Optical Properties of a Particle above a Dielectric Interface: Cross

Article pubs.acs.org/JPCC

Optical Properties of a Particle above a Dielectric Interface: Cross Sections, Benchmark Calculations, and Analysis of the Intrinsic Substrate Effects Jean Lermé,* Christophe Bonnet, Michel Broyer, Emmanuel Cottancin, Delphine Manchon, and Michel Pellarin Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS, Université de Lyon, Bât. A. Kastler, 43 Bld du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France S Supporting Information *

ABSTRACT: We show that the optical properties of a particle above a plane dielectric interface differ dramatically from those of the same particle embedded in a homogeneous matrix. Calculations for gold and silver spheres have been carried out in using the exact multipole expansion method, providing thus benchmark results for testing the accuracy of the available numerical methods. For silver spheres, the dependence of the extinction cross-section has been studied in detail as a function of the parameters characterizing the particle/ interface system, namely, the radius of the sphere, the particle-surface distance, and the dielectric index of the substrate, as well as those characterizing the light excitation, that is, the angle of incidence and the polarization. Throughout this study we have separated the effects arising from the inhomogeneity of the applied field (interference between the incoming and reflected plane waves) from the intrinsic substrate effects resulting from the interaction with the induced surface charges on the surface. These last effects are, in the present formalism, encoded in the reflected scattered field impinging on the particle. For particles close to the interface, a rich multipolar plasmonic structure is observed, which can be described in the frame of a hybridization scheme similar to that developed for dealing with layered particles or dimers. Comparison with approximate models is also provided. single optically probed nano-object. 14−33 Among these techniques, the spatial modulation spectroscopy (SMS) method is, to our knowledge, the only far-field quantitative technique allowing the absolute extinction cross-section of small single particles to be measured.22,34−37 However, detailed intrinsic information about a specific nano-object can be gained from any single-particle optical spectrum on the condition that precise experimental conditions are taken into account in the theoretical analysis, in addition to the intrinsic parameters of the particles (size, shape, ...), namely, the irradiation conditions and the particle environment. The influence of the spatial distribution of the illuminating convergent light beam (often diffraction-limited) that is used to address optically a single nano-object has been tackled in detail by several authors within the generalized Mie theory.38−42 Second, the 3D material environment in the vicinity of the particle, an issue that is addressed in this work, may have a strong influence on the optical response, especially with regard to the spectral location of the LSPR bands in the case of metallic species (referred to as environment/dielectric/effective matrix effects). In most experiments, the particle under

1. INTRODUCTION Nanostructured materials and metal nanoparticles are now routinely used in a wide range of technological and medical areas, for instance, catalysis, sensing and optical applications, diagnosis and therapy in medicine.1−8 For any potential application, tailoring and optimizing the properties of devices comprising nanoparticles, it is required to know the intrinsic properties of the individual nano-objects. Among the various available characterization tools, optical techniques have received particular attention for a long time.9,10 Noble metal nanoparticles are of particular importance due essentially to the collective excitations of their conduction electrons, known as localized surface plasmon resonances (LSPR).9 The strong dependence of the LSPR frequencies on the size, shape, and local particle environment explains the prolific literature devoted to these metallic species at the nanoscale.8−12 These dependencies, highly desirable for developing powerful optical, sensing, and monitoring applications, make the standard spectroscopic methods on ensembles13 unsuitable for investigating intrinsic single particle properties. These last years, sensitive near/far-field techniques have been developed in order to probe properties of individual nano-objects. Moreover, many research groups have successfully combined optical measurements and high-resolution imaging techniques in order to correlate the optical response with the exact morphology of the © 2013 American Chemical Society

Received: December 11, 2012 Revised: February 26, 2013 Published: March 15, 2013 6383

dx.doi.org/10.1021/jp3121963 | J. Phys. Chem. C 2013, 117, 6383−6398

The Journal of Physical Chemistry C

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morphology of the nanosystem. Indeed, loss of accuracy is expected when very short length scales are present in the system, especially in the presence of close dielectric interfaces. In such cases, for example, in single nanodimers near the conductive contact limit,27 the complex and enhanced local electromagnetic fields that are induced in the interparticle region in the plasmon spectral ranges result in an unavoidable decrease of the numerical accuracy.79 These unavoidable drawbacks are at the origin of the unceasing literature reporting on tests of numerical accuracy, respective advantages and shortcomings, and comparative analyses of the mostly used numerical methods.75−77,80−82 In this context, benchmark computations obtained from exact analytical theories on various systems (geometry, optical indexes, irradiation conditions) are of tremendous interest for assessing the accuracy of the numerical methods. The main purpose of this paper is to provide benchmark results in the case of a metal sphere on or near a plane dielectric surface for which an exact solution can be obtained within the generalized Mie theory.44−46 Owing to the high speed of the computations, the respective influence of many parameters ruling the optical response is reported on and analyzed through numerous figures (in the main text, section 3, and the Supporting Information): (1) material (silver and gold particles, dielectric index of the substrate), (2) particle/interface distance, (3) sphere radius, (4) angle of incidence, and (5) polarization of the incoming plane wave. The theory, which has been developed in previous works by several authors,45,46 is summarized in the Supporting Information where compact formulas for the integrated scattering, absorption, and extinction cross sections are also provided. Because previous theoretical works on the particle/interface system have focused on the influence of the interface on the angular distribution of the scattered field in the far-field region,45,46,83 the benchmark results reported on in this work will be restricted to integrated extinction cross sections. In addition, “exact” results have been compared with the predictions from the HM, multipolar image charge, and exact electrostatic models, as well as with those obtained from DDA calculations where the substrate is represented in the DDA target by a cylindrical dielectric slab of appropriate radius and thickness.8,52,55,62 A brief summary of these comparative tests is reported in section 4. Finally, this work is summarized in section 5.

investigation is deposited on a substrate, the surface of the particle thus experiencing different “dielectric environments” (embedding medium and substrate, characterized by different optical indexes).43 This issue will be studied in detail in this work in the case of a spherical particle in the frame of an exact multipole expansion based method44 published independently by Fucile and co-workers45 and Wriedt and Doicu,46 following the pioneering work by Bobbert and Vlieger.47 The substrate effect has received great attention these last years. A widespread simplifying hypothesis is to assume that the substrate effect can be mimicked through that from an effective homogeneous embedding medium characterized by an appropriate refractive index Neff (approach referred to as the “homogeneous matrix (HM)” model). This crude assumption, allowing the LSPR frequency to be shifted in a very simple way, is rooted in the observation that, in many experiments on ensembles, the main conspicuous substrate-induced effect is a systematic red shift of the LSPR with respect to “airembedded/particle” calculations.48 In order to quantify this red-shift, several phenomenological models, differing in the respective embedding medium and substrate index weightings selected for setting the effective index Neff, have been introduced in the literature.9,49−51 Assessing the relevance of the crude HM model is clearly of outstanding importance in the context of the development of nanoplasmonics-based sensing applications and for gaining reliable physical information from optical measurements on single particles. Recently, numerous experimental and theoretical works have tackled this important issue.8,18,52−70 Most experiments have been carried out at the single particle level (dark-field scattering spectroscopy), although reliable results can also be obtained from regular arrays of nanoparticles produced by lithographic techniques (UV−vis extinction spectroscopy measurements).52,53,55 Results on single particles of various sizes and shapes involving different particle/substrate geometries or irradiation conditions have shown that the substrate effects are complex, highly systemdependent,71 and may lead to optical responses noticeably different, quantitatively and qualitatively, from those predicted within the crude HM model. These specific effects are essentially due to the interaction with induced surface charge distributions on the embedded-medium/substrate interface and the strong inhomogeneity of the local field experienced by the particle.72 In some cases, these specific features can be qualitatively understood and analyzed within the “molecular-like” hybridization model introduced by Prodan and Nordlander73 and the well-known image charge (IC) approach.9,74 These electrostatic models (retardation effects neglected) provide intuitive physical frameworks for rationalizing the optical data obtained from different nanosystems, in particular, the multimode pattern of the spectra. When arbitrarily shaped particles are involved, the optical data must be simulated using numerical electrodynamics methods, such as the discrete-dipole approximation (DDA), the finite element method (FEM), the finite difference time domain method (FDTD), or the boundary element method (BEM).75−78 When using the numerical electrodynamics methods, the accuracy of computations is unavoidably far to be perfect, owing to memory size and computational time constraints, and their respective efficiency may depend noticeably on both the system in hand or the property to be computed.75−77 For instance, in the FEM and FDTD methods, the numerical accuracy depends crucially on the number and the size of the small-volume cells filling the finite computational domain, as well as on the

