Li-ion diffusion in single crystal LiFePO$_4$ measured by muon spin spectroscopy

Muon spin spectroscopy ($\mu^+$SR) is now an established method to measure atomic scale diffusion coefficients of ions in oxides. This is achieved via the ion hopping rate, which causes periodic change in the local magnetic field at the muon site(s). We present here the first systematic study on a single crystalline sample. The highly anisotropic diffusion of Li-ions in the battery cathode material LiFePO$_4$, combined with the extensive investigation of this material with $\mu^+$SR and other techniques make it a perfect model compound for this study. With this experiment we can confirm that Li diffusion in the bulk LiFePO$_4$ is measurable with $\mu^+$SR. Hence, surface/interface effects, which might play a crucial role in case of powders/nano crystals, are less significant for macroscopic single crystals where bulk diffusion is in fact present. We observe that the internal magnetic field fluctuations caused by the diffusing Li-ions are different depending on the crystal orientation. This is not obviously expected based on theoretical considerations. Such fluctuation rates were used to estimate the diffusion coefficient, which agrees well with values estimated by first principle calculations considering anisotropic diffusion.