Solar Cells Lecture 2: Physics of Crystalline Solar Cells
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Transcript: English(auto-generated)
00:16
So welcome back. So we change topics now and we continue the discussion on solar cells.
00:25
So this afternoon we're going to have two. I'm going to give the first one and what I'm going to do is to talk a bit about the internal physics of solar cells using a nice clean well
00:40
characterized system as an example, so silicon, and I'm going to use a lot of examples that come from simulations and that particular simulation program is called ADEPT and Professor Gray after the break will give us a lecture on what ADEPT is and how it works and what equation it solves. It's available on the nano hub so some of you may want to use it.
01:05
So solar cells are all about recombination and generation. So let's dive into this. Again I want to thank my students who worked really hard trying to get this lecture together over the last couple of days. So remember how simple this device is. It's just a PN
01:21
junction with sunlight shining on it and to maximize efficiency it's really very simple. We want to generate as many electron hole pairs as we can and we want to make sure that as few as possible of them recombine before we get them out to contacts and collect them. So this is just a short introduction, I think it's short, hopefully it's less
01:42
than an hour, to the physics of crystalline solar cells and we're going to use silicon. So here are the topics. I'll first of all look at what happens under short circuit conditions and then I'll look at what happens under open circuit conditions. That's really all I'm going to talk about because in between everything is either some
02:02
combination of those two characteristics. And let me just remind you about how we think about these devices. So here's my simple PN junction. First of all in the And remember the idea is that the contact on the left is an electron injecting contact.
02:21
That means it sends electrons into the conduction band of the n-type semiconductor. The contact on the right, well first of all on the left, if we apply a forward bias to this we lower the barrier, electrons get over the potential barrier, that means I end up
02:40
with an excess concentration of electrons on the p-side and if they recombine current flows. At the same time, holes get injected to the n-side. Contact on the right side is a metal contact that is connected to the valence band. So we'll call that a hole injecting contact or that's electrons come in and out at the
03:03
valence band. So the idea is what happens is if I inject an electron into the p-side, first of all it leaves behind the missing electron on the n-side so the n-side is not neutral anymore. When the electron gets over on the p-side it's an excess carrier.
03:24
There are more electrons than there are supposed to be in equilibrium. The system always reacts by trying to restore equilibrium which means it'll tend to recombine. If it recombines with a hole it means now I have a missing hole on the p-side and the p-side is no
03:41
longer electrically neutral. So an electron comes out of the valence band of the p-side and out in the metal wire and that creates a new hole. So I've replaced the hole, it goes around the power supply and comes in the n-side, comes in at the conduction band and replaces the missing electron in the conduction band. And that's how the
04:02
whole process works. Anytime an electron and a hole recombine anywhere within the p-side one electron flows in the external circuit. And we can look at it on an energy band diagram and we can see these electron, excess electrons on the p-side, they could
04:22
recombine through defects. That's what we call Shockley-Read-Hall recombination after the people who first worked this out. There's a classic paper that anyone who works on semiconductor devices still reads because it's still relevant. So recombines, we send the electron around the external circuit, but the electrons, remember the property
04:45
of these contacts is we would like this electrode on the right side to allow electrons to come in and out of the valence band. But we would prefer if they just blocked minority carriers and kept them away from the contact. A real contact is usually very
05:04
bad for minority carriers. If the minority carrier electron gets over to the contact it usually finds a lot of defects to recombine with and that's bad because if it recombines anywhere, it gives us one electron in the external circuit. So people will go to great
05:20
lengths to engineer contacts to try to keep the minority carriers away from them but allow the majority carriers to come in and out so that it's an ohmic contact for the majority carriers. Okay that was the die out under the dark. Under light what happens? We shine sunlight, most of the electron hole pairs are generated in the thick base
05:43
absorbing region. So now we have one extra hole in the p-region, we have one extra electron. The electron can diffuse to the p-n junction, fall down in energy and then go out the n side. Now it's flowing in the wrong direction so the current flows in the opposite direction, goes around the external circuit, comes back in the contact. You know
06:09
I have one extra hole meaning I have one missing electron in the valence band in the p side. This electron comes in the valence band and fills that up and now we're back to where we started from. So any time an electron hole pair is generated, one
06:25
electron flows in the external circuit. Now this is important. You know you might, I remember reading a paper a long time ago by a very bright person, I read a lot of his papers, he wrote very good papers. But he wrote one where he said ah, but look at what would happen if the electron hole pair were generated in the depletion region.
