Solar Cells Lecture 4: What is Different about Thin-Film Solar Cells?
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Alam, Muhammad
Thin film solar cells promise acceptable efficiency at low cost. This tutorial examines the device physics of thin-film solar cells, which generally require a different type of analysis than crystalline solar cells.
 Nanohub.org Online simulation and more for Nanotechnology. You can follow along with this presentation by going to Nanohub.org and downloading the corresponding slides. Enjoy the show. You can hear in the back. What I am going to talk about today is thin film solar. If you think about the summer school, it has covered a lot of topics. You must be tired by now. But essentially it has two broad groups of things. One is for transistors, spintronics, memory elements. These were for information processing. That is, if you learn those things, you can get a job at Intel. On the other hand, the other topic, set of topics, were the grand challenges of today, which is not information processing necessarily. This has to do with energy conversion. We have talked about thermoelectronics and PV, the photovoltaics. Of photovoltaics, there were five lectures. These five lectures had to do with the basics of solar cell. By now you know that. What this is all about? That if you have a semiconductor and light comes in and shines on it, it may come from the sun or it may come from some other source, essentially a photoelectric effect. Then you essentially have electron hole pair generated. Professor Lunsstrom explained that. When you connect a load to it or a light bulb to it, the light bulb lights up without having to connect to your power socket. That is what the PV is all about. Then once that was done, the very basics, you learned a lot about crystalline solar cell. You learned about how thick this region has to be. Because it does not absorb very well, you remember that. You also remember perhaps that 80% of the market today is due to crystalline silicon. This is the dominant technology. That is for sure. If that was all there is to it, of course we could go home. That would be nice. But unfortunately it looks like that there are in fact very interesting additional trends. That counts mainly because of the cost. Because at the end of the day, when somebody says that your iPhone costs $300, you say, okay, here is the $300. You buy it. But for solar, for energy, it is very cost sensitive. Because in one hand you have oil and gas, which are very inexpensive. On the other hand, if the PV people say that it costs twice as much, but look, this is very environmentally friendly, you would say, no thank you, I don't want it. So therefore it's very cost sensitive. And this is the cost aspect of it. That makes this thin film organic solar cell, plastic solar cells, so interesting. But also the physics is very beautiful. And I am going to explain to you how that's going to work. So two talks. This talk is about thin film solar, the organic photovoltaics. We'll be the next lecture. Now, here is a brief outline. I'm going to talk a little bit about the background information. Give you some background information. But I'm going to talk a lot about photo current. And this aspect you might find a little bit interesting. Because Professor Lundstrom, one thing of all the things he said, one thing you might remember, he said, that when he was a student, he wanted to go home and explain to his wife that why he doesn't work on the photo-generated carriers for the solar cells. Because it was flat. And so therefore he could focus on the dark characteristics. So if you remember one thing from this lecture, that is what you should remember, is that he cannot give this explanation to his wife for thin film solar. So I'd like to then explain a lot, explain how this is done. But I'm not going to do it by using any differential equation. I'm going to use very simple things that just counting, counting electrons going left and right. That's all you need in order to find how this electrons move around and how these photo currents are generated. Very simple. If you just don't fall asleep, then you'll be able to understand them all. And then I'm going to talk about dark current. Very briefly, Professor Lansstrom has already explained to you, Professor Gray, a lot about dark current. So I'll not really do much about it. Instead, what I want to tell you about is that once you put these cells in a module or connect them up so that you can use them, variability and reliability becomes fundamentally important. And this is the aspect that you didn't really hear about much. But at the end of the day, it matters a big deal. If your solar technology lasts for 20 years versus 40 years, because it's free energy, you bought your panel for $1,000, let's say from Sun Power, and that's your initial investment. If it gives you free energy for 40 years versus free energy for two months, it makes a huge difference in terms of cost per kilowatt hour, the amount of energy. So reliability is very important. And I'm going to explain to you how that works. So very briefly, there are different types of solar cells. You have heard about this crystalline solar cells. And you may remember that the green N plus region is very thin. You remember that we cannot make it any thinner, because otherwise there will be big resistance drop when you want to collect it to the electrodes on the top, this plug-like structure on the top. But the main problem is that the crystalline silicon, the P region, the P base is very thick, 200 micron. And why is it 200 micron? It's because it's a very weak absorber, indirect band gap. That's what crystalline silicon is. And so therefore it takes many, many bounces on the order of 10,000 micron before everything is absorbed, 50 bounces maybe, before it's absorbed. So it's a thick material compared to what you could do with gallium arsenide. Or any other thin film technology. For example, you could take a PIN structure. For example, based on amorphous silicon, where you have a P region, that's the yellow, and the intrinsic I region, that's in the middle, 250 nanometer. You see, on one side you have 200 micrometer, in another case you have 250 nanometer, and then you have an N region on the bottom which is green, and the top and bottom you have the two electrodes. You see, you could put the whole thing just almost inside the N plus region. It is that thin. So maybe just the device itself without the encapsulation or anything, maybe a factor of 500 smaller, thinner, compared to the other one. So you can immediately see that if this material, the two things cost the same, then this would be way less expensive compared to the silicon. Of course, there are, this is not the only version. The other version involves metal insulator metal, and that will be the next lecture. And that has to do with flexible organic solar cells, so we'll talk about that. Now you might ask why is it PIN here and PIN over there. In this case, in both cases you have a given electric field, and once you have that electric field, the electrons can be pulled out on the both, from the both contacts. So you can have either type of the technology depending on your need. So what I'm going to tell you about is this, basically the middle one, PIN structure, thin film amorphous silicon, and explain the physics of it, and how it differs from the crystalline silicon. It's actually very different. But of course, you will see how this comes about. But of course, this is not all great news. If the thin film was so good, why is the market 80% crystalline silicon? It's because many times the material may not be abundant in the earth. You know, silicon is everywhere. So silicon in various forms, it may not be, many of the things may be very rare. So once lots of people start buying it, then you, they mean the suppliers may not be able to sell it. Some of them are toxic. So there are other problems, and generally the substrate may be more expensive. So although you are using far fewer material, far smaller amount of material, it may be more expensive per unit. And so therefore you may not end up really saving a lot of money. So if you make a quick comparison, and this is by the way, I take no responsibility for these numbers. The numbers are very sensitive. But my main point is to show that for crystalline silicon, the material is about like say $200 per meter square, the process cost $100. And when you put them all together, it is cost per watt without installation. It's about close to $1.75. And compared to that, other thin film technologies generally typically have lower cost. But in addition, there is another problem that generally crystalline silicon is heavy because you have to put it on top of some sort of substrate. If the rock has to come to your home and then put it up, you cannot go to Walmart and buy a plastic or cheap wafer and bring it on yourself and put it on the top of your roof. Not that you should try. But in principle, that would make things much more expensive. It turns out one-third of the cost of PV is the installation cost, half or one-third. So therefore this silicon technology generally is very expensive. So a thin film PV might actually be much better. So let me explain to you then what the physics is. You see it's thin. So when things are very thin, lots of trouble can happen because things can short from one side to another. Thin is not always necessarily a good thing. So for example, this region, do you remember, in the middle region, this I region, this we said, this is on the order of 2 to 250 or 300 nanometer. Very thin. Then doping regions, the dope regions are on the order of 10, 20 nanometers. So again, very, very thin. This material often grown in low temperature. And so therefore there can be a lot of other problems also. For example, one problem it could be that since these regions are very thin, the contact might diffuse inside the device. Now you might say, well, a little contact diffusing inside, what harm is it going to do? It's a very big, a little bit. But what I'm going to show, this is like impact crater, that it may enter small, but its impact will be huge. A micron size hole can essentially drain away carriers from a millimeter size region. So we are going to talk about that. And then, of course, many of these materials have horrendous grain boundaries. Nothing beautiful like silicon, where all the atoms are like an army, right? All regimented and all, every atom in their right place. Instead, here you have grain boundaries going from left and right. And the question you ask yourself, how in my life I'm going to actually solve this problem? Of course, there is a simple answer, is that we can close our eyes, forget about the defects, and put some effective parameters in the silicon simulators, and whatever result we get, we don't tell anybody else and go home. Now, this is the four things I'm going to talk about, but the fifth thing is equally important, then that has to do with series connection. Solar cells are always series connected, because each cell is like one volt battery, just like in a toy. Many times you have to put three, four batteries in order to get the right voltage. In the solar cell, many times you have to connect, let's say as much as 100 of them in series, so that you can get 100 volt out, if it's one voltage. And the way people do it, that if you have a glass substrate, if you put the mirror, TCO is type of a electrode, highly conducting electrode, then you put the material that you want to put in. This is your solar cell material, amorphous silicon, cadmium telleride, whatever. And then you do the laser stripe, open up these holes, and then put the second contact on top of. Now, do you see what it just did? Instead of having a simple PIN junction, this successive layers of things, now currents are going to do this, that it will enter through one end, go up through this contact, come down, so this is one solar cell, then go down, come up, and again come down, that's another solar cell. And so, although initially you had thin planar structure, you have just by striving and connecting it up properly, you have just made it a series of solar cells, which are connected in series. Now, this is very important, many times you will see if you Google, I hope nobody is doing it now, but if you Google, then you would see many of these lines in here, and essentially these are scribe lines, essentially that are showing you that, that's how you cut it up, so that when you string them together, it gives you a bigger voltage. So these five things, we are going to think about as we go through the rest of the talk. So now let me remind you, let's go back, take a step back, that what did we learn from Professor Lundstrom talk? I'm sitting in the back, so this is what I learned. Essentially what he talked about, that when a photon comes in, a light particle comes in, if it has energy larger than the band gap, then an electron hole pair will be generated. The red electron will go to the left contact, the white hole will go to the right contact, and they will eventually go through the load, and therefore whatever is delivered, the power there. And you remember what we talked about also in that lecture, that there is something called a dark current on the right hand plot, there is something called a dark current which is essentially the diode, and you can see on the bottom that we have a symbol for this diode, and that is ID, we have written ID next to it, so that was the dark current, and he essentially explained to you that the photo current is essentially a constant in the graph, it's a photo IPH, that's a constant, so therefore it's like a current source, and when you combine them together, the total current essentially is a dark current, and the photo current together, and that's what you get. And so the voltage at which there is no current is called open circuit voltage, and the point where you want to operate is the maximum power point, how do you find that point? Well if you remember from your circuit days, that if you have one element that has this particular IV characteristics, and if you have a load line, then wherever that one intersects that red point, that is where the power would be. Now what is the maximum power? Maximum power would be whatever corresponds to the red point, with a voltage that is little less than VOC, and photo current which is little less than the ideal photo current, the short circuit current. Now solar cell people are lazy, so instead of really calculating, what they do is that they say okay, this is my original box, VOC multiplied by I photo, but of course that box is too big, what I should really have calculated is green box, now I don't know really what this green box to this one ratio is, so I'll call that field factor, and so that is what they will, this is how they will write this formula, that this IPH multiplied by VOC is the bigger box, and the field factor is the factor that you multiply so that you get the green box, which is the power that you really want. Now this is ideal solar cell, because in this case, you can take that current, I'm sorry, you can take essentially this current, and add it up to this dark current to see at what point the VOC is, so you can calculate a VOC, you can get a maximum power and you will be done, but it turns out for solar cells, this is not really, for thin film solar, that's not really the full model or the full story, what you need is two more things, one is this extra current source that only occurs when there is light in there, so instead of having a voltage independent current source as you heard before, now I have this combination which depends on voltage, and I am going to explain to you very simple, there's nothing funny about this whole thing, and then there is another thing in solar solar, in particular called this shunt, now somebody asked professor Lansstrom the other day that what is shunt, and he mentioned that there could be pinholes and other things, didn't really go into the details, I want to explain to you that this can be understood very simply as well, so we will talk about that, so let's, the main point I want to make is that superposition doesn't hold, so therefore if you see a thin film solar, most of the time you shouldn't try to use copy textbook formula from your undergraduate textbook and try to apply it because you may not get right result, okay, so let's get started in terms of photo current, now before I get started let me tell you something simple, and this is something you understand very easily, no differential equation or anything, just look at what the electrons go, assume that I have a semiconductor and then two metal contacts, and assume that I have six photons coming in, and that has generated this six red electrons and six white holes, now for some reason, and we will understand that reason later, four of the electrons are going to the left and two of them going to the right, and similarly two of the holes are going to the left and four of the holes are going to the right, how many currents, how much current do you have, now you have no differential equation to hide behind, so therefore you must be able to answer this clearly, the answer obviously is not 4q, because you could look at this electron and you could say four of the electrons coming out, so I will say it's 