 Does the microphone work? Does it work well? All right. Oh, does it work well? OK. No? Higher? Up here? Is it better now? Better? Thank you very much. Well, thanks for coming today. I'd love to talk about the work we've done with my group on phone-assisted optical properties in materials and new implementation on ETW. So first of all, a few references for you if you're in the field of optics. I recommend the following books for you. The OxoMaster series by Mark Fock is a wonderful introduction to the topic of optics in general. Another very interesting for technical aspects is the Bassanian Paravitini book. It's his very daily equations for everything, as long as these three papers are very handy. Oh, thank you. Thanks so much. Well, I think Mike made something. All right. So first of all, the motivation. Why should we worry about phone-assisted optics? And a very strong motivation is silicon itself, perhaps one of the most important materials in history, and definitely the most important semiconductor. If you look at its direct absorption, a direct absorption of here's the occupied valence band, conduction band occupies an empty state. When we send the photon in, electrons absorb the photon and go from the valence to the conduction band. That's a direct absorption. And it's well understood. Today, we have very good spectra. And here's an example from the Berkeley GW code. The dots you see here are experimental data for the absorption of silicon, the dielectric function. It has a function of photon energy. And you see that the case without the excitons, that's the GW level. You get the right peak positions. The peak heights don't come out as correctly. But once you get the excitons, you get an excellent agreement for the peak position and the peak heights at the same time. And not only that, also this work by Marine also shows that you can also determine the spectra, same spectra as a function of temperature. These are, if I can broaden the absorption and tell you how the spectra ultraviolet I change with temperature as well. So, silicon would do understand very well. If you notice though, all this absorption calculations occur in the ultraviolet. And while in the visible spectrum, the absorption you would get from it, if you do direct calculation, would be zero. And the reason is that we know though that silicon does absorb visible light. It's not transparent. It's in the gap is 1.2 electron volts while the meaning direct gap is 3.4. So you only get direct transitions when your photon energy is in the ultraviolet. It's impossible for silicon to absorb in the visible. To absorb across the indirect gap, you also need the assistance of phonon. You need some extra mechanism to provide momentum and conserve the overall energy and momentum in the process. And actually this kind of mechanism is what enables silicon solar cells to work. Without phonon, silicon will be transparent like glass. So I want to understand this mechanism in materials in general. All right, so when you come back to a little bit about linear optics just to introduce the coefficients that we're going to calculate today, linear optics primarily deals with refraction and absorption, right? And here's an example of Snare's law. The refractive index tells you how a light bends when it crosses one material to the other. And absorption has example of a laser pointer pointed to a solution and you see this exponential decay of the intensity of the beam as it gets absorbed in the material, okay? And absorption is a function of position and in this case the depth in this solution that decays exponentially and the exponent is alpha times the position. And alpha is the coefficient where it is in, the absorption coefficient. It's units as you see it multiplies length so it has to be inverse length and usually measure in inverse centimeters. When you talk about the material as a strong absorber like a gallium arsenide solar cell or so, values you get about 10 to the 5 to 10 to the 6. And if you invert that, you get the characteristic length scale over which light absorption happens. So for a direct gap material, typically one micron is a distance over which you absorb almost of the light that passes through it. Okay, so these are the coefficient we're focused on and these are numbers to keep in mind about what is strong and what is weak, okay? All right, so to generalize, to combine those two we let them together in what we call the complex refractive index. It has the real and imaginary part. The real part dictates propagation of light. The imaginary part, imaginary component dictates the dissipation of light, the absorption, okay? You also talk about the complex electric function. Again, the real part presents a screening and propagation while epsilon two, the imaginary component is the absorption that we saw earlier in those plots. The two of them are of course connected. In the simplest approximation, one is square root of the other, but if you want the full equation, it's right here. So N and K depend on epsilon two through these two equations, right? And last, if you want to find the absorption coefficient, again, in the case of inverse length, it's directly proportional to the kata, the imaginary part of the refractive index. Okay, so you multiply by omega and C and then you have this coefficient and you can then go and measure experimentally in an optical experiment. Okay, so these are the parameters we'll focus on in our calculations. Okay, now, if you go back to the early theories of light absorption materials, in this case, a metal, you look into the semi-classical theory divided by Drude that are actually older than quantum mechanics. And the way you would do this is you would write down a Newton equation, effective mass standard acceleration is equal to, here's the electric material then you apply an electric field and they feel a force by the field. And at the same time, they scatter, right? Every