1 Introduction

The surface plasmon (SP) has received much attention in nano-optics, e.g. biochemical sensors [1–3], optical nanoantennas [4, 5] and solar cells [6, 7]. These parts are used in applications that can exploit the local electric field enhancement of a metal surface by means of SP. The electric field enhancement by SP has the effect of improving a light excitation efficiency of applications. Such an electric field enhancement is observed particularly in metal nanoparticles and structures, with a resonance wavelength that depends on the material [8], size [8, 9], shape [8–10] of the metal, and the refractive index of the surrounding medium [11]. Sharp nanoparticles are much more conducive to a large enhancement of the electric field at their corners. Several groups have already reported optical characteristics, i.e., enhancement and distribution of the electric field in close interaction with the corner shape of a metallic nanoparticle [12–14]. Especially, the surface charge density at a triangular shape with sharp angle corners can be extremely high because of the stronger localization of the electromagnetic field. Meanwhile, a larger electric field was formed in the gap between the two nanoparticles in a dimer. It is expected that the enhancement of a dimer structure reaches ~105 times that of the incident intensity [13, 15, 16]. The bowtie structure, which consisted of a pair of triangular nanoprisms with large enhancement, has been reported [17–19]. The 2.0 nm narrow gap between two diagonally aligned gold nanoparticles has been achieved by lift-off method using electron beam lithography. Additionally, surface-enhanced Raman scattering (SERS) signals were obtained from these nanoparticles [20]. Moreover, the quantum regime of tunnelling plasmonics with controllable subnanometer (<1 nm) separation has been revealed by the quantum corrected model simulations and experiments [21].

However, it is difficult to fabricate a nanoprism or rod with sufficiently sharp corners by physical processing method [22]. Additionally, the relationship between the optical characteristics and the problem of fabrication is not well understood. In this research, we analyzed optical characteristics of the single silver nanoprism and the bowtie structure, and investigated their dependence on corner shape. Moreover, we elucidated the interaction with the curvature radius of a nanoprism and the gap distance of a bowtie structure. These results provide information for basic optical characteristics of silver nanoprisms with rounded corners.

2 Numerical methods

In the analysis of this research, we used the three-dimensional finite-difference time-domain (3D-FDTD) method. The basic calculation structure of a single nanoprism was set up on a glass substrate of refractive index of 1.515 in air (n air: 1.00) as shown in Fig. 1a. The schematic of a cross section represented in Fig. 1b is that of a nanoprism along the x–z plane. A nanoprism with rounded corners, whose curvature radius (CR) is indicated by the solid line, is shown in Fig. 1b. The height (H), side length (L), and angle (θ) of the nanoprism structure were 50, 100 nm, and 60°, respectively. The permittivity of the silver was expressed by the combined Drude and Lorentz model. The transverse-electric (TE)-polarized light (intensity: 1 × 10−10 W/μm2) was parallel to the substrate surface and directed toward the corner of the nanoprism. The incident direction was determined to be applied to the part of integrated optical circuits, e.g., optical branching filter and switching elements. When the optical circuit devices are assembled and packaged by using those elements, it is most suitable the incident light is parallel to the substrate. Moreover, we chose the TE-polarized light capable of obtaining a high the electric field intensity enhancement. Then, the electric field intensity enhancement was measured at an observation point (the black dot shown in Fig. 1b). The observation point was located 1.0 nm away from the nanoprism corner immediately above the glass substrate. The distributions of light intensity were calculated using an SP resonance wavelength. The electric field intensity enhancement and distributions of light intensity were calculated with mesh size of 1.0 nm due to accurate calculation in three dimensions. The value of 1.0 nm is reasonably satisfactory to us considering that an accurate calculation and a computational time. The schematic of a cross section represented in Fig. 1c is that of the bowtie structure in the x–z plane. The CR of a nanoprism was 8.66 nm, which was derived from experimental results [22]. The gap distance (G) is the distance between opposite nanoprisms. The black dot shown in Fig. 1c is an observation point in the bowtie structure. The observation point was located at the midpoint of distance between opposite nanoprisms immediately above the glass substrate.

