What a metal actually is

Ask what makes a metal a metal and four answers come back. It is shiny. It conducts electricity. It is malleable. It is an element from the left-hand side of the periodic table.

The first and third are consequences with several causes, the fourth is a chemical classification that predates any explanation, and the second is closest but still not it — a doped semiconductor conducts, and a metal at absolute zero with no field applied conducts nothing at all.

The property that actually separates a metal from an insulator can be stated without the word conduct appearing, and checked on a sequence of finite molecules.

A metal has excitations of arbitrarily small energy.

Why “arbitrarily small” and not “zero” #

The phrasing is careful and the care is the content.

No finite system has an excitation of zero energy — a finite matrix has finitely many eigenvalues and adjacent ones are, generically, different. So a metal cannot be defined as something with a zero-energy excitation, since nothing anybody can hold has one.

What a metal has is a sequence: take larger and larger pieces, and the cheapest excitation of each falls without limit. Beyond a certain size that cheapest excitation is smaller than the thermal energy available at any temperature the material survives, smaller than the energy an applied field supplies, smaller than the linewidth of anything used to look at it — and at that point the distinction between “very small” and “zero” stops having consequences.

An insulator’s sequence does the opposite. The cheapest excitation settles on a number and stays there however large the piece is, and that number is the band gap.

So the definition is about the behaviour of a family rather than the property of an object, which is uncomfortable and is also why it works. Every other candidate definition breaks on some case; this one does not, because it is stated at the level where the difference actually lives.

The measurement #

The check is direct. Take rings of increasing size, fill each with one electron per site, and find the distance between the highest occupied level and the lowest empty one.

For a ring of twelve, that distance is exactly one $\beta$. For twenty-four it is $0.518$; for forty-eight, $0.261$; for ninety-six, $0.131$; for a hundred and ninety-two, $0.065$. Every doubling at least halves it, which is the $1/n$ behaviour a partly filled band requires.

Nothing about that measurement mentions conduction, current, mobility or scattering. It is a property of a level ladder.

The insulating case, for contrast #

Fill the same ring with two electrons per site and the picture changes completely. Every level is doubly occupied and there is nothing above to excite into within the band at all. The cheapest excitation is across to whatever the next band is, which in this model is not represented and in a real material is a fixed energy set by the chemistry.

That fixed energy does not fall as the sample grows, because it is not a level spacing — it is the distance between two groups of levels, and adding sites adds levels inside each group without moving the groups apart.

The two behaviours are therefore distinguished by a question about a limit rather than by a question about a number. A gap of $0.065\beta$ that is still halving is a metal caught at a finite size; a gap of $0.065\beta$ that has stopped moving is an insulator with a small gap. The number alone does not say which.

What decides which case a material is in #

In this model, exactly one thing: how many electrons there are per site.

An odd number of electrons per repeating unit leaves a band half filled and gives the metallic sequence. An even number can fill bands completely, and whether it does depends on whether the bands are separated. That is the whole of the elementary rule, and it works remarkably often: the alkali metals have one valence electron per atom and are metals, and the noble gases and the covalent solids have closed shells and are not.

It also fails in ways worth knowing, and the failures are more instructive than the successes.

It fails when the counting is right and the answer is wrong. Beryllium has two valence electrons per atom and should by this argument be an insulator. It is a metal, because two of its bands overlap in energy so neither is full — a fact about where bands sit relative to each other, which needs more than a count.

It fails when the structure changes to avoid the metallic state. A half-filled one-dimensional chain does not stay half filled and metallic; it distorts, opens a gap, and becomes an insulator. That is not a failure of the counting but a failure of the assumption that the structure stays put, and it is the subject of a chain cannot stay even.

It fails when the electrons repel each other. This is the deepest of the three and the one this model cannot see at all. A half-filled band is not always a metal sets it out.

Where the chemist’s rules come from #

The counting rule above is the ancestor of several familiar chemical statements, and seeing them as one rule is worth the paragraph.

