Free to Move, Yet No Current — Bloch, Wilson and the Electrons in Solids, 1928–1931

Last time, Dirac's equation led to the idea that an empty seat can behave like a particle with positive charge (Part 3 is here). Part 4 moves from the vacuum to the ordinary solids around us. A copper wire carries electricity well; a diamond carries almost none. Both are full of electrons. So what is the difference? The answer took three years and came in two steps. First the question "why do metals conduct?" was solved — and that made "why does anything not conduct?" a mystery.

A metal wire is not a hollow tube

Metals conduct because they contain electrons that are free to roam. The German physicist Paul Drude gave this idea its shape in 1900: treat the electrons in a metal like the particles of a gas flying about inside a box. It is a simple picture, but it accounted rather well for how easily metals pass electricity and heat.

There was one awkward point. If so many electrons are flying about, they ought to take up some of the heat when a metal is warmed. Yet when the heat was measured, their share was almost nowhere to be found.

That point was settled at the end of 1927. Arnold Sommerfeld, in Munich, applied the brand-new quantum way of counting to the gas of electrons. The key was Pauli's exclusion principle, which we met in Part 4 of the previous series: no more than one electron can occupy a given state. The electrons fill the seats from the lowest energy upwards, and most of them cannot change seats when the metal is warmed a little. Only the handful near the top can take up heat.

But Sommerfeld made no new assumption about how the electrons deal with the atoms of the metal. He took the premises of the old theory and worked them through again with the new counting — he says so himself in the summary of his paper. The plainest question of all was left standing. A metal is packed with atoms that have given up electrons and carry positive charge (ions). Why do the electrons not keep bumping into them? Felix Bloch, who went on to solve the problem, later put it this way: a metal wire, crammed with ions, is nothing like a hollow tube.

Leipzig, 1928: to a wave, a regular row is no obstacle

Bloch arrived in Leipzig in the autumn of 1927 to work under Werner Heisenberg, who suggested the theory of metals as a subject for his doctoral thesis. The main problem, Bloch felt, was to explain how the electrons could "sneak by" all the ions.

He began with the easiest case: ions set out in a single row at equal distances, and the wave of an electron among them, worked out with quantum mechanics. He found "to my delight" — the words are his — that the answer was almost the same as the wave of a free electron travelling through empty space. The only difference was a fine pattern laid over the wave, repeating at the same spacing as the ions. When he showed it to Heisenberg, the reply came at once: "That's it!"

Think of the electron as a particle, and the densely packed ions are obstacles. Think of it as a wave, and things set out in a regular pattern are no obstacle at all. The wave adjusts its shape a little to fit the pattern and travels on without end. In a perfectly regular crystal, electrical resistance should be zero.

Where, then, does the resistance of a real metal come from? From whatever disturbs the pattern. One cause is the jiggling of the ions with heat; the other is impurities and flaws in the crystal. Bloch's paper, received on 10 August 1928, showed that once the thermal jiggling of the ions is taken into account, the size of a metal's resistance and the way it changes with temperature come out in rough agreement with experiment. A metal conducts better when it is cooled because the ions jiggle less.

Metals in which positive charge seems to flow

Bloch's theory also answered an older puzzle. Bring a magnet near a metal plate that is carrying a current and the flow is pushed sideways, so that a small voltage appears between the edges of the plate. This is the Hall effect, found in 1879. The direction of that voltage tells you whether the things that are flowing carry negative or positive charge.

Electrons are negative, so the result should have been a foregone conclusion. Yet in some metals the voltage points the way it would if positive particles were flowing. Sommerfeld's theory could not produce that sign at all.

At the end of 1928 Rudolf Peierls, also in Leipzig, showed that Bloch's calculations give both signs (his paper was received on 24 December). An electron in a crystal, when it sits near the top of the range of energies open to it, behaves as though it moved against the push it is given. If the seats are almost all taken and only a few are empty, the empty seats appear to move like particles with positive charge.

Readers of Part 3 have heard this before. Dirac wrote that an empty seat in a full house looks like a positive particle in December 1929. At almost the same moment, an idea of the same shape was growing inside solids. In 1931 Heisenberg worked it into the concept of the "positive hole".

Solved too well: why are there insulators?

Here the trouble starts. In Bloch's theory every electron in a perfect crystal is free to move. In that case every substance ought to conduct. Before Bloch, the existence of metals was hard to understand. After Bloch, it was the existence of insulators — substances that do not conduct — that needed explaining.

The usual assumption at the time was that an insulator is just a very poor metal, one in which electrons find it hard to pass from atom to atom. A difference of degree, in other words.

In the first week of January 1931, Alan Wilson, aged 24, arrived in Leipzig from Cambridge. Heisenberg promptly asked him to give a talk explaining Peierls's work — in German. It was while preparing for it, at the end of January, that Wilson saw the point.

The energies open to an electron in a crystal do not run on without a break. Allowed "bands" alternate with forbidden gaps (Peierls and others had shown this by about 1930). And the electrons, obeying the exclusion principle, fill the seats of the lowest band first and work upwards.

