Optical Response of Lorentzian Nanoshells in the Quasistatic Limit

Article pubs.acs.org/JPCB

Optical Response of Lorentzian Nanoshells in the Quasistatic Limit Demet Gülen* Department of Physics, Middle East Technical University, Ankara 06531, Turkey ABSTRACT: Recently, plexcitonic systems consisting of a plasmonic nanoshell or a core covered by an excitonic shell are engineered. Such systems hold promise for tunable nanophotonic devices for imaging, chemical sensing, and resonance energy transfer. Their plasmonic response is grasped well, while understanding of their excitonic response remains to be improved. To this end, we have developed a methodology in which the functionalities of the dispersive properties of the spherical shell and the nanoenvironment in tuning the optical response are clearly separated. Using this methodology, we have studied the response of the Lorentzian/excitonic nanoshells with optically inactive core and embedding medium and compared it with the well-known properties of the Drude/ plasmonics nanoshells. Contrary to Drude nanoshells exhibiting a resonance pair red-shifted with respect to the bulk, Lorentzian nanoshells are identified by a resonance pair blue-shifted with respect to the in-solution excitonic resonance. While the Drude red-shifting is more effective at increasing dielectric constants (core, shell, and embedding medium), the Lorentzian blue-shifting is governed by the excitonic strength and is suppressed at increasing dielectric constants. The implications of the results for manipulating the optical response of plexcitonic systems are briefly discussed.

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INTRODUCTION Confinement of an optically active medium in a “nanoscale” volume offers the ability to effectively engineer the optical resonance(s) of nanoparticles against the dielectric constant(s) of optically inactive/nonabsorbing surrounding media. For particles much smaller than the excitation wavelength, size is not an effective control parameter for tuning the resonances. However the systems of nanoshells have two important facilities to surpass this scale invariance. First, they allow structural manipulation of the optical response through the changes in their internal geometry without changing the overall size of the nanoparticles. Second, their layered structure allows integration of different core, spacer, and shell materials and increases their ability for effective dielectric induced tuning of the response further.1−4 In the past decade these two facilities have been extensively exploited for tuning the localized surface plasmon resonance (LSPR) properties of a variety of systems containing spherical noble metal nanoshells. The systems of nanoshells with LSPRs anywhere between the UV and the mid-IR have been engineered.4−7 This flexibility has greatly improved numerous applications of the LSPRs in a wide variety of areas involving surface enhanced Raman scattering, fluorescence, and dielectric sensing.8−11 The hybrid systems composed of an optically active dielectric nanoshell supporting excitons coating a noble-metal core or a shell supporting LSPRs have raised considerable recent interest. Current literature covers quite a few examples of such hybrid nanosystems, generically referred to as the plexcitonic systems.12−21 The excitonic shell is typically composed of J© XXXX American Chemical Society

aggregates self-assemblies of organic molecules of cyanines or chlorophylls.22,23 Such hybrids being capable of both plasmonic and excitonic transitions and their hybridized/mixed states are considered to be potentially very functional in designing tunable nanophotonic devices for molecular imaging, chemical sensing, and plasmonic resonance energy transfer.24−27 Theoretical understanding of the optical response of noble metal nanoshells has also been grounded rather well through analytical as well numerical methods.2,4,28 A conceptual explanation of the response has been conveniently provided through the hybridization of the primitive (the cavity and the sphere) LSPR modes of the nanoshell.2,4 There exists a lack of discussion on the nanoscale response of the optically active dielectric media. For example, in most of the interpretations of plexcitonic systems, while the nanoscale control of the plasmonic states has been included within the current understanding, the nanoscale optical response of the excitonic shell has been ignored at large, as we have already remarked in our recent notes.29,30In these recent notes we have remarked on the significance of using a suitable dispersion for dielectric medium, that is, Lorentzian dispersion. The primary purpose of the current contribution is to investigate the resonances of the Lorentzian nanoshells systematically using an approach to improve the understanding on the dependences of the resonances and their hybridization Special Issue: Rienk van Grondelle Festschrift Received: February 28, 2013 Revised: May 8, 2013

