Spin-glass behavior in single crystals of hetero-metallic magnetic warwickites $MgFeBO_4$, $Mg_{0.5}Co_{0.5}FeBO_4$, and $CoFeBO_4

Magnetic properties of heterometallic warwickites $MgFeBO_4$, $Mg_{0.5}Co_{0.5}FeBO_4$, and $CoFeBO_4$ are presented, highlighting the effect of Co substitution on the magnetic properties of these compounds. The analysis of magnetization and heat capacity data has shown that these compounds exhibit a spin-glass transition below $T_{SG}$=10, 20 and 22 K, respectively. Using zero field ac susceptibility as entanglement witness we find that the low dimensional magnetic behavior above $T_{SG}$ show quantum entanglement behavior $\chi(T)\varpropto T^{-\alpha(T)}$ up to $T_E$ = 130K. The $\alpha$ parameters have been deduced as a function of temperature and Co, indicating the existence of random singlet phase in this temperature region. Above $T_E$ the paramagnetism is interpreted in terms of non-entangled spins giving rise to Curie-Weiss paramagnetism. The different intra- and inter-ribbon exchange interaction pathways have been calculated within a simple indirect coupling model. It is determined that the triangular motifs in the warwickite structure, together with the competing interactions, induce frustration. The spin-glass character is explained in terms of the substitutional disorder of the Mg, Fe and Co atoms at the two available crystallographic sites, and the frustration induced by the competing interactions. The Co substitution induces uniaxial anisotropy along the b axis, increases the absolute magnetization and increases the spin-glass freezing temperature. The entanglement behavior is supported in the intermediate phase irrespective of the introduction of anisotropy by the Co substitution.