Solving the disordered structure of β-Cu_2-xSe using the three-dimensional difference pair distribution function

High-performing thermoelectric materials such as Zn₄Sb₃ and clathrates have atomic disorder as the root to their favorable properties. This makes it extremely difficult to understand and model their properties at a quantitative level, and thus effective structure-property relations are challenging to obtain. Cu_2-xSe is an intensely studied, cheap and non-toxic high performance thermoelectric, which exhibits highly peculiar transport properties, especially near the β-to-α phase transition around 400 K, which must be related to the detailed nature of the crystal structure. Attempts to solve the crystal structure of the low-temperature phase, β-Cu_2-xSe, have been unsuccessful since 1936. So far, all studies have assumed that β-Cu_2-xSe has a three-dimensional periodic structure, but here we show that the structure is ordered only in two dimensions while it is disordered in the third dimension. Using the three-dimensional difference pair distribution function (3D-ΔPDF) analysis method for diffuse single-crystal X-ray scattering, the structure of the ordered layer is solved and it is shown that there are two modes of stacking disorder present which give rise to an average structure with higher symmetry. The present approach allows for a direct solution of structures with disorder in some dimensions and order in others, and can be thought of as a generalization of the crystallographic Patterson method. The local and extended structure of a solid determines its properties and Cu_2-xSe represents an example of a high-performing thermoelectric material where the local atomic structure differs significantly from the average periodic structure observed from Bragg crystallography.