Finite-Size effects in the Optical and Magnetic Properties of MnCo$_{\mathrm{2}}$O$_{\mathrm{4}}$ Nanostructures

MnCo$_{\mathrm{2}}$O$_{\mathrm{4}}$ nanoparticles (NPs) find extensive applications in energy sectors such as in fuel cells, Li-ion batteries, supercapacitors, etc. Here we present a detailed study of the surface and finite size effects on the optical and magnetic properties of MnCo$_{\mathrm{2}}$O$_{\mathrm{4}}$ NPs synthesized by the sol-gel method. MnCo$_{\mathrm{2}}$O$_{\mathrm{4}}$ particles of various sizes (5.4 nm $\le $ d $\le $ 112 nm) were prepared by varying the heat treatment conditions of the oxalate precursor. The optical absorption spectra of these samples were recorded using a diffuse reflectance accessory. The optical bandgap (E$_{\mathrm{g}})$ values, determined using the Kubelka-Munk analysis, reveal a strong confinement induced blue shift, increasing E$_{\mathrm{g}}$ from 1.73 to 2.4 eV with decreasing size from 112 to 5.4 nm. Also, the role of crystallite size on the crystal field transitions e.g. ligand-to-metal (p-d at 3.10 eV) and intra-band metal-to-metal charge transfer transition (2.6 eV) within d-states has been analyzed. The ferrimagnetic ordering temperature (T$_{\mathrm{C}})$ determined from temperature dependence of dc-magnetic susceptibility $\chi $(T) measurements decreases to 140 K from the bulk value of 185 K with decreasing the crystallite size to 5.4 nm. Such size dependent variation of T$_{\mathrm{C}}$ follows the finite-size scaling relation T$_{\mathrm{C}}$(d) $=$ T$_{\mathrm{C}}(\infty )$[1-($\xi _{\mathrm{o}}$/d)$^{\mathrm{\lambda }}$, with shift exponent $\lambda $ $=$ 0.81 and microscopic correlation length $\xi_{\mathrm{o}} \quad =$ 1.48 nm that is almost twice the lattice parameter (8.27 {\AA}), confirming its microscopic nature.