Intrinsic magnetoelectric coupling describes the interaction between magnetic
and electric polarization through an inherent microscopic mechanism in a single
phase material. This phenomenon has the potential to control the magnetic state
of a material with an electric field, an enticing prospect for device
engineering. We demonstrate 'giant' magnetoelectric cross-field control in a
single phase rare earth titanate film. In bulk form, EuTiO3 is
antiferromagnetic. However, both anti and ferromagnetic interactions coexist
between different nearest neighbor europium ions. In thin epitaxial films,
strain can be used to alter the relative strength of the magnetic exchange
constants. Here, we not only show that moderate biaxial compression
precipitates local magnetic competition, but also demonstrate that the
application of an electric field at this strain state, switches the magnetic
ground state. Using first principles density functional theory, we resolve the
underlying microscopic mechanism resulting in the EuTiO3 G-type magnetic
structure and illustrate how it is responsible for the 'giant' cross-field
magnetoelectric effect