1. Introduction

Recently, much interest has been placed on tuning the local surface plasmon resonance (LSPR) of nanostructured materials to intensely scatter and redirect optical fields [1]. Upon resonance, an external incident driving field interacts with free electrons within a nanostructure and constructive interference between displacement and Coulomb restoring forces result in collective oscillation of its electron cloud. Of particular interest are nanoparticles comprised of noble metals such as Au and Ag, where highly efficient interactions with wavelengths near and within the visible spectrum can occur [2]. As such, nanoparticles at their LSPR frequency can have relative scattering cross-sections (Q_sca) many times larger than their physical cross-sectional area and have potential to impact many technologies including sensing [3], lazing [4], telecommunications [5], and photovoltaics [6].

Plasmonic resonant nanoparticles have a first-order LSPR in the form of a quasi-static dipole. At this first-order resonance, displacement of an electron cloud gives rise to a dipole-field about the particle. Moreover, incident energy that is utilized to displace the electron cloud is subsequently re-radiated from the nanoparticle in the form of a dipole-field. Under certain circumstances, in addition to a first-order dipole resonance, it is also possible for higher-order LSPR modes to occur [7]. For a given wavelength of incident light, as the size of a nanoparticle increases, the incident external driving field becomes non-uniform about the particle and higher-order resonant modes occur as shown in Fig. 1a .

 

Fig. 1 The relative cross-sectional scattering area, Q_sca, as indicated by color, of a single Ag nanosphere with radius ranging from 25 to 100 nm. A nanosphere 25 – 60 nm in radius supports a first-order dipole LSPR while a sphere larger than 60 nm in radius exhibits both first-order dipole and second-order quadrupole LSPR modes.

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Nanoparticles with highly efficient LSPR modes are of specific interest for light capturing applications in thin-film Si photovoltaic (PV) cells because they can aid in the collection of light by scattering targeted wavelengths within the solar spectrum. Ongoing work in this field is focused on tuning plasmonic response and integration of nanoparticles into PV cells to increase photon capture and subsequently boost device efficiency [8,9]. One of the most likely candidates for forming nanoparticles in an industrial setting is to heat PV cells coated with a thin metal film, typically 12-22 nm in effective thickness, to 200-300 °C for about an hour [10]. This process offers a rapid, low-cost, highly scalable means of forming random nanoparticle arrays with a targeted average diameter ranging from 30 - 300 nm. Despite these advantages, one potential challenge is controlling the distance between nanoparticles. If two adjacent nanoparticles in the array are in close proximity to one another they can become coupled. This work will show that interactions among coupled plasmonic nanoparticles must be accounted for because their scattering behavior is drastically different from that of an array of largely spaced, independent, nanoparticle scattering elements. We demonstrate that with proper tuning, coupled nanoparticles can be advantageous and scatter light over a broader portion of the solar spectrum compared to that of a single, uncoupled particle of identical size and shape. To this end, if the distance between particles is controlled, the scattering behavior of an array can be optimized to scatter a broader range of wavelengths within the solar spectrum.

Much work has been reported on first-order resonant scattering modes in coupled metallic nanoparticles. Zhang et. al. investigated Au ellipsoidal nanoparticles with a maximum long-axis length of 104 nm. This work showed that a shift in the dipole plasmon resonance decays approximately exponentially with increased particle spacing and also that coupling becomes negligible at particle gaps approximately greater than 2.5 times the particle’s short-axis length [11]. Works of El-Sayed et.al. have addressed coupling effects from size and shape of Au nanostructures [12], orientation of Au nanorods [13], and gap between surfaces of Au nanodiscs [14]. El-Sayed also reports on universal scaling in coupled nanostructures and shows that the relative fractional shift in the dipole plasmon resonance decays over an inter-particle gap with the same universal trend independent of particle size, shape, metal type or surrounding medium [15–17]. These reports provide understanding of first-order dipole LSPR shifts in coupled nanoparticles, however more investigation addressing LSPR behavior in coupled nanostructures exhibiting both first-order dipole and second-order quadrupole resonance is needed.

Our work expands upon previous reports of coupled nanostructures that discus nanostructures typically smaller than 100 nm in their long-axis dimension. We investigate large nanospheres, which are commonly used for enhanced light collection in PV cells, with diameters greater than 100 nm, where both dipole and quadrupole LSPR modes are present. Using exact Mie solutions [18] and finite-difference time-domain (FDTD) simulations [19–21] we determine the shift in dipole and quadrupole LSPR modes in coupled particles and demonstrate tuning of the particle-pair’s relative scattering cross-section (Q_sca) within the solar spectrum. This has important implications on advancing plasmon enhanced light trapping in thin film PV cells where little work has addressed improving Q_sca via tuning inter-particle coupling to control dipole and quadrupole LSPR modes.

