In the last two videos, we have looked at how solid materials are formed, the concept of electronic band structures, the photovoltaic effect, and what determines the maximum voltage of a solar cell. However, we still haven't considered how the charge carriers are actually transported out of the solar cell to generate the electric current. And as a matter of fact, we don't have an infinitely long time to get them out. As semiconductors are able to absorb light, they also have to emit light again, thus losing the chemical energy we wanted to convert into electricity. Emission of light is the result of electrons and holes recombining. And this may actually happen in a variety of ways. Some we have to live with, others we may engineer our way out of. In this video, we will look at the various recombination mechanisms and address the time we have before the charge carriers are lost. We will then study the transport mechanisms and finally revisit the working principles of a solar cell. In the bulk of a semiconductor, electrons and holes may recombine in three ways. The first is known as radiative recombination, or band-to-band recombination. And it is the spontaneous transition of an electron in the conduction band into a hole in the valence band. And as a result, a photon with an energy equal to the band gap energy is emitted. As both electrons and holes participate in this reaction, the recombination rate increases with increasing concentration of electrons in the conduction band and holes in the valence band. Here B is the coefficient of radiative recombination. The second way is Auger recombination, which involves three particles, two electrons and one hole or two holes and one electron. The energy emitted from the recombination of an electron and a hole is transferred to a third particle, either an electron or a hole, and is subsequently lost as heat to the crystal lattice in the process we know as thermalization. The recombination rate increases with the concentration of electrons in the conduction band and holes in the valence band. But as the third particle can either be a hole or an electron, we actually need two coefficients of Auger recombination. If the third particle is an electron, their rate would increase even more with an increasing concentration of electrons in the conduction band. We therefore have to multiply the rate coefficients in the parenthesis by the concentration of the third particle as well. The third recombination mechanism is known as impurity recombination, where defects in the semiconductor crystal introduces electronic states in the middle of the band gap. As an electron moves around with some thermal velocity, there is a probability that it is captured in the impurity state. There must be a hole in the impurity state for the electron to be captured in the first place. And this is all we need to know to write up the rate equation for impurity recombination of electrons. Here we have the probability of the electron being captured. Here we have the thermal velocity of the electrons, the concentration of the electrons in the conduction band, and the concentration of impurity states occupied by a hole. Similarly, if we swap the n's and p's around, we would obtain the rate equation for impurity recombination of holes. Radiative and Auger recombination are intrinsic mechanisms, meaning that we have to live with them even in a perfect pure crystal. But we can limit the impurity recombination by making semiconductor crystals with fewer defects and impurities. If we look at the surface of our semiconductor, we see a large density of states in the middle of the band gap. Such states will promote recombination, and it's definitely something we need to fix when making a solar cell. As the crystal lattice is disrupted at the surface, the atomic bonds of the outermost atoms may now be dangling, and such dangling bonds form recombination centers. Therefore, we must find some element to saturate the dangling bonds at the surface that doesn't introduce as many states in the middle of the band gap. This fix where we reduce the number of dangling bonds at the surface is known as surface passivation. In amorphous silicon, most of the atoms are not bonded to other atoms in an ordered fashion. And therefore, many of the bonds are actually dangling, even in the bulk. As the dangling bonds form recombination centers, the material is completely unsuitable for a solar cell. However, by introducing hydrogen to saturate and thus reduce the number of dangling bonds, amorphous silicon has actually been a commercial success in PV for certain applications. Sadly we cannot saturate all the dangling bonds in a bulk piece of amorphous silicon, as this would change the optical properties off the material. And what is even worse, the irregular crystal structure strains the silicon silicon bonds that are actually present. Such strained bonds may be split open as charge carriers recombine over time, thus forming new dangling bonds. This effect is known as the stable of onesky effect. Finally, we must study the interface between materials. Once again it is undesirable to have stakes in the middle of the back up. But as we need metallic contacts on both sides of our solar cell To transport the carrying out an intro load, there will be two semiconductor to metal interfaces at some point. Now, there are several engineering tricks to avoid recombination at such interfaces, but the objective is always the same. Make sure only one type of carrier reaches a metal contact. If both an electron and a hole gets close to such an interface, recombination is practically unavoidable. This