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Molecular Dynamics Simulations of Chemically Disordered Ferroelectric (Ba,Sr)TiO$_3$ with a Semi-Empirical Effective Hamiltonian

We present a semi-empirical effective Hamiltonian to capture effects of disorder associated with Ba and Sr cations occupying $A$ sites in (Ba$_{x}$Sr$_{1-x}$)TiO$_3$ on its ferroelectric phase transition. Averaging between the parameters of first-principles effective Hamiltonians of end members BaTiO$_3$ and SrTiO$_3$, we include a term with an empirical parameter to capture the local polarization and strains arising from the difference between ionic radii of Ba and Sr. Using mixed-space molecular dynamics of the effective Hamiltonian, we determine $T$-dependent ferroelectric phase transitions in (Ba$_{x}$Sr$_{1-x}$)TiO$_3$ which are in good agreement with experiment. Our scheme of determination of semi-empirical parameters in effective Hamiltonian should be applicable to other perovskite-type ferroelectric solid solutions.