A full band is not an insulator

The rule is one of the tidiest in the subject and it is the first thing anybody learns about why some materials conduct. Count the electrons. If they exactly fill a band, there is a gap above them, and moving an electron costs a finite amount of energy: an insulator. If they half fill one, there are empty levels immediately above the occupied ones: a metal.

Magnesium has two valence electrons per atom. It fills its band. It is one of the better conductors in the periodic table.

The rule that is not being broken #

It is worth being precise about which half of the argument fails, because the count itself is not in question.

A structure with two orbitals per site and two electrons per site really does put two electrons into every one of the lower set of levels and none into the upper set. That is arithmetic and it is right. Counting electrons in an extended structure is the essay that shows how far such counting goes, and nothing here contradicts it.

What fails is the step after the count: the assumption that “the lower set” and “the upper set” occupy separate ranges of energy. A band is an interval, two intervals need not be disjoint, and if they overlap then the highest occupied level is not below the lowest empty one by any margin at all — because they are drawn from different sets that share the same stretch of energy.

There is a second reason the failure is easy to miss, and it is about how the picture is usually drawn. A band diagram shows one band with a gap above it, and the gap is on the page because whoever drew it knew the material was an insulator. The count did not put it there, and a reader who takes the diagram as an argument has been shown the conclusion as a premise.

The model, and why the two sets cannot mix #

The system is a ring whose every site carries two orbitals: one pointing along the ring — a σ function — and one perpendicular to it, a π function.

That choice is not arbitrary and the reason is the sharpest one available. The two sets do not interact at all. The overlap between a σ function on one centre and a π function on another is zero — not small, zero — because the integrand is odd in the coordinate perpendicular to the axis joining them, and the contributions cancel in pairs. Exactly zero is the essay about that vanishing and about why computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

So the Hamiltonian is exactly block diagonal: a σ block and a π block, with nothing joining them. Each block is a ring of one orbital per site, which is the object this whole field is built on, and the only new parameters are how far apart the two orbitals sit in energy and how strongly each kind couples to its neighbours.

Where the two bands lie #

The arithmetic is trivial and that is the point. The σ band runs from Δ/2 − 2t_σ to Δ/2 + 2t_σ and the π band from −Δ/2 − 2t_π to −Δ/2 + 2t_π. They are disjoint when Δ > 2(t_σ + t_π), and the gap between them is then Δ − 2(t_σ + t_π) exactly.

Four numbers decide it: two site energies and two couplings. None of them is the number of electrons.

What the bands look like when they are drawn #

The picture makes plain what the algebra says, and it also makes plain why the failure is so easy to miss when a band is drawn as a single interval with the electrons written inside it. The usual diagram has one band and a gap above it, and the gap is drawn because the artist knew the answer. Nothing in the electron count put it there.

The test a finite calculation can make #

Whether a system is a metal is not a question about any one finite ring, and this field settled the point some essays ago. What a metal actually is defines it as a system with excitations of arbitrarily small energy, and checks the definition on a sequence of rings rather than on one.

The same test applies here and gives an unambiguous answer.

The separated case gives 1.000 at twelve sites, at twenty-four, at forty-eight, at ninety-six and at a hundred and ninety-two: the gap is Δ − 2(t_σ + t_π), it does not know how large the ring is, and it will not shrink however large the ring is made.

The overlapping case gives 0.098, then 0.098, then 0.073, then 0.0056, then 0.0056. That sequence is not smooth, and the un-smoothness is honest rather than a defect: the two sets of discrete levels are incommensurate, so which pair happens to straddle the boundary depends on the size in a way that has no pattern. What it does not do at any size is grow. What is left in the limit is nothing.

Where the threshold sits, and what moves it #

The threshold is 2(t_σ + t_π), so anything that widens either band pushes it out. Two consequences follow and both are chemical rather than formal.

The first is that the sum of the couplings enters, not either one separately. A material with a very wide σ band and a narrow π band behaves, for this purpose, like one with two moderate bands — which means a single wide band is enough to close a gap on its own.

The second is that the couplings both grow as the atoms are pushed closer together, while the separation of the site energies does not: the site energies are properties of an atom and the couplings are properties of a pair. So compression always pushes towards the metal, and the direction is not a matter of which material it is.

The count was never the whole of the argument #

The step this essay is about has a name in the textbook version of the story, and it is instructive that it does: band overlap. It is stated as a caveat, usually in a sentence, and the caveat is doing all the work.

The reason it can be reduced to a caveat is that the count is memorable and the inequality is not. Two electrons per atom is something a reader can carry around. Δ > 2(t_σ + t_π) is four quantities that have to be looked up for each material, and a rule with four inputs does not survive being taught alongside one with one.

But the asymmetry is misleading, because the inequality is not the exception to the count — it is a separate and independent question, and the count has no bearing on it whatever. The electron count decides whether the lower set is full. The inequality decides whether being full means anything.

What makes the alkaline earths the standing case #

Beryllium, magnesium and calcium each bring two valence electrons, and each is a metal. It is worth saying what makes them the case rather than an accident.

In an isolated atom the s and p levels of the same shell are separated by several electronvolts, so at large separation the two bands are far apart and the count would give the right answer: two electrons, s band full, insulator. Bringing the atoms closer widens both bands, because the couplings grow as the orbitals overlap more, and the widening is what closes the gap. At the observed separations $2(t_s + t_p)$ exceeds the atomic separation of the levels and the two bands overlap.

So the same material, stretched, would be an insulator, and compressed, is more firmly a metal. That is not a hypothetical: it is the reason a great many elements change their electrical character under pressure, in both directions.

The other direction of the same reasoning is a chemical one. What separates the s and p levels of an atom is the screening one electron feels from the others, which grows down a group as the inner shells multiply — so the counting rule fails in a way that is systematic across the periodic table rather than randomly across materials.

Two ways a count can fail, and this is the second #

There are now two essays about the electron count giving the wrong answer, and they fail in opposite directions and for unrelated reasons. Keeping them apart is worth a paragraph.

A half-filled band is not always a metal is the case where the count says metal and the material is an insulator. The reason is the repulsion between electrons, which this model does not contain at all: where the repulsion is strong enough, the electrons stay one to a site and moving one costs energy even though there are empty levels available.

This essay is the case where the count says insulator and the material is a metal. The reason has nothing to do with repulsion — it is one-electron arithmetic throughout — and everything to do with the geometry of two intervals.

So the count is unreliable at both fillings, and a reader who has learnt only the first correction will apply it in exactly the wrong place: they will suspect a half-filled band of being an insulator, which is a live possibility, and will trust a filled one to be an insulator, which is equally unsafe and for a reason the first correction never mentions.

There is a third way to be an insulator that has nothing to do with either: a half-filled band with a repulsion in it opens a gap band theory has no term for. That mechanism is orthogonal to this essay’s — it makes an insulator out of a system band theory calls a metal, where this essay makes a metal out of one band theory calls an insulator.

The gap is not the width, and here they compete #

The inequality Δ > 2(t_σ + t_π) is a competition between two quantities of quite different origin, and this field has already separated them.

The gap is not the band width makes the point in a different setting: the width is set by how many neighbours an atom has and the gap by how unequal they are, and a structure can have a wide band and no gap or a narrow band and a large one. Here the two are in direct competition — the widths of the two bands are what close the gap between their centres — so a material with strong coupling is less likely to be an insulator at a filled band, other things equal.

That is worth stating because it inverts a plausible instinct. Strong bonding does not make an insulator; it makes wide bands, and wide bands overlap.

The same trap in a molecule #

The extended case has a molecular counterpart, and it has already appeared without the connection being named.

A closed-shell molecule is the molecular version of a filled band: every occupied orbital holds two electrons and the next one up is empty. Whether that makes the molecule unreactive is decided not by the count but by how far the highest occupied orbital lies below the lowest empty one — which is the gap conjugation shrinks as a chain of double bonds is lengthened, and which goes to nothing in the limit for exactly the reason the bands here overlap.

So a long enough polyene has a closed shell and no gap worth the name — and where a molecule stops being one is the essay about how large it has to be before that matters — and calling it closed-shell tells a reader nothing they can use. The count was a proxy for the gap, in molecules as in solids, and it is a proxy that fails whenever the levels crowd.

What would have to be true for the rule to work #

It is worth ending the positive part of the argument by stating the conditions under which the count is sufficient, because they are not empty and a reader should be able to recognise them.

The count settles the question when the separation between the two sets of orbital energies is large compared with the widths of the bands they form. That is the situation in an ionic compound, where the occupied levels are essentially the anion’s and the empty ones the cation’s and the two are several electronvolts apart with narrow bands between them; it is the situation in a molecular crystal, where the bands are narrow because the molecules barely interact; and it is the situation in a covalent solid with a strong bonding–antibonding split, where the separation is itself a bonding energy and is large by construction.

What those cases have in common is not that the electrons were counted correctly. It is that the inequality happens to hold with room to spare, so the count is not doing the work and appears to be.

The four numbers can be changed with a press #

The comparison this turns on is between two level separations and two band widths, and it is worth noticing that one of the four is adjustable from outside — which converts the argument into an experiment anybody with a diamond anvil can perform.

Squeezing a solid brings its atoms closer. Closer atoms overlap more, and more overlap means wider bands. The separations between atomic levels are properties of the atoms and barely move; the widths grow steadily. So pressure drives the comparison in one direction only: towards overlap, and therefore towards metallic.

Iodine is the clean demonstration. At ordinary pressure it is a molecular solid — diatomic molecules packed together, a filled band from the bonding levels, an empty one from the antibonding, and a gap between. It is a black-purple insulator. Compress it to around sixteen gigapascals and it becomes a metal: the bands have broadened until they overlap, with no change in the count of electrons anywhere.

Push further and the pattern continues down the group and across the table. Xenon, a closed-shell atom whose bands are as narrow as bands get, metallises above a hundred gigapascals. Oxygen becomes a metal near a hundred and is superconducting there. Hydrogen is the standing case that has not yet been settled.

None of those materials gained or lost an electron. Their counts are the counts they always had, and the count said insulator in every case — correctly, at one pressure, and incorrectly at another.

That is the essay’s argument stated as an experiment rather than as an inequality. The count is a fact about the material and the comparison is a fact about the material at a density, and a variable that changes only one of the two separates them completely.

What is left #

The model has two orbitals per site and no coupling between them, which is exact for a σ and a π set in a linear structure and is not exact in general. In a three-dimensional structure a σ and a π function on different centres are related by neither a mirror plane nor an axis in the general case, and the two sets mix. Where they mix the crossing becomes an avoided one, a gap opens where the bands would have overlapped, and the story is different — genuinely different, not a correction.

That is the situation in most real materials, and the reason the alkaline earths still conduct is one this model cannot reach: the two bands overlap in some directions and not others, so a gap opens along one direction and not along another and the total density of states has no gap anywhere. That is a statement about a wavevector, and there is no wavevector here — see a band with no structure in it for the standing account of what this field gives up.

So what is demonstrated here is the mechanism at its cleanest, in the one case where the two sets provably cannot mix, and what is claimed is the mechanism rather than the material. The claim is that a filled band does not settle the question. It is not a claim about magnesium’s band structure, which this cannot compute.