# Interrogating Quantum Nonlocal Effects in Nanoplasmonics through Electron-
Beam Spectroscopy

P. A. D. Gonçalves<EMAIL_ADDRESS>ICFO – Institut de Ciències
Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Barcelona, Spain  F. Javier García de Abajo
<EMAIL_ADDRESS>ICFO – Institut de Ciències Fotòniques,
The Barcelona Institute of Science and Technology, 08860 Castelldefels,
Barcelona, Spain  ICREA – Institució Catalana de Recerca i Estudis Avançats,
Passeig Lluís Companys 23, 08010 Barcelona, Spain

###### Abstract

A rigorous account of quantum nonlocal effects is paramount for understanding
the optical response of metal nanostructures and for designing plasmonic
devices at the nanoscale. Here, we present a scheme for retrieving the quantum
surface response of metals, encapsulated in the Feibelman $d$-parameters, from
electron energy-loss spectroscopy (EELS) and cathodoluminescence (CL)
measurements. We theoretically demonstrate that quantum nonlocal effects have
a dramatic impact on EELS and CL spectra, in the guise of spectral shifts and
nonlocal damping, when either the system size or the inverse wave vector in
extended structures approach the nanometer scale. Our concept capitalizes on
the unparalleled ability of free-electrons to supply deeply subwavelength
near-fields and, thus, probe the optical response of metals at length scales
in which quantum-mechanical effects are apparent. These results pave the way
for a widespread use of the $d$-parameter formalism, thereby facilitating a
rigorous yet practical inclusion of nonclassical effects in nanoplasmonics.

The optical response of few-nanometer-scale plasmonic structures, such as
those engineered with state-of-the-art nanofabrication techniques, can exhibit
substantial quantum nonlocal effects associated with the inherently quantum
mechanical nature of the plasmon-supporting electron gas in the involved
materialsZhu et al. (2016); Feibelman (1982); Kreibig and Vollmer (1995);
Cirací et al. (2012); Savage et al. (2012); Scholl et al. (2012); Raza et al.
(2013, 2015); Teperik et al. (2013); Varas et al. (2016); Zhang et al. (2014);
Christensen et al. (2017); Campos et al. (2019); Yang et al. (2019); Gonçalves
et al. (2020); Baghramyan et al. (2021). Broadly speaking, the impact of
nonclassical effects becomes non-negligible when either the characteristic
size of the system falls below $\sim 1020\text{\,}\mathrm{nm}$ or the optical
response is mediated by field components of large momenta such as those
produced by confined near-field confinement. Hence, a quantum nonlocal
description of the underlying plasmon-mediated light–matter interaction is
required in order to explain experimental data as well as to draw insight into
the elementary processes governing that interaction in the few-nanometer
regime.

Since an all-encompassing quantum-mechanical treatment of the many-electron
system [e.g., using time-dependent density-functional theory Marques et al.
(2006) (TDDFT)] is severely constrained to few-atom clusters much smaller than
the typical nanoplasmonic systems exploited in experiments, in practice it is
necessary to resort to quantum-informed models that incorporate dominant
quantum effects to leading-order Zhu et al. (2016); Pitarke et al. (2007);
David and García de Abajo (2011); Mortensen (2021). Among these, the Feibelman
$d$-parameter formalism Feibelman (1982) is particularly appealing because it
simultaneously incorporates electron spill-out/spill-in, nonlocality (i.e.,
momentum-dependent response), and surface-enabled Landau damping through the
introduction of two microscopic surface-response functions,
$d_{\perp}(\omega)=\int\mathrm{d}z\,z\,\rho_{\mathrm{ind}}(z,\omega)/\int\mathrm{d}z\,\rho_{\mathrm{ind}}(z,\omega)$
and
$d_{\parallel}(\omega)=\int\mathrm{d}z\,z\,\partial_{z}J_{\parallel,\mathrm{ind}}(z,\omega)/\int\mathrm{d}z\,\partial_{z}J_{\parallel,\mathrm{ind}}(z,\omega)$,
corresponding to the centroids of the induced charge density along the surface
normal $\mathbf{\bf\hat{z}}$ and of the normal derivative of the current
parallel to the interface, respectively. Once they are known for the planar
dielectric–metal interface(s) of interest, these parameters allow the
incorporation of the above-mentioned nonclassical effects in the optical
response of metallic nanostructures using standard electromagnetic solvers
upon replacing the macroscopic boundary conditions Jackson (1975) by their
$d$-parameter-corrected counterparts Yang et al. (2019); Gonçalves et al.
(2020); Echarri et al. (2021); Zhou et al. (2022); Yan and Qiu (2022);
Hohenester and Unger (2022); Babaze et al. (2022); Tao et al. (2022).
Naturally, this procedure relies on our ability to compute the $d$-parameters
in the first place using, for example, linear-response TDDFT. However, while
simple metals (e.g., alkali metals or aluminum) can be well-described by
jellium-level TDDFT, for which accurate $d$-parameter data exist Feibelman
(1982); Liebsch (1987); Kempa et al. (1988); Liebsch (1997); Christensen et
al. (2017), noble metals such as gold and silver require a more demanding
atomistic treatment beyond the jellium approximation due to valence-electron
screening from the lower-lying bands Liebsch (1997); Suto et al. (1989). While
valence-band screening may be semiclassically included using screened jellium
models Liebsch (1993, 1997) containing a polarizable background contribution,
such approaches still lead to _quantitatively_ unsatisfactory predictions for
the $d$-parameters (see, for instance, the discussion in the supplementary
information of refs. 12 and 14). As a result of this, and despite the
relevance of noble metals in nanoplasmonics, quantitatively accurate
$d$-parameter data remains elusive, thus limiting the widespread use of the
$d$-parameter framework.

