  The thermoelectric properties of 54 different group 4 half-Heusler (HH)
alloys have been studied from first principles. Electronic transport was
studied with density functional theory using hybrid functionals facilitated by
the $\mathbf{k} \cdot \mathbf{p}$ method, while the temperature dependent
effective potential method was used for the phonon contributions to the figure
of merit $ZT$. The phonon thermal conductivity was calculated including
anharmonic phonon-phonon, isotope, alloy and grain-boundary scattering. HH
alloys have an ${\it XYZ}$ composition and those studied here are in the group
4-9-15 (Ti,Zr,Hf)(Co,Rh,Ir)(As,Sb,Bi) and group 4-10-14
(Ti,Zr,Hf)(Ni,Pd,Pt)(Ge,Sn,Pb). The electronic part of the thermal conductivity
was found to significantly impact $ZT$ and thus the optimal doping level.
Furthermore, the choice of functional was found to significantly affect
thermoelectric properties, particularly for structures exhibiting band
alignment features. The intrinsic thermal conductivity was significantly
reduced when alloy and grain boundary scattering were accounted for, which also
reduced the spread in thermal conductivity. It was found that sub-lattice
disorder on the ${\it Z}$-site, i.e. the site occupied by group 14 or 15
elements, was more effective than ${\it X}$-site substitution, occupied by
group 4 elements. The calculations confirmed that ZrNiSn, ZrCoSb and ZrCoBi
based alloys display promising thermoelectric properties. A few other n-type
and p-type compounds were also predicted to be potentially excellent
thermoelectric materials, given that sufficiently high charge carrier
concentrations can be achieved. This study provides insight into the
thermoelectric potential of HH alloys and casts light on strategies to optimize
thermoelectric performance of multicomponent alloys.
