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