2. Simulation Setup

All scattering behavior presented in this work is from Ag spheres ranging in radius from 10 to 122 nm surrounded in air with an index of 1. For the case of single particles, Mie theory was used to solve for the scattered field and determine the subsequent Q_sca. All particle-pairs were simulated in three-dimensions (3D) using symmetric and anti-symmetric perfectly matched layer (PML) boundary conditions with the FDTD method. Unless noted otherwise, a cubic mesh cell measuring 0.7 × 0.7 × 0.7 nm was used over the entire simulation space surrounding the particle-pair. Plane wave light ranging from 300 – 1100 nm was used in all calculations and for the case of particle-pair simulations, with light polarized parallel to the pair’s axis was used. The real and complex index of refraction for Ag was taken from the work of Johnson and Christy [2]. For FDTD simulations, an eight-coefficient expression was fitted to the refractive indices to best represent the Ag material model. Lastly, it should be noted that the Q_sca calculated in particle-pairs was defined by the scattering cross-section (C_sca) normalized by the cross-sectional area of both spheres, where as the Q_sca of a single particle was found by normalizing the cross-sectional area of the single sphere of interest.

3. Results and Discussion

3.1 Quadrupole modes in coupled nanoparticles

As previously seen in Fig. 1, an individual Ag particle < 60 nm in radius exhibits a single first-order dipole resonance. However, when two identical particles < 60 nm of the same size are coupled, they can exhibit both first-order dipole and second-order quadrupole LSPR modes. Figure 2 shows an independent 45 nm radius particle with only a first-order plasmon resonance. When coupled to an identical particle, the pair exhibits both a first- and second-order resonance at 470 and 360 nm respectively.

 

Fig. 2 The relative cross-sectional scattering area, Q_sca of coupled particle-pairs with a gap between the surface of each particle measuring 110, 60 and 10 nm. The particle radius for all simulations is 45 nm. As the two particles are brought closer together and the gap becomes smaller, the dipole resonance red-shifts. At a gap of 10 nm, a quadrupole resonant mode is induced. A single uncoupled particle of identical size is included for reference; note this only exhibits a first-order dipole mode.

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The presence of the quadrupole mode in coupled nanospheres can be attributed to a non-uniform, incident field driving the LSPR. If the particle-pair has a large center-to-center distance, L, the plasmon coupling strength is similar to that of the interaction energy between two classical dipoles where the coupling potential decays as (1/L3) [14,22]. In this state, the radiative dampening of the induced dipole dominates over that of dynamic depolarization and results in a single dipole resonance. At very small L however, coupling of the near-by dipole fields intensifies and is expressed by the superposition of the external driving field and according near-field dipole fields. Through such superposition it is seen that the coupled spheres greatly confine the net field between the particles and results in a non-uniform external driving field across each particle [23]. If L becomes small enough, the non-uniform field results in dynamic depolarization in each particle and a quadrupole LSPR is induced [11,24]. As L decreases, the dynamic depolarization strengthens and the LSPR red-shifts as a result of heightened dephasing and mixing of higher-order multipole oscillations within each nanoparticle. Figure 2 demonstrates this trend as the gap between the surfaces of two identical nanoparticles decreases, a strong red-shift of the first-order dipole resonant mode from ~380 – 470 nm occurs. Our simulation is similar to the experimental results reported by El-Sayed where the LSPR frequency of Au nanodiscs red-shifted as the gap between them was reduced [14].

3.2. Plasmon resonance in coupled particles with fixed center-to-center spacing

To investigate the scattering behavior of nanoparticles with a fixed center-to-center distance, light polarized along the inter-particle axis of identically paired Ag spheres ranging in radius, r, from 10 to 122 nm were simulated using the FDTD method. The center-to-center spacing of the sphere-pairs was fixed at L = 55, 100, 150, 200 or 250 nm as the radius of the spheres were varied. Results from particle-pairs held at L = 200 nm are shown in Fig. 3 . At small r/L conditions, corresponding to largely spaced particles, a single dipole resonance occurs between ~370 - 415 nm. These pairs exhibit a single dipole resonance because the incident field driving the LSPR is uniform across the particle volume and minimal coupling is occurring. The maximum Q_sca for a particle-pair that exhibits a single dipole LSPR is 4.7 and occurs at an r/L of 0.250.

