Ferroelectric and multiferroic materials present a nonlinear variation in their permittivity due to domain wall motion. Currently, this variation is described either by the Rayleigh law for fields above a threshold or by a power law for soft ferroelectrics. We propose a hyperbolic law based on the contributions of domain walls and intrinsic lattice which includes the two classic approaches. The threshold field is clearly defined by considering reversible and irreversible components of the permittivity. A good agreement between the hyperbolic law and experimental data is obtained. Moreover, we show that the threshold field obeys to the Volgel–Fulcher law.

In the present paper, the influence of manganese doping on the dielectric properties of BaSrTiO3 thin films is presented. The real and imaginary parts of the material's permittivity have been measured in a large frequency range (100 Hz–1 MHz) and as a function of the electric field. The tunability and the figure of merit of the material have been obtained from the measurement of the permittivity under an applied DC bias electric field. For the undoped material, the dielectric losses become important for a large DC bias which leads to breakdown. At a suitable dopant rate, this effect disappears. In order to better understand the origin of the related phenomena, we measure the permittivity as a function of the AC excitation amplitude and we decompose the obtained permittivity with the hyperbolic law. This enables to extract the different contributions of the bulk (low frequency diffusion and high frequency lattice relaxation) and of the domain wall motions (vibration and pinning/unpinning) to the material's dielectric permittivity and to understand the effect of manganese doping on each contribution. Knowledge of the related mechanisms allows us to establish the optimum dopant rate (mainly conditioned by the lattice contribution) and to reduce the domain wall motion, which finally is beneficial for the desired properties of the ferroelectric thin film. A particular attention is paid to low frequency diffusion, an especially harmful effect when a DC biasing is mandatory (tunable electronic component in mobile telecommunication devices for example).

Two Pb(Zr_0.20Ti_0.80)O₃ samples of different thickness and domain configuration have been studied. The c-domain sample was found to have a higher coercive field E _c and higher dielectric losses than the other which presents approximately 60% of c-domains and 40% of a-domains as observed by piezo force microscopy (PFM) characterization. Hyperbolic law measurements reveal that the higher coercive field is due to domain wall pinning in deeper defects and hence a higher field E _th is required for unpinning. The dissipation factors due to domain wall motion, however, are similar in both samples since the domain wall density is low and there is almost no interaction between domain walls. The higher dielectric losses in the c-domain oriented sample are a result of a greater contribution from the lattice and seem to be due to strain from the substrate, which is not relieved in a thin sample. PFM and dielectric characterization are complementary methods which provide a better understanding of the domain wall motion.