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dx.doi.org/10.1021/jp402109p | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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properties on the aspect ratio (“the shell thickness”) in addition to the dielectric constants of the core, the shell, and the embedding medium. Our prototype system will be a spherical core−shell nanoparticle with a homogeneous optically inactive core concentric with an optically active shell. The shell has an arbitrary dispersion, and the nanoparticle is suspended in a homogeneous nonabsorbing embedding medium. The nanoshell resonances will be examined in the context of quasistatic limit of classical electrodynamics. We will base our approach on the observation that there are two key factors that act separately for tuning the optical response of a nanoscale system. The first one is the tuning by the effective dielectric environment or tuning of the nanoenvironment. This tuning is ruled by the geometry of the system and the dielectric constants of the optically inactive materials surrounding this particular geometry. The second one is the dispersive character(s) of the optically active nanoscale component(s) of the system. Our primary objective will be to convey the distinctive significance of each of these key factors in tuning the nanoshell resonances and to delineate their roles in hybridization of resonances. While the Lorentzian shell resonances are our main focus, the Drude shells will also be included to provide a ready comparison and to highlight the basis of hybridization common to both dispersions. Thereby, convenient rulers to guide the resonance nanoengineering will be provided for both types of nanoshells. For a controllable bottom-up design of the plexcitonic hybrid systems it is crucial to have an understanding of the resonances of optically active dielectric nanoshells through theory and simulations in addition to the existing understanding of the LSPRs.

α(ω) ∝

(2)

Here εeff(ω) is the effective dielectric function of the spherically symmetric system suspended in the embedding medium. The imaginary part of the polarizability follows as α(ω , im) ∝

εMεeff (ω , im) (εeff (ω , re) + 2εM)2 + (εeff (ω , im))2

(3)

For the core−shell system of interest, εeff(ω) can be expressed within the Mie formalism1as15,19 εeff (ω) = εshell(ω)

A(ω) − 2ρ3 B(ω) A(ω) + ρ3 B(ω)

(4a)

where A(ω) = 3εcore[2εshell(ω) + εcore]

(4b)

and B(ω) = 3εcore[εshell(ω) − εcore]

(4c)

The dielectric function of “the excitonic” shell is described using a homogeneously broadened one-oscillator model, that is, the Lorentzian dispersion ∞ εL,shell(ω) = εshell −

fω02 (ω 2 − ω02) + iγ0ω

(5a)

which has the following real and imaginary parts ∞ εL,shell(ω , re) = εshell − fω02

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MATERIALS AND METHODS Spherical core−shell nanoparticles with a homogeneous optically inactive core and an optically active shell are suspended in a homogeneous nonabsorbing embedding medium. The core and the embedding medium have the respective dielectric constants of εcore and εM, and the shell is described by the dielectric function, εshell(ω). The nanoshell geometry is defined by the aspect ratio ρ = (Rc/R), where Rc and R are, respectively, the inner and the outer radii of the shell. The size of the nanoparticles is assumed to be much smaller than the wavelength of the incident light, and the particles are assumed to be predominantly absorptive. It then becomes sufficient to consider only the dipolar response to incident light; furthermore, the quasistatic approximation can be employed to simplify the calculation of the absorption crosssection. The absorption cross-section σ(ω) is quite generally expressed as σ(ω) ∝ ωα(ω , im)

εeff (ω) − εM εeff (ω) + 2εM

(ω 2 − ω02) (ω 2 − ω02)2 + γ02ω 2

(5b)

and εL,shell(ω , im) = fω02

γω (ω − ω02)2 + γ02ω 2 2

(5c)

ε∞ shell

Here is the high-frequency component of the dielectric function, ω0 is the “in-solution” transition frequency, γ0 is the excitonic transition line width, and f is the excitonic oscillator strength. The dielectric function of “the plasmonic” shell is described using the Drude dispersion ∞ εD,shell(ω) = εshell −

ωP2 ω(ω + iγP)

(6a)

which has the following real and imaginary parts ∞ εD,shell(ω , re) = εshell −

ωP2 ω 2 + γP2

(6b)

and (1)

εD,shell(ω , im) =

Throughout the text, all of the complex numbers are defined in the format C(x, y, ...) = C(x, y, ..., re) + iC(x, y, ..., im), where x, y, and so on indicate the parametric dependences. The use of proportionalities in the formulas below is adequate if one is only interested in the resonance frequencies/wavelength or the relative intensities of these resonances. In the limit defined above the polarizability for a spherically symmetric system (no matter how complicated, e.g., concentric nanoshells, etc.) reads

γωP2 ω(ω 2 + γP2)

(6c)

where ε∞ shell is defined above, γP is the plasmonic transition line width, and ωP is the bulk plasmon frequency.

