  Layered two-dimensional (2D) materials have highly anisotropic thermal
properties between the in-plane and cross-plane directions. In general, it is
thought that cross-plane thermal conductivities ($\kappa_z$) are low, and
therefore c-axis phonon mean free paths (MFPs) are small. Here, we measure
$\kappa_z$ across MoS$_2$ films of varying thickness (20 to 240 nm) and uncover
evidence of very long c-axis phonon MFPs at room temperature in these layered
semiconductors. Experimental data obtained using time-domain thermoreflectance
(TDTR) are in good agreement with first-principles density functional theory
(DFT). These calculations reveal that ~50% of the heat is carried by phonons
with MFP >200 nm, exceeding kinetic theory estimates by nearly two orders of
magnitude. Because of quasi-ballistic effects, the $\kappa_z$ of nanometer thin
films of MoS$_2$ scales with their thickness and the volumetric thermal
resistance asymptotes to a non-zero value, ~10 m$^{2}$KGW$^{-1}$. This
contributes as much as 30% to the total thermal resistance of a 20 nm thick
film, the rest being limited by thermal interface resistance with the SiO$_2$
substrate and top-side aluminum transducer. These findings are essential for
understanding heat flow across nanometer-thin films of MoS$_2$ for
optoelectronic and thermoelectric applications.
