  A series of LiFe$_{1-x}$Co$_{x}$As compounds with different Co concentrations
have been studied by transport, optical spectroscopy, angle-resolved
photoemission spectroscopy and nuclear magnetic resonance. We observed a Fermi
liquid to non-Fermi liquid to Fermi liquid (FL-NFL-FL) crossover alongside a
monotonic suppression of the superconductivity with increasing Co content. In
parallel to the FL-NFL-FL crossover, we found that both the low-energy spin
fluctuations and Fermi surface nesting are enhanced and then diminished,
strongly suggesting that the NFL behavior in LiFe$_{1-x}$Co$_{x}$As is induced
by low-energy spin fluctuations which are very likely tuned by Fermi surface
nesting. Our study reveals a unique phase diagram of LiFe$_{1-x}$Co$_{x}$As
where the region of NFL is moved to the boundary of the superconducting phase,
implying that they are probably governed by different mechanisms.
