 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. So welcome back. So we change topics now and we continue the discussion on solar cells. So this afternoon we're going to have two. I'm going to give the first one and what I'm going to do is to talk a bit about the internal physics of solar cells using a nice clean, well-characterized system as an example, so silicon. And I'm going to use a lot of examples that come from simulations. And that particular simulation program is called ADEPT and Professor Gray after the break will give us a lecture on what ADEPT is and how it works and what equation. It solves, it's available in the Nanohub so some of you may want to use it. So solar cells are all about recombination and generation. So let's dive into this. Again I want to thank my students who worked really hard trying to get this lecture together over the last couple of days. So remember how simple this device is. It's just a PN junction with sunlight shining on it. And to maximize efficiency it's really very simple. We want to generate as many electron hole pairs as we can. And we want to make sure that as few as possible of them recombine before we get them out the contacts and collect them. So this is just a short introduction. I think it's short, hopefully it's less than an hour, to the physics of crystalline solar cells and we're going to use silicon. So here are the topics. Now first of all look at what happens under short circuit conditions and then I'll look at what happens under open circuit conditions. That's really all I'm going to talk about because in between everything is either some combination of those two characteristics. And let me just remind you about how we think about these devices. So here's my simple PN junction. First of all in the dark. And remember the idea is that the contact on the left is an electron injecting contact. That means it sends electrons into the conduction band of the N-type semiconductor. The contact on the right, well first of all on the left if we apply a forward bias to this we lower the barrier electrons get over the potential barrier. That means I end up with an excess concentration of electrons on the P side and if they recombine current flows. At the same time holes get injected to the N side. Contact on the right side is a metal contact that is connected to the valence band. So we'll call that a whole injecting contact or that's electrons coming in and out at the valence band. So the idea is what happens is if I inject an electron into the P side. First of all it leaves behind a missing electron on the N side. So the N side is not neutral anymore. When the electron gets over on the P side it's an excess carrier. There are more electrons than they're supposed to be in equilibrium. The system always reacts by trying to restore equilibrium which means it'll tend to recombine. If it recombines with a hole it means now I have a missing hole on the P side. And the P side is no longer electrically neutral. So an electron comes out of the valence band on the P side and out in the metal wire and that creates a new hole. So I replace the hole, it goes around the power supply and comes in the N side, comes in at the conduction band and replaces the missing electron in the conduction band. And that's how the whole process works. Anytime an electron and a hole recombine anywhere within the diode one electron flows in the external circuit. So we can look at it on an energy band diagram and we can see these electron excess electrons on the P side. They could recombine through defects. That's what we call Shockley-Reed-Hall recombination after the people who first worked this out. There's a classic paper that anyone who works on semiconductor devices still reads because it's still relevant. So recombines, we send the electron around the external circuit but the property of these contacts is we would like this electrode on the right side to allow electrons to come in and out of the valence band. But we would prefer if they just blocked minority carriers and kept them away from the contact. A real contact is usually very bad for minority carriers. If the minority carrier electron gets over to the contact it usually finds a lot of defects to recombine with and that's bad because if it recombines anywhere it gives us one electron in the external circuit. So people will go to great lengths to engineer contacts to try to keep the minority carriers away from them but allow the majority carriers to come in and out so that it's an omic contact for the majority carriers. That was the diode under the dark. Under the light what happens? We shine sunlight most of the electron hole pairs are generated in the thick base absorbing region so now we have one extra hole in the P region, we have one extra electron. The electron can diffuse to the P injunction, fall down in energy and then go out the end side. Now it's flowing in the wrong direction so the current flows in the opposite direction. Goes around the external circuit, comes back in the contact. I have one extra hole meaning I have one missing electron in the valence band in the P side this electron comes in the valence band and fills that up and now we're back to where we started from. So anytime an electron hole pair is generated one electron flows in the external circuit. So this is important. I remember reading a paper a long time ago by a very bright person, I read a lot of his papers, he wrote very good papers. But he wrote one where he said look at what would happen if the electron hole pair were generated in the depletion region. Both the electron and the hole would be collected and you'd get two amounts of charge flowing in the external circuit. But try it out as a little exercise, draw this little diagram if that electron hole pair is generated in the depletion region and you'll see that only one electron flows in the external circuit. It doesn't matter where the electron hole pair is generated. So of course he discovered that shortly after his paper was published and then following paper he pointed out oops I made a mistake. That happens. Alright. But recombination can also occur under light. So if we're illuminating these, let's say we create three electron hole pairs. One of the electrons might diffuse to the p-in junction and get collected, that's what we want to happen. But some of them might recombine through defects then I don't get any electron out the external circuit or it might recombine at the back contact. Those are all bad things in terms of if I'm trying to get the maximum current out I'm losing current because they're recombining. So three were generated, one was collected. So my collection efficiency would be the number that came out or the number that were generated. The collection efficiency would be one third. Not very good. So carrier recombination does two things. It lowers the short circuit current and it reduces the open circuit voltage and if we want to optimize solar cell performance we really need to understand this carrier recombination we need to understand how many are recombining and then we need to understand where they're recombining so we know how to engineer the device to reduce the number that recombine. So we need a quantitative