Equivalence of direct and converse magnetoelectric coefficients in strain-coupled two-phase systems

We demonstrate an equivalence between direct and converse magnetoelectric effects in two-phase systems comprising piezoelectric and magnetostrictive materials. This was achieved by recasting the Maxwell relation in terms of the effective electrical and magnetic dipole moments for the system and comparing coupling strengths at the same electrical and magnetic d.c. biases. Our findings therefore apply to magnetoelectric systems comprising more than two phases and correct the two widely held but opposing views that are compromised by incorrect parameterization or inconsistent bias conditions.