Size-dependent electric voltage controlled magnetic anisotropy in multiferroic heterostructures: Interface-charge and strain comediated magnetoelectric coupling

We present a phenomenological scheme to study the size-dependent electric voltage-controlled magnetic anisotropy in ferromagnetic (FM) and ferroelectric (FE) heterostructures. The FM layers are either metallic [Fe(001), Ni(001), Co(0001)] or half-metallic [(La, Sr)MnO${}_{3}$] films. Two magnetoelectric mechanisms, i.e., interface-charge and strain-mediated couplings, are considered. We show that the interface-charge-mediated coupling is the main mechanism for the magnetoelectic coupling when the FM film thickness is below a certain transition thickness ${d}_{\mathrm{tr}}$ while the strain-mediated coupling dominates above ${d}_{\mathrm{tr}}$.