Fig. 1
figure 1

Configuration for numerical simulation of a nanoprism and a bowtie structure with mesh size 1.0 nm. a Whole structure of a nanoprism. The x–z cross sections of b a nanoprism and c a bowtie structure. The nanoprism has height H [as drawn in (a)], curvature radius CR [as drawn in (b)], side length L [as drawn in (b)], and angle θ [as drawn in (b)]. The bowtie structure has gap distance G [as drawn in (c)]. The observation point was located immediately above the glass substrate. The inset of (c) is magnified view in the G

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3 Results and discussion

The optical characteristics depending on the curvature radius (CR) of the metal nanoparticle are shown in Figs. 2 and 3. Figure 2 shows wavelength dependencies of the electric field enhancement for a silver nanoprism with several sizes of rounded corners and for a cylinder without corners. The largest electric field enhancement was 240 times, obtained with the 0 nm CR. When the CR increased, the electric field enhancement was rapidly decreased. The electric field enhancement of the cylinder (which has the largest CR, i.e. that for non-edge corners) is 13 times. Furthermore, it was clear that the resonance peak at 520.67 nm (for the 0 nm CR) was blue-shifted when the CR became large. At the same time, the resonance peak at 384.64 nm (the 0 nm CR) did not appear. In contrast, when the spectral shape was compared between the nanoprism with the 20 nm CR and the cylinder, the resonance peak at 380.02 nm (the 20 nm CR) was nonexistent in the cylinder. This result is due to the corner shape becoming rounded. These results were also realized by the distributions of light intensity at the resonance peak, as shown in Fig. 3. When the light intensity distribution is compared between the 0 nm CR nanoprism (Fig. 3a) and the 20 nm CR nanoprism (Fig. 3c), it is seen that the electric field enhancement was largest at the corner of the nanoprism with the 0 nm CR. This is because of the surface electric charge induced by localized surface plasmons at the corner points. It is well known that the surface charge density at a sharp corner can be extremely high because of the stronger localization of the electromagnetic field [12]. In contrast, the electromagnetic field is less able to localize at the rounded corners because of coarse surface charge density at those corners. Therefore, the electric field enhancement decreased when the CR was increased. Figure 3a–d shows the distributions of light intensity at the long-wavelength resonance peak. These light intensity distributions resemble two-part electric fields localized at the corners, a so-called dipole mode. The symmetry of the light intensity distribution was broken due to the localization at corners. Figure 3e–g shows the light intensity distributions of nanoprisms at the short-wavelength resonance peak. Figure 3f, g resembles quadrupole mode in the form of combining two dipoles. Figure 3e resembles high-order mode. Namely, a nanoprism has the optical multi-mode. The high mode depends on the size and shape. The resonance peak at 384.64 nm (for the 0 nm CR) disappeared as the shape of the nanoparticles became more rounded. The quadrupole mode denotes the same tendency of the high mode.

Fig. 2
figure 2

Wavelength dependencies of the electric field enhancement for the nanoprism with rounded corners (curvature radius: 0, 10, and 20 nm), and for the cylinder. The secondary Y-axis exhibits the electric field enhancement of the 0 nm CR nanoprism

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Fig. 3
figure 3

Light intensity distribution dependence upon curvature radius (CR) for the nanoprism and the cylinder at localized surface plasmon (LSP) resonance peak wavelengths in a cross section of the x–z plane. a CR: 0 nm, λ inc.: 520.67 nm, b CR: 10 nm, λ inc.: 478.57 nm, c CR: 20 nm, λ inc.: 436.94 nm, d Cylinder λ inc.: 433.93 nm (diameter: 75 nm), e CR: 0 nm, λ inc.: 384.64 nm, f CR: 0 nm, λ inc.: 423.50 nm, g CR: 20 nm, λ inc.: 380.02 nm

