  The general approach for the consideration of the magnetoelectric effects in
ferroic nanorods is proposed in the framework of the phenomenological theory.
The intrinsic surface stress, magneto- and electrostriction as well as
piezoelectric and piezomagnetic effects are included into the free energy. The
intrinsic surface stress under the curved nanoparticle surface is shown to play
an important role in the shift of ferroelectric and ferromagnetic transition
temperatures and built-in magnetic and electric fields appearance, which are
inversely proportional to the nanorod radius. We consider the case of quadratic
and linear magnetoelectric coupling coefficients. The linear coupling
coefficient is radius independent, whereas the quadratic ones include terms
inversely proportional to the nanorod radius and thus strongly increase with
decrease of the radius. The predicted giant relative dielectric tunability in
the vicinity of ferromagnetic and ferroelectric phase transition points induced
by quadratic magnetoelectric coupling increases by 2-50 times. The quadratic
magnetoelectric coupling dramatically changes the phase diagrams of ferroic
nanorods when the radius decreases. In particular the second order phase
transition may become a first one, the triple point state characterized by
continuous set of order parameters appears at zero external electric and
magnetic fields and the tricritical points appear under external fields.
