Short-circuit boundary conditions in ferroelectric PbTiO${}_3$ thin films

We examine the application of short-circuit electrical boundary conditions in density functional theory calculations of ferroelectric thin films. Modeling PbTiO${}_3$ films with Pt electrodes, we demonstrate that under periodic boundary conditions of supercells, short-circuit conditions for the electrodes are equivalently satisfied in two repeated slab geometries: a PbTiO${}_3$/metal superlattice geometry and a periodic metal/PbTiO${}_3$/metal/vacuum geometry, where the metal is Pt or SrRuO${}_3$. We discuss the benefits of each geometry in the study of ferroelectricity in thin films.