so often, at this time scale tau, those electrons scatter the material and change the momentum. And you must have seen this problem if you apply a static electric field, you can derive the conductive of the material that relates to mobility calculation we discussed earlier in the school with parameter tau. If you also studied the same problem under the influence of an alternating field, an AC field, you can also derive an equation for the absorption coefficient, okay? And again, directly proportional to tau. And the reason is that if those electrons can't scatter, then they would not absorb light, okay? You can see this because this is a free electron, basically, parabolic band. There's no way to conserve energy and momentum in light absorption of parabolic band. You must have a phonon to give the overall conservation of energy. And also when this was derived, this tau parameter was phenomenological. You basically measure a spectrum, say absorption in a metal, and then you find the tau parameter that makes your equation fit the spectrum. But today, of course, we can calculate this tau and that's what we are going to do. We need to calculate, we need to study this scattering of electrons by phonons, by defects, by any impurities you might have in your material and be able to predict absorption. To do it right, of course, you have to go back to quantum mechanics. You can't just do the theory for a predictive theory. Again, so the way we study optics is we use perturbation theory. We assume the light perturbation is weak. We work in the regime of linear optics where the field is not too strong to damage the material or causing linear effects. We assume that we have an arbitrary state, our starting point, is what comes out of a DFT calculation. And you also may correct it to get GW corrections to your band gaps, but that will be your starting point. You have a set of initial non-interacting electrons occupy the quantum orbitals and the eigenvalues. And then what you do is you act of them with the electron photon Hamiltonian. You assume you're sending an electromagnetic field. So it's characterized by the vector potential A times the momentum operator P. Okay. Sometimes this is the dipole approximation. It's like the electric field coupled to the dipole moments in your material, but it's easier to work in the momentum space because extended solids don't have a well-defined position operator. It needs more complex math to work it out. Okay. So how we find the optical transitions, the rate for optical transitions as a function of time. So it comes out from the well-known famous golden rule. The probability per unit time that the optical transition will occur from state I to state F, your initial to final state is to pi over h bar, the square of the matrix element of the turbine Hamiltonian times the delta function which ensures that we conserve energy. The initial and the final state must be the same. Okay. Now, what are the initial and final states? Okay. In the initial state, you can assume that you have electron in some valence band on a wave vector K. Your final state, now your electron has gone to a different band, the conduction band, and also you have the energy of the photon. This is the case of emission. In case of absorption, you have the h bar omega in the initial state. Electron plus photon gives you the excited state electron. So how then do you connect this microscopic quantum property to a macroscopic measurable quantity? Okay, the absorbed power, the power of your system absorbs per unit time. This is the probability for a transition of a given electron from a given initial state to a given final state. Of course, these states have to be, the initial state has to be occupied and the final state has to be empty according to a Fermi statistics. So you need this pre-factor. And they need to sum over all possible initial and final states in your system. And that basically, this part tells you how many electrons go from the valence, the conduction band per unit time. You multiply by h bar omega, the energy of each of the photons getting absorbed. And that's how you get how much power is coming into your system. Okay, so this is how much power is absorbed by your system. Of course, this quantity P contains the electron photon Hamiltonian, which then depends on the amplitude of the electric field. So to normalize, we also have to factor in what's the incident power coming in. So here you know the refractiveness of the material, the amplitude of the vector potential and the optical parameters. So once you know the power coming in, the power getting absorbed, you can derive the equation for the absorption coefficient. It's simply this ratio. You can. And we translate this to quantum mechanics. Here what we get. We get this pre-factor that involves constants, optical parameters of the material, like the refractive index. And then here's the quantum mechanics. We have a sum over initial state band, final state band and the region zone index K. The parameters that attend there are the fermi-occupation factors. Again, we can only go from an occupied to an empty state. If both are occupied, the power exclusion principle forbids an optical transition. And then you have this property, the optical matrix elements between your initial and your final state. But also you need to take into account the polarization of the light you're sending in for a plane wave polarized light. The lamb dies along the x or the y or the z direction. So you correspondingly take the x or the y or the z component of the momentum operator. Now you have to multiply by the energy concerning delta function. So your final state has to be the initial plus the photon energy. So this acts like a density of states. The more states you have, the more probabilities to absorb the photon. Okay. And of course, if you know the absorption coefficient, you can also find the imaginary part of the dielectric