brings us to two very important and useful parameters, the carrier lifetime and the diffusion length. If I shine light onto a semiconductor, I'm generating electrons and holes. But if I now turn off the light, the electrons and holes vanish after a mean lifetime. This is the result of the recombination mechanisms. We've just looked at Carry a lifetime, some sub the contributions of the individual recombination mechanisms and is therefore a good indicator of material quality. If we were to calculate the bulk lifetime, we need to use the excess carrier concentration. That is how many extra carriers have we generated using light. Divided by the sum of the recombination rates in the bulk in silicon carrier lifetimes may be as high as one millisecond, which is considered extremely high. The diffusion length describes the average distance to charge carriers move around as a result of diffusion before recombining. To calculate the diffusion length, we need to use the carrier lifetime and the diffusivity also known as the diffusion constant D, which is the rate at which the carrier spreads out in the material. The diffusion length and silicon is typically in the range of a few 100 microns, which is very impressive as well. But notice one thing, I have never specified which type of charge carrier I was talking about in this slide. The lifetime of holes and the lifetime of electrons are different. But when talking about lifetime and diffusion length, it is usually never explicitly stated which type of carrier we're talking about. It is implied that we are always talking about the carrier, we have the least of which is also known as the minority carrier. Consider this a rule of thumb. Whenever you don't know which type of carrier people are talking about. assume it's the minority carriers So, charge carriers have a limited lifetime, and we therefore only have a limited time to transport them out of the semiconductor. But how do we get a charge carrier to move? The two basic transport mechanisms are drift and diffusion. And in the last slide we briefly introduced the concept of diffusivity. That is the rate at which particles spread out in a given material. Let's look at how electrons spread from the right hand side of the material, and let's also plot the electron concentration, which right now is a step function. Diffusion is a process where particles spread out from regions of high particle concentration to regions of low particle concentration. Therefore, after a short time, a few of the electrons must have made it to the left hand side and the electron concentration is now more even than it was before. However, there still is a gradient and the electrons will therefore continue to spread out until the concentration is the same everywhere. The diffusion charge current for electrons is the electron charge multiplied by the diffusivity and the electron concentration gradient. This means that the charge becomes smaller for smaller concentration gradients. The exact same principle applies for holes but with an opposite sign, as the whole charge is positive. If we add the two charge currents, we arrive at the total diffusion charge current. Now, how does this all look in our band diagram, as the quasi Fermi levels represent the water level of electrons and holes, which is just another way of saying the concentration of electrons and holes. The quasi-fermi level of electrons must be closer to the conduction band Edge when we have more electrons and the quasi-fermi level four holes must be closer to the valence band Edge where we have more holes. The other transport mechanism is known as drift as electrons and holes are charged particles, forces. Act on them in an electric field. Let's set up an electric field to see how this works. We charge two parallel plates with opposite polarities. The first one is going to be positively charged and the second one is going to be negatively charged. In between the two plates, we have now set up an electric field that is indicated by these field lines. If we place our carriers in between the two plates, we see that the negatively charged electrons will be attracted to the positively charged plate and repulsed by the negatively charged plate. We say that the electric field induces a force onto the electrons, driving them against the field lines. As the holes are positively charged, they will be attracted to the negatively charged plate. However, this acceleration of the charge carriers is frequently disturbed as the electrons and holes bump into thermally vibrating atoms in the crystal as well a Zionist impurity atoms. The carrier mobility is a measure of how easily the carrier drifts through the material in the presence of an electric field. We may write the electron drift charge current as the electron charge multiplied by the electron concentration, the electron mobility and the electric field strength. The whole charge current may be written in a similar way, but notice that the sign is the same. This is because the holes have an opposite charge of the electrons, and they move in the opposite direction and minus times minus gives plus. You may see the electronic charge multiplied by the carrier concentration and the mobility. Written as the Greek letter Cigna, these three fundamental parameters multiplied together is what we know as the conductivity of the material or, more specifically, the electron and hold conductivity. We get the total drift charge current by summing the two contributions. But how does an electric field look in our band diagram? If the particle concentration is constant throughout our semiconductor, the distances between