A band with no structure in it

Every band in these essays is produced the same way: write down a matrix of neighbours, diagonalise it, look at where the eigenvalues fall. Nothing repeats, nothing is assumed to repeat, and no wave is written down.

That is a real method. It gives the band edges, the density of states, the exact relation between the second moment and the coordination, the gap, the filling, the defect levels and the surface states — every one of them checked against a computation rather than quoted.

It is also a method with a boundary, and the boundary has a name. This route produces a band and cannot produce a band structure.

The two words, and what separates them #

A band is a set of levels crowded into an interval. It is a fact about a spectrum: how many states there are at each energy, where the edges are, whether there is a gap.

A band structure is a function. It assigns an energy to each value of a wavevector — a label that says how a state’s phase advances from one repeat unit to the next — and it is drawn as a curve over a path through the space of those labels.

Everything the first has, a finite matrix computes. The second requires the label, and the label requires that there be repeat units between which phase can advance.

The distinction is not merely presentational. A very large amount of what is actually predicted about a semiconductor is a statement about the shape of those curves rather than about the set of energies: whether the top of one band lies above the bottom of another in the same place, how sharply a curve bends, which direction an electron moves easily along. None of those is a question about a count of levels, and none of them can be asked here.

The clearest casualty: direct and indirect gaps #

The sharpest example is the one with the largest practical consequence.

Silicon and gallium arsenide both have a band gap of a little over one electronvolt. Gallium arsenide is used to make light-emitting devices and silicon essentially is not, and the reason is not the size of the gap. It is that in gallium arsenide the highest occupied state and the lowest empty one carry the same wavevector, so an electron can cross between them by absorbing or emitting a photon, which carries almost no momentum. In silicon they carry different wavevectors, so the crossing additionally requires a lattice vibration to make up the difference — a three-body process, orders of magnitude less likely.

To a finite-matrix calculation those two materials have the same description: a gap of about one electronvolt with states below it and states above it. The distinction that decides whether a material can be made to emit light is not present in the data, because the data is a list of energies and the distinction is about labels the list does not carry.

Saying so plainly is more useful than working around it. A reader who has followed this field to here can compute a great deal and should know precisely what has been left on the other side of the line.

What periodicity actually buys #

It is worth being equally precise about the other side, because the usual account gets the causation backwards.

Periodicity does not cause bands. Amorphous silicon has no long-range order in any direction, and it has a band gap, a density of states with recognisable band edges, and semiconducting behaviour. Liquid metals conduct. A glass is transparent for the same reason its crystalline counterpart is. If periodicity were what produced bands, none of that would be true.

What periodicity buys is a labelling. When a structure repeats, there is an operation — translate by one repeat — that commutes with the Hamiltonian, so states can be chosen to be eigenstates of it as well. The eigenvalue of a translation is a phase factor, its exponent is the wavevector, and every state can then be sorted into a family indexed by it.

That is exactly the argument this site makes about point groups, one dimension of generality up. Degeneracy is a group theorem shows that a symmetry operation commuting with the Hamiltonian forces states into irreducible representations, and that the dimensions of those representations are the only degeneracies available. A translation group does the same job with an infinite group instead of a finite one, and the wavevector is its label the way $a_1$ and $t_2$ are a point group’s.

So the missing thing here is a quantum number, not a phenomenon. This site’s states are perfectly good states; they simply have not been sorted, because there is no operation to sort them by.

A test that separates the two, and can be run here #

There is a way to see the missing label without leaving the finite matrix, and it is worth doing because it makes the abstraction concrete.

Take a ring, which is the closest thing to a periodic system available here, and look at its eigenvectors rather than its eigenvalues. Each one has amplitudes that go around the ring as a sampled sine or cosine, and the number of times the sign changes on the way round is a perfectly good label — it counts the nodes, and it is what a wavevector reduces to for this system. Levels come in degenerate pairs precisely because a given node count can be traversed either way round.

