  The strong exchange interaction between 3$d$-4$f$ magnetic sublattice in
rare-earth perovskites introduces a variety of complex magnetic states hosting
fascinating electronic ground states with exotic properties. Especially when it
comes to rare-earth nickelate and cobaltite perovskites, tuning their rich
magnetic phase diagram and spin-state transitions make them potential
candidates for spintronic applications. Here, we report the observation of
antiferromagnetic coupling between Pr 4$f$ and Ni/Co 3$d$ magnetic sublattices
and its tunability with strain in PrCo$_{0.5}$Ni$_{0.5}$O$_{3-\delta}$ (PCNO)
thin films. SQUID magnetization measurements reveal ferromagnetic (FM) ordering
around 25 K, followed by a spin glass transition at low temperatures subject to
spin reorientation. Competing magnetic interactions arise owing to the
3$d$-4$f$ antiferromagnetic (AFM) coupling between Pr and Co/Ni sublattice as
revealed by the X-ray absorption spectroscopy (XAS) and X-ray magnetic circular
dichroism (XMCD) at the Pr $M_{4,5}$ and Co/Ni $L_{2,3}$ absorption edges.
Strain dependence on these AFM coupling reveals an increase (decrease) in the
AFM exchange interaction for tensile (compressive) strained films, leading to a
net decrease (increase) in the magnetization of PCNO films at low temperatures.
The relative increase in low-temperature negative magnetoresistance for
compressively strained films also reflects the enhanced ferromagnetic ordering
in the system. The angle-dependent magnetoresistance measurements reveal a
two-fold anisotropic magnetoresistance (AMR) in tensile strained PCNO films. In
contrast, temperature-dependent switching of AMR accompanied by a two- to
four-fold symmetry crossover is observed for LaAlO$_3$-grown compressive
strained films.
