Where the states pile up

A solid here is a molecule that did not stop, and a band has been treated in this field as an interval: it runs from one energy to another, it is so many β wide, and the width is a count of neighbours. That last identity is exact and was the subject of an essay of its own — the mean of the squared level energies is the average coordination, for any structure, with no limit taken and no periodicity assumed.

An interval is not much of a description. Two bands of the same width can hold their levels in entirely different arrangements, and this essay is about how different, and about what the difference decides.

The comparison has to divide the width out #

The obstacle to asking the question at all is that structures with more neighbours have wider bands. A chain is two-connected and its band is 4β wide; a square net is four-connected and 8β; a cubic structure is six-connected and 12β. Put the three densities on one axis and almost all of what is seen is three widths.

So the width is divided out. Every structure’s levels are scaled to run from −1 to +1, which leaves exactly one thing varying — where the levels sit inside the band — and the whole of this essay is that one thing.

The three structures are chosen to have the same number of levels — 4,096 apiece — so that even the number of things being counted is held fixed. What differs between them is the dimension and nothing else.

The three shapes, in numbers #

Cut each band into three equal parts by width and count what is in them.

More than half of a chain’s levels are in the two outer thirds of its band. Fewer than an eighth of a cubic structure’s are. That is not a shading of one distribution into another; it is a reversal.

The last sentence is the whole mechanism and it is worth stating plainly, because it explains the pattern without any physics in it. A wrapped structure of several directions is a product: its levels are sums of one contribution per direction. A level near the bottom of the band needs every one of those contributions to be near its own minimum simultaneously, and independent quantities are rarely all extreme together. So the more directions there are, the more the sum piles up in the middle — which is the same statement as the central limit theorem, arriving in a place where nobody is expecting it.

That is a strong claim to make in passing, so it is made as a number. A distribution’s departure from a Gaussian is read off its fourth moment, normalised by the square of its second so that the width drops out, and for a sum of $d$ independent contributions the limit theorem fixes what that number must be.

Three is where the sequence is heading and no lattice arrives there, which is the honest form of the statement: the pile-up in the middle is the limit theorem operating on a sample of size three, and a sample of three is not large.

In one dimension there is no sum to take. The single contribution is 2cos(2πk/n), and the density of a cosine is largest where the cosine is flattest, which is at its turning points — the two band edges.

Where the closed form comes in #

Everything above could be a histogram of a badly converged diagonalisation, so it is not one. The spectrum of a wrapped structure is available in closed form: the levels are sums of 2cos(2πk_i/L_i), one term per direction, and every combination occurs exactly once.

That is a derivation, and derivations of that shape have a characteristic way of failing — they get the level positions right and the multiplicities wrong, which is invisible in a band edge and fatal in a density of states. So the closed form is checked against a diagonalisation of an actual 4×4×4 structure, sorted spectrum against sorted spectrum, entry for entry: sixty-four levels agreeing to 10⁻¹³, multiplicities included. Only then are the four-thousand-level cases computed from it.

The chain has a second check available, and it is the sharper one, because it is a number rather than a comparison. At half filling a long chain’s binding per site tends to 4/π, and its band is 4β wide, so its binding per unit of band width tends to 1/π = 0.31831. That is checked against the computed value to five decimals.

The consequence: a wide band is not a strong one #

The shape is not a curiosity about how levels are distributed. It decides how much binding a band supplies, because filling a band to the halfway mark means occupying the upper half of it, and how much energy that is worth depends on where the upper half’s levels are.

A chain gets nearly twice as much binding out of a band of given width as a cubic structure does. The reason is visible in the shapes: the chain’s occupied half is concentrated at the far edge, a long way from the middle, while the cubic structure’s occupied half is mostly just above the middle, where the levels are worth very little.

The same shapes decide something at the other end of the band, and it is worth naming because it is the question a solid-state text asks first. What decides whether a structure has excitations of arbitrarily small energy is the spacing between the levels at the boundary between occupied and empty — which is one over the density of states there, not one over the width. A chain of 384 sites and a cubic structure of the same count have their levels crowded at completely different places, so the same total number of levels in the same interval leaves them with different spacings where it matters.

This is the missing half of an earlier measurement #

Three essays ago this field measured how a solid’s binding grows as neighbours are added, and found something it could not fully explain. The bond that weakens as neighbours multiply reports binding per site of 1.272, 1.611 and 1.979 for two, four and six neighbours, and observes that the exponent relating binding to coordination is not the square root that the standard argument predicts: it runs from about 0.34 between the chain and the square net to about 0.51 between the square net and the cubic structure.

