Unraveling heat transport and dissipation in suspended MoSe$_2$ crystals from bulk to monolayer

Understanding thermal transport in layered transition metal dichalcogenide (TMD) crystals is crucial for a myriad of applications exploiting these materials. Despite significant efforts, several basic thermal transport properties of TMDs are currently not well understood. Here, we present a combined experimental-theoretical study of the intrinsic lattice thermal conductivity of the representative TMD MoSe$_2$, focusing on the effect of material thickness and the material's environment. We use Raman thermometry measurements on suspended crystals, where we identify and eliminate crucial artefacts, and perform $ab$ $initio$ simulations with phonons at finite, rather than zero, temperature. We find that phonon dispersions and lifetimes change strongly with thickness, yet (sub)nanometer thin TMD films exhibit a similar in-plane thermal conductivity ($\sim$20~Wm$^{-1}$K$^{-1}$) as bulk crystals ($\sim$40~Wm$^{-1}$K$^{-1}$). This is the result of compensating phonon contributions, in particular low-frequency modes with a surprisingly long mean free path of several micrometers that contribute significantly to thermal transport for monolayers. We furthermore demonstrate that out-of-plane heat dissipation to air is remarkably efficient, in particular for the thinnest crystals. These results are crucial for the design of TMD-based applications in thermal management, thermoelectrics and (opto)electronics.