Origins of superstructure ordering and incommensurability in stuffed CoSn-type phases

The CoSn structure type contains large interstitial void spaces that frequently host electropositive guest atoms, such as rare earth elements. In this stuffing process, an intriguing ordering occurs between the neighboring void spaces leading to a family of long-period superstructures comprising intergrowths of the ScFe₆Ge₆ and ScFe₆Ga ₆ structure types. This superstucture ordering culminates in incommensurability in the REFe₆Ge_6-δGa _δ systems with RE = Sc, Tb, and Lu. In this work, we derive a 3 + 1D superspace model encompassing this series of structures and investigate the origins of the structural trends in this family with electronic structure calculations, at both the LDA-DFT and extended Hückel levels. Using our 3 + 1D model, we refine the structures of four new ErFe₆Ge _6-δGa_δ (0 ≤ δ ≤ 6) phases, two commensurate and two incommensurate, from powder X-ray diffraction data. The refinement results confirm trends observed in the Sc-, Tb-, and Lu-based series: a gradual lengthening and, eventually, turning of the q-vector as Ge is progressively exchanged for Ga. These trends, and the incommensurate ordering as a whole, are traced to a tension between two modes by which the host lattice responds to stuffing atom insertion: (1) an atomic charge modulation enhancing the anionic character of the cavity walls around the guest atoms, and (2) a positional modulation expanding the cavities occupied by guest atoms. These two modes direct the stuffing atom ordering pattern toward opposite ends of the ScFe₆Ge₆-ScFe₆Ga₆ intergrowth series. The full series of structures, complex and incommensurate, reflects various degrees of balance between these two factors.