1. Introduction

Focusing and confining electromagnetic waves into nanoscale dimensions by employing unique properties of surface plasmon polaritons are of great fundamental and practical interest [1–7]. As a result of nanofocusing/nanoconfinements, local electromagnetic field can be enhanced by as high as 3 orders of magnitude leading to a number of extraordinary effects on the materials and molecules at the focus. For example, nonlinear optical effects [3, 8, 9], fluorescence emission [10, 11], Raman signals [12, 13], and optical transmission through subwavelength holes can be enhanced significantly. Especially, recent surface enhanced Raman scattering (SERS) experiments show that the enhancements can reach up to 14 orders of magnitude, that not only makes SERS an attractive single molecule technique with promising applications but also posts fundamental questions regarding its physical mechanisms[5, 14–16].

Early experimental systems where single molecule SERS was observed were based on metallic nanoparticle aggregates [5, 14, 15]. In such colloidal systems, hotspots were normally found by chance from a large ensemble of samples which make the experiments not controllable. The theoretical illustration of hotspots with nanoparticle dimmers [4, 17] has stimulated new nanoantenna designs [9, 18–21], and the state-of-the-art techniques have been applied to fabricate these dimmers with nanogaps [22–24]. However, it is still challenging to build these nanoantenna dimmers with gaps below 5nm in a reproducible fashion and in large scale. In order to decipher the physical mechanisms of single molecule SERS and for practical industrial applications, a plasmonic system allowing for reproducible fabrication and well controlled EM enhancements is highly desired. Detailed characterization of the EM enhancements in such a model system will facilitate detailed comparison and validation of various theories. Recently, metal nanodisks separated with dielectric spacers have been proposed and demonstrated as tunable plasmonic resonators for SERS applications [25–27]. One great advantage of these metal-dielectric-metal nanodisks is that the dielectric gap may be controlled with sub-nm accuracy by using advanced deposition techniques. One issue of such plasmonic nanodisks is that Raman molecules can only be attached to the metal disk circumferences where the electrical field enhancement is still low in comparison to the field enhancement within the gap of nanoparticle dimmers.

In this paper, we propose a new design of optical nanoantennas and study their optical properties by numerical simulations. The nanoantennas are composed of two metal cylinders stacked vertically with a circular dielectric disk spacer. The diameter of the dielectric disk is designed smaller than that of the metal cylinder to accommodate SERS molecules. We show that when the dielectric disk is thinner than 5nm, such nanoantennas exhibit two types of resonances: one corresponding to antenna resonance, the other corresponding to cavity resonances. The antenna resonance generates a peak in scattering spectra, while the cavity resonances lead to multiple dips in the scattering spectra. The cavity resonances cause giant electrical field enhancements inside the nanocavity. The enhancement of electrical field inside the nanocavity is maximized when the diameter of the dielectric disk is roughly half that of the rod and when the cavity and antenna resonant frequencies coincide with each other. An advantage of such nanoantenna design is that the gap size can be accurately controlled below 1nm by using atomic layer deposition, making it a good candidate for single molecule SERS applications.

2. Nanoantenna design and simulation methods

A schematic of the nanoantenna is shown in Fig. 1. The diameter of the dielectric disk is designed to be smaller than that of the metal cylinders to accommodate SERS molecules. Ag is chosen as an exemplary material for the metal cylinders to achieve wide tuning range of the antenna resonant frequency. SiO₂ is chosen for the dielectric disk layer because atomic layer deposition process for SiO₂ is readily accessible with sub-1nm accuracy in film thickness control [28, 29]. The optical properties of this new type of nanoantennas are studied by employing the commercial software package CST Microwave Studio (MWS). The software is based on finite-difference time domain (FDTD) method, and it has been shown that the simulation results of surface plasmon waves with this software agree well with analytical solutions [30].

 

Fig. 1. Schematic side view (left) and top view (right) of the metal-dielectric-metal nanoantenna. The wave vector k of the incident light is indicated by the arrow; the polarization is along the nanoparticle axis. The blue spot indicates the probe position where the local field spectra are calculated.

