In 1954, Wolff proposed a theory as to how carriers attain enough energy for impact ionization. It's actually opposite to Shockley's lucky electron theory. In Wolff theory, impact ionizations are predicted to arise from the diffusion of carriers upward in energies. Carriers are assumed to undergo many collisions with phonons while gaining energy. Yet, the energy lost per collision is very small. The carrier distribution becomes nearly isotropic and is described as a drifted Maxwellian, with an effective temperature, Te. Impact ionization occurs from the carriers and high-energy tail of the distribution, whose energies exceed the threshold energies. As the applied electric field is increased, the distribution shifts towards higher energy, resulting in more carriers in the high energy tail and thus higher impact ionization rates. So, we can go ahead and derive the impact ionization rate, under the diffusion theory. Here, what we can assume as we can assume steady-state, so the energy gain from the field is balanced by the energy loss by collisions with phonons. So the rate equation we can write, is that dE, dt where E is energy is equal to q F, Vd minus Ep, V theta over lambda equals zero. So, U d is the drift velocity in the field direction. q F Vd is the power input by the field into the system. So, this is power from field into the system. Ep is the phonon energy. V theta is the average thermal velocity defined by the Maxwellian distribution, and lambda is the mean free path between collisions. So, we can go ahead and solve. So V theta equals q F Vd lambda over Ep. We can now write an expression for the rate of change of momentum, dp dt equals q F. So this is essentially the momentum gained from the field minus mVd, V theta over lambda. This is essentially the momentum loss from collisions. So in steady state, we know that dp, dt equals zero. In this case, Vd is going to equal qF lambda over mV theta. So we can substitute in the drift velocity, the drift velocity is nothing more than the square root of Ep over m. In a Maxwellian distribution, we know that three-halves, kB, Te equals one and a half mv theta squared. So here, on this side, this is the thermal energy and on the other side, what we have is we essentially have the mean thermal velocity. So if we substitute expressions for V theta and Vd, what we find is that K sub B, Te equals q F lambda squared over three E p. So the electron distribution is Maxwellian and it's characterized by an effective electron temperature T e. So, the probability that a carrier reaches the threshold is given by the Boltzmann factor. So, this probability of reaching E threshold is basically going to be proportional to exponential of minus E threshold over k sub B, T e. So, we can go ahead and substitute k sub d, Te into the Boltzmann factor. So, what we find is that p, E threshold equals exponential of minus three E threshold E p over q F lambda squared. So, the impact ionization rate will be the inverse of the mean free path. So, either alpha will equal p E threshold q F over E threshold and so that equals q F, E threshold exponential of minus three E threshold E p over q F lambda squared. So, you can see here that the dependence on the field is extremely different than the case of the lucky electron model, in the case of the diffusion model, until we have to think about what's going on. In the case of the lucky electron theory, the impact ionization rate is proportional to the exponential of one over F. So, you have electrons with no collisions that initiate the impact ionization. In the case of diffusion theory, alpha is proportional to one over exponential of minus one over F squared. All the electrons get their energy through collisions, so there's the thermalized distribution. So, there's a puzzle here, and I'll talk more about it in a second.