The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

Eric Cancès (1, 2) , Mathieu Lewin (3) , Gabriel Stoltz (1, 2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Science des matériaux [cond-mat.mtrl-sci]; Equations aux dérivées partielles [math.AP]

Eric Cancès, Mathieu Lewin, Gabriel Stoltz. The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint. Björn Engquist, Olof Runborg and Yen-Hsi R. Tsai. Numerical Analysis of Multiscale Computations. Proceedings of a Winter Workshop at the Banff International Research Station 2009, Springer, pp.87--125, 2011, Lecture Notes in Computational Science and Engineering, ⟨10.1007/978-3-642-21943-6⟩. ⟨hal-00527022⟩