The octet rule is the closed-shell version. Eight electrons per centre fills the available valence orbitals, leaves nothing partly occupied, and produces a system whose cheapest excitation is finite. That is the same statement as “an insulator”, made about a molecule rather than about a solid, and hypervalency without d orbitals shows what happens when the counting is applied where its derivation does not reach.

Hückel’s 4n + 2 is the ring version. A ring whose π shell is exactly closed has a finite gap; one whose shell is open has a partly filled degenerate level and is unstable. Aromaticity as a computed shell closure derives it here rather than quoting it, and the derivation is the one above with a small $n$.

The 8 − N rule for the structures of the main-group elements is the extended version. An element with $N$ valence electrons forms $8 - N$ bonds to its neighbours, which is exactly the number needed to close the shell — selenium’s two, arsenic’s three, silicon’s four. The rule predicts chains for group six, sheets for group five, three-dimensional networks for group four, and it is right about all three. What it is predicting is the structure that turns a partly filled situation into a filled one, which is to say the structure that avoids being a metal.

Each of those is the same instinct: an open shell is unstable, and a system will change its structure or its bonding to close one. The metallic state is what is left when no available rearrangement manages it, which is why metals are common on the left of the table, where there are too few electrons for any amount of bond formation to fill a shell.

Why conduction follows, and why it is not the definition #

The chain from “arbitrarily cheap excitations” to “conducts electricity” is short and worth walking, because it explains why the consequence is a bad definition.

An electric field does work on electrons and can only do so by changing their state. If every available state costs a finite energy to reach, a weak field cannot supply it and nothing happens — the material polarises slightly and no current flows. If states are available at arbitrarily small cost, an arbitrarily weak field moves electrons into them, and a current flows.

So conduction follows from the definition. It does not imply it, for three reasons.

A doped semiconductor conducts well and has a large gap; what it has is a few carriers in a band that is nearly empty, which is a different situation entirely and behaves differently with temperature.

Conductivity is set by scattering, not by the electronic structure. A perfectly periodic metal with no defects and no vibrations would have infinite conductivity at any temperature. Real values are decided by how often an electron is scattered, which is chemistry and metallurgy rather than band theory.

And temperature reverses the trend. A metal conducts worse when heated, because there is more to scatter off. A semiconductor conducts better, because more carriers are excited across the gap. That opposite sign is the practical test used in a laboratory, and it is a test of the definition above rather than of conductivity itself.

The density of states at the Fermi level #

There is a second formulation of the same idea that connects it to almost everything else measurable, and it is worth stating because it is what most experiments actually report.

Excitations of arbitrarily small energy exist exactly when there are states arbitrarily close to the last occupied one — which is to say, when the density of states does not vanish at the Fermi level. Metals have a finite density of states there and insulators have zero.

That single number then controls a long list of properties, all for the same reason: each of them is a sum over states within about a thermal energy of the Fermi level, and how many such states there are is $g(E_{\mathrm{F}})$ times that energy. The electronic heat capacity is proportional to it. So is the Pauli magnetic susceptibility. So is the electronic thermal conductivity.

This is where the density of states is exactly the right instrument, and it is worth contrasting with a density of states is not a spectrum, where it is the wrong one. The difference is that these quantities are sums over states with no pairing and no matrix element, and a spectrum is not.

The smallest system with the same feature is cyclobutadiene, whose half-filled degenerate pair is what makes it unstable rather than metallic. A molecule that small escapes by distorting; a chain long enough that the escape costs more than it saves does not, and the boundary between those two is what separates a molecule from a metal.

The half-filled chain is where this field’s argument turns #

Everything above has treated the structure as given. That assumption is doing more work than it looks.

The metallic case in this model is the half-filled band, and the half-filled band is exactly the case in which a one-dimensional chain is unstable — the levels at the Fermi energy are the ones a distortion can split, and splitting them lowers the occupied ones and raises the empty ones. So the model’s own prediction is that its metallic state, in one dimension, does not survive.

