Fig. 5: Energy barriers from NEB simulations.

a Calculated total internal energy profile along the NEB paths connecting states with different numbers of bubble domains N_b. Differently colored curves correspond to different values of the applied bias field E_z. For each E_z value, ΔE corresponds to the difference between energies of the state on the transition path and the homogeneously polarized state (N_b = 0). Schematic representations of states corresponding to different N_b values are shown in subpanels below the graph. b The total internal energy variation along the transition paths between the single bubble states where initial (state A) and final bubble locations (state B) are separated by Δx unit cells along [001] p.c. direction. All curves are obtained at the bias field E_z = 50 × 10⁷ V/m. c Variation of the barrier for bubble displacement with the distance Δx at different bias magnitudes. In panels, a and b, ξ denotes the normalized distance along the NEB path with ξ = 0 and ξ = 1 corresponding to initial and final states, respectively.