A beat is not a period

Put a sheet of graph paper on another and turn it a few degrees. What appears is not two grids; it is a third, much coarser pattern of bright patches where the lines nearly coincide and dark ones where they do not. It has a spacing anyone can measure with a ruler, and the spacing gets larger as the twist gets smaller.

That pattern is a moiré, and this essay is about two questions it invites and one answer that is usually given to both.

The first question is: what is the spacing? It has a short answer, and the answer is exact.

The second is: does the superposition repeat? That answer is not short, is usually no, and when it is yes the period is often not the spacing.

The beat, in one line of algebra #

Work in reciprocal space, where a lattice is a set of vectors and a superposition is a union of two of them. One layer contributes g; the other, turned by R, contributes Rg. What beats — what appears at a coarse scale — is the difference of two nearby vectors, g − Rg = (I − R)g.

So the moiré reciprocal lattice is the original one acted on by I − R. For a rotation through θ, I − R is itself a rotation composed with a scaling by 2 sin(θ/2), so the moiré direct lattice is the original scaled by 1/(2 sin(θ/2)) and turned through a right angle plus half the twist. Its period is

λ = a / (2 sin(θ/2))

which is the formula quoted everywhere, and which is checked here against the construction rather than taken on trust: the module builds the basis from (I − R)⁻¹ and requires its vectors to have exactly that length.

Nothing in that derivation asked whether the superposition is periodic. It asked what happens to two sets of plane waves when one is rotated, and the answer is a beat, and a beat is defined whether or not anything repeats.

When the superposition repeats #

The union of two lattices is periodic exactly when they share a sublattice of finite index — when L ∩ RL is a lattice rather than the origin alone. That is the coincidence site lattice condition, which this collection builds elsewhere for grain boundaries, and it is the same condition here for a reason that is not a coincidence: a twist boundary between two grains is two lattices at an angle, and a moiré is what a transmission electron microscope sees when it looks through one.

The condition has a test that needs no formula. R is a coincidence rotation exactly when it carries some lattice vector onto another lattice vector — necessarily of the same length, since a rotation preserves length. So the enumeration here is over pairs of primitive lattice vectors of equal norm: each pair gives an angle, the rotation is written in the lattice’s own basis where it must come out as an integer matrix over the vector’s norm, and the index Σ is then counted as the size of a kernel rather than divided out of a formula.

The angles come out as the ones a reader of the twisted-bilayer literature will recognise — 21.7868° at Σ = 7, 13.1736° at Σ = 19, 9.43° at Σ = 37 — and they are produced rather than looked up, which is the whole habit of this collection.

The two lengths, and when they differ #

Now the point. At a commensurate twist there are two lengths in the picture:

• λ = a / (2 sin(θ/2)), the beat, which is what a ruler on the photograph measures;
• a√Σ, the edge of the coincidence cell, which is the actual period of the structure.

At 21.7868° they are equal: √7 = 2.6458 and λ = 2.6458. At 27.7958°, where Σ = 13, they are not: the cell edge is 3.606 and the beat is 2.082, a ratio of exactly √3. The pattern a reader sees repeats three times inside the true period, and the three copies are not identical — they differ in a way the coarse pattern does not show, because the coarse pattern is a beat between the two lattices and not a picture of the structure.

The ratio is never less than one, and that is not an accident either. The coincidence lattice is where the two layers agree exactly; the moiré lattice is where they agree at a coarse scale. Exact agreement is a stronger condition, so the coincidence lattice is a sublattice of the moiré lattice, and its cell is a whole number of moiré cells.

A square net never shows its own period. That is a fact about the square lattice rather than about measurement, and the arithmetic behind it is short enough to give.

The ratio squared is a whole number at every commensurate twist, on both lattices — which is the statement that the coincidence cell holds a whole number of moiré cells, and is checked rather than argued. Write it out: the ratio squared is Σ · 2(1 − cos θ), and for the square lattice with a coincidence rotation from the pair (m, k) that comes to 4k² divided by the power of two dividing m² + k². Since m and k are coprime and cannot both be odd — a coincidence index in the plane is always odd is the same observation — that power of two is at most the first, so the ratio squared is 4k² or 2k². Both are even, and one is not.

