The present manuscript completes the study presented in two recent research articles [K. Koumpouras and I. Galanakis, textit{J. Magn. Magn. Mater.} 323, 2328 (2011); textit{ibid}, textit{J. Spintron. Magn. Nanomater.} 1, in press]. Preliminary first-
principles calculations using the QUANTUM-espresso package [P. Giannozzi et al textit{J. Phys.:Condens. Matter} 21, 395502 (2009)] on the magnetic behavior of ultra-thin epitaxial multilayers between the BiFeO$_3$ magnetoelectric compound and various types of spacers are presented. As spacer we have considered i) InP semiconductor, ii) Fe which is a ferromagnet, and iii) metallic V. In all cases under study the growth axis of the multilayer was the [001]. Our results indicate that the magnetic properties are seriously downgraded for the ultrathin BiFeO$_3$ multilayers independent of the nature of the spacer and in some cases under study magnetism even vanishes. More extensive calculations are needed to establish a more clear view of the physical properties of the interfaces involving the BiFeO$_3$ compound.
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$_6$.
