Problem 1. Spontaneous emission & radiative recombination (20 pts) In today's lecture I related the emission in LED under charge injection is essentially spontaneous emission. The spontaneous emission rate can be calculated as Tsp(E) = Pem Nj(E) fn(62) [1 - fp(1)] where Pem is a rate constant, and the total spontaneous emission rate is the integration Rap = Jo sp (E) de Under low injection conditions and using Boltzmann statistics, show that Rep co np, where n and p are injected carrier densities. Hint: This exercise nicely relates the spontaneous emission rate to the band-to-band recombination rate introduced earlier in the class. Express n and p using quasi-Fermi levels. This question only asks to show Rep is proportional to np product so the prefactor can be lumped into constant. Low injection implies quasi-Feimi levels are within the bandgap and away from the band edges so Fermi- Dirac distribution can be approximated by Boltzmann distribution