Surface Plasmon Resonance in a metallic nanoparticle embedded in

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Surface Plasmon Resonance in a metallic nanoparticle embedded in a semiconductor matrix: exciton-plasmon coupling Rui M. S. Pereira, Joel Borges, Georgui V. Smirnov, Filipe Vaz, and M. I. Vasilevskiy ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b01430 • Publication Date (Web): 18 Dec 2018 Downloaded from http://pubs.acs.org on December 19, 2018

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Surface Plasmon Resonance in a metallic nanoparticle embedded in a semiconductor matrix: exciton-plasmon coupling Rui M. S. Pereira,∗,†,§ Joel Borges,† Georgui V. Smirnov,†,§ Filipe Vaz,†,‡ and M. I. Vasilevskiy∗,†,‡,¶ Centro de Física, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal, Departamento de Física, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal, and International Iberian Nanotechnology Laboratory, Braga 4715-330, Portugal E-mail: [email protected]; [email protected]

Phone: +351 253 510 436; +351 253 604 069. Fax: +351 253 510 401 ; +351 253 604 061

Abstract We consider the effect of electromagnetic coupling between localized surface plasmons in a metallic nanoparticle (NP) and excitons or weakly interacting electron-hole pairs in a semiconductor matrix where the NP is embedded. An expression is derived for the NP polarizability renormalized by this coupling and two possible situations are analyzed, both compatable with the conditions for Fano-type resonances: (i) a narrow bound exciton transition overlapping with the NP surface plasmon resonance (SPR), and (ii) SPR overlapping with a parabolic absorption band due to electron-hole transitions in the semiconductor. The absorption band ∗ To

whom correspondence should be addressed de Física, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal ‡ Departamento de Física, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal ¶ International Iberian Nanotechnology Laboratory, Braga 4715-330, Portugal § Departamento de Matemática, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal † Centro

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lineshape is strongly non-Lorentzian in both cases and similar to the typical Fano spectrum in the case (i). However, it looks differently in the situation (ii) that takes place for gold NPs embedded in a CuO film and the use of the renormalized polarizability derived in this work permits to obtain a very good fit to the experimentally measured LSPR lineshape.

Keywords: Nanoparticle, Localized surface plasmon, Semiconductor, Exciton, Composite, Dielectric function, Fano resonance. Plasmonic nanoparticles (NPs) have received considerable attention of researchers owing to their high sensitivity to the dielectric environment, which is based on the Localized Surface Plasmon Resonance (LSPR) phenomenon 1,2 . The characteristic LSPR band in the light absorption spectra can be tailored by controlling the size, shape and spatial distributions of the NPs and by choosing the host dielectric matrix where the NPs are embedded or even the substrate on which they are deposited 1,3–6 . It opens a wide range of possibilities for designing novel nanomaterials with the targeted applications including detection of bio- 7,8 and gas 9 molecules, plasmonic nano-antennas 10,11 , surface-enhanced spectroscopy 12–14 , LSPR-enhanced solar cells 15 as well as systems for nanometrology 16 , nanolithography and photocatalysis 17 . A century ago, Gustav Mie deduced a solution of Maxwell’s equations for a sphere, which permit to calculate the scattering and extinction spectra of a metallic nanoparticle 1 . The extinction cross-section depends of the dielectric functions of the metal (εM ) and host matrix (εh ), and the sphere’s radius (a) and the wavelength of the incident electromagnetic (EM) wave (λ ). In the limit a/λ 1 (relevant to NPs), Mie’s expressions are greatly simplified because the sphere’s response is dominated by its dipole contribution (compared to higher multipoles). In this limit, the particle’s polarizability becomes independent of its size (unless εM is size dependent because of quantum confinement or other effects) and given by 1

α0 (ω) =

εM (ω) − εh 3 a , εM (ω) + 2εh

(1)

where ω is the EM field frequency. The dipolar resonance (i.e. LSPR) frequency is determined

