The metal a thermometer cannot find

What a metal actually is takes some care over the definition, because the word is used loosely and the loose uses contradict one another. What it settles on is this: a metal is a system with excitations of arbitrarily small energy, and that is a statement about a sequence of finite systems rather than about any one of them. A uniform chain’s gap goes as one over its length and an alternating one’s settles on a constant; the first sequence is a metal and the second is not, and neither individual chain is either.

The definition is exact and it is also incomplete, because nothing in a laboratory is at zero temperature. As soon as there is one, the question is not whether there are excitations of arbitrarily small energy but whether there are excitations at an energy comparable with kT, and that is a different question with a different answer.

This essay takes both limits and finds that they do not commute.

What a carrier is here #

The measurement is deliberately the simplest one that means anything. Fill the levels with a Fermi–Dirac distribution at temperature kT, with the chemical potential found by bisection at fixed electron number, and count

$$N_{\text{carriers}} = 2\sum_i f_i (1 - f_i).$$

That is the number of electrons sitting in a state they would not occupy at zero temperature, counted once for each electron and once for each hole. It is not a conductivity — nothing here has a scattering time or a mass in it — and it is the quantity a conductivity is proportional to in any model where the carriers are what move.

Two systems are compared at every size: a uniform ring, whose gap is a level spacing and shrinks, and an alternating ring at δ = 0.15, whose gap tends to 4δ and does not. Those are the two sequences the words are defined with.

The two corners where the words mean nothing #

The cold, small corner. A ring of forty-two at kT = 0.002 carries 1.69 × 10⁻¹⁶ electrons per site whether it is uniform or alternating: the ratio is one, to the last bit a double holds. The uniform ring’s level spacing there is 0.29892, which is a hundred and fifty times the temperature, so nothing is excited across it either. Its spacing is a gap, and a gap is a gap whatever its origin.

That is the finding the definition needs and does not have. A finite piece of metal is not a metal at a low enough temperature, and the temperature it stops at is set by its size. Where a molecule stops being one asked the same question about the size alone, with the temperature held at nothing. There is nothing paradoxical in it — a nanoparticle with a level spacing above kT is a well-known object, and the end of a chain is where its states go when it is small enough for ends to matter — but it does mean that the sequence definition and the thermal one are not two ways of saying one thing.

The hot corner. At kT = 0.1 the ratio is 4.2 at forty-two sites and 4.4 at three hundred and twenty-two. The alternating ring’s gap of 0.6688 is only six times the temperature, so plenty is excited across it, and the two systems differ by a factor a measurement could easily miss. That is the ordinary statement that a small-gap semiconductor is metallic when it is hot enough, and it is here for completeness. A half-filled band is not always a metal made the opposite point about the same pair of systems at zero temperature.

And the middle. At three hundred and twenty-two sites and kT = 0.005 the ratio is 1.9 × 10¹². That is what the words are for.

The temperature at which it goes #

The obvious quantity is the temperature at which the distinction is half gone, and it can be found rather than estimated. Bisecting for the temperature at which the alternating ring carries half what the uniform one does, at three hundred and twenty-two sites, gives

$$kT^* = 0.15474$$

against a gap of 0.60124 — a quarter of the gap, which is the right order and is worth having as a number rather than as an order.

A quarter rather than a half or a whole because the gapless ring is not carrying an unlimited number either: both curves are rising, and the ratio between two rising exponentials crosses a half before the smaller one has caught up.

The size that answers at every temperature #

There is one case where the answer does not depend on the temperature at all, and it is a count rather than a structure.

A half-filled ring of 4m has a degenerate pair of levels exactly at the Fermi level. Each is occupied with probability one half at any temperature above zero, so each contributes 2 × ¼ to the carrier count, and the total is exactly one electron pair — independent of the temperature and independent of the size.

A half-filled ring of 4m + 2 has a closed shell and a gap that shrinks as one over its length. Its carrier count depends on both.

This is Hückel’s rule arriving in a place it is not usually invited. The 4m rings are the antiaromatic ones — an open shell, a degenerate pair half filled — and what an open shell means for a spectroscopist is that there is something at the Fermi level to excite, which is exactly what a metal is. So the same count that decides whether a ring is aromatic decides whether a finite ring is a conductor, and the two statements have the same content.

