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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