 You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. But so I'm testing your ability to listen to me for a few hours and also my ability to speak for a few hours but I'm switching subjects now so we're going to talk about something completely different. We're going to talk about solar cells. So we have a few different faces in the audience including some experts like Professor Agarwal and solar cells so it'll be a little more challenging but what I'm going to try to do is to give a very basic introduction to solar cells. So some of this material for some of you will seem a little too basic because I'm going to explain what an energy band diagram is and if you didn't know what an energy band was you probably haven't been following the last few days so but there won't be a lot of that and I mentioned to my wife this morning that I had 69 slides and she said all of those poor people but you know this is a high level I'm it's a high level look and we have four more lectures where we're going to delve into it in a little more detail so it's an introduction to solar cells and so let me start out by thanking my students some of whom I see in the audience here they've been working very hard over the last few days to get my plots for me so I want to thank them for all of their help in putting this together. Okay so you know the idea of a solar cell has a long history you know in the 1800s people discovered the photovoltaic effect you could shine light on materials and get a current to flow and you know Albert Einstein got a Nobel Prize for figuring out what the photoelectric effect was you know how the absorption of photons translated into electrical current but it really wasn't until the 1950s after the invention of the transistor and semiconductor technology was perfected at Bell Labs that people started to make reasonably good solar cells and the field started to be taken seriously as a means of generating electricity so the first work was done in 1954 by a team at Bell Labs oh I forget now I mean this solar cell was five or six percent efficient I think basic silicon solar cell and there were high hopes for the technology now if you're interested you can see what's happened in efficiencies if you go to the this is a plot that's maintained by NREL the national renewable energy lab so you can find this somewhere on their web page or you could go to the Wikipedia site on solar cells this is a plot showing efficiencies over time and I don't have a laser pointer here but you know you can see what's happened there there's an interesting plot down there about 1976 it says IBM TJ Watson Research Center an efficiency of about 22% so that was a record efficiency done by Jerry Woodall who's now a professor here in Purdue some of you know about and that efficiency was not exceeded for something like 15 or almost 20 years by other technologies that was a gallium arsenide aluminum gallium arsenide solar cell with a heterojunction it was very nice work let's see if we look at crystalline silicon so crystalline silicon is still the mainstream technology and when I do examples and things I'm going to be talking about crystalline silicon because it's just the easiest to explain what this is all about crystalline silicon single crystal are the black squares and you can see around 19 late 70s 1980 that blue line is about 15% or so that's when I left my job at Hewlett Packard and came back to do a PhD on solar cells and a lot of very good work got done then people really dove into the device physics at that time people thought it was a mature technology it had been developed from five or six percent for when Bell Labs first demonstrated it it had been increased to 15% or so when it was used to power satellites you know the efficiency gains had saturated and then there was our first energy crisis in the late 1970s and people got serious about solar cells and a lot of very good work was done we dove in not only in crystalline silicon but other materials and figured out ways to raise the efficiency and you can see from that blue line with the with the rectangles there that the efficiency the record efficiencies now are up around 24% so that's that's a very big increase from 15% what was thought to be relatively mature to 24% it'll be very hard to keep going on that we've extracted almost every percentage point of efficiency that we can get out of crystalline silicon now but you do see some lines you see a what is that 42.4% up there how's that done that's a material that that's a solar cell that uses a combination of band gaps you know we'll talk about this a little later so it uses different band gaps that are optimized to different parts of the solar spectrum and if you can hook those all in series appropriately and take out the photons that need to go to the right band gaps to my conductor they're much more expensive to produce but the efficiencies can be remarkably high and then there are a whole variety of other techniques in their various types of thin film and polycrystalline which have respectable efficiencies but are much lower costs and we'll talk a little bit about that later on okay so if you would like a recent view from industry Mark Pinto is that applied materials and he has a very nice talk from an industrial perspective about what the future prospects look like for photovoltaics so you can find that on the nano hub so I would refer you to his talk I'm going to be talking by and large about some just some very basics how do these devices work and they're incredibly simple so you might think there's not much to say it's just a P in junction with light shining on it or some kind of junction with light shining on it but you need a very good knowledge of P in junctions to figure out how to extract every percentage point of efficiency out of it so we have sunlight shining on a diode we make a solar cell and if we want to understand how this works we first of all have to understand how a P in junction works so we can talk about that in the dark first and then we have to understand a little bit about how you absorb light in a semiconductor and then we can put the two together and talk about how a solar cell works all right so let's do you know we we treat P in junctions in about the first third of a course here and semiconductor devices but let's just talk very briefly about what a P in junction work is and how it works because it's not too difficult to understand the basic principles so you remember if you have silicon you have atomic number 14 you have these discrete energy levels if you put silicon atoms together in a crystal and all of their wave functions overlap then all of those energy levels broaden into a band and each one of them has one for every atom that it that it originally came from so we have a band of energy levels that are so closely spaced that the electrons can move between energy levels without thinking about discrete hops now the only bands that really matter to us are the ones near the top the top most filled level or band which called the valence band and the top most empty band because under operating conditions we can take a few electrons out of the valence band or we can put a few electrons into the conduction band the deeper levels the core levels are just going to remain unperturbed and shielded from us and nothing is going to change there so if we draw an energy band diagram we only worry about the bottom of the top band which at zero degrees is empty and the top of the filled band and at zero degrees the all of those states below the top of the band are filled and the band gap is a material parameter for silicon it's 1.1 electron volts for gallium arsenide it's 1.4 depends on what semiconductor we use all right and then just to acquaint you with some basic terminology and semiconductors if we just have pure silicon you know in silicon these days is probably the purest material that human beings make it's been refined to enormous precision very small densities of defects if we take an intrinsic pure piece of silicon ideally we shouldn't have any empty states in the valence band and we shouldn't have any electrons in the conduction band but at room temperature there's some thermal energy that can break some bonds and can promote an electron from the valence band to the conduction band now we've got a missing an empty state in the valence band that we think of as a positive charge carrier a hole and we have an electron in the conduction band okay that's an intrinsic semiconductor equal numbers of electrons and holes we crank the temperature up we have more thermal energy to break bonds we get more holes in the valence band and more electrons in the conduction band okay now what's an n-type semiconductor so if I intentionally introduce an impurity in a semiconductor like phosphorus which has a valence of five silicon has a valence of four and there's one electron left over and we can it's weakly bound into the phosphorus atom and at room temperature we can easily break that bond so for every phosphorus atom that I put in if I put in Nd of them I break a bond and that extra electron is now a free electron to move around in the conduction band so we would say that that material is doped n-type and the number of electrons is equal to the number of phosphorus atoms that we put into the silicon