It also means the two families never converge. As n grows, a 4m + 2 ring’s gap goes to zero and its carrier count rises towards the 4m ring’s — but the 4m ring’s is pinned at one pair, so the ratio approaches one from below without ever being fixed at any finite size. The sequence definition is recovered only in the limit, and every member of the sequence disagrees with it.

What this does to the third way #

The third way to be an insulator finds a third mechanism — disorder, localising states without opening a gap anywhere — and it is worth asking what a temperature does to that one, because the answer is different in kind.

A disordered system has states at the Fermi level and they do not conduct, because they are localised. Warming it does not remove the localisation; it adds states further away in energy that are also localised. So the carrier count rises and the conductivity need not, and the quantity measured here stops being a proxy for the quantity of interest.

How far a state spreads as disorder is added is the measurement the carrier count is blind to: two systems with the same number of thermally excited electrons, one of them with states that reach across the sample and one with states on a handful of sites, are not the same material and are the same number.

That is a real limitation of this calculation rather than an aside. Everything above is a count of occupations, and an occupation is not a current.

The order of the limits, stated plainly #

Two limits are in play: the size going to infinity and the temperature going to zero. The results above say what happens in each order.

Size first, then temperature. Take the sequence to infinity at fixed temperature and the gapless system’s carriers per site tend to a positive constant while the gapped system’s tend to a smaller one. The ratio is finite, and the two are distinguishable at any temperature. This is the ordinary picture.

Temperature first, then size. Cool a fixed ring towards zero and both carrier counts go to nothing, for both systems, at every size. There is no metal in this order at all.

The two answers are different, so the limits do not commute, and the word metal has to be told which order is meant. In practice it means the first — a material is large before it is cold — and the definition given above is the correct one for that order. What is added here is that the second order is not a pathological corner: a small enough particle at a low enough temperature is genuinely in it, and the carrier counts here are what that looks like.

What a measurement of this would look like #

It is worth asking which of the numbers above a laboratory could reach, because the table is not a set of predictions and the difference matters.

The carrier count itself is not directly measurable. What is measurable is a conductivity, a Hall coefficient, an optical absorption edge, or a heat capacity — and the last of those is the closest relative of the quantity computed here, because an electronic heat capacity is a count of states within kT of the Fermi level, weighted by energy. The 4m ring’s pinned pair would show as a heat capacity that does not go to zero linearly, which is what a degenerate ground state does.

A thermal excitation reaches only the neighbourhood of the top of the occupied set, so everything above depends on the shape of the density of states there and not on the band as a whole. That is why the answer is a property of the filling rather than of the width.

The two sequences the words were defined by are a uniform ring’s gap falling as one over its length and an alternating one’s settling. The entire content of this essay is what happens when a horizontal line — a temperature — is drawn across those two curves.

A third kind of gap comes from a calculation with repulsion in it: the insulator band theory cannot see has a half-filled band and a gap that no one-electron model has a term for. A thermometer cannot tell that case from either of the two here, which widens the difficulty rather than resolving it.

The other thing a laboratory has that this does not is a length. A ring of three hundred and twenty-two sites at a hopping of a few electronvolts has a level spacing of tens of millielectronvolts, which is room temperature — so the sizes in the table are not a mathematical abstraction. They are the sizes at which the question is actually asked.

What this cannot say #

No conductivity. A carrier count is not a current. There is no mass, no scattering, no field, and a system with many carriers and no way to move them looks identical to a conductor here.

No repulsion. The insulator band theory cannot see is a system with a half-filled band and a gap, and the gap is made by an interaction this model does not have. The whole of the arithmetic here would apply to it unchanged given its levels, and getting its levels needs a different calculation.

One dimension, one filling. Half filling puts the Fermi level at the band centre for both systems, which is what makes the comparison clean and is not general.

And the temperatures are in units of the hopping. Turning kT = 0.15 into kelvin needs a value for the hopping, which this model does not have and which the essay declines to invent.

What was checked #

Both systems carry more as they warm, at every size, monotonically — the check that the occupations are occupations.

A uniform ring carries almost nothing at the coldest temperature, at every size: below 10⁻³ per site, against a level spacing that is a hundred times kT.

