Abstract
We present a review of plasmonic nanostructures in which the constituent materials are coupled together by optical interactions. The review first provides a comprehensive coverage of theoretical framework where the optical interactions are described by the multiple scattering among the nanostructures. We then discuss the two limiting cases of weak and strong interactions. The weak interaction regime is described well by the effective medium theory while the strong interaction regime requires theoretical tools that can describe the new eigenmodes delocalized over the entire system. Weakly interacting plasmonic nanostructures have been studied extensively in the metamaterials research, which has been a major research thrust in photonics during the past decade. This review covers some of the latest examples exhibiting perfect absorption and invisibility. Strongly coupled systems started to receive attention recently. As a representative example, plasmonic molecules exhibiting Fano resonance are discussed in detail. Plasmonic nanostructures offer an excellent platform to engineer nanoscale optical fields. With the recent progress in nanofabrication technologies, plasmonic nanostructures offer a highly promising pathway to discovering new phenomena and developing novel optical devices.
Introduction
In the Drude theory, metal is modeled as a system of free electron gas undergoing forced oscillation with some damping. Other than some notable exceptions such as the strong interband absorption involving the d band in transition metals, the Drude model generally provides good descriptions of the optical properties of metals. It predicts that in the low frequency region the induced polarization oscillates 180° outofphase with the external field, resulting in negative permittivity and consequently high reflectivity. At certain frequencies, the electron gas can also undergo collective oscillations whose quantum is called plasmon. The collective oscillation may involve volume charge density (volume plasmon) or surface charge density (surface plasmon). Volume plasmons are longitudinal oscillations and cannot be excited by electromagnetic waves which are normally transverse. Surface plasmons, however, interact strongly with light. By storing a significant portion of its energy in oscillating electron gas, surface plasmon offers an effective means to localize light down to length scales much smaller than wavelength. For this reason, the past decades have seen an explosive growth in research on surface plasmon nanophotonics.
One of the topics of high interest is nanoscale waveguiding with potential applications in integrated photonics for which a large body of literature exists [1]. Another important class of plasmonic nanostructures more relevant to this review are those supporting localized surface plasmon (LSP) modes. These structures include metal nanoparticles of various shapes, nanoholes in metal films, and some variations of these. The LSP resonances typically manifest themselves with strong scattering and absorption. The characteristic color arising from the LSP resonances has enabled a wide range of applications from the medieval stained glasses to modern optical sensors. For example, colorimetric sensing of biomolecules and ions has been demonstrated [2,3]. Also, strongly scattering nanoparticles have been used to enhance contrast in darkfield microscopy and optical coherence tomography [4] while thermal ablation sing strongly absorbing nanoparticles is being actively developed for targeted therapies of various diseases [5,6].
LSP naturally provides a highly localized and intense optical field which consequently enhances a variety of optical processes. The bestknown example is surface enhanced Raman scattering (SERS). Using a rough silver surface, Raman scattering by a single molecule has been observed with enhancements up to a factor of 10^{14}[7]. Much of the enhancement is believed to arise from the local field enhancement due to the highly localized hot spots produced by the silver nanostructure [8]. Surface plasmon resonance can also be used to enhance luminescence [9] and Förster energy transfer process [10,11]. Since the optical processes depend on the local field strength, strong emphasis has been placed on achieving higher local field enhancement using novel nanostructure geometries. These structures are generally termed nanoantennas analogous to conventional antennas in the microwave frequency region and have been the subject of extensive research in the past decade [12]. Since hot spots are typically formed in the nanogap between plasmonic nanoparticles, the nanoantenna research naturally directed attention to the interaction between LSP modes, which is the main subject of this review. An important recent development in this field is the concept of plasmon hybridization [13]. In this picture, the individual LSP modes in closely spaced nanoparticles hybridize to form a set of new modes delocalized over the entire structure, in much the same way atomic orbitals hybridize to form molecular orbitals in real molecules. For this reason, these plasmonic nanostructures are also called plasmonic molecules. The hybridized modes in plasmonic molecules have been extensively studied. The simplest form of plasmonic molecule is a dimer or a pair of nanoparticles in which the plasmon resonance of single nanoparticle hybridizes to produce symmetric and antisymmetric dimer plasmons [14]. Hybridization in more complex structures has also been studied, e.g. trimer [15], quadrumer [16], hexamer [17] and most widely heptamer [1822]. In these complex structures, symmetry properties of the hybridized plasmon modes should be described by the group representation theory [23]. In this formalism, how each plasmon mode transforms under the symmetry operations is described by the irreducible representations of the symmetry group. This allows for a systematic description of modes with complex field profiles and also their interaction with incident optical fields, providing a complete picture when combined with the numerical techniques that can precisely calculate the hybridized LSP modes. It is noted that the group theoretical approach classifying the symmetry properties of plasmon modes does not require any approximations. For example, in nanostructures with features sizes much smaller than the wavelength, the plasmon modes exhibit predominantly electric dipole nature and are often approximated as point dipoles. The group theoretical approach, however, does not require dipole approximation and is equally applicable to larger and more complex structures exhibiting higher order mulitpole components.
The strength and nature of the optical interaction between plasmonic nanostructures varies with the distance. For long distances, the interaction is mainly radiative and involves propagating electromagnetic waves undergoing multiple scattering among the plasmonic nanostructures. In this case, the original character of the individual LSP modes should largely be preserved. For short distances, evanescently decaying nearfield components should be taken into account. In principle, the nearfield interaction can be treated the same way as the farfield interaction in the general framework of multiple scattering theory. That is, the evanescent modes undergo multiple scattering between nanostructures as well as the propagating modes. In extreme short distances, where the individual LSP modes have significant overlaps with one another, the nearfield interaction will dominate and hybridization resulting in fully delocalized modes would occur. This review is aimed at providing a unified view of plasmonic nanostructures in which the strength of the interaction among individual metallic components is varied. We will first present a theoretical framework followed by examples of nanostructures and their optical properties in the weak and strong coupling regimes.
Review
Theoretical framework
Classical description of surface plasmon resonance
Surface plasmon, while inherently a quantum mechanical entity, is generally well described by the classical theory where surface plasmon presents itself as a surface resonance. There are of course quantum mechanical characteristics that cannot be fully described by the classical theory such as nonlocal response of electrons [24,25], tunneling and screening effect [26], spontaneous and stimulated emission of photons [27] and plasmons [28,29], to name a few. Nevertheless, the classical theory has proved sufficient for a wide range of systems. The classical theory is based on the Maxwell’s equations where the boundary conditions are set up by the material properties specified by the macroscopic permittivity and permeability. For the planar geometry, surface wave solutions with fields decaying exponentially away from the interfaces can be found. In the simplest example of a single planar interface between semiinfinite metal and dielectric, the surface wave vector, q, is given by the dispersion relation [30],
Multiple scattering theory
We now consider a cluster of nanoparticles or nanostructures which individually support LSP resonances. The key additional consideration needed to properly describe the cluster is the multiple scattering between the individual components. Since analytical solutions are available for spheres, it is instructive to consider a cluster of nanospheres. In this case, the scattering by the individual spheres can be described by the standard Mie theory. The fundamental difference from the single sphere case is that for each sphere in the cluster the incident field is the combination of the actual incident field impinging on the cluster and the scattered fields by all other spheres in the cluster [3638]. To briefly provide an overview of this process, we first remind the reader of the standard Mie theory for a single sphere where all fields are described in the coordinate system with the sphere located at the origin. The incident and scattered fields are then expressed in terms of the vector spherical harmonics,
Once the scattering coefficients, and, are found from the above equations, then the total scattered field by the entire cluster can be obtained by using the addition theorem one more time to express all fields in the common coordinate system of j_{0}th sphere,
The scattering cross section can be written in a similar fashion to the single sphere Mie scattering case,