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by the pole of the expression Eq. (1). Obviously, it depends on the host dielectric constant and it is the basic principle of the NPs’ use for environmental sensing. Of course, NP’s shape has a major impact on the polarizability and resonance frequencies 2 , in particular, the former becomes a tensor and the three-fold degeneracy of the single dipolar resonance in a sphere is lifted for less symmetric shapes. However, for the sake of simplicity, here we shall confine our consideration by spherical NPs. Within the dipole approximation, further effects can be taken into account, such as the EM interactions between the particles, which are relevant since in practice one almost always deals with an ensemble of them. These interactions were considered for the first time by J. C. MaxwellGarnett 18 and are one of the most important effects influencing the LSPR lineshape, sensitive to the spatial distribution of the particles (such as short-range clustering or formation of larger aggregates, eventually of fractal dimension) 19–21 .Not only the spectral position but also the width and the shape of the LSPR band is important for applications. For instance, it has been noticed that the response of LSPR band curvature to εh changes is superior to peak shifting in terms of signal-to-noise ratio 22 . A broad SPR band facilitates the requirement to couple with the excitation wavelength used in Surface Enhanced Raman Scattering (SERS) detectors and can provide a double resonance at both excitation and Stokes-shifted frequencies 23 but, on the other hand, steep resonances are better for detection of spectral shifts in the refractive index change–based molecular sensing. Both situations can be achieved with plasmonic NPs, moreover, strongly asymmetric bands due to Fano resonances 24 can be achieved in finite clusters (oligomers) of plasmonic particles for sufficiently small interparticle separations 25,26 . Interaction with other excitations in surrounding bodies that may be in resonance with the localized surface plasmons should also affects the characteristics of the LSPR band. It has been demonstrated that strong coupling between a surface plasmon in a metallic film and an exciton in an organic semiconductor leads to a large Rabi splitting of the coupled modes 27 and that excitonic emission in complexes composed of semiconductor and metal NPs is enhanced or suppresed compared to the purely semiconductor system. 28 More recently, plasmon-exciton coupling effects

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were observed in hybrids consisting of 2D semiconductors of the transition metal dichalcogenide (TMD) family and silver NPs 29,30 , in particular, leading to a better efficiency of light-driven chemical reactions. 29,31 Some oxide semiconductors are potentially interesting materials for using as a matrix for embedding gold NPs. For instance, CuO films with embedded nanoparticles have been demonstrated to be good for CO sensing 9,32 . CuO is a semiconductor with a band gap of ≈ 1.5 eV 33 . That is, such a matrix absorbs light itself, via excitonic and inter-band transitions and there are dynamical dipoles associated with these transitions, which must couple to the localized surface plasmons. How will it affect the LSPR band position and line shape due to embedded NPs? Can we simply replace the constant εh by an appropriate complex function εh (ω) in the the expression Eq. (1) for NP’s polarizability as it was suggested long ago 34 or the effect of an absorbing matrix requires a more sophisticated description? Light scattering by spherical particles embedded in an absorbing matrix has been considered by several researchers 35–38 . In earlier works 35,36 the extinction produced by the particles was obtained by analyzing the energy flow through a large integration sphere, concentric with the particle, which led to an unphysical dependence of the particle’s extinction cross-section upon the integration sphere radius. A different (namely, local) definition of the extinction was used in Refs. 37,38 , which yielded results resembling the original Mie expressions for the scattering efficiency where p the size parameter, x = 2πa/λ , is replaced by a complex value, x0 = 2π εh (ω)a/λ . Here we are interested in NPs with a λ and need not the complex full Mie-type expression including all of the multipole orders of scattering. However, if the host matrix contains a resonant (e.g. excitonic) p transition, both the real and imaginary parts of εh (ω) may become large, therefore taking the limit x 1 in order to obtain the generalization of Eq. Eq. (1) may not be correct. In this work we follow a different approach and show that the NP’s polarizability is renormalized because of the electromagnetic interaction between elementary excitations in the particle (localized surface plasmons) and in the matrix (bound or unbound excitons). Our approach is analogous to the random phase approximation (RPA) in the theory of interacting elementary excitations 39 . We derive a compact formula that generalizes Eq. (1) for this case. Its application

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to the situation where a sharp excitonic transition is superimposed on a broad LSPR band yields a Fano-type resonance. We also analyze the calculated LSPR lineshape in the case of CuO matrix where the interband transition is little affected by the electron-hole interaction and compare it with experimental results for the Au-CuO system.

