A self-consistent theory for the classical description of the interaction of light and matter at the nanoscale is presented, which takes into account spatial dispersion. Up to now, the Maxwell equations in nanostructured materials with spatial dispersion have been solved by the introduction of the so-called additional boundary conditions which, however, lack generality and uniqueness. In this paper, we derive an approach where nonlocal effects are studied in a precise and uniquely defined way, thus allowing the treatment of all solid-solid interfaces (among metals, semiconductors or insulators), as well as solid-vacuum interfaces in the same framework. The theory is based on the derivation of a potential energy for an ensemble of electrons in a given potential, where the deformation of the ensemble is treated as in a solid, including both shear and compressional deformations, instead of a fluid described only by a bulk compressibility like in the hydrodynamical approach. The derived classical equation of motion for the ensemble describes the deformation vector and the corresponding polarization vector as an elastodynamic field, including viscous forces, from which a generalized nonlocal constitutive equation for the dielectric constant is derived. The required boundary conditions are identical to that of elastodynamics and they emerge in a natural way, without any physical hypothesis outside the current description, as is commonly required in other nonlocal approaches. Interestingly, this description does not require the discontinuity of any component of the electric, magnetic, or polarization fields and, consequently, no bounded currents or charges are present at the interface, which is a more suitable description from the microscopic point of view. It is shown that the method converges to the local boundary conditions in the low spatial dispersion limit for insulators and conductors, quantified by means of a parameter defined as the characteristic length. A brief discussion about the inclusion of the spill out of electrons across surfaces is discussed. Finally, the planar interface is studied and numerical examples of the behavior of the different fields at the interfaces are presented, showing the limiting situations in which the local limit is recovered, reinforcing the self-consistency of this description.

• Received 19 December 2019
• Revised 10 September 2020
• Accepted 15 September 2020

DOI:https://doi.org/10.1103/PhysRevB.102.115308