06:44
Both the electron and the hole would be collected and you'd get two amounts of charge flowing in the external circuit. But try it out as a little exercise, draw this little diagram if that electron hole pair is generated in the depletion region and you'll see that only one electron flows in the external circuit. It doesn't matter where the electron
07:04
hole pair is generated. So of course he discovered that shortly after his paper was published and then following paper he pointed out oops I made a mistake. That happens. Alright, so you know but recombination can also occur under light. So if we're illuminating
07:24
these, let's say we create three electron hole pairs. One of the electrons might diffuse to the p-n junction and get collected, that's what we want to happen. But some of them might diffuse, some of them might recombine through defects, then I don't get any electron out the external circuit or it might recombine at the back contact. Those
07:44
are all bad things in terms of if I'm trying to get the maximum current out, I'm losing current because they're recombining. So three were generated, one was collected, so my collection efficiency would be the number that came out or the number that were generated, the collection efficiency would be one third. Not very good. Okay, so carrier recombination
08:11
does two things. It lowers the short circuit current and it reduces the open circuit voltage and if we want to optimize solar cell performance we really need to understand this
08:23
carrier recombination. We need to understand how many are recombining and then we need to understand where they're recombining so we know how to engineer the device to reduce the number that recombine. So we need a quantitative relationship between recombination and generation and solar cell performance. And I'm just going to state it because it's kind of obvious
08:44
and I have it in the handout notes, I have a mathematical derivation of this, but you really don't need any mathematical derivation. All you need are these little pictures of electrons flowing around in the external circuit. So the diode current J sub D is equal to,
09:04
I argued that any time an electron and hole recombine in the semiconductor, one electron flows in the external circuit. Any time an electron hole pair is generated, if I collect it, it flows. If it recombines, it subtracts from that. So that first expression
09:24
says, to understand the IV characteristic, all I have to do is to understand how carriers are recombining inside the device and how they're being generated. So I'll ask what is the total amount of recombination? So I'll just integrate across the entire thickness
09:41
of the solar cell from 0 to L, the recombination rate, the number per cubic centimeter per second that are recombining and I'll integrate across the thickness. But I have to be very careful. I have to include the contacts because at x equals 0, if there's any flow
10:00
of minority carriers into that contact, they'll immediately recombine. So the recombination rate will be the net flow of minority carriers into the n type contact. If they're not recombining, there will be no flow of current there. And similarly, on the p contact, if there are any minority carrier electrons flowing to that contact, they'll recombine
10:23
immediately. So that's part of the total recombination in the device. It's the recombination in the volume of the device plus the recombination at the contacts. And the total generation is just the total number of electron hole pairs that are generated inside that solar
10:42
cell. Alright, so let's talk about short circuit then. I'm going to go back. Here's a generic crystalline silicon solar cell. Not super high efficiency, but reasonably high efficiency. Remind you again, we have a very thick n plus, a very thin n plus layer on the top. Actually 3 tenths of a micron is relatively thick. We have a thick p type
11:06
absorbing layer to absorb as many of the photons in this indirect gap semiconductor that we can. And then we have a back surface field that we'll talk about a little bit more, but its role is to try to keep minority carriers away from the back contact
11:21
because if they get there they'll recombine. And that's not good. So here are some of the dimensions and lifetimes and things. So I'll show some results and I'll illustrate how things happen quantitatively just by doing the calculations on this particular cell. So some of the questions that we ask. First of all, what is the total number of electron
11:44
hole pairs that are generated if I shine an AM 1.5 G spectrum on this cell? We'd like to know where they're generated. What percent of them are generated in the n plus emitter? What percent of them are generated in the p type base? We'd also like to know how
12:01
many of them are recombining? What's the total recombination? We'll be careful to include the recombination at the contacts in that. And then I'll ask how good is this back surface field? If I put it in or take it off, how do things change? So these are calculations that we can do. So first of all,
12:22
if this were an infinitely thick slab of silicon, then every photon above the band gap would be absorbed. And if I do the integral of the generation rate versus distance from zero to infinity, I would get the maximum number of
12:41
photons in the solar spectrum that have an energy big enough to create an electron hole pair. And that number is 2.97 times 10 to the 17th per square centimeter per second. Now I have a finite thickness sample. And actually in this one we assume that if the light goes down and
13:00
it's not absorbed, that it will bounce off the back metal contact and go back through a second time. So I'm assuming that that might not be quite right because there might be some roughness there and it might get bounced off at an angle. But we're just going to make the assumption that it makes two passes through. It goes through once if there's anything left over it reflects off
13:22
the bottom metal and goes through again. So I do the integral from zero to 2L. And you can see that I don't quite get every photon that I could. Some of the advanced cell structures where they do this complicated light trapping and they try to get as many bounces as possible, they can get a little more than that. But we only get 2.79 instead of 2.97. So we're giving
13:43
up some that are weakly absorbed and don't get absorbed in this finite thickness. Now the short circuit current in this particular cell, we can simulate this cell. Professor Gray will tell you how we do the simulation. The short
14:00
circuit current is 39.4 milliamps per square centimeter. If I divide that by charge on the electron, I find the flux of electrons that's getting collected. That's a little bit less. That's 2.46 times 10 to the 17th. And that means that some are recombining, so the number that are recombining is 3.31 times 10
14:22
to the 16th. So the ratio of the short circuit current divided by Q to the total number of electron hole pairs that are generated is the collection efficiency. It's the ratio of what comes out over what was generated, and that's 88 percent. That's a pretty
14:40
good number, but for the ultimate record efficiency silicon cells, people do better over 90 percent. Now we can do these plots. So we can run a simulation and we can look at these plots and we can try to see what's going on inside this solar cell. So if I look on
15:01
the left, this is a plot. The blue line is the number of electron hole pairs per cubic centimeter per second that are generated. And I'm going from zero to the end of the cell. So you can see that the high energy photons are very steeply absorbed, and a lot of them
15:20
are absorbed right near the top surface. The green line is the integrated total, normalized to 100 percent. So it tells me what fraction of the photons were absorbed up to so you can see if I go to 25 microns it looks like I'm close to 90 percent of them are absorbed or 85 percent
15:40