Here, we propose and demonstrate a scheme in which electron-beam (e-beam)
spectroscopies García de Abajo (2010); Polman et al. (2019) are employed to
determine the quantum surface response (i.e., the $d$-parameters) of metals
directly from experimental spectra (Fig. 1). To that end, we present a
quantum-corrected theory of electron energy-loss spectroscopy Egerton (1996);
García de Abajo (2010); Polman et al. (2019) (EELS) and cathodoluminescence
Kociak and Zagonel (2017); García de Abajo (2010); Polman et al. (2019) (CL)
based on the aforementioned quantum surface-response formalism and use it to
infer $d_{\perp}$ and $d_{\parallel}$ from the measured spectra by quantifying
the size- or wave-vector-dependent spectral shifting and broadening due to
quantum nonlocal effects. Crucial to this is the ability of e-beams to produce
broadband and highly confined near-fields García de Abajo (2010), which may be
tailored by, for example, varying the electron kinetic energy or controlling
the e-beam trajectory. Such fields contain evanescent components that allow
free electrons to efficiently couple to strongly confined optical excitations
in materials and retrieve sub-nanometer spatial information, thus rendering
them first-class probes of nonclassical effects in nanoplasmonics Scholl et
al. (2012); Raza et al. (2013, 2015); Campos et al. (2019). Our work opens an
powerful route toward a better quantitative understanding of the nonclassical
optical response of metallic nanostructures, which is instrumental from a
fundamental viewpoint and constitutes a key ingredient in the design of
nanophotonic devices operating at the few-nanometer scale.

Figure 1: Probing quantum effects in nanoplasmonics with electron-beam
(e-beam) spectroscopy. (a) Conceptual approach underpinning the Feibelman
$d$-parameter formalism, wherein a microscopic, quantum mechanical description
of a dielectric–metal interface is mapped onto a mesoscopic one that is
tantamount to a classical treatment augmented by a set of quantum surface-
response functions, $d_{\perp}$ and $d_{\parallel}$, encapsulating the
leading-order corrections to classicality. (b) Schematics of metallic quantum
surface response encoded in the $d$-parameters and probed via EELS and CL
spectroscopies.

We begin our analysis by considering the canonical scenario of a swift
electron moving with constant velocity $v$ along a straight-line trajectory
$\mathbf{r}_{e}(t)$ parallel to a metal surface placed at $z=0$. Taking
$\mathbf{v}=v\,\mathbf{\hat{x}}$ and $\mathbf{r}_{e}(t)=(v\,t,0,b)$, with $b$
defining the electron–surface separation, and assuming that the medium
adjacent to the metal is a lossless dielectric with relative permittivity
$\epsilon_{\mathrm{d}}$, the spectral EELS probability experienced by the
electron after traveling a length $L$ reads García de Abajo (2010)

