Band theory of solids
Why atomic theory doesn't work for solids
- [Teacher] To figure out how we use semiconductors to build all these awesome computing devices, we're going to start from scratch, all the way down to even understanding why semiconductors are semiconductors. I mean, why is it that certain materials behave like conductors, which are very good at passing electricity through them while others are not? To understand this, we need to look at the atomic level. Now we might have some intuition about these atoms, but guess what? Turns out that our knowledge of the atomic structure is not enough. And so in this video, we're just gonna recapitulate all the stuff that we might already know from the previous videos. And we'll see why the current knowledge or the current theory of the atoms is not sufficient to talk about solids in general, which we'll be interested in. For starters, you may already have some intuition. For example, you may know that all matter is made of atoms. And if you were to pick any one of them and zoom in, then you might know that the atoms themselves are made of even smaller things. At the center, we have this thing called as the nucleus, which have a positive charge, and the electrons which are negatively charged are attracted by the nucleus and end up going around the nucleus in different orbits just like the solar system and how the planets go around the sun. Now this is not a very accurate model, we'll get back to that. But as of now, let's use this model. But the important thing is there are some electrons like these, which are tightly bound to the nucleus. We call them as bound electrons. Bound electrons, and these are not responsible for conduction. Whereas there are other electrons which are not strongly attracted by the nucleus and they are free, as in, they're free to move from one atom to another. And it's these electrons which we call as conduction electrons or free electrons, which are really responsible for conduction. And in some materials, it's very easy to get these free electrons. And so they end up having a lot of them, and we call these materials as good conductors or conductors. On the other hand, some materials, well, it's extremely difficult to get these free electrons. And as a result, you have extremely negligible amount. And as a result, they are bad conductors or insulators. And of course we have the intermediate ones which we end up calling semiconductors. So I think the most important question that we have to ask ourselves over here, is how does an electron become free? I mean, what makes it free and what does that depend on? That's the thing that we need to figure out. And we have to look at, look at this whole thing for a solid, because our semiconductors are solids. So we need to find out, or we need to figure out what makes an electron free in solids. And to do that, we need to get past this solar system model of the atom, as I mentioned before, it's not very accurate. And we need to look at a more accurate model of the atomic structure. So let's do that. Now, you may have already learned about this in chemistry. It turns out that instead of thinking of where the electrons are and what orbits or what path they take, it's much better to think about them in terms of energies. It's better think about what are the energies that the electrons can take up. And you may have already studied in chemistry that the inside of any atoms, so if I draw over here energies, inside any atom, electrons can have only some specific energy values, only some specific energy values. And so maybe the lowest energy that electron can have maybe somewhere over here. We're not gonna write down the numbers over here. We're not gonna look at it very quantitatively, don't worry about it. So maybe this is the lowest energy that an electron can possess. The next higher energy an electron can possess might be somewhere over here, and maybe next higher might be somewhere over here, and so on and so forth. And we give names to these energy levels. We call the lowest one as the 1S energy level. The next higher one becomes 2S, the one that comes above that would be 2P. Then we have 3S and 3P and so on and so forth. And again, if this looks very new to you and you have no idea what S and P are, it would be a great idea to pause this over here, go back and watch the electron configuration videos on chemistry, and then come back over here. But anyways, it turns out electrons cannot take up these energy levels randomly. There's a particular rule using which electrons sort of fill up these energy levels. And that rule, again, you may have studied about them. We call that as the Pauli's exclusion principle. Pauli's exclusion, exclusion principle, or rule. And it simply says that no two electrons, no two electrons can have identical, can have identical energies. Now, again, this is not the accurate statement of Pauli, but this will help us, this is enough for us. So let's take a concrete example. Suppose we take, say, a sodium atom, then it has, it has 11 electrons inside it. There are 11 electrons. And now these 11 electrons can only have these specific energy levels. And the way these electrons are going to fill up the energy levels will be using the exclusion principle. So the first electron, well, remember, electrons always want to take the lowest energy possible. So the first electron would go over here, over here, and then you might think, well, the next electron can't go over here because that's what Pauli's telling us. No arguing with Pauli. Second electron, if it comes over here, it might have identical energy, but not really, because it turns out that electrons can have up spin and down spins. So if the first electron goes into the 1S tier, and suppose it takes up the up spin, then another electron can actually take up the same energy level and now be down spin because turns out these two spins have slightly different energy. So these two electrons are strictly speaking, still being Pauli, because they're not exactly identical because of their spins. But the next electron, the third electron, well, it cannot take up the 1S energy level anymore, because if it does and then up spin, then it'll be identical to this one. If it does with a down spin, then it'll be identical to this one. So it can't take the that up anywhere. So it has to take up now the next higher energy level available that's over here. It can take up anywhere in between as well. The energy levels in between are inaccessible to these electrons. So the next energy it will take up would be 2S, again, it might take up with an up spin. The fourth electron might go over with a down spin. The next electron will take up over here, up spin, and the next one will be down spin. Now here's the thing. It turns out that in P, in P energy level, there are three ways in which electrons can occupy that energy level. We call them as orbitals, right? It turns out that in the S energy levels, there's only one way. So there's only one orbital, but in P there are three orbitals. So another electron can take up the 2P energy level by being in a different orbital. So this electron and this electron will be in different orbitals, or different configuration, we could say, don't have to worry about it too much. And so they'll still not be identical. And so another electron can take up that same orbital with a down spin. Another electron, the third orbital of P with an up spin, and then down spin. And now the 2P is completely filled. There are no more orbitals available. And so the last electron, we're down to one, two, three, four, five, six, seven, eight, nine, 10, the last electron will be over here in the 3S up spin. But this is for a single atom of sodium. What if we have say, two atoms of sodium, very close to each other, what happens then? Somewhat like this, what if they form some kind of a molecule? How would the electrons of this molecule fill up the energy levels? Can we say that now each atom will have something like this. Each atom will have electrons filled up accordingly. Well, that won't work, that can't be possible. And the way we can think about it, is we can say that, if you do it this way, Pauli's rule will be violated. Remember, Pauli says no two electrons, and when we say no two electrons, it can be no two electrons inside an atom, or no two electrons inside a molecule, or maybe no two electrons inside an entire solid. No two electrons can have identical energies. So if the two atoms have these electron configurations then I hope you can see that this electron and this electron will, they will be identical. This one, and this one will be absolutely identical. And so all of them will have identical pairs and Pauli will be very, very sad, so that can't be possible. And if we have an entire solid, which is made of sodium, where we have like 10 to the 23 atoms packed very close to each other, and if we used this model for each atom, then there would be about 10 to the 23 identical copies of electrons in each level. And that would make Pauli extremely sad, extremely sad. So the key takeaway is that this structure that we have learned for a single atom cannot be extended when we go all the way to the solids. We require a new theory to understand what's going on and how electrons are arranged or how to think about them when it comes to solids. And we'll explore them in the future videos.