In its most general form, the magnetoelectric effect (ME) denotes any coupling between the magnetic and the electric properties of a material. The first example of such an effect was described by Wilhelm Röntgen in 1888, who found that a dielectric material moving through an electric field would become magnetized.  A material where such a coupling is intrinsically present is called a magnetoelectric.
Historically, the first and most studied example of this effect is the linear magnetoelectric effect. Mathematically, while the electric susceptibility and magnetic susceptibility describe the electric and magnetic polarization responses to an electric, resp. a magnetic field, there is also the possibility of a magnetoelectric susceptibility which describes a linear response of the electric polarization to a magnetic field, and vice versa:
The tensor must be the same in both equations. Here, P is the electric polarization, M the magnetization, E and H the electric and magnetic fields.
The first material where an intrinsic linear magnetoelectric effect was predicted theoretically and confirmed experimentally is Cr2O3. This is a single-phase material. Multiferroics are another example of single-phase materials that can exhibit a general magnetoelectric effect if their magnetic and electric orders are coupled. Composite materials are an another way to realize magnetoelectrics. There, the idea is to combine, say a magnetostrictive and a piezoelectric material. These two materials interact by strain, leading to a coupling between magnetic and electric properties of the compound material.
Some promising applications of the ME effect are sensitive detection of magnetic fields, advanced logic devices and tunable microwave filters.
The first example of a magnetoelectric effect was discussed in 1888 by Wilhelm Röntgen, who showed that a dielectric material moving through an electric field would become magnetized. The possibility of an intrinsic magnetoelectric effect in a (non-moving) material was conjectured by P. Curie in 1894, while the term "magnetoelectric" was coined by P. Debye in 1926. A mathematical formulation of the linear magnetoelectric effect was included in L. D. Landau and E. Lifshitz famous book series on theoretical physics. Only in 1959, I. Dzyaloshinskii, using an elegant symmetry argument, derived the form of a linear magnetoelectric coupling in Cr2O3. The experimental confirmation came just few months later when the effect was observed for the first time by D. Astrov. The general excitement which followed the measurement of the linear magnetoelectric effect lead to the organization of the series of MEIPIC (Magnetoelectric Interaction Phenomena in Crystals) conferences. Between the prediction of I. Dzialoshinskii and the MEIPIC first edition (1973), more than 80 linear magnetoelectric compounds were found. Recently, technological and theoretical progress triggered a renaissance of these studies and magnetoelectric effect is still heavily investigated.
If the coupling between magnetic and electric properties is analytic, then the magnetoelectric effect can be described by an expansion of the free energy as a power series in the electric and magnetic fields and : 
Differentiating the free energy will then give the electric polarization and the magnetization . Here, and are the static polarization, resp. magnetization of the material, whereas and are the electric, resp. magnetic susceptibilities. The tensor describes the linear magnetoelectric effect, which corresponds to a polarization induced linearly by a magnetic field, and vice versa. The higher terms with coefficients and describe quadratic effects. For instance, the tensor describes a linear magnetoelectric effect which is, in turn, induced by an electric field.
The possible terms appearing in the expansion above are constrained by symmetries of the material. Most notably, the tensor must be antisymmetric under time-reversal symmetry.  Therefore, the linear magnetoelectric effect may only occur if time-reversal symmetry is explicitly broken, for instance by the explicit motion in Röntgens' example, or by an intrinsic magnetic ordering in the material. In contrast, the tensor may be non-vanishing in time-reversal symmetric materials.
There are several ways in which a magnetoelectric effect can arise microscopically in a material.
In crystals, spin-orbit coupling is responsible for single-ion magnetocrystalline anisotropies (https://en.wikipedia.org/wiki/Magnetocrystalline_anisotropy) which determine preferential axes for the orientation of the spins (such as easy axes). An external electric field may change the local symmetry seen by magnetic ions and affect both the strength of the anisotropy and the direction of the easy axes. Thus, single-ion anisotropy can couple an external electric field to spins of magnetically ordered compounds.
Symmetric Exchange striction
The main interaction between spins of transition metal ions in solids is usually provided by superexchange. This interaction depends on details of the crystal structure such as the bond length between magnetic ions and the angle formed by the bonds between magnetic and ligand ions. Symmetric exchange can be both positive and negative and is the main responsible of magnetic ordering. As the strength of symmetric exchange depends on the relative position of the ions, it couples spins to collective lattice distortions, called phonons. Coupling of spins to collective distortion with a net electric dipole can happen if the magnetic order breaks inversion symmetry. Thus, symmetric exchange can provide a handle to control magnetic properties through an external electric field.
