Multi-order and structural mechanism of atom nanocluster formation in rhombohedral Cu–Zr-thiospinels

A structural mechanism for the formation of nanoclusters and the rhombohedral modification of CuZr$_{1.86(1)}$S$_{4}$ has been proposed. Group-theoretical and crystal-chemical methods are used to study the features of the crystal structure and multi-order in Cu–Zr-thiospinel. A multi-order is a structural organization of a material, in the formation of which different degrees of freedom of a crystal participate. It has been proven that the rhombohedral structure of CuZr$_{1.86(1)}$S$_{4}$ is formed as a result of displacements and orderings of all types of atoms. It is theoretically shown that the formation of a rhombohedral phase is associated not only with the lattice, but also with the charge and, possibly, orbital degrees of freedom of the crystal. Theory predicts that the rhombohedral phase must be an improper ferroelastic. An important feature of the rhombohedral structure is metal nanoclusters: “bunch” of dimers, trimers, and tetrahedra. Bunch of dimers are an unusual type of self-organization of atoms in rhombohedral spinel-like structures, formed by atoms of two (tetrahedral and octahedral) frustrated crystal sublattices.