  Silicon carbide (SiC) is a wide band gap semiconductor with a variety of
industrial applications. Among its many useful properties is its high thermal
conductivity, which makes it advantageous for thermal management applications.
In this paper we present \textit{ab initio} calculations of the in-plane and
cross-plane thermal conductivities, $\kappa_{\text{in}}$ and
$\kappa_{\text{out}}$, of three common hexagonal polytypes of SiC: 2H, 4H and
6H. The phonon Boltzmann transport equation is solved iteratively using as
input interatomic force constants determined from density functional theory.
Both $\kappa_{\text{in}}$ and $\kappa_{\text{out}}$ decrease with increasing
$n$ in $n$H SiC because of additional low-lying optic phonon branches. These
optic branches are characterized by low phonon group velocities, and they
increase the phase space for phonon-phonon scattering of acoustic modes. Also,
for all $n$, $\kappa_{\text{in}}$ is found to be larger than
$\kappa_{\text{out}}$ in the temperature range considered. At electron
concentrations present in experimental samples, scattering of phonons by
electrons is shown to be negligible except well below room temperature where it
can lead to a significant reduction of the lattice thermal conductivity. This
work highlights the power of \textit{ab initio} approaches in giving
quantitative, predictive descriptions of thermal transport in materials. It
helps explain the qualitative disagreement that exists among different sets of
measured thermal conductivity data and provides information of the relative
quality of samples from which measured data was obtained.