function. Again, a similar equation with a different prefactor. Otherwise the same coin is coming in. And once the imaginary part, you can switch to the real part and find the screening properties, all the refractive index through the Kramer-Scroning relation. And if you do that, you get this equation. So it's not the delta function. Now you have these energy denominators. And one thing to note here is that in the imaginary part, because you conserve energy, you don't need to worry about states close to your photon energy. If your photon energy doesn't match an energy of a transition, you won't have absorption. But when it comes to the refraction, this sum can basically go all the way to infinity. You have to sum over states that are occupied even to high energies. So this sum is not limited to near the bandage. You have to do a summation of many empty states. So here we'll focus on absorption. We'll focus on the near-edge properties. However, you can do this calculation as well. You just have to include a lot more bands than you would do for absorption. All right. Now, how do we put phonons in the picture? So far, we just talked about a single photon for near material and look at direct optics. The photon phonons will need to go to second approximation theory because now we have two perturbations coming in. We have the electron-photon interaction but also the electron-photon interaction. And we need to consider both of them at the same time. So we have to generalize a Frames-Gottend rule to go to second order. Again, two pi over each bar as before, the energy conserving delta function. And now the matrix element, we have to generalize it. We have to take our perturbation to second order. And this perturbation and also divide. So this course transition from our initial state to some intermediate state and from the intermediate state of perturbation course to the final state. And we have to sum over all those possible the immediate state to get a total rate. Now, each of these H perturbation Hamiltonians contains the electron-photon part and the electron-photon part. So here what we have here are really four terms. Electron-photon two times or electron-photon two times or the cross terms, well, one of them is the photon and that is the phonon. Like this one. So if you want to study phonon-assisted absorption, we need to keep the cross terms on. If the other two terms is either two-photon absorption or two-phonon absorption. Okay, so here's what you get. Either your electron emits the phonon first and then absorbs the photon or the other way around. First the photon and then the phonon. And because we can't distinguish these two processes and the microcopter can distinguishable, we need to take into account the quantum interference between them as well. So we first sum and then square. Just like a double slit experiment first, we need to sum the amplitude and then square the total amplitude combined. Okay, all right. So then how does this translate to equations relating to quantum eigenstates and energies? Well, here it is. Again, algorithm coefficient alpha is given in terms of constant prefactors and then a sum over the initial and final states. And now notice we go from band i to band j starting from wave vector k and we go to wave vector k plus q. And here's how we conserve energy, right? Your final state, the electron at j and k plus q is equal to the initial electron energy plus the photon energy or plus minus the energy of the phonon because it can be the phonon absorption or phonon emission. And what changed? So this change with respect to direct absorption. The algorithm that changed is also the matrix element, right? Now our matrix element is not just a momentum operator, we have to generalize it to account also for electron phonon coupling. And again, because we talk about two ways to start from initial state to go to the final state, we have to account for two possible parts where in part the label SS1, the electron absorbs the photon first and then emits the phonon or emits the path S2, the electron emits the phonon first and then absorbs the light, okay? So what comes in here is, you have the obstacle matrix element between the initial state i and band m times the phonon coupling in the conduction band divided by this energy of the intermediate state. And the energy here is the total energy combined energy of electrons plus photons plus phonons. So your initial state, your intermediate state is here, your initial is the electron plus the photon that has been absorbed yet. For path S2, you first emit the phonon followed by light absorption at the other k plus q and another intermediate state involves the absorption remission of the phonon. Okay, there's also a statistic factor P here which accounts for the occupation numbers. Again, we must ensure that we go from occupied state to an unoccupied state. You can't go within two occupied states, of course, or two empty states. And but what we need to multiply by the phonon occupation numbers, the Bose Einstein occupations. The upper sign corresponds to phonon emission. So here we have n plus one. So phonon emission is possible even at zero temperature, even if you don't hear phonons, you can always emit a phonon. The minus sign cancels this term so you just get simply nq. So in that case, the absorption of phonons is proportional to how many phonons you have. Okay, and last one more comment about this sum m over the immediate states. So you start from an unoccupied state, you go to an empty state, what should the m state be? There is now the should be all states, both occupied and empty. So an electron can even go down in the venous band and then electron from the cathode to the conduction band. How is this possible? It's possible because this is just a virtual process. Now this is a mathematical process that for convenience we draw it this way. In fact, the immediate state does not have to conserve energy