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Hot-electron cooling by acoustic and optical phonons in monolayers of MoS$_2$ and other transition-metal dichalcogenides

We study hot-electron cooling by acoustic and optical phonons in monolayer MoS$_2$. The cooling power $P$ ($P_e = P/n$) is investigated as a function of electron temperature $T_e$ (0-500 $\mathrm{K}$) and carrier density $n$ ($10^{10}$-$10^{13}$ $\mathrm{cm}^{-2}$) taking into account all relevant electron-phonon (el-ph) couplings. We find that the cross over from acoustic phonon dominated cooling at low $T_e$ to optical phonon dominated cooling at higher $T_e$ takes place at $T_e \sim 50$-$75$ $\mathrm{K}$. The unscreened deformation potential (DP) coupling to the TA phonon is shown to dominate $P$ due to acoustic phonon scattering over the entire temperature and density range considered. The cooling power due to screened DP coupling to the LA phonon and screened piezoelectric (PE) coupling to the TA and LA phonons is orders of magnitude lower. In the Bloch-Grüneisen (BG) regime, $P\sim T_e^4$ ($T_e^6$) and $P\sim n^{-1/2}$ ($P_e\sim n^{-3/2}$) are predicted for unscreened (screened) el-ph interaction. The cooling power due to optical phonons is dominated by zero-order DP couplings and the Fröhlich interaction, and is found to be significantly reduced by the hot-phonon effect when the phonon relaxation time due to phonon-phonon scattering is large compared to the relaxation time due to el-ph scattering. The $T_e$ and $n$ dependence of the hot-phonon distribution function is also studied. Our results for monolayer MoS$_2$ are compared with those in conventional two-dimensional electron gases (2DEGs) as well as monolayer and bilayer graphene.