  Half Heusler (HH) thermoelectric alloys provide a wide platform to choose
materials with non-toxic and earth abundant elements. This article presents an
ab-initio theoretical evaluation of electrical and thermal transport properties
of three Bismuth-based most promising thermoelectric alloys, selected out of 54
stable HH compounds. These are brand new compounds which are recently proposed
to be stable (Nature Chem. 7, 308 (2015)) and may have interesting properties.
The calculated band structure of the three compounds, namely HfRhBi, ZrIrBi and
ZrRhBi, served as a hint for their promising thermoelectric properties. To gain
confidence on the theoretical predictions of these unreported systems, we first
checked our calculated results for a well studied similar compound, ZrNiSn, and
showed reasonable agreement with the measured ones. HfRhBi and ZrIrBi turn out
to be narrow band gap while ZrRhBi is a moderate band gap semiconductor. A
detailed study of the carrier concentration and temperature dependance of the
Seebeck coefficient (S), Power factor (S$^2 \sigma$), lattice ($\kappa_L$) and
electronic ($\kappa_e$) thermal conductivity and hence the figure of merit (ZT)
is carried out. In contrast to most promising known thermoelectric materials,
we found high power factor for these materials (highest S$^2 \sigma\sim$17.36
mWm$^{-1}$K$^{-2}$ for p-type ZrIrBi). All the three systems (specially p-type)
show high figure of merit, with ZT value as high as 0.45 for ideal crystal.
Maximum ZT and the corresponding optimal n- and p-type doping concentrations
($n_c$) are calculated for all the three compounds, which shall certainly pave
guidance to future experimental work.
