Upper Bound on the Magnetoelectric Susceptibility

The change in the free energy that occurs when electric and magnetic fields are simultaneously applied to a magnetoelectric medium is calculated. It is shown that a quadratic form related to this change in the free energy is positive definite, from which it follows that all elements of the magnetoelectric-susceptibility tensor must be smaller than the geometric mean of appropriate elements of the magnetic- and electric-susceptibility tensors. It is pointed out that the diamagnetic contribution to the magnetic-susceptibility tensor is negligible in materials in which the magnetoelectric effect is allowed. It is concluded that the magnetoelectric susceptibility should be small compared with unity, except possibly in ferroelectric or ferromagnetic materials.