Self-consistent evaluation of effective Coulomb interaction of V atom and its importance to understand the comparative electronic behaviour of vanadium spinels

In present work, we try to understand the importance of effective Coulomb interaction ($U_{ef}$) between localized electrons of V atom to understand the comparative electronic behaviour of AV$_{2}$O$_{4}$ (A=Zn, Cd and Mg) compounds. The suitable values of $d$-linearization energy ($E_{d}$) of impurity V atom for calculating the $U_{ef}$ for these compounds are found to be $\geq$44.89 eV above the Fermi level. Corresponding to these values of $E_{d}$, the self-consistently calculated values of effective $U_{LSDA}$ ($U_{PBEsol}$) for ZnV$_{2}$O$_{4}$, MgV$_{2}$O$_{4}$ and CdV$_{2}$O$_{4}$ are $\sim$5.73 ($\sim$5.92), $\sim$6.06 ($\sim$6.22) and $\sim$5.59 ($\sim$5.71) eV, respectively. The calculated values of $\frac{t}{U_{ef}}$ ($t$ is the transfer integral between neighbouring sites) increases with decreasing V-V distance from CdV$_{2}$O$_{4}$ to MgV$_{2}$O$_{4}$ to ZnV$_{2}$O$_{4}$ and are found to be consistent with experimentally reported band gap. The values of $\frac{t}{U_{ef}}$ for ZnV$_{2}$O$_{4}$, MgV$_{2}$O$_{4}$ and CdV$_{2}$O$_{4}$ are found to be $\sim$0.023, $\sim$0.020 and $\sim$0.018, respectively. Hence, CdV$_{2}$O$_{4}$ with small (large) $\frac{t}{U_{ef}}$ (experimental band gap) as compared to ZnV$_{2}$O$_{4}$ and MgV$_{2}$O$_{4}$ is found to be in localized-electron regime, while ZnV$_{2}$O$_{4}$ and MgV$_{2}$O$_{4}$ are intermediate between localized and an itinerant-electron regime. The calculated values of lattice parameters $a_{LSDS}$ ($a_{PBEsol}$) are found to be $\sim$1.7\%, $\sim$2.0\% and $\sim$2.4\% ($\sim$0.6\%, $\sim$0.7\% and $\sim$0.7\%) smaller than $a_{exp}$ for CdV$_{2}$O$_{4}$, MgV$_{2}$O$_{4}$ and ZnV$_{2}$O$_{4}$, respectively, which indicates that the PBEsol functional predicts the lattice parameters in good agreement with the experimental data.