 You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. So our next speaker is Professor Jeff Grave, saying Professor Grave has been a researcher in portable tags for a long time. Not as long as you. No, maybe the same. Probably. Yeah, yeah. That's true. He's an old timer, but he's worked on a lot of different solar cell technologies. Primarily in modeling and simulation. The results that I showed you in my talk, all of these lines and plots about where the recombination will return, that all came from his simulation tool. So Jeff is going to talk to us, I guess, about simulation and simulation more generally and about something, about what goes on inside the simulation program. Yeah, a little bit. Okay. All right, thank you, Mark. So can everyone see the mouse pointer? Okay, because I'm going to use that rather than the laser so I can see better what's going on. All right, so first I'll just go over what I plan to talk about this afternoon. So I'm going to talk about, so when we're doing modeling and simulation, you know, why we're doing it. Then I'll talk a little bit about single junction or just PV device modeling, a single, just the device itself. I'll talk a little bit about fundamental limits. Then I'm going to talk a little bit about system modeling, and that's something that I've gotten into relatively recently. And then I'll talk about detail numerical simulation, and those are the results. That's basically what Professor Lunstrom was referring to when he was talking about the plots and stuff that he generated. He came from a detail numerical simulator adept. And I'll try to give you just a kind of quick overview of what goes, what's under the hood of a simulation program like adept. And it'll be pretty much universal in terms of whatever simulation program you might be using. You have some basic idea of what's going on and what to believe and what not to believe when you see your results. All right, so objectives of PV modeling and simulation. And a lot of times you'll hear modeling and simulation, the term modeling and simulation used interchangeably. Really, I'd like to try to think about it more as we have models. So we create models, and that's the modeling part. The simulation is when we apply those models to either analyze a device or a system or make predictions about what's going to happen. So we use the models and the simulations for two main purposes, at least from one perspective. And that's one is to sort of understand what's going on with the terminal characteristics. And I think Professor Lundstrom went over the solar cell figures of merits with you of open circuit voltage, source circuit current, fill factor, efficiency. And we'd like to, you know, okay, how does the device design or the device structure, how are those choices affect those figures of merit? And the things that go into that are the size and dimensions of the device, thickness of the layers, the doping of the emitter compared to the doping in the base layer, whether we put window layers on it or not, all sorts of things like that. Plus also in terms of the choice of materials in this, the material parameters, you know, mobility, lifetimes, etc. And some of those things like lifetime in particular, you can have a device or a material that's identical in every respect except for its lifetime. And that tends to be more of a process type of things. So some things are fundamental properties of materials, and other things are process properties of materials where if you refine your process, you can see some improvements in performance of solar cells or other devices. And then often, then what you really would like to do then and once you have that is think you have a good model. Use the model to make some predictions either by say, well, if I change this, how will the performance improve or change? Or if you're testing in a laboratory at 25 degrees Celsius and you take a solar cell out into the wild and put it under the sun, it's going to heat up significantly from 20 Celsius to 40 or even more Celsius and how is this performance going to be under those conditions? And so those are all sorts of things that modeling and simulation can be used for. And hopefully the use of the modeling and the simulation will lead to improved designs. So I'm going to talk about some various levels of modeling and we'll sort of end up with the detailed numerical modeling. But simpler models have a lot of use, too. And so I'm going to talk... So compact models are basically things that are based on the terminal characteristics. They may be more empirical in form. They'll be based on some fundamental physics like the Shockley-Diode equation, which is what I'm going to show here in a second. And they will give you some idea of what's going on. So they'll use lump circuit elements and maybe semi-analytical models, which we'll see here in the next slide. And so a fairly common representation of a solar cell is what I've shown here. The sun is represented by this current source. There's two junctions or it looks like it's a two-diode model in this particular case. One diode is sort of representing... The diode labeled one is sort of representing the bulk behavior of the device. So I think when Professor Lanstrom hopefully talked to you about quasi-neutral regions and depletion regions in the device, so the diode one is going to sort of reflect the behavior of the quasi-neutral region of the device and diode two is going to reflect the behavior in the depletion region of the device. And then there's some parasitics that get attached to that. And here we're just talking about steady states. I'm not going to... There's no capacitances here. But if you're doing this under time-varying conditions then you would also include some capacitors. And there's a shunt resistor which is basically a leakage maybe around the edge of the device. It would be one way you could get some shunt. And then series resistance. And series resistance typically comes from the sheet resistance of the emitter layer because you have a front grid which you'll see here in a minute where the current has to flow laterally and that gives you some resistive losses. The grid fingers themselves, the metallic grid that you attach to make electrical content. Contact will also add some series resistance. And so it's very common to use a model of this form to basically be able to predict what the IV characteristic looks like. So let's just take a look at all the terms. Corresponding to the short circuit current is the current source. This term is diode number one. The second term is diode number two. And there's a shunt resistor here and a series resistance there. And you can see that the voltage across the junction is not the same as the terminal voltage and that's why the series resistance appears in the exponentials of the basically the Shockley diode equation. I-01 and I-02 are sort of fundamental parameters of the material that the diode is made of and of the recombination process is involved. I-01 is typically a much smaller number than I-02 but doesn't necessarily have to be that way. Okay, so then we move on to analytic models and those are based on the relevant device physics. Simplified though and typically minority carried diffusion equation and I'm guessing that Professor Lundstrom talked or one of the lecturers talked about the minority carried diffusion equation. We're going to talk about it some more today so if you're not sure what I'm talking about we'll see that again later. And the analytic models provide a little bit deeper insight into the device operation and design dependencies. Much better than a lumped system, a compact model where you have to make some inferences to lumped resistors and very gross sort of things. You can look a little more carefully at things with an analytic model. The device and material characterization methods are based on these models but they are limited in order to be able to basically get a closed form solution to the minority carried diffusion equation you have to make quite a lot of simplifying assumptions. So we're going to talk, like I said, we'll talk about the minority carried diffusion equation here for a moment and let me talk a little bit about my notation in the equation right here. So the little m's are referring to the minority carriers so if this was a... and the capital M is referring to the majority carrier. So if we were in a p-type region, let's say over here in this p-type base region m would be a capital P and the minority carries electrons the m would become a little n but it's the same equation otherwise between the two. And then a simple way to apply the minority carried diffusion equation is to solve it in these regions independently. So we would solve the minority carried diffusion equation in the emitter region here and a couple of minutes ago I had mentioned that current flow you'd see some lateral current flow and actually this is, let me just digress a moment