So a ring does have the label after all, and this is not a contradiction: a ring is periodic. What it demonstrates is the shape of what is missing elsewhere. Do the same on a chain and the eigenvectors are standing waves rather than travelling ones — each is a superposition of the two directions, forced by the ends, and no single one of them carries a direction at all. Put one atom out of place anywhere in the chain and even the node count stops being a clean label, because the states mix.

The lesson is that the label is a property of the symmetry rather than of the physics, and it disappears the moment the symmetry does. Every real solid has defects, surfaces and thermal disorder, so the label is always approximate; it is an extremely good approximation in a good crystal and a poor one in a glass, and the useful question is never whether it exists but how far it can be pushed.

Why the line is drawn where it is #

There is a second reason for the boundary beyond what the method can reach, and it belongs to the collection rather than to the physics.

This site’s subject is molecular shape and bonding. The neighbouring subject — lattices, space groups, reciprocal space, diffraction, and the classification of periodic structures — is a large subject in its own right, and treating it here in passing would be treating it badly. The field’s whole design is therefore to take the molecular argument as far as it goes and stop where the periodic one begins, rather than to produce a thin second account of both.

The line is easy to state: anything that requires the structure to repeat is not claimed here. Wavevectors, Brillouin zones, structure factors, systematic absences, the classification of the periodic groups. What is claimed is what a large molecule does, which is a great deal, and every essay in the field is a statement about a finite system.

An infinite group worked in a finite one is the same move made earlier and in a smaller setting: a linear molecule’s group is infinite, its character table cannot be generated by closing operations under multiplication, and the analysis is done in a finite subgroup instead — with the resulting loss recorded rather than hidden.

The size at which a cluster stops being a cluster #

A related question is worth answering with a number, because it is the one a reader who wants to use this route will actually face: how large does a finite system have to be before its properties are those of the extended one?

The answer is not a single number, because different quantities converge at different rates, and the field has measured two of them.

The energy per site of a chain approaches its limit as one over the length, and the difference between a ring and a chain of the same length settles at about $0.71\beta$ divided by the length. Reaching a thousandth of a $\beta$ per site therefore takes something like seven hundred atoms.

The band edges converge much faster, as one over the square of the length. A chain of forty already has its top level within a quarter of a per cent of the limit, and a chain of two hundred is within a part in ten thousand.

The density of states converges slowest of all near the edges, where the true curve diverges and no finite system can, which is why the check on it in this site’s library compares the interior and excludes the outer three-tenths of a $\beta$.

That spread — a factor of several hundred between the fastest and slowest quantity — is the practical content of the boundary, more so than any statement about wavevectors. It also explains an old frustration in cluster calculations, where the total energy converges smoothly and the gap oscillates alarmingly with size: those are different quantities with different convergence, not a computation misbehaving.

There is an exact identity available on this route that is worth naming, because it holds on structures no band-structure calculation can be run on at all: the mean coordination counted off the bond list equals the second moment of the spectrum, to arithmetic precision, whatever the structure is. A quantity that needs no periodicity is a quantity a disordered solid still has.

What survives the restriction, and it is most of the field #

It is worth listing what does not depend on the missing label, because it is more than a reader might expect.

Thermodynamic quantities. Anything that is a sum over all states with no pairing — total energy, heat capacity, the number of electrons below an energy, magnetic susceptibility in the simplest cases. All of these are integrals over the density of states, which is exactly what this route produces.

Whether there is a gap, and how big. A gap is the distance between the top of one set of levels and the bottom of the next, and both are computable from a spectrum. What is not computable is where in the zone each of them sits.

The exact identities. The second moment is the mean coordination, at every size and for every structure, as the width of a band is a count of neighbours works through. That result does not merely survive the absence of periodicity — it is cleaner without it, since it holds for structures a band-structure calculation cannot be performed on at all.

