Dielectric Polarization of Materials: A Modern View
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Dielectric Polarization of Materials: A Modern View Ra aele Resta Istituto Nazionale di Fisica della Materia (INFM) and Dipartimento di Fisica Teorica, Universita di Trieste, I-34014 Trieste, Italy; Max-Planck-Institut fur Festkorperforschung, D-70506 Stuttgart, Germany.
ABSTRACT The concept of macroscopic polarization is the basic one in the electrostatics of dielectric materials: but for many years this concept has evaded even a precise microscopic de nition, and has severely challenged quantum-mechanical calculations. This concept has undergone a genuine revolution in recent years (1992 onwards). It is now pretty clear that|contrary to a widespread incorrect belief|macroscopic polarization has nothing to do with the periodic charge distribution of the polarized crystal: the former is essentially a property of the phase of the electronic wavefunction, while the latter is a property of its modulus. An outline of the modern viewpoint is presented. Experiments invariably address polarization derivatives (permittivity, piezoelectricity, pyroelectricity,...) or polarization di erences (ferroelectricity), and these di erences are measured as an integrated electrical current. The modern theory addresses this same current, which is dominated by the phase of the electronic wavefunctions. First-principle calculations based on this theory are in spectacular agreement with experiments, and provide thorough understanding of the behavior of dielectric materials.
INTRODUCTION The dipole moment of any nite N {electron system in its ground state is a simple and well de ned quantity. Given the single{particle density n(r) the electronic contribution to the dipole is: Z eN hri = e dr r n(r); (1) where e is the electron charge. This looks very trivial, but we are exploiting here an essential fact: the ground wavefunction of any nite N {electron system is square{ integrable and vanishes exponentially at in nity; the density vanishes exponentially as well. Considering now a macroscopic solid, the related quantity is macroscopic polarization, which is a very essential concept in any phenomenological description of dielectric media [1]: this quantity is ideally de ned as the dipole of a macroscopic sample, divided by its volume. The point is that, when using Eq. (1), the integral is dominated by what happens at the surface of the sample: knowledge of the electronic distribution in the bulk region is not enough to unambiguosly determine the dipole. This looks like a paradox, since in the thermodynamic limit macroscopic polarization must be an intensive quantity,
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insensitive to surface e ects. Macroscopic polarization in the bulk region of the solid must be determined by what \happens" in the bulk as well. This is the case if one assumes a model of discrete and well separated dipoles, a la Clausius-Mossotti: but real dielectrics are very much di erent from such an extreme model. The valence electronic distribution is continuous, and often very delocalized (particularly in covalent dielectrics). Most textbooks attempt at e
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