LETTER pubs.acs.org/NanoLett
Trimeric Plasmonic Molecules: The Role of Symmetry Lev Chuntonov* and Gilad Haran* Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
bS Supporting Information ABSTRACT: Artificial plasmonic molecules possess excitation modes that are defined by their symmetry and obey group theory rules, just like conventional molecules. We follow the evolution of surface-plasmon spectra of plasmonic trimers, assembled from equal-sized silver nanoparticles, as gradual geometric changes break their symmetry. The spectral modes of an equilateral triangle, the most symmetric structure of a trimer, are degenerate. This degeneracy is lifted as the symmetry is lowered when one of the vertex angles in opened, which also leads to a subtle transition between bright and dark modes. Our experimental results are quantitatively explained using numerical simulations and plasmon hybridization theory. KEYWORDS: Surface plasmon, nanoparticles, trimers, dark-field spectroscopy, symmetry, plasmon hybridization
C
ollective oscillations of the conduction electrons on the surface of metallic nanostructures of various compositions and geometries, referred to as surface plasmons,1,2 provide the means to manipulate light on the nanoscale.35 This ability is exploited in experimental techniques such as surface-enhanced Raman spectroscopy,58 surface-enhanced fluorescence spectroscopy,9 surface-enhanced infrared spectroscopy,10 and sensing,11 as well as in various applications, such as for biomedicine.12 Plasmonic molecules13 (PMs) are nanostructures in which individual plasmon modes strongly interact and show distinct collective behavior. Interactions between such modes themselves and with external perturbations may lead to phenomena analogous to those found in atomic and molecular systems, including electromagnetically induced transparency,14 slow light,15 and Fano resonances.1618 Recognition of the analogy mentioned above paves the way to the application of concepts developed in chemical physics to PMs, which may reveal new insights on the physics of the light-surface plasmon interaction. Examples are the application of the quantum state hybridization formalism,19 dressed-states picture,20 and group theory.21,22 Spherical nanoparticles of the noble metals are building elements for PMs, which can be constructed by controlled aggregation of individual nanoparticles into small clusters.1,5,2326 To date, the most studied example of coupled nanoparticles is a pair of nearly touching spheres, a dimer.2732 Strong interaction between the surface charges induced on each of the particles results in coupled plasmon modes and tight focusing of the light into the nanogap formed between them. More complex arrangements of the nanoparticles (trimers, tetramers, etc.) not only redistribute the light within their gaps22 but may open the window into numerous novel fascinating phenomena.18,23,25,33 The plasmonic states of an individual nanoparticle, which correspond to multipole orders of the Mie solution for the r 2011 American Chemical Society
interaction with an electromagnetic field,1,2 are analogues to the electron states of an atom.19 When the particles are arranged into a cluster, these states strongly interact to form the corresponding cluster states24,34 in analogy to molecular orbitals built up by linear combination of atomic orbitals. Just like molecular orbitals, the plasmonic states can be designated as bonding or antibonding, depending on whether the induced charge is distributed in the low- or high-energy configuration. The exact arrangement of the nanoparticles within a cluster therefore has an essential impact on the symmetry of the resulting plasmonic states. Symmetry-adapted linear combinations (SALCs) of individual particle states, derived with the aid of the irreducible representations (irreps) of the relevant point group, are known to provide a good basis for understanding the physical states of a symmetric system.3537 Symmetry breaking in plasmonic nanostructures can introduce effects that are not present in the symmetrical configurations. These effects, arising from coupling between otherwise uncoupled plasmon modes of the system, were explored, for example, with individual nanoshells,38 nanoparticle dimers30 and trimers,25 nanocavities,39 and split-ring resonators.40 Here, we present experimental and theoretical analysis of a homologous series of trimeric silver PMs and show how a systematic configuration transformation leads to changes in plasmonic spectra, which can be understood with the help of group theory. We employed single-cluster optical plasmon spectroscopy26,41,42 to probe the optically active transitions, and classified them using group theoretical arguments. The highest-symmetry form of a trimeric PM is the equilateral triangle Received: March 15, 2011 Revised: May 2, 2011 Published: May 10, 2011 2440
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Figure 1. Scattering spectra of silver nanoparticle trimers showing mode splitting due to gradual opening of the vertex angle. Thin black lines, raw experimental data; color lines, low pass filter applied. The clusters are excited by either nonpolarized light (green) or light linearly polarized along the longitudinal (red) and transverse (blue) directions (which are indicated by arrows of matching colors). The polarized spectra are normalized to the corresponding peaks of the nonpolarized spectra. Plasmon mode symmetries of the bonding modes are marked. TEM images of the clusters are shown in the insets (the bars are 50 nm). Calculated surface charge distributions of the selectively excited bonding modes are shown below the spectra.
