 So, what we have discussed, we wanted to discuss drift and diffusion and generation and recombination. So, these are the 4 events taking place in solar cell. If you know these 4 events, we can put them together in what is called continuity equation and we can find out what is happening with the charge, where are they going, their concentration decreasing, not decreasing, flowing, not flowing, what is happening. So, once you know those 4 phenomena, I can actually put them together and actually derive the equation for the device. Is yesterday we are also running late today, but it is better to learn something rather than finish something, right. You are happy with it or you want to finish it? No. You are happy with the pace we are going, right, right, there is no point in finishing, fine. Generation we have seen that generation also occurs because of the thermal excitation, ok. Room temperature there is always some electron which can go from valence band to electron band, but the generation we are interested is because of the optical excitation and we know that in my solar spectrum, there are enough photons of high energy which can excite electron from conduction valence band to conduction band because the energy of those photons is from something like 3.5 electron volt up to some like 0.3 electron volt. So, as long as the energy of the photon is equal to or greater than the band gap, it can absorb, it can give energy to one of the electron which can go there. Normally you have in the room temperature the field conduction band, all energy levels are occupied by electron and empty valence band and no electrons here. Thus we have seen yesterday if photon energy is less than the band gap, the material is transparent and we lose something like 23 percent of our energy because of this 4th silicon, ok. So, what is the solution? If you want to reduce the transparent this losses, what is the solution? What I am saying is silicon is having band gap of 1.12 electron volt because of this all photons having energy lower than 1.12 electron are transmitted. Now, we want to reduce the transmission losses. So, what is the solution? Use other material of what type? Use other materials of what type? Reduce band gap. Reduce band gap. Use other material with the reduced band gap. So, that so what is the reduced band gap ideally 0.3 electron volt. If I use the material with 0.3 electron volt band gap, all photons are going to absorb. Wonderful. Everything is getting absorbed nothing is transmitted, but we do not do that. Why? We will come to that. This is one question to be answered, right. So, the lowest possible band gap material suitable for our solar spectrum, our sun. If you go to some other galaxy things will change. Our sun, this is a spectrum. For that sun our the lowest possible band gap material that we should use is 0.3 electron volt. And there are materials which are having 0.3 electron, but we do not do it. Why? We will come back to that. We will be getting less potential. You know how much you shift potential energy that is you are giving the potential. If you are only giving rise to this much energy then your voltage is less and the overall the effect on the efficiency will be less. We will come back to that. Sorry, that all other effects are there. Possilientary combination N i will be higher, your situation current will be higher, things like that. All other effects are there, but the most important effect is your voltage is going to be very less. So, generation happens there is a what is called absorption coefficient. I was telling talking about absorption coefficient is the probability of absorption. Higher absorption coefficient means good probability. Absorption coefficient is given in per unit centimeter. And you can see the absorption coefficient can vary up to 1000, 10000 or 1045, 100000. And we have discussed that high absorption coefficient is good, desired. High absorption coefficient is desired. Why? Then there is a higher probability absorption and you require thin material. So, one over absorption coefficient is called absorption length 1 over alpha. Let us do the quick calculation. Absorption length is the length of the material or thickness of the material that would require to absorb the whole spectrum. That is what we want, right? Absorption length is the length that would require to absorb the whole spectrum. If my absorption. So, in this incident, if my absorption coefficient is 1, what is my absorption length? If my absorption coefficient is 1 per centimeter, what is the absorption length? 1 centimeter. 1 centimeter. Have you ever seen a solar cell with 1 centimeter thickness? That would be a disaster, isn't it? That would be a disaster to use 1 centimeter thick silicon of high quality, 99.99999 percent. By the silicon is the single largest material and purest material produced on the earth in such a large quantity. No other metal is produced in that quantity. So, look at silicon, where is silicon? Absorption coefficient 1 for this wavelength, what does it mean? Actually, we need 1 centimeter thick silicon. Actually, we need it if you want to absorb everything, but then life is all about compromises, right? You have to make compromises. I cannot use 1 centimeter thick silicon. There is no money to actually do that. So, I do not care if some of the radiation do not get absorbed. Fine for me, okay? But this is only for that radiation. It is a function of wavelength, right? Absorption coefficient is a function of wavelength. Only for that particular radiation, where the wavelength is about 1.2 micrometer close to the band gap of silicon, we do not absorb efficiently. No problem. Here, the absorption coefficient is tens for 5. For what wavelength? 