A similar multiple scattering theory has also been developed for an infinitely periodic array of spheres [4042]. In this case, since the system has infinite extent, the main concern is the propagation of light inside the periodic array. Due to the similarity to the KohnKorringaRostocker (KKR) method widely used for electronic energy band structure studies, this technique is sometimes called the vector KKR method. Once again, vector spherical harmonics are used to take advantage of the spherical geometry. Following the original development by Stefanou et al. [40,43], we begin with the general expression for spherical wave expansion given in equation (6) for a twodimensional (2D) periodic array of spheres. When a plane wave is incident on the 2D array, the scattered wave must be expressed as a sum of the waves scattered by all spheres in the 2D plane. Due to the periodicity, the wave scattered by a sphere at r = R_{n} differs only by a simple phase factor, exp(ik_{} ⋅ R_{n}), from the wave scattered by a sphere at the origin. Here, k_{} represents the tangential component of kvector parallel to the plane of spheres. The scattered field can then be written as,
Here k_{} represents the reduced wave vector defined as k_{tangential} = k_{} + g, where g is the reciprocal lattice vectors of the 2D lattice, so that k_{} now represents a vector within the unit cell of the 2D reciprocal lattice or the surface Brillouin zone. The sign in the superscript indicates forward and backward propagating waves in the direction perpendicular to the 2D plane. Now we construct the same plane wave expansion for the incident wave as well and obtain a matrix equation relating the coefficients of the incident wave and the total scattered wave,
Here the matrices Q are the scattering matrices defined as
Effective medium theory
Now we explore the limiting cases where the multiple scattering theory presented above can either be simplified significantly or is inadequate. An obvious possibility of simplification is found when the interaction among the scatterers is weak. In this case, the multiple scattering effect will be small and may be ignored in the extreme case. A commonly employed approximation for such a case is the effective medium theory which attempts to obtain an average response of a nonuniform medium composed of multiple constituents. The most widely used effective medium theory is the Maxwell Garnett theory [46], which is derived from the Lorentz local field relation covered in many textbooks. Briefly, Lorentz considered an imaginary spherical cavity in a dielectric medium and stated that the local field is simply given by the sum of the dielectric response of the medium and the additional fields due to the surface charge on the spherical cavity and due to the atoms inside the cavity. The lattermost contribution cancels out to zero in a highly symmetric crystal structure. This leads to the famous expression,
We now make a connection with the more rigorous multiple scattering theory presented earlier. In the Mie theory for a single spherical scatterer, the scattering coefficients a_{n} and b_{n} are generally expressed as infinite series of spherical Hankel functions. Using the power series expansions of the spherical Hankel functions, it can be shown that the leading terms of these coefficients are a_{1} ~ x^{3}, b_{1} ~ x^{5}, a_{2} ~ x^{5}, b_{2} ~ x^{7}, etc., where x = 2πa/λ. Therefore, for small spheres, we may retain only the leading term in a_{1}. Since the scattering and absorption cross sections are proportional to a_{1}^{2} and Im(a_{1}) (imaginary part of a_{1}), respectively, the scattering and absorption efficiencies defined as the ratio of scattering and absorption cross sections to the physical cross section depend on x^{4} and x, respectively, yielding the wellknown Rayleigh scattering results. More specifically, the leading term of coefficient a_{1} is
The relationship between the a_{1} coefficient and the polarizability is apparent,
This relationship reveals one of the key approximations involved in the Maxwell Garnet effective medium theory: retaining only the leading term of the leading coefficient a_{1} in the general solution of the Mie theory. This is essentially the electrostatic approximation in which the polarizability of small spheres is modeled by the static polarizability given in equation (15). A modest improvement on this approximation may be achieved by the extended Maxwell Garnett theory in which the full expansion of a_{1} is used to calculate the polarizability [47]. The extended Maxwell Garnett theory has been used successfully for metamaterials [48,49]. However, if higher order terms in coefficient a_{1} are to be included, it is logical to include b_{1} and a_{2} terms as well. These terms represent the contributions of the magnetic dipole and electric quadrupole terms, respectively. The inclusion of b_{1} can be done straightforwardly by defining effective permeability in exactly the same way effective permittivity is calculated except that a_{1} is replaced by b_{1} to define magnetic polarizability. It has been shown that the optical properties of small particle composites may exhibit magnetic permeability even when the constituent materials are all nonmagnetic [50]. Thus, the use of extended Maxwell Garnett theory is more appropriate when both effective permittivity and permeability are considered. It has also been attempted to improve the Maxwell Garnett theory using dressed polarizability to better account for the resonant effect [51]. In addition to neglecting higher multipole contributions to the light scattering and absorption, which is appropriate for small particles, the Maxwell Garnett effective medium theory also ignores multiple scattering among particles. This can be seen from the facts that the polarizability is defined by the single sphere Mie coefficient and that the effective permittivity is constructed based on the Lorentz local field concept. There are other issues related to the Maxwell Garnett effective medium theory such as the Lorentz catastrophe and nonsymmetry and there exist other effective medium theories that improve on those shortcomings [52]. However, it can be said in general that the effective medium theory is appropriate for small particles with low volume fraction where both the higher order multipole contributions and interparticle interaction by multiple scattering can be ignored.
Plasmon hybridization
We now us consider the opposite extreme where the particles are closely spaced and thus strongly interacting with one another. In this case, especially near plasmon resonances, strongly overlapping evanescent fields exist, leading to significant modification in the nature of the resonance. In principle, this nearfield interaction can be described by the multiple scattering theory presented earlier. However, there is a practical difficulty in that it requires the inclusion of a large number partial waves and thus makes the numerical computation extremely challenging. Instead it is more profitable to look for new eigenmode solutions described by the collective modes delocalized over the entire structure. Prodan et al. proposed the plasmon hybridization model [13]. To be sure, the concept of plasmon hybridization itself is not new. A classic example is coupling of two surface plasmon modes in a metal thin film and the resultant formation of symmetric and antisymmetric modes [53,54]. But Prodan’s model is highly useful as it provides a complete set of eigenmode solutions in complex nanostructures. The plasmon hybridization model considers the electron gas as incompressible fluid subject to Coulombic interaction. The plasmon resonance is then represented by selfsustained oscillations of the electron fluid which may be obtained from the EulerLagrange equation for the Lagrangian,
For an arbitrary geometry, the boundary integral equation (20) can be solved numerically to obtain the eigenvalue γ and eigenfunction σ(r). The eigenvalue γ is related to the permittivity ε(ω) of the metal as
It should be noted that both of the two techniques described above employ electrostatic approximation and thus cannot describe the retardation effect. Their applicability is therefore limited to extremely small particles or low frequencies. Nevertheless, these methods proved highly useful in the exploration of coupled plasmon modes in complex nanostructures consisting of multiple nanoparticles. The main reason is the applicability of elegant group theoretical method in a fashion similar to the molecular orbital theory. Just as the molecular orbital can be constructed from the symmetry adapted linear combination of atomic orbitals, the hybridized plasmon modes can be constructed from the symmetry adapted linear combination of individual nanoparticle plasmon modes. This enables a highly effective and systematic exploration of plasmon modes in a complex, strongly coupled nanostructure. Here we briefly describe this approach using the boundary integral method for a collection of nanoparticles. First, the plasmon modes of an isolated nanoparticle can be found from the eigenvalue equation given in (20). For the collection of N nanoparticles, the surface charge distribution can be expressed as a linear combination of the individual nanoparticle eigenmodes,
Using equation (20), we may write
The total surface charge distribution in equation (22) must satisfy the boundary integral eigenvalue equation in (20) where the integral is conducted over the surfaces of all nanoparticles. However,
the problem now has been transformed to a matrix eigenvalue equation given in (24). Physically, the matrix K represents the interaction between the surface dipole density of the mth mode of
Weakly coupled plasmonic nanostructures
As discussed in the previous section, when the plasmonic nanostructures are only weakly coupled, they may be described well by the effective medium theory. There is a large body of literature on metamaterials which are composed of deep subwavelength scale structures and are generally described well by the effective medium theory. The initial development of metamaterials was driven largely by the negative index materials for which excellent reviews are available [58,59]. In this section, we focus on the latest development of plasmonic nanostructures in the weak coupling regime.
Perfect absorber
An interesting application of plasmonic nanostructures is the perfect absorber which absorbs all incident light with minimal reflection and transmission. Such a material would find a wide range of applications in, for example, coatings for optical instruments, sensors, and photothermal or photovoltaic energy conversion devices. An inspection of Fresnel’s coefficients immediately reveals that perfect absorption is achieved when the real part of the refractive index is matched with the incident medium and the imaginary part is small. This condition would result in negligibly small reflection and, if the material is thick enough, vanishing transmission as well. This is, in fact, how perfectly absorbing metamaterial was obtained using aligned carbon nanotube arrays [60]. This example illustrates clearly that a perfect absorber does not have to involve a plasmonic nanostructure [60,61]. It is, however, of great technological importance to achieve perfect absorption in thin films and plasmonic nanostructures are ideally suited for that purpose.
