Chem 2017 Solid State Chemistry Lecture Notes PDF

SPEAKER 0 Okay, let's make a beginning. Uh, first of all, announcement for any of you that missed yesterday's lecture, is the tutorial. A workshop is now as a tutorial. Worksheet is now online and the due date for that is next Wednesday. Uh, and of today, uh, the workshop sheet for um, the workshee...

SPEAKER 0 Okay, let's make a beginning. Uh, first of all, announcement for any of you that missed yesterday's lecture, is the tutorial. A workshop is now as a tutorial. Worksheet is now online and the due date for that is next Wednesday. Uh, and of today, uh, the workshop sheet for um, the worksheet for the workshop, uh, will be online and associate chemistry workshops in two weeks on Monday, our two weeks from yesterday. Uh, and the thermodynamics workshop is going to be two weeks from today. Uh, your tutorial takes precedence over the in the workshop, but to be good, if you look over the worksheet online before, uh, the workshop itself, I'm, I, uh, Doctor Wheatley tells me that he should get the his worksheet online and next week, if it's not already there. Um, we're now into lecture eight of chem 27 cam 2017 as part of the solid state chemistry course. And today we're going to be talking about the electronic structure or the electronic properties of materials. In particular, we're going to be looking at conductance and um, and resistance. We're going to introduce a band theory to understand what's happening here, specifically the linear combination of atomic orbitals for solids. You should have already done the missing atomic here. Apologies if you've already done linear combination of atomic orbitals when you're doing your molecular courses, but we're now going to extend it into solids. Um, and we're going to talk about the differences between conductors, semiconductors and insulators. Uh, just to remind you from the lecture yesterday, uh, we were discussing different kinds of X-ray sources. Um, in particular, we introduced the x ray anode, which is the most common type of actually source, uh, on this planet. Um, at least on this planet, uh, And it's going to be the actual source you'll find in a laboratory at the university is going to be the actual source you find in a medical lab, uh, or in an airport when you're going through security. And it basically works by having a very high voltage on your anode. This then strips electrons out of a hot filament. These electrons are accelerated towards the anode where you'll have a target material. The electrons will kick out electrons from the core levels of the target material. Electrons from higher Angela will fall down and then actually light with a well-defined energy. These are the emission lines that you get from, uh, anode actually sources. Um, we also heard about synchrotron light sources, which instead of working on the emission lines that come out from, uh, from atomic species, is working on, uh, oscillating or changing the direction of an electron and causing that to emit X-rays as a result, and from using these sorts of devices using very energetic electrons about the 99.999% the speed of light, we get far more intense light, far more brilliant light that can be used for a wider range of experiments. I am these kind of sources. The synchrotron sources emit light from many different kinds of wavelengths, many different colours, many different energies, depending on what way you want to think about this. But for our experiments, we often just want one colour. We want to monochromatic light. I mean, I find out actually that we can do that using a very similar principles that we've been talking about throughout this lecture of diffraction, but instead we have a case where we know the structure of a material, we know the energy we want, and therefore we can choose our diffraction material and the orientation of the diffraction material to select out a different kind of light. This is used pretty much universally in synchrotron light sources. You can also use them in lab sources. But there you don't need the second crystal because you're only looking at it. Need a single bounce because you only have the single primary energy that you want to take out from your, um, your emission line. Um, and because we have these monochrome metres in our synchrotron sources, uh, we can choose the energy we want to get. And so the different beam lines that you get at synchrotron will all cover a different range of energies. We can choose the energy we want for our specific experiment, uh, rather being beholden to what the atoms give us. But today we're going to go on to electronic properties of materials. In our first lecture we talked about some physical properties of materials. Uh, so things like metallic compounds, um, can be malleable. You can bend them and twist them. Um, and ionic compounds are very stiff and fragile. you can shatter them, whereas you get very flaky components when you're working with van der Waals materials. We similarly talk about hardness that when you have a covalent material, it is very stiff, it's very strong. You can't just shatter it by poking. It is very hard to break up equivalent material. Um, and when we're talking about thermal stability, we learn that the ionic compounds and the covalent compounds have very high melting temperatures. Metallic compounds have a range of melting temperatures, and the materials are typically very low. But what we're going to talk a lot more about in this lecture, and the next one is going to be the optical and electronic properties of materials. Optical properties are related to electronic properties. They come out of the electronic properties of the material. Uh, one of the most common