Plasmonic Nanoparticles Array for High-Sensitivity Sensing: A

Article pubs.acs.org/JPCC

Plasmonic Nanoparticles Array for High-Sensitivity Sensing: A Theoretical Investigation Ophélie Saison-Francioso,† Gaeẗ an Lévêque,*,† Abdellatif Akjouj,† Yan Pennec,† Bahram Djafari-Rouhani,† Sabine Szunerits,‡ and Rabah Boukherroub‡ Institut d’Électronique, de Microélectronique et de Nanotechnologie (IEMN, UMR CNRS 8520), Université Lille 1, Cité Scientifique 59655 Villeneuve d’Ascq, France ‡ Institut de Recherche Interdisciplinaire, Parc de la Haute Borne, 50 Avenue de Halley, BP 70478, 59658 Villeneuve d’ Ascq, France †

ABSTRACT: In this paper, we investigate theoretically the localized plasmon resonance mode of a periodic system of bidimensional gold nanostructures coated with a dielectric layer of variable thickness. When illuminated by a plane wave in normal incidence, the interaction between the Fabry−Pérot modes established inside the layer with the particle plasmon leads to a thickness-dependent shift of the absorption maximum. Combining the Green’s tensor and finite difference time domain methods, we propose first a simple description of the physical phenomenon responsible for the wavelength shift, and then analyze the effect of the background structure refractive indexes on the characteristics of the maximum wavelength evolution.

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INTRODUCTION The past decade has witnessed significant interest in the phenomenon of localized surface plasmon resonance (LSPR) for the development of label-free sensor platforms.1 The excitation of localized surface plasmon resonances in noble metal nanostructures creates sharp spectral absorption and scattering peaks as well as strong electromagnetic near-field enhancement.2,3 The sensitivity of the position and amplitude of the LSPR signal to the presence of organic and biological molecules as well as to the refractive index of the adjustment medium is the fundamental basis of LSPR-based sensors. Despite the steadily increasing number of reports emphasizing the interest of surface linked metallic nanostructures as transducers for plasmonic based biosensors, their large scale implementation has remained limited. Significant issues concern the development of LSPR interfaces with stable and reproducible optical properties. The poor adhesion of metal island films to the substrate induces morphological changes upon exposure to solvents and analytes, causing the degradation of the optical properties of the LSPR transducer and thus uncertainty in any detection scheme based on refractive index sensitivity.4,5 We6−14 and others15,16 have shown that the morphology and optical response of gold and silver island films, formed by high temperature annealing of evaporated thin metal films, can be stabilized by postcoating the metal islands with a dielectric matrix. The presence of the coating also allowed developing innovative strategies for the coupling of organic and/or biological molecules onto the LSPR transducer.8,11,13,14 The presence of the overcoating has important consequences on the position, amplitude, and line width of the plasmonic signal as well as on the bulk refractive index sensitivity.7,17,18 The magnitude © 2012 American Chemical Society

of the LSPR shift is however not only determined by the thickness of the dielectric overlayer but additionally by the contrast between the refractive indexes of the underlying substrate and the bulk environment, and the decay length of the electromagnetic field around the nanostructure. The near-field and thus molecular sensitivity of the LSPR signal has been demonstrated by measuring the LSPR shift upon the formation of self-assembled monolayers of alkanethiols on silver nanotriangles fabricated via nanosphere lithography.18,19 It was found that the near-field LSPR sensitivity falls off with distance from the metal surface and levels off for films thicker than approximately 20 nm.18,19 For dielectric-coated metallic particles, it was assumed that the LSPR shift exhibits a similar decay until saturation sets in for dielectric layer thicknesses comparable to the particle size.20,21 However, long-range effects could be recorded on short-range-ordered gold nanodisks and nanoholes on glass in air when modified with multilayers of 22-tricosenoic acid (n = 1.53) using the Langmuir−Blodgett technique.22 In accordance with these results, we have shown that the LSPR position and line-shape are drastically affected when gold nanostructures are coated with thick (100−300 nm) dielectric layers.8−10 While the interest of these hybrid LSPR surfaces to fine-tune the plasmonic signal9,11,12 and for sensing applications has been demonstrated,6,11,14 the origin of the theoretically and experimentally observed oscillations of the position of the LSPR band for increasingly thick overcoatings has not yet been investigated. Received: May 31, 2012 Revised: July 19, 2012 Published: July 24, 2012 17819

