Wang-Landau Monte Carlo formalism applied to ferroelectrics

The Wang-Landau Monte Carlo algorithm is implemented within an effective Hamiltonian approach and applied to ${\mathrm{BaTiO}}_3$ bulk. The density of states obtained by this approach allows a highly accurate and straightforward calculation of various thermodynamic properties, including phase transition temperatures, as well as polarization, dielectric susceptibility, specific heat, and electrocaloric coefficient at any temperature. This approach yields rather smooth data even near phase transitions and provides direct access to entropy and free energy, which allow us to compute properties that are typically unaccessible by atomistic simulations. Examples of such latter properties are the nature (i.e., first order versus second order) of the phase transitions for different supercell sizes and the thermodynamic limit of the Curie temperature and latent heat.

Figure 1

Temperature behavior of the supercell average of the local mode (a) and average dielectric susceptibility (b) for a $16\times 16\times 16$ periodic ${\mathrm{BaTiO}}_3$ supercell, as predicted from the Wang-Landau method (crosses) and as computed using the Metropolis algorithm (squares).

Figure 2

Specific heat [(a) and (b)] and electrocaloric coefficient (c) versus temperature for periodic ${\mathrm{BaTiO}}_3$ supercells, as computed within the Wang-Landau algorithm. (a) shows the specific heat $C$ of a $16\times 16\times 16$ periodic ${\mathrm{BaTiO}}_3$ supercell for an interval of temperatures covering the three phase transitions, while (b) depicts $C$ close to the paraelectric-to-ferroelectric transition for $L\times L\times L$ supercells having different $L$ lateral sizes. The inset of (b) shows $C_L^{\max }$ (see text) as a function of $L^3$, with the linear fit being represented by a straight line. A $14\times 14\times 14$ supercell is used for the results depicted in (c). Error bars are smaller than the dimensions of the symbols.

Figure 3

Properties related to the nature of the paraelectric-to-ferroelectric transition of ${\mathrm{BaTiO}}_3$ bulks, as obtained from the Wang-Landau algorithm. (a) represents the free-energy-like quantity $A$ versus the internal energy for temperatures above, below, and at the transition point for a $16\times 16\times 16$ periodic ${\mathrm{BaTiO}}_3$ supercell. Panel (b) provides similar data but for a $10\times 10\times 10$ supercell. Panel (c) displays the the finite-volume latent heat plotted against the lattice size $L$, with the fit to $\mathrm{\Delta }e\left(L\right)=\mathrm{\Delta }e\left(\infty \right)+aexp(-bL)$ being shown via a dashed line.