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The electric fields, which appear in the nanometer-sized void between two nanoparticles, are on a so-called hotspot. These have been demonstrated to provide much greater enhancement of the local electric field in the hotspot than a single particle does. Therefore, we investigated the bowtie structure, which consists of a pair of nanoprisms and was expected to obtain a huge electric field enhancement. The bowtie structure shown in Fig. 1c was compared to a single nanoprism. Here, the CR of nanoprisms constituting the bowtie structure was 8.66 nm, as determined from experimental results [22]. Figure 4 shows wavelength dependencies of the electric field enhancement the nanoprism with rounded corners (CR: 0, and 8.66 nm) and the bowtie structure [gap distance (G): 1 nm, and CR: 8.66 nm]. The electric field enhancement at the bowtie structure reaches about 360 times that of the incident intensity. It was found that the electric field enhancement of the bowtie structure is 1.5 times larger than for the 0 nm CR nanoprism, and that in turn is 10 times larger than for the 8.66 nm CR nanoprism.

Fig. 4
figure 4

Wavelength dependencies of the electric field enhancement for the nanoprism with rounded corners [curvature radius (CR): 0, and 8.66 nm] and the bowtie structure [gap distance (G): 1 nm, and CR: 8.66 nm]

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Finally, we compared the curvature radius (CR) of a nanoprism with the gap distance (G) of a bowtie structure, as shown in Fig. 5. As with the curvature radius’s influence, when the gap distance is enlarged, the electric field enhancement decreases rapidly. Importantly, the behavior of the induced polarizations depends strongly on gap distance. The dipole–dipole interaction in the interfacial region between the nanoprisms is strongly influenced by interparticle coupling when the nanoprisms are close together. As a result, the electric field enhancement of bowtie structure is very sensitive to separation distance. This gap distance dependency of electric field enhancement was expressed by representing the experimental and calculation results reported by our group and Yokota et al. [20, 22]. When the CR and G increase, the surface charge density of nanoprism corner and the interparticle interaction of bowtie structure decrease, respectively. This means that it was a tendency becoming weak both the localization of the electromagnetic field. Therefore, the changes of the CR and G have the same influence on the electric field enhancement. In order to obtain larger electric field enhancement, the CR and the G are required to be minimized. In our group, the 8.66 nm CR single nanoprism and the 20 nm G bowtie structure were achieved by using focused ion beam milling in an experiment, as shown in Fig. 5 (dashed blue line [22] and dashed red line [17]). Recently, the dimer type gold nanoblock with 2.0 nm gap has been reported using electron beam lithography/lift off methods by Yokota et al. as shown in Fig. 5 (dashed red line [20]). It is now realistic to optimize bowtie structure with the advancement of physical techniques only.

Fig. 5
figure 5

The dependence of electric field enhancement on the curvature radius (CR) (shown in blue), and the gap distance (G) (shown in red). The dashed lines show the experiments [Reference No.]

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4 Conclusions

We clarified numerical optical characteristics controlled by the shape of corners of nanoscopic metallic structures. The electric field of a single nanoprism with sharp corners was enhanced by a factor of 240 compared to the incident light. However, the electric field enhancement rapidly decreased when the curvature radius of the corners increased. We also indicated that the light intensity distribution depends on the roundness of corners at resonance wavelengths. The multi-mode was changed to single dipole mode due to the nanoprism corners gaining a rounded corner shape. Finally, we showed that the electric field enhancement of the bowtie with rounded corners and G = 1 nm was 1.5 times larger than for the 0 nm CR nanoprism. The change of the CR and the change of the G have the same influence on the electric field enhancement. It tended to linearly increase the electric field enhancement, as the CR (10–0 nm) and the G (8–2 nm) were small. Our results were evaluated to decide how the curvature radius and the gap distance related to the electric fields of the nanoprism and the bowtie structure.