or something. And in fact it looks like a lot is happening right close to the surface that I can't see in that plot. So over here on the right we just blow it up. Remember the emitter is three tenths of a micron thick, so if you look at that green line you can see at three tenths of a micron it
16:01
looks like maybe 35 percent or so of the photons are absorbed in that thin n plus layer. Those are the high energy photons that are absorbed very quickly. So we can just look at those numbers. These are the kind of questions a solar cell designer would ask themselves. Where are things being generated? We
16:22
can see 37 percent are absorbed in that top thin layer. 14 percent are absorbed in the depletion region around the pn junction and then the remaining half of them are absorbed in the rest of that thick p type layer. Now we worry about this because
16:41
the heavily doped region is a low lifetime region. I gave up a lifetime there, a nice long lifetime of 34 microseconds. That was of the p-type region, moderately doped. The very heavily doped region typically has shorter lifetimes and in this particular case it might be because
17:01
there are some defects that have been introduced as a result of the doping but it also can be some fundamental processes. One that's important in heavily doped regions is Auger recombination. It could also be the front surface too. In any photon that is collected in the depletion
17:20
region we don't worry about too much because there's a strong electric field that's just going to sweep them across to the right side and collect them. So we don't worry about the fact that those might recombine. But in the base we have to worry about do we have a long enough minority carrier lifetime so
17:40
that those that are generated deep in the base can diffuse to the junction and get collected. Have we protected the back surface so that they don't diffuse to the back surface and recombine? So we worry about those. So let's look at the recombination. This is where we're losing those photo generated carriers. So again the blue line is the recombination rate per
18:05
cubic centimeter per second and the dashed green line is the integrated total so it goes from 0 to 100 percent. So you can see very quickly there that I start at about, it looks like about 75 percent
18:20
of them. So about 75 percent of them recombine very close to the beginning. So close I can't resolve it there. And then another looks like maybe 10 percent or so recombine in that thick P region and then you get to the back contact and it takes a jump up. That looks like it might
18:41
be another 15 percent or something recombine at the back contact. So that's where we're losing them all. If I blow up that surface region because a lot is happening there you can see the blue line is the recombination rate and you can see that it's very high in that N plus region because the lifetime
19:01
is very short so they recombine very quickly. And if I look at the integrated total that dashed line you can see that I quickly get up 75 percent of the carriers that I'm missing that I'm losing are recombining in that thin three-tenths micron layer. So if I
19:25
look at these numbers 75 percent of them are occurring, I'm losing it in the N plus emitter. I hardly lose anything in the depletion region because the electric field is so strong it just sweeps them across before they get a chance to recombine and then I
19:42
lose some 23 percent in the base and at the back. So you can compare those to where the carriers were generated and you can see that although only 37 percent of the carriers were generated in that thin top layer, 75 percent of them that recombine anywhere
20:00
in the device are recombining there. So we're losing a lot in that top layer and it's because the lifetime is so low. All right so I'll just point out, I just want to mention one of the things you do when you run a simulation is
20:21
to make sure that you can believe the results. Usually the program is right, sometimes it isn't, but it's easy on these programs to set them up in a way it's not quite the way you expected to set them up at. Maybe you're simulating the correct solution to a different problem, so it's always nice to check and see other results and getting reasonable. So
20:43
one of the things we do in traditional semiconductor courses is we solve for the distribution of minority carriers in p regions and if I want to solve for the distribution of electrons in the p region I would start with my continuity equation which says divergence of the electron
21:01
flux is minus the recombination rate and I'd say that this is a region that doesn't have an electric field so the current is only a diffusion current and if I plug that current into the equation above it I'll get this minority carrier diffusion equation. So this is an equation that would tell me the profile of
21:21
the excess carriers inside the device. L sub n is the minority carrier diffusion length. It tells me the average distance that an electron can diffuse until it recombines and then I would apply boundary conditions and I would solve that differential equation and I could find the profile and I would expect it to look
21:42
something like this dashed line. I would expect it to be very low if I go near the edge of the depletion region because I'm sweeping them out there just as fast as I can. I would expect it to have some gradient that would give me carriers to diffuse towards the back contact but if I keep the recombination at the back
22:02
contact so it's not too high I won't have much of a gradient there and I won't get too many that are diffusing back there. So this is the kind of analytical solution that I would get. If you go in and look at the numerical solution that doesn't make any of those approximations. It's the green line. You see it has the expected
22:22
shape. If I just take delta n and divide by 34 nanoseconds I would get the blue line which is the recombination rate. So everything behaves the way it should. When you're running a simulation program you always stop and do some tests like this to make sure that I can make sense of the
22:40
results that I'm getting and the numbers that are coming out are about what I expect the numbers to be. OK so let's look at that back surface field because you know and see if it's doing what it's supposed to do. So on the top left there is my energy band and I dope
23:00
the region right adjacent to the contact very heavily p-type. That means that the Fermi level, the red line there has to be very close to the valence band. The p-type base layer is more moderately so the Fermi level is further away from the valence band. So my energy band diagram looks like this. Now if
23:22
I don't have that back surface field then my energy band diagram is just flat all the way to the back contact. The back contact is a place where there are lots of defects. If any minority carrier gets there it can recombine immediately but really the fastest that can happen is going to be limited by how
23:41
fast they come to that contact and the quickest that they can come to that contact is at the thermal velocity. So I would you know that boundary condition is going to be characterized by the velocity at which they are diffusing to that junction and that velocity is going to be the thermal velocity of electrons which is about one times ten to
24:00
the seventh. Now I have a little bit of a barrier there. I wish it were bigger but this is what we're able to do with doping. The barrier height is 0.13 electron volts. So if I take e to the minus barrier height over kT I'm going to knock down that voltage by a little
24:21
bit and you know what we end up there's some probability that an electron will hop over that barrier is e to the minus barrier height over kT so I'm going to expect the effective velocity that they're diffusing to the contact to be the thermal velocity reduced by the
24:40
probability that it can get over that barrier. So it knocks it down a little bit. If I really wanted a good barrier I might try to do something like a heterojunction, put a wider band gap there, get a much bigger barrier and I would really keep things away much better. But we could ask ourselves is this really performing any function in our cell?
