Phase field simulations of ferroelectrics domain structures in PbZr _xTi_1-xO₃ bilayers

Domain stability and structures in Pb(Zr_0.3Ti _0.7)O₃/Pb(Zr_0.7Ti_0.3)O₃ bilayer films under different substrate strains are studied using the phase field method. It is demonstrated that the domain structure of the bilayer film is very different from those of the corresponding single layer films grown on the same silicon substrate with an incoherent interface. Moreover, the predicted rhombohedral domains in the Pb(Zr_0.7Ti_0.3)O₃ layer of the bilayer film have smaller sizes than those in the single layer case. These results are compared with experimental observations and previous thermodynamic analyses. The polarization distributions of the ferroelectric-paraelectric bilayer are analyzed as a function of the thickness of the bilayer film, where there is a "ferroelectric proximity effect" due to dipole-dipole interactions. The phase diagrams for both the bilayer and single layer films as a function of temperature and effective in-plane substrate strain are constructed.