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RESULTS AND DISCUSSION Implementation of the general formalism presented above in the small gamma limit is very instructive and offers a useful guide for nanoengineering of the optical response of the B



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Figure 1. Quantification of the aspect ratio dependencies of ε+shell(re) (a), ε−shell(re) (b), V+ and V− (c), and [ε+shell(re) − ε−shell(re)] (d) for different core materials with εcore ranging between 1.0 and 12.0. The embedding medium is water (εM = 1.768) in panels a, b, and d. See the color-coding box for εcore (for all panels) and the special εcore−εM coding for panel c. In panels a and b the black dashed vertical lines show the positions of the primitive modes. The insert of panel b displays the dependence for large ρ. In panel d, the dashed vertical markers stand for the difference between the positions of the primitive modes.

spherical nanoshells. εeff(ω,im) scales with the transition line width (see, for example, eqs 5c and 6c) and therefore the εeff(ω,im) ≪ εeff(ω,re) condition is usually satisfied for typical shell materials of current interest, for example, the noble metals such as Ag and Au and the J-aggregates of organic molecules. Typical line width to transition energy ratio of the LSPR mode of the Au and the Ag nanoparticles and the excitonic J-band of typical organic molecules (e.g., cyanines and chlorophylls) is ∼10−2.12−15,23,31−33 Therefore, εeff(ω,re) dominantly determines the resonance frequencies and εeff(ω,im) decides the transition line widths. In this limit, the resonances for the spherical shells occur at the frequencies satisfying the condition, εeff(ω,re) = −2εM. For the system of interest, the nanoshell is the only optically active medium, and the values of the real part of the shell dielectric function at which the optical resonances happen are ± εshell (re) = −

1 1 X ± [X2 − 4Y ]1/2 2 2

± ωL,shell

(8a)

and ± ∞ ± ωD,shell = ωP[εshell − εshell (re)]−1/2

(8b)

The effective dielectric nanoenvironment experienced by the shell is defined by ρ, εcore, and εM. The “variable” aspect ratio of the core−shell complex tunes the effective dielectric constant of this nanoenvironment by mixing the two dielectric media adjacent to the shell, the core, and the embedding media. Alternately, the changes in the core or the embedding media can be used to further tune this environment at a fixed aspect ratio. Viewed in this way the dependence of the resonance frequencies on the dispersive properties of the shell material and the functionality of the nanoenvironment in tuning these resonances are clearly separated. This separation provides a convenient control for a bottom-up manipulation of the optical response. In particular, the active control leading to the frequency splitting apparent in eqs 8a and 8b is provided through the hybridization of ε±shell(re) and is common to all shell materials with any given dispersion relationship. The optical response subsequent to this predetermined ε±shell(re) hybridization is unique for a specific dispersion. Next we will systematically study the tuning of the optical resonances for the spherical nanoshells on the basis of ε±shell(re) hybridization. The primary focus will be on the Lorentzian nanoshells because the resonances of this particular dispersion

(7)

with X=

⎡ ⎤1/2 f = ω0⎢1 + ∞ ⎥ ± εshell − εshell (re) ⎦ ⎣

1 [εcore(1 + 2ρ3 ) + 2εM(2 + ρ3 )] 3 2(1 − ρ )

and Y = εcore . εM

The resonance frequencies for an optically active shell with a specific dispersion can then be found using the condition εeff(ω,re) = −2εM. For example, the respective resonance frequencies for the Lorentzian and the Drude dispersions are C



