1. Introduction

Metallic nanostructures support plasmon resonance modes [1]. They attracted intensive interests for the roles in the surface enhanced spectroscopy and local surface plasmon resonance (LSPR) sensing [2,3].The tunability and local field enhancement (LFE) of plasmonic structure are the two most important properties. Firstly, plasmon resonance wavelength must be tuned to the exciting light wavelength. For surface enhanced fluorescence spectroscopy, the exciting light is usually in the visible region [3,4], and for surface plasmon enhanced spectroscopy, in the visible and near infrared regions [3, 5]. The LFE factor and the line-width of plasmon resonance modes are two crucial factors affecting sensitivities. For example, in surface enhanced Raman scattering, the LFE factor above 1000 is needed to push the sensitivity to the level of single-molecule detection [6], and in LSPR sensing, the refractive index sensitivity is inversely proportional to plasmon resonance line-width [3].

Recently, many groups concentrated their interests on sharp-tip and particle-particle structures to study the enhancement of local electric field [7–9]. Crescent is a typical sharp-tip structure. By changing the geometrical parameters, the dipole peak was red shifted to the near infrared region [10–12]. The concentric and non-concentric ring/disk cavities were examples utilizing particle-particle interaction. The sharp resonance peak at the near-infrared region was due to the antisymmetric coupling of the disk and the ring dipolar plasmons [13–15]. In our earlier paper [16], we suggested a naocrescent/nanodisk (NCND) structure. The LFE factor of quadrupole peak reached to 700 by coupling of the gap and tip modes. However, how to enhance the LFE factors of the dipole and octupole modes, which are very important to enhance the tunability in the visible and near infrared regions, was not resolved. In addition, the LFE factor of the quadrupolar peak was not high enough for single molecule detection [6].

In this paper, we study the non-concentric nanocrescent/nanodisk (NNCND) structure made of silver. The crescent gap modes are similar with the crescent plasmon modes, and so, they can match perfectly. Both the LFE factors of the dipolar and octupole modes can reach to 700. In addition, the LFE factor of the quadrupole mode further enhances to 1400. Besides, the resonance modes can be changed in the visible and near infrared regions.

2. Methods

The NNCND structure, as shown in Fig. 1(a) , consists of a disk with radius D, a nanocrescent with inner and outer radii r and R and the centre to centre distance ∆₁, a crescent-gap (which is assumed to be free space) with least gap between the crescent tip and disk d. The distance between the centers of disk and crescent inner circle is ∆₂. The tip-to-tip distance W and the tip radius s are 30 and 1 nm, respectively. We suggest a method of fabricating NNCND structure based on the wafer-scale nanosphere lithography [17,18]. The crescent gap can be formed by changing the etching angle as rotating the sample.

 

Fig. 1 (a) The sketch of the NNCND structure and the incident light (R = 60 nm, r = 40 nm, ∆₁ = 19 nm, w = 30 nm, s = 1 nm, and d = 1 nm). (b) LFE spectra of the NNCND structure with different D. (c-e) The LFE factors of NCND (squares) and NNCND (circles) structures at (c) dipolar, (d) quadrupolar, and (e) octupolar resonance wavelengths as a function of the disk radius.

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The finite element method (COMSOL) adopting an adaptive mesh is used to solve the time-harmonic Maxwell equations. The computation domain is consisted of a NNCND structure, a region of free space (larger than a half of the light wavelength) surrounding it, and a perfectly matched layer eliminating the reflections at the domain boundaries. A transverse magnetic (TM) plane wave insides from the left as shown in Fig. 1(a). In the calculations, the permeability of silver is μ = 1, the complex permittivity comes from reference 19. The mesh is refined until it doesn’t change the simulation results.

3. Results and discussion

Figure 1(b) presents the LFE factors changing with the disk radii. LFE factor is defined as the ratio of the maximum local electric field amplitude to the value of incident light. The crescent inner radius is 40 nm. When D is 39 nm, the disk and inner ring of the crescent is concentric. Then, with decreasing the disk radius, a crescent gap with different shape is formed. As D decreases to 14 nm, the disk moves among the two crescent tips. Here the least gap d between the crescent tip and disk keeps as 1 nm.