Download Full Size | PDF

The Drude model was utilized to describe the frequency dependence of silver permittivity: ε_Ag(ω)=ε_∞-ω² _p/ω(ω-iδ), where these parameters used for simulations, ε_∞=3.57, ω_p=1.388×10¹⁶ rad/s, and δ=1.064×10¹⁴ Hz, are obtained by fitting the Drude model to the measured permittivity of bulk Ag material [31]. Bulk dielectric constant for SiO₂, i.e. ε=2.13 is used for the SiO₂ disks for all frequencies of simulations.

The nanoantenna is excited by a plane light wave with the electrical field along the nanoantenna axis and the wave vector perpendicular to the nanoantenna axis (Fig. 1). Four geometric parameters are needed for characterizing the nanoantenna: diameter and length of the metal cylinders, diameter and thickness of the dielectric disk. In our simulations, the diameter and length of the metal cylinders are fixed at 60nm and 80nm respectively. All local field spectra shown in this paper are for the electrical field at the probe position indicated in Fig. 1, and have been normalized by the strength of incident field to represent the local field enhancement. The far-field intensities are calculated for a position about 100 wavelengths away from the nanoparticle, and are rescaled to have unity maximum intensity for the sake of better comparison.

3. Cavity and Antenna Resonances

Local field and far field spectra for different SiO₂ disk diameters and thicknesses are shown in Fig. 2; interesting resonance features can be observed. In the local field spectra, the resonant wavelength and the peak height vary considerably with the SiO₂ disk sizes. When the SiO₂ diameter is equal to that of the metal cylinder, no major resonant peaks exist with field enhancements larger than 50 even for 1nm thickness of SiO₂ [Fig. 2(d)]. Reduction of the SiO₂ disk diameter leads to dramatic changes in the local field spectra. When the SiO₂ disk thickness is less than 5nm, the dominant resonance appears as a narrow peak. The bandwidth at the half field strength of this resonance is between 40nm to 100nm and increases with the SiO₂ disk diameter [see Figs. 2(a)–2(c)]. When the gap is larger than 5nm the narrow peak shifts to a higher frequency. Meanwhile, the broad peak, which is less prominent at small gaps, becomes dominant.

 

Fig. 2. Local electrical field spectra (a–d) and far field scattering spectra (e–h) for the SiO₂ disk diameters at 20nm (a, e), 30nm (b, f), 40nm (c, g) and 60nm (d, h).

Download Full Size | PDF

 

Fig. 3. (a). Simulated (symbols) and calculated (lines) cavity resonant wavelength as a function of the SiO₂ disk thickness for different SiO₂ disk diameters. (b) The electrical enhancements at the probe position as a function of the SiO₂ disk thickness at different SiO₂ disk diameters. (c) The local E-field enhancements as a function of SiO₂ disk diameters for 1nm disk thickness. The solid lines are the guidance to the eye.

Download Full Size | PDF

The broad peaks observed at large gaps are fundamentally different from the narrow resonant peaks at small gaps. Close to the wavelengths of these broad peaks, far-field scattering spectra show resonant peaks as well. In contrast, at the wavelengths of these narrow peaks, the far-field spectra show dips. This indicates that the narrow peaks in local field spectra are resulted from cavity resonances, while the broad peaks are resulted from the plasmon resonances of the whole antenna.

Regardless these dips caused by the cavity resonances, the far-field scattering spectra consist of only single peaks [Figs. 2(e)–2(h)]. These far field resonances, denoted as antenna resonances, are similar to those observed previously in various nanoantenna dimmers [19].

It is important to note that the cavity and nanoantenna resonance frequencies are not always coincident with each other and can be widely tuned by varying the SiO₂ disk sizes. Both the cavity and antenna resonant wavelengths decrease appreciably with the increase of the SiO₂ disk thickness, while increase only slightly with the increase of SiO₂ disk diameter [Fig. 3(a)]. This indicates that the accurate control of the nanogaps between two nanorods is critical to controlling cavity resonances and the local field enhancements, while the control of the SiO₂ disk diameter is insignificant.

The local field enhancement at the dominant cavity mode frequencies increases when the SiO₂ disk thickness decreases. Especially a sharp rise can be seen when the thickness is less than 5nm [Fig. 3(b)]. Meanwhile, the local field enhancements are affected by the diameter of the SiO₂ disk. With the increase of SiO₂ disk diameter, the local field enhancement goes up first and then declines. It can be found that the local field enhancement reaches its maximum when the SiO₂ disk diameter is approximately half the diameter of the metal cylinder [Fig. 3(c)].

