In this video, we will discuss continuity equation. Continuity equation is an equation that tells us the rate of change in carrier concentrations. So let's consider a one dimensional semiconductor and then we consider a small slice of width dx located at x. Now, what are the things that can impact the carrier concentration within this small volume? Well, you could first consider an electron flowing in. And the rate of electrons flowing in is represented by this current going to the right. So current flowing out represents electron flux coming in. Now, there could be a current coming in and that current coming in represents the electron flux going out. So that flux, it tends to reduce the carrier concentration here. And then, of course, you could have a generation and recombination within this volume. So the generation could be a generation due to light illumination, light absorption, or there could be some other mechanism that could increase the carrier concentration. And then, there will be recombination. And we talked about multiple recombination mechanisms in the past several videos. There could be radiative recombination, there could be Shockley-Hall-Read process, surface recombination, and Auger recombination. And all of those things will give you a recombination rate. And if there are multiple recombination mechanism at work, total recombination rate will be the sum of all those recombination rates, of course. So, you can set up a rate equation in saying that rate of increase in electron concentrations within this volume here is the number of electrons flowing in, that will be this component one, current density here, minus number of electrons flowing out, that will be this guy here. And then, the rate at which electrons are generated, there will be a generation n, minus the rate at which the electrons are annihilated or recombined. That will be the recombination process four. So this is basically the continuity equation. So, the rate of change in the electron times total number of electrons within that slice is given by the rate of change in the electron concentration times the volume. A here is the cross-sectional area and dx is the width. So that represents the volume of this small slice. And then the electron flux is basically the current density here. Electron flux is the current density divided by the electronic charge. So this first term here represents the electron flowing out. And then the second term here represents the electrons flowing in. And this is current density. So the total current is, of course, times the cross-sectional area. And then of course, there is the generation rate and the recombination rate times the volume. So if you cancel out all the cross-sectional area and you come up with a very nice differential equation, and this here is the continuity equation for electron. And you can go through the same process and derive a similar expression for holes. The only difference is the sign here because hole carries the opposite sign of charge then the electron. Now, we know that there can be two different types of currents in semiconductor in general. And the first type is the drift current and then the second type is the diffusion current. So this is the equation for general drift and diffusion equation, current equation for holes and the same thing for electrons. Now plug that into your continuity equation in the slide before, then you get these two equations here. And this is the full form, full explicit form of continuity equation. The one at the top here, this guy, is the continuity equation for electron, and here is the continuity equation for hole. If you can solve this, then you have the general expression for carrier concentration as a function of position and also as a function of time in both equilibrium and non-equilibrium situation. So again, just quickly recap, there can be multiple recombination and generation mechanism and we talked about band-to-band recombination. This could be radiative or non-radiative. And we talked about radiative recombination extensively. We also talked about trap-assisted recombination, Shockley-Hall-Read process. And if this process occurs at the surface or an interface, then it will be a surface recombination process. And then of course, we talked about the Auger recombination process. Now all of these processes could be active simultaneously. If that's the case, the total recombination rate will be the sum of these individual recombination rate. Usually, the most efficient one, the fastest one, would dominate and determine the total recombination rate. Now for the generation, we talked about light emission. If your incoming light has an energy greater than the band gap, then the valence band electron can absorb the energy and get promoted into the conduction band, generating electron in hole pair. So that's the photo generation process. And this is something that we haven't talked about. It's called the impact ionization process. It is an inverse process of Auger recombination. And we will talk about this in much more detail later when we talk about junction breakdown. So, let's consider a simple example of continuity equation and how this can play out in a real problem. So let's consider a p-type silicon uniformly doped and the accepter density is here, 10 to the 17th. Now, you shine a red light and that is on a ball pen get light. And this light is then generate carriers, will generate carriers and the generation rate is given here, 10 to the 19th, per cubic centimeter, per second. And the minority carrier lifetime is given