In this video, we will continue our discussion on non-radiative transitions. So now we're ready to a quantitative discussion, now let's consider an acceptor-like center in an n-type semiconductor. Now in an n-type semiconductor, your Fermi level is located at the upper half of the band. And in general, you can expect the r1 and r2 are much greater than r3 and r4, because the electron concentration are much greater than the whole concentration. And r1 and r2, if you remember, are the electron capture rate and electron emission rate. And r3 and r4 are hole capture and hole emission rates, respectively. Now, consider a low level excitation or a low level injection condition. What that means is that whatever you're doing to drive your semiconductor away from equilibrium. Whether you're shining light or injecting current, the resulting changes in carrier concentration, which we call excess carrier concentration, remains small compared to the equilibrium majority carrier concentration, so what does that mean? That means your majority carrier concentration, electron concentration in this case, remain largely the same. While the minority carrier concentration, hole concentration, changes by many orders of magnitude. So the electron capture rate therefore remains largely the same, because electron concentration doesn't really change. But the hole capture rate, r3, will increase by a large amount, by a large factor, because hole concentration changes by a lot. Now, the net electron capture rate and net hole capture rate should remain the same, and we require r1- r2 = r3- r4. Now we define the net recombination rate, U, as this difference r1- r2 = r3- r4. And we use those expressions for these r1, r2, r3, r4 in the previous video. And plug that all in, you can derive an expression for U, the net recombination rate, as shown here. Now if you divide numerator and denominator by this quantity here, ntvth sigma n sigma p, then you can write the same expression in terms of tau lifetimes. And these lifetimes are defined as this, so this is a lifetime that are used to characterize the effectiveness of carrier capture. So this tau sub p0, this shows how effective this defect or localized state is in capturing holes. Likewise, tau n0 characterizes the effectiveness of electron capture by this localized state. Now, the numerator is pn- ni squared, so the net recombination rate is positive when Np product, or p times n, is greater than ni squared, which is the np product at thermal equilibrium, according to the law of mass action. So if you're doing something to the semiconductor to drive the carrier concentration higher, so np product becomes higher than its former equilibrium value, then you have a net recombination. Meaning that the semiconductor is responding by increasing the recombination rate, and therefore trying to reduce the carrier concentration back to the equilibrium level. On the other hand, if somehow the semiconductor is driven away from the equilibrium in such a way that the carrier concentration's np product is lower than the equilibrium value, ni squared. Then you have a net generation, you have a negative net recombination, which means you have a net generation. So again, the semiconductor is responding by increasing the carrier concentration, and therefore trying to restore equilibrium. Now, in the simple case that the electron and hole capture cross section sigmas are the same, then the tau sub n0 and tau sub p0 defined in the previous slide become the same quantity, and we call that tau 0. And it's a simpler expression so that we can easily see their behavior. And you can see that this cosine hyperbolic function reaches minimum when Et is equal to Ei, intrinsic Fermi level. And therefore, the whole quantity net recombination rate reaches maximum when Et, the defect state energy level or the localized state energy level, is equal to E sub I, which is at the middle of the band gap. So those defect levels that are located at the mid gap Is the most effective recombination center. On the other hand, those defect states that are away from the mid gap, they tend to become more of carrier traps rather than recombination center. Now, suppose that you then shined a light pulse onto your semiconductor, and you created some extra electrons in the holes due to light absorption. And we assume low level injection once again, and what that means again is the excess carrier concentration we call delta n is small compared to the equilibrium majority concentration n0. Now, you set up a simple rate equation, the rate of change in the carrier concentration, majority carrier concentration in this case, is equal to negative U. The net recombination rate, negative sign because U here is the recombination rate, it tends to reduce the carrier concentration. Now, n here, the total carrier concentration, is n0 plus delta n by definition, and so n here is n0 plus delta n. And n0 here is equilibrium quantity, by definition, it is not dependent on time. So the only thing that depends on time is the second term, excess carrier concentration. And hence, the time derivative is simply the time derivative of the excess carrier concentration. Now once again, assume that the tetra cross section for both electron and hole are the same, to use that simplified form for mu. And then, again, use the low level injection assumption, then you can write this equation here, the original form, into this. And what this does is that, if you look at all the