Strained ${\mathrm{NaNbO}}_3$ films of different thicknesses are epitaxially grown on (110) ${\mathrm{NdGaO}}_3$ substrates. A detailed analysis of the permittivity of these films demonstrates that strain not only leads to a modification of the permittivity and the ferroelectric transition temperature, it also results in a pronounced relaxor-type behavior and allows a direct estimation of the size and mobility of the polar nanoregions (PNRs). The compressive strain reduces the transition temperature to 125 K and enhances the corresponding permittivity up to ${\varepsilon }^{\prime }\approx 1500$ for the thinnest film. Since the strain relaxes with increasing film thickness, both effects, reduction of phase transition temperature and enhancement of ${\varepsilon }^{\prime }$, depend on the thickness of the film. The films show a characteristic frequency and electric field dependence of ${\varepsilon }^{\prime }$, which is discussed in terms of the Vogel-Fulcher equation and Rayleigh law, respectively. Using the electric field dependence of the resulting freezing temperature $T_{\mathrm{VF}}$, allows a direct estimation of the volume of the PNRs at the freezing temperature, i.e. from 70 to $270\mathrm{n}{\mathrm{m}}^3$. Assuming an idealized spherical shape of the PNRs, diameters of a few nanometers (5.2–8 nm) are determined that depend on the applied ac electric field. The irreversible part of the polarization seems to be dominated by the presence and mobility of the PNRs. It shows a characteristic peak at low temperature around $T_{\mathrm{VF}}$, vanishes at a temperature where the activation energy of the PRNs extrapolates to zero, and shows a frequency dispersion that is characteristic for relaxor-type behavior.

• Received 13 October 2015
• Revised 22 April 2016

DOI:https://doi.org/10.1103/PhysRevB.93.224107