I'd like to spend some more time talking about the superlattice APD. If we think about most compound semiconductors, they have nearly equal hole and electron ionization rates, which as we know, they're super undesirable as far as noise is concerned, and also as for avoiding avalanche breakdown. So, for example, if we take gallium arsenide and indium phosphide, we find that the Alpha equals Beta over a large range of allowed fields. So, the bottom line here is that conventional APDs with gallium arsenide and indium phosphide will provide completely unsatisfactory noise and speed performance. One exception is silicon, where the electron ionization rate is significantly higher than that of the whole ionization rate by a factor of 20 at the same field strength. But unfortunately, it's got an indirect band gap. So, it's got smaller absorption than gallium arsenide or phosphide. The large band gap is really not suitable for infrared detection applications for which indium arsenide, mercury cadmium telluride, and indium antimonide are used. So, we'd really like a way to get the direct band gap materials with a large difference between Alpha and Beta, and it's hard to find materials like this. So, instead what we can do, is we can go to an artificial structure that's made so to exhibit highly asymmetric impact ionization rates, improved noise, and bandwidth behavior. So, the way to do this is with a superlattice or multi-quantum well where the intrinsic region consists of alternating high and low band gap materials. So, as the electron moves across the depletion region, they cross from one layer to the next, and when they enter the narrow band gap material from the wide band gap layer, the electrons get a boost in kinetic energy due to the conduction band discontinuity. So, even if the overall applied field is less than that's necessary to initiate impact ionization in the bulk, impact ionization can still occur in the vicinity of the band discontinuity. So, the band discontinuity in this case really supplies sufficient energy periodically to cause impact ionization that wouldn't occur otherwise. So, the total field exerted on the carriers and the superlattice, APD, is the sum of the overall bias field that's zero and a field due to the periodic potential of the superlattice structure. So, we can go ahead and write down the impact ionization rate for a superlattice structure. We're going to have Alpha equals P of t_0 over L_b. Okay, so, remember that P of t_0 is the probability of not having any scattering in the region from zero to t_0 and L_b is the mean distance for ballistic impact ionization. If we want, we can write this as P of t_0, and then qF of Z over E threshold. So, essentially, we can write that P of t_0 is equal to the exponential of minus zero to t_0 of dt prime over Tau of E. It's really more convenient to place this time integral with an energy integral by using the definition of the velocity. The velocity is equal to dl, dt, or if we want, we can write this as one over h bar dE, dk. So, if you think about dl, dl Is actually the distance traversed in the direction of the applied field. So dl can be written as dE over qF of Z. So, now, we can go ahead and write dt, dt is nothing more than dl over V, which is equal to dE, qF of Z h bar, dE, dk to the minus one. So, we can go ahead and substitute this into the expression for P of t_0. So, what we get is that P of t_0 equals exponential of minus h bar over qF of Z, and then the integral from zero to E threshold of dE, dk to the minus one, and then we have dE over Tau of E. We can write down now an expression for Alpha of Z. So, Alpha Z is going to be equal to qF_0 over E threshold exponential of minus h bar over qF of Z, and then multiplied by the integral from zero to E threshold of dE, dk to the minus one, and then dE over Tau E. So, we're only including the Z dependence of the field and the exponential and we're neglecting the prefactor. So, we can use the Fourier series to solve. So, for a square potential, we can write that V of Z is equal to V nought over two plus 2V nought over Pi cosine of Pi Z over L minus 2V nought over three Pi cosine of three Pi Z over L and then et cetera. So, V nought is the conduction band discontinuity, and then, essentially, 2L is the period of the superlattice. So, the total field, essentially, is going to be equal to F_0 minus dV of Z, dZ. So, we can essentially find the average impact ionization rate by integrating Alpha Z all over Z. So, what we find is we get that Alpha is equal to q over LE threshold, and then here, we have zero to L, F_0 exponential, and here, we've got minus h bar over qF of Z, zero to E threshold. Then here, we've got dE, dk to the minus one. Then here, we have dE over Tau of E. So, as a first approximation, we can retain the first two terms in the Fourier series. So, in that case, essentially, my impact ionization becomes nothing more than Alpha equals q over LE threshold, and then we're going to have zero to L, F_0 of exponential, and here, this becomes minus c over F_0 plus 2V nought over L sine of Pi Z over L, and then this whole thing. So, this is integrated over dZ. Okay, I need to define for you what c is. So, let me do that. So, c is nothing more than h bar over q, and then multiplied by the integral from zero to E threshold of dE, dk to the minus one dE over Tau of E. So, here that's just simplifying this expression.