The square-root rule comes from the second moment. If a band’s shape is fixed and only its width scales, then the width scales as √⟨x²⟩ = √Z, and so does the binding. The measurement departed from that, and the explanation offered at the time was that the chain is the outlier because its density of states piles up at both edges.

That explanation was correct and was not drawn. The figures above are it. The chain’s shape is not the cubic structure’s shape, so the premise of the square-root argument — a fixed shape — is false, and the departure is exactly the factor of 1.91 in binding per unit width that the last figure measures.

There is a general lesson in the shape of that correction, and it is worth having: a rule derived from one moment of a distribution is a rule that assumes the distribution’s shape. The second moment fixes a width and says nothing about how the weight is arranged inside it. Where the shape is common to the things being compared, moment arguments are exact and free; where it is not, they are approximations whose error is a fact about shape and can be as large as a factor of two.

The same caution applies to every quantity in this field that is quoted as a single number. A band gap is one number about a spectrum and is not convertible into a bond energy; a band width is one number about a distribution and is not convertible into a binding energy. Each of them summarises something real and discards the thing that would have been needed.

What the chain’s divergences do to the sum #

The chain’s density is the one case with a closed form to compare against, and it is worth putting the two side by side because the agreement is what licenses reading the histograms at all.

The check is run on a chain of two thousand sites, binned and compared against 1/(π√(4 − x²)) bin by bin. The interior agreement is checked; the outer region at each end is excluded from the comparison, and excluded for a stated reason rather than a convenient one — the closed form diverges there and a histogram of finitely many levels cannot, so a comparison including the edges would be a comparison against infinity.

An integrable divergence is an awkward thing to draw and an easy thing to sum. The density goes as (1 − |x|)^(−1/2) near each edge, whose integral converges; so a finite fraction of the levels really is arbitrarily close to the edge, and the binding integral 4/π is finite and exact. Nothing about the divergence is pathological. What it does is put weight where it is worth the most, which is the whole of this essay’s mechanism in one clause.

The comparison with the three-dimensional case is then stark. There the density goes to zero at the edges, so the levels that would have been worth the most are the ones the structure has fewest of.

It is worth remembering where the histograms came from, since a density of states can look like an object in its own right. Draw every level of a chain of two, four, eight, sixteen and forty sites and the band edges do not move: every level lies inside ±2β at every size, and what changes is how the levels crowd inside a fixed interval. The same is true of rings. A solid is a molecule that did not stop draws both sequences; growth adds levels to an interval rather than widening it, and a density of states is what that crowding becomes in the limit.

The band width does not change what a structure is #

One consequence deserves separating out, because it undoes a piece of intuition that is otherwise hard to shake.

A band’s width is set by the hopping between neighbours — how strongly two adjacent orbitals interact — and by how many neighbours there are. Both of those are chemistry: they change with the element, the bond length, the orbital in question. So it is natural to think of the width as the chemically interesting quantity and the shape as a detail.

The shape is not chemistry at all. Every structure here is a graph: its matrix has ones where two sites are joined and zeroes everywhere else, and no element, bond length or orbital enters. Vary the hopping and every level scales together — the width changes and the shape does not move by a thousandth. Vary the dimension and the shape changes completely.

So of the two numbers a chemist can influence, one is the width and the other is the coordination, and the coordination influences the shape only through the dimension it implies. A four-connected structure that is a flat net and a four-connected structure that is a three-dimensional framework have the same second moment and therefore the same width, and they do not have the same band.

That claim is a pair of coordinates, so it can be plotted rather than stated.

The last of those is the sharpest of the three, because it is a structure the whole account above cannot place. The hypercubic sequence runs from 1.5 up towards three and stops short; a chain with third-neighbour hopping is past three. Adding reach along one direction is not a weak version of adding a direction, and the two operations do not move a band along the same axis at all.

The filling matters as much as the shape #

One more variable belongs in the account, because holding it fixed is what made the comparison possible and varying it changes the answer.

Everything above is at half filling. That is the case where the binding is greatest — a band swept from empty to full supplies binding that rises to a maximum at the halfway mark and returns to exactly zero when every level is occupied — and it is also the case where the shape matters most — half filling is where the question of whether a structure is a metal is decided — because the boundary between occupied and empty sits where the states are densest in two of the three structures.

At a quarter filling the comparison between dimensions is different again, and in the direction the shapes predict: the chain’s occupied quarter is at the far edge where its density is largest, so it does even better, while the cubic structure’s occupied quarter reaches only into the thin lower tail. The shape does not merely change the size of the answer; it changes how the answer responds to the electron count, which is the quantity a chemist actually varies.

