How Dark Are Radial Breathing Modes in Plasmonic Nanodisks?

Due to a vanishing dipole moment, radial breathing modes in small flat plasmonic nanoparticles do not couple to light and have to be probed with a near-field source, as in electron energy loss spectroscopy (EELS). With increasing particle size, retardation gives rise to light coupling, enabling probing breathing modes optically or by cathodoluminescence (CL). Here, we investigate single silver nanodisks with diameters of 150–500 nm by EELS and CL in an electron microscope and quantify the EELS/CL ratio, which corresponds to the ratio of full to radiative damping of the breathing mode. For the investigated diameter range, we find the CL signal to increase by about 1 order of magnitude, in agreement with numerical simulations. Due to reciprocity, our findings corroborate former optical experiments and enable a quantitative understanding of the light coupling of dark plasmonic modes.


Data normalization
Comparing intensities of dierent spectrum images requires stable conditions during the whole microscope session. High beam current stability is important but often dicult to maintain for cold eld emitters. As the deposition of residual gas molecules reduces the emitted current, the emitter is ashed from time to time to restore the initial beam current.
Therefore, care has to be taken when intensities of consecutive measurements are compared.
Ideally, one uses a reference that allows to render the signal independent of the beam current.
For the EELS data from the disk center and from half radial distance (R/2) we use the bulk plasmon peak at 3.75 eV as such a reference for the spectra extracted, i.e., we normalize the spectra to the integrated signal from 3.63.9 eV. The peak intensity of the bulk plasmon mainly depends on the particle thickness (but not on its lateral dimensions), which we assume to be homogeneous over the disk (30 nm). Alternatively, we could also use the zero-loss peak (elastically and quasi-elastically scattered electrons) as reference, which gives almost identical results (not shown). A modied normalization procedure was applied to the EELS spectra from the edge region, as the assumption of homogeneous thickness does not hold here (the extraction region is partly on and partly o the particle). We apply thus the bulk plasmon normalization factor extracted for the disk center, weighted by the ratio of pixel numbers corresponding to the edge and center extraction areas. The normalization procedure demands constant beam current during the acquisition of a single spectrum image, which seems to be justied for the given acquisition times of approximately 2.5 min per spectrum image.
In case of the CL measurements, there exists no spectral feature that can be used as reference. Therefore, we ashed the tip before each spectrum image acquisition to achieve similar conditions. The spectra of the three dierent regions (center, R/2, edge) are normalized by the number of included pixels × pixel time (50100 ms). Furthermore, the CL spectra I(λ)/∆λ (with ∆λ=0.7 nm/channel being the spectrometer dispersion) are multiplied by λ 2 , according to the conversion from wavelength (nm) to energy units (eV). 1 Figure S1: CL-to-EELS ratio of the RBM. (a) Experimental EELS (top row) and CL (bottom row) position dependent spectra for dierent disk diameters as indicated. The blue areas correspond to the signal used for the experimental CL-to-EELS ratio in Figure 4 and to set the contrast range of the CL and EELS maps in Figure 1 of the main text. (b) Simulated spectra, the yellow regions correspond to the signal used for the simulated CL-to-EELS ratio in Figure 4 of the main text.

CL-to-EELS ratio
The EELS and CL signals of the RBM for both experiment and simulation used to extract the CL-to-EELS ratios in Figure 4 are depicted in Figure S1. As we discuss in the main text, we take as the experimental EELS RBM intensity the values I center -I R/2 , integrated from -0.25 eV to +0.25 eV around the RBM peak energy (blue areas in the upper row of Figure S1a). For the experimental CL data we integrate directly the center spectra over the same energy width around the RBM peak energies (blue areas in the lower row of Figure S1a).
A subtraction similar to EELS is not required, as the SP 1 peak is missing in the CL data S3 Figure S2: Center-to-edge EELS and CL ratio. (a) Disk-diameter-dependent ratio of the CL (circles) and EELS (crosses) experimental signals I center /I edge . I center and I edge are extracted from the areas indicated by the blue and green squares in the CL and EELS maps (exemplarily shown for the 500 nm disk), and integrated from 1.43.9 eV (CL) and 14 eV (EELS) respectively. (b) Corresponding data for simulated results, the arrows indicate the positions where I center and I edge are extracted. and there is no spectral overlap with the RBM. For the simulated CL-to-EELS ratio we t Lorentzian functions to the EELS and CL spectra and use the peak area as indicated by the yellow areas in Figure S1b. This integration area is dened by the spectral full-width-at-half maximum, corresponding to half of the Lorentzian peak area.
Center-to-edge ratio An alternative option to analyze the disk-size-dependent RBM CL intensity independently of the beam current is via the center-to-edge signal ratio. Therefore, we rst sum up the full spectral response covered by the experiment (14 eV for EELS, 1.43.9 eV for CL) for the center and edge spectra, marked by the green and blue squares in the right panel of Figure S2a (experiment) and by the green and blue dashed arrows in Figure S2b (simulation). While the center spectra mainly include the RBM, the signal extracted from the edge region is primarily due to edge modes, as illustrated by the CL and EELS maps of the dipole (left) and RBM (right) in Figure S2a. Then, we calculate the ratio for each disk size for CL and EELS as plotted in Figure S2, thereby eliminating beam current variations. We clearly recover the S4 strong CL size dependence, as opposed to the largely size-independent behavior of the EELS signal. However, this ratio is not straightforward, as the edge signal not only includes the (bright) dipole signal but also the higher order modes, which also show a dark character with decreasing particle size. A substantial dierence between experiment and simulation in Figure S2 is found for the absolute values of the CL center-to-edge ratio. This is mainly because of a much lower CL edge signal in the experiment than in the simulation. Higher order edge modes are emitting in large angles (see Figure 5 and