you know if we go if we cool it down they might freeze out because it might not be enough thermal energy to break that weak bond to the phosphorus atom we could do this we could also make it p-type so we could introduce a material like boron boron has a valence of three in a crystal in a silicon lattice it wants to be surrounded by four nearest neighbors so it's got one two few electrons so you know so in the in the boron atom then there'll be what we can do is to satisfy its bonding to its four nearest neighbors we can pull an electron out of the valence band and satisfy that bond and now we've introduced a hole in the valence band so for every boron atom that goes in we have one empty state in the valence band and it's the energy to do that is very weak so the number of holes is equal to the number of boron atoms that we put in okay so now the concept of a Fermi level and those of you that have been with me for the previous lectures you know all about Fermi levels and Fermi functions but for those of you that are brand new to this lecture let me remind you of what it is we sort of think the Fermi level tells us how the states are filled up it's like you can think of it as a water of a liquid water level things below the level are filled things above are empty so if the Fermi level is up near the conduction band it means that at t equals zero there's an abrupt drop-off all the states below the Fermi energy are filled all the states above the Fermi energy are empty but if there's some thermal energy then it means there's a small probability of the states above the Fermi level to be occupied so that gives us electrons in the conduction band there's a small probability of states below the Fermi level to be unoccupied but the valence band is way down below it so there's virtually none right so in my p-type semiconductor I would draw the Fermi level down near the valence band it means that most of the states below it are filled but close to the Fermi energy there's some probability that a few of them will be empty so the Fermi level is near the conduction band for n type near the valence band for p type the Fermi function is given by this expression and you can see that if the energy is equal to the Fermi energy the probability that the state will be occupied is exactly one half but if the Fermi energy is located in the band gap where there are no states then it's probability of one half time zero so there's nothing there okay all right and if you go if you look at that expression and you go to energies that are way less than the Fermi energy that expression will go to one if you go to energies that are way above the Fermi energy that expression will go to zero all right okay so now back to intrinsic semiconductors again if I don't have any of those boron or phosphorus atoms then I've got an equal number of electrons and holes just thermal energy that is broken a bond gives me one electron in the conduction band one hole in the valence band that number will call n sub i it's the intrinsic density of carriers so that's an important number for semiconductors and the product of the number of electrons times the number of holes is ni squared because each one is equal to ni each one came from that same broken bond and you might guess that the energy that it takes to do that is related to the band gap and the probability that you'll break one of those bonds is e to the minus band gap over k t so in silicon the band gap is 1.1 electron volts and k t at room temperature is 0.025 electron volts so the probability that you'll break one of those bonds is e to the minus 40th so it's very small but there are a large number of silicon atoms 10 to the 22nd per cubic centimeter or something so what you'll find is that the number of electrons and holes is about it's almost very close to 10 to the 10th per cubic centimeter in intrinsic silicon minuscule fraction of the total number of atoms that are there if you heat it up to a very high temperature you can get much more thermal energy and you can break many more and you can get lots of carriers okay now this is the interesting you know so I'm giving you a synopsis of semiconductor physics in a whirlwind and then we'll talk about solar cells you know what if we dope this by n type then the number of electrons at room temperature is about equal to the number of phosphorus atoms that we put in N D okay now the thing that we need to remember is this law n times p is equal to n I squared that always holds in equilibrium that's like a chemistry law of math action so I know what the number of electrons is if I want to find out what the number of holes is it's just n not is N D so it's just the number of holes P not is n I squared over N D so if you take a look at this in this example if I put 10 to the 17th phosphorus atoms per cubic centimeter in that's a moderate doping you know not too heavy not too light if we put that in and I is about 10 to the 10th so the number of holes is 10 to the 10th squared over 10 to the 17th so number of holes is 10 to the third now we've got even fewer than we did in equilibrium and it works the same way if I have a P type semiconductor now I know that the number of holes is equal to the number of boron atoms that we intentionally put in but N P is still n I squared so if I put in 10 to the 17th boron atoms I'll end up with 10 to the third electrons such that the product of the two is equal to n I squared okay so now we can get to a P n junction a solar cell is a P n junction right so we have to talk about what a P n junction is so a P n junction we just bring an n type and P type semiconductor together so this isn't the way you make a P n junction but it's a nice way to conceptually think about it we have an n type semiconductor we have a P type semiconductor we bring the two together and we ask what will happen well this Fermi level if we think about this is like the like the level of a liquid that's filling up the states and the top surface is the Fermi level if I bring the two together it's like you've got two separate lakes at different levels you dig a channel between the two the water will flow and the water level will equalize in equilibrium there's only one Fermi level in one position so when I bring these two together I can only have one Fermi level right when they're separate I have two isolated systems I have two different Fermi levels so if we bring them together that's what happens so if we want to draw the energy band diagram for this P n junction we start by drawing a straight line for the Fermi level because we know it's constant and I'll just ground it over here and I need some reference somewhere it can be arbitrary so I'll say it stays where it did what was that side was that my P side of the semiconductor okay now if I draw the energy band diagram I'll draw something like this if I get way to the right then I won't know that I've made a junction and it'll just be a p-type semiconductor with the Fermi level down by the valence band if I get way to the left it's just an n-type semiconductor that doesn't know that there's a P n junction over there to the right somewhere so the Fermi level has to be by the conduction band and then I just smoothly draw a line from one to the other that's the energy band diagram of my P n junction okay so you can see that in the region in the transition region the Fermi level is a long way from the conduction band which means there are very few electrons and it's a long way from the valence band which means there are very few holes so people call that a depletion region and physically you can think that the holes have diffused in one direction and the electrons have diffused in another direction and they've left behind this depletion region where there aren't very many carriers okay now how did this all happen you know what we had for an electron electron has a minus charge so if you apply a potential you'll lower the electron energy by minus Q times the voltage so this movement of charge set up an electric field that set up a potential difference between the n-type and p-type that lowered the energy on the n-side and pulled the n-side down so that the conduction band will come down and be where it's supposed to be with respect to the Fermi energy so there must be a positive voltage that developed on the n-side okay that positive voltage is called the built-in potential of the P n junction and this is a really interesting voltage you know that we talk about a bit when we do semiconductor courses you have a P n junction you own the lab you put your voltmeter across the P side and the n-side what do you measure you know nothing but the voltage is there alright and this takes a little bit of discussion that you know that maybe we can do at the break or something what you argue is that what the voltmeter really measures is the difference in Fermi levels it's not me it doesn't measure the difference in the electrostatic potential what it really measures is a difference in the Fermi energies and in some cases those are the same things but when you have junctions and you have built-in voltages in equilibrium they're not the same thing you know you can see what would happen if that was a real voltage I would just attach a resistor across there I would there would be voltage across that resistor current would