$\displaystyle\Gamma_{\text{{EELS}}}(\omega)=\frac{2\alpha L}{\pi
c\,\beta^{2}}\int_{0}^{\infty}\frac{\text{d}k_{y}}{q^{2}}\operatorname{Re}\left\\{\mathrm{e}^{\mathrm{i}2k_{z,\mathrm{d}}b}\left[\frac{k_{y}^{2}\,\beta^{2}}{k_{z,\mathrm{d}}}r_{s}-\frac{k_{z,\mathrm{d}}}{\epsilon_{\mathrm{d}}}r_{p}\right]\right\\},$
(1)

where $\beta=v/c$ is the normalized electron velocity, $\alpha\simeq 1/137$
the fine-structure constant, and $q=\sqrt{\omega^{2}/v^{2}+k_{y}^{2}}$ and
$k_{z,\mathrm{d}}=\sqrt{\epsilon_{\mathrm{d}}\,\omega^{2}/c^{2}-q^{2}}$ (with
$\operatorname{Im}\\{k_{z,\mathrm{d}}\\}\geq 0$) stand for the in-plane and
out-of-plane wave vector components, respectively. The quantum surface
response enters Eq. (1) through the reflection coefficients for $s$\- and
$p$-polarized waves, $r_{s}\equiv r_{s}(q,\omega)$ and $r_{p}\equiv
r_{p}(q,\omega)$, respectively. The EELS probability expressed in the form of
Eq. (1) is thus particularly convenient to incorporate quantum nonlocal
effects by simply employing the $d$-parameter-corrected reflection
coefficients Feibelman (1982); Yang et al. (2019); Gonçalves et al. (2020);
Gonçalves (2020) (see Methods) instead of their classical counterparts, which
are reinstated in the $d_{\perp,\parallel}\to 0$ limit.

Incidentally, $d_{\parallel}$ vanishes for charge-neutral surfaces Feibelman
(1982); Liebsch (1997), thereby leaving $d_{\perp}$ as the only quantity
embodying quantum mechanical corrections in the present context, where we take
$d_{\parallel}=0$. We consider both jellium-like and noble metals (as their
nonclassical optical response is distinct), herein represented, respectively,
by a jellium with density parameter $r_{s}=4$ (corresponding to the plasma
energy $\hbar\omega_{\mathrm{p}}\approx$5.89\text{\,}\mathrm{eV}$$ for sodium
Ashcroft and Mermin (1976)) and silver. For the former, we use the frequency-
dependent $d_{\perp}$ calculated from TDDFT Christensen et al. (2017) for an
air–jellium interface (see SI), whereas for silver we incorporate a
surrounding dielectric with $\epsilon_{\mathrm{d}}=2$ (simulating
$\mathrm{SiO}_{2}$, which protects it from oxidation) and take
$d_{\perp}=(-0.4+0.2\,\mathrm{i})\,\mathrm{nm}$. This value is estimated by
fitting its real part to experimental measurements of size-dependent resonance
shifts Raza et al. (2013), while its imaginary part is set so that it
reproduces the phenomenological Kreibig damping Kreibig and Fragstein (1969)
(see SI for details). The classical optical response of silver is modeled
through a Drude-type dielectric function
$\epsilon_{\mathrm{m}}(\omega)=\epsilon_{\mathrm{b}}(\omega)-\omega_{\mathrm{p}}^{2}/(\omega^{2}+\mathrm{i}\gamma\omega)$,
where $\hbar\omega_{\mathrm{p}}=$9.02\text{\,}\mathrm{eV}$$ and
$\hbar\gamma=$22\text{\,}\mathrm{meV}$$ describe the conduction electrons,
whereas screening due to bound electrons is included via
$\epsilon_{\mathrm{b}}(\omega)=\epsilon_{\mathrm{m}}^{\mathrm{exp}}(\omega)+\omega_{\mathrm{p}}^{2}/(\omega^{2}+\mathrm{i}\gamma\omega)$
with $\epsilon_{\mathrm{m}}^{\mathrm{exp}}(\omega)$ taken from experimental
data Johnson and Christy (1972).

Figure 2: Nonclassical corrections to the EELS for an aloof electron parallel
to a planar metal surface. (a) Schematics of the configuration under
consideration. (b) Classical and quantum EELS spectra for an electron
traveling in air ($\epsilon_{\mathrm{d}}=1$) above a planar jellium surface
($r_{s}=4$, with $\hbar\omega_{\mathrm{p}}\simeq$5.89\text{\,}\mathrm{eV}$$
and $\gamma=\omega_{\mathrm{p}}/50$) for selected values of the kinetic energy
$E_{k}$. (c,d) Classical (c) and quantum (d) EELS spectra for the same
air–jellium interface as a function of $E_{k}$. The classical SPP result at
$q=\omega/v$ is indicated by the white-dashed curves, while the vertical gray-
solid lines indicate the classical nonretarded surface plasmon frequency
$\omega_{\mathrm{SP}}^{\mathrm{cl}}=\omega_{\mathrm{p}}/\sqrt{2}$. (e)
Dispersion relation of SPPs from classical and quantum treatments of the
planar air–jellium interface in (b–d). (f–i) Same as (b–e), but for a silver
surface (screened plasma frequency
$\hbar\omega_{\mathrm{p}}^{*}=$3.82\text{\,}\mathrm{eV}$$) capped with a
dielectric of permittivity $\epsilon_{\mathrm{d}}=2$ (representative of
$\mathrm{SiO}_{2}$). We take $b=$5\text{\,}\mathrm{nm}$$ in all cases.

