Tailoring properties of Heusler alloys by elemental substitution and electron counting: (Co$_{2-\alpha}$Mn$_\alpha$)FeGe, Co$_2$(Fe$_{1-\beta}$Mn$_{\beta}$)Ge, and (Co$_{2-\alpha}$Fe$_\alpha$)MnGe
Rational material design by elemental substitution is useful in tailoring materials to have desirable properties. Here we consider three non-equivalent substitutional series based on Co$_2$FeGe, viz; (Co$_{2-\alpha}$Mn$_\alpha$)FeGe, Co$_2$(Fe$_{1-\beta}$Mn$_{\beta}$)Ge, (Co$_{2-\alpha}$Fe$_\alpha$)MnGe ($0\!\le\!\alpha\!\le\!2, 0\!\le\!\beta\!\le\!1$), and study how material properties evolve with the interchange of Mn, Fe, and Co in Co$_2$FeGe. In all three schemes, single-phase compounds can be obtained over a wide range of compositions: $0.125 < \alpha < 1.375 $ for (Co$_{2-\alpha}$Mn$_{\alpha}$)FeGe, $0 \!\le\! \beta \!\le\! 1$ for Co$_2$(Fe$_{1-\beta}$Mn$_{\beta}$)Ge, and $0 \!<\! \alpha \!<\! 1.50$ for (Co$_{2-\alpha}$Fe$_\alpha$)MnGe. All the single-phase compounds crystallise in fcc structure with chemical ordering consistent with the ``4-2'' rule of Butler et al. The compounds are soft ferromagnets with low temperature saturation magnetisation agreeing with the Slater-Pauling rule. Very high Curie temperatures are measured, with values up to 1000 K for lower Mn concentrations. First principle calculations indicate, in the most stable atomic configuration, Mn prefers sharing sublattice with Ge, also consistent with the 4-2 rule. The calculations further predict half-metallic behaviour for (Co$_{1.625}$Mn$_{0.375}$)FeGe, while finding other compositions to be nearly half-metallic. Upon comparing the results of the three series, it is found that single-phase alloys occur for a specific range of valence electrons per unit cell ($\sim\!28.5\!-\!29.75$), and that even for multi-phase samples the structural, magnetic, and electronic properties depend primarily on the number of valence electrons and not on the specific substitution scheme employed.