For the case of 2-2 laminated composites (piezo-magnetostrictive)  models are well developed in several paper published by Srinivasa, Laletin, and Bichurin separately and individually, and match pretty closely with experimentally observed magneto electric voltage coefficients with several combinations of piezoelectric and magnetostrictive materials.

However models for other composite based material systems still need to be verified.

K. Sreenivas 

Dear M. Sreenivas,

I am glad that you enlighten me about those models of which I suspected the existence but didn't know about their good agreements in experiments.. as soon as this is not my field of research.

However, concerning the mechanisms I wish to understand, it concerns "Natural Magnetoelectrics" such as specific metal oxides (MgO) or other crystals that could show this behavior. I would add that, unfortunately, the magnetoelectric coefficient of lamellar composites are so highly unbeatable by natural ME that it is not so much an investigation interest for most researchers in this field so that this mechanism still seems obscure to me.

Indeed, composites are a very good Route toward understanding subtelties of this coupling but the pathways and interactions as well as chemical stability are, I guess, fundamentally different from the possible origins of ME behavior in case of Direct processes (interaction between magnetic field applied, the magnetic moments perturbed and then the ferroelectric components appearing..)

Nevertheless, I totally agree with you on something that was also an outcome of my simulations results which is that the MagnetoStrictive phenomenon as you mentionned is a key process in the step:

Magnetic moments perturbed ===> Ferroelectric component appearing

as soon as this should happen through the motion of atoms which make us come back to the spin orbit as fundamental origin... And we thus would obtain such process:

Magnetic field applied==>Magnetic moments perturbed ==>Ions/atoms slight motion triggered by SO coupling ==> change of opposite charges barycenters ==> Ferroelectric component appearing

Finally, I should say it clearly:

I wonder if the real mechanism of ME is Magnetostrcition + PiezoElectricity or is it inherent coupling in matter.

Thank you again for your feedback and please feel free to point out any problem you see in my reasoning because my objectif is to deeply understand rather than to publish or to be right :)

Best,

CHAKER Ziyad

There are magnetoelectric materials where the magnetoelectric effect is not based on spin orbit coupling but primarily an exchange phenomenon. In Cr2O3 for example there are both effects present, but the non-relativistic is dominating. I sketch the physics of the linear magnetoelectric effect  here from  the perspective of E-field induced magnetization, the inverse effect is described the exactly the same tensor and thus the same physics. In Cr2O3 the presence of an applied elecrtic field moves the Cr3+ ions along the threefold c-axis. This lifts the degeneracy of the sublattices by two effects . The dominant is effects is the non-relativistic effect just due to the change of the exchange parameters associated with this motion. The second so called single ion effect is relativistic in origin and related to spin-orbit coupling in the sense that it originates from the inequivalent change of the g-tensor due the different crystal field changes of Cr3+ ions on different sublattices. This effect is however, smaller than the exchange effect. Again, the exchange based magnetoelectric effect does not depend on spin orbit coupling. 

One can derive general formulae for (linear) magnetoelectric couplings in terms of the microscopic conductivity tensor, which are based only on the Maxwell equations. From these general formulae one can also derive conditions under which magnetoelectric coupling may occur. The microscopic conductivity tensor can in turn be evaluated in the Kubo formalism, which, of course, still requires a concrete material model.

May I humbly point out this recent article, where we have worked out the most general formulae for linear magnetoelectric couplings in terms of the conductivity tensor? I hope this might be of help in answering your question.

http://www.sciencedirect.com/science/article/pii/S1569441015000127

Best regards, Giulio Schober

thanks M. Binek for your fruitfull answer and thanks for pointing out This article to me M. Schober , it is exactly the type of approaches I wanted.

Thanks again to all of you and I I will add more to this question once I will understand more about this.