Polarization and distance dependent coupling in linear chains of gold nanoparticles

Electromagnetic interactions in chains of noble metal nanoparticles are of interest for plasmonics, nanooptics,¹ photovoltaics,² and biochemical applications.^3–7 The electromagnetic coupling between nanoparticles in a chain results in a tunable collective Surface Plasmon Resonance (SPR).⁴ Recently, Kravets et al. showed that the sensitivity of ordered-nanoparticles-array-based sensors can be an order of magnitude higher than that of conventional sensors.⁵ Plasmonic coupling between nanoparticles has also been used as a molecular ruler in biology.^6,7 Based on the distance-dependent shift of the SPR wavelength, this method has demonstrated a spatial resolution down to a few nanometers. In this paper, we study the simultaneous polarization and distance dependence of the electromagnetic coupling between regularly arranged nanoparticles on a substrate. This could potentially be of use for molecular rulers. Other promising applications of ordered plasmonic nanostructures include use in active thin-film polarizers and filters.⁸ Nanoparticles in conductive polymers can form a polarizer with tunable efficiency or a filter with a tunable frequency range.

Collective SPR in a chain of metal nanoparticles is modeled as a system of coupled dipoles.⁹ The frequency of the collective oscillations is derived using a quasi-free electron model.¹⁰ According to this model, an electron in a nanoparticle is exposed to the internal electric field as well as the scattered field from neighboring nanoparticles.^10,11 Interactions with the scattered electric fields are dominant in the near-field, middle-field, or the far-field, depending on the distance between the neighboring nanoparticles.^10,11 The near-field interaction between pairs of flat (17 nm height and 150 nm diameter) particles can be described by their dipolar interactions.¹² Lamprecht et al.¹³ experimentally studied far-field interactions in 2D arrays of flat (14 nm height and 150 nm in diameter) particles and found agreement with a theory presented by Meier et al.¹⁴ 

Here, we study the near, middle, and far-field coupling in arrays of nearly spherical (50 nm height, 50 nm diameter) nanoparticles with an inter-particle separation between 100 nm and 1000 nm. We have found a strong agreement between the theoretical dipolar interaction model and our experiments. Experimental peak positions were determined to be within 7 nm (1.3%) or less of the simulated wavelengths. We interpret this small discrepancy due to the fact that the theory only takes into account dipolar interactions and neglects higher order multipoles. The theory we used here does not include any fitting parameters in the calculations.

The focus of this work was to simultaneously study the polarization and distance dependence of the collective SPR extinction spectra in chains of gold nanoparticles. Previous literature presents the polarization dependence of the SPR wavelength when the angle of the light incidence is varied.¹⁵ However, the polarization dependence of the SPR wavelength in-plane with the sample at the normal light incidence has been studied much less. Earlier experiments were performed only for parallel and perpendicular polarizations of the incident light relative to the axis connecting the nanoparticles in the chain.^16,17 There are no reports regarding the whole range of intermediate polarization orientations. The majority of previous reports focused on near-field interactions,^16,18–21 with only a few focused on the far-field.¹³ It is, however, important to study the whole range of interactions for applications such as the “molecular ruler” and wave guiding. It is important to know at which distances between the particles the dominate interaction changes from the near- to the far-field. The in-plane polarization dependence of the SPR wavelength is an indicator of this transition. No experimental results have been reported on simultaneous in-plane polarization and distance dependence of collective SPR in a chain of nanoparticles. In this work, the range of distances between nanoparticles, for which the polarization dependence was studied, spans over the entire range of interactions from near to far-field. The angle between polarization and chain of particles is varied in 10° increments. Finally, our work also presents a simple model for (1D) chains of spherical nanoparticles, where the interaction between chains is negligible. Previous reports either study 2D arrays,^15,22 or non-spherical nanoparticles (L-shaped,²² rectangles,¹⁷ rings,¹⁶ discs,¹⁹ ellipsoids,²⁰ and hollow spheres).²¹ 

Understanding the distance dependence of nanoparticles' collective modes in these ordered nanostructures is crucial for the development of biochemical sensors,^5,19,23–25 sub-wavelength wave guiding,^3,26,27 surface enhanced Raman spectroscopy,^28,29 optical tweezing,¹ and enhancement of light absorption by solar cells.^2,30,31 The effect of polarization is especially crucial for “molecular ruler” applications,³² where the orientation of the nanoparticles is unknown.

