Antisite-disorder driven tuning of magnetic properties and exchange-bias in Nd$_{2-x}$Sr$_{x}$CoMnO$_{6-\delta}$ $(0 \leq x \leq 1)$ ($\delta \sim 0.5$) double perovskites
We demonstrate precise control of exchange bias (EB) in the Nd$_{2-x}$Sr$_x$CoMnO$_{6-\delta}$ ($0 \leq x \leq 1$) double-perovskite series through Sr$^{2+}$ induced hole doping, unveiling a remarkable transition between normal and inverse EB states. Employing neutron powder diffraction and X-ray absorption spectroscopy, we reveal a structural evolution from a B-site-ordered monoclinic ($P2_1/n$) phase to a disordered rhombohedral ($R\overline{3}c$) phase with increasing $x$, accompanied by a shift in the effective Co valence from +2 toward +3, while the Mn valence remains essentially unchanged. DC magnetization measurements indicate a gradual suppression of ferromagnetism with hole doping, whereas AC susceptibility measurements at $x = 0.75$ reveal pronounced cluster-glass behavior and the highest EB field of $\sim 4$ kOe at 8 K under a 6 T cooling field. After correcting for minor-loop effects, we identify robust inverse EB at $x = 0.75$, persisting even under a cooling field of 6 T. We attribute this phenomenon to competing ferromagnetic--antiferromagnetic and ferromagnetic--glassy interfaces, governed by strong magnetic frustration and the magnetocrystalline anisotropy associated with rare-earth 4$f$ electrons. These findings elucidate the pivotal role of doping-induced structural and magnetic competition in tailoring EB behavior in rare-earth double perovskites, providing new insights for the design of advanced magnetic materials.