applications of this throughout history is the colour or pigmentation. The colour of pigmentation that you get from these sorts of materials. I've got feeling much better interacting with the microphone. Let's hopefully fix it with that. And so pigmentation is a very big part of a of our industries throughout the history of, uh, humanity. We've always wanted to have colourful things colourful clothes, colourful textiles, colourful, uh, windows, colourful glasses, colourful, um precious objects. And so this has been the first way that we've really used electronic structure of materials in order to gain, in order to, to make things and build things around us. In fact, the first nano science, sorry, first nano scale industry in the world was actually doing pigmentation from gold nanoparticles that they used gold nanoparticles to make stained glass windows. Uh, back in the medieval era, they didn't know they were nanoparticles per se, but they knew that they took small amounts of gold to mix the end with, uh, a solvent, some kind of other. They got these interesting colours that you could put into glass. Similarly, glasses are very important, not just because we want to be able to see out of the buildings we're in or see through walls. But also if you think about your mobile phone, you have your LED screen on there. You don't want to get that wet, you want it to be sealed against the outer environment, but you also want to be able to see through it. So it needs to be transparent. Going even more complicated than this. I'm if you're going to like photovoltaics, you need to have something that's transparent on top of it so that the light goes through it to make your solar cell work. But you also need to have a contact and electrical contact on there. So it needs to be something that is both transparent and also conducting, which is as it would turn out, a very challenging thing to do. Um, refractive index is also something that's been very, uh, important through history, you know, with the advent of glasses apart from anything else, but also anything that uses lasers or anything like this. So you're talking things like, uh, and maybe showing age a bit here, but CDs and DVD players and, but anything that uses a laser has to use refractive index material in order to direct where the light is going. It's very useful to control where we are sending light using the refractive index of materials. And then we're going to get to the three things that are what we're going to spend the most time today talking about, which is conductivity. Uh, everyone here has used a has used electricity. Electricity is transmitted along conductors. Uh, you will perhaps in your day to day have used a conductor, a wire that is maybe metres in length. But you'll also know that like in high voltage transmission lines, you get wires. They're kilometres in length. And we need these to work in order to get the electricity from where we generate it into our houses and to our places of work and stuff like that, But perhaps even more important than electrical conduction is semiconductors. This is semiconductors. This is why we have computers. This is why we have the transistor. This is why we can have the font of all human knowledge in our pockets and our mobile phones. Semiconductors is what revolutionised the world over the last hundred years. And without semiconductors, it is not possible to have logic in the way that we have it in, um, inanimate objects like computers. And then since we have things to conduct, since we have things in semiconductor, we also need to be able to separate them from things that we don't want to conduct or semiconductor. We need to have electrical insulators. You wouldn't want the outer part of your, uh, of your mobile phone to be solid metal and, and connect your battery or your gas shock every time you pick it up. You need to have something that separates the charges from you so you can operate these devices. So we need to talk about a new term to us Today, which is electrical conductivity. Um, so for conductivity in our arrays we define a few new terms, the first of which is resistance. You may well have come across resistance before, especially if you did A-level physics or if you're doing a physics and chemistry course. Are we doing anything electrical engineering and you'll have come across before resistance is what slows down current going through a wire when you apply voltage to it. So when you apply voltage to a conductor, you'll have current flowing through it. How much current you get is the voltage divided by the resistance? Um, resistance is measured in ohms. And you can think of it in terms of when you pass current through material, it is going to scatter off the material. And you know, if the material is very resistive, the current cannot pass through very easily. It's going to struggle to conduct. You will get therefore less electrical current. The conductance of the same piece of material is given by C, and it's just the inverse of R in units of. So resistance is in ohms. That should be an ohm not a w. Apologies. Uh is in ohms I conductance is in inverse ohms. Inverse ohms are typically given the unit of Siemens. But the um the problem with both resistance and conductance is they are very specific to the particular item you're dealing with. They are unique to the item you're dealing with. And we want to talk about general properties of materials. And so we need to define new terms which are the resistivity and the conductivity. And these are properties of a given material. Every piece of copper will have the same resistivity and conductivity. Every piece of iron will have the same resistivity and activity. Every specific alloy of steel