Theory of bulk electron-hole recombination in a medium with energetic disorder

The kinetics of bulk recombination between electrons and holes in semiconductors that have energetic disorder is studied theoretically. The effect of energetic disorder in the medium is taken into account by using a multiple trapping model in which charges repeat trapping into and detrapping from trap sites with different trapping energies. We assume that recombination between an electron and a hole can occur only when at least one of them is detrapped. The distribution of trap sites with different trapping energies E is assumed to be exponential: similar to exp(-E/E(0)). A general theory for the kinetics of bulk recombination between electrons and holes is formulated which is valid for an arbitrary ratio of the recombination rate constant k(r) and the trapping rate constant k(t). In the cases of k(r)=k(t) and k(r) << k(t) the kinetics of bulk recombination is solved analytically. The analytical result for the case of k(r)=k(t) agrees with the simulation results by Nelson [Phys. Rev. B 67, 155209 (2003)] for the same case. Our theory predicts that the number density of charges decays as t(-alpha) at long times, where alpha=k(B)T/E(0) and T is the temperature. This result explains recent experimental observations on bulk recombination between electrons and holes in organic solar cells.