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 stateoftheart nanoplasmonic studies, exemplars of extremely
interfacelocalized 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 surfacerelated, 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
leadingorder 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 frequencydependent 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 outofplane and inplane
directions, respectively. The d parameters enable a
leadingorderaccurate incorporation of nonlocality, spillout, and
surfaceenabled Landau damping (tunnelling and size
quantization, which are not incorporated in the d parameters, are nonnegligible
at feature sizes below about 1 nm). Mesoscopic boundary conditions The d parameters drive an
effective nonclassical surface polarization with contributing an outofplane surface
dipole density π(r) and contributing an inplane 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 .
Numerical solver We implemented the mesoscopic boundary
conditions in a standard fullwave numerical solver COMSOL Multiphysics. Our
implementation and a few numerical examples are available at https://github.com/yiymit/nanoEM In the numerical implementations, we
include planewave scattering solutions for cylinders, bowtie antennas,
spheres, and filmcoupled nanodisks. They can be
generalized to other electromagnetic problems such as normal/quasinormal mode problems, spontaneous emission, nearfield
scanning microscopy, electron energy loss spectroscopy, and more.
We
establish a systematic approach to measure the d parameter
dispersion of a general twomaterial interface, and illustrate it for Au–AlO_{x}
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 darkfield 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 selfconsistent 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
filmcoupled Au nanodisk. e. Observation of large nonclassical
corrections (spectral shift ~400 nm) in filmcoupled 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/s4158601918031
Related works The following works are particularly
related: [1]
P.J. Feibelman,
Prog. Surf. Sci. 12,
287 (1982). A.
Liebsch, Electronic Excitations at Metal Surfaces, Physics of Solids and Liquids (Springer, 1997). [2]
P. Apell
and A. Ljungbert, Physica Scripta 26, 113 (1982). [3]
N. A. Mortensen, S. Raza,
M. Wubs, and S. I. Bozhevolnyi,
Nat. Commun. 5, 3809 (2014). [4]
W. Yan, M. Wubs, and N. A. Mortensen, Phys. Rev. Lett. 115, 137403
(2015). [5]
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). [6]
T. Christensen, W. Yan,
A.P. Jauho, M. Soljačić, and N.A. Mortensen, Phys. Rev. Lett. 118, 157402 (2017). [7]
Y. Yang, O.D. Miller, T.
Christensen, J.D. Joannopoulos, and M. Soljačić, Nano Lett. 17, 3238 (2017). [8]
P. Lalanne,
W. Yan, K. Vynck, C. Sauvan,
and J.P. Hugonin, Laser Photonics Rev. 12, 1700113 (2018). 