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Enhanced surface plasmon resonances in a silvershell nanocylindrical pair connected by a different type of nanobar that interacts with incident plane wave of transverse magnetic polarization are simulated by use of the finite element method. Arrays of silver nanoshells connected by silver nanobars are also investigated. The proposed structure exhibits a red-shifted localized surface plasmon that can be tuned over an extended wavelength range by varying the width of the nanobar and the dielectric constant in dielectric holes (DHs). The increase in the scattering cross sections is attributed to the effects of surface plasmon on the nanobar surface and a larger effective size of DH that is filled with a higher refractive medium. The predictive character of these calculations allows one to tailor the shape of the nanoparticle to achieve excitation spectra on demand with a controlled field enhancement.
©2010 Optical Society of America
1. Introduction
Surface plasmon resonance (SPR) [1–3] has been employed as an important spectroscopic tool for sensitive and specific detection chemical, biological, and medical analysis; and is of great interest to many fields such as biology, medicine, and biomedical engineering, environmental and industrial monitoring, as well as defense and security [4–7]. Recently, experiments which show the extraordinary high transmission of light through the metallic subwave-length hole array are of considerable interest [8]. When the size of metallic hole is much smaller than the wavelength of incident light, it was found that the maximum transmission is about 2-3 times higher than the hole porosity of the structure [9]. Because of the localized field amplification that occurs, current-carrying devices or biomolecule sensors based on the surface plasmon excitation at metal nanoparticles can be expected to lead to new and interesting optoelectronic phenomena and applications.
In a recent experimental work, Wang et al. have prepared gold nanorods [10] and observed a large two-photon lumininescence (TPL) efficiency in the scattering spectra. Several numerical studies of this phenomenon have been performed on different particle shapes [11] and particles surrounded by shells [12]. Arrays of metallic nanoparticles are also used to transfer electromagnetic energy in the near field regime [13]. However, the effects of the nanoshell connected by a nanobar of metallic nanoparticles arrays have not been considered. The nanoshells can be constructed by forming an air hole inside the metallic nanocylinders arrays. In recent years, the physical and chemical properties of metallic nanoshells have received particular attention [14–16]. Nanoshells possess several attractive features which make them interesting as nanoscale optical components [17]. They possess a plasmon-derived tunable optical resonance controlled by the dimensions of the air hole in the nanocylinder and the thickness of the metallic shell, spanning much of the visible and infrared regions of the optical spectrum [18]. Additionally, nanoshells and other nanoscale metallic structures have been shown to greatly enhance local electromagnetic fields in certain regions near their surfaces at specific wavelengths of light, controlled by nanostructure geometry [19]. This subwavelength structure could provide a tool for manipulating light below the diffraction limit.
In this paper, the near field response of a silvershell nanocylindrical pair connected by a nanobar interacting with transverse magnetic (TM) mode incident plane wave is simulated by using the finite element method (FEM), which includes the investigation of particle-particle interaction. The enclosure of a silvershell nanocylindrical pair with dielectric hole (DH) forms an open cavity model and the electromagnetic field is effectively confined in the gap of the pair to generate high local-field enhancement. The nanobar which connects between two identical silvershell nanocylinders and the dielectric constant in DHs are the key factors which provide additional variables to explore, for the tuning of the near field optical properties between the nanocylindrical pairs. We shall compare the optical response in the near field zone of a silvershell nanocylindrical pair to that of a solid silver nanocylindrical pair. The influences of wavelength of incident light, different types of nanobars, the width of nanobar and the refractive index of DHs in a silvershell nanocylindrical pair on local-field enhancement are discussed in our simulations. In addition, arrays of the silver nanoshells connected by the nanobars are also investigated.