400 nanometer, 0.4 micrometer. What is the absorption length? Find out quickly. Transform minus 5 centimeter, which is how many microns? Come on. It is a one division. There cannot be so many answers. There has to be one answer. 10 to the power minus 5 centimeter is equal to one particular micron. It cannot be all micron. How much it is? 0.1 micron, 100 nanometer. Just 100 nanometer, okay? What does it mean? If I am putting a photon of 0.4 micrometer or 400 nanometer wavelength, how much material thickness are required for silicon? Only 100 nanometer. That is it. If I have 100 nanometer, it will absorb everything, not everything, everything of 0.4 micrometer wavelength. So, material thickness requirement for a different wavelength photon is different. First thing, okay? Material thickness requirement for high or short wavelength photon is very small. Material thickness requirement for a long wavelength photon is large and it can varying from 100 nanometer to 1 centimeter. Is that point clear to everybody? And that is why we want always absorption coefficient to be low, right? High. That is why always we want absorption coefficient to be high, so that our material requirement is small, right? Is that point clear to everybody? Our material absorption coefficient should be high, so that our material requirement is lower. All thin film solar cell materials have very good absorption coefficient. All thin film solar cell materials, what are the thin film solar cell materials? So, thin film materials for solar cell, amorphous silicon, CDT, PIGS, gallium arsenide, organics, dyes. Yeah, so that comes in some of it. So, all these materials have very good absorption coefficient and all of this material will require if absorption coefficient is tens for 5 means absorption length of 100 nanometer. Absorption coefficient is tens for 4 means absorption length of 1000 nanometer, which is 1 micrometer. So, most of this material within 1 micrometer everything will get absorbed because their absorption coefficient is always more than tens for 4 percent. So, 1 micrometer is enough for most of them, but not for silicon, not for silicon. And that is the only one bad thing about silicon that has to be thick amorphous silicon commercially available, commercially available, commercially available, not so much available commercially, not so much available. But there is a big hope from these two materials cadmium telluride, cadmium telluride, copper indium gallium selenide or it can be sulphide telluride, telluride. So, let me write. So, you have amorphous silicon, you have cadmium telluride, your copper indium gallium selenide, your gallium arsenide, organics actually there are many materials, some of the names I also don't know and the dye sensitize, because they are thin, simple. No, I mean really that is the case because they are thin. You know why because as I said their absorption coefficient is high, most of the time more than tens for 4 per centimeter and therefore, absorption length is 1 is about 1 micrometer, that is it. And they are so when the material requirement is so thin, you need to always deposit on some other substrate. So, all of them are deposited, not grown like silicon. Silicon wafer is grown in the ingot and the dyes, this is never done like this, you always deposit on some substrate. And because they are very thin, I mean normally for a thin film solution. It is a quaternary compound, this is a copper, there is indium, there is gallium, there is selenide sitting there, all of them. So, it is a compound semiconductor put together. So, as I said yesterday there are people use elemental semiconductor like silicon or people use compound semiconductor like 2 of them cadmium and tellurium together, gallium and arsenic together or people put 4 of them together or 3 of them together or sometimes 5 of them together to get the best optical and electrical property. So, it is a compound semiconductor with many many elemental semiconductor are there in various quantities. Is organic useful for solar material? Yes. How useful is different question? It is useful. So, these are all. So, when people started working on thin film technology, by the way the all thin film technology materials are very bad as compared to crystalline silicon, very bad. But yeah, because they have the potential to be low cost because they are very thin in the amount. Imagine crystalline silicon at 180 micron and thin film at 1 micron. So, the material requirement difference is huge right and therefore, all the thin film technology are potentially low cost technology. They are not necessarily low cost technology, they are potentially low cost technology. Yeah, so that is why there is this potential if you solve some problem, you become billionaire and you solve the stability. So, there are many ways to become billionaire if you are working in solar fossil. And in my lectures to my students, every day I give them one problem which they can solve, they become billionaire. Just one problem, there are many of such problems are there. So, you solve stability problem of morpho-silicon. I guarantee you that you become billionaire. Okay fine. So, absorption coefficient for look at the gallium arsenide, where it is? Gallium arsenide is this. This is the band gap of gallium arsenide and then after that it is always about in so far which is very thin, gallium phosphate very high at the band gap. So, there are many materials, thin film materials actually require very small quantities to absorb everything. One reason silicon is bad is because of this nature of its electronic