2. THEORY: FORMAL SOLUTION AND CROSS-SECTION FORMULAS The geometry of the particle/substrate system is displayed in the top of Figure 1 in the case of a spherical particle. In the bottom of this figure, the arrows symbolize the various electromagnetic fields that are involved. The black arrows are associated with the incoming, reflected, and transmitted plane waves (Ei, Ei,R, and Ei,T, respectively). The gray ones correspond to the (“primary”) scattered field Es, the (“secondary”) reflected scattered field Es,R (the scattered field reflecting off the surface and striking the particle), and the transmitted scattered field Es,T. The computations reported on in this work have been carried out thanks to a home-developed computer code based on the exact multipole expansion method.44−46 The formal, and quite general, solution of the problem in hand is outlined in the Supporting Information (section A), to which the reader is referred for the notations and the field expressions. It should be emphasized that the convenient VSHs used (the same as those defined in our previous papers, refs 39 and 40) differ from 6384



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and reflected plane waves) and of the scattered field Es, respectively, and E0 is the electric field amplitude of the incoming plane wave. It should be pointed out that the compact formula for extinction, eq 1, is indeed identical to the general formula established in the case of a particle embedded in a homogeneous dielectric medium (see, for instance, eq 5.18a, page 119 in ref 77). Note, however, that both the incoming and the reflected plane waves enter the expression. On the other hand, the formulas for the scattering and absorption contributions (eqs 31 and 34 in the Supporting Information) differ from those derived for homogeneous matrix-embedded particles (eq 5.18b in ref 77) and involve explicitly the expansion coefficients of the reflected scattered wave. It is also worthwhile emphasizing that, depending on the VSHs used, “different” general formulas can be found in the literature (see, for instance, the widespread formulas eqs 4.61 and 4.62 given in the textbook by Bohren and Huffman, ref 10, in the case of a spherical particle).

3. RESULTS In this section we report on extinction cross sections computed in the frame of the analytical formalism summarized in the Supporting Information, section A. The results point out that the optical response of a single metallic nanosphere (radius R) on or near a dielectric interface depends on numerous extrinsic parameters: (i) the “relative” geometry of the system (d/R is actually the relevant parameter; see Figure 1), (ii) the dielectric index ratio N2/N1 that governs the magnitude of the Fresnel reflection coefficients (see Figure 2 in the Supporting Information), (iii) the incoming beam-related parameters, namely, the angle of incidence θi, and the polarization of the incident electric field with respect to the plane of incidence (p- or s-character). We will see that in most cases of physical interest, that is when d ≈ R, the extinction spectra exhibit spectral features that are not expected when the particle is embedded in an effective homogeneous medium. Prior to presenting these results, some remarks deserve to be stated. (1) First, it should be stressed that the respective effects induced by the previously listed parameters are in general strongly entangled. In particular the evolution of the extinction spectrum that is observed when a given parameter is varied may depend on the values of the other fixed parameters. Only the major systematic trends or conspicuous spectral pattern changes are exemplified throughout this work. On the other hand, only simple intuitive arguments are brought forward for interpreting the observed trends. Actually, for gaining a comprehensive physical interpretation of the multimodal pattern of the extinction cross sections and its evolution as a function of the varying parameter, each spectrum ought to be analyzed in the frame of the electrostatic models quoted in the introduction and through field intensity or surface charge distributions computed for suitably selected wavelengths. Nevertheless, we have used an efficient alternative strategy for unambiguously analyzing the multimodal extinction spectra (subsections 3.4 and 3.5). (2) Second, the results reported in this work are restricted to the case of a spherical particle and, thus, to a specific geometry of the effective capacitor formed by the facing surfaces of the particle and the medium/substrate interface. Recent works have shown that, in the case of

Figure 1. Upper figure: geometry of the embedded-particle/substrate system. k(θi,ϕi) is the wavevector of the incoming plane wave in medium 1. θi is the angle of incidence. The Nis are the optical indexes of the three homogeneous media. In the lower figure, the arrows symbolize the various electromagnetic fields that are involved in the problem.

those involved in refs 45 and 46, and the respective explicit formula cannot be strictly compared (for instance, eqs 15−18 in the Supporting Information). Let us remark that other relevant works, some of them based also on the generalized Mie theory, have been devoted to this problem.83−88 However, simplifying assumptions (image-charge or electric-dipole approximations), avoiding the numerical difficulties associated with the integration in the complex plane (eq 19 in the Supporting Information), yield only approximate, and often crude, solutions in many cases of physical interest, as it will be discussed in section 4. Let us quote also the work by Stefanou and Modinos who investigate the scattering of light by a periodic two-dimensional array of spherical particles adsorbed on a homogeneous dielectric slab,89 as well as those by Doicu and co-workers (approximate solution for nonaxisymmetric particles)90 and by Takemori and co-workers (spherical particle on a metal substrate).91 Compact formulas for the integrated scattering, absorption, and extinction cross sections are reported on in the Supporting Information. The detailed derivation of the relevant formulas appropriate for the particle/interface system is rather technical and is detailed in section A of the Supporting Information (eqs 31, 33, and 34 correspond to the time-averaged powers (Wi) dissipated by scattering, extinction, and absorption, respectively; the cross sections (Ci) are obtained by dividing the dissipated powers by the intensity (I0) of the incoming plane wave). For extinction, the relevant formula is 4π 1 * + Q bnm *] Cext = Cabs + Csca = − 2 Re ∑ [Pnmanm nm k |E0|2 n,m (1)

where the (Pn,m,Qn,m)s and the (an,m,bn,m)s are the expansion coefficients of the total applied field Einc (sum of the incoming 6385



