A bigger cell, a smaller zone

The reflections a superlattice adds computes what ordering two kinds of atom onto a sublattice does to a diffraction pattern: exactly n − 1 new reflections per parent cell, at the wavevectors the ordering introduced, with intensities proportional to the difference between the two scattering factors rather than to their sum.

The same ordering does something to the levels. It is the same arithmetic, at the same wavevectors, and the two statements are worth putting side by side because neither is usually told with the other.

Folding is a relabelling and nothing else #

A state in a crystal is labelled by a wavevector, and the labelling is only defined modulo the reciprocal lattice — the reciprocal lattice is precisely the set of vectors that change nothing. Which wavevectors count as different is therefore a fact about the cell, and doubling the cell halves the zone.

The states do not care. Doubling the cell adds a reciprocal lattice vector Q — the one the ordering introduced — and now k and k + Q are the same label. Every wavevector of the reduced zone carries two of the parent’s states, and what was one band becomes two.

That is worth checking rather than asserting, and it is cheap: at every wavevector of a grid, the pair of energies the folded description gives must be exactly the pair the parent gives at k and at k + Q. They agree to four parts in ten to the sixteen, which is machine precision for a cosine.

The state count is exact and it is a counting argument. Adding Q to a wavevector is adding half the grid in both directions, which has no fixed point, so the grid pairs up exactly: 256 parent wavevectors fall into 128 classes of exactly two. Nothing has been created and nothing lost, which is what makes the description honest — a folding that produced an odd class somewhere would be a folding by the wrong vector.

What the ordering does #

Folding on its own leaves two bands touching at the new boundary. The ordering is what separates them.

The two states a reduced wavevector now labels differ by Q, so the matrix element that connects them is the Fourier component of the potential at Q — which is zero if the two kinds of site are identical and non-zero exactly when they are not. Write that component V and the problem at each wavevector is two-by-two: the two parent energies on the diagonal and V off it.

At the zone boundary the two parent energies are equal, so the matrix is a multiple of the identity plus V times an off-diagonal, and its eigenvalues are that energy plus and minus V.

The gap is 2V exactly, and that is not an approximation that happens to be good. It is degenerate perturbation theory in its simplest instance, and the agreement is to machine precision at every value tested.

The exactness is easiest to believe after watching the same picture at two more strengths, because what changes with V is not only the size of the gap but how much of the zone notices it. Turn the ordering down and the two bands separate at the boundary and nowhere else; the curves away from the boundary lie on top of the faint parent bands, and a measurement that did not look at the boundary would report no ordering at all.

Turn it up instead and the perturbation stops being one. The gap is still exactly twice the potential — that is what makes the statement a theorem about a two-by-two matrix rather than a limit — but the two bands are now visibly bent away from the parent’s along their whole length, because the coupling is no longer small compared with the difference of the two parent energies anywhere in the zone.

Three values are three values, so the claim is worth making over a range rather than at points chosen by hand. Solving the two-by-two matrix at the boundary for a ladder of potentials and comparing each gap against twice its own potential is a one-line test, and it comes out equal to the last digit at every rung — not close, equal, because the matrix at that wavevector is a multiple of the identity plus V times an off-diagonal and its eigenvalues are written down rather than approximated.

That component is not an abstraction either. It is the Fourier component of the potential at the wavevector the ordering introduced, and the ordering is a fact about which atom sits where.

One fact, two experiments #

The two consequences of ordering are the same computation read in two places, and the sameness is worth stating plainly because they are usually taught in different courses.

In diffraction, the ordering contributes a Fourier component to the scattering density at Q, and a Fourier component at Q is a reflection at Q. Its amplitude is the difference of the two scattering factors, which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons: the difference is what matters, not the sum.

In the levels, the ordering contributes a Fourier component to the potential at Q, and a Fourier component at Q couples states differing by Q. Its effect is a gap of twice that component at the zone boundary.

Same Q, same component, two observable consequences. A crystal whose superlattice reflections are strong has a large gap; one whose ordering is weak has faint reflections and a small gap; and a disordered crystal has neither. The intensity of a superlattice reflection goes as the square of the ordering, and so does the order parameter it measures — which is why the two are so often plotted on the same axes against temperature.

Why this is the mechanism behind so much #

The arrangement above is one of the most productive in solid state physics, and its productivity comes from one feature: a gap at the Fermi level lowers the energy of the occupied states.

If the electrons happen to fill the band exactly up to the wavevector where the gap opens, then every occupied state has been pushed down and every empty state pushed up, and the crystal has gained energy for free. That is an argument that a crystal should order — the ordering pays for itself — and it is the mechanism named after Peierls in one dimension and after Hume-Rothery in alloys.

