by Brian Cox
How so? Let’s begin by considering a chain of atoms (as ever modelled by a chain of potential wells), but now suppose that each atom has several electrons bound to it. This, of course, is the norm – only hydrogen has just the one electron bound to a single proton – and so we are moving from a discussion of a chain of hydrogen atoms to the more interesting case of a chain of heavier atoms. We should also remember that electrons can come in two types; spin up and spin down, and the Pauli principle informs us that we can drop no more than two electrons into each allowed energy level. It follows that for a chain of atoms each containing just one electron per atom (i.e. hydrogen) the n = 1 energy band is half-filled. This is illustrated in Figure 8.7, where we have sketched the energy levels for a chain of 5 atoms. This means that each band contains 5 distinct allowed energies. These 5 energy states can accommodate a maximum of 10 electrons, but we only have 5 to worry about so, in the lowest energy configuration, the chain of atoms contains the 5 electrons occupying the bottom half of the n = 1 energy band. If we had 100 atoms in the chain then the n = 1 band could contain 200 electrons, but for hydrogen, we only have 100 electrons to deal with and so once again the n = 1 band is half filled when the chain of atoms is in its lowest energy configuration. Figure 8.7 also shows what happens in the case that there are 2 electrons for every atom (helium) or 3 electrons per atom (lithium). In the case of helium, the lowest-energy configuration corresponds to a filled n = 1 band, whilst for lithium the n = 1 band is filled and the n = 2 band is half filled. It should be pretty clear that this pattern of filled or half-filled continues such that atoms with an even number of electrons always lead to filled bands whilst atoms with an odd number of electrons always lead to half-filled bands. Whether a band is full or not is, as we shall very soon discover, the reason why some materials are conductors whilst others are insulators.
Figure 8.7. The way electrons occupy the lowest available energy states in a chain of five atoms when each atom contains one, two or three electrons. The black dots denote the electrons.
Let’s now imagine connecting the ends of our atomic chain to the terminals of a battery. We know from experience that if the atoms form a metal then an electric current will flow. But what does that actually mean, and how does it emerge from our story so far? The precise action of the battery on the atoms within the wire is, fortunately, something we don’t really need to understand. All we need to know is that connecting up the battery provides a source of energy that is able to kick an electron a little, and that kick is always in the same direction. A good question to ask is exactly how a battery does that. To say ‘it is because it induces an electric field within the wire and electric fields push electrons’ is not entirely satisfying, but it will have to satisfy us as far as this book is concerned. Ultimately, we could appeal to the laws of quantum electrodynamics and try to work the whole thing out in terms of electrons interacting with photons. But we would add absolutely nothing to the current discussion by doing this, so in the interests of brevity, we won’t.
Imagine an electron sitting in one of those states of definite energy. We will start by assuming that the action of the battery can only provide very tiny kicks to the electron. If the electron is sat in a low energy state, with many other electrons above it on the energy ladder (we have Figure 8.7 in mind when using this language), it will be unable to receive the energy kick from the battery. It is blocked, because the energy states above it are already filled. For example, the battery might be capable of kicking the electron up to an energy state a few rungs higher, but if all the accessible rungs are already occupied then our target electron must pass up on the opportunity to absorb the energy because there is simply nowhere for it to go. Remember, the Exclusion Principle prevents it from joining the other electrons if the available places are taken. The electron will be forced to behave as if there is no battery connected at all. The situation is different for those electrons with the highest energies. They are lying close to the top of the heap and can potentially absorb a tiny kick from the battery and move into a higher energy state – but only if they are not sitting at the very top of an already full band. Referring back to Figure 8.7, we see that the highest-energy electrons will be able to absorb energy from the battery if the atoms in the chain contain an odd number of electrons. If they contain an even number, then the topmost electrons still cannot go anywhere because there is a big gap in their energy ladder, and they will only overcome this if they are given a large enough kick.
This implies that if the atoms in a particular solid contain an even number of electrons, those electrons may well behave as if the battery had never been connected. A current simply can’t flow because there is no way for its electrons to absorb energy. This is a description of an insulator. The only way out of this conclusion is if the gap between the top of the highest filled band and the bottom of the next empty band is sufficiently small – we shall have more to say about that very soon. Conversely, if the atoms contain an odd number of electrons then the topmost electrons are always free to absorb a kick from the battery. As a result they hop up into a higher energy level and, because the kick is always in the same direction, the net effect is to induce a flow of these mobile electrons, which we recognize as an electrical current. Very simplistically, therefore, we might conclude that, if a solid is made up from atoms containing odd numbers of electrons, then they are destined to be conductors of electricity.
