  Electron spin diffusion is investigated in monolayer MoS$_2$ in the absence
of external electric and magnetic fields. The electron-impurity scattering,
which is shown to play a negligible role in spin relaxation in time domain in
this material, has a marked effect on the in-plane spin diffusion due to the
anisotropic spin precession frequency in the spatial domain. With the
electron-impurity and inter-valley electron-phonon scatterings separately
included in the scattering term, we study the intra- and inter-valley diffusion
processes of the in-plane spins by analytically solving the kinetic spin Bloch
equations. The intra-valley process is found to be dominant in the in-plane
spin diffusion, in contrast to the case of spin relaxation in time domain,
where the inter-valley process can be comparable to or even more important than
the intra-valley one. For the intra-valley process, we find that the in-plane
spin diffusion is suppressed with the increase of impurity density but
effectively enhanced by increasing electron density in both the degenerate and
nondegenerate limits. We also take into account the electron-electron Coulomb
scattering in the intra-valley process. Interestingly, we find that in the
nondegenerate limit, the intra-valley spin diffusion length presents an
opposite trend in the electron density dependence compared to the one with only
electron-impurity scattering.
