fig5

Figure 5. (A) Free energies of the pristine c-domain (ΔF_c (V); black line) and newly formed a-domain (ΔF_a (V) + ΔF_w; red line) as a function of applied voltages. Coercive voltages for c- to a- (V_c→a) and c- to c- (V_c→c) domain switching and their corresponding free energies ΔF_c→a (orange dashed line) and ΔF_c→c (green dashed line) are indicated. Energy required for formation of a second a-domain (ΔF_c→2a); brown arrow) is also shown; (B) Elastic energy ($$ \Delta U_{w}^{e l}\left(\varepsilon_{\mathrm{a}}\right)+U_{w} $$) as a function of a-domain width (w) for the first (black solid line) and second (black dotted line) a-domain formation in the c-domain matrix, considering surface energies of U_w and 2U_w, respectively. Blue rectangles and yellow parallelograms of varying width represent the c-domain and newly formed a-domain, respectively. w_a indicates the average a-domain width (20 nm), and ΔF_w corresponds to the elastic energy at w_a. (C) Schematic energy barriers for the first (orange line) and second (brown line) a-domain formation and c^- to c⁺ ferroelectric switching (green line), with corresponding free energies of ΔF_c→a, ΔF_c→2a, and ΔF_c→c. The a-domain energy state is ΔF_w, and the energy barrier for a- to c-domain switching is ΔF_a→c at V_a = 0.