Uniqueness of the Polarization in Crystals and Nanostructures*

Ionic crystals such as solid electrolytes and complex oxides are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum scale modeling of these materials is the question of defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell is found to depend qualitatively and quantitatively on the choice of the unit cell.
We examine this issue using a rigorous approach based on the framework of two-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we prove that accounting for the boundaries consistently provides a route to uniquely compute electric fields and potentials despite the non-uniqueness of the polarization. Specifically, different choices of the unit cell in the interior of the body leads to correspondingly different partial unit cells at the boundary; while the interior unit cells satisfy charge neutrality, the partial cells on the boundary need not; the net effect is that these changes compensate each other.