If a band is only partly filled, there are empty seats just above the highest electrons. Apply a voltage, the electrons move into the neighbouring empty seats, and a current flows. That is a metal. But what if there are exactly enough electrons to fill a band completely? Then no electron has a seat in its own band to move to. To reach the next band it would have to jump the gap, and a weak voltage cannot make it do that. Every electron is free to move, and yet, taken together, nothing flows. That is an insulator.

Wilson called it a "rather curious result" in his paper: bound electrons can conduct, and free electrons can fail to. The difference was not one of degree. It was a matter of how the seats are filled.

Heisenberg understood the idea at once and fetched Bloch. Bloch did not believe it at first. If it were right, he objected, metals such as calcium, whose atoms each supply two electrons, would fill a band exactly and be insulators. Wilson replied that in a three-dimensional crystal bands can overlap, so an element with an even number of electrons can still be a metal. Bloch soon came round. The paper reached the Royal Society on 18 June 1931.

In diamond, the gap between the full band and the next one is about 5.5 eV (eV, the electronvolt, is a unit for the energy of a single electron). The figure was measured much later, from the way diamond absorbs light. The energy that the warmth of a room can hand to one electron is only about a two-hundredth of that. In practice, no electron gets across.

Silicon was filed under "metals"

The story is often summed up as "Wilson explained semiconductors in 1931, and the transistor followed". A semiconductor is a substance that conducts, though not as well as a metal, and conducts better, not worse, when it is warmed. What actually happened was a good deal less sure-footed.

The experimental picture at the time was confused. Every substance that clearly behaved as a semiconductor was also very impure. Bernhard Gudden, at Erlangen, held that no pure substance is ever a semiconductor, and that the conduction always comes from impurities. Wilson's first paper left open whether "true" semiconductors, free of impurities, exist at all.

In a second paper two months later (received 25 August), Wilson accepted Gudden's view and built a theory in which impurity atoms hand their electrons up into the higher band. His evidence was cuprous oxide, a much-studied semiconductor of the day. The gap worked out from how it conducts was about 0.6 eV; the gap worked out from how it absorbs light was about 2 eV. The two do not match — so what carries the current must be electrons supplied by impurities, not the electrons of the substance itself. Controlling conduction with tiny amounts of impurity is the foundation on which semiconductors are made today.

And yet in that same paper Wilson classifies silicon as follows: it looks like a semiconductor when impure, but it is a good conductor when pure, and so belongs with the metals. Germanium, he says, probably belongs to both classes. Silicon, on which today's integrated circuits rest, was not counted among the semiconductors when the theory began.

Nor did things run straight afterwards. Wilson went on to give a theory of why current passes more easily one way across the boundary between a metal and a semiconductor, but the direction he predicted was the opposite of what later experiments found. Around 1938, seeing little interest in solids at Cambridge, he moved on to nuclear physics and cosmic rays. The first transistor was made in 1947, out of germanium. The rest of that story is in the semiconductor timeline.

What could be done at this point, and what could not

Here is what was understood by 1931. As a wave, an electron travels through a regular crystal without bumping into anything. Resistance comes from disturbances of the pattern. Whether a substance conducts depends not on whether its electrons are free, but on how far its energy bands are filled. And an empty seat in an almost full band behaves like a positive particle. Metals, insulators and semiconductors could now be set side by side in a single picture.

What could not be done is just as clear. The theory assumes that each electron moves on its own, in a regular field that averages out the influence of all the other particles. The way electrons keep out of one another's way is not in the calculation. In 1937 Jan Hendrik de Boer and Evert Verwey drew attention to substances such as nickel oxide. Its band is only partly filled, and yet it does not conduct. By the band picture it ought to be a metal. Their suggestion was that an electron setting out to move is drawn back to the atom it started from. Within six years it was plain that some substances are not settled by the band picture alone.

Superconductivity was left over too: cool certain metals and, at a definite temperature, their resistance suddenly drops to zero. Bloch himself tried again and again to explain it and failed, to the point of joking that "all theories of superconductivity can be disproved". The explanation came in 1957.

In Part 3, Dirac's full house was a picture that produced a correct prediction and was then given up. This time the same picture has turned up again — and here it has not been given up. Inside a solid, a band really is full of electrons, and an empty seat really does move like a positive particle. A picture that was retired as an account of the vacuum is still in daily use as an account of solids. Whether a picture is right is not settled by the picture alone. It depends on where you apply it.

One more thing. The answer this time was less a result of calculation than a way of seeing: noticing that being "free to move" and "carrying a current" are two different things. The equations were already in Bloch's paper, and the bands were already there in his result. Bloch later admitted that he had not thought of bands in the sense now common, and that this was why he had misjudged the real difference between metals and insulators. The answer was on the page. It had not yet been read.

Next time we return to questions about quantum mechanics itself. At the Solvay conferences of 1927 and 1930, what did Einstein ask Bohr — and how did Bohr reply?