The double exchange (DE) model is systematically applied for studying the coupling between ferroelectric (FE) and magnetic orders in several prototypical types of multiferroic manganites. The model itself was constructed for the magnetically active Mn $3d$ bands in the basis of Wannier functions and includes the effect of screened onsite Coulomb interactions in the Hartree-Fock approximation. All model parameters were derived from the first-principles electronic-structure calculations. The essence of our approach for the FE polarization is to use the Berry-phase theory, formulated in terms of occupied Wannier functions, and to evaluate the asymmetric spin-dependent change of these functions in the framework of the DE model. This enables us to quantify the effect of the magnetic symmetry breaking and derive several useful expressions for the electronic polarization $\mathbf{P}$, depending on the relative directions of spins. The spin dependence of $\mathbf{P}$ in the DE model is given by the isotropic correlation functions ${\mathbf{e}}_i·{\mathbf{e}}_j$ between directions of neighboring spins. Despite formal similarity with the magnetostriction mechanism, the magnetoelectric coupling in the proposed DE theory is not related to the magnetically driven FE atomic displacements and can exist even in compounds with the centrosymmetric crystal structure, if the spatial distribution of ${\mathbf{e}}_i·{\mathbf{e}}_j$ does not respect the inversion symmetry. The proposed theory is applied to the solution of three major problems: (i) the magnetic-state dependence of $\mathbf{P}$ in hexagonal manganites, using ${\mathrm{YMnO}}_3$ as an example; (ii) the microscopic relationship between canted ferromagnetism and $\mathbf{P}$ in monoclinic ${\mathrm{BiMnO}}_3$; (iii) the origin of FE activity in orthorhombic manganites. Particularly, we will show that for an arbitrary noncollinear magnetic structure, propagating along the orthorhombic $\mathbf{b}$ axis and antiferromagnetically coupled along the $\mathbf{c}$ axis, the polarization is induced by an inhomogeneous distribution of spins and can be obtained by scaling the one of the E-type antiferromagnetic (AFM) phase with the prefactor depending only on the relative directions of spins and being the measure of this spin inhomogeneity. This picture works equally well for the twofold (${\mathrm{HoMnO}}_3$) and fourfold (${\mathrm{TbMnO}}_3$) periodic manganites. The basic difference is that, even despite some spin canting of the relativistic origin and deviation from the collinear E-type AFM alignment, the twofold periodic magnetic structure remains strongly inhomogeneous, which leads to large $\mathbf{P}$. On the contrary, the fourfold periodic magnetic structure can be viewed as a moderately distorted homogeneous spin spiral, which corresponds to much weaker $\mathbf{P}$.

• Received 3 August 2014
• Revised 11 October 2014

DOI:https://doi.org/10.1103/PhysRevB.90.184425