Modeling of transport properties of interfacially controlled electroceramics: Application to n-conducting barium titanate

Grain boundary regions in n-conducting barium titanate (BaTiO₃) are re-oxidized during the cooling process after sintering the ceramics in air. The kinetics of this re-oxidation process is determined by rapid transport of oxygen along the grain boundaries and slow (rate-determining) diffusion of cation vacancies from the grain boundaries into the grains until the diffusion process is frozen-in. Based on numerical calculations of frozen-in diffusion profiles of cation vacancies at grain boundary regions for various cooling rates, a modified Schottky-barrier model is introduced in order to calculate the grain boundary resistivity as a function of temperature from the Curie-point up to 900°C. A change of the activation energy at approximately 500°C is predicted owing to an enrichment of holes in the space charge layers at elevated temperatures. The modeling results are compared with experimental data for BaTiO₃-based positive temperature coefficient resistors (PTCRs).