The impact of quantum nonlocal effects imparted on the EELS spectrum of an
electron traveling parallel to a planar metal surface is presented in Fig. 2
(see panel (a) for a sketch of the geometry). Notably, while at large electron
kinetic energies $E_{k}$ the EELS spectra are well-described by classical
dielectric theory, such a description progressively deteriorates as $E_{k}$ is
reduced. More precisely, we find that for
$E_{k}\lesssim$20\text{\,}\mathrm{keV}$$ the impact of nonclassical effects
becomes substantial, imprinting considerable spectral shifts and resonance
broadening on the EELS spectra. The broadening is a direct consequence of
surface-assisted Landau damping, entering via
$\operatorname{Im}\\{d_{\perp}\\}$, whereas the observed resonance shifts are
produced by the displacement of the induced charges relative to the
classically defined abrupt interface, which is encoded by
$\operatorname{Re}\\{d_{\perp}\\}$. The sign of
$\operatorname{Re}\\{d_{\perp}\\}$ dictates the direction of the frequency
shift: toward the red if positive, reflecting the electron spill-out
characteristic of jellium metals (Fig. 2b–e) Liebsch (1987); Tsuei et al.
(1989, 1991); Sprunger et al. (1992); Reiners et al. (1995); or toward the
blue if negative, signaling the electron spill-in observed in silver (Fig.
2f–i) and other noble metals Liebsch (1993); Tiggesbäumker et al. (1993);
Cottancin et al. (2006); Cirací et al. (2012); Scholl et al. (2012); Raza et
al. (2013, 2015); Yang et al. (2019); Campos et al. (2019). Furthermore, since
the peak in the EELS spectrum is associated with the excitation of surface
plasmon polaritons (SPPs), the observation that the impact of quantum nonlocal
effects grows with decreasing $E_{k}$ can be understood as follows: (i) the
main contribution to the EELS probability arises at a lost energy
$\hbar\omega$ for which the wave-vector transfer threshold $q=\omega/v$
intersects that of the SPP; (ii) lower electron velocities lead to
intersections occurring at correspondingly larger wave vectors (Figs. 2e and
2i), which is precisely where quantum nonlocal effects become sizable (with
resonance frequency shifts $\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
0.5pt\cr\sim\cr\kern-1.0pt\cr}}}q\operatorname{Re}\\{d_{\perp}\\}$ and
nonlocal broadening $\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
0.5pt\cr\sim\cr\kern-1.0pt\cr}}}q\operatorname{Im}\\{d_{\perp}\\}$)
Christensen et al. (2017); Gonçalves et al. (2020); Gonçalves (2020).
Together, (i) and (ii) provide a simple and intuitive explanation underpinning
the main features observed in Fig. 2.

Figure 3: Nonclassical optical response of metallic spheres probed through
EELS and CL spectroscopies. (a) Illustration of the aloof configuration under
consideration. (b,c) Calculated EELS (b) and CL (c) spectra for jellium
spheres with different radii in air, contrasting the classical (black dashed
curves) and quantum (color-filled solid curves) treatments. (d,e) Same as
(b,c), but for silver spheres in a host dielectric with
$\epsilon_{\mathrm{d}}=2$. We take $E_{k}=$50\text{\,}\mathrm{keV}$$ and
$b=R+$5\text{\,}\mathrm{nm}$$ in all cases.

