Nonfrustrated magnetoelectric with incommensurate magnetic order in magnetic field

We discuss a model nonfrustrated magnetoelectric in which strong enough magnetoelectric coupling produces incommensurate magnetic order leading to ferroelectricity. Properties of the magnetoelectric in magnetic field directed perpendicular to wave vector describing the spin helix are considered in detail. Analysis of classical energy shows that in contrast to naive expectation the onset of ferroelectricity takes place at a field $H_{c1}$ that is lower than the saturation field $H_{c2}$. One has $H_{c1}=H_{c2}$ at strong enough magnetoelectric coupling. We show that at H=0 the ferroelectricity appears at $T=T_{FE}<T_N$. Qualitative discussion of phase diagram in $H-T$ plane is presented within mean field approach.