2. Numerical method
The dispersion properties of the metal must be considered here since the absorption and permittivity of the metallic material are frequency dependent. The Drude model is used to describe the dependence of the metallic permittivity on frequency [20,21]. The FEM that we applied to metal nanoparticles was comprehensively described in Refs [20,21]. We have used the silver permittivity data obtained from Johnson and Christy [22] and corrected with the Drude model, which includes the size effect [23]. In our formulation we used triangular high order edge elements. To model an infinite simulation region with a two-dimensional finite geometry model (i.e., to enclose the computational domain without affecting the numerical solution), it is necessary to use anisotropic perfectly matched layers (PMLs) [20,21] that are placed before the outer boundary. This formulation can be used to deal with anisotropic material in terms of both dielectric permittivity and magnetic permeability, allowing anisotropic PMLs to be implemented directly.
For a single metal nanoshell, the SPRs can be obtained analytically by the extended Mie theory within the quasi-static approximation [24]. Here we used the FEM to investigate the SPRs for different nanoshells from the study of the scattering cross section (SCS) [25,26] of the nanoparticles. SCS indicates the amount of light scattered in the far field. The SCS is defined from the scattered and incident field via Poynting’s vector, and it is essentially the ratio of the outgoing radial flux to the incoming flux associated with the planewave solution. Taking advantage of variable geometries and mode hybridization, metallic nanoshells can provide large tuning ranges for SPRs, spanning from the visible wavelength range to the mid-infrared even for a simple two-layered nanoshell sphere [27,28].
3. Results and discussions
First, we investigate the difference of SPR on a solid silver nanocylindrical pair connected by a dielectric nanobar (structure #1) and a silvershell nanocylinder pair connected by a dielectric nanobar (structure #2) which are illuminated with a TM electromagnetic plane wave (see the insets of Fig. 1 ). The medium of the dielectric nanobar is InGaAsP (ε = 2.25) and its widths are varied from 4 to 40 nm. The direction of the electric field E is perpendicular to the chain axis. The wavelength of incident light varies from 300 nm to 1200 nm. The gap (interparticle distance) is set to g = 20 nm. Near field intensities of local fields in the gaps between two nanocylinders are also quite sensitive to the radius of the nanocylinders. For our simulations, we set the radius of a solid silver cylinder a = 50 nm; and the outer and inner radii of a silvershell a = 50 nm and b = 40 nm (thickness d = a-b = 10 nm), respectively.
Fig. 1 SCSs on a solid silver nanocylinder pair and a silvershell nanoscylinder pairs with different width of dielectric nanobar (w) as a function of wavelengths of TM incident light, where w = 4,5,6,7,8,9,10,11,15,20, and 40 nm, respectively.
Figure 1 shows the SCSs on structure #1 and #2 with different width (w) of dielectric nanobar as a function of wavelengths of TM incident light. In structure #1 (see the left side of Fig. 1, enclosed by a blue-dashed line), the resonance peaks occur around λ = 350 nm as w<10 nm and a little red-shifted as w>10 nm. Note that the decreasing of w results in a higher SCSs and larger width of resonance peaks. The near field intensity is the same tendency with the SCS. In structure #2 (see the right side of Fig. 1, enclosed by two red-dashed lines), an obvious red-shifted and larger width of peaks can be seen when a solid silver nanocylinder pair is replaced by a nanoshell one. There are two peak wavelengths (around λ = 350 nm and 600 nm) for w = 4 nm and only one peak for w≧5 nm which occur around λ = 600nm and show a lower SCSs with the increase of the width of the dielectric nanobar. The phenomenon of decreasing SCSs is due to the absorption of field intensity by the dielectric nanobar in the gap of the nanocylindrical pair. Note that a pair of silver nanoshell connected by a dielectric nanobar can tune the SPRs by changing the width of nanobar.
