1. Introduction

The illumination of plasmonic-metal nanostructures and nanoparticles with light gives rise to collective oscillations of electrons at the interface of the nanostructures if the incident electromagnetic field matches the resonant frequency of collective electrons. This phenomenon generates intense electromagnetic fields known as localized surface plasmon resonance (LSPR). Recently, surface-plasmon-based nanostructures have been employed to concentrate and manipulate light, enhance light-matter interaction, and couple light energy to photonic devices at the nanoscale regime [1–3].

The excited plasmonic resonances undergo radiative and nonradiative decays. The far-field light scattering leads to radiative loss. In turn, plasmonic enhancement of the near electromagnetic field results in the interband and intraband transitions, which generate energetic (hot) electrons with very high kinetic energies [3]. In many cases, the generation of hot electrons in plasmonic nanostructures is attributed to the nonradiative decay of surface plasmon resonance through Landau damping [3,4]. Plasmon-generated hot electrons have attracted increased attention due to their ability to improve the efficiency of photovoltaic devices and photodetectors, photocatalysis for green fuels, photochemistry and electrochemical processes, photothermal heating, as well as their potential applications in optoelectronics and nonlinear optics [5–11].

The development of injection of plasmonic hot electrons into vacuum, solid, or liquid environments has been reported both numerically and experimentally by many research groups [12–18]. For instance, Dombi et al. reported numerical investigation of highly directional and monoenergetic plasmonic hot electrons into vacuum for ultrafast and high spatial resolution applications [19]. Knight et al. experimentally investigated plasmonic hot electron injection into a semiconductor (silicon) for photodetection [20]. Boulais et al. reported investigations of plasma-mediated off-resonance plasmon-enhanced nanocavitation by the injection of plasmonic hot electrons from a gold nanosphere into water, induced by ultrafast laser radiation [21]. Zilio et al. used both numerical and experimental techniques to demonstrate that the energy transfer of plasmonic hot electrons from gold nanoelectrodes into water is more efficient in water than in vacuum (17 times higher), because free-electron clouds are more confined when injected hot electrons from the gold nanoelectrodes make contact with water molecules [22].

A periodic array of metallic nanoparticles has been shown to support LSPR and possess diffractive behavior in the same spectral region [23–25]. The Rayleigh anomaly is defined as wavelengths at which a diffracted order appears or disappears at a grazing angle, and the diffractive effects are attributed to the interparticle lattice period. By tuning the interparticle lattice period, it is possible to excite lattice modes resulting from the hybridization of the LSPRs and lattice resonances in the proximity to the Rayleigh anomaly induced by the periodic arrangement of the nanostructure. When the lattice period is altered, one can achieve a rapid increase in the amplitude of the diffracted spectral orders resulting from the Rayleigh anomaly, and thereby generating intense electric field enhancement [26–29]. The very high quality factors associated with plasmonic nanoparticle arrays due to diffractively coupled plasmon resonances can offer potential applications in the development of optoelectronics, photovoltaics, data storage, and biosensing [30].

Nanoelectrodes involving the injection of hot electrons have many potential applications in electrochemical sensing of single nanoparticles, chemical imaging of samples at the ultrahigh spatial resolution, development of plasmonic nanobubbles, and energy storage such as batteries and capacitors. Most importantly, such nanoelectrodes can be used, for example, as an in vitro platform for delivering a broad range of molecules into the intracellular compartment [18]. Injected hot electrons can be accelerated in the nanoantennas near-field and produce nanoscopic shockwaves opening membrane pores. This process allows avoiding nanobubbles, which are dangerous for biological cells. The microfluidic chip underneath the nanostructure allows bringing the required components to the targeted cell part.

Nanoelectrodes can be optical antennas (nanotubes) made up of either metals or semiconductors of nanometer dimensions. Plasmonic nanoantenna with a high aspect ratio can support the excitation of multiple resonances because of its complex shape and possibility of excitation of higher-order modes [31]. Such three-dimensional vertical plasmonic gold nanoantennas (nanoelectrodes) can radiate in all directions and act as a metal reservoir [32,33]. This configuration also offers a stronger plasmonic response, higher field enhancement, longer carrier lifetime, and more efficient carrier generation and recombination.