flow would be given by ohm's law I would be delivering power to a load I'd have a perpetual motion machine right something would be wrong but there is a potential drop there and it's central to the operation of this device and you know we can measure it with special techniques and things and another thing to think about is you can see that that potential drop is about the band gap divided by Q the band gap in electron volts because the Fermi level in the valence band in the P side was near the valence band the Fermi level in the N side was near the conduction band I have to establish a voltage that lines those two so it's roughly the band gap that it takes to move those two into alignment and you can get this simple little expression if you derived exactly what it is okay and you know one of the other things that we that we teach students in beginning semiconductor courses is that the variation of that energy band is occurring because there's a variation in the local electrostatic potential so the slope of that is the gradient of the electrostatic potential it's the electric field so if you look at the gradient of that you can get the electric field you can see it's positive there so you can think of the electric field as exerting a force on holes to the right which stops them from diffusing and stops current flowing and exerts a force on electrons to the left which stops them from diffusing away from the end region that's what establishes equilibrium and stops the current from flowing and then we draw this little circuit diagram here where the arrow points from the P side to the N side always alright that's our convention now let me look about this in in equilibrium a little bit more so I have this potential barrier that got set up this is what keeps the electrons on the N side and keeps the holes on the P side that when this charge sloshed around and moved apart and set up an electric field it set up a potential energy barrier that holds all the electrons on the N side and all the holes on the P side but still things are happening at a microscopic basis if I look at what's really happening electrons have a probability of hopping over that potential barrier and then some of them will just drop down the potential barrier and those two processes just balance in equilibrium so that we get the right number of electrons in equilibrium on the P side and the right number of holes on the N side now what is that probability well you know there are lots of problems in physics where you ask what's the probability that you can get over a barrier or you ask the things like if you know the density of of oxygen molecules on the surface what's the density 50,000 feet higher well it's e to the minus gravitational potential mgh e to the minus barrier height over kt in this case the barrier height is that q times V bi so the probability that an electron will get over that is e to the minus 40 because V bi is about one electron bolt very small but a few of them will get over there a few of them will drop down just enough to balance everything out and keep it all in equilibrium okay now we go under forward bias because our solar cell is going to develop a voltage and deliver power so if we apply a positive voltage to the P side positive voltage lowers electron energy and these are electron energy band diagrams so a positive voltage on the P side is going to pull everything down and if I pull everything down it means that that potential energy barrier is not as high as it was in equilibrium it's the built in potential of about one volt minus whatever forward bias I put on it if I put on a half a volt then the potential energy barrier is a half a volt okay now the probability of getting over that barrier is much higher it's still e to the minus barrier height over kt but the barrier is much smaller so the probability is exponentially bigger so in fact the probability is the equilibrium probability times e to the q applied voltage over kt right which kt is small so you apply just a few tenths of a volt that can be a very big number what it means is it's much easier now for electrons to get over that barrier so we're going to be out of equilibrium on the inside we're going to have many more minority carrier electrons on the inside than we did in equilibrium because they can hop over that barrier from the inside and get to the P side same thing we'll have many more holes on the inside than we did in equilibrium because holes the whole energy goes down so the whole energy hop over a barrier by going down okay so the important point here then is that there is a lot of what we would call excess charge on the inside there are excess electrons that weren't there in equilibrium and we're going to ask the question how much what is the total concentration of excess electrons in the P region so we would just integrate from the beginning of the P region to the end that total concentration and if I want to find how much current flows current is charged divided by time so that excess carriers it's the current I know it's going to be charged divided by some time and I'll have to figure out you know what is that time mean but current is always charged divided by time so I'll just have to figure out what the time means now at the same time I'm injecting excess holes onto the inside so there's some excess hole charge Q sub P and it will give me current also when I just add the two up okay now as we'll discuss in a minute here this time is actually the average time it takes for one of these electrons or holes to recombine and I'll talk about that in a minute or it's the average time that it takes to diffuse to a contact where it will recombine quickly so I'm kind of suggesting that every time an electron whole pair recombines I get current through my diode so let's see how that works so this is an important point that then I use to try to understand solar cells it's recombination leads to current so we have a forward bias diode we've lowered that barrier electrons can get injected over and now we have a population of excess minority carrier electrons in the P side now they can recombine you know in a number of different ways but one way they they could recombine is that there could be a defect even though silicon is very very pure well for solar cells you like to use less expensive silicon so it's not not as pure as it could be but it's still very good but there are there are defect energy states inside the band gap and electron could hop down into that and then hop down again and fill up a whole state so that's a dominant way you know it could just hop directly from the conduction band to the valence band and and that happens more more you know in some semiconductors that's the dominant way in semiconductors like gallium arsenide and 3 5 that's the way most of them occur and then that extra that energy that they lost in dropping down in energy is given off as light that's how you make a light emitting diode in materials like silicon that's not the strongest recombination path it's through these defects and the excess energy is given up to phonons and it just heats up the lattice okay so let's let's see what happens you know when that happens the electron drops down fills up the hole we've now lost the hole the p-side was happy it was electrostatically neutral you know it had just the right number of holes and just the right number of boron atoms that were ionized that's where the holes came from everybody was happy now we've destroyed a hole there's an electrostatic imbalance the system doesn't like that so it reacts immediately by kicking an electron out of the valence band to create a hole now the p-side is happy again you know but the inside isn't happy because it was electrostatically neutral it had just the right number of electrons to become to be electrostatically neutral and balance the charge of the phosphorus atoms that were ionized to give us that electron well that one electron flows through the power supply around the other side comes in the conduction band of the n-side and replaces the missing electron so you can see anytime an electron or a hole recombine one electron flows in the external circuit so diode current is all about recombination now the electron might be able to diffuse all the way to the contact and the contact is usually a highly defective region there's a lot of perhaps an interface states a lot of ways it can recombine if an electron gets to the contact and recombines quickly they're the same thing happens doesn't matter whether it recombines in the bulk whether it recombines at the contact all that matters is that it recombines and we send one electron around the external circuit okay so here's a summary of how the forward bias diode works we lower the potential energy barrier by applying a positive voltage on the p-side that allows us to eat more easily electrons can more easily hop over the barrier we now have an excess population of electrons on the p-side those excess electrons know the system always reacts it's out of equilibrium it always reacts by trying to restore equilibrium which means the system will try to promote recombination to get rid of those excess electrons when they recombine then one electron flows in the external circuit okay so now we can get our we can develop an equation for the IV