Comprehensive landscape and simple rules for transition metal Heusler semiconductors

Heusler alloys, renowned for their multifunctionality and capacity for vast elemental customization, are primarily classified into half-Heusler (XYZ) and full-Heusler (${\mathrm{X}}₂\mathrm{YZ}$) structural types. Typically, the 18-electron half-Heusler and the 24-electron full-Heusler alloys are recognized as semiconductors, following the Slater-Pauling rule. Semiconductors are desired for many applications, but they represent a minor portion compared to the predominantly metallic and half-metallic members of the Heusler family. Recently, vacancy-filling off-stoichiometric Heuslers of ternary ${\mathrm{X}}_{1+\ensuremath{\beta}}\mathrm{YZ}$ ($0\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}1$) and quaternary ${\mathrm{X}}_{\ensuremath{\alpha}}{{\mathrm{X}}^{\ensuremath{'}}}_{\ensuremath{\beta}}\mathrm{YZ}$ ($1\ensuremath{\le}\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{\le}2$) have emerged as a versatile strategy to broaden the scope of Heusler semiconductors. However, the flexibility associated with off-stoichiometry inevitably leads to complications, including issues with fractional filling ratios and complex site occupations. This work presents a comprehensive landscape of transition metal-containing Heusler semiconductors, focusing on the off-stoichiometric Heuslers but seamlessly encompassing the integer-stoichiometric systems. The structural and electronic properties can be theoretically understood through a few simple rules. Many systems have been experimentally validated, showcasing their potential for applications such as thermoelectric converters.