Agreement between simulation and experiment
The main discrepancy between experiment and simulation is the number of observed plasmon resonances, which is much larger in the simulation. This is especially true for the edge spectra, which show a few broad peaks in the experiment (only one in CL) and many narrow peaks in the simulation ( Figure 3 from the main text). According to numerical simulations ( Figure S4), a number of eects may contribute to this discrepancy: (i) the spatial averaging on the experimental data, (ii) defects within the sample and (iii) the exact angular detection S5 Figure S3: Simulated disk-size dependent light emission. (a) CL spectra of 30 nm thick Ag disks, with diameters of 100, 150 and 200 nm, calculated at the disk edge (green) and center (blue). Peaks 1, 2 and 3 correspond to the dipole, quadrupole and RBM, respectively. (b) Corresponding emission patterns of the modes 13 for the dierent disk diameters with the color reecting the absolute value of the energy ux density, each normalized to its maximum (red color). The relative intensity scale is given by the factors as depicted, indicating the lowest overall emission for the RBM (3) of the 100 nm disk.
range of the experiment. The simulations of the EELS and CL edge spectra used in this work are performed for one discrete electron position located 5 nm away from the disk edge.
By contrast, the experimental data are averaged around the edges to increase the signal-tonoise ratio. To investigate the eect of spatial averaging, we calculate CL spectra for several electron positions around the disk edges ( Figure S4a). Obviously, the ratio between the dipolar mode peak and the quadrupolar mode peak intensities depends on the exact electron S6 Figure S4: What inuences the CL signala simulation study on a single silver nanodisk. The disk has a diameter of 150 nm and a thickness of 30 nm and is placed on a 15 nm thick Si 3 N 4 membrane. (a) Eect of the electron beam position around the disk edge as marked in the inset on the CL spectra: The spectra correspond to electron positions 25 nm (dashed) and 5 nm (solid, position used for simulations in the main manuscript) away from the edge inside the disk and 5 nm (dash-dotted) and 25 nm (dotted) outside the disk edge. (b) Eect of damping: The spectra correspond to an edge excitation (5 nm away from the edge inside the disk). An extended Drude model 2 was used to t the experimental dielectric function of reference 3 (solid line, Γ=1). The dashed line corresponds to increased damping with Γ=2.
(c) Eect of angular detection range: CL spectra are integrated over the full space of 4π sr (solid lines) and over a 1.2π sr large angular window of the upper half space (approximating the experimental condition, dashed lines) upon edge and center excitation (green and blue curves, respectively). The spectra in (a) and (b) are normalized to the maximum value by the given multiplication factors. Thereby the intensity of the dipolar mode (lowest energy peak in (a) and (b)) can be compared at the same time with the relative change of the higher order modes. In (c) the edge spectra (green) are normalized to their own maximum, while the center spectra (blue) are normalized by the factors of the corresponding edge spectra.
position. In particular, it increases when moving away from the edge.
The simulations are performed by using the dielectric function of high-quality silver lms. 3 However, samples obtained by EBL contain adhesion layers and show dense grain boundaries and surface roughness. Furthermore, silver oxidizes quickly. All these parameters aect the optical properties of the sample in a complex way. To pinpoint the eect of defects on the spectra, we use the simple modeling proposed by Bosman et al., 2 considering all the above parameters by a single phenomenological damping term contained within the optical properties of silver. The optical properties of silver obtained from Palik 3 are tted to the sum of a Drude response term and two Lorentzian response terms. Defects are then introduced by multiplying the damping term within the Drude response term by a given factor Γ, Figure S5: Deconvolution of EELS spectra. EELS spectra of a silver disk, 500 nm in diameter, for a dierent number of iterations using a Richardson-Lucy deconvolution routine described in Ref. 4 (a) Zero-loss peak: The energy spread of the electron beam (elastically scattered electrons) is shown in dependence on the number of iterations, leading to reduction from approximately 0.3 eV (raw data) to 0.11 eV (100 iterations) with respect to the full-width-athalf-maximum (FWHM) of the zero-loss peak. (b) RBM peak: EELS spectra, originating from the center of the disk, highlight the stable intensity of the RBM peak for a dierent number of iterations. The peak amplitude varies below 5% between 15 to 100 iterations, as indicated by the black arrows.
corresponding to α in equation 1 of Ref. 2 (Figure S4b, green dashed line). This leads to a broadening of the resonances and a decrease of the quadrupolar mode peak relative to the dipolar mode peak.
In the simulations, the experimental detection range is only approximated. To identify the eect of the detection range, we summed up the emitted light intensities within dierent angular windows ( Figure S4c). Reducing the detection range decreases the quadrupolar mode peak relative to the dipolar mode peak (green curves). The RBM (blue curves) is also strongly aected as the majority of the RBM emission goes into the substrate (see Figure 5 and Figure S3). Finally, the modeling is obviously not perfect. It relies on dielectric functions that only approach the real material optical properties and simplied object shapes.
However, the overall agreement is very good and the size dependent EELS to CL ratio of the RBM as the main experimental nding is well reproduced by the simulations. S8