In this video, we will continue our discussion on avalanche breakdown and finish with the Zener breakdown. So, we discuss that the fundamental physical mechanism that leads to avalanche breakdown is impact ionization, and the breakdown condition, and the multiplicative factor that characterizes the number of impact ionization events, all depend on this physical constant called the impact ionization coefficient Alpha, which represents the number of impact ionization event per unit distance of propagation of electrons or holes. If you think about the physical mechanism, there are two competing processes. The electrons get accelerated by electric field or holes get accelerated by the electric field. This is a process of increasing kinetic energy. However, these electrons and holes collide with phonons, lattice vibrations, or impurities. These are the collision processes in which these carriers lose their energy. So, while the carriers are accelerated by the electric field, they also collide with phonons and impurities and lose energy. The average distance between these two successive collisions with phonons or impurities, is characterized by mean free path and when we call that Lambda. So, this distance characterizes the energy loss process for electrons and holes. On the other hand, the energy gain is due to the acceleration by the electric field. So, you have to propagate a certain distance on the electric field in order to gain some sufficient kinetic energy. So, that distance d that an electron must be accelerated over without phonon collision, without losing any energy in order to acquire the threshold energy E_th for impact ionization is represented by the ratio E_th to the qE. This is the force produced by the electric field. So, there is a competition between Lambda and d. So, Lambda represents the energy loss process, d represents the energy gain process. So, in order to have an impact ionization process occurring, you have to have an electron propagating distance d without colliding with the phonons or impurities. Now, so, you have to talk about the probability of an electron not colliding with phonons or impurities over a distance d and that simply is given by this exponential factor, exponential minus d over Lambda. Therefore, the number of electrons or the density of electrons not colliding with phonons over a distance d is simply given by the total electron density times that exponential factor. So, n star is the concentration of electrons that have acquired energy greater than the threshold energy for impact ionization. These are the guys that who can impact ionize. So, let's assume that the impact ionization occurred immediately after an electron acquires threshold energy, then the number of impact ionization event is simply given a number of impact ionization event within a small width interval dx is simply given by dx over d, the distance over which the electron must be accelerated in order to gain energy, E_th. So, the dn, the concentration of electrons produced by impact ionization process is given by this n star, which is the concentration of electrons that have acquired threshold energy, E_th, times the number of impact ionization event that can occur within distance dx, which is dx over d. So, this if you plug in the definitions for this equation here for n star and also the equation for d in the previous slide, then you get this equation here and because Alpha is defined as one over n times dn, dx. This is simply the definition of impact ionization coefficient. From this equation, you can derive an expression for Alpha. So, K and B are constants that depends on the mean free path, and the threshold energy, carrier concentration. All of those are material-specific parameters and the key dependence that we have identified here is the electric field. So, Alpha is proportional to primarily exponential negative one over electric field. This is the main result from a theory called Shockley's lucky electron theory. So, if you recall the way we describe is that the impact ionization event occurs by those lucky electrons that have avoided phonon collision or impurity collision long enough to acquire high enough kinetic energy to initiate impact ionization events. In that theory, the impact ionization coefficient is proportional to exponential negative one over the electric field strength. Now, the electric field is not something that we can directly measure, rather we apply voltage. So, if we know something about the voltage, then it will be a more straightforward parameter that we control. So, it's important to know something about the breakdown voltage and for this we consider a one-sided junction and we write down the maximum electric field for a one-sided step junction, which is this, and we write it out in terms of the applied voltage and the approximate equations are given by this V_R here is the breakdown voltage. If this quantity the E_max becomes equal to the threshold electric field, which is the minimum electric field required to initiate impact ionization. So, if you plot the breakdown field, which is E_1 here, as a function of the doping density, then the breakdown field depends on doping density because as you increase your doping density, impurity scattering becomes more and more prevalent and it tends to reduce Lambda, the mean free path. So, you tend to lose energy more efficiently, and therefore, it requires higher field. So, as you increase your carrier concentration, your breakdown field increases. At some point, avalanche breakdown gives way to Zener breakdown. Another breakdown mechanism that is commonly observed in semiconductor-based diode, is Zener breakdown and the Zener breakdown occurs due to electrons tunneling across the band gap from p side to n side. So, here is an exaggerated energy band diagram for a p-n junction under very largely reverse bias. So, the band bending here is very large and some of these electrons in the valence band of the p-type side has a high enough energy to tunnel across the band gap junction, and show up in the conduction band of the n-type side. When this tunneling probability becomes substantial, then you have a Zener breakdown. So, the tunneling probability can be calculated by using some standard techniques in quantum mechanics. So, to simplify problems typically, you model this slanted energy band diagram with a triangular energy barrier. So, for this, you can adopt again various techniques in quantum mechanics, for example, WKB approximation to calculate the tunneling probability and we're showing the result here without derivation. The important thing is that the tunneling probability is proportional to exponential negative one over electric field. B here is a constant that depends on the material parameters such as band gap and the effective mass. So, just to give you some ideas, let's consider the tunneling current, which is simply given by the total current is given by the electronic charge cross-sectional area and the number of valence band electrons. Now, this capital N is not conduction band electron that we are familiar with now, it is the valence band electrons. So, this is close to the atomic density, very large number and the V here is the thermal average velocity and therefore, it will be the thermal velocity. The capital Theta at the end is the tunneling probability we just calculated in the previous slide. So, for a typical diode, you have a cross-sectional area of 10 to the negative fifth square centimeters, atomic density is 10 to the 22nd cubic centimeters, average thermal velocity is 10 to the seventh centimeters per second. So, with these parameters in order to have a current of 10 milliamp, the tunneling probability needs to be 10 to the negative seventh. The field required to generate that much current is about a megavolt per centimeter. So, for a semiconductor with a one electron volt band gap, which is silicon band gap is pretty close to this one electron volt, and the depletion region width has to be about four nanometers. So, if this is the case, then your field approach is about a megavolt per centimeter and that requires a doping density of about 10 to the 18th per cubic centimeters. So, you can see that in order to have a substantial tunneling current and therefore Zener breakdown, the doping density has to be pretty high. So, at mild doping density at low doping level, avalanche breakdown generally takes place first and dominates. But, for diodes made of very heavily doped p-side or n-side, then you can expect Zener breakdown. Now, how do I tell precisely which breakdown mechanism is at work? You can look at the temperature dependence. So, with increasing temperature, you're tunneling current will increase. Why? Because if you recall the tunneling current equation involves a thermal velocity. So, the thermal velocity increases and therefore, your tunneling current increases. On the other hand, impact ionization rate will decrease with increasing temperature. Why? Because at high temperature, phonon collisions will become more and more efficient, and therefore, your Lambda mean free path decreases, and therefore the probability of electrons not colliding with phonons long enough to acquire threshold energy for impact ionization, that probability will decrease, and therefore, the breakdown current will decrease with increasing temperature. So, this temperature dependence is a deciding factor that you could use to distinguish the breakdown mechanism.