25:03
So I could run a simulation with and without that back surface field and if you look at those two green lines the dotted line is what we had before and you can see that's the integrated total recombination rate. So when I get to the end of the cell that has to be a hundred percent of the recombination. That's
25:21
all of the recombination that occurred in the device. It looks like about fifteen percent of the total recombination is occurring at the back contact when I had that back surface field there. And under those conditions I had a total recombination of if I multiply it by Q which
25:41
corresponds to 5.3 milliamps per square centimeter. If you remove that heavily doped region then I get the solid green line and you can see now first of all if you look at the number you can see that the total recombination corresponds to 6.5 milliamps per centimeter
26:01
squared. I have more recombination so it means more electrons are getting to the back and you can see that in that solid green line there's a step it looks like it's maybe about thirty percent or something of the carriers are now recombining at the back surface field. The percentage is twice as high as
26:22
it was before. So the back surface field is helping but it hasn't shut off that back surface recombination totally and you see that the total collection efficiency without the back surface field is eighty five percent with the back
26:40
surface field it's just a little bit higher it's eighty eight percent. Now here's another common measurement that people will do is that they will take the device at short circuit and they'll shine an incident flux of photons at a specific wavelength at the cell and then they'll measure
27:00
the current that they get. The ratio of the current that comes out divided by the charge to the incident flux that came in is the quantum efficiency it's the percentage of electron hole pairs that were collected and now if you sweep that as a function of wavelength you'll get some information about what's going on because the short wavelength has high
27:22
energy they're absorbed very near the surface and that will tell me something about recombination near the surface. The long wavelengths are absorbed more deeply and they'll be more sensitive to recombination in the base or at the back surface contact and if the wavelength is too long it won't be absorbed at all and it won't get anything out so it drops to zero. So you
27:42
can see what's happening here with and without the back surface field. With the back surface field I get a little bit more current at the long wavelengths because the back surface field is helping reduce the recombination and keep them away from the back field and helping them be collected a little bit more. So this is a
28:01
measurement that people do frequently. You can see that the internal quantum efficiency is dropping down at the short wavelengths. That's because they're being absorbed in this n plus layer where the lifetime is very short and they're recombining very quickly. So this is a standard type of characterization that people will do when
28:21
they're trying to understand what's going on in their solar cells. So some questions you can think about. You could actually go into the simulation results and use and deduce what the actual surface recombination velocity is. You could think about doing things like putting heterojunctions in and reducing the
28:40
surface recombination velocity at the back to essentially zero so that no carriers can recombine at the back contact. How much would that improve our efficiency? The importance of the back surface field is going to depend on what the minority carrier lifetime is. If the minority carrier lifetime is
29:01
very long then nothing will recombine in the base and the importance of the back surface field will increase and you can ask all kinds of questions like this. Now the last question here I want to just mention, since most of our recombination is occurring in this thin N plus region, why not just make it a lot thinner so that fewer carriers are
29:24
absorbed there and they'll be absorbed in a region that they can be collected. So here I have to point out that there's a whole set of effects that I'm not going to spend any time on. I've been thinking about this device as a 1D device but it's really a two-dimensional device. I've done one section
29:41
of it here. There's a metal grid, the fingers on the top contact. This is repeated over here so I can illuminate maybe 90% of the area. My metal grid obscures 10% or less of the area. So what happens is that when my minority carrier electrons are generated in the base they diffuse towards the
30:03
PN junction and are collected. Once they get collected they have to travel laterally out the contact and as I go towards the contact the magnitude of that current just builds up because I'm just adding all of them that are collected along that contact. So if the electrons are
30:21
flowing in the direction of the arrow then the current will be flowing in so I'll have a current flowing in the opposite direction and that means I'll have a lateral voltage drop along that. So the voltage along my PN junction will not be the same everywhere. When I was thinking of this as
30:41
1D I wasn't accounting for this effect and the fact that the voltage is less means I'm going to have a lateral voltage drop and the magnitude of that drop is something that can be computed. In effect it's going to appear when I look at the IV characteristic it's going to act like a series resistance but it's a little funny series resistance
31:03
it's a current dependent series resistance because this is a more complicated current flow problem but it's going to degrade the performance of the cell. So people will be very careful about trying to make sure that that layer is not too thin because if it's too thin
31:21
its resistance is very high and then these lateral voltage drops get very big. So when you're designing a grid you'll be careful about the spacing between these metal fingers and about the sheet resistance of that top layer such that you don't introduce these lateral volt drops. All right so that's how things happen at short
31:45
circuit. Let's look at what happens under open circuit conditions. So under open circuit conditions the top expression there I've argued that that's exact. The diode current is recombination minus total generation.
