Nonlocal Dipolar Interactions in Complex Geometries for Quantum Embedding

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 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 – allowing even a change in the sign! – 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 typically do not, and the net effect is for these changes to compensate for each other.

Broadly, the view advocated in this work is in the spirit of classical continuum mechanics; the polarization field is a multiscale mediator that captures some information from the atomic level, and one can take any choice as long as consistent transformations between energy, kinematics, and boundaries are respected. The immediate analog in continuum mechanics is the freedom in the choice of reference configuration and the corresponding value of the deformation field and strain energy density response function as long as care is taken to define suitable transformations between different choices. The inadequacies in the definition of polarization have received much attention over the years. This has given rise to alternative definitions which have attempted to present a unique definition for polarization. The first, popular amongst physicists,goes by the Modern Theory of Polarization and defines the rate of polarization as the macroscopic current flowing in the crystal. Assuming the change is adiabatic, the current is evaluated as the Berry phase of the wave function. The latter, popular amongst mechanicians, goes by the name of the energetic definition of polarization. Here the polarization is the derivative of the energy density with respect to the electric field. The above definitions of polarization do not suffer from non-uniqueness, which motivates us to connect the three definitions of polarization. Namely, we seek to connect the dipole definition of polarization to the energetic and Modern Theory of Polarization separately. We show that the energetic definition of polarization comes with a surface energy term that describes the surface charge distribution, compensating for the non-unique polarization analogous to the way surface charge distribution compensates for the polarization in the dipole definition of polarization. The Berry-phase definition of polarization evaluates to the same bound charge as the classical definition of polarization. Having defined polarization for the general situation, we extend the definition of polarization to 2D bodies. We are interested in the flexoelectric response of 2D materials. To this end, we consider the thermodynamic limit of Thomas-Fermi Density Functional Theory model and show that the Coulomb energy functional has a well-defined limit in terms of the polarization in the system. We show that the energy functional is uniquely defined irrespective of the non-uniqueness in polarization. The main advantage of this result is that it sheds light on embedding methods.