Happily, the real world is not that simple. Diamond, a crystalline solid made up entirely of carbon atoms which have six electrons, is an insulator. Graphite, on the other hand, which is also pure carbon, is a conductor. In fact, the odd/even electron rule hardly ever works out in practice, but that is because our ‘wells in a line’ model of a solid is far too rudimentary. What is absolutely true, though, is that good conductors of electricity are characterized by the fact that the highest-energy electrons have the headroom to leap into higher energy states, whilst insulators are insulators because their topmost electrons are blocked from accessing the higher energy states by a gap in their ladder of allowed energies.
There is a further twist to this tale, and it is a twist that matters when we come to explaining how the current flows in a semiconductor in the next chapter. Let us imagine an electron, free to roam around an unfilled band of a perfect crystal. We say a crystal because we mean to imply that the chemical bonds (possibly covalent) have acted so as to arrange the atoms in a regular pattern. Our one-dimensional model of a solid corresponds to a crystal if all of the wells are equidistant and of the same size. Connect a battery, and an electron will merrily hop up from one level to the next as the applied electric field gently nudges it. As a result, the electric current will steadily increase as the electrons absorb more energy and move faster and faster. To anyone who knows anything about electricity, this should sound rather odd, because there is no sign of ‘Ohm’s Law’, which states that the current (I) should be fixed by the size of the applied voltage (V) according to V = I × R, where R represents the resistance in the wire. Ohm’s Law emerges because as the electrons hop their way up the energy ladder they can also lose energy and drop all the way down again – this will only happen if the atomic lattice is not perfect, either because there are impurities within the lattice (i.e. rogue atoms that are different from the majority) or if the atoms are jiggling around significantly, which is what is guaranteed to happen at any non-zero temperature. As a result, the electrons spend most of their time playing a microscopic game of snakes and ladders as they climb up the energy ladder only to fall down again as a result of their interactions with the less than perfect atomic lattice. The average effect is to produce a ‘typical’ electron energy and that leads to a fixed current. This typical electron energy determines how fast the electrons flow down the wire and that is what we mean by a current of electricity. The resistance of the wire is to be seen as a measure of how imperfect the atomic lattice is through which the electrons are moving.
But that is not the twist. Even with
out Ohm’s Law, the current doesn’t just keep increasing. When electrons reach the top of a band, they behave very oddly indeed, and the net effect of this behaviour is to decrease the current and eventually reverse it. This is very odd: even though the electric field is kicking the electrons in one direction, they end up travelling in the opposite direction when they near the top of a band. The explanation of this weird effect is beyond the scope of this book, so we shall just say that the role of the positively charged atomic cores is the key, and they act to push the electrons so that they reverse direction.
Now, as advertised, we will explore what happens when a would-be insulator behaves like a conductor because the gap between the last filled band and the next, empty, band is ‘sufficiently small’. At this stage it is worth introducing some jargon. The last (i.e. highest-energy) band of energies that is completely filled with electrons is referred to as the ‘valence band’, and the next band up (either empty or half-filled in our analysis) is referred to as the ‘conduction band’. If the valence and conduction bands actually overlap (and that is a real possibility), then there is no gap at all and a would-be insulator instead behaves as a conductor. What if there is a gap but the gap is ‘sufficiently small’? We have indicated that the electrons can receive energy from a battery, so we might suppose that, if the battery is powerful, then it could deliver a mighty enough kick to project an electron sitting near to the top of the valence band up into the conduction band. That is possible, but this is not where our interest lies because typical batteries can’t generate a big enough kick. To put some numbers on it, the electric field within a solid is typically of the order of a few volts per metre, and we would need fields of a few volts per nanometre (i.e. a billion times stronger) in order to provide a kick capable of making an electron jump the electron volt7 or so in energy needed to leap from the valence band to the conduction band in a typical insulator. Much more interesting is the kick that an electron can receive from the atoms that make up the solid. They are not rigidly sitting in the same place, but rather they are jiggling around a little bit – the hotter the solid the more they jiggle and a jiggling atom can deliver far more energy to an electron than a practical battery; enough to make it leap a few electron volts in energy. At room temperature, it is actually very rare to hit an electron that hard, because at 20°C the typical thermal energies are around of an electron volt. But this is only an average, and there are a very large number of atoms in a solid, so it does occasionally happen. When it does, electrons can leap from their valence band prison into the conduction band, where they may then absorb the tiny kicks from a battery and in so doing initiate a flow of electricity.
Materials in which, at room temperature, a sufficient number of electrons can be lifted up from the valence to conduction band in this way have their own special name: they are called semiconductors. At room temperature they can carry a current of electricity, but as they are cooled down, and their atoms jiggle less, so their ability to conduct electricity diminishes and they turn back into insulators. Silicon and germanium are the two classic examples of semiconductor materials and, because of their dual nature, they can be used to great advantage. Indeed, it is no exaggeration to say that the technological application of semiconductor materials revolutionized the world.
