The length at which levels become a band

A particle in a box the alloy made found that the states a random alloy puts in its gap are particles in boxes: 38 of 40 of them matched the closed form for a run of low sites to 3.1 per cent, with nothing fitted. It then named the question it had not asked:

Two runs share a state when they are close enough, and close enough is a distance already measured for two impurities. Working out at what chain length the typical gap between neighbouring long runs falls below that distance would give the size at which the defect levels become a defect band.

The question has a defect in it, and finding the defect is most of the answer. The typical gap between neighbouring runs does not depend on the chain length at all. Runs of a given length occur at a density fixed by the composition, so their mean spacing is one over that density whatever the chain is — four hundred sites or four million.

What does depend on the length is the closest pair.

What two runs are worth to each other #

Two runs of six low sites in an otherwise uniform chain, at separations from one site to twenty, with the splitting of the level they share:

A straight line on a logarithmic axis, with a decay length of 2.003 sites and an amplitude of 3.94 × 10⁻². Both come out of the fit. It is a tunnelling amplitude through the barrier between the two boxes, and the straightness is what says so.

It never reaches zero, which is why close enough is not a distance. Two runs twenty sites apart still split their level, by 10⁻⁵; whether that matters is a question about what else is going on at that energy. That is the same shape of answer a healing length turned out to have — a decay with no natural end, so any length taken out of it is a length somebody chose — and it is why the criterion below is a comparison between two computed quantities rather than a cutoff.

The criterion, which is a comparison rather than a threshold #

A set of levels is a band when the coupling between neighbours exceeds the spacing between them. That is what every account of a band says, and it is the right criterion here because both quantities are computable.

The coupling is the splitting of the closest pair. With $m = N\rho$ runs of a given length scattered over $N$ sites, the closest pair is about $N/m^2 = 1/(N\rho^2)$ apart, so it shrinks as the chain grows and the coupling rises towards its short-range value.

The spacing is the mean separation in energy of the gap levels. There are $N$ times a density of them and they lie between two fixed box energies, so the spacing falls as $1/N$.

One rises with $N$ and one falls, so they cross once, and the crossing is the onset.

Both halves are worth stating in the form that makes them checkable. The coupling at the onset is a splitting between two particular runs, which the diagonalisations above measure directly. The spacing is a mean over the gap states, which the run densities give in closed form. Neither is fitted, and the crossing is found by bisection on their difference — so the number that comes out is a consequence of two measurements rather than a parameter of a third.

The run density, which is the input #

Everything above needs $\rho(L)$, and it is a closed form rather than a measurement: a site is low with probability $x$ independently, so a run of exactly $L$ needs $L$ lows bounded by two highs, at a density of $x^L(1-x)^2$ per site.

Counted over sixty thousand sites of generated chain:

The answer #

A factor of about three per site of run length, and the reason is arithmetic: the density enters the closest-pair distance squared, so halving it quadruples the chain length needed, and the exponential coupling then asks for a little more.

What this says about the box analysis #

The box analysis worked on chains of four hundred. Every number in the table is larger, and the shortest is larger by a factor of two and a half.

A vacancy is not an impurity and a band with no structure in it were both established on chains of the same size, so the point applies to them too: everything those measurements say about states in a gap belongs to the discrete regime.

So it was working entirely in the isolated-level regime, and its finding is exactly what that regime produces: 38 of 40 gap states matching a closed form for a single box. The two that did not are the two that were shared between runs, which the box analysis identified by their participation ratios and set aside — and those two are the whole of what a longer chain would have had more of.

There is a small vindication in that. One defect is a level and many are a band was established by adding defects to a fixed chain — which raises the density and shortens the spacing between them directly. The same thing happens at fixed density if the chain is made longer instead, which is a different route to the same place and a much slower one: a factor of two in the density is worth a factor of four in the length.

Neither result is wrong and the pair of them is the useful statement. A chain of four hundred is a set of isolated levels in the gap; a chain of ten thousand is a band in its shorter runs and still isolated levels in its longer ones. That last clause is the finding worth carrying: one chain is in both regimes at once, at different energies, and there is no single length at which “the levels become a band”.

Why the answer is not a single number #

The onset depends on the run length because the density does, and the run length sets the energy. So the gap states of a long chain are a band at the shallow end — many short runs, close together, strongly coupled — and a set of discrete levels at the deep end, where the runs are rare.

That is a structure produced by counting rather than by any argument about localisation — an edge in the spectrum with nothing at it, which a band gap that is not a bond energy is the neighbouring warning about, and it is worth being careful about the difference. Every state of a disordered one-dimensional chain is localised, which the box analysis’s control established and which no amount of chain length changes. What the crossing above locates is where the levels stop being resolvable as individual boxes, which is a spectroscopic statement rather than a transport one.

What a chain of ten thousand looks like #

Putting the two ends together gives a picture of a long chain that neither result draws alone.

At ten thousand sites, runs of four and five are past their onset — seventy-eight runs of five, their closest pair under two sites apart, coupled at 2.8 × 10⁻² against a mean level spacing of 7.9 × 10⁻⁴. That is a coupling thirty-five times the spacing, which is a band by any reading. Runs of seven are not: twenty of them, closest pair twenty-six sites apart, coupled at 2.4 × 10⁻⁷ against the same 7.9 × 10⁻⁴ — three thousand times smaller than the spacing, which is a set of isolated levels by any reading.

So the gap of that chain holds a band at the shallow end and a comb of discrete levels at the deep end, with the boundary somewhere near a run of six, and the two ends differ by a factor of a hundred thousand in the ratio that decides it. Nothing about the chain is different at the two energies — the same disorder, the same composition, the same hopping — and the difference is entirely how many of each kind of run there are.