For these cavity resonance modes, we calculated the electrical field distributions at the plane through the middle of the SiO₂ disk and the results for the SiO₂ disk diameter at 30nm are presented in Fig. 4. It can be seen that when the SiO₂ thickness is less than 5nm, the field distribution pattern for the major cavity mode does not vary with SiO₂ thickness, while the field distribution patterns of other modes show significant change when the SiO₂ thickness is changed. In addition, the electrical field distribution for the broad peak is very different from that of the major cavity mode at small gap, which can also be seen from the snapshots of local electrical field vectors. For the narrow band cavity mode, the electrical field direction inside the SiO₂ disk is opposite to the field direction outside the disk (Fig. 4). While for the broadband mode, the electrical fields inside the gap between two Ag cylinders are parallel to the direction of incident field [Fig. 4(c)].

 

Fig. 4. (a). Local electrical field amplitude distributions in the plane through the middle of the SiO₂ disk at three primary cavity resonant modes for SiO₂ disk thickness at 1nm (first row), 2nm (second row), 5nm (third row) and 10nm (fourth row). The wavelengths indicated in each picture are the wavelengths of excitation. (b) and (c) Snapshots of local electrical field vectors for the major cavity mode for the SiO₂ thickness at 2nm (b) and 5nm (c) respectively. For the sake of clarity, logarithmic color scales are used. Red color represents the maximal field strength of individual pictures, and thus does not indicate the same field strength for pictures of different modes.

Download Full Size | PDF

In order to understand these cavity resonances, it is necessary to recall the surface plasmon modes present in the dielectric gap of two metal films. For an interface between semi-infinite dielectric and metal materials, there exists only one surface plasmon mode whose wave vector is given by k_sp=(ω/c)[ε_mε_d/(ε_m+ε_d)]^1/2 Here ε_m and ε_d are the permittivity of the metallic and dielectric materials, ω and c are the radial frequency and speed of light. When two metal-dielectric interfaces are brought to close proximity, the surface plasmons at two interfaces couple with each other, which results in separation of the degenerate mode into a symmetric mode and an anti-symmetric mode. The anti-symmetric mode only exists at high frequency ranges, while the symmetric mode can sustain at all frequencies for very small gaps. For the small dielectric thicknesses considered here, the anti-symmetric mode is out of range, and will not be considered. The dispersion relationship for the symmetric mode can be expressed as [32, 33]:

where t is dielectric gap thickness, k_m=(k ² _sp-ε_m ω ²/c ²)^1/2, k_d=k ² _sp-ε_dω ²/c ²)^1/2, and k_sp is the wave vector for the gap surface plasmon. In addition, the permittivity of the dielectric disk affects the local field distribution. The electrical field decays with the distance from the metal-dielectric interface. For the field inside the dielectric gap, the decay length, also called the skin depth [32], can be given n as ${\delta }_d=\frac{1}{\mathrm{Re}\left[{\left(\frac{k_{\mathrm{sp}}^2-{\epsilon }_d{\omega }^2}{c^2}\right)}^{\frac{1}{2}}\right]}$ . Given that an increase of ε_d leads to an increase of k_sp, the skin depth δ_d decreases with the increase of the permittivity of the dielectric layer. When δ_d is larger than the dielectric thickness, the electrical field inside the gap will be homogeneous in the direction perpendicular to the gap plane.

In fact, the major cavity resonance observed above corresponds to formation of a standing wave of surface plasmons, as the symmetric distribution of electrical field shown in Fig. 4(b) is in agreement with the symmetric surface plasmon wave. Considering that the electrical field is maximal at the edge of the gap, the resonant condition can be written as:

where D and d are the diameter of the metal cylinder and dielectric disk respectively, k′_sp and k″_sp are the surface plasmon wave vectors for SiO₂ and air portions of the cavity respectively. Simple numerical calculations yield that the cavity resonant wavelengths obtained from this condition agree well with the simulation results [Fig. 3(a)]. Some discrepancies at 10nm thickness of SiO₂ are due to the fact that the simulated resonant wavelengths are assigned to these broad peaks. At this large gap size (10nm), the wave vectors for the symmetrical plasmon wave is decreased, thus Eq. (2) leads to the cavity resonant wavelength being much smaller than that of antenna resonances [small peaks for 10nm gap size in Figs. 2(a)–2(d)]. Moreover, at the antenna resonant frequency, k′_spd+k″_sp (D-d)≪π, so the electrical field inside the gap for the broad resonant peak [Fig. 4(c)] is always in the same direction.