One of the simplest examples is a line grating. It is well known that a line grating on a metal surface can couple the free space light into surface plasmons. An ideal grating presents infinitesimal modulation of surface so that the plasmon modes of the original flat surface are well preserved and the grating only provides the extra momentum for coupling. However, it was found that, even in a grating as shallow as 20 nm, additional surface plasmon modes strongly localized at the grating lines exist and interact with the surface plasmonpolariton modes. Lu et al. investigated silver nanograting structures fabricated on the backelectrode of a PCDTBT:PCBM organic photovoltaic device [62]. The system supports two surface plasmon modes when excited by transversemagnetic (TM, electric field perpendicular to the grating lines) polarized incident light. One is the localized surface plasmon (LSP) mode whose field profile is strongly localized at the grating lines and wavelength insensitive to the incident angle and active layer thickness. The other is the surface plasmonpolariton (SPP) mode strongly interacting with the grating period and thus highly sensitive to incident angle and active layer thickness. Figure 1(a) shows the wavelength of surface plasmon induced absorption peaks as a function of active layer thickness, revealing two branches with distinct dependence on active layer thickness. In general, as the active layer thickness is increased, the refractive index of the dielectric side of the surface is increased, resulting in a red shift in surface plasmon wavelength. The surface plasmon dispersion relation for a dielectric loaded metal surface can be calculated numerically [34]. As shown by the light blue crosses in Figure 1(a), the surface plasmon wavelength exhibits a red shift with increasing active layer thickness and this line coincides well with the SPP branch of the silver nanograting overcoated with PCDTBT:PCBM active layer. This indicates that the 20 nm tall silver nanograting acts as an ideal grating for the SPP modes which retain the characteristics of those in the flat dielectric loaded metal surface. The other branch representing LSP modes is unique to the grating structure and not observed in flat interfaces. Due to the strongly localized nature, it is insensitive to the thickness of the dielectric layer. For thicknesses near 30 nm, the two branches exhibit anticrossing, indicative of strong mixing between the LSP and SPP modes. The field profiles in Figure 1(b) and (c) show the two contrasting patterns for LSP and SPP modes while those in Figure 1(d) and (e) reveal considerable mixing. Since these modes strongly affect the absorption spectra, the knowledge of the existence of two different modes and their interaction is essential to designing light trapping plasmonic grating structures for photovoltaic devices.
Figure 1. Interaction of surface plasmon.(a) Surface plasmon peak positions as a function of PCDTBT:PCBM active layer thickness exhibiting anticrossing behavior. Blue and red solid dots represent LSP and SPP peak positions for various PCDTBT:PCBM thicknesses. Blue crosses represent the SPP peak positions calculated from the effective index of the SPP mode supported by the flat silverPCDTBT:PCBMair multilayer structure. (b, c) Electric field profiles for the surface plasmon modes in the 80 nm thick PCDTBT:PCBM on silver grating and (d, e) electric field profile for the surface plasmon modes in the 30 nm thick PCDTBT:PCBM on silver grating. To clarify the structure the bottom grating is the silver nanograting fabricated by nanoimprint lithography. The top surface is also corrugated simply by the result of depositing on a corrugated surface (From Ref. [62]).
Interaction of surface plasmon.(a) Surface plasmon peak positions as a function of PCDTBT:PCBM active layer thickness exhibiting anticrossing behavior. Blue and red solid dots represent LSP and SPP peak positions for various PCDTBT:PCBM thicknesses. Blue crosses represent the SPP peak positions calculated from the effective index of the SPP mode supported by the flat silverPCDTBT:PCBMair multilayer structure. (b, c) Electric field profiles for the surface plasmon modes in the 80 nm thick PCDTBT:PCBM on silver grating and (d, e) electric field profile for the surface plasmon modes in the 30 nm thick PCDTBT:PCBM on silver grating. To clarify the structure the bottom grating is the silver nanograting fabricated by nanoimprint lithography. The top surface is also corrugated simply by the result of depositing on a corrugated surface (From Ref. [62]).
When the grating height is increased, the surface plasmons lose their original characteristics of flat interfaces and the LSP modes become the dominant features. Kravets et al. studied a gold grating with 130 nm height fabricated by depositing gold on a 130 nm thick poly(methyl methacrylate) (PMMA) grating [63]. As shown in Figure 2, this structure exhibited extremely low reflectance and transmittance for TM polarized light, achieving blackbodylike absorption in the visible spectrum with a thickness much smaller than the wavelengths. A theoretical analysis is needed to understand this remarkable behavior. A rigorous theoretical treatment would require full consideration of two sets of gold gratings, one formed in between the PMMA lines and another formed on top of PMMA lines. However, since the grating lines are well separated (by more than 100 nm), nearfield coupling between adjacent grating lines should be small and therefore it should be possible to use a much simpler effective medium theory. To generalize the Maxwell Garnett results for spherical particles given in equations (14) ~ (16) to nonspherical shapes, we introduce depolarization factor, L. For a general ellipsoidal shape, L is defined as [64]
Figure 2. Reflectance and transmittance spectra of the gold nanostripe array with period = 320 nm and linewidth = 150 nm. Insets show the polarizationcontrast optical micros copy images (From Ref. [63]).
Reflectance and transmittance spectra of the gold nanostripe array with period = 320 nm and linewidth = 150 nm. Insets show the polarizationcontrast optical microscopy images. The inset in (b) shows reflectance spectra for s and ppolarizations at normal incidence (From Ref. [63]).
A question of both scientific and technological interest is how much absorption is
possible in ultrathin films. A simple solution may be found by using total internal
reflection. Using the Kretschmann geometry, an ultrathin film absorber can be excited
at glancing angle, effectively extending the optical path to infinity and consequently
achieving nearperfect absorption. Using this method, 94% absorption in a 4.5 nm thick
NbN nanowire array has been demonstrated [67]. It should be noted that this method does not require surface plasmon or any other
types of surface resonances, although they could certainly help. While this approach
clearly offers a quick solution to achieve perfect absorption, it is not always possible
or feasible to use total internal reflection geometry and it would be much more desirable
to achieve high absorption at normal incidence or over a broad range of incident angles.
To tackle this problem, let us consider a thin film with thickness d and refractive index m = n + i
Figure 3. Absorptance of thin film with complex refractive index.(a) Thin homogeneous film with complex refractive index m = n + iκ and thickness d is embedded between two semiinfinite dielectrics of refractive indices n_{i} and n_{s}, respectively. In (be), absorptance in the film is shown as a function of its optical constants for normal incident light. The wavelength is 900 nm, and the dielectric environment has n_{i} = n_{s} = 1. The film thicknesses d are 2, 10, 50, and 250 nm in panels b, c, d, and e, respectively. The dotted lines mark n = κ about which the shown absorption is nearly symmetric for the two thinnest films (b,c). In these cases, the absorptance reaches a peak value of close to 50% on the symmetry line. For the two thicker films (d,e), the peak height increases and additional maxima appear along the naxis due to FabryPerot modes. (From Ref. [68])
Absorptance of thin film with complex refractive index.(a) Thin homogeneous film with complex refractive index m = n + iκ and thickness d is embedded between two semiinfinite dielectrics of refractive indices n_{i} and n_{s}, respectively. In (be), absorptance in the film is shown as a function of its optical constants for normal incident light. The wavelength is 900 nm, and the dielectric environment has n_{i} = n_{s} = 1. The film thicknesses d are 2, 10, 50, and 250 nm in panels b, c, d, and e, respectively. The dotted lines mark n = κ about which the shown absorption is nearly symmetric for the two thinnest films (b, c). In these cases, the absorptance reaches a peak value of close to 50% on the symmetry line. For the two thicker films (d, e), the peak height increases and additional maxima appear along the naxis due to FabryPerot modes (From Ref. [68]).
Also,
The maximum absorptance is given as
Equations (31) and (32) show that the maximum absorption occurs at n =
It is obvious that no natural materials would exhibit the optimum refractive index given in equation (31). It is therefore necessary to create a composite medium whose effective refractive index is close to the optimum. It was shown that 50% absorptance could be realized over a broad range of wavelengths using an array of ellipsoidal silver shells with various geometry [68]. This scheme can be extended to a thin film with a back reflector. The reflector automatically eliminates transmission and the problem is reduced to minimizing reflectance. A general system to consider is an ultrathin film with a complex refractive index m and thickness d, separated from a perfect reflector by a spacer layer with refractive index n_{s} and thickness h. The incident medium index is n_{i}. The reflection coefficient is then given as
The critical wavelength λ_{c} is defined as the wavelength satisfying the quarter wavelength condition, λ_{c} = 4n_{s}h Equation (33) has the same form as equation (30) and thus
When λ = λ_{c},
Figure 4. Absorptance of gold nanoparticle embedded thin films.(a) Finite element method (FEM) calculated absorption rates, determined for the sample with the highest absorbing SnS_{x} coated Au array at its peak wavelength. A plane wave with incident electric field strength E_{0} propagates along the normal (x) direction, with its electric field vector along z. The rectangular unit cell used for the hexagonal lattice is shown below the plot (dashed lines). (b) Analogous for ZnO coated sample. (c,d) Sample reflectance (R), array absorptance (A), and absorptance in the Al reflector (AAl, including intermix layer), as calculated by FEM and obtained from the spectroscopic ellipsometry (SE) model. (e) FEM calculated absorptance in the Au dots of the SnS_{x} coated sample, in the SnS_{x} coating, and in the SiO_{2} spacer, respectively. (f) Analogous for the ZnO coated sample. (From Ref. [70])
Absorptance of gold nanoparticle embedded thin films.(a) Finite element method (FEM) calculated absorption rates, determined for the sample with the highest absorbing SnS_{x} coated Au array at its peak wavelength. A plane wave with incident electric field strength E_{0} propagates along the normal (x) direction, with its electric field vector along z. The rectangular unit cell used for the hexagonal lattice is shown below the plot (dashed lines). (b) Analogous for ZnO coated sample. (c, d) Sample reflectance (R), array absorptance (A), and absorptance in the Al reflector (AAl, including intermix layer), as calculated by FEM and obtained from the spectroscopic ellipsometry (SE) model. (e) FEM calculated absorptance in the Au dots of the SnS_{x} coated sample, in the SnS_{x} coating, and in the SiO_{2} spacer, respectively. (f) Analogous for the ZnO coated sample (From Ref. [70]).
