Design of novel artificial materials based on ferroelectric perovskitesrelies on the basic principles of electrostatic coupling and in-plane latticematching. These rules state that the out-of-plane component of the electricdisplacement field and the in-plane components of the strain are preservedacross a layered superlattice, provided that certain growth conditions arerespected. Intense research is currently directed at optimizing materialsfunctionalities based on these guidelines, often with remarkable success. Suchprinciples, however, are of limited practical use unless one disposes ofreliable data on how a given material behaves under arbitrary electrical andmechanical boundary conditions. Here we demonstrate, by focusing on theprototypical ferroelectrics PbTiO3 and BiFeO3 as testcases, how suchinformation can be calculated from first principles in a systematic andefficient way. In particular, we construct a series of two-dimensional mapsthat describe the behavior of either compound (e.g. concerning theferroelectric polarization and antiferrodistortive instabilities) at anyconceivable choice of the in-plane lattice parameter, a, and out-of-planeelectric displacement, D. In addition to being of immediate practicalapplicability to superlattice design, our results bring new insight into thecomplex interplay of competing degrees of freedom in perovskite materials, andreveal some notable instances where the behavior of these materials depart fromwhat naively is expected.
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