relationship between recombination and generation and solar cell performance and I'm just going to state it because it's kind of obvious and I have it in the handout notes I have a mathematical derivation of this but you really don't need any mathematical derivation. All you need are these little pictures of electrons flowing around in the external circuit. So the diode current J sub D is equal to I argued that anytime an electron and hole recombine in the semiconductor one electron flows in the external circuit. Anytime an electron hole pair is generated if I collect it it flows and if it recombines it subtracts from that. So that first expression says to understand the IEB characteristic all I have to do is to understand how carriers are recombining inside the device and how they're being generated. So I'll ask what is the total amount of recombination. So I'll just integrate across the entire thickness of the solar cell from 0 to L the recombination rate the number per cubic centimeter per second that are recombining and I'll integrate across the thickness but I have to be very careful I have to include the contacts because at x equals 0 if there's any flow of minority carriers into that contact they'll immediately recombine so the recombination rate will be the net flow of minority carriers into the n type contact. If they're not recombining there will be no flow of current there. And similarly on the p contact if there are any minority carrier electrons flowing to that contact they'll recombine immediately. So that's part of the total recombination in the device. It's the recombination in the volume of the device plus the recombination at the contacts. And the total generation is just the total number of electron hole pairs that are generated inside that solar cell. Alright, so let's talk about short circuit then. I'm going to go back. Here's a generic crystalline silicon solar cell. Not super high efficiency but reasonably high efficiency. Remind you again we have a very thick n plus, a very thin n plus layer on the top. Actually 3 tenths of a micron is relatively thick. We have a thick p type absorbing layer to absorb as many of the photons in this indirect gap semiconductor that we can. And then we have a back surface field that we'll talk about a little bit more but its role is to try to keep minority carriers away from the back contact because if they get there they'll recombine and that's not good. So here are some of the dimensions and lifetimes and things. So I'll show some results and I'll illustrate how things happen quantitatively just by doing the calculations on this particular cell. So some of the questions that we ask. First of all, what is the total number of electron hole pairs that are generated if I shine an AM 1.5G spectrum on this cell? We'd like to know where they're generated. What percent of them are generated in the n plus emitter? What percent of them are generated in the p type base? We'd also like to know how many of them are recombining? What's the total recombination? We'll be careful to include the recombination at the contacts in that. And then I'll ask how good is this back surface field if I put it in or take it off, how do things change? Okay, so these are calculations that we can do. So first of all, if this were an infinitely thick slab of silicon then every photon above the band gap would be absorbed and if I do the integral of the generation rate versus distance from 0 to infinity I would get the maximum that could ever, the maximum number of photons in the solar spectrum that have an energy big enough to create an electron hole pair. And that number is 2.97 times 10 to the 17th per square centimeter per second. Now I have a finite thickness sample and actually in this one we assume that if the light goes down and it's not absorbed that it will bounce off the back metal contact and go back through a second time. So I'm assuming that that might not be quite right because there might be some roughness there that get bounced off at an angle but we're just going to make the assumption that it makes two passes through. It goes through once, if there's anything left over it reflects off the bottom metal and goes through again. So I do the integral from 0 to 2L. And you can see that I don't quite get every photon that I could. Some of the advanced cell structures where they do this complicated light trapping and they try to get as many bounces as possible they can get a little more than that. But we only get 2.79 instead of 2.97. So we're giving up some that are weakly absorbed and don't get absorbed in this finite thickness. Now the short circuit current in this particular cell, we can simulate this cell. Professor Gray will tell you how we do the simulation. The short circuit current is 39.4 milliamps per square centimeter. If I divide that by charge on the electron and the flux of electrons that's getting collected, that's a little bit less. That's 2.46 times 10 to the 17th. And that means that some are recombining, so the number that are recombining is 3.31 times 10 to the 16th. So the ratio of the short circuit current divided by Q to the total number of electron whole pairs that are generated is the collection efficiency. It's the ratio of what comes out over what was generated. And that's 30%. That's a pretty good number. But for the ultimate record efficiency silicon cells, people do better over 90%. Now we can do these plots. So we can run a simulation and we can look at these plots and we can try to see what's going on inside this solar cell. So if I look on that look on the left, this is a plot. The blue line number of electron whole pairs per cubic centimeter per second that are generated. And I'm going from 0 to the end of the cell. So you can see that the high energy photons are very steeply absorbed and a lot of them are absorbed right near the top surface. The green line is the integrated total, normalized to 100%. So it tells me what fraction of the photons were absorbed up to. So you can see if I go to 25 microns it looks like I'm close to 90% of them are absorbed or 85% or something. And in fact it looks like a lot is happening right close to the surface that I can't see on that plot. So over here on the right we just blow it up. Remember the emitter is 3 tenths of a micron thick. So if you look at that green line you can see at 3 tenths of a micron it looks like maybe 35% or so of the photons are absorbed in that thin n plus layer. Those are the high energy photons that are absorbed very quickly. So we can just look at those numbers. These are the kind of questions a solar cell designer would ask themselves. Where are things being generated? We can see 37% are absorbed in that top thin layer. 