band arrangement. So, in silicon arrangement is like this, what is called indirect and the other material like gallium arsenide is called direct. Basically, if you look at the energy versus, so all our energy band diagram we plot energy versus displacement. This is the same diagram, but it is plotted energy versus momentum. So, if you look at the energy versus momentum diagram it looks like this and if you want to excite electron from here to here, you need only change in energy. You do not require change in momentum. So, there is one interaction, one action to take this, but if you need to do this here, you need to change energy as well as momentum. And it is a series step. There are two actions to be taken place. This should take and that should take. Then only the transition will occur and you know in practice what happens. Whenever we rely on two people, what happens you know very well. So, process becomes slow, it requires more thickness and things like that. So, because of the indirect band gap nature of the silicon crystalline silicon, it has to be thicker. It cannot be thin. So, the current industry is using only 180 micron at the cost of losing some of the spectrum unabsorbed, which potentially could have been absorbed if you go for 1 centimeter thick silicon. So, absorption probability is low and therefore, it has to be thicker. High absorption probability therefore, it has to be thinner. The recombination as I said there are three mechanism bend to bend, trapezoid state, auger recombination. This is less common in solar cell. It can happen only at high concentration because three particles are involved. Three people are involved. This has to be there, this has to be there, that has to be there. So, more people are there, then only. So, more particles are there, then only this happens, which means more doping is there, then only this can happen. In silicon, this cannot happen because direct transition cannot happen, direct recombination also cannot happen or very low probability. So, this can happen. So, this is one of the most common way of coming down the recombination. And you can very simply way you can write the expression for the recombination. For example, if you are writing the expression for the bend to bend recombination, it is proportional to, it is proportional to how many electrons are there to recombine and how many holes are there, which can be taken place. So, it depends on both number of these and number of these. So, n and p and minus n i square is the opposite reaction that is taking place, that is the generation that is taking place. Augeric combination will depend on three people. So, it depends on n and p square, two of the holes are n square and p, two electrons and one hole, either case, because it depends on three particle, therefore, three terms, depends on two particle, therefore, you have two terms. And then you have the TREP assisted recombination, how many defect energy levels are there, that will determine. So, it will depend on this, it will depend on this, it will depend on how many TREP energy levels and so on. And you can write the rate of recombination as a function of the capture cross section, the number of defects, etc. Again, there is a very complicated theory, people have been doing it over a period of time to give actually the rate of recombination. We quickly give an idea of the junction. So, normally when we start, we start with the uniformly doped substrate, we start with some uniformly doped substrate and then we dope with the other atom. So, this becomes then this is acceptor atom which is p type, donor atom which is n type. So, this part becomes n, that part becomes p and that is how we get a p-enjunction. The band diagram we have seen yesterday, this is the p-type semiconductor, this is n-type semiconductor put together, you get a p-enjunction. What do you see here? Straight line, straight line, what do you see here? Not straight line. What is the meaning of not straight line? Electric field. Electric field. Electric field. Electric field. Non-flat, whenever bands are banded, which means electric field, which means there is an electric field exist at the junction, not everywhere else. Only at the junction, everywhere else are flat. What does it mean? There is zero electric field, what does it mean? What happens to the drift current here? Zero. No electric field, drift current zero. So, mainly diffusion current. What is happening here? Drift current zero, mainly diffusion current. What will happen here? There should be a drift current, there is an electric field. In reality, this is not the case. In reality, what happens? There is a very small electric field. So, we do not call it a neutral region, it is called quasi-neutral region. It is not really charged region, but it is called quasi-neutral region. There has to be some drift happening, very, very small. But still, because the majority carrier concentration is so high, you know, what is the equation for drift current? q n mu e. Because n is very high, a small e will also cause a lot of current to flow. Got it? The expression you have the charge carrier concentration and electric field. Even if electric field is negligible is zero, negligible is zero, but not zero. Very close to zero, but not zero. Then, also, there can be significant drift current because of the, because of the large carrier concentration. But the diffusion current can only flow when there is a concentration gradient. Minority carrier will not carry the diffusion current because of the drift, but they will carry the diffusion current because of the diffusion. Actually, when we go into the details of the