Here, we report on a numerical study of a strong plasmonic field enhancement in a gold nanoelectrode array applied for hot-electron generation in a water environment. The schematic of the unit cell of the nanostructure under study is illustrated in Fig. 1(a), with corresponding parameters used in numerical simulations. We show an excitation of the unique optical modes induced by the Rayleigh anomalies supported by the periodicity of the nanostructure and thereby enhancing the generation of plasmonic hot electrons. The modes are observed in the resonant absorptance and field enhancement in the nanostructure. We demonstrate the effect that can serve as a guideline for improving electric field enhancement and consequently stimulate the generation of plasmonic hot electrons from the nanoelectrodes in an aqueous environment.

 

Fig. 1. (a) Unit cell of the nanostructure under consideration. The period is P = P_x= P_y in the x- and y-directions. The cylindrical gold nanoelectrode (nanotube) has a height of h = 1800nm and an internal radius of 60 nm. The nanotube has 30-nm-thick walls, and it is connected to a planar gold electrode of 30-nm thickness. Both the nanotube and planar electrodes are immersed in water (upper half-plane) and are placed on a silicon nitride substrate. The wave propagates in the z-direction and is polarized in the x-direction. (b) Schematic of the mesh curvature in the numerical simulations and position of the probe for detecting electric field. Assuming the center of the nanotube is at x_o = 0 and y_o = 0, the probe is positioned at x_p = 75 nm and y_p = 0 and at the height of z_p = h + 1 nm = 1801nm from the substrate surface or 1 nm above the nanotube.

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2. Results

In our design, both planar and nanotube electrodes are placed on a silicon nitride substrate, as shown in the schematic in Fig. 1(a). To simulate the nanostructure, we consider a cylindrical gold nanoelectrode with an internal radius R and a height h. The nanoelectrode walls are 30-nm-thick. The nanoelectrode is positioned on a planar gold film of thickness 30 nm. We choose the same thickness for the walls of the plasmonic gold nanoantenna and planar electrode because of the specifics of the possible fabrication procedures [32]. For the same reason, the material of the cylindrical and planar electrodes is the same. In the fabrication of the cylindrical gold nanoantenna (nanoelectrode), a layer of insoluble resist is deposited on a silicon nitride substrate. A thin layer of gold, whose shape is determined by the insoluble resist, is then deposited, covering the whole sample surface. As a result, the planar electrode is deposited at the same time as the nanotube, and because of it, they have a comparable thickness and the same material. The planar electrode also acts as a heat sink to decrease thermal effects resulting from electromagnetic heating of the plasmonic gold nanoantenna, generated due to resistive losses.

In the near-field zone, the electric field radiates in all directions at the tip of the nanoelectrode. To analyze the nanostructure's resonant behavior, we carry out full-wave numerical simulations using the finite-element method (FEM) implemented in CST Studio Suite frequency-domain solver. We use periodic boundary conditions in the x- and y-directions and domain with the same periodicity P. The structure is illuminated with a plane wave propagating along the z-direction and with the electric field polarized in the x-direction (electric field along the x-axis) at normal incidence. The top edge of the nanoelectrode is curved (Fig. 1(b)) to avoid artificial hot spots from artifacts of numerical simulations. We use data from Refs. [34,35] to define complex permittivities of gold and silicon nitride, and the refractive index of water is n_w = 1.33.