14% are absorbed in the depletion region around the p-n junction and then the remaining half of them are absorbed in the rest of that thick p-type layer. Now we worry about this because the heavily doped region is a low lifetime region. I gave a lifetime there a nice long lifetime of 34 microseconds that was of the p-type region, moderately doped. The very heavily doped region typically has shorter lifetimes and in this particular case it might be because there are some defects that have been introduced as a result of the doping but it also can be some fundamental processes. One that's important in heavily doped regions is OJ recombination. Could also be the front surface too. In any photon that is collected in the depletion region we don't worry about too much because there's a strong electric field that's just going to sweep them across the right side and collect them. So we don't worry about the fact that those might recombine. But in the base we have to worry about do we have a long enough minority carrier lifetime so that those that are generated deep in the base can diffuse to the junction and get collected. Have we protected the back surface so that they don't diffuse to the back surface and recombine? So we'll worry about those. So let's look at the recombination. This is where we're losing those photo-generated carriers. So again the blue line is the recombination rate per cubic centimeter per second. And the dashed green line is the integrated total so it goes from 0 to 100%. So you can see very quickly there that I start at about, it looks like about 75% of them. Out 75% of them recombine very close to the beginning. So close I can't resolve it there. And then another it looks like maybe 10% or so recombine in that thick P region. And then you get to the back contact and it takes a jump up. That looks like it might be another 15% or something recombine at the back contact. So that's where we're losing them all. If I blow up that surface region because a lot is happening there, you can see the blue line is the recombination rate. And you can see that it's very high in that N plus region because the lifetime is very short. So they recombine very quickly. And if I look at the integrated total, that dashed line, you can see that I quickly get up 75% of the carriers that I'm missing, that I'm losing are recombining in that thin 3 tenths micron layer. So if I look at these numbers, 75% of them are occurring, I'm losing it in the N plus emitter. I hardly lose anything in the depletion region because the electric field is so strong it just sweeps them across before they get a chance to recombine. And then I lose some 23% in the base and at the back. So you can compare those to where the carriers were generated. And you can see that although only 37% of the carriers were generated in that thin top layer, 75% of them that recombine anywhere in the device are recombining there. So we're losing a lot in that top layer. And it's because the lifetime is so low. Alright, so I'll just point out, I just want to mention one of the things you do when you run a simulation is to make sure that you can believe the results. Usually the program is right. Sometimes it isn't. But it's easy on these programs to set them up in a way it's not quite the way you expected to set them up at. Maybe you're simulating the correct solution to a different problem. So it's always nice to check and see how the results are getting reasonable. So one of the things we do in traditional semiconductor courses is we solve for the distribution of minority carriers in P regions. And if I want to solve for the distribution of electrons in the P region, I would start with my continuity equation which says divergence of the electron flux is minus the recombination rate. And I'd say that this is a region that doesn't have an electric field so the current is only a diffusion current. And if I plug that current into the equation above it, I'll get this minority carrier diffusion equation. So this is an equation that would tell me the profile of the excess carriers inside the device. L sub n is the minority carrier diffusion length. It tells me the average distance that an electron can diffuse until it recombines. And then I would apply boundary conditions and I would solve that differential equation and I could find the profile. And I would expect it to look something like this dashed line. I would expect it to be very low if I go near the edge of the depletion region because I'm sweeping them out there just as fast as I can. I would expect it to have some gradient that would give me carriers to diffuse towards the back contact but if I keep the recombination at the back contact so it's not too high, I won't have much of a gradient there and I won't get too many that are diffusing back there. So this is the kind of analytical solution that I would get. If you go in and look at the numerical solution that doesn't make any of those approximations, it's the green line. You see it has the expected shape. If I just take delta n and divide by 34 nanoseconds I would get the blue line which is the recombination rate. So everything behaves the way it should. When you're running a simulation program you always stop and do some tests like this to make sure that I can make sense of the results that I'm getting and the numbers that are coming out are about what I expect the numbers to be. So let's look at that back surface field because see if it's doing what it's supposed to do. So on the top left there is my energy band and I dope the region adjacent to the contact very heavily p-type. That means that the Fermi level, the red line there has to be very close to the valence band. The p-type base layer is dope more moderately so the Fermi level is further away from the valence band. So my energy band diagram looks like this. Now if I don't have that back surface field then my energy band diagram is just flat all the way to the back contact. The back contact is a place where there are lots of defects. If any minority carrier gets there it can recombine immediately. But really the fastest that that can happen is going to be limited by how fast they come to that contact. And the quickest that they can come to that contact is at the thermal velocity. So that boundary condition is going to be characterized by the velocity at which they're diffusing to that junction and that velocity is going to be the thermal velocity of electrons. Which is about 1 times 10 to the 7th. Now I have a little bit of a barrier there. I wish it were bigger but this is what we're able to do with doping. The barrier height is 0.13 electron volts. So if I take e to the minus barrier height over kT I'm going to knock down that voltage by a little bit. What we end up, the probability that an electron will hop over that barrier is e to the minus barrier height over kT so I'm going to expect the effective velocity that they're diffusing to the back contact to be the thermal velocity reduced by the probability that it can get over that barrier. So it knocks it down a little bit. If I really wanted a good barrier I might try to do something like a hetero junction. Put a wider band gap there and get a much bigger barrier. I would really keep things away much better. Now what we can do with our cells is this really performing any function in our cell so I could run a simulation with and without that back surface field and if you look at those two green lines the dotted line is what we had before and you can see that's the integrated total recombination rate. So when I get to the end of the cell that has to be 100% of the recombination. That's all of the recombination that occurred in the device. It looks like about 15% of the total