Role of dilution on the electronic structure and magnetic ordering of spinel cobaltites

By means of the first-principles density functional theory ($\mathrm{DFT}+U$) calculations and experiments, we investigate the role of dilution on the structural, magnetic, electronic, and optical properties of the antiferromagnetic (AFM) spinel ${\mathrm{Co}}_{3}{\mathrm{O}}_{4}$ having N\'eel temperature $({T}_{\text{N}})\ensuremath{\sim}30$ K. As the octahedral cobalt site of spinel lattice is diluted with Ge, Al, Ti, Ru, and Sn cations, we observe a substantial increase in the size of the unit cell as well as destruction of the long-range magnetic ordering with a spin-orbit compensation effect. The ferrimagnetic ordering in diluted inverse spinels such as ${\mathrm{Co}}_{2}\mathrm{\ensuremath{\Sigma}}{\mathrm{O}}_{4}$ ($\mathrm{\ensuremath{\Sigma}}$ = Ti and Sn) emerges due to the difference in the magnetic moments of two sublattices A (3.87 ${\ensuremath{\mu}}_{\text{B}}$) and B (4.16 ${\ensuremath{\mu}}_{\text{B}}$ for ${\mathrm{Co}}_{2}{\mathrm{SnO}}_{4}$ and 5.19 ${\ensuremath{\mu}}_{\text{B}}$ for ${\mathrm{Co}}_{2}{\mathrm{TiO}}_{4}$). Experiments and DFT calculations indicate antiferromagnetic configuration for ${\mathrm{Co}}_{3}{\mathrm{O}}_{4}$, ${\mathrm{Co}}_{2}{\mathrm{AlO}}_{4}$ (${T}_{\text{N}}\ensuremath{\sim}4.8$ K) spinels with an equal and opposite moment of $\ensuremath{\sim}2.60$ ${\ensuremath{\mu}}_{\text{B}}$ in tetrahedral sites of divalent Co ions and negligible contribution from trivalent B-site Co due to the complete filling of ${t}_{2g}$ levels having a giant crystal field of $\ensuremath{\sim}2.5$ and 1.8 eV, respectively. However, in ${\mathrm{Co}}_{2}{\mathrm{GeO}}_{4}$ (${T}_{\text{N}}\ensuremath{\sim}20.4$ K) case AFM behavior originates due to the opposite spins at octahedral sites of divalent Co ions. The remaining spinels ${\mathrm{Co}}_{2}{\mathrm{TiO}}_{4}$ (${T}_{\text{N}}\ensuremath{\sim}47.8$ K), ${\mathrm{Co}}_{2}{\mathrm{RuO}}_{4}$ (${T}_{\text{N}}\ensuremath{\sim}16$ K), and ${\mathrm{Co}}_{2}{\mathrm{SnO}}_{4}$ (${T}_{\text{N}}\ensuremath{\sim}41$ K) are more favorable to ferrimagnetic structure as evident from our magnetization measurements with a different temperature dependence of magnetic moments A(T) and B(T) at tetrahedral A and octahedral B sites, respectively. The variation in the energy band gap (${E}_{g}=1.68\ensuremath{\rightarrow}3.28$ eV for ${\mathrm{Co}}_{2}{\mathrm{RuO}}_{4}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}{\mathrm{Co}}_{2}{\mathrm{GeO}}_{4}$) obtained from $\mathrm{DFT}+U$ calculations are in good agreement with our experimental results (${E}_{g}=1.52\ensuremath{\rightarrow}3.16$ eV) obtained from the diffusive reflectance spectroscopy. The extent of exchange splitting ${\mathrm{\ensuremath{\Delta}}}_{\text{EX}}^{{e}_{g}}$ of tetrahedral ${\mathrm{Co}}^{2+}$ varies between 1.8 and 1.3 eV for ${\mathrm{Co}}_{3}{\mathrm{O}}_{4}$ and ${\mathrm{Co}}_{2}{\mathrm{AlO}}_{4}$, respectively. However, ${\mathrm{\ensuremath{\Delta}}}_{\text{EX}}^{{t}_{2g}}$ exhibits a decreasing trend ($5.2\ensuremath{\rightarrow}3.6$ eV for ${\mathrm{Co}}_{3}{\mathrm{O}}_{4}\ensuremath{\rightarrow}{\mathrm{Co}}_{2}{\mathrm{SnO}}_{4}$) with increasing the lattice parameter, except for cobalt-orthogermanate ${\mathrm{Co}}_{2}{\mathrm{GeO}}_{4}$.