We expand our study on cubic BiFeO$_3$ alloys presented in [K. Koumpouras and I. Galanakis, textit{J. Magn. Magn. Mater} 323, 2328 (2011)] to include also the BiMnO$_3$ and Bi$_2$MnFeO$_6$ alloys. For the latter we considered three different cases of distribution of the Fe-Mn atoms in the lattice and six possible magnetic configurations. We show that Fe and Mn atoms in all cases under study retain a large spin magnetic moment, the magnitude of which exceeds the 3 $mu_B$. Their electronic and magnetic properties are similar to the ones in the parent BiMnO$_3$ and BiFeO$_3$ compounds. Thus oxygen atoms which are the nearest-neighbors of Fe(Mn) atoms play a crucial role since they mediate the magnetic interactions between the transition metal atoms and screen any change in their environment. Finally, we study the effect of lattice contraction on the magnetic properties of Bi$_2$MnFeO$_6$.
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