Figure 2 illustrates the spectra maps of the absorptance and the electric field enhancement at a particular probe position indicated in Fig. 1(b), calculated for the plasmonic gold nanoantenna with a varying periodicity of the nanostructure P. Absorptance is defined as a ratio of optical power lost in the nanotube and planar electrode to the incident power. Absorptance in each nanostructure element is extracted from the simulation software package directly (settings “Specials” → “Calculate material power loss” → “Store per solid”). It does not require additional simulations as the software stores each solid's characteristics and material separately. While a significant portion of the absorbed incident power goes into heating rather than generating hot carriers [11], the latter is higher for higher absorptance. In our simulations, the part of absorptance that is due to the silicon nitride substrate is negligible for the entire lattice period range under consideration. Some optical power is absorbed in the nanotube, and some in the planar electrode. The planar electrode absorptance is negligible for the lattice period larger than 677 nm (not presented here). On one side, with an increase of lattice period, a larger portion of the unit cell is occupied by the planar electrode, so its contribution to the absorptance increases proportionally with the increase of lattice period. However, on another side, the lattice with a period larger than 677 nm is sparse, and the nanotubes are positioned far away from each other. Absorptance in the planar electrode is defined mainly by the tails of the near field around the nanotubes. The sparse lattice result in weaker nanotube resonances, a smaller electric field around the nanotube and in the planar electrode, and therefore smaller absorptance in the planar electrode. Below those values, the absorptance due to the planar electrode is about 10 to 15% of the total. The absorptance in the nanostructure due to the nanotube is about 90%, leading to intense electric field enhancement and hence high generation of plasmonic hot electrons.

 

Fig. 2. (a) Absorptance in the array of gold nanoelectrodes for different lattice periods P. The absorptance increase indicates spectral positions of modes excited in the nanostructure. The solid magenta lines indicate Rayleigh anomalies (1,0) and (1,1). The dashed white lines indicate periods P = 602 and 677 nm, analyzed in the figure that follows (see Fig. 3). See Fig. 8 in Appendix for the simulation results in a broader range of periods and wavelengths. (b) The electric field enhancement E/E₀ in the array of gold nanoelectrodes for different lattice periods P at a particular probe position (see Fig. 1(b)). The enhancement reaches the values of up to 50 times in resonances following Rayleigh anomalies. The internal radius is R = 60 nm, and the height is h = 1800nm.

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Absorptance is an integral characteristic that determines the spectral position of field enhancement in the whole structure (Fig. 2(a)). Alternatively, one can analyze an enhancement of the field at the specific point of the nanostructure (Fig. 2(b)), and one can identify places with the higher field enhancement. Whenever electric field enhancement is presented in our work, it is detected by a probe positioned as shown in Fig. 1(b). To calculate the enhancement E/E₀ in the array, the electric field E at the probe position is normalized to the magnitude of the incident electric field in water E₀. We compare Figs. 2(a) and 2(b) as scan maps to Figs. 3(a) and 3(b) as linear profiles, and we see that peaks in both characteristics are in good agreement. It confirms that enhancement of the electric field at the probe position close to the nanotube edge spectrally coincides with increases in absorptance for the whole structure. It also confirms that both absorptance and field enhancement can be used for identifying most efficient regimes for hot electron generation.

 

Fig. 3. (a) Absorptance and (b) electric field enhancement E/E₀ at a particular probe position (shown in Fig. 1(b)) in the array of gold nanoelectrodes for periods P = 602 and 677 nm. The dashed lines indicate Rayleigh anomalies (1,0): λ_RA = 800 nm for period P = 602 nm (blue dashed line) and λ_RA = 900 nm for period P = 677 nm (magenta dashed line). The spectra are shifted in the ordinate axis for clarity. Four resonances for P = 602 nm and three resonances for P = 677 nm are labeled for further discussion. (c)-(f) Field distributions averaged over the wave oscillation cycle at the resonances #1-4 for P = 602 nm. The maps are shown in the xz-coordinate cross-section (for y = 0). (g) Color scale that corresponds to all resonances in panels (c)-(f). The internal radius of the nanotube is R = 60 nm, and the height is h = 1800nm.

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Tracing absorptance peaks is the most reliable way to interpret the nanostructure spectrum and identify the mode spectral position. Being an integral characteristic of field enhancement in the nanostructure, the absorptance profiles are accurate indicators of nanostructure modes and their resonances. If nanostructure modes do not couple with each other (e.g., uncoupled dipole, quadrupole, etc., eigenmodes of a single nanoparticle), the absorptance profile is not subjected to interference effects, as opposed to reflection or transmission.