Everything about defects and surfaces. This is the part where the finite-matrix route is straightforwardly better. A defect breaks periodicity, so a band-structure calculation has to either give it up or restore it artificially by repeating the defect in a large cell. Here a defect is one changed entry in a matrix, and the state it binds is computed directly. A defect is a level in the gap and the end is the hardest place to bind are both natural in this framework and awkward in the other.

And the whole of the field’s chemistry. The questions a chemist arrives with — why this arrangement rather than that one, what an extra electron does, why one material is coloured and another is not — are questions about occupation and about local structure, and neither needs the label. That is not a coincidence: the reason the tight-binding picture became a chemist’s tool while the wave picture became a physicist’s is that they answer the questions each discipline asks.

The honest summary of the model, not just the method #

Two limitations run together in this field and it is worth separating them, because only one of them is about periodicity.

The first is the missing label, which is what this essay has been about. It costs the dispersion, the direct–indirect distinction, and the effective masses.

The second is the one-electron approximation, and it is older and more serious. Every calculation in this field puts each electron in a fixed average potential and lets it ignore the others. Orbitals are not where the electron is makes the general case; here the specific failure is sharp enough to have its own essay. A one-electron calculation predicts that a half-filled band conducts, and there are materials with a half-filled band that are excellent insulators because the electrons repel one another strongly enough to sit still. A half-filled band is not always a metal is where that is set out.

Restoring periodicity would fix neither of those. A band-structure calculation at the same level of approximation gets the correlated insulator wrong in exactly the same way, and for the same reason.

The materials for which this route is the right one #

The boundary has been drawn as a restriction, which is the honest way round for a site whose subject is molecules. It is worth turning it over once, because there is a large class of real solids for which the missing label is not missing from the calculation — it is missing from the material.

A wavevector is a label supplied by translational symmetry. A glass has none. Neither does an amorphous semiconductor, a metallic glass, a polymer, or any of the disordered solids that make up a great deal of what is actually manufactured. For those materials $k$ is not a poor approximation or an expensive one; it is not a quantum number at all, and a band-structure calculation cannot be performed on them even in principle. What can be performed is exactly what this field does: write down who is bonded to whom, diagonalise, and read the levels.

So the finite-matrix route is not merely a cheaper approach to a crystal. It is the correct framework for a solid with no periodicity, and the crystal is the special case where a better tool exists.

The clearest evidence that this is physics rather than bookkeeping is what happens to the direct–indirect distinction, which this essay names as the route’s clearest casualty. Crystalline silicon has an indirect gap, so absorbing a photon near the gap needs a lattice vibration to supply the missing momentum, the process is weak, and a solar cell made of it has to be a hundred micrometres or more thick to catch the light. Amorphous silicon has the same atoms and roughly the same local coordination, and no momentum to conserve. Its absorption in the visible is one to two orders of magnitude stronger, and a cell made of it works at a thickness of well under a micrometre.

That is the missing label being genuinely absent from the material, with a consequence measured in metres of deposited film. A calculation with no wavevector in it gets the amorphous case right for the same reason it gets the crystal wrong.

The restriction therefore has a shape rather than only a size. What is lost is every question that begins at which point in the Brillouin zone, which is most of what is asked about a crystalline semiconductor. What is not lost is anything about a solid that has no zone to have a point in — and the count of such materials, weighted by how much of them gets made, is not small.

What a boundary is worth #

A field that stated no limits would be more comfortable to read and less useful. The reason to write this essay in the middle of the field rather than at the end is that everything after it inherits the boundary, and a reader who knows where it falls can use the rest with confidence instead of with suspicion.

The claim this field makes is narrow and, within its width, checked: a large molecule has a band, the band’s scale is a count of its neighbours, its filling decides whether it conducts, its ends and defects carry states that can be computed and localised, and an evenly spaced chain of it is not stable. Every one of those is arithmetic on a matrix that is written down, and every one is checked against the matrix it comes from.

What it does not claim is anything requiring a direction in a crystal. That is not a gap to be apologised for; it is the shape of the argument, chosen at the start and stated here in the middle so that nothing later has to pretend otherwise.