Multiferroics, materials where spontaneous long-range magnetic and dipolar orders coexist, represent an attractive class of compounds, which combine rich and fascinating fundamental physics with a technologically appealing potential for applications in the general area of spintronics. Ab initio calculations have significantly contributed to recent progress in this area, by elucidating different mechanisms for multiferroicity and providing essential information on various compounds where these effects are manifestly at play. In particular, here we present examples of density-functional theory investigations for two main classes of materials: (a) multiferroics where ferroelectricity is driven by hybridization or purely structural effects, with BiFeO3 as the prototype material, and (b) multiferroics where ferroelectricity is driven by correlation effects and is strongly linked to electronic degrees of freedom such as spin-, charge-, or orbital-ordering, with rare-earth manganites as prototypes. As for the first class of multiferroics, first principles calculations are shown to provide an accurate qualitative and quantitative description of the physics in BiFeO3, ranging from the prediction of large ferroelectric polarization and weak ferromagnetism, over the effect of epitaxial strain, to the identification of possible scenarios for coupling between ferroelectric and magnetic order. For the second class of multiferroics, ab initio calculations have shown that, in those cases where spin-ordering breaks inversion symmetry (e.g. in antiferromagnetic E-type HoMnO3), the magnetically induced ferroelectric polarization can be as large as a few µC cm−2. The examples presented point the way to several possible avenues for future research: on the technological side, first principles simulations can contribute to a rational materials design, aimed at identifying spintronic materials that exhibit ferromagnetism and ferroelectricity at or above room temperature. On the fundamental side, ab initio approaches can be used to explore new mechanisms for ferroelectricity by exploiting electronic correlations that are at play in transition met al oxides, and by suggesting ways to maximize the strength of these effects as well as the corresponding ordering temperatures.
Macroscopic electric polarization is a fundamental concept in the physics of matter, upon which the phenomenological description of dielectrics is based (Landau and Lifshitz, 1984). Notwithstanding, this concept has long evaded even a precise microscopic definition. A typical incorrect statement — often found in textbooks — is that the macroscopic polarization of a solid is the dipole of a unit cell. It is easy to realize that such a quantity is neither measurable nor model-independent: the dipole of a periodic charge distribution is in fact ill defined (Martin, 1974), except in the extreme Clausius-Mossotti model, in which the total charge is unambiguously decomposed into an assembly of ocalized and neutral charge distributions. One can adopt an alternative viewpoint by considering a macroscopic and finite piece of matter and defining its polarization P as the dipole per unit volume: 1 We consider the change in polarization P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation.