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We report AC-impedance studies on a series of high-qualityLiMn$_{1-x}$Fe$_x$PO$_4$ single crystals with $0\leq x \leq 1$. Our resultsconfirm quasi-one-dimensional transport in LiFePO$_4$ with fast Li-diffusionalong the $b$-axis. The conductivities along the crystallographic $b$-, $c$-and $a$-axis differ by a factor of about 10, respectively. Whereas, theactivation energy $E_{\rm A}$ of the effective diffusion process isparticularly large for the $b$-axis and smallest for the $a$-axis. Remarkably,the $b$-axis ionic bulk conductivity of LiMn$_{0.5}$Fe$_{0.5}$PO$_4$ is of thesame order of magnitude as in undoped LiFePO$_4$, which implies similarly fastLi-transport even upon 50~\% Mn-doping which, owing to the higher redoxpotential of the Mn$^{3+}$/Mn$^{2+}$-couple, yields enhanced energy density inlithium-ion batteries. The overall results of our impedance studies draw a farmore complex picture than it would be expected from a simple one-dimensionalionic conductor. Our results suggest a strong contribution of crystal defectsin real materials.

Anisotropic ionic conductivity of LiMn$_{1-x}$Fe$_x$PO$_4$ ($0\leq x \leq 1$) single crystals