## 3 Results and Discussion
Hysteresis loops were calculated by applying dynamic electric potentials on top of the film, following the order \(\,0\,\mathrm{V}\to 5\,\mathrm{V}\to-5\,\mathrm{V}\to 5\,\mathrm{V}\,\). Consequently, electric fields were generated in directions downward/upward perpendicular to the film surface. Applying positive potentials on top of the film would result in a downward electric field, which further gave rise to downward polarizations (represented with negative signs in Figure 4). On the contrary, negative potentials led to the expansion of upward variants at the expense of downward ones. Domain configurations close to saturation polarizations (Points A and C) and coercive fields (Points B and D) were shown alongside the \(P\)-\(E\) loops. At Saturation Point A, domains were mainly formed by the four downward variants: \(\,\mathrm{R}_{1}^{-}\,,\,\mathrm{R}_{2}^{-}\,,\,\mathrm{R}_{3}^{-}\,,\,\mathrm{R}_ {4}^{-}\,\). As the period of electric field gradually decreased from positive maximum to zero, polarization dropped slowly, suggesting high remnant polarization, especially in thick films. When the reverse electric field grew, particularly near the coercive field (Point B), opposite variants started to emerge around the domain walls, and their areas expanded over domains with the strengthening of electric potential, which was the result of the electric-field-induced polarization switching. As opposite polarizations reached a maximum (Point C), domains were mainly formed by the four upward variants: \(\,\mathrm{R}_{1}^{+}\,,\,\mathrm{R}_{2}^{+}\,,\,\mathrm{R}_{3}^{+}\,,\,\mathrm{R}_ {4}^{+}\,\). In the process of a gradually changing electric potential from \(-5\,\mathrm{V}\) (Point C) to \(5\,\mathrm{V}\) (Point A), polarizations in films switched from upward to downward via the turning Point D. As film thickness decreased from \(32\,\mathrm{nm}\) in Figure 4a to \(4\,\mathrm{nm}\) in Figure 4d, domain width decreased remarkably, which is a size effect--also reported in the literature [38; 39]. Another important effect of the thickness is the changed coercive field. In Figure 4, it is clear that the coercive potential, at which total polarizations become zero, decreases as film thickness decreases. In order to obtain the critical effect of thickness, coercive field is derived by dividing the coercive potential by the film thickness, as follows: \(E_{\mathrm{c}}=\frac{U_{\mathrm{c}}}{h}\), in which \(E_{\mathrm{c}}\) is the coercive field, \(U_{\mathrm{c}}\) is coercive potential obtained from the \(P\)-\(E\) loop, and \(h\) is the corresponding film thickness. From Figure 5, the relation between coercive fields and film thicknesses can generally be expressed as \(E_{\mathrm{c}}\propto h^{-\frac{3}{2}}\), which has a similar trend as the well-known Kay-Dunn law (\(E_{\mathrm{c}}\propto h^{-\frac{2}{3}}\)) [40], but a greater value in power. This is because the Kay-Dunn law was derived from films with millimeter scale thickness, in which a switching process originated from nucleation. Yet in our work, the systems we studied are ultrathin films, and the polarization switching mechanism can be quite different due to the size effect [31; 41]. We can see from the calculated domain configurations in Figure 5 that the domain width of the \(4\,\mathrm{nm}\) film is much smaller than the width of the \(32\,\mathrm{nm}\) film, which means that there are more domain boundaries in the \(4\,\mathrm{nm}\) film. As seen in the figure, domain boundaries would block the process of domain switching, so thinner film will have a higher coercive field. The results that the coercive field decreases with increasing film thickness, and that thicker films have faster switching processes, confirm the general relation between switching time and coercive field [42]: \(t_{\mathrm{s}}\propto\frac{1}{E-E_{\mathrm{c}}}\), where \(t_{\mathrm{s}}\) is switching time, \(E\) is applied electric field, and \(E_{\mathrm{c}}\) is corresponding coercive field.