to talk about that. You're going to have current flowing out of the p-side of the device so current will be flowing in this direction. The black here represents the contact so the current will flow in through the contact and then it'll spread out laterally and flow pretty much uniformly especially by the time it gets to the base region as it exits the base contact. So this emitter region right here is a source of the series resistance. So let me get back to that. So everyone here I'm sure has had differential equations and they have differential equations. You have a region in which you're going to solve that difference equation and you need two boundary conditions. So here we actually have three different regions. We can only apply the minority carried diffusion equation to two of them. The so-called quasi-neutral regions. So here's the N plus region emitter. We'll assume it's quasi-neutral and we'll also assume the base is quasi-neutral. And for the sake of this discussion we're going to assume that the depletion region isn't really adding anything to the device. So there's very little recombination or generation going on in the depletion region. And in good solar cells especially operating at a sufficiently high illumination level or forward bias condition you can pretty much ignore what's going on in the emitter let's say in a good silicon solar cell. The diode characteristic is typically an n-factor equals one which is your hint that you don't really have to worry about the depletion region very much. You can include it but you don't use a minority carried diffusion equation to account for the current generated in the depletion region. And for the sake of this discussion we're basically going to assume that generation recombination or zero in the depletion region so that all the current and recombination that we see are going to be in the emitter and in the base. At the edge of the depletion regions we're going to apply the law of the junction and I believe I saw in Mark's presentation a discussion of the law of the junction so I'm not going to belabor that point. But that gives us a boundary condition that depends on the junction voltage. And then at the back contacts we also have boundary conditions and at the front contact we call that we use a surface recombination boundary condition and SS effective. And the reason it's SS effective is because there's actually even though you can do two dimensional and even three dimensional simulations if you're going to solve the minority carried diffusion equation though for the most part if you're going to get an algorithmic solution you're going to limit yourself to one dimension. So you have to basically treat this surface as a single point and come up with a single boundary condition that will account for what's going on in that interface and right now, right here I'm just illustrating that with SS effective. So this is an effective front surface recombination velocity. At the back contact you see I have two different boundary conditions. If you had, if this back surface field here or this P plus region here wasn't there at all we would use this boundary condition that it's a perfect element contact and the excess carrier concentration at the contact is identically equal to zero. And if you took a simple device physics in your very first thing to look at a device physics course solving the minority carried diffusion equation that's probably the assumption that you made. However, in a good solar cell you're going to put a more heavily doped region here at the back to basically prevent the minority carriers from reaching the contact because recombination is bad and hopefully we'll see some illustrations of that later. And in that case there would be a back surface recombination velocity that we would use to represent that. And I'm going to show some slides that will show how the predicted performance of a solar cell is going to vary, depend on some of these parameters and give you a little bit better intuitive understanding. Okay, so it's worth knowing, like I already mentioned, that the front surface recombination velocity is not just from a single it's not just from the contact it's not just from the passivated surface here, which is where we let the light in. It's actually a combination of the two. And you can derive a fairly simple, I mean it doesn't look that simple, but it really is a fairly simple expression for what that effective front surface recombination velocity is. And I'm not going to go through all the terms here right now, other than just to point out a few key ones. This little small s here represents the percentage of front contact coverage. So that's basically the area of this small finger right here compared to the total area, surface area of the device. Because that will end up being a key parameter. G is sort of the average generation rate in the emitter. And you probably recognize all the other terms, you see the diffusion length and the thickness of the device as such in there. So it's hard to look at this and really understand what's going on, but you can look at a few special cases just to get a feel for what's going on. So let's suppose that you were able to make the grid size so small that it was essentially negligible, or getting very close to zero. Then the front surface, effective front surface recombination velocity would just be whatever characteristic velocity of the surface of the, in this case, silicon of a semiconductor with its anti-reflective coating. And typically you can get those numbers down quite small. So as low as one or two, I think, in some of the best silicon solar cells, maybe even lower than that now. If you have full metal coverage, the effective surface recombination velocity goes to infinity. And that's equivalent to, go back here, to saying that the excess care of concentration goes to zero. In fact, if you look at this equation, this goes to infinity, the only way to keep the derivative finite is to go to zero. So infinite surface recombination velocity basically corresponds to an ideal ohmic contact. In the dark, you get a effective surface recombination velocity that's some function of the coverage of the device. It'll be the actual one plus another term depending on the diffusion coefficient and the thickness of the, in this case, the emitter layer. At short circuit, in terms of the modeling, effectively, when you're at short circuit, assuming there's no series resistance anyway, it's as if the metal contact isn't even there. What you would model it with would be whatever the free surface of the semiconductor were to look like. But at open circuit voltage, it would be a different number which would be a combination of whatever that free surface looked like. And basically this term here is sort of the effect of the metal on the surface recombination velocity. What we found when we did simulations was is that you're pretty well off, even though you're not going to be right at short circuit exactly, recombination at short circuit in a good solar cell really isn't typically all that important. And if you model it with whatever it should be near open circuit voltage, you'll do a pretty good job. So you don't really need to use your analytic model with a lot of different values and trying to figure out you can say, this is characteristic of the front surface and that's what you'll use. But I digressed a little bit here, so we'll go back to the minority carrier diffusion equation and just not worry about SF effect of how it varies. We'll just treat it as a constant when we look at the solution to it. We can learn a lot by solving the minority carrier diffusion equation and I'm not going to do a lot with it other than just show you the basic scheme for solving it. Anyone who took their in their first differential equations course is the second order differential equation with a forcing function which is the generation rate you basically take the homogeneous solution add a particular solution invoke the boundary conditions and solve for basically all of these coefficients and we're not going to take the time to do that and I'm sure everyone breathes a sigh of relief not to have to go through and derive this but if you want to you can certainly do that on your own. It's tedious but it's very straightforward in fact you don't even have to really know differential equations because once you recognize what the solution is it just becomes a messy algebraic equation. We're going to assume that we've solved this, solved for all the coefficients and now we can actually look at a lot of things so we can for instance we could plot the minority carry concentration if we wanted to. I'm not going to show that here but we can basically from this derive the terminal characteristics and see how the terminal characteristics depend on some key parameters and that's what I'm going to show you here in the next few slides. So if we take sort of a typical silicon solar cell this is