derivation of the IV equation for the p-n junction, we actually look at all those parameter which we do not have time. In our December course, we will do everything step by step. We will go step by step from one step to other step till we get hold of solar cell equation. So, this we have seen. Electric field will be there. Then, there is a, because of that, there is voltage. You can find out the built-in voltage across the junction. If people have might have done in somewhere else. This is again important to understand. So, now, you can understand what is happening to this carriers. This guy can go there because of the drift that we have seen yesterday. This will go, come here because of the diffusion, concentration gradient. This holes will go there because of the diffusion. This hole will come there because of the drift. This is what we have seen yesterday. Again, corresponding currents. So, when it is in thermal equilibrium means there is no electric field, external applied, there is no light, there is no magnetic field, nothing. Then, all these current components are equal in opposite in direction. There has to be right. There is no net current flowing from the device. So, therefore, all the motion is anyway happening. So, net current has to be 0. The current particle flow direction, you can see from here and here and the corresponding current direction. You can do it yourself. Now, we have spent enough time. Which direction the hole will diffuse? What is the direction of the hole current? Which direction the current will, electron will diffuse and what is the corresponding direction? And the total current has to be 0 because there is no net connection from the device. Then, what you do when you do the forward bias? You know, remember what happens in a forward bias? Lot of current flows. Why lot of current flows? What you are doing in forward bias? You are making this like this. So, earlier barrier was still here. Now, you brought down the barrier. So, what happens now? Because you brought down the barrier, what will happen to this electron? Easily diffuse. Well, easily diffuse now. Straight road for them. Same thing, this hole will easily diffuse. As a result, what will happen? The electrons are going this way. Current is in this direction. Right? Holes are going that way. Current is also in their direction. So, from P to N. So, when you make the forward bias, lot of current flow from P to N that we know from our device diode characters. This current, is it changing with the forward bias? This current, it does not change because of the drift because this electron depends on the excitation, thermal excitation. So, diffusion current can change significantly because of the bias. The drift current is unaffected. It does not matter how much you bias. Even if you reverse bias does not matter. Got the point? I will let me show when you put. So, what happened because of the bias? Your barrier has come down and now there are more electrons which can cross the junction and cause the diffusion current. Now, distribution of this electron here from this energy, the distribution of the electrons is exponential. What does it mean? If you linearly, if you linearly decrease this barrier, the electrons which can now cross the junction are increasing exponentially. Therefore, the corresponding current is also increasing exponentially. So, linear decrease in the barrier will result in exponential increase in the current because the electrons are distributed exponentially. Again we do not have time, other we could show that why it is exponential. And this diffusion current is always from P to N for the holes, holes are going that way. So, current is from P to N electrons are coming that way. So, therefore, current is also P to N. So, the diffusion current is always from P to N and it increases exponentially because of the uncovering of those electrons. But the drift current for the electron which are the minority carrier. So, basically minority carrier current which is in what direction? Opposite direction to the diffusion current. This electron is going that way. So, current is this way N to P, this hole is coming from N to P. So, therefore, current is also N to P. So, the minority carrier current is always N to P, majority carrier current is always P to N. What is the other point to note? Minority carrier current is not changing with the bias, majority current increases exponentially with the bias, fine with it. And similarly, if you apply the reverse bias when similarly, when you apply the reverse bias you made the barrier very high, diffusion will stop and this current does not change, still because it is going on the same way. So, the minority carrier current does not depend on the bias that is what I am saying. And the majority current which actually when it is the reverse bias becomes zero and its forward bias increases exponentially. This is the condition when there is no bias. What is happening? Diffusion current is equal to the drift current, they are equal, they have to be equal because the net current is zero, right, got it. So, now, we can actually write on a qualitative discussion where we can write the expression for the diode. Let us try. Is that clear? At the no bias condition, net current is zero, but the drift current and diffusion current are flowing, it cannot be zero, but they are equal and opposite to each other. So, at v equal to zero, there are two current components, right. I drift and I diffusion, they are equal and opposite to each other. Drift current is because of what? Which carrier? Minority carrier, diffusion current is because of the majority carrier. When v is greater than