characteristic so current is charged divided by time that Q sub n is the total number of excess electrons on the p-side Q sub p is the total number of excess holes on the n-side and tau is the characteristic time this is either the time it took to recombine through the defect or the time it took to diffuse to the contact and immediately recombine at the contact whichever one is shorter okay so the charge is going to be exponentially proportional to e to the Qv over kt in equilibrium and we have this relation np is equal to ni squared the equilibrium charge of electrons on the p-side was ni squared over over the p-type dopant concentration when I lower the barrier I increase that by e to the QvA over kt the minus one is so that when I'm in equilibrium I don't have any excess charge Q is only the extra charge not the equilibrium charge okay so my ideal diode equation you can see that if I just take t and the RQ and divide it by some t I'm going to get an equation that has some constant out front times e to the Qv over kt minus one so people call that either the Shockley diode equation or the ideal diode equation that's the simplest form of the description of the ideal diode equation now more generally if you work with p-unjunctions a little bit you know that you more generally you put a factor in the denominator and you write the diode current is e to the QvA over nkt so we've described a process that leads to a current that has an n equals one there are other processes that lead to n equals two and I don't remember whether I discussed that in this talk but I surely discuss it in the next one I'll have to remind myself about what's on board here so in practice if you measure a diode you should expect to see a diode ideality factor between n equals one and n equals two if you get something that's like n equals one you say I have an ideal diode okay so dark IV let's see okay so let's take a quick look at here this is a generic solar cell so this is a generic you know not not a record efficiency silicon solar cell but something that might be manufactured at reasonable cost so that dimensions and doping are fairly reasonable it's built on a p-type layer it has a very heavily doped p-type layer on the back and I'll discuss that in a minute and then a bottom ohmic contact it has a very thin n-type layer in the bottom actually point three microns is not all that thin but it's relatively thin the whole wafer is about 200 micrometers thick and you know it has I can't put a top contact I can't put a metal contact across the whole top surface where I can't get I wouldn't let light in right and I'm trying to make a solar cell so there's a metal grid the fingers and you try to design the grid such that you obscure as less than 10% of the of the area with the metal grid if you get it too fine you'll start adding some resistances and things okay so if I draw an energy band diagram you know if I look near the front region it's just like that energy band diagram that I sketch this is a simulated one with a program called adept and you'll hear about that tomorrow professor gray will talk about it but you can see it looks just like the one we sketched the n-type layer is doped pretty heavily so the Fermi level is actually inside the conduction band the p-type layer is doped moderately now if I go way back to the end remember that there was this heavily doped p-type region at the end so if I go way back at the end you know the whole structure is 200 micron stick if I go back in the final one micron or eight tenths of a micron you can see that when the valence band gets closer to the Fermi level it means that I have more holes so you can see the heavily doped region right at the back and then you can see it's a little bit more lightly doped there and actually this is something that we'll talk about in my lecture tomorrow you can see an energy barrier there we're going to insert excess electrons in the conduction band and you can see that there's an energy barrier there that they have to hop over if they are going to get to the contact that's designed to keep them from getting to the contact because the contact is a defect where they recombine we want it's called a minority carrier mirror so if you look at the IV characteristic again this is a simulated one using realistic material parameters and lifetimes and things if you look to the left you're on a linear plot and you might remember it it takes about six tenths of a volt to turn a silicon diode on and for significant current to start to flow but if you look at it on a log plot that's the green plot you can see even below six tenths of a volt there's a lot of current flowing and you can see it on a log plot if you look very carefully here you'll see that there's a region in the middle where N is almost exactly one and I explained the physics of what that was with those simple arguments earlier if you look down near at lower voltages you can see a region where it's starting to increase you know and depending on the band gap and the lifetimes and things it could go all the way to two in this case it's just you're just beginning to hint see a hint that something else is going on we'll talk about that tomorrow and then you see the dip down and what you're seeing there is just that minus one in the e to the qv over kt minus one if you go up to higher voltages you can see it's starting to roll off just a little so N is a little bit bigger than one and that can happen for a couple of reasons the most common one is that there's a series resistance and we'll talk about that later on also okay so these are plots that I'm going to go through carefully tomorrow but I just want to make a point here that I argued that the current is related to recombination so in order to understand the device you'd like to understand recombination inside the device so the blue line is the recombination rate now you'd also like to know where are those carriers recombining because if you want to change the performance or improve the performance you might want to go in and re-engineer the cell to shut off some recombination mechanisms the green line is the integrated total now you can see that there's a bunch of stuff happening right at the beginning at that first three tenths of a micron that you can't see you know so and there's a bunch of stuff happening at the end where it looks like the green line plot goes from zero to one that's a hundred percent of the recombination you can see that an awful lot of the recombination is occurring at that back contact you know that little barrier we put there isn't keeping the electrons away and if I look in the front I can see that there's a significant amount of recombination in the front two orders of magnitude bigger than what's happening in the P region that's because the lifetime is very very short and that very heavily dope material so it's recombining very quickly it's so there's a lot of information that we can get out of that and that's sort of the subject of the talk tomorrow I will mention that roll-off region I've described this ideal diode but you put metal contacts on P and N type silicon you're going to introduce some contact resistance so really any real device that you measure in the lab you'll have a series resistance and you'll only be able to apply your voltage to those two white terminals there meaning that when some of the current flows it'll be lost across that series resistance and the actual voltage that gets applied across the diode V sub a the applied voltage across the junction will be less than the voltage that you apply across the terminals of the diode and as you get higher and higher in current you'll get more and you'll lose more and more of that voltage you're applying in that series resistance and that's what causes things to roll-off so here's a simulation of that same solar cell when you have a series resistance of no ohms you get that red dashed line that we showed earlier if you put a hundred ohms in you'll get that blue line and you can see it starting to roll off so one of the things that people try to do in solar cells is to minimize that series resistance as much as possible so we'll talk about that later on too okay so that's how the IV characteristic works in the dark when I did my PhD thesis I basically worked on dark current of solar cells and I would go home and my wife would say what did you do in the lab all day and I said I'm working on the dark current of a solar cell and she thought this was the silliest thing to do you shine the whole point of a solar cell is shine light on it and why are you working on the dark current now I'll explain in a minute here but let's let's talk about optical absorption so just a few fundamentals so you know the idea is this light comes in it's got some wavelength for it it's got some energy that is Planck's constant times its frequency if the energy is bigger than the band gap it can promote an electron from the valence band to the conduction band that will leave behind an empty state in the valence band so now I've got a hole and will give me an electron in the conduction band so whether or not that happens will depend on the energy of the light that comes in so the energy must be big enough or the wavelength must be short enough such that I've got enough energy to create an electron hole