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(including εM = 1) is also shown in this panel. Figure 1d shows + − the aspect ratio dependence of [εshell (re) − εshell (re)] corresponding to Figure 1a,b. Both interactions are positive and V− is larger than V+ at all nonzero aspect ratios for a given set of εcore and εM. At increasing aspect ratios the tuning of the nanoenvironment or the mixing of the core and the embedding medium gets more effective, and the larger the values of the dielectric constants, εcore and εM, the larger the interaction strengths. The ρdependence of V− becomes very rapid and starts to differ widely from V+ at increasing aspect ratios. The formula, ω±D,shell = (ωP/√2)[1 ± (1/3)(1 + 8ρ3)1/2]1/2, often used to describe the interaction scheme for the plasmon hybridization of Drude nanoshells in the literature,2 follows from eqs 9a and 9b as a special case for εcore = εM = ε∞ shell = 1. Similarly, a recent treatment the optical response of systems of nanoshells using the transfer matrix method is analytically explicit only for εcore = εM = 1.34 The interaction strengths and the positions of the “unperturbed/primitive” modes are bound to act in opposite directions in tuning ε±shell(re). As the difference between the primitive modes gets larger, the effects of interactions are expected to be reduced. However, owing to the very rapid increase in V− at increasing aspect ratios and the enhancement of V+ and V− at increasing εcore and εM one can expect a crossover between the effects of these two factors. This crossover is clearly observed through the behavior of the difference between the two primitive modes shown in Figure 1d. The trends observed in Figure 1c,d are reflected in the aspect ratio dependences of ε±shell(re): owing to the positive hybridization interactions, the two modes are tuned in contrasting directions by the dielectric constants, and hybridization of the two primitive modes gets highly asymmetrical at increasing values of ρ; the ε−shell(re)’s are subjected to faster and more substantial changes compared with the ε+shell(re)’s at all values of εcore and εM. Primitive Mode Resonances of the Nanoshells. For the primitive modes the nanoenvironment is defined by εM(sphere) and εcore(cavity). The Lorentzian shell parameters f and ε∞ shell tune the frequencies of the primitive modes as

have not been discussed in detail the previous literature. In addition, the resonances of the Drude nanoshells will be briefly discussed to highlight the basis of hybridization common to both dispersions and to allow comparison between the two cases. Thereby, convenient rulers to guide the nanoengineering of the optical resonances will be provided for both Lorentzian and Drude nanoshells. Tuning of the Nanoenvironment. For the brevity of discussion, we will discuss the implications of eq 7 in relation to the results given in Figure 1. In this Figure, the aspect ratio dependence of the two modes and their hybridization characteristics are examined at several different sets of dielectric constants for the core and the embedding medium. This examination is carried out by changing the aspect ratio from close to 0 to close to 1. The results for ε+shell(re) and ε−shell(re) for various core materials with dielectric constants ranging between 1.0 (magenta) and 12.0 (light green) and in a specific embedding medium (water, εM = 1.768) are, respectively, given in panels a and b. In addition the results at εcore = 1 and at several different εM are included in panel c. Both ε±shell(re) values satisfying resonance are naturally negative at all possible aspect ratios. + Furthermore, εshell (re) is always the upper branch (less negative). In the limit ρ ≪ 1 (or ρ → 0), the two primitive modes, the sphere and the cavity modes, can be defined. In this limit, ε+shell(re) and ε−shell(re) assume the respective values of −2εM (sphere) and −0.5εcore (cavity). That is, the εshell(re) value of each primitive mode depends only on the dielectric medium adjacent to it. As a result, both branches evolve from the degenerate cavity and sphere modes for εcore = 4εM. The + modes start to evolve with the cavity mode for εcore > 4εM and with the sphere mode for εcore < 4εM, whereas the − modes evolve conversely. With the increasing aspect ratio the mixing/ hybridization of the two primitive shell modes occurs (to be discussed below in more detail) that eventually leads to the appearance of two hybridized nanoshell resonances for a specified dispersion. In the limit of extreme mixing (ρ → 1, which in practice is achieved by shells much thinner than the core radius) the ε+shell(re) and ε−shell(re) values, respectively, approach to 0 (zero) and “−∞”. The behavior between the two limiting cases can be understood in an asymmetric two-state hybridization scheme. Any two-state hybridization is controlled by two main parameters: the interaction strengths and the positions of the “unperturbed/primitive” modes.22 One can define two separate interactions that lead to the asymmetrical evolution of the hybridized branches as an admixture of the primitive nanoshell modes ± (re) = − εshell