recombination is occurring at the back contact when I had that back surface field there. And under those conditions I had a total recombination if I multiply it by Q which corresponds to 5.3 milliamps per square centimeter. If you remove that heavily doped region then I get the solid green line. And you can see now, first of all if you look at the number you can see that the total recombination corresponds to 6.5 milliamps per centimeter squared. I have more recombination so it means more electrons are getting to the back. And you can see that in that solid green line there's a step it looks like it's maybe about 30% or something of the carriers are now recombining at the back surface field. Maybe the percentage is twice as high as it was before. So the back surface field is helping but it hasn't shut off that back surface recombination totally. And you see that the total collection efficiency without the back surface field is 85%. With the back surface field it's just a little bit higher, it's 88%. Now here's another common measurement that people will do is that they will take the device at short circuit and they'll shine an incident flux of photons at a specific wavelength at the cell and then they'll measure the current that they get. The ratio of the current that comes out divided by the charge to the incident flux that came in is the quantum efficiency. It's the percentage of electron hole pairs that were collected. And now if you sweep that as a function of wavelength you'll get some information about what's going on because the short wavelength has high energy and they're absorbed very near the surface and that'll tell me something about recombination near the surface. The long wavelengths are absorbed more deeply and they'll be more sensitive to recombination in the base or at the back surface contact. And if the wavelength is too long it won't be absorbed at all and it won't get anything out so it drops to zero. So you can see what's happening here with and without the back surface field. With the back surface field I get a little bit more current at the long wavelengths because the back surface field is helping reduce the recombination and keep them away from the back field and helping them be collected a little bit more. So this is a measurement that people do frequently. You can see that the internal quantum efficiency is dropping down at the short wavelengths. That's because they're being absorbed in this N plus layer where the lifetime is very short and they're recombining very quickly. So this is a standard type of characterization that people will do when they're trying to understand what's going on in their solar cells. So some questions you can think about. You could actually go into the simulation results and deduce what the actual surface recombination velocity is. You could think about doing things like putting heterojunctions in and reducing the surface recombination and velocity at the back to essentially zero so that no carriers can recombine at the back contact. How much would that improve our efficiency? The importance of the back surface field is going to depend on what the minority carrier lifetime is. If the minority carrier lifetime is very long, then nothing will recombine in the base and the importance of the back surface field will increase. And you can ask all kinds of questions like this. Now the last question here I want to just mention. Since most of our recombination is occurring in this thin N plus region, why not just make it a lot thinner? So that fewer carriers are absorbed there and they'll be absorbed in a region that they can be collected. So here I have to point out that there's a whole set of effects that I'm not going to spend any time on. I've been thinking about this device as a 1D device, but it's really a two-dimensional device. I've done one section of it here. There's a metal grid of fingers on the top contact. This is repeated over here. So I can illuminate maybe 90% of the area. My metal grid obscures 10% or less of the area. So what happens is that when my minority carrier electrons are generated in the base, they diffuse towards the p-enjunction and are collected. Once they get collected, they have to travel laterally out the contact. And as I go towards the contact, the magnitude of that current just builds up because I'll just add all of them that are collected along that contact. So if the electrons are flowing in the direction of the arrow, then the current will be flowing in. So I'll have a current flowing in the opposite direction and that means I'll have a lateral voltage drop along that. The voltage along my p-enjunction will not be the same everywhere. When I was thinking of this as 1D, I wasn't accounting for this effect. And the fact that the voltage is less means I'm going to have a lateral voltage drop and the magnitude of that drop is something that can be computed. In effect, it's going to appear when I look at the IV characteristic. It's going to act like a series resistance. But it's a little funny series resistance. It's a current dependent series resistance because this is a more complicated current flow problem, but it's going to degrade the performance of the cell. So people will be very careful about trying to make sure that that layer is not too thin because if it's too thin, its resistance is very high. And then these lateral voltage drops get very big. So when you're designing a grid, you'll be careful about the spacing between these metal fingers and about the sheet resistance of that top layer such that you don't introduce these lateral voltage drops. Alright, so that's how things happen at short circuit. Let's look at what happens under open circuit conditions. So under open circuit conditions, the top expression there, I've argued that that's exact. The diode current is recombination minus total generation. Under open circuit conditions, no current is flowing. So I just solved that equation under open circuit conditions. So the voltage will adjust such that the total recombination is equal to the total generation. That will give me open circuit conditions. So open circuit happens when the diode forward biases itself such that the recombination exactly balances the generation and no electron hole pairs come out of the contacts. Now we frequently think about superposition and dark current. I'm going to say just a word about this. The top equation is exact. Now in the dark, I have a dark diode current. I have no optical generation, so G total is zero, but I have carriers recombining in the dark. When I'm illuminated, I have carriers being generated and I also have some recombination occurring under illuminated conditions. Now superposition says I'm going to take my short circuit current and I'm going to assume that it's voltage independent. This is yesterday why that's reasonable for a silicon cell. And I'm going to take my dark current and I'm going to get a diode characteristic like this. And then I'm going to assume that what happens when I have both bias applied and light applied is just the superposition of those two. And my open circuit condition, for example, will occur when the diode current in the positive direction is equal and opposite to the light generated