4q, not really, because the two of the holes are coming out through the wrong contact, so as soon as this four comes out they will recombine and only two can go to the load, and so therefore the current will always be the current at any given junction, the electron current to the left contact minus the hole current to the left contact, and when you sum them up that's the total current, and so in this case I have only 2qV0, V0 is the velocity at which electrons are going, now of course you could write it slightly differently, you could say that these four are really out of the six, because six photon came in, so four over six, so it's really the fraction of the total that is going to the left contact, and these are the ratio of the probability that you will escape to the left contact, if I have two doors then the probability that I will escape through that one versus the probability that I will escape through this one, so it's normalized, so that's it, so you have, so I will write a formula which is something like this, which is generation rate, probability that you, the electrons escape through the right contact, the proper contact, and minus the probability that the holes escape through the wrong contact, and that gives me the total current, now what happens if you have recombination, remember if you have defects and all, you might have recombination, we don't have to worry too much about it, because that's really again very simple, see here, I have three coming out, two electrons, two holes coming out through the wrong contact, so the total current is just one electron, that's going around the circuit, right? Now what happened all through all this recombination and all, why did I forget them? Not really, I didn't really forget them, because if you think about it, that if I wanted to write what fraction of the total electrons are getting out, then I would write three divided by six, three actually went out through here, and six were generated, and so when I write the ratio, my recombination is sitting right here, because the total rate had the recombination in there, of that only a fraction escaped, and so therefore these two formulas, if you sort of remember that sort of makes sense, then you will see how very complicated things will get very very simple, as you will see. Now before I get there, let me tell you about this semi-classical probability of escaping through the contacts. So for example, if you have a barrier, and this is something Professor Lunsstrom also mentioned, of a certain height, and if you wanted to know what is the probability that the electron will escape there, you will say that this probability is proportional to the height, right? Proportional to the height of the barrier, if the barrier is big, nobody is escaping, and if the barrier is very low, everybody will go away very quickly, so that's something you know, and this KT tells you the temperature, so the more the thermal vibration is, easier it is for it to jump over the barrier. Now you can of course make the same argument if you had a left contact, if you have a device in the middle, I am trying to see whether how can I get to the left, and how can I get to the right, because then I can do something called a partitioning. So in general therefore, what I do that if I have two barriers, then essentially depending on the barrier height, the left and the right, I will escape with two different probabilities. Now this turns out to be very important, as we try to understand why thin film solar is different from crystalline silicon. As you will see next. So assume this very simple case, you remember this simple formula from before, that this is if I injected a particle here, a photon has just come in, generated two electron-hole pair, one electron-one-hole pair, and I want to know the fate of this electron. What is it going to do based on this formula? Well it has a probability that it will escape through the left contact, but it has also some probability that it will escape through the right contact, and so you can see this Ln and Rn are the probabilities for the electrons escaping through the left contact and the right contact. And similarly for the holes, you also have the same probability, it can go to the left or it can go to the right. Now if I ask you, what is the probability of electrons going to the left? Well in that case there is no barrier, it's just fall, it's like water flowing down a hill, and in that case, I will, there is no barrier, so whatever is the flux, it will just get out from there. On the other hand, if it wants to go through the other contact, let's say this is a very, this electron likes torture, so it wants to go out to the wrong contact. Now if it wants to, of course, it can, if the electric field is E, and if this distance is X and whole thing is W, then the barrier it has to climb is E, W minus X over KT, that's the barrier height, right? Electric field multiplied by the distance, and so that's the barrier height. All you have to do is to insert these few quantities in here, because you see I have everything, gamma L, gamma R, I will just put it in here. By the way, why do I do this integration? Because of course sunlight is going to come at every point, right? It's going to generate at every point, and so therefore when I want to know the fate of all the electrons that are in there, I'll have to integrate them up. Now if I do that, I'll put this formula, you recognize this one is here, this V naught is here, you put them in, and very soon you get a formula which looks something like this. Now you, this integration exponentian on all, if you don't remember, you can Google it, the integration formula, but this is something that is that you'll get. And in fact, it turns out that this formula, it is originally derived in 1981, it probably takes eight pages, six to eight pages, with lots of differential equation, left and right and boundary condition. At the end you get this, but you do not know what happened to the electrons in between. You are so lost in differential equation that the electrons here have forgotten about the electron. But you can see very simply what these electrons are doing in here. What this is all about is essentially that if you are close to this region where electrons can skip, in that case there will be a loss of certain number of electrons will not come out through the right contact, and so therefore you will have a reduction in current. Your current cannot be voltage independent because you see as I am pushing this voltage up and up, right, biasing it, as I am pushing it up and up, the fraction that will begin to go through the other contact, the wrong contact, their fraction will increase. And eventually there may be a point, and as I will show in the next slide, there may be a point where this is flat, half going the wrong contact, half going the right contact, and essentially you have zero current. Voltage independence cannot hold in this particular case, right, it is simple. So you know why this voltage independent requires a lot of discussion, but why it is voltage dependent should be simple to understand. So you will think that well this is a simple formula, may not be very good, who knows. So you have this formula, and here is this green line, and I am going to talk about this red and the, this is the dark current, the blue one is the dark current, red is the total current, this is photo current. This is not only not constant, it is going through zero and going to the other side, flipping its sign, right. Photo current is not only constant, but it is changing its sign. And you can, if you put it in a numerical simulator like ADEPT or any other device simulator that Professor Gray talked about yesterday, the results will be right on top of it, and so therefore you can trust this result, and you can easily see the physics. What happens here is that you have in the right, at low voltages, this partitioning occurs, so most of the current is going in the right way, the way you want it to be. When the voltage is VBI, the built-in voltage, the partition equally, the current is zero, right. So this point is current is zero, and if you flip it on the other side, then of course the currents are going to go the wrong way, instead of electron flowing this way, now the electron is going to flow to the right side, and so the current is going to flip. That's it. So voltage independence doesn't really hold in this case. Now how do you want to make it if you really wanted to make it voltage independent, then what you could do, you could put a big barrier here, and put a big barrier here, so that nobody is allowed to go through the wrong way, right. You close one door. If you close one door, then even if I try, I cannot escape through the wrong contact. So in that case, of course this formula immediately knows that that has to be voltage independent. Why? Because the right contact, this would be zero now, because right contact you cannot escape anymore, that has been blocked. And similarly, this holes, this cannot also escape, and so therefore this will become one, and when you, this is very happy integration, zero to W of dx multiplied by one gives you W, and that means every photon that has been, every electron that has been created, will come out through the right contact, independent of voltage. Now under this circumstances for sure, you are going to have a voltage independent, voltage independent photo current. Okay, so blocking layer is very good. Yesterday you heard about this electron mirrors and other things, right, in the lecture, if you remember. And so this blocking layer is therefore very important. Now only if there is low recombination, as I am going to tell you. Now one thing I have not been telling you so far, is that this material is full of defects, right. Remember low temperature deposition, all those grains sort of crisscrossing, full of defects. Now when you have things full of defects, then life is a little bit more complicated, because even in that case what I want to show, that even if you put a blocking layer, it is not going to help you in terms of making the current voltage independent. You know I am very trying very hard to make it voltage independent, it does not want to stay that way. Why not, what happens in this case, why can I not make it happen? Because as soon as an electron hole pair is generated, this electron hole pair is going to randomly walk around. It is very soon going to find a hole that has also been generated, it is going to recombine. How far is it going to go before it recombines? Well you would say this time to recombination that depends on how many defects I have, and the velocity of it. And the electrons movement will be mu multiplied by the electric field, that will be the velocity, mobility multiplied by the electric field. And it is going to stay alive for tau, so that is the amount of distance over which it is going to recombine. And if it recombines and if it does not come out, is it as if it were never generated, right. Because I do not really know whether it was generated and then recombined or essentially it generated, it sort of, it was not generated to begin with. Okay, again this formula, would this formula work? Well generally it should work because you can see how this formula should work. Gamma LP the whole zero, because I have put a blocking layer. This is zero, gamma Rn is also zero, because I have put a blocking layer, so that is not going to go anywhere. Now the fraction of the electrons that is injected at a given point and comes out because just going through this recombination process is this ratio. Gamma ln, gamma ln divided by plus gamma R. You know, let us say the chances of going from here to over there, there are a lot of shards between here and over there. And if I have to swim through this, my chances of getting out on the other side is exponentially suppressed, that ratio. And so what I am going to put is zero to Wdx and this ratio is e to the power x over LC, LC is that distance. Because that is how far I can go before I am eaten up. So I can integrate this simple thing up and again I can write the photo current. Again, if you look at this paper, the original paper which derives this, that will be another six page. But this is essentially, the essence of it is just this, that you cannot get out without getting through this recombination processes. So therefore you can rewrite it a little bit. Qgw was my total generation, voltage independent, right? Independent because if I could collect everything, independent of voltage, that would be the thing. But of course this blue one is telling me what fraction of it is recombining inside the device before getting out. And so therefore this whole quantity is essentially this voltage dependent current source. Again, in the presence of recombination in this region, the current will be voltage dependent. Now, if we understood so much, now it's good to go back and ask the question that what happened to the crystalline cell then? In that case things were much nicer. So what happened in that case? Well, you can easily see what happened in this case and why if the quality of the material is bad or poor, many of the crystalline silicon, if you want to pay less money, then not the champion cells of Martin Green. But if you want to do a commercial cells, their quality is generally lower because that's less expensive. So in that case, many times this barrier may do you no good, very counterintuitive why barrier wouldn't do any good. Again, this formula will tell you that why the barrier doesn't do any good. Look at this formula simply says that all you have to keep track of is the number of electrons that are going, getting out through a junction. So I have the electron current moving to the left and the whole current moving to the left. But I have a potential barrier here, it's a p-n junction. So my whole current is essentially zero. The amount of electron that can get out through that contact. So if that is the case from the previous slide you remember, so it is as if these electrons are getting out through this region and that's what I showed in the last slide. And when you put this thing in, put this same formula, copy this same formula, you find that only the electrons within a diffusion length away from this region, only those will be collected, rest of them will essentially recombine here. Now you may say that let's take out this potential barrier, but this barrier must be doing me some good because after all here is sort of reflecting all the carriers, electron mirror, this should be doing me some good, but actually it doesn't do any good because you can see for the following reason. Because if you remove it, of course these electrons will be coming out to the wrong contact. But because the electron number is a little bit less now, the recombination of the holes that are coming out here will also be a little less. And so therefore what's going to happen, more holes are going to come out here. And when this current is added to that current, it will show there is no difference whatsoever between these two cases. Because who cares whether you recombine right here or you recombine two nanometers into the contact, there's no difference. Wrong contact, wrong recombination, so it makes no difference. And you can, this seems very counterintuitive, the blocking layer in one case whether it makes no difference. So in one case you see this is the electron current, the red one and the blue one is the hole current. In the other case, this is the electron current through the right contact, this is through the wrong contact, it's going the wrong way. But what will happen, the hole will also escape in a larger number and the total current will remain exactly the same. So this sounds as I said counterintuitive, but it will turn out that this is exactly consistent with numerical simulation. You can see that this is the current, electron current going out through here, the hole current is very small. And then if you remove this blocking layer, there will be a spike in the hole current. I'm sorry, this is spike in the hole current, but there will be a little bit negative electron current and when you sum them up, the total current will be absolutely flat. Total sum of the current will be absolutely flat. So again, as I'm saying that this view, I haven't solved a single differential equation, but you can see most of the things that would be non-intuitive will come out if you just think about which contracts or which doors the electrons are coming out through. Differential equations are good capabilities, but not necessary. And if you combine, I don't want to go through this point too much, but if you combine, if you have both recombination and electric field, if you have this, don't put a blocking layer, don't want you feel to complicate your life without putting the blocking layer, the life would be a little bit more complicated. But once you have done this, your numerical simulations and this simpler form, most of the time will give you good results. I'm glossing over this point a little bit, but it requires a little bit more discussion. So long you get the basic idea, I think that should be fine. These numerical simulations are the symbols and then the analytical formula is essentially this green line. So you can see that how accurate this simple formula. Okay, so