The spin-1/2 square-lattice Heisenberg model is predicted to have a quantum disordered ground state when magnetic frustration is maximized by competing nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ interactions ($J_2/J_1 approx 0.5$). The double perovskites Sr$_2$CuTeO$_6$ and Sr$_2$CuWO$_6$ are isostructural spin-1/2 square-lattice antiferromagnets with Neel ($J_1$ dominates) and columnar ($J_2$ dominates) magnetic order, respectively. Here we characterize the full isostructural solid solution series Sr$_2$Cu(Te$_{1-x}$W$_x$)O$_6$ ($0 leq x leq 1$) tunable from Neel order to quantum disorder to columnar order. A spin-liquid-like ground state was previously observed for the $x$ = 0.5 phase, but we show that the magnetic order is suppressed below 1.5 K in a much wider region of $x approx$ 0.1-0.6. This coincides with significant $T$-linear terms in the low-temperature specific heat. However, density functional theory calculations predict most of the materials are not in the highly frustrated $J_2/J_1 approx 0.5$ region square-lattice Heisenberg model. Thus, a combination of both magnetic frustration and quenched disorder is the likely origin of the spin-liquid-like state in $x$ = 0.5.
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