Commensuration torques and lubricity in double moiré systems

We study the commensuration torques and layer sliding energetics of twisted trilayer graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) that have two contiguous superposed moir\'e interfaces. Lattice relaxations for typical graphene twist angles of $\ensuremath{\sim}{1}^{\ensuremath{\circ}}$ in t3G or t2G/BN are found to break the out-of-plane layer mirror symmetry, give rise to layer rotation energy local minima dips of the order of $\ensuremath{\sim}{10}^{\ensuremath{-}1}$ meV/atom at double moir\'e alignment angles, and have stacking energy minima of comparable magnitude between the next-nearest top-bottom layers. Thus, in t3G, the top and bottom layers tend to align when one twisted interface angle is fixed, whereas in t2G/BN, the alignment of the two moir\'e patterns favors t2G with $\ensuremath{\theta}\ensuremath{\simeq}1.{12}^{\ensuremath{\circ}}$ near the magic angle when the G/BN interface is rotated at $\ensuremath{\theta}\ensuremath{\simeq}0.{56}^{\ensuremath{\circ}}$. Precedence of rotation over sliding during the moir\'e commensuration is confirmed for periodic boundary systems where the sliding energy barriers drop to $\ensuremath{\sim}{10}^{\ensuremath{-}4}$ meV/atom for physical misalignment angles as small as $\ensuremath{\sim}0.{03}^{\ensuremath{\circ}}$. For finite graphene flakes of diameter $D,$ we find enhanced friction forces for a wider range of angles $\mathrm{\ensuremath{\Delta}}{\ensuremath{\theta}}_{\mathrm{FWHM}}\ensuremath{\sim}C/D$ both near the zero alignment angle in t2G and commensurate double moir\'e angles in t3G.