When we shine line of an appropriate wavelength at a metal, e.g. gold, such that there is sufficient energy to promote an electron from the valence band to the conduction band, we'll generate with some probability a set of electron-hole pairs. We can then, to my understanding, rough approximate the electrons and holes as Brownian particles with distinct three-dimensional diffusion coefficients for movement in the metal (conditional on lattice defects / etc. being randomly or uniformly distributed).
Now, when a diffusing electron comes with some critical Euclidean distance $r$ of a hole, the "particles" should recombine with some probability and annihilate one-another.
My question is... how can we characterize this process in principle in some common metals or semiconductors like silicon? Is there a known distance-dependent relationship / probability distribution for recombination? Is this relationship conserved over different metals / metalloids? To be clear, I'm not talking about some overall probability distribution for recombination as a function of time - I'm strictly asking about how the physical distance between the electron and hole effects the annihilation probability.
Here's a cartoon drawing from Wikipedia of an electron and a hole diffusing to the cathode and anode, respectively, of a silicon-based photovoltaic chip: http://en.wikipedia.org/wiki/File:Silicon_Solar_cell_structure_and_mechanism.svg. Of course, here there's a built-in potential to induce the appropriate directed Brownian motion of the electron and hole, but this should of no consequence for my above question.