That is worth putting beside what a defect level in a gap established at the other extreme, one defect at a time. The two pictures are the same physics counted at two densities, and the whole of this essay is the arithmetic that says where one turns into the other.

In a real sample both answers are true at once #

The onset is quoted for two run lengths — about a thousand sites for runs of four, about eighty thousand for runs of eight — and the ratio between them is worth extrapolating, because it says something about a macroscopic sample that neither number says on its own.

Eighty times for four extra sites is a factor of very nearly three per site, which is what the competition demands: the coupling falls exponentially with separation, so the chain has to grow by a fixed factor to bring the closest pair one site nearer. Continuing it,

The last row is the interesting one. A mole of material is about $10^{23}$ sites, so in a real sample the runs up to roughly forty-six sites long have formed bands and the longer ones have not.

And longer ones exist. The longest run in $10^{23}$ sites is near seventy-six, and every run length between forty-six and seventy-six is present in the sample, at a density falling by a half per site. Those runs are further apart than their states can reach, so each of them sits in the gap as an isolated box level of the kind a short chain shows.

So a macroscopic alloy has both descriptions at once, and they are not competing accounts of the same states. Its short runs — which are almost all of them — have merged into a defect band. Its longest runs, of which there are a handful, remain individual levels deeper in the gap than the band reaches, because a longer run gives a level nearer the gap centre and there is nothing nearby to mix with.

That is a testable shape rather than a philosophical point: the gap of a real random alloy should contain a band of states with a defined edge, and below that edge a sparse set of isolated levels, thinning out as the gap centre is approached. Which is exactly what a deep-level spectroscopy of a disordered semiconductor finds, and it is usually attributed to chemistry — to particular impurities with particular energies — rather than to the arithmetic of how long a run of like atoms a large sample happens to contain.

What is quoted, and what is computed #

Nothing is quoted. There is no material: a chain of stated length, two site energies, one hopping, and a composition.

The splittings are diagonalisations of chains with the runs placed rather than found, which is the difference between this calculation and the box analysis — a run at a chosen separation can be varied, and a run found inside a random chain cannot. The decay length and amplitude come from a least-squares fit on the logarithms over the separations where the decay is clean.

The run densities are counted over sixty thousand sites and compared against a closed form; the closest-pair distance is the standard estimate for $m$ points scattered over $N$ sites, which is an expectation rather than a measurement and is the one approximation in the chain of reasoning.

The onset is found by bisection on the crossing of two curves, and the check is on the crossing rather than the answer: at the length returned, the coupling and the spacing agree to a part in a million.

What this cannot say #

The closest-pair estimate is an average. A particular chain of ten thousand sites might have its two nearest runs of six twice as far apart as the estimate, or half. What the number locates is where the typical chain crosses over, and the spread around it is not computed here — it would need the distribution of the minimum spacing rather than its expectation.

The coupling is measured between two runs of equal length. Two runs of different lengths have levels at different energies and couple more weakly for the same separation, so the estimate above is the most favourable case. A proper treatment would sum over pairs of unequal runs, and would move the onset to longer chains rather than shorter.

The gap states of runs longer than nine are ignored. The mean spacing is computed over runs between three and nine, which is where the density is appreciable, and a chain of eighty thousand sites holds runs of twelve. Including them widens the energy range slightly and lowers the density slightly, and the two corrections act in opposite directions on the spacing. Moving the upper limit from nine to fifteen moves every onset by under two per cent — 985 to 1,012 for runs of four, 81,508 to 82,981 for runs of eight — so the numbers are robust to the choice rather than dependent on it, which was checked rather than argued.

And there is no electron repulsion in it. A defect band a tenth of an electronvolt wide is exactly the situation where a one-electron picture is least safe, since a band narrower than the repulsion is a band that does not conduct. Every band argument carries that caution and this is where it bites hardest.

What was checked #

The splitting falls at every step, and falls exponentially with a length between half a site and six — checked as a monotone sequence and as a fitted length, because a sequence that fell without being exponential would still pass the first.

The run density is the closed form, at every length separately, to within twelve per cent over sixty thousand sites.

The onset grows with the run length, and at the onset the coupling equals the spacing to a part in a million — the second is the check on the bisection rather than a result, and without it a bisection that converged to an endpoint would report a length.

The run densities are counted, and the counting has its own error. Sixty thousand sites hold fifty-six runs of eight, so that row’s density carries about thirteen per cent of counting noise — which is why the check is at twelve per cent rather than at one, and why the ratio for runs of seven comes back at 0.930 rather than at one. A tighter check would need a longer chain rather than a better argument.

And the refusal is a single run. One run of six puts one level near the box energy with the next one four times further off, so there is no splitting to measure — and the level is not at the box energy either, because a barrier of four is not infinite.

Still open: the density of gap states, and the band’s width #

The obvious open question is the density of states the whole picture implies. Every run contributes a level at an energy the closed form gives, so the density of gap states is computable from the run-length distribution with no diagonalisation anywhere — and comparing it against a direct count of the levels a long chain actually puts in the gap would test the box description on a chain far too large to solve. That was the box analysis’s own next question and the missing piece is now supplied: the length at which the comparison stops being between levels and starts being between densities.

The nearer question is the width. A band has one, and the crossing above locates the onset without saying how wide the band is when it forms. The width is set by the coupling at the typical spacing rather than at the closest, which is a different average over the same distribution — and the two together would say whether a defect band ever becomes wide enough to overlap the host band, which is the question that decides whether the gap closes.