Unraveling the magnetic interactions and spin state in insulating Sr$_{2-x}$La$_x$CoNbO$_6$

We investigate the structural, magnetic and spin-state transitions, and magnetocaloric properties of Sr$_{2-x}$La$_x$CoNbO$_6$ ($x=$ 0--1) double perovskites. The structural transition from tetragonal to monoclinic phase at $x$ $\geqslant$ 0.6, and an evolution of (101)/(103) superlattice reflections and Raman active modes indicate the enhancement in the B-site ordering with $x$. The magnetic susceptibility data reveal the transition from weak ferromagnetic (FM) to antiferromagnetic (AFM) ordering for $x$ $\geqslant$ 0.6 with T$_{\rm N}$$\approx$9--15~K. Interestingly, the La substitution drives towards more insulating state due to increase in high-spin Co$^{2+}$, whereas a spin-state crossover is observed in Co$^{3+}$ from high-spin to intermediate/low-spin states with $x$. We discuss the correlation between complex magnetic interactions and the presence of various Co spin-states in the system. Moreover, the emergence of metamagnetic nature due to the competition between FM and AFM interactions as well as crossover from conventional to inverse magnetocaloric effect have been demonstrated by detailed analysis of temperature and field dependent change in magnetic entropy.