current, which is negative. And I'll add the two and I'll get zero current. That seems very reasonable, but it's really a little hard to justify. Under illuminated conditions, the total recombination in the light is equal to the total optical generation. Superposition says that, well, I'll look in the dark and when the recombination in the dark gives me the short circuit current, then I'm under open circuit conditions. So I'm basically going to assume that the recombination in the dark at VOC is similar to the recombination in the light at VOC. It's not obvious that that should be the case. And in some cases it's really a very poor approximation. It tends to be a poor approximation in materials that have lots of defects where when you shine light on them, you can change the population of these defects in their charged states and now you change all of the internal electric fields and distributions and things can be dramatically different under illuminated conditions and under dark conditions. In a high quality crystalline silicon where you have very few defects, it generally works very well just to superimpose these two. So it's not easy to justify this but for silicon solar cells in practice it works very well and that's the reason that we think a lot about dark current because if we can understand dark current we can deduce what the open circuit voltage can be. So I mentioned yesterday that this dark current has a characteristic that typically under low bias it has a characteristic that looks like n equals 2 under moderate bias it has a characteristic that looks like n equals 1. Under higher biases it starts to go back towards n equals 2 and maybe even bigger and that might be series resistance or it might be high level injection or it might be a number of other things. So here's the simulated IV characteristics of that silicon diode that we're looking at that green solid line. The green solid line is on the log plot on the right the blue line is the linear plot and you can see a nice clear n equals 1 region. You can see a hint that there's a region where it's trying to go above n equals 1 at low bias the way we expect and you can see a little bit of roll off at the high end which we expect also. So what determines that J0 and the ideality factor and the answer is that the amount of recombination is going to determine J0 but the location of the recombination is going to determine the n factor depending on where they recombine I'll get different diode ideality factors. So let's forward bias this diode in the dark at about 7 tenths of a volt because that's about where the open circuit voltage is and let's ask how does the recombination work in this particular device? So here's the recombination rate and I'm showing it here very near the surface the first 8 tenths of a micron remember it's 200 micron stick there's a lot of recombination in the emitter the recombination rate is very high but if I integrate that through the emitter it's only about 20% of the total recombination a little different. Now if I continue to integrate across the entire device then you can see that maybe 10% of it occurs in the p-type base and then I get to the back surface field and there it looks like about 70% of the total recombination is occurring at the back surface field So one of the points here is that under short circuit conditions and under open circuit conditions different parts of the device are controlling the performance and if you want to increase the short circuit current you'll work on the parts that are controlling it if you want to increase the open circuit voltage you might have to work on a different part So these are the numbers that we just saw that's where things are recombining and the point I was just making here is if you compare that to short circuit conditions you can see it's quite different Now one of the reasons is that under short circuit conditions there were a lot of electrons that were minority carrier holes that were generated in the N plus emitter so they had a chance to recombine there that's why I had a lot of recombination there When you forward bias a p-enjunction there's this thing called the one-sided junction you tend to inject most of the excess carriers in the lightly doped side the heavily doped side it's hard to inject many excess holes in it so even though the lifetime is still short we just aren't injecting many excess holes in there to recombine much more of the recombination is occurring on the lightly doped side and this is just an example to show you the back surface field now the back surface recombination is much more important than it was under short circuit conditions so you can see even with the back surface field it's not a very good back surface field it's not keeping the minority carriers away if you look at that green line and you see the big jump it takes at the end of the device it goes from 30% total recombination and then it jumps up to 100% so 70% of the recombination is occurring at that back contact and the back surface field is not keeping them away if you look without the back surface field it's even worse there it looks like it's probably 85% of the recombination is occurring back there and your total current meaning your total recombination is significantly higher alright so let's talk a little bit about the physics why is the end current 1 and I did this yesterday so I'll just remind you we can have recombination occurring in a couple of different places it can occur in the neutral N or P type region those regions are almost neutral we injected a few excess carriers but it's a small amount so we call them quasi-neutral regions so when I inject an electron over the barrier into the quasi-neutral region and it recombines it can recombine at the back contact that's also within the quasi-neutral region and the same thing I can inject holes into the quasi-neutral N type region and it can recombine in the N type region or at the N type contact but that's not the only place the electron hole pairs can recombine they actually have a chance of recombining in that barrier before they get over it alright so let's talk about the first one let's talk about recombination in quasi-neutral regions the diode current is the total recombination the total recombination is just the total amount of excess charge in the quasi-neutral region the P region say divided by a characteristic time that characteristic time is the average time it takes to recombine through a defect or the average time it takes to diffuse to the back contact and recombine there the excess carrier concentration goes exponentially with the forward bias because it lowers the barrier and makes it easier for electrons to get over so if I just take Q sub N over tau you can see that I'm going to get a current that goes as e to the QVA over 1 kT so the point is if the carriers recombine in a neutral region they're going to give me an N equals 1 diode current so we've explained that N equals 1 part there let's lower the bias and go down and see what happens so if I go down and looking at two tenths of a volt things are a little bit different when I look near the junction on a linear scale the only recombination I see is a sharply peaked recombination remember