as 10 microseconds. So we don't know exactly what recombination mechanism that are going on, ban to ban, Shockley-Read-Hall process, or surface, whatever that might be. The whole combined effect of those recommendations are characterized, they can be characterized by this simple lifetime number 10 microsecond, okay? So, now we want to calculate the carrier concentration in this general non-equilibrium situation. How do we do that? By solving continuity equation. So you write down the full continuity equation here. Now, let's look at each term. This term here in the left hand side, represents the time derivative. Now, we're shining a light steadily, and we are here considering a steady state. And steady state by definition has no time dependence. It is time independent. So, this left hand side goes to zero. Now, there is no electric field. Nobody is exerting an electric field voltage here. So, anything that contains electric field will go to zero. So this go to zero. These two drift current terms go to zero. Now, the doping is uniform, and therefore equilibrium carrier concentrations are uniform. Also, the light illumination is done uniformly. So, the light it generating carriers uniformly everywhere. So, the carrier concentrations are position independent, they're uniform. So this term, the diffusion current term goes to zero as well. Well then, you're only left with these two. So, your continuity equation in this simple case says that generation rate should be equal to recombination rate. Now the recombination rate here can be expressed by the excess carrier concentration divided by the excess carrier lifetime. This we have shown multiple times for different mechanisms in the previous few videos. So, the equation, the continuity equation gets reduced to this equation here. Generation rate is given here, 10 to the 19th, per cubic centimeter, per second. Lifetime given as 10 micro-second. Therefore, we can calculate for these excess carrier concentration, which is simply given by the products of the generation rate and the minority of the lifetime, and that turns out to be 10 to the 14th, per cubic centimeters. Now, you can do the same for holes and you will get the same result, 10 to the 14th because the carriers are generated as pairs, electron and old pair. Now let's check the compare that with the equilibrium carrier concentration, p-type silicon with a doping density of 10 to the 17th per cubic centimeters. Your doping density is much greater than the intrinsic carrier concentrations, so you have an extrinsic semi-conductor. Moderate carrier concentration is equal to the doping density. So, equilibrium moderate carrier concentration, whole concentration is 10 to the 17th. Now the excess carrier concentration is 10 to the 14th. Three others are magnitudes smaller than the moderate carrier concentration. So moderate carrier concentration doesn't change. It changes only by point one percent. But the minority carrier concentration, equilibrium minority carrier concentration is given by ni squared, from the law of mass action if you recall. So, n naught is given by ni squared divided by p naught. And ni square, ni in silicon is of the order of 10 to the tenth. So, this quantity will give you something of the order of 10 to the third. On average, you have about a thousand electron per cubic centimeters at equilibrium. Now you're creating a tenth to the 14th, many many many orders of magnitude greater. So, you can see that a small change for majority carriers could mean really dramatic change in minority carriers, and a lot of time because of this reason, minority carrier dynamics determines the whole dynamics. Now let's do one more example here. Let's say that suddenly we have turned off the light. We have reached the steady state, and the steady state carrier concentration was 10 to the 14th, as shown in the previous slide. Excess carrier concentration was there is. Now suddenly, at t equal zero, you turn the light off. What happens then? Now, write down the continuity equation once again, okay? Now you can see that you still don't have any electric field, so this term is zero. This term is zero. You still have no carrier concentration gradient. The doping is still uniform and the light was turned off. So there is nothing that causes non-uniform carrier concentration, so this thing is still zero. The light was turned off. The thing that was generating carriers went away. So your generation rate is also zero. Now, however, your rate of change of carriers is no longer zero. You no longer have steady state. The carriers are decaying, and you can imagine the carrier concentration gradually decrease and eventually approach the equilibrium value. So, the continuity equation gets reduced to this simple equation. The rate of change in n is equal to minus the recombination rate. And once again, write down your recombination rate as the excess carrier concentration divided by the lifetime. And the n rate of change in n is simply equal to the rate of change in delta n because n here is equal to n naught plus delta n. N naught equilibrium value doesn't change over time by definition. So, the rate of change in n, is equal to rate of change in excess carrier concentration. This is very simple. First order differential equation will give you a simple exponential solution. What does this mean? This means that as you turn your light off your excess carrier, minority care concentration decays exponentially. And the time constant with which that excess carrier concentration decreases is given by the carrier lifetime.
