  We discuss the possibility of light scalar dark matter generated by
right-handed neutrino in a $L_{\mu}-L_{\tau}$ model, in which the dark matter
$\phi_{dm}$ carries $U(1)_{L_{\mu}-L_{\tau}}$ charge but it is a singlet in the
Standard Model. We discuss the case that dark matter production mainly comes
from scattering associated with a pair of right-handed neutrinos non-thermally
while other related processes are highly suppressed. A feasible parameter space
is considered and we found the correct dark matter relic density can be
obtained without influencing the result of leptogenesis result. The heavier
right-handed neutrino will induce colder dark matter production and the allowed
dark matter mass region is $[\rm 10^{-5}\ GeV,0.1\ GeV]$.
