Universal time-temperature scaling of conductivities in random site energy and associated random barrier model

TL;DR

The paper addresses whether universal time-temperature scaling of conductivity spectra, well described by the Random Barrier Model, can account for site-energy disorder. It develops a mapping from many-particle hopping in a disordered site-energy landscape (RSEM) to independent particles hopping over barriers (A-RBM) and validates the mapping with kinetic Monte Carlo and velocity autocorrelation analyses. A key result is that the associated barrier model reproduces the RBM scaling at low $T$, while the RSEM exhibits scaling over a broader temperature range in agreement with experiments; differences arise from barrier distributions, correlations, and forward-backward hopping. The work extends the framework to multicomponent systems, predicting how partial conductivities in mixed alkali glasses may scale differently and offering a practical route to interpret time-temperature superposition in real materials.

Abstract

Universal time-temperature scaling of conductivity spectra in disordered solids has been explained by thermally activated hopping of noninteracting particles over random energy barriers. An open problem is whether the random barrier model accounts for site energy disorder in real materials. Through mapping many-particle hopping in a disordered site energy landscape to that of independent particles in a barrier landscape, we show that time-temperature scaling is correctly described by the associated barrier model in the low temperature limit. However, the site energy model displays good scaling behavior at substantially higher temperatures than the barrier model, in agreement with experimental observations. Extending the mapping to different types of mobile charge carriers allows us to understand why time-temperature superposition can be absent in mixed alkali glasses.

Figures (6)

• Figure 1: (a) Real parts $\sigma'(\omega,T)$ of conductivity spectra for five different temperatures. Symbols refer to the RSEM and lines to the A-RBM. The inset shows the imaginary part $\sigma"(\omega,T)$ for $T=1/12$, where $\Delta\epsilon(T)$ is extracted from the low-frequency behavior $\sigma"(\omega,T)\sim \Delta\epsilon\,\omega$. (b) Scaled spectra for frequencies one order of magnitude below the attempt frequency, $\omega\le\nu/10$. The inset shows the correspondingly scaled imaginary parts of the conductivity. Data for the RSEM and A-RBM were obtained by averaging over 100 realizations of site energies. The solid line represents the master curve given by Eq. \ref{['eq:sigma-scaling-RBM']}.
• Figure 2: (a) Arrhenius plots of dc-conductivities in the RSEM and A-RBM. A least squares fit to the data (solid line) yields an activation energy $E_{\rm dc}\cong1.06$. (b) Dielectric strength $\Delta\epsilon$ as a function of temperature for the two models. Least squares fits to power laws (solid lines, fit for $T\le1/6$ for the A-RBM) yield $\Delta\epsilon\sim T^{-1.50}$ (RSEM) and $\Delta\epsilon\sim T^{-2.78}$ (A-RBM).
• Figure 3: Histogram of energy barriers $U_{ij}$ [Eq. \ref{['eq:uij']}] in the A-RBM for $T=1/12$. It can be described by a truncated Gaussian (solid line) with mean $\bar{U}$ and standard deviation $\Delta_U$ of the barrier distribution. Since $T=1/12$ is in the low-temperature regime (see text), the truncated Gaussian represents also the barrier distribution in the limit $T\to0$. At higher $T$, slight changes occur, as demonstrated by the truncated Gaussian describing the histogram at $T=1/4$. The table gives $\bar{U}$, $\Delta_U$, and the correlation $C_U=\langle U_{ij}U_{kl}\rangle-\langle U_{ij}\rangle^2$ between barriers at neighboring links $(ij)$ and $(kl)$.
• Figure 4: Arrhenius plot of dc-conductivites of the A-RBM calculated from Eq. \ref{['eq:sigma-VAC']}. The activation energy $E_{\rm dc}\cong0.97$ obtained from a least squares fit in the range $16\le \beta\le 50$ (orange line) is in agreement with a percolation analysis based on the barriers in the limit $T\to0$ [Eqs. \ref{['eq:UijT=0']},\ref{['eq:Edc-percanalysis']}]. A least squares fit in the range $4\le \beta\le 15$ (blue line) yields the apparent activation energy $E_{\rm dc}\cong1.06$ that agrees with the activation energy of the RSEM in Fig. 2a of the main text.
• Figure 5: Scaled real parts of conductivity spectra in the A-RBM for temperatures $T\ge1/15$, yielding no data collapse onto the master curve given by Eq. \ref{['eq:sigma-scaling-RBM']} (solid line). The scaled data in the inset show that scaling behavior sets in at lower temperatures $T\lesssim1/40$.