ωL,sphere

(10)

and ⎡ ⎤1/2 f ωL,cavity = ω0⎢1 + ∞ ⎥ εshell + 0.5εcore ⎦ ⎣

⎞ 1 1 ⎛⎜ 1 1 2εM + εcore⎟ ± [(2εM − εcore)2 ⎝ ⎠ 2 2 2 2

+ 4(V±)2 ]1/2

⎡ ⎤1/2 f = ω0⎢1 + ∞ ⎥ εshell + 2εM ⎦ ⎣

(11)

Both primitive mode resonances are blue-shifted with respect to ω0, the in-solution “excitonic” resonance. The oscillator strength is the source of the blue shift and is dominant in controlling the magnitude of the shift for both resonances. In contrast, all of the dielectric constants suppress the f-induced blue shifts. In the same nanoenvironment both primitive mode resonances of the Drude nanoshells are red-shifted with respect to ωP, the bulk “plasmon” frequency. The suppression of ωP is entirely attributed to the dielectric constants and the primitive modes are well known to occur at

(9a)

or V+ and V− can be identified as, V± = ⎤1/2 ⎡ ± ⎛ ⎞ 1 2 ± ⎜ ⎟ ⎢⎣(εshell(re)) − εshell(re)⎝2εM + εcore⎠ + εMεcore ⎥⎦ 2 (9b)

Figure 1c displays the interaction strengths corresponding to the ε±shell(re) studied in panels a and b. The aspect ratio dependence of V+ and V− for εcore = 1 at several different εM

∞ ωD,sphere = ωP[εshell + 2εM]−1/2

D

(12)



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and ∞ ωD,cavity = ωP[εshell + 0.5εcore]−1/2

(13)

As previously discussed, each primitive mode frequency is tuned by the dielectric medium adjacent to the respective optically active nanosystem, and whether the higher energy one is a sphere or a cavity mode depends on the relative values of 4εM and εcore. Equations 10−13 show that the sphere mode is of higher energy for εcore > 4εM, the two modes have the same energy for εcore = 4εM, and the cavity mode is the mode of higher energy for εcore < 4εM for both dispersions. Once again, apart from the difference between the two dispersions, the dielectric constants of the core, the shell, and the embedding medium have the same functionality, that is, the frequency suppression. It is the difference between the two dispersions that yields their contrasting tuning and defines the Drude nanoparticles as red-shifters and the Lorentzian nanoparticles as blue-shifters of the bulk/the in-solution resonance. Likewise the dielectric-suppression of the primitive mode energies is experienced in the opposite directions. The Drude primitive modes are more effectively red-shifted at increasing dielectric constants and the Lorentzian primitive modes are more effectively blue-shifted at decreasing dielectric constants. Resonances of the Lorentzian Nanoshells. In Figure 2 the aspect ratio dependences of the fractional change in the resonance frequencies of Lorentzian nanoshells are examined for various core materials with dielectric constants ranging between 1.0 (magenta) and 12.0 (light green). The fractional changes in the resonance frequencies are defined as ΔωL,± = (ω0 − ωL,±)/ω0 to set a ω0-independent ruler for the blue shifts. In each panel the corresponding wavelength values λL,± for ℏω0 = 1.75 eV (λ0 ≈ 708.5 nm) are also quantified. The results at two different oscillator strength values (f = 0.1 and 2.5) and at a fixed dielectric constant of the shell (ε∞ shell= 1.5) are displayed in the first and the third panel. In the second panel the results for two different values of the shell dielectric ∞ constant (ε∞ shell = 1.5 (solid curves) and εshell = 2.5 (dashed curves)) are compared at another oscillator strength (f = 1.0). In these panels the embedding medium is water (εM = 1.768). In the last panel the effects of εM on the aspect ratio dependences of ΔωL,± and λL,± are given for several typical cores (Si, Ag, and Au) at fixed values of f and ε∞ shell (f = 1 and ε∞ =1.5). The line coding for the ε values is: 1 (short shell M dashes), 1.768 (dashed-dotted), 2.5 (long dashes), and 3.5 (solid). The spectral signature of the Lorentzian nanoshells is a pair of resonances, both of which are blue-shifted with respect to ω0. The Lorentzian shell resonances are more effectively blueshifted at decreasing values of all dielectric constants. The dependencies of λL,+ and λL,− on the shell parameters f and ε∞ shell on the hybridization properties originating from ε±shell(re)and the consequences of the λL,±-ε±shell(re) correlation implied by eq 8a are clearly manifested throughout Figure 2. Already blue-shifted primitive modes hybridize strongly with the increasing aspect ratio (“decreasing shell thickness”). The + modes are further blue-shifted over the primitive mode of higher energy/lower wavelength. The hybridized modes are red-shifted over the primitive mode of lower energy, yet λL,−’s still remain blue-shifted with respect to λ0, the in-solution transition wavelength, which is also the limiting value of λL,− at −1/2 ρ=1. The limiting value of λL,+ at ρ = 1 is (λ0[1 + f/ε∞ ). shell]

