Localized polarons and conductive charge carriers: understanding CaCu$_{3}$Ti$_{4}$O$_{12}$ over a broad temperature range

CaCu$_{3}$Ti$_{4}$O$_{12}$ (CCTO) has a large dielectric permittivity that is independent of the probing frequency near the room temperature, which complicated due to the existence of several dynamic processes. Here, we consider the combined effects of localized charge carriers (polarons) and thermally activated charge carriers using a recently proposed statistical model to fit and understand the permittivity of CCTO measured at different frequencies over the whole temperature range accessible by our experiments. We found that the small permittivity at the lowest temperature is related to polaron frozen, while at higher temperatures the rapid increase is associated with the thermal excitation of polarons inducing the Maxwell-Wagner effect, and the final increase of the permittivity is attributed to the thermally activated conductivity. Such analysis enables us to separate the contributions from localized polarons and conductive charge carriers and quantify their activation energies.