The condition is a coincidence between two quite different things: the wavevector where the ordering opens a gap, which is geometry, and the wavevector at which the electrons stop filling, which is counting. Where they coincide the ordering is stabilised, and where they do not it costs energy like any other distortion.

This collection has the geometry half in several places — the zones above the first is where the boundaries are, and the star of a wavevector is how the group acts on them — and nothing at all about the electron count, which is physics rather than symmetry. Where the two meet is worth marking as a boundary of what this site computes.

The geometry underneath that is the zone itself. Doubling the cell replaces the first zone with one of half the area, and the states that lived in the outer half of the old one are relabelled into it — the same cut-and-move operation the zones above the first performs on a fixed lattice, done once rather than repeatedly and for a different reason. The last section of this essay returns to the difference, which is worth keeping straight: there the lattice does not change and the zones are repacked; here the lattice changes and the labels move once.

What the two bands are made of #

The two bands of the doubled cell are not two different kinds of state; they are two combinations of one kind, and which combination is which changes as the wavevector moves.

At a general wavevector inside the reduced zone the two parent energies are far apart, the off-diagonal coupling is small compared with their difference, and each folded band is almost entirely one parent state. The bands are the parent band, drawn in two pieces, with a small admixture.

At the boundary the two parent energies are equal, the coupling dominates however small it is, and the two folded states are the sum and difference of the parent states in equal measure. The lower one concentrates its density on the deeper of the two kinds of site and the upper one on the shallower, which is the physical content of the gap: the electrons in the lower band have arranged themselves to sit preferentially on one sublattice.

That transition — from almost purely one parent state to an equal mixture — happens over a range of wavevectors set by the ratio of the coupling to the band width, so a weak ordering mixes the states only very near the boundary and a strong one mixes them everywhere. It is the same crossover a coincidence the group did not ask for turns on: a degeneracy that symmetry does not force is lifted by any perturbation at all, and the size of the effect is set by how nearly degenerate the states were to begin with.

The same fold, in a real experiment #

Zone folding is not only a bookkeeping convenience; it is something a measurement sees, and what it sees is worth naming because it is the reason the word is used at all.

A measurement that maps the bands sees the folded ones. Angle-resolved photoemission on an ordered alloy finds bands in the reduced zone, with the extra branches that folding produced, and the branches that came from folding are systematically weaker than the original ones — because their intensity is proportional to the admixture, which is proportional to the ordering. So the folded copies are called shadow bands, and their strength is a measurement of the order parameter by a route with no diffraction in it at all.

A measurement of the vibrations sees the same thing. A phonon dispersion in a doubled cell has branches folded back in exactly this way, and neutron scattering finds them. The folded branches are the ones that were at the zone boundary of the parent and are now at the centre of the reduced zone, which is why an ordering transition makes new Raman-active modes appear: a mode at the boundary is invisible to light, and the same mode after folding sits at the centre where light can reach it.

That last consequence is the one this collection has the machinery for. What a group forbids to happen is the selection rule, and the reason folding changes what is visible is that light carries almost no momentum — so only a zone-centre mode can absorb it, and folding is what puts a mode there.

One fact, two powers #

The two experiments read the same Fourier component, and they do not read it with the same sensitivity. The difference is a single exponent and it decides which measurement finds an ordering first.

Let $\eta$ be the order parameter — how completely the two kinds of atom have sorted themselves onto the two sublattices, running from zero in the disordered crystal to one when the sorting is perfect. Both the scattering density and the potential acquire their component at $Q$ in proportion to $\eta$, since both are linear in how much of each atom sits where.

The gap is $2V$, so the gap is linear in $\eta$.

The superlattice reflection is an intensity, so it goes as $\eta^2$. A diffraction pattern measures $|F|^2$, and the structure factor at $Q$ is what is proportional to $\eta$.

So a weakly ordered crystal shows its ordering much more readily in the levels than on a film. At a tenth of full order the gap is a tenth of its final value and the superlattice reflection is a hundredth of its final intensity — the first is a shift a spectroscopy resolves comfortably, the second is a spot competing with background.

This is the ordinary situation just below an ordering transition, where $\eta$ rises continuously from zero. The transition is visible in transport and in the levels before the extra reflections can be seen at all, which is a recurring reason for disagreement about where a transition temperature actually is: two techniques answering the same question with two different powers of the same small number.

What folding is not #

Two confusions are worth heading off, because both are about the word fold being used for two different operations in this collection.