dx.doi.org/10.1021/jp305310v | J. Phys. Chem. C 2012, 116, 17819−17827

The Journal of Physical Chemistry C

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substrate and overcoating layers n1 = 2.0 (indium tin oxide, ITO), n2 = 1.45 (silica), and n3 = 1.0 (air) as a reference system. In the first part of this paper, we study the absorption of light by a single gold nanostructure, using Green’s tensor approach. In a second part, we extend these results to an array of particles using a finite-difference time-domain method (FDTD).

In the present paper, we focus on a numerical study based on the Lorentz−Drude model to describe the origin of these oscillations. A single isolated gold nanostructure deposited on a dielectric such as indium tin oxide (ITO) and coated with silicon dioxide (SiOX) is investigated first numerically with Green’s tensor method. These simulations are then extended to an ordered array of gold nanostructures. After a presentation of a simple analytical model allowing qualitatively the reproduction of the observed oscillation phenomena, several quantitative results about the influence of the refractive indices of substrate, dielectric overcoating, and immersing medium on the position of the LSPR band, the oscillation amplitude, the oscillation period, and the average oscillation wavelength are discussed. Description of the Investigated Structure. Figure 1 displays the schematic illustration of the structure investigated

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SINGLE NANOSTRUCTURE SYSTEM In this section, the absorption of a single gold nanostructure is computed as a function of the wavelength and the thickness of the intermediate layer. The Green function method, well documented in several articles,24,25 is particularly convenient for the study of localized objects placed in a multilayered environment. For convenience, we remind that it relies on the resolution of the following Lippmann−Schwinger equation: E(r, ω) = E0(r, ω) + k 0 2

∫V dr′G(r, r′, ω)Δε(r′, ω)E(r′, ω) (2)

where ω is the light pulsation, k0 = ω/c is the wavevector modulus of the light in vacuum, c is the velocity of light in a vacuum, E is the total electric field at location r, E0 is the unperturbed electric field (electric field at r without the particle), and G(r, r′) is the Green function of the multilayered environment between a source point r′ and the observation point r. The integral is performed on the volume V of the particle. We compute the absorption inside the particle, given by A abs(ω , d) =

in this work. Gold nanostructures (NSs) are deposited on a dielectric substrate (refractive index n1) and coated with a second dielectric (refractive index n2, thickness d). The top medium is air (refractive index n3). We compute the transmitted signal in air emitted by a light source placed inside the substrate. In the whole article, the gold dielectric constant is described using a Lorentz−Drude model: M

∑ m=0

fm ωp 2 ωm 2 − ω 2 + jω Γm

∫V drε″(r, ω)|E(r, ω)|2

(3)

