First-principles study of Exchange Interaction in Ising-type Multiferroic Ca$_3$CoMnO$_6$

We perform first-principles calculations of multiferroic Ca$_3$CoMnO$_6$ and evaluate the exchange coupling constants using the Green's function method. We clarify the effect of intra-chain and inter-chain exchange interactions on magnetic stability. We find that inter-chain exchange coupling constants are antiferromagnetic and that there are geometrical frustrations in the triangular lattices of magnetic chains in Ca$_3$CoMnO$_6$. The magnetic transition temperature is evaluated using effective Hamiltonian with calculated exchange coupling constants. We obtain the transition temperature 5.80K. The value has the same order as that of experimentally observed.