1. Introduction

The interaction between the incident light and metallic materials gives rise to collective and coherent oscillations of conduction electrons. The collective oscillation leads to either propagating surface plasmons (SPs) across the metal-dielectric interface or localized surface plasmon resonances (LSPRs) in nanostructures [1–3]. These features localize and amplify light emission in the vicinity of metallic nanostructures with sub-wavelength scale [1,2]. The localized field enhancement further increases to several orders of magnitude larger than the incident light, at the nanogap between plasmonic nanoparticles (NPs). Particularly, the strong radiative coupling between NPs [4,5] generates strong field emission known as hotspots [5,6]. Understanding of hotspot systems contributes to the development of several possible practical applications such as surface enhanced Raman scattering (SERS) [7], nanoantenna [8,9], molecule detection [10,11], biological sensing [12], solar-cell design [13] and nanofocusing devices [14].

Recently, homogenous metallic NP dimers are intensively investigated for their near- and far-field plasmonic properties and SERS performances. In homogenous dimers, the near- and far-field resonances are dominated by bonding dipole plasmon (BDP) modes [15]. In addition to the BDP modes, weak coupled higher-order bonding resonances appear at shorter wavelengths [16]. The peak position of BDP modes shows an exponential relationship with the interparticle separation between dimers [17,18]. The localized near-field Raman radiation is proportional to the fourth power of the electromagnetic field enhancement [11,19]. The peak resonance and field-enhancement efficiency of such dimers are also shown to be effectively modulated by various mechanisms, such as the shape, size, and surrounding dielectric properties [18,20–23]. Furthermore, NP homodimers suffer from a high ohmic loss [24,25], which is damping the surface Raman enhancement factor from electron-surface scattering in small NPs and generates the radiation damping in large NPs [26,27].

In this aspect, dramatical modification of the near- and far-field plasmonic properties is further demonstrated through manipulating surface morphologies [28], adding conductively bridged structures [29], using dielectric or metallic mirror surfaces [30,31], and applying asymmetric homogenous dimers [24,26,27]. In addition, a simplified system containing NP heterodimers (bimetallic), such as Al-Au and Al-Ag heterodimers, has been demonstrated with possibilities of generating nonvanishing dipole moment and multiple Fano resonances in a wider wavelength range [32–34]. However, most works are dedicated to characterizing the hotspot effect and spectra tunability based on far-field scattering and extinction spectra [35–39].

In this contribution, we propose alternative simple heterodimers composed of metal nanosphere hotspot systems in order to optimize the near- and far-field resonances in a broad energy range. Numerically, the finite element method (FEM) is implemented to simulate LSPRs for three heterodimer pairs consisting of Au, Ag, or Al nanospheres. We present the dependence of the near- and far-field resonance spectra on the particle size and gap width. The gap dependent Raman model and plasmon ruler equation are applied to analyze the resonance peak wavelengths and intensities. Three-dimensional (3D) distributions of local electric fields and surface charges indicate that three heterodimers have distinct bonding conditions of dominant resonance modes. The investigation of the heterodimers can provide a general picture of the plasmonic hybridizations and the related SERS effects of metallic nanostructures.

2. Computational methods

Numerical electrodynamic simulations for heterodimers were conducted by frequency-domain finite element method (FEM) using a commercial COMSOL Multiphysics software. Al, Au and Ag were chosen as prototypical materials due to their ability to support plasmonic resonances and interband transitions at distinct energies [40]. Interpolated data obtained for Al, Au and Ag was used in the simulation [41,42]. Identical radius pairs of Au-Ag, Al-Au, and Al-Ag surrounded by air were designed in the simulation at which incident light propagates from the side of the particle and polarizes along the dimer axis. A perfectly matched layer (PML) enclosing the surrounding scattering medium was applied to avoid the back-scattering effect. For better accuracy of the simulation, triangular meshing elements of minimum meshing size 0.1 nm were used to discretize the NP surface. The far-field extinction cross-section, ${\sigma _0}$, was obtained by integrating the time-averaged extinction Poynting vector, ${S_{ext}}$, over a surface enclosing the heterodimer [2]: ${\sigma _{ext}} = {\; } - {\int\!\!\!\int }{S_{ext}}dS/{I_0}{\; }$. Here the expression ${I_0} = \left|{\frac{1}{2}c{\varepsilon_0}E_0^2} \right|$ represents the power flow per unit area of the incident light, where ${E_0} = 1{\; }V/m$ is the amplitude of the incident electric field, ${\varepsilon _0}$ is the permittivity of vacuum, and c is the velocity of light. The average near-field Raman enhancement, $\overline {EF} ,$ was calculated by averaging the volume integral of${\; }{|E |^4}/{|{{E_0}} |^4}$ [43,44], where $\overline {EF} = \; \frac{{\int\int\int {{|E |}^4}/{{|{{E_0}} |}^4}dV}}{V}$ and V is the volume within 2 nm above the surface of NPs [44]. In this model, the minimum sub-NP separation was set at 2 nm to avoid the quantum tunneling and non-local effects [45,46].