the result for a typical silicon solar cell with nominal choices of parameters and I'm just going to vary a few of them so what I'll vary in this particular case is everything else will be the same but I'm just going to vary the base lifetime and the surface combination velocity will be sort of nominal ones. I'll vary those later so you can see what the dependence of those kind of things are. But if you look at base lifetime and this is basically this is most solar cells are going to be dominating what's going on in the base. You know the emitter does add some effects to it but the primary you know the first order behavior of the device is going to be dominating what's going on in the base. So I'm going to focus on those parameters here and this dashed line down the middle here is representing the point at which the diffusion length is either smaller is equal to the thickness of the base. And those of you who have taken a device physics course probably remember there's a short base diode approximation and a long base diode approximation. So when you're on this side of the curve especially you know so that this ratio is about a third of the thickness of the device you would call that a long base diode because the diffusion length is much smaller than the thickness of the device and over here once you have the diffusion length two or three times the thickness of the device that would be considered a short base diode. So there's sort of two regimes of behavior that you can see here and as you might expect if you think about it a long diffusion length corresponds to a long lifetime and so you have relatively little recombination going on and therefore if you have a little recombination the voltage should be higher you should get higher short circuit current and basically because the open circuit voltage is going up fill factor also goes up and we'll see that a little bit the relationship between open circuit voltage and fill factor later on but really the two key ones then are really the open circuit or short circuit current behavior here and open circuit voltage behavior here and as you might say well as you might well guess the better your lifetime the better those figures of merit are in this particular case we picked a nominal base lifetime and I don't remember quite remember what it is but it's basically it's going to be in the short base diode approximation which means it would be a good solar cell and this is showing what the effect of the back surface field is so this is with you can see that with or without a back surface field what the behavior would be so basically a back surface field would be over here with essentially a surface recombination velocity of zero which means no there's no flow of minority carriers to the back contact instead they're basically being able to be collected and add to the output of the device and over here basically 10 to the 7th is that's the thermal velocity and that's essentially equivalent to being infinite surface recombination velocity and this particular curve would tell you how well you would have to be able to do to get effectively you know what the S would have to be to be an effective barrier to the minority carriers and then you could tie that to in this particular case a doping level for the back surface field to see how close you can get to the ideal case but clearly there's a lot of improvement that you can gain from by having a back surface field the short circuit current can drop by see that's like three parts out of 36 so it's almost a 10% effect on the short circuit current and you can also see it's a significant effect on the open circuit voltage over here another thing that's useful to look at when you're looking at a solar cell is the spectral response which is basically the looking at what the short circuit current is as a function of monochromatic light so if you think about absorption high energy photons are going to be absorbed more quickly than lower energy photons are and so if you'll which means that the front surface is going to have much more of an effect on the collection of high energy photons than it would have on the lower energy photons and you can kind of see that in this particular discussion so the solid line is I guess I don't have it labeled but I believe it's a number right around a thousand or so for the front surface recombination velocity and if you increase that even more basically which you can see it's in terms of the in terms of the high energy photons even when you increase the back surface this is with the back surface on here so it's not doing very good with the high energy photons but if you were able to passivate that front surface and get the front surface recombination velocity in this case down to a hundred centimeters per second then you basically get a flat response out of this and this also shows the effect of the back surface field on the spectral response what this is showing is that the longer wavelength or lower energy photons are better collected when the back surface field is reasonably good back surface field of 10 to the 7th is basically like having no back surface field so you don't have a back surface field at all and your red response which is the longer wavelength photons they get zort back there and because they're right next to a contact they recombine immediately and can't be collected by the junction but if you put a back surface field on as in this case and I don't remember exactly what number I used here but let's assume it's zero just for the sake of argument then you've got a good chance of collecting many of those photons much better chance than you would if there was no back surface field there at all so this gives you some insight into why you might want to change the design of the device in such a way so that you can improve these things alright so that kind of gives you a sense, a general sense of what's going on with how a model might be used so the only thing I try to do right now is use some basically use some very general modeling to let's just take a look at what makes a good solar cell and see if we can come up with maybe some design rules, some general design rules for a good solar cell just based on some very simple modeling considerations and I know for a fact that Prof. Lundstrom showed you the equation much like this and this particular one but maybe I better go up here so let's so first the key thing is always open circuit voltage you know if you're making a solar cell you're going to be able to probably get a good short circuit current easily unfortunately that's only you're not even halfway there once you have a good short circuit current and I think Prof. Lundstrom here he's sitting there in the back and he knows that from his organic solar cells they probably have much better short circuit current than they do open circuit voltage and it's much easier to get the short circuit current but at open circuit voltage that's basically where all the warts show up so a good solar cell if you can get a good open circuit voltage you probably have everything else handled in the device as well it's not 100% sure but very good case for that so in this particular example that's what I'm going to assume is that we're going to look at the open circuit voltage and then we're going to consider a solar cell which has that perfect back surface field so minority carers will not combine at the back contact they have the best chance possible of being collected or contributing to the open circuit voltage and a very thin emitter so again we have almost no recombination going on at the front surface and so a very good emitter not only includes you know having a very thin emitter but that implies something about the front surface too because that's connected to the emitter so we're going to assume that we have somehow made a perfect back surface reflector and have an emitter that basically contributes nothing to the recombination now those are hard things to accomplish but in this particular we're going to try to look to see you know where we could get with things and so let's assume that we were there and then all that's left then to consider is the base region of the device so and it also had an open circuit the minority carrier in the base is going to be constant with position so and the way to think about that is that okay at open circuit there's no current flowing and the current flow remember is the gradient of the carrier concentration so if the current is zero that means that the gradient is zero and I said a boundary condition at each side one is that the current is zero and one is that there's no minority carrier current no minority no minority carrier current at the back of the device so the excess carrier concentration is going to be exactly constant which makes means that when I perform this integral it's very easy because r is constant g is and I'm going to assume g is constant even though it really isn't but it's very easy to treat it