Every NNCND structure supports three resonance modes as shown in Fig. 1(b). These modes are the results of the hybridization of disk and crescent plasmons [13–15]. The crescent supports multipolar resonance modes. Symmetry breaking can enable admixture of dipolar plasmon modes and dark multipolar modes which makes the dark multipolar modes dipole active [20]. The dipole, quadrupole and octupole modes of the NNCND structures are bonding combination of the dipole disk modes and dipole, quadrupole and octupole crescent modes. Because the crescent inner and outer wall must keep the same charges, the three modes are all subradiant (except the gap is very narrow) [13].

As the disk radii decrease from 39 to 38.2 nm, the LFE factor of the quadrupole mode increases abruptly from 31 to 1406, and the octupole one from 155 to 815. Then, the LFE factors of the two peaks decrease with the disk radii as shown in Fig. 1(b), however, the dipole peak always increases. When the disk radius decreases to 14 nm, the dipole peak reaches to the largest value of 759. The widths at half maximum intensity of the largest dipolar, quadrupole and octupole peaks are of 96, 4 and 2 nm, respectively. When the disk radius decreases, the three resonance peaks blue shift all the time, which are resulted from the decreasing of the interactions between the crescent and the disk modes [14]. For the traditional particle-particle scheme with gap of 1 nm, the LFE factor is at the level of 100 and the width at half maximum intensity is about 100 nm [8]. In the NNCND structure, the sharp tip, narrow gap and antiparallel orientation coupling of the disk and crescent modes contribute all to increasing the LFE factor and decreasing the line width [13].

Figure 1(c-e) compares the LFE factors of NCND and NNCND structures with different disk radii. Obviously, the largest LFE factors of dipolar and octupole peaks are enhanced by 4 times than the concentric one, and the quadrupole is about doubled.

Figure 1(d-e) shows that when the disk radius increases from 38.2 to 39 nm, the LFE factors of the quadrupole and octupole modes decrease abruptly. In order to understand the mechanisms, we calculate the distribution of electric field amplitude and orientation of the quadrupole mode of the NNCND structures with the disk radius of 38.2 and 39 nm, respectively. Figure 2 shows that when the disk radius change from 38.2 nm to 39 nm, the charges in crescent inner and outer walls become different for the strong interaction of the disk and crescent modes. The crescent quadrupole modes cannot be excited, and the NNCND mode changes to superradiant. The strong radiative damping results in a drastic decrease in the LFE factor. The radiative mode change prevents the LFE factor to further increasing when decrease the gap width [13]. For octupole mode, the case is similar. If we design proper geometrical parameters of the NNCND structure, for example, magnifying the crescent structure, the quadrupole and octupole peaks can further increase before the radiative mode change [13].

 

Fig. 2 The distribution of electric field amplitude and orientation of the quadrupole modes of the NNCND structures with disk radius of (a) 38.2 and (b) 39 nm, respectively.

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From the viewpoint of transformation optics [21–23], the crescent can be transformed from periodic metallic strips which support slow surface plasmon polaritons being boundded to and propagating along the metal surface [24]. The crescent multipole modes are analogous to Fabry–Pérot (FP) modes arising from the reflections between the two tips. The nanogap also supports such modes reflected by the two gap entrances. Changing the gap width can alter the mode index and so do the resonance modes [25]. In the NNCND structure, the LFE factor is decided by the coupling between the crescent and gap modes.

