Theoretical Methods in the Magnetoelectric Effect. I. Formal Treatment

We consider a two-sublattice model of an antiferromagnet lacking inversion symmetry and possessing interaction between the magnetic atoms of the same and different sublattices, i.e., nearest-and next-nearest-neighbor coupling. Correlation functions are derived by the Green's-function method, which is linearized by the random-phase and Callen's decoupling approximation. Formal expressions for the parallel and perpendicular magnetic and magnetoelectric susceptibilities are then obtained. Comparison of these results with those gained in the molecular-field approximations elucidates the shortcomings of the latter method. The theory is applied to ${\mathrm{Cr}}_2$ ${\mathrm{O}}_3$ with the simplification that this crystal is regarded as having two cations in the unit cell instead of (as in reality) four.