a moire superstructure
/ mwah-RAY /
Lay two fine window screens on top of each other and rotate one by a hair: at once a bold pattern of light and dark bands sweeps across, far larger than the tiny mesh of either screen. That large pattern is a moire (pronounced mwah-RAY), and it appears whenever two regular patterns overlap slightly out of step. Stack two atomic sheets — say two layers of graphene — with a small twist between them, and their two honeycomb lattices beat against each other in exactly the same way, producing a moire superstructure: a new, much larger periodic pattern riding on top of the atomic lattices.
The geometry is precise. If each sheet has lattice spacing a and you twist one by a small angle theta (in radians), the moire pattern repeats over a period L that is roughly a divided by theta — far bigger than a, and it grows without limit as the twist shrinks toward zero. For graphene, with a about 0.246 nm, a twist of 1.1 degrees (about 0.019 radians) gives a moire period near 13 nm — dozens of atoms across. The moire is not made of any new atoms; it is a pattern of how well the two lattices happen to line up as you move across the sample, alternating between locally-matched and locally-offset regions.
This matters because that long-wavelength moire acts on the electrons like a gentle new periodic landscape, on top of the atomic one — and near special twist angles (the famous magic angle near 1.1 degrees in twisted bilayer graphene) it flattens the electron bands so dramatically that new collective states, including superconductivity, appear. The whole field of twistronics turns the twist angle into a structural design knob. Honestly, though, a generic twist gives an incommensurate structure that never exactly repeats; the moire is a real, visible near-periodicity, not always a true crystallographic lattice.
In a scanning tunnelling microscope image of two graphene sheets twisted by about 1 degree, the individual carbon atoms are almost too fine to see, but a bold triangular pattern of bright and dark patches stands out, repeating every 13 nm or so. That large pattern is the moire — dozens of atoms wide — and its size alone tells you the twist angle between the sheets.
Twist two lattices by theta and they beat into a moire of period L ~ a/theta — a huge superstructure from a tiny angle.
The moire is a beat pattern of how two lattices overlap, not a lattice of new atoms — the atoms never move to form it. And a moire is only a true periodic superlattice at special commensurate twist angles; at a generic angle the stacking is incommensurate and the pattern is only approximately periodic.