First-principles free energies and Ginzburg-Landau theory of domains and ferroelectric phase transitions in BaTiO<SUB>3</SUB>

We present a method based on combination of (a) constrained polarization molecular dynamics and (b) thermodynamic integration to determine the free-energy landscape relevant to structural phase transitions and related phenomena in ferroelectric materials, bridging the gap between first-principles calculations and phenomenological Landau-type theories. We illustrate it using a first-principles effective Hamiltonian of BaTiO3 to (a) uncover the quantitative features of the free-energy function that are responsible for its first-order ferroelectric transitions, (b) calculate the minimum free-energy pathways for the polarization switching and (c) evaluate temperature-dependent free energy of domain walls, and a minimum free-energy pathway to formation of ferroelectric domains. We use our method within a variational mean-field theory to connect with Landau theory and show through comparison with numerically exact simulations that (a) the state constrained to have vanishing order (away from the equilibrium) below the transition temperature is highly degenerate due to fluctuations that drive the phase transition first order, and (b) certain terms need to be added to the phenomenological Landau-Devonshire free-energy functions to capture the physics of spatial fluctuation in order parameter. Our method can be readily generalized to any classical microscopic Hamiltonian and ensembles characterized with a given constraint