Two bands, if the chain is short enough

Put impurities into a chain and turn up their concentration, and the levels they contribute stop being separate and become a band of their own. That picture stops where the word impurity stops meaning anything: at half and half there is no host, and what is left is an alloy of two components in which neither is the defect.

The question it left was whether the alloy’s band is the average of its two components’ — which is what the simplest picture says — or something else, and if something else, whether the crossing between the two behaviours is sharp.

Both halves have answers. The average is wrong in a way that a second moment cannot see and a fourth moment can. And the crossing is sharp for two arrangements of the components, exactly sharp, with closed forms on both sides of it — and for the third arrangement it is not sharp at all, because where it sits depends on how long the chain is.

The system, which has one number in it #

A chain of $n$ sites, every neighbouring pair joined by the same resonance integral, and half the sites raised by $\delta$ with the other half lowered by the same amount. Nothing else: no length, no repulsion, no relaxation. It is the same finite matrix a solid was made of in the first place, with two kinds of diagonal entry in it. The one parameter is the contrast $\delta$ between the two components, measured in units of the resonance integral, and the composition is fixed at equal parts throughout.

What is not fixed is which site is which, and three arrangements are compared:

Ordered. Every site’s neighbours are all of the other kind — the alternating compound. On a chain of 2,048 this makes 2,047 unlike bonds out of 2,047.

Random. The raised sites are chosen by shuffling the indices with a stated seed, so exactly half of them are raised and their positions carry no pattern. That arrangement gives 996 unlike bonds of 2,047, which is the half a shuffle is expected to give.

Segregated. The first half raised, the second half lowered — two pure blocks meeting at one boundary. One unlike bond, out of 2,047.

Those three counts are the same quantity that orders three cohesive energies, where the arrangement with the most unlike bonds was the most bound. It will order these three too, and it will order them in the same direction, which is not obvious in advance and is one of the two surprises here.

What the composition fixes, and what it does not #

The mean of the band and its variance are both traces, and a trace does not know which site is where.

The mean is $\operatorname{tr}H/n$, which is the mean site energy: zero at equal parts, and $(2x-1)\delta$ at a composition $x$. The variance is $\operatorname{tr}H^2/n$ less the square of the mean, and $\operatorname{tr}H^2$ counts the squared site energies along the diagonal and twice the bond count off it. Neither sum can tell one arrangement from another, so all three must return the same two numbers exactly.

They do. At $\delta = 1.5$ on a chain of 240 sites all three arrangements give a variance of 4.241666666667, and the value composition alone predicts — two bonds a site less the chain’s two ends, plus $\delta^2$ — is 4.241666666667.

The fourth moment divided by the square of the second runs from 1.1107 for the ordered arrangement through 1.5859 for the random one to 2.1028 for the segregated one. A single chain’s band has 1.5 exactly and a bell curve has 3, so all three sit below a Gaussian and they are nowhere near each other. The fourth moment is where a band keeps the information about where its states pile up, and the second moment is where it does not.

That is the first answer. An alloy’s band is not the average of its components’: the average gets the first two moments right, which is all it can get right, because those two are the only ones that do not depend on the arrangement.

Two arrangements with closed forms #

The ordered arrangement is not disordered at all: it is a two-site repeating structure, which is the same object as two orbitals per site with the roles of the indices exchanged. Its levels are $\pm\sqrt{\delta^2 + 4\cos^2 k}$, and the two branches never meet, so the gap is $2\delta$ and it opens the instant $\delta$ is not zero.

The segregated arrangement is two pure chains that happen to share a bond. Each block has a band four wide centred on its own site energy, so the two bands run from $-\delta-2$ to $-\delta+2$ and from $\delta-2$ to $\delta+2$. They overlap until $\delta = 2$ and separate after it, and the gap is exactly $2\delta - 4$.

Both are drawn as dashed lines through the figure at the head of this essay, and the computed points sit on them: the ordered gap agrees with $2\delta$ to better than a part in ten thousand at every contrast tested, and the segregated one with $2\delta - 4$ to within one level spacing.