and doesn't have to even be occupied and empty. I have a question. You have a more than one phonon in the process. You mean this term, right? Yeah, so the plus minus one over two is, if it's plus one, if it's the plus sign, this becomes one, right? If it's the minus sign, it becomes zero. So it's either one or zero. It's just a mathematical compact notation. You have two choices, right? Either it's zero for phonon absorption or one for phonon emission, right? And so to make the binary choice, we write like this. So if this is either one or zero, it's never one half, you're right. And of course you can generalize to have more than one phonon, more than one of photon. You can go to third or higher orders, but the Bayesian theory, we can generalize this method, okay? So basically, but these are the key equations we want to implement to study phonon-assisted optics. All right, now I want to emphasize why this sum is challenging. Why it's not as routine as it has been. There are few calculations for phonon-assisted optics and here's the reason. If you want to study direct absorption, then you have a single sum over a k-point, right? So you have to, for each k-point, you sum over bands and then you do a sum over your entire breathing zone, right? And that you can do quite easily these days. You can even interpolate them with 1 and 90 and both the energy and the matrix elements and do it very efficiently for very fine meshes. When it comes down to indirect absorption, though, you have to sum both over all possible initial states and all final states. This double k-sum is what makes it completely very expensive. So try to approach with a brute force way, just brute force calculate matrix elements is gonna be very, very challenging. To give an idea, for a case of silicon, if you want to use an energy resolution of 30 milliV to get a good spectrum to converge with a grid of 24 by 24 by 24 k-point, that is both for the initial and the final k-points. And if you account for possible combinations, that gives you about 200 million combinations. So even though calculating one of these diagrams is actually easy, takes less than an hour for silicon, if you multiply by 200 million, that's a lot. Yeah, that's why kind of approaching the brute force way is basically hopeless. And that's where one interpolation comes in. This is really one of the examples where the power of one interpolation evident itself. So again, here we use maximally localized way of linear functions to convert to the localized basis. And this way we can have a localized basis set that doesn't interact in the long range. And we can use it interpolate energies and optical matrix elements. That's what happens in one N90. But the power of EPWs can also do the same thing for the electron-formal coupling matrix elements. So let me go back one step again. So the ingredients we need here is we need optical matrix elements, electron-formal matrix elements, phonon frequencies, and electron eigenvalues. Okay, the rest is summing them up. So having it from the two codes allows us to implement these equations in a very efficient and combinationally not very demanding way. One note here is that the current version will implement the code and the optical matrix elements are those of the momentum operator. And I want to make a comment here in that we'll hear a pseudo-potential momentum operator. It's not the right decorator to use. We have to use the velocity operator that is independent of your pseudo-potential. And we're working on changing that. The two of them typically differ by a small factor, but we will have a venture of velocity implemented soon as well. Because the velocity can be interplayed with the Rwanya basis as well. All right, let me make a comment also about what you expect to find when you do a calculation for direct versus phonon-assisted absorption. And I want to make a note here about experiment. How does experiment know, if you don't experiment, how would you know if your material has a direct gap or an indirect gap? And the answer is you do what we call a TAOQ plot. TAOQ is the name of the person who invented the method. So if you look at absorption very near the bandage, you will find that for direct absorption, the absorption coefficient is proportional to, photon energy minus the bandage up to the one half power, this is basically the density of states, right? And divided by omega. So what experimentals do is they measure the absorption coefficient, multiply it by omega, and then take the square of the whole thing. And the answer is you get a linear relationship with respect to the photon energy. So if you plot alpha omega square versus h bar omega, you should get a straight line. Okay, and here's an example here. Here's an experiment for team selenite nano sheets. I like this plot because it shows both approaches. So on the one hand, you have this point f of r, which is approach for the electron coefficient times h bar omega. You square the whole thing, you get this straight line, so you extend it to well across the zero of the x-axis, and that will be your direct band gap, right? Where this line intersects the x-axis, that's the direct gap. Now, when you have an integral absorption, though, this absorption coefficient is proportional to, again, the photon energy minus the indirect gap plus minus a phonon frequency. And this whole thing now, it is square. It's not a square, but a square here, because the density of states are independent, and divided by omega. So now, if you combine alpha times omega and raise not to the square, but to the one-halves power, you will get your straight line again. So here's what happens in these curves. These are the exact same data as before, except now we plot the absorption times h bar omega and take the square root of that. And again, you see these straight line segments, extend to zero, allow us to get the indirect gap. And of course, it has to be smaller than