that way and we'll talk a little bit more about that when we look at the result but I'm going to assume that well I guess here I don't even really have to because I'm just saying the integral of the generation rate is the light generated current and I guess even if you have exponential relationship there you can integrate that relatively easily but this is the key part right here so you have the minority carrier minority carrier density divided by the lifetime which is the basically recombination rate times the thickness of the base times q to get current and that should be equal to has to be equal to the light generated current now we'll go back and look at the law of the junction because we want to relate this to a voltage so we take the minority carrier current and calculate an equivalent open circuit voltage or minority carrier concentration I mean calculate an equivalent open circuit voltage and with this from the previous page just a little bit of algebraic manipulation we'll give us that the open circuit voltage is KT and there should be I'm probably missing a Q but it's not a big deal and this should be Q this should be KT over Q sorry natural log of the doping in the base the light current divided by the charge and the intrinsic carrier concentration squared and divided by the thickness of the base so that's the open circuit voltage and just for completeness you really need the other figures of merit here also and we're going to assume that you're basically any light that's created in the device you're going to be able to collect and that's really a pretty good assumption it's very easy to get very close to 100% of the photons are generated actually collected and with at least within 1 or 2% and maybe even closer to that fill factor expression here this is actually quite exact you can actually calculate the fill factor exactly it turns out to be a transcendental equation I thought about deriving that for you today but that would have taken the rest of the time probably to do that but you can check out this reference and he talks about how that comes up but this is quite good all the way down for band gas down to about two-tenths of a volt so for all intents and purposes this form for the fill factor ignoring series resistance anyway is quite exact and so with the open circuit voltage you can calculate the current fill factor you can calculate the efficiency of the device but like I said we're going to focus here on the open circuit voltage because that will be the key thing one thing you can maybe not see obviously here but it's true is that the higher the open circuit voltage the higher the fill factor so increasing the open circuit voltage is always good in this case we're assuming that we're getting all the short circuit current is basically just constant so improving the open circuit voltage will automatically improve the fill factor and then that in combination will automatically improve the efficiency which is obviously our ultimate goal but we've broken it down to the key parameter which is open circuit voltage so let's say let's take a look at this and say so high open circuit voltage yields a high fill factor and high short circuit current and therefore high efficiency and repeated the open circuit voltage equation here let's just take a quick look at this so JL is basically ideally it would be the number of photons in the solar spectrum whose energy is at or above the band gap of the semiconductor now it's hard to do that because as you get close to the band edge the absorption isn't perfect you can't have an infinitely thick semiconductor and so some of the photons are going to pass right through the semiconductor without ever being absorbed but you can get close to that and there's some tricks you can pull it's called light trapping for instance if you texturize the front surface in other words instead of the light coming straight in it comes in it gets diffracted it comes in at different angles more likely than to get total internal reflection if you put a mirror on the back you can get the light that hits the back that mounts back and you can get multiple passes and you can get dozens of multiple passes in a good light trapping design so what you want is you want it very thick optically and that will make the short circuit current or JL very large you want it mechanically thin and the reason you want it mechanically thin is just from this equation this equation tells you that the smaller W is the bigger the open circuit voltage will be and the way to think about that physically is that you have a certain number of electron hole pairs or excess minority carriers and you want the density so you have a raw number of those and they're going to be combined to whatever volume the base is well the density is going to go up as the thickness of the base decreases and so as the density of X carous goes up that means the open circuit voltage is going to go up so you want a high number number of high excess carrier density number and that means that you really want to keep things mechanically thin now obviously these two are competing mechanisms you can't make it arbitrarily thin because then the light trapping becomes too difficult to do but you can trade off on those two and get some improvement there now the other thing this tells you is that it looks like you want a higher doping in the base in order to get a higher open circuit voltage and that's somewhat true although you have to be a little bit careful here because the assumptions that I made to get to this point assumed low level injection our light trapping may actually drive us into high level injection which means that I have to go back and re-derive the equation for those conditions and also lifetime tends to be a fairly severe function of doping in silicon especially and the higher the doping is the lower the lifetime you can get plus there's other other recombination mechanisms that will kick in in silicon OJ recombination will kick in once you get excess care concentrations somewhere around 10 to the 17th and again then you'd have to redo this derivation making some different assumptions to see what the limitations are let's see so but you typically would in the bases of typical solar cells silicon solar cells are 10 to the 16th, 10 to the 17th and the reason they do that is to get a pretty good high open circuit voltage there's also another regime which is actually very lowly doped and that's where you get in high level injection but again you would have to derive for that particular condition and I squared you say well I can't do much about that but you can because while silicon has a particular Ni squared a wider band get material is going to have a lower Ni squared and so gallium arsenide has a wider band gap it's going to have a lower Ni and therefore it's going to have a higher open circuit voltage now it suffers though from the fact that it's because it has a higher energy band gap it can't absorb as many photons from the solar spectrum as silicon because it can only absorb the photons with energy above 1.42 electron volts whereas silicon can absorb everything from 1.12 electron volts so there again there's going to be a tradeoff between current and voltage in that particular case so that's the tradeoff with the light current that's mentioned right here and just remind you that we assume perfect back surface reflector and essentially an ideal emitter and then you have to like I said several times you have to modify this expression to account for some other recombination mechanisms but you can do that relatively easily and it turns out you know some cases for instance if you were in high level injection and you basically had very low lifetimes silicon basically radiated radiative lifetime isn't very important because it's indirect band gap material but OJ is important if you had an OJ recombination dominated device you would actually see a two thirds out here in front it would actually have an n factor less than one I don't think I've seen that in anything yet but if you had a really really efficient silicon solar cell where you would probably see n factors of two thirds in that device so this is going to illustrate some of the points that I mentioned before so this is what this plot is available short circuit current as a function of band gap so very small band gaps basically can collect the whole spectrum so there's and I think this is a AM 1.5 direct spectrum if you want to see what this particular spectrum is so you can get a lot of current but not very much voltage and as you go up the current will will decrease and but the available the band gap there for the opposite voltage is going to increase so silicon is right around here and so silicon you can probably if you had perfect light trapping and be able to absorb all the photons in a silicon device you could get as much as 40 or more milliamps per score centimeter of short circuit current and you know Gallium arsenide