Low-dimensional frustrated magnetic square networks feature a variety of unconventional phases with novel emergent excitations. Often these excitations are intertwined and manifest into intriguing phenomena, an area that has remained largely unexplored in square-lattice systems, especially, double perovskites (A2BB՛O6). In this study, we explore these interactions between the fundamental excitations such as phonons and magnons in square-lattice Sr2CuTeO6, Sr2CuWO6, and Ba2CuWO6 isostructural double perovskites that exhibit both short-ranged (TS) as well as long-ranged Néel antiferromagnetic (TN) transitions. Our Raman measurements at variable temperatures reveal an intriguing broad peak (identified as 2-magnon (2M)) surviving beyond TSfor W-based compositions contrary to the Te-based system, suggesting a key role of diamagnetic B՛-site cation on their magnetism. The thermal response of 2M intriguingly shows signatures of correlation with phonons and control over their anharmonicity, depicting magnon-phonon interaction. Further, a few phonons exhibit anomalies across the magnetic transitions implying the presence of spin-phonon coupling. In particular, the phonon modes at ~ 194 cm-1 of Sr2CuTeO6 and ~ 168 cm-1 of Sr2CuWO6, that show a strong correlation with the 2M, exhibit the strongest spin-phonon coupling suggesting their roles in mediating magnon-phonon interactions in these systems.
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