(D3h symmetry),22,33,43 which possesses two degenerate optically active bonding plasmon resonances. Gradual symmetry breaking lifts mode degeneracy and enables mode-selective excitation with polarized light. We investigated spectroscopically and theoretically this symmetry breaking and the corresponding evolution of the plasmon modes of trimeric clusters. The series studied included clusters with a gradually increasing vertex angle, spanning all configurations between an equilateral triangle and a trimeric linear chain. Our experiments were performed on clusters of silver nanoparticles having mean diameter of 60 nm (nanoComposix Inc., San Diego). Individual clusters were generated through the application of capillary forces acting during solvent evaporation from droplets of aqueous solution (approximately 500 pL each), sprayed upon the support layer of transmission electron microscopy (TEM) grids. Optical measurements were made using an inverted microscope equipped with a dark-field condenser, a variable numerical aperture objective and a 75 W Xe lamp. A polarizer combined with a selecting sector30 restricted illumination to s-polarized light only and enabled rotation of the direction of linear polarization with respect to the cluster of nanoparticles. Spectroscopic measurements were correlated with TEM images (obtained with a Phillips CM120 electron microscope). We used a comprehensive array of numerical and analytical methods to analyze the data. First, the Multiple Multipole Program (MMP),44,45 based on boundary discretization, was employed to simulate electromagnetic fields within particle clusters. This semianalytical technique is best suited to follow gradual changes in a cluster’s geometry, owing to the analytically defined boundaries of the nanostructures studied. The calculated electromagnetic fields were used to evaluate the surface charge distribution induced on the nanoparticles. The dielectric functions of ref 46 were used in the simulation, assuming in addition that the particles are immersed in a layer of condensation water.47 The results were found to barely depend on the direction of incidence of light onto the sample. Therefore, normal incidence was used in all calculations, allowing us to take advantage of the symmetry planes of the cluster with respect to the light polarization and reduce the computational efforts. The good agreement between experiment and numerical results (see below) indicated that light retardation effects did not
play an important role in this study. Gaps between particles were kept at 1 nm, since our experimental observations (based on tilt series of TEM images ranging from 40 to þ40 degrees) revealed typical gaps of this size and below. Obviously, numerical calculations using the MMP were based on pure electrodynamics and did not include any effects due tunneling of the conduction electrons across the nanoparticle junctions.28,48 Group theoretical analysis was used to find SALCs, which were employed as a basis for decomposition of the physical plasmon bands, that is, of the induced surface charge distributions reconstructed from the exact numerical calculations. Finally, in order to find the explicit forms and energies of the modes for arbitrary cluster geometry, we resorted to a full theoretical description, using the hybridization theory of Nordlander and coworkers.19,22,27 This theory obtains the eigenstates of the system as solutions of the EulerLagrange equation for the interacting charge distributions induced on the individual nanoparticles. Considering for simplicity only the dipolar (l = 1) modes of the multipole expansion, the solutions can be represented by point dipoles positioned in the center of each nanoparticle.13,19,22,27 This dipolar model is valid in the long wavelength limit (ka, 1, where k is the scattering wave vector and a is the radius of the nanoparticle), which holds approximately for silver nanoparticles excited with visible light. The SALCs discussed in the present work are eigenmodes of the bare Lagrangian, which does not include the coupling to the excitation electric field of light. The light can excite the eigenstates of the system, depending on their symmetry and resonance frequency. A representative cluster of D3h symmetry is shown in Figure 1A, together with measured spectra and numerical results. To simplify further discussion, we define two principle in-plane directions with respect to the geometry of the cluster: the transverse and longitudinal directions. While the choice of these directions remains arbitrary in the case of a D3h cluster, it becomes natural when the symmetry is reduced (Figure 1B,C); the longitudinal direction is parallel to the base of the isosceles triangle formed by the nanoparticles, and the transverse direction is perpendicular to it. As we have experimentally restricted the excitation to s-polarized light, only the in-plane modes are excited and discussed in this work. 2441
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Figure 2. Correlation table of dipolar plasmon eigenmodes of trimers for varying vertex angles, obtained using the plasmon hybridization theory. Upper part, antibonding modes; lower part, bonding modes. The symbols of the corresponding irreps are shown. The thickness of halos surrounding the particles represents the relative “brightness” of plasmon modes, while their color corresponds approximately to the resonance wavelength.
