  We report AC-impedance studies on a series of high-quality
LiMn$_{1-x}$Fe$_x$PO$_4$ single crystals with $0\leq x \leq 1$. Our results
confirm quasi-one-dimensional transport in LiFePO$_4$ with fast Li-diffusion
along the $b$-axis. The conductivities along the crystallographic $b$-, $c$-
and $a$-axis differ by a factor of about 10, respectively. Whereas, the
activation energy $E_{\rm A}$ of the effective diffusion process is
particularly 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 the
same order of magnitude as in undoped LiFePO$_4$, which implies similarly fast
Li-transport even upon 50~\% Mn-doping which, owing to the higher redox
potential of the Mn$^{3+}$/Mn$^{2+}$-couple, yields enhanced energy density in
lithium-ion batteries. The overall results of our impedance studies draw a far
more complex picture than it would be expected from a simple one-dimensional
ionic conductor. Our results suggest a strong contribution of crystal defects
in real materials.
