We have now seen how metals, semiconductors, and insulators are formed and we have touched upon the concept of electronic band structures. In this video, we will look at how light is absorbed in semiconductors and how light exposure generates a voltage. However, there are a couple of losses associated with the absorption of light, and we therefore need to figure out the optimal bandgap of a photon absorbing material in order to maximize a power conversion efficiency. After looking into the theoretical efficiency limit of a solar cell, we're going to review some of the most important semiconductors in the field of photovoltaics and finally, we're going to address the maximum voltage of a solar cell. First, we need to consider how light interacts with the semiconductor and learn some basic terminology. Remember, the concept of an energy bandgap with no allowed electronic states. For an electron in the valence band of a semiconductor to be excited by a photon into the conduction band, the energy of the photon must be, at least, that of the bandgap energy. Let us consider an incident photon with an energy greater than the bandgap energy. The photon is absorbed and excites the electron far up into the conduction band. Notice how the electron leaves behind an empty state in the valence band. Such an empty state is called a hole and it carries a positive charge. The fact that holes that charge is very important because they contribute to the total charge current in a solar cell. Another thing to notice is that the electron is excited higher into the conduction band relative to the conduction band edge. This is a result of the excess photon energy. However, this excess energy is lost as heat in a process known as thermalization, and this happens very quickly. The electron thermalizes down to the conduction band edge because these states are lower in energy. Ultimately, the electron will annihilate with the hole to once again fill the empty state in the valence band. As this transition takes place, a photon with an energy corresponding to the bandgap energy is emitted. This electron-hole annihilation is also known as recombination and happens on a much larger timescale relative to the very small energy transitions during thermalization. The time before the electron and hole recombines is actually enough for the charge carriers which we now know includes both the electrons and holes to be transported out of the solar cell. We just need to design the device properly. Let us look at the entire energy conversion process. We have learned that the sun is extremely hot and that this thermal energy is radiated from the surface of the sun. We quantified this radiant energy as photons and the solar spectrum describes the energy distribution. We have also seen that semiconductor is able to absorb photons if the photon energy is greater than or equal to the bandgap energy. This, however, means that the highly energetic photons will excite electrons high up into the conduction band and photons with energy smaller than the bandgap energy will not be absorbed. The excess energy of highly energetic photons is lost as heat in a process known as thermalization. Actually, we have now seen two of the most fundamental losses in a solar cell that is, below the bandgap, absorption losses, and thermalization losses. So far, we have only converted solar energy into chemical energy by exciting and thermalizing carriers. The pure semiconductor also known as an intrinsic semiconductor is therefore not enough. To convert chemical energy into electrical energy, we need the charge carriers to be transported out of the solar cell. We have just learned that both holes and electrons carry a charge, but the electron charge is negative whereas the charge of the hole is positive. For the charges not to cancel each other out, the electrons and holes must be transported in opposite directions out of the solar cell. This will generate an electric current and because the charges are at different energy potentials, there is an electric potential difference between them, or in other words, we have a voltage. The greater the bandgap is, the greater the voltage will be. Multiply the two together and the solar cell is generating electrical power. This entire conversion process where light is absorbed to generate an electric current and the voltage is also known as the photovoltaic effect. As we have already considered two of the most fundamental losses in a solar cell, we can already begin to optimize our devices by finding the ideal width of the energy bandgap. We would, of course, like both the voltage and the current to be as high as possible and we already know that the voltage increases with increasing bandgap. However, the low bandgap material is able to absorb both high, medium, and low energy photons. As the bandgap increases, the low energy photons are no longer absorbed and the high bandgap material is only able to absorb high-energy photons. The more photons and material absorbs, the more electron-hole pairs are generated and may contribute to an electric current. The current is therefore greater for the low bandgap material. If we calculate the electric power, we see that there is an optimum in-between the low energy bandgap and the high energy bandgap. But we already know that the solar spectrum isn't simply one blue, one green, and one red photon. We, therefore, need to consider how many photons a semiconductor is able to absorb from the solar spectrum as the bandgap changes. As we wander, solar panels would be on the surface of earth, we need to consider the atmospheric effects on the spectrum as well. This brings us to one of the most important contributions to the field of photovoltaics, namely the Shockley-Queisser limit. The Shockley-Queisser limit is the theoretical efficiency limit of a solar cell as a function of the bandgap