you can see it's right at 1.42 it's under 30 milliamps per square centimeter so that's the tradeoff between voltage and current that I talked about now I guess I'm going to use this right here right now but you can see if I say I use silicon I'm only going to use this part of you know of the spectrum and these higher energy photons I'm probably not using very effectively I suppose I have a 2EV photon I'm almost using only the average of that photon in the device so coming up in the talk we're going to talk about multi-junction PV systems in other words where there's going to be two different band gaps or three different band gaps or four different band gaps used simultaneously to get a higher conversion efficiency but this kind of shows you the tradeoff between current and voltage the other thing is is that the leakage current is a sort of and this is a plot that shows a number of things so the black line is room temperature that's 300k and this shows how J0 which is proportional to the Ni squared varies with band gap and you can see that white band gap materials have a very low leakage current whereas low band gap materials have a fairly high leakage current and of course solar cells are going to be operating if we're putting it in a power plant we're subject to the whims of mother nature in a day like today outside they're going to be operating even at ambient temperature they're going to be operating at a much higher than room temperature and they absorb energy as well and heat up just in the sun so they can get very hot and you can see what happens as the temperature increases the leakage current increases and therefore the voltage goes down so and this has become more important probably the first 20 years we were doing modeling all the metrics were at room temperature measurements so we didn't really worry a lot about what happened we did worry about a lot but did worry about it but not as much as we're doing today now we're talking about deploying these devices in power plants out in the desert where the temperatures are high in different places and the temperature behavior is very very important and so this kind of a curve will give you kind of a general idea of where you are now this isn't you know this doesn't mean that all silicon devices are going to be right on this line this particular line as I mentioned our first state of the art devices so all the device let's say take the black line here although I probably have to revise this from since the last PVSE because I think some of these numbers have changed but this straight line is pretty good representation for all of the best devices irrespective of band gap but there's plenty of devices that show up all over down here the poor devices will be higher up on this curve there's something called the Shockley-Kaiser limit and it's a line that is just under the at room temperature is actually probably not too far off of where this red line is and as we get closer we'll be approaching that Shockley-Kaiser limit with the very best solar cells but this formula is a pretty good function of making predictions about what's going on with basically where the technology is today and what you might be able to do in terms of producing a solar cell alright so I'm going to talk a little bit about not a lot but a little bit about fundamental limits of solar cells and here's the reference to the paper I mentioned the Shockley-Kaiser paper detailed balance limit of efficiency of p-injunction solar cells and he defines something called an ultimate efficiency so that would be if you could collect all the current that's the JSC equals JL somehow had a perfect fill factor of one and the only way to get that is either have the device operate at absolute zero or concentrate the sunlight down to infinitely high concentration and get the open circuit voltage to be equal to the band gap and again it pretty much takes going taking the cell operating temperature down to absolute zero to get that but it does give you a very good idea of what the potential and how much energy there is available for conversion you know you're not going to be able to convert all of this but you should be able to convert you know a good fraction of it if you can get fill factor as high as possible and open circuit voltage as close to as close to EG as possible in this particular plot I've plotted efficiency and this is for just a single junction solar cell and there happens to be a peak efficiency for that and you can see in this particular for this particular spectrum and I believe again this is the AN 1.5 direct spectrum it peaks almost exactly where silicon is so it's somewhat fortunate that you know silicon is so abundant and so popular material but also for a single junction gives us the potential most energy potential now this doesn't mean that I haven't since I assume that the band gap or the old circuit voltage is the band gap this is a little bit misleading and if you were to correct this curve to account for realistic voltages you'd actually see a double hump and it turns out that both silicon and gallium arsenide are very near ideal materials for terrestrial solar cells silicon probably wins out for terrestrial applications right now because it's so much cheaper than gallium arsenide is but either one of them uses the photons that absorbs completely well you know gallium arsenide lets everything with 1.142 EV and less it doesn't even use it at all so that kind of gives you the hint that gee if I could put another solar cell underneath gallium arsenide had a lower band gap I would be able to capture some of that energy back and that's more or less what we're leading into here in a minute so a fundamental limit is the Carnot limit if you took a thermodynamic class maybe even from your high school physics class you're probably familiar with the Carnot limit refrigeration uses this limit a lot to do its calculation instead of the temperature of the solar cell it would always be the temperature inside your refrigerator and instead of the sun's temperature you'd be using the room temperature of your house to calculate an efficiency we can do much better than a refrigerator can because the sun is quite hot 1500K approximately and a solar cell is approximately 300K and so the Carnot fundamental limit with the assumptions that we've made is around 95% efficiency now we're not even close to that today obviously and this doesn't tell you how to do it just as this is what might be theoretically possible now there are some more detailed calculations that put the limit closer to 87% as you add more and more junction I talked about adding one under gallium arsenide to collect that energy but you can put another one or that and put one over gallium arsenide to more efficiently collect those and you can basically keep stacking and stacking and stacking and there's been a lot of work looking at well what this fundamental limit is and many of them say that many of those calculations point to an acetotic efficiency somewhere around 87% it varies depending on what exact assumptions they've made I looked at this a few years ago and I realized that well you probably can't just keep adding junctions eventually there's going to be you think about every time you add a junction you're taking away current from the other devices because you can't there's only so much current available so you have an infinite number of junctions all the devices have essentially zero current although there's an infinite number of infinitesimal currents but they also simultaneously have an infinitesimal voltage and if you look at the limit of what that is from a mathematical perspective it actually will approach zero as you go to the as you do go to an infinite number of junctions and if you do the calculation again do use this analytic model to do the to do it you'll get an efficiency curve that looks something like this so these different lines are for different solar concentrations start at one sun go all the way up to 10,000 suns which is getting close to the maximum concentration that you could get on earth because you can't concentrate the light anymore so then it's concentrated on the sun so you can't get an effective temperature any higher than the effective temperature of the sun's surface and I've seen numbers ranging from somewhere anywhere from 50 to 70,000 suns probably depending on the assumptions in the calculation so 10,000 is a pretty good number I've seen concentrators at 1,000 suns I haven't seen anything at 10,000 suns yet at least for solar applications but you can see that as you would expect as you add junctions the efficiency goes up but there seems to be some diminishing return going from 1 to 2 junctions there's a big jump 1 junction is I don't think it's on the plot unless it started at 2 but 2 to 3 you get an improvement 3 to 4 etc but then you start getting into the fact that you keep diminishing the current in the sense that it's a current voltage product