A heuristic approach to steady-state semiconductor filament theory, with applications

Abstract

We present a heuristic approach for solving the current-voltage characteristic of a semiconductor filament in the steady state excited by a time-independent and space-independent carrier density injected at the front surface. The treatment assumes a model having infinite surface recombination velocity (zero excess carrier density) on the lateral surfaces, a model used by previous workers for both transient and steady-state conditions. For the steady state, we demonstrate that the exact solution for total current crossing the injecting face diverges. Use of partial sums derived from truncating the Fourier or Bessel-function series of the periodic extension in space of the constant carrier density on the injecting face yields closed-form analytic solutions for the current-voltage characteristic. From these solutions, a definition of the effective filament lifetime emerges. Its dependence on lateral dimensions nearly agrees with the dependence of the transient filament lifetime defined by Shockley. The approach advanced here enables treatment of several semiconductor devices containing grain boundaries, including one in which the model employing infinite surface recombination velocity admits use of the superposition principle to build a total solution by summing more elementary solutions. A partial justification for the heuristic approach taken here derives from regarding the partial-sum approximation as a means of moving the model in the direction toward a physically realizable system. Brief discussion appears of electrostatic and thermal analogs of the semiconductor boundary-value problem treated.