It is not the folding of the higher zones. The zones above the first cuts the second, third and seventh zones into pieces and translates each piece by a reciprocal lattice vector into the first — a repacking of one lattice’s zones, with the lattice unchanged. The folding here changes the lattice, and it happens once. The two operations share a word because both are translations by reciprocal lattice vectors, and they answer different questions.

And it is not a change in the crystal. Nothing in the first figure has been done to any atom. A crystal described in a doubled cell is the same crystal; the description merely has more bands and fewer wavevectors. That is the same freedom the cell is a choice, the lattice is not is about, met in reciprocal space, and it is why the folded description at V = 0 has to reproduce the parent exactly — a description that changed an answer would be a wrong description rather than an interesting one.

The physics enters only with V, which is a statement about the atoms rather than about the cell.

Where the exactness stops #

Computed here: a nearest-neighbour tight-binding band on the square lattice; its folding under a doubling with wavevector at the zone corner; the agreement between the folded energies and the parent’s, at 144 wavevectors, to machine precision; the exact pairing of a discrete parent zone into classes of two; and the gap at the new boundary at six ordering strengths, against twice the potential.

A model, and it says so. One orbital per site and one hopping integral is the simplest band a crystal can have. Every statement above is a statement about that model; what carries over to a real material is the structure of the argument — folding conserves states, the coupling is the Fourier component at Q, the gap at a degeneracy is twice the coupling — and not any number.

And the folding here is by one vector. Ordering into a cell n times as large folds by n − 1 vectors and gives n bands, with a matrix n by n at every wavevector. The doubling is the case where that matrix is small enough to solve in a line, which is why it is the one drawn.

Reading the folded picture back #

There is a practical question hidden in the first figure that is worth answering, because it is what makes folding a nuisance as well as a convenience.

Given a band structure drawn in a reduced zone, can the parent’s be recovered? Sometimes, and the sometimes is the interesting part.

When the ordering is weak the recovery is easy. Each folded band is nearly a piece of the parent band, the shadow branches are faint, and undoing the fold is a matter of shifting the weak pieces by Q and seeing that they join up. This is what an experimentalist does with a photoemission map of an ordered alloy, and the joining-up is the evidence that the folding interpretation was right.

When the ordering is strong it is not. The two folded bands mix completely across the whole zone, neither is a piece of anything, and there is no parent band to recover — the doubled cell is simply the cell, and calling it a doubling of something smaller is a historical statement about how the material was made rather than a fact about it.

Between the two is a continuum, and where a given material sits on it is a quantitative question the symmetry cannot answer. That is the usual boundary: symmetry says which description is available and the material decides which one is useful.

Who found it, and when #

The reduced-zone description is Brillouin’s, from 1930, and the folding is implicit in it: a zone is defined by a lattice, so a change of lattice is a change of zone.

The energetic argument is Rudolf Peierls’s, from the 1930s, in a form he described as too trivial to publish and which turned out to describe a whole class of transitions in low-dimensional materials. The alloy version is Hume-Rothery’s rule, which explains why certain alloy phases occur at certain electron-to-atom ratios and not others — a numerical coincidence between a zone boundary and a Fermi surface, stated in the 1920s from empirical regularities and explained by exactly this arithmetic afterwards.

The arithmetic in the reciprocal lattice #

It is worth writing the folding down as a statement about lattices rather than about pictures, because in that form it is the same sentence as the diffraction result.

The parent has a lattice L and a reciprocal lattice L*. Ordering onto a sublattice replaces L by a sublattice S of index n — this is thinning a lattice, and there are as many ways to do it as the index has divisors. The reciprocal of a sublattice is a superlattice: S* contains L* with index n.

So the new reciprocal lattice has n times as many points, of which n − 1 per parent cell are new. Those new points are the superlattice reflections. They are also the vectors by which the zone folds. They are the same set, and the two consequences of ordering are two readings of one containment of lattices:

• read as positions in reciprocal space, the new vectors are places where scattering can now occur;
• read as identifications of wavevectors, they are the labels that have become equivalent, which is folding.

The Brillouin zone of S is smaller than that of L by the factor n, because a zone’s area is the reciprocal cell’s area and S* is n times denser. Everything in this essay is that one fact with a physical interpretation attached at each end.

Where the ladder goes next #

Back, to the zones themselves: the zones above the first, which are the same first zone cut up and moved, and have exactly the same area for the same reason folding conserves states.

Sideways, to the diffraction half of this essay: the reflections a superlattice adds, where the same Fourier component is measured rather than deduced.

And to the case where the degeneracy is not accidental but forced: where two levels must meet is about the wavevectors at which a whole point group survives and a degeneracy cannot be lifted by any perturbation the crystal permits — which is exactly the opposite situation to the one here, where the touching was an artefact of the description and the ordering removed it.