Ferroelectric to incommensurate fluctuations crossover in PbHfO3−PbSnO3

Perovskite solid solutions ${\mathrm{PbHfO}}_{3}\text{\ensuremath{-}}{\mathrm{PbSnO}}_{3}$ offer valuable opportunities for studying the formation mechanisms of incommensurate phases, owing to the presence of an intermediate (between cubic and incommensurate) phase, which is stabilized in ${\mathrm{PbHfO}}_{3}$ upon ${\mathrm{PbSnO}}_{3}$ admixture. Here, an x-ray diffuse scattering signal is used to quantify the evolution of susceptibilities related to different modes of distortion (ferroelectric, incommensurate, antiferrodistortive) as a function of temperature and the results are critically compared to the predictions of a minimal symmetry-based Landau-like model with two coupled order parameters (ferroelectric and antiferrodistortive) and an incommensurate order parameter being interpreted as inhomogeneous polarization. Experimentally, we observe a Curie-Weiss-like linear dependence of ferroelectric stiffness (inverse of susceptibility related to homogeneous polarization fluctuations) in the cubic phase down to about 50 K above the transition to the intermediate phase, where this dependence nearly saturates. Upon cooling down to the intermediate phase, the maximum of susceptibility shifts gradually to the nonzero wave vector, where another Curie-Weiss-like linear stiffness trend is established, but with respect to the incommensurate order parameter. Symmetry of diffuse scattering distributions indicates an orthorhombic symmetry of the intermediate phase. A notable temperature dependence of the constant that describes the energy of polarization inhomogeneities is observed experimentally, which is in disagreement with the model expectations. The specifics of this dependence suggest the presence of a nearly temperature-independent characteristic length scale for inhomogeneities across several phases. Other differences with the model suggest that the incommensurate order parameter cannot be straightforwardly identified with weakly inhomogeneous ferroelectric polarization.