the direct gap in general. Again, so in this material, here's the direct gap, and here it's indirect gap, and that's how we can tell them experimentally. Again, notice though here that the absorption, in the case of interabsorption, has two onsets. You can have the phonon absorption term, as well as the phonon emission term, and the two of them will take off at different energies, plus minus the energy of the phonon that dominates the absorption, okay? Because you have these two separate terms. All right, so after discussing this, I want to show you the calculated results for the absorption onsite of silicon, okay? Again, here's the absorption coefficient of silicon as a function of the photon energy, and now we have the indirect band gap, okay? So again, absorption starts around the value of the indirect gap around 1.2 electron volts. And notice here, this is the talk that I discussed earlier, and we take the absorption coefficient, multiplied by the photon energy, and raise to the one-half power. So this is why we do this, and this combination should give us straight lines. And you see that. Let's go first to low temperature. So again, you see a straight line coming from phonon emission, and another straight line here coming from phonon absorption. When you're at low temperature, of course, what dominates is the phonon emission, that's why it's much stronger, and phonon absorption is a weak contribution. And the difference is two onsites is the phonon energy involved in the transition. And what happens now when you increase the temperature, phonon emission, you see, remains kind of constant, not very sensitive, right? It's proportional to N plus one. At low temperature, the plus one term dominates. Okay, but the phonon absorption term becomes progressively stronger and stronger with increasing temperature. The reason is that this term is proportional to NQ, and the number of phonons increase with temperature approximately linearly for acoustic phonons. So when you go to a zero temperature, you don't have phonon absorption, and in the limit of infinite temperature, both of them are equally important and equal to each other, because of phonon emission, phonon emission temperature is equally probable. So that gives you an example of a comparison of freeway to experiment as a function of temperature for the spectra, and I also wanted to make a note here that when we do this calculation, we didn't account for the fact that the banger depends on temperature, so we have to shift those onsites to match the onsite of the spectra. Now you see here the scales and the order of 0.1 electron volts, and that's, as we discussed yesterday, this is the characteristic scale of how temperature changes your band gap over this range. Okay, so besides this constant shift, and the spectra are very good agreement with what we see in experiment, and explain the temperature dependence and the frequency dependence. Okay, now this calculation you can also do by using simple models, let's say, but you can get a constant optical matrix element and a constant electron phonon matrix element. You'll be able to do this even without here the calculations, but where you didn't need heavy calculations for is when it comes to calculating the absorption spectrum in the visible range, okay? And now we're talking about photon energies between the Indian gap almost to a direct gap, and that covers the entire visible spectrum. And here it's really when the power of the one interpolation comes in, because now we have transition from every possible to every possible final state, and that means we can very hard to account for with simple models, and that's why the power of Abinish emerges. And again, here we see a very good agreement with experiment over many orders of magnitude for all those frequencies. I want to mention again that we had to shift the origin because we don't account for temperature dependence of the bond structure, but besides this constant factor, you see that the spectra agree with experiment. And today we have the tools to account for this energy shift, and we'll get much more predictive power. All right, so this is about silicon. And I want to mention another system where phonesis absorption matters, and that is laser diodes, like the point that I'm showing you. For example, blue-ray lasers, which are actually violet, I mean that's 405 nanometers, based on gallium nitride we're using for optical storage, for laser projectors. And today we want to develop a powerful green laser so we can combine a red, a green, a blue laser to make projects that can fit in one cubic centimeter, say inside your cell phone instead of this bulky object. And one limitation there is this kind of absorption, I'll show you why. So here's a theorist's view of an LED or a laser. So we make them, so what we have is we start with a quantum well of the material that we want to emit light, in this case, indium gallium nitride alloys with a band of, say, green or violet, and we sandwich between T and N-type conduct, like gallium nitride. We connect the battery and the battery takes an electron from the valence band of the T-type material and moves it to the conduction band of the N-type. Now those electron holes fall into the quantum well because it's a low energy state, whether they combine and emit light. So that's what we want to happen in our LED or laser. So in the actual structure, we have more than one quantum well between the T-type and the N-type layers. So in LED we want to create those photons, get them out as fast as possible so we design our structure to get light out easily. But when you make a laser, what you want to do is you want to those photons to bounce back and forth many, many times and get amplified, right? What want to happen is the following. We know about absorption, right? You send photons in a material and the photons dissipate by exciting electrons, whole pairs across the band gap or