  Double half-heusler alloys are the new class of compounds which can be seen
as transmuted version of two single half-heusler with higher flexibility of
tuning their properties. Here, we report a detailed study of thermoelectric
(TE) properties of two double half-heusler (HH) alloys X$_2$Ni$_2$InSb
(X=Hf/Zr), using first-principles calculation. These alloys exhibit a rich
phase diagram with the possibility of tetragonal, cubic and solid solution
phase at different temperature range. As such, a comparative study of TE
properties of all these phases is performed. The ordered phases show quite
favorable electronic transport as compared to the disordered ones in both
compounds. Lattice thermal conductivity of double HH alloys is lower than their
ternary counter-part, making them most promising for TE application. Simulated
band gap, obtained using hybrid functional, of ordered phases of
Hf$_2$Ni$_2$InSb and Zr$_2$Ni$_2$InSb lie in the range 0.24-0.4 eV and
0.17-0.59 eV respectively, while for disordered phase, it is 0.05- 0.06 eV.
Hf$_2$Ni$_2$InSb shows a reasonably high ZT value of $\sim$ 2.19, while
Zr$_2$Ni$_2$InSb yields 2.46 at high temperature for n-type conduction in
tetragonal phase. The ZT value for p-type conduction is also quite promising
($\sim$ 1.35 and $\sim$ 2.19 for Hf- and Zr-based compounds). In both the
compounds, electronic transport (Seebeck and electrical conductivity) plays the
dominant role for the high ZT-value. Keeping in mind the promising TE
performance, we propose immediate attention from experimentalists to synthesize
and cross validate our findings for these new candidate materials.