that's all I had to say about photocurrent and the bottom line is that you cannot tell someone that current is flat and therefore you're working on dark current. All right. Now the dark current and very quickly, the dark current, I'll just show you two slides. This has some math. Just want to show you that if you do this partitioning argument, the dark current will also come out, right? And all the features will come out, right? But I don't want to go through too much detail because I want to show you a few more things. So if you wanted to know the dark current, no light generation, then again you will do the same thing. You will calculate the electron current by taking, looking at how many electrons you have on the left side and how many of them can go to the right hand side. And similarly, you will see how many electrons on the right hand side and that I should have put a R here, sorry, gamma R and gamma L. And then correspondingly, how many electrons are going to the opposite side? So divided highway or at least two-lane highway that professor that I talked about. Remember that there was these two fluxes and you want to subtract them essentially to get the total flux. Now at zero bias, this current must be zero and if this current is zero, you can get a ratio of gamma L to gamma R because that's the only unknown. And if you do that, you remember the probability of going from the left to the right is exponentially suppressed because there is a big barrier here, right? Big barrier, p-n junction barrier and so that is exponentially suppressed. And correspondingly, this one on the other hand, if somebody is trying to come from here and go to this side, of course there is no barrier. So that's one. And you can just insert this in this formula. Okay, so by the way, so if you have this gamma L and gamma R, put it in this equation that will tell you this concentration on the left hand side. How is it related to the concentration on the right hand side? Now do you remember what this relationship is? This just turns out to be the law of the junction. Essentially telling you that two sides, the electron concentration are exponentially off and equal to this VBI or whatever the barrier is, barrier height is. Now once you insert these equations, again gamma R and gamma L is the same. You insert this equation under bias condition. In that case, this will remain the same, the gamma L0, n L0, so you take this expression in. And the gamma L now, the barrier is a little smaller because you have applied a bias and it is smaller by V, minus V, VBI minus V, because by applying a voltage V, this has been suppressed. So you do a few lines of algebra and you get an expression here. And that expression tells you essentially that what is the dark current for a PIN structure like this. And if you want to get the total current, you can just sum up the electron current and the whole current and you can get it all. So again, no differential equation, but you can see that the final result comes out reasonably okay. And this I0 you have seen many times, that's the pre-factor that we just use it as a constant. So now the question is, my dark current should look like this. I have a beautiful formula on when I'm applying a voltage on the positive side, it should go up exponentially. And you remember why this is saturating off because of that series resistance and other effects, right? That's why it saturates off. Now when V is positive, it goes up exponentially. When V is negative, what should it do? It should be minus QV over KT and that will go to zero very rapidly. And so if I plot the absolute value, this height will be equal to I0. This is what a textbook will tell you what is supposed to happen. But if you go and make a measurement like a show rough dead, then you are going to get something which doesn't look like a solar cell or a diode at all. And this is how it looks like. Now I do not know whether you have made enough measurements. When we were younger in college, we had to make a lot of diode measurements. It used to have these two pins or three legs, transistors. We made a lot of measurements and I have never seen in my life that the measurement, a diode characteristics looks like this. First of all, look at this. This is supposed to go down exponentially. It hasn't. Then it has these two wings which looks like a butterfly wing, completely symmetric with respect to each other. Now if you do the theory which I do it so beautifully, partitioning and all, and then try to compare with experiment, you will see that you are in for a big surprise. And because this doesn't really look like a ideal diode characteristics at all. So what is this? So this is something I want to talk to you about next. And this has to do with this contact diffusion. What the contact diffusion will essentially do in a summary, that originally I had a PIN structure. The contact diffusion in this place localized, it will make this region P type because aluminum when it diffuses in, that will make this convert this region to P type and therefore this region will become a PIP structure. It will not remain a junction anymore. And in that case, those butterfly wings will come out of this immediately. Let's see how it works. And in fact, there are experiments in which you can find out this dark spot, this light spots correlating to this shunt regions. So how does it work? Well, if you had a P region which sort of diffused through this, through this n layer, then you can see you do not have a diode anymore. What you have is a PIP, a shunt region where the current can leak through. At low voltages, what's going to happen? Most of the current is going to flow through this low resistance structure. At high voltages, of course, the diode will turn on and then everything is fine. So at low voltages, this is the butterfly wing associated with this region. And at high voltages, of course, the, finally, the diode behaves like a diode and its value generally comes up. The reason being this is a much larger, larger diode. These are tiny, few tiny regions. So therefore, it can be higher only up to a certain point. Beyond that certain point, of course, this larger structure will show up. Now, this is a good theory, but of course, how do you know these theories at all, correct? Or are you daydreaming that if I had this type of thing, that would be very good to have. But you can go and try to explain that how this works. So what the current transport through this structure, if it's a PINP structure, then I'll not go through the details, but you can again calculate the current through a structure like this. And that current, I should have a P here because whole, primary, but this is equally well, works for electrons. So if you had just electron current or if you had the whole current, then this would be proportional to the electric field, majority carrier region. And once you solve for this, and I'm not going to go through the details, but this is something in the notes, you can go through essentially if you combine this transport through this region with the Poisson equation associated with these charges. And then solve what you're going to see is that the current in these tiny structures is going to have a very special property. It is going to go as square of the voltage and be inversely proportional to the thickness of this region, I'm sorry, thickness to this region to the power cube. That is how this particular current is going to behave. The butterfly wing is going to have this property. So if this theory is correct, this is how this component of the current is going to behave. In general, of course, if you do generally this exponent is between 2 and 3 and this could be on the order of 3 to 4. So essentially that is what we'll be looking for. So there are more general theories, but essentially we're looking for whether the exponent is between 2 and 3 or none. And this is the butterfly wing. This all things are about butterfly wings because you can see that if you take this two points, take a point here and take a point here. And take its ratio. If it really was that conduction, then it will ratio be absolutely equal to 1. So you can see it's an exponential curve and there you are taking two linear quantities on top of each other and dividing them. And when they become one, you can realize that how close to perfect this butterfly wings are, this shunt conduction are. Then you can also look at the temperature dependence, but most interestingly you can look at the nonlinearity of this shunt between 2 and 3. You can see and this is about close to 60 devices. So it's not just one or two accidental happening. 60 devices all showing essentially the same characteristics. And here is the 1 over LQ. I believe this is also 60 devices, right? Or is it 400 devices? 280 devices. So this is based on 280 devices with different eye layer thickness. This is the region in here. Again you can get approximately a 1 over LQ type dependence showing that this type of shunt conduction is parasitic, but at the same time can be accurately and quantity described. Okay, so I'm coming towards the end. I have a few more slides. So we saw voltage dependent photo current, which is good. We understood the shunt which people say, well, I really do not know what it is, but essentially say it could be a linear thing. It could be a nonlinear conductance, but you can clearly see how to think about the physics in a consistent way. Now let's talk about the final piece and then we'll be done. The final piece has to do with grain boundaries and how current these grain boundaries affect essentially the response of it. And it will be very brief. Assume, and what I want to show you here that a small defect, a micron size defect can take away a big region and make them ineffective. So sort of take them out of the circuit and how it does so. It's like the example would be like a sink, you know, in a faucet or in a tub you have the sink and the sink or the opening in the bottom. And that may be a very small space, but the amount of region it affects is significantly larger. So that is what I'm trying to do through all this equation. That's the goal I have. Assume that you have a solar cell and I have divided the solar cell into many pieces and one of them has grain. Now that means all the solar cells have the same characteristics except I have this black ship. Oh, I should have put, I'm sorry, this is a diode. So this line is missing, but I have a black ship diode, meaning that that diode has a high recombination. So I marked it black. And so you can easily see that if you have a current, I, total current is I, this is the dark current and this is the photo current. And we say that it's a voltage dependent photo current. Then I can say that if the, at the VOC point when there's no current flowing in the external circuit, then at that point I is zero. And so therefore I can solve for VOC because I can put the VOC in here, I can put the VOC here and I can solve for the VOC. Now what you can do is you can write this I naught in terms of this VOC, you see, because this I photo and if you say this is big compared to minus one, you can bring this up exponentially on top here and then can rewrite the whole diode equation solely in terms of the photo current. You see here, here I had the dark current and the dark current of course depends on the recombination. All I did was to convert it in terms of VOC so that I could write everything in terms of the photo current. Now think about what will happen. What will happen if there's one diode that has a lot of recombination or because you have a lot of recombination, low VOC. If you have a low VOC then what's going to happen that it is going to sink instead of this current going to the useful load to your battery or to your light bulb. These are all going to supply current and sink in here, right, all of them. And so the number of those diodes that have been affected, that has been affected is this number. They have all VOC one and this black one has VOC two lower and if you solve for it, what you find that the number of diodes that have been affected by it is exponentially dependent on the VOC difference. And if you have a defective region, it will take away a whole set of region around it. And so therefore, if you all had the same VOC, you didn't have any one that is defective, of course there will not be any penalty. But if you have one of them bad, that will really cause a huge amount of trouble. Okay, so now towards this end, let me come to this final thing. Many times you have seen this beautiful pictures where a satellite is gradually rising and many times there you can see the boom of that satellite casting a long shadow on its wing, on the solar cell wings. And you may not really worry about it that well, that looks beautiful. And many times the solar cars are going next to trees with a shadow on them. And there may be other sources of shadow, some bird droppings and other things and other bad things may happen. And you may not think or a leaf maybe, that's a better example. That may fall on top of it. And you may not really think much about it, but it turns out that that is inside the solar cell when things are in shadow, there's a havoc inside happening inside. And that's because of this serious connectedness of solar cells. So I told you in the beginning, right, that was the fifth topic, that the solar cells like this dolphins coming up and going down, coming up and going down. And these are of course the diode and the current sources, each sort of facing opposite and the continuous current is flowing around it. Beautiful, no problem. But think about for a second what will happen when this long shadow of the boom is on the solar cell wing. Then all of a sudden one of them has just lost his power because that doesn't have any photon. And so therefore just lost his power. Now of course everybody else doesn't like it. They have their current and they want to supply their current. Serious connected cells, you like it or not, you have to play with this. This diode has to supply current. Now in order for this diode to supply current, what is it going to do? It's going to go in the reverse bias, break itself down and then supply its current, go in date and then essentially make sure everybody is satisfied. Very simply I can show you through this diagram. Think about this blue line, blue dashed line, a single diode and then this blue solid line being a set of diodes. These are in series so therefore your VOC has gotten multiplied by that number. All of a sudden the middle one loses power. If it loses power then it must go down, it doesn't have any photo generation anymore. It must go down and get in the reverse bias, break down and supply the current that the circuit dictates that it must have. And therefore the remaining two must now give up a little bit of voltage to help it break down even more so that it can give some current in the reverse bias and your total output will be dramatically reduced. It's like this, analogy is like this. Let's say somebody you are in a battlefield, five of you are going in somewhere. When one person gets hit, it's not that just one person gets hit and the four can run. You have to carry that person around with you. So essentially with one person getting hit, essentially it may take out the five person. And you may be in a very bad situation. So this is the same issue with the series connected thing and this is called shadow degradation. Shadow on a solar cell is a very bad thing. And you can show that as a function of time if you put things in shadow that this is going to degrade significantly. The dark current is going to degrade significantly because more and it's one of them in the reverse bias. And not only the output power will temporarily go down, but the whole power, that shadowed one which you are thinking is just resting. That one will essentially be in the reverse bias and a significant amount of defect will be generated. Very counter intuitive. Shadow you think good, it can rest but not when you just take one of them and don't give everybody else rest as well. My final two slides are on something that is called light induced degradation. Very important for amorphous silicon. Everybody in the field knows it so therefore you should too. What light induced degradation is that solar cell when you put it in operation, you don't even have to put it in operation. If you go to many companies what you will see many times they have a test set up right in front. And what they are trying to do is trying to show that as soon as you put a solar cell, especially amorphous silicon solar cell in light, it begins to degrade. And people knew it from 1977 almost from the very early days. The solar cell degrades very quickly. So the champion solar cells you hear about in the newspaper or in various journals. Well if you asked about them about two months later what the efficiency is, the efficiency might have been significantly lower. So this is light induced degradation and it has a very beautiful physics because if you look at the efficiency of the solar cells, they drop as a function of time. But the most interesting thing is that if you put it as a function of time on a log-log plot then you will see this has an exponent of t to the power one-third. Time to the power one-third. And this is very strange. We know exponential degradation, we