  Magnetoelectric effects are investigated ab-initio at the interface between
half-metallic and ferroelectric prototypes: Heusler Co$_2$MnSi and perovskite
BaTiO$_3$. For the Co-termination ferroelectricity develops in BaTiO$_3$ down
to nanometer thicknesses, whereas for the MnSi-termination a paraelectric and a
ferroelectric state energetically compete, calling for a full experimental
control over the junction atomic configuration whenever a ferroelectric barrier
is needed. Switch of the electric polarization largely affects magnetism in
Co$_2$MnSi, with magnetoelectric coupling due to electronic hybridization at
the MnSi termination and to structural effects at the Co-termination.
Half-metallicity is lost at the interface, but recovered already in the
subsurface layer.