will have the same resistivity in cold activity. So the item you're using may have a different resistance or conductance, and so resistivity is measured in metres and conductivity is the inverse of resistivity. So it's in Siemens per metre. And you may be wondering why is it in. Why is metres playing a role in here. Well great. My clicker is not working. Well I'll come round to you and tell you then. So, uh, why it matters is that the formula to convert conductance back into resistance is the length of the wire that you're conducting along, divided by the cross-sectional area that it's going through. So your resistance equals length divided by area divided by conductivity. And if we just work through the units you'll see it comes out that you've got resistance. Ohms length is metres area's inverse metres. So conductivity needs to be per room per metre. But that's not really a very interesting answer that the formula works out. The reason for this is pretty easy to imagine that you have your material here, and this has a certain amount of resistivity as a thermal conductivity. You make it longer, and there's going to be more material that's resisting your current flowing through it. The longer your wire is, the more force you'll need to shove an electron through the whole thing. It's the longer it is, the more resistant the material has to be. Similarly, if we think about the cross-sectional area, you may think, oh, we have more material so that not give us more resistance. But if you think about water flowing through a pipe, think about rainfall landing on the ground and then going down into a drain. If you have a very narrow pipe, it's going to be very quickly back up. It's going to be very hard to force all the water through. If you have a very broad pipe, it's going to be much easier for the water to go through it. It's the same with conductivity. The larger the area, the more channels you have for the electrons to travel through. The easier it is for um, For electrical current to conduct down through the wire. And so this is why you have the area is inversely proportional to the resistance and the length is proportionate to it. So we now get onto conductors themselves. And we're going to give a definition for conductors. And it's going to be very reminiscent to you because effectively identical to the definition we gave for a metallic species. An electrical conductor consists of iron cores embedded in the sea of free electrons. This is the definition we gave for metals. Something being metallic is synonymous with it being a conductor. Something cannot be metallic unless it is a conductor. If you have an element that has one phase that is conducting and one phase is not conducting like we saw for tin a couple of lectures ago, then the phase that is conducting is generally known as the metallic phase. The one that is non non-conductive, as known as the non-metallic phase. Metallic means more than anything else that it conducts. So we have our electrical sector that's in, uh, has anchors in the sea of electrons. When we apply a potential difference across the conductor, the electrons can flow from the stationary iron cause from the negative end to the positive end, and hence carry the electric currents. So we apply a negative voltage at one side, a positive voltage to the other, and the electrons will want to flow from the negative end to the positive end. So at this point it's worthwhile talking about what we mean by negative and positive end. What does it mean that we've applied a negative bias to something. We apply the negative voltage. Essentially all it really means is we have put our conducting element in contact with something that has too many electrons on the negative end, and then the positive end is something that is too few electrons. If you think about the water case, you're basically connecting a pipe between the place that's got a really high pressure of water in one side and no pressure of water in the other, so it's going to want to flow down this potential down this hill. And the same way the electrons will flow down the electric potential to go from the negative end to the positive end. The conductivity of, as I wanted to know, though, is that the electric field that causes these electrons to move the field itself is going to travel at close to the speed of light. It's something like two times ten to the eight metres per second, but the individual electrons will go far more slowly, more like the speed of sound, something like 300m/s. And this is because the electrons will the field will go very quickly. That field will have experienced by all the electrons almost immediately from from our perspective. But the electrons as they travel along are going to scatter off the iron, cause they're not truly free. They will still see that in chorus. They'll still want to go towards the end. Cause so as they travel through this material, they're going to scatter off them. And so this is what you see in terms of resistivity, that every time the electron scatters and bounces back your current is going to decrease. And this is what resistivity really means. And so as you heat up a conductor, as you put more energy into the system, the individual atoms are going to bounce around a lot more, which means it's more likely that an electron passing through is going to be scattered by one of them. And so this means that the conductivity of a metallic compound will decrease as the temperature increases. Or in other words, the resistance of a piece of metal of a kind of a conductor, a metal will increase as its temperature goes up. Now, you've written this is working. Um, and so that is the case for conductors. One thing I should mention that I