9. The Modern World
In 1947, the world’s first transistor was built. Today, every year the world manufactures over 10,000,000,000,000,000,000, which is one hundred times more than the sum total of all the grains of rice consumed every year by the world’s seven billion residents. The world’s first transistor computer was built in Manchester in 1953, and had ninety-two of them. Today, you can buy over a hundred thousand transistors for the cost of a single grain of rice and there are around a billion of them in your mobile phone. In this chapter, we are going to describe how a transistor works, surely the most important application of quantum theory.
As we saw in the previous chapter, a conductor is a conductor because some of the electrons are sitting in the conduction band. As a result, they are quite mobile and can ‘flow down’ the wire when a battery is connected. The analogy with flowing water is a good one; the battery is causing current to flow. We can even use the ‘potential’ concept to capture this idea, because the battery creates a potential within which the conduction electrons move, and the potential is in a sense, ‘downhill’. So an electron in the conduction band of a material ‘rolls’ down the potential created by the battery, gaining energy as it goes. This is another way to think about the tiny kicks we talked about in the last chapter – instead of a battery inducing tiny kicks that accelerate the electron along the wire, we are invoking a classical analogy akin to water flowing down a hill. This is a good way to think about the conduction of electricity by electrons, and it is the way we will be thinking throughout the rest of this chapter.
In a semiconductor material like silicon, something very interesting happens because the current is not only carried by electrons in the conduction band. The electrons in the valence band contribute to the current too. To see that, take a look at Figure 9.1. The arrow shows an electron, originally sitting inert in the valence band, absorbing some energy and being lifted up into the conduction band. Certainly the elevated electron is now much more mobile, but something else is mobile too – there is now a hole left in the valence band, and that hole provides some wriggle room for the otherwise inert valence band electrons. As we have seen, connecting a battery to this semiconductor will cause the conduction band electron to hop up in energy, thereby inducing an electric current. What happens to that hole? The electric field created by the battery can cause an electron from some lower energy state in the valence band to hop into the vacant hole. The hole is filled in, but now there is a hole ‘deeper’ down in the valence band. When electrons in the valence band hop into the vacant hole, the hole moves around.
Figure 9.1. An electron-hole pair in a semiconductor.
Rather than bother keeping track of the motion of all the electrons in the almost-full valence band, we can instead decide to keep track of where the hole is and forget about the electrons. That book-keeping convenience is the norm for those working on the physics of semiconductors, and it will make our life simpler to think in that way too.
An applied electric field induces the conduction band electrons to flow, creating a current, and we should like to know what it does to the holes in the valence band. We know that the valence band electrons are not free to move, because they are almost completely trapped by the Pauli principle but they will shuffle along under the influence of the electric field and the hole moves along with them. This might sound counterintuitive, and if you are having trouble with the idea that if electrons in the valence band shuffle to the left then the hole also shuffles to the left, perhaps the following analogy will help. Imagine a line of people all standing in a queue 1 metre apart, except that somewhere in the middle of the line a single person is missing. The people are analogous to electrons and the missing person is the hole. Now imagine that all the people stride 1 metre forwards so that they end up where the person in front of them was standing. Obviously the gap in the line jumps 1 metre forwards too, and so it is with the holes. One could also imagine water flowing down a pipe – a small bubble in the water will move in the same direction as the water, and this ‘missing water’ is analogous to a hole in the valence band.
But, as if that wasn’t enough to be going on with, there is an important added complication; we now need to invoke the piece of physics that we introduced in the ‘twist’ at the end of the last chapter. If you recall, we said that electrons moving near to the top of a filled band are accelerated by an electric field in the opposite direction to electrons moving near to the bottom of a band. This means that the holes, which are near the top of the valence band, move in the opposite direction to the electrons, which are near the bottom of the conduction band.
The bottom line is that we can picture a flow of electrons in one direction and a correspon
ding flow of holes in the other direction. A hole can be thought of as carrying an electric charge that is exactly opposite to the charge of an electron. To see this, remember that the material through which our electrons and holes flow is, on average, electrically neutral. In any ordinary region there is no net charge, because the charge due to the electrons cancels the positive charge carried by the atomic cores. But if we make an electron–hole pair by exciting an electron out of the valence band and into the conduction band (as we have been discussing), then there is a free electron roaming around, which constitutes an excess of negative charge relative to the average conditions in that region of the material. Likewise, the hole is a place where there is no electron and so it corresponds to a region where there is a net excess of positive charge. The electric current is defined to be the rate at which positive charges flow,1 and so electrons contribute negatively to the current and the holes contribute positively, if they are flowing in the same direction. If, as is the case in our semiconductor, the electrons and holes flow in opposite directions, then the two add together to produce a larger net flow of charge and hence a larger current.