## Abstract

In this work, we describe finite element simulations of the plasmonic resonance (PLR) properties of a self-similar chain of plasmonic nanostructures. Using a broad range of conditions, we find strong numerical evidence that the electric field confinement behaves as ${\left(\Xi /\lambda \right)}_{PLR}\propto EF{E}^{-\gamma}$, where *EFE* is the electric field enhancement, $\Xi $is the linear size of the focusing length, and *λ* is the wavelength of the resonant excitation. We find that the exponent *γ* is close to 1, i.e. significantly lower than the 1.5 found for two-dimensional nanodisks. This scaling law provides support for the hypothesis of a universal regime in which the sub-optical wavelength electric field confinement is controlled by the Euclidean dimensionality and is independent of nanoparticle size, metal nature, or embedding medium permittivity.

© 2012 OSA

Scaling laws are ubiquitous in many natural and engineering systems and have been a rich source of physics insight for over 30 years [1,2]. An especial challenge exists when the equations of the underlying physical system are unknown and one must then rely on numerical simulation or data to make predictions. On the other hand, precise control over the energy confinement has proven pivotal for the study of plasmon resonance (PLR) properties in nanostructures. Recent advances in the development of nanoplasmonics and light harvesting [3] suggest that the spatial confinement of plasmon excitations at metal-dielectric interfaces or in hybrid (metallodielectric) nanostructures can be applied to enable entirely new structures with versatile applications ranging from highly sensitive sensors to photovoltaics. These can typically be shaped on subwavelength scales. Particular interest [4–21] has been focused on linear chains of *N* resonantly coupled plasmonic nanostructures because these systems are very sensitive to an applied electric field, giving rise to extremely high electric fields (hot spots), i.e. nanosphere cascade nanolenses yielding electric field enhancement in the nanogap between two nanoparticles which can exceed the excitation field by a factor of 10^{3} at the smallest particle.

Presently, efforts are underway to understand the relationship between the electric field enhancement (*EFE*), the linear size of the focusing length ($\Xi $), and the geometry of the plasmonic (metal) phase in nanoplasmonic systems. Not long ago it was hypothesized that the locality of plasmon dispersion in a self-similar chain of magnetoplasmonic core-shell two-dimensional (2D, equivalently, circular infinite cylindrical) nanostructures embedded in a host matrix might be governed by a scaling law [6]. This is ${\left(\Xi /\lambda \right)}_{PLR}\propto EF{E}^{-1.5}$, where *λ* denotes the free space wavelength of the resonant excitation. Fundamental questions remain unanswered about such a law, (i) whether self similar chains of three-dimensional (3D) particles can be characterized within a simple universal scaling behavior, (ii) whether it is valid for any particle geometry, or dependent on physical characteristics of the chain, and (iii) whether these observations are consistent with the PLR characteristics. We stress that a complete understanding of the scope of universality of this law still evades our grasp, e.g. for chains of *N*>5 nanostructures the applicability of scaling law is debatable because nonlocal (quantum confinement) effects lead to significant plasmon broadening in metal nanoparticles of diameter smaller than 10 nm. We note that DNA has been used to design plasmonic nanostructures, such as plasmonic molecules, polymers and crystals [14]. While numerous investigations have shown that *EFE* can reach $\ge $ 10^{3} (hot spots) in nanosystems [19], there are only a few calculations on predicting $\Xi $ in 2D [8] and 3D [4–7] systems.

In this work, we report on a systematic numerical study of the sub-optical wavelength electric field spatial confinement in a series of self-similar chains of plasmonic (full and core-shell (CS)) nanospheres. The results are then used as fitting data to demonstrate that *EFE* and $\Xi $are related by a scaling law. The study focuses on two important theoretical issues. Firstly, although we expect the physical origin of the scaling law to be similar to the situations discussed in [8], the present results differ from prior work in revealing an exponent different from 1.5. Secondly, we provide numerical evidence that the PLR characteristics play a key role on the observed scaling law.

