A Real Space Approach to Uniqueness in Polarization

A fundamental issue in the atomic and quantum scale modeling of dielectric materials is the question of defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the dipole of the charge in a unit cell depends on the choice of the unit cell. We examine this issue using a rigorous approach based on the framework of two-scale convergence. Starting with a periodic charge density on a compact domain, we examine the continuum limit of lattice spacing going to zero. We prove that accounting for the boundaries consistently provides a route to uniquely compute electric fields and potentials, despite the non-unique polarization. Specifically, there are partial unit cells at the boundary which, not being charge-neutral, give rise to a surface charge. Different choices of the unit cell in the interior of the body leads to different partial unit cells at the boundary; the net effect is that these changes compensate each other. We also explain how the aforementioned polarization is connected to the “Free Energy Density” and the “Modern Theory of Polarization”/“Berry Phase” definitions of polarization. We show that using both these definitions, the potentials and bound charges are the same.