A larger uniform ring carries more per site at fixed temperature, because its spacing is shrinking — while the alternating ring’s carrier count is the same per site at every size to within a factor of two, because its gap is not.

The erasure temperature is of the order of the gap, found by bisection rather than declared, and between a twentieth and five times it.

The refusal is the cold corner: at the smallest size and lowest temperature the two systems must be within a factor of forty of one another, so the table contains its own counterexample.

And a half-filled ring of 4m carries exactly one pair at four temperatures spanning three orders of magnitude, while a ring of 4m + 2 four sites larger carries less than 10⁻⁹ at all of them.

The half of a conductivity not computed here #

A carrier count is one of two factors, and it is worth being explicit about the other, because for the material this whole field is named after the other one carries the entire temperature dependence.

A conductivity is a number of carriers times how freely each of them moves. The first is what is computed here. The second is a mobility, and it falls as a material is warmed, because a warmer lattice vibrates more and a vibrating lattice scatters electrons more often.

The two factors behave completely differently in the two kinds of material, and the difference is the criterion a laboratory actually uses.

In a metal the carrier count does not move. The states near the Fermi level are already there at every temperature and warming the sample changes their number by a negligible fraction. So the whole temperature dependence of the conductivity is the mobility’s, and the conductivity falls as the sample is warmed. Copper’s resistivity is roughly eight times larger at room temperature than at liquid-nitrogen temperature, and its carrier density is the same at both.

In a semiconductor the carrier count moves enormously. It rises exponentially with temperature — the count of thermally excited pairs across a gap, which is exactly what is computed here — and it does so far faster than the mobility falls. So the conductivity rises as the sample is warmed, by orders of magnitude over a few hundred kelvin.

That inversion is the operational test. A resistance that rises on warming is a metal; one that falls is a semiconductor or an insulator; and neither statement requires knowing a gap, a carrier density or a band structure.

It also says precisely where the arithmetic here applies and where it is silent. The carrier count is the whole story for a gapped material and none of the story for a metal. The essay’s alternating rings, which have a gap and carry nothing below a temperature of order that gap, are the case the count describes completely. Its uniform rings, which have no gap and whose carriers are present at every temperature, are the case where the count is nearly constant and the conductivity is decided entirely by a quantity not computed here.

Which sharpens the essay’s own difficulty rather than dissolving it. The two limits that do not commute are limits on a carrier count, and a carrier count is the right diagnostic for one of the two kinds of material and the wrong one for the other. A uniform ring’s count being small and slowly varying is not evidence that it conducts badly — for a metal, a count that barely varies is what conducting well looks like.

The missing factor is also the reason the disordered case is invisible here. Localised states can be present at every energy, in any number, and carry no current at all: their contribution to a count is the same as an extended state’s and their contribution to a mobility is zero. So a calculation of counts cannot distinguish the third kind of insulator from a metal, which is the blindness the disorder calculation measures from the other side.

One consolation is worth recording, because it says the calculation is not measuring nothing. The carrier count is the factor that varies by orders of magnitude and the mobility is the factor that varies by ones — a semiconductor’s carriers change by a millionfold over a few hundred kelvin while its mobility changes by a factor of a few. So for any material with a gap, the count is not merely the right factor, it is overwhelmingly the larger one, and the mobility can be wrong by a factor of two without touching the conclusion. It is only for the gapless case that the small factor becomes the whole answer.

Still open: a velocity on a ring, and the ring that distorts #

The obvious open question is the quantity the carrier count is a proxy for. A carrier count becomes a conductivity when it is multiplied by a mobility, and a mobility in a tight-binding band is a curvature — the second derivative of the energy with respect to the wavevector — which is not available without a lattice. What is available instead is a velocity that can be defined on a finite ring by threading a flux and differentiating, which is the same construction the ring-current calculation already uses. That would turn a count of excited electrons into a count of moving ones, and would separate the disordered case from the clean one, which is exactly where a carrier count is blind.

The nearer question is about the 4m rings. A degenerate pair at the Fermi level is not a stable situation: a chain cannot stay even for the same reason, and such a ring distorts. So the carrier count pinned at one pair is the carrier count of a structure that will not hold still, and computing the temperature at which the distortion is thermally undone — against the carrier count it would restore — is a competition between two effects measured separately and never together.