Fig. 2: Thermodynamic calculation of switching free energy for BFO.

a. (e.) and b. (f.) show 109°, out-of-plane, and 71°, in-plane, switching-energy landscapes, respectively, calculated using Landau coefficients obtained from DFT (used in the phase-field model, Supporting Information Section 1) for the strain-clamped and membrane cases. c. (g.) and d. (h.) show 109° and 71° double-well potentials, respectively, calculated using the Landau potential from DFT (from the phase-field model) for strain + tilt clamped and membrane cases. To obtain the “clamped” (solid orange curves) results in panels a., b., e., and f., the in-plane strains are fixed, modeling the effect of strain clamping from the substrate. “Clamped (Weak)” (solid orange) curves in c. and d. (g. and h.) represent switching potentials derived from DFT parameters (phase-field parameters), but subject to “weak strain + tilt clamping” constraints, where all nonswitching polarization and tilt components are held fixed. “Clamped (Strong)” (dashed orange) curves in c. and d. (g. and h.) show switching potentials derived from DFT parameters (phase-field parameters), but subject to “strong strain + tilt clamping” constraints, where, all the nonswitching polarization and ferrodistortive components are held fixed and the switching components of tilts are linearly interpolated between the values corresponding to the minima of the free-energy curves. Percentages listed are reductions in maximum energy barrier for membrane vs. clamped films in each scenario. Calculations correspond experimentally to the thickest, fully relaxed, films. In the “strain clamping” case, the in-plane strains are fixed to their values in the initial state, before the 109° switch occurs. Hence, the fixed strains cannot adapt to the new polarization state after the 109° switch. The resulting state has a higher energy than the initial one, resulting in the asymmetric shape of the free-energy curves. This effect is more pronounced in the simulations using the phase-field model-parameter set (panels e.–h.) compared with the DFT (a.–d.) due to the relative magnitude of model parameters in the phase-field set being higher compared with the DFT set, which leads to the stronger-predicted clamping effect.