The influences of the symmetry-breaking on the plasmon resonance couplings in the isolated gold nanotube and the gold nanotube dimer have been investigated by means of the finite element method. It is found that the core offset of gold nanotubes leads to the red-shifts of the low energy modes and the enhanced near-field on the thin shell side of the symmetry-broken gold nanotube (SBGNT). In the weak coupling model of the SBGNT dimer, the interference of the bonding octupole mode of the dimer with the dipole modes causes a strong Fano-like resonance in scattering spectrum. The Fano dip shows a red-shift and becomes deep with the increase of the offset-value. In the strong coupling model of the SBGNT dimer, the coupling between two SBGNTs induces giant electric field enhancement at the gap of the dimer, which is much larger than that in the symmetry gold nanotube dimer. The SBGNT with larger offset-value exhibits stronger near-field at the “hot spot”.
©2012 Optical Society of America
Metallic nanoparticles (MNPs) have attracted considerable interest because of their special electronic and optical properties, which are determined by the localized surface plasmon resonance (LSPR) [1–3]. The LSPR can induce strongly enhanced electric fields (E-fields) on the surfaces of the MNPs. Such enlarged E-fields can lead to many applications such as surface enhanced spectroscopy, nonlinear optics, and nanotrapping [4–6]. The LSPR is also highly sensitive to the geometry of the MNP. Varying the geometry enables to modulate the LSPR of the MNP for different requirements. In the past decade, many kinds of MNPs, such as nanorods, nanoshells, nanocups, nanorings, and nanostars , have been developed and applied in various areas. Another important design proposal is the symmetry breaking in the MNP, which causes the increased surface area and giant enhancement of near-field [8–11]. Thus, the symmetry-broken MNPs are firstly of considerable interest in surface enhanced Raman scattering.
Recently, it is found that the symmetry breaking in the MNPs can induce prominent Fano resonance signals [12–19]. Fano resonances in plasmonic nanostructures have received extensive attention because of their potential applications in chemical and biological sensors, near field imaging, optical waveguides, and nonlinear optical devices [20–22]. Hao et al. [12, 13] found that the symmetry breaking in the concentric ring/disk cavity enables the coupling between plasmon modes of differing multipolar order, resulting in a tunable Fano resonance. An asymmetric inner Au core in an Au nanoshell can allow the interference between superradiant and subradiant plasmon modes and hence a distinct Fano resonance in its optical response . In a dual-disk ring symmetry-broken structure, the octupole mode of the ring interacts with the dipole mode of the disks to generate the asymmetric spectral feature of Fano resonance . Peña-Rodríguez and Pal have found that Au@Ag core–shell structures with a thin silver layer can produce a well defined and measurable Fano profile . A Fano resonance is found in a nanodisk with a missing wedge, which has ascribed to the coupling between the modes supported by the disk and wedge . Brown et al.  further reported that the heterodimers can induce the Fano resonances and the “optical nanodiode” effect.
The nanoegg, a simple symmetry-reduced nanoparticle, is modeled as a nanoshell with a nonconcentric core. Compared to the symmetry nanoshell, the cavity and sphere plasmons of all multipolar orders interact in the nanoegg, which results in strong hybridization and large plasmon energy shifts [8, 9]. The nanoegg particle also can exhibit large E-fields on its thin shell side and could be an effective substrate for surface enhanced spectroscopy . Yun et al.  further investigated the optical properties of nanoegg dimers and found the much great enhanced near-fields. However, the plasmon coupling properties of the symmetry-broken nanotube and its dimer were seldom reported .
In this paper, we study the plasmon resonance properties of the symmetry-broken gold nanotubes (SBGNT) and their dimers. The far-field spectra and near-field enhancements have been calculated by using a two-dimensional finite element method (2D-FEM). We focus on the influences of the symmetry breaking on the plasmon couplings in the SBGNTs and the dimers of SBGNT. It is found that the weak plasmon coupling between two SBGNTs can lead to a distinct Fano-like resonance in the scattering spectra. The strong coupling model of the SBGNT dimer induces giant E-field enhancements at the gap between two SBGNTs. We further investigate the effects of the various core offsets on the Fano resonances and near-field enhancements in the SBGNT dimers.