know from circuit and other things. But a time the degradation that is going as power law is actually very unusual. So we want to know a little bit about how that happens and with that we will be able to end. The basic idea is this. You know amorphous silicon is a random, it's not crystalline, it's a random structure. And many of the bonds are essentially not saturated. And so therefore what happens is that you need to passivate them by hydrogen. If you bring it hydrogen that sort of ties up and then therefore you can have good performance. But as soon as you put them on the light, inside in light, then these bonds begin to break. And therefore something that you make easy also breaks easily, right? Cheap things. And so therefore what happens that this defects begins to come up and as you have more and more recombination centers, your shorts are this dark current begins to increase. And so you can write a very simple equation. You can say the number of broken bonds is proportional to the light amount of light you have. But of course once they are broken, some of them can come back and repassivate. So that repassivation probability is a number of hydrogen that has been liberated and the number of broken bonds. So you can write one equation like that. And you can write another equation that accounts for the diffusion of hydrogen within this region. And the dimerization of it. So just two little equations. And if you solve this, remarkably it will tell you that this thing should degrade as T to the power one-third. It will go as T to the power one-third. And most more importantly it will also tell you it goes as light intensity to the power two-third. That's also been shown in experiment that it goes as light intensity in a particular way. So what I'm trying to tell you, although recombination and reliability, this type of thing seems very esoteric. But they are good fundamental theories that can account for it. Let me skip this one. And then in the final one, so let me tell you that what I told you so far about is all intrinsic degradation. This is light induced degradation, shunt leakage, shadow degradation, weak diodes. I just told you all about material degradation. But of course solar cells are not something that you carry in your pocket. You may. But generally it's not like your iPod where you protect it very nicely. If there's water falls on it, you clean it up. You don't treat solar cells so nicely. You hang it out there and you get the power out of it. And when things are left to its elements, then of course a whole lot of other things can happen. There can be corrosion, there can be moisture coming in, and there can be, and the most important delamination, the layers can peel off. And so that can, if you are in the middle list, then there is sandstorms and it can be coated with sand and essentially no power output. So there are a whole lot of extrinsic reliability issues that people also have to worry about. And the most important reliability concern these days when people say they cannot have a 25 year reliable solar cell is the inverter reliability. Because you have to put an inverter in order to make the DC output, make it to AC so that you can connect to the grid. And most of them don't have more than five years of reliability. So here is a 25 year cell which you have gone away home hoping that this will be reliable and no problem. Five years down the road, inverter breaks and your solar cell is out. And so you have to go send someone to fix that inverter. And that's the huge cost. So one of the most important reliability is not with solar cell at all. It has to do with power electronics and power conditioning. Okay, so very briefly then. These are my conclusions. Economic incentives, right? We want cheap solar cells so that we can have good economic performance. I told you about thin film PV and its unique features, right? No voltage independent, voltage independent current. That's the important thing I said. Wig diodes, shadow degradation, all sorts of things I told you about. And extrinsic reliability is of course important. But the intrinsic reliability people are spending a lot of time on because this is something a good understanding of physics can help you really make a lot of progress. The things that are outside your control or the control of mechanical engineers, they will think about it. But we as electrical engineers, the equations that are in our control, we address those. And turns out shadow degradation, light induced degradation, those we can contribute along. And this reliability and variability are key concerns. And this is something, anytime you have a new advertisement, let's say, that I have such and such solar cell and big news release and all, let's say in technology, MIT technology reviewers and the most important thing is not to read that part. The most important thing is to read the comments underneath. Because in the comments, the real experts in the field, they make most interesting comments about the technology. And most of the time what people show is their best, highest performance. As soon as you put it in a module, 4 to 5% of the efficiency is gone because of the variability. And so therefore things that look really very impressive on paper, right? Or in the first laboratory, it's not always good. But it turns out that this type of variability, until they are addressed, there's very little hope that this type of thin film solar will actually be commercially viable. Okay, so that's it. Thank you for your attention. Is this characteristic of crystalline solar cell? Yeah, the one slide I skipped is the one slide you're asking the question about. So maybe I go back in order to respond to this. Let me go back to this one. In principle, the way I described light-induced degradation, right? Do you remember a random network? You have to tie them up with hydrogen and therefore when light comes in, the light dissociates them and all the bonds are again broken. Something that you sort of patched up with duct tape, you know, as soon as light comes in, those are again broken and so again you have problem. For long people thought that amorphous silicon are most susceptible and which is true. That everybody knows that. But it turns out that if you have this boron drop or low quality chakraski crystals, that also show light-induced degradation. But here the boron essentially diffuses around. This is not to do with dangling hydrogen bond. Boron diffuses around and form complexes which acts like trap. So therefore in order to get rid of this, if you can show that this is really related to light by replacing this boron with gallium, in that case you see all of a sudden the light-induced degradation disappears. Or you can do float zone. Float zone is a much purifier version and in that case if you use that one, that also suppresses light-induced degradation. But it adds to cost. So generally light-induced degradation is there also. So the rate is slower and it creates line silicon. Yes. So these numbers I cannot be exact but let me give you a general idea that crystalline silicon is still the dominant technology. I'd say 80% probably is crystalline silicon. In part because for long the price of silicon was high and especially in the last crisis around 2008 or so or 2009, there was a spike in price. So there was a lot of thin-film solar cell companies jumped in at that time. But now it's really dropping very fast. Especially from the Chinese companies, they have very high quality silicon at relatively lower price. But in terms of thin-film, the most successful company is first solar. And that is the only company that can supply now power, that is below $1 per watt. It is more like 70, 74 cents. And they have consistently done so. So when the Middle East I have seen also when China decided to have this 2GW plant in the Gobi Desert I guess. So in that case when they decided, they had to think about which technologies to choose, they chose actually the Scadmium Terrarium based technology over silicon. In part because of price, but there was always a persistent question that will it be reliable for 25 years? And in many places in the Middle East already the things are running for 12 years or so. No problem, 10, 12 years, no problem. So thin-film has hope, but crystalline silicon, you know, the price keeps going down and there will be a strong competition always. So you showed us that nice derivation of the I.D. characteristic and transitive transmission to the left and the right. So when you measure the Zyoth factors and you get out of the Schott's regime, what's the end factor for the Zyoth? Is it one? Tell me one more time. When you measure the dark I.D. characteristic and you show us the Schott's current, but when you get the higher voltages where you just have the junction current, what's the diode factor? So the diode factor I'd assume between 1 and 2 and very robustly about 1.5. 1.5 to 1.8 in that. So it's in between 1 and 2, nothing, not perfect 1 and neither perfect 2. Especially for this structure. And you interpret that like recombination and spatial region that's giving you some Yeah, I think if you do numerical simulation, many features of it will probably, because it's not really like where you can separate out the recombination and the injection over barrier into separate components and add them. Because at every point, just like the transmission picture shows, that at every point you have a probability of recombination and probability of going up in the barrier or down in the barrier. So at every point, so when you add them up, unlike a PN junction, where it's very sharp and small, when you add them up, you expect more complicated N values. Let's assume. So many of the general ones, I have heard it's not installing the other solar panels. What is this all crystalline or the thin film at the consumer level? So which of the... Let me see whether you can immediately tell which one it is. Because they look different. Do you see a difference in these two pictures? Which one do you think is crystalline? Of course the answer is given. The ones that you have wafer found like this, because you've got that structure and then you've cut it into slices. And so this would be crystalline, crystalline silicon. In many cases, the thin film will look like this, because it's roll to roll, or essentially vacuum deposited. So it will look continuous. You cannot see it here, but there are striped lines, which allows them to connect it in series. So if it looks beautiful blue, then you know that's the silicon solar cells that you have. If you have it like this, then continuous, then you know that this is... Is that right? Generally, that's more thin film structures. Yes. So if you find that basically, for the crystalline silicon, it has about 50 times something like that before you get much of a life. And you said that the reason why we are doing these thin silicon is because of cost. But I was wondering, I mean, we've seen some visitation in the past because of these thin silicon. So when you compare the current and how it comes out of these soil, I mean, does it really save that much money? And the other question I had is also, you know, you've also learned that silicon is basically, you know, most of the solar cells is based on silicon, right? So why not use a head infrastructure where you can essentially harness much of the life from the sun? You know, does it cost really...is it that big that, you know, we have these solar cells because they're cheap and not, you know, although people can use a head infrastructure to treat like that? I mean, the main reason for thin film, as Professor Lundstrom also mentioned, that it can absorb light within a very short distance, right? So gallium arsenide will be a few microns, and it will be all that you need in order to get 90% of the...1995% of the light in. Now, remember the 50 bounces that I told you about, it's not on all wavelength. The wavelength that are close to band gap, those are the most difficult to absorb. And they don't want to get absorbed. You keep bouncing, and you have to wait a long time before they get fully absorbed. So the main thing is that you require a smaller amount of material in part because you have easier absorption. It's the direct band gap material. Most of them thin films are in direct band gap material, so therefore it absorbs things easily. Now, in terms of cost, if you go to one of these company websites and ask them that what is the install cost for some of these companies, it's actually as high as $5 because although in specific numbers you get in various industry positions is one thing, but essentially a 200 watt module will cost you about $900 when you buy this thing. So silicon is already very expensive. Now, the price is coming down, that's true, and the thin film is in that especially for cadmium telleride, it's significantly cheaper. Now, whether all thin film can go there, whether you can have a reliability problem or not, that will sort of balance the cost, that's the second issue. I'm not sure that question can be definitely answered today. Which one is the ultimate winner? But very difficult to count out silicon. Silicon, despite all sorts of problems, is very resilient. We hear a lot about grid factory, can you comment on number one on that? Number two, solar is ultimately a DC power source. What's the extra cost here being in order to generate the reactor power to run the grid? Very important. So anytime people talk about solar energy, they talk about the cost issue is a very important one. I mean if you have gas based power generation stations, it's about what do they call it, maybe four or five cents per kilowatt hour, something like that? One tenth of that. So it's very inexpensive, darts cheap, because in some way God has already put the solar cell prefabricated the energy and put it in the gas. So you don't pay anything for the equipment to generate the gas. So therefore you are just bringing it up and using it. So the question is that whether you can get below ten cents, that's the goal per kilowatt hour in terms of energy use. And it's very difficult. Very difficult. People are trying all sorts of ways to get the cost down. But even the best effort, it's not really without incentive from the government that you have tax breaks and other things. It's very difficult to see that how it can be directly competitive even next two, three, four years. So grid parity is an important question. And the reactive power in terms of variability, how do you do load balancing? All those things are very important questions. But what people from sun power, what they argue that this variability, because sometimes there is sun like now, and sometimes there is no sun in the night. And then the way that the question is whether that would be a significant variation. And if you have that level of variation, generating power is very, very difficult. What they show that if you connect a lot of solar cells, solar power generating units into a grid, the variability is essentially all eliminated. Because different regions have different time pattern for the solar cells. And then they show if you combine with the wind power, and this is all these things about smart grid technology, then essentially you can have a very stable power source. So there's a lot of work going on in terms of this circuit or the power transmission system aspect of it. But reactive power generation is to be done even when you have regular power. Exactly. That's right. So the cost includes the power. So this is the power electronics cost. So this is already included. When sun power or somebody will sell you a module, the module has everything, all the inverters, it has all those pieces in. You have to come in when the person comes with a truck. All they do essentially, these are contractors. They don't know anything about solar cells, no shadow degradation, nothing. So they essentially, they should be able to hook it up without asking too many other questions. So when you buy that system, that thousand dollar has all those pieces in. You don't have to buy them separately and install them. Right? Yes. If I have a product, I know the water like this, the inverter part of it, not this one. No, no, no. The inverter is very inexpensive. How much did you say, 22 cents? Yeah, I don't take your money. It was like 20 percent. You see, very inexpensive. Those are normal silicon technology. I mean, you don't really have to, national semiconductor produces those parts. Many companies produce the parts. That's not the problem. The problem is it fails. After three, four, five years, it fails. So you put it on the backside. But you cannot, and you try to encapsulate it. But still, if you leave something like this over a period of time in the exposed air, think about it, what would happen if you lost your cell phone and the cell phone was sitting in front of the park, then it's not going to survive very long. So that is what we are asking that park and what to do. Very cheap, but very problematic. Okay. So, yeah, thanks.