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gold and silver dimers, the plasmonic couplings are strongly sensitive to the exact geometry of the effective interparticle capacitor,27,31 and this feature is expected to hold true in the case of the particle/surface system near the contact limit. In consequence, specific effects might be expected in the case of flat nano-objects, as nanoprisms or nanocubes, which present large facing surfaces in the quasi and strict contact limit (supported particles). (3) Third, it should be emphasized that, as compared to the simple system consisting of a particle embedded in a homogeneous matrix, the presence of a dielectric interface separating two homogeneous media gives rise to two specific electromagnetic processes leading to quite different effects. The first process corresponds to the reflection of the incoming plane wave and results in a strongly inhomogeneous applied field intensity distribution (|Einc(r)|2) in the embedding medium, especially for large N2/N1 ratios (see Figures 3 and 4 and eq 9 in the Supporting Information). From our previous works devoted to the optical properties of nanospheres in a tightly focused light beam39,40 it is expected that the absolute magnitudes of the energy rates dissipated by absorption and scattering will be correlated to the local applied field intensity |Einc(d)|2 (eq 9 in the Supporting Information) and will follow the so-called “local intensity approximation”39 in the case of particle sizes noticeably smaller than the wavelength (for small particles, the absolute cross sections, normalized to the local intensity, are close to those computed within the standard Mie theory). This “trivial” scaling effect, which will be referred to as the “extrinsic substrate effect”, is indeed the major effect for large d/R values, namely, when the reflected scattered field Es,R can be neglected in the vicinity of the particle. The second process corresponds to the reflection of the “primary” scattered wave (Es) and results in the presence of a “secondary” scattered wave (Es,R), which contributes to the effective applied field impinging on the particle. The specific effects rooted in the reflected scattered wave will be referred to as the “intrinsic substrate effects”. Indeed, the reflected scattered wave Es,R underlies all the physical ingredients that are explicitly involved in the popular electrostatic models (induced surface charge distributions, interaction with the image charges/dipoles, plasmon hybridization, mode coupling and splitting, plasmonic excitations of higher orders, ...). The intrinsic substrate effects, which give rise to a complex spectral response, are observed in many cases of physical interest (large N2/N1 ratios and d ≈ R). In this work, we use the experimental bulk metal dielectric functions of gold and silver given by Johnson and Christie,92 and linear interpolation between consecutive data has been applied. All the spectra correspond to absolute extinction cross sections and can be used as benchmark results for testing numerical methods (the numerical accuracy of the present calculations is discussed in the Supporting Information, section C). 3.1. Extinction at Normal Incidence. Extinction spectra at normal incidence of the incoming beam (θi = 0) for N1 = 1 and refractive substrate indexes in the range N2 = 1 → N2 = 2.4, for a gold sphere of radius R = 10 nm located just above the interface (d = 10.1 nm) are shown in Figure 2a. A strong decrease in the magnitude of the extinction cross-section and a slight red-shift of the plasmon band with increasing N2 values can be noticed. This last feature, quite consistent with the HM

Figure 2. (a) Extinction cross sections Cext of a gold sphere of radius R = 10 nm in air (N1 = 1) located just above the interface (d = 10.1 nm), at normal incidence of the incoming field (θi = 0), for several refractive substrate indexes in the range N2 = 1 (no interface, black full line curves) → N2 = 2.4 (black dashed line curves). Color code: red (N2 = 1.2), blue (N2 = 1.4), magenta (N2 = 1.6), green (N2 = 1.8), gray (N2 = 2), violet (N2 = 2.2). (b) Normalized cross sections Cext|E0|2/|Einc(d)|2. (c) Results obtained when the reflected scattered field Es,R is disregarded in the formalism.

model predictions, was expected because increasing the refractive substrate index amounts to enlarging the average effective dielectric index of the particle environment (the results of the HM model for gold and silver spheres of radii R = 10 and 50 nm are displayed in Figure 7 in the Supporting Information). On the other hand, the first feature disagrees with the HM model prediction (strong increase). Actually, an enhancement of the plasmon band should be observed because the red-shift results in a weakening of the interband-induced damping contribution. This rather unexpected trend is a direct consequence of the extrinsic substrate effect which is not specifically discussed when the optical response of the particle/ interface system is addressed. In Figure 2b are displayed the normalized spectra Cext|E0|2/|Einc(d)|2, namely, the spectra plotted in Figure 2a multiplied by the wavelength-dependent “local intensity factor”, which takes into account the change of the local applied electric field resulting from the interference between the incoming and the reflected waves (see eq 9 and Figure 4 in the Supporting Information). Extinction crosssection spectra Cext(λ) multiplied by the local intensity factor 6386



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will be referred to as “normalized extinction spectra” throughout this work.93 The displayed trends as a function of the N2 value are now in total accordance with those of the HM model with respect to both the absolute magnitude of the cross-section and the spectral shift of the plasmon band. The slight red-shift is indeed rooted in the reflected scattered field Es,R impinging on the particle (intrinsic substrate effect). The Figure 2c shows the results when Es,R is neglected in the formalism, that is, when the matrix R is set equal to zero in eq 20 in the Supporting Information. In this case, no spectral shift of the plasmon band occurs when N2 is varied and, moreover, all the curves normalized in including the “local intensity factor” are perfectly superimposed (not shown), except in the very short-wavelength range where the N2-dependent applied field-inhomogeneities induce particle radius-dependent multipolar excitations (differences of the order of a few percents for R = 10 nm). Actually, for gold, the intrinsic substrate effects are strongly blurred by the coupling and damping of the free electron collective excitations with the interband transitions. For silver the free electron plasmon excitations occur below the interband energy threshold (on the order of 4 eV) and extrinsic as well as intrinsic substrate effects are expected to be much more conspicuous. This statement is clearly evidenced in Figure 3

phenomenological HM model. In both approaches, this spectral shift can be merely interpreted as resulting from the increase of the dielectric environment-induced screening of the Coulombic restoring force between the conduction electron gas and the positively charged ionic background.94 For large N2 values, the reflected scattered field Es,R contributes noticeably to the effective applied field impinging on the particle (large Fresnel reflection coefficients). This gives rise to large local intensity inhomogeneities which, consequently, lead to the emergence of multipolar excitations and substrate-induced mode splittings, which are not observed in the case of small particles subject to a plane wave in the IR-visible-near UV spectral range. The shoulders and multipeak patterns computed for large N2 values are the signatures of these intrinsic substrate effects and underlie complex near field and polarization surface charge distributions. Concerning the magnitude of the absolute cross sections a large part of the N2-dependence has to be attributed to the local intensity factor (compare Figure 3a and b). Despite the taking into account of the local intensity correction, and as compared to gold, the absolute magnitude of the spectra remains noticeably N2-dependent. This large residual N2-dependence, which is actually opposite to that predicted in the HM picture (if one excepts the range N2 < 1.4, where only a tiny red-shift of the dipolar plasmon is observed in Figure 3b), is the consequence of the enlargement of the plasmonic spectral range due to the enhancement of the Es,R-related intrinsic substrate effects. This spectral broadening with increasing N2 values stems from (i) the sequential emergence and concomitant shift of additional multipolar excitations (increase of the spatial inhomogeneity of the local near-field intensity), (ii) the increase of the substrateinduced energy splitting of degenerate modes, and (iii) the sharing of the oscillator strength corresponding to bright modes (essentially of dipolar character) with several spectrally distinct other, initially dark, modes, for instance, the quadrupolar modes or eventually higher-order multipolar modes, depending on the N2 value, which cannot be excited in a homogeneous field intensity distribution. These effects could be described in the frame of the popular electrostatic models in terms of interactions of the particle plasmon modes with the charge distributions of the particle image or substrate-induced plasmon mode hybridization. The spectral red-shifts and complex multipolar patterns with increasing N2 values are rubbed out when the reflected scattered field Es,R is disregarded in the modeling, as shown in Figure 3c. As for a gold sphere, all the curves in Figure 3c are superimposed when the cross sections are normalized to the local intensity factor. The extinction cross sections for gold and silver spheres of radius R = 50 nm located just above the interface (d = 50.1 nm) are plotted in the Figures 8 and 9 in the Supporting Information. Except for some features related to the broadening of the plasmonic spectral band arising from the radiation damping and to the enhancement of the multipolar excitations, the substrate-induced effects are similar to those discussed previously. In the following only results obtained on silver spheres will be reported, because, in the case of gold spheres of small and moderate sizes, most intrinsic substrate effects are blurred by the coupling of the free electron plasmon excitations with the interband transitions. Let us remark that the general conclusions that will be drawn from the investigations on silver spheres are expected to hold in the case of gold nanoshells, strongly elongated gold particles, or gold spheres embedded in a matrix of very high refractive index N1, because these systems can exhibit red-shifted plasmon modes below the interband