32:04
Under open circuit conditions no current is flowing. So I just solved that equation under open circuit conditions. So the voltage will adjust such that the total recombination is equal to the total generation. That will give me open circuit
32:20
conditions. So open circuit happens when the diode forward biases itself such that the recombination exactly balances the generation and no electron hole pairs come out of the contacts. Now we frequently think about superposition and dark current and I'm going to
32:42
say just a word about this. The top equation is exact. Now in the dark I have a dark diode current. I have no optical generation so G total is zero but I have carriers recombining in the dark. When I'm illuminated I have carriers being generated and I
33:04
also have some recombination occurring under illuminated conditions. Now superposition says I'm going to take my short circuit current and I'm going to assume that it's voltage independent. I argued yesterday why that's
33:20
reasonable for a silicon cell and I'm going to take my dark current and I'm going to get a diode characteristic like this and then I'm going to assume that what happens when I have both bias applied and light applied is just a superposition of those two and I'll get and my open circuit condition
33:43
for example will occur when the diode current in the positive direction is equal and opposite to the light generated current which is negative and I'll add the two and I'll get zero current. That seems very reasonable but it's really a
34:00
little hard to justify. Under illuminated conditions the total recombination in the light is equal to the total optical generation. Superposition says that well I'll look in the dark and the recombination when the recombination in the dark gives me the short
34:21
circuit current then I'm under open circuit conditions. So I'm going to say I'm basically going to assume that the recombination in the dark at VOC is similar to the recombination in the light at VOC. It's not obvious that that should be the case and in some cases it's really a very poor
34:40
approximation. It tends to be a poor approximation in materials that have lots of defects where when you shine light on them you can change the population of these defects and their charge states and now you change all of the internal electric fields and distributions and things can be dramatically different under illuminated conditions and
35:02
under dark conditions. In a high quality crystalline silicon where you have very few defects it generally works very well just to superimpose these two. So it's not easy to justify this but for silicon solar cells in practice it works very well and that's the reason that we
35:23
think a lot about dark current because if we can understand dark current we can deduce what the open circuit voltage can be. So I mentioned yesterday that this dark current has a characteristic that typically under low bias it has a characteristic that looks like n equals two. Under moderate bias it has a characteristic
35:44
that looks like n equals one. Under higher biases it starts to go back towards n equals two and maybe even bigger and that might be series resistance or it might be high level injection or it might be a number of other things. So here's the simulated IV characteristics of
36:01
that silicon diode that we're looking at, that green solid line. The green solid line is on the log plot on the right, the blue line is the linear plot and you can see a nice clear n equals one region. You can see a hint that there's a region where it's trying to go above n equals one at low bias the way
36:21
we expect and you can see a little bit of roll off at the high end which we expect also. So what determines that J zero and the ideality factor and the answer is that the amount of recombination is going to determine J zero but the location
36:43
of the recombination is going to determine the n factor. Depending on where they recombine I'll get different diode ideality factors. So let's forward bias this diode in the dark at about seven tenths of a volt because that's about where the open circuit voltage is and let's ask
37:01
how does the recombination work in this particular device. So here's the recombination rate and I'm showing it here very near the surface, the first eight tenths of a micron, remember it's 200 microns thick. There's a lot of recombination in the emitter, recombination rate is very high but if
37:22
I integrate that through the emitter it's only about 20% of the total recombination. Now if I continue to integrate across the entire device then you can see that maybe 10% of it occurs in the p-type base and
37:40
then I get to the back surface field and there it looks like about 70% of the total recombination is occurring at the back surface field. So one of the points here is that under short circuit conditions and under open circuit conditions different parts of the device are controlling the performance and if you want to engineer, if you want
38:00
to increase the short circuit current you'll work on the parts that are controlling it. If you want to increase the open circuit voltage you might have to work on a different part. So these are the numbers that we just saw, that's where things are recombining and the point I was just making here is if you compare that
38:21
to short circuit conditions you can see it's quite different. Now one of the reasons is that under short circuit conditions there were a lot of electrons that were minority carrier holes that were generated in the n plus emitter so they had a chance to recombine there, that's why I had a lot of recombination there. When you forward
38:41
bias a pn junction there's this thing called the one sided junction, you tend to inject most of the excess carriers in the lightly doped side. The heavily doped side it's hard to inject many excess holes in it. So even though the lifetime is still short we just aren't injecting any many excess holes in there to
39:01
recombine, much more of the recombination is occurring on the lightly doped side. This is just an example to show you what the back surface field now is, the back surface recombination is much more important than it was under short circuit conditions. So you can see even with the back surface field it's not a very
39:22
good back surface field, it's not keeping the minority carriers away. If you look at that green line you see the big jump it takes at the end of the device it goes from 30 percent total recombination and then it jumps up to 100 percent. So 70 percent of the recombination is occurring at that back contact and
39:41
the back surface field is not keeping them away. If you look without the back surface field it's even worse, there it looks like it's probably 85 percent of the recombination is occurring back there and your total current meaning your total recombination is significantly higher.
40:01
All right so let's talk a little bit about the physics, why is the end current one and I did this yesterday so I'll just remind you, we can have recombination occurring in a couple of different places. It can occur in the neutral N or P type region, those regions are almost neutral. We injected a few
40:21
excess carriers but it's a small amount so we call them quasi neutral regions. So when I inject an electron over the barrier into the quasi neutral region and it recombines it can recombine at the back contact that's also within the quasi neutral region and the same thing I can inject holes into the quasi neutral N
40:42
type region and it can recombine in the N type region or at the N type contact. But that's not the only place the electron hole pairs can recombine. They actually have a chance of recombining in that barrier before they get over it. So let's talk about
41:01
the first one. Let's talk about recombination in quasi neutral regions. The diode current is the total recombination. The total recombination is just the total amount of excess charge in the quasi neutral region, the P region say, divided by a characteristic time. That characteristic time is the average time
41:23
it takes to recombine through a defect or the average time it takes to diffuse to the back contact and recombine there. The excess carrier concentration goes exponentially with the forward bias because it lowers the barrier and makes it easier for electrons to get over. So if I
41:43
just take q sub n over tau you can see that I'm going to get a current that goes as e to the q va over one kt. So the point is if the carriers recombine in a neutral region they're going to give me an n equals one diode current. So we've explained that n
42:03
equals one part there. Let's lower the bias and go down and see what happens. So if I go down and look at two tenths of a volt things are a little bit different. When I look near the junction on a linear scale the only recombination I
42:22
see is a sharply peaked recombination. Remember the n plus layer ends at three tenths of a micron and then the depletion region is about three tenths of a micron thick so it goes out to six tenths of a micron. So I have a sharply peaked recombination right
42:42
in the middle of the depletion region. And you can see that even though it's sharply peaked it's not contributing-this is a very high lifetime silicon solar cell so it doesn't contribute a whole lot to the dark current but I'm beginning to see it. So the green line is the integrated total and
43:01
you can see it step up as we integrate through that recombination. It looks like about twenty percent of the total recombination is coming from electrons and holes that are recombining in the depletion region. Twenty percent, I was right. So here I'm just comparing where the recombination occurred. So the point
43:21
here is that under different biases, the first point was under the light and the dark conditions they recombine in different places. Under different voltages they recombine in different places. So you can see what happened under high forward bias. I didn't have much going on in the depletion region.