3. Results and discussion

To fully investigate the plasmonic response of each heterodimer, the calculated average near-field Raman enhancement and far-field extinction spectra of Au-Ag, Al-Au, and Al-Ag heterodimers are indicated in Fig. 1. We fix R at 60 nm with a gradually increasing gap, g = 2 to 20 nm, and their homodimers have been studied in [44,47]. According to Fig. 1(a, c, and e), the calculated extinction spectra with several resonant modes are investigated for different gap sizes. The source of these resonance modes is more demonstrated by the surface charge density and electric field (EF) enhancement in Fig. 2. For all heterodimers, the dominant peak (${\lambda _{1E}}$) is identified as the BDP mode, which appears at the longer wavelength (visible regime) and has a strong blueshift with the increasing separation gap. For the same gaps, in the extinction cross-section of the Al-Ag, the BDP mode shows a broader linewidth than the Al-Au and Au-Ag. Higher-order modes (broad and nearly overlapping) are known as bonding quadrupole plasmon (BQP) modes, which are formed by the antisymmetric field coupling. They are marked as ${\lambda _{2E}}$ and appear in the lower wavelength. The BQP peak, ${\lambda _{2E}}$, also shows a shift but with a weak dependence on the gap size. In contrast, the Au-Ag heterodimer adopts an additional static multipolar mode, ${\lambda _{3E}}$, which is identified in the lower wavelength. The differences in relative positions of dipole-mode spectral peak and the number of peaks in the heterodimer are related to the optical properties and gap between the dimers.

 

Fig. 1. A dimer structure composed of heterodimer nanospheres with the radius of R = 60 nm, and the inter-particle separation of g = 2-20 nm. Calculated extinction cross-section, ${\sigma _{ext}}$, for heterodimers made of (a) Au-Ag, (c) Au-Al, and (e) Al-Ag, respectively. And the corresponding average near-field Raman enhancement spectra, $\overline {EF} $, for (b) Au-Ag, (d) Au-Al, and (f) Al-Ag heterodimers.

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Fig. 2. Local electric field distributions at resonance wavelengths in the form of logarithmic ${|E |^4}/{|{{E_0}} |^4}$ for the Au-Ag heterodimer with R = 60 nm and g = 2 nm. From left to right, (a) λ = 660 nm, (b) λ = 470 nm and (c) λ = 370 nm. k is the wave vector, and ${E_0}$ is the incident polarization. (d)–(f) Corresponding 3D surface charge distributions. The red color represents the positive charge while the blue relates to the negative charge.

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Figure 1(b, d, and f) shows the calculated near-field $\overline {EF} $ spectra. Similar to the far-field resonance, several peaks are observed depending on the gap and dimer combination. As expected, the $\overline {EF} $ intensity obtains a high sensitivity as the gap size increases, leading to a degeneration of the hotspot [47]. Looking at the Au-Al heterodimer, the $\overline {EF} $ intensity shows two distinct enhanced and extended wavelength range in Fig. 1(c and d). The first region ranges from λ = 370 nm to λ = 510 nm and the second is above λ = 520 nm. The former is from higher-order modes (leading to an upper limit of 5 × 10³) and the latter is the BDP effect (leading to an upper limit of 10⁶) for small gaps. Within the similar wavelength scale, the BDP mode is weaker for heterodimers consisting of Al, which have a maximum electric field intensity about ∼ 10⁵. Most notably, the Au-Ag BDP mode emission has narrower full width at half maxima (FWHM) than the Al-Au and Al-Ag BDP modes since the plasmonic figure of merits (FOMs) of Au and Ag show better plasmonic properties than Al after λ = 650 nm [48]. Because of the FOMs of Ag and Al are higher than Au at λ = 350-550 nm [48], the strongest emission of the BQP mode is achieved in the Al-Ag heterodimer system. As a result, the higher-order modes in Al-Ag demonstrate the enhancement that extends into the visible region matching with the contribution of electric dipole coupling. Below λ = 380 nm, the average near-field surface Raman enhancement with a scale larger than 10² is observed in the Al-Au, while the Au-Ag and Ag-Al record 10. In other words, the near-field enhancement of heterodimers has more possibilities to be tuned by selecting the composition of metallic pairs than conventional homodimers.