Every value the search produces bears that out: on the square lattice the ratios squared are 2, 4, 16, 18, 36, 50, …, all even. On the triangular lattice they are 1, 3, 4, 12, 16, 25, 27, …, and the ones there are the twists at which the eye and the structure agree.

The same formula, from the other direction #

There is a second place in this collection where a / (2 sin(θ/2)) appears, and it is worth putting the two side by side because they look nothing alike.

A small angle is a row of dislocations computes the spacing of the dislocations in a low-angle grain boundary. Two grains meeting at a small misorientation do not meet along a continuous mismatch; the mismatch is concentrated into a periodic array of dislocations, with good crystal between them, and Frank’s formula gives their spacing as the Burgers vector over twice the sine of half the angle.

That is the same expression. It is not a resemblance: the dislocation spacing and the moiré period are the same length measured two ways. Where the two lattices nearly agree, the boundary is good crystal and the moiré is bright; where the accumulated mismatch reaches a whole lattice vector, the boundary must insert a dislocation and the moiré is dark. The dislocations sit at the nodes of the beat because the beat is exactly the accounting of how much mismatch has accumulated.

That identity is also why the second half of this essay matters to a materials scientist rather than only to an arithmetician. A dislocation network is a real object with an energy, and its period is the beat. The structure’s period — the coincidence cell — is a different and usually larger length, and it is the one that decides whether the boundary is a special boundary in the sense that governs its energy and its mobility. A boundary can have a fine dislocation network and a coarse structural period, and the two are read from the same picture only if the reader knows which is which.

Why reciprocal space is the right place to do it #

The derivation above took four lines and never drew a lattice. It is worth saying why that is not an accident, because the direct-space version of the same argument is a mess.

In direct space, “where do the two lattices nearly coincide” is a statement about the distance from a point of one lattice to the nearest point of the other, which is a function with no simple form: it is small in patches, the patches have soft edges, and where exactly a patch begins depends on how near is near enough. Anything computed from it inherits that arbitrariness.

In reciprocal space the same question is a subtraction. Two sets of plane waves, one turned; the coarse structure of their sum is governed by the differences of nearby wavevectors; the differences form a lattice, and a lattice has a period with no tolerance in it. The beat is exactly defined even though “nearly coincide” is not, and that is the whole reason the formula is clean.

The same trade appears throughout this collection. The reciprocal lattice turns a question about spacings into a question about vectors; a zone is a vanishing dot product turns a question about which faces share an edge into an equation with no metric in it. Going to reciprocal space is what converts a statement about approximation into a statement about arithmetic, and the moiré is a particularly clean case because the object it produces — a lattice of differences — is what a reader was looking at all along.

What a picture cannot tell #

The practical version of all this is a warning about instruments, and it is worth stating in the terms an experimenter would use.

A moiré measures the twist and does not measure the period. From λ and the known lattice constant, θ follows immediately and accurately — that is why moiré fringes are a good angle gauge, and why they are used as one at interfaces and in twisted stacks. What does not follow is whether the structure repeats at all, or on what cell if it does.

That last row is the trap. A five-degree twist has a beat of eleven and a half lattice constants and shares no vector with its partner, and the closest any vector comes to landing on one is a fiftieth of a cell. In a micrograph a few hundred atoms across, that is indistinguishable from an exact coincidence. The picture cannot tell the difference and no better picture can, because the difference is not about resolution — it is about whether an infinite pattern closes, which no finite patch decides.

This is the same distinction a translation that is nearly there draws for pseudosymmetry and near-symmetry and the tolerance draws for a structure that is almost more symmetric than it is. Nearly periodic is not periodic, the gap between them is not a matter of degree, and every instrument that measures a finite region is on the wrong side of it.

The angle everybody talks about #

The twist that made this subject famous is about 1.05°, in a stack of two graphene sheets, and it is worth walking through it because it is a clean case of what the arithmetic here does and does not decide.