 Nanohub.org  Online simulation and more for Nanotechnology.  You can follow along with this presentation by going to  Nanohub.org and downloading the corresponding slides.  Enjoy the show.  You can hear in the back.  What I am going to talk about today is thin film solar.  If you think about the summer school, it has covered a lot of topics.  You must be tired by now.  But essentially it has two broad groups of things.  One is for transistors, spintronics, memory elements.  These were for information processing.  That is, if you learn those things, you can get a job at Intel.  On the other hand, the other topic, set of topics,  were the grand challenges of today, which is not information  processing necessarily.  This has to do with energy conversion.  We have talked about thermoelectronics and PV, the photovoltaics.  Of photovoltaics, there were five lectures.  These five lectures had to do with the basics of solar cell.  By now you know that.  What this is all about?  That if you have a semiconductor and light comes in and shines on it,  it may come from the sun or it may come from some other source,  essentially a photoelectric effect.  Then you essentially have electron hole pair generated.  Professor Lunsstrom explained that.  When you connect a load to it or a light bulb to it,  the light bulb lights up without having to connect to your power socket.  That is what the PV is all about.  Then once that was done, the very basics,  you learned a lot about crystalline solar cell.  You learned about how thick this region has to be.  Because it does not absorb very well, you remember that.  You also remember perhaps that 80% of the market today is  due to crystalline silicon.  This is the dominant technology.  That is for sure.  If that was all there is to it, of course we could go home.  That would be nice.  But unfortunately it looks like that there are in fact very  interesting additional trends.  That counts mainly because of the cost.  Because at the end of the day, when somebody says that your iPhone costs  $300, you say, okay, here is the $300.  You buy it.  But for solar, for energy, it is very cost sensitive.  Because in one hand you have oil and gas, which are very inexpensive.  On the other hand, if the PV people say that it costs twice as much,  but look, this is very environmentally friendly, you would say,  no thank you, I don't want it.  So therefore it's very cost sensitive.  And this is the cost aspect of it.  That makes this thin film organic solar cell, plastic solar cells,  so interesting.  But also the physics is very beautiful.  And I am going to explain to you how that's going to work.  So two talks.  This talk is about thin film solar, the organic photovoltaics.  We'll be the next lecture.  Now, here is a brief outline.  I'm going to talk a little bit about the background information.  Give you some background information.  But I'm going to talk a lot about photo current.  And this aspect you might find a little bit interesting.  Because Professor Lundstrom, one thing of all the things he said,  one thing you might remember, he said, that when he was a student,  he wanted to go home and explain to his wife that why he doesn't  work on the photo-generated carriers for the solar cells.  Because it was flat.  And so therefore he could focus on the dark characteristics.  So if you remember one thing from this lecture,  that is what you should remember, is that he cannot give this  explanation to his wife for thin film solar.  So I'd like to then explain a lot, explain how this is done.  But I'm not going to do it by using any differential equation.  I'm going to use very simple things that just counting,  counting electrons going left and right.  That's all you need in order to find how this electrons move around  and how these photo currents are generated.  Very simple.  If you just don't fall asleep, then you'll be able to understand them all.  And then I'm going to talk about dark current.  Very briefly, Professor Lansstrom has already explained to you,  Professor Gray, a lot about dark current.  So I'll not really do much about it.  Instead, what I want to tell you about is that once you put these cells  in a module or connect them up so that you can use them,  variability and reliability becomes fundamentally important.  And this is the aspect that you didn't really hear about much.  But at the end of the day, it matters a big deal.  If your solar technology lasts for 20 years versus 40 years,  because it's free energy, you bought your panel for $1,000,  let's say from Sun Power, and that's your initial investment.  If it gives you free energy for 40 years versus free energy for two months,  it makes a huge difference in terms of cost per kilowatt hour,  the amount of energy.  So reliability is very important.  And I'm going to explain to you how that works.  So very briefly, there are different types of solar cells.  You have heard about this crystalline solar cells.  And you may remember that the green N plus region is very thin.  You remember that we cannot make it any thinner,  because otherwise there will be big resistance drop when you want to  collect it to the electrodes on the top, this plug-like structure on the top.  But the main problem is that the crystalline silicon, the P region,  the P base is very thick, 200 micron.  And why is it 200 micron?  It's because it's a very weak absorber, indirect band gap.  That's what crystalline silicon is.  And so therefore it takes many, many bounces on the order of 10,000 micron  before everything is absorbed, 50 bounces maybe, before it's absorbed.  So it's a thick material compared to what you could do with gallium arsenide.  Or any other thin film technology.  For example, you could take a PIN structure.  For example, based on amorphous silicon, where you have a P region,  that's the yellow, and the intrinsic I region, that's in the middle,  250 nanometer.  You see, on one side you have 200 micrometer,  in another case you have 250 nanometer,  and then you have an N region on the bottom which is green,  and the top and bottom you have the two electrodes.  You see, you could put the whole thing just almost inside the N plus region.  It is that thin.  So maybe just the device itself without the encapsulation or anything,  maybe a factor of 500 smaller, thinner, compared to the other one.  So you can immediately see that if this material, the two things cost the same,  then this would be way less expensive compared to the silicon.  Of course, there are, this is not the only version.  The other version involves metal insulator metal,  and that will be the next lecture.  And that has to do with flexible organic solar cells,  so we'll talk about that.  Now you might ask why is it PIN here and PIN over there.  In this case, in both cases you have a given electric field,  and once you have that electric field, the electrons can be pulled out  on the both, from the both contacts.  So you can have either type of the technology depending on your need.  So what I'm going to tell you about is this, basically the middle one,  PIN structure, thin film amorphous silicon, and explain the physics of it,  and how it differs from the crystalline silicon.  It's actually very different.  But of course, you will see how this comes about.  But of course, this is not all great news.  If the thin film was so good, why is the market 80% crystalline silicon?  It's because many times the material may not be abundant in the earth.  You know, silicon is everywhere.  So silicon in various forms, it may not be, many of the things may be very rare.  So once lots of people start buying it, then you, they mean the suppliers  may not be able to sell it.  Some of them are toxic.  So there are other problems, and generally the substrate may be more expensive.  So although you are using far fewer material, far smaller amount of material,  it may be more expensive per unit.  And so therefore you may not end up really saving a lot of money.  So if you make a quick comparison, and this is by the way,  I take no responsibility for these numbers.  The numbers are very sensitive.  But my main point is to show that for crystalline silicon,  the material is about like say $200 per meter square, the process cost $100.  And when you put them all together, it is cost per watt without installation.  It's about close to $1.75.  And compared to that, other thin film technologies generally typically have lower cost.  But in addition, there is another problem that generally crystalline silicon is heavy  because you have to put it on top of some sort of substrate.  If the rock has to come to your home and then put it up,  you cannot go to Walmart and buy a plastic or cheap wafer and bring it on yourself  and put it on the top of your roof.  Not that you should try.  But in principle, that would make things much more expensive.  It turns out one-third of the cost of PV is the installation cost, half or one-third.  So therefore this silicon technology generally is very expensive.  So a thin film PV might actually be much better.  So let me explain to you then what the physics is.  You see it's thin.  So when things are very thin, lots of trouble can happen  because things can short from one side to another.  Thin is not always necessarily a good thing.  So for example, this region, do you remember, in the middle region,  this I region, this we said, this is on the order of 2 to 250 or 300 nanometer.  Very thin.  Then doping regions, the dope regions are on the order of 10, 20 nanometers.  So again, very, very thin.  This material often grown in low temperature.  And so therefore there can be a lot of other problems also.  For example, one problem it could be that since these regions are very thin,  the contact might diffuse inside the device.  Now you might say, well, a little contact diffusing inside,  what harm is it going to do?  It's a very big, a little bit.  But what I'm going to show, this is like impact crater,  that it may enter small, but its impact will be huge.  A micron size hole can essentially drain away carriers from a millimeter size region.  So we are going to talk about that.  And then, of course, many of these materials have horrendous grain boundaries.  Nothing beautiful like silicon, where all the atoms are like an army, right?  All regimented and all, every atom in their right place.  Instead, here you have grain boundaries going from left and right.  And the question you ask yourself, how in my life I'm going to actually solve this problem?  Of course, there is a simple answer, is that we can close our eyes,  forget about the defects, and put some effective parameters in the silicon simulators,  and whatever result we get, we don't tell anybody else and go home.  Now, this is the four things I'm going to talk about,  but the fifth thing is equally important,  then that has to do with series connection.  Solar cells are always series connected,  because each cell is like one volt battery, just like in a toy.  Many times you have to put three, four batteries in order to get the right voltage.  In the solar cell, many times you have to connect, let's say as much as 100 of them in series,  so that you can get 100 volt out, if it's one voltage.  And the way people do it, that if you have a glass substrate,  if you put the mirror, TCO is type of a electrode, highly conducting electrode,  then you put the material that you want to put in.  This is your solar cell material, amorphous silicon, cadmium telleride, whatever.  And then you do the laser stripe, open up these holes,  and then put the second contact on top of.  Now, do you see what it just did?  Instead of having a simple PIN junction, this successive layers of things,  now currents are going to do this, that it will enter through one end,  go up through this contact, come down, so this is one solar cell,  then go down, come up, and again come down, that's another solar cell.  And so, although initially you had thin planar structure,  you have just by striving and connecting it up properly,  you have just made it a series of solar cells, which are connected in series.  Now, this is very important, many times you will see if you Google,  I hope nobody is doing it now, but if you Google, then you would see many of these lines in here,  and essentially these are scribe lines, essentially that are showing you that,  that's how you cut it up, so that when you string them together, it gives you a bigger voltage.  So these five things, we are going to think about as we go through the rest of the talk.  So now let me remind you, let's go back, take a step back,  that what did we learn from Professor Lundstrom talk?  I'm sitting in the back, so this is what I learned.  Essentially what he talked about, that when a photon comes in, a light particle comes in,  if it has energy larger than the band gap, then an electron hole pair will be generated.  The red electron will go to the left contact, the white hole will go to the right contact,  and they will eventually go through the load, and therefore whatever is delivered, the power there.  And you remember what we talked about also in that lecture,  that there is something called a dark current on the right hand plot,  there is something called a dark current which is essentially the diode,  and you can see on the bottom that we have a symbol for this diode, and that is ID,  we have written ID next to it, so that was the dark current,  and he essentially explained to you that the photo current is essentially a constant in the graph,  it's a photo IPH, that's a constant, so therefore it's like a current source,  and when you combine them together, the total current essentially is a dark current,  and the photo current together, and that's what you get.  And so the voltage at which there is no current is called open circuit voltage,  and the point where you want to operate is the maximum power point, how do you find that point?  Well if you remember from your circuit days, that if you have one element that has this particular IV characteristics,  and if you have a load line, then wherever that one intersects that red point,  that is where the power would be.  Now what is the maximum power?  Maximum power would be whatever corresponds to the red point,  with a voltage that is little less than VOC,  and photo current which is little less than the ideal photo current, the short circuit current.  Now solar cell people are lazy, so instead of really calculating,  what they do is that they say okay, this is my original box, VOC multiplied by I photo,  but of course that box is too big, what I should really have calculated is green box,  now I don't know really what this green box to this one ratio is, so I'll call that field factor,  and so that is what they will, this is how they will write this formula,  that this IPH multiplied by VOC is the bigger box, and the field factor is the factor that you multiply  so that you get the green box, which is the power that you really want.  Now this is ideal solar cell, because in this case, you can take that current,  I'm sorry, you can take essentially this current,  and add it up to this dark current to see at what point the VOC is,  so you can calculate a VOC, you can get a maximum power and you will be done,  but it turns out for solar cells, this is not really, for thin film solar,  that's not really the full model or the full story, what you need is two more things,  one is this extra current source that only occurs when there is light in there,  so instead of having a voltage independent current source as you heard before,  now I have this combination which depends on voltage,  and I am going to explain to you very simple, there's nothing funny about this whole thing,  and then there is another thing in solar solar, in particular called this shunt,  now somebody asked professor Lansstrom the other day that what is shunt,  and he mentioned that there could be pinholes and other things,  didn't really go into the details, I want to explain to you that this can be understood very simply as well,  so we will talk about that, so let's, the main point I want to make is that superposition doesn't hold,  so therefore if you see a thin film solar, most of the time you shouldn't try to use  copy textbook formula from your undergraduate textbook and try to apply it  because you may not get right result, okay, so let's get started in terms of photo current,  now before I get started let me tell you something simple,  and this is something you understand very easily, no differential equation or anything,  just look at what the electrons go, assume that I have a semiconductor and then two metal contacts,  and assume that I have six photons coming in, and that has generated this six red electrons and six white holes,  now for some reason, and we will understand that reason later,  four of the electrons are going to the left and two of them going to the right,  and similarly two of the holes are going to the left and four of the holes are going to the right,  how many currents, how much current do you have, now you have no differential equation to hide behind,  so therefore you must be able to answer this clearly, the answer obviously is not 4q,  because you could look at this electron and you could say four of the electrons coming out,  so I will say it's 4q, not really, because the two of the holes are coming out through the wrong contact,  so as soon as this four comes out they will recombine and only two can go to the load,  and so therefore the current will always be the current at any given junction,  the electron current to the left contact minus the hole current to the left contact,  and when you sum them up that's the total current,  and so in this case I have only 2qV0, V0 is the velocity at which electrons are going,  now of course you could write it slightly differently,  you could say that these four are really out of the six, because six photon came in,  so four over six, so it's really the fraction of the total that is going to the left contact,  and these are the ratio of the probability that you will escape to the left contact,  if I have two doors then the probability that I will escape through that one versus the probability  that I will escape through this one, so it's normalized, so that's it,  so you have, so I will write a formula which is something like this,  which is generation rate, probability that you, the electrons escape through the right contact,  the proper contact, and minus the probability that the holes escape through the wrong contact,  and that gives me the total current, now what happens if you have recombination,  remember if you have defects and all, you might have recombination,  we don't have to worry too much about it, because that's really again very simple,  see here, I have three coming out, two electrons, two holes coming out through the wrong contact,  so the total current is just one electron, that's going around the circuit, right?  