To address this influence of plasmonic coupling, Fig. 2 and Fig. 3 present typical local electric fields and 3D surface charge distributions at near-field resonance peaks obtained in Fig. 1. The EF mapping and surface charges of the heterodimers, R = 60 nm and g = 2 nm, are in the form of logarithmic ${\; }{|E |^4}/{|{{E_0}} |^4}$, under the y-polarization field (${E_0}$) and the z-polarization propagation ($k$). The red and blue colors represent the strength of charge distributions for positive and negative charges, respectively. The electric field and surface charge distributions of the Au–Ag heterodimer at λ = 660 nm are shown in Fig. 2(a and d). The field distribution is symmetric around both NPs accompanied by a strong hotspot at the gap region. The surface charge density indicates a similar distribution at this peak related to the BDP mode. For the other two lower resonances, λ = 470 nm and λ = 370 nm, the EF intensities cover small volumes of the gap and sharply decrease around Au and Ag, as indicated in Fig. 2(b and c). The intensity due to higher-order modes gradually localizes in a small volume, unlike the BDP mode, as both the positive and negative surface charges tend to accumulate within the dimer gap region. Relatively, the field around the Ag NP is stronger than the Au NP and that breaks the spatial symmetry of the localized electric field. This is due to the accumulation of more surface charges on the Ag NP than on the Au NP. Hence, the near-field Raman enhancement suddenly decreases due to the weak coupling of the higher modes below λ = 520 nm.

 

Fig. 3. Local electric field distributions and surface charge distributions of Al-Au and Al-Ag heterodimers with R = 60 nm, and g = 2 nm. Calculated results for the Au-Al heterodimer at λ = 632 nm (a and e), and the Al-Ag heterodimer at λ = 622 nm (c and g); their corresponding near-field enhancements at λ = 460 nm (b and d) and λ = 410 nm (f and h) are dominated by BQD modes.

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Similarly, Fig. 3(a-h) presents electric field mappings and surface charge distributions of the Au-Al and Ag-Al heterodimers at the resonance peaks in Fig. 1. The Au-Al heterodimer at λ = 632 nm in Fig. 3(a and e) and the Al-Ag heterodimer at λ = 622 nm in Fig. 3(c and g) show a strong near-field enhancement due to the BDP modes. Both the electric field and surface charge form a spatial symmetry around the dimer. On the other hand, the Au-Al heterodimer at a lower wavelength λ = 460 nm in Fig. 3(b and f) and the Al-Ag heterodimer at λ = 410 nm in Fig. 3(d and h) correspond to the near-field enhancement caused by higher-order BQP modes. This enhancement is the result of the asymmetric surface charge distribution. The electric field intensity of the Al-Au heterodimer is relatively lower than that of the Al-Ag heterodimer. As for the Al-Ag heterodimer, the relative field distribution and surface charge density are stronger at lower wavelengths. This explains the continuous arising of the BDP and BQP modes with comparable electric field intensities. Given these results, using the Al-Ag heterodimer structure should be considered as a priority for the broadband near- and far-field SERS enhancements.