Consider the schematic of the typical array of *N* metallic nanospheres in Fig. 1
. The dielectric properties of the embedding medium can be assimilated to water. We assume ideally smooth interfaces between rigid phases. The spheres are obtained using scaled-down copies of an initial geometry. A recursive algorithm, i.e. ${R}_{i+1}=k{R}_{i}$ and ${d}_{i,i+1}=\ell {R}_{i+1}$ can be developed to generate any occurrence of such self-similarity for a given iteration *i*. Here, *R _{i}* denotes the radius of the

*i*-th iteration and

*i*parameterizes the iteration process (as illustrated in Fig. 1). In all simulations the radius corresponding to the first iteration is held fixed at

*R*

_{1}= 185 nm. The choice of this parameter and the number of iterations should be consistent with the overall volume ${L}^{3}$ of the cubic cell. Clearly, the process cannot be carried up to a high number of iterations. In this paper, we were able to perform calculations up to four iterations for the geometric parameter ranges:

*k*= 0.30-0.35 and $\ell =0.3-0.6$. We will assume throughout that the time dependence of the electric field excitation, assumed to be directed along the

*x*direction, is proportional to $\mathrm{exp}(-2\pi jct/\lambda )$, where

*c*is the speed of light in vacuum. At long wavelength the physics of the system turns out to allow a further simplification. It must be borne in mind that the validity of this long-wavelength behavior is rooted in the fact that all length scales must be much smaller than

*λ*. To ensure that this constraint was satisfied, we use

*L*= 1226 nm. We employ a continuum modelling approach built upon constitutive equations which can capture the material behavior on experimentally relevant scales. That is, when the local electrical response in terms of a position dependent permittivity. The consistency and validation of this procedure was verified by agreement (not shown) of our calculations with those obtained from Kramers-Kronig causality relationships [22].

Calculations are performed using the finite element method as implemented in the COMSOL Multiphysics code [12] to study the effective complex permittivity $\epsilon ={\epsilon}^{\text{\'}}-j{\epsilon}^{"}$ as in [8] and [22]. The method is a straightforward generalization of that used recently in Ref [21]. for studying self-similar chains of 2D plasmonic nanostructures. In practice, the calculations use homogeneous Dirichlet-Neumann boundary conditions (see Ref [22]. and references cited therein). Simulations were run with 4-iteration chains since technically, a higher iteration number is computationally demanding. In this context, it is also salient to note that a fundamental limiting factor for how small systems can be designed is due to the quantized character of plasmons which can be significant for *N*$\ge $5, i.e. the thickness of the metallic shell should be larger than the Fermi wavelength (${\lambda}_{F}\approx $0.5 nm for Au) [3]. In our setup, typical running times for one set of fixed *k* and $\ell $ parameters required approximately 1 h using a personal computer with a Pentium IV processor (3 GHz). Additional technical details of secondary importance are given in [8]. To quantify the focusing length at the different iterations $\Xi $ was specified according to a previous report [8] as the distance over which *EFE* attains 85% of its maximum value. We checked (not shown) that shifting the threshold up from 85% to 90% of the maximum value of the *EFE* does not drastically affect the −1.5 exponent. An alternative definition of $\Xi $ may be obtained from the first moment of the field intensity, $\Xi ={\displaystyle \int xE\left(x\right)}dx/{\displaystyle \int E\left(x\right)}dx$which characterizes the average transport distance in the field direction. Both methods give similar results. As is evident from Fig. 1, the maximum *EFE* is achieved close to the points of the surface of the nanoparticle with maximal curvature, i.e. localized near the surface of the smaller particle, in good agreement with the observations in Refs [6–8,23–33]. That is the optical excitation generates a local electric field in the vicinity of the largest nanosphere which plays the role of the excitation field for the smaller particle, and so on. We explicitly checked that the hot spots are localized to the *x* axis.