The trimer cluster belonging to the symmetry group D3h has two in-plane doubly degenerate irreps of symmetry E0 , as dictated by the presence of the symmetry operations constituting the rotational subgroup C3. The plasmon hybridization calculation (see also ref 22) shows that one pair of these doubly degenerate modes corresponds to high-energy antibonding modes, while another pair corresponds to low-energy bonding modes. The shapes of the corresponding modes as calculated by plasmon hybridization are shown explicitly in the correlation table (Figure 2), left column. Each pair of degenerate antibonding and bonding modes includes one mode having its total dipole moment aligned along the longitudinal direction of the cluster, and another mode with its total dipole moment along the transverse direction. The nonzero total dipole moments ensure that the E0 modes are optically active, allowing for efficient coupling to the excitation light at the resonant wavelength. Indeed, two pairs of nearly degenerate plasmon resonance bands appear in the spectrum shown in Figure 1A. The resonances are located at 490 and 650 nm. The small deviation from full degeneracy between the spectra excited with light polarized along the transverse and longitudinal directions of the cluster is likely to originate from slight shape and size irregularities of the
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polycrystalline nanoparticles and can be ignored. Our numerical simulations, conducted for the case of ideally spherical and equalsized nanoparticles, show that the scattering spectra corresponding to excitation with two different light polarizations are indeed identical. Although our spectra include also the high-energy spectral region where the antibonding states appear, we focus the following discussion only on the properties of the bonding states. The surface charge distributions calculated numerically for the selectively excited degenerate transverse and longitudinal bonding resonances of the D3h cluster are shown in lower panel of Figure 1A. These charge distributions represent physical plasmon states of the cluster with the corresponding symmetry. The physical states mostly involve bonding E0 dipolar SALCs, which contribute more than 60% of the charge distributions. Some additional contributions, responsible for the deviation between the calculated physical states of Figure 1 and the corresponding SALCs shown in Figure 2, may originate from the admixtures of terms of orders higher than dipole.22 This coupling to higher order multipoles is beyond the scope of the present paper. The two degenerate E0 bonding modes are not the lowest energy eigenmodes of the D3h cluster. Rather, the state of lowest energy is an A0 2 bonding mode having a ringlike arrangement of dipoles, as shown in Figure 2. This state is dark for excitation with linearly polarized light, but breaking the D3h symmetry perturbs its dipole arrangement and provides a route for excitation. Let us discuss symmetry breaking introduced by opening the vertex angle of the cluster. The degree of the symmetry breaking is parametrized with φ, the vertex angle of the triangle: φ = π/3 for the equilateral triangle of symmetry group D3h, π/3 < φ < π for the case of the Λ-shaped cluster, corresponding to the group C2v, and φ = π for a linear chain of three particles, corresponding to the group D¥h. Measured plasmonic spectra, combined with theoretical analysis, allow us to follow in detail the evolution of the plasmon modes of the symmetry-broken clusters. As shown on Figure 1BD, an increase of φ beyond π/3 lifts the degeneracy of the plasmon resonances and results in new plasmon resonance bands, selectively excited by light polarized along the longitudinal and transverse directions of the cluster, respectively. The positions of the new bands as a function of φ, as obtained from experimental spectra, are shown in Figure 3A. The splitting of the two bands increases monotonically with the vertex angle. Figure 3B reproduces these results based on the numerical simulation. The ratio between the resonance intensities also depends on φ, as shown in Figure 3C,D; the intensity of the redshifted longitudinal resonance increases, whereas that of the blue-shifted transverse resonance decreases. To understand the origin of the experimentally observed trends, let us turn to an in-depth theoretical analysis based on numerical simulations and plasmon hybridization calculations. The symmetry group C2v lacks the symmetry operations constituting the rotational subgroup C3. Therefore, when φ gradually increases, the degeneracy of the E0 modes is lifted. The transverse and longitudinal in-plane E0 irreps transform to irreps A1 and B2 of C2v, respectively, which correspond to symmetric and antisymmetric modes, as shown on Figure 2. The structure of the new modes gradually diverges from that of the D3h modes as φ increases, a phenomenon which can be seen as a synchronous counter-clockwise and clockwise rotation of the individual dipoles on two of the particles. The orientation of the dipole on the vertex nanoparticle is not altered by the symmetry breaking. The total dipole of the A1 state is oriented along the transverse 2442
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Figure 3. Effect of gradual opening of the vertex angle of trimeric PMs on spectral features. (A) Resonance wavelengths of the A1 (blue) and B2 (red) bonding plasmon states, as extracted from experimental spectra similar to those of Figure 1. (B) Corresponding resonance wavelengths obtained from spectra simulated by the MMP method. (C) Ratios between the peak scattering intensities of the bonding A1 and lowest energy B2 plasmon modes in experimental spectra. (D) Corresponding ratios from simulated spectra.
