General approach for the description of size effects in ferroelectric nanosystems

We propose general analytical approach for the description of size effect influence on polarization and dielectric susceptibility in ferroelectric nanosystems based on the two-parametric direct variational method and Landau–Ginzburg–Devonshire phenomenology. The essence of the approach is to solve Euler–Largange boundary problem for polarization distribution exactly in paraelectric phase without ferroelectric nonlinearity and then to use the linearized solution for derivation of the approximate analytical expression for spontaneous polarization distribution in ferroelectric phase with the average polarization and characteristic spatial scale as variational parameters. Corresponding polarization distributions calculated within the approach in thin ferroelectric films, nanowires and nanotubes were compared with the available exact solution of Landau–Ginzburg–Devonshire equation or approximate results obtained earlier from the one parametric solution. Perfect agreement between the exact solution and obtained approximate ones is demonstrated. The realization of the proposed scheme of the two-parametric direct variational method seems even simpler than the one-parametric scheme based on the Landau–Ginzburg–Devonshire free energy expansion with renormalized coefficients, while the validity range of two-parametric solution is much wider and the accuracy is higher. So, obtained analytical results have methodological importance for calculation of the phase diagram size effects, polarization distribution, all related polar, dielectric, piezoelectric and pyroelectric properties of single-domain ferroelectric nanoparticles and thin films. The proposed method is applicable to different ferroic nanosystems.