43:41
Under lower forward bias I start to have a significant percentage. Now depending on how this all maps out, what the bandgaps is, what the lifetimes are, you know the percentages here can change and I could easily have a lot higher percentage in the depletion region. So how do we understand that characteristic? So we have to focus
44:01
on what's happening in the depletion region and the way we think about this is an electron might not make it all the way over a barrier or the hole might not make it all the way over its barrier and get into the other side. On the way there they might recombine and if my electron
44:22
gets into the p-region then there are lots of holes to recombine with and I have more than enough holes. I can always find a hole to recombine with. If the hole gets injected into the n-region there are lots of electrons to recombine with so that's not a limiting factor. There are plenty of electrons around. But if I'm in the depletion
44:42
region I have a small number of electrons and holes. At one edge I have a lot of electrons, at the other edge I have a lot of holes, but I'm not going to have much recombination there because I need both an electron and a hole to recombine. So the place that I'm going to get maximum recombination is the place where I have
45:01
both electrons and holes. So at the point in there where I have equal populations of electrons and holes, that's where I'll get the maximum recombination. That's that peak that we saw. And one can show, just by a little bit of semiconductor statistics here, that inside the region
45:22
there the product of nP was ni squared in equilibrium and it's just exponentially bigger because of the applied bias. So it's ni squared e to the qv over kT. But the peak occurs when n is about equal to p, so
45:40
I can just take the square root of that expression. So the peak concentration scales as ni times e to the qv over 2kT. The recombination rate then is going to be the concentration divided by some effective time that now will include both the sum of the minority carrier
46:03
electron recombination time and hole recombination time and it will be proportional now instead of qv over 1kT like it was in the previous case, now it's qv over 2kT. So the point is that if most of the carriers are recombining in a depletion region, I should
46:22
expect that my measured ID characteristic to see an n equals two. And here we can go through a few numbers. In order to get the total amount of current, I have to take the area under that blue line. I have to integrate the total recombination across that blue
46:41
line. And one way to get an estimate of that is to take the peak value, which is just the peak number divided by the time and multiply it by the width of that. And the width of that is the region over which n is about equal to p. And since the quasi Fermi
47:01
level is flat and the conduction band is changed by the electric field, the width of that region is on the order of kT over the electric field. You can check it dimensionally. Volts divided by volts per centimeter. That gives me the distance over which the conduction band varies by about kT and that will
47:23
lower the electron density by kT. Now you can argue about should this effective width be 1 kT or 2 kT, it's something on the order of that. So you go into this simulation, you pull off the electric field at that peak, you calculate the effective width. It's about 11 nanometers. That's
47:41
maybe a little bit thinner than what we're seeing there. A tenth of a micron is 100 nanometers, so the distance between the tick marks is 200 nanometers. But maybe the effective width is 2 kT over the field. That's not bad. It gives you the right trends. And it shows you that if you apply a forward bias, you're going to lower the electric field in the depletion region
48:02
and this peak is going to spread out and become broader. So it gives you some physical insight. Okay, so what's the point of all of this? It's that your dark current, if you measure a dark current, you'll find that it's J zero times e to the qv over n kT minus one. You might
48:24
get an n of 1.4. There is no physical mechanism that gives you an n of 1.4. It's some combination of a physical mechanism that gives you an n equals one, which is carriers recombining in a neutral region, and
48:41
n equals two. And depending on the relative weights between those two, you can get an n factor between one and two. So you really have to separate them out. The important point was the n equals two component is proportional to ni. So it goes as e to the minus eg over two kT. The n
49:01
equals one component is proportional to ni squared, so it goes as e to the minus eg over kT. So when you have small band gaps and high temperatures, ni squared gets very big and you can just see an n equals one component. If you go to bigger band gaps and smaller temperatures, that tends to promote the n equals two component. So people
49:23
will sometimes measure IV characteristics over temperatures. They'll try to pull out the n equals two component at lower temperatures and the n equals one at higher temperatures. OK, so there are a number of things, experiments, you're going to learn about a depth and you can get online and run it and you
49:41
can play with some things and see what happens to develop your intuition. Alright, so now I've got a few things to discuss and then we'll wrap up and see if you have some questions. So how do you, the whole objective in solar cell design is to increase
50:02
the generation rate, absorb as many electron hole pairs as you can, so you do that by minimizing reflections and by trapping all of the photons in the structure, and then to minimize the recombination rate, the smaller it is the better off you are, and you minimize the recombination rate by getting as
50:22
high a quality material as you can with very long lifetimes. Sometimes you might do it by making the layer very thin, so it just doesn't have a lot of volume to recombine in, and if you can trap all of the photons in that thin layer such that they're all absorbed, then that's helpful. You
50:41
might induce built in fields, so if you're building solar cells out of less expensive material that has low lifetimes, then you would like to engineer the structure such that most of the photons are absorbed in a depletion region, where there's a strong electric field that will quickly sweep them out. You can try to build back surface fields and see if you
51:02
can get a better back surface field than the one we had. Another thing you can do is just reduce the area of the contact. So if we get back to this Martin Green cell, you can see some of the things that they're doing. If you look at that back contact, initially people just diffused a P plus layer over
51:20
the entire back. That was their back surface field, and then you put a metal layer on the back. But that isn't an especially effective back surface field, we've seen. But all I need is an ohmic contact to that P type layer, so I could take that heavily doped region and I could make it