mentioned to this slide. The conductor of a conductor is in the order of greater than a million Siemens per metre. We then jump to semiconductors. The conductivity is lower. I basically it's going to go from 100,000 Siemens per metre down to like ten to the minus four, down to a very low number. Uh, in the semiconductor we still have conductivity happening. We still have. If we apply a bias, we still get electrons going from one end to the other end. It's just much, much less in the sort of materials point of view. You can there is you can imagine an almost continuum of materials going from being very poorly semiconducting all the way up to beautifully conducting, and they will all exist. And an insulator is just a special kind of a semiconductor where the conductivity becomes really low, down to the point that's almost immeasurable. So this would be less than ten to the minus four or Siemens per metre. Um, one thing to note, though, is unlike conductors, where the conductivity decreases as you increase the temperature, the conductivity of semiconductors increases as you increase the temperature. And we'll get into this a bit later. Uh, we'll get into why this is a bit later in this lecture. And it's actually the case that an insulator, you can definitely turn into a semiconductor by putting temperature into the system. And in theory, a semiconductor, you could turn into a conductor by putting energy into the system. But the problem is that most are certainly most insulators. If you put enough energy into them to turn them into conductor, they're going to stay decompose or phase change into different material. It will no longer be the same material. By the time you get to the temperature, there might become a conductor. And you can also see that certain point these two lines will end up merging. Though you may turn into a conductor, it might not be a particularly good conductor. At that point the temperature will be too high. So we want to develop a way to talk about why we get conductivity, why we get semi connectivity. And for this we want to talk about the electronic structure of our materials. So we're going to get into band three. Um, there are broadly two sort of theoretical models which are commonly used to explain the bands that we see in solid materials. Note that in both these cases, we still have our individual orbitals that are associated to our individual elements. They are still localised around their element. But we also need to have the, uh, electronic bands that are present in our whole material in a similar way that we have our molecular orbitals that are present in their molecules. Um, the most mathematically correct model for this and some other we're not going to really talk about at all today, uh, is in the early free electron model, uh, what the newly free electron model is, is, uh, in case you may not guess the name, there's another model which is the free electron model. The free electron model is you basically very similar to what you get in like orbitals. You bring your two ion And of course, together they sort of meet a certain point in potential. And then you have an additional band that sits above that potential. And so this is your free electrons that your sea of electrons are conducting in your metallic material. But the problem is this is not a very good descriptor of what's actually happening in a, in the physical sense, in, say, the crystal, it does not end up describing semiconductor states, for example. Um, and rather the more precise theory is the nearly free electron model, where you don't actually have this flat top in the potential. Instead, there's a bit of a spill over the potential. So this an electron that's in this state can't just freely travel from one end to another. Instead, there's a little potential has to tunnel through. So this is electron tunnelling through a potential, um, and this can happen. It's not a very high potential. There's a probability an electron make it through, but there's also a probability that the electron will be backscattered. And so this ends up becoming very mathematical very quickly. And then in order to understand it, you need to go through the math. The math ends up being very similar to those in particle in the box. The same sort of idea. You have an electron wave, its wave function scattering off the potential wall, going back and interfering with itself. Um, and it ends up describing a lot of what we're going to see very precisely. But this is what we want to pursue for wanting a mathematical description of what we're doing. But that's not really what we're after today. What we're after is an understanding of what is happening. And so for this we're going to go to the linear combination of atomic orbitals. Uh, our local uh, theory. You I believe, were introduced to this probably last year to even talking about molecular orbitals. Um, but it's a case where you start with discrete or atomic orbitals. You then bring the atomic orbitals close to one another and they make new molecular like orbitals. When you're talking about solids, you're taking many, many of these atoms to make a much larger Orbital, which cannot be called a band, hence the band theory. The big advantage of using a linear combination of atomic orbitals to explain this is, first of all, that the is not that bad if you work through it mathematically, which we're not going to do today. But if you do work through mathematically, it's close. It's just not great. Um, but because of its strong relationship to molecular orbitals, to the work that you've already gone through, it becomes a very chemically intuitive way to think about what's happening. And so it's very useful to understand the electrical