## Abstract

We present a simple and intuitive picture, an electromagnetic analog of molecular orbital theory, that describes the plasmon response of complex nanostructures of arbitrary shape. Our model can be understood as the interaction or “hybridization” of elementary plasmons supported by nanostructures of elementary geometries. As an example, the approach is applied to the important case of a four-layer concentric nanoshell, where the hybridization of the plasmons of the inner and outer nanoshells determines the resonant frequencies of the multilayer nanostructure.

The fabrication of materials on a nanoscale can be used to enhance and exploit properties that become stronger under conditions of reduced dimensionality. In metallic systems, the conduction electron charge density and its corresponding electromagnetic field can undergo plasmon oscillations. Because of the nature of the optical constants for noble metals, the charge oscillations can propagate along the surface (rather than vanish evanescently) at optical frequencies. These surface plasmons can be excited by incident light in a process that depends on the dielectric constant of the material at the metal's surface, an effect that is exploited in surface plasmon resonance spectroscopy. In particles of dimensions on the order ofthe plasmon resonance wavelength, this surface plasmon dominates the electromagnetic response of the structure.

Recent advancements in the chemical synthesis of metal nanostructures have led to a proliferation of various shapes such as rods (*1*, *2*), shells (*3*–*5*), cups (*6*, *7*), rings (*8*), disks (*9*, *10*), and cubes. These developments, in addition to deep submicrometer lithographic methods for fabricating nanostructure grids and arrays, have provided the tools for realizing experimental studies of plasmon properties of metal nanostructures of arbitrary geometry. To fully exploit these new fabrication capabilities, accurate numerical methods (*11*–*13*) for calculating the electromagnetic properties of nanoscale structures are essentially defining the new field of “plasmonics,” providing an understanding of how to manipulate light at the nanometer scale with metal nanostructures as nano-optical components.

We now describe a way in which the plasmon response of metal-based nanostructures can be understood as the interaction or “hybridization” of plasmons supported by metallic nanostructures of more elementary shapes. The plasmon hybridization picture can be used to describe the sensitive structural tunability of the plasmon resonance frequency of the nanoshell geometry as the interaction between plasmons supported by a nanoscale sphere and cavity (*14*). This simple and intuitive picture can also be used to understand the plasmon resonance behavior of composite metallic nanostructures of greater geometrical complexity. The plasmon hybridization picture is important because it provides the nanoscientist with a powerful and general design principle that can be applied, both qualitatively and quantitatively, to guide the design of metallic nanostructures and predict their resonant properties.

A hollow metallic nanosphere, or nanoshell, supports plasmon resonances with frequencies that are a sensitive function of the inner and outer radius of the metallic shell (*15*). Recent experimental realization of this topology by a variety of methods has demonstrated the sensitive dependence of the plasmon resonance frequency on nanostructure geometry and embedding medium (*3*, *16*–*19*). Although this property can be calculated with electromagnetic theory, it has also recently been shown that ab initio electronic structure methods result in exact agreement with Mie scattering theory (*20*) for nanoshells in the dipole limit (*21*). This convergence between electronic and electromagnetic theory for predicting the plasmon response of metal nanostructures provides the rigorous foundation for the plasmon hybridization picture.

For the specific case of nanoshells, the highly geometry-dependent plasmon response can be seen as an interaction between the essentially fixed-frequency plasmon response of a nanosphere and that of a nanocavity (Fig. 1). The sphere and cavity plasmons are electromagnetic excitations that induce surface charges at the inner and outer interfaces of the metal shell. Because of the finite thickness of the shell layer, the sphere and cavity plasmons interact with each other. The strength of the interaction between the sphere and cavity plasmons is controlled by the thickness of the metal shell layer. This interaction results in the splitting of the plasmon resonances into two new resonances: the lower energy symmetric or “bonding” plasmon and the higher energy antisymmetric or “antibonding” plasmon (Fig. 1). To describe the geometry of a nanoshell, we adopt the notation (*a,b*) to indicate the inner radius *a* and the outer radius *b* of the shell.