the N plus layer ends at three tenths of a micron and then the depletion region is about three tenths of a micron thick so it goes out to six tenths of a micron so I have a sharply peaked recombination right in the middle of the depletion region and you can see that even though it's sharply peaked it's not contributing this is a very high lifetime silicon solar cell so it doesn't contribute a whole lot to the dark current but I'm beginning to see it so the green line is the integrated total and you can see it step up as we integrate through that recombination it looks like about 20% of the total recombination is coming from electrons and holes that are recombining in the depletion region 20% that was right so here I'm just comparing where the recombination occurred so the point here is that under different biases the first point was that under the light and the dark conditions they recombine in different places under different voltages they recombine in different places so you can see what happened under high forward bias I didn't have much going on in the depletion region under lower forward bias I start to have a significant percentage now depending on how this all maps out what the band gaps is what the lifetimes are the percentages here can change and I could easily have a lot higher percentage in the depletion region so how do we understand that characteristic so we have to focus on what's happening in the depletion region and the way we think about this is an electron might not make it all the way over a barrier or the hole might not make it all the way over its barrier and get into the other side on the way there they might recombine and if my electron gets into the P region then there are lots of holes to recombine with and I have more than enough holes I can always find a hole to recombine with if the hole gets injected into the N region there are lots of electrons to recombine with so that's not a limiting factor there are plenty of electrons around but if I'm in the depletion region I have a small number of electrons and holes at one edge I have a lot of electrons at the other edge I have a lot of holes but I'm not going to have much recombination there because I need both an electron and a hole to recombine so the place that I'm going to get maximum recombination is the place where I have both electrons and holes so at the point in there where I have multiple populations of electrons and holes that's where I'll get the maximum recombination that's that peak that we saw and one can show just by a little bit of semiconductor statistics here that inside the region there the product of Np was Ni squared in equilibrium and it's just exponentially bigger because of the applied bias so it's Ni squared e to the qV over kT but the peak occurs when N is about equal to P so I can just take the square root of that expression so the peak concentration is scales as Ni times e to the qV over 2 kT the recombination rate then is going to be the concentration divided by some effective time that now will include both the sum of the electron minority carrier electron recombination time and hole recombination time and it will be proportional now instead of qV over 1 kT like it was in the previous case now it's qV over 2 kT so the point is that if most of the carriers are recombining in a depletion region I should expect on my measured ID characteristic to see an N equals 2 and here we can go through a few numbers in order to get the total amount of current I have to take the area under that blue line I have to integrate the total recombination across that blue line and one way to get an estimate of that is to take the peak value which is just the peak number divided by the time and multiply it by the width of that and the width of that is the region over which N is about equal to P and since the quasi Fermi level is flat and the conduction band is changed by the electric field the width of that region is on the order of kT over the electric field you can check it dimensionally volts divided by volts per centimeter that gives me the distance over which the conduction band varies by about kT and that will lower the electron density by kT now you can argue about should this effective width be 1 kT or 2 kT it's something on the order of that so you go into this simulation you pull off the electric field at that peak you calculate the effective width it's about 11 nanometers that's maybe a little bit thinner than what we're seeing there a tenth of a micron is 100 nanometers so the distance between the tick marks is 200 nanometers but maybe the effective width is 2 kT that's not bad it gives you the right trends if you apply a forward bias you're going to lower the electric field in the depletion region and this peak is going to spread out and become broader so it gives you some physical insight ok so what's the point of all of this it's that your dark current people if you measure a dark current you'll find that it's j0 times e to the qv over n kT minus 1 there is no you might get an n of 1.4 there is no physical mechanism really that gives you an n of 1.4 it's some combination of a physical mechanism that gives you an n equals 1 which is carriers recombining in a neutral region and n equals 2 and depending on the relative weights between those two you can get an n factor between 1 and 2 so you really have to separate them out the important point was the n equals 2 component is proportional to n i so it goes as e to the minus e g over 2 kT the n equals 1 component is proportional to n i squared so it goes as e to the minus e g over kT so when you have small band gaps and high temperatures n i squared gets very big and you can just see an n equals 1 component if you go to bigger band gaps and smaller temperatures it's a n equals 2 component so people will sometimes measure IV characteristics over at temperatures they'll try to pull out the n equals 2 component at lower temperatures and the n equals 1 at higher temperatures ok so there are a number of things experiments you're going to learn about a depth and you can get online and run it and you can play with some things and see what happens to develop your intuition ok so now I've got a few things to discuss and then we'll wrap up and see if you have some questions so how do you the whole the whole objective in solar cell design is to increase the generation rate absorb as many electron hole pairs as you can so you do that by minimizing reflections and by trapping all of the photons in the structure and then to minimize the recombination rate the smaller it is the smaller off you are and you minimize the recombination rate by getting as high a quality material as you can with very long lifetimes sometimes you might do it by making the layer very thin so it just doesn't have a lot of volume to recombine in and if you can trap all of the photons in that thin layer such that they're all absorbed then that's helpful you might induce built in fields so if you're building solar cells out of less expensive material that has low lifetimes then you would like to engineer the structure such that most of the photons are absorbed in a depletion region where there's a strong electric field that will quickly sweep them out