Application of Heywang model to the apparent PTCR effect in water-adsorbing layered materials

Surface proton/hydroxide conduction predominates at ambient conditions prior to levelling off at elevated temperatures due to water evaporation and subsequent loss of charge carriers. This water-induced charge transport results in the “apparent” positive temperature coefficient of resistivity (PTCR) effect. Herein, we show that Heywang model typical of classical ferroelectric PTCR ceramics is applicable to a wide range of water-adsorbing layered materials (0.17–4.76 wt% H₂O). Several examples include layered alkali titanates with negatively-charge sheets; one van der Waals material (g‒C₃N₄) with neutral sheets; and a NiFe layered double hydroxide with positively-charge sheets. The linear log ρ_DC vs (ε′_DCT)^-1 plots (ρ_DC = static resistivity, ε′_DC = static dielectric permittivity, and T temperature) are observed from 25 to 250 °C where resistivity and dielectric permittivity varied up to five orders of magnitude. Using Cs₂Ti₆O₁₃ as a representative sample, the density of acceptor states at the grain boundary N_s (the exact nature to be elucidated) is ∼10¹⁰-10¹¹ cm^-2, slightly dependent on the heating/cooling rates (0.5, 2 and 5 °C·min^-1). Complex plane analyses show that capacitances at grain/grain boundaries alike are constant regardless of temperatures, but resistances in both cases peak at 150–200 °C. While rigorous theoretical basis is yet to be constructed, the observed linearity suggests that there could be a common foundation between these two classes of PTCR materials which have been treated separately so far.