Origin of Curie temperature non-linearity in ferroelectric BiBO$_3$-PbTiO$_3$ solid solutions

We use first-principles density functional theory calculations to investigate compositional trends in ferroelectric BiBO$_3$-PbTiO$_3$ solid solutions. We find that cation displacements which give rise to ferroelectricity in these materials follow a linear relationship with the average displacive characteristics of the B-site. The transition temperatures in these materials are well correlated with polarization magnitudes computed from our calculations. The coupling between composition of the B-site and A- and B-site displacements explains the highly nonlinear and sometimes non-monotonic dependence of the Curie temperature ($T_c$) on composition observed in these materials.