  Spinel oxides are an ideal setting to explore the interplay between
configurational entropy, site selectivity, and magnetism in high entropy
oxides. In this work we characterize the magnetic properties of the spinel
(Cr,Mn,Fe,Co,Ni)$_3$O$_4$ and study the evolution of its magnetism as a
function of non-magnetic gallium substitution. Across the range of compositions
studied here, from 0% to 40% Ga, magnetic susceptibility and powder neutron
diffraction measurements show that ferrimagnetic order is robust in the spinel
HEO. However, we also find that the ferrimagnetic order is highly tunable, with
the ordering temperature, saturated and sublattice moments, and magnetic
hardness all varying significantly as a function of Ga concentration. Through
x-ray absorption and magnetic circular dichroism, we are able to correlate this
magnetic tunability with strong site selectivity between the various cations
and the tetrahedral and octahedral sites in the spinel structure. In
particular, we find that while Ni and Cr are largely unaffected by the
substitution with Ga, the occupancies of Mn, Co, and Fe are each significantly
redistributed. Ga substitution also requires an overall reduction in the
transition metal valence, and this is entirely accommodated by Mn. Finally, we
show that while site selectivity has an overall suppressing effect on the
configurational entropy, over a certain range of compositions, Ga substitution
yields a striking increase in the configurational entropy and may confer
additional stabilization. Spinel oxides can be tuned seamlessly from the
low-entropy to the high-entropy regime, making this an ideal platform for
entropy engineering.