Invisibility
In addition to enhancing absorption, controlling or suppressing light scattering is also of great interest for many applications. It has been shown that the dominant dipolar contribution to the scattering from a spherical or cylindrical dielectric object could be greatly reduced when coated by a layer of material with negative or very low values of the real part of permittivity [72]. This technique is inherently nonresonant and thus the bandwidth of the scattering cancellation can be quite broad and limited only by the material dispersion of the coating. It was shown that the operational bandwidth could be further extended by use of multiple layers of coatings [73]. Potential applications of this type of cloaking include cloaked sensors [74], cloaked nearfield probes [75,76] and reduction of optical forces [77]. For implementation, a natural choice for the coating material is a metal, which exhibits negative permittivity values at frequencies below its plasma frequency. However, in the optical frequency region, most metals tend to exhibit large negative permittivity values. Consequently, an unrealistically thin layer is required to cancel the dipole moment of a dielectric object whose permittivity is typically a small positive number. To circumvent this issue, an effective medium formed by plasmonic nanostructure is needed. Recently, Tamma et al. reported scattering cancellation by a gold nanograting fabricated on the sidewall of a silicon nanorod [78]. As shown in Figure 5(a), the cloaked object was a silicon nanorod fabricated by electronbeam lithography. The gold nanograting was then fabricated on the sidewall by a combination of gold evaporation, focused ion beam milling and silica protective coating deposition by sputtering. The final structure was coated with SU8 to form a waveguide layer and the structure was excited by 1550 nm light coupled into the SU8 guiding layer. The scattering properties were then investigated by nearfield scanning optical microscopy (NSOM). The waveguide geometry used in this study makes it impossible to directly measure the scattered light intensity. Instead, NSOM was used to map the interference fringes produced by the incident light and backscattered light (Figure 5b). It was clear that the cloaked silicon nanorod scattered much less because the interference fringes were weak and required a long integration time and high gain to obtain reliable data whereas the reference nanorod showed a strong interference pattern. Tamma et al. then used the effective medium theory to relate the effective index of the coating with the fringe curvature. Here the gold nanograting coating was described as a multilayer structure for which the effective medium theory simple yields [79]
Figure 5. Scattering cancellation by plasmonic nanostructure.(a) Scanning electron micrograph (SEM) of the silicon nanorod with gold nanograting fabricated on the side wall. (b) Nearfield scanning optical micrograph (NSOM) of scattered light by the cloaked silicon nanorod measured with λ = 1550 nm. The fringe curvature contains the information on scattering cross section. (c) Scattering cross section extracted from the analysis of fringe curvatures. A reduction of 9.8 dB was achieved (From Ref. [78])
Scattering cancellation by plasmonic nanostructure.(a) Scanning electron micrograph (SEM) of the silicon nanorod with gold nanograting fabricated on the side wall. (b) Nearfield scanning optical micrograph (NSOM) of scattered light by the cloaked silicon nanorod measured with λ = 1550 nm. The fringe curvature contains the information on scattering cross section. (c) Scattering cross section extracted from the analysis of fringe curvatures. A reduction of 9.8 dB was achieved (From Ref. [78]).
In concluding the discussion on weakly coupled system that can be described well by the effective medium theory, it is worth noting the robustness of effective medium theory. As discussed in the previous section, the effective medium theory is valid only when the feature size is small compared to the wavelength and the interaction among the nanostructures is weak. When these two conditions are met, the optical properties of the composite structure are determined by the dipole moment of the individual nanostructure without having to include higher order resonances and any higher order interaction. These conditions certainly sound very restrictive but in reality the effective medium theory has been successfully applied to a wide variety of systems which at first glace do not seem to satisfy the conditions for validity. An excellent example is the silicon nanorod cloaked by an array of gold nanowires described above. The individual gold nanowire was 13 nm thick and 20 nm wide and the spacing between two adjacent gold nanowires was 63 nm. This system was modeled as a multilayer composed of 20 nm thick gold and 43 nm thick silica films. Despite this seemingly gross simplification, the effective medium theory provided a remarkably accurate result in good agreement with experiments. It can be argued that the effective medium theory, although oversimplified, has captured the essential part of the lightmatter interaction, which in this case concerns the induced surface charge and resultant screening effect. The effective medium theory for multilayers resulting in effective permittivity given in equation [35] considers the induced polarization in each layer. When the incident electric field is perpendicular to the multilayer interfaces, it produces a net interface charge due to the discontinuity in permittivity at the interface. These interface charges then produce a polarization field that counteracts the incident electric field, resulting in a permittivity given as ε_{⊥} in equation [35]. On the other hand, when the incident electric field is parallel to the interface, no interface charge is formed and thus the effective permittivity is simple given as an arithmetic mean of the constituent permittivities, as given for ε_{} in equation [35]. The experiments in the cloaked silicon nanorod were done for TM waveguide mode which has electric field parallel to the gold nanowires. This corresponds to the no interface charge case and the multilayer effective medium theory for no screening case is applicable. If the experiments were done for perpendicular polarization, it would have been important to accurately account for the interface charge density and thus the multilayer effective medium theory could have been inaccurate.
Another case worth mentioning is the nanoparticle array. It is well established that small nanoparticles have dominantly dipole character with negligible contributions by the high order components. Thus, as long as the nanoparticles are separated well enough, effective medium theory should work well. But defining a precise validity limit is difficult because unlike the perturbation theory the effective medium theory does not allow quantitative estimation of errors. For plasmonic nanoparticles, the literature suggests volume fraction of 10 ~ 20% would be the upper limit where the effective medium theory begins to break down [49,81]. In a study of collective magnetic resonance supported by a cluster of plasmonic nanoparticles, Tamma et al. presented a comparison between the rigorous multiple scattering theory and the extended Maxwell Garnett effective medium theory [49]. Figure 6 shows the dispersion curves and effective permittivity for silver nanoparticle arrays with silver volume fractions of 0.1, 0.3 and 0.5. As shown, even at a very high volume fraction of 0.5, the effective medium theory agrees reasonably well with the rigorous multiple scattering theory. The discrepancy is found near the band edge in all cases where the discrepancy remains small for f = 0.1 and becomes significant for higher volume fractions. This discrepancy observed near the band edge captures the nature of approximation in the effective medium theory. Unlike the effective medium theory, the multiple scattering formalism requires the complete knowledge of the nanoparticle positions in the array. For the calculations in Figure 6, the nanoparticles were assumed to be in the facecentered cubic (fcc) structure and the light propagation along the (111) direction was calculated. If the calculations were made for other crystallographic directions, the resulting dispersion curves would be different and the differences would be most pronounced near the band edge where the Bragg resonance condition is met. By construction, the effective medium theory does not require the knowledge of nanoparticle arrangement. This implies the effective medium theory would represent an averaged response of all different crystallographic orientations. It is therefore not surprising to see the major difference between multiple scattering theory and effective medium theory near the band edge. This however does not necessarily represent the deficiency of effective medium theory because most nanocomposite structures do not have a definite crystal structure. A few exceptions are DNAdriven selfassembly of plasmonic nanoparticles that lead to single crystalline fcc and bodycentered cubic structures [82,83]. But even in those cases, the achievable structures remain small and multiple crystallographic orientations may be in play. Therefore, although it is advisable to compare with rigorous multiple scattering or other methods whenever possible, the effective medium theory remains highly useful in an extremely wide range of plasmonic nanostructures.
