Multiferroicity with coexisting isotropic and anisotropic spins in Ca$_3$Co$_{2-x}$Mn$_x$O$_6$

We study physical properties in Ca$_3$Co$_{2-x}$Mn$_x$O$_6$ (CCMO) by magnetization, magnetostriction, electric polarization, and magnetocaloric measurements under high magnetic fields. On the controversial topic of the spin state of Co, we find evidence for high spin state with $S$ = 3/2 with no field-induced spin-state crossover up to 97 T. In addition, our data also indicate that Mn spins are quasi-isotropic and develop components in the $ab$-plane in applied magnetic fields of 10 T and cant until saturation at 85 T whereas the Ising Co spins saturate by 25 T. We also find transient steps that appear only within a range of magnetic field sweep rates. This feature resembles the behavior observed in isostructural compound Ca$_3$Co$_2$O$_6$ (CCO) where metastable steps appear in magnetization in non-equilibrium condition. However, CCMO has more complex magnetic interactions due to Mn spins and a different ground state compared to CCO. Our results give a different view to the magnetoelectric coupling in this material, namely the spin-flop of Mn$^{4+}$ destabilizes electric polarization instead of a spin flip. The origin of the the transient behavior in CCMO is also discussed.