Influence of depolarization field on polarization states in epitaxial ferroelectric thin films with nonequally biaxial misfit strains

Depolarization field and misfit strain are two important factors that greatly influence the properties of ferroelectric thin films. The present work studies the effect of depolarization field on polarization states in a single-domain ferroelectric thin film with nonequally biaxial in-plane misfit strains. Three coupled Euler-Lagrange equations are derived from the minimization of the total free energy and solved numerically. Misfit-strain–misfit-strain phase diagrams are constructed to represent the polarization states. Under the short-circuit boundary condition, the depolarization field shifts the phase boundaries in the phase diagrams, leading to the contraction and expansion of the regions of out-of-plane and in-plane phases, respectively. Under the open-circuit boundary condition, there exists an out-of-plane electretlike state under high compressive misfit strains when a surface polarization $P_m$ is included.

Figure 1

(Color online) Schematic illustration of nonequally biaxial misfit strains of ${\epsilon }_{11}$ and ${\epsilon }_{22}$ in a single-domain ferroelectric thin film coherently deposited on an infinitely thick substrate.

Figure 2

(Color online) Misfit-strain–misfit-strain phase diagrams for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the short-circuit boundary condition with and without depolarization field (DF) at room temperature, where ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 3

(Color online) Misfit-strain–misfit-strain phase diagram for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the short-circuit boundary condition at room temperature with extrapolation lengths of ${\delta }_1=3\mathrm{nm}$ and ${\delta }_3=4\mathrm{nm}$.

Figure 4

(Color online) Misfit-strain–misfit-strain phase diagrams for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the short-circuit boundary condition with and without depolarization field (DF) at room temperature when the additional surface polarization is ignored and ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 5

(Color online) Profiles of the polarization magnitude along the thickness direction in the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the short-circuit boundary condition at room temperature with and without the additional surface polarization, where ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 6

(Color online) Misfit-strain–misfit-strain phase diagram for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the open-circuit boundary condition at room temperature, where ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 7

(Color online) Misfit-strain–misfit-strain phase diagram for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the open-circuit boundary condition at room temperature when the additional surface polarization $P_m$ is ignored, where ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 8

(Color online) Misfit-strain–misfit-strain phase diagram for the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin films with partial surface charge compensation at room temperature when the additional surface polarization $P_m$ is ignored, where ${\delta }_1={\delta }_3=3\mathrm{nm}$.

Figure 9

(Color online) Thickness-averaged polarization components (a) $\overline{P_3}$ and (b) $\overline{P_1}$ as a function of equally biaxial misfit strain under the short-circuit boundary condition $\alpha =1$ and under the open-circuit boundary condition $\alpha =0$. The inset in (a) is the rescaled plot of $\overline{P_3}$ for $\alpha =0$.

Figure 10

(Color online) Profiles of polarization components (a) $P_1$ and (b) $P_3$ along the thickness direction in the $8\text{−}\mathrm{nm}$-thick $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ thin film under the open-circuit boundary condition at room temperature.