Figure 6. Comparison of multiple scattering theory and effective medium theory.(a), (b), (c) Dispersion curves obtained from multiple scattering (M.Scat) and extended MaxwellGarnett effective medium theory (EMT) for silver nanoparticle array with volume fractions of 0.1, 0.3 and 0.5, respectively. (d), (e), (f) Real and imaginary parts of the effective permittivity calculated by the extended MaxwellGarnett effective medium for silver volume fractions of 0.1, 0.3 and 0.5, respectively (From Ref. [49]).
Comparison of multiple scattering theory and effective medium theory.(a), (b), (c) Dispersion curves obtained from multiple scattering (M.Scat) and extended MaxwellGarnett effective medium theory (EMT) for silver nanoparticle array with volume fractions of 0.1, 0.3 and 0.5, respectively. (d), (e), (f) Real and imaginary parts of the effective permittivity calculated by the extended MaxwellGarnett effective medium for silver volume fractions of 0.1, 0.3 and 0.5, respectively (From Ref. [49]).
Strongly coupled plasmonic nanostructures
Dimers
Strongly coupled plasmonic nanostructures have been studied extensively during the past decade. For example, the hot spot leading to enhanced Raman scattering is generally attributed to the coupling between two adjacent silver nanoparticles producing strong local field in the gap between the two nanoparticles [84]. A similar phenomenon has been observed in a coupleddipole nanoantenna composed of two nanorods aligned endtoend and separated by a small gap [85]. Also, early experimental studies on the simple model system of two closely spaced metal nanoparticles, a system often dubbed as dimer, showed the plasmon resonance exhibits red shift with decreasing gap between the nanoparticles [8689]. Both the formation of hot spots and the red shift with decreasing gap size are due to the hybridization of individual nanoparticle plasmons. When the gap between the nanoparticles is small enough to allow significant overlap of the plasmon modes of the individual nanoparticles, the strong nearfield interaction consequently leads to hybridization of the individual single particle plasmon modes to form a new set of coupled modes delocalized over the entire structure. In a dimer, the single particle plasmon modes couple with each other to form symmetric and antisymmetric combinations. This is the standard coupled oscillator problem with numerous analogous examples in physics [90]. In the plasmonic dimer, the oscillating entity is the free electron charge in the metal. When the charge density in the two particles oscillates in phase along the axis of the dimer, we have a symmetric mode with strong electric field in the gap. When the two oscillations are 180° out of phase, we have an antisymmetric mode whose electric field profile has a null in the gap. In molecular analogy, the symmetric mode corresponds to the bonding orbital whereas the antisymmetric mode is the antibonding orbital. Also, the symmetric mode has a lower energy and the antisymmetric mode has a higher energy, just as the bonding orbital has a lower energy and the antibonding orbital has a higher energy. As illustrated in Figure 7 for a nanorod dimer, this hybridization model offers a good explanation for both the hot spot formation and the red shift in plasmon resonance wavelength.
Figure 7. Plasmon coupling in nanorod dimer.(a) Schematic drawings of a coupleddipole nanoantenna made of two plasmonic nanorods with a small gap and its mechanical analog, a coupled mechanical oscillator. (b) Energylevel diagram and simulated nearfield intensity spectra for 30 nm high, 50 nm wide and 110 nm long symmetric twonanorod antennas with 6 nm (blue dashed) and 16 nm (green solid) gap, as well as singlenanorod antennas with the same dimensions (black dotted) (From Ref. [90]).
Plasmon coupling in nanorod dimer.(a) Schematic drawings of a coupleddipole nanoantenna made of two plasmonic nanorods with a small gap and its mechanical analog, a coupled mechanical oscillator. (b) Energylevel diagram and simulated nearfield intensity spectra for 30 nm high, 50 nm wide and 110 nm long symmetric twonanorod antennas with 6 nm (blue dashed) and 16 nm (green solid) gap, as well as singlenanorod antennas with the same dimensions (black dotted) (From Ref. [90]).
The hybridized plasmon modes in a dimer can be monitored by directly solving Maxwell’s equations numerically using techniques such as finitedifference timedomain (FDTD) [91] or finiteelement method (FEM) [92]. Nevertheless, it is always profitable to seek for a theoretical model that could provide deeper insights on the physics governing the hybridization. Also, it is worth mentioning the inability of most brute force numerical simulations to excite antisymmetric modes. This deficiency stems from the typical simulation geometry in which the nanostructure under investigation is excited by incident light. The simulation then calculates the resultant absorption or scattering which reveals the characteristics of the hybridized plasmon modes. This unfortunately works well only for the symmetric mode, which typically possesses a large dipole moment and thus interacts strongly with light. The antisymmetric mode, however, represents a state in which the induced dipole moments in the individual nanoparticles align antiparallel and thus only produce a small net dipole moment. This mode therefore does not interact with light and remain invisible in most experiments and simulations. For this reason, the symmetric mode is often referred to as the bright mode and the antisymmetric mode as the dark mode. The antisymmetric mode is nonetheless important as, for example, it can interact with the bright mode to produce Fano resonance as described later. It is therefore useful to have a theoretical model that can describe both the symmetric and antisymmetric modes equally well. The simplest model employs the electrostatic approximation where the retardation effects are neglected. In this formalism, the polarizability, α, of individual nanoparticles is given by equations (15) and (27) for spheres and ellipsoids, respectively. The induced dipole moment in each particle is then given as μ = αE. In a dimer, the field, E, contains both the applied field and the field produced by the neighboring particle. Since an individual nanoparticle is treated as a point dipole, we can write
Here the subscripts 1 and 2 index the two particles in the dimer, ε_{m} is the permittivity of the background medium and g is the geometry dependent parameter given as g = 2 for endtoend coupling and g = 1 for sidetoside coupling [93,94]. The coupled equation in [36] can be solved to find the polarizability of the bonding mode,
The plasmon resonance condition is found by the maximum polarizability and, if the two particles are identical, is given as,
Using the polarizability given in equation (27), we find
Since the permittivity of gold has almost linear dependence on wavelength in the visible region, equation (39) specifies the dependence of the surface plasmon wavelength of a gold dimer on the centertocenter spacing. This simple model correctly predicts the red shift of the symmetric mode for endtoend aligned dipole pair and the blue shift for sidetoside aligned dipole pair. The opposite behavior originates from the opposite sign of g in equations (37) ~ (39). In plasmonic dimer made of two identical particles, the endtoend dipole alignment corresponds to the excitation of dimer with incident polarization along the dimer axis while the sidetoside coupling case corresponds to incident polarization perpendicular to the dimer axis. The opposite peak shifts in the two cases have been experimentally observed in the extinction spectra of gold nanodisk pairs fabricated by electronbeam lithography (Figure 8a) [88]. Furthermore, Jain et al. reported the fractional plasmon peak shift, ∆λ/λ_{o}, exhibits a universal scaling against the normalized gap width defined as gap/diameter, irrespective of the detailed geometrical parameters, as shown in Figure 8(b) [95]. This universal scaling was also found for dimers of noncircular shapes and nanoshells and also in trimers [96,97]. The origin of the universal scaling behavior may be found in equation (39), which predicts that the fractional peak shift would behave as (d/D)^{− 3} or [(s + D)/D]^{− 3} where s = d – D is the gap width between the particles and D is the diameter. When the gap width becomes extremely small, the simple dipoledipole interaction model breaks down and one must take into account the higher order multipole terms, leading to an exponential dependence observed in Figure 8. This omission of multipole interaction terms is one of the key limitations in the dipoledipole interaction model, which leads to a significant underestimation of plasmon coupling at extremely small gaps. Additionally, the dipole model is essentially based on the electrostatic approximation which ignores any retardation effect. To properly describe the interaction at small gaps including the retardation effects would require the use of numerical modeling techniques such as FDTD and FEM. Before moving on, it is worth mentioning that the universal exponential dependence has led to the derivation of the socalled plasmonic ruler equation, which enables precise determination of distances in biological systems, opening many opportunities for biosensing [95].
Figure 8. Plasmon coupling in gold nanoparticle dimer.(a) Spectral position of the extinction maximum vs. interparticle distance for seven gold nanodisk pairs fabricated by electronbeam lithography. (From [88]) (b) Calculated fractional plasmon peak shift vs the ratio of interparticle gap to nanodisk diameter, showing that the scaled data points for the different disc sizes follow a common trend, which can be fit together to the singleexponential decay (solid curves) y = ae^{− x/τ} with and τ = 0.23 ± 0.03 (From Ref. [95]).
Plasmon coupling in gold nanoparticle dimer.(a) Spectral position of the extinction maximum vs. interparticle distance for seven gold nanodisk pairs fabricated by electronbeam lithography (From [88]). (b) Calculated fractional plasmon peak shift vs the ratio of interparticle gap to nanodisk diameter, showing that the scaled data points for the different disc sizes follow a common trend, which can be fit together to the singleexponential decay (solid curves) y = ae^{− x/τ} with and τ = 0.23 ± 0.03 (From Ref. [95]).
