A time-dependent charge-collection efficiency for diffusion

The diffusion equation has some applications relevant to charge collection from ion tracks in silicon devices. Textbook solutions for the diffusion equation are available only for a few simple boundary geometries and special types of boundary conditions. A broader class of geometries was previously treated via a charge-collection efficiency function, but this applies only to total (integrated in time from zero to infinity) collected charge. The earlier work took advantage of the fact that Laplace's equation can be solved for a broad class of geometries. This paper extends the earlier work so that it applies to charge collected up to an arbitrary time. A time-dependent charge-collection efficiency function can be estimated for any geometry such that Laplace's equation has been solved. In particular, the analysis permits a comparison between diffusion calculations and a computer simulation of charge collection from an ion track. This comparison supports an earlier model in which charge collection, including a so-called "prompt" component, is driven by diffusion. The analysis applies to arbitrary track locations and directions. It also provides the option of treating a device geometry as two-dimensional in rectangular coordinates (if desired) while simultaneously treating the track as a line instead of a plane. Simulation codes having such flexibility regarding geometry are difficult to use, so the analysis makes the study of geometry effects accessible to a larger number of investigators.