8.4 Magnetoelectric Coupling

Landau theory describes the magnetoelectric effect in a single phase material through expansion of the free energy expression as

	(8.1)

where E and H are the electric and magnetic field respectively. Here ε and µ are the dielectric permittivity and magnetic permeability respectively.  The second and the third term in Equation (8.1) are the temperature dependent electrical polarization, P_i^s, and magnetization, M_i^s. Fourth and fifth terms describe the effect of electrical and magnetic field on the electrical and magnetic behavior respectively, while sixth term consisting of α_ij describes linear magnetoelectric coupling. The next two terms consisting of β_ijk and γ_ijk are third rank tensors and represent higher order coupling coefficients.

Differentiation of Equation (8.1) with respect to electric and magnetic fields respectively leads to polarization and magnetization which are as follows:

	and

	(8.2)

In most cases, we are interested to know about the linear magnetoelectric coefficient, α_ij, as magnetoelectric effect is linear in most compounds. This coefficient basically quantifies the dependence of polarization on magnetic field or of magnetization on the electric field. In case of multiferroics, although many linear magnetoelectric effects are expected because these materials often possess large susceptibility and permeability respectively, this is not a necessary condition as some ferroelectrics and ferromagnets do show small dielectric susceptibility and magnetic permeability.

In addition to direct coupling, there may be instances of indirect coupling mediated by strain. This is likely to arise in two phase systems where two components are couple via strain. However, more recently, in cubic SrMnO₃ and EuTiO₃, strain mediated ME effect is observed in single phase.

Indirect measurements of magnetoelectric coupling include measurement of changes in the magnetization near the magnetic transition temperatures or changes in dielectric constant near the magnetic transition temperature. However, such measurements do not provide any mechanistic insight into the coupling constant. Direct measurements measure magnetic response of material to an applied electric field or electric response to an applied magnetic field.