Introduction

Metal nanoantennas support localized surface plasmon resonances (LSPR), which are bound collective oscillations of surface electron density [1]. LSPR can be excited with free space light, and its resonance frequency depends on the geometrical shape/size and chemical composition of a nanoantenna, as well as on the environment [2]. Excitation of LSPR results in electromagnetic confinement and field intensity enhancement of up to 105 times in the vicinity of a nanoantenna [3, 4]. The LSPR field in such a hot spot decays rapidly away from the nanoantenna surface. Thus, LSPR can be used to locally enhance light-matter interaction [5]. LSPR have been demonstrated to have great potential for surface enhanced label-free bio-chemical sensing, in particular, surface-enhanced infrared absorption (SEIRA) [6–8]. When a resonantly excited nanoantenna is brought in contact with an analyte, containing molecules, which are in resonance with LSPR, an enhancement of a molecule specific absorption occurs. It is generally accepted that SEIRA efficiency increases with increasing hot spot field intensity. SEIRA is especially demanded for the important infrared (IR) spectral region, containing fundamental molecular vibration resonances (wavenumbers below 4000 cm−1), and even more so for the fingerprint region (range 500–1500 cm−1) [9, 10]. Since SEIRA action is limited to the hot spot area [5], a meaningful SEIRA sensor design requires an ordered array of nanoantennas with high spatial density of the latter [11, 12]. Consequently, hot spots density is a crucial parameter, which strongly impacts SEIRA sensor efficiency.

Realization of IR LSPR with conventional metals (CM), such as Au and Ag, requires μm-sized nanoantennas [7], due to large fixed free electrons concentration (ne ∼ 5 × 1022 cm−3). Strong interaction of proximal nanoantennas results in a spectral shift of LSPRs frequency and decrease of field intensity enhancement, as has been observed in CM arrays [13–15]. This effect has been suggested to limit sensor performance [12, 16]. Alternatively, degenerately doped wide band gap transparent conducting oxides (TCO), such as Al:ZnO, Ga:ZnO, and In:ZnO, offer IR plasmonic response at ne as low as 1 × 1020 cm−3 [17–21]. Due to the lower ne, the same IR LSPR frequency can be realized with geometrically smaller TCO nanoantennas, as compared to CM, which may result in higher density and less interaction. Other advantages offered by TCO are their plasmonic response is widely adjustable (up to near-IR) by doping, chemical stability and non-cytotoxicity, cost-effectiveness, and compatibility with industrial fabrication standards [8, 22]. On the other hand, lower ne results in weaker field intensity enhancement in hot spots. The two groups of materials, CM and TCO, offer functionality required for SEIRA, each having its own strengths and disadvantages. Up to now, the discussion of the proximity effect in plasmonic arrays on SEIRA performance has been limited to a couple of publications [12, 15, 16]. Even in the relevant works, no account of effect of material of nanoantennas on hot spots density has been provided.

In this paper, we theoretically investigate proximity effects in dense arrays of plasmonic rectangular nanoantennas, corresponding to two material groups: CM and degenerate TCO. Firstly, we model and compare optical response of single nanoantennas of CM and TCO. Then, interactions in the near-field zone of nanoantennas are studied by investigating the dependence of calculated optical extinction and field intensity enhancement of arrays on nanoantenna-to-nanoantenna distance. We report on two regimes of the near-field interaction for both CM and TCO arrays, each of which is important at a specific separation of nanoantennas. We discuss the effect of the interaction on the important parameters for sensing: hot spots density and signal to noise ratio.

Theoretical Method

The optical response of a metal nanoantenna is given by its frequency-dependent polarizability. Exact analytical expression for polarizability is only available for spherical metallic nanoantennas and, to some extent, spheroid metal structures, as elaborated in Mie theory [23]. In this work, we employ full 3D calculations of Maxwell’s equations with the finite element method (FEM). In FEM, the entire simulated domain is divided into small elements (meshes). Local solutions of differential Maxwell’s equations are found for each of the elements and then added to obtain a solution for the whole domain. We use a commercial JCM Wave software FEM solver [24], which has been successfully applied to many relevant problems in the field of nanoplasmonics [25]. Modeling in JCM Wave is done by arranging geometry and mesh size within calculation domain, setting up boundary conditions of the domain, assigning materials, by defining respective permittivity and permeability, and choosing excitation conditions, e.g., wave-type, wavelength, incidence conditions, and intensity.

