Lattice-Geometry Effects in Garnet Solid Electrolytes: A Lattice-Gas Monte Carlo Simulation Study

Ionic transport in solid electrolytes can often be approximated as ions performing a sequence of hops between distinct lattice sites. If these hops are uncorrelated, quantitative relationships can be derived that connect microscopic hopping rates to macroscopic transport coefficients. At non-dilute concentrations the relationships between hopping frequency, diffusion coefficient, and ionic conductivity deviate from the random walk case, with this deviation quantified by single-particle and collective correlation factors, $f$ and $f_\mathrm{I}$. These factors vary between materials, and depend on the concentration of mobile particles, the nature of the interactions, and the host lattice geometry. Here we study these correlation effects for the garnet lattice using lattice-gas Monte Carlo simulations. We find that for non-interacting particles (volume exclusion only) single-particle correlation effects are more significant than for any previously studied three-dimensional lattice. This is attributed to the presence of two-coordinate lattice sites, which causes correlation effects intermediate between typical three-dimensional and one-dimensional lattices. Including nearest-neighbour repulsion and on-site energies produces more complex single-particle correlations and introduces collective correlations. We predict particularly strong correlation effects at $x_\mathrm{Li}=3$ (from site energies) and $x_\mathrm{Li}=6$ (from nearest-neighbour repulsion). Both effects are consequences of ordering of the mobile particles. Using these simulation data, we consider tuning the mobile ion stoichiometry to maximise the ionic conductivity, and show that the "optimal" composition is highly sensitive to the precise nature and strength of the microscopic interactions. Finally, we discuss the practical implications of these results in the context of lithium garnets and other solid electrolytes.