another between two states. And that's the, and they are lambda low discussed earlier. But here though in the quantum wells, we have lots of exciting electron whole pairs and what you get there is gain. And one photon gets amplified by stimulating a recombination. And so this positive quantity, G is a positive number. It tells you how your signal gets amplified as it propagates through the material. In a laser, both of these things happen, both absorption and gain. Of course, gain is what we want, but often we don't want it. So in the quantum wells, you have primarily gain. But what happens is your photons don't live here. They have a distribution over the structure. This is the distribution of the electric field of the optical mode. And so they leak into these doped regions where they may get absorbed. And that's what we want to discuss. I want to discuss the process of free carrier absorption. Okay, now this is not across the band gap, right? So here we have the wide band gap semiconect like gallium nitride. And the light of the laser is not enough to excite across the gap. What can happen though, is that here you have lots of free carriers, either free or letting by the bottom of the conduction band or free holes at the top of the valence band. And in this case, direct absorption is not possible. For holes, for example, there are no states for them to be excited to. For example, there is a state, but it's a type of familiar transition. So it's a very weak absorption. Once you factor in phonons though, then those free electrons at the conduction band can absorb the light and go to a different state in this intra band process mediated by both a phonon and a photon. And that's possible for every photon energy. So that can be a source of loss if you have a doped semiconductor. So again, we did the calculation for phonon assisted free carrier absorption. Again, now we're focusing on either kind of intra band processes within the bottom of the band to higher states. And we found the absorption cross section. Absorbent cross section, why? The absorption coefficient here is proportional to how many carriers you have. And the proportionality constant is this cross section. This is a weak process to give an idea how weak it is. If you have a 10 to 19 carrier, which is actually quite high. It's typical for ladies under operating conditions. Your coefficient is not of manganese about 10 inverse centimeters. To translate this into a length, it means that you expect a strong absorption over a length scale of one over 10 centimeters, about one millimeter. That's the scale you would absorb light at. And contrast this to a direct gas material where it's 10,000 or 105,000 stronger absorption like absorbing a micron. So this course of course is weak. And one of the reasons of course is that you don't have so many electrons, right? You don't have 10 to 19 electrons to absorb light. It turns out that because photons in lasers bounce back and forth many, many times, the mean field path becomes hundreds of microns. So it starts to become comparable to the one millimeter we discussed earlier. So this process tends turns out to be quite important for lasers. And one last comment that you can absorb light by your free electrons. But you can also absorb light by bound holes or bound electrons like dopants in a semiconductor. And the important thing about gallium nitride is that it has lots of these nonionized holes which are still bound to their acceptors. It's very hard to P-type-dom gallium nitride. It's done, but actually that was what held the field back for many years. That was the discovery that led to the Nobel Prize for blue LEDs, the actually the ability to P-type-dom gallium nitride. So what we found in calculations is that because you have these holes bound to magnesium, they can cause obstruction loss in the P-type contacts of lasers. And that is the reason why you lose a lot of those photons inside the laser. All of my estimates tells you that in a green laser pointer you would lose about half the light in this absorption instead of getting it out of your laser. So it's a significant source of loss. And it's more important as you go to longer wavelength and that's not the reason that it's still challenging to make powerful green lasers with direct emission. Most green laser pointers they use the frequency doubling of the infrared laser to give you green light. But the work has been done to make direct laser that can be as efficient as red or violet ones. Okay, another thing where light absorption can, another case of pharmacist absorption is transparent conductors. Again, this is another case of absorption by free carriers. So in conducting a transparent, you want to make a transparent conductor when you make an electrical device like a solar cell, an LED or a cell phone display that conducts electricity but it's transparent to light. So if you want to use a smartphone you can touch the screen and send commands to your system while you're still being able to see your screen. It's because it's coated by a transparent contact like indium tin oxide or tin oxide or zinc oxide. So what special is materials? They have wide band gaps. The band gap is at least equal to the UV energy of three point four electron volts, but they're doped. They're heavily doped oxides. So three electrons in the conduction band are those what give you conductivity. So if you look at the band diagram, of course it's impossible to absorb visible light across the band gap because the band gap is too wide. There's no states for the electrons to go to. And if you look at direct absorption of these doped carriers, again, most of the time there's no states to go to. But if you factor in phonons, then light absorption becomes possible. And that's important because it tells us the fundamental transparency limit of these materials. So it tells you how transparent can a transparent contact in oxide be. What is