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{Effect of on-site Coulomb interaction (\textit{U}) on the electronic and magnetic properties of Fe$_{2}$MnSi, Fe$_{2}$MnAl and Co$_{2}$MnGe

The electronic band structures, density of states plots and magnetic moments of Fe$_{2 }$MnSi, Fe$_{2 }$MnAl, and Co$_{2 }$MnGe are studied by using the first principles calculations. The FM solutions using LSDA without \textit{U} show the presence of half-metallic ferromagnetic (HFM) ground state in Fe$_{2 }$MnSi, whereas the ground state of Fe$_{2 }$MnAl is found to be metallic. In both compounds the maximum contribution to the total magnetic moment is from the Mn atom, while the Fe atom contributes very less. The electronic structures and magnetic moments of Fe-based compounds affected significantly by \textit{U}, whereas its effect is very less on Co$_{2}$MnGe. The magnetic moment of Fe atom in Fe$_{2 }$MnSi (Fe$_{2 }$MnAl), increased by $\sim$ 70 % ($\sim$ 75 %) and in Mn atom it decreases by $\sim$ 50 % ($\sim$ 70 %) when the value of \textit{U} is increased from 1 to 5 eV. The Hund's like exchange interactions are increasing in Fe atom while decreasing in Mn atom with increase in \textit{U}. The Fe and Mn moments are ferromagnetically coupled in Fe$_{2 }$MnSi for all values of \textit{U}, whereas in Fe$_{2 }$MnAl they coupled antiferromagnetically below \textit{U} = 2 eV and ferromagnetically above it. Above \textit{U} = 2 eV the metallic ground state of Fe$_{2 }$MnAl changes to semiconducting ground state and the ferromagnetic coupling between Fe and Mn atoms appears to be responsible for this.