you can try to build back surface fields and see if you can get a better back surface field than the one we had another thing you can do is just reduce the area of the contact so if we get back to this martin green cell you can see some of the things that they're doing if you look at that back contact initially people just diffused the P plus layer over the entire back that was their back surface field and then you put a middle layer on the back but that isn't an especially effective back surface field we've seen but all I need is an only contact to that P type layer so I could take that heavily doped region and I could make it a very small area just little points and I could make contact just in those small areas any electrons that get to those small areas will recombine but the total area is a small fraction of the area of the back surface I'm inserting a silicon dioxide layer there which will passivate the back oxide tie up all the dangling bonds and give me a low surface give me a low back surface recombination velocity let's see now if I design the thickness of that oxide just right it can promote the reflection of photons back in so it can benefit the optical design so people are just enormously careful about how they design these cells they run simulations and calculations like the ones I've talked about they find out where is most of the recombination occurring they try to figure out a structure to press that recombination then you look at it again and now the dominant recombination might be somewhere else in the cell then you work on that and see if you can lower that that's the process that people went through from the late 1970's when silicon efficiencies were 15% until now when they're approaching 25% just very carefully getting as many electron hole pairs generated and reducing the recombination as much as they could now another thing that I wanted to get back to is just this discussion of how good is super position and sort of the assumption there is that if I look at what the internal recombination remember at open circuit the total recombination is equal to the total generation so I can run the simulation under illuminated conditions and on the right there I'm seeing what the total recombination looks like under illuminated conditions on the left what you're seeing is what the total recombination looks like in the dark at the same voltage and you can see that they're reasonably close so it looks like that approximation might work here's an actual confirmation I asked Dionysus to do this here and I didn't know how good it would be we always make this assumption for silicon and even I was surprised by how good it is because you can't find a simple one line justification for when it works and when it doesn't so what you're seeing in the blue line there that's just the simulated performance of this solar cell under illumination the black line is the simulated performance of that diode when it's dark if you just take that black line and add that current to the short circuit current there which is almost minus 40 milliamps and add the two, you get the red dots and so the red dots are superposition and you can see that they fall just right on top now there are a lot of solar cells for which this isn't even close but if you can make use of superposition it's a wonderful thing because it allows you to do a lot of your diagnostics on what's controlling recombination just by simple measurements in the dark then you can be confident that you'll know how to predict what's happening in the light ok, so we'll summarize and then we're ready to break so important point I'm trying to make is that the diode current is simply Q times the total recombination minus the total generation that's really exact that's not a superposition argument that's exact at open circuit voltage equal to the generation such that no electron hole pairs come out and no current flows if I have any recombination under short circuit conditions that lowers the short circuit current of the collection efficiency when you can use superposition then the dark current tells you an awful lot about what to expect under open circuit conditions and as I mentioned, solar cell design is all about maximizing generation and minimizing recombination and in the process of this talk the final point that I was trying to illustrate is a lot of people make use of simulation tools they'll just simulate it look at the IEV characteristic and see if it's close to what they measure if that's all you do you're really not extracting much value from the simulation program and what the simulation program allows you to do if you've got the right physics in it and if you've got the right mobilities and diffusion coefficients look inside and understand why the device is performing that way and help you understand how you can make it perform better so don't just look at the computed IEV characteristics when you run a simulation look inside at recombination rates and carrier profiles and energy band diagrams and see if you can figure out how to make the device better so in the handouts and by the way I've been meaning to mention there's a website here where all of the handouts for the summer school are available if you haven't got them in the handout for this I've got first of all a mathematical derivation of the fact that current is Q times recombination minus generation although it's intuitive you don't really need a derivation and an attempt at a mathematical justification of superposition but I'm not very happy with it because it's not easy to justify in general alright so I don't want to go there, that's the appendix so I'll stop and see if there are any questions and we've got this fellow here with the mic and yeah yes, we were talking about doping with these impurities what would you recommend if you said you're having a problem with efficiency so when you're doping I know you're changing the characteristics of this semiconductor so have you ever thought about some type of inhibitor that you can actually place or to actually suppress the recombination an inhibitor that could, yeah the way I think of ways to inhibit recombination is things like minimizing defects engineering structures that keep carriers away from places where there are lots of defects keeping minority carriers away from contacts I'm not sure what you're getting at in terms of inhibitor there are some besides the heterojunction yeah heterojunction is a wonderful so if you use a heterojunction as a back surface field it's very helpful because you can get very big energy barriers it's difficult with doping to get big energy barriers so as you saw the numbers it's not being very effective in keeping the carriers away from the contact but with a heterojunction you can have a very big energy barrier there and you can essentially reduce the recombination to zero so that was causing the inefficiency of the solar cell right because that was adding recombination so it was inhibiting both the short circuit current but it was hurting the open circuit voltage even more I just thought something was developed to actually increase the efficiency of this yeah increasing the efficiency of whatever technique you can find to reduce recombination we'll do that