Orbital Ordering in Fe$_{1-x}$Mn$_x$V$_2$O$_4$: A First Principles Study

Long range orbital order has been investigated in Fe$_{1-x}$Mn$_x$V$_2$O$_4$ as a function of doping (x) using first principles density functional theory calculations including the effects of Coulomb correlation and spin-orbit interaction within GGA+U and GGA+U+SO approximations. Through a detailed analysis of corresponding Wannier orbital projections of the Vanadium d bands, we have clearly established that for x$\le$0.6, the orbital order at V sites consists of a linear superposition of d$_{xz}$ and d$_{yz}$ orbitals of the type d$_{xz}\pm$d$_{yz}$. Within each ab-plane a ferro-orbital ordering of either d$_{xz}$+d$_{yz}$ or d$_{xz}$-d$_{yz}$ is observed which alternates in successive ab-planes along the c-direction. On the contrary, for x$>$0.6, it is the d$_{xz}$ or d$_{yz}$ orbital which orders at V sites in successive ab-planes along c-direction (so called A-type ordering). At Fe sites, we observe an orbital ordering of d$_{x^2-y^2}$ orbitals for x$\le$0.6 and d$_{z^2}$ orbitals for x$>$0.6. Effect of spin-orbit interaction on orbital ordering is found to be not significant in the entire range of doping studied.