Now what happened all through all this recombination and all, why did I forget them?  Not really, I didn't really forget them, because if you think about it,  that if I wanted to write what fraction of the total electrons are getting out,  then I would write three divided by six, three actually went out through here,  and six were generated, and so when I write the ratio, my recombination is sitting right here,  because the total rate had the recombination in there,  of that only a fraction escaped, and so therefore these two formulas,  if you sort of remember that sort of makes sense,  then you will see how very complicated things will get very very simple, as you will see.  Now before I get there, let me tell you about this semi-classical probability of escaping through the contacts.  So for example, if you have a barrier, and this is something Professor Lunsstrom also mentioned,  of a certain height, and if you wanted to know what is the probability that the electron will escape there,  you will say that this probability is proportional to the height, right?  Proportional to the height of the barrier, if the barrier is big, nobody is escaping,  and if the barrier is very low, everybody will go away very quickly, so that's something you know,  and this KT tells you the temperature, so the more the thermal vibration is,  easier it is for it to jump over the barrier.  Now you can of course make the same argument if you had a left contact,  if you have a device in the middle, I am trying to see whether how can I get to the left,  and how can I get to the right, because then I can do something called a partitioning.  So in general therefore, what I do that if I have two barriers, then essentially depending on the barrier height,  the left and the right, I will escape with two different probabilities.  Now this turns out to be very important, as we try to understand why thin film solar is different from crystalline silicon.  As you will see next.  So assume this very simple case, you remember this simple formula from before,  that this is if I injected a particle here, a photon has just come in, generated two electron-hole pair,  one electron-one-hole pair, and I want to know the fate of this electron.  What is it going to do based on this formula?  Well it has a probability that it will escape through the left contact,  but it has also some probability that it will escape through the right contact,  and so you can see this Ln and Rn are the probabilities for the electrons  escaping through the left contact and the right contact.  And similarly for the holes, you also have the same probability, it can go to the left or it can go to the right.  Now if I ask you, what is the probability of electrons going to the left?  Well in that case there is no barrier, it's just fall, it's like water flowing down a hill,  and in that case, I will, there is no barrier, so whatever is the flux, it will just get out from there.  On the other hand, if it wants to go through the other contact, let's say this is a very, this electron likes torture,  so it wants to go out to the wrong contact.  Now if it wants to, of course, it can, if the electric field is E, and if this distance is X and whole thing is W,  then the barrier it has to climb is E, W minus X over KT, that's the barrier height, right?  Electric field multiplied by the distance, and so that's the barrier height.  All you have to do is to insert these few quantities in here, because you see I have everything,  gamma L, gamma R, I will just put it in here.  By the way, why do I do this integration? Because of course sunlight is going to come at every point, right?  It's going to generate at every point, and so therefore when I want to know the fate of all the electrons that are in there,  I'll have to integrate them up.  Now if I do that, I'll put this formula, you recognize this one is here, this V naught is here, you put them in,  and very soon you get a formula which looks something like this.  Now you, this integration exponentian on all, if you don't remember, you can Google it, the integration formula,  but this is something that is that you'll get.  And in fact, it turns out that this formula, it is originally derived in 1981, it probably takes eight pages,  six to eight pages, with lots of differential equation, left and right and boundary condition.  At the end you get this, but you do not know what happened to the electrons in between.  You are so lost in differential equation that the electrons here have forgotten about the electron.  But you can see very simply what these electrons are doing in here.  What this is all about is essentially that if you are close to this region where electrons can skip,  in that case there will be a loss of certain number of electrons will not come out through the right contact,  and so therefore you will have a reduction in current.  Your current cannot be voltage independent because you see as I am pushing this voltage up and up,  right, biasing it, as I am pushing it up and up, the fraction that will begin to go through the other contact,  the wrong contact, their fraction will increase.  And eventually there may be a point, and as I will show in the next slide,  there may be a point where this is flat, half going the wrong contact, half going the right contact,  and essentially you have zero current. Voltage independence cannot hold in this particular case, right, it is simple.  So you know why this voltage independent requires a lot of discussion,  but why it is voltage dependent should be simple to understand.  So you will think that well this is a simple formula, may not be very good, who knows.  So you have this formula, and here is this green line, and I am going to talk about this red and the,  this is the dark current, the blue one is the dark current, red is the total current, this is photo current.  This is not only not constant, it is going through zero and going to the other side, flipping its sign, right.  Photo current is not only constant, but it is changing its sign.  And you can, if you put it in a numerical simulator like ADEPT or any other device simulator that Professor Gray talked about yesterday,  the results will be right on top of it, and so therefore you can trust this result, and you can easily see the physics.  What happens here is that you have in the right, at low voltages, this partitioning occurs,  so most of the current is going in the right way, the way you want it to be.  When the voltage is VBI, the built-in voltage, the partition equally, the current is zero, right.  So this point is current is zero, and if you flip it on the other side, then of course the currents are going to go the wrong way,  instead of electron flowing this way, now the electron is going to flow to the right side, and so the current is going to flip.  That's it. So voltage independence doesn't really hold in this case.  Now how do you want to make it if you really wanted to make it voltage independent, then what you could do,  you could put a big barrier here, and put a big barrier here, so that nobody is allowed to go through the wrong way, right.  You close one door. If you close one door, then even if I try, I cannot escape through the wrong contact.  So in that case, of course this formula immediately knows that that has to be voltage independent.  Why? Because the right contact, this would be zero now, because right contact you cannot escape anymore, that has been blocked.  And similarly, this holes, this cannot also escape, and so therefore this will become one, and when you, this is very happy integration,  zero to W of dx multiplied by one gives you W, and that means every photon that has been, every electron that has been created,  will come out through the right contact, independent of voltage.  Now under this circumstances for sure, you are going to have a voltage independent, voltage independent photo current.  Okay, so blocking layer is very good. Yesterday you heard about this electron mirrors and other things, right, in the lecture, if you remember.  And so this blocking layer is therefore very important. Now only if there is low recombination, as I am going to tell you.  Now one thing I have not been telling you so far, is that this material is full of defects, right.  Remember low temperature deposition, all those grains sort of crisscrossing, full of defects.  Now when you have things full of defects, then life is a little bit more complicated, because even in that case what I want to show,  that even if you put a blocking layer, it is not going to help you in terms of making the current voltage independent.  You know I am very trying very hard to make it voltage independent, it does not want to stay that way.  Why not, what happens in this case, why can I not make it happen?  Because as soon as an electron hole pair is generated, this electron hole pair is going to randomly walk around.  It is very soon going to find a hole that has also been generated, it is going to recombine.  How far is it going to go before it recombines?  Well you would say this time to recombination that depends on how many defects I have, and the velocity of it.  And the electrons movement will be mu multiplied by the electric field, that will be the velocity, mobility multiplied by the electric field.  And it is going to stay alive for tau, so that is the amount of distance over which it is going to recombine.  And if it recombines and if it does not come out, is it as if it were never generated, right.  Because I do not really know whether it was generated and then recombined or essentially it generated, it sort of, it was not generated to begin with.  Okay, again this formula, would this formula work?  Well generally it should work because you can see how this formula should work.  Gamma LP the whole zero, because I have put a blocking layer.  This is zero, gamma Rn is also zero, because I have put a blocking layer, so that is not going to go anywhere.  Now the fraction of the electrons that is injected at a given point and comes out because just going through this recombination process is this ratio.  Gamma ln, gamma ln divided by plus gamma R.  You know, let us say the chances of going from here to over there, there are a lot of shards between here and over there.  And if I have to swim through this, my chances of getting out on the other side is exponentially suppressed, that ratio.  And so what I am going to put is zero to Wdx and this ratio is e to the power x over LC, LC is that distance.  Because that is how far I can go before I am eaten up.  So I can integrate this simple thing up and again I can write the photo current.  Again, if you look at this paper, the original paper which derives this, that will be another six page.  But this is essentially, the essence of it is just this, that you cannot get out without getting through this recombination processes.  So therefore you can rewrite it a little bit.  Qgw was my total generation, voltage independent, right?  Independent because if I could collect everything, independent of voltage, that would be the thing.  But of course this blue one is telling me what fraction of it is recombining inside the device before getting out.  And so therefore this whole quantity is essentially this voltage dependent current source.  Again, in the presence of recombination in this region, the current will be voltage dependent.  Now, if we understood so much, now it's good to go back and ask the question that what happened to the crystalline cell then?  In that case things were much nicer.  So what happened in that case?  Well, you can easily see what happened in this case and why if the quality of the material is bad or poor, many of the crystalline silicon, if you want to pay less money, then not the champion cells of Martin Green.  But if you want to do a commercial cells, their quality is generally lower because that's less expensive.  So in that case, many times this barrier may do you no good, very counterintuitive why barrier wouldn't do any good.  Again, this formula will tell you that why the barrier doesn't do any good.  Look at this formula simply says that all you have to keep track of is the number of electrons that are going, getting out through a junction.  So I have the electron current moving to the left and the whole current moving to the left.  But I have a potential barrier here, it's a p-n junction.  So my whole current is essentially zero.  The amount of electron that can get out through that contact.  So if that is the case from the previous slide you remember, so it is as if these electrons are getting out through this region and that's what I showed in the last slide.  And when you put this thing in, put this same formula, copy this same formula, you find that only the electrons within a diffusion length away from this region,  only those will be collected, rest of them will essentially recombine here.  Now you may say that let's take out this potential barrier, but this barrier must be doing me some good because after all here is sort of reflecting all the carriers,  electron mirror, this should be doing me some good, but actually it doesn't do any good because you can see for the following reason.  Because if you remove it, of course these electrons will be coming out to the wrong contact.  But because the electron number is a little bit less now, the recombination of the holes that are coming out here will also be a little less.  And so therefore what's going to happen, more holes are going to come out here.  And when this current is added to that current, it will show there is no difference whatsoever between these two cases.  Because who cares whether you recombine right here or you recombine two nanometers into the contact, there's no difference.  Wrong contact, wrong recombination, so it makes no difference.  And you can, this seems very counterintuitive, the blocking layer in one case whether it makes no difference.  So in one case you see this is the electron current, the red one and the blue one is the hole current.  In the other case, this is the electron current through the right contact, this is through the wrong contact, it's going the wrong way.  But what will happen, the hole will also escape in a larger number and the total current will remain exactly the same.  So this sounds as I said counterintuitive, but it will turn out that this is exactly consistent with numerical simulation.  You can see that this is the current, electron current going out through here, the hole current is very small.  And then if you remove this blocking layer, there will be a spike in the hole current.  I'm sorry, this is spike in the hole current, but there will be a little bit negative electron current and when you sum them up, the total current will be absolutely flat.  Total sum of the current will be absolutely flat.  So again, as I'm saying that this view, I haven't solved a single differential equation, but you can see most of the things that would be non-intuitive will come out if you just think about which contracts or which doors the electrons are coming out through.  Differential equations are good capabilities, but not necessary.  And if you combine, I don't want to go through this point too much, but if you combine, if you have both recombination and electric field, if you have this, don't put a blocking layer,  don't want you feel to complicate your life without putting the blocking layer, the life would be a little bit more complicated.  But once you have done this, your numerical simulations and this simpler form, most of the time will give you good results.  I'm glossing over this point a little bit, but it requires a little bit more discussion.  So long you get the basic idea, I think that should be fine.  These numerical simulations are the symbols and then the analytical formula is essentially this green line.  So you can see that how accurate this simple formula.  Okay, so that's all I had to say about photocurrent and the bottom line is that you cannot tell someone that current is flat and therefore you're working on dark current.  All right.  