The electron gas deformations can be decomposed as spherical harmonics of order *l*. We can show (supporting online text) that the interaction of the plasmons on the inner and outer surfaces of the shell gives rise to two hybridized plasmon modes |ω_{+}〉 and |ω_{–}〉 for each *l* > 0. The frequencies of these modes are (1) The |ω_{+}〉 mode corresponds to antisymmetric coupling between the sphere and cavity modes and the |ω_{–}〉 mode corresponds to symmetric coupling between the two modes (Fig. 1). The validity of this expression for the nanoshell plasmon energies has been explicitly verified with fully quantum mechanical calculations (*20*). The inclusion of a dielectric core and/or embedding medium in the formalism is straightforward. Although the resulting plasmon energies are the same as what would be obtained from a Drude dielectric function and classical Mie scattering in the dipole limit, the present treatment clearly elucidates the nature of the nanoshell plasmon resonances and, in particular, the microscopic origin of their tunability.

The plasmon hybridization picture can be used to understand the multifeatured plasmon response of more complex metallic nanostructures, such as dimers or other nanoparticle aggregates. The structure that we examine here is that of a concentric nanoshell, a four-layer nanoparticle consisting of a dielectric core, metal shell, dielectric spacer layer, and a second metallic shell. This “nano-matryushka” structure is shown in Fig. 2A. The plasmon response of this structure can be understood as an interaction and hybridization of the plasmons of the two individual metal shells (supporting online text). To illustrate this concept, we consider two concentric nanoshells with geometries (*a*_{1},*b*_{1}) and (*a*_{2},*b*_{2}). In this system, there are four linearly independent, incompressible charge deformations (plasmons). Their interaction results in four hybridized plasmon resonances, as schematically depicted in Fig. 2B. The thickness of the dielectric spacer layer |*a*_{2} – *b*_{1}| controls the strength of the coupling between the inner and the outer nanoshell. The resulting plasmon energy shifts will depend on the coupling strength and the energy between the plasmons on the inner and outer shell. The plasmon energies of the concentric nanoshell structure can therefore be tuned both by changing the dielectric spacer layer and by tuning the plasmon energies of the individual nanoshells.

Concentric nanoshells were synthesized on the basis of the gold nanoshell fabrication chemistry previously reported (Fig. 3) (*3*). Figure 4 shows the hybridized plasmon response of concentric nanoshells for the cases of strong coupling (Fig. 4A), weak coupling (Fig. 4B), and uncoupled plasmons (Fig. 4C). In this figure, the spectra denoted (1) show the experimental and theoretical extinction spectra for the isolated inner-shell plasmon |ω_{–,NS1}〉. The spectra denoted (2) are the theoretical extinction spectra of the isolated outer-shell plasmon |ω_{–,NS2}, calculated as though the inner-shell structure were replaced wholly by a dielectric (silica) core. The spectra denoted (3) are the experimental and theoretical extinction for the concentric nanoshell. In these spectra, the |ω^{+}_{–,CS}〉 and |ω^{–}_{–,CS}〉 plasmons are clearly apparent.

In Fig. 4A, the strong coupling case, the plasmon hybridization and splittings are quite strong because of the small (28 nm) interlayer spacing between inner– and outer–metal shell layers and because the inner |ω_{–,NS1}〉 and outer |ω_{–,NS2}〉 nanoshell plasmons are nearly resonant with each other. The hybridization of the plasmon appears to be strongly asymmetric; we attribute this asymmetry primarily to phase-retardation effects (*22*). A second contributing factor to the asymmetry is the small but finite interaction with the higher energy |ω_{+,NS1}〉 and |ω_{+,NS2}〉 plasmon modes. Figure 4B depicts a concentric nanoshell with a weak plasmon coupling between the inner and outer shell. In this case, the inner- and outer-plasmon resonances are detuned from each other in energy, and the spacing between inner and outer metal layers is slightly increased (39 nm). Because the hybridization is weak, the concentric shell plasmon modes show only small shifts relative to the isolated shell plasmons. In Fig. 4C, a concentric nanoshell with a fully decoupled plasmon response is shown. In this case, the intershell spacing is 236 nm, effectively isolating the inner-shell and outer-shell layers. The concentric nanoshell response (3) appears to be almost indistinguishable from the calculated nanoshell response for the outer shell. Because of the large intershell spacing and the finite penetration depth of the light, the inner-nanoshell plasmon is not excited.

We have shown that the plasmon response of metal-based nanostructures can be viewed as the collection of plasmons arising from simpler geometries to form an interacting system. The plasmonics of the metallic nanostructure is determined by the electromagnetic interaction between these “free” plasmons, which leads to mixing (hybridization), splittings, and shifts of the plasmon energies.