pair so we have to think about this this spectrum you know the the solar spectrum is roughly I think a black body of about 6,000 degrees Kelvin I guess you know roughly approximately if you go in outer space and measure the solar spectrum so this is the amount of solar power in each little wavelength integral this is called air mass zero because there's no atmosphere for it to go through and be absorbed so the solar spectrum will look like this you can see some sharp lines here that have to do with things in the solar atmosphere and if you integrate the power you get 136 milliwatts per square centimeter of solar power and when you make a solar cell you want to compare that to the electrical power that you get out and maximize the efficiency now if you have if you look on the on the surface of the earth you'll get a slightly different solar spectrum and it will depend on how much atmosphere you have to go through so that's going to depend on the latitude that you're at and what angle the Sun is at so people measure this thing they call it air mass it's one over cosine of the angle to the normal and air mass 1.5 is sort of a typical condition for mid-latitudes in the US or mid-latitudes anywhere and it corresponds to a latitude of 42.8 degrees right so you have to go if you have to equate it and you're going directly through you don't have to go through as much atmosphere if you're up here at our latitude you have to go through and there are various you know gases species water vapor in the atmosphere that have absorption at particular frequencies so if you look at the blue line you can see that we have these notches where the solar spectrum that came in is strongly absorbed in these certain bands so it's the blue line that's going to be the incident power for our solar cell if we're making a terrestrial solar cell and the G there means global so when people do these measurements there's a lot of diffuse scattering in the atmosphere so the beam that's coming down is not just a direct beam there's some diffuse scattering too so when they do these measurements they include that so AM 1.5 G means it includes all of those global and diffuse effects as well and the total integrated power is exactly 100 watts milliwatts per square centimeter so you may ask how can it be exactly 100 milliwatts per square centimeter well it isn't but it's just a standard so people have adjusted this spectrum so that it's exactly 100 milliwatts per square centimeter and everybody compares their efficiencies to a spectrum like this its intensity has been adjusted so it's exactly 100 milliwatts per square centimeter so if you get 10 milliwatts out of your solar cell you know you have a 10% efficient solar cell okay so then you can ask yourself well how many photons can be absorbed so if your solar cell is made out of silicon the band gap is 1.1 EV so only photons with an energy above 1.1 EV can be absorbed or that means only photons with a wavelength below something that turns out to be close to 1.1 micrometers so if I look at that same solar spectrum I can only absorb the photons that have a short enough wavelength those are the ones in yellow right the rest of them are totally wasted they contain power they contain part of that hundred milliwatts per square centimeter but I can't take advantage of that in silicon just goes through so the total number of photons per square centimeter per second that I can get if I collect every one of those is 2.761 times 10 to the 17th and that corresponds to a current of 44.24 milliamps per square centimeter these high efficiencies silicon cells are getting remarkably close to that okay so what happens to all of those a lot of those photons have an energy more than the band gap so they put an electron way up in the conduction band somewhere you know then what happens well what happens is it emits optical phonons sheds that energy and it just heats up the solar cell that's all wasted right so that's one of the problems that you have in these solar cells is that you're just wasting the energy of any photon that has more energy than the band gap so there are a lot of people that think about you know are there ways to prevent this from happening one way is to use a lot of different band gaps and just take a slice of the solar spectrum and use the right band gap for that right part of that spectrum that's how those over 40 percent efficiency solar cells are made but people also think about you know is there some scheme that I could get that electron out before it loses all of its energy and then I wouldn't waste it you know those other ideas that fall under this category of third generation PV okay now the question we have potentially every photon with an energy above the band gap could be absorbed but if I have some finite thickness of solar cell not everyone will actually get absorbed some of them won't have a chance to so here I'm just pointing out that the incident flux is going to be the k as e to the minus alpha x so to the k exponentially with position and alpha is the optical absorption coefficient if it's greater than zero it means electrons are being absorbed so I can compute the the generation rate because as the flux decays the reason it's decaying is because electron hole pairs or are photons are being converted into electron hole pairs so I can just differentiate that position dependent flux and I get an expression for the generation rate versus position now that's at a specific wavelength so then I have to integrate over that complicated solar spectrum but all of that's easy to write a mat lab script these are things that my students did for me in a few of these plots that I'm going to show you now so your question is what determines alpha and this gets into some semiconductor physics that we won't be able to go into deeply but this gets into the details of the band structure so remember for a classical particle energy is momentum squared divided by two times math now in a semiconductor crystal the momentum is really the crystal momentum h bar k and we have some complex band structure but it often time looks something like that now when the minimum of the conduction band and the maximum of the valence band occur at the same momentum at zero we call that a direct band gap semiconductor now it turns out photons carry very little momentum so I can make vertical transitions there come in with an energy bigger than h new I can make a transition up and I can conserve momentum and it's no problem materials like this absorb light very efficiently now if you have silicon it's what's called an indirect gap semiconductor the minimum in the conduction band is at a different momentum than the maximum in the valence band the photon doesn't have much momentum so in conserving momentum in exciting an electron from the valence band to the conduction band what you have to do is to find a lattice vibration with the right momentum so that it's a it's an extra interaction with a lattice vibration that's less probable so the absorption coefficient is going to be weaker so it's not going to be as efficient in absorbing as a direct band gap semiconductor is so here's an example so sigs is a material copper indian gallium diselinaide is an is a material that is widely used for photovoltaic applications professor agarals lab does a lot of work on it over here at Purdue silicon is also the most common commercial technology still today although some thin film materials are making significant inroads if you look at the percentage these are the percentage of photons absorbed you look at just the photons above the band gap what percentage of those are absorbed versus thickness of the absorbing layer you can see that for sigs that's a direct gap material one micron will absorb 90% of all photons that can possibly be absorbed for silicon you know I have to get closer to 10,000 microns of silicon if I want to absorb every photon because the absorption coefficient is much weaker it's an indirect gap semiconductor so if you're looking to minimize material costs for low-cost photovoltaics you'd like to use a direct gap semiconductor because a very small amount of material can absorb all of the photons silicon it takes a very much thicker layer okay so to get as many electron hole pairs generated we have to get the light in so people do things like anti reflection coatings to make sure you get as many photons in and then you want to make the solar cell as thick as possible but sometimes there are tricks that you can use to make it effectively thick and this is one of the ways so this is a record efficiency silicon cell from martin greens group at university of new south wales it was over 24% efficient it's only got 300 to 400 microns of thickness of silicon so it can't absorb all the photons but if you look at the structure it has a set of etchings there along 111 planes so it gets inverted pyramids it creates this structure in silicon such that when the light comes in if it doesn't get absorbed if it doesn't get transmitted into the silicon it gets bounced off into another one it's got a chance to get transmitted there so it has a very low reflection coefficient once the light gets in if it goes all the way to the back and it hasn't been absorbed yet you can see that most of them on the back surface is a thin layer they're