Nonlinear ionic conductivity of thin solid electrolyte samples: Comparison between theory and experiment

Nonlinear conductivity effects are studied experimentally and theoretically for thin samples of disordered ionic conductors. Experimentally observed nonlinear effects can be expressed by a length scale, which, in the literature, has often been interpreted as an effective hopping distance. This interpretation implies that it is possible to map a disordered hopping model system on a coarse-grained regular model and to identify the effective hopping distance with the distance between neighboring sites of the coarse-grained model. With an improved experimental setup, using ac electric fields and higher-order harmonic current detection, the results of older dc field experiments are checked and any undesired effects arising from Joule heating are excluded. The values we obtain for the length scale are up to 43 Å and are thus higher than typical values extracted from dc field experiments. Additionally, we explore the influence of temperature and sample thickness on the nonlinearity. Studying a simple disordered hopping model analytically we find that a mapping of a disordered system on a regular system is, in general, not justified. This invalidates the standard interpretation of the nonlinear conductivity experiments in terms of effective hopping distances. A possible future extension of the disordered hopping model is suggested to closer recover properties of real ion conductors and thus to improve the understanding of the information content of nonlinear conductivity experiments.

Figure 1

(Color online) Sketch of the 1D hopping model used in the theoretical analysis. Periodic boundary conditions are employed. The model is point symmetric.

Figure 2

(Color online) Comparison of the numerical solution for $a_{app}$ with the asymptotic analytical solution.

Figure 3

(Color online) Dependence of $a_{app}$ on the size parallel and orthogonal to the electric field.

Figure 4

Apparent jump distance of mobile ${\mathrm{Na}}^{+}$ ions, $a_{app}$, in NCAS25 glass samples with different thicknesses.

Figure 5

Base current density $j^{\prime }\left(\nu \right)$ (closed symbols) and higher harmonic current density $j^{\prime }\left(3\nu \right)$ (open symbols), measured after applying an ac field with amplitude $E_0=41.6\mathrm{kV}∕\mathrm{cm}$ to the NS25 glass. Both Fourier components of the current density are normalized by the field amplitude $E_0$ and are plotted vs frequency $\nu$ at two different temperatures, $T=303\mathrm{K}$ and 313 K. In addition, the quantity $j^{\prime }\left(3\nu \right)∕E_0$ is multiplied by a factor of $-4$.

Figure 6

Frequency dependence of the imaginary part of the impedance, $Z^{″}$ and of the real part of the permittivity, ${\epsilon }^{\prime }$, for the NS25 glass. The data were taken at $T=313\mathrm{K}$ and different voltage amplitudes. Above 1 Hz, the spectra are determined by bulk ionic conduction, while below one 1 Hz, sample-electrode interface impedances govern the frequency and field dependence of the electrical data. The interfacial impedances are related to additional barriers for the diffusion of the sodium ions from the glass samples into the NaCl solution and vice versa.