In this work, we model rectangular metal nanoantennas, characterized by length (L), width (W), and height (H). This nanoantenna shape has been shown to be a particularly useful plasmonic design for sensing applications as well as easy to fabricate in direct [26] and inverse (nanoslit) [27] geometries. High aspect ratio of the nanoantenna type ensures larger local field intensity enhancement, as compared to, e.g., spherical particles [28]. We use transparent and periodic boundary conditions to model single nanoantennas and arrays, respectively. Discretization mesh size was chosen to be substantially smaller than the excitation wavelength (IR range). In particular, mesh side length constraint away from a nanoantenna was set to 200 nm, while in hot spots it was set to 1 nm.

The frequency-dependent permittivity of a nanoantenna is described by the Drude dielectric function:

$$ {\varepsilon}_{metal}\left(\omega \right)={\varepsilon}_{\infty }-\frac{\omega_P^2}{\omega^2+i\times \gamma \times \omega } $$

where ε∞ is the background dielectric constant, ωp is plasma frequency and γ is electron damping. The ωp is given by \( {\omega}_P=\sqrt{q_e{N}_e/{\varepsilon}_0{m}_e^{*}} \), where N e is the free electron density, m * e is the effective electron mass, q e is the electron charge, and ε 0 is the vacuum permittivity. The permittivity of a TCO in IR is well described by the Drude function, as justified in Refs. [29] Drude parameters used to model CM and TCO materials are listed in the Table 1. For CM we take ε∞ = 5.3, which is an in-between value of silver (ε∞ =1) [30] and gold (ε∞ = 9.6) [31] background dielectric constants. Parameters used to describe ε TCO fit well heavily doped ZnO:Ga and In2O3:Sn [8, 29]. Large values of γ were chosen to account for the increased electron damping at the surface [32, 33]. In this work, we model nanoantennas in air (εd = 1), disregarding dielectric substrate effect, which is known to affect LSPR in a general matter: a red shift and damping of the LSPR with increasing magnitude of substrate refractive index [34–36]. A similar “real-life” situation is highly desirable and, in fact, is realized in the case of plasmonic nanostructures on pedestals [36].

Table 1 Drude parameters used to model the permittivity of CM and TCO nanoantennas
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LSPR excitation was simulated by a plane-wave at normal incidence to the W × L plane of a nanoantenna with the electric field always parallel to the L-dimension of a nanoantenna. The electric field amplitude of exciting wave was set to 1.

Results and Discussion

First, we consider single CM and TCO nanoantennas. As expected, nanoantennas, much smaller than the exciting wavelength, exhibit a strong dipolar LSPR resonance, Fig. 1a. A comparison of calculated extinction spectra of single CM and TCO nanoantennas, Fig. 1a, shows that the same spectral position of LSPR peak, corresponding to a CM nanoantenna, is realized with a 4 times shorter TCO nanoantenna. The geometrical aspect ratio of a TCO nanoantenna, L/W = 300/80 = 3.75, is closer, than that of a CM nanoantenna, L/W = 1200/80 = 15, to the experimentally determined optimum aspect ratio of 3, for which the highest value of the quality factor (ratio of the resonance peak energy to the resonance linewidth) for a given peak resonance energy is expected [37]. Increasing the length of the nanoantenna results in a red shift of the resonance peak. In the case of a TCO nanoantenna, LSPR can be tuned by doping over a very wide range. Indeed, IR to NIR LSPRs are realized for ne in the range 1 × 1020 cm−3 to 9 × 1020 cm−3, Fig. 1b. The required ne has been experimentally realized in ZnO:Ga and In2O3:Sn [19].

Fig. 1
figure 1

a Calculated extinction of CM (L = 1200 nm) and TCO (L = 300 nm and 1200 nm) nanoantennas of identical width (W) and height (H) of 80 nm. b Calculated extinction spectra of a single TCO nanoantenna (L = 300 nm) of identical width (W) and height (H) of 80 nm, corresponding to the denoted free electrons concentration. In all cases, nanoantennas are excited with light polarized along the nanoantenna length

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A 2D spatial profile of electric field near the edge of single TCO and CM nanoantennas is shown in Fig. 2a and Fig. 2b, respectively. It can be seen that a CM nanoantenna provides an order of magnitude larger electric field intensity enhancement, due to the larger plasma frequency. In both cases, a noticeable field intensity enhancement occurs only within a small volume adjacent to the nanoantenna, a hot spot. Mapping of the field intensity away from the nanoantenna surface, Fig. 2c, shows an exponential decay of the field, with the decay constant of about l HS   = 10 nm for both TCO and CM nanoantennas. This value gives the length-scale of SEIRA action and is in good agreement with experiment [5]. Therefore, hot spots density is expected to have a dramatic influence on sensor efficiency.