Note that the previous discussion of dipoledipole coupling focused only on the symmetric mode where the two individual dipoles are parallel to each other. As stated earlier, the antisymmetric mode with antiparallel dipoles has a small net dipole moment and thus interacts only weakly with light. This is not true when the antisymmetric mode possesses a large magnetic moment. In this case, the antisymmetric mode interacts strongly with light through its magnetic field. A dimer of small spherical nanoparticles does not support strong magnetic resonance but a pair of nanorods can. A pair of gold nanorods was shown to support a strong magnetic resonances [98,99] and soon after used to construct negative index material [100]. A similar behavior was also observed in a pair of nanoholes in two closely spaced metal films, which is expected to exhibit the same properties as the nanorod pair through the Babinet principle [101]. The nanohole pair design was then adapted to the fishnet structure which is known to be the most effective design for negative index [102]. Later, nanorod pairs with various different arrangements were investigated [103]. The hybridization scheme for various geometries is shown in Figure 9 together with the surface change densities for various plasmon modes. Here the hybridization scheme was obtained by the plasmon hybridization model and the surface charge density by the boundary integral method, both of which are described in Section 2.4. As shown in Figure 9(a), the single nanorod supports two resonances, longitudinal and transverse modes where the transverse mode has a higher energy due to the smaller transverse dimension. When the two nanorods are placed close together, the nearfield interaction leads to hybridization and mode splitting where the symmetric and antisymmetric combinations form bonding and antibonding modes. In the perfectly symmetric arrangements of endtoend and sidetoside pairing, the antisymmetric combination results in perfect cancellation of dipole moments. Thus the resultant hybridized mode is dark and does not respond to the incident light. Note that, unlike the nanorod pair used to observe magnetic resonance, which has one nanorod on top of the other so that the induced magnetic moment can align with the incident magnetic field, these nanorod pairs are all fabricated on the same plane and thus no magnetic interaction is expected. The energies of the hybridized modes are easily understood by the electrostatic interaction between electronic charge densities. For example, in the endtoend pair, the antisymmetric mode has the same charge concentrated near the gap, resulting in high energy while the symmetric mode has the opposite charge across the gap, leading to lower energy. Thus in this case, the symmetric mode becomes the bonding mode and the antisymmetric mode forms the antibonding mode. For the sidetoside pair, the opposite is the case and the symmetric mode has higher energy than the antisymmetric mode, making the antisymmetric mode the bonding mode and the symmetric mode the antibonding mode. For other nonsymmetric arrangements, the hybridized mode energy depends on the exact geometry. It should be noted that both the plasmon hybridization model and the boundary integral method used to generate Figure 9 take into account only the electrostatic interaction. The energy levels in Figure 9(a) and (b) are thus different from the experimentally measured plasmon peak positions, generally blue shifted from the experimental values due to the neglect of retardation effect.
Figure 9. Plasmon hybridization schemes for (a) nanorod dimers in different geometric arrangements and (b) nanorods initially arranged sidetoside and then increasingly longitudinally offset as a function of the centertocenter offset. (c) Surface charge density of interacting gold nanorods placed 1.5 nm apart and with dimensions 78 nm × 24 nm, calculated using the boundary integral method (From Ref. [103]).
Plasmon hybridization schemes for (a) nanorod dimers in different geometric arrangements and (b) nanorods initially arranged sidetoside and then increasingly longitudinally offset as a function of the centertocenter offset. (c) Surface charge density of interacting gold nanorods placed 1.5 nm apart and with dimensions 78 nm × 24 nm, calculated using the boundary integral method (From Ref. [103]).
Heptamers
To go beyond the simple dimer structure to more complex nanostructures, one needs a systematic way to classify various eigenmodes with complex symmetry properties. The group representation theory provides a power means to accomplish that [23]. For a given eigenvalue equation, Ax_{n} = a_{n}x_{n}, we can define a group of symmetry operators that leaves the operator A invariant. Note that the discussion here is generally applicable to any types of eigenvalue equations which could be the Schrödinger’s equation in quantum mechanics, Maxwell’s equation in optics and electromagnetics, or the fluid dynamics and surface charge density equations used in the plasmon hybridization model and boundary integral formalism, respectively, discussed in Section 2.4. If a symmetry operator, S, leaves the operator A invariant, it is straightforward to show that the two operators commute with each other and share the same eigenstates. If all symmetry operators that leave the operator A invariant are found, then it is possible to classify all eigenstates of operator A in terms of how they transform under the various symmetry operators. In the language of group theory, all eigenstates of operator A can be indexed by the irreducible representations of the symmetry group. Now let us proceed to the plasmon modes in a trimer using the group representation theory. For a trimer composed of three identical nanoparticles placed at the corners of an equilateral triangle, the symmetry operators that leave the system invariant form the point group D_{3h} and thus the eigenmodes of a trimer can be classified by the irreducible representations of D_{3h}. For simplicity, we treat the plasmon mode of individual nanoparticle as a dipole mode, which is a good approximation for small particles. Then, the individual plasmon mode belongs to the D^{(l=1)} irreducible representation of the full rotation group. For a trimer, we should consider the triple direct product of D^{(l=1)} irreducible representations and their subsequent reduction into a combination of irreducible representations of point group D_{3h}, which yields [104],
The transformation properties of the various irreducible representations of point
group D_{3h} can be found in the character table available in the literature [23]. From the character table, one can then construct the projection operators and generate
the symmetryadapted basis functions, as shown in Figure 10, which transform according to the symmetry properties of the irreducible representations.
The onedimensional irreducible representations,
Figure 10. Irreducible representations and symmetric linear combinations of thel = 1 dipole modes as predicted by group theory and plasmon hybridization model (From Ref. [[104]]).
Irreducible representations and symmetric linear combinations of thel = 1 dipole modes as predicted by group theory and plasmon hybridization model (From Ref. [[104]]).
The group theory provides a powerful and effective way to describe more complex structures and extensive studies, both theoretical and experimental, have been conducted on quadrumers [56],[104106], pentamers [20,107], and hexamers [17,18,20]. The structure that received by far the most attention was the heptamer which is constructed by adding a central particle to a hexamer. The main reason for the interest has been the Fano resonance the heptamer supports. In most systems exhibiting resonance, the lineshape function describing the resonance is typically Lorentzian. However, when a discrete state interacts strongly with a continuum, a new resonance characterized by a distinctly asymmetric lineshape function may arise, as first discovered by Fano [108]. Since the origin of Fano resonance is the interference between the discrete and continuum quantum states, it is fundamentally a quantum phenomenon. However, the interference phenomenon is commonly observed in classical optics and Fanolike resonances have been observed in many systems. The oldest example is the Wood’s anomaly arising from the interference between the tangentially diffracted wave and the incident wave [109]. It has also been observed in coupled waveguidecavity systems, photonic crystals and plasmonic nanostructures for which many reviews are available [110112]. In addition to the rich physics it reveals, Fano resonance also has many technological applications thanks to its high sensitivity to the environmental parameters [19]. Naturally, this led to the demonstration of optical sensing based on Fano resonances [113,114]. Here we review Fano resonances in heptamers in detail.
A heptamer is composed of seven nanoparticles in a geometry with 6fold rotational symmetry. Six peripheral nanoparticles form a hexamer where the nanoparticles are placed at the corners of a regular hexagon. One central nanoparticle is located at the center of the hexagon. The heptamer has the same point group symmetry as hexamer, D_{6h}. Since the central nanoparticle is not connected to any other nanoparticles by the symmetry operators of point group D_{6h}, the irreducible representations for a heptamer can be found by the union of those for the hexamer formed by the peripheral nanoparticles and the central nanoparticle, Γ_{hep} = Γ_{hex} + Γ_{cent}. Similarly to the trimer case, considering only the dipole modes of the individual nanoparticles, the irreducible representations of the hexamer, Γ_{hex}, are found by taking the direct product of six l = 1 irreducible representations, Γ^{(l = 1)}, of the full rotation group and reducing it into the irreducible representations of the D_{6h} symmetry group. Likewise, the irreducible representation of the central nanoparticle is found by simply reducing Γ^{(l = 1)}. The results are as follows
Figure 11. Silver nanoparticle heptamer.(a) Symmetry adapted basis functions of heptamer inplane modes belonging to the D_{6h} point group. (b) Scattering and (c) absorption spectra calculated with the finite element method for heptamers with various sizes. The dashed curves show the redshift of the dark (green) and bright (blue) modes with increasing size. (From Ref. [115])
Silver nanoparticle heptamer.(a) Symmetry adapted basis functions of heptamer inplane modes belonging to the D_{6h} point group. (b) Scattering and (c) absorption spectra calculated with the finite element method for heptamers with various sizes. The dashed curves show the redshift of the dark (green) and bright (blue) modes with increasing size (From Ref. [115]).