terms in this second expression, none of them depend on the time. The only thing that depends on time is delta n, so the U is something, some constant times delta n. So you can simply solve this simple first-order differential equation, and get a solution, which is a simple exponential function, so that is this solution here. So, this delta n0 is whatever the excess carrier concentration at t = 0. And that would be determined by your light illumination condition. And then when the light is turned off, then the carrier concentration will then decrease exponentially. And then the time constant with which the carrier decays is this tau sub n, and that is given by this, from the equation for U that was in the previous slide. This here, tau sub n, Is the excess carrier lifetime, or the minority carrier lifetime. We have defined a similar quantity for the radiative transition before, and this is the same thing. This is the perfect analogue of that minority carrier lifetime, or the excess minority carrier lifetime. And this one here is defined for a different physical mechanism for the recombination. Previously, the recombination mechanism was the light emission. Here, the recombination mechanism is through a defect located in the mid gap. Now, when Et = Ei, that's the case when U becomes maximum, we talked about it. Then, this complex equation simplifies into this, so this whole thing, you can ignore this term and cancel these two things out. And the tau sub n, the minority carrier lifetime, simply becomes equal to tau 0, which is defined thus. Now, this lifetime, tau 0, if you recall, is a parameter that characterized the effectiveness of carrier capture by the defect. This tau sub n is the minority carrier lifetime, it is the average time your excess minority carrier hangs around, before they decay through these recombination process. So when Et = Ei, then those two are equal quantities. And so in this case, your minority carrier lifetime actually is limited, is the same as the minority carrier lifetime that characterizes the carrier capture process. And because your majority carrier is always plentiful, the whole process then becomes limited by the availability of the minority carriers. And this is why we call this quantity often minority carrier lifetime. Now, you can imagine that the same process can occur through defects located on the surface, or an interface. This actually is pretty common, and is a pretty major technological issue in many semiconductor devices. Surface by definition is defect, that's where your crystalline periodicity stops. If you have an interface between two different materials, that interface is also by definition a defect. So there will be some energy levels created by those surface defects or interface defects that are located in the middle of the band gap, as shown here in this figure on the right. And these mid gap states or the defect states can mediate the carrier recombination, just like the Shackley-Hall-Reed process we just talked about. The equation is exactly the same because the physical mechanism is the same. The only difference is that the concentration or the density of these defect states is now a two dimension density, because they are located on the surface rather than spread out in the three dimensional bulk. So this Nst is now a two dimensional density, has a unit of inverse square centimeter, and everything else is the same. And so, instead of the lifetime, this quantity here, if you compare this expression with the analogue of the three dimensional bulk case, this quantity gave you the lifetime with a unit of inverse time. And that was the quantity that characterized the effectiveness of carrier capture in the 3D defect. In this case, if you do the dimensional analysis, this whole thing have the dimensional velocity, because this guy is a two dimensional density. And so you now have a surface recombination velocity, to characterize the effectiveness of carrier capture by surface defects or interface defects. Please note that this is not a physical velocity, nothing is moving at this velocity. This is a parameter that characterizes the effectiveness of carrier capture by the surface defects. Now lastly, I want to talk about Auger recombination. And the Auger recombination process schema [INAUDIBLE] is shown here. So this is a process in which initial condition, there are two electrons in the conduction band, one hole in the valence band. Electron here in the conduction band jumps down to recombine with the hole in the valence band. In doing so, it gives an energy to the nearby electron, and this electron goes up, becomes a hot electron, high energy electron. Eventually, they thermalize back to the bottom of the conduction vent, just losing its energy to heat. So this process is called the Auger recombination. And because this is a three particle process, you need two electrons and one hole, so the net recombination rate is a triple product of concentration, okay, so this is npm. This is the process that is depicted here, and you can imagine that a similar process is possible with two holes and one electron, and that rate is represented by the second term in this equation. So this process is Auger recombination process, and the Auger recombination process becomes very efficient when your carrier concentration is very high. So Auger recombination is a major recombination mechanism that in a heavily doped semiconductor commonly used in, for example, semiconductor lasers