2. Electromagnetic scattering model
The gold nanotube consists of a dielectric core with radius r1 coated with a gold shell with radius r2. For a long nanotube, the values of r1 and r2 are much small compared to the length. Therefore, the gold nanotube can be considered as an inðnitely long cylinder. The schematic of the infinitely long SBGNT dimer is shown in Fig. 1 . Throughout the paper, r1 and r2 are fixed at 60 and 85 nm, respectively. The dielectric constants of the core, gold shell, and embedding medium are ε1, ε2, and ε3, respectively. The separation between two SBGNTs is D. At C point, we will calculate the electric field enhancements in the SBGNT dimers for discussing the variation of the “hot spot”. The core offset in left nanotube is denoted as d1 and the core offset in the right nanotube is d2. The shift of the inner core is along the axis of the dimer. If the inner core moves to the left, the value of the offset is assumed to be positive. Contrarily, the offset-value is negative. The inner core is SiO2 with ε1 = 2.04. The gold permittivity data are obtained by fitting the experimental results from Johnson and Christy . The numerous simulations have been carried out based on 2D-FEM, which is used to solve Maxwell’s equations on a discretized spatial grid . The FEM has recently been shown to be highly useful in the study of the optical properties of metallic nanostructures [28, 29]. The SBGNT dimer is illuminated by an electric field polarization vector along the axis of dimer propagating in the direction perpendicular to the axis of dimer. To avoid nonphysical reflections of outgoing electromagnetic waves from the grid boundaries, the perfectly matched layers of absorbing boundaries were applied around the targets. All of the systems we study are assumed to be in a vacuum.
3. Results and discussion
Firstly, we investigate the optical properties of isolated infinitely long gold nanotube. In Fig. 2(a) , the solid and dashed lines show the scattering and absorption spectra of the coaxial gold nanotube. The dotted line shows the scattering spectrum calculated by employing Mie scattering theory . The result of FEM shows perfect agreement with the analytical solution of Mie theory.
A strong scattering peak appears at 646 nm, which is due to the dipole plasmon resonance. In absorption spectrum, the dipole resonance mode appears at 613 nm. Figure 2(b) shows the far-field spectra of the gold nanotube with an offset core. Here the offset-value d is fixed at 23 nm. As shown in the inset of Fig. 2(b), the direction of the offset of inner core is along the incident polarization. In the symmetry-broken structure, the cavity and sphere plasmons of all multipolar modes will interact and form bonding () and antibonding () plasmons [8, 9]. For large offset of inner core, the bonding mode shows a strong red-shift and the antibonding mode shows a blue-shift compared to the concentric nanotube. In the scattering spectrum shown in Fig. 2(b), the peak at 820 nm is the quadrupole peak of bonding mode () and the peak at 648 nm shows the octupole peak of bonding mode (). At the shoulder of peak, the peak at 880 nm is the dipole peak of bonding mode (). Figure 2(c) shows the scattering spectra of the SBGNTs with various offset-values. With the increase of the offset-value, all of the dipole, quadrupole, and octupole peaks of bonding mode show red-shifts while the quadrupole and octupole peaks become strong due to the additional hybridizations of all multioplar indices.
Figures 3(a) -3(c) show the E-field enhancement distributions in SBGNT, which are calculated at the , , and mode peaks, respectively. Here the offset-value d is fixed at 23 nm and the offset of inner core is along the incident polarization. The E-filed focusing on SBGNT can be increased by offsetting the core and exciting the SBGNT at a higher-energy multipolar resonance . In Figs. 3(a)-3(c), higher energy hybridized modes increase the larger E-field enhancement. For all plasmon resonances, the E-field enhancement occurs around the outer shell surface and shows strongest on the thinnest part of the SBGNT. In Fig. 3(d), the solid line shows the variation of the E-field enhancement maximum in the coaxial gold nanotube. The dashed and dotted lines represent the variations of the near-field enhancement in the SBGNT, which are obtained at the A and B points in the inset of Fig. 2(b), respectively. It is obviously that the SBGNT provides larger enhancements than the concentric nanotube for all plasmon modes investigated.