why we get conductors and semiconductors and why we get these bands in materials. So just a little flashback to what actually is to what this combination actually means. Imagine you have two hydrogen atoms and they all have A1S orbital. And that's orbital can either be a positive phase or a negative phase. Both of those are energetically equivalent. It doesn't matter which one it is. It's just you know, they are equivalent to one another, where they become equivalent as when you start to bind them together. If you get two orbitals being pushed together, they are of the same phase. This will form a bonding orbital, and you'll have both a plus plus and the minus minus version of this. As long as they're at the same phase, this will be energetically favourable. You get nice overlap of the wavefunctions. You get increased density in the wavefunction between the two orbitals. It makes a bonding state. We're very happy. However, if we instead take one, the positive is a negative phase and we push them together, we get an antibonding orbital. The wave function, instead of having extra dents in the bond area, has decreased density in the bond area. It pushes start to fill. It pushes the two atoms apart, is not energetically favourable, and so we end up with their molecular orbitals where one is lower than the energy, our bonding orbital one is higher up in the energy our antibonding orbital. And in the case of hydrogen, where each has one electron being donated into the new orbital, we have the two electrons for then the lowest level and the antibonding orbital remains empty. When we go to solids, we are basically doing the same thing, but many, many more times. So first of all, let's imagine the bonding orbital case. We have our orbitals that have the nice positive positive overlap or negative negative same difference. And we have an Avogadro's number of them. So we have six times ten to the 23 atoms that all have the same our orbitals I should say that all have the same phase lining up together. This is very energetically favourable. This makes a new orbital that is extended over the entire material because they're all sharing this electron charge. Um, and you get this very low energy for this new band. And then on the exact opposite side, you have the case where every other orbital is of the opposing phase and this will be very antibonding. You'll have six times ten to the 23 orbitals that do not want to overlap with one another. If you populate this with the electrons, it's going to be very energetically unfavourable. And so it's going to have a very high energy. Then we can think about the edge cases, the in-between positions. So imagine you have six times ten to the 23 in-phase orbitals and then one that's out of phase. This will clearly be a higher energy than the one where they're all in phase with each other, but not really that high. You have, what, ten to the 23 more atoms that are all aligned with one another and only one orbital. Sorry, they're all aligned with one another and only one that's not. So this will be just a tiny bit higher in energy. And then you have the case where you have n Avogadro's number minus two orbitals are in phase and two they're out of phase. Clearly the simulation less energetically favourable again but not by that much, there will be another ten step up. And this will happen exactly the same way in the anti phase as they do. You have n orbitals, Avogadro numbers, orbitals there and face each other. And then two there have to be in phase. This will of course be a bit more energetically favourable than the totally anti phase one, but not by much. And so these are all combined together. Every single one of these combinations will exist. And you'll get this near continuum. Continuum means there's a any value go to there is something sitting there. You'll have this near continuum of states where you can potentially find unavailable electron orbital. This is a big distinction for what happens in molecular systems. Molecular systems, by their definition, have discrete electronic states. Atoms have discrete electronic states. There's only certain energies which can occupy, but in a solid material there is a band where effectively the differences in energies are so small you can occupy any of them, any potential energy across there. You can also see the same thing replicated in 2D and in 3D. And so you get bands the same, very similar sort of bands in two dimensions and three dimensions. And when you have a case where you have. So these are all drawn as if they are s orbitals. This is where electrons have their sphere charged. They're going to donate into a metallic system. They'll have one s electron in their outermost shell. In general at least when we're talking about the transition metals. And so they'll be donating one electron into this sort of bonding configuration. And this will have fill the band that you're creating. Um. Pardon me. Sorry. Uh, you'll also get the same thing for p d bands that they will similarly overlap to make these sort of bonds. They're just a bit harder to draw, but you. SPEAKER 1 Get. SPEAKER 0 Basically. SPEAKER 1 Exactly the same. SPEAKER 0 Thing. You'll have the one where you have the most bonding orbital, the bottom, and the most antibonding orbital at the top. Um, and the pad bands will fill based on the number of electrons you actually put into it. And the way we'll talk about in a couple of slides. And so we get to the idea that we now have these continuum of states. It's no longer worthwhile to talk about individual orbitals. You know, if we're talking about a molecule, we talk about the sigma bond to the pi bond or like the A to G, um, orbital, things like this. This makes no