Effective macrospin model for $C{o}_{x}F{e}_{3-x}{O}_4$ nanoparticles: decreasing the anisotropy by Co-doping?

Abstract

$Co$-doping of $F{e}_3{O}_4$ magnetic nanoparticles is an effective way to tailor their magnetic properties. When considering the two extreme cases of the $C{o}_{x}F{e}_{3-x}{O}_4$ series, i.e. the $x=0$ and $x=1$ values, one finds that the system evolves from a negative cubic-anisotropy energy constant, $K_C^{-}<0$, to a positive one, $K_C^{+}>0$. Thus, what happens for intermediate $x$-compositions? In this work we present a very simple phenomenological model for the anisotropy, under the \textit{macrospin} approximation, in which the resultant anisotropy is just directly proportional to the amount of $Co$. First, we perform a detailed analysis on a rather ideal system in which the extreme values have the same magnitude (i.e. $|K_C^{-}|=|K_C^{+}|$) and then we focus on the real $C{o}_{x}F{e}_{3-x}{O}_4$ system, for which $|K_C^{+}|\sim 18|K_C^{-}|$. Remarkably, the approach reproduces rather well the experimental values of the heating performance of $C{o}_{x}F{e}_{3-x}{O}_4$ nanoparticles, suggesting that our simple approach may in fact be a good representation of the real situation. This gives rise to an intriguing related possibility arises: a $Co$-doping composition should exist for which the effective anisotropy tends to zero, estimated here as 0.05.

6 pages, 7 figures