  Topological semimetals are currently attracting increasing interest due to
their potential applications in topological qubits and low-power electronics,
which are closely related to their thermal transport properties. In this work,
by solving the Boltzmann transport equation based on first-principles
calculations, we systematically investigate the phonon transport properties of
topological semimetal WC and WN. The predicted room-temperature lattice thermal
conductivities of WC (WN) along a and c directions are 1140.64 (7.47)
$\mathrm{W m^{-1} K^{-1}}$ and 1214.69 (5.39) $\mathrm{W m^{-1} K^{-1}}$.
Considering the similar crystal structure of WC and WN, it is quite interesting
to find that the thermal conductivity of WC is more than two orders of
magnitude higher than that of WN. It is found that, different from WN, the
large acoustic-optical (a-o) gap prohibits the
acoustic+acoustic$\rightarrow$optical (aao) scattering, which gives rise to
very long phonon lifetimes, leading to ultrahigh lattice thermal conductivity
in WC. For WN, the lack of a-o gap is due to soft phonon modes in optical
branches, which can provide more scattering channels for aao scattering,
producing very short phonon lifetimes. Further deep insight can be attained
from their different electronic structures. Distinctly different from that in
WC, the density of states (DOS) of WN at the Fermi level becomes very sharp,
which leads to destabilization of WN, producing soft phonon modes. It is found
that the small shear modulus $G$ and $C_{44}$ limit the stability of WN,
compared with WC. Our works provide valuable information for phonon transports
in WC and WN, and motivate further experimental works to study their lattice
thermal conductivities.