Regular arrays of nearly-spherical 50 nm diameter gold nanoparticles were fabricated using electron beam lithography at Argonne National Laboratory. The area of each array was 2 × 4 mm and contained linear chains of nanoparticles with a specific pitch, d. The chains in the arrays of the nanoparticles were separated by the same distance of 1000 nm, which provided a high enough density of nanoparticles while still avoiding any electromagnetic coupling between adjacent chains. Figures 1(a)–1(d) show scanning electron microscopy (SEM) images of the chains with different pitch, and Fig. 1(e) shows a sketch of the sample.

The optical density of each array was measured using light extinction experiments at normal incidence. Light from a Xenon lamp was linearly polarized and focused onto a 1 mm spot on the array. The sample was fixed on a rotating platform, and the angle between the sample and the polarization of light was changed in a 0°–180° range with 10° increments (insets on Fig. 2(a) show orientation of the electric field with respect to the chains of nanoparticles).

For small distances between nanoparticles (pitch), the near-field electromagnetic coupling was dominant. A red shift in the resonance wavelength was observed as the polarization changed from perpendicular (s-pol) to parallel (p-pol) with respect to the chain (Fig. 2(a)). The opposite trend was observed for large distances between nanoparticles, where the far-field interaction was dominant (Fig. 2(b)).

The optical density spectra on Fig. 2 were smoothed using the Savitzky-Golay³³ algorithm and then normalized by height via adding a constant in order to illustrate the shift of the resonance wavelength. Substrate-only extinction was used as the reference. The extinction spectra were measured for all 15 arrays, with pitch varying from 100 nm to 620 nm. Results for all arrays are summarized in Fig. 3.

To confirm that the effect observed here was a result of the inter-chain interactions (1D), a control experiment was performed. We studied a square lattice reference array with a pitch of 1000 nm. The spectral position of the resonance peak for the reference array was in the 529–530 nm range for all polarization orientations (Fig. 3(d)). This wavelength is very close to the SPR wavelength of a single 50 nm spherical nanoparticle (533 nm), which was calculated using the Mie theory.³⁴ Hence, the interaction between nanoparticles at this distance is very weak and its effects on the resonance wavelength are negligible.

In our recent publication, we have shown analytical expressions for the modified frequencies of collective SPR modes for two angles (0° and 90°) between the polarization of incident light and the chain of nanoparticles.¹¹ These frequencies can be easily modified for an arbitrary angle. For a normal incidence, the angular dependence of the near, middle, and far-fields projected on a z-axis is given by

In Eqs. (1)–(3), p is the z-projection of an induced dipole moment in a nanoparticle, $k$ is the wave-vector of the dipole field, d is distance between nanoparticles, and $θ$ is the angle between chains of nanoparticles and the incident light's polarization vector. It can be seen that near and middle-field depend on polarization as $cos2 θ$ and the far-field as $sin2θ$⁠. The frequency of the surface plasmon oscillations in a chain of nanoparticles therefore is described by

For p-polarized light, the far-field component vanishes and by using, $ω=kc=2πλc$⁠. We can express the wavelength of the collective mode as

where $λSPR$ is the resonance wavelength of an isolated spherical nanoparticle, R is the radius of the nanoparticle, and d is the pitch in a chain of 3 nanoparticles. $ΛSPR$ was calculated for N nanoparticles, where N varied from 3 to 1001 particles in the chain. Results varied from the $ΛSPR$ for the chain of 3 nanoparticles by maximum of 1%. Resonance SPR wavelengths, obtained using the theory described here, are illustrated in Fig. 3(a). It was shown that at a distance of 320 nm between nanoparticles, the interaction changes from near and middle-field to the far-field. The distance at which the interaction shifts can be viewed as an isotropic point.¹⁰ At this distance, the combined effects of all components of the interacting fields result in the same $ΛSPR$ for all polarizations. This distance (red dashed line) is unique and depends on the dielectric constant of the medium in which nanoparticles are placed. Hence, it can be used for sensing. When the dielectric constant of the medium changes, the iso-point will correspond to some new distance between the nanoparticles.