Metal nanoparticles constitute another quintessential architecture in which
e-beam spectroscopies have played an important role (e.g., to map plasmonic
fields in real-space with nanometric resolution Nelayah et al. (2007); Kociak
and Stéphan (2014); Kociak and Zagonel (2017)). As we show below, localized
surface plasmon (LSP) resonances in small metal nanoparticles investigated
with EELS and/or CL can be used to quantitatively probe the nonclassical
optical response of metals. Focusing on metal spheres, the spectrally resolved
EELS and CL probabilities associated with an aloof e-beam passing near a
sphere of radius $R$ with impact parameter $b>R$ (see Fig. 3a) are given by

$\displaystyle\Gamma_{\text{{EELS}}}(\omega)=\frac{\alpha}{\omega\sqrt{\epsilon_{\mathrm{d}}}}$
$\displaystyle\sum_{l=1}^{\infty}\sum_{m=-l}^{l}K_{m}^{2}\left(\frac{\omega
b}{v\,\gamma_{\epsilon_{\mathrm{d}}}}\right)$ $\displaystyle\times$
$\displaystyle\left[C_{lm}^{\text{{E}}}(\beta_{\epsilon_{\mathrm{d}}})\,\operatorname{Im}\left\\{t_{l}^{\text{{E}}}\right\\}+C_{lm}^{\text{{M}}}(\beta_{\epsilon_{\mathrm{d}}})\,\operatorname{Im}\left\\{t_{l}^{\text{{M}}}\right\\}\right],$
(2)

and

$\displaystyle\Gamma_{\text{{CL}}}(\omega)=\frac{\alpha}{\omega\sqrt{\epsilon_{\mathrm{d}}}}$
$\displaystyle\sum_{l=1}^{\infty}\sum_{m=-l}^{l}K_{m}^{2}\left(\frac{\omega
b}{v\,\gamma_{\epsilon_{\mathrm{d}}}}\right)$ $\displaystyle\times$
$\displaystyle\left[C_{lm}^{\text{{E}}}(\beta_{\epsilon_{\mathrm{d}}})\,\left|t_{l}^{\text{{E}}}\right|^{2}+C_{lm}^{\text{{M}}}(\beta_{\epsilon_{\mathrm{d}}})\,\left|t_{l}^{\text{{M}}}\right|^{2}\right],$
(3)

respectively, where $K_{m}$ is a modified Bessel function of the second kind
Abramowitz and Stegun (1972),
$\gamma_{\epsilon_{\mathrm{d}}}=(1-\beta_{\epsilon_{\mathrm{d}}}^{2})^{-1/2}$,
and we have defined
$\beta_{\epsilon_{\mathrm{d}}}=\sqrt{\epsilon_{\mathrm{d}}}v/c$. Here, the
quantities $C_{lm}^{\text{{E}}}$ and $C_{lm}^{\text{{M}}}$ are coupling
coefficients that, for a given pair of angular momentum numbers $(l,m)$,
depend uniquely on $\beta_{\epsilon_{\mathrm{d}}}$ (see Ref. García de Abajo
(2010) for explicit expressions). Equations (2) and (3) extend the previously
derived results for the interaction of a fast electron with a sphere in vacuum
García de Abajo (1999, 2010) to a configuration in which the sphere is
embedded in a lossless dielectric medium with arbitrary
$\epsilon_{\mathrm{d}}$. The optical response of the sphere enters these
equations through the Mie scattering coefficients $t_{l}^{\text{{E}}}$ and
$t_{l}^{\text{{M}}}$ for transverse magnetic (TM) and transverse electric (TE)
waves, respectively. In analogy to the planar interface considered above,
quantum mechanical corrections in the optical response are straightforwardly
accounted for by adopting the generalized Mie coefficients containing the
$d$-parameters Gonçalves et al. (2020) (see Methods).

Figure 3 compares classical and quantum predictions for the EELS probability
(Figs. 2b,d) and CL (Figs. 2c,e) spectra from metallic spheres with different
radii. In many ways, they echo the general conclusions discussed above for the
planar interface, but in this instance $R^{-1}$ takes the role previously
played by the in-plane wave vector $q$. Specifically, the nonclassical
spectral shifts and broadening increase when reducing the particle
radius—qualitatively following $\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
0.5pt\cr\sim\cr\kern-1.0pt\cr}}}l(l+1)\operatorname{Re}\\{d_{\perp}\\}/R$ and
$\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
0.5pt\cr\sim\cr\kern-1.0pt\cr}}}l(l+1)\operatorname{Im}\\{d_{\perp}\\}/R$,
respectively Gonçalves et al. (2020)—, ultimately leading to pronounced
differences in the spectral peak corresponding to the dipolar ($l=1$) LSP for
$R\lesssim$10\text{\,}\mathrm{nm}$$. In passing, we note that higher-order
multipoles in larger spheres can still display deviations from classicality
(profiting from the $l(l+1)$ factor noted above, which reflects the faster
surface oscillations as $l$ increases), albeit much less recognizable in
comparison to those observed for the dipolar LSP in small spheres. Indeed,
aside from being quenched by nonlocal broadening, dipolar LSP resonances in
jellium (silver) spheres of a few nanometers in size are dramatically red
(blue) shifted (by as much as $\sim 200\text{\,}\mathrm{meV}$) with respect to
the classical nonretarded result
$\omega^{\text{cl}}=\omega_{\text{p}}/\sqrt{\epsilon_{\mathrm{b}}+2\epsilon_{\mathrm{d}}}$.
The breaking of scale-invariance characterizing the classical nonretarded
limit is thus lifted within this investigated regime due to the introduction
of the inherently quantum-mechanical length-scale associated with
$|d_{\perp}|$.