the answer? If you calculate this, this probably you will find the ultimate limit. And what it is for n-type doped tin oxide and what we got is you've got a curve that looks like this. Of course, the crystal is anisotropic. So we have two different spectra for two different directions. And what you see is that surprising this material is most transparent in the visible, which is a very nice coincidence. As you go towards the UV, you get more light absorption. But as you go to the infrared, this form-assisted absorption of bifurc carriers becomes exponentially more important as well. So it is a power law as well. This is log scale. So this tells what the fundamental transparency limits in those devices. So if you want to dope them or to get more conductivity, you have to pay a price that they're going to be stronger absorbers as well. So it may contribute to a loss in the device. All right. And going back to silicon. This process, this free carrier absorption process can also occur in silicon. And that may actually matter for solar cells because to make a solar cell made of silicon, you have to make a PN junction. So before your light actually gets absorbed in the junction itself, it has to pass through a heavily dope P or N-type region. Well, again, you have three carriers and those three carriers can absorb the light. And so there's a competition here. Will your photons get absorbed across the band gap and give you current? Or will they be absorbed within the conduction band and then eventually become heat? Okay. And so what happens here is you can have actually both processes in silicon. You have both a direct process between these two bands or you can have a form-assisted process through this intra-band mechanism. Okay. So here are the results. We examine many mechanisms, we examine the direct absorption, form-assisted, impurity-assisted as well as the dissipated behavior. It turns out all of them contribute. All of them give you some contribution in the visible and in the infrared. And when we sum them all up to compare to experiment in the infrared range because experiment, you can distinguish, here you have a person across the gap and masks the experimental data. But in the infrared, we have reliable data. We see that our field is again in very good agreement of experiment and also explains why this shoulder feature happens. This shoulder is coming basically from the direct heat. At this energy, this energy here corresponds to the difference of energy between the two conduction bands that gives you this slight shoulder behavior as well as the monotonic increase. So overall, we trust that these methods give very good results compared to experiment and can explain what happens in the dope silicon solar cell. All right. Other groups have often used this method to study metals because in a metal as well, again, in a metal, I'm not sure this is silver, I'm not sure what the banter, which of the metals is this one. But again, you see that direct absorption becomes possible only from the D band to the S band. And only once you factor in a phonons, you can have absorption in the infrared and longer wavelength. And this kind of absorption mechanism is what gives you the dissipative flasmon energy loss in metals, right? The dissipation rate directly proportional to the emissive post-directive function. And again, there is a strong phonon contribution. So you see here similar equations to the one of matter for a phonon assist optics. And they do contribute a lot for metal because as you can imagine, when you go to long wavelengths, you primarily have the phonons contributing to this absorption. So the details coming from this manuscript. And last of course, I would like to emphasize a very promising alternative method to calculate phonon assist optics developed by the Korea Saint Justino. And it's a very smart method, is instead of doing second perturbation theory where your two perturbations are photons and phonons, what they discovered is that you can do a direct calculation and incorporate the phonons into your static Hamiltonian, into the stature itself. So they discovered that a single supercell, single opium supercell, where you multiply your unit cell by an integer number and then you displace the atoms according to specifically a combination of the vibrational modes. So your structure now incorporates the electron-phonon Hamiltonian. And then, so because you incorporate it, you only need to do a special perturbation with direct absorption. And again, you get excellent specs that are much experiment to an amazing degree as well. I want to make sure some advantages here compared to the secondary method is that this method also avoids the divergence that we have in phonon-assisted optics when you are in the direct cap region. The previous equations have this problem that if your intermediate state becomes real, then the equation is diverged and you have to incorporate somehow the life of those states to avoid it. It also avoids the influence interpolation and also gives you the temperature dependence of the eigenvalues and the band gaps as well as this long urbach tail at long wavelengths. One other person that's very appealing to me at least is that we have many methods to calculate cross-sectional direct optics, but they can be generalized for other functionals, for some which, for example, we may not have the best function but the version is very implemented. For example, you can do hybrid functionals, you can put van der Waals functionals, you can do excitons. So it's a very general method and the details is very nice technical paper and physical review beam. All right, so this is the summary of the story so far. And of course, I'd like to acknowledge the funding sources as well as the collaborators both in doing the science and implementing the code and we'll practice more this afternoon to do some calculations yourself. And with that, thank you very much.