things will it tends to flatten out and it turns out the flat part is somewhere around 55 junctions now no one is even contemplating 55 junction solar PV arrays right now although there's a company called MCOR in Albuquerque they're right now working on a 6 junction PV system so this is kind of a good lead in so once you have more than one junction you kind of have to change how you look at things a little bit you know, it's, you know, fill factor Opal circuit voltage and solar circuit current work real well for understanding a single junction a little bit harder to use those to understand what's going on when you have 2, 3, 4, or 5 junctions and one of the research projects I'm working on right now is a DARPA funded project we're actually looking at a 4 or 5 junction system and so you can see one of the conceptions of that system is shown here so you have light coming in of all different colors have some kind of collecting optics here and it may be concentrating it as well in this particular concept you have a say a very wide band gap semiconductor right at the top and you'll let the light go through it and all the energy below its band gap passes through and then you have a basically an optical splitter here and this particular, this is meant to be a represented dichroic mirror some of the energy is reflected and goes to what we call mid band gap devices in this case there's two devices stacked on top of another and the light that's transmitted through the longer wavelength light goes to another two cell stack and the efficiency or the output is going to be the efficiency is going to be the sum of all of the power output of all those different junctions but it's kind of hard to look at that and try to decide what's going on in terms of have I made the right choice of junctions and things like that maybe not as clear of what's going on so there's another way to look at things is to write the system efficiency as this quantity and you can derive this from first really from first principles and you can take a look at this paper to see how it's done but it's a relatively simple expression and it kind of gives you some insight into some of the difficulties that you might get. So this eta ultimate is what I'd refer to in that previous graph on the Shockley Kaiser call if you have a particular set of band gaps and assume that you get the open circuit voltage from each of those band gaps that would be the ultimate efficiency that you could get for that set of band gaps. The photon efficiency here is basically making sure that every photon goes to the right band gap semiconductor. So ideally you wouldn't send a 2EV photon to the silicon device. Silicon device would be able to use it but it wouldn't be able to use it as efficiently as say a 1.9EV device. So how well you're able to do that is the photon efficiency. In a real system you have to connect all these things you got 5 junctions you got to be able to get that power into an electrical inverter or something there's going to be some electrical losses associated with that so that's what the eta IC interconnect efficiency is. And then there's a term that looks a little bit like our standard figures of merit there's this weighting factor here which is basically just a ratio of how much energy each of the PV devices is seeing the sum of these betas is equal to one but other than that it's really not the key thing to look at right here. Fill factor is the same as fill factor as you already know but instead of looking at open circuit voltage we look at the voltage efficiency which is the ratio of the open circuit voltage to the band gap. So its efficiency would be one if we could somehow get to the band gap voltage. And this actually when you look at this equation and stick in typical numbers what sticks out at you is that this ratio is much smaller. This collection efficiency is very close to one. The voltage efficiency is nowhere near one. In fact for low band gap devices it's quite small and so this is definitely a key area for researchers on how to get the open circuit voltage as high as you can in collection efficiency which is typically one and it would be great if we could convert half the energy in the solar spectrum into electricity. But if you take a look and say this is really the product in some sense of six terms the ultimate efficiency, photon efficiency, interconnect efficiency, fill factor, voltage efficiency if we had just one junction it would just be those six terms and now you got six numbers multiplied together. In order to get 50% you have to have about 90% efficiency in each of those steps. Well the ultimate efficiency for a single band gap is not there it's roughly a half so you have to go to multiple junctions in order to get this number even close to 89 a lot of junctions to get up to that high and the other key thing that jumps out at you is that the voltage efficiency is just very low. Alright so I'm going to sort of shift gears right now and get back to basically where my roots came from and actually Professor Lundstrom's too my PhD thesis was on detailed numerical simulation sort of following in Professor Lundstrom's footsteps he did a one-dimensional device simulator and I did a two-dimensional device simulator and then these programs have evolved over time here but the detailed numerical simulation is based on much more rigorous device physics. I think Professor Lundstrom I thought his slides and I think he showed you the summing and ductuary equation, the Poisson's equation the continuity equations so we're going to basically solve those equations with making as few simplifying assumptions as possible which means we're going to need the computer to do that and what it does is if you can do that then you can basically generate everything that we did with the analytic models but without having to make the the same simplifying assumptions. We can generate terminal characteristics and that's sort of like I mentioned before sort of the predictive capability of it and then also make do diagnostic capability because not only can we generate the external terminal characteristics we can also take a look at what's going on inside the device and Professor Lundstrom mentioned that he showed you lots and lots of pictures of what was going on inside and that's probably the main difference between at least one of the most powerful features of the detailed numerical simulation is like taking a microscope and looking inside the device but it also gives you, there's still a lot to be said for analytic models because they're simple to use and give you a little bit more intuition of what's going on but this is a convenient way to make sure that the assumptions that you've made are really valid. So before I get talking more about what's under the hood in terms of detailed numerical simulation they kind of give you a very brief not comprehensive overview of solar cell simulation at Purdue and I already mentioned that Professor Lundstrom started with SCAP 1D with his program back in about 1979. We both had the same major professor, Dick Schwartz and I followed on a couple years later with the 2D simulator SCAP 2D, very original title for the name and then Professor Lundstrom actually kept working with modeling for a while and he had students working in the mid-80s on a program called PUFS and that was mainly for 3-5 heterostructures at about that same time Dick Schwartz and I started looking at amorphous silicon solar cells and we created a program TFS it's called Thin Films Semiconductor Simulation Program and that program is probably the main source of where ADEPT has been derived from but it started out trying to do amorphous silicon solar cells and we had to include a lot more device physics to do the amorphous silicon and that's when we realized that if you write the program right so that it can handle lots of different materials you don't have to write a new program every time you look at a new material and so you can see here we look and yes I'm not going to go through the whole list but you can see there's just a number of materials that we've been looking at and ADEPT 2.0 I think it's called is on the nano hub that's the original Fortran version of it we have multiple and it's just as one dimensional simulation there's multiple C versions of the code both 1D, 2D and even 3D actually it's a single code that can do 1D, 2D and 3D simultaneously and that allows you the benefit of using some of the same code over especially for device models whether I have to rewrite things and I'd say relatively recently MATLAB which is basically written in C has become sophisticated enough and fast enough that I'm right now working to basically convert the code into a MATLAB environment and that'll make it a lot more customizable by an individual user because it'll just be a MATLAB toolbox you'll call a function and then you can manipulate the results in any way that you want and not be at the mercy of whatever the