Figure 3. Same as Figure 2, except for a silver sphere.

where similar computations are reported on for silver. The redshift of the dipolar plasmon is much more pronounced as compared to gold when N2 is enlarged, as also predicted in the 6387



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transition threshold. It should be noted that recent experiments have been carried out on large supported gold nanoshells with a silica core,61,64 to which the present multipole expansion formalism can be straightforwardly applied. 3.2. Influence of the Electric Field Polarization at Oblique Incidence. At oblique incidence of the incoming beam (θi > 0), the optical response is expected to be noticeably modified relative to the normal incidence case. These changes will affect both the extrinsic and intrinsic substrate effects and simple physical arguments can be advanced from the outset. First, the applied field intensity distribution |Einc(r)|2 near the particle, which depends on the N2 and θi values through the interference between the incoming and reflected waves, is strongly modified. This distribution depends on the polarization of the incoming electric field E0ei, on the one hand, because the respective Fresnel reflection coefficients for p- and s-polarized light excitations are different and, on the other hand, because the distribution depends on the Ep/Es ratio, even if rp ≈ rs (see eq 9 in the Supporting Information). Second, p-polarized incident light excitation at a given oblique incidence will produce much stronger intrinsic substrate effects as compared to a s-polarized incoming light. This feature has been observed and discussed recently in the case of gold nanoshells deposited on dielectric substrates.61,64 Two quite distinct relevant factors [labeled (i) and (ii) below] in support of this allegation can be brought forward, the first of them [factor (i)] being easily set out in the frame of the simple quasistatic IC model.56,67,70,74 In the IC model the presence of the substrate breaks the spherical symmetry of the initial problem and lifts the 3-fold degeneracy of the dipolar plasmon mode (frequency ω0) of the point-like polarizable particle through the interaction with the image dipole. The degeneracy removal yields a nondegenerate Dz mode (frequency ω1; dipole moment normal to the interface; oscillation of the conduction electron gas along the z-axis) and a 2-fold degenerate Dx,y mode (frequency ω2; dipole moment parallel to the interface; oscillation of the conduction electron gas along the x or y axis). Both new modes are red-shifted relative to the initial 3-fold degenerate mode and one has ω1 < ω2 < ω0 when N2 > N1, with a splitting all the larger that the ratios N2/N1 and R/d are larger. (i) First, for s-polarized light excitation, only the degenerate Dx,y mode can be excited, whereas the two spectrally distinct Dz and Dx,y modes can be excited in the case of p-polarized light excitation [the respective weights of both excitations depend to a large extent on the angle of incidence θi and on the Fresnel reflection coefficient rp(θi)]. A broader plasmonic band and a bimodal pattern are expected in this case for large enough N2/N1 ratios and θi values. However, it should be emphasized that when N2 is large, this simple analysis might be thought to fail. When the reflected scattered field Es,R contributes noticeably to the effective local field Eeff (= Einc + Es,R) impinging on the particle, the polarization of Eeff(r) near the particle may become unrelated with that of the applied field Einc(r). Therefore, one might conjecture that both Dz and Dx,y modes can be excited for any polarization of the incoming field. In fact, a thorough analysis of the symmetry of the problem indicates that only the Dx,y mode is coupled to the electromagnetic field for normal incidence or s-polarized light excitation at oblique incidence, as it will be proved in the following subsections.

(ii) The second relevant factor requires taking into account the finite size of the particle (and d/R not too large) and is related to the localization of the expected maxima of the induced charge densities on the particle surface. These zones of large charge density are strongly dependent on the angle of incidence θi and the polarization of the incoming field. In a first approximation, suitable for moderate N2/N1 ratios and, thus, a minor Es,R contribution in Eeff(r) [that is, when the polarization of the effective local field Eeff(r) is close to the one of the incoming field Ei(r) or Einc(r) if the reflected plane wave contributes noticeably], the charge density maxima and the magnitude of the induced substrate effects can be straightforwardly inferred in drawing the particle and its mirror image (see Figure 10 in the Supporting Information). For normal incidence, as well as for s-polarized light excitation at any oblique incidence, the two surface charge maxima at both ends of the particle diameter directed along the Ei(d) [or Einc(d)] direction are located at the distance d above the interface. On the other hand, for p-polarized light excitation, the polarization of the local applied field depends on the angle of incidence. The z-coordinates of the maxima at both ends of the particle diameter directed along the Ei(d) [or Einc(d)] direction become θi-dependent, and one of them approaches the interface when θi is increased. In particular, when θi ≈ 90° and d ≈ R, this maximum is very close to the surface and will strongly interact with the corresponding image charge located just below the interface. In consequence, for particles close to the interface, strong particle/substrate effects are expected in the case of p-polarized light excitation at large angles of incidence. This finite size effect can be paralleled with that responsible for the breakdown of the Δ−3-scaling law which is predicted in the popular quasistatic dipolarcoupling model (model assuming point-like polarizable particles) when applied to metal particle pairs near the conductive contact limit (Δ is the center-to-center distance).95−97 Obviously strong interactions occur in the present context only for large enough N2/N1 ratios because the image charges q′ are related to the real charges q in medium 1 by the equation q′ = −q(N22 − N21)/(N22 + N21). These preliminary comments are clearly supported by calculations carried out for a silver sphere of radius R = 10 nm located just above the interface (d = 10.1 nm) and subject to p- and s-polarized light excitations at oblique incidence (θi = 45°; see Figure 4). Note that, at this angle of incidence and in the p-polarized case, the applied field intensity at the particle center Einc(d) is independent of the wavelength [cos(2θi) = 0] and, moreover, rather close to unity for any N2 value in the range N2 = 1 → N2 = 2.4 (see eq 9 in the Supporting Information). So the normalized p-polarized spectra (not shown) are quantitatively on the same order of magnitude as those plotted in Figure 4a. The s-polarized spectra in Figure 4b are quite similar to those computed for the normal incidence (Figure 3). Actually all the observed differences in magnitude are rooted in the differences in the local applied field intensity. When the local intensity factor is taken into account the s-polarized spectra for θi = 45° are found identical to those computed at normal incidence (compare Figures 3b and 4c). This feature demonstrates that, for small spheres, the intrinsic 6388



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3.3. Influence of the Angle of Incidence for p- and s-Polarizations. In Figure 5 are displayed the normalized

Figure 5. Normalized extinction cross sections Cext|E0|2/|Einc(d)|2 of a silver sphere of radius R = 10 nm in air (N1 = 1) located just above the interface (d = 10.5 nm, N2 = 2.4) for several angles of incidence of the incoming wave: θi = 0° (black curve), 25° (red), 45° (blue), and 75° (magenta). The incident electric field is p-polarized.

