Small power-law dependence of ionic conductivity and diffusional dimensionality in β-alumina O. Kamishima a, ⁎, Y. Iwai b , J. Kawamura b a Faculty of Science and Engineering, Setsunan University, Osaka 572-8508, Japan b Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan abstract article info Article history: Received 28 May 2015 Received in revised form 16 September 2015 Accepted 16 September 2015 Available online 6 October 2015 Keywords: Super ionic conductor Power-law conductivity β-Alumina Scale-invariance Fractal 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) ∝ ω n with frequency exponent n = 0.11–0.15, not σ(ω) ∝ ω 0 = 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. © 2015 Elsevier B.V. All rights reserved. 1. Introduction Dynamical processes of random walks in disordered media differ from those of the simple Brownian motion. The dynamics is usually known as an anomalous diffusion. The mean square displacement (MSD) 〈r(t) 2 〉 ∝ t k characterized by k ≠ 1 was explained by the continu- ous time random walk (CTRW) method, which has been extended to the dynamics on fractal structures [1]. In frequency-domain, the fre- quency dependence of electric conductivity does not follow a well known Drude-type; instead, it is well expressed by power-law functions [2]. The power-law dependence conductivity σ(ω) ∝ ω n has been widely observed for disordered materials with frequency exponent n = 0.6 ap- proximately, called the ‘Jonscher regime’, and the n approaches to 1.0 at higher frequencies, called the ‘nearly constant-loss (NCL) regime’ [3–6]. Typically the σ(ω) varies with frequency as shown in Fig. 1. Recently starting from the master equation within random barriers for hops, and building on the relaxational mode theory, Ishii and Abe [7] theoretically succeeded to describe the universal behavior of the power-law dependence conductivity σ(ω) ∝ ω n . The introduced ran- domness to the hopping events was ascribed to the structural disorder or the many-particle interactions. Molecular dynamics simulation of ionically conducting glasses has been performed [8]. It was found that the first jump freely or indepen- dently to neighboring sites was a dominant role in the NCL regime. The backward jump after the first jump will occur frequently under influence of the already occupied other ions. Then Jonscher regime was corresponding to the correlated jump process of the next jump affected by the other ions. Subsequently the long time diffusion yields as frequency independent. Jonscher has proposed a universality of the Jonscher regime such as a “scale-invariance” behind the dynamics in the disordered systems [9]. There has been a good deal of interest in fractal structures, mainly due to their dilation symmetry (scale-invariance) in contrast to translationally invariant systems [10]. A random walk [11], polymers [12] and percolation clusters [13] are several examples of fractals with the spatial and temporal aspects. It has been recognized that many dis- ordered structures in condensed matter physics appear to be of fractal nature. From this view point, fractals are expected to bridge the gap be- tween crystalline materials and disordered ones coupled with random potentials by its structural disorder or ion–ion interactions. One may define the fractal dimensionality (d f ) to the fractal struc- tures upon the Euclidean dimensionality (d). A universality of physical property is naturally characterized by the indices d f and d. If a “scale- invariance” holds true behind the ion dynamics, the role of the geomet- rical disorder in different d-space will give rise to a different power-law dependence σ(ω), which depends on its dimensionality. In real solids, most of the ionic conductors belong to the three dimension (d = 3) for the ionically conduction paths. However, there is a real crystal having a two-dimensional conduction path as introduced below. Na β-alumina is composed of a rigid layered framework of alumini- um oxide (spinel blocks), interleaved and separated by a loosely packed plane containing a hexagonal arrangement of bridging oxygen ions, to- gether with the mobile cation Na + (Fig. 2). Because of the loose packing, Solid State Ionics 281 (2015) 89–95 ⁎ Corresponding author. E-mail address: kamishima@mpg.setsunan.ac.jp (O. Kamishima). http://dx.doi.org/10.1016/j.ssi.2015.09.011 0167-2738/© 2015 Elsevier B.V. All rights reserved. Contents lists available at ScienceDirect Solid State Ionics journal homepage: www.elsevier.com/locate/ssi