Gold and silver are chosen as model shell materials and were modelled using the Drude model, i.e. ${\epsilon}_{2}\left(\omega \right)={\epsilon}_{2\infty}^{\text{'}}-{\omega}_{p}^{2}/\omega \left(\omega -j{\omega}_{c}\right)$. For Au, plasma frequency ${\omega}_{p}/2\pi =$2228 THz, collision frequency ${\omega}_{c}/2\pi =$6.0 THz [16], and ${\epsilon}_{\infty}^{\text{'}}=7$. For Ag, plasma frequency ${\omega}_{p}/2\pi =$2149 THz, collision frequency ${\omega}_{c}/2\pi =$12.2 THz [31–33], and ${\epsilon}_{\infty}^{\text{'}}=2.48$. Notice that the penetration depth of electromagnetic waves at optical frequencies is about 20 nm for Au. While nonlocality turns out to be a generic feature of small nanoobjects, our calculations show that if we only consider the conventional Drude’s form *EFE* can be significantly modified (Fig. 2(d)
) compared to the case including the finite-size correction (FSC) leading to an enhanced rate of electron scattering. The FSC was included by changing *ω*_{c} in Drude’s model of permittivity by ${\omega}_{c}+A{v}_{F}/R$, where *v*_{F} is the Fermi velocity for bulk Au, *R* is the radius of the particle, and *A* is a constant of order unity [9,17,19,21]. This is consistent with earlier studies [19,20,30,34–38]. We later return to discuss this point for the effective permittivity. An immediate consequence of the Drude’s model when nonlocal surface effects are neglected is that in the near-field region *EFE* is scale invariant.

Our main results are summarized in the ${\left(\Xi /\lambda \right)}_{PLR}$vs *EFE* diagram shown in Fig. 2. The main feature of the present study is that this diagram displays a scaling law ${\left(\Xi /\lambda \right)}_{PLR}\propto EF{E}^{-\gamma}$with an exponent $\gamma \approx 1$. This relationship is found to be independent of the values of *k* and $\ell $ investigated, the metal’s composition and the dielectric medium that surrounds the metal nanoobject, suggesting that it could be universal. With regard to the actual value of *γ* = 1, it is interesting to observe that it is significantly lower than the 1.5 found for nanodisks. The panels in Fig. 2 present the simulations for a range of model parameters *k* and $\ell $, two kinds of noble metals, and two kinds of embedding medium (water and air in order to compare with Refs [4–7].). In all cases, we see from Fig. 2 that there is only a weak sensitivity of the scaling behavior upon the various sets of parameters. Remarkably, we find that the ratio of the *γ* values found between the 2D [8] and the current 3D cases is $\approx $ to the inverse ratio of their Euclidean dimensions. With a further increase of *N* we find that *EFE* is much larger (Fig. 3
) than ${g}_{N}=Q{\left({R}_{N}/{R}_{1}\right)}^{\mathrm{ln}Q/\left|\mathrm{ln}k\right|}$, where $Q\approx -{\epsilon}_{2}^{\text{\'}}/{\epsilon}_{2}^{"}$ denotes the quality factor of the surface plasmon resonance and *ε*_{2} is the permittivity of the metal at the surface-plasmon resonance frequency, as was suggested by Stockman *et al*. [4,5]. It is also interesting to note that Dai and associates [21] had hinted at the possibility that cascade amplification produces a local field enhanced by a factor of ${\overline{g}}_{N}={Q}^{N}$ at the Nth particle. Figure 3 compares also the *EFE* values achieved as the structure geometry and metal nature are varied and Dai *et al*.’s estimate.

Figure 3 shows that *EFE* continues to grow for powers far above the Dai *et al*. estimate. A similar analysis reveals that $\gamma \approx 1$ persists when core-metal shell particles are considered at a specific of the metal layer’s thickness (${e}_{i}=t{R}_{i}$ with *t* = 0.2) instead of full particles (Fig. 2). The simulations for Fe_{3}O_{4}-Au CS nanospheres were procedurally similar from the 2D ones reported in [8]. Initially, it has been expected that the value of *γ* could be accounted for through the fractal dimension of these arrays, i.e. ${d}_{f}=\mathrm{ln}2/\left|\mathrm{ln}k\right|$, but this hope was dashed when it became clear our simulations that probe the ${\left(\Xi /\lambda \right)}_{PLR}$vs *EFE* scaling behavior are incapable of distinguishing between types of nanoparticle, metal, and embedding medium but are typically controlled by the Euclidean dimensionality. It is therefore tempting to suggest that it is a universal and robust property of self-similar chains of plasmonic nanoparticles and, most likely, which comes from the resonant excitation of a damped plasmon mode.

