The Potential Due to a Charged Metallic Strip on a Semiconductor Surface

We solve numerically the problem of finding the potential and electric field around a negatively charged metallic contact on the surface of an n-type semiconductor. The semiconductor, which has permittivity ei , fills the half-space y < 0. The contact is an infinitely long strip of width 2a, defined by y = 0, 0 ^ x ^ 2a, -- » < z < Q . The region y > 0 is O vacuum with permittivity c0 · In suitable dimensionless coordinates the potential (f> satisfies Laplace's equation in y > 0 and the equation V2<£ = e* -- 1 in y < 0. On the boundary y = 0, = 0 < 0, 0 ^ x ^ 2a, and the usual electromagnetic boundary conditions at the remainder of the interface. Finite difference schemes are used to solve the resulting boundary value problem. In most practical cases |0O| » 1 and 77 = e0/ei 0| » 1. In case the |0O| 1 we show that our numerical solution agrees well with the exact analytical solution of a linearized version of the problem. For |<£0| 1, we give plots of the equipotential curves, curves of equal charge density, and curves of constant electric field amplitude. These results also yield expressions for the capacitance of both a strip and a circular electrode. The modifications of these results when rj > 0 are also given in some detail. Finally, we discuss the numerical calculations at some length.