Depending on the temperature, the examination of complex impedance and complex electrical modulus revealed a non-Debye types relaxation process. Impedance measured data analyzes that a capacitance phase element perfectly modulates the fabricated samples with the equivalent circuits. All prepared samples have a hexagonal wurtzite structure, and the average crystalline size varies between 46 and 56 nm, according to the XRD spectra. The effects of copper (Cu) doping on structure, morphology, impedance, dielectric, and electrical conductivity were investigated using an XRD, SEM, and LCR meter, respectively. The nanostructures of ZnO are studied in this research utilizing a simple, eco-friendly, and cost-effective co-precipitation have a formula Zn 1 − x Cu ( x ) O with (x = 0.0, 0.03, 0.06, and 0.10).

Finally, simulations of the AC dispersion region at high temperatures and frequencies suggest the dominance of polaron tunneling mechanisms instead of the classical ion hopping mechanism assumed for NaSICON, in line with the latest insights on superionic conduction. With Cole–Cole analysis establishing non-Debye relaxation behavior, dissipation analysis is employed to identify relaxation bands, used for extracting initial condition parameters for the Jonscher power law. Activation energy anomalies and scaling distortions inherent to the Arrhenius approximation are investigated, and an alternative formulation based on linearized difference equations is proposed to remedy these issues.

We thoroughly analyze broadband conductivity, dielectric permittivity, and electric modulus data over the extended temperature range. We apply this technique to non-stoichiometric NaSICON based on the canonical NZSP formula with 5% excess sodium, synthesized by an augmented solid-state reaction (SSR) method.

Less constrained by bandwidth limitations and sampling scarcity, broadband profiling in a wide temperature range, starting at the cryogenic threshold at −150 ☌ and extending to 200 ☌, can be used to derive parameters of minimal variance for the Jonscher power law for ionic conductivity these are employed to model the superionic regime over elevated temperatures and frequencies beyond the limits accessed by contemporary electrochemical impedance spectroscopy (EIS) equipment.

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