We wanted to avoid any fitting parameters or numerical calculations to keep the theory as simple as possible. To derive an analytical solution, we made the assumption that the wavenumber of the dipole field radiated by each nanoparticle in a chain is $k=2πλSPR$⁠. In fact, a more accurate wavenumber would be $k=2πΛSPR$⁠, but with this expression, Eqs. (4) and (5) can only be solved numerically. This assumption has a negligible effect on the resonance frequency; hence, the analytical expression accurately describes the resonance wavelength. A numerically solved equation led to only a 10⁻²nm change in $ΛSPR$ in comparison to the analytical solutions.

The experimental data presented in Fig. 3(b) show the dependence of the SPR wavelength on the distance between the nanoparticles and on polarization of the incident light. First, it can be seen that the maximum wavelength deviation was observed for the smaller distances between the particles (red and purple areas). This can be explained by the greater density of field lines with small distances between the dipoles. The maxima for $ΛSPR$ corresponded to the parallel polarization of light (0° and 180°). As the distance between the nanoparticles increased, the maxima were replaced by minima. For the perpendicular polarization (90°) at short distances, there is a minimum (purple) which is replaced by a maximum for larger pitch (light blue area for 500 nm pitch). Next, the distance between the nanoparticles at which the transition from the near and middle to the far-field was observed as a contour of constant $ΛSPR$(red dashed line on Fig. 3). Experimentally, it was found at 280 nm, while according to the theory it should be around 320 nm. This discrepancy was due to the discrete nature of the experimental data. Experimental pitch was changed every 40 nm. While in production of the theoretical graph, the pitch was varied by 1 nm. Additionally, the fact that no fitting parameters were included in the theory could also lead to some disagreement.

For better visualization of the data presented in Figs. 3(a) and 3(b)), we have shown two cross-sections of Figs. 3(a) and 3(b) at the pitch 100 nm (red and grey curves) and 420 nm (blue and black curves) on Fig. 3(c). We observed a $cos2 θ$ dependence for the array with 100 nm pitch, which was a signature of the near- and middle-field interaction. The $sin2 θ$ dependence for the array with a 420 nm pitch was a signature of a far-field interaction.

In summary, we experimentally determined the particle separation at which the resonance frequency was polarization-independent to be 320 nm. It was close to the theoretically predicted 280 nm and was rationalized by the equilibrium of the near- and far-field interactions between the particles. The discrepancy between the theory and experiment was attributed to the discreetness of the experimental data (particle separation was varied by 40 nm). We have shown that the peak of the resonance extinction can be tuned by both adjusting the in-plane light polarization and the distance between nanoparticles in the chains. This was rationalized by considering the electromagnetic interaction between nanoparticles via all three components of a dipole field (near, middle, and far). Theoretical calculations of the resonance wavelength were obtained via solutions of the equation of motion for a simple harmonic oscillator (free electrons) and fit well with the experimental observations. The $cos2 θ$ near-field dependence was observed for chain pitches less than 300 nm, while there was a $sin2 θ$ far-field dependence for larger pitches (340 nm–620 nm). We have also found that the coupling became negligible when nanoparticles were 1000 nm apart. These results will help further understanding of the polarization and distant dependent coupling between nanoparticles regularly arranged on a substrate and will provide utility for the development of biochemical sensors, sub-wavelength waveguides, surface enhanced Raman spectroscopy, optical tweezers, enhancement of light absorption by solar cells, and “molecular rulers.”

The authors acknowledge partial support from the Biofrontiers Institute, CRDF (UKC2-7071-CH-12) and NATO (SFPP-984617) grants. Use of the Center for Nanoscale Materials was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357 (CNM proposal 1112). We also want to thank Dr. Zbigniew Celinski, Dr. Robert Camley, Dr. Anatoliy Glushchenko, Kyle Culhane, Nickolas Anderson, Sara Goldman, and Kristen Petersen for their insight and stimulating discussions of this work.