  The formation of a skyrmion crystal and its phase transition are studied
taking into account the Dzyaloshinskii-Moriya (DM) interaction at the interface
between a ferroelectric layer and a magnetic layer in a superlattice. The
frustration is introduced in both magnetic and ferroelectric films. The films
have the simple cubic lattice structure. Spins inside magnetic layers are
Heisenberg spins interacting with each other via nearest-neighbor (NN) exchange
$J^m$ and next-nearest-neighbor (NNN) exchange $J^{2m}$. Polarizations in the
ferroelectric layers are assumed to be of Ising type with NN and NNN
interactions $J^f$ and $J^{2f}$. At the magnetoelectric interface, a DM
interaction $J^{mf}$ between spins and polarizations is supposed. The spin
configuration in the ground state is calculated by the steepest descent method.
In an applied magnetic field $\mathbf H$ perpendicular to the layers, we show
that the formation of skyrmions at the magnetoelectric interface is strongly
enhanced by the frustration brought about by the NNN antiferromagnetic
interactions $J^{2m}$ and $J^{2f}$. Various physical quantities at finite
temperatures are obtained by Monte Carlo simulations. We show the critical
temperature, the order parameters of magnetic and ferroelectric layers as
functions of the interface DM coupling, the applied magnetic field and $J^{2m}$
and $J^{2f}$. The phase transition to the disordered phase is studied in
details.\\ Keywords: kyrmions; phase transition; frustration; superlattice;
magneto-ferroelectric coupling; Dzyaloshinskii-Moriya interaction; $J_1-J_2$
model; Monte Carlo simulation.