zero, positive, what is happening? Diffusion is happening, right. Now, as I said diffusion will happen, but diffusion and let us say this is equal to some current I zero at the zero bias condition. This current some current I zero. When diffusion will happen, I said diffusion current increases exponentially with the voltage. Why? Because electrons are distributed at very exponential currents. So, at v equal to greater than zero, your diffusion current will be equal to some constant proportionality constant or let us say it is proportional to q v by k t, some voltage and some constant and the proportionally constant because at the equilibrium it is at I zero. So, proportionally constant can be I zero. So, your drift current is diffusion current is this, sorry, for voltage v equal to greater than zero. This is diffusion current and what I said drift current does not change with the bias, right. So, what is the net current? Net current in the forward bias case, net current when v equal to zero, greater than zero, this current minus the drift current because they are in opposite direction. So, the I zero e q v by k t minus I zero. That is my drift current, right. Isn't it? My net current at condition v equal to zero, let us say net current is, let me call it I total current. Total current is equal to this. This is diffusion current and this is drift current. So, therefore, my I total is nothing, but I zero e restore q v by k t minus one. Is this expression familiar to you? Is this expression familiar to you? Nothing, but a diode equation, right. So, this is nothing, but I equal to I zero e restore q v by k t minus one. This is a diode equation and v is equal to very negative reverse bias. This term will become zero and you will only minus I zero. So, this equation very well satisfy what happens in my diode and you have plot this equation, you have this kind of a this is current voltage. So, by putting just argument what will happen to the electron flow and hole flow and what will happen to the drift and diffusion as a function of bias, we can simply write that the device should behave as we have written here. At v equal to zero, both the currents are equal in opposite and we have given equal name some called I zero. In the forward bias, it is proportional to the voltage and some constant. So, q v by k t and therefore, forward bias diffusion current is this and this current is always flowing, but in the opposite direction always. Drift current is always flowing in the opposite direction. So, therefore, net current is this current minus drift current and therefore, the overall equation is this. So, this is what I said and this is what we have derived exactly the same thing. Now, let us put some light on it. So, once you put a light additional things happens, additional things happens in addition to this, these things are extra. So, carriers will be generated everywhere and this will go there and you will get the lot of electrons here, you will get the lot of holes here and one important thing to note that this electron because this band is flat. This electron has to go there before it gets separated, but why it should go there? There is no there is no electric field, why it should go there? It can go here also, but it cannot go I mean. So, this band is very long, I have to. So, when it is generated, it cannot go anywhere, it can go this direction or that direction. If it comes here, definitely it will go there, no doubt about it. So, what does it mean? Electron, if it is very far from the junction, may not actually go there and before going there, it may better come down here, recombination. If this process does not occur in seconds, microseconds, not seconds, microseconds time then it recombines. So, there is a two process happening. One is charge separation, other is recombination. It depends on which one is faster, that will win. If the carriers are generated very far from the junction, it may take more time to go there and if the time is longer than the recombination time, it will recombine. So, therefore, only carriers generated only is certain length, then only it is separated and contributes to the current. It is called the diffusion length. If the carriers are separated in which carriers, minority carriers. This is the n type that I am talking p. Only when the minority carriers are separated within the diffusion length, then only it separates. Otherwise, it die out, it recombines, it becomes dead and that photon is vested. So, not electrons generated everywhere will contribute. I think it is not clear, but if I draw the appropriate diagram, it will become extremely clear. P N junction, I am trying again the P N junction in my Fermi level. As I told you, my P N junction overall thickness is 180 micron. And my junction is, yesterday I told you only n type is only 300, 400 micron, 300 nanometers. This is how my junction in reality looks like. The p region is very big, n region is very small. Now, if electrons get generated here, it has to come all the way till here before it separates. Now, there is a good chance that before coming here, it can recombine here. So, therefore, if the electron is generated within certain length, then only it will reach there. Otherwise, the reaching probability becomes lower and lower and that is given by that length. So, that is what will happen and this is the diffusion length for electrons, diffusion length for the holes. Diffusion length is the length, electrons or holes travel before it dies, before it recombines. You get positive. So, what is happening? Your junction is automatically getting positive, positive bias, forward bias. What does it mean? If