Effects of disorder and bond angle on the magnetic properties of ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$-like materials

The mineral herbertsmithite, ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, contains two-dimensional Kagome layers of Cu$^{2+}$ (S=1/2) ions. It is a candidate for the long-sought-after spin-liquid ground state as no long range magnetic order is found above T = 50 mK despite strong antiferromagnetic interactions, $\theta _{CW} \quad \sim $ -300 K. However, it is difficult to determine how Zn-Cu site mixing affects the ground state properties, due to difficulties in quantifying the degree of Zn-Cu disorder. Here the structural and magnetic properties on chemically related systems in which Zn$^{2+}$ is replaced by Mg$^{2+}$ and Cd$^{2+}$ are presented. These results permit direct identification of the effect of disorder on the observed magnetic behavior, and highlight the importance of the Cu-O-Cu bond angle in determining the magnetic coupling between Cu$^{2+}$ ions within each layer.