase of the absolute value of the current starting at a certain reverse basis. We interpret the avalanche model as a-nonlinear eigenvalue problem (with the current as eigenparameter) 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 for every negative voltage. Therefore, the solution branch does not “break down” at a certain threshold voltage (as expected on physical grounds). We discuss the current-voltage characteristic and prove that the absolute value of the current increases at most (and at least) exponentially in the avalanche case as the voltage decreases to minus infinity.