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This work develops a multiscale modeling framework for defects in crystals with general geometries and boundary conditions in which ionic interactions are important, with potential application to, e.g., ionic solids and electric field interactions with materials. The overall strategy is posed in the framework of the Quasicontinuum multiscale method; specifically, the use of a finite-element inspired kinematic description enables a significant reduction in the large number of degrees of freedom to describe the atomic positions. The key advance of this work is a method for the efficient and accurate treatment of nonlocal electrostatic charge-charge interactions without restrictions on the geometry or boundary conditions. Electrostatic interactions are long-range with slow decay, and hence require consideration of all pairs of charges making a brute-force approach computationally prohibitive. The method proposed here accounts for the exact charge-charge interactions in the near-field and uses a coarse-grained approximation in the far-field. The coarse-grained approximation and the associated errors are rigorously derived based on the limit of a finite body with a small periodic lengthscale, thereby enabling the errors in the approximation to be controlled to a desired tolerance. The method is applied to a simple model of Gallium Nitride and it is shown that electrostatic interactions can be approximated with a desired level of accuracy using the proposed methodology.
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This content will become publicly available on September 19, 2026
Nonlocal electrostatics and boundary charges in continuum limits of two-dimensional materials
Two-dimensional (2D) electronic materials are of significant technological interest due to their exceptional properties and broad applicability in engineering. The transition from nanoscale physics, which dictates their stable configurations, to macroscopic engineering applications requires the use of multiscale methods to systematically capture their electronic properties at larger scales. A key challenge in coarse-graining is the rapid and near-periodic variation of the charge density, which exhibits significant spatial oscillations at the atomic scale. Therefore, the polarization density field—the first moment of the charge density over the periodic unit cell—is used as a multiscale mediator that enables efficient coarse-graining by exploiting the almost-periodic nature of the variation. Unlike the highly oscillatory charge density, the polarization varies over lengthscales that are much larger than the atomic, making it suitable for continuum modeling. In this paper, we investigate the electrostatic potential arising from the charge distribution of arbitrarily-deformed 2D materials. Specifically, we consider a sequence of problems wherein the underlying lattice spacing vanishes and thus obtain the continuum limit. We consider three distinct limits: where the thickness is much smaller than, comparable to, and much larger than the in-plane lattice spacing. These limiting procedures provide the homogenized potential expressed in terms of the boundary charge and dipole distribution, subject to the appropriate boundary conditions that are also obtained through the limit process. Furthermore, we demonstrate that despite the intrinsic non-uniqueness in the definition of polarization, accounting for the boundary charges ensures that the total electrostatic potential, the associated electric field, and the corresponding energy of the homogenized system are uniquely determined.
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- Award ID(s):
- 2108784
- PAR ID:
- 10637268
- Publisher / Repository:
- Sage
- Date Published:
- Journal Name:
- Mathematics and Mechanics of Solids
- ISSN:
- 1081-2865
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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