A near-field study on the transition from localized to propagating plasmons on 2D nano-wedges

In this manuscript we report on a near-feld study of two-dimensional plasmonic gold nano-wedges using electron energy loss spectroscopy in combination with scanning transmission electron microscopy, as well as discontinuous Galerkin time-domain computations. With increasing nano-wedge size, we observe a transition from localized surface plasmons on small nano-wedges to non-resonant propagating surface plasmon polaritons on large nano-wedges. Furthermore we demonstrate that nano-wedges with a groove cut can support localized as well as propagating plasmons in the same energy range.


Introduction
Focusing light down to nanometric volumes is of great interest for various applications. For instance, nanoscale spectroscopy and imaging techniques such as tip-enhanced Raman scattering, apertureless near-field optical microscopy (A-NSOM), and ultrafast photoemission of electrons make use of highly confined electromagnetic fields [1][2][3][4][5]. Many of these techniques rely on the excitation of plasmonic modes in suitable metallic nano-structures, where the collective oscillations of the conduction band electrons in the metal lead to strongly enhanced electromagnetic near-fields [6].
In metallic nano-structures that are small compared to the relevant free-space wavelength, e.g. nano-antennas [7], plasmonic oligomers [8], or split-ring resonators [9], the boundary conditions give rise to standing wave patterns of the charge carrier oscillations. The corresponding resonant modes are the so-called localized surface plasmons (LSPs), which typically exhibit hot-spots of the electromagnetic field in the vicinity of the surface. The spectral and spatial properties of the LSPs can be controlled by designing the shape, size, or surrounding media of the nano-structure [6].
Extended metal nano-structures support propagating surface plasmon polaritons (SPPs), which are formed by charge carrier density waves moving along the surface. On tapered nano-structures such as nano-wedges, the SPP dispersion relation varies along the propagation direction. This can be utilized to compress SPPs and to achieve strong local-field enhancements. For instance, a SPP propagating along a nano-wedge towards the apex is gradually slowed down as it reaches the apex region [10,11]. For most experimentally accessible wedge parameters, the SPP is however not brought to a complete stop but rather is reflected at the apex [12][13][14].
Precise knowledge of the spatio-spectral distribution of the plasmonic near-field is of utmost importance for the applications mentioned above. A very powerful experimental method that allows to map localized as well as propagating plasmon modes is electron energy loss spectroscopy (EELS) in combination with scanning transmission electron microscopy (STEM) [9,12,[15][16][17][18][19][20][21][22]. In STEM-EELS, a highly focused electron beam scans over the sample. Passing nearby or through the structure, the electrons can excite plasmon modes. As a result, the fast electrons lose kinetic energy by interacting with the self-induced electric field in the vicinity of the nano-structure. The probability for this process, the so called electron energy loss probability (EELP), is related to the plasmonic local density of states (LDOS) and can thereby be used to characterize the plasmonic nano-structure [23][24][25].
In this work, we report on a near field study on two-dimensional plasmonic gold nano-wedges using STEM-EELS. With increasing nano-wedge size, we observe a transition from LSPs on resonant nano-wedges to non-resonant propagating SPPs on large nano-wedges. Furthermore we demonstrate that nano-wedges with a groove cut can support both, localized and propagating plasmon modes.
The experimental data is compared to numerical computations based on the Discontinuous Galerkin Time-Domain (DGTD) method. system (Cs-corrector). An in-column omega-type energy filter, fully corrected for second-order aberrations, is integrated into the microscope. The spectra are recorded using a Gatan UltraScan 2k x 2k CCD camera with an acquisition time for each spectrum of 5 ms and a dispersion of the spectrometer of 0.016 eV per channel. The energy resolution, which is defined by the FWHM of the spectrum's zero-loss peak measured through the silicon nitride membrane, is 0.1 eV in the center of the scan area. In the present STEM-EELS experiments, the electron beam raster scans over the sample with a step width ranging between 5 nm and 20 nm, depending on the size of the investigated area. All spectra are recorded for normal incidence of the electron beam. For data postprocessing, each spectrum is normalized to its total number of electron counts.