Although nonclassical effects permeate EELS and CL spectra in similar ways,
there are some important differences. Being the result of spontaneous light
emission following e-beam excitation, CL is only sensitive to bright LSP
modes, whereas EELS grants us access to dark multipolar LSPs Losquin and
Kociak (2015); Losquin et al. (2015) (cf. the EELS and CL spectra in Fig. 3).
In addition, the CL signal drops considerably for small nanoparticles due to
the realization of the dipole limit and the concomitantly smaller scattering
cross section. Therefore, EELS is better suited for measuring the optical
response at very small sizes, with EELS measurements of silver particles down
to $\sim 2\text{\,}\mathrm{nm}$ in diameter having been reported Scholl et al.
(2012); Raza et al. (2015); Campos et al. (2019).

Figure 4: Retrieval of $\bm{d}_{\perp}(\bm{\omega})$ from (simulated)
experimental EELS data of metallic jellium spheres in air. (a) EELS spectra
for selected radii (indicated in each panel), showing the theoretically
calculated result (corresponding to the “ground truth”, blue curves), the
simulated experimental data (by adding noise to the previously calculated
ones, green curves), and the reconstructed spectra by fitting $d_{\perp}$
(dashed red curves). (b–c) Extracted $d_{\perp}$-parameter data from EELS
spectra of jellium nanospheres of different radii. Each dot (one for the real
part and another one for the imaginary part) corresponds to the fitted value
of $d_{\perp}$ for a specific radius $R=1,2,3,4,6\,\mathrm{nm}$ (each
corresponding to a specific LSP resonance frequency $\omega_{0}$). The dashed
lines are obtained from a linear fit for $d_{\perp}(\omega)$ using all the
data for different radii. In (b), for each radius, $d_{\perp}$ is fitted to a
constant, whereas in (c) it is fitted to a linear frequency dependence around
each $\omega_{0}$. The solid lines represent the original TDDFT data (same as
in Fig. S1) in the spectral window of interest for LSPs in jellium sheres. In
all cases, we take $b=R+$5\text{\,}\mathrm{nm}$$ and
$E_{k}=$50\text{\,}\mathrm{keV}$$.

Furthermore, focusing on metallic jellium spheres for concreteness, we
explicitly show in Fig. 4 a scheme to extract $d$-parameter data from EELS
measurements (for CL the procedure and results would be the same) using the
$d$-parameter-corrected theory introduced above (further details on the
implementation are provided in Supplementary Section 2). We emulate
experimental EELS data by adding noise to the theoretically calculated spectra
(Fig. 4a, green curves), while also accounting for the fact that the EELS
signals from smaller nanoparticles exhibit more noise than those from larger
ones due to the smaller signal-to-noise ratio. We stress that the simulated
experimental spectra mimics well those obtained from actual EELS measurements
Campos et al. (2019). We then use our $d$-parameter-corrected theory to
extract $d_{\perp}$ following two alternative approaches: a simple one in
which we fit a constant $d_{\perp}$ value for each particular radius (Fig.
4b), and a more refined one in which we infer the $d_{\perp}$ dispersion near
the LSP peak to leading order via a linear fit (Fig. 4c). Finally, the
combination of data from spheres of different radii enables the reconstruction
of the complex-valued, frequency-dependent $d_{\perp}$ across a broad spectral
range with extremely good accuracy (Fig. 4b–c, dashed lines), thereby
underscoring the viability of our proposed scheme. Importantly, such a scheme
is extremely valuable to unravel the quantum surface response of metals
directly from experimental data with high accuracy. This is particularly
relevant for noble metals, for which quantitatively accurate $d$-parameter
calculations are not yet available. In addition, even for simple metals, for
which TDDFT data exist, the determination of the $d$-parameters using the
approach developed here provides a new path for benchmarking first-principles
calculations (inasmuch as the fidelity of such methods ultimately depends on
the chosen implementation or functional Lejaeghere et al. (2016); Mata and
Suhm (2017)) directly against experiment. Incidentally, although we have
employed a standard least-squares fitting procedure to extract the
$d$-parameters (see SI), our approach could benefit from machine-learning
methods, which have been applied in similar settings, namely, in transmission
electron microscopy Spurgeon et al. (2021), scanning near-field optical
microscopy Chen et al. (2022), ellipsometry Liu et al. (2021); Wang et al.
(2022), and others Peurifoy et al. (2018).