energy and considers only the two fundamental losses we have already considered. Let us look at the assumptions in detail. The solar cell is illuminated with the air mass 1.5 global solar spectrum and all photons with an energy greater than or equal to the bandgap energy are absorbed. It should be noted that Shockley-Queisser originally used the airmass zero spectrum, but the air mass 1.5G spectrum is more relevant for solar panels on earth. The next assumption is that every photon generates exactly one electron and one hole. In principle, an electron in the conduction band could absorb another photon and be excited even further into the conduction band. But this energy would be lost as heat, which is of course not ideal. The third assumption is that the excess photon energy is lost as we've already learned. But a very important side note here is that the solar cell doesn't heat up. Later, we will see that solar cells actually performed worse as they heat up. The fourth assumption is that electrons and holes are allowed to recombine and emit a photon. But in the next video, we will see that this may actually happen in a variety of ways. The recombination mechanism with the electron annihilates with the hole in a transition across the bandgap is the only one allowed and as this transition happens from the conduction band to the valence band, it is also called band-to-band recombination. Finally, the last assumption is that the electrons and holes are transported without any losses in the solar cell. Therefore, the conversion of chemical to electrical energy is assumed to happen without any losses at all. The result of modeling all of these assumptions, is this efficiency graph, where I can see that the optimal bandgap for achieving the maximum efficiency is somewhere between 1.1 and 1.4 electron volts. The efficiency limit in this range is around 33 percent and this is very important to remember. Now things become a little bit more tricky. In the previous video, I said we have to borrow some results from quantum mechanics when the symbol Flat-band model fails to describe certain phenomena. Now, some materials absorb more strongly than others and this we cannot explain without a little help from more advanced models. Semiconductors are either direct or indirect, which refers to the position of the valence band maximum and the conduction band minimum relative to each other in this type of graph. We still have energy on the y-axis, but now we have momentum on the x-xis, which is equivalent to lattice vibrations. In the case of a direct semiconductor, the band edges are at the same position in momentum space. Therefore, to excite the electron into the conduction band, all we need is a photon. Such a transition is known as a direct transition. In the case of the indirect semiconductor, the band edges are no longer at the same point in momentum space. Therefore, to excite the electron into the conduction band, we need not only a photon, but also a lattice vibration to move the electron in momentum space. Such a lattice vibration is also known as phonon and this type of transition is known as an indirect transition. For an indirect transition to take place two particles are required to be present at the same time, namely a photon and a phonon. But this is not always the case. Therefore, an indirect transition is less probable than a direct transition. This is very important as indirect semiconductors therefore must be thicker to absorb all the useful photons, whereas direct semiconductors absorb photons very effectively. An example of a direct bandgap material is gallium arsenide, and silicon is an example of an indirect semiconductor. Let us take a look at some of the most important photovoltaic materials. Silicon is the most widely used photo absorber and with a bandgap in-between 1.1 and 1.4 electron volts, it is very close to being ideal according to the Shockley-Queisser limit. But silicon has an indirect bandgap, meaning that it absorbs light very poorly. We therefore need to use a fair amount of silicon combined with light trapping techniques to make sure all useful photons are absorbed. Researchers are looking for direct bandgap materials, as an alternative to silicon to reduce the material consumption. This field is known as thin-film photovoltaics, and the commercially dominant thin-film PV photo absorber is cadmium telluride with a direct bandgap of 1.45 eV. Another thing to notice is that I've written silicon three times in this table. In the last video, we saw that solid materials may be classified as being either single crystals, poly crystalline, which is also known as multi-crystalline or amorphous. The highest record efficiency has been achieved with a single crystal. But it is interesting that the bandgap has increased in amorphous silicon to 1.75 eV and it is now direct. This is, however, still far from the Shockley-Queisser optimum and the maximum attainable efficiency is therefore lower. Other direct bandgap materials include gallium arsenide, where the incredible record efficiency is 29.1 percent. We also have copper indium gallium selenide and copper zinc tin selanide, and these materials are often discussed in the field of PV. So it is a good idea to get acquainted with them. The voltage we get out of a solar cell is not exactly the width of the band gap. Actually, we need a deeper understanding of how electrons and holes are distributed in the conduction and valence band should evaluate the maximum voltage of a solar cell. The probability of a state being occupied by an electron is described by the Fermi-Dirac distribution function, and since there are extremely many electrons, this statistical description is very accurate. First, let us consider the case where the temperature is zero Kelvin. The distribution function becomes a step function that goes from one to zero at the Fermi