51:41
a very small area, just little points, and I could make contact just in those small areas. Any electrons that get to those small areas will recombine, but the total area is a small fraction of the area of the back surface. I'm inserting a silicon dioxide layer there
52:00
which will passivate the back oxide, tie up all the dangling bonds and give me a low surface, give me a low back surface recombination velocity. If I design the thickness of that oxide just right, it can promote the reflection of photons back in so it can
52:21
benefit the optical design. So people are just enormously careful about how they design these cells. They run simulations and calculations like the ones I've talked about, they find out where is most of the recombination occurring, they try to figure out a structure that will suppress that recombination then you look at it again and now the
52:41
dominant recombination might be somewhere else in the cell, then you work on that and see if you can lower that. That's the process that people went through from the late 1970s when silicon efficiencies were 15% until now when they're approaching 25%. Just very carefully getting as many electron hole pairs
53:02
generated and reducing the recombination as much as they could. Now another thing that I wanted to get back to is just this discussion of how good is superposition. Sort of the assumption there is that if I look at what the internal recombination, remember at open circuit the total recombination
53:24
is equal to the total generation. So I can run the simulation under illuminated conditions and on the right there I'm seeing what the total recombination looks like under illuminated conditions. On the left what you're seeing is what the total recombination looks like in the dark at the same
53:41
voltage and you can see that they're reasonably close so it looks like that approximation might work. Here's an actual confirmation. I asked Dionysus to do this here. I didn't know how good it would be. We always make this assumption for silicon and even I was surprised by how
54:00
good it is because you know you can't find a simple one line justification for when it works and when it doesn't. So what you're seeing in the blue line there, that's just the simulated performance of this solar cell under illumination. The black line is the simulated performance
54:21
of that diode when it's dark. If you just take that black line and add that current to the short circuit current there which is almost minus 40 milliamps and add the two you get the red dots. So the red dots are superposition and you can see that they fall just right on top. Now there
54:41
are a lot of solar cells for which this isn't even close but if you can make use of superposition it's a wonderful thing because it allows you to do a lot of your diagnostics on what's controlling recombination just by simple measurements in the dark and then you can be confident that you'll know how to predict what's
55:02
happening in the light. OK so we'll summarize and then we're ready to break. Important point I'm trying to make is that the diode current is simply Q times the total recombination minus the total generation. That's really exact, that's not a superposition argument, that's exact. At open circuit voltage
55:24
the recombination is equal to the generation such that no electron hole pairs come out and no current flows. If I have any recombination under short circuit conditions that lowers the short circuit current to the collection efficiency, when you can use superposition then the dark current tells you
55:42
an awful lot about what to expect under open circuit conditions and as I mentioned solar cell design is all about maximizing generation and minimizing recombination and you know in the process of this talk the final point that I was trying to illustrate is a lot of people make use of simulation tools. They'll just
56:02
simulate it, look at the IV characteristic and see if it's close to what they measure. If that's all you do you're really not extracting much value from the simulation program and what the simulation program allows you to do if you've got the right physics in it and if you've got the right mobilities and diffusion coefficients and things is to
56:21
look inside and understand why the device is performing that way and start to help you understand how you can make it perform better. So don't just look at the computed IV characteristics when you run a simulation. Look inside at recombination rates and carrier profiles and energy band diagrams and see if you can figure out how to
56:41
make the device better. So in the handouts and by the way I've been meaning to mention we've written there's a website here where all of the handouts for the summer school are available if you haven't got them. In the handout for this I've got first of all a mathematical derivation of the fact that current is Q times recombination
57:02
minus generation although it's intuitive you don't really need a derivation and an attempt at a mathematical justification of superposition but I'm not very happy with it because it's not easy to justify in general. Alright so I don't want to go there that's the appendix so I'll stop
57:21
and see if there are any questions and we've got some questions. Yes you were talking about doping with a decent period. What would you recommend
57:43
you said you're having a problem with efficiency so when you're doping I know you're changing the characteristics of the semiconductor and so have you ever thought about some type of inhibitor that you can actually replace or to actually suppress
58:01
the recombination? An inhibitor that could, yeah so the way I think of ways to inhibit recombination is things like minimizing defects, engineering structures that keep carriers away from places where there are lots of defects like keeping minority carriers
58:23
away from contacts. I'm not sure what you're getting at in terms of inhibitor, there are some. Yeah heterojunction is a wonderful, so you know if you use a heterojunction as a back surface field
58:43
it's very helpful because you can get very big energy barriers. It's difficult with doping to get big energy barriers so as you saw the numbers it's not being very effective in keeping the carriers away from the contact, but with a heterojunction you can have a very big energy barrier there and you can essentially reduce the recombination
59:03
to zero. Because that was adding recombination so it was inhibiting both the short circuit current but it was hurting the open circuit voltage even more.
59:28
Increasing the efficiency, whatever technique you can find to reduce recombination will do that. And I think Professor Alam will be talking later about thin film
59:41
cells and thin film solar cell designs are quite different because the lifetimes tend to be very short so you have to design them differently than you would design a cell like this where you can get very long lifetimes.