extinction cross sections of a silver sphere of radius R = 10 nm in air (N1 = 1) located just above the substrate (d = 10.5 nm; refractive index N2 = 2.4), subject to a p-polarized light excitation, for several angles of incidence in the range [0°,75°]. Increasing further the angle of incidence does not change drastically the θi = 75° spectrum. The normalized spectra computed for a s-polarized light excitation (not shown), which are found identical to the black curve (θi = 0°), are perfectly superimposed, supporting therefore the “correlation rule” discussed previously. The four spectra exhibit a multimodal spectral pattern. Except for the spectrum at normal incidence, which consists of a single band centered at 365 nm with a noticeable shoulder near 355 nm on its blue rising side, all the curves display three distinct spectrally fixed bands that have to be considered as those characterizing the geometry defined by the length parameters R and d. As expected, the respective weights depend on the angle of incidence, namely, on the polarization of the local applied field Einc(d) impinging on the particle. Examination of the two extreme spectra (θi = 0° and θi = 75°) suggests strongly that the intermediate mode at 365 nm corresponds to the dipolar Dx parallel-mode and the most red-shifted mode at 379 nm to the dipolar Dz normalmode. When θi is increased, the respective weights of both modes evolve in total accordance with the relative contributions of the parallel and normal components of the applied field, thus supporting the previous mode assignment. Note however that, because of the overlapping of the broad bell-shaped modes, a reliable quantitative estimation of the respective weights cannot be achieved (see subsection 3.5). The blue mode near 355 nm has to probably be attributed to the quadrupolar or higherorder excitations. Similar results are displayed in Figure 6 for a silver sphere of radius R = 50 nm (d = 50.5 nm). The normalized spectra for a s-polarized light excitation are also shown, confirming the “correlation rule” previously inferred. The differences in the blue spectral range, below 375 nm, result from the changes of the spatial inhomogeneities of the applied field intensity |Einc(r)|2 when the angle of incidence is varied (changes of the magnitude of the multipolar contributions). Note that for large particles the radiation damping makes the precise labeling of

Figure 4. Extinction cross sections Cext of a silver sphere of radius R = 10 nm in air (N1 = 1) located just above the interface (d = 10.1 nm) at oblique incidence of the incoming field (θi = 45°) for several refractive substrate indexes in the range N2 = 1 (no interface, black full line curves) → N2 = 2.4 (black dashed line curves). The color code is given in the caption of Figure 2. (a) p-Polarized incident electric field. (b) s-Polarized incident electric field. (c) Normalized cross sections Cext|E0|2/|Einc(d)|2 for the s-polarized incident electric field.

substrate effects depend on the incoming beam parameters essentially through the polarization of the local applied field, and their magnitude can be inferred from the expected z-localization of the maxima of the induced charge density on the particle surface. On the other hand, the p-polarized spectra are much more broadened and structured, and the evolution as a function of the N2-value much more pronounced as compared to the case of the s-polarized light excitation. In this case very large red-shifts and conspicuous multimodal patterns are observed, confirming again the previous analysis. These spectacular polarization and intrinsic substrate effects, observed for large N2/N1 ratios, stress the inadequacy of the HM model for analyzing the optical properties of a single particle on or near a dielectric substrate. The unambiguous assignments of the multipeak patterns will be given in the following subsections, for both s- and p-polarized light excitations. Results for a large silver sphere of radius R = 50 nm located just above the interface (d = 50.1 nm), subject to p- and s-polarized excitations at oblique incidence, are displayed in Figure 11 in the Supporting Information, where specific comments are also provided. 6389



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Figure 7. Normalized extinction cross sections Cext|E0|2/|Einc(d)|2 of a silver sphere of radius R = 10 nm in air (N1 = 1) located above a substrate of high refractive index (N2 = 2.4), as a function of the distance d. The particle is subject to a p-polarized light excitation with an angle of incidence equal to θi = 45°. The successive spectra have been shifted along the vertical coordinate (shift equal to 500 nm2). In the d = 10.1 nm spectrum are indicated the dipolar Dx (shoulder) and Dz excitations.

Figure 6. Same as Figure 5, for a silver sphere of radius R = 50 nm and d = 50.5 nm. (a) p-Polarized incident electric field. (b) s-Polarized incident electric field.

the observed broad peaks or shoulders difficult in terms of multipolar excitations. Both figures clearly demonstrate that the beam-related parameters, especially the ratio between the perpendicular and parallel components of the applied field Einc close to the particle (eqs 10−12 in the Supporting Information), are of great importance for interpreting the optical properties of the particle/substrate system. 3.4. Influence of the Distance d. As emphasized in the beginning of section 3, the larger the particle/interface distance d, the smaller will be the intrinsic substrate effects, for a given refractive index ratio N2/N1. Indeed, for very large distances, the dissipated powers appropriately normalized to take into account the extrinsic effects, are expected to be identical to those of an isolated particle embedded in a homogeneous medium of refractive index N1. In particular, in the case of a small silver sphere, a single Lorentzian curve-shaped dipolar plasmon band will be observed. Besides, for moderate distances and small particles (d/R sufficiently large), the trends of the quasistatic IC model are expected to be verified. Thanks to these features, which are predicted for large enough d/R ratios, the evolution of the optical response as a function of the distance d can be used as a powerful tool for analyzing the structured multimodal spectra corresponding to the cases of physical interest (d ≈ R). These preliminary remarks are strongly supported by the normalized extinction spectra plotted in Figure 7, which correspond to a silver sphere of radius R = 10 nm above a substrate of high refractive index (N2 = 2.4; N1 = 1), subject to a p-polarized light excitation (θi = 45°). The normalized spectrum for d = 100 nm (λLSPR ≈ 356 nm; bandwidth ≈ 10 nm) is identical to that of an isolated sphere embedded in a homogeneous medium of index N1 = 1, and this figure shows that the convergence toward the HM model is almost effective for d ≥ 2R. When the particle approaches the interface the predictions of the IC model are clearly observed. On the one hand, the LSPR-band is red-shifted and, on the other hand, the

bandwidth is progressively enlarged and a shoulder appears on the blue side of the LSPR band. These last features are the signature of the onset of the lifting of the spectral degeneracy between the Dx and Dz dipolar modes. For N2 = 2.4, d = 10.1 nm, and θi = 45°, the square modulus of the ratio E⊥inc/E∥inc is close to 3 (see Figure 5 in the Supporting Information). This explains why the Dx band is much lower than the Dz band. When the distance d is decreased further, additional peaks which correspond to higher-order excitations induced by the inhomogeneous field distribution created by the image dipole appear in the blue spectral range. For instance, between 350 and 370 nm, a single and two additional multipolar modes are clearly visible in the d = 10.5 nm and d = 10.1 nm spectra, respectively. In the following subsection we will see that each of these extra peaks is in general underlain by several spectrally distinct higher-order multipolar excitations. This figure proves that the labeling of a complex structured spectrum may be reliably determined, without computing 2D/3D field distributions or charge densities at several wavelengths, in studying its steady evolution as a function of a suitably selected parameter (here, the distance is d). 3.5. Plasmon Mode Analysis of the Intrinsic Substrate Effects. Sets of extinction spectra similar to those plotted in Figure 7 can be computed for analyzing and labeling the multipolar patterns of the spectra displayed in previous figures. A powerful method, allowing to bring out the multipolar substructures underlying most of the broad bell-shaped bands in the near UV spectral range, consists in suitably modifying the Drude-like dielectric function [εs(ω)] corresponding to the conduction electrons.98 We explain briefly the “strategy” used. For large particles, the broadening of the individual collective excitations is due mainly to the radiation damping mechanism, but for smaller sizes, typically for R < 20 nm, the line width Δλ (≡ℏΔω on the photon energy scale) is exhausted essentially by the intrinsic electron energy losses characterizing the bulk material. In the Drude parametrization of εs(ω), the dissipation 6390