labeled oxide that oxide layer is about a half wavelength thick so that when the photons go down reflect off the metal back contact and come back they interfere in phase so it's a layer that's designed to maximize reflection they go back through and they make a second pass when they make a second pass they hit most of those inverted pyramids at an at an angle that's above the critical angle so they're internally reflected and they stay in and they go back down and they reflect out again that's called light trapping so it can make a physically thin layer of silicon appear to be very very thick so we get a chance to absorb the photons. Okay so we've generated the electron hole pairs we have to collect them if we collect them then then we have a p-n junction so the way we collect them is we just see that if we create a minority carrier electron on the p side near the junction it'll just fall down in energy and go over to the inside so the p-n junction collects the carriers that's why we use a p-n junction. So we have a layer usually the top layer is relatively thin that was three tenths of a micron in my example the absorbing layer is thicker so we can absorb all of the photons and what we want to be careful of is that the photons that we absorb the minority carriers that we generate have to diffuse to the junction so that they can be collected and go out to contact. Now some of them might recombine at a defect before they get to the junction some of them might diffuse backwards to the back contact where they can recombine so you want to engineer the structure so that most of them diffuse towards the front and are collected. So a parameter that people will talk about is the collection efficiency so it's the if JL max so if I know how many electron hole pairs are generated I multiply that by the charge on the electron that's the maximum current I could ever get the current that I actually get out J light is a little bit less than that and that ratio is called the collection efficiency for a high efficiency solar cell you want that to be well over 90% okay and I'll just mention one of the nice things about a silicon cell is that you don't expect this collection to have much to do with the voltage it's just going to fall down that barrier you know if I apply a forward bias and I make the barrier a little smaller it'll still fall down that smaller barrier and go out the end type so the collection is relatively insensitive to the voltage that I apply across the diode okay and I'll just mention briefly these electrons that are generated away from the junction they typically diffuse before they recombine they can diffuse on the order of the diffusion length which is the square root of diffusion coefficient times time so we want a material that has a high diffusion coefficient or high mobility or material that has a very long lifetime so it has plenty of time to diffuse to the junction get collected now that's one of the reasons that you can't make the layer as thick as you want because if you make it too thick it'll be more than a diffusion length away from the junction and it won't get there so that's one of the reasons that when people design cells like this you try to keep the absorbing layer physically thin so that once you generate an electron hole pair it can diffuse to the junction and get collected okay so now we can talk about a solar cell we've done it in the light and we've done it in the dark so here's how it works the light creates an electron hole pair the p-n junction collects the electron hole pair the electron goes out the end contact electron going out the end contact runs around the external circuit and comes back in the p-type and that gives us a current that flows opposite to the direction of the arrow so the direction of the arrow tells us the direction that the current flows when we apply a positive voltage to the p-side that's a forward bias junction the light generated current flows in the opposite direction so current flows and flowing in the opposite direction of the arrow it flows through that resistor R which is the load that we're delivering the power to we're trying to do something useful with it that creates a positive voltage that gets applied to the positive side that forward bias is the diode and gives me a current that goes in the other direction so the net result is that I get some combination of those two so that forward bias lowers the barrier and now some electrons are being collected but other electrons are hopping over the barrier in the opposite direction and going to the to the p-side so what I get is the combination of the two and the simplest way I can think about this is it just behaves like a superposition so I had the current in the dark we discussed where that comes from now we have a current in the light and I discussed that it's more or less independent of voltage it doesn't matter how big that barrier is once an electron gets close to it just falls down the barrier goes out the n-type contact if I want the total IV characteristic of the solar cell I just add the two and I'll get a characteristic that looks like that so that's my that's why I've worked on dark current from my PhD thesis because then if somebody tells me what the light current is I just have to add it to the dark current and we can understand how the solar cell works so if we look at this again you can see that first of all you can see that the current is negative the voltage is positive that means that the power which is the product of the two is negative what does a negative power mean it means I'm not dissipating power in the diode I'm generating power now if you look down at v equals zero the power is zero we have a lot of current but no voltage so the power is zero if I look at the open circuit voltage where no current flows I have a lot of voltage but no current so there's no power there somewhere in between I get maximum power that's where you want to operate the device and that's called the max power point and you can see that it's less than the product of the short circuit current and the open circuit voltage and it's less by a factor people call the field factor and the field factor is something that you can't do a whole lot about it depends on the shape of the diode characteristic which is the Shockley diode equation you can make it worse but you can't make it better very easily and then the efficiency then is just a power out which is a short circuit current times the open circuit voltage times the field factor divided by the power in so I'll just mention briefly I told you I was going to fly through these and I'll do it a little more slowly tomorrow but this idea of superposition it's not immediately obvious why this should work you know so the idea is we take this light generated current we take this dark IEV characteristic of the diode and we just add the two and we say that's how the solar cell works so I'll say a little bit more about that in a minute or two now there are going to be some non-idealities you know we're going to have a large PN junction because we want to collect a lot of sunlight and generate a lot of power there may be shorts here in their shunt passage leakage mechanisms defects in the diode so that would be a resistor a shunt resistor that's in parallel with the junction that's not good it's going to lower the performance the contacts themselves are going to introduce some extra resistance that's RS the series resistance so if you do that you can see here's the effect of taking that blue curve is the simulation of the device without any extra series resistance if you add some extra series resistance in you can see that it's lowering the current in the voltage at the max powerpoint or it's lowering the fill factor so it's bad you know people worry a lot about that the shunt resistance does the same thing so you worry about these kind of defects a lot when you're trying to make high efficiency solar cells okay so just we run a simulation so you can see what the IV characteristic looks like this is not a record 24% efficient cell but it's not bad it's 20% efficient this is a typical thing you could do without lots of fancy expensive processing open circuit voltage is about 616 millivolts short circuit current remember what was that number anybody remember we said the maximum current that you could get if you absorbed every photon above the band gap and and collected every one of those was that 42 or something 44 okay so we get 39 so if we had a better design we might be able to get a higher current fill factor is point eight three we haven't added any extra have we added any extra series resistance in the simulation one ohm so you know one ohm people people worry you know even one ohm is a significant series resistance for a solar cell and that lowers the fill factor a little bit we could get a little bit higher if we didn't have any okay then I'm just going to wrap up here and say a few things and you know we'll get a chance to discuss these in more detail in some of the other talks so this idea of super position you know I told you that this was not maybe intuitively obvious as to why this should work it really works well for silicon and I'll show you this in my lecture tomorrow but it's not so easy to explain why you know and we just take this light current we take