Figure 2. Quantification of the aspect ratio dependencies of ΔωL,± = (ω0 − ωL,±)/ω0 and the λL,± for ℏω0 = 1.75 eV (λ0 ≈ 708.5 nm) at ∞ different sets of f, ε∞ shell, εcore, and εM. The values of f, εshell, and εM are given in each panel. The εcore values are the same as those in Figure 1 and range between 1.0 and 12.0. The color coding for εcore in the first three panels is the same as that in Figure 1. In the last panel the effects of εM are shown for three typical core materials: Si (green), Ag (gray), and Au (yellow).

The exhibition of asymmetry between the two modes at increasing aspect ratios is a consequence of the correlation E



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implied by eq 8a between ε±shell(re) and λL,±. Accordingly another manifestation of the Lorentzian dispersion is the larger sensitivity of λL,+ to the variations in the aspect ratio. That is, λL,+ resonances are not only more blue-shifted with respect to λ0 but also can be tuned in a wider wavelength range compared with the λL,‑ resonances. Figure 3 displays the characteristics of ΔωL,± and λL,± discussed above as a function of εcore at different f studied in a wide range of different aspect ratios.

Figure 4. f dependencies of ΔωL,± = (ω0 − ωL,±)/ω0 and λL,± for ℏω0 = 1.75 eV (λ0 ≈ 708.5 nm) at different ρ for three typical cores (Si, Ag, and Au with the respective εcore values of 2.04, 4.8, and 9.84. The embedding medium is water (εM = 1.768) in all panels. The color coding for different ρ is shown at the bottom.

ΔωL,± is examined at various aspect ratios for three typical core materials (Si, Ag, and Au). The resonance wavelengths λL,± for ℏω0 = 1.75 eV (λ0 ≈ 708.5 nm) are also given. In all panels the embedding medium is water (εM = 1.768) and ε∞ shell = 1.5. For f = 0, ω0 is the limiting value at any aspect ratio (although it will not carry any intensity), and as ρ → 0 the primitive mode frequencies indicated by the black lines in the gap are approached. The characteristics of ΔωL,± and λL,± discussed above in relation to Figures 2 and 3 can be identified again. The control of the primitive mode frequencies by εcore, εM, and f, the enhancement of hybridization with increasing aspect ratio, and the suppression of the blue shift with the increasing shell dielectric are further quantified as a function of f. Owing to the strong f dependence of the primitive mode frequencies and the strong dependence of the hybridization on the aspect ratio, the plots illustrated in Figure 4 can be very practical in guiding the nanoengineering of the optical response. Resonances of the Drude Nanoshells. In Figure 5a the aspect ratio dependence of the fractional change in the resonance frequencies of two typical Drude nanoshells (Ag and Au) is examined for various core materials with εcore ranging between 1.0 and 12.0. In panel a the embedding medium is water (εM = 1.768) and the color coding for

Figure 3. Quantification of the εcore dependencies of ΔωL,± and the λL,± rulers at different ρ and different f (0.1, 0.5, and 5.0). εM = 1.768 (water), ε∞ shell = 1.5, and ℏω0 = 1.75 eV (λ0 ≈ 708.5 nm).