What this cannot see #

The line this field holds is worth restating here, because the argument above is one of the places where the cost of holding it is largest.

There is no wavevector anywhere in this calculation. What is computed is a list of eigenvalues of a finite matrix, and a density of states is a histogram of that list. So the shapes drawn above are correct as densities and carry none of the information that a band structure carries — see a band with no structure in it for what that costs and why it is a decision rather than an omission.

In particular, the divergences at the edges of the chain’s band and the spike at the centre of the square net’s are what a solid-state text calls van Hove singularities, and derives from the geometry of a surface in reciprocal space. Nothing here derives them. They are measured — the histogram’s outer bars are large, the closed form behind them diverges — and the mechanism given for them is the one available in this language: a sum of independent contributions piles up in the middle, and a single cosine piles up at its turning points.

The one-electron caution applies with its usual force. There is no repulsion in any of this, and a half-filled band is not always a metal is the standing reminder of what that omits.

Whether the shape is visible, which it partly is #

A shape that no experiment can see would be a shape worth doubting, so it is worth setting out what a measurement actually returns. Three things do, and they return different amounts of the curve.

One number, from a heat capacity. The electronic contribution to a metal’s low-temperature heat capacity is linear in temperature, and its coefficient is proportional to the density of states at the Fermi level and to nothing else — only the electrons within about $k_BT$ of the top of the filled set have anywhere to go. So a heat capacity measured below a few kelvin reads off $g(E_F)$ directly, at one energy, with no model between the measurement and the number beyond the assumption that the electrons are independent.

The values say the shape matters. Sodium’s coefficient is about 1.4 millijoules per mole per kelvin squared; palladium’s is about 9.4. Seven times as many states at the Fermi level, in a metal whose atoms are not seven times anything — because sodium is filling a broad s band where the levels are thinly spread and palladium is filling a narrow d band where they are piled up. That is this essay’s argument arriving in a calorimeter.

The same number again, from a magnet. The Pauli paramagnetic susceptibility of a metal is also proportional to $g(E_F)$, by the same argument applied to spin rather than to energy. Two measurements with nothing in common — a thermometer and a magnetometer — return the same quantity, and their ratio is a fixed number if the one-electron picture holds. Where the ratio departs from it, the departure is a measurement of the interactions the model leaves out, which is a more useful failure than a disagreement with a calculation.

And the curve itself, from a tunnelling junction. The current through a tunnel junction at a given bias is an integral over the states available at that energy, so its derivative with respect to voltage is proportional to the density of states at the energy the voltage selects. Sweeping the bias sweeps the energy, and what comes back is $g(E)$ as a function of $E$ — the shape rather than one of its values, measured on a real material, including the peaks.

So the honest answer to the question is three-quarters yes. The value at one energy is measured routinely and by two independent routes; the whole curve is measured by a third, at the price of an instrument that reads a surface rather than a bulk. What is not measured is the quantity this essay actually compares — the shape with the width divided out — because no experiment normalises anything, and separating a shape from a scale is an operation performed on data rather than by an apparatus.

That is enough to keep the argument from being about an unobservable. The claim here is that two bands of one width can differ by a factor of two in what they bind, and the mechanism is where the levels sit. A metal whose states pile up at its Fermi level has a large heat capacity coefficient, a large susceptibility and a peak in its tunnelling spectrum, and those three facts are the shape being visible one energy at a time.

What is left #

The three structures compared here differ in dimension and in nothing else, which is what makes the comparison clean and also what makes it narrow. A real solid is not a hypercubic lattice: it has several orbitals per site, unequal hoppings, and a coordination that is not twice its dimension. Each of those changes the shape, and none of them is a change of dimension.

The other thing this leaves open is the relation between the shape and what a spectrum shows. A density of states is not a spectrum already separates the two — a count of levels at an energy is not a count of transitions to it, because the transitions have their own rules and their own strengths — and everything in this essay is on the count side of that line. The section above takes up whether the shape is visible in any measurement and gets three-quarters of an answer; what it cannot supply is a measurement of the normalised shape, since dividing a width out is something done to data rather than by an instrument.

The question that is now askable and was not before is whether the shape is a more useful summary of a band than the width — whether, given two structures, knowing where their states pile up predicts more than knowing how far apart their extreme levels are. On the evidence here it does: the width was held fixed by construction and the binding still varied by nearly a factor of two.