Plasmon Modes of Nanosphere Trimers and Quadrumers Daniel W. Brandl, † Nikolay A. Mirin, ‡ and Peter Nordlander* ,† Department of Physics and Astronomy M.S. 61, and Chemistry M.S. 60, and the Laboratory for Nanophotonics, LANP, Rice UniVersity, Houston, Texas 77251-1892 ReceiVed: March 3, 2006; In Final Form: April 18, 2006 Using the plasmon hybridization method, we investigate the plasmon frequencies and optical absorption spectra of symmetric configurations of nanosphere trimers and quadrumers. Plasmon hybridization allows us to express the fundamental plasmon modes of these multinanosphere systems as linear combinations of the plasmons of individual nanospheres in a manner analogous to molecular orbital theory. We show how group theory may be used to interpret the plasmon modes of each multiparticle system as specific structure-dependent symmetric combinations of the plasmon modes of the individual nanoparticles. We compare the optical absorption spectra calculated using plasmon hybridization with the spectra obtained using finite difference time domain simulations. I. Introduction The discovery of surface-enhanced Raman spectroscopy (SERS) has led to a number of studies of the plasmonic properties of nanoparticles. 1-7 This interest is due in part to large local electric field enhancements that are induced near nanoparticle surfaces upon excitation of their plasmons. SERS cross sections can depend of the fourth power of the electric field across a target molecule when the wavelengths of the incident and scattered photons are near the plasmon resonance of a substrate. Thus, high field enhancements near nanoparticles can potentially provide for greatly increased SERS efficiency, as has been demonstrated recently for dimers. 8-10 The electric-field enhancements induced by surface plasmons depend greatly on the shape of the nanoparticle. 11 Field enhancements of single dipolar particles are typically of an order of magnitude, whereas those in the junctions of nanoparticle aggregates can be as large as 3 orders of magnitude. 12 In the systems that will be investigated in the present study, the maximum electric field enhancements are at least 2 orders of magnitude for both the symmetric trimer and symmetric quadrumer. In contrast, an individual nanosphere has a maxi- mum electric field enhancement of only around 20. The collective effects associated with the interactions of plasmons in different nanoparticles thus lead to very strong enhancements. Because of these large field enhancements, nanoparticle ag- gregates are of particular importance as substrates in SERS. Despite the existence of large electric field enhancements and their relevance to SERS, there have been few theoretical studies on the nature of the plasmon modes in multi-particle systems. 13-18 In a recent paper, we extended the plasmon hybridization method to treat the plasmonic properties of nanoparticle dimers including the effects of dielectric backgrounds. 19 In that ap- plication, we focused on symmetric nanosphere and nanoshell dimers and found excellent agreement with results from other theoretical methods. The present paper extends the formalism developed previously for nanoshell dimers 19 to arbitrary structures composed of a finite number of nanoparticles. We apply this formalism to study the plasmons of an equilateral nanoparticle trimer and an equilateral nanoparticle quadrumer. The results show that as the nanopar- ticle separation decreases, the plasmons of individual nanopar- ticles begin to interact and hybridize with the plasmons of the other particles. These interactions are not diagonal in the multipolar order of the individual nanoparticle plasmons, and for small interparticle separation, all plasmon modes of the combined system contain finite amounts of all multipolar order plasmons of the individual nanoparticles. This hybridization introduces a finite dipolar component into many of the plasmon modes of the combined system, making them dipole active and visible by light in the small-particle limit. This hybridization behavior is reminiscent of the situation for nanoparticle dimers, but the complexity of the hybridization is much more involved in the multinanoparticle case. We show how group theory can be used to classify the symmetry of each plasmon mode of the interacting system and help identify the modes with large dipole moments and large electromagnetic field enhancements. The role of symmetry in the present plasmonic problem is entirely analogous to its role in molecular orbital theory and represents a significant simplification in the analysis and understanding of extinction spectra. In section II, we extend the plasmon hybridization formalism to take into account more than two nanoparticles that do not necessarily lie on the same axis. In section III, we will discuss the use of group theory to predict the linear combinations of individual nanoparticle plasmons that compose a plasmon of a symmetric nanoparticle aggregate. In section IV, we will show the results of plasmon hybridization for a symmetric nanosphere trimer and quadrumer configurations. In this section, we will use group theory to classify the plasmon modes using the irreducible representations of point groups. Finally, section V contains a comparison of optical absorption spectra calculated by the plasmon hybridization and finite difference time domain (FDTD) methods, electric field enhancements of each system calculated using FDTD, and a discussion of the effect of symmetry breaking on the plasmon resonances. * To whom correspondence should be addressed. Phone: (713)348-5171. Fax: (713)348-4150. E-mail: nordland@rice.edu. † Department of Physics and Astronomy. ‡ Department of Chemistry. 12302 J. Phys. Chem. B 2006, 110, 12302-12310 10.1021/jp0613485 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/07/2006