direction of the cluster, while that of the B2 state is oriented along the longitudinal direction of the cluster. The lowest energy A0 2 ringlike mode also transforms to a mode of B2 symmetry. However, while the A0 2 mode is dark and not excited by linearly polarized light, the B2 mode originating from it has a nonzero dipole moment in the longitudinal direction. This is also the case for the other B2 mode, which originates from the degenerate E0 mode. Interestingly, as φ increases the dipole moment of the lowest energy B2 mode originating from A0 2 gradually increases, whereas the dipole moment of the B2 mode originating from E0 decreases. This gradual interchange of the total dipole moments of the B2 modes of the Λ-shaped cluster is reflected in a corresponding change of the scattering intensity of the two plasmon resonances; the intensity of the former resonance increases, while the intensity of the latter decreases. The energy diagram of the plasmon states obtained in the hybridization calculation for different values of φ is shown in Figure 4. Such a diagram is analogous to the Walsh diagrams for the energy states of bending triatomic molecules.49 It is worth noting that energy curves of the same shape as those on Figure 4 can be obtained from a simple model that takes into account only electrostatic interaction between point dipoles located at the center of the particles. The diagram shows that the energy of the B2 mode originating from E0 increases with φ, while the energy of the B2 mode originating from the A0 2 mode increases at the small values of φ up to 80 and decreases at higher values. In contrast to the electron tunneling effect occurring at a small distance between particles,28 which leads to a change in the direction of plasmon shift from blue to red as the gap between the particles increases, the change seen here is a purely classical effect of electrostatic origin, fully consistent with the energies of point dipoles of the corresponding configurations, and independent of the interparticle distances. It is evident from Figure 4 that the indicated change in plasmon shift direction of the bonding B2 modes and their coupling to the excitation light occurs within the small range 60 < φ < 80. As a consequence of such a “fast” evolution of the plasmon modes, the B2 mode originating from E0 cannot be observed in the spectrum beyond these values of φ and the only
Figure 4. Energy diagram of the plasmon modes of a trimer for different values of φ, as calculated from the plasmon hybridization theory. The symmetry symbols of the modes are shown. The dotted line corresponds to the plasmon resonance of a single particle. The energies of the antibonding modes (thin lines) are higher than the single-particle resonance, while the energies of the bonding modes (thick lines) are lower than that value.
bonding longitudinal resonance observed corresponds to B2 mode originated from A0 2. This is the reason that only a single red-shifting mode is seen in the experimental spectra, as can be observed Figure 3, in which the first experimental point above φ = 60 is at φ ∼ 80. The full development of the different bands can be seen in the numerical spectra shown in Figure 5A, which are in complete agreement with the plasmon-hybridization energy curves of Figure 4. The trends discussed above are emphasized by the dotted vertical lines connecting the resonance peaks in Figure 5A. As φ increases further, the cluster geometry finally attains the symmetry of the group D¥h, which corresponds to a linear chain of nanoparticles with φ = π. Here, the higher energy B2 mode of C2v, which at this point is already dark, transforms to the mode Πg of D¥h, which is oriented perpendicular to the chain axis 2443
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Figure 5. Effect of gradual opening of the vertex angle in trimeric PMs on the spectrum: numerical simulations with MMP. (A) Plasmon spectra excited with longitudinally polarized light (B2 modes are excited). (B) Spectra excited with transversely polarized light (A1 modes are excited). Specific values of φ are indicated near each spectrum.
