A mixture is not the average of its ends

Two elements are mixed and the mixture is quoted as though its properties were somewhere between the two. Half germanium and half silicon should have half the band gap of one and half of the other; half copper and half gold should have a cohesive energy in proportion. The assumption is so ordinary that it is not usually written down, and it is wrong in a direction that has a name in one field — bowing — and no name at all in the others.

Here the departure is computed, in the smallest model that has one.

The line is arithmetic, not a fit #

A structure’s sites are given one of two energies, +δ or −δ, with every hop between them the same. The two pure ends need no calculation: a structure with every site raised has every level shifted by +δ, so its binding per site is the pure band’s plus δ, and the other end is the same with a minus sign. The line between them at composition x is

$$E_{\text{line}}(x) = E_{\text{band}} + (2x - 1)\,\delta$$

and it is known before anything is diagonalised. That is what makes the departure a measurement rather than a comparison of two fits.

Every intermediate composition lies above the line — more bound. That is not a coincidence of the parameters: raising some sites and lowering others splits the levels, half filling takes the lower half, and taking the lower half of a wider distribution is worth more than taking half of a narrower one. The only way to get nothing would be for the splitting to do nothing, which happens only when the two components are identical.

What the departure actually is #

There is a temptation to call this an interaction between unlike atoms, and it is worth resisting, because there is no interaction term anywhere in the model. Every hop is the same number. What has changed is the diagonal, and the electrons have rearranged themselves to sit preferentially on the low sites.

That rearrangement is charge transfer, and it has been priced before: a difference in site energy moves a definite amount of charge, and how much depends on what resists it. Here nothing resists it — there is no on-site repulsion — so the transfer is as large as the band allows, and the energy it buys is the departure being measured. The insulator band theory cannot see is the case where something does resist, and it is a different structure entirely.

The composition does not decide it #

The sharper result is what happens at one composition.

Half and half can be arranged three ways on a square net: on the two sublattices, so that every bond joins unlike sites; at random, so that about half do; or in one block, so that only the boundary does. The three have identical compositions and different bindings.

The ordering is by a count. That matters because a count is independent evidence: the three structures could have been ranked by their bindings alone and the ranking would be a fact about three diagonalisations, whereas ranking them by a property of the graph and finding the bindings agree is two independent statements about the same three objects.

The gaps go the same way and more dramatically. The ordered arrangement opens a gap of exactly 2δ at half filling — every bond joins a high site to a low one, which is the two-sublattice problem and has a gap in closed form. The random arrangement leaves 0.0997 and the segregated one 0.0704, both of which are level spacings rather than gaps and would shrink as the structure grew.

So an ordered compound is not a random alloy with the entropy taken out. It is a different electronic structure, more bound before any entropy is counted, and the difference is what drives ordering in the first place.

The whole of the departure is a statement about where the electrons sit in a band at half filling: splitting the levels moves weight out of the middle towards both ends, and filling half of them collects the lower half — so the gain is a property of the shape near the middle of the band rather than of its width.

Not a fixed share of the contrast #

The next question is whether the departure is proportional to anything. A bowing parameter is usually quoted as a single number for a pair of components, which would require it to be a fixed multiple of the difference between them.

It is not.

The figure above measures it across a factor of five hundred in the contrast. At the small end the departure is about a third of the site-energy difference that made it; at the large end it is the whole of that difference, because the electrons take the low sites and ignore the rest. There is no regime in which it is a fixed share, so a bowing parameter quoted as one number is a number quoted at one contrast without saying which.

The large-contrast limit is easy to see and worth stating, because it is where the intuition about averages comes from and it is exactly where the intuition is most wrong. When δ is much larger than the band width, half filling puts every electron on a low site; the binding per site is then −δ plus a small band term, and the line at the same composition is the pure band. So the departure is δ itself, and the fraction goes to one: 0.98766 at δ = 128.

At small contrast the fraction falls, but it does not fall to zero linearly. Fitting the departure against the contrast over the range 0.01 to 0.32 gives an exponent of 1.7887 for a chain and 1.2801 for a square net. Neither is two.

Why the exponent is not two, and why the two differ #

A second-order argument would give two. The site energies are a perturbation, the first-order term cancels between the raised and lowered sites, and the second-order term goes as δ² over an energy denominator — so δ² is what a reader would predict.

