A general theoretical and experimental framework for nanoscale electromagnetism
Background on nonclassical corrections in nanophotonics
Local, bulk response functions, e.g. permittivity, and the macroscopic Maxwell equations completely specify the classical electromagnetic problem, which features only wavelength and geometric scales. They have proven extremely successful at macroscopic length scales, across all branches of photonics. Even state-of-the-art nanoplasmonic studies, exemplars of extremely interface-localized fields, rely on their validity.
This classical description, however, neglects the intrinsic electronic length scale associated with interfaces. This omission leads to significant discrepancies between classical predictions and experimental observations in systems with deeply nanoscale feature sizes, typically evident below ∼10 – 20 nm.
The d parameters, first introduced
by Feibelman, are a convenient mathematical
parametrization for surface-related, quantum corrections. They can be derived
from a careful analysis of the reflection of an external potential off a
planar interface by going beyond the conventional assumptions of local and
stepwise material response: the d parameters then introduce the
leading-order corrections to the classical reflection coe
The Feibelman and parameters play a role analogous to the local bulk permittivity, but for interfaces between two materials. and are equal to the frequency-dependent centroids of the induced charge and the normal derivative of the tangential current, respectively, at an equivalent planar interface. Mathematically, their definitions are given by
where is the induced charge, is the induced current, and and are out-of-plane and in-plane directions, respectively.
The d parameters enable a leading-order-accurate incorporation of nonlocality, spill-out, and surface-enabled Landau damping (tunnelling and size quantization, which are not incorporated in the d parameters, are non-negligible at feature sizes below about 1 nm).
Mesoscopic boundary conditions
The d parameters drive an effective nonclassical surface polarization with contributing an out-of-plane surface dipole density π(r) and contributing an in-plane surface current density K(r).
These surface terms can be equivalently incorporated as a set of mesoscopic boundary conditions (here without external interface currents or charges) for the conventional macroscopic Maxwell equations, as summarized below. Evidently, the mesoscopic boundary conditions reduce to the classical boundary conditions in the limit .
We implemented the mesoscopic boundary conditions in a standard full-wave numerical solver COMSOL Multiphysics. Our implementation and a few numerical examples are available at https://github.com/yiy-mit/nanoEM
In the numerical implementations, we include plane-wave scattering solutions for cylinders, bowtie antennas, spheres, and film-coupled nanodisks. They can be generalized to other electromagnetic problems such as normal/quasinormal mode problems, spontaneous emission, near-field scanning microscopy, electron energy loss spectroscopy, and more.
We establish a systematic approach to measure the d parameter dispersion of a general two-material interface, and illustrate it for Au–AlOx interfaces.
We translate the mesoscopic d parameter directly into observables—spectral shifting and broadening—and measure them in specially designed plasmonic systems that exhibit pronounced nonclassical corrections. Our experimental testbed enables a direct procedure to extract d parameters from standard dark-field measurements, in a manner analogous to ellipsometric measurements of the local bulk permittivity.
Moreover, by investigating a complementary hybrid plasmonic setup, we discover and experimentally demonstrate design principles for structures that are classically robust—i.e. exhibit minimal nonclassical corrections—even under nanoscopic conditions.
Framework, experimental structure, measured nonclassical shifts, and surface response
dispersion. a. Equilibrium and induced densities. is the centroid of induced charge.
b. Nonclassical corrections can be formulated as self-consistent surface polarizations, representing effective surface dipole density π(r) and current density K(r). c. Experimental structure. d. Nonclassical surface dipole density π(r) of the fundamental dipolar gap plasmon of a film-coupled Au nanodisk. e. Observation of large nonclassical corrections (spectral shift ~400 nm) in film-coupled Au nanodisks. Measured frequencies (circles) of the resonance blueshift relative to the classical prediction (dashed line) and quantitatively agree with nonclassical calculations. f–g. Measured (markers) dispersion of (f), (g), and their linear fits (lines).
Yi Yang, Di Zhu, Wei Yan, Akshay Agarwal, Mengjie Zheng, John D. Joannopoulos, Philippe Lalanne, Thomas Christensen, Karl K. Berggren, Marin Soljačić, arXiv: 1901.03988. Nature. doi:10.1038/s41586-019-1803-1
The following works are particularly related:
 P.J. Feibelman, Prog. Surf. Sci. 12, 287 (1982).
A. Liebsch, Electronic Excitations at Metal Surfaces, Physics of Solids and Liquids (Springer, 1997).
 P. Apell and A. Ljungbert, Physica Scripta 26, 113 (1982).
 N. A. Mortensen, S. Raza, M. Wubs, and S. I. Bozhevolnyi, Nat. Commun. 5, 3809 (2014).
 W. Yan, M. Wubs, and N. A. Mortensen, Phys. Rev. Lett. 115, 137403 (2015).
 W. Zhu, R. Esteban, A.G. Borisov, J.J. Baumberg, P. Nordlander, H.J. Lezec, J. Aizpurua, and K.B. Crozier, Nat. Commun. 7, 11495 (2016).
 T. Christensen, W. Yan, A.-P. Jauho, M. Soljačić, and N.A. Mortensen, Phys. Rev. Lett. 118, 157402 (2017).
 Y. Yang, O.D. Miller, T. Christensen, J.D. Joannopoulos, and M. Soljačić, Nano Lett. 17, 3238 (2017).
 P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Laser Photonics Rev. 12, 1700113 (2018).