Rules governing Jahn-Teller order in Prussian blue analogues

Jahn-Teller distortions of transition-metal coordination environments link orbital occupancies to structure. In the solid state, such distortions can be strongly correlated through the propagation of strain and/or through orbital interactions. Cooperative Jahn-Teller (CJT) order of this kind affects the electronic, magnetic, and structural properties of the materials in which it occurs. Conventionally studied in dense ceramics, CJT order also occurs in hybrid materials, albeit the underlying phenomenology is not well established. Here we use synchrotron powder X-ray diffraction measurements to identify the compositional factors governing cooperative Jahn-Teller order in a series of Prussian blue analogue (PBA) families. We develop a simple microscopic model based on the dual considerations of strain and crystal-field stabilisation that rationalises the stability, extent of CJT order, and crystallite strain measured experimentally. This model shows how PBA compositions might be tuned to control the emergence and nature of CJT effects, and predicts universal phase behaviour for JT-active PBAs more generally. Our results establish a microscopic framework for understanding and controlling CJT effects in PBAs, and reveal an interplay between compositional, structural, and orbital degrees of freedom closely analogous to that of the manganite perovskites.