Polarization states of polydomain epitaxial $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ thin films and their dielectric properties

Ferroelectric and dielectric properties of polydomain (twinned) single-crystalline $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ thin films are described with the aid of a nonlinear thermodynamic theory, which has been developed recently for epitaxial ferroelectric films with dense laminar domain structures. For $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ (PZT) films with compositions $x=0.9$, 0.8, 0.7, 0.6, 0.5, and 0.4, the “misfit strain-temperature” phase diagrams are calculated and compared with each other. It is found that the equilibrium diagrams of PZT films with $x⩾0.7$ are similar to the diagram of ${\mathrm{PbTiO}}_3$ films. They consist of only four different stability ranges, which correspond to the paraelectric phase, single-domain tetragonal ferroelectric phase, and two pseudotetragonal domain patterns. In contrast, at $x=0.4$, 0.5, and 0.6, the equilibrium diagram displays a rich variety of stable polarization states, involving at least one monoclinic polydomain state. Using the developed phase diagrams, the mean out-of-plane polarization of a poled PZT film is calculated as a function of the misfit strain and composition. Theoretical results are compared with the measured remanent polarizations of PZT films grown on ${\mathrm{SrTiO}}_3$. Dependence of the out-of-plane dielectric response of PZT films on the misfit strain in the heterostructure is also reported.

Figure 1

Phase diagrams of (001)-oriented single-crystalline $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ films epitaxially grown on dissimilar cubic substrates. The composition $x$ of the solid solution equals 0.9 (a), 0.8 (b), 0.7 (c), 0.6 (d), 0.5 (e), and 0.4 (f). The second- and first-order phase transitions are shown by thin and thick lines, respectively.

Figure 2

Mean out-of-plane polarization $⟨P_3⟩$ of poled $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ films at $T=25°\mathrm{C}$ as a function of the Zr content. The misfit strain $S_m$ in the epitaxial system is taken to be $-3\times {10}^{-3}$, which corresponds to thick PZT films grown on ${\mathrm{SrTiO}}_3$. Circles and triangles (connected by continuous lines) show the theoretical values calculated for polydomain and single-domain films, respectively. Squares denote the remanent polarization measured by Foster et al. (Ref. 6) in submicron-thick PZT films deposited on ${\mathrm{SrTiO}}_3$ at $T_g=600°\mathrm{C}$. For the PZT $65∕35$ film, the remanent polarization is taken from Ref. 31.

Figure 3

Out-of-plane $\left(a_3\right)$ and in-plane ($a_1$, $a_2$) lattice constants in $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ films grown on cubic substrates providing different misfit strains $S_m$ in the epitaxial system. The temperature equals $25°\mathrm{C}$. For the $c∕a∕c∕a$ and $c{a}^{*}∕a{a}^{*}∕c{a}^{*}∕a{a}^{*}$ states, only the lattice parameters in the $c$ and $c{a}^{*}$ domains are shown. In the case of the $a_1∕a_2∕a_1∕a_2$ and $c{a}_1∕c{a}_2∕c{a}_1∕c{a}_2$ structures, the in-plane lattice constants correspond to the $a_1$ and $c{a}_1$ domains, respectively.

Figure 4

Out-of-plane dielectric response ${\epsilon }_{33}$ of epitaxial $\mathrm{Pb}\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right){\mathrm{O}}_3$ films calculated as a function of the misfit strain at $T=25°\mathrm{C}$. The composition $x$ of the solid solution equals 0.6 (a), 0.5 (b), and 0.4 (c).