# The probability of electron-hole pair recombination as a function of physical proximity

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.

• Not my area, but I would have guessed that both the electron and hole are delocalised so there is no well defined physical distance between the two. Recombination would be more a matter of matching the momenta. If the momenta of the two states don't match the recombination will require a lattice interaction to shed/gain momentum. Mar 27 '14 at 9:09
• @JohnRennie Agreed regarding the matching of momenta. However, I don't quite understand what you mean in saying that there isn't a well-defined distance between the electron and hole? I thought (from photovoltaics) that charges (which I'm treating as particles with distinct diffusion coefficients) diffused quite some distance apart? Mar 27 '14 at 9:15
• The expectation value of the distance between the electron and hole will have some value, but this is very different to claiming a well defined distance between any one electron and hole. I'd guess the Wikipedia diagram isn't meant to be taken literally. Mar 27 '14 at 9:24
• @JohnRennie When I say "precise distance" I mean plus-or-minus a few nanometers (a cluster of several atoms) rather than angstroms (at the bond length scale). The diffusion distances in question here, though, are on order microns to millimeters. Is my understanding of the spatial probability distribution of a delocalized charge incorrect? This, of course, seems more reasonable at the limit of a large number of defects, but what happens in a perfect crystalline lattice? Mar 27 '14 at 9:27
• I don't know how localised an electron in the conduction band is. I guess it would be related to the mean free path. I think this is of the order of tens of nanometres in most metals but I don't know what it is in semiconductors. Mar 27 '14 at 9:31

The electrons and holes are quasiparticles which can be hardly represented by a free-particle model due to their exchange-correlation coupling. For instance, at the extremely low densities, when low-energy electron and hole arrive at very short distance to each other, they do not necessary recombine, they probably form exciton, a stable bound state of the electron-hole pair. After some time, exciton can recombine. At high densities, the exciton lifetime is very small due to Coulomb interactions with other particles and it recombines faster. This time can be computed or be estimated from experimental absorption spectra from broadening of the excitonic lines. At highly dense electron-hole plasma, the recombination probability is computed using simple Fermi's Golden Rule. Here you have to know recombination mechanism in order to compute a matrix element: luminescence, phonon-assisted, through traps. Each of these processes has own probability. In all these cases, dependence on the density is determined via the Fermi-Dirac distribution of electrons and holes in k-space that enters Fermi's Golden Rule.

• I don't quite understand why we can't use a free-particle model at the low density limit? Yes, the electron and hole will recombine to form an exciton, and this will lead to an annihilation event, but now we just concern ourselves with the exciton formation probability as a function of electron-hole distance, right? We're just adding another (reversible) step to the annihilation process by considering the exciton pair. Mar 27 '14 at 10:06
• Yes, I guess we can relate the probability of the exciton formation to the average distance between particles. But the exciton formation is not a recombination, since exciton do not necessary annihilate afterward, but, another possibility, it can dissociate returning free electron and hole. The probabilities of these two options do not depend on the inter-particle distance in the small density limit. Mar 27 '14 at 10:26
• In any case, the recombination depends on the density, but this dependence is rather determined by the overlap of the distribution functions in k-space for electrons and holes (that is related to the momentum conservation laws). Mar 27 '14 at 10:30