A diffusion problem in semiconductor technology

Summary

This paper deals with a mathematical model of a SEM-EBIC experiment devised to measure the diffusion length of semiconductor materials. In the model the semiconductor material occupies a half-space, of which the plane bounding surface is partly covered by a semi-infinite current-collecting junction, the Schottky diode. A scanning electron microscope (SEM) is used to inject minority carriers into the material. It is assumed that injection occurs at a single point only. The injected minority carriers diffuse through the material and recombine in the bulk at a rate proportional to their local concentration. Recombination also occurs at the free surface of the material, not covered by the junction, where its rate is determined by the surface recombination velocity v. The mathematical model gives rise to a mixed-boundary-value problem for the diffusion equation, which is solved by means of the Wiener-Hopf technique. An analytical expression is derived for the measurable electron-beam-induced current (EBIC) caused by the minority carriers reaching the junction. The solution obtained is valid for all values of v, and special attention is given to the limiting cases v=∞ and v=0. Asymptotic expansions of the induced current, which are usable when the injection point is more than a few diffusion lengths away from the edge of the junction, are derived as well.