  The lattice thermal conductivity of the candidate thermoelectric material
Mg$_3$Sb$_2$ is studied from first principles, with the inclusion of
anharmonic, isotope, and boundary scattering processes, and via an accurate
solution of the Boltzmann equation. We find that the anomalously low observed
conductivity is due to grain-boundary scattering of phonons, whereas the purely
anharmonic conductivity is an order of magnitude larger. Mass disorder due to
alloying and off-stoichiometry is also found to contribute significantly to its
decrease. Combining ab initio values vs sample size with measured grain-size
distributions, we obtain an estimate of $\kappa$ vs T in nano-polycrystalline
material in good agreement with typical experiments, and compute the ZT figure
of merit in the various cases.
