  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.
