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.

Figure 1

(a) Sketch of the lowest period electronic superlattices with period $p+q$ along the perpendicular direction. They are labeled by the ratio $p/q$ between the phase minority crystal planes (light blue) and the phase majority crystal planes (red). (b) Schematics of the modified Farey tree construction. Each phase minority to phase majority ratio $\left(p/q\right)$ is generated by its ancestors $p^{\prime }/q^{\prime }$ and $p^{\prime \prime }/q^{\prime \prime }$, where the superscripts specify the originating yellow boxes, as $p/q=(p^{\prime }+p^{\prime \prime })/(q^{\prime }+q^{\prime \prime })$. The $0/1$ fraction corresponds to the homogeneous phase.

Figure 2

Phase diagram in the uniform density approximation for $a_3\le \sqrt{6}$. The colors encode the periodicity $p+q$ of superlattices. The $▴$ indicate the triple points separating a second-order transition line (above) from a first-order one (below). The strength of the Coulomb interaction has been measured in units of $Q_c=2/\sqrt{\pi a_3}$. The appearance of the larger periodicity tongues goes at the expense of the lower periodicity tongues, which acquire a funnel shape. We have restricted to structures with $p+q\le 11$. Allowing for arbitrary periodicities would shrink the lower part of the funnel to one point.

Figure 3

Phase diagram including in-plane electronic microemulsions for a layered system with layer spacing $a_3=3\xi$. The strength of the Coulomb interaction has been measured in units of the maximum coupling strength $Q_c=0.6623...$. The phase diagram displays two symmetric Lifshitz points (LP) and additional triple points (TP) at lower couplings.

Figure 4

Phase diagram for layer spacing $a_3=5\xi$. As before, the strength of the Coulomb interaction has been measured in units of the maximum coupling strength $Q_c=0.5999...$. The phase diagram displays triple points from in-plane unidirectional to triangular modulated structures (TP1) and additional triple points (TP2) marking the disappearance of in-plane charge modulations.