(see Figure 2, right column). On the other hand, the lower energy B2 mode transforms into a bright mode of symmetry ∑þ u, corresponding to a parallel alignment of the individual dipoles. The MMP calculation shows that in the spectral region discussed here the solely populated mode of D¥h symmetry is indeed the ∑þ u mode, which is the only bright bonding mode for this symmetry group. The total dipole moment of the ∑þ u mode is larger than the dipole moment of the E0 modes of the equilateral triangle. Indeed, both the experimental results and numerical simulations show a gradual decrease in the plasmon energy and a simultaneous increase in the scattering cross-section for the corresponding plasmon resonance excited with longitudinal light polarization (see Figure 3 and Figure 5A). In the case of excitation with transverse polarization, opening of the vertex angle shifts the peak of the plasmon resonance to the blue. As φ becomes larger than π/3, the intensity of the A1 bonding mode gradually decreases, as seen from Figures 1B,C and 3C,D. The Πu bonding mode of D¥h, which correlates with this A1 mode (see Figure 2) is of higher energy than the corresponding bonding E0 mode of D3h and has zero total dipole moment. Indeed, as φ increases, the plasmon bonding resonance excited with transverse polarization shifts to the blue, and its intensity gradually decreases, completely vanishing at the linear chain limit. These experimental observations and plasmon hybridization theory results are in good agreement with MMP simulations shown in Figures 3 and 5B. The trimers shown in Figure 1B,C are examples of the intermediate stage in the gradual transition between the symmetry groups. When the value of φ is high as in Figure 1C, the intensity of the A1 state of C2v is weak but the corresponding peak is still distinguishable in the spectrum. According to the trend shown in Figures 3A,B and 5A, the position of this mode converges to 600 nm. The numerically calculated charge distribution for the excitation with transverse polarization, shown in the bottom panel of Figure 1C, demonstrates the presence of an additional plasmon mode at this wavelength. This is the A1 antibonding bright mode of the cluster, which correlates on one hand to the A0 1 mode of D3h and on the other hand to the Πu mode of D¥h group as shown on Figure 2. The peak maximum of
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this mode is located in the blue region of the spectrum at 450 nm, while its tail is still significant at 600 nm. This spectral overlap is the reason for the difference between the charge distribution shown in the bottom panel of Figure 1C for the A1 mode and that of Figure 1B, where the relative contribution of the antibonding mode is significantly smaller. For completeness, in Figure S1 of the Supporting Information we show simulated electric field plots corresponding to each charge plot of Figure 1. This data can be important, for example, in the context of surface-enhanced Raman spectroscopy, where the plasmon-induced enhancement of the electric field allows for single-molecule sensitivity. In conclusion, we have demonstrated the impact of the cluster symmetry on plasmon mode spectroscopy of trimeric plasmonic molecules. The experimental results show how symmetry breaking can be induced by gradual opening of the vertex angle. Selective excitation by polarized light of nondegenerate plasmon modes was demonstrated. The evolution of the plasmon modes across different symmetry groups was monitored through the scattering spectra of the symmetry-broken trimers, and was explained with the aid of group correlation tables and plasmon hybridization theory. Our findings are of high importance for both fundamental understanding of the interaction between light and plasmonic nanostructures, as well as for the design of new devices that will efficiently control the light on the nanoscale.
’ ASSOCIATED CONTENT
bS
Supporting Information. Simulated electric field plots corresponding to each charge plot of Figure 1. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: (L.C.)
[email protected]; (G.H.) gilad.haran@ weizmann.ac.il.
’ ACKNOWLEDGMENT The authors acknowledge valuable discussions with Professors Adi Stern (Weizmann Institute) and Peter Nordlander (Rice University). This research was made possible by a grant from the Israel Science Foundation (450/10). ’ REFERENCES (1) Kreibig, U. Optical properties of metal clusters; Springer-Verlag: Berlin Heidelberg, 1995. (2) Bohren, C. F.; Huffman, D. R. Absorption and scattering of light by small particles; Wiley: New York, 1998. (3) Maier, S. A., Plasmonics.: Springer Science: New York, 2007. (4) Gramotnev, D. K.; Bozhevolnyi, S. I. Nat. Photonics 2010, 4 (2), 83–91. (5) Haran, G. Acc. Chem. Res. 2010, 43 (8), 1135–1143. (6) Camden, J. P.; Dieringer, J. A.; Zhao, J.; Van Duyne, R. P. Acc. Chem. Res. 2008, 41 (12), 1653–1661. (7) Etchegoin, P. G.; Le Ru, E. C. Phys. Chem. Chem. Phys. 2008, 10 (40), 6079–89. (8) Jiang, J.; Bosnick, K.; Maillard, M.; Brus, L. J. Phys. Chem. B 2003, 107 (37), 9964–9972. (9) Lakowicz, J.; Ray, K.; Mustafa Chowdhury, M.; Szmacinski, H.; Fu, Y.; Zhang, J.; Nowaczyk, K. Analyst 2008, 133, 1308–46. 2444
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