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light excitation at the oblique incidence θi = 45°. Results for p-polarized light excitations at θi = 25° and θi = 75° are shown in Figures 12 and 13 in the Supporting Information. Prior to analyzing the results let us point out that the analytical Drudefunction parametrization of εs(ω), as well as the KK extraction of εib(ω) from the experimental data, are not perfect, explaining why the spectra computed in using the modified dielectric function εmod(ω) are slightly red-shifted relative to those computed with the experimental one ε(ω) = N23(ω) (compare, for example, Figures 7 and 9 where a global shift of about 10 nm is clearly evidenced).102 Nevertheless, one can see that the multipolar spectral patterns are identical, except for this global shift on the wavelength scale and, of course, the different curve smoothing. Obviously the magnitude of the cross sections is strongly enlarged in using the modified Drude function since the integrated oscillator strength of each resonance is now concentrated in a spectral range five times narrower. These figures clearly prove that broad structures correspond in many cases to several spectrally distinct multipolar electronic excitations. Detailed information can be gained from these four figures (Figures 8 and 9 in the main text; Figures 12 and 13 in the Supporting Information).

is characterized by the electron scattering rate Γ, which is directly related to the intrinsic LSPR line width through the equation ℏΔω = ℏΓ.98 The strategy consists in reducing the intrinsic spectral widths of the plasmon bands. First, the dielectric function corresponding to the interband transitions has been extracted from the experimental complex dielectric function of silver by a Kramers−Kronig (KK) analysis, as described for gold in a previous work.99 The values of the free electron density nc and of the effective optical electron mass mopt in bulk silver, allowing to set the value of the plasma frequency entering the Drude parametrization (ωp ≈ 9 eV; see note 98), have been taken from available data.100,101 Finally we have chosen the value ℏΓ = 19 meV for setting the electron scattering rate (a narrow line width Δλ ≈ 2 nm will be thus ensured). Figures 8 and 9 show the results obtained for a small sphere of radius R = 10 nm in air above a substrate of refractive index N2 = 2.4 for, respectively, normal incidence and p-polarized

(1) First, the predictions of the IC model (degeneracy removal and d-dependence of the spectral locations of the parallel and normal dipolar modes) are clearly put to the fore for d-values not too close to R. This allows to label unambiguously the two most red-shifted bands, namely, the Dx and Dz modes when θi > 0, which evolve steadily when d is decreased. For normal incidence, only the Dx mode is excited. Thanks to the narrowing of the individual excitations, which prevents their overlapping, the predictions of the quasistatic IC model as a function of the parameters involved (Ni, d, R) can be investigated in details. (2) Prior to the strong development of the high-order multipolar structure in the near-UV spectral range (see, for instance, the d = 11 and 12 nm curves in Figure 9) the respective magnitudes of the Dx and Dz bands are clearly correlated to the square modulus of the ratio E⊥inc/E∥inc close to the particle (see eqs 10−12 and Figure 5 in the Supporting Information, keeping in mind that in Figure 5 the ratios have been computed for three selected distances d). (3) When d is decreased further, a rich high-order multipolar structure emerges in the near-UV spectral range. Each additional growing band corresponds to the sequential emergence of a new contribution in the induced surface charge distribution on the particle surface and is characterized by a specific (and increasing) order S , with S > 1 (multipole of order S ). In the HM model, each S -multipole has a (2S + 1)-fold degeneracy (m = 0, ±1, ..., ±S ). The |m|-degeneracy is removed in the presence of a close interface. Except for the normal incidence case θi = 0 (and s-polarized light excitation), a progressive splitting of each multipolar excitation into two substructures is clearly displayed in the three other figures. This splitting is the signature of the |m|-degeneracy removal. These high-order excitations are induced by the strong spatial inhomogeneities of the electric field created by the image particle inside and around the particle, as also pointed out by several authors.61,64

Figure 8. Normalized extinction cross sections of a silver sphere of radius R = 10 nm in air (N1 = 1) located above a substrate of high refractive index (N2 = 2.4), at normal incidence of the incoming wave, as a function of the distance d. The dielectric function of silver has been modified in order to reduce the intrinsic LSPR bandwidth (see text). The successive spectra have been shifted along the vertical coordinate (shift equal to 2500 nm2).

Figure 9. Same as Figure 8, for p-polarized light excitation at the oblique incidence θi = 45°. 6391



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number m = 0 (a bonding low energy bright mode and an antibonding dark mode of larger energy) and two 2-fold degenerate transverse modes associated with |m| = 1 (a bonding low energy dark mode and an antibonding bright mode of larger energy).97 In both systems, degeneracy arises from the equivalence of the transverse axes x and y. In view of the respective dipole orientations, the red-shifted m = 0 and |m| = 1 modes of the particle/interface system can be related to some extent to the longitudinal bright mode and the transverse dark mode of the dimer system, respectively, but this parallel is misleading because both modes of the particle/interface system are coupled to the applied electromagnetic field. In the dimer system, the (red or blue) shifts of the plasmon modes and the hybridization mechanism are caused by the Coulomb interaction of the modes of a given particle with the modes of the other particle of similar symmetry (same |m|; x either y, if |m| > 0). In the case of the particle/interface system, the red-shifts and the hybridization mechanism result from the coupling of |m|-modes belonging to the real particle, although the coupling terms involve two distributions of charge localized, respectively, on the surface of the real particle and that of the mirror partner. All these remarks point out that the modes of the particle/interface system are not straightforwardly related to a subset of those of the dimer system.103 From the above discussion it is thus clear that the two independent high-order multipolar structures displayed in the four figures correspond, respectively, to the S ≥ 2-plasmon excitations characterized by, respectively, m = 0 (correlated to the Dz excitation) and |m| = 1 (correlated to the Dx excitation). Similar computations using the modified Drude parametrization have been also carried out for a large sphere of radius R = 50 nm. Extinction spectra for p-polarized light excitation at the oblique incidence θi = 45° and normal incidence are provided in Figure 10 (main text) and in Figure 14 (Supporting

(4) For a given distance d, the spectral locations of the observed excitations are identical in the four figures, indicating that these collective modes are set by the intrinsic parameters defining the particle/interface system, that is, the dielectric indexes of the three materials [Ni(λ)] and the geometrical parameters d and R. On the other hand, the excitation of these modes and the relative intensities depend on the parameters defining the incoming wave (angle of incidence and polarization). (5) Examination of the high-order multipolar patterns shows that, in each couple of components arising from a given multipole S > 1, the red component is correlated to the excitation of the Dz mode, whereas the blue component is correlated to the excitation of the Dx mode (in other terms, two independent high-order multipolar structures are present). The development of both multipolar structures are thus directly related to the polarization of the incoming wave, with a relative weight that is, as for the most red-shifted dipolar bands Dx and Dz, ruled by the square modulus of the ratio E⊥inc/E∥inc in the vicinity the particle. From a qualitative point of view, it is tempting, and quite justified, to discuss these features in terms similar to those used within the plasmon hybridization model developed by Nordlander and co-workers.97 This approach has been embraced in refs 61 and 64, where large gold nanoshells with a silica core deposited on a substrate of high refractive index have been investigated experimentally (dark-field scattering spectroscopy) and theoretically through numerical approaches (FDTD and FEM-COMSOL Multiphysics package). However, it is worthwhile emphasizing that the plasmonic particle/dielectric substrate system cannot be paralleled strictly with the particle dimer system in many respects, even if some intrinsic substrate effects can be described in using a similar terminology (mode coupling, hybridization). The physics underlying the plasmonic particle/ dielectric substrate system is different from that of a particle dimer because the dielectric substrate does not sustain intrinsic plasmon modes, unlike the dimer system where self-sustained deformations of both incompressible free-electron clouds (plasmon modes) can be defined for each particle of the pair. Nevertheless, since in the frame of the approximate multipolar IC approach the particle/plane interface system can be replaced by a system consisting of two spheres (see the Supporting Information, section A), appealing to the hybridization approach seems quite natural. However, it should be emphasized that the surface charge distribution of the mirror particle [σ(r″)] is related to that of the particle in medium 1 [σ(r)] by the scaling law σ(r″) = −σ(r)f(N2,N1). Because of this imposed condition, the Lagrangian of the system and the resulting Euler−Lagrange equations, allowing to determine the plasmon eigenfrequencies, are different. In particular, the number of modes of the particle/ interface system is twice smaller than the dimer system (only the modes of the real particle above the interface are involved). From this major difference some important concepts introduced in the case of dimers become irrelevant, for instance, the bonding/ antibonding and bright/dark characters of the hybridized modes built from two modes of given (S ,m) values belonging, respectively, to each particles of the pair. For instance, for S = 1, only three modes exist in the particle/interface system, that is, the nondegenerate mode Dz (m = 0) and the degenerate modes Dx and Dy (|m| = 1). In a dimer six individual modes exist: two longitudinal modes associated with the azimuthal quantum