Methods and instrumentation
Subsequently, the first moment of the zero-loss peak is centered to 0 eV and a background spectrum is subtracted. In the presented EEL maps, the signal is normalized to the maximum value found in the relevant area for the given electron loss energy.
To support our experimental findings we have performed numerical computations for selected structures using the DGTD method [26,27]. The geometrical parameters for the nano-wedge are taken from the corresponding HAADF micrographs. The nano-wedges are embedded into vacuum and have a dielectric permittivity, which is approximated using a Drude-Lorentz-model for gold similar to Ref. [12]. The swift electron's speed is set to v = 0.77 c. We obtain the induced polarization of a relativistic moving electron passing the gold nano-wedges and determine the back-action of the induced field on this electron [24,27,28].
The expansion of the electric and the magnetic field into Lagrange polynomials of third order is carried out for each mesh element of the structure. The mesh's geometry is determined by inspection of the micrograph of the corresponding structure and the mesh's elements have side lengths down to 5 nm (see figure 1).   A qualitatively different behavior is expected for metal wedges whose lateral dimensions are large compared to the plasmon decay length. In this case, a discrete standing wave pattern of the charge carrier oscillations can not form since multiple reflections from the nano-wedges's ends are suppressed by absorption.

Results
Hence, long nano-wedges are expected to support a continuum of propagating SPPs instead of discrete LSPs.  Naturally the question arises for which nano-wedge size a transition from a set of discrete LSPs to a continuum of SPPs takes place in the considered energy range. By investigating nano-wedges with different sizes, we find that the crossover between these two regimes occurs for a length between 2.0 µm and 2.5 µm. Figure 5(b) depicts the relative EELP recorded parallel to the edge of a 2.0 µm long nano-wedge. This nano-wedge clearly shows a resonant behavior with several distinct LSP modes. In contrast, a qualitatively different behavior is observed for a 2.5 µm long nano-wedge. In addition to the maxima at the ends, the corresponding EELP (see Figure 4(d)) features two pronounced continuous bands. With increasing electron energy loss, they shift towards the apex and base of the wegde, respectively. These bands can be interpreted as maxima of the two standing wave patterns resulting from the reflection of SPPs at the apex and the base, respectively. Adding grooves to a long nano-wedge leads to interesting new effects. Figure 6(e) shows the apex region of a 15 µm long nano-wedge. 600 nm away from the nanowedge's apex two 50 nm broad and 55 nm deep grooves are cut, leaving a fillet with a width of 100 nm, separating the front section from the rear section. The front section has the same dimensions as the short wedge discussed above. For low electron loss energies, one expects that the grooves have a minor effect on the SPP propagation. Hence, the structure should behave similar to the long nano-wedge without grooves. Our experimental data confirms this prediction.
The EEL map with an electron loss energy of 0.5 eV (see Figure 6(a)), as well as the low energy region (< 0.85 eV) of the EELP distribution recorded along the edge (see Figure 6(f)) show a clear SPP like behavior. In particular, we observe that the maxima in the EELP distribution continuously shift towards the apex with increasing energy loss. This is a clear signature of the formation of a standing wave pattern as also observed in figure 4. In contrast to this, for loss energies above approximately 0.85 eV, a clear influence of the grooves can be observed. In this case, the front section of the structure (z < 600 nm) shows a similar behavior as the small nano-wedge. More specifically, we can identify two distinct resonances in the front section at 1.0 eV and 1.35 eV with three and four maxima, respectively. These resonances correspond to the second and third LSP mode of the short nano-wedge (cf. figure 2). Furthermore, we observe in the EELP distribution a pronounced maximum at the right side of the grooves for loss energies above 0.85 eV. Additionally, a second maximum occurs in the rear section that continuously shifts towards the grooves with increasing electron loss energy. This indicates the formation of a standing wave pattern, where the reflection of the SPPs takes place at the grooves.

Conclusion
In this work we demonstrated the transition from LSP-supporting nano-structures to a SPP-supporting structures by prolongating the structural dimensions from some hundred nanometers to some ten micrometers. For the shorter samples, multiple reflections at the ends lead to the formation of LSPs. For the longer samples, also propagating plasmons can be found. These are SPPs that are re-