01:00:07
So that's a good question, why not just increase the thickness? There is some effect there, let's take a look at that structure.
01:00:29
We can actually do a calculation of what that back surface field is. So the carriers have to get over that barrier that's created and then they diffuse
01:00:41
to the back surface and recombine. So if I'm thicker, it might make it harder for them to diffuse to the back surface field and recombine. Now the trade-off is that it's heavily doped, so its lifetime tends to be very short. So that increases the volume of low lifetime material where they can recombine.
01:01:02
So there is some trade-off there, there's some thickness where there's some sweet spot. You can do a little bit about it, you can't do anything very dramatic. I think there was a question down here. What's the status on transparent metal contacts?
01:01:28
So the question is, what is the status on transparent metal contacts? Ashraf, are you going to talk about this? You planning on talking about this? If so, I'll defer it to you.
01:01:42
Because Ashraf is more up on it than I am. So in a lot of cases, what would be nice is if you had a transparent conductive material. So instead of having to make a contact in a metal grid and have these lateral volt drops, the metal grid shadows some of it, so it obscures some of the junction and
01:02:04
the lateral volt drops are bad. What if you could just put a transparent metal across the entire layer? You wouldn't have any lateral volt drops. There are transparent conductors, things like indium tin oxide, and they're not perfect.
01:02:21
Probably you get transmittances, Rakesh probably knows, on the order of 90% or so. And you get sheet resistances that are 10 ohms per square or something, so they're not perfect metals either, there's some compromise. But people frequently use those for solar cells. There's a lot of interest these days in whether you can, well first of all indium is rare,
01:02:43
so that's kind of expensive, so you'd like to find other ways to make transparent conductors. There's a lot of interest right now in graphene, because graphene has a lot of very high electrical conductance and a single layer of graphene has very low optical absorption.
01:03:00
So there are a lot of people that are doing research on whether graphene could make a transparent top conductor for solar cells. It's not a slam dunk, one layer of graphene doesn't have a low enough sheet resistance for solar cell applications, so you have to put multiple layers in. When you put multiple layers in, you get a little bit more absorption.
01:03:22
But there's some promise of doing something there. Any other questions? Yes. How effective? Can you make it more clear?
01:03:40
Which parameters does it have? Well, so the question was what is this tau effective that I talked about. Now the right answer to that is to learn Shockley-Reed-Hall statistics. There's a famous paper that was written probably in the 1960s about recombination
01:04:01
through defects, and that's usually the starting point for this. But the way to think about this is that this defect has to capture an electron and that's described by an electron capture time tau sub n. But then to recombine it has to capture a hole, and that's described by a hole capture
01:04:21
time. So the time it takes to do the entire recombination process is tau sub n plus tau sub p. So the effect of that would be the sum of the two lifetimes. Now if you go out in the neutral region, I only use the electron lifetime. I had an excess electron density, and I just said that the proper time to use
01:04:43
there was the electron lifetime, and that's because there are so many holes that it's always quickly possible for it to find a hole to recombine with.
01:05:04
So you said that the high energy photons can be absorbed very quickly, because there's more density states up there, so I'm not going to say. If the devices are going to be much thinner, they might be used more, I don't know for reasons or other things. I'm wondering if people really spend a lot of effort on the optical design, maybe making
01:05:24
multi-layer basically mirrors out of thin film structures, specifically to make the device into an optical resonator for the wavelengths that take a long time, or things
01:05:41
like that. How much effort is spent on the software design basically? I think that there's a considerable amount of effort that's spent on optical design. One example of that is this Martin Green cell that I've continued to show.
01:06:01
Let's see if I can find that again. So this is one example of optical design where people have spent a lot of care, and there are various ways that you can texturize the top surface. They've got a mirror at the back that they've carefully designed to try to maximize their
01:06:23
reflection. They'll do ray tracing studies to see how many times a ray would bounce around inside this structure. So there is a lot of thought, and the idea is to try to make the silicon layer as thin as you can possibly make it, but absorb every photon that you possibly can.
01:06:42
Now it's not as important when you have direct gap materials, because things are absorbed so quickly there, but in these indirect materials it's extremely important. Even in the direct gap materials, the idea of texturizing the top surface to get all of the photons in is a thing that is widely used.
01:07:09
Okay, we have another question. In describing the recombination in the space charge regions, why does it go as just n
01:07:24
over tau and not a product of n times p because it depends on both concentrations? Does it come into the thing that you're talking about with the lifetime?
01:08:07
Let me see if I can answer that. Yeah, if you have something like radiated recombination where you need both an electron and a hole to recombine, then you would write the recombination rate proportional
01:08:21
to the product of the two, and your lifetime would depend on that one. Now this is recombination through a defect, and I'm simply arguing that the recombination is occurring everywhere. It's not a bimolecular one. The electron is falling down into a trap.
01:08:44
When you have a radiated recombination process, you need both an electron and a hole, and they recombine together, and the recombination rate is proportional to the product of the two. In this case, you have a trap that's collecting one and a trap that's collecting the other, and the rate at which electrons recombine is the rate at which
01:09:05
that whole process occurs, and we're arguing that that process is going to be maximum at the place where n is about equal to p. So if I look at the recombination rate, you can see
01:09:21
I couldn't have an n squared there because its rate is number per cubic centimeter per second. But in these bimolecular processes, when you do radiated recombination, you do get np, and what that means is that the tau is dependent on the other quantity.
01:09:41
The tau might be dependent on, I would write it as n over tau, but the tau would have a one over p dependence in it. We'd find out the carrier density. Okay, are we ready for a break?
01:10:02
Okay, let's take a break then. Thank you.