 

Fig. 3 The relative cross-sectional scattering area, Q_sca of Ag nanosphere-pairs held at a fixed center-to-center spacing. Each curve is labeled by the according r/L of the particle-pair, that is the ratio of particle radius, r, to center-to-center spacing, L. Particle radius was varied from 20 – 90 nm while particle spacing was fixed at 200 nm.

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Particle-pairs with r/L > 0.350 exhibit well-defined dipole and quadrupole LSPR modes. As the r/L of particle-pairs becomes larger than 0.350, the dipole and quadrupole modes red-shift. In the most extreme case shown in Fig. 3 at r/L = 0.450, two particles 90 nm in radius with a 20 nm gap between them are strongly coupled and exhibit a dipole and quadrupole LSPR at 719 and 396 nm respectively. This corresponds to a dipole and quadrupole red-shift of 216 and 18 nm respectively from that of the particle-pair at r/L = 0.350. Furthermore, the red-shift in the dipole resonance is much larger than that of the quadrupole response. It is also noteworthy that the Q_sca in the dipole resonance of the particle-pair with the highest r/L (r/L = 0.450) was the second strongest in magnitude and higher than 1 for wavelengths 300 – 1100 nm. This has important implications for Si based PV cells where the collection of such wavelengths must occur with Q_sca > 1 to maximize cell efficiency. Similar results were also seen in simulations with fixed c-t-c distances of 100, 150, and 250 nm. Lastly, particle-pairs with r/L > 0.45 were also simulated, however not shown in Fig. 3. Particle pairs at such r/L conditions experience complex field confinement and a series of many higher order resonant modes occur. This is related to interference of complex polarizations and charge distributions caused by resonances that are spaced closer than their physical oscillation length [25].

As the particle radius becomes larger in fixed c-t-c simulations, two factors must be considered; first, the incident external driving field about the particle will become less uniform, and second, the gap between the surfaces of the particles becomes smaller resulting in stronger inter-particle coupling. As mentioned previously, both factors increase dynamic depolarization, which ultimately gives rise to the quadrupole LSPR mode. Control of dynamic depolarization is realized in this case by adjusting the r/L condition of the particle-pair. This can be applied to previously mentioned metallic nanoparticle arrays made for enhanced absorption in thin film PV cells where optimizing Q_sca (i.e. Q_sca > 1, for all wavelengths) and targeting a specific spectral range (i.e. λ = 700 - 1100 nm for Si thin film cells) are critical to maximize photon capture. Although the r/L simulations reported here are between two inter-coupled particles, it may be possible to use the plasmon hybridization method to further examine the plasmon LSPR of more complex multiparticle arrays [23,26]. To this end, a multiparticle array could be broken down into liner combinations of plasmons from individual nanoparticles with a defined r/L and the scattering behavior could be known.

3.3. Onset and relative strength of nanoparticle coupling

To determine the onset and relative strength of coupling between sphere-pairs in Fig. 3, the first- and second-order LSPR modes in coupled particle-pairs, λ₁ and λ₂ respectively, are compared to that of the corresponding first- and second-order LSPR modes from a single, independent, uncoupled nanosphere of the same size, λ_o1 and λ_o2 respectively.

The LSPR modes of an individual particle seen in Fig. 4 are obviously unaffected by any factors arising from inter-particle coupling. More importantly they reveal shifts of the LSPR due to dynamic depolarization in the particle solely as a result of an increase in particle size. Thus, in the single particle simulations, it is the particle size that determines the presence of the second order mode and red-shift of first- and second-order LSPR modes. For the case of the particle-pair simulations, any deviation from the individual particle’s resonance is a result of inter-particle coupling. As seen in Fig. 4, a deviation from the individual particle dipole LSPR is seen for r/L ≥ 0.150. An r/L = 0.150 corresponds to spheres with a radius of 30 nm and a gap between the surfaces of the particle pair of 140 nm. As such, the gap between the two nanoparticles is 2.33 times the particle diameter and is similar to the results of Zhang et al. who have experimentally shown the onset of inter-particle coupling occurring when the gap between two nanoparticles is ~2.5 times a particle’s short-axis [11]. It is also noted that coupling between particles does not affect the quadrupole LSPR as significantly as the dipole LSPR. These results agree well with our findings in coupled nanoparticles of the same size separated by different gaps in Fig. 2 as well as coupled particles held at a fixed L with increasing radius in Fig. 3.