Renormalization of NP’s polarizability in absorbing matrix Let us consider a metallic nanoparticle (NP) embedded in a medium that contains some resontantly polarizable entities randomly dispersed everywhere arround the NP (see Figure 1). These entities can be excitons or weakly interacting electron-hole pairs that can be created in the matrix if the incident EM field is in resonance with appropriate electronic transitions. They are responsible for absorption of the EM field in the matrix. For clarity we shall call these polarizable entities by excitons and our purpose is to evaluate their influence on the NP’s polarizability. If there were no excitons, NP would be described by the bare polarizability, in the electrostatic limit given by Eq. (1) with εh replaced by ε∞ ≡ η 2 = const, which is the (real) dielectric constant of the medium without excitons and η is its refractive index.

Figure 1: Nanoparticle embedded in an absorbing medium containing randomly dispersed entities (excitons) that are polarized in resonance with the applied EM field. On the other hand, the medium containing excitons under illumination but free from NP inclusions has a dispersive dielectric function, ε(ω), characteristic of a bulk semiconductor and its 5


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imaginary part, ε 00 (ω) = Im ε(ω), represents light absorption owing to bound and unbound exciton states 40 .This dielectric function is related to the exciton polarizability by an equation similar to the Clausius-Mossotti (CM) formula 41 .Before considering a NP, we shall derive this equation by considering the excitonic polarization of the semiconductor medium that is polarized by an external EM field in resonance with the excitonic transition. We shall assume that the external EM field has the electric component of the form, ~0 ei~q~r ; E~0 (~r,t) = E

~0 ∼ e−iωt , E

(2)

√ where ~q is the wave vector and ω = qc/ ε∞ . 42 The polarization (i.e. the dipole moment per unit of volume) can be described by the following integral equation similar to the so called LippmannSchwinger equation adapted to the context of EM field scattering 5 . Z 0 0 0 ~ ~ ~ ~P(~r) = χe (ω) E~0 + ~ Tˆ (~r − r , ω)P(r )d r , V0

(3)

where χe is the excitonic susceptibility. Although it is apparently classical, Eq. (3) corresponds to the quantum-mechanical description of the excitonic polarization within the mean-field approximation known as RPA. 39 This equation is also analogous to the coupled dipole equations used to describe ensembles of polaizable NPs 19–21 . In the last term of Eq. (3), we should exclude the self-action, therefore V 0 is the volume of the system excluding a small region arround ~r, which is "occupied" by the exciton. Let us denote the radius of this spherical region by b. In Eq. (3) the dipole-dipole interaction tensor is 43 : iqR

ˆ 2e Tˆ (~R, ω) = Iq

R

iqR

e + ~∇ ⊗ ~∇(

R

),

(4)

where Iˆ is the cartesian unit matrix, and ⊗ means the direct product. Eq. (3) can be solved by performing Fourier transform (see Appendix A in Supporting Infor-

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mation) and it yields the polarization, χe E~0 . 1 − 4π χ e 3

~P(~r) =

(5)

Since the electric displacement vector is ~D = ε∞ (E~0 + 4π ~P), from Eq. (5) follows the CM-type relation for the dielectric function, ε = ε∞

1 + 8π 3 χe 1 − 4π 3 χe

.

(6)

Now we shall insert one NP in the medium. The dipole moment of the NP can be written as

~0 + ~p0 = α0 E

Z |~r|>a

Tˆ (~r, ω)~P(~r)d~r,

(7)

where the second term represents the field created by the excitonic polarization of the medium at the origin where the NP is placed. Eq. (3) for the polarization now has to include an additional term due to the field created by the NP: Z ~P(~r) = χe E~0 +

|~r−~r0 |>b

0 0 0 ~ ~ ~ Tˆ (~r − r , ω)~P(r )d r + Tˆ (ˆr, ω)~p0 .