sense in the solid. You have a very large number of these orbitals. You do not want to identify 6 to 23 of them. You will go quite insane. Instead, what we define is a density of states that we have an orbital over a specific energy range. Uh, you can have electrons in there. And so there's a certain density of states in electron volts per atom that you can define. And as the density is defined, it is the number of orbitals in the band per EV per atom in the band structure. So we have an example here of a rectangular band, which in reality you don't really see. But let's assume you do, uh, you have a rectangular band that has a total, a maximum density states of a third and spread over a third over three EV. So you have one third of an orbital per EV per atom, which would be 1 or 2 per atom in the whole band. When we then make up a material, we will have our full bands that are that will have bands are fully occupied. There will be a very low energy. We can have bands where, um, we can only partially fill them, so those will be half full and half empty. We can also bands are completely empty, but they will all have their own individual density states. And in this case down here, it's a very narrow range of energy. The band goes over, there's going to be very little of an overlap between the different elements, so there's less of a spread, uh, between the sorry, between the orbitals of the different elements in the solid. So you're less of a spread in energy, and so you have a higher density of states, but ultimately the same number of electrons are potentially in there. Whereas for half filled one we see here we have a much lower density state spread over a larger range of um, energies. And so it's got a lower density, though it's spread over a larger energy range. You'll also notice that we drew this, uh, this half full band. When you have such a half a band, you don't have enough electrons to fill it. As you mentioned before, you're going to fill from the lowest energy upwards. Um, and every orbital that you make into here is going to contribute. So there's going to be, uh, two electrons per every orbital. This is from parallel, um, parallel exclusion principle, from the alpha principle. And so we'll eventually get up to a stage where we have the maximum Angie that's an electron is occupying. And when we define this or when we find this Angie at zero Kelvin, we have found the Fermi edge. And the associated energy is called the Fermi energy. I sorry, the Fermi level, not the Fermi edge. And this is the Fermi energy. Um, this is the maximum energy that an electron will have at zero Kelvin around the solid. I note that quite often this Fermi H is set to zero on the energy field, zero electron volts on the energy scale, because it is the barrier between where all the states are occupied and where all the states are unoccupied. So you have ones that are bound and ones that are quasi or totally unbound from the ionic core. And so you'll often see that the binding energy that you get down here is going to be technically negative. So the um is going to be negative as you go down. So we talked briefly about filling of these new bands and how this that occurs is vaguely similar. Broadly similar to what happens when you have molecular orbitals that if you have a band from this, that's coming from s orbitals overlapping, if you have the s orbital from the individual elements is completely empty of electrons, then your new forming s band will also be completely empty. If instead it's half filled with one electron per atom, then your new band will be half filled. And if you have the case where you have two electrons in the original s orbital per atom, then you have a completely filled S band and you see the same thing across here for the P band that you go from zero electrons for a completely empty band, up to six electrons for a full band with the individual numbers in between. And so this is how you fill up the bands. So now that we have our idea of this band structure, we can now talk about the conductivity, The total band width. The total width of these I electron bands is going to be a numbers of EV, something like six five EV. It's not going to be a thing, you know, less than an electron volt really in its width. But the energy difference between the individual states in this band is going to be about ten to the -20, and this comes out straight away, if you think about it, that we have a total of, let's say, 610 EV of range for one of these bands, and in it there are ten to the 20 something atoms in it, then the orbitals in it, sorry. Then the energy difference between them must be very, very small. And thermal energy is a couple of MeV. So at room temperature everything has got about 25 MeV of energy on average. So it's completely trivial for an electron to hop up into one of these states. And once it's hopped up into one of these states, it can then move to the next state and the next state. And every time it does this, it will be moving across atoms. If you think about the full band, however, you cannot have the electron move from this state over to this one. There's already an electron there. It has to hop up to where you have this sea of empty bands, and after the hop along and say the material. And so as long as you have this gapless sort of structure, you will have a conductor. So you may immediately find out or have figured out where we're going with this. When we have a semiconductor, it is not gapless. Suddenly we have a band that is completely full and a band that is completely empty, and there's an energy gap between them. So at zero Kelvin, you will have no electrons in this upper band and no anti states in this lower band. So none of