  Design of novel artificial materials based on ferroelectric perovskites
relies on the basic principles of electrostatic coupling and in-plane lattice
matching. These rules state that the out-of-plane component of the electric
displacement field and the in-plane components of the strain are preserved
across a layered superlattice, provided that certain growth conditions are
respected. Intense research is currently directed at optimizing materials
functionalities based on these guidelines, often with remarkable success. Such
principles, however, are of limited practical use unless one disposes of
reliable data on how a given material behaves under arbitrary electrical and
mechanical boundary conditions. Here we demonstrate, by focusing on the
prototypical ferroelectrics PbTiO3 and BiFeO3 as testcases, how such
information can be calculated from first principles in a systematic and
efficient way. In particular, we construct a series of two-dimensional maps
that describe the behavior of either compound (e.g. concerning the
ferroelectric polarization and antiferrodistortive instabilities) at any
conceivable choice of the in-plane lattice parameter, a, and out-of-plane
electric displacement, D. In addition to being of immediate practical
applicability to superlattice design, our results bring new insight into the
complex interplay of competing degrees of freedom in perovskite materials, and
reveal some notable instances where the behavior of these materials depart from
what naively is expected.