What breaks it is the denominator. The states that mix most strongly are those separated by the smallest energy, and at half filling those are the states at the Fermi level, where the denominator goes to zero. The second-order sum then diverges logarithmically and the true dependence is δ² times a logarithm — which over a finite range of δ looks like a power below two, and does.

How far below two depends on how many states are piled up at the Fermi level, and that is a property of the density of states. A chain has an inverse-square-root divergence there and comes out at 1.79. A square net has a logarithmic divergence at the band centre — the van Hove singularity that makes two dimensions special — and comes out at 1.28, much further from quadratic.

Where a band’s curve is high at its centre, half filling has many states to mix and the departure from the average is large; where it is low, the departure is small. The three densities of states differ most at exactly that point, which is why the effect is a function of the structure and not only of the composition.

That connection is the same one the Peierls argument rests on: a chain cannot stay even because the energy it gains by opening a gap at the Fermi level goes as δ² log δ rather than δ², and beats any elastic cost that goes as δ². Here the perturbation is a site-energy difference rather than a bond-length one, but the arithmetic is the same arithmetic and it says the same thing. Ordering is a Peierls distortion in a different coordinate.

The gap, and what a mixture does to it #

The composition dependence of a gap is where bowing was first named, and this model has one to look at.

At half and half on the ordered arrangement the gap is exactly 2δ, which is the two-sublattice closed form: every bond joins a raised site to a lowered one, the problem is the same one a heteroatomic ring poses, and the answer needs no diagonalisation. Move away from half and half and the arrangement can no longer be perfectly alternating — some raised sites must sit next to each other — and the gap closes rapidly. By a quarter and three quarters it is a level spacing again.

So the gap does not interpolate between the two pure ends either. Both ends have no gap at all — a pure structure of one kind is a uniform band — and the middle has a large one. That is the opposite of a weighted average in the strongest sense available: the two endpoints agree exactly and the interior does not lie between them.

The distinction between a gap and a level spacing has to be made by looking at a sequence rather than at one structure, and the two behave quite differently as the system grows. The ordered arrangement’s gap survives; the random arrangement’s shrinks, and a single computed spectrum would not have said which was which.

Half filling is doing a great deal of the work #

Every number above is at one electron per site, and it is worth saying what that assumption is carrying.

At half filling the Fermi level sits at the band centre, which is where a two-sublattice splitting opens its gap and where the density of states of a square net diverges. Both of the results here — the one-signed departure and the sub-quadratic exponents — are properties of that coincidence. Fill the band a quarter full instead and the Fermi level moves off the centre, the splitting no longer opens a gap at it, and the departure becomes an ordinary second-order quantity going as δ².

Binding against filling has its maximum at half filling, which is where every mixture in this essay is evaluated. That choice is deliberate: it is where the effect is largest, and quoting it anywhere else would understate it.

That is not a limitation so much as a location. Real alloys of the kind this argument is about — the semiconducting ones where bowing was measured, and the ordering intermetallics — are the ones with a filling that puts the Fermi level in an interesting place. An alloy whose Fermi level sits in a featureless part of its band has a small and quadratic bowing, and nobody writes it down.

What it says about a cohesive energy #

Two structures with identical coordination have different bindings, and a bond is not a fixed quantity that can be counted. This adds the third member of that family: a composition is not a fixed quantity either.

Put together, the three say that the quantity a chemist calls a cohesive energy is not decomposable into anything much. It is not a sum of bond energies, it is not a function of coordination, and it is not an average over components. What it is is a property of a spectrum, and a spectrum is a property of a matrix, and the matrix depends on which atom is where. What holds a solid together draws the boundary of what a finite calculation can say about that question; this is inside it, and the parts that are not — the lattice sum, the relaxation — are outside it for the same reasons.

The same lesson holds at a boundary rather than in a bulk: a surface is not a count of broken bonds either, and for the same reason. A site that has lost neighbours has a different local density of states, and what it costs is a property of that shape rather than of the count.

The same departure, measured, and the sample it depends on #

Bowing has a name in one field because that field has to fit it, and the fitting exposes both halves of this essay’s finding in a place where somebody is paid to care about the answer.

Semiconductor alloys are described by

$$E_g(x) = x E_A + (1-x)E_B - b\,x(1-x),$$

where the first two terms are the straight line this essay draws and $b$ is a bowing parameter taken from measurement. Its values are of the right size to matter: a few tenths of an electronvolt for the common III–V alloys, against gaps of one to two. A designer choosing an alloy composition to hit a wavelength gets it wrong by a visible amount if the line is used instead of the curve, which is why the parameter is tabulated at all.