That is not a defect in the argument. It is a real effect with a name and a great deal of experimental support, and it is why one-dimensional metals are rare and why polyacetylene has alternating bonds. A chain cannot stay even computes the balance.

In three dimensions the same instability exists and is far weaker, because a distortion that splits levels along one direction does nothing for the states travelling in others. That is one of the few places in this field where dimensionality changes an answer rather than an exponent — and, honestly, it is a statement a chain calculation cannot check, since a chain has only one direction.

The four wrong answers, revisited #

With the definition in hand the four opening answers can be given their due, because each of them is tracking something real.

Shiny is a statement about reflectivity across the whole visible range, and it follows from the definition fairly directly: a system with excitations at every small energy can absorb and re-emit light of any frequency below its plasma frequency, so it reflects rather than transmits. It fails as a definition because several non-metals are reflective for other reasons and because some metals are coloured — copper and gold absorb in the blue, from transitions between filled $d$ levels and the partly filled band above them, which is a band-structure fact rather than a metallic one.

Malleable is a statement about how a structure responds to shear, and it follows from the bonding being non-directional. Metallic bonding is delocalised over many neighbours, so sliding one plane past another does not break specific bonds. It fails as a definition because it is about the arrangement of atoms as much as about the electrons, and because several metals are brittle.

Conducts is the closest and is dealt with above.

On the left of the periodic table is an empirical classification that long predates any of this, and it works because the counting rule works: few valence electrons means no accessible way to close a shell. It fails at the boundaries, which is exactly where the interesting chemistry is — tin has a metallic form and a semiconducting one differing by a structural change, and the elements along the diagonal from boron to polonium are called metalloids precisely because the classification stops being decisive there.

Every one of the four is a consequence of a partly filled band in a one-electron picture. None of them is the property itself, and each fails on a different case, which is a reasonable working definition of a symptom.

What the definition corresponds to in a real material #

Defining a metal by a limit makes it sound like a property no experiment can address. It has a direct physical counterpart, and the counterpart is one of the most precisely measured objects in solid-state physics.

Excitations of arbitrarily small energy require occupied states arbitrarily close to empty ones, which means the boundary between the two must exist as a surface rather than as a gap. That surface — the set of states at the energy where the filling stops — is the Fermi surface, and it is what a metal has and an insulator does not.

It is measurable, and to remarkable accuracy. Placing a metal in a magnetic field and watching its magnetisation oscillate as the field is varied gives the cross-sectional areas of that surface directly, and the technique has been mapping the shapes of Fermi surfaces since the 1930s. An insulator produces no such oscillations, because there is no surface to have cross-sections.

That makes the definition operational for a single sample after all — not from its level ladder, which is the thing that cannot decide the question, but from a measurement that asks whether the boundary is a surface or a gap.

Bismuth is the case that shows the definition earning its keep at a boundary. Its carrier density is about five orders of magnitude below copper’s, its conductivity is correspondingly poor, and by the symptom definitions it is a marginal case that could be argued either way. It has a Fermi surface — a very small one, whose cross-sections were among the first ever measured — so by the Fermi-surface definition it is a metal, without qualification and without a threshold anybody had to choose.

That is the difference between a definition and a symptom stated once more, in a material rather than in a sequence. A conductivity has to be compared against a number somebody picked. A surface either exists or does not.

What the definition costs #

Defining a metal by a limiting property rather than by an observable has one real cost, and it should be stated plainly.

It makes the question undecidable for any single system. Given a molecule with a level ladder, no amount of examination of that ladder says whether the material it is a piece of is a metal, because the answer is about the sequence and one member of a sequence contains no information about its limit.

In practice that is exactly the difficulty faced by anyone computing clusters, and it is why the figures in this field show sequences. The claim that a half-filled ring is metallic rests on five rings rather than one, and the claim would not have been available from any of them alone.

The cost is worth paying because the alternative definitions each fail on a case somebody cares about, and because this one states the thing that all the consequences follow from. A definition that requires a sequence is awkward; a definition that quietly excludes doped semiconductors, or that makes a metal at absolute zero not a metal, is wrong.