Effective macrospin model for $Co_{x}Fe_{3-x}O_{4}$ nanoparticles: decreasing the anisotropy by Co-doping?

$Co$-doping of $Fe_{3}O_{4}$ magnetic nanoparticles is an effective way to tailor their magnetic properties. When considering the two extreme cases of the $Co_{x}Fe_{3-x}O_{4}$ 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 $Co_{x}Fe_{3-x}O_{4}$ system, for which $|K_{C}^{+}|sim 18|K_{C}^{-}|$. Remarkably, the approach reproduces rather well the experimental values of the heating performance of $Co_{x}Fe_{3-x}O_{4}$ 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.