Figure 3(a-d) shows the distribution of electric field amplitude at the dipolar resonance wavelength of crescent, gap, and NNCND structure. Figure 3(e) shows the normalized LFE factors (the value at tip apex is assumed to be 1) along the arc in the inner wall of the cavities. For the crescent dipole mode, the maximum LFE is at the tip apex, and the LFE decreases rapidly with the increase of arc length. For the ring gap, the maximum value is in the neighbouring of the gap entrance, and the field amplitude decreases slowly in the gap. The gap and crescent modes can’t match well. However, when the disk deviates from the cavity center, the ring gap becomes a crescent gap and the field amplitude decreases faster in the gap. When the disk contains in the two tips as shown in Fig. 3(d), Fig. 3(e) shows that the field in the crescent inner wall of the NNCND structure changes similarly with the single crescent. The dipolar modes of the gap and crescent match perfectly. In addition, we found that the point of maximum LFE approaches to the tip apex with deviating the disk. The two factors contribute both to enhance the LFE factor to the highest value of 759.

 

Fig. 3 (a-d) The distributions of electric field amplitudes and (e) the normalized LFE factors in the cavity inner walls at dipole resonance wavelength. The lines (a-d) in figure (e) represent the cases of figures (a)-(d), respectively. The tip apex is set as the start point of the arc length.

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In a single nano-tip, there were multiple nodes with the electric field converging and diverging vertically [10]. The electric field amplitude on the central point between adjacent two nodes is minimum [16], as shown by the black arrows in Fig. 4(a) . In the octupole mode of the crescent structure, every tip is cut into three parts by two minimum points. The length of every part is different. In the octupole mode of the ring gap, five minimum points cut it into four equal parts as shown in Fig. 4(b), so the gap and crescent modes can’t match well in NCND structure. However, in the crescent gap, the octupole mode is similar with that of crescent as shown in Fig. 4(c), and they can match well. Figure 4(d) shows the distribution of electric field amplitude of the NNCND structure. For quadrupole mode, the case is similar and described in detail in Ref. 16, and we don’t repeat here.

 

Fig. 4 The distribution of electric field amplitude of octupolar resonance modes of (a) crescent, (b) ring gap, (c) cresent gap and (d) NNCND structure.

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In Fig. 5(a-b) , we show the influences of the least gap d between the crescent tip and disk on the resonance LFE peaks. When least gap d increases, the interaction between the crescent and the disk modes decreases correspondingly, which results in the decrease of LFE factor and blue shifts of the resonance wavelength. The interactions between the crescent and disk in the quadrupole and octupole modes are stronger than that of the dipole one, so the LFE factors decrease faster. When d increases to 6 nm, the quadrupole peak is still about 400, and the dipolar and octupole ones are above 200. All of them are still very high.

 

Fig. 5 (a) LFE factors and (b) resonance wavelengths of the NNCND structures as a function of the least gaps d between crescent tip and disk.

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Figure 5(b) shows that the dipole peak is in the near infrared region. The dipolar mode is a high tunable peak which has been studied intensively in the concentric and non-concentric ring/disk cavities [13–15]. The octupole resonance wavelength is shorter than the quadrupole mode, and it is easily tuned in the visible region. In one words, the dipole and octupole modes enhance greatly the tunability of the NNCND structure. The three resonance peaks cover the wavelength from 400 nm to 1500 nm just by changing the least gap d. If want to tune the resonance wavelength with higher LFE factor, we can enlarge the overall size of the structure and keep the least gap with small size.

In the NNCND structure, the coupling of tip and gap modes is strongly influenced by the refractive index around the structure. In Fig. 6 , we compare the LFE spectra of NNCND with two different dielectric insertions, where the disk radius is of 30 nm. Figure 6(a) shows that when the refractive index changes from 1 to 2, the dipole, quadrupole and octupole plasmon resonance peaks shift by 2170, 568 and 349 nm, respectively. When we just change the refractive index in the gap as shown in Fig. 6(b), the resonance peaks also shift strongly.

 

Fig. 6 LFE spectra of the NNCND structure with dielectric insertions in all space (a) and in the gap (b) with the refractive indexes of 1 (black solid line), 1.5 (red dashed line) and 2 (blue dash-dotted line), respectively.

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We study the influences of the surface imperfection of real structures on the local field enhancements. Figure 7(a) shows the geometry configuration of the NNCND structures with 24 randomly distributed surface faults. These faults are of 50% bumps and 50% dimples with radii ranging from 1 to 3 nm. Figure 7(b) shows the LFE factors of the NNCND with 0, 12, and 24 randomly distributed surface faults. Obviously, the surface faults influence slightly on the LFE factors.

