Unfolding method for periodic twisted systems with commensurate Moiré patterns

: We present a general unfolding method for the electronic bands of systems with double-periodicity. Within density functional theory with atomic orbitals as basis-set, our method takes into account two symmetry operations of the primitive cell: a standard expansion and a single rotation, letting to elucidate the physical effects associated to the mutual interactions between systems with more than one periodicity. As a result, our unfolding method allows studying the electronic properties of vertically stacked two-dimensional homo- or heterostructures. We apply our method to study [Formula: see text] single-layer graphene, [Formula: see text] twisted single-layer graphene, and [Formula: see text] graphene- [Formula: see text] tungsten disulfide heterostructure with an interlayer angle of [Formula: see text]. Our unfolding method allows observing typical mini gaps reported in heterostructures, as well as other electronic deviations from pristine structures, impossible to distinguish without an unfolding method. We anticipate that this unfolding method can be useful to compare with experiments to elucidate the electronic properties of two-dimensional homo- or heterostructures, where the interlayer angle can be considered as an additional parameter.