Steady-State Junction-Current Distributions in Thin Resistive Films on Semiconductor Junctions (Solutions of ▿2v = ±ev)

A collection of exact and approximate solutions to the nonlinear equation ▿2v = ±ev is presented. This equation appears in an extremely wide variety of physical contexts, which are summarized. In particular, the equation is shown to describe the nonuniform junction voltage v resulting from current-spreading resistance when the p and/or n layer of a p-n-junction device is thin and only partly covered by a contact. Outlines are given for the application of the solutions to the design of electroluminescent diodes, lasers, transistors, and metal-insulator-metal or Schottky-barrier structures.