the dark current we add the two and we say super position says that this will give us the response when it's illuminated now you know how can you justify this if you have a system that described by a differential equation if the differential equation is linear you can add solutions okay let's look at the differential equation to describe a semiconductor device or a solar cell right here they are the first one is divergence D is equal to rho the charge density this is Gauss's law the second one is divergence current density is equal to generation minus recombination the third one is divergence of whole current is generation minus recombination you know the currents are given by drift diffusion equations we have expressions for the recombination rate the point is we have three coupled nonlinear partial differential equations it is really not obvious that you could take the solution in the light you could take the solution in the dark and add them there's no reason to expect that would happen from these equations now it turns out that you can and I'm still searching for a simple explanation of why this works so well in silicon we'll show you some simulations that show how beautifully it works but to try to explain why it works is takes a little bit of work there have been a few papers about this I I haven't been able to boil it down to a one line explanation you know they can establish some conditions under which if these conditions hold you can expect to see superposition but there are many solar cells especially thin film solar cells contain more defects than high-quality crystalline silicon for which superposition does not apply now we should talk briefly about efficiency limits you know what determines the limit efficiency so we have only three things to think about short circuit current open circuit voltage and fill factor we can understand those three factors we can understand efficiency the fill factor is just determined by the shape of the IV characteristic which is the diode characteristic that's pretty pretty fundamental we can't change that very much it we can make it worse if we have too much series resistance so we'll spend a lot of time trying to reduce series resistance now the short circuit current well the smaller the band gap the more photons I can absorb from the from the solar spectrum so the short circuit current strongly depends on the semiconductor that I've chosen if I've chosen a given semiconductor then I have to work hard to minimize the reflection of photons from the top surface to maximize their absorption in the layer and to minimize their recombination now the open circuit voltage so the open circuit voltage has something to do with VBI and and VBI is it goes as proportional to the band gap so what you'll find is that as you increase the band gap the open circuit voltage increases so there's a very classic paper that if you work on solar cells everyone needs to study the Shockley quizer paper because this is a very famous paper where back in 1961 they did a calculation to try to determine the theoretical upper limit the efficiency of a solar cell and they did it so well that everybody has used it ever since and but the results are kind of intuitively easy to see if you have a smaller band gap the smaller your band gap the higher the short circuit current so what you're seeing on the bottom axis on the bottom that's a normalized band gap on the top you can see it in electron volts as you increase the band gap the efficiency gets better and better because you have more and more photons that you can absorb okay now larger band gaps give higher voltage so as you in I'm sorry did the opposite didn't buy as you increase the band gap you have fewer and fewer photons that you can absorb but the volt the open circuit voltage is increasing so that's why the efficiency goes up as you keep increasing the band gap the voltage keeps going up but the number of photons that you can absorb goes down so the current drops so there's some peak there there's some optimum band gap where you're extracting for that particular band gap you're getting the most photons that you can get at the highest voltage that you can get it turns out that that peak look how nice that is that peak is just a little above one electron volt you know that silicon it turns out that silicon is very very close you just calculation a little more carefully it might be a little bit above gallium arsenide is also very close to the optimum so if you're going to pick one band gap you'd pick something around just a little above one bolt give you the best efficiency and you can see the numbers here the best that you could possibly get is is a little over 40% I think people have refined these calculations and they're the optimum is felt to be a little bit lower than that but it's up around 40% so when you're doing solar cells you know there are three things you're trying you're basically trying to reduce the cost of producing electricity because that's what's needed in order to make solar cells economically viable so one thing you can do is to have very high efficiencies but it's usually very expensive you saw that martin green cell that I showed you processing that is very expensive you can try to produce cells that have good enough efficiency but at very very low cost and then you would use cheaper materials thin film materials polycrystalline materials that you don't have to epitaxially grow and they're very expensive and take high-temperature processing you know a third approach which is really related to the first is you could use concentration and the idea there is you spend a lot of money on a very high efficiency small cell and then you have a very big set of optics to collect lots of solar energy and focus it down on that so most of your expenses in is in the optics to focus it down onto a smaller cell that you can afford to make more expensive so those are the three general approaches and the thing that people are after these days and trying to solve energy challenges is something you call grid parity and they go through these economic analyses and in order to get electricity that's at five to six cents per kilowatt hour you need to have a system that you can and you need to install a PV system at about a dollar a watt so if you build a hundred megawatt system it will cost you a hundred million dollars and if you can do that then all of the costs will translate you can charge five to six cents per kilowatt hour and it can be competitive with other sources and the system includes more than just the solar cell so you package all of these solar cells and you wire them up into a module and then you have to have power conditioning electronics to take that fluctuating DC that comes out and put it in into AC and put it on to the grid and you have all of these you have to install them and you have to you have to clean them and you have to do everything else that it takes to maintain them and operate the system. Now the current costs 2011 are about three dollars and forty cents per watt so that's a long where a long ways from where it needs to be in order to be economically competitive if you just look at what's happening in the improvement and manufacturing processes by 2016 it should be down to two dollars and twenty cents a watt but that's still more than double of where it needs to be to be competitive so this is a big research challenge that people are looking at these are some numbers from the Department of Energy and they're sort of showing you in 2010 this is where we're at in 2016 that's where we're projected to be and this is where we need to be to start being competitive and you can see how it's broken down into the power conditioning electronics are very efficient that only contributes ten cents or contributes twenty two cents now and it has to drop in half but the cell part has to get down to fifty cents per watt for the module so that's not just one cell you hook a whole set of cells together in a module you package them encapsulate them so that they'll last for thirty years the efficiency drops a bit after you do all of that you've got to be able to do that module that fifty cents per watt we're at a dollar seventy a lot now so more than a factor is three reduction is needed so that's the challenge it's different from those of us who work on integrated circuit chips and things where you know the value added comes because you put it in an expensive product the cost of the chip is not that much you know the manufacturing you know you add the value in the design of a complex microprocessor or something and the manufacturing costs are relatively small fraction of the overall cost in we're dealing with a PN junction here the manufacturing costs are really crucial okay so a summary the solar cell is really very simple light is absorbed produces electron hole pairs the junction separates those electron hole pairs sends the electron out the end contact the hole out the P contact that current source as it's flowing in the external circuit induces a forward bias voltage that reduces the total current the output power is short circuit current times open circuit voltage times fill factor and as I just mentioned everything is about cost in photovoltaics okay so I can point you to a couple of references if you want to get a broader