Fig. 2
figure 2

Electric field spatial profile at resonance conditions of TCO (a) and CM (b) nanoantennas. The insets, schematics of the area of the nanoantenna shown in panels (a) and (b) (c) electric field intensity enhancement spatial profile of single TCO (filled squares) and CM (filled triangles) nanoantennas, measured away from a nanoantenna edge along the dashed arrows in panels (a) and (b)

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As discussed above, nanoantennas behave like an electric dipole with far field (radiating) and near-field (evanescent) characteristics. The antenna behavior becomes significant when considering arrays of spatially close nanoantennas [16, 38]. The far field coupling has been shown to play a significant role in 3D (stacked) arrays of nanoantennas [39]. In planar dense arrays of nanoantennas, for which the distance between adjacent antennas is comparable with a half of the LSPR wavelength (λ LSPR /2), the near field interaction is most important [15]. We consider this essential regime further.

Below, we demonstrate that a near field zone, within approximately λ LSPR /2 distance form a nanoantenna, can be additionally separated into two regimes of very short-range (VSR) and quasi long-range (QLR) interactions. We start with VSR interaction. When two nanoantennas are so close spatially that their hot spots overlap in the gap between nanoantennas, the hot spot electric field intensity is substantially enhanced, Fig. 3a, b. The enhancement depends on the gap width and becomes noticeable at a separation below 100 nm, Fig. 3c. It can be seen that an order of magnitude enhancement of the hot spot field intensity can be achieved by reducing gap width to 10 nm, corresponding to a complete overlap of the hot spots. The characteristic length-scale of VSR interaction l VSR   ≈ 100 nm, determined by our modeling, is in good agreement with experimentally reported values [40, 41].

Fig. 3
figure 3

Spatial distribution of electric field intensity enhancement in the vicinity of a single and b two TCO nanoantennas at resonance. Two-sided cyan arrow depicts direction of light polarization. c Electric field intensity enhancement in the gap between nanoantennas vs. distance. Field values are taken along the gray dashed lines in the panels (a) and (b)

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The QLR interaction results in a spectral shift of the LSPR frequency of an array. We analyzed spectral position of LSPR peak with respect to the distance between adjacent nanoantennas in arrays. We considered two orthogonal directions: along the nanoantennas length (Dy) and perpendicular to it (Dx), as schematically illustrated in Fig. 4a. In both cases, TCO and CM arrays Fig. 4b, c, respectively, decreasing Dy results in a red shift of LSPR peak, while a blue shift is apparent for decreasing Dx spacing. In the case of the TCO array, spectral shifts are symmetric with respect to Dx and Dy. However, in the case of the CM array, the blue shift dominates as both Dy and Dx are decreased. CM-nanoantennas have larger induced longitudinal (Dy direction) dipole moment as compared to shorter TCO-nanoantennas. Thus, for the induced longitudinal dipole, the interaction is the strongest between CM-nanoantennas in the transverse (Dx) direction. The latter results in the asymmetric shape resonance shift in the case of a CM-nanoantenna, in agreement with our simulations (Fig. 4c) and experimental data of, for example, Bagheri et al. [12].

Fig. 4
figure 4

a A schematic representation of an array element. b and c Nanoantenna array resonance peak position vs. nanoantenna-to-nanoantenna distance in array in vertical (Dy) and horizontal (Dx) direction for fixed distance along the orthogonal direction for TCO and CM arrays, respectively

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A larger length of CM nanoantennas results in enhanced QLR interaction, as compared to TCO nanoantennas. The onset range of QLR of a CM array is about 2400 nm, while for a TCO array the interaction becomes important below 600 nm for both Dy and Dx. The 4 times reduced onset length is in good agreement with 4 times smaller geometrical cross section of a TCO nanoantenna as compared to a CM nanoantenna. Therefore, a relation between the onset distance of QLR interaction and nanoantenna size can be approximated as l QLR   ≈ 2 × L, where L is the nanoantenna length. A similar relation has been experimentally observed in Refs. [15] for two Au nanoantennas of L1 = 2 × L2. The spectral shift of LSPR frequency due to the QLR interaction is explained well by LSPR hybridization model [42, 43].