Strain driven magnetoelectric heterostructured effect
Because materials exist that couple strain to electrical polarization (piezoelectrics, electrostrictives, and ferroelectrics) and that couple strain to magnetization (magnetostrictive/magnetoelastic/ferromagnetic materials), it is possible to couple magnetic and electric properties indirectly by creating composites of these materials that are tightly bonded so that strains transfer from one to the other.
Thin film strategy enables achievement of interfacial multiferroic coupling through a mechanical channel in heterostructures consisting of a magnetoelastic and a piezoelectric component. This type of heterostructure is composed of an epitaxial magnetoelastic thin film grown on a piezoelectric substrate. For this system, application of a magnetic field will induce a change in the dimension of the magnetoelastic film. This process, called magnetostriction, will alter residual strain conditions in the magnetoelastic film, which can be transferred through the interface to the piezoelectric substrate. Consequently, a polarization is introduced in the substrate through the piezoelectric process. The overall effect is that the polarization of the ferroelectric substrate is manipulated by an application of a magnetic field, which is the desired magnetoelectric effect (the reverse is also possible). In this case, the interface plays an important role in mediating the responses from one component to another, realizing the magnetoelectric coupling. For an efficient coupling, a high-quality interface with optimal strain state is desired. In light of this interest, advanced deposition techniques have been applied to synthesize these types of thin film heterostructures. Molecular beam epitaxy has been demonstrated to be capable of depositing structures consisting of piezoelectric and magnetostrictive components. Materials systems studied included cobalt ferrite, magnetite, SrTiO3, BaTiO3, PMNT.
Magnetically driven ferroelectricity is also caused by inhomogeneous magnetoelectric interaction. This effect appears due to the coupling between inhomogeneous order parameters. It was also called as flexomagnetoelectric effect. Usually it is describing using the Lifshitz invariant (i.e. single-constant coupling term). It was shown that in general case of cubic hexoctahedral crystal the four phenomenological constants approach is correct. The flexomagnetoelectric effect appears in spiral multiferroics or micromagnetic structures like domain walls and magnetic vortexes. Ferroelectricity developed from micromagnetic structure can appear in any magnetic material even in centrosymmetric one. Building of symmetry classification of domain walls leads to determination of the type of electric polarization rotation in volume of any magnetic domain wall. Existing symmetry classification of magnetic domain walls was applied for predictions of electric polarization spatial distribution in their volumes. The predictions for almost all symmetry groups conform with phenomenology in which inhomogeneous magnetization couples with homogeneous polarization. The total synergy between symmetry and phenomenology theory appears if energy terms with electrical polarization spatial derivatives are taking into account.
- Fiebig, M. (2005). "Revival of the magnetoelectric effect". Journal of Physics D: Applied Physics. 38: R123. doi:10.1088/0022-3727/38/8/R01.
- ETH Zürich, Laboratory for Multifunctional Ferroic Materials, "Magnetoelectric Effect". Accessed July 2017.
- Röntgen, W.C. (1888). "Ueber die durch Bewegung eines im homogenen electrischen Felde befindlichen Dielectricums hervorgerufene electrodynamische Kraft". Ann. Phys. 35: 264. doi:10.1002/andp.18882711003.
- L. Landau & E. Lifshitz, "Electrodynamics of continuous media", Pergamon press, 1960
- Dzyaloshinskii, I. (1960). "On the Magneto-Electrical Effect in Antiferromagnets" (PDF). Zh. Exp. Teor. Fiz. 37: 881.
- Astrov, D. (1960). "The magnetoelectric effect in antiferromagnetics" (PDF). Sov. Phys. JETP. 11: 708.
- Nan, C.W.; Bichurin, M.I.; Dong, Shuxiang; Viehland, D.; Srinivasan, G. (2008). "Multiferroic magnetoelectric composites: Historical perspective, status, and future directions". J. App. Phys. 103: 031101. doi:10.1063/1.2836410.
- P. Curie J. Physique, 3ième série III (1894)
- P. Debye, Z. Phys. 36, 300 (1926)
- Cardwell, M.J. (1969). "Measurements of the magnetic field dependent electric susceptibility of yttrium iron garnet". Philosophical Magazine. 20: 1087. doi:10.1080/14786436908228077.