We further investigate the dielectric effects arising from the dielectric holes (DHs) in the nanoshells connected by a dielectric nanobar. As observed from our simulation results (results not shown here), filling a medium with a higher refractive index in DHs increases the effective size of the DH and also increases the effective index neff . Now we investigate one case of structure #2 with width of a dielectric nanobar w = 8 nm (we name it case 1). The refractive medium is filled in the DHs, varying fromε = 1.00, 1.77, 2.31, 2.66 to 3.06 [8]. It can be clearly seen in Fig. 2 that the SCS is incremental and the maximum SPR mode (peak) tends toward a higher wavelength. It’s unlike the case with no refractive medium filled in DHs (w = 8, see Fig. 1, we name it case 2), where only one peak in case 2 but more than one peak and much higher SCSs in case 1 corresponding to the SPR. The dielectric effects arising from the DHs in nanoshells show more peak wavelengths, higher SCSs and red-shifted extension. Note that increasing the refractive index of the medium in DHs results in higher SCSs and a narrower peak width, together with a trend of red-shifts with the increase of the dielectric constant in DHs. The TM-mode near field distributions at their corresponding resonant peak wavelength are (a) λ = 580 nm forε = 1, (b) λ = 620 nm forε = 1.77, (c) λ = 680 nm forε = 2.31, (d) λ = 720 nm forε = 2.66 and (e) λ = 780 nm forε = 3.06, respectively, as shown in Fig. 2(b).
Fig. 2 Upper: Results of SCSs vs. incident wavelengths. The parameters are maintained as: the thickness of silver-shell d = 10 nm, the interparticle distance g = 20 nm and the DHs with dielectric constant ε = 1 (air-hole), 1.77, 2.31, 2.66 and 3.06, respectively. Bottom: The TM-mode near-field distributions at their corresponding resonant peak wavelengths: (a) λ = 580 nm forε = 1, (b) λ = 620 nm forε = 1.77, (c) λ = 680 nm forε = 2.31, (d) λ = 720 nm forε = 2.66 and (e) λ = 780 nm forε = 3.06, respectively.
Base on our previous analysis, it is expected that the use of two identical silver nanoshells connected by a silver nanobar may provide additional degrees of freedom for the design of chain waveguides compared to their dielectric nanobar counterparts. Now we investigate the difference of SPR on a solid silver nanocylinder pair connected by a silver nanobar (structure #3, see the inset of Figs. 3(a) ) and a silvershell nanoscylinder pair connected by a silver nanobar (structure #4, see the inset of Figs. 3(b)) with different widths of silver nanobar (w), which are illuminated with a TM electromagnetic plane wave and propagate in the nanochain direction. The widths of silver nanobar are varied from 4 to 98 nm. The gap is set to g = 20 nm and the thickness of silvershell is d = 0 nm. It can be seen in Fig. 3 that the performance of structure #4 is quite different from that of structure #3. In Fig. 3(a), an obvious red-shifted and SCSs enhancement can be found as the width of silver nanobar is smaller than 8 nm. This phenomenon can be attributive to the combination of SPR and a narrower silver nanobar whose width is smaller than its skin depth (near 10 nm). Due to the light is absorbed by more volume of metal in the gap, the intensity of SCSs of structure #3 is decreased as the width of the silver nanobar is larger than 10 nm, as shown in Fig. 3 (a).
Fig. 3 The difference of SCSs on (a) a solid silver nanocylindrical pair connected by a silver nanobar and (b) a silvershell nanoscylindrical pairs connected by a silver nanobar of different width (w), which are illuminated with a TM electromagnetic plane wave and propagated in the nanochain direction.
Turning to the silvershell nanocylinder pair connected by a silver nanobar as shown in Fig. 3(b) (structure #4), we have smaller volume of structure #4 compared to structure #3. Due to the symmetries of the charge distributions in a silver shell nanocylindrical pair, the dipole or quadrupole resonances can be induced. Furthermore, the dipole moments of the inner and outer surfaces are aligned. There is a strong electromagnetic coupling between the inner and outer shell surfaces when the thickness d is small compared to the nanocylinder radius a. This leads to a new scheme of polarization and results in mode splitting similar to the case of a thin metallic slab, which is characterized by symmetric and asymmetric mode branches. In Fig. 3(b), two clear resonance peaks can be found as the width of a silver nanobar is less than 10 nm. The SPR effect obtained from structure #4 (Fig. 3(b)) is superior to that of structure #3 (Fig. 3(a)). For example, for the case of w = 4 nm, the peak wavelength can be shifted from λ = 930 nm (see Fig. 3(a)) toλ = 1000 nm (see Fig. 3(b)).