Figure 4(a) shows the scattering and absorption spectra of the gold nanotube dimer (the model I). Here the separation D between two nanotubes is fixed at 10 nm. It is well known that, with the decrease of the dimer separation, the longitudinal plasmon resonance of the nanoparticle dimer shows a red-shift . Compared to the isolated gold nanotube [shown in Fig. 2(a)], the dipole peak of the gold nanotube dimer show a red-shift and broaden. Figure 4(b) shows far-field spectra of the SBGNT dimer with D = 10 nm. Here d1 and d2 are fixed at −23 and 23 nm, respectively. We assume this SBGNT dimer as the model II. It is surprising that a distinct Fano dip can be observed at 765 nm in the scattering spectrum (solid line) while a weak Fano dip appears at 620 nm. According to the plasmon hybridization theory, the dimer plasmon modes can be understood as the bonding (low energy) and antibonding (high energy) modes of the plasmons of the constituent nanoparticles . In the absorption spectrum (dashed line), the peak at 908 nm corresponds to the bonding coupling () of the quadrupole modes of the SBGNT and the peak at 784 nm is due to the bonding octupole mode (). A weak peak appears at 627 nm, which arises from the antibonding of the quadrupole mode (). In the model II, on the both sides of the dimer gap is the thick shell of the SBGNT. As discussed above, the near-field on the thick shell side of the SBGNT is weak, which should lead to the weak near-field coupling between two SBGNTs. Therefore, the bonding and antibonding modes of the SBGNT dimer are close. The interference among the sharp mode, the broad mode, and the mode leads to the strong Fano dip at 765 nm. The Fano resonance at 620 nm arises from the coupling between and modes.
Figure 4(c) shows the scattering and absorption spectra of the SBGNT dimer with D = 10 nm. Here d1 and d2 are fixed at 23 and − 23 nm, respectively. This SBGNT dimer is defined as the model III. In this structure, the thin shell side of the SBGNT locates on both sides of the dimer gap. The large near-field on the thin shell side of the SBGNT should cause the very strong near-field coupling between two SBGNTs, and hence the bonding modes of the SBGNT dimer are far away from the antibonding modes. Thus, we cannot find the distinct Fano profile in the scattering spectra of the model III. However, the giant E-field enhancements can be found at the gap of the SBGNT dimer (“hot spot”) in the model III. Figure 4(d) shows the variations of the E-field enhancements, which are obtained at the C point in Fig. 1. The solid, dashed, and dotted lines show the variations for the model I, II, and III, respectively. Compared to the other two models (the model I and II) of dimers, the model III exhibits the very stronger “hot spot” enhancement. The maximum E-field enhancement of the model III is about 3 times larger than that in the model I. The E-field enhancements in the model II are weaker than those in the other two modes (the model I and III), especially at positions of Fano dips. The electric field enhancements at the Fano dips are greatly depressed because of the destructive interferences in the Fano resonances of the model II [33, 34].
Figure 5(a) shows the E-field enhancement distribution of the SBGNT dimer at the wavelength of 675 nm, which represents the plasmon resonance of the mode. Figure (b) and (c) shows the E-field enhancement distributions of the SBGNT dimers at 765 nm (Fano minimum) and 827 nm (subradiant dark mode ), respectively. In Fig. 5(d), the E-field enhancement distribution in the SBGNT dimer at about 955 nm presents the plasmon resonance of mode. Compared to the E-field enhancement at mode, the decrease of the E-field enhancement at the gap of the dimer is mainly due to the destructive interferences of the subradiant dark mode with the and modes. The variations of the E-fields at the two end points of the SBGNT dimer mainly depend on the plasmon resonances in the SBGNT. Figure 5(e) and 5(f) show the surface charge distributions of SBGNT dimers at the Fano minimum and the mode, respectively. In Fig. 5(e), compared to the mode as shown in Fig. 5(f), the additional surface charges on two sides of the dimer gap arise from the effects of the and modes. We further investigate the surface charge distributions of SBGNT dimers for the subradiant dark mode and the plasmon resonance of mode, as shown in Fig. 5 (e) and (f), respectively.
Furthermore, we discuss the influences of the various offset-values on the Fano-like resonance in the model II and the near-field enhancement in the mode III. In Fig. 6(a) , the solid, dashed, dotted, dash-dot and dash-dot-dot lines show the scattering spectra of the mode II with offset-values of 0, 10, 15, 20, and 23 nm, respectively. It is found with the increase of the offset-value that the Fano dip in the scattering spectrum of the model II becomes broad and deep while the position of Fano dip shows a distinct red-shift. As discussed above, the increased offset-value causes the red-shifts of the low energy modes in isolated SBGNT. With increase in the offset in SBGNT dimer, both the bonding and antibonding couplings between the two SBGNTs show red-shifts, while the interaction of all multipolar modes enhances the strengths of the bonding and antibonding coupling peaks. Thus, the Fano dip shows a red-shift and becomes deep. Figure 6(b) shows the variations of the E-field enhancement at C point of the model III. With the increase of the offset-value, the E-field enhancement maximum shows a significant increase from ~7.9 at offset-value of 0 to ~25.54 at offset-value of 23 nm while the position of the maximum E-field enhancement shows a red-shift. This is mainly because the near-field on the thin shell side of isolated SBGNT should enhance with the decrease of the thin shell thickness.
Finally, we briefly discuss the conditions where the core offset is perpendicular to the incident polarization. In Fig. 7(a) , the dashed and solid lines show the scattering spectra of the SBGNTs with core offsets along and perpendicular to the incident polarization. In solid line, the peaks at 915 and 660 nm represent the and mode peaks, respectively. The large red-shifts of the dipole and octupole peaks of bonding mode are due to the additional hybridizations of all multioplar indices. It can be seen that the additional hybridizations of all multioplar indices show little effect on the quadrupole mode. Figure 7(b) and 7(c) show the scattering spectra of the SBGNT dimers with core offset perpendicular to the incident polarization. In Fig. 7(b), the core offsets are in the same directions. The peaks at 988 and 675 nm show the and modes, respectively. In Fig. 7(c), the core offsets of dimer are in the different directions. The and mode peaks appears at about 980 and 677 nm. In this case, the plasmon coupling between two SBGNTs becomes weaker compared to that shown in Fig. 7(b).
In addition, we should note that, if the length of the gold nanotube is comparable to the radius of nanotube, the above results should have some differences. For finite length cylinder, when the incident polarization is not perpendicular to the axis of the gold nanotube, the collective motions of the conduction electrons in two cross sections will happen, which should induce great influence on the plasmon resonances in gold nanotube. When the incident polarization is perpendicular to the axis of the gold nanotube, the localizations from two cross sections also affect the plasmon resonances in the gold nanotube. On the other hand, in practice, the models I, II, and III always coexist in the samples. If these three configurations are very far away from each other, the plasmon couplings among them cannot happen. In this case, the far-field spectrum of the sample shows the properties of the overlap among the spectra of three models, and then the near-filed enhancements in each model almost have no any variation. When these three configurations are very close, the near-field couplings among them take place. Then, the far-field spectrum and near-field enhancements should have greatly changes.
We have investigated the plasmon resonances in the SBGNTs and the dimers by using the 2D-FEM. It is found with increase in the offset-value that the low energy modes of isolated SBGNT show red-shifts due to the additional hybridizations of all multioplar indices while the near-field on the thin shell side of the SBGNT enhances. In the model II of the SBGNT dimer, the interference of the sharp mode with the broad dipole modes leads to a strong Fano dip in the scattering spectrum, which shows a red-shift and becomes deep with the increase of the offset-value. In addition, the coupling between and mode induces a weak Fano resonance at 620 nm. In the model III, the strong coupling between the two SBGNTs leads to giant E-field enhancement at the gap of the dimer. Such an enlarged E-field enhancement may be useful for the enhancement of SERS signal. The tunable strong Fano-like resonance in the SBGNT dimer also can be helpful for plasmon induced transparency.
This work was supported by the National Basic Research Program of China under Grant No. 2012CB921504, National Natural Science Foundation of China under Grant Nos. 11174113, 10904052, 11204129, 11274171, and 11104319, and Project Funded by the Priority Academic Program Development of Jiangsu higher education institutions.
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