The presentations exemplified in Figures 2 and 3 are practical as rulers for guiding the design of the optical response of the Lorentzian nanoshells. Such rulers would be rather precise given that the order of magnitude of f is known since the remaining parameters are either known (εcore and εM) or can be determined independently to a good degree of accuracy (ε∞ shell and ρ). Conversely, for a shell material specified by ω0 and ε∞ shell the experimentally determined resonance wavelengths can be used to determine f. In this respect, the presentations given in Figure 4 may be more practical in which the f dependence of F



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are more effective red-shifters at increasing values of all dielectric constants: for example, the Au nanoshells, with larger ε∞ shell, are more effective red-shifters than their Ag counterparts; the core−shell systems with a larger core dielectric and embedded in media with larger dielectrics are better redshifters. For the Ag nanoshells, λD,+ is tunable in the UV and the blue part of the visible whereas the tuning of λD,−’s extends from blue to IR. For the Au nanoshells, λD,+ is mainly tunable in the blue and the tuning of λD,−’s extends from red to IR. ∞ The dependencies of λD,+ and λD,− on εshell and the correlation between the spectral tuning and the ε±shell(re) hybridization are clearly shown in these Figures, in compliance with the nanoenvironmental tuning of ε±shell(re) by εM, εcore, and ρ. The primitive mode energies, which are already red-shifted by the conduct of Drude dispersion, hybridize strongly with the increasing aspect ratio. A pair of resonances, in which both peaks are red-shifted with respect to ωP, emerge. The relative ordering of their energies and their shift tendencies over the respective primitive modes are the same as in the case of Lorentzian nanoshells. That is, independent of the dispersion, the + modes are always of higher energy and progress as a result of the blue shifts over the respective primitive mode. Despite the blue shifting, λD,+’s still remain on the red side of λP at all aspect ratios. While at the lower energies, the − modes progress as a result of the red shifts over the respective primitive mode, and the limiting value of λD,− at ρ = 1 is infinite. A pronounced asymmetry between the two modes is observed at increasing aspect ratios. Following the ε±shell(re)λD,± correlation implied by eq 8b λD,−‑ shows larger sensitivity to the variations in the aspect ratio and can be tuned in a wider wavelength range compared with the λD,± resonances. Figures 5 and 6 provide convenient rulers for engineering of the optical response of the Drude nanoshells that to the best of our knowledge have not been provided in that systematic elsewhere in the literature. Implications for the Plexcitonic Systems. Excitonplasmon coupling has been customarily characterized by the exhibition of a pair of absorption bands split around the nearly degenerate in-solution exciton and core/shell plasmon resonances.12−20 In a recent contribution we have studied the classical electrodynamics-based response of a plexcitonic system with a prolate ellipsoidal noble-metal core coated by a uniform Lorentzian shell.29 In that contribution we have illustrated that the LSPR of the core couples to the blue-most resonance of the Lorentzian/excitonic shell, while the lower energy excitonic resonance remains uncoupled. In the current study we explicitly show that the excitonic doublet is a generic property of spherically symmetric Lorentzian/excitonic shells. The absorption spectra of the Lorentzian nanoshells and the means of coupling between a plasmonic core or a shell and the excitonic resonances of the shell remain to be studied further. The following statements can be made if the abovementioned mode of exciton-plasmon coupling holds generally for the spherically symmetric core−shell complexes. First, the absorption spectra would quite generally display three peaks: an exciton-plasmon hybrid pair and an uncoupled excitonic band. The pair should experience band narrowing owing to its coupled nature.22,29 Second, the coupling would be maximized if the plasmon and the blue-most excitonic transitions were in

Figure 5. (a) Aspect ratio dependencies of ΔωD,± = (ωP − ωD,±)/ωP ∞ for the Ag (decorated, ε∞ shell = 4.8) and the Au (εshell = 9.84) nanoshells at different εcore for the core−shell particles in water, εM = 1.768 for ℏωP= 9 eV (λP ≈ 138 nm). The color-coding for εcore is the same as in Figures 1 and 2. (b) λD,± ruler for λP ≈138 nm. (c) Variations in the aspect ratio dependencies of ΔωD,± for the Ag (gray) and the Au (yellow) nanoshells coating a silica core (εcore = 2.04) for the core− shell particles embedded in different media: εM values are 1 (air, short dashes), 1.768 (water, dashed-dottted), 2.5 (long dashes), and 3.5 (solid).

