Interfacial Strain Effects on Lithium Diffusion Pathways in the Spinel Solid Electrolyte Li-Doped MgAl$_2$O$_4$

(Li,Al)-co-doped magnesium spinel (Li$_x$Mg$_{1-2x}$Al$_{2+x}$O$_4$) is a solid lithium-ion electrolyte with potential use in all-solid-state lithium-ion batteries. Interfaces with spinel electrodes, such as Li$_y$Mn$_2$O$_4$ and Li$_{4+3z}$Ti$_5$O$_{12}$, may be lattice-matched, with potentially low interfacial resistances. Small lattice parameter differences across a lattice-matched interface are unavoidable, causing residual epitaxial strain. This strain potentially modifies lithium diffusion near the interface, contributing to interfacial resistance. Here we report a density functional theory study of strain effects on lithium diffusion pathways for (Li,Al)-co-doped magnesium spinel, for $x_\mathrm{Li} = 0.25$ and $x_\mathrm{Li} = 0.5$. We have calculated diffusion profiles for the un-strained materials, and for isotropic and biaxial tensile strains of up to 6%, corresponding to {100} epitaxial interfaces with Li$_y$Mn$_2$O$_4$ and Li$_{4+3z}$Ti$_5$O$_{12}$. We find that isotropic tensile strain reduces lithium diffusion barriers by as much as 0.32 eV, with typical barriers reduced by ~0.1 eV. This effect is associated with increased volumes of transitional octahedral sites, and broadly follows local electrostatic potentials. For biaxial (epitaxial) strain changes in octahedral site volumes and in lithium diffusion barriers are much smaller than under isotropic strain. Typical barriers are reduced by only ~ 0.05 eV. Individual effects, however, depend on the pathway considered and the relative strain orientation. These results predict that isotropic strain strongly affects ionic conductivities in (Li,Al)-co-doped magnesium spinel electrolytes, and that tensile strain is a potential route to enhanced lithium transport. For a lattice-matched interface with candidate spinel-structured electrodes epitaxial strain has a small, but complex, effect on lithium diffusion barriers.