As another example, we choose the case of structure #4 whose width w = 4 nm. The refractive medium is filled in the DHs, varying fromε = 1.00, 1.77, 2.31 to 3.06. It can be also seen in Fig. 4 that three resonance peaks are obtained and the SCS is obviously increased compared to the same case but no refractive medium filled in the DHs as shown in Fig. 3(b). The maximum SPR mode (peak) tends toward a higher wavelength and the SCSs is nearly the same as the wavelengths become greater thanλ = 1000 nm. The TM-mode near-field distributions at their corresponding resonant peak wavelengths are λ = 1000 nm forε = 1.77, λ = 600 nm forε = 2.31 and λ = 650 nm forε = 3.06, respectively, as shown in the inset of Fig. 4.
Fig. 4 Results of SCSs vs. incident wavelengths. The parameters are maintained as: the width of silver nanobar w = 4nm, the thickness of silvershell d = 10 nm, the interparticle distance g = 20 nm and the DH silvershell nanocylindrical pair with dielectric constant ε = 1, 1.77, 2.31, 2.66 and 3.06, respectively. Inset: The TM-mode near field distributions at their corresponding resonant peak wavelengths.
The field distributions can help us to understand how the light distributes through the silver nanoshell with a nanobar. Figure 5 shows the near field intensity versus a different refractive index of DH at their relative peak wavelengths along chain axis in the range of [-60, 60] nm. The width of the nanobar is kept at w = 4 nm, and the thickness of nanoshell is kept at d = 10 nm. It can be observed in Fig. 5 that higher refractive index of DH results in higher field intensities. This phenomenon can be explained as follows. Filling holes with higher refractive medium makes the effective size of holes increased and will also enhance the effective refractive index neff. For the atoms along the surfaces of the silvershell nanocylinder pair and the silver nanobar can be considered as many dipoles at the symmetry positions around the circumference of the DH and the surface of nanobar. The strongest field intensity peak value is found at the central part of the gap between the nanocylinder pair and decreases rapidly inside the silver nanoshells as shown in Fig. 5. Note that the localized electric field enhancement at the central part of the gap due to the particle-particle interaction and the diopole moments of the inner and outer surfaces of the nanoshell pair, and the surface of the nanobar. The field enhancement of the silver nanocylinder pair originates mainly from the localized surface plasmon mode excited by the evanescent field. These near field optical phenomena of the metallic nanostructure correspond to the near field optical properties that were found in previous simulations and experiments [29,30].
Fig. 5 The near field intensity versus different refractive index of DH at their relative peak wavelengths along chain axis in the range of [-60, 60] nm. The thickness of the nanoshell is kept at d = 10 nm.
From the analysis and discussion of the above cases, we can summary that highly efficient SCSs enhancement can be achieved in the silver nanocylinder pairs connected by a silver nanobar through a proper design of the dielectric constant and the width of silver nanobar. The SCSs can be enhanced by several orders of magnitude and the resonance of the spectrum of the SCSs depends on the DHs and the width of the silver nanobar.
As deduced from the previous simulation results, now we investigate the transmission properties of chain waveguides made of the silver nanoshells connected by the silver nanobars. One commonly used approach for evaluating the transmission properties of a chain waveguide is to use the SNOM tip to drive the first particle at the near end of the chain [31–33]. The energy transport in the chain is then realized by the field coupling between the modes of the neighboring particles [32,33]. The propagation length is determined based on the energy decay within the particle chain. In the following simulations, we will focus mainly on the cases when the chains are driven at the resonances of two identical silvershells connected by a silver nanobar as discussed above.