Figure 10. Normalized extinction cross sections of a silver sphere of radius R = 50 nm in air (N1 = 1) located above a substrate of high refractive index (N2 = 2.4), for p-polarized light excitation at the oblique incidence θi = 45°, as a function of the distance d. The dielectric function of silver has been modified in order to reduce the intrinsic LSPR bandwidth (see text). The successive spectra have been shifted along the vertical coordinate (shift equal to 15000 nm2).

Information). The comparative analysis of these spectra leads to the same conclusions as previously, although some features are much more difficult to assess with reliability because of the large radiation damping for this particle size. In particular, the 6392



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broadening of the dipolar bands Dx and Dz are so large that they cannot be resolved in Figure 10, even at short distances. Nevertheless, the broad and rather flat structure observed for d ≤ 51 nm is undoubtedly underlain by both dipolar bands. It seems moreover that the Dz band is much more broadened than the Dx band, the bandwidth of which can be estimated from the d = 300 nm spectrum (or the θi = 0 spectrum; Figure 14 in the Supporting Information). This statement is also supported by Figure 5 in the Supporting Information where the square modulus of E⊥inc/E∥inc at θi = 45° is seen to be significantly larger than unity in the spectral range λ > 400 nm in the region where is located the particle (d-values < 50 nm). This suggests that the integrated oscillator strength associated to the Dz band is diluted over a very large spectral range. To confirm this hypothesis and label unambiguously both dipolar bands in the broad flat structure located between 425 and 600 nm in Figure 10, extinction spectra for p-polarized light excitation at oblique incidence θi = 45° have been computed for sphere radii in the range R = 15−45 nm and a distance d = R + 0.3 nm (see Figure 15 in the Supporting Information). The striking differences observed between the respective bandwidths of the Dx and Dz modes can be easily interpreted in terms of interferences between two radiating dipoles (we have checked that the scattering contribution in the extinction is by far the dominant dissipation mechanism for large sizes, typically for R > 20 nm). For normal polarization (z-component of the applied field Einc), the induced dipoles in the particle and its mirror partner oscillate in phase, whereas for in-plane polarization (x- or y-component of the applied field Einc), the two induced dipoles oscillate in opposition of phase. Because the distance between the two dipoles is much smaller than the wavelength in the visible spectral range the scattered fields of both dipoles interfere constructively in the first case and, destructively, in the second case, in the far-field region. This explains why the broadening by radiation damping is much larger for the Dz-band.

why the HM approximation can be suitably assumed in introducing an adjustable effective surrounding-medium index for simulating the spectra (Neff). On the other hand, such a modeling is unsuitable for analyzing optical data from experiments involving both high substrate indexes and large angles of incidence. Unlike the crude HM model, the multipolar image−particle approximation (section A.2 in the Supporting Information) preserves the intrinsic substrate effects. However, from a quantitative point of view, the intrinsic effects have been found noticeably underestimated in most cases of physical interest, typically when d < 2R. This indicates that the usual simple ray-tracing analysis for justifying the approximation used [r(θ) = r(0)] fails for dealing with particles close to an interface and that the angular-dependence of both Fresnel reflection coefficients is of main importance for computing accurately the intrinsic substrate effects. Nevertheless, for large distances, this approximation is quantitatively quite justified. Third, the “slab model”, in which the infinite plane interface is replaced by a finite cylindrical slab (diameter D = 4R; height H = 2R) of similar refractive index N2, has been investigated through DDA calculations104 in involving various irradiation conditions. In most cases, the integrated absorption and scattering cross sections differ strongly from the “exact” results. Concerning this comparative investigation, it should be stressed that the distance d to be chosen in the exact multipole formalism is to some extent “problematic” when d ≈ R, because d is not necessarily strictly related to the thickness of the layer of the “empty” cubic cells separating the upper slab surface and the rough DDA-sphere surface.79 In fact, the disagreement is rather mainly rooted in an incorrect modeling of the applied field Einc (giving rise, in particular, to the extrinsic substrate effects). In response to the incoming plane wave, the finite slab behaves as a finite scatterer, producing a specific D- and H-dependent scattered field. In consequence, a finite slab of small dimensions relative to the wavelength does not ensure the perfect specular reflection of the infinite interface and, thus, the specific intensity distribution arising from the interference between the incoming and reflected plane waves. In particular, for large angles of incidence, both Fresnel reflection coefficients approach the value −1 (see Figure 2 in the Supporting Information) and the applied field intensity is close to zero just above the interface, explaining why the DDA cross sections we have computed within the slab model are found much larger than the exact ones. In our opinion, only qualitative trends with respect to the LSPR substrate-induced shifts can be ensured by this approach when D and H are too small. Finally, results from the exact multipole expansion formalism have been compared against those obtained from the exact analytical electrostatic solution (retardation effects neglected). It should be emphasized that only absorption is addressed in electrostatic models, which are, a priori, quantitatively suitable for very small spheres only. Despite these intrinsic limits, these approaches are currently invoked in the literature for analyzing optical data involving rather large particles. Therefore, the errors made deserve to be quantified. The exact solution of the electrostatic problem for the particle/interface system has been derived by Ruppin a long time ago, by separation of variables in bispherical coordinates.105 In this approach the electrostatic potential in the three homogeneous media is expanded upon Legendre polynomials PSm. In applying the boundary conditions on both dielectric interfaces, a set of linear homogeneous equations relating the

4. COMPARISON WITH APPROXIMATE MODELS During the course of this work the predictions of some approximate models have been compared with those obtained from the exact multipole expansion method. Here, we briefly summarize these investigations which will be reported in a forthcoming paper. The comparison with the HM model (Figure 7 in the Supporting Information) has been addressed in some places in the previous Section, pointing out that this simple model wipes out all the intrinsic substrate effects. Actually this ad-hoc phenomenological model is appropriate only for small N2/N1 and R/d ratios, that is, when the removal of the dipolar mode degeneracy can be disregarded. In such cases, the LSPRs exhibit small red-shifts and slight “apparent” broadenings for p-polarized light excitation at oblique incidence. It is worthwhile emphasizing that in single-particle SMS experiments involving a tightly focused beam22,34−37 the electromagnetic field distribution near the focus reflects that of the incoming plane wave and the polarization is essentially parallel to the substrate, or the film, on which is deposited the particle (this statement does not hold for dark-field scattering experiments since large angles of incidence are involved). So, only the inplane degenerate modes Dx or Dy are excited. Moreover, in the case of a very thin dielectric film of moderate refractive index (typically N2 = 1.5), the reflection coefficients are small and the intrinsic substrate effects are strongly weakened. This explains 6393



