  Understanding the coupling between spin and phonons is critical for
controlling the lattice thermal conductivity ($\kappa_l$) in magnetic
materials, as we demonstrate here for CrX$_3$ (X = Br and I) monolayers. We
show that these compounds exhibit large spin-phonon coupling (SPC), dominated
by out-of-plane vibrations of Cr atoms, resulting in significantly different
phonon dispersions in ferromagnetic (FM) and paramagnetic (PM) phases. Lattice
thermal conductivity calculations provide additional evidence for strong SPC,
where particularly large $\kappa_l$ is found for the FM phase. Most strikingly,
PM and FM phases exhibit radically different behavior with tensile strain,
where $\kappa_l$ increases with strain for the PM phase, and strongly decreases
for the FM phase -- as we explain through analysis of phonon lifetimes and
scattering rates. Taken all together, we uncover the very high significance of
SPC on the phonon transport in CrX$_3$ monolayers, a result extendable to other
2D magnetic materials, that will be useful in further design of thermal spin
devices.
