60 electrons in 60 levels

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

Eight essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

What a metal actually is

Where the filling stops, for a ring of sixty with one electron per site: in the middle of the band, where the level spacing is smallest, and the gap above the last occupied level is printed. Everything above turns on that one number and how it behaves as the ring grows.

Forty levels with eighty electrons: every one occupied, and the figure reports that there is no level above the filling at all. This is the completely filled case, and its cheapest excitation is not a level spacing but the distance to a band that this calculation does not contain.

The same ring at exactly half filling rather than at the two counts above. The highest occupied and lowest empty levels are adjacent members of a crowd, and the spacing between them falls as the ring grows — which is the condition for a metal, and it is a statement about the filling rather than about the band.

A chain cannot stay even

The state the distortion destroys: a half-filled band with the filling stopping where the levels are densest. Every one of those closely spaced levels near the top of the occupied set is one the distortion can move down, and the number of them is what makes the gain large enough to beat the springs.

A half-filled band is not always a metal

The picture this essay is about. A half-filled band, with the filling stopping in the middle where the levels are densest and the cheapest excitation costing almost nothing. In a one-electron model this is a metal, necessarily and by construction. Nickel oxide has exactly this band and is an insulator with a gap of about four electronvolts.

The quarter-filled case, where nobody expects a metal and the same arithmetic gives one. The filling stops in a crowd of levels exactly as it does at half filling, so whatever makes a half-filled band conduct in this model makes a quarter-filled one conduct too — and the model has no term that could distinguish them.

The same half filling on twice as many sites. The spacing at the Fermi level has halved and the shape of the picture is unchanged, which is exactly one side of the comparison: a one-electron calculation can compute how the spacing shrinks and has no quantity in it that could say whether the electrons decline to move past one another.

Half filled is as bonded as it gets

The occupied-level sum per site of a ring of sixty, swept from an empty band to a full one, with a chain of sixty beside it and the closed form behind both. The curve rises to a maximum at half filling and comes back to exactly zero when every level is occupied.

The same curve on twice as many sites and at twice the resolution. The maximum is at half filling to the accuracy the steps allow, and the two ends are exactly zero: an empty band and a full one bind nothing whatever the shape between them is, which is the only part of this curve that is a theorem rather than a computation.

The same sweep on a ring of twelve. The shape is unmistakably the same and the numbers are visibly coarser — and the tie at ten electrons against twelve is the case that made the maximum an inequality rather than an argmax.

The bond that weakens as neighbours multiply

The filling half of the same question. Binding against how many electrons the band holds, with the maximum at half filling — which is the point at which every number in this essay was measured.

Where the states pile up

The binding a band supplies as it is filled, from empty to full. The maximum is at half filling and the return to zero at complete filling is exact: a full band binds no more than a filled shell of one atom binds another. Every number in this essay is read off the top of this curve.

A full band is not an insulator

And the reason the count feels like it should be enough: filling a band to the top really does exhaust what it can supply, so a completely filled band is doing no more bonding. What that does not settle is whether there is anywhere for an electron to go, which is a question about a different band.

The arrangement a count cannot pick

Why the filling matters: binding against how many electrons a band holds, with the closed form drawn through it. Everything in this essay is computed at the maximum of this curve, which is also where the count of unlike bonds is doing the most work.