  Transition Metal Dichalcogenide (TMD) monolayers have most widely studied due
to their unique physical properties. Recently, Janus TMD Monolayer MoSSe with
sandwiched S-Mo-Se structure has been synthesized by replacing the top S atomic
layer in $\mathrm{MoS_2}$ with Se atoms. In this work, we systematically
investigate the phonon transport and lattice thermal conductivity ($\kappa_L$)
of MoSSe monolayer by first-principles calculations and linearized phonon
Boltzmann equation within the single-mode relaxation time approximation (RTA).
Calculated results show that the $\kappa_L$ of MoSSe monolayer is very lower
than that of $\mathrm{MoS_2}$ monolayer, and higher than that of
$\mathrm{MoSe_2}$ monolayer. The corresponding sheet thermal conductance of
MoSSe monolayer is 342.50 $\mathrm{W K^{-1}}$ at room temperature. These can be
understood by phonon group velocities and lifetimes. Compared with
$\mathrm{MoS_2}$ monolayer, the smaller group velocities and shorter phonon
lifetimes of MoSSe monolayer give rise to lower $\kappa_L$. The larger group
velocities for MoSSe than $\mathrm{MoSe_2}$ monolayer is main reason of higher
$\kappa_L$. The elastic properties of $\mathrm{MoS_2}$, MoSSe and
$\mathrm{MoSe_2}$ monolayers are also calculated, and the order of Young's
modulus is identical with that of $\kappa_L$. Calculated results show that
isotope scattering leads to 5.8\% reduce of $\kappa_L$. The size effects on the
$\kappa_L$ are also considered, which is usually used in the device
implementation. When the characteristic length of MoSSe monolayer is about 110
nm, the $\kappa_L$ reduces to half. These results may offer perspectives on
thermal management of MoSSe monolayer for applications of thermoelectrics,
thermal circuits and nanoelectronics, and motivate further theoretical or
experimental efforts to investigate thermal transports of Janus TMD monolayers.
