Bonding properties of Rubik's-cube-like Slater-Pauling Heusler semiconductors for thermoelectrics

Heusler compounds [XYZ for half-Heusler (HH) and $\mathrm{X}{\mathrm{Y}}_{2}\mathrm{Z}$ for full-Heusler (FH)] with semiconducting behaviors have been long suggested as promising high-performance thermoelectric (TE) materials. Recently, some off-stoichiometry Heusler compounds (such as the mixture of HH and FH compounds) were predicted as semiconductors as well, which disobey the traditional valence electron count rule and are named Slater-Pauling (S-P) semiconductors. However, the bonding behaviors in these S-P semiconductors and the relationship between the geometry and TE properties are still unclear. Therefore, focusing on the Ti-Fe-Sb and $M$-Co-Sn ($M=\mathrm{Ti}$, Zr, Hf) Heusler systems, we theoretically predict several thermodynamically stable off-stoichiometry Heusler compounds ($\mathrm{Ti}{\mathrm{Fe}}_{1.5}\mathrm{Sb}$ and $M{\mathrm{Co}}_{1.33}\mathrm{Sn}$) and clarify the bonding behaviors of those S-P semiconductors. From the geometry analysis of these compounds, in addition to the HH and FH local geometries, the disorder occupation of Y atoms (Fe or Co) in the lattice also induces the defective-HH and defective-FH substructures, and the stacking style forms second- and third-order Rubik's-cube-like structures. This unique stacking arrangement (or the Y occupations) not only plays an essential role in the electron redistribution in the lattice to form the band gap but also lowers the acoustic Debye temperature and strengthens the anharmonic to suppress the lattice thermal conductivity. Owing to the high power factor and low thermal conductivity, the calculated $zT$ value of $p$-type $\mathrm{Zr}{\mathrm{Co}}_{1.33}\mathrm{Sn}$ can reach 0.54 at 1000 K. This value could be further optimized using defect engineering. Thus, we not only elucidate the underlying mechanism behind the emergence of S-P semiconductors in Heusler systems but also offer valuable insight into experimental exploration and discovery of S-P semiconductors exhibiting superior TE performance.