  We present an ab initio study of the structural and electronic transport
properties of tetrahedrite, Cu$_{12}$Sb$_4$S$_{13}$, in its high-temperature
phase. We show how this complex compound can be seen as the outcome of an
ordered arrangement of S-vacancies in a semiconducting fematinite-like
structure (Cu3SbS4). Our calculations confirm that the S-vacancies are the
natural doping mechanism in this thermoelectric compound and reveal a similar
local chemical environment around crystallographically inequivalent Cu atoms,
shedding light on the debate on XPS measurements in this compound. To access
the electrical transport properties as a function of temperature we use the
Kubo-Greenwood formula applied to snapshots of first-principles molecular
dynamics simulations. This approach is essential to effectively account for the
interaction between electrons and lattice vibrations in such a complex crystal
structure where a strong anharmonicity plays a key role in stabilising the
high-temperature phase. Our results show that the Seebeck coeffcient is in good
agreement with experiments and the phonon-limited electrical resistivity
displays a temperature trend that compares well with a wide range of
experimental data. The predicted lower bound for the resistivity turns out to
be remarkably low for a pristine mineral in the Cu-Sb-S system but not too far
from the lowest experimental data reported in literature. The Lorenz number
turns out to be substantially lower than what expected from the free-electron
value in the Wiedemann-Franz law, thus providing an accurate way to estimate
the electronic and lattice contributions to the thermal conductivity in
experiments, of great significance in this very low thermal conductivity
crystalline material.
