I'd like to discuss a different modality for photodiodes. Under high reverse bias in a photodiode, photogenerated currents can undergo a multiplication process via impact ionization. This mode of operation is called an avalanche photodiode. Impact ionization is a non-radiative process, where an energetic electron or hole, collides with an electron in the valence band, creating an electron-hole pair. It's really the inverse process of Auger recombination. Here you can see Auger recombination, where you have an electron and a hole that recombine and the excess energy is used to knock either electron higher in the band or a hole deeper in the valence band. In the case of impact ionization, you start with a very energetic electron, and then it collides with an electron in the valence band creating an electron hole pair. It's really the inverse process of Auger recombination. So, one of the important things to think about with APD is to think about the transition rate for impact ionization. It can be determined by a quantum mechanical transition rate calculation, similar to that, that you have for laser transitions. The transition rate is really subject to k-selection rules, i.e., momentum conservation, which requires that the sum of the k vectors be unchanged before and after the transition process. So, for parabolic energy bands, we can write that E threshold is equal to E_g, two Gamma plus one over Gamma plus one, where Gamma is the ratio of the effective mass for an electron to the effective mass of a hole. So, if the effective masses of the electron and the hole are the same, then E threshold equals three-halves E_g. So, what this is telling you is that, in order to get impact ionization in that case, you have to have an electron with one and half times the band gap energy. We can go ahead and write down the transition rate for impact ionization. Essentially, it's defined as W_ii. So, here we've got the charge on the electron, the dielectric constant, the effective mass of an electron, h bar cubed, so Planck's constant, and then E, the energy, E, threshold, the minimum energy that an electron needs to undergo impact ionization and the band gap energy. So, I_c and I_v are overlap integrals for the conduction and the valence band. You can get these by integrating the wave functions of the conduction band electrons and the valence band hole. But, in order to do this, you really need full knowledge of the band structure to evaluate I_c and I_v. So, to get around this, we often determine this prefactor experimentally. We can write empirically that W_ii is nothing more than P, which is this empirical parameter that's adjusted to fit experimental data, and then Tau E threshold, the scattering rate at the threshold energy times E minus E threshold over E_g squared. So, W_ii determines the rate at which impact ionization will occur as a function of the initiating carrier's energy one-to-one reaches threshold. So, it's also important that in order to know the total impact ionization rate, one has to know the rate at which the carriers attain threshold energy, not just the transition rate of the impact ionization as a function of energy. So, effectively, what we are going to go after next is the survival rate of the high energy carriers.