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Systematic understanding of half-metallicity of ternary compounds in Heusler and Inverse Heusler structures with 3$d$ and 4$d$ elements

Employing {\it ab initio} electronic structure calculations we extensively study ternary Heusler compounds having the chemical formula X$_2$X$^\prime$Z, where X = Mn, Fe or Co; Z = Al or Si; and X$^\prime$ changes along the row of 4$d$ transition metals. A comprehensive overview of these compounds, addressing the trends in structural, electronic, magnetic properties and Curie temperature is presented here along with the search for new materials for spintronics applications. A simple picture of hybridization of the $d$ orbitals of the neighboring atoms is used to explain the origin of the half-metallic gap in these compounds. We show that arrangements of the magnetic atoms in different Heusler lattices are largely responsible for the interatomic exchange interactions that are correlated with the features in their electronic structures as well as possibility of half-metallicity. We find seven half-metallic magnets with 100\% spin polarization. We identify few other compounds with high spin polarisation as "near half-metals" which could be of potential use in applications as well. We find that the major features in the electronic structures remain intact if a 3$d$ X$^{\prime}$ constituent is replaced with an isoelectronic 4$d$, implying that the total number of valence electrons can be used as a predictor of half-metallic nature in compounds from Heusler family.