 

Fig. 4 The dipole and quadrupole LSPR resonance peak position (indicated by squares) of particle-pairs with radius, r, varied from 20 to 99 nm held at a fixed center-to-center spacing, L, of 200 nm compared to that of corresponding dipole and quadrupole LSPR modes of individual, uncoupled Ag spheres of the same size (indicated by triangles).

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3.4. Universal scaling law of dipole resonance in coupled nanoparticles

As previously stated, decay of fractional shifts in dipole plasmon resonance wavelength occurs over an inter-particle gap with the same universal trend independent of particle size [14]. This trend was shown in particles < 103 nm in diameter where radiative dampening dominated over that of dynamic dephasing and only dipole modes occurred. To expand upon this, we show the universal scaling law is also valid for dipole LSPR shifts in particle-pairs with larger particles, > 100 nm in diameter, where dynamic dephasing conditions induce quadrupole modes. This is done by calculating the shift of the first-order plasmon resonances, Δλ, in coupled sphere-pairs by taking the position of LSPR modes at each r/L value in Fig. 3 and normalizing by the corresponding first-order LSPR wavelengths, λ_o1, from a single, independent, uncoupled nanosphere of the same size, respectively (i.e. Δλ / λ_o1 = (λ₁ - λ_o1) / λ_o1). When plotted against the ratio of inter-particle gap scaled by the nanoparticle diameter, 2r, as seen in Fig. 5 , the relative fractional shift of the dipole resonance follows a trend of exponential decay that is independent of particle size.

 

Fig. 5 The shift of the first-order dipole plasmon resonance, Δλ, normalized by the corresponding first-order LSPR wavelengths, λ_o1, from a single, independent, uncoupled nanosphere, in coupled sphere-pairs plotted against the ratio of inter-particle gap scaled by the nanoparticle diameter, 2r. A fitted exponential decay of y = 0.01 + 0.6 × exp(-x / 0.22), where x = L / 2r is fitted to the data to show the universal scaling trend.

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For large gaps between particles, gap/2r > 1, relative fractional shifts in the dipole LSPR are very small. As the gap between particles becomes smaller, corresponding to larger particles and smaller gap/2r conditions, an increase in particle coupling results in large fractional shifts. From this we note the magnitude of the shift is proportional to the particle volume; larger particles exhibit larger shifts.

A single-exponential decay function of the form y = y_o + A × exp(-x / τ), where x = L / 2r is fitted to the data in Fig. 5. The decay constant, τ, in this fit is 0.22 and agrees well with previously reported universal decay constants for both Au and Ag particles exhibiting only a single dipole LSPR mode. Despite good agreement with our decay constant in other works, the fitted magnitude, A, in our exponential function in significantly larger that that of other reports using Au nanoparticles. Exponential fits to scaling behavior in nanoparticles comprised of Au typically exhibit a magnitude, A = 0.04 – 0.21 [12,14,15], whereas silver nanoparticles used herein resulted in A = 0.6. A previous report on universal scaling in Ag nanoparticles by Gunnarson et. al. [27], have shown an experimentally observed magnitude that is also larger than that found in Au. From this we note the decay constant is universal across systems comprised of different materials however, it is clear that the magnitude of this trend varies from one material system to the next. The difference in magnitude is a result of higher efficiency plasmon resonances in Ag over that of Au. It is well known that the lower efficiency observed in Au nanoparticles originates from inter-band electron excitations and transitions in sp and d bands, which in turn compete with and dampen the plasmon resonance [2,28,29].

As previously shown in Fig. 4 the dipole LSPR of particle-pairs approaches that of an identical single particle when the gap between the particles is approximately > 2.5 times the particle diameter. As such, the relative fractional shift in the dipole LSPR plotted against the gap between particles normalized by particle diameter in Fig. 5 should equal zero when gap/2r is approximately > 2.5. On the contrary, a small offset of y_o = 0.01 is needed to fit the single-exponential decay for conditions when gap/2r > 2.5. Such an offset is likely due to a mismatch between results from particle-pair FDTD simulations and exact single-particle Mie solutions used to calculate Δλ / λ_o1. Additionally, it should be noted that as the gap between particle-pairs became smaller, the volume between the spheres decreased to a level such that the density of modeling elements needed to be increased to achieve accurate results. Simulations for gap/2r conditions < 0.25 used a mesh as small as 0.4 × 0.4 × 0.4 nm to achieve accurate results.

