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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 without 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.