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The formation of a skyrmion crystal and its phase transition are studiedtaking into account the Dzyaloshinskii-Moriya (DM) interaction at the interfacebetween a ferroelectric layer and a magnetic layer in a superlattice. Thefrustration is introduced in both magnetic and ferroelectric films. The filmshave the simple cubic lattice structure. Spins inside magnetic layers areHeisenberg spins interacting with each other via nearest-neighbor (NN) exchange$J^m$ and next-nearest-neighbor (NNN) exchange $J^{2m}$. Polarizations in theferroelectric layers are assumed to be of Ising type with NN and NNNinteractions $J^f$ and $J^{2f}$. At the magnetoelectric interface, a DMinteraction $J^{mf}$ between spins and polarizations is supposed. The spinconfiguration in the ground state is calculated by the steepest descent method.In an applied magnetic field $\mathbf H$ perpendicular to the layers, we showthat the formation of skyrmions at the magnetoelectric interface is stronglyenhanced by the frustration brought about by the NNN antiferromagneticinteractions $J^{2m}$ and $J^{2f}$. Various physical quantities at finitetemperatures are obtained by Monte Carlo simulations. We show the criticaltemperature, the order parameters of magnetic and ferroelectric layers asfunctions of the interface DM coupling, the applied magnetic field and $J^{2m}$and $J^{2f}$. The phase transition to the disordered phase is studied indetails.\\ Keywords: kyrmions; phase transition; frustration; superlattice;magneto-ferroelectric coupling; Dzyaloshinskii-Moriya interaction; $J_1-J_2$model; Monte Carlo simulation.

Skyrmion Crystal and Phase Transition in Magneto-Ferroelectric Superlattices: Dzyaloshinskii-Moriya Interaction in a Frustrated $J_1-J_2$ Model