Now the dark current and very quickly, the dark current, I'll just show you two slides.  This has some math.  Just want to show you that if you do this partitioning argument, the dark current will also come out, right?  And all the features will come out, right?  But I don't want to go through too much detail because I want to show you a few more things.  So if you wanted to know the dark current, no light generation, then again you will do the same thing.  You will calculate the electron current by taking, looking at how many electrons you have on the left side and how many of them can go to the right hand side.  And similarly, you will see how many electrons on the right hand side and that I should have put a R here, sorry, gamma R and gamma L.  And then correspondingly, how many electrons are going to the opposite side?  So divided highway or at least two-lane highway that professor that I talked about.  Remember that there was these two fluxes and you want to subtract them essentially to get the total flux.  Now at zero bias, this current must be zero and if this current is zero, you can get a ratio of gamma L to gamma R because that's the only unknown.  And if you do that, you remember the probability of going from the left to the right is exponentially suppressed because there is a big barrier here, right?  Big barrier, p-n junction barrier and so that is exponentially suppressed.  And correspondingly, this one on the other hand, if somebody is trying to come from here and go to this side, of course there is no barrier.  So that's one.  And you can just insert this in this formula.  Okay, so by the way, so if you have this gamma L and gamma R, put it in this equation that will tell you this concentration on the left hand side.  How is it related to the concentration on the right hand side?  Now do you remember what this relationship is?  This just turns out to be the law of the junction.  Essentially telling you that two sides, the electron concentration are exponentially off and equal to this VBI or whatever the barrier is, barrier height is.  Now once you insert these equations, again gamma R and gamma L is the same.  You insert this equation under bias condition.  In that case, this will remain the same, the gamma L0, n L0, so you take this expression in.  And the gamma L now, the barrier is a little smaller because you have applied a bias and it is smaller by V, minus V, VBI minus V,  because by applying a voltage V, this has been suppressed.  So you do a few lines of algebra and you get an expression here.  And that expression tells you essentially that what is the dark current for a PIN structure like this.  And if you want to get the total current, you can just sum up the electron current and the whole current and you can get it all.  So again, no differential equation, but you can see that the final result comes out reasonably okay.  And this I0 you have seen many times, that's the pre-factor that we just use it as a constant.  So now the question is, my dark current should look like this.  I have a beautiful formula on when I'm applying a voltage on the positive side, it should go up exponentially.  And you remember why this is saturating off because of that series resistance and other effects, right?  That's why it saturates off.  Now when V is positive, it goes up exponentially.  When V is negative, what should it do?  It should be minus QV over KT and that will go to zero very rapidly.  And so if I plot the absolute value, this height will be equal to I0.  This is what a textbook will tell you what is supposed to happen.  But if you go and make a measurement like a show rough dead, then you are going to get something which doesn't look like a solar cell or a diode at all.  And this is how it looks like.  Now I do not know whether you have made enough measurements.  When we were younger in college, we had to make a lot of diode measurements.  It used to have these two pins or three legs, transistors.  We made a lot of measurements and I have never seen in my life that the measurement, a diode characteristics looks like this.  First of all, look at this.  This is supposed to go down exponentially.  It hasn't.  Then it has these two wings which looks like a butterfly wing, completely symmetric with respect to each other.  Now if you do the theory which I do it so beautifully, partitioning and all, and then try to compare with experiment,  you will see that you are in for a big surprise.  And because this doesn't really look like a ideal diode characteristics at all.  So what is this?  So this is something I want to talk to you about next.  And this has to do with this contact diffusion.  What the contact diffusion will essentially do in a summary, that originally I had a PIN structure.  The contact diffusion in this place localized, it will make this region P type because aluminum when it diffuses in,  that will make this convert this region to P type and therefore this region will become a PIP structure.  It will not remain a junction anymore.  And in that case, those butterfly wings will come out of this immediately.  Let's see how it works.  And in fact, there are experiments in which you can find out this dark spot, this light spots correlating to this shunt regions.  So how does it work?  Well, if you had a P region which sort of diffused through this, through this n layer, then you can see you do not have a diode anymore.  What you have is a PIP, a shunt region where the current can leak through.  At low voltages, what's going to happen?  Most of the current is going to flow through this low resistance structure.  At high voltages, of course, the diode will turn on and then everything is fine.  So at low voltages, this is the butterfly wing associated with this region.  And at high voltages, of course, the, finally, the diode behaves like a diode and its value generally comes up.  The reason being this is a much larger, larger diode.  These are tiny, few tiny regions.  So therefore, it can be higher only up to a certain point. Beyond that certain point, of course, this larger structure will show up.  Now, this is a good theory, but of course, how do you know these theories at all, correct?  Or are you daydreaming that if I had this type of thing, that would be very good to have.  But you can go and try to explain that how this works.  So what the current transport through this structure, if it's a PINP structure, then I'll not go through the details,  but you can again calculate the current through a structure like this.  And that current, I should have a P here because whole, primary, but this is equally well, works for electrons.  So if you had just electron current or if you had the whole current, then this would be proportional to the electric field, majority carrier region.  And once you solve for this, and I'm not going to go through the details, but this is something in the notes,  you can go through essentially if you combine this transport through this region with the Poisson equation associated with these charges.  And then solve what you're going to see is that the current in these tiny structures is going to have a very special property.  It is going to go as square of the voltage and be inversely proportional to the thickness of this region, I'm sorry,  thickness to this region to the power cube.  That is how this particular current is going to behave.  The butterfly wing is going to have this property.  So if this theory is correct, this is how this component of the current is going to behave.  In general, of course, if you do generally this exponent is between 2 and 3 and this could be on the order of 3 to 4.  So essentially that is what we'll be looking for.  So there are more general theories, but essentially we're looking for whether the exponent is between 2 and 3 or none.  And this is the butterfly wing.  This all things are about butterfly wings because you can see that if you take this two points, take a point here and take a point here.  And take its ratio.  If it really was that conduction, then it will ratio be absolutely equal to 1.  So you can see it's an exponential curve and there you are taking two linear quantities on top of each other and dividing them.  And when they become one, you can realize that how close to perfect this butterfly wings are, this shunt conduction are.  Then you can also look at the temperature dependence, but most interestingly you can look at the nonlinearity of this shunt between 2 and 3.  You can see and this is about close to 60 devices.  So it's not just one or two accidental happening.  60 devices all showing essentially the same characteristics.  And here is the 1 over LQ.  I believe this is also 60 devices, right?  Or is it 400 devices?  280 devices.  So this is based on 280 devices with different eye layer thickness.  This is the region in here.  Again you can get approximately a 1 over LQ type dependence showing that this type of shunt conduction is parasitic, but at the same time can be accurately and quantity described.  Okay, so I'm coming towards the end.  I have a few more slides.  So we saw voltage dependent photo current, which is good.  We understood the shunt which people say, well, I really do not know what it is, but essentially say it could be a linear thing.  It could be a nonlinear conductance, but you can clearly see how to think about the physics in a consistent way.  Now let's talk about the final piece and then we'll be done.  The final piece has to do with grain boundaries and how current these grain boundaries affect essentially the response of it.  And it will be very brief.  Assume, and what I want to show you here that a small defect, a micron size defect can take away a big region and make them ineffective.  So sort of take them out of the circuit and how it does so.  It's like the example would be like a sink, you know, in a faucet or in a tub you have the sink and the sink or the opening in the bottom.  And that may be a very small space, but the amount of region it affects is significantly larger.  So that is what I'm trying to do through all this equation.  That's the goal I have.  Assume that you have a solar cell and I have divided the solar cell into many pieces and one of them has grain.  Now that means all the solar cells have the same characteristics except I have this black ship.  Oh, I should have put, I'm sorry, this is a diode.  So this line is missing, but I have a black ship diode, meaning that that diode has a high recombination.  So I marked it black.  And so you can easily see that if you have a current, I, total current is I, this is the dark current and this is the photo current.  And we say that it's a voltage dependent photo current.  Then I can say that if the, at the VOC point when there's no current flowing in the external circuit, then at that point I is zero.  And so therefore I can solve for VOC because I can put the VOC in here, I can put the VOC here and I can solve for the VOC.  Now what you can do is you can write this I naught in terms of this VOC, you see, because this I photo and if you say this is big compared to minus one,  you can bring this up exponentially on top here and then can rewrite the whole diode equation solely in terms of the photo current.  You see here, here I had the dark current and the dark current of course depends on the recombination.  All I did was to convert it in terms of VOC so that I could write everything in terms of the photo current.  Now think about what will happen.  What will happen if there's one diode that has a lot of recombination or because you have a lot of recombination, low VOC.  If you have a low VOC then what's going to happen that it is going to sink instead of this current going to the useful load to your battery or to your light bulb.  These are all going to supply current and sink in here, right, all of them.  And so the number of those diodes that have been affected, that has been affected is this number.  They have all VOC one and this black one has VOC two lower and if you solve for it, what you find that the number of diodes that have been affected by it is exponentially dependent on the VOC difference.  And if you have a defective region, it will take away a whole set of region around it.  And so therefore, if you all had the same VOC, you didn't have any one that is defective, of course there will not be any penalty.  But if you have one of them bad, that will really cause a huge amount of trouble.  Okay, so now towards this end, let me come to this final thing.  Many times you have seen this beautiful pictures where a satellite is gradually rising and many times there you can see the boom of that satellite casting a long shadow on its wing, on the solar cell wings.  And you may not really worry about it that well, that looks beautiful.  And many times the solar cars are going next to trees with a shadow on them.  And there may be other sources of shadow, some bird droppings and other things and other bad things may happen.  And you may not think or a leaf maybe, that's a better example.  That may fall on top of it.  And you may not really think much about it, but it turns out that that is inside the solar cell when things are in shadow, there's a havoc inside happening inside.  And that's because of this serious connectedness of solar cells.  So I told you in the beginning, right, that was the fifth topic, that the solar cells like this dolphins coming up and going down, coming up and going down.  And these are of course the diode and the current sources, each sort of facing opposite and the continuous current is flowing around it.  Beautiful, no problem.  But think about for a second what will happen when this long shadow of the boom is on the solar cell wing.  Then all of a sudden one of them has just lost his power because that doesn't have any photon.  And so therefore just lost his power.  Now of course everybody else doesn't like it.  They have their current and they want to supply their current.  Serious connected cells, you like it or not, you have to play with this.  This diode has to supply current.  Now in order for this diode to supply current, what is it going to do?  It's going to go in the reverse bias, break itself down and then supply its current, go in date and then essentially make sure everybody is satisfied.  Very simply I can show you through this diagram.  Think about this blue line, blue dashed line, a single diode and then this blue solid line being a set of diodes.  These are in series so therefore your VOC has gotten multiplied by that number.  All of a sudden the middle one loses power.  If it loses power then it must go down, it doesn't have any photo generation anymore.  It must go down and get in the reverse bias, break down and supply the current that the circuit dictates that it must have.  And therefore the remaining two must now give up a little bit of voltage to help it break down even more  so that it can give some current in the reverse bias and your total output will be dramatically reduced.  It's like this, analogy is like this.  Let's say somebody you are in a battlefield, five of you are going in somewhere.  When one person gets hit, it's not that just one person gets hit and the four can run.  You have to carry that person around with you.  So essentially with one person getting hit, essentially it may take out the five person.  And you may be in a very bad situation.  So this is the same issue with the series connected thing and this is called shadow degradation.  Shadow on a solar cell is a very bad thing.  And you can show that as a function of time if you put things in shadow that this is going to degrade significantly.  The dark current is going to degrade significantly because more and it's one of them in the reverse bias.  And not only the output power will temporarily go down, but the whole power, that shadowed one which you are thinking is just resting.  That one will essentially be in the reverse bias and a significant amount of defect will be generated.  Very counter intuitive.  Shadow you think good, it can rest but not when you just take one of them and don't give everybody else rest as well.  My final two slides are on something that is called light induced degradation.  Very important for amorphous silicon. Everybody in the field knows it so therefore you should too.  What light induced degradation is that solar cell when you put it in operation, you don't even have to put it in operation.  If you go to many companies what you will see many times they have a test set up right in front.  And what they are trying to do is trying to show that as soon as you put a solar cell, especially amorphous silicon solar cell in light, it begins to degrade.  And people knew it from 1977 almost from the very early days.  The solar cell degrades very quickly. So the champion solar cells you hear about in the newspaper or in various journals.  Well if you asked about them about two months later what the efficiency is, the efficiency might have been significantly lower.  So this is light induced degradation and it has a very beautiful physics because if you look at the efficiency of the solar cells, they drop as a function of time.  But the most interesting thing is that if you put it as a function of time on a log-log plot then you will see this has an exponent of t to the power one-third.  Time to the power one-third. And this is very strange. We know exponential degradation, we know from circuit and other things.  