Small power-law dependence of ionic conductivity and diffusional dimensionality in β-alumina

AC conductivity σ(ω) of single crystals of Ag β-alumina and Na β-alumina has been studied in the temperature range from approximately 100 K to room temperature. The DC regime of σ(ω), which had a close relation to the ion dynamics for the long-range diffusion, was found to have the small power-law dependence σ(ω) ∝ ωⁿ with frequency exponent n = 0.11-0.15, not σ(ω) ∝ ω⁰ = const. In higher frequencies, the σ(ω) was monotonically increasing with Log-frequency and was put into Jonscher's universal law (σ(ω) ∝ ω^0.6). The β-alumina is an ideal two-dimensional super ionic conductor. Its low-dimensionality and the scaling theory for a random walk enable us to understand the small power-law dependence of ionic conductivity. We suggested that a "scale-invariance" would hold true behind the super ionic conduction in the self-similarity point of view.