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three expansion coefficient sets, which depend on R, d, and the dielectric functions εi(ω) = N2i (ω), is derived. The eigenfrequencies are then computed by searching the frequencies ensuring nonvanishing solutions of the linear homogeneous equation set. Due to the axial symmetry, the problem separates with respect to the m-values, as in the hybridization approach discussed previously. To each m-value corresponds, therefore, a specific set of eigen-modes {ωi,m, i = 1, 2, 3, ...}. In the presence of an external uniform electric field, the optical absorption can also be computed analytically in expressing the square modulus of the electric field inside the sphere.106 Analysis of the electrostatic solution indicates that the eigen-mode spectrum {ωi,m} and the absorption cross section Cabs(E = ℏω) scale as follows: ωi,m ≡ ωi,m(d/R, εi) and σabs (E) ≡ R3 f(E,d/R, εi). In addition R-independent bandwidths are expected in the electrostatic model (no radiation damping). Comparison between the exact and the electrostatic predictions can thus be carried out through sets of full electrodynamic computations where, for each set, the radius R is varied for given d/R ratio and dielectric functions. For this purpose, a simple parametrization of the silver dielectric function, leading to a dipolar plasmon peak at 355 nm (3.493 eV) for an isolated small sphere in vacuum, has been used here by way of illustration. Besides, a very small value for the intrinsic parameter Γ has been selected for separating the various excited eigen-modes on the energy scale. Results obtained for N1 = 1, N2 = 2.4, and N23 = ε3(ω) = ε∞ − ω2P /[ω(ω + iΓ)], with ℏωP = 9 eV, ε∞ = 4.64, d/R = 1.05, and ℏΓ = 0.002 eV, is displayed in Figure 11. With these input

3.692 (3.694) eV. In the high energy side, the absorption spectrum has been strongly magnified in order to make visible to the eye the high S -order modes, which are only weakly hybridized with the dipolar excitations. Let us remark that such tiny details cannot be reproduced by any numerical methods. For the lowest (unphysical) radius, R = 0.1 nm, the electrostatic eigen-mode spectrum is perfectly recovered, emphasizing the accuracy of our calculations. For very small radii (R = 0.1 and 1 nm), the absorption spectra follow the scaling R3 law of the electrostatic solution, as expected. For the largest radii (R ≥ 5 nm), large discrepancies with the electrostatic predictions are clearly observed: (i) the plasmon frequencies are red-shifted, (2) the bandwidths become S -dependent and are strongly enlarged, especially those of the lowest S -modes, and (3) the relative contributions of the various modes in the absorption spectra are modified, whereas all the “electrostatic” spectra should be related from each other by a mere R3-scaling factor. It should be pointed out that, in order to spectrally separate the eigen-modes, a small damping parameter Γ has been used, and the relative scattering contribution in the extinction is large for R ≥ 5 nm (see the Figure 16 in the Supporting Information). Actually, when the Γ-value is strongly increased, the absorption to scattering contribution ratio is increased and the discrepancies with regard to both the S -dependence of the bandwidth and the relative contributions of the various excitations are found less dramatic. On the other hand, the spectral locations of the eigenmodes do not depend on the Γ-value, indicating that the observed shifts are not directly correlated to the scattering efficiency, but result from the retardation effects, which are disregarded in any electrostatic approach. Another set of absorption spectra is displayed in Figure 17 in the Supporting Information. In conclusion, electrostatic solutions (Ruppins’s or hybridization approaches) are of main importance for analyzing qualitatively optical spectra and providing an appealing physical picture of the particle−substrate interaction. However, from a quantitative point of view, electrostatic solutions are restricted to very small particles in the case of negligible relative light scattering.

5. CONCLUSION In this work the extinction cross sections of a spherical metal particle above a plane dielectric interface has been computed in the frame of the exact multipole expansion method, thus providing benchmark spectra for testing the various available numerical methods currently used in the nanoplasmonics literature. The formal theory has been outlined in the Supporting Information and the compact expressions of the integrated extinction, absorption and scattering cross sections have been derived. The dependence of the extinction cross sections on the parameters defining the particle/interface system, as well as on the irradiation conditions, has been analyzed in detail. In order to bring out more clearly the intrinsic substrate effects, the changes arising from the inhomogeneity of the applied field, resulting from the interference between the incoming and reflected plane waves, have been, in most cases, rubbed out through an appropriate spectrum normalization. The intrinsic effects which, in the present formalism, result from the action of the reflected scattered field impinging on the particle, can be paralleled with those observed in the case of metal dimers. Besides the dependence on the geometrical parameters (R and d) and on the material refractive indexes [Ni(ω), i = 1−3], the optical response of particles close to the interface depends dramatically on the irradiation conditions, namely, the angle of

Figure 11. Absorption cross sections of spheres (radius R) in air (N1 = 1) located at the distance d above a substrate of refractive index N2 = 2.4, for p-polarized light excitation at the oblique incidence θi = 45°. The radio d/R = 1.05 is kept constant. The dielectric function of the sphere is N23 = ε3(ω) = ε∞ − ω2P/[ω(ω + iΓ)], with ℏωP = 9 eV, ε∞ = 4.64, and ℏΓ = 0.002 eV. The gray curve in the lower figure is a magnification of the high energy part of the spectrum (1000 × σabs(E) + 0.02 nm2).

parameters the five lowest m = 0 (m = 1) electrostatic modes are: 3.223 (3.372), 3.496 (3.524), 3.605 (3.615), 3.66 (3.664), 6394



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incidence of the incoming plane wave and the electric field polarization with respect to the plane of incidence. This strong dependence is a consequence of the breaking of the spherical symmetry that is induced by the presence of the nearby interface (axial symmetry) and the resulting removal of the 3-fold spectral degeneracy (isolated sphere) of the normal and parallel dipolar modes. For large enough distances, these intrinsic substrate effects can be suitably accounted for within the well-known image charge/dipole picture. When the metal particle approaches the interface, the Coulomb interaction with the polarization charges induced on the dielectric interface results in a strong coupling between the collective electronic excitations of the conduction electrons inside the particle. This substrate-induced hybridization mixes the different multipolar (S ,m)-electronic excitations of the isolated particle. Due to the axial symmetry, only modes of the same |m|-value interact together. The (2S + 1)-fold degeneracy of the S -mode of the isolated sphere (frequency ωS ) is lifted, giving birth to (S + 1) spectrally distinct modes that are red-shifted relative to ωS and characterized by a given |m|-value (a 2-fold degeneracy remains for |m| > 0 because of the equivalence of the x and y axes). Concomitantly with this hybridization/splitting mechanism on the energy/frequency scale, a large part of the oscillator strength of the dipolar modes (the only modes that are strongly coupled to the applied electromagnetic field for particle sizes noticeably smaller than the wavelength) is transferred to the new (S ,|m| = 0) and (S ,|m| = 1) modes. The red-shifts of these substrate-induced modes, as well as the rates of transfer of the dipole oscillator strengths, are all the larger that the particle is closer to the interface. All these effects have been clearly evidenced: first, in modifying the Drude dielectric function associated with the conduction electrons in order to reduce the intrinsic linewidths of the various excitations, and second, in involving different incoming field polarizations. Finally, the limits of various approximate models for tackling the optical response of a particle located above a plane interface have been discussed. We hope that future experimental studies involving various substrates of high refractive indexes and more complex particle geometries will be stimulated by this work.

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ASSOCIATED CONTENT

* Supporting Information S

Section A: theory (multipole expansion method and cross sections) and image charge approximation; section B: Fresnel reflection coefficients and some characteristics of the applied field; section C: numerical accuracy of the results; section D: extinction cross sections; section E: comparison with exact electrostatic results. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Optical Properties of a Particle above a Dielectric Interface: Cross

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