Magnetostructural transitions and metamagnetism induced by Ising spins in spinel-rock salt intergrowth Co$_{10}$Ge$_3$O$_{16}$

Co$_{10}$Ge$_3$O$_{16}$ crystallizes in an intergrowth structure featuring alternating layers of spinel and rock salt, making it related to GeCo$_2$O$_4$. Variable-temperature synchrotron X-ray powder diffraction, magnetometry, and heat capacity experiments reveal a magnetostructural transition at antiferromagnetic $T_N$ = 205 K. This rhombohedral-to-monoclinic transition involves a slight elongation of the CoO$_6$ octahedra. Curie-Weiss analysis suggests that the Co$^{2+}$, with $S$ = 3/2 and $L$ = 3, acts as a Kramer's doublet due to spin-orbit coupling. Below $T_N$, the Ising-like Co$^{2+}$ causes spin reorientation at high applied magnetic field that is first seen as an upward kink in $M$-$H$ near $H_C$ = 3.9 T. A ``butterfly'' loop emerges when $T <$ 150 K, with the transition causing hysteresis at high fields while linear and reversible behavior persists at low fields. $H_C$ decreases as temperature is lowered and the loops at positive and negative fields merge beneath $T$ = 20 K. The low-temperature behavior is complicated by a field-induced first-order transition that is observed in temperature-dependent measurements for $H >$ 1000 Oe. We discuss the $H$-$T$ phase diagram with reference to other measurements including neutron powder diffraction and high-field magnetometry.