Electrostatics of superlattices by first principles

A complete theory of epitaxial perovskite superlattices requires an understanding of both epitaxial strain effects and of electrostatic boundary conditions. Here, focusing on the latter issue, we have carried out first-principles calculations of the nonlinear dielectric properties of short-period BaTiO$_3$/SrTiO$_3$ and PbTiO$_3$/SrTiO$_3$ superlattices having the in-plane lattice constant of SrTiO$_3$. In particular, we have mapped the energy $E$ vs.~polarization $P$ for these superlattices, and for the parent bulk materials, by extending the $E(P)$ mapping algorithm of Di\'eguez and Vanderbilt to handle strain relaxation in the $c$-direction. We then test whether the superlattice results can be understood quantitatively in terms of the $E(P)$ information for the bulk materials and similar information on the interfaces. We reformulate the theory in terms of the electric displacement field $D$ (since this is uniform throughout the superlattice) and the corresponding internal energy $U(D)$, and investigate how to model $U(D)$ in terms of ab-initio derived bulk and interface information. We stress that once the $U(D)$ is obtained for a given superlattice, all the electrostatic properties, both linear and non-linear, can be predicted.