Cooperative Ion Conduction Enabled by Site Percolation in Random Substitutional Crystals

TL;DR

The paper addresses the challenge of achieving high ionic conductivity without sacrificing stability in solid electrolytes by exploring random substitutional rocksalt crystals. Using molecular dynamics on the Li$_x$Pb$_{1-2x}$Bi$_x$Te system, it shows that $ ext{conductivity } \sigma$ rises sharply once the Li$^+$ concentration exceeds a critical threshold near $x_c \approx 0.2$, corresponding to the onset of Li$^+$ site percolation. The conductive transport is dominated by cooperative knock-on migration within a system-spanning Li$^+$ cluster, and this percolation-driven mechanism is isotropic with respect to the applied field direction. The findings imply a universal design principle for solid electrolytes: engineer carrier concentrations above the percolation threshold while maintaining structural stability, with potential applicability to other multicomponent ionic systems; the reported $6.8 \times 10^{-3}$ S cm$^{-1}$ at 295 K demonstrates competitive performance relative to liquid electrolytes.

Abstract

Efficient and safe energy storage technologies are essential for realizing a sustainable and electrified society. Among the key challenges, the design of superionic conductors for all-solid-state batteries often faces a fundamental trade-off between stability and ionic conductivity. Random substitutional crystals, where atomic species are randomly distributed throughout a crystal lattice, present a promising route to overcome this trade-off. Although the importance of cooperative motion in ion conduction has been pointed out, there is a lack of understanding of the relationship between mesoscale structural organization and macroscopic conductivity, limiting the rational design of optimal compositions. Here, we systematically investigate the ionic conductivity of rock salt random substitutional ionic crystals Li$_x$Pb$_{1-2x}$Bi$_x$Te as a function of Li concentration $x$ using molecular dynamics simulations. We find that ionic conductivity increases sharply once the $x$ exceeds a critical threshold, without disrupting the underlying crystal structure. Strikingly, this threshold aligns with the site-percolation threshold predicted by percolation theory. Our findings establish ion percolation as a universal design principle that reconciles the trade-off between conductivity and stability, offering a simple and broadly applicable strategy for the development of robust, high-performance solid electrolytes.

Figures (5)

• Figure 1: The $x$-dependence of ionic conductivity $\sigma$ (left axis). Square, triangle, and diamond symbols represent $\sigma$ under electric fields applied along the (001), (101), and (111) directions, respectively. Symbols and error bars indicate time-averaged values and their standard deviations. Independent of the electric field direction, $\sigma$ begins to increase sharply from $x = 0.2$. This increase can be attributed to changes in the internal Li$^+$ configuration occurring at $x = 0.2$. The $x$-dependence of the length $L_c$ of the largest Li$^+$ cluster is also shown (right axis). $L_c$ is normalized by the maximum simulation box length, with $L_c = 1$ indicating percolation. Circle symbol corresponds to time-averaged values. Error bars correspond to maximum and minimum values of $L_c$. $L_c$ begins to increase from $x = 0.2$, indicating the onset of percolation, and $L_c$ = 1 beyond $x = 0.25$.
• Figure 2: The trajectories of ions that moved more than a distance of $\Delta r = 3$ during a time interval $\Delta t = 300$, at $x = 1/3$ under an electric field $E = 2.1$ applied along the (a) (001), (b) (101), and (c) (111) direction respectively. Different colors represent different Li$^+$. For clarity, only ions located near the center of the simulation box are shown. Those mobile Li$^+$ follow the same trajectory aligned along the direction of the applied electric field. (d) Schematic image of single-ion hopping and cooperative knock-on migration. (e) A snapshot of displacement vectors at $x = 1/3$ under an electric field $E = 2.1$ applied along the (101) direction during time interval $\Delta t = 1$. For clarity, displacement vectors crossing the simulation box boundaries are omitted. Red and blue arrows indicate cooperative knock-on migration and single-ion hopping, respectively.
• Figure 3: Snapshots of the three largest Li$^+$ clusters at $x$ = (a) 0.15, (b) 0.2, and (c) 0.25. Clusters are color-coded by size (blue, red, green). Clusters are defined by Li$^+$ ions occupying nearest-neighbor cationic sites. At $x = 0.15$, only small isolated clusters are observed. At $x = 0.2$, clusters intermittently percolate, coinciding with the onset of the increase in $\sigma$. At $x = 0.25$, a fully percolated single cluster spans the system.
• Figure 4: (a) Time evolution of the mean displacement $\Delta \xi$ of Li$^+$ ions along the $E$ direction over a time interval $\Delta t = 10$ at $x = 0.2$. Circles and squares represent Li$^+$ ions belonging to the largest cluster and the other clusters, respectively. At $x=0.2$, Li$^+$ ions in the largest cluster exhibit higher mobility, suggesting that the percolated clusters contribute to ionic conduction. (b) $x$-dependence of the time-averaged displacement $\langle \Delta \xi \rangle$, with error bars indicating standard deviations. Circles and squares correspond to Li$^+$ ions in the largest cluster and in the other clusters, respectively. Above $x = 0.2$, $\langle \Delta \xi \rangle$ increases sharply for ions in the largest cluster, indicating that Li$^+$ percolation enhances ionic conduction.
• Figure 5: The $x$-dependence of ionic conductivity $\sigma$ in X$_x$Pb$_{1-2x}$Bi$_x$Te (X = Ag, Na) under electric fields applied along the (101) direction. Circle and square correspond to X = Ag and Na, respectively. symbols and error bars indicate time-averaged values and their standard deviations. For Ag, crystals collapse when an electric field is applied for $x > 1/3$. Both roughly start to increase $\sigma$ at $x\sim0.2$. However, the absolute value of $\sigma$ is one order of magnitude smaller than in the Li$^+$ case, and the rise is more gradual.