 

Fig. 7 (a) The geometry of the NNCND structure with 24 randomly distributed surface faults, and (b) the LFE factors of the NNCND with 0, 12 and 24 surface faults.

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The size dependent dielectric response of single nanoparticle has been studied intensely. For metal nanoparticle with size smaller than the mean-free-path of electrons (~10nm), the surface scattering starts to contribute to the losses of light field amplitude. A simple way to treat the additional scattering losses is to modify the imaginary part of the metal permittivity as ${\epsilon }^{″}={{\epsilon }^{″}}_{bulk}+k{\lambda }^3$ [26], where ${{\epsilon }^{″}}_{bulk}$is the imaginary permittivity of bulk metal, k is a constant inversely proportional to the particle radius a.

Large metal structures with sharp tips have been studied experimentally and theoretically for the huge LFE factors [27–29]. The sharp tips in such structures are mostly smaller than 10 nm, and the sharper the tips, the higher local field enhancement for the stronger lightning rod effect. Y. Lu et. al reported Raman scattering spectrum of Rhodamine 6G molecules adsorbed on a gold nanocrescent moon with sharp edge of sub-10 nm excited by a diode laser [27]. The Raman enhancement factor was estimated to be larger than 10¹⁰, which suggested the local electric field enhancement was no less than 10^2.5 (316.2). They also calculated the corresponding field enhancement by finite element method using bulk permittivity. At the resonant wavelength, the maximum LFE factor reached to 10^2.6 (398.1) and accorded well with the experimental suggestion (316.227). However, if the permittivity of nanoparticle with diameter of 10 nm is used, our calculated results show that the LFE factor is less than 100, which is much less than the experimental suggestion.

D. Ward et al. reported that gold electrodes separated by subnanometer gap resulted in electron transfer, but there was no obvious evidence that it decreased the LFE factor. They found that the LFE factor above 1000 was induced in their experiment [28]. They calculated the corresponding field enhancement by finite difference time domain (FDTD) method using bulk permittivity [29], and explained well the experimental results.

These experimental and theoretical results indicate that for large metal structures with sharp tips, the permittivity is nearly equal to the value of bulk metal. Therefore, our calculation results of NNCND structures with permittivity of bulk metal are reasonable.

Electron tunneling between particles with gap of 0.1-1.5 nm has been studied in theory. Recently, J. Zuloaga et. al presented a full quantum mechanical theory of the plasmon resonances as a function of gap of a nanoparticle dimer [30]. They compared the calculation results by classical theory and quantum theory, and found that the quantum effects began to reduce greatly the LFE factors as the gap less than 0.42 nm. L. Mao et. al studied the influence of the quantum effects of nanogap on the surface-enhanced Raman scattering [31]. They found that when the gap was smaller than 0.6 nm, the quantum tunneling began dramatically reduce the field amplitude between nanoparticles. For nanogap of 1 nm, the field amplitude was only decreased by less than 20%.

The least gap in the NNCND structure is 1 nm, so the LFE factor will decrease by less than 20% if consider the quantum effects. We adjust the imaginary part of the metal permittivity ${\epsilon }^{″}={{\epsilon }^{″}}_{bulk}+k{\lambda }^3$, so that the decrease of the LFE factor is about 20%, and study its influence on the match of the resonant modes. Figure 8 shows that the resonance wavelength keeps nearly un-moved. So we think the coupling of tip and gap modes can still be described by classical electromagnetic theory.

 

Fig. 8 The LFE factors for k taken as 0 and 0.6.

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3. Conclusion

In summary, by deviating the nanodisk in the NNCND structures, the coupling of crescent and crescent gap enhances the LFE factors of the dipole, quadrupole and octupole resonance peaks. These resonance peaks shift in the visible and near infrared regions by adjusting the NNCND geometries. In addition, the LFE factor of the quadrupole mode is more than 1000, which suggests a new way to the level of single molecular detection.