the atoms can move between the different places they are stuck. Reality is we don't live or work in zero Kelvin. We live and work in the room temperature, so there will be some energy available to them that will allow a small fraction of the electrons to hop over into the conduction band. And as these electrons that will conduct and what we'll talk about that a bit later. And so in the case of say, silicon germanium and tin each have four valence electrons. So the valence band holds exactly four electrons per atom. You may I pointed this and wonder, shouldn't this be, you know, 6 or 8 to fill up all the mass band? But what? We'll get to that in a few slides. The one thing I did want to highlight, though, is the band gap in the semiconductors is under about an EV, anything more than an EV, and the population of electrons that can hop across here is going to be too low. And so the definition of an insulator compared to a semiconductor is the band gap is simply larger. The probability of an electron hopping across there goes down to zero. So we had this case of diamond and silicon. Diamond is an insulator. Silicon is a semiconductor. When both these cases they're making an S-band where they're contributing one electron per band, one electron band. And we have a pea band with the contributing three electrons per band. So should we end up with a conducting stage? Then we end up with a band that has a, um, there's completely half filled because we have four, uh, four out of the potential eight electrons fill up. Yes. I mean, from what we describe, that is indeed what you'd expect. But that's because our current description of the linear combination of orbitals is too simple. What we have assumed so far is that when we combine these orbitals together, um, the energy difference between the antibonding orbitals and the bonding orbitals that are formed is comparatively small compared to the size of the band itself. If instead the difference between the antibonding and the bonding band is very large. It's much larger than the size band itself. Then they will stay split and you'll end up with a bonding band and an antibonding band. This is what we call the valence band. The valence band is the bonding one where we have filled our valency. We have made our nice covalent compound, and the top one is the conduction band which is empty as the antibonding band. If we filled it up, then our, um, our atomic structure would fall apart. And so this is why you get these band gaps, especially in covalent materials. We have the really strong interactions and overlaps between the orbitals. So the antibody orbitals become very energetically unfavourable and the bonding orbitals become very energetically favourable. So we get to where we talk about the conductivity and semiconductors. And in semiconductors as I mentioned, what happens is you have just enough energy from thermal to hop an electron up from the valence band in the conduction band. I think it's pretty clear from what we talked about for the metallic system, the ones you have an electron up in this conduction band, it can conduct away. It's very happy to move. It has all these density states available to it. They can hop between to hop between the different atoms it can conduct. What may not be immediately obvious is when you're kicking the electron out the valence band, you suddenly have a hole in the valence band and an electron can hop into that hole. Now electrons are all identical to one another, so you're going to find it completely impossible to differentiate between this electron, this electron, this electron. But what you end up seeing is the whole moves and the whole moves in the opposite direction to the electron movement. And so when you're talking about semiconductors, what we end up talking about is we have our electrons conducting our conduction band. And this is n type semiconducting. And we have our holes conducting in our valence band. This is p p for positive type semiconducting. And this is going to become a lot more important in uh, next week's lecture where we're going to be talking about the difference between intrinsic and expensive semiconductors, where you can drive the semiconductors to be either primarily and type conducting or primarily p type conducting and an intrinsic inductor intrinsic just means we have it's a one made from a single element. We have an altered at all. It is a simple semiconductor where the Fermi energy lies halfway between the collection and the villain's band. And this you'll have an equal number of holes, as you have electrons conducting an equal number of charge carriers in your valence band as charged carriers in your conduction band. So with that, we come to the end of our electricity. After this lecture, you should be able to recall the trend and in conductivity of conductive semiconductor insulators with respect to temperature so that conductors get worse. The introduction of you and the temp as you decrease the temperature. Semiconductors get better because you can hop across the band gap more. You must be able to explain the origin of energy bands, as opposed to discrete orbital energies and solid materials from our linear combination of orbitals. As you have to understand the terms density of states and Fermi level. And next Monday will be our last lecture that has new material in it, uh, with the lecture and next Tuesday being a revision lecture on, uh, what we've done so far. If there's anything like me to revise, please send me an email and I will go over it. Uh, otherwise, I am just going to work through some, uh, previous exam questions in the revision lecture. Uh, please all remember to do your tutorial for next week and have a good week.