That last qualification is not a hedge, it is the arithmetic. A finite chain has no gap below its threshold; it has a nearest pair of levels straddling the centre, and the distance between them is about $2\pi/n$ — a thousandth on a chain of 2,048, which is what the curves sit at before they lift off. Every threshold quoted below is therefore a threshold at a stated floor, and the two closed forms check that the floor is behaving: at a floor of 0.02 the ordered arrangement must split at $\delta = 0.01$ and the segregated one at $\delta = 2.01$. Measured: 0.00997 and 2.00999.

The arrangement with no closed form #

The random one splits at 1.933 on that same chain, at that same floor. It is between the other two and it is nearer the segregated end, which is what a first guess would say.

What a first guess would not say is what happens above the threshold. The random arrangement’s gap is wider than the segregated one’s at every contrast where both are open — 0.495 against 0.400 at $\delta = 2.2$, 1.093 against 1.000 at 2.5, 2.092 against 2.000 at 3, 4.091 against 4.000 at 4. The offset is roughly constant at about a tenth, and it is in the direction nobody expects: mixing the two components more thoroughly has helped the gap.

The reason is a statement about runs. A sub-band is made of states living on the raised sites, and a state cannot spread further than the raised sites are contiguous. In the segregated arrangement they are contiguous for a thousand sites and the sub-band reaches its full width of four. In the random arrangement the longest stretch of like sites is fourteen, and a chain of fourteen has a band $4\cos(\pi/15)$ wide rather than four. A narrower sub-band leaves a wider gap.

That is a mechanism rather than a fit, and it makes a prediction that can be checked: the sub-band edge should sit where an isolated run of the longest length puts it, at $\delta - 2\cos(\pi/(L+1))$ for a run of $L$ sites.

The gap that goes away when there is more of the material #

If the gap is set by the longest run, then it is set by a quantity that grows with the size of the system. The longest run of heads in $n$ tosses of a coin grows like $\log_2 n$, without bound and without hurry, and a longer run means a wider sub-band and a narrower gap.

Measured at a contrast of 4, over four chains at four sizes and four samples apiece:

The gap falls by more than a factor of two across a factor of sixty-four in length, the longest run rises by half, and the isolated-run estimate accounts for the measurement at every size — always a little low, which is right, because a run embedded in a chain is coupled at both ends and is therefore a little less confined than an isolated one.

So the answer to the question the impurity band leaves is: the crossing is sharp for the ordered arrangement and for the segregated one, and for the random one there is no crossing in the thermodynamic limit at all. The random alloy’s gap is a finite-size quantity. Take enough of the material and it contains a run long enough to put a state in the gap, and the split-band description — the one every textbook draws for an alloy with a large enough contrast — describes a sample rather than a substance.

That is the same shape of result as the impurity band, where the count of levels stopped matching the count of impurities above a concentration set by how localised the states were. Both are cases where the received picture is right about a typical configuration and the atypical ones are the whole story.

How it was computed, and why not by diagonalising #

A chain of 32,768 sites has a matrix with a billion entries in it, and a general dense diagonalisation is a cubic-time sweep that would take days.

It is not needed. The question asked here is never what are the levels but only how many are below this energy, and that has a linear-time answer: factor $H - E$ as $LDL^{\mathsf{T}}$ and count the negative pivots. The number of negative pivots of a symmetric matrix is its number of negative eigenvalues, whatever the pivots’ sizes are, so the count is exact rather than approximate. Two counts at $\pm E$ and their difference is the number of levels in a window, and a bisection on that difference finds the gap.

The whole of it is one loop over the sites, and a chain of thirty-two thousand is a hundred and thirty thousand arithmetic operations rather than a billion. Counting levels below an energy is also exactly what filling a band requires, so the same routine answers both questions.

A routine like that produces a plausible number when it is wrong, which is exactly the failure this collection is built against — so it is not trusted. On a chain of 100 sites, small enough to diagonalise, the count is compared against a sort of the computed eigenvalues at seven probe energies, on all three arrangements, and the two must return the same integer twenty-one times. An integer is a good thing to check against: a tolerance cannot absorb a disagreement.

What this cannot say #

There is no repulsion. Each site holds one level and the electrons neither see nor charge one another, so nothing here can produce the insulator that band theory cannot see — and an alloy at half filling with a moderate contrast is exactly the situation where that omission is most likely to matter.

