In plain words
Whether two repeating patterns fit each other. If one pattern repeats after exactly a whole number of steps of the other – every three atoms, say – the two are commensurate, and together they repeat again after a short distance. If the ratio is not a simple fraction, they are incommensurate and never line up exactly. The question comes up whenever a crystal grows a second pattern of its own – a , a magnetic spiral – and whenever two different layers are stacked, as in a .
Going deeper
Fitting two periods
A crystal is a pattern repeated with one period. When a second pattern appears – a ripple in the electron density, a twist of the , a second crystal laid on top – its period can relate to the first in two ways. If it is a whole multiple or a simple fraction of it, the two coincide again after a short distance and together form a new, larger crystal: commensurate. If the ratio cannot be written as a fraction, the second pattern slides steadily against the first and every position along it is different: incommensurate. Diffraction tells them apart, because the extra spots of an incommensurate modulation sit at positions unrelated to those of the lattice.
Which one forms is a competition. The second pattern has a period it prefers for reasons of its own – the shape of the for a charge density wave, competing for a spin spiral – while locking onto the lattice lowers the energy further. Often the preferred period wins at high temperature and the lattice wins on cooling, in a lock-in transition.
Discommensurations and 1T-TaS2
Between the two lies a common compromise: domains in which the modulation is commensurate, separated by narrow walls – discommensurations – in which it catches up with its preferred period. -TaS2 shows the whole sequence. Its charge density wave is incommensurate below about 550 K, breaks into nearly commensurate domains a few nanometres across below about 350 K, and locks into a single commensurate Star-of-David pattern below about 180 K, where the crystal also turns . McMillan described such lock-in transitions with a Landau theory in 1975.
Thin change the sequence: the commensurate phase is suppressed as the flake gets thinner and can vanish altogether, and a gate or a current can switch between the phases – which is what makes 1T-TaS2 interesting for devices.
Two lattices: moiré and lock-in
Stacking two crystals poses the same question. The of graphene and hBN differ by under 2 percent, so when the two are aligned, graphene stretches locally to match hBN over large domains, separated by strained walls where the mismatch piles up – a commensurate state – while at larger the layers stay incommensurate and keep their own lattices. Twisted bilayers of one material likewise relax towards commensurate stacking, which reshapes the moiré pattern into triangular domains at very small angles.
Incommensurate contact has a useful side too. Two lattices that never line up cannot lock into each other, so the between them can almost vanish – superlubricity, seen between graphite flakes and at graphite–hBN interfaces that are rotated out of register.
For specialists
A modulation is commensurate when its wavevector is a rational fraction of a reciprocal lattice vector, so that crystal and modulation share a finite supercell, and incommensurate when the ratio is irrational, leaving no exact common period. The charge density wave of 1T-TaS2 passes from incommensurate (below about 550 K) to nearly commensurate – commensurate domains separated by discommensurations – and locks into the commensurate √13 × √13 Star-of-David phase below about 180 K; RTe3 keeps an incommensurate wave, and NiI2 an incommensurate spin spiral. For two lattices – graphene on hBN, twisted bilayers – the same competition between elastic energy and interlayer adhesion decides whether the layers stretch into commensurate domains or keep their own lattices.