Finite temperature properties of (Ba,Sr)TiO$_3$ disordered alloys and BaTiO$_3$/SrTiO$_3$ superlattices from first-principles

We develop and use a first-principles-based scheme to predict properties of (Ba,Sr)TiO$_3$-based systems at finite temperature. This scheme yields a composition-versus-temperature phase diagram of disordered (Ba$_{1-x}$Sr$_{x}$r)TiO$_3$ solid solutions that is in rather good agreement with experimental data. We further use this scheme to reveal and understand the strain-versus-temperature phase diagram of several BaTiO$_3$/SrTiO$_3$ superlattices. A wide variety of dipole patterns, including homogeneous ferroelectric phase and periodic stripe patterns, are predicted to occur depending on the interplay between temperature, strain and superlattice periods.