The effect of symmetry lowering on the Fano resonance is of interest to both the fundamental physics and potential applications. Cui et al. fabricated gold heptamers on a flexible poly(dimethylsiloxane) (PDMS) membrane and applied mechanical stress while monitoring the evolution of the Fano resonance [21]. As shown in Figure 12, the Fano dip exhibited a blue shift with increasing mechanical stress for the polarization along the direction of stress while it shifted to lower frequencies for the perpendicular polarization. Also, for the parallel polarization, an additional Fano dip shows up at large mechanical strain values. Figure 12 also shows scattering spectra obtained by FEM simulations which agree very well with the experimental spectra. The same behavior was also confirmed by the generalized multiparticle Mie calculations following the theory described in Section 2.2. To gain insight into the observed behavior, the eigenmodes of the heptamer structure were obtained by solving the boundary integral eigenvalue equations for the charge distribution, as described in Section 2.4. Note that this technique is valid only in the static limit and thus the obtained plasmon energies are not the same as the experimentally observed peak positions. Nevertheless, this study allows for the analysis of the eigenmodes with the group representation theory and thus correlate the symmetry of the system with that of the eigenmodes. As discussed earlier, the heptamer structure has the symmetry of point group D_{6h} and, among the irreducible representations of D_{6h}, E_{1u} is the only one with a net dipole moment and thus optically active. Furthermore, we only consider two lowest energy E_{1u} modes in the unstressed gold heptamer structure because all higher modes are masked by strong absorption by gold. In the first column of Figure 13, we show the charge distribution of the two lowest energy E_{1u} modes in an unstressed gold heptamer structure composed of seven identical gold spheres where the sphere diameter is 150 nm and gap between the spheres is 25 nm. Here the modes shown in Figure 13(a) and (b) belong to the lowest energy E_{1u} mode and (d) and (e) to the second lowest E_{1u} mode. The twodiomensional E_{1u} irreducible representation represetns a doubly degenerate mode with two orthogonal states having net dipole moment in the x and y directions, respectively. Accordingly, the four charge distributions shown in the first column of Figure 13 possess net dipole moment where (a) and (d) are xdipoles and (b) and (e) are ydipoles. Depending on the energy and the relative alignment of dipole moment of the center sphere to those of the six satelite spheres, the E_{1u} modes can be classified as dark or bright modes. The lower energy E_{1u} mode shown in Figure 13(a) and (b) is a dark mode where the dipole moment of center particle aligns against the dipole moments of satelite particles, making the total dipole moment small. On the other hand, the higher energy E_{1u} mode shown in Figure 13(d) and (e) is a bright mode where the dipole moments align together and add. The energies of these two E_{1u} modes were found to be 2.394 eV and 2.459 eV or 517.8 nm and 504.0 nm, respectively. These mode energy values would be accurate only for heptamers made of very small nanoparticles, as the boundary integral method is valid in the static limit only. For larger sizes, retardation effect will shift and broaden the modes. The resultant overlap and interference between the two modes lead to the Fano resonance. The bright mode will broaden much more significantly than the dark mode, resulting in Fano resonance which manifests itself in the form of a dip in the extinction spectrum as observed in Figure 12.
Figure 12. Mechanical tuning of gold nanoparticle heptamer.(a)(b) show the experimentally measured and simulated extinction spectra for mechanical stress along horizontal direction. Stretching direction and polarizations are shown in the insets. Solid line shows experiment results and dashed line shows simulation results. (From Ref. [21])
Mechanical tuning of gold nanoparticle heptamer.(a)(b) show the experimentally measured and simulated extinction spectra for mechanical stress along horizontal direction. Stretching direction and polarizations are shown in the insets. Solid line shows experiment results and dashed line shows simulation results (From Ref. [21]).
Figure 13. The evolution of charge distribution with uniaxial stress along the x direction. The first column shows the two lowest E_{1u} modes of unstressed heptamer and the B_{1u} mode. Second and third columns show B_{2u} and B_{3u} modes of the heptamer under 10% and 30% strains, respectively. The resonance wavelengths of the heptamers are also shown. (From Ref. [21])
The evolution of charge distribution with uniaxial stress along the x direction. The first column shows the two lowest E_{1u} modes of unstressed heptamer and the B_{1u} mode. Second and third columns show B_{2u} and B_{3u} modes of the heptamer under 10% and 30% strains, respectively. The resonance wavelengths of the heptamers are also shown in paraenthesis (From Ref. [21]).
When the heptamer is subject to a uniaxial mechanical stress, the symmetry of the system is lowered to D_{2h}. The doubly degenerate E_{1u} mode splits into two nondegenerate modes belonging to B_{2u} and B_{3u} irreducible representations of the point group D_{2h}. Figure 13 shows the evolution of charge distribution as the mechanical stress is applied along the x direction. It clearly shows the original doubly degenerate modes split into xdipole (B_{3u}) and ydipole (B_{2u}) modes. Remarkably, the nature of the modes are preserved. That is, the bright E_{1u} mode splits into bright B_{2u} and B_{3u} modes while the dark E_{1u} mode spawns dark B_{2u} and B_{3u} modes. Also, all modes shift to shorter wavelengths with increasing mechanical strain values. However, the B_{3u} modes which have dipole moment along the direction of mechanical stress shift more than the B_{2u} modes with dipole moment perpendicular to the mechanical stress. This leads to the distinct polarization dependence observed in Figure 12 as the B_{3u} modes interact with xpolarized light and B_{2u} with ypolarized light. Therefore, as the heptamer is stretched along the x direction, xpolairzed light would show resonance features at shorter wavelengths than the ypolarized light. Even when the retardation effects are included and the resonance peaks broaden and shift, this general behavior survives and leads to the experimental observation in Figure 12: the dip in the extinction spectrum due to the Fano resonance blue shifts for polarization parallel to the direction of mechanical stress but red shifts for polarization perpendicular to it. The exact numerical simulations by FEM and generalized multiparticle Mie theory also confirmed this.
In addition to the splitting of E_{1u} modes leading to the poalrization dependence, the scattering spectra for x polarization also show an additional dip at shorter wavelengths. The additional dip is apparent in the spectra for 30% and 45% strain for x polarization but is clearly missing in all spectra for y polarization. The origin of this second dip can be found by the group theoretical analysis. Briefly reiterating, in the unstressed heptamer structure possessing D_{6h} symmetry, only the E_{1u} modes have nonzero dipole moments and thus are optically active. When the symmetry is lowered to D_{2h} by uniaxial stress, the E_{1u} modes split into B_{2u} and B_{3u} modes which interact with x and ypolarized light, respectively. Since the unstressed heptamer has two E_{1u} modes in the frequency range of interest, we obtain two B_{2u} and two B_{3u} modes, producing Fano resonance just as in the original unstressed heptamer. However, what is missing in this narrative is that the optically inactive B_{1u} mode in the unstressed heptamer becomes optically active B_{3u} mode under uniaxial stress along the x direction. As shown in Figure 13(c), the charge distribution reveals that this B_{3u} mode is also a dark mode where the dipole moment of the center sphere aligns antiparallel against those of the satelite spheres, thereby producing a second Fano dip in the scattering spectra. In constrast, there are no other modes of the unstressed heptamer evolving into B_{2u} mode within the frequency range we investigated and thus we do not see any additional dip for the y polarization.
Now we can take a step further and study which dipole components contribute to the formation of Fano resonance. For this, we consider nanorod heptamers composed of a circular center particle surrounded by six nanorods [22]. A circular nanoparticle supports a degenerate dipole mode with two orthogonal dipole moments having the same energy. The anisotropy of nanorod, however, lifts this degeneracy and the transverse mode has a substantially different energy than the longitudinal mode. This allows us to probe how the each dipole components hybridize separately. This design principle is schematically shown in Figure 14(a). The doubly degenerate dipole modes of the circular particle can be arranged into azimuthal dipoles and radial dipoles according to the sixfold rotational symmetry of the heptamer structure. We then replace the circular satellite particles with nanorods arranged azimuthally and radially. By designing the nanorod dimensions so that only the longitudinal modes are in the frequency range of interest, we can construct heptamers with only the azimuthal dipoles and radial dipoles. Subsequent numerical simulations revealed that the azimuthal nanorod heptamer exhibits a strong and welldefined Fano resonance (Figure 14(b)) but the radial nanorod heptamer does not support Fano resonance (Figure 14(c)). We then conducted a series of numerical simulations for a wide range of structural parameters to find that the azimuthal nanorod heptamers always support Fano resonance but the radial nanorod heptamers do not. These results indicate that the Fano resonance observed in circular heptamers are the result of hybridization of azimuthal dipole modes of the satellite particles with the center particle and that the radial dipole modes do not contribute.