Two things about that expression are worth reading against what has been computed here.

The assumed shape is $x(1-x)$, which is symmetric about equal composition and exactly quadratic in it. Nothing here produces that. The departure measured above is not a fixed share of anything, it depends on the arrangement by a factor of 1.81 at one composition, and its dependence on the contrast is a power between one and two rather than the two the formula’s derivation assumes. So $b$ is not a constant of a pair of materials being measured; it is a one-parameter fit to a curve of a different shape, and its value depends on the composition range the fit was taken over.

And the formula has no room for an arrangement at all. It takes a composition and returns a gap, which is precisely the assumption this essay refuses.

That second point is the one with a direct measurement behind it, and it is the strongest available evidence that the arrangement dependence computed here is real rather than a feature of a model. Some III–V alloys grown at one composition can be made either substantially ordered — with the two cations alternating on particular planes — or nearly random, depending on the growth temperature and the substrate. The two are the same alloy by every compositional measure. Their band gaps differ by of order a hundred millielectronvolts, with the ordered material the narrower, and the size of the difference tracks how complete the ordering is.

One composition, two arrangements, two gaps, in a material grown deliberately both ways. That is the arithmetic of this essay’s three arrangements arriving in a spectrometer, and it also explains a long-standing untidiness in the tables: published bowing parameters for the same alloy disagree between laboratories by more than their stated uncertainties, and part of that spread is that the samples were not equally ordered.

The direction agrees too. The ordered arrangement here has the most unlike bonds and departs furthest from the line — 0.42987 per site against the segregated arrangement’s 0.23690 — and the ordered alloy is the one whose gap falls furthest below the interpolation.

None of which makes the interpolation useless. It makes the fitted parameter a property of a sample rather than of a pair of elements, and quoting it without the growth conditions is the same species of half-number as a delocalisation energy without its reference or a projected density of states without its basis. The line is arithmetic; everything interesting is the distance from it, and that distance is not a function of composition alone.

What this cannot say #

There is no repulsion. Charge moves freely here because nothing charges for it, and in a real mixture it does not: an on-site repulsion opposes the transfer and reduces the departure, by an amount an exact Hubbard calculation could measure on four sites and not on four hundred.

There is no lattice relaxation. Two components of different sizes distort the structure they share, and that costs elastic energy the departure would have to pay out of. Nothing here has a length in it.

The arrangements are three of very many. A random arrangement is one deterministic sample rather than an average over samples, and its number would move by a little from sample to sample. The ordering of the three would not: two hundred unlike bonds against twenty is not a sampling question.

And the exponents are effective ones over a finite range. A δ² log δ dependence has no single exponent; 1.79 and 1.28 are what a straight line through six points of the logarithms gives, and they are quoted as a comparison between two dimensions rather than as a law.

What was checked #

The ends are arithmetic: a structure of one kind only binds at the pure band’s value shifted by exactly ±δ, to eight decimal places, for all three arrangements — because at those compositions there is no arrangement.

Every mixture is above the line, at eight intermediate compositions and for all three arrangements.

The three arrangements are ordered by their unlike-bond counts, which is a count of edges and not a spectrum, and they span a factor of 1.81 — so composition decides neither.

The departure is a rising share of the contrast, at five contrasts spanning a factor of two hundred and fifty, and it goes to the whole of it: 0.98766 at δ = 128.

The exponents differ between one dimension and two, both between one and two, and the square net’s is at least 0.2 below the chain’s.

And the tripwire is δ = 0: with nothing to mix, the departure must be zero at every composition, to twelve decimal places.

Still open: ordering against lattice strain #

The natural open question is the thing this model has no term for. A real mixture pays for its ordering with lattice strain, because two components of different sizes cannot both sit at their preferred spacing, and the competition between an electronic gain that goes as δ² log δ and an elastic cost that goes as the square of a length mismatch is what decides whether an alloy orders or separates. Adding a bond-length term to this model would put both on one axis, and the line between ordering and segregation would be an output rather than a piece of received chemistry.

The nearer question is about the arrangement itself. Three arrangements were chosen here and ranked by a count of unlike bonds; a fourth would be the arrangement that maximises the binding at a given composition, which is a combinatorial optimisation over a finite set and is answerable directly for a small net. Whether the answer is always the one with the most unlike bonds — or whether at some composition a different structure wins — is a question a direct search could settle, and the count being a perfect predictor for three cases is much weaker evidence than it looks.