overview of the field and I'll stop there and see if we if I haven't warned you out if there are if there's a question or two I'll try to answer it we have one down here in the front thank you which slide slide 24 let's take a look at slide 24 see I got through all 69 of my slides so and most of you are here I'm gonna tell my wife that oh that's not the one which oh I might have done some editing since I passed that out yeah okay yes yeah so what it you know I will say a little more about this in my next lecture yeah you know I'm not so this is what people I mean this is what people call the ideal diode equation and you know what they mean by that is that all of the recombination processes are the kinds of reprints combination processes that I talked about carriers are recombining in the neutral n or p region carriers could also recombine in the depletion region and that gives an n equals 2 now there are lots of other non ideologies that we could talk about tunneling through defects and things and things could get very complicated but but you see how much it would be effect that our simulations how much it would be change percent percent or 50 percent oh no I mean you know especially when you look at these low-cost photovoltaic materials where you have thin films of polycrystalline material processed at low temperatures there can be high levels of defects and there can be many of these non ideologies and it can be impossible to see an n equals one component anywhere I mean that can be the whole game yeah well we'll see well maybe we'll come back to you if we have some yes we have a question in that it depends on short circuit and voltage so like how the carrier the current connection across the so yeah so so so I guess the question is you know what what role does the charge collection efficiency play and if it's less than a so we can easily take the solar spectrum and we can say we can compute from that solar spectrum how many electron whole pairs are generated inside the solar cell so the best we can do is to collect every one of those and that would give us a short circuit current of 44 milliamps per square centimeter or something for silicon now in practice you know we lose some by recombination and I'm going to say a little more about that in my next lecture so different types of cells you know sometimes people design a cell where all of the generation comes in this junction region where there's a strong electric field which can take everything and quickly sweep it out to the right contact there are other things you do to try to have minority carrier mirrors so you make sure that the electrons don't diffuse to the contact and recombine there you turn them around and have them diffuse towards the front but there's a lot of a lot of solar cell design is all about trying to maximize that collection efficiency that's going to be my message on my next lecture it's just recombination and generation that's all there is to the solar cell I'm not sure if I exactly got solar energy is free I heard that yeah so I you know I may ask Professor Agarwal to comment on here because I think he's more of an expert than I am but I know people that do this economic analysis they'll make some assumption for what is the lifetime of this solar array and you know and when you back all of that out they say okay if we can build the initial system for one dollar per watt we can build a hundred megawatt system for a hundred million dollars and if it has their assumed lifetime which is going to be 20 or 30 years and they do all of the other cost analysis then they can make an economic case that they could charge electricity for five to six cents per kilowatt hour so there are a lot of assumptions reliability is a really important factor in in these solar cells because if it if it lasts for 30 years instead of 20 years it's much easier to make the economic case you know you just made that one time investment and then if it's the longer at last the cheaper the electricity you produce is is that on yes it is if you hold it close enough to your mouth so when the energy protons are absorbed by a proton to exalt some of the energy is lost to a proton to exalt that's the heat inside the device right people work on like using thermoelectric principles to extract some of the excess heat well you should know that you're from MIT right I just saw I didn't I just I just saw a paper in was it nature or nature materials maybe last week from the gang Chen group which is on my laptop to try to read so my understanding I just looked at the title and I thought oh this sounds interesting so my understanding is is that he's trying to take this waste heat that for a solar cell this is just waste right and to take that waste heat and couple the solar cells to a thermoelectric device and convert that into electricity and get a little more power out of it I don't know yeah I'm a little you know the efficiency of thermoelectric devices is not that great but if you get it for free you know you've got that heat there and if it doesn't increase your manufacturing cost by much and you can get another one percent or even two percent out of it that would be very useful yeah so that's a good question right so it connects this lecture with the previous two lectures on thermoelectric devices one question I have is like how does this high energy carrier affect the efficiency of the electric current that we get it's just the high energy that we lose yeah so you know what happens we talked about energy relaxation time in the last lecture so what happens is these carriers very quickly in picoseconds or a few tenths of a picosecond shed all of this excess energy and just relax down to the bottom of the conduction band and you know generally that happens so fast that when we think about current flow we don't even have to account for those processes when we're looking at steady state currents and things they very quickly thermalize yeah you get the electrical current but but you don't get all of the energy because they've dropped down you know they've dropped down to the band gap right now you've lost all of that you've lost some portion of that energy that they had you know that's the whole idea of using tandem junction solar cells multiple band gaps I was just wondering have you thought about just in the size of your p&n junction is there a certain standard that you that's regulated I don't know the size of the p&n junction well I mean so you need large solar cell to pass me what you're holding p&n junction is there like a certain standard well I mean basically you you want to collect light from as big an area as possible so that that you would tend to think I want as big a p&n junction as I can get now that the difficult so there is no not that I know of but but but you know there's something related to that maybe I could you know I could point out it might come up in the lecture tomorrow I'm looking for my Martin Green cell here now this is really a nice example of of solar cell engineering you'll you'll notice notice how small that there's a small p-type contact on the end there's also another variety of solar cell that gets also gets near record efficiencies where the end layer is a small thing like that they call them point contact cells so there's a big piece of silicon that might be their n type or p type but the p-n junctions the n region and the p-type contact are very small so they're trying to minimize the size of the p-n junction so it's mostly absorbing materials these p-n junctions are very small so the carriers can diffuse to them and get collected but there's a lot of recombination and things that happen in the p-n junction so if you can minimize that you can those are called point contact cells now I mean the area of the cell can still be very big but the area of the p-n junction itself is can actually be quite small all right are we ready for a break oh no oh yes okay see you're keeping all of these people from a break so I don't want to put pressure on you but it better be a great question pardon me 38 side 38 all right let's take a look at slide 38 now is that the right one yeah yeah yeah yeah yeah yeah I'm not not sure I exactly understand the question. You know, there actually is a direct band gap in silicon, right? There is a conduction band that's up there in higher energy, and the higher energy photons can be absorbed by that direct process. Yeah? My question is, direct band gap is better or indirect band gap? Which one is better? Oh, I think direct band gap would be better. Yeah. Because it would take less, you know, everything is about cost. You know, even if you can minimize the materials cost and you can absorb all of the photons in a very thin layer, it's always best to have a direct gap. And the other question is, what is shunt resistance? Pardon me? Shunt resistance. Yes, yeah. You know, it's, it is some parallel current flow path. You know, it might be a defect. You know, you might have a crystal defect that goes right through the junction and just kind of shorts it out. You might have had a pin hole in your process, so when you deposited the metal contacts, a little bit of metal went down and shorted out. You know, there can be all kinds of extraneous current paths, yeah, that are, you know, can only be bad, right? Thank you. Yeah.