Investigating the electronic structure of MSi (M = Cr, Mn, Fe $\&$ Co) and calculating $\textit{$U_{eff}$}$ $\&$ $\textit{J}$ by using cDFT

We have investigated electronic energy band structures and partial density of states of intermetallic compounds $\textit{viz.}$ CrSi, MnSi, FeSi and CoSi, by using density functional theory (DFT). CrSi \& MnSi, FeSi and CoSi have metallic, indirect band gap semiconducting (band gap $\sim$ 90 meV) and semi-metallic ground state, respectively. On studying the band structures while going across the series Cr-Co, the occupied bands around the Fermi level are getting narrower while the unoccupied bands are getting wider. Similarly, band edge in partial density of states is shifting away from the Fermi level due to increased hybridizations. The effective mass of holes for FeSi is found to be much larger than that of electrons, giving rise to positive Seebeck coefficient and negative Hall coefficient, which is consistent with experimental results. For different ionic states of 3\textit{d}-metal, the values of \textit{U_{eff}} and $\textit{J}$ are evaluated by using constrained DFT method. \textit{U_{eff}} ($\textit{J}$) for $2^{+}$ ionic state across the series are $\sim$ 3.3 eV ($\sim$0.65 eV), $\sim$ 3.7 eV ($\sim$0.72 eV), $\sim$ 4.4 eV ($\sim$ 0.82 eV) and $\sim$ 4.5 eV ($\sim$ 0.87 eV). $\lambda$ and $\textit{J}$ are also calculated by considering Yukawa form of Coulomb interaction. $\lambda$ values for $2^{+}$ ionic state along the series are $\sim$ 1.97 a.$u^{-1}$, $\sim$ 2.07 a.$u^{-1}$, $\sim$ 2.07 a.$u^{-1}$ and $\sim$ 2.34 a.$u^{-1}$. 4$\textit{s}$ electrons are found to be contributing more in screening the 3$\textit{d}$ electrons as compared to 4$\textit{p}$ electrons of 3$\textit{d}$ metals.