3.5. Dependency of particle size at a specified gap for quadrupole resonance in coupled nanoparticles

The universal scaling law can be used to describe the relative fractional shift in dipole LSPR modes for particle-pairs spaced at various distances from one another independent of particle size. With respect to shifts in quadrupole LSPR modes in particle-pairs however, the universal scaling law does not apply. Fractional shifts of the quadrupole mode are not universal for all particle sizes; rather it’s dependent on a given set of particle sizes within a fixed L. In Fig. 6 , the relative fractional shift in quadrupole resonances is plotted against the gap normalized by particle diameter.

 

Fig. 6 The fractional shift of the second-order quadrupole plasmon resonance, Δλ, normalized by the corresponding second-order LSPR wavelength λ_o2 from a single, independent, uncoupled nanosphere, in coupled sphere-pairs plotted against the ratio of inter-particle gap scaled by the nanoparticle diameter, 2r. The relative fractional shifts take on bands based on the center-to-center spacing of 150, 200, and 250 nm.

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In a similar fashion to relative fractional shifts observed in dipole resonances, large shifts in the quadrupole LSPR of particle-pairs occur at small gap/2r conditions. In contrast however, it is seen that the onset of shifts in the quadrupole mode take on bands that are dependent on L. This can be explained by considering the onset of coupling for particle-pairs held at small and large L. For a small L, as the particle radius is increased the gap between the particles will become smaller sooner compared to that of particles spaced at a much larger L. That is, for a small L for particle size can be small before particles are close enough together for coupling to shift the quadrupole mode. At large L, larger particles are necessary to make the gap small enough for coupling to shift the quadrupole mode. This in turn is quantified in the gap/2r value of a particle-pairs consisting of a given particle size at different L. For example, a particle-pair with a particle radius of 75 nm has a gap/2r of 0.67, 0.33, and 0 for an L of 250, 200 and 150 nm respectively.

The onset of the quadrupole mode in Fig. 6 for an L of 250, 200 and 150 nm occurs in particle-pairs with radius 85, 75 and 65 nm respectively. It is not surprising the onset of the quadrupole mode is occurring in smaller particles spaced closer together (smaller L). As previously shown in Fig. 2, particles that don’t normally exhibit a quadrupole mode can be brought closer together to induce a quadrupole mode. In the case of Fig. 6, particle-pairs that are closest together at L = 150 nm support a well-defined quadrupole mode in the smallest sized particle-pairs of radius 65 nm; the L is small and subsequently the gap is small (gap = 20 nm). If the same sized particles were set at an L = 250 nm, the gap would be 120 nm and the coupling would not be strong enough to shift the quadrupole mode. However, at L = 250 nm, when the particle radius is 85 nm in radius, significant coupling occurs and the quadrupole mode is shifted.

4. Summary

In summary, we investigate coupling conditions between Ag nano-spheres where both first-order dipole and second-order quadrupole LSPR modes exists. A single particle that may only exhibit a single dipole resonance can be coupled to another identical particle to induce a second-order quadrupole mode. The quadrupole mode can be tuned and, with proper coupling (i.e. r/L conditions), its Q_sca can be made > 1 for wavelengths 300 – 1100 nm. These are especially important scattering conditions that can be used in designing Ag nanoarrays for enhanced light capture in Si based thin film PV cells. Furthermore, the onset of coupling was determined in Ag particle-pairs where the LSPR position of coupled particles was compared to that of a comparable independent particle. This allowed for the LSPR shift to be quantified independently of dynamic dephasing resulting from particle size and coupling. It was shown that subsequent first-order dipole shifts in particle-pairs with both dipole and quadrupole LSPR modes follow universal scaling independent of particle size. While the decay constant was similar to that of coupled particles consisting of different materials, it is noted that the magnitude of the exponential fit was significantly larger. This is attributed to the higher efficiency of Ag that allows larger relative fractional shifts of the dipole LSPR mode. Lastly, the relative fractional shift of the quadrupole mode in coupled nanospheres was investigated. Rather than being independent of particle size much like universal scaling of the dipole mode, the quadrupole mode is dependent on a predetermined set of particle sizes that is governed by the center-to-center spacing of the particle-pair. As such the relative fractional shift manifests bands based on these center-to-center distances. These conclusions have critical implications on the design of nanoparticle arrays for plasmon enhanced PV cells and may also help to advance other plasmon assisted applications where coupling among nanoparticles in an array can be tuned to increase scattering within a targeted region of an incident spectrum.