In addition, we study the gap effect on both near- and far-field BDP resonance peak wavelengths, ${\lambda _{1E(R )}}$. Figure 4 illustrates the relationship between the gap space and the resonant wavelengths, which is fitted to a single exponential decay (plasmon ruler) equation. To be specific, the model is calculated as ${\lambda _1}(\textrm{g} )= a \cdot {e^{ - ({g/D} )l}} + {\lambda _0}$, where a, l, and ${\lambda _0}$ are fitting parameters [11,31,32,47] with the diameter D = 2R. Due to the gap dependence, we obtain the near-field BDP mode shift in the wavelength, which is weak for Al-Au, λ ∼ 40 nm and strong for Au-Ag, λ ∼ 60 nm. The shifted BDP-mode curves are quite similar to the near-field enhancements. On the other hand, the near- and far-field BDP resonance peaks for Au-Ag at g = 2 nm almost remains at the same wavelength. In the case of Al-Au and Al-Ag heterodimers, the fitting results deviate by λ = 15 nm and λ = 30 nm, respectively. When g increases to 20 nm, distinctive deviations between the near- and far-field resonances can be identified: the largest value λ ∼ 100 nm for Al-Ag, λ ∼ 45 nm for Al-Au and λ ∼ 40 nm for Au-Ag heterodimers. From the fitting, an exponential relation is shown in each heterodimer, while the decay length l is within the range of 10 nanometers previously but higher than Au-Au and Ag-Ag [32,47]. The lowest decay length l is observed for Al-Au and the highest for Al-Ag heterodimers. In addition to similar behavior trend with their corresponding plasmonic properties of homodimers [28,47], this power-law relationship gives valid results which are largely affected by heterodimer-materials combination.

 

Fig. 4. Dependence of the BDP mode resonant wavelength on the gap for heterodimers with R = 60 nm. The peak wavelengths were taken from the near-field (red dotes) and the far-field (blue dotes) of the individual heterodimers. The figure shows a strong red-shift for decreasing gaps. The solid lines (red and blue) indicate an exponential fit using the universal scaling law. The fitting values (near-field, far-field) are a_Au-Ag = (87.7, 129.8), a_Au-Al = (57.8, 88.8), a_Al-Ag = (71.9, 142.2); l_Au-Ag = (6.5, 6.7) nm, l_Au-Al = (7.4, 6.9), l_Al-Ag = (10.7, 9.8) nm; and λ_{0, Au-Ag} = (594, 562) nm, λ_{0, Au-Al} = (586, 552) nm, λ_{0, Al-Ag} = (560, 483) nm.

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Suggested by previous reports about hotspot systems, the relationship between the maximum average near-field intensity (${\overline {EF} _{max}}$) and the gap size (g) is generalized as ${\overline {EF} _{max}} = \; {10^A} \cdot {g^n}$. Parameters A and n relate to the bonding ability of BDP modes [47]. Accordingly, ${\overline {EF} _{max}}$ of the BDP mode as a function of g for three heterodimers decreases linearly in the logarithm scale, as shown in Fig. 5. The fitting factors are found to be A_Au-Ag = 6.8, A_Al-Au = 6.3, and A_Al-Ag = 6.03, while n_Au-Ag = -2.82, n_Al-Au = -2.71, n_Al-Ag = -2.59. Owing to the rise of the BQD mode in the violet and blue range, the BDP mode is weakened most for Al-Ag heterodimers. For example, ${\overline {EF} _{max}}$ of the Ag-Au heterodimer at g = 20 nm is ∼ 2.3 times larger than Au-Al and Ag-Al heterodimers for the spacing consideration. The scales of the fitting factor n are generally within the same scale of previous results obtained from homogeneous dimers [11]. Despite these ${\overline {EF} _{max}}$ values are, in fact, smaller than the corresponding homodimers [47,49], these results provide an insight of optimized modeling of heterodimers for near-field SERS in practical applications.

 

Fig. 5. Log-log plots of the ${\overline {EF} _{max}}$ enhancement of BDP modes as a function of the gap size for R = 60 nm. Blue for Au-Ag, red for Au-Al and pink for Al-Ag heterodimers. The solid lines indicate fitted model.

