Commensuration torques and lubricity in double moire systems

We study the commensuration torques and layer sliding energetics of double moire systems exemplified through twisted trilayer graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN). Lattice relaxations show that the twist angles leading to commensurate moire patterns are energetically favored typically by a fraction of meV/atom when the t2G interface angle is around ~1°. For aligned moire patterns, we find sliding energy barriers of similar magnitude and minimization is achieved in alternating twist t3G when top-bottom layers are AA stacked and breaks mirror symmetry, while the favored geometries for t2G/BN have AA top-bottom layer local stacking for type I and BA stacking for type II. When the moire patterns are misaligned we restore moire superlubricity where the sliding energy landscape amplitude is suppressed by several orders of magnitude with energy barriers of the order of ~10^-4 meV/atom, indicating that aligning moire pattern angles through rotation precedes over sliding in the double moire commensuration process.