The sharp resonances are observed due to diffraction at the Rayleigh anomaly, resulting from collective plasmonic resonances in the nanostructure. By tuning the periodicity of the nanostructure, we can excite lattice resonances at the wavelength close to the Rayleigh anomaly. Since the nanoantennas are immersed in water with the refractive index n_w = 1.33, the resonance positions of absorptance and field enhancement occur at the wavelength close to λ_RA = n_wP, where λ_RA is the Rayleigh anomaly wavelength. We denote this Rayleigh anomaly (1,0), and because the lattice is square in our case, the notations (1,0) and (0,1) can be used interchangeably. The next Rayleigh anomaly for the square lattice appears at λ_RA = n_wP/√2, and we denote it as (1,1) Rayleigh anomaly. In the general case of rectangular lattice with periods P_x and P_y. and with normal light incidence, the spectral positions of Rayleigh anomalies can be found from the equation ${\left( {\frac{{2\pi }}{{{P_x}}}{n_x}} \right)^2} + {\left( {\frac{{2\pi }}{{{P_y}}}{n_y}} \right)^2} = k_w^2$, where n_x and n_y are the integers 0, ±1, ±2,…, and k_w is the propagation constant in the surrounding medium that corresponds to the Rayleigh anomaly (n_x, n_y), that is k_w = 2π/λ_RA. The case of an oblique incidence and different polarizations is discussed in [34].

Lattice resonances provide an additional degree of freedom and great flexibility in tuning the resonant response of the nanostructure. Because of the excitation of lattice resonances in the proximity to Rayleigh anomaly wavelength, one can adjust resonance wavelength by changing the period of the structure. One can see in Fig. 2(a) that resonances closely follow Rayleigh anomalies (1,0), (1,1), and higher-order (not presented here). In Fig. 3, we demonstrate the linear plots of the absorptance and field enhancement in the structure at two different lattice periods, P = 602 and 677 nm, and we study the effects on the nanostructure spectral response at these lattice periods. We observe that in the case of the smaller lattice period of 602 nm, there are slightly higher absorptance peaks compared to those for the lattice period of 677 nm. Also, we notice a slightly higher field enhancement for the lattice period of 602 nm than for the lattice period of 677 nm. Examples of spectra for two different periods, P = 602 and 677 nm, are shown in Fig. 3(a). We deliberately choose lattice periods that result in multiple resonances, and we show the change in resonance position with the shift of Rayleigh anomalies, λ_RA = 800 and 900 nm for the periods P = 602 and 677 nm, respectively (Figs. 3(a),(b)).

Multiple resonances are excited because the nanotube supports modes of different orders due to its elongation. In the periodic array, these modes experience hybridization, which results in multiple lattice resonances. Figures 3(c)-(f) show the simulated electromagnetic field distribution around the plasmonic gold nanoantenna of lattice period P = 602 nm, where we observe an electric field enhancement in the proximity to Rayleigh anomaly λ_RA = 800 nm. Namely, there are four peaks with field profiles that correspond to the excitation of different modes. The results show strong electric field enhancement around the plasmonic nanoantenna along its walls and at the tip. Figures 3(c)-(f) depict the electric field distribution around the nanoantenna due to higher-order modes resulting from the coupling of these modes of plasmonic gold nanoantenna into diffractive orders of the lattice period. These resonances can be controlled by the lattice period and shifted altogether (see Fig. 2 and 3).

Most of the lattice resonances, whether they are excited in the arrays of nanoparticles of simple shapes, such as a sphere, or nanoantennas with rather complex modes, possess similar features with respect to Rayleigh anomaly tuning. When the Rayleigh anomaly is close to the resonance of a single nanoscatterer (nanoparticle, nanorod, nanotube, etc.), the lattice resonance is relatively broad and spectrally positioned at a distance from the Rayleigh anomaly [30]. When the array period increases, the lattice resonance gets narrower and moves spectrally closer to the Rayleigh anomaly. Higher-order resonances approach the Rayleigh anomaly faster, while lower-order resonances often move parallel to the Rayleigh anomaly in a large spectral range. One can see in Figs. 2(a),(b) and 3(a,b) that resonances #1 and #2 shift parallel to the Rayleigh anomaly (i.e., remain at the same spectral distance for periods P = 602 and 677 nm), resonances #3 moves spectrally much closer to the Rayleigh anomaly for period P = 677 nm in comparison to its spectral position for period P = 602 nm, and resonance #4 disappears for periods P = 677 nm. A similar trend can be seen for higher-order Rayleigh anomaly (1,1) in Fig. 2.

