Consider a solar cell that is based on the inversion and accumulation of surfaces

Consider a solar cell that is based on the inversion and accumulation of surfaces. Assume that the front surface is inverted and the rear surface is accumulated. The material is n-type silicon. In calculation of the recombination, only consider surface SRH recombination for the surfaces and bulk SRH, auger and radiative recombination for bulk silicon. Under the consideration of flat quasi-fermi level and high level injection, derive a mathematical relationship between minority carrier lifetime and excess carrier concentration for the device