Electrical polarization and orbital magnetization: the modern theories

Abstract

Macroscopic polarization P and magnetization M are the most fundamental concepts in any phenomenological descriptionof condensed media. They are intensive vector quantities that intuitivelycarry the meaning of dipole per unit volume. But for many years bothP and the orbitalterm in M evaded even a precise microscopic definition, and severely challenged quantum-mechanicalcalculations. If one reasons in terms of a finite sample, the electric (magnetic) dipole isaffected in an extensive way by charges (currents) at the sample boundary, due to thepresence of the unbounded position operator in the dipole definitions. ThereforeP and the orbitalterm in M—phenomenologically known as bulk properties—apparently behave as surfaceproperties; only spin magnetization is problemless. The field has undergone agenuine revolution since the early 1990s. Contrary to a widespread incorrect belief,P 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. Analogously, the orbital term inM has nothing to do with the periodic current distribution in the magnetized crystal. Themodern theory of polarization, based on a Berry phase, started in the early 1990s and isnow implemented in most first-principle electronic structure codes. The analogous theoryfor orbital magnetization started in 2005 and is partly work in progress. In theelectrical case, calculations have concerned various phenomena (ferroelectricity,piezoelectricity, and lattice dynamics) in several materials, and are in spectacularagreement with experiments; they have provided thorough understanding of the behaviourof ferroelectric and piezoelectric materials. In the magnetic case the very firstcalculations are appearing at the time of writing (2010). Here I review both theorieson a uniform ground in a density functional theory (DFT) framework, pointingout analogies and differences. Both theories are deeply rooted in geometricalconcepts, elucidated in this work. The main formulae for crystalline systems expressP andM in terms of Brillouin-zone integrals, discretized for numerical implementation. Ialso provide the corresponding formulae for disordered systems in a singlek-point supercellframework. In the case of P the single-point formula has been widely used in the Car–Parrinello community to evaluateIR spectra.