As a first step, we compare the field intensity for the chain waveguides with 5 silver nanoshells without the silver nanobars (structure #5) and with the silver nanobars (structure #6, w = 4 nm). The chain waveguides of structure #5 and structure #6 shown in Fig. 6(a) and Fig. 7(a) with a linear array of silver nanoshells are illuminated with a TM electromagnetic plane wave along the chain axis and their parameters are shown in Figs. 6(a) and 7(a), respectively. The electric field distributions along the chains at their resonance wavelengths of structure #5) and structure #6 are shown in Figs. 6(b)-(d) and Figs. 7(b)-(d) for different DHs, and their corresponding resonance wavelengths, i.e., λ = 1000 nm for ε = 1.00, λ = 1000 nm for ε = 1.77, λ = 600 nm for ε = 2.31, and λ = 650 nm for ε = 3.06, respectively. From the field distributions shown in Fig. 7 we can observe that the mesoscopic nature of the finite chain waveguide is clearly visible, as the structural DHs dramatically change with the increase of the DHs, e.g., compared to structure #5 which is without a silver nanobar as shown in Fig. 6. It can be clearly seen in Fig. 6 that the field intensity decays gradually along the chain even with a higher index dielectric medium filled in the DHs (for example, ε = 3.06). The field confined along the surface of the silver nanoshells, and an exponential decay of the intensity are clearly observed. Turning to the case of structure # 6 which has silver nanobars connected between two identical silver nanoshells as shown in Fig. 7, the field not only is well confined effectively along the surface of the silver nanoshells, but also in the region of silver nanobars. For the chain waveguides with 20 silver nanoshells, the near field intensities measured along the central line of chain axis is shown in Fig. 8 for different DHs, with (see (1) and (2) in Fig. 8) and without (see (3) and (4) in Fig. 8) the silver nanobars. It can also be expected from the near field intensities shown in Fig. 8 that the resonances of structures with the silver nanobars dramatically change with the increase of the DHs, e.g., compared to the resonances of nanoshells N = 20 without connecting the silver nanobars.
Fig. 6 (a): Chain waveguides with numbers N = 5 of silver nanoshells without a silver nanobar. (b)-(e): Field intensities corresponding to their peak wavelengths with different DHs.
Fig. 7 (a): Chain waveguides with numbers N = 5 of silver nanoshells with a w = 4 silver nanobar. (b)-(e): Field intensities corresponding to their peak wavelengths with different DHs.
Fig. 8 Near field intensity of chain waveguides with numbers N = 20 of silver nanoshells for different DHs, with (see (1) and (2)) and without (see (3) and (4)) the silver nanobars.
4. Conclusion and Applications
In conclusion, we have shown that a silvershell nanocylinder pair connected by a silver nanobar exhibits tunable plasmon resonances and enhanced SCSs in the near field zone that are not observed for a solid silver nanocylinder or a silvershell nanocylinder pair without a silver nanobar of similar size. The volume confined by the silvershell nanocylinder pair with a silver nanobar is filled with refractive medium in DHs and is, therefore, accessible to various sensing and spectroscopy applications at the nanometer scale. As observed from numerical simulations, the main features can be qualitatively understood from a simple silvershell nanocylindrical pair model. Silvershell nanocylinder pairs could serve as resonant nanocavities to hold and probe smaller nanostructures, such as biomolecules or quantum dots. The predictive character of these calculations allows one to tailor the shape of the nanoparticle to achieve excitation spectra on demand with a controlled field enhancement.
Acknowledgements
The authors are thankful for the financial support from National Science Council, Taiwan, ROC, under Grant number NSC 96-2112-M-231-001-MY3 and NSC-98-2120-M-002-004-.
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