We study in this paper effects of Dzyaloshinskii-Moriya (DM) magnetoelectric coupling between ferroelectric and magnetic layers in a superlattice formed by alternate magnetic and ferroelectric films. Magnetic films are films of simple cubic lattice with Heisenberg spins interacting with each other via an exchange J and a DM interaction with the ferroelectric interface. Electrical polarizations of ±1 are assigned at simple cubic lattice sites in the ferroelectric films. We determine the ground-state (GS) spin configuration in the magnetic film. In zero field, the GS is periodically non collinear and in an applied field H perpendicular to the layers, it shows the existence of skyrmions at the interface. Using the Green's function method we study the spin waves (SW) excited in a monolayer and also in a bilayer sandwiched between ferroelectric films, in zero field. We show that the DM interaction strongly affects the long-wave length SW mode. We calculate also the magnetization at low temperature T . We use next Monte Carlo simulations to calculate various physical quantities at finite temperatures such as the critical temperature, the layer magnetization and the layer polarization, as functions of the magnetoelectric DM coupling and the applied magnetic field. Phase transition to the disordered phase is studied in detail.

line: magnetization of interface magnetic layers, green line: magnetization of interior magnetic layers, blue line: polarization of interface ferroelectric layers, purple line: polarization of interface ferroelectric layers ; (c) Susceptibilities versus T , with the same color code. J m = 1, J mf 1 = −0.15, J mf 2 = −0.135 corresponding to the GS with energy E 1 . B. Particular Case J mf 1 = J mf 2In this section we present the results of the MC simulations in the particular case where J mf 1 = J mf 2 . We will compare these results with results from the MF theory.Results for the temperature dependence of layer magnetizations, polarizations, and susceptibilities of the system are shown in Fig. 6 for J m = J f = 1, J mf = J mf 1 = J mf 2 = −0.15, −0.55. For J mf , these curves present a sharp second-order transitions at T m c 1.32 for magnetic layers and T f c 1.84 for ferroelectric layers.The results with an external magnetic field H z = 0.7 are shown in Fig. 7 for the order

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