In conclusion, we have demonstrated that EELS and CL spectroscopies constitute
powerful tools to probe quantum-mechanical corrections in nanoplasmonics,
which here we have calculated by augmenting the classical, local-response
theory with the Feibelman $d$-parameters. In particular, we have shown that
quantum effects in the response of metallic surfaces lead to substantial
nonclassical shifts and nonlocal broadening of the EELS and CL spectral
features associated with surface plasmon resonances. In extended planar metal
surfaces, such deviations from classicality become non-negligible for electron
kinetic energies below $\sim 20\text{\,}\mathrm{keV}$ due to the contribution
from large wave vector components associated with free-electrons, which
increases as the electron energy is lowered. In metallic spheres, the relevant
length scale is instead determined by the particle size, and thus, the impact
of nonclassical corrections is weakly dependent on the electron kinetic energy
(see Fig. S3 in SI). Specifically, we find that quantum nonlocal effects
become substantial for spheres with radii $\lesssim$10\text{\,}\mathrm{nm}$$,
in-line with experimental observations Scholl et al. (2012); Raza et al.
(2015); Campos et al. (2019).

Our work provides a viable, concrete scheme for interrogating the nonclassical
optical response of metals in a quantitative fashion through the retrieval of
the $d$-parameters associated with the involved dielectric–metal interfaces
from EELS and CL measurements. In practice, as demonstrated here, this is
achieved using the $d$-parameter-corrected theory to infer such parameters
from fits of experimental spectra (Fig. 4), as all other experimental
parameters can be well-characterized using currently available techniques.
Additionally, we underscore that our proposal based on e-beam spectrocopies is
superior to optics-based techniques because it can effectively address
individual particles rather than an ensemble (which invariably introduces
inhomogeneous broadening due to particle size and/or morphology variations).
Yet another advantage of e-beams is that, by varying their orientation,
tomography-based methods Nicoletti et al. (2013); Atre et al. (2015); Anton
Hörl and Hohenester (2015); Li et al. (2021) can be exploited to precisely map
the exact shape of the nanoparticle (and even account for surface roughness, a
classical effect that could be erroneously mistaken for nonclassical spectral
shifts and broadenings).

We envision that the scheme presented in this work will stimulate experimental
endeavors for measuring the Feibelman $d$-parameters for relevant combinations
of dielectric–metal interfaces. Indeed, a systematic compilation of a
“$d$-parameter catalogue” would allow the full deployment of this formalism
across the board in nanophotonics, with key implications not only for
understanding the fundamentals of plasmon-based light–matter interactions at
the nanoscale but also for optimizing and designing nanoplasmonic devices with
nanometer-sized footprints.

## Methods

Mesoscopic scattering coefficients. The mesoscopic, $d$-parameter-corrected
scattering coefficients for a planar metal surface and for metallic spheres
have been previously introduced by Feibelman Feibelman (1982) and Gonçalves et
al. Gonçalves et al. (2020), respectively. Here, we reproduce them for
completeness.

For the planar dielectric–metal interface, the nonclassical version of the
Fresnel reflection coefficients for $p$\- and $s$-polarized waves read
Feibelman (1982); Liebsch (1997); Gonçalves et al. (2020); Gonçalves (2020)