Lattice resonances are narrow and appear as sharp features in the spectra. Besides them, one can see broader features, e.g., minima at wavelengths around 800, 950, and 1180 nm (vertical dark regions in Fig. 2) and maxima at the wavelengths around 880 and 1080 nm for any period. These features do not have a resonant nature and are not of the Lorentzian line shape (see, e.g., Fig. 3(b), blue line for P = 602 nm for the wavelength range 960 - 1200 nm). Instead, these are interference patterns related to the reflection from the substrate and planar gold electrode of 30 nm thick. The interference brings an envelope function and modulates lattice resonance excitations.

Surface and volume photoelectric effects are competing causes of hot electron generation [15,36]. As discussed above, the total loss of optical power in the nanoelectrode defines the efficiency of the volume mechanism. Similarly, the normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface defines the efficiency of the surface mechanism [15,36–38]. Absorptance in the nanostructure is chosen in Figs. 2 and 3 to predict the efficiency of hot electron generation. As a counterpart, in Fig. 4, we calculate the wavelength dependence of (E_n/E₀)² averaged over the nanoelectrode surface in the structure at two different lattice periods, P = 602 and 677 nm. Normal electric field E_n is an E field component normal to the nanotube's surface at each coordinate point. We further average its normalized square by integrating this characteristic over the entire nanotube surface and dividing this integral by the total nanotube surface. The total nanotube surface includes the outer and inner walls of the tube.

 

Fig. 4. Normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface in the array of gold nanoelectrodes for periods P = 602 and 677 nm. The dashed lines indicate Rayleigh anomalies (1,0): λ_RA = 800 nm for period P = 602 nm (blue dashed line) and λ_RA = 900 nm for period P = 677 nm (magenta dashed line). The spectra are shifted in the ordinate axis for clarity. The internal radius of the nanotube is R = 60 nm, and the height is h = 1800 nm.

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The normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface (in Fig. 4) is about one order of magnitude smaller than the magnitude of E/E₀ at the probe position (in Fig. 3(b)). The reason is that the probe position is chosen at the nanotube edge, where the field enhancement is very strong. It is common for near fields in a plasmonic nanostructure to be significantly enhanced at the curved surfaces of the nanostructure. In contrast, the normal electric field enhancement square (E_n/E₀)² is an average characteristic and accounts for the regions in nanotube with smaller field enhancement.

In Fig. 4, peaks #1 and #2 are very close to each other, and they are indistinguishable because of their broad linewidth. Nevertheless, results in Figs. 3 and 4 have the qualitative agreement, and we see that the peaks are spectrally positioned at the same wavelength. It confirms that the lattice resonances can increase the efficiency of hot-electron generation regardless of whether the surface or volume mechanism dominates in the photoelectric effect. Excitation of lattice resonances results in a more prominent field enhancement in the nanostructure, which, in turn, produces more hot electrons.

Lattice resonances are stronger in the case of a uniform environment or when substrate and superstrate have close or equal refractive indices [30]. For this situation, we choose three common examples of the substrate and corresponding index-matching liquid [39]: fused silica, acrylic, and BK 7 glass (Fig. 5). One can see from the simulation results that even a slight change in the refractive index (1.45, 1.48, and 1.51 for fused silica, acrylic, and BK 7 glass, respectively) of the surroundings results in a significant shift in the nanoelectrode resonance.