A comparison of an average field intensity in a hot spot of a TCO nanoantenna at different Dx and Dy, shown in Fig. 5, visualizes the implications of VSR and QLR regimes on field intensity enhancement. Firstly, that VSR-induced field intensity enhancement scales as ∼1/Dy 3 [16]. The QLR interaction in the Dx direction results in a decrease of the field intensity enhancement with decreasing Dx separation. It can be seen that the decrease of the field intensity enhancement becomes noticeable at Dx ≈ 2 × L, which in agreement with value of l QLR derived from Fig. 4b.

Fig. 5
figure 5

Average field intensity enhancement of a TCO nanoantenna at different distance between nanoantennas in Dx and Dy directions (see Fig. 4a). The solid blue line is the 1/Dy 3 fit to the data. The vertical dashed line marks the shortest calculated separation between nanoantennas. In the inset, the solid shape is a nanoantenna, and patterned area represents the field averaging region

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Let us consider a unit cell of an array of nanoantennas, schematically shown in Fig. 6. For a planar sensor, this work, and because HTCO = HCM, we only consider the area in the L × W plane. We define an area density of nanoantennas as: ρ = 1/S cell , where S cell  = P x  × P y is the area of a unit cell, P x  = W + D x and P y  = L + D y are lattice constants in orthogonal directions. We estimate the maximal ρ in the absence of QLR interaction. In the case of a TCO array, assuming D x  = D y  = 2 × L TCO  = 600 nm and W = 80 nm, we find ρ TCO  = 1.63 × 10− 6 nm − 2. For a CM array, setting D x  = D y  = 2 × L CM  = 2400 nm and W = 80 nm, we obtain ρ CM  = 1.12 × 10− 7 nm − 2. Thus, a TCO array of an optimum geometry (LSPR conditions are unaffected by interactions) supports ρ TCO / ρ CM  = 14.6 times higher nanoantennas density. Since each nanoantenna has 2 hot spots, a TCO array of non-interacting nanoantennas supports about 2 × ρ TCO  / ρ CM  ≈ 30 times higher density of hot spots, as compared to a CM array, tuned to the same LSPR frequency.

Fig. 6
figure 6

A schematic of an array unit cell. Patterned light-pink area represents the background space. Gray area is a nanoantenna. Violet area represents hot spot region

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Due to the fact that the strongest interaction between the LSPR near field and environment occurs only within a hot spot, we suggest that the signal-to-noise ratio (SNR) of a 2D LSPR array sensor can be defined as SNR = 2 × S HS  / S BG , where S HS  = W × l HS is the hot spot area and S BG  = (L + D y ) × (W + D x ) − L × W − 2 × S HS is the background area, where there is no field intensity enhancement. Assuming l HS  = 10 nm, W = 80 nm, and D x  = D y  = 2 × L TCO  = 600  nm, and D x  = D y  = 2 × L CM  = 2400  nm for TCO and CM arrays, respectively, we obtain respective SNR values: SNRTCO ≈ 2.7 × 10−3 and SNRCM ≈ 1.8 × 10−4. Therefore, for the case of non-interacting nanoantennas, a TCO array sensor has SNR TCO  / SNR CM  ≈ 15 times higher SNR, as compared to a CM array sensor. Consequently, a large area sensor based on a dense array of TCO nanoantennas can potentially outperform an array of CM nanoantennas, despite an order of magnitude lower hot spot field intensity enhancement of individual TCO nanoantennas.

In the present paper, we focus on the effect of nanoantennas material and density on the hot spots density. Recently, Huck et al. [44] demonstrated experimentally that plasmonic nanoantennas dimers with dimer separation below a few nanometers provide even larger signal enhancement [44]. In the same paper, the authors proposed that such small gaps can provide additional morphological information. Modeling of the field enhancement within sub-10-nm gaps and the actual magnitude of enhancement of IR-signals may be a logical continuation of the present work.

Conclusions

In conclusion, using electromagnetic FEM modeling, we showed that a single CM nanoantenna provides an order of magnitude higher near-field intensity enhancement as compared to a degenerately doped TCO nanoantenna. On the other hand, the same LSPR frequency is realized with a 4 times shorter TCO nanoantenna. By modeling dense arrays of plasmonic nanoantennas, we have demonstrated that a TCO nanoantennas array provides 30 times larger hot spots density resulting in a 15 times higher signal-to-noise ratio as compared to CM nanoantennas array.