There is no relaxation. Two components of different sizes distort the chain they share, which changes the resonance integrals as well as the site energies, and every resonance integral here is 1. A missing site is a different perturbation again, and a vacancy is not an impurity even when the two are given the same depth.

It is one dimension, and a run of like sites in one dimension is a much more effective confinement than a cluster of like sites in three, where a state can go round. The size dependence found here is therefore an upper bound on the effect rather than a number to carry across.

And it is one sample per size, four times over. The spread between samples at 32,768 sites is 4.063 to 4.104, which is smaller than the fall between sizes and is quoted rather than hidden.

What a measurement would see #

A measured optical gap for a disordered alloy is a report on the sample. Not on its composition, which fixes the mean and the variance and neither of the two thresholds; and not on the material, since the answer moves with how much of it there is. Two samples of the same composition prepared with different degrees of ordering have different gaps, and the more ordered one has the larger.

The ordering is what the gap measures, and it measures it steeply. Between the fully ordered arrangement at $2\delta$ and the segregated one at $2\delta-4$ there is a difference of four resonance integrals — the whole band width — at every contrast. Nothing else in this model moves the gap by anything like that.

And a second-moment measurement settles nothing about either. A band width from a photoelectron spectrum, or a variance from any line shape, is a statement about the composition. Two materials with identical second moments here differ by a factor of two in fourth moment and by the whole band width in gap.

What is quoted, and what is computed #

Nothing is quoted. There is no measurement in this essay and no material named in it. The chain, the contrast, the composition and the three arrangements are the model’s parameters; every level, gap, moment, threshold and run length is computed from a matrix that was written down.

The two closed forms — $2\delta$ and $2\delta - 4$ — are derived above and are then used as checks on the computation rather than as substitutes for it. The isolated-run band edge is derived the same way and is used as a check on the mechanism, which is a different and weaker kind of claim, and it is quoted with the twelve per cent it is good to.

What the calculation requires #

The Sturm count agrees with the diagonalisation, as an integer, at seven probe energies on three arrangements. A tolerance would have hidden a count that was off by one.

The ordered arrangement’s gap is exactly twice the contrast at every contrast tested, to a part in ten thousand — the residual is a finite-size effect that falls as the square of the length.

The segregated arrangement’s gap is exactly the contrast less the band width, to within one level spacing, and the tolerance is computed from the chain’s length rather than chosen, so a longer chain is held to a tighter standard.

All three arrangements have the same mean and the same variance to ten decimal places, and the values are the ones composition alone predicts.

And they do not have the same shape: the fourth moments differ by more than 0.2 across the three, which is the claim the second moments cannot support.

The random arrangement’s gap is the wider of the two at every contrast above 2.2.

The random arrangement’s gap falls as the chain grows and its longest run rises, at every step of the size scan — and the run accounts for the gap to within thirty-five per cent, which is the tolerance the mechanism is claimed to.

And the refusal is a contrast of zero: with nothing to distinguish the two components there is no alloy, and all three arrangements must return one level spacing rather than a gap.

Still open: controlling the longest run, and the states in the gap #

The defect picture runs from one impurity to a concentration of them to a composition where neither component is the defect, and the last step finds a length in the problem that none of the earlier ones had. That length — the longest run of like sites — is a property of the arrangement rather than of the chemistry, and the obvious next step is to control it rather than to let a shuffle decide it.

Short-range order is the name for that in a real alloy: a tendency for unlike neighbours, or for like ones, that falls short of the full ordering. It is one parameter, it is measurable by diffraction, and it interpolates between exactly the three arrangements compared here. Computing the gap against it would turn a comparison of three cases into a curve, and the interesting question is whether the curve is monotone — whether every step towards ordering widens the gap, or whether the ordered arrangement’s advantage arrives suddenly near the end.

The nearer question is about the states in the gap rather than the gap itself. A run long enough to put a level near the centre puts a localised level there, living on that run and nowhere else, and how many sites a state occupies is directly measurable. Whether those levels are as localised as the runs that carry them, or whether they leak into the surrounding material, decides whether they behave like a single defect’s levels — and the answer would say whether a very large alloy is best described as a split band with defects in it, which is not a description anybody draws.