  Magnetoelastic coupling, i.e., the change of crystal lattice induced by a
spin order, is not only scientifically interesting, but also technically
important. In this work, we propose a general microscopic model from
first-principles calculations to describe the magnetoelastic coupling and
provide a way to construct the microscopic model from density functional theory
calculations. Based on this model, we reveal that there exists a previously
unexpected contribution to the electric polarization induced by the spin-order
in multiferroics due to the combined effects of magnetoelastic coupling and
piezoelectric effect. Interestingly and surprisingly, we find that this lattice
deformation contribution to the polarization is even larger than that from the
pure electronic and ion-displacement contributions in BiFeO3. This model of
magnetoelastic coupling can be generally applied to investigate the other
magnetoelastic phenomena.