A couple of remarks on our results are in order. Interestingly, we find that they show qualitatively good consistency with recently reported identical universal scaling behavior of plasmon coupling in metal nanoshells and that in metal nanoparticles [39–41]. As another point of comparison, it is important to note that finite-difference time-domain computations demonstrated a nanolens effect which can convert a diffraction limited Gaussian beam into a sub-wavelength focus as small as *λ*/10 for self-similar Ag nanosphere array embedded in glass [42–44]. Our finding is consistent with recently published works that have highlighted the importance of carefully considering the issue of meshing the computational domain when calculating *EFE* and *ε* [6,7,38,45]. From a practical point of view, the scaling law is very useful, because all the quantities involved can be measured experimentally and do not rely on microscopic details. Only very recently, promising measurements have been reported of the optical-field enhancement from well controlled plasmonic arrays [23–28,45–49].

We end with a brief discussion of the effective permittivity for these self-similar chains of nanospheres which, to the best of our knowledge, was not considered in the majority of past works.

Figure 4
shows the imaginary parts of the effective permittivity which consist of many resonant peaks across a wide spectral region and are characterized by an intricate interplay between them (a full discussion of the permittivity spectra is beyond the scope of this study and will be the object of a future work). With respect to the maximum field enhancement, one can see a systematic redshift of the PLR (marked by the asterisk in Fig. 4) when the iteration number increases. The electrostatic resonance spectrum of nanoscale particles can be complex due to their multiband nature. This issue has been recently addressed in a particularly pointed fashion by Mayergoyz and associates [50]. However, an analytical framework of general applicability for the collective PLR behavior of an arbitrary (non-translation invariant) plasmonic array of nanostructures is lacking. Although our calculations are performed without the nonlocality correction of *ε* (black line in Fig. 4), there is no significant change in the ${\epsilon}^{"}$ results when the FSC is taken into account (green line in Fig. 4) with the exception of the 4th iteration for which the low-frequency modes vanish (Fig. 4(d)). To put the result shown in Fig. 2 into perspective, we provide also the ${\epsilon}^{"}$ data for Fe_{3}O_{4}-Au CS nanosphere arrays (Fig. 5
).

Interestingly, the data in Fig. 5 show that the PLR characteristics (blue line in Fig. 5) corresponding to the maximum field enhancement show a different spectral profile than the corresponding case of full Au nanoparticles. Although huge, tunable responses to electric perturbations are possible in this system, i.e. by varying *t* and the CS phases, experimental difficulties to fabricate self-similar plasmonic arrays will prevent perfect tuning [19,20,23–28,49].

In summary, our findings evidence a scaling relation between two fundamental properties of self-similar chains of plasmonic nanostructures: the field enhancement *EFE* and the linear size $\Xi $ of the hot spot. This law holds robustly for all simulation data. Taken together, this scaling law provides support for the hypothesis of a universal regime in which the sub-optical wavelength electric field confinement in nanoplasmonic systems is controlled by the Euclidean dimensionality and is independent of nanoparticle size, metal nature, or embedding medium permittivity. The present results suggest to us that the universal features of this scaling law are strongly related to the widespread multiple resonant (PLR) peaks and the shifting of the intensity centre as a function of the number of nanoparticles. This scaling law is a significant step towards controlling and designing plasmonic materials with desired sub-optical wavelength electric field confinement properties, e.g. 3D optical near-field trapping [51,52]. It is our intention to probe the scaling relation between *EFE* and $\Xi $ by including the influence of a coupling mechanism for electric and magnetic fields in magnetoplasmonic heterostructures for which the PRL is controlled using a weak magnetic field [53,54]. While there may be subtleties that would only be manifest were we able to study more complex (nonconvex) nanoparticles, we speculate that this scaling law likely applies to other plasmonic architectures coupling the metallic components through nanogaps, e.g. chain of nanocrescents [55–61].

## Acknowledgments

We acknowledge financial support from the Ph.D. funding programme (grant programme 211-B2-9/ARED) of the Conseil Régional de Bretagne. Lab-STICC is UMR CNRS 6285.

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