designer of the particular software that you're using decided was important to look at in terms of analysis so when you do a detailed numerical simulation you gotta give us some inputs and I was gonna put this on a slide but I forgot to do it and this is sort of but this is sort of key I remember the first programming class I took almost the first day of class the instructor said remember this I'm not a hacker in them, guy go garbage in, garbage out the computer is gonna do exactly what you tell it to do and if you tell it wrong it's gonna give you exactly the wrong answer it'll give you the answer that's right for the input that you got but it may not be what you thought you had told it so never you know even though I do numerical simulation all the time I don't trust the output at all until I've done some sort of sanity check on the result and tested it and that's the key thing you know just because someone wrote a program and it spits out a result doesn't mean it's right and remember you're also giving it inputs and you may have given it some wrong inputs and I know some of our students have run into that very fact I'll get some very strange results and they'll either discover that it's that it's something that they did in terms of the inputs aren't behaving the way that they thought they were there's some honest questions about what assumptions the program is making whether they're valid or not but you'll give it the structure of the device you gotta set up basically the geometry of the device you gotta describe all the materials all the material properties because those are all inputs to the program and then you gotta tell it what operating condition that you're at so the light that's on it, the operating temperature whether it's DC small signal transient all sorts of things like that and you've already probably seen I don't know if professionalism showed you a typical input but a depth basically uses doesn't use a GUI although you could write a GUI to generate this type of an input file that's basically just a simple set of dictas that sort of describe the device and the material the properties of each layer in the device or each region in the device outputs are probably the thing that everyone is mainly most interested in and a professionalism showed you lots and lots of pictures of that but once you've got the solution then you can look at anything you want to try to understand what's going on and as I mentioned a couple times before the terminal characteristics are sort of like a predictive sort of you use that maybe as predictive and then when you look inside the device that's sort of a diagnostic use of the code and I've just got a few plots that sort of illustrate that this is for a three junction we talked about multiple junctions and so these are the output characteristics IV characteristic for three different junctions generated by a depth simulations and then if they're connected in series you would get a total IV characteristic that looked like this and you notice that the short circuit current is limited by the short circuit current of the lowest one which happens to be the green curve in this particular case since they're series connected you're always limited by the the lowest current you can look internally to the device what I find the most useful plot is is looking at the recombination of the device because that tells you an awful lot about what's going on just sort of an example right here you see actually you see two spikes in the recombination rate right here and right here and you might say does anything else say that gee something really bad is going on there in terms of recombination this is something I really need to fix but you really need to look at something else as well which is how much does that recombination contribute to the total recombination going on the device which is the dashed line here and you'll notice that as that line goes through these points it barely even budges which means these spikes in recombination which look like they're very important aren't important at all and actually what this says is that because this dashed line starts out almost at the surface something very close to the front surface of the device is creating a lot of recombination and at the back it looks like 15% of recombination is just occurring right at the back contact and that these these spikes in recombination um really aren't important at all because they're not adding anything to the recombination current okay so now we're going to get under the hood the under hood pod so this is the semiconductor equation so we have Poisson's equation, the continuity equations and then the transport equations for this and you know so it's relatively you know three partial differential equations with some auxiliary equations for the for the currents and you know the operating conditions material properties and other physics are in the boundary conditions and the parameters that appear throughout there's temperature dielectric constant um traps which includes doping generation rate recombination rates mobilities, VP and VN are called band parameters and they account for the fact that the band gap may not be constant with position it may have a heterostructure and that's that um those properties are captured in the band parameters so all the physics really is in these things in terms of the specific physics are in these things so we have these differential equations which we can't solve analytically without making a lot of simplifying assumptions which that's where the minority carrier diffusion came from but we don't want to do that we want to try to solve these as exactly as possible so the first step is to do this numerically is to transform the differential equations into difference equations and that means creating a spatial grid either in 1D, 2D or 3D and changing the derivatives into differences so think back to the fundamental definition of the derivative which is delta y over delta x and you know have those approach 0 well we're just going to have those small we're going to approximate the derivative essentially by using differences and you can go through and do that so you'll create a difference equation for every point in the grid when you when you do that now results is is a very large set of nonlinear difference equations typically in a 1D simulation um you'll use 250 1000 mesh points so that means you have 250 times 3 because there's 3 separate equations at each mesh point so 750 1000 or 3000 equations um and unfortunately they're nonlinear which means you can't just make a matrix and solve for the solution but I'm sure you're all familiar with Newton's method for solving you'll find the root of a nonlinear equation so you can apply a generalized Newton method to solving this large system of nonlinear equations and iterate to find its zero or its solution essentially and that's what's represented here in this equation where k is representing an iteration number and I think for a typical number for that is probably 10 to 20 iterations is probably what you can typically do sometimes even fewer than that some complex materials may take a couple hundred iterations but typically a single crystalline is pretty well behaved and relatively quickly and so the programs can run very quickly I've had my grad students run literally hundreds of thousands of run simulations overnight so obviously it's got to be pretty quick to to be able to do that and what we'll solve for are the carry concentrations and the electrostatic potential at every point in the device now you think well gee if I have a thousand nodes that's three thousand equations that's a matrix that's three thousand by three thousand fortunately though it's very sparse and so this is a representation of the sparsity of that matrix in fact in one dimension it's block tri diagonal so that you know there's nine times see this is three thousand by three thousand so this is a thousand nodes of the six elements in the matrix so nine million elements but only thirty three thousand of them are non zero and also since it's banded I only have to store the non zero ones and it's very efficient to solve that equation 2D even bigger equation let's suppose that you had a hundred by a hundred by a hundred mesh times three equations per mesh that's thirty thousand by thirty thousands that's nine times ten to the eight elements half a million of them are non zero but that's still a small fraction of one percent are non zero so sparse matrix now we used to spend a lot of time using with mess as a solve that MATLAB now almost handles this automatically for you you just use sparse matrix and it'll figure out the best way to invert it and solve it for you so I used to spend a lot of time worrying about that I spent almost zero time worrying about that anymore I just spent all nine billion or ninety billion nine tenths of a billion elements and that's basically it so let me just kind of go through this under