The Flat Plate Problem for a Semiconductor

In this paper we calculate the electrostatic potential in a currentfree semiconductor surrounding a semi-infinite flat plate carrying a small potential. Our objective is to determine the nature of the singularity in the field at the plate edge. Such information should be useful in the application of finite difference methods to similar, but more complicated, problems. We find that the field near the edge has the same weakly singular behavior as in the classical potential problem, being inversely proportional to the square root of the distance from the edge. It turns out, more or less fortuitously, that the present boundary value problem has a closed form solution, in terms of exponentials and error functions, so that it has some intrinsic mathematical interest. The solution was originally obtained by a very tortuous path. The problem was first attacked by the Wiener-Hopf technique. Then it was recognized that the inverse of the Fourier transform of the x1483 1484 T H E BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER