A half-filled band is not always a metal

Every calculation in this field puts each electron in a fixed average potential and lets it move without noticing where any other electron is. That approximation is stated in the caption of every figure and it is easy to read past.

It should not be read past here, because in this field it fails harder than anywhere else on the site — not by getting a number somewhat wrong, but by predicting that a material conducts when it is an excellent insulator.

The case that breaks it #

Nickel oxide has nickel in the $\mathrm{Ni}^{2+}$ state with eight $d$ electrons in five $d$ orbitals. Counting as this field counts, the $d$ band is partly filled, so nickel oxide should be a metal.

It is a transparent green insulator with a band gap of about four electronvolts. Cobalt oxide, manganese oxide and iron oxide behave the same way. So does the undoped parent compound of every copper-oxide superconductor, which band theory confidently predicts to be a metal and which is an antiferromagnetic insulator.

This is not a small quantitative failure that better parameters would fix. It is a qualitative one, in the direction that matters, on a class of materials large enough to have its own name.

The size of the discrepancy is worth dwelling on. A four-electronvolt gap is not a marginal case at the edge of a model’s range; it is a larger gap than most semiconductors have, on a material the theory says has no gap at all. Nothing about the band calculation is arithmetically wrong — the levels are where the calculation says they are — and the conclusion drawn from them is as wrong as a conclusion can be.

Why the model cannot see it #

The missing ingredient is the energy cost of putting two electrons on the same atom.

In a one-electron picture an electron occupies an orbital spread over the whole structure, and whether another electron happens to be nearby is not part of the description. In reality two electrons on the same atom repel each other strongly — for a $3d$ orbital the cost is several electronvolts — and that cost is not in the matrix at all.

Now compare two energies.

The band width measures what an electron gains by spreading out. It is the quantity the width of a band is a count of neighbours computes, set by how many neighbours each atom has and how strongly they interact.

The on-site repulsion measures what it costs for two electrons to be on the same atom at once. Moving an electron from one site to its neighbour means one atom temporarily has two and another has none, so the repulsion is paid every time an electron moves.

If the width is larger, spreading out wins and the material is a metal. If the repulsion is larger, the electrons stay one per site — they cannot move without paying more than they gain — and the material is an insulator with one electron localised on every atom.

The band is the same in both cases. What differs is a quantity the band does not contain.

What the insulating state looks like #

The picture on the other side of that comparison is a genuinely different one, and it is worth drawing in words because no one-electron figure can draw it.

Each atom holds exactly one electron. Moving one costs the repulsion energy, so at low temperature nothing moves and the material does not conduct. The cheapest excitation is not a band-to-band transition at all — it is moving one electron onto a neighbour that already has one, creating a doubly occupied site and an empty one, and its cost is roughly the on-site repulsion.

That is a gap, and it is not a band gap. It exists because of a correlation between where the electrons are, not because of any structure in the level ladder.

The spins then arrange themselves, because a pair of neighbouring electrons with opposite spins can briefly hop onto each other’s sites and back while a pair with the same spin cannot — the Pauli principle forbids it. The brief excursion lowers the energy slightly, so antiparallel is favoured, and the material becomes antiferromagnetic. Every one of the oxides named above is antiferromagnetic, and the mechanism is a direct consequence of the localisation.

The transition, which is a genuine one #

The comparison between two energies suggests something stronger than a classification: if the ratio can be changed, a material should be able to cross from one behaviour to the other. It can, and the crossing is one of the sharpest phenomena in the subject.

By pressure. Squeezing a crystal shortens the distances between atoms, increases the overlap and widens the band while leaving the on-site repulsion essentially alone. So a narrow-band insulator put under enough pressure becomes a metal, and several do — the transition is abrupt, and it is accompanied by a large change in volume because the metallic state binds more strongly.

By doping. Adding or removing electrons takes the filling away from exactly one per site. With one electron per site, moving anything creates a doubly occupied site; with slightly fewer, there are empty sites available and an electron can move into one at no repulsion cost at all. So a lightly doped narrow-band insulator conducts, and this is the route by which the copper-oxide superconductors are made from their insulating parents.

By temperature. Several oxides show a transition on heating, with the resistance falling by orders of magnitude over a few degrees. Vanadium dioxide’s is at 68 °C and is used in switchable coatings.

The existence of the transition is the best evidence that the two energies are what matter. A classification that merely sorted materials into two lists could be a coincidence; a knob that moves a single material across the boundary, three different ways, is a mechanism.

The name, and the honest history #

The failure was noticed early and correctly. In 1937 de Boer and Verwey reported that nickel oxide and several similar compounds were insulators, at a meeting where band theory was being presented; Peierls responded immediately with the essential argument, that a strong repulsion between electrons on the same atom would hold them apart and that band theory omitted it.

Mott developed the argument over the following decades and the insulating state carries his name. The model that captures it in its simplest form — a band width and an on-site repulsion, and nothing else — is Hubbard’s, and it is still not solved in more than one dimension, which is a fair indication of how much is thrown away when the repulsion is dropped.

What is worth taking from that history is that this was never a hidden flaw discovered late. It was pointed out at the first presentation, by the same Peierls whose instability a chain cannot stay even computes, and the theory was adopted anyway because it worked so well for the materials people were studying. That is the ordinary way a good approximation is used, and the ordinary risk that comes with it.

The clue that the model is being asked too much #

There is a rule of thumb for when to distrust the band picture and it follows directly from the comparison above.

A narrow band is a warning. The on-site repulsion is roughly fixed by the atom — a few electronvolts for a $3d$ orbital, less for a diffuse one. The band width is not: it depends on overlap. So the danger zone is wherever the band is narrow, which means compact orbitals and long distances between the atoms carrying them.

That picks out exactly the materials where the failures happen. The $3d$ orbitals of the first transition series are compact, and in an oxide the metal atoms are separated by oxygens — so the bands are narrow, the repulsion is large, and the comparison goes the wrong way. In a metal like copper the $s$ band is wide and the picture works.

It also explains why the failure is essentially unknown in main-group chemistry and in the organic conjugated systems Hückel theory was built for. A carbon $2p$ orbital is diffuse, carbon–carbon distances are short, and $\beta$ is comparable with the repulsion rather than much smaller. The band picture is doing reasonably there, which is why the calculations in this field’s other essays are worth something.

What the one-electron picture is actually assuming #

It is worth stating the approximation precisely rather than as a slogan, because “electrons are treated independently” undersells how much is being assumed.

Every calculation here writes the many-electron state as a single arrangement of one-electron orbitals — each electron in an orbital, the orbitals fixed, and the only correlation between electrons the one the Pauli principle forces on pairs of the same spin. The probability of finding one electron somewhere is then independent of where the others are, apart from that.

Reality is not like that. Electrons avoid each other, and the avoidance is not describable by any single arrangement of orbitals: it requires a superposition of arrangements, weighted so that configurations with two electrons close together get smaller amplitudes.

That superposition is what the word correlation names, and it has appeared before under another heading. Molecular orbital and valence bond is exactly the comparison between a description that lets both electrons occupy the same delocalised orbital and one that keeps them apart, and the essay’s conclusion — that the two are limits of one description and neither is the truth on its own — is the same conclusion the solid-state version reaches.

The same failure, met earlier and smaller #

This is not the first time a one-electron description breaks in this particular way, and the earlier case makes the mechanism concrete.

Stretch a hydrogen molecule. The molecular orbital description keeps both electrons in the bonding orbital, which is an equal mixture of the two atoms — so at large separation it predicts that half the time both electrons are on the same atom, giving $\mathrm{H}^+$ and $\mathrm{H}^-$. That is badly wrong, and it is wrong by exactly the amount of the on-site repulsion.

The valence bond description gets the dissociation right, because it puts one electron on each atom by construction. Molecular orbital and valence bond works through what the two descriptions are and why they differ.

A material with a half-filled narrow band is a whole crystal in the position of a stretched hydrogen molecule. The band picture is the molecular orbital description; the localised, one-electron-per-site insulating state is the valence bond one; and which is right depends on the same comparison between delocalisation energy and on-site repulsion.

That correspondence is exact rather than an analogy, and it is why the fix for this failure has always come from the chemistry side of the subject.

What survives #

It would be wrong to leave the impression that band theory is unreliable in general, because it is one of the most successful theories in physics. Its predictions for the simple metals, the covalent semiconductors and the ionic insulators are quantitatively excellent, and every device built in the last seventy years relies on them.

What survives is:

Everything about geometry and counting. The exact identity between the second moment and the coordination, the relation between structure and band width, the electron counting that decides whether a shell can close. None of those involves the repulsion.

Everything about wide-band materials. Where the band width is the larger of the two energies, the one-electron picture is the right starting point and corrections to it are small.

The classification itself. The question “is this material’s band partly filled?” is well posed and answerable, and the answer is necessary information even in the cases where it is not sufficient. A material with a completely filled band is an insulator whatever the repulsion does, since the repulsion can only make electrons less mobile; the failure is one-directional, and knowing that is worth something.

What does not survive is the inference from partly filled to metallic, in the narrow-band case. And there is no repair available within a one-electron model — the missing quantity is not a parameter that has been set wrongly, it is an interaction the model does not represent.

The sequence that defines a metal in this model is a level spacing at the Fermi energy falling to zero as the ring grows. It is correct arithmetic about a model, and whether the model applies is a separate question the arithmetic cannot answer.

The measurement that tells the two kinds of insulator apart #

A material can be an insulator because its band is full or because its electrons refuse to pass one another, and the two look identical in a conductivity measurement. They are not identical, and the property that separates them is magnetic.

A band insulator has its electrons paired in filled levels. There are no unpaired spins anywhere, no local moments, and nothing to order: such a material is diamagnetic and stays diamagnetic to the lowest temperature reached.

An insulator of the other kind is insulating because each site holds one electron and none of them can move. One electron per site is one unpaired spin per site — the moments are still there, all of them — and although charge cannot flow, the spins can still talk to one another by the virtual hopping that makes an antiferromagnetic coupling.

So the material has two gaps of very different sizes. Moving charge costs the repulsion, which is electronvolts. Flipping a spin costs the coupling, which is a small fraction of it — hundredths of an electronvolt for the same parameters. An insulator with a large charge gap and small spin excitations is not what a band picture describes at all.

The consequence is measurable and unambiguous. Nickel oxide has a gap of about four electronvolts and orders antiferromagnetically at 523 kelvin; manganese oxide does the same at 118. Both are insulators with a lattice of local moments and a magnetic transition. Silicon and sodium chloride are insulators with neither.

That gives the criterion the band picture cannot supply. Ask whether the insulator is magnetic. A full band has no spins left over to order, so magnetic order in an insulator is direct evidence that the electrons are localised by their own repulsion rather than by a filled shell.

It is also the reason this failure was noticed at all. The transition-metal oxides were being studied for their magnetism long before anybody asked why they were not conducting, and it was the combination — insulating and magnetically ordered — that had no place in the one-electron account.

Why this belongs in the field rather than at the end of it #

A reader could reasonably ask why a field would build its whole apparatus and then devote an essay to the apparatus failing.

The answer is that the alternative is worse. A collection that computes bands, states the metallic criterion, and never mentions the class of materials where it gives the wrong answer has taught something that will be believed in the cases where it is false. The failure is not obscure — it covers the transition-metal oxides, which is most of the interesting magnetism and all of the high-temperature superconductors.

There is a further reason, and it is about what a reader should take away rather than about what is true. The most useful thing a model can carry is a statement of where it stops, because a model without one gets applied everywhere and its user has no way to tell a result from an artefact. This field’s other essays are worth more with this essay in them than without it, and the arithmetic in them is unchanged either way.

It also fits the discipline this site runs on. Orbitals are not where the electron is makes the general case that a one-electron orbital is a description rather than a fact, and this is that argument arriving with a measurable consequence attached: a material that is green and transparent, which the description says should be shiny and conducting.