  The ferroelectricity of multiferroics induced by spin order is commonly
explained by considering either purely electronic or ion-displacement
contribution. However, there is no general model which includes both effects
simultaneously. Here, we suggest a realistic model to describe the
ion-displacement part of the ferroelectricity based on the spin-lattice
coupling Hamiltonian. Combining this model with our previous pure electronic
model for spin-order induced polarization, we propose a unified model that
includes both effects. By applying the unified model to representative
multiferroics where the electronic and ion-displacement contributions vary
widely, we find that this model can not only reproduce the first-principles
results, but also provide insight into the origin of ferroelectricity.