different εcore curves follows that of Figures 1 and 2 and is indicated at the bottom. In Figure 5c the Au (yellow) and Ag (gray) nanoshell resonances at several different εM and a fixed εcore (Si) are shown. The εM values used are: 1 (short dashes), 1.768 (dashed-dotted), 2.5 (long dashes), and 3.5 (solid). The fractional changes in the resonance frequencies are defined as ΔωD,± = (ωP − ωD,±)/ωP which sets a ωP-independent ruler for the red shifts. The corresponding wavelength values (λD,±) are quantified in panel b. In Figure 6 part of data of Figure 5 is re-evaluated to display the εcore-dependence of ΔωD,± and ΔλD,± in a wide range of different aspect ratios. The properties communicated through Figures 5 and 6 have already been discussed in the literature extensively.2−7 Here we merely remark on these previous results to highlight the functioning of the hybridization scheme and to allow comparison with the Lorentzian nanoshells. The spectral signature of the Drude nanoshells is a pair of resonances, both of which are red-shifted with respect to the bulk resonance of the shell material, ωP. The Drude nanoshells G



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∞ Figure 6. Quantification of the εcore dependencies of ΔωD,± and the λD,± rulers for the Ag (ε∞ shell = 4.8, a and b) and the Au (εshell = 9.84, c and d) nanoshells at different ρ for the core−shell particles in water, εM = 1.768 for ℏωP = 9 eV (λP ≈ 138 nm). The color coding for different ρ is also given. Note that the core is always optically inactive.

resonance as opposed to the usual discussion in the literature that it is maximized when the plasmon and the in-solution exciton bands are in resonance. It is promising to recognize that these two features can in fact be accommodated into some of the data existing in the literature.18,20

This separation allows a convenient control for a bottom-up manipulation of optical response of nanoshells with different dispersions. The resonances of the Lorentzian/excitonic shells are evaluated in comparison with the Drude/plasmonic shell resonances. Contrary to Drude nanoshells well-known to be characterized by a pair of resonances that are red-shifted with respect to the bulk plasmon resonance, the spectral signature of the Lorentzian nanoshells is a pair of resonances that are blueshifted with respect to the in-solution excitonic resonance. As is well known, Drude red-shifting is enhanced at increasing aspect ratios (“decreasing shell thickness”) and Drude nanoshells are more effective red-shifters at increasing values of the dielectric constants (core, shell, and embedding medium). For the Lorentzian shells primitive modes that are already blue-shifted by the excitonic oscillator strength hybridize strongly with the increasing aspect ratio, while the blue-shifting is suppressed at increasing dielectric constants. Practical resonance rulers that cover the blue-shifting and the red-shifting dependencies on the experimentally controllable parameters are provided for both cases. It is suggested that the results on the resonances of the Lorentzian shells can be instrumental for manipulating the optical response of plexcitonic nanophotonic devices.

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CONCLUSIONS Hybridized optical resonances of the spherical nanoshells with optically inactive core and embedding medium are examined in the quasistatic approximation of classical electrodynamics. It is emphasized that the origin of the hybridization is the nanoenvironmental tuning by the core, the embedding medium, and the aspect ratio of the core−shell system. The hybridization interactions that mix the two primitive modes of the spherical shells (cavity and sphere) for arbitrary values of these parameters are identified in an asymmetric two-state scheme. A natural consequence of the nanoenvironmental tuning is that resonance hybridization should hold for any shell material that can support different excitations such as plasmons and excitons. Thereby a methodology is developed in which the dependence of the resonance frequencies on the dispersive properties of the nanoshell material and the functionality of the nanoenvironment in tuning these resonances are separated. H



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone/Fax: +90 (312) 210-5099. Notes

The authors declare no competing financial interest.

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Optical Response of Lorentzian Nanoshells in the Quasistatic Limit

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