and I think Professor Alam will be talking later about thin film cells and thin film solar cell designs are quite different because the lifetimes tend to be very short so you have to design them differently than you would design a cell like this where you can get very long lifetimes okay okay so can we increase the thickness of the back to decrease the recombination so that's a good question why not just increase the thickness you know there is some effect there let's take a look at that structure you know we can actually do a calculation of what that back surface field is so the carriers have to get the carriers get over that barrier that's created and then they diffuse to the back surface and recombine so if I'm thicker it might make it harder for them to diffuse to the back surface field and recombine now the trade off is that it's heavily doped so it's lifetime tends to be very short so that increases the volume of low lifetime material where they can recombine so there is some trade off there there's some thickness where there's some sweet spot you can do a little bit about it you can't do anything very dramatic I think there was a question down here what's the status on the transparent little top contact is there a question? yeah so the question is what is the status on transparent metal contacts Ashraf are you going to talk about this are you planning on talking about this if so I'll defer it to you no no because Ashraf is more up on it than I am you know so in a lot of cases what would be nice is if you had a transparent conductive material so instead of having to make a contact in a metal grid and have these lateral bolt drops the metal grid shadows some of it so it obscures some of the junction and the lateral bolt drops are bad what if you could just put a transparent metal across the entire layer you wouldn't have any lateral bolt drops well there are transparent conductors things like indium tin oxide and you know they're not perfect they transmit probably you get transmittances Rakesh probably knows 90% or so and you get sheet resistances that are 10 ohms per square so they're not perfect metals either there's some compromise but people frequently use those for solar cells there's a lot of interest these days in whether you can well first of all indium is rare so that's kind of expensive so you'd like to find other ways to make transparent conductors there's a lot of interest right now in graphene has a lot of very high electrical conductance and a single layer of graphene has very low optical absorption so there are a lot of people that are doing research on whether graphene could make a transparent top conductor for solar cells you know it's not a slam dunk you know one layer of graphene doesn't have a low enough sheet resistance for solar cell applications so you have to put multiple layers in when you put multiple layers in for absorption but there's some promise of doing something there any other questions yes how effective yeah can you make it more clear on which part it does depend well so there is so the question was what is this how effective that I talked about now the right answer to that is to learn Shockley-Reed Hall statistics so there's a famous paper that was written probably in the 1960s about recombination through defects and that's usually the starting point for this but the way to think about this is that this defect has to capture an electron and that's described by an electron capture time tau sub n but then to recombine it has to capture a hole and that's described by a hole capture time so the time it takes to do the entire recombination process is tau sub n plus tau sub p so the effective then would be the sum of the two lifetimes now if you go out in the neutral region I only use the electron lifetime I had an excess electron density and I just said that the proper time to use there was the electron lifetime and that's because there are so many holes that it's always exactly possible for it to find a hole to recombine with what oh ok so you said that the high energy photons can we get absorbed correctly because there's more than 6 days after or something I said if the devices if you make much better it might be useful for reasons or other things I wondered if you really spent a lot of effort on the optical design maybe making like multilayer basically mirrors out of thin film structures specifically to make the device into a optical resonator for the wavelengths that take a long time or things like that and how much effort has been on the soccer design yeah I think that there's a considerable amount of effort that's spent on optical design and there are very one example of that is this Martin Green cell that I continued to show let's see if I can find that again so this is one example of optical design where people have spent a lot of care and there are various ways that you can texturize the top surface they've got a mirror at the back that they carefully designed to try to maximize their reflection they'll do ray tracing studies to see how many times a ray would bounce around inside this structure so there is a lot of thought and the idea is to try to make the silicon layer as thin as you can possibly make it but absorb every photon that you possibly can now it's not as important when you have direct gap materials because things are absorbed so quickly there but in these indirect materials it's extremely important even in the direct gap materials the idea of texturizing the top surface to get all of the photons in is a thing that is widely used okay, we have another question describing the recombination in the space charge region why does it go as just in over tau and not a product of in times p because it depends on both concentrations yeah, let's does it come into the thing they are talking about with the lifetime yeah yeah yeah yeah, let's see here let me see if I can answer that yeah, if you have something like radiated recombination both an electron and a hole to recombine then you would write the recombination rate proportional to the product of the two and your lifetime would depend on that one now this is recombination through a defect you know, so and I'm simply arguing that the recombination is occurring everywhere it's not a bimolecular one the electron is falling down into a trap when you have a radiated recombination process you need both an electron and a hole and they recombine together and the recombination rate is proportional to the product of the two in this case you have a trap that's collecting one and a trap that's collecting the other and the rate at which electrons recombine is the rate at which that whole process occurs and we're arguing that that process is going to be maximum where at the place where n is about equal to p so if I look at the recombination rate you can see I couldn't have an n squared there because its rate is number per cubic centimeter per second but in these bimolecular processes when you do radiative recombination you do get np and what that means is that the tau is dependent on the other quantity the tau might be dependent on I would write it as n over tau but the tau would have a 1 over p dependence in it ok are we ready for a break ok let's take a break then