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In order to further investigate the dependence of the particle size, we evaluate the near- and far-field resonance with a fixed gap, g = 2 nm, and gradually increasing the radius of nanospheres from R = 10 to 80 nm [21,50]. As shown in Fig. 6, the calculated $\overline {EF} $ and ${\sigma _{ext}}$ spectra of the three heterodimers are indicated from the near-ultraviolet (UV) through the visible wavelength range. Particularly, the Au-Ag heterodimer shows two pronounced resonant peaks for the radius below 40 nm and three resonance peaks above, as indicated in Fig. 6(a). In Al-Ag and Au-Al dimers, however, two strong and two weak peaks are observed above R = 30 nm, as presented in Fig. 6(c and e). As R increases progressively, the light extinction coupling effect of the NP heterodimer increases, leading to the formation of enhanced local fields. The far-field BDP modes of Au-Ag and Au-Al heterodimers exist above λ = 500 nm. This effect is quite similar to those obtained from Au-Au and Ag-Ag homodimers [47]. On the other hand, in Ag-Al heterodimers, the BDP mode occurs at shorter wavelengths at the beginning and increases rapidly towards higher wavelengths. Moreover, it can be observed in Fig. 6(c) that the Ag-Al heterodimers obtain the largest blueshift of the BDP mode. The evolution of hybridized plasmon modes, mainly merging into a single peak in small dimer structures, is attributed to the continuous suppression of the BDP mode together with additional higher-order resonances [28]. When it comes to the near-field $\overline {EF} $ spectra, the highest intensity can be dominated by the BDP resonance or higher-order modes, depending on the heterodimer radius. Particularly, for Au-Ag heterodimers of R = 10 and 20 nm, the BQP modes contribute the most of $\overline {EF} $ with the order of 10³ and 10⁴, as shown in Fig. 6(b). The BDP modes begin to completely dominate the $\overline {EF} $ as R reaches 30 nm. The maximum $\overline {EF} $ intensity is achieved in the order of 10⁶ for R = 50 nm. For Al-Ag and Al-Au heterodimers, the BDP modes dominate the $\overline {EF} $ spectra. The maximum $\overline {EF} $ intensities are found at R = 30 nm for Au-Al heterodimers and at R = 50 nm for Au-Ag heterodimers. For the Al-Ag heterodimer at R = 50 nm and above, the $\overline {EF} $ trend becomes almost flat in the tail, leading to the broadband near-field enhancement from λ = 380 to 900 nm. It is also important to notice the similarity between the Au-Al and Ag-Al heterodimers up to R = 30 nm that only dipolar peaks exist in the $\overline {EF} $ and far-field spectrum as the BQP resonance is completely suppressed. This weak higher-order $\overline {EF} $ contribution as R increases is due to the decomposing of multipole moments. Though the Al-Au and Al-Ag heterodimers exhibit better performances from the visible to near-infrared spectrum range, below λ = 340 nm the $\overline {EF} $ intensity undergoes a drastic reduction while the Al-Ag heterodimer $\overline {EF} $ scales 10 to 10³ as the radius changes. These values indicate that heterodimer nanospheres are possible to provide a choice for hotspot systems to enhance both the near-field and far-field plasmonic resonance from the near UV through the visible range.

 

Fig. 6. Calculated size-dependent results of the far-field extinction spectra and average near-field Raman enhancement as a function of the wavelength for heterodimers made of Au-Ag (a and b), Au-Al (c and d), and Al-Ag (e and f) with g = 2 nm and R = 10–80 nm.

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4. Conclusions

We introduced a heterodimer system made of metallic nanospheres. Taking advantage of the differences of optical properties, gap size, and nanosphere radius, both the resonance and local electric field enhancement exhibited significant changes. Considering the Au-Ag heterodimer, more than two resonant peaks were observed compared to the Al-Au and Al-Ag counterparts in the UV-visible optical range. Strong electric field enhancement was observed in the Au-Ag heterodimer for the higher-wavelength range (λ > 510 nm). The best Raman enhancement between 370 nm and 510 nm was revealed in the Al-Ag heterodimer. While the combination of Au-Al heterodimers displayed a weak light coupling effect with a significant decrease of electric field enhancement above λ = 370 nm. The physical origin of the hybridized plasmonic modes was dominated by BDP modes in all heterodimers based on the distributions of surface charges and local electric fields while Al-Ag supported higher-order modes. The maximum intensity of BDP modes of the near-field and far-field resonance decayed exponentially as the gap size increased in the heterodimers. In addition, the maximum intensities of the near- and far-field resonances exhibited a weak power-law dependence with the gap size. This work offers a possible direction to facilitate the hotspot system made of selected heterodimers, which should be considered for tuning the spectral resonance and the surface Raman enhancement in a broad optical range, from the near UV through the visible range.