 

Fig. 5. Lattice resonances for nanoelectrodes in a different environment. The uniform surrounding can be realized by means of the substrate and index-matching liquid in the upper half. We show examples for fused silica, acrylic, and BK 7 glass, using Cargille datasheets for the refractive indices [40]. The dashed lines indicate Rayleigh anomalies (1,0), and they are shifted because of the different material indices. The period P = 677 nm and other geometrical parameters are the same as in Figs. 2 and 3. The spectra are shifted in the ordinate axis for clarity. The internal radius of the nanotube is R = 60 nm, and the height is h = 1800 nm.

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The enhancement of the near field produced by the lattice resonances of arrays of metallic nanoparticles has been investigated extensively in recent years. In particular, it has been shown that for infinite arrays, the enhancement can be made arbitrarily large by appropriately designing the geometrical characteristics of the array [41]. Here, we aim to show the general mechanism of enhancing hot electron generation with the lattice resonances. We refrain from optimizing the nanostructure, as this routine task can be performed at a later stage of engineering a particular practical application. Figures 6 and 7 indicate that the nanotube height and radius variations can result in more optimal hot electron generation. Varying the internal radius of the nanotube, we also change the external radius so that their difference remains at 30 nm. In Fig. 6, we see that using a larger radius, e.g., R = 60 or 80 nm, results in a higher absorptance (panel (a)) and consequently more efficient volume photoemission. However, a smaller radius, e.g., R = 40 nm, results in a larger normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface and multiplied by the nanotube surface area (panel (c)). We use this characteristic for an accurate comparison of surface photoemission efficiency in the case of nanotubes with different radii and correspondently different surface areas. In Fig. 7, we see that a larger nanotube length facilitates more intense surface photoemission. Further optimizations, potentially involving machine learning algorithms, can be performed to identify specific nanostructure parameters targeting applications with a particular wavelength, material composition, and so on.

 

Fig. 6. Change in absorptance and averaged field enhancement under variations of the internal radius of nanotube R. (a) Absorptance; (b) Normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface; (c) Same as (b) but multiplied by nanotube surface area for an accurate comparison of surface photoemission efficiency. The legend is the same for all three panels. The period P = 616 nm. The dashed lines indicate Rayleigh anomalies (1,0): λ_RA = 820 nm for period P = 616 nm. The nanotube height is h = 1800nm.

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Fig. 7. Change in absorptance and averaged field enhancement under variations of nanotube height h. (a) Absorptance; (b) Normal electric field enhancement square (E_n/E₀)² averaged over the nanoelectrode surface; (c) Same as (b) but multiplied by nanotube surface area for an accurate comparison of surface photoemission efficiency. The legend is the same for all three panels. The period P = 616 nm. The dashed lines indicate Rayleigh anomalies (1,0): λ_RA = 820 nm for period P = 616 nm. The internal radius of the nanotube is R = 60 nm.

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3. Conclusions

We have numerically studied the collective modes observed in the electric field enhancement and absorptance spectra in the plasmonic gold nanostructure. We have demonstrated that by selecting the array period comparable to the resonance wavelength, one can achieve high and narrow bandwidth absorptance and electric field resonances in the proximity of the Rayleigh anomaly in the nanostructure. We have shown intense electric field enhancement close to the Rayleigh anomaly wavelengths, consequently enhancing the generation of plasmonic hot electrons in the plasmonic gold nanoantenna (nanoelectrodes). We believe this novel approach for designing nanoelectrodes will be useful in researching and developing enhanced plasmonic hot-electron generation.

Appendix

The results below show the lattice modes in a broader range of wavelengths and periods.

 

Fig. 8. Absorptance in the array of gold nanoelectrodes for different lattice periods P. Compared to Fig. 2, the range of wavelengths and periods is larger. The white rectangle shows the range of results in Fig. 2. (a) Absorptance map. (b) Electric field enhancement map.

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Funding

Office of Naval Research (N00014-19-1-2117); U.S. Navy (HBCU-MI, N00014-18-1-2739).

Acknowledgments

D.B. acknowledges the support from grant N00014-18-1-2739 (the Navy HBCU-MI Program) and grant N00014-19-1-2117 (the Office of Naval Research). V.E.B. acknowledges the support from the University of New Mexico Research Allocations Committee, award RAC 2022, and WeR1: Investing in Faculty Success Program for the computational resources.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

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