energy level. Here I have drawn a metal and as you can see, all the electron states below the Fermi level are occupied, whereas all the electron stays above the Fermi level are empty. Now let's turn up the temperature. As you can see, the distribution function broadens. If we look at the metal from before, you can see that a few electrons are now thermally excited above the Fermi level. However, the high on energy we look, the fewer electrons we find, and this is in agreement with the distribution function that states, the high on energy we go, the lower the probability of a state being occupied. Finally, let's turn the temperature way up. You can see that the distribution function broadens even more, and so does the distribution of electrons and holes. We can actually use the probability of finding an electron to describe the probability of finding a hole as well. Remember a hole is merely a state where there is no electron. If we have a 30 percent probability of finding an electron in a state, the probability of finding a hole in that same state must be 70 percent. We may write the probability of finding a hole as one minus the Fermi-Dirac distribution function. Now that we know the probability of each state being occupied, which is given by the Fermi-Dirac distribution function, we would like to know the concentration of electrons and holes. For this, we need to know how many states are there. This is given by the density of states, which could look something like this. If we have 10 states at a given energy, and the probability of each state being occupied is 30 percent, we simply have three electrons at this specific energy level. To calculate the total concentration of electrons in the conduction band, we have to integrate the density of states multiplied by the probability of each state being occupied, over all the energy states in the conduction band. The same concept applies for the hole concentration in the valence band, but notice that the probability of each state being unoccupied, is given by one minus the Fermi-Dirac distribution function. Until now we have only talked about perfect pure semiconductors, which is also known as intrinsic semiconductors, and if we calculate the electron concentration in the conduction band, and the hole concentration in the valence band, they are actually equal. Therefore, if we multiply the two intrinsic carrier concentrations, the results should be the same as if we just square the intrinsic electron concentration. I know this may seem quite obvious, but this equation is actually super important. Let's say I somehow pour a bucket full of electrons into a semiconductor. We have previously considered the Fermi level as an electron water level. As a result, the Fermi level must shift up. But what about the product equation then? The electron concentration n has increased, but the concentration of empty states p has decreased as a result, and therefore the product is still equal to the intrinsic electron concentration squared. Similarly, had I put holes into the semiconductor, the electron concentration would decrease. But what happens if we shine light onto the semiconductor? When light is absorbed, we generate both electrons and holes, and therefore both n and p must be increasing. As a result, the equation is now an inequality. But why is this important? When this equation becomes an inequality a voltage is generated. The water level has increased as we increased the concentration of electrons, but as we have also increased the concentration of holes, the water level should also have decreased. Therefore, we need a Fermi level now to describe the concentration of electrons, and another one to describe the concentration of holes. We say that the Fermi level splits up into two quasi-Fermi levels. The difference between these two quasi-Fermi levels is exactly the maximum voltage of the solar cell. This concept has of course only been covered very vaguely, but the main point is that the Fermi level splits up into two quasi-Fermi levels when the solar cell is illuminated. One for the electron water level and one for the hole water level. The difference between the quasi-Fermi levels is the maximum voltage of a solar cell, and to maximize this quasi-Fermi level splitting, we can either increase the band gap or the concentration of both electrons and holes. As we transport electrons and holes out of the semiconductor, the carrier concentrations of course decrease. We need to sacrifice a bit of voltage to get an electric current. In the next video we will look at how to transport the charge carriers before they recombine and the energy is lost. But first we need a summary. In this video, we have learned that a photon with energy equal to or greater than the band gap energy, may be absorbed in a semiconductor and excite an electron up into the conduction band. The empty state left in the valence band is known as a hole, and it carries a positive charge. Any excess photon energy is lost as heat to the lattice in a process known as thermalization. We also looked at the photovoltaic effect that describes the entire conversion process of light into a voltage and an electric current. We also looked at the maximum theoretical efficiency limit of a solar cell as a function of it's band gap, known as the Shockley-Queisser limit. Now this is a compromise between the number of photons in the solar spectrum the semiconductor is able to absorb and the maximum voltage. We also saw that indirect semiconductors require the participation of a phonon or a lattice vibration to absorb photons in the first place. Therefore, direct semiconductors absorb much more strongly. Finally, we look into the light exposure, which splits the Fermi level of a semiconductor into two quasi-Fermi levels. The difference between the quasi-Fermi levels is the maximum voltage of the solar cell.