the two quasi-Fermi levels and the band that just don't change. But the band and just themselves actually shift downwards towards the side where we placed the positively charged plate. Now we're going to add the two contributions drift and diffusion together to get the total charge current. You may see this equation in many other forms as the defensive it ease or diffusion constants are closely related to the carrying mobility's. This relationship is described by the Einstein relation. Another important conclusion to make is that the electric field forces may in fact be counteracted by the diffusion forces. So even if you have an electric field built into your device at grading in the carrier, concentrations could potentially cancel out the forces from the electric field, resulting in a net force equal to zero and thus no current. Let's look at our band diagram. In this special case, we have placed the two opposite. We charged plates to set up the electric field, and we do see the band that just bending down towards the positively charged plate. Now introduce more electrons on the left hand side and more holes on the right hand side of the semiconductor, such that we have a gradient in the carrier concentrations. The drift forces want to move the electrons to the left, and the diffuse of forces want to move them to the right. As we have more electrons on the left hand side of the semiconductor the qualified for my level of the electrons should be closer to the conduction band edge on the left side and further from the bandage on the right side. The same applies for holes where the quasi-Fermi level should be closer to the valence band on the right side and further from the bandage on the left side. Interestingly, the two quasi-Fermi levels are constant throughout the semiconductor when no current is flowing and this brings us to the main conclusion for charts transport in a solar cell, the derivation and concepts used to derive the Fulham relation. It's much too complex for this course, but the result is super practical and important. The total current can be described using only, the carrier conductivity, ease and the gradients of the quasi-Fermi levels. So to transport electrons and holes out of the semiconductor, we need to create a gradient in the quasi-Fermi levels. In the next few videos, we will look into how this is realized in practice. But first, let's revisit all the different concepts in a single slide to put them into a broader context. Thermal energy is radiated from the surface of the sun as black body radiation. A single quantum of this radiant energy is known as a photon and photons may be absorbed in a semiconductor. That is, if the energy of the photon is at least that of the band Gap Energy. Any excess votes on energy is very quickly lost as heat to the crystal lattice in the process we call thermalization. The shark requires a limit, predicts the maximum theoretical efficiency limit by considering the losses from thermalization, and the nun absorbed photons with energies below that of the band Gap Energy. The Fermi levels describes the concentration of electron. Electrons and holes, but as we would generate both types of carriers on the illumination, the Fairmont level splits up into two cross fair my levels. And we have now converted through radiant energy, into chemical energy. We need to do one more conversion step, but now the clock is ticking. Electrons and holes will recombine after and mean lifetime by either radiative or a share or impure to recombination. Especially the surface and interfaces require special attention where's radius and sheer recombination are intrinsic mechanisms and therefore unavoidable. So if we manage to set up drift and diffusion correctly the total charge current will transport the current out of the device which is our electric current. Voltage was achieved by splitting the costs I fare my levels, but we actually do have to sacrifice a bit of voltage to get the current flowing. Remember we need a gradient in the quasi fair my levels to transport the electrons and holes out. As we have now set up a voltage and gotten an electric current to flow, we have successfully converted radiant energy from the sun into electrical power. In this video we have learned about the three bulk recombination mechanisms radiative or shear and impurity re combination, while radiative and shear recombination are intrinsic mechanisms, impurity recombination is something we can limit by growing semiconductors with fewer defects and impurities. Especially the surfaces and interfaces require special attention as the dangling bonds must be passivated at the surface and then each metallic contact only one type of carrier should be allowed. By summing up the contributions of the different recombination mechanisms we arrive at the carrier lifetime and the diffusion length, which are important parameters as they are used to assess the quality of the semiconductors used in PV. Knowing that the charge carriers don't exist infinitely long in their excited state, we looked at the two basic transport mechanisms to get them out of our device and into an external load. The first transport mechanism is diffusion, which is a result of a gradient in the carrier concentrations. The second transport mechanism is drift, which is the result of an electric field being present in our semiconductor. Adding the two contributions, we arrived at the total charge current, where we dealt with a special case of drift and diffusion cancelling each other out. From this we learned that for a current to flow, we must have a gradient in the crossfire for my energy levels.
