There are several theories as to how electrons attain the threshold energy needed to initiate impact ionization. One such theory is the Lucky Electron Theory. In 1961, Shockley suggested that impact ionization is mainly due to "lucky" electrons which suffer no collisions while they're accelerated to reach the threshold energy. In this model, carriers that start at or near the band edge drift under the applied electric field to energies at which they can initiate impact ionization. So during the acceleration process, most carriers encounter a lot of collisions with phonons which are lattice vibrations and they lose their energy, but the lucky carriers do not. Therefore, the lucky carriers eventually attain the threshold energy needed to initiate impact ionization. In this theory, we'll derive and see that the impact ionization rate goes exponentially with the reciprocal of the applied field. So, that is denoted as F. So, let's go ahead and do a derivation of this impact ionization rate in the case of the Lucky electron theory. We're going to start by writing down the rate of scattering for carrier with energy E, we're going to denote this as one over T of E. So, the probability of a carrier being scattering in a given time is going to be delta or dt over TE, and the probability of not being scattered in a given time interval zero to T is going to be denoted as P of t. So, this is the probability of not scattering in time interval zero to t. We can write that P of t plus dt equals p of t, and then here, one minus dt over T of E. So, that's the probability of not being scattered in the interval between t and t plus dt. So, now let's go ahead and rearrange. P of t plus dt minus P of t over dt. So essentially, we're just looking at the probability of not scattering in the interval of t and t plus dt, and the probability of not scattering and integral of zero to t, this is equal to nothing more than minus p of t over TE. So, what we can write now is we can write that dp of t over dt equals minus P of t over TE. So we can now integrate, and so if we integrate, what we find that P of t is equal to the exponential, and then here we're going to have the integral minus integral from zero to t of dt prime over TE. So, this is the probability that the carrier will not experience any scattering in the zero to t interval. So, the probability of no scattering in zero to t. So we can also define t zero, t zero is essentially the average time between scattering events. So, it's essentially equal to the mean free time. We can write Newton's second law as h bar k equals h bar dk dt equals q times F, where F here is the applied E field. So, we can use this to find the final case state. We find that h bar kf minus ki equals qFt zero, where t zero is the time it takes for the carrier to drift to the impact ionization threshold. So, if the carrier starts to drift from the band edge, then ki equal zero. So, if drift starts at band edge, ki equals zero. So therefore, then we can write that kf equals qFt zero over h bar. We're going to assume parabolic energy bands. So if we do that, we have E threshold equals h bar squared, kF squared over 2m. We can go ahead and we can substitute for kF from the expression above. So if we do that, what we get is that E threshold equals q squared F squared over 2m t zero squared, where t zero equals 2mE threshold over qF squared. So essentially, this is the average time it takes for the carrier to acquire enough energy to initiate impact ionization.