Understanding order in compositionally graded ferroelectrics: Flexoelectricity, gradient, and depolarization field effects

A nonlinear thermodynamic formalism based on Ginzburg-Landau-Devonshire theory is developed to describe the total free energy density in (001)-oriented, compositionally graded, and monodomain ferroelectric films including the relative contributions and importance of flexoelectric, gradient, and depolarization energy terms. The effects of these energies on the evolution of the spontaneous polarization, dielectric permittivity, and the pyroelectric coefficient as a function of position throughout the film thickness, temperature, and epitaxial strain state are explored. In general, the presence of a compositional gradient and the three energy terms tend to stabilize a polar, ferroelectric state even in compositions that should be paraelectric in the bulk. Flexoelectric effects produce large built-in fields which diminish the temperature dependence of the polarization and susceptibilities. Gradient energy terms, here used to describe short-scale correlation between dipoles, have minimal impact on the polarization and susceptibilities. Finally, depolarization energy significantly impacts the temperature and strain dependence, as well as the magnitude, of the susceptibilities. This approach provides guidance on how to more accurately model compositionally graded films and presents experimental approaches that could enable differentiation and determination of the constitutive coefficients of interest.