Dielectric properties and spin-phonon coupling in multiferroic double perovskite Bi$_2$CoMnO$_6$

First-principles electronic structure calculations have been performed for the double perovskite Bi$_2$CoMnO$_6$ in its non-centrosymmetric polar state using generalized gradient approximation plus Hubbard U approach. We find that while Co is in a high spin state, Mn is in an intermediate spin state. The calculated dynamical charge tensors are anisotropic reflecting a low-symmetry structure of the compound. Magnetic structure dependent phonon frequencies indicate the presence of spin-phonon coupling. Using Berry phase method, we obtain a spontaneous electronic polarization of 5.88 ${\mu}C/cm^2$ which is close to the experimental value observed for a similar compound, Bi$_2$NiMnO$_6$.