the hood sort of thing for you so just one last time just don't lose track of the equations that you're solving you know when you're using the computer program you know and don't think of it as magic you know adept isn't magic centaurs isn't magic PC1D isn't magic they all basically solve the same set of equations and they all solve them pretty much the same way with slight variations but fundamentally they're the same concept and the key thing is the assumptions that go into it so you know they have a set of equations they'll discretize them solve them with some type of newton type method then for the nonlinear part and then plot the results and give you some you know they'll give you some control over the inputs which is good and bad if you're not sure what to do you can make some very bad choices for what the input should be but then again relying on what they think the best numbers are can lead to some problems as well so when you think about that you should know now have a basic idea of what goes on under the hood and any type of simulation program but the main thing I want you to come out of here is with question every result make sure it makes sense to you because until it makes sense to you you really can't believe it because these are things that are very hard to test without having some intuition involved in terms of yes that makes sense in terms of how it behaves alright so any questions for a multi-adjusted cell it seems like just putting in the series you said something about if the series if the short circuit currents don't match and you have problems I would see that could cause some cells to be sort of input into forward bias to you so is that part of the optimization well yeah yeah picking the bandgap so that's perfect but you know there's there's debate about the many of the cell manufacturers they're much simpler to make a lot of times they'll be built in a stack all basically a monolithic stack and so it's much easier not to have to worry about getting extra context but in this although I want to make them in series but then you have to optimize to pick the exact of the right bandgaps and let me go back to this graph in this particular graph I show a number of junctions but in each case the bandgaps of the junctions were picked to be the ideal bandgaps to give you the maximum efficiency for that number of junctions so it is very sensitive and it turns out that if you put them in series it gets very sensitive to the operating condition because you got to be exactly matched and if you look the efficiency curves like just for two junctions it's a very steep curve here's your exactly matched if you change let's say the spectrum of the light just a little bit either way you fall off of that whereas if they're electrically independent it's a much flatter curve so there's the meeting that I came back at there was two camps there one that believed that we can pick the bandgaps it's not going to be sensitive enough that we worry about it another camp that really insisted that you really should have multiple terminals now the people have to worry about all of those electrical lines really don't want to have to worry about all that because it makes the manufacturing a lot more complex actually I've had graduate students look at and other people have looked at this as well what's the sensitivity sensitivity to exact match of the spectrum how much can you get away with by putting them in series with one another but the key thing is though is to match that bandgap and not only match the bandgap but make sure you're matched at what the anticipated operating temperature is because it's also sensitive to that bandgap changes with temperature as the temperature goes up the bandgap goes down and so you're matching changes with temperature as well so you have to anticipate what your operating temperature is going to be but clearly that's not going to be a constant you know in the winter especially in Indiana it's going to be much lower than it would be in the summertime yeah I mean there can be in thin film materials actually they do try to do that partly because they have the advantage there comes from if you have trouble getting the base lifetime high enough and so that the diffusion length is shorter or on the order of the thickness of the device then that built-in gradient of the field of the doping can help you enhance collection but if you have a very long lifetime then you're probably not going to see much benefit from it so again it depends on the particular situation this is the electron of some of the photons and the whole energy is added like that and the electron distributes that whole energy to the whole device so is the self heating problem very important so you really need to especially in concentrator devices where you're instead of just having one heat sinking is very important I mean that's the basic answer and with concentration it becomes even more you want to basically be able to pull the heat out of the excess heat out of it as fast as possible sort of like a fault can you get anything out of spectrally concentrating the life what do you mean by spectrally concentrating doing oh grabbing the other words yeah and there people are looking at some of those things so that you somehow convert one photon into two maybe use it more effectively or conversely combine two low energy photons together to get a higher energy photon together not even something that's so complicated is that just something where like you maybe take like a 1.1 eV photon turn into 1 eV photon so that point 1 eV is dissipated in some optical piece instead of like something you need to heat sink really well I'm just wondering if like if you can change where the heat dissipation is enough to win I don't know and I guess it would probably depend on the efficiency of the process to do that yeah but yeah that's an interesting idea actually what do you primarily do with amulets are you able to look at the simulation for amulets on time only very little with professor along I know he's doing some simulations so I'm not sure I could but ask if you have a particular question I'll do my best answer I don't think I don't think we can use at least right now and I'm sure you can use simulations to decide whether organics are better than inorganics you can certainly use simulations to help understand what's going on with organic devices to see how to make them better right now they're not competitive efficiency wise with inorganic material so I'm just going to I'm just going to say I'm just going to say efficiency wise with inorganic materials but modeling can certainly help improve that situation and do things but I'm not sure I'm going to tell you that inorganic is better than organic because there's so many things in terms of the simulation that aren't part of that equation especially the the most important thing whether technology is going to be when or not is its cost and how much does the electricity that comes out of it cost you and detailed numerical simulations will contribute to understanding of that but they're not going to give you the answer because they're really going to be the economic assumptions that are going to dictate that I think we have a whole lecture on organics right? We save the best of us and have lots of other I'm sorry When you solve the the equation in the boundary I think it makes to use the law of junction Were you talking about detail numerical modeling? No In detail numerical modeling no All we solve for is basically recombination and charge things at the interfaces A lot of junction actually comes naturally out of the solution of the detailed equations So the boundary equation for your model is just the boundary from the Contacts and the surfaces yes Interface No Those are interfaces you can treat those as actually bulk properties so they're not really treated as boundary conditions But as the beginning of your slides some slides show the law of junction Oh sure but that was for solving the minority carry diffusion equation And that's a simpler analytic model where then you have to apply that but when you do the full detailed model you don't have to do that any longer You're welcome If you have a really great lifetime of history such that you're in a short diet regime Is it to your advantage to make the base longer at that point? No. In fact you go back to this equation that we showed earlier That's the wrong way It's exactly the opposite Your advantage is to make it It's in the denominator So the advantage is to make it narrower Now that's assuming that you can trap all the photons and basically get generation from all the photons in there And the reason that you want to make it smaller is that it's a density open circuit voltage related to the density of excess carriers If they're combined to a smaller region and it's the same number of excess carriers the density is going to be higher If the density is higher then the voltage is going to be higher