(8)

Repeating the same steps as for pure matrix (see Appendix A in Supporting Information) and Fourrier transforming the last term in Eq. (8) forward and back, we obtain:

~P(~r) =

h i χe ~0 + Tˆ (~r)~p0 , E 1 − 4π 3 χe

(9)

where the argument ω was skipped for clarity. Finally, substituting Eq. (9) into Eq. (8), we get: R ~0 + χ4πe ~p0 = α0 E Tˆ (~r)E~0 d~r 1− 3 χe |~r|>a R χe + 1− 4π χ |~r|>a Tˆ (~r) Tˆ (~r)~p0 d~r 3

(10)

e

The second term in the brackets in Eq. (10) vanishes in the limit ~q → 0 by virtue of the angular

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integration, while that in the second line is evaluated to [see Appendix B in Supporting Information, Eq. (S8)]: 2-nd line = α0

χe 8π ~p . 3 0 3a χ 1 − 4π e 3

Thus, we obtain:  ~p0 = 

 1−

α0 E ~0 . χe α0 8π ( ) 3 1− 4π χe a3 3

(11)

The proportionality coefficient in Eq. (11) may be called renormalized polarizability. We can express the excitonic susceptibility, χe , in terms of the matrix dielectric function, εh (ω) = ε∞ + 4π χe (ω), which yields: α0 (ω) εh (ω) − ε∞ −1 α(ω) = α0 (ω) 1 − 2 . a3 εh (ω) + 2ε∞

(12)

This is our main result: the NP’s polarizability is renormalized because of the back action of the excitonic polarization in the semiconductor matrix, induced by the polarized NP.

Discussion and comparison with experiment The renormalized polarizability determines the effective response of the particle to external electric field and incorporates the environment effect. 44 The denominator of Eq. (12) represents the influence of the EM wave’s dispersion and absorption in the host material, which is measured by [εh (ω) − ε∞ ]. In a non-absorbing matrix with εh = ε∞ = const we have α = α0 ; also if the dispersion and absorption are small, they have little impact on the NP’s polarizability. However, if a sharp excitonic resonance overlaps with the surface plasmon resonance in the NP, the denominator of Eq. (12) has a strong and non-trivial dependence upon the frequency in the vicinity of the overlap range. Indeed, in this spectral region the imaginary parts of α0 and [ε(ω) − ε∞ ] both are large, while the real parts may change sign and one may expect peaks and dips in the imaginary part of α(ω). This situation is characteristic of Fano-type resonances which occur when a discrete state

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(e.g. bound exciton) interferes with a continuum band of states (e.g. the relatively broad LSPR band) 24,25,45,46 . The famous Fano formula for resonances in photonics is written in the form 45 :

σ (E) = C

(Q + x)2 , 1 + x2

(13)

where C is a constant, x = (E − E0 )/Γ, E0 is the resonance energy, Γ denotes the damping and Q is the Fano parameter determined by the phase shift of the continuum states. In describes the absorption cross-section but can also be applicable to different optical spectra including transmission and scattering 45 . For a spherical particle characterized by a dipolar polarizability α, the absorption cross-section is given by 19 : σabs = 4πq|α|2 [−Im(α −1 ) − 2q2 /3] ,

(14)

where q is the real part of the wavevector. Taking [εh (ω) − ε∞ ] as a simple Lorentzian centered at E0 = hω ¯ 0 = 1.45 eV and α0 (ω) ≈ const in Eq. (12) and substituting it into Eq. (14), we calculated the absorption cross-section spectrum presented in Figure 2. As it can be seen from this figure, it is rather well fitted by the Fano formula Eq. (13). 15 Σabs, arb. un.