- Srinivasan, G. (2002). "Magnetoelectric effects in bilayers and multilayers of magnetostrictive and piezoelectric perovskite oxides". Phys. Rev. B. 65: 134402. doi:10.1103/physrevb.65.134402.
- Scott, J.F. (2007). "Data storage: Multiferroic memories". Nature Materials. 6: 256-257. doi:10.1038/nmat1868.
- Xie, S.; Cheng, J.; et al. (2008). "Interfacial structure and chemistry of epitaxial CoFe[sub 2]O[sub 4] thin films on SrTiO[sub 3] and MgO substrates". App. Phys Lett. 93: 181901–181903. doi:10.1063/1.3006060.
- Bibes, M.; Barthélémy, A. (2008). "Multiferroics: Towards a magnetoelectric memory". Nature Materials. 7: 425–426. doi:10.1038/nmat2189.
- Yang, J. J.; Zhao, Y.G.; et al. (2009). "Electric field manipulation of magnetization at room temperature in multiferroic CoFe[sub 2]O[sub 4]/Pb(Mg[sub 1/3]Nb[sub 2/3])[sub 0.7]Ti[sub 0.3]O[sub 3] heterostructures". Applied Physics Letters. 94: 212504. doi:10.1063/1.3143622.
- Baryakhtar, V.G.; L'vov, V.A.; Yablonskiy, D.A. (1983). "Spin reversal in 180 domain walls of the spin-flop phase of ease-axis antiferromagnets". JETP Lett. 37 (12): 673–675.
- Pyatakov, A.P.; Zvezdin, A.K. (2009). "Flexomagnetoelectric interaction in multiferroics". Eur. Phys. J. B. 71: 419–427. doi:10.1140/epjb/e2009-00281-5.
- Mostovoy, M. (2006). "Ferroelectricity in Spiral Magnets". Phys. Rev. Lett. 96: 067601. doi:10.1103/physrevlett.96.067601.
- Tanygin, B.M. (2011). "On the free energy of the flexomagnetoelectric interactions". Journal of Magnetism and Magnetic Materials. 323 (14): 1899–1902. doi:10.1016/j.jmmm.2011.02.035.
- Kimura, T.; et al. "Magnetic control of ferroelectric polarization". Nature. 426: 55–58. doi:10.1038/nature02018.
- Logginov, A.S.; Meshkov, G.A.; Nikolaev, A.V.; Nikolaeva, E.P.; Pyatakov, A.P.; Zvezdin, A.K. (2008). "Room temperature magnetoelectric control of micromagnetic structure in iron garnet films". Applied Physics Letters. 93: 182510. doi:10.1063/1.3013569.
- Pyatakov, A. P.; Meshkov, G. A. "Electrically stabilized magnetic vortex and antivortex states in magnetic dielectrics". arXiv: .
- Pyatakov, A. P.; Meshkov, G. A.; Zvezdin, A.K. (2012). "Electric polarization of magnetic textures: New horizons of micromagnetism". JMMM. 324: 3551–3554. doi:10.1016/j.jmmm.2012.02.087.
- Dzyaloshinskii, I. (2008). "Magnetoelectricity in ferromagnets". EPL. 83: 67001. doi:10.1209/0295-5075/83/67001.
- Baryakhtar, V.; L'vov, V.; Yablonsky, D. (1984). "Magnetic symmetry of the domain walls in magnetically ordered crystals". Sov. Phys. JETP. 60 (5): 1072–1080.
- V.G. Baryakhtar, V.A. L'vov, D.A. Yablonskiy, Theory of electric polarization of domain boundaries in magnetically ordered crystals, in: A. M. Prokhorov, A. S. Prokhorov (Eds.), Problems in solid-state physics, Chapter 2, Mir Publishers, Moscow, 1984, pp. 56-80
- Tanygin, B.M. (2011). "Symmetry theory of the flexomagnetoelectric effect in the magnetic domain walls". JMMM. 323 (5): 616–619. doi:10.1016/j.jmmm.2010.10.028.
- Tanygin, B.M. (2010). "Inhomogeneous Magnetoelectric Effect on Defect in Multiferroic Material: Symmetry Prediction". IOP Conf. Ser. Mater. Sci. Eng. 15: 012073. doi:10.1088/1757-899x/15/1/012073.