Figure 14. Gold nanorod heptamers.(a) Schematic showing the splitting of doubly degenerate modes in circular heptamer to azimuthal and radial modes in nanorod heptamer. The white colored arrows represent the radial modes and gold colored arrows represent azimuthal modes. Simulated extinction spectra for (a) azimuthal and (b) radial nanorod heptamers for the same structural parameters: individual nanorods have a length of 95 nm, width of 50 nm and thickness of 40 nm. The center particle has a diameter of 147 nm and gold thickness of 40 nm. The gaps between the center particle and the nanorods are 22 nm. (From Ref. [22])
Gold nanorod heptamers.(a) Schematic showing the splitting of doubly degenerate modes in circular heptamer to azimuthal and radial modes in nanorod heptamer. The white colored arrows represent the radial modes and gold colored arrows represent azimuthal modes. Simulated extinction spectra for (b) azimuthal and (c) radial nanorod heptamers for the same structural parameters: individual nanorods have a length of 95 nm, width of 50 nm and thickness of 40 nm. The center particle has a diameter of 147 nm and gold thickness of 40 nm. The gaps between the center particle and the nanorods are 22 nm (From Ref. [22]).
The Fano resonance supported by the azimuthal nanorod heptamer was further investigated for their behavior under mechanical stress and symmetry lowering. As before, the mechanical stress was applied along the x (horizontal) direction. Figure 15 shows the simulated extinction and absorption spectra at the induced mechanical strain values of 0%, 13%, 26%, and 42%. For the polarization parallel to the mechanical stress (x direction), the Fano resonance exhibited a moderate blue shift while for the perpendicular polarization it showed a small blue shift. In addition to the spectral shifts with applied mechanical stress, the absorption spectra for the perpendicular polarization showed an additional feature at shorter wavelengths when the heptamer was under mechanical stress. This feature located near 620 nm was particularly noticeable at high strain values of 26% and 42%. The behavior of the spectral features in Figure 15 can be explained by analyzing the eigenmodes obtained by the boundary integral method. The evolution of charge distribution is dictated mainly by the symmetry reduction from D_{6h} to D_{2h} and is similar to that observed in the circular heptamer under mechanical stress described earlier. As was the case in the circular heptamer, the doubly degenerate E_{1u} mode of the azimuthal nanorod heptamer splits into two nondegenerate modes belonging to B_{2u} (ydipole) and B_{3u} (xdipole) irreducible representations of the point group D_{2h}. Furthermore, the bright E_{1u} mode was found to split into bright B_{2u} and B_{3u} modes and the dark E_{1u} mode into dark B_{2u} and B_{3u} modes. Both the dark and bright B_{2u} and B_{3u} modes shift to shorter wavelengths with increasing mechanical strain values. However, the B_{3u} modes, which have dipole moment along the direction of mechanical stress, shifts more than the B_{2u} modes whose dipole moment is perpendicular to the mechanical stress, leading to the polarization dependence observed in the extinction and absorption spectra in Figure 15. Once again, the retardation effect results in overlap and interference between the dark and bright modes thus causing the Fano resonance to be observed as a dip in the extinction spectrum.
Figure 15. Simulated extinction and absorption spectra for the azimuthal nanorod heptamer for different induced mechanical strain values of 0%, 13%, 26% and 42%. (a) and (c) plot the extinction spectra for parallel and perpendicular polarizations (orientations in inset) respectively while (b) and (d) plot the absorption spectra for parallel and perpendicular polarizations respectively. (From Ref. [22])
Simulated extinction and absorption spectra for the azimuthal nanorod heptamer for different induced mechanical strain values of 0%, 13%, 26% and 42%.(a) and (c) plot the extinction spectra for parallel and perpendicular polarizations (orientations in inset) respectively while (b) and (d) plot the absorption spectra for parallel and perpendicular polarizations respectively (From Ref. [22]).
The additional absorption peak observed under stress can be explained by noting that the optically inactive B_{2u} mode in the unstressed heptamer becomes an optically active B_{2u} mode under uniaxial stress along the x direction. Here we have an unfortunate coincidence of having the same label for two very different modes. The B_{2u} irreducible representation in D_{6h} symmetry has no net dipole moment and thus represents an optically inactive mode. However, the B_{2u} irreducible representation in D_{2h} symmetry has a finite dipole moment along the ydirection and thus represents an optically active mode that can interact with ypolarized light. The energy of this B_{2u} mode is higher than the lowest dark B_{2u} mode originating from the dark E_{1u} mode of unstressed heptamer, thereby producing an additional feature at a higher energy. The azimuthal nanorod heptamer differs from the circular heptamer in the behavior of this optically active mode. In the circular heptamer, the optically inactive B_{1u} mode was located between the two lowest energy E_{1u} modes in the unstressed heptamer and became optically active B_{3u} mode under uniaxial stress along the x direction. As a result, in the circular heptamer, the additional mode was observed for xpolarized light. The difference between the circular heptamer and azimuthal nanorod heptamer is readily understood by comparing the eigenmode charge distributions. In the B_{1u} mode observed in the unstressed circular heptamer, the satellite particles have their dipole moments aligned along the radial direction (Figure 13c). In the azimuthal nanorod heptamer, this configuration would have had a much higher energy because the radial dimensions of the nanorods are much smaller than the circular particles used in circular heptamer. On the other hand, the B_{2u} mode in which the satellite particles have their dipole moments aligned along the azimuthal direction has a lower energy because the nanorods have larger dimensions along that direction. For these reasons, we find the B_{2u} mode is located between the two lowest energy E_{1u} modes in the azimuthal nanorod heptamer structure. These behaviors predicted by the theoretical modeling studies have been confirmed by experiments on gold nanorod heptamers fabricated by electronbeam lithography [22].
Conclusion and outlook
Plasmonics in mutually interacting nanostructures is an exciting field with rich physics and numerous potential applications. The recent progress clearly demonstrates that plasmonic nanostructures allow us to precisely control the nanoscale optical modes and how they interact with one another. These capabilities make the coupled plasmonic nanostructures an ideal platform to study nanoscale optical phenomena and build optical devices with novel functionalities. Though fundamentally a quantum mechanical entity, surface plasmons can be described well by classical electrodynamics in most cases. The theoretical framework presented in this review provides a firm foundation for designing interesting structures and analyzing their properties. Also, the strong field enhancement near the metal surface and the complex field profiles resulting from the interaction among the nanostructures makes the optical properties highly sensitive to the environment and consequently leads to exotic properties such as perfect absorption, invisibility and Fano resonance.
Some of the latest developments in the coupled plasmonic systems deal specifically with the quantum mechanical nature of the phenomenon. In surface enhanced Raman scattering and many other applications, it is generally advantageous to achieve high local field strength. In the simplest geometry of a dimer, a smaller gap supports higher field. However, when the two nanoparticles in a dimer are nearly touching, the electrons may tunnel through the gap, resulting in a dramatic decrease of local field strength and shift of plasmon energy [26,116]. Furthermore, at such a small length scale, the abrupt interface between two materials is unrealistic and the nonlocality of the dielectric function has to be considered [117]. These considerations have led to the development of a quantumcorrected model that could correctly reproduce the fully quantum mechanical calculations [118]. Unveiling the quantum mechanical nature of the gap plasmons in extremely small gaps remains an active research field with hopes of discovering new physics and also with potential for developing novel quantum devices.
Another interesting subject of fundamentally quantum nature is the coupling between plasmon and atomic or molecular exciton. The interaction between plasmon and exciton is of great fundamental interest with strong technological implications in organic electronics and photovoltaics. It has been shown that the surface plasmon can be used to control the relaxation pathways of molecular excitons [119]. Furthermore, in a fashion similar to plasmon hybridization, atomic or molecular exciton can hybridize with plasmon, forming a new quantum mechanically entity sometimes called plexciton [120,121]. The strong coupling between plasmon and exciton has recently been observed in an organic photovoltaic material [122]. The strong interaction could result in dramatic changes in excitation and relaxation dynamics of the molecular excitons, opening new opportunities in controlling and engineering the optical processes in organic materials.
In closing, surface plasmons provide a powerful means to gain access to the near field components that could result in a wide range of unconventional properties. Controlling and manipulating the mutual interactions among the plasmonic nanostructures offer design flexibility and engineering freedom unparalleled by any other photonics technologies. Obtaining high quality materials with low loss in precise nanoscale geometry will continue to be a challenge but the remarkable progress in nanofabrication technologies in the past decades makes it a perfect time to invest in research on the plasmonic nanostructures.
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