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Eqs.H12L,H14L Fano Eq.H13L

10 5 0 1.2

1.3

1.4

1.5

1.6

1.7

E, eV

Figure 2: Absorption cross-section calculated using Eq. (12) and Eq. (14) (full curve) and the Fano formula Eq. (13) with Q = −4 (dashed curve). The full curve was scaled to fit the Fano formula.

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As a more realistic example, in Figure 3 we present the renormalized polarizability calculated for a gold NP embedded in GaAs. At low temperatures, the optical response of GaAs includes a sharp peak due to the bound exciton (in its ground state) and the interband absorption band, which is also modified by the exciton effect. Using the parametrization presented in Appendix C in in Supporting Information, the dielectric function is shown in Figure 3a. The major effect in the renormalized polarizability comes from the bound exciton transition, it produces a sharp dip in the imaginary part of α(ω) and a corresponding jump in its real part ( Figure 3b and c). The figure also presents the absorption coefficient, A, of a hypothetical composite film of GaAs containing gold inclusions; the Maxwell-Garnett approximation 18,47 was used for this purpose, which roughly √ leads to A = (3α/a3 ) f ε∞ ω/c, where f is the volume filling fraction of gold. The spectrum of the absorption coefficient presents the typical asymmetric lineshape (inset in Figure 3d) characteristic of Fano resonances that have been observed for different systems, see e.g. Ref 45 . Interband transitions in the matrix material also influence the renormalized polarizability and the absorption coefficient, although in a less drastic way. In order to look closer at this effect, we considered the situation of higher temperatures where bound exciton states are dissolved and the interband absorption in the semiconductor follows the simple parabolic law 40 . In the absence of the bound exciton contribution, we shall take the background dielectric constant as the square of the refractive index, ε∞ = η 2 , and ε = η 2 + iε 00 (ω). Therefore Eq. (12) becomes:

α(ω) =

α0 (ω) iε 00 (ω)

h 1 − 2( α0a(ω) 3 ) 3η 2 +iε 00 (ω)

.

(15)

h

Here the imaginary part of the host dielectric function just increases monotonically with ω above the absorption edge, εh00 (ω) ∝

p hω ¯ − Eg , ω2

and the situation, in a sense, is the opposite compared to the previous case, i.e. LSPR is superimposed on a continuum of interband transitions. The effect can already be noticed Figure 3d (above Eg ) but it is illustrated better in Figure 4a for the case of Au NPs in CuO matrix. The optical 10


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b)

a)

d)

c)

Figure 3: (a): Dielectric function of GaAs including bound exciton and interband transitions (real part, full curve and imaginary part, dashed curve) [see Appendix C in Supporting Information for details]; (b) and (c): Real ans imaginary parts of the NP polarizability, bare (dashed curves) and renormalized (full curves); (d) absorption coefficient of a composite material containing f = 0.1 volume fraction of NPs, calculated using the modified Maxwell-Garnett approximation 47 .

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density of such a composite material was calculated using its effective dielectric function obtained by solving coupled dipole equations for an ensemble of identical NPs randomly distributed over sites of a mesh with the occupation probability that guarantees the required volume fraction of the metal (see Refs. 20,48 for details of this procedure). In this figure we observe an asymmetric broadening of the LSPR band caused by its superposition with the parabolic absorption band of the semiconductor but the line shape is not like in the usual case of Fano-type resonances. Next we present the results obtained from our experiment with a film (about 40 nm in thickness) with approximadetly 15% of gold (by volume fraction, determined by the Rutherford Backscattering Spectrometry technique), dispersed in a CuO matrix. The nanocomposite film was deposited by magnetron sputtering and then annealed to stimulate the NP growth. The average size of Au NPs in the sample, estimated by X-Ray Diffraction peak fitting was ≈ 12 nm. More details concerning the deposition and characterization of Au/CuO films can be found in Ref 32 . The experimental transmittance spectrum is very well reproduced by the theory using the NPs’ parameters evaluated experimentally by independent techniques . This fit is much better than any one which we could obtain using the bare polarizability Eq. (1) and eventually prescribing an imaginary prescribing and imaginary part to εh . We emphasize that such a good fit could not be obtained even adjusting, say, the NPs’ filling fraction. Therefore we are convinced that it demonstrates the importance of the renormalization effect in matrices where interband light absorption, overlapping with the LSPR, takes place.

