  Under normal conditions, bulk crystals of BiScO$_3$ , BiCrO$_3$, BiMnO$_3$,
BiFeO$_3$, and BiCoO$_3$ present three very different variations of the
perovskite structure: an antipolar phase, a rhombohedral phase with a large
polarization along the space diagonal of the pseudocubic unit cell, and a
supertetragonal phase with even larger polarization. With the aim of
understanding the causes for this variety, we have used a genetic algorithm to
search for minima in the surface energy of these materials. Our results show
that the number of these minima is very large when compared to that of typical
ferroelectric perovskites like BaTiO$_3$ and PbTiO$_3$ , and that a fine energy
balance between them results in the large structural differences seen. As
byproducts of our search we have identified charge-ordering structures with low
energy in BiMnO$_3$ , and several phases with energies that are similar to that
of the ground state of BiCrO$_3$. We have also found that a inverse
supertetragonal phase exists in bulk, likely to be favored in films epitaxially
grown at large values of tensile misfit strain.
