Self-organized electronic superlattices in layered materials

We show that in layered systems with electronic phase separation tendency, the long-range Coulomb interaction can drive the spontaneous formation of unidirectional superlattices of electronic charge in a completely homogeneous crystalline background. In this self-organized electronic heterostructure, the ratio among the number of crystalline planes in the minority and majority electronic phases corresponds to Farey fractions with the superlattice period controlled by the background charge density and the frustrating Coulomb interaction strength. The phase diagram displays Arnold tongues obeying a modified Farey tree hierarchy and a devil's staircase, typical of systems with frustration among different scales. We further discuss the competition of these electronic superlattices, recently observed in iron-based superconductors and mixed valence compounds, with in-plane electronically modulated phases.