Conclusion In summary, we have shown that the optical properties of a nanocomposite system composed by plasmonic nanoparticles (e. g. Au nanospheres) embedded in a semiconductor matrix (e.g. CuO) results from the coupling between localized surface plasmons and excitons or free-carrier interband transitions. In the latter case, characteristic of CuO at room temperature (because of the small exciton binding energy), the use of the renormalized polarizability derived in this work

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Figure 4: (a) Optical density calculated using the renormalized polarizability (red curve) and the bare one (black curve); also shown is an experimental spectrum of host matrix (blue line). (b) Transmittance of the composite Au/CuO film described in the text, experimental (black dashed curve) and calculated numerically (red curve) using Eq. (15) and Eq.(S9); also shown is the transmittance of the host matrix CuO (blue line). The parameters of gold were taken from Ref 49 . Eq. (12) permits to obtain a very good fit to the experimentally measured LSPR lineshape. For the theoretically considered situation where a sharp exciton transition exists in the matrix, spectrally overlapping with the LSPR band, the exciton interaction with localized surface plasmons results in a Fano-type resonance with its characteristic non-Lorentzian lineshape ( Figure 3). A steep spectral feature (the Fano resonance) combined with a broad plasmonic band is intersting from the point of view of complementarity of spectral shift assays related to refractive index changes and SERS molecular identification 7 . Indeed, such a dual-sensor idea based on Fano resonances originating from the interaction between a (broad) dipole mode and a (narrow) quadrupole one, produced by two different plasmonic components of the nanostructure (e.g. spheres and rings) put in a contact 46 . A tunable Fano resonance in a resonator combining metallic NPs with a dielectric microcavity was demonstrated in Ref 50 . It is also interesting to explore a structure combining plasmonic NPs with a two-dimensional semiconductor such as the TMDs, 6,51 which are known for their strong excitonic transitions 52 . Such nanostructures, beyond the enhanced catalytic properties, 29,31 also promise the advantages of enhanced sensitivity and a better figure of merit compared to dipole-based localized surface plasmon resonance (LSPR) sensors 46 .

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Acknowledgement Financial support from the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2013 is acknowledged. This research was also partially sponsored in the framework of FCT Project 9471 - RIDTI, co-funded by FEDER (Ref. PTDC/FIS-NAN/1154/2014). Joel Borges thanks FCT for Postdoctorate Grant (SFRH/BPD/117010/2016).

Supporting information contains: i) Derivation of the relation between excitonic polarization and external field in NP free matrix; ii) Evaluation of NPs self-induced dipole moment via polarization of the matrix; iii) Parametrization of the semiconductor dielectric function.

References (1) Bohren, C. F.; Huffman, D. R. Absorption and scattering of light by small particles; Wiley, NY, 1998. (2) Stockman, M. Nanoplasmonics: The physics behind the applications. Physics Today 2011, 64, 39. (3) Garcia de Abajo, F. J. Light scattering by particle and hole arrays. Rev. Mod. Phys. 2007, 79, 1267 – 1290. (4) Lachaine, R.; Boulais, E.; Rioux, D.; Boutopoulos, C.; Meunier, M. Computational Design of Durable Spherical Nanoparticles with Optimal Material, Shape, and Size for Ultrafast Plasmon-Enhanced Nanocavitation. ACS Photonics 2016, 3, 2158–2169. (5) Amorim, B.; Gonçalves, P. A. D.; Vasilevskiy, M. I.; Peres, N. M. R. Impact of Graphene on the Polarizability of a Neighbour Nanoparticle: A Dyadic Green’s Function Study. Appl. Sci. 2017, 7, 1158.

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E, eV

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1.7

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b)

a)

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Surface Plasmon Resonance in a metallic nanoparticle embedded in

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