Overall, the thickness and diameter of the dielectric disk determine the resonant wavelengths of the major cavity mode, providing a way of tuning the cavity resonance wavelength. There exist other cavity modes [Fig. 4(a)], which correspond to more complicated patterns of surface plasmon standing waves. From simulations, these high order cavity modes do not affect far field scattering much, as no dips can be seen in the far field spectra [Figs. 2(e)–2(h)].

These seemingly complicated cavity and antenna resonance behaviors can be qualitatively understood in the concept of optical circuit elements [34, 35]. It should be noted that the optical circuit element concept has been developed for very small plasmonic particles, extending this concept to large particles like ours implies that phase retardation has been omitted. Following the same analogy as the optical circuit elements, the metal cylinders can be considered as inductors (L₁), while the nanocavity can be considered as two capacitors (C₁ and C₂) in parallel connections representing the cavity regions of SiO₂ and air respectively. C_fringe represents the capacitance between the metal cylinders outside the cavity region. It is also important to note that the outside capacitor ring C₂ and the inside capacitor C₁ of SiO₂ disk are connected through the Ag cylinder. Therefore, to take this into account, two inductors, L₂ and L₃, need to be added (Fig. 5). For simplicity, losses are ignored here. Such a circuit network contains two LC circuits and thus exhibit two resonant frequencies. L₂, L₃, and C₁, C₂ form a small LC circuit whose resonance corresponds to the cavity resonance. While all elements together form a big LC circuit whose resonance corresponds to the antenna resonance.

 

Fig. 5. Optical circuit element representation of the optical nanoantenna.

Download Full Size | PDF

At the cavity resonance, the current circulating inside the small loop should lead to opposite phases for C₁ and C₂, which explains the field directions for the cavity mode [Fig. 4(b)]. When the SiO₂ thickness is increased or the capacitances of C₁ and C₂ are decreased, the resonant frequency of the small LC circuit increases. Especially, when the frequency of the input signals are close to the resonant frequency of the whole circuit while lower than that of the small LC circuit, the L₂, L₃, C₁, C₂ loop acts as more capacitive, and C₁ and C₂ should have the same phase. This agrees well with the above simulation results.

In addition to the losses of metallic materials, the local field enhancements are affected by two design factors: radiation loss and the cavity/antenna coupling. When the gap size is reduced, it becomes difficult for the electrical fields to leak out of the cavity, or radiation loss is reduced for the cavity modes. When the resonant frequencies of cavity mode and antenna resonance coincide, the whole structure will be optimized to receive energy at the cavity mode frequency, and should leads to optimized local field enhancements. We simulated nanoantennas with increased Ag cylinder lengths. When the Ag cylinder is 120nm in length, the resonant frequency of the major cavity mode coincides well with the antenna resonance frequency at 384THz. The results show that the local field enhancements can increase by ~50% in comparison with these nanoantennas of 80nm length shown above. Therefore, by tuning the antenna resonance frequency and cavity resonance frequency to be coincident, the local field enhancements can be maximized.

It is worth pointing out that the major contributions to the large local field enhancements originate from the cavity resonances. Even when the cavity and antenna resonance frequencies do not match each other, reasonably high local field enhancements can be obtained. This may be relevant to previous experimental observations that the plasmon resonance wavelengths of hotspots do not correlated well with the laser wavelengths [36]. This also indicates that matching cavity resonant wavelengths with the excitation laser wavelength are critical to achieving large SERS signals.

4. Summary

In summary, we have proposed and numerically studied a new design of plasmonic nanoantennas which are composed of two metal cylinders stacked with a dielectric spacer. It is shown that by making the diameter of the dielectric disk smaller than that of the metal cylinder, cavity resonances can be generated inside the gap. This cavity resonance is different from the plasmon resonances of the whole antenna, and generates dips in stead of peaks in far field scattering spectra. Moreover, this cavity resonance is more critical to the local field enhancements within the nanocavity. Experimental testing of SERS using such nanoantennas may be helpful understand the potential roles of cavity resonances in SERS enhancements.