But a time the degradation that is going as power law is actually very unusual.  So we want to know a little bit about how that happens and with that we will be able to end.  The basic idea is this. You know amorphous silicon is a random, it's not crystalline, it's a random structure.  And many of the bonds are essentially not saturated. And so therefore what happens is that you need to passivate them by hydrogen.  If you bring it hydrogen that sort of ties up and then therefore you can have good performance. But as soon as you put them on the light, inside in light, then these bonds begin to break.  And therefore something that you make easy also breaks easily, right? Cheap things.  And so therefore what happens that this defects begins to come up and as you have more and more recombination centers, your shorts are this dark current begins to increase.  And so you can write a very simple equation. You can say the number of broken bonds is proportional to the light amount of light you have.  But of course once they are broken, some of them can come back and repassivate.  So that repassivation probability is a number of hydrogen that has been liberated and the number of broken bonds.  So you can write one equation like that. And you can write another equation that accounts for the diffusion of hydrogen within this region.  And the dimerization of it. So just two little equations. And if you solve this, remarkably it will tell you that this thing should degrade as T to the power one-third.  It will go as T to the power one-third. And most more importantly it will also tell you it goes as light intensity to the power two-third.  That's also been shown in experiment that it goes as light intensity in a particular way.  So what I'm trying to tell you, although recombination and reliability, this type of thing seems very esoteric. But they are good fundamental theories that can account for it.  Let me skip this one. And then in the final one, so let me tell you that what I told you so far about is all intrinsic degradation.  This is light induced degradation, shunt leakage, shadow degradation, weak diodes.  I just told you all about material degradation. But of course solar cells are not something that you carry in your pocket. You may.  But generally it's not like your iPod where you protect it very nicely. If there's water falls on it, you clean it up.  You don't treat solar cells so nicely. You hang it out there and you get the power out of it.  And when things are left to its elements, then of course a whole lot of other things can happen.  There can be corrosion, there can be moisture coming in, and there can be, and the most important delamination, the layers can peel off.  And so that can, if you are in the middle list, then there is sandstorms and it can be coated with sand and essentially no power output.  So there are a whole lot of extrinsic reliability issues that people also have to worry about.  And the most important reliability concern these days when people say they cannot have a 25 year reliable solar cell is the inverter reliability.  Because you have to put an inverter in order to make the DC output, make it to AC so that you can connect to the grid.  And most of them don't have more than five years of reliability.  So here is a 25 year cell which you have gone away home hoping that this will be reliable and no problem.  Five years down the road, inverter breaks and your solar cell is out.  And so you have to go send someone to fix that inverter.  And that's the huge cost.  So one of the most important reliability is not with solar cell at all.  It has to do with power electronics and power conditioning.  Okay, so very briefly then.  These are my conclusions.  Economic incentives, right?  We want cheap solar cells so that we can have good economic performance.  I told you about thin film PV and its unique features, right?  No voltage independent, voltage independent current.  That's the important thing I said.  Wig diodes, shadow degradation, all sorts of things I told you about.  And extrinsic reliability is of course important.  But the intrinsic reliability people are spending a lot of time on because this is something a good understanding of physics can help you really make a lot of progress.  The things that are outside your control or the control of mechanical engineers, they will think about it.  But we as electrical engineers, the equations that are in our control, we address those.  And turns out shadow degradation, light induced degradation, those we can contribute along.  And this reliability and variability are key concerns.  And this is something, anytime you have a new advertisement, let's say,  that I have such and such solar cell and big news release and all, let's say in technology, MIT technology reviewers and  the most important thing is not to read that part.  The most important thing is to read the comments underneath.  Because in the comments, the real experts in the field, they make most interesting comments about the technology.  And most of the time what people show is their best, highest performance.  As soon as you put it in a module, 4 to 5% of the efficiency is gone because of the variability.  And so therefore things that look really very impressive on paper, right?  Or in the first laboratory, it's not always good.  But it turns out that this type of variability, until they are addressed,  there's very little hope that this type of thin film solar will actually be commercially viable.  Okay, so that's it. Thank you for your attention.  Is this characteristic of crystalline solar cell?  Yeah, the one slide I skipped is the one slide you're asking the question about.  So maybe I go back in order to respond to this. Let me go back to this one.  In principle, the way I described light-induced degradation, right?  Do you remember a random network?  You have to tie them up with hydrogen and therefore when light comes in,  the light dissociates them and all the bonds are again broken.  Something that you sort of patched up with duct tape, you know, as soon as light comes in,  those are again broken and so again you have problem.  For long people thought that amorphous silicon are most susceptible and which is true.  That everybody knows that.  But it turns out that if you have this boron drop or low quality chakraski crystals,  that also show light-induced degradation.  But here the boron essentially diffuses around.  This is not to do with dangling hydrogen bond.  Boron diffuses around and form complexes which acts like trap.  So therefore in order to get rid of this,  if you can show that this is really related to light by replacing this boron with gallium,  in that case you see all of a sudden the light-induced degradation disappears.  Or you can do float zone.  Float zone is a much purifier version and in that case if you use that one,  that also suppresses light-induced degradation.  But it adds to cost.  So generally light-induced degradation is there also.  So the rate is slower and it creates line silicon.  Yes.  So these numbers I cannot be exact but let me give you a general idea that crystalline silicon  is still the dominant technology.  I'd say 80% probably is crystalline silicon.  In part because for long the price of silicon was high  and especially in the last crisis around 2008 or so or 2009,  there was a spike in price.  So there was a lot of thin-film solar cell companies jumped in at that time.  But now it's really dropping very fast.  Especially from the Chinese companies, they have very high quality silicon  at relatively lower price.  But in terms of thin-film, the most successful company is first solar.  And that is the only company that can supply now power,  that is below $1 per watt.  It is more like 70, 74 cents.  And they have consistently done so.  So when the Middle East I have seen also when China decided to have this 2GW plant  in the Gobi Desert I guess.  So in that case when they decided, they had to think about which technologies to choose,  they chose actually the Scadmium Terrarium based technology over silicon.  In part because of price, but there was always a persistent question that  will it be reliable for 25 years?  And in many places in the Middle East already the things are running for 12 years or so.  No problem, 10, 12 years, no problem.  So thin-film has hope, but crystalline silicon, you know, the price keeps going down  and there will be a strong competition always.  So you showed us that nice derivation of the I.D. characteristic  and transitive transmission to the left and the right.  So when you measure the Zyoth factors and you get out of the Schott's regime,  what's the end factor for the Zyoth?  Is it one?  Tell me one more time.  When you measure the dark I.D. characteristic and you show us the Schott's current,  but when you get the higher voltages where you just have the junction current,  what's the diode factor?  So the diode factor I'd assume between 1 and 2 and very robustly about 1.5.  1.5 to 1.8 in that.  So it's in between 1 and 2, nothing, not perfect 1 and neither perfect 2.  Especially for this structure.  And you interpret that like recombination and spatial region that's giving you some  Yeah, I think if you do numerical simulation, many features of it will probably,  because it's not really like where you can separate out the recombination  and the injection over barrier into separate components and add them.  Because at every point, just like the transmission picture shows,  that at every point you have a probability of recombination  and probability of going up in the barrier or down in the barrier.  So at every point, so when you add them up, unlike a PN junction,  where it's very sharp and small, when you add them up, you expect more complicated N values.  Let's assume.  So many of the general ones, I have heard it's not installing the other solar panels.  What is this all crystalline or the thin film at the consumer level?  So which of the...  Let me see whether you can immediately tell which one it is.  Because they look different.  Do you see a difference in these two pictures?  Which one do you think is crystalline? Of course the answer is given.  The ones that you have wafer found like this,  because you've got that structure and then you've cut it into slices.  And so this would be crystalline, crystalline silicon.  In many cases, the thin film will look like this, because it's roll to roll,  or essentially vacuum deposited.  So it will look continuous. You cannot see it here, but there are striped lines,  which allows them to connect it in series.  So if it looks beautiful blue, then you know that's the silicon solar cells that you have.  If you have it like this, then continuous, then you know that this is...  Is that right? Generally, that's more thin film structures.  Yes.  So if you find that basically, for the crystalline silicon,  it has about 50 times something like that before you get much of a life.  And you said that the reason why we are doing these thin silicon is because of cost.  But I was wondering, I mean, we've seen some visitation in the past because of these thin silicon.  So when you compare the current and how it comes out of these soil,  I mean, does it really save that much money?  And the other question I had is also, you know,  you've also learned that silicon is basically, you know,  most of the solar cells is based on silicon, right?  So why not use a head infrastructure where you can essentially harness much of the life from the sun?  You know, does it cost really...is it that big that, you know,  we have these solar cells because they're cheap and not, you know,  although people can use a head infrastructure to treat like that?  I mean, the main reason for thin film, as Professor Lundstrom also mentioned,  that it can absorb light within a very short distance, right?  So gallium arsenide will be a few microns,  and it will be all that you need in order to get 90% of the...1995% of the light in.  Now, remember the 50 bounces that I told you about, it's not on all wavelength.  The wavelength that are close to band gap, those are the most difficult to absorb.  And they don't want to get absorbed. You keep bouncing,  and you have to wait a long time before they get fully absorbed.  So the main thing is that you require a smaller amount of material in part  because you have easier absorption. It's the direct band gap material.  Most of them thin films are in direct band gap material,  so therefore it absorbs things easily.  Now, in terms of cost, if you go to one of these company websites  and ask them that what is the install cost for some of these companies,  it's actually as high as $5 because although in specific numbers  you get in various industry positions is one thing,  but essentially a 200 watt module will cost you about $900 when you buy this thing.  So silicon is already very expensive.  Now, the price is coming down, that's true,  and the thin film is in that especially for cadmium telleride,  it's significantly cheaper.  Now, whether all thin film can go there,  whether you can have a reliability problem or not,  that will sort of balance the cost, that's the second issue.  I'm not sure that question can be definitely answered today.  Which one is the ultimate winner?  But very difficult to count out silicon.  Silicon, despite all sorts of problems, is very resilient.  We hear a lot about grid factory,  can you comment on number one on that?  Number two, solar is ultimately a DC power source.  What's the extra cost here being in order to generate the reactor power to run the grid?  Very important. So anytime people talk about solar energy,  they talk about the cost issue is a very important one.  I mean if you have gas based power generation stations,  it's about what do they call it, maybe four or five cents per kilowatt hour,  something like that?  One tenth of that.  So it's very inexpensive, darts cheap, because in some way  God has already put the solar cell prefabricated the energy and put it in the gas.  So you don't pay anything for the equipment to generate the gas.  So therefore you are just bringing it up and using it.  So the question is that whether you can get below ten cents,  that's the goal per kilowatt hour in terms of energy use.  And it's very difficult. Very difficult.  People are trying all sorts of ways to get the cost down.  But even the best effort, it's not really without incentive from the government  that you have tax breaks and other things.  It's very difficult to see that how it can be directly competitive  even next two, three, four years.  So grid parity is an important question.  And the reactive power in terms of variability, how do you do load balancing?  All those things are very important questions.  But what people from sun power, what they argue that this variability,  because sometimes there is sun like now, and sometimes there is no sun in the night.  And then the way that the question is whether that would be a significant variation.  And if you have that level of variation, generating power is very, very difficult.  What they show that if you connect a lot of solar cells,  solar power generating units into a grid,  the variability is essentially all eliminated.  Because different regions have different time pattern for the solar cells.  And then they show if you combine with the wind power,  and this is all these things about smart grid technology,  then essentially you can have a very stable power source.  So there's a lot of work going on in terms of this circuit  or the power transmission system aspect of it.  But reactive power generation is to be done even when you have regular power.  Exactly. That's right.  So the cost includes the power.  So this is the power electronics cost.  So this is already included.  When sun power or somebody will sell you a module,  the module has everything, all the inverters,  it has all those pieces in.  You have to come in when the person comes with a truck.  All they do essentially, these are contractors.  They don't know anything about solar cells, no shadow degradation, nothing.  So they essentially, they should be able to hook it up  without asking too many other questions.  So when you buy that system, that thousand dollar has all those pieces in.  You don't have to buy them separately and install them.  Right?  Yes.  If I have a product, I know the water like this, the inverter part of it, not this one.  No, no, no. The inverter is very inexpensive.  How much did you say, 22 cents?  Yeah, I don't take your money.  It was like 20 percent.  You see, very inexpensive.  Those are normal silicon technology.  I mean, you don't really have to,  national semiconductor produces those parts.  Many companies produce the parts.  That's not the problem.  The problem is it fails.  After three, four, five years, it fails.  So you put it on the backside.  But you cannot, and you try to encapsulate it.  But still, if you leave something like this over a period of time in the exposed air,  think about it, what would happen if you lost your cell phone  and the cell phone was sitting in front of the park,  then it's not going to survive very long.  So that is what we are asking that park and what to do.  Very cheap, but very problematic.  Okay.  So, yeah, thanks.
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