The beat is immediate. At 1.05° the moiré period is 54.6 lattice constants; graphene’s is 0.246 nanometres, so the pattern is 13.4 nanometres across — a factor of fifty-five, and large enough that a scanning probe resolves it comfortably. That number is the formula and nothing else, and it is right.

Whether the stack is commensurate at that angle is a separate question with a discouraging answer. Sorting the commensurate twists by size, the small ones come from large Σ: 9.43° at Σ = 37, and going lower means going up the index, fast. Below about two degrees the search inside any reasonable bound finds nothing at all — not because such twists do not exist, but because their coincidence cells contain thousands of atoms, and a cell with thousands of atoms is a period in the same sense that a very long word is a sentence.

So the interesting angle is not commensurate in any useful sense, and the interesting property is not one symmetry decides. What makes 1.05° special is a statement about electrons: the bands flatten there, the kinetic energy nearly vanishes, and interactions that would be invisible take over. That is a calculation about a Hamiltonian, and no argument in this collection produces it or could.

Symmetry supplies the stage and not the play. It says what the moiré period is at every angle, exactly, and it says which angles admit a period at all. It does not say which of them a material prefers, which of them does something interesting, or how large any effect is — the same boundary what the count does not say draws for every count in this collection, arriving here in a subject where the temptation to cross it is unusually strong, because the moiré is so visible and the number is so easy.

There is one thing the arithmetic does contribute to the physics, and it is worth ending on. A calculation that assumes a periodic supercell has assumed something false at almost every twist. Modelling a twisted stack normally requires a repeating cell, so the angle is moved to the nearest commensurate one and the cell taken there — and the results are then about that angle rather than the one asked about. The move is usually small and usually harmless. Knowing that it is a move, and how far, is what the enumeration above is for.

What this does not say #

It says nothing about energies. Which twists a real stack prefers is a question about how the two layers interact — whether they relax, whether the commensurate patches grow at the expense of the rest, whether the structure locks to a nearby commensurate angle. Those are the questions the physics of twisted stacks is about, and they need forces this collection does not compute. What is established here is which angles are available to lock to, which is the arithmetic half of the question and the half a symmetry argument can answer.

And it is two layers of one lattice. Two different lattices at an angle — the epitaxy problem, where the lattice constants differ as well — is a richer question. Epitaxy is a measurement takes it up, and the coincidence condition there involves the ratio of the constants as well as the angle, so the set of commensurate configurations is larger and messier. The equal-lattice case is the one where the arithmetic is clean enough to enumerate.

What the coincidence cell is for #

It is worth saying why the larger of the two lengths is the one a materials scientist wants, since the smaller is the one the picture shows.

A boundary between two grains is described by the coincidence lattice rather than by the beat, because the coincidence lattice is where the two crystals genuinely share atoms. The fraction of shared sites is one in Σ, so a Σ = 3 boundary — the coherent twin — has a third of its sites in common with both grains and is correspondingly cheap; a Σ = 37 boundary shares one site in thirty-seven and is not special in any useful sense. That ordering by Σ is the whole of why the coincidence index is the number quoted about a boundary, and it is an ordering the beat knows nothing about: two twists with wildly different Σ can have almost the same moiré period, and two with the same Σ can have periods differing by a factor of two.

The lattice of displacements that leave the coincidence lattice alone — the DSC lattice, which the dislocations a boundary allows computes — is finer than either, and it is what fixes the Burgers vectors a boundary dislocation may have. So a twist boundary carries three lattices at once: the beat, which the microscope shows; the coincidence lattice, which decides how special the boundary is; and the DSC lattice, which decides what can move along it. They are nested, they are computed from the same rotation, and only the first is visible.

The one number worth carrying #

A moiré is an amplifier. The beat has period a / (2 sin(θ/2)), so a tenth of a degree of twist on a lattice constant of a quarter of a nanometre gives a pattern a seventh of a micrometre across — a factor of six hundred, from an angle to something an optical microscope can nearly see.

That amplification is why the technique is useful and why it misleads. It magnifies the angle faithfully. It magnifies nothing else: the period of the structure, the size of its cell, whether it has one — all of those are on the other side of the arithmetic, and a bigger picture of the beat contains no more information about them than a small one does.