  We use rigorous group-theoretic techniques and molecular dynamics to
investigate the connection between structural symmetry and ionic conductivity
in the garnet family of solid Li-ion electrolytes. We identify new ordered
phases and order-disorder phase transitions that are relevant for conductivity
optimization. Ionic transport in this materials family is controlled by the
frustration of the Li sublattice caused by incommensurability with the host
structure at non-integer Li concentrations, while ordered phases explain
regions of sharply lower conductivity. Disorder is therefore predicted to be
optimal for ionic transport in this and other conductor families with strong Li
interaction.