$\displaystyle r_{p}$
$\displaystyle=\frac{\epsilon_{\mathrm{m}}k_{z,\mathrm{d}}-\epsilon_{\mathrm{d}}k_{z,\mathrm{m}}+(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\left[\mathrm{i}q^{2}d_{\perp}-\mathrm{i}k_{z,\mathrm{d}}k_{z,\mathrm{m}}d_{\parallel}\right]}{\epsilon_{\mathrm{m}}k_{z,\mathrm{d}}+\epsilon_{\mathrm{d}}k_{z,\mathrm{m}}-(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\left[\mathrm{i}q^{2}d_{\perp}+\mathrm{i}k_{z,\mathrm{d}}k_{z,\mathrm{m}}d_{\parallel}\right]},$
(4a) $\displaystyle r_{s}$
$\displaystyle=\frac{k_{z,\mathrm{d}}-k_{z,\mathrm{m}}+(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\mathrm{i}k_{0}^{2}d_{\parallel}}{k_{z,\mathrm{d}}+k_{z,\mathrm{m}}-(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\mathrm{i}k_{0}^{2}d_{\parallel}},$
(4b)

where $q$ is the in-plane wave vector, $k_{0}=\omega/c$, and
$k_{z,j}=\sqrt{\epsilon_{j}k_{0}^{2}-q^{2}}$ with
$j\in\\{\mathrm{m},\mathrm{d}\\}$ denoting the out-of-plane wave vector
components.

For a metallic sphere of radius $R$, the generalized, nonclassical transverse
magnetic (TM) and transverse electric (TE) Mie coefficients are given by
Gonçalves et al. (2020); Gonçalves (2020)

$\displaystyle t_{l}^{\text{{e}}}$
$\displaystyle=\mathrm{i}\,\frac{\epsilon_{\mathrm{m}}j_{l}(x_{\mathrm{m}})\Psi^{\prime}_{l}(x_{\mathrm{d}})-\epsilon_{\mathrm{d}}j_{l}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})+(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\left\\{j_{l}(x_{\mathrm{d}})j_{l}(x_{\mathrm{m}})\left[l(l+1)\right]d_{\perp}+\Psi^{\prime}_{l}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})\,d_{\parallel}\right\\}/R}{\epsilon_{\mathrm{m}}j_{l}(x_{\mathrm{m}})\xi^{\prime}_{l}(x_{\mathrm{d}})-\epsilon_{\mathrm{d}}h_{l}^{(1)}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})+(\epsilon_{\mathrm{m}}-\epsilon_{\mathrm{d}})\left\\{h_{l}^{(1)}(x_{\mathrm{d}})j_{l}(x_{\mathrm{m}})\left[l(l+1)\right]d_{\perp}+\xi^{\prime}_{l}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})\,d_{\parallel}\right\\}/R},$
(5a) $\displaystyle t_{l}^{\text{{m}}}$
$\displaystyle=\mathrm{i}\,\frac{j_{l}(x_{\mathrm{m}})\Psi^{\prime}_{l}(x_{\mathrm{d}})-j_{l}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})+\big{(}x_{\mathrm{m}}^{2}-x_{\mathrm{d}}^{2}\big{)}j_{l}(x_{\mathrm{d}})j_{l}(x_{\mathrm{m}})\,d_{\parallel}/R}{j_{l}(x_{\mathrm{m}})\xi^{\prime}_{l}(x_{\mathrm{d}})-h_{l}^{(1)}(x_{\mathrm{d}})\Psi^{\prime}_{l}(x_{\mathrm{m}})+\big{(}x_{\mathrm{m}}^{2}-x_{\mathrm{d}}^{2}\big{)}\,h_{l}^{(1)}(x_{\mathrm{d}})j_{l}(x_{\mathrm{m}})\,d_{\parallel}/R},$
(5b)

in terms of the dimensionless wave vectors $x_{j}\equiv
k_{0}\sqrt{\epsilon_{j}}R$. Here, $j_{l}(x)$ and $h^{(1)}_{l}(x)$ stand for
the spherical Bessel and Hankel functions of the first kind Abramowitz and
Stegun (1972), $\Psi_{l}(x)\equiv xj_{l}(x)$ and $\xi_{l}(x)\equiv
xh_{l}^{(1)}(x)$ are Riccati–Bessel functions Abramowitz and Stegun (1972),
and primed functions denote their derivatives.

## Supporting Information

Supporting information is available free of charge at DOI:
10.1021/acs.nanolett.3c00298.

  * Details on the Feibelman $d$-parameter data, comprehensive description of the scheme to extract the $d$-parameters from EELS spectra, and explicit demonstration of the robustness of the EELS and CL peak position with varying e-beam kinetic energy.

## Acknowledgments

This work has been partly supported by the European Research Council (Advanced
Grant No. 789104-eNANO), the Spanish Ministry of Science and Innovation
(PID2020-112625GB-I00 and CEX2019-000910-S), the Generalitat de Catalunya
(CERCA and AGAUR), and the Fundaciós Cellex and Mir-Puig.

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