it is a junction, it is a forward bias. Will the forward bias current will flow in solar cell? Will the forward bias current flow at normally flows in the P N junction diode? It will not flow. And if you think like that, you are wrong. It has to flow, it is a junction. How does it care? As long as it is getting forward bias, physics should follow the rules. Physics only said, whenever you get forward bias, there has to be diffusion current. So, even in solar cell, because it is automatically getting forward bias, nobody is doing externally. Why it is getting forward bias? Because of the light falling on it. So, because of the light falling on it, it is getting forward bias and therefore, forward bias diffusion current must flow and it does flow. It is just diode. So, forward bias diode. So, forward bias diffusion current is flowing, but this current, which is because of the light now is larger than that. So, forward bias current will flow from there to where? P to N. And this current is because of what? This is extra carriers which are coming in because of the light and the direction of that current is N to P. So, because it is forward bias, forward bias current does flow, but the reverse current is bigger. The reverse current which is because of the light and it is called light generated current is bigger. So, now, I can actually draw. I can write my expression for the solar cell. My solar cell should behave like a P N junction. There should not be any exception. It is P N junction. It is getting forward bias. So, my solar cell should also behave. It must follow this. Ultimately, it is a junction. How we need to modify this expression? We need to modify this expression. What we need to do? You have to take care of the extra current that is being generated now, which is because of the light. In my diode, it is all encapsulated. There is no light. So, no extra current. And this light, because it is just getting forward bias like a P N junction. So, it should follow the same forward bias current. Same is here, but that additional current component, which is because of the light should take care of. And that additional current component is in what direction? In opposite direction. So, I should actually put a negative sign here and put another component, which is L, light generated current. And that is it. This expression becomes solar cell equation, nothing else. You derive it with whatever your fundamental principle eventually will come to the same expression that this extra current component, which is in the reverse direction of the normal diode current flow, needs to be added when the light is falling on it. This current component is coming because of the light. This voltage is also coming because of the light, but it is as if my diode is forward bias. So, this current component is nothing, but a forward bias current in my device. And this current component is a light generated current. So, that is what I said. This current component, now this is again, the whole expression is I 0. The whole expression I 0, we have not done the derivation. If we do the derivation, we will find out this. But this is a very important part of the equation. Normally, when diode equation is I 0 e q pi k t, I 0 is the most important part because that determines everything. The whole efficiency game depends on these parameters. So, what is the parameter? Diffusion length, very important. Diffusion coefficient, very important. Diffusion length, very important. Then your minority carrier concentration is very important. Then you get the lifetime of the carrier is very important. So, all material parameters are incorporated in this expression, all material parameters. Whatever will change from one manufacturer to manufacturer, whatever will change from one material to other material, everything is part of this expression. So, this is very important expression. In short, what we call it? I 0 or reverse saturation current. And this is normal expression q by k t. And this is your I L, light generated current. This is your I L, light generated current. It is very simple to derive this expression. G is the generation rate, the number of electron hole pair generating in a volume. What is the volume? A is the cross section area and that volume, right? This volume. So, cross section area is A and this is the volume. L n plus volume is L p. These are the cadets which are contributing. So, A times this, this is the volume. So, rate of generation, rate of generation, G is the number per unit, volume per unit time. You multiply with the q. So, you get the charge per unit volume per unit time. And you multiply the volume, which is A times this. So, you get the current, light generated current. And therefore, you get this is your equation, normal diode equation. And I said we discussed this is your solar cell equation. And solar cell because it operates in the fourth quadrant, current negative, voltage positive, power is negative. Therefore, it is a power generating device if it operates here. So, the solar cell equation is same because our expression is also same. Only thing is there is a large negative current flows. And therefore, because of the large negative current, it goes in the reverse case and should operate somewhere here. So, you understand what so, we still we missed some points. So, we have not done the actual derivation. If you do the actual derivation, you will find each and everything, void diffusion, how much diffusion where it is happening. And eventually, you will of course, have the efficiency. So, you write this in the fourth quadrant and you can write the expression for this we have discussed yesterday.