where ε″(r, ω) is the imaginary part of the particle dielectric constant at pulsation ω. As an important parameter in this work, the silica layer thickness d is explicitly added as a variable in Aabs. First, the nanostructure has a small square-shape cross section of 4 nm side, small enough to be described as a single polarizable cell. It lies on the ITO substrate, and is separated from the air by a silica layer of varying thickness d. Figure 2a shows in black solid line the evolution of λmax, the wavelength of the absorption maximum of the particle, when excited by a plane wave in normal incidence from the ITO substrate (the electric field being parallel to the x-axis) as a function of the silica layer thickness d. This curve is qualitatively very similar to what was reported in a previous publication by our group, for a periodic structure.9 We observe first a rapid red-shift of the plasmon wavelength within the first few nanometers from the interface (see inset), followed by strongly anharmonic oscillations. The amplitude of these oscillations is very large in the first 500 nm, and then slowly decreases for larger values, where the maximum wavelength λmax simply jumps from a wavelength to a shorter wavelength. In order to understand the origin of these oscillations, several absorption spectra are plotted in Figure 2b and c for different thicknesses of the silica layer. As we can see, for a thickness lower than 500 nm, the absorption exhibits essentially one maximum in the investigated range of wavelengths. A fast redshift happens within the first 20 nm, followed by a slower drift in the same direction for d up to 100 nm, together with a decrease of the absorption maximum (Figure 2b). However, several peaks start to appear when d increases: the thicker the layer, the larger the number of peaks (Figure 2c). The comparison of two curves computed for d = 1280 nm (cyan) and d = 1284 nm (magenta) in Figure 2c shows that every peak

Figure 1. Schematic representation of the structure investigated in this work. Metallic nanostructures are characterized by their height h = 15 nm, width l = 25 nm, and period a = 70 nm. The thickness of the second dielectric is labeled d. As a reference system, we have chosen n1 = 2.0, n2 = 1.45, and n3 = 1.0.

ε(ω) = εr, ∞ +

k0 2π

(1)

where εr,∞ is the dielectric constant at infinite frequency, ωp is the plasma frequency, and ωm, f m, and Γm are, respectively, the resonance frequency, the strength, and the damping frequency of the mth oscillator. We used M = 5 damped harmonic oscillators in order to take interband transitions of gold into account. The parameter values were taken from Rakic et al.23 In the whole work, and without any contrary indications, the metallic nanostructures are characterized by the following dimensions: the height h = 15 nm, width l = 25 nm, and period a = 70 nm (in the case of periodic systems). This value for the period has been chosen according to the results obtained in a previous work,9 where it has been shown that the effect of a small value for a is essentially a red-shift of the LSPR mode. This shift starts to be negligible when a > 70 nm, even in the case of an ITO substrate. Finally, we have chosen for the 17820



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rapidly “stabilizes“ for a distance of about the height of the particle. The amplitude of this shift will depend on the three indexes n1, n2, and n3. Moreover, a red-shift occurs when n3 < n2, whereas a blue-shift is obtained in the opposite case, as the average index above the particle decreases. Thick Layer Behavior. For a larger silica layer thickness, the shift of the absorption maximum is explained by the modulation of the particle plasmon response by the partially stationary unperturbed light field E0 (see eq 2) created inside the Fabry−Pérot cavity in between the two dielectric interfaces. This effect can be evidenced by considering the following quantity: A̅ abs (ω , d) =

k0 2π

∫V drε″(r, ω)|E(r, ω)|2 /∫V dr |E0(r, ω)|2

= A abs(ω , d)/⟨|E0|2 ⟩d

which represents the absorption inside the nanostructure normalized to the average intensity of the unperturbed electric field inside the nanostructure (which is dependent on the silica layer thickness d). The wavelength λ̅max of the maximum of A̅ abs as a function of d corresponds to the red line in Figure 2a. This curve shows the same rapid shift for thin layers as for Aabs but is however completely flat for thicker layers, which is a consequence of the fact that A̅ abs is d-independent for d > 2h. Hence, the several maxima in the absorption spectra of Figure 2c correspond to the different Fabry−Pérot modes created by the incident plane wave inside the dielectric layer (|E0|2), as the absorption is given by A abs(ω , d) ≈ A̅ abs (ω , d)⟨|E 0|2 ⟩d = A̅ abs (ω)⟨|E0|2 ⟩d when

d > 2h

(4)

When an antinode of the stationary wave in the dielectric layer coincides with the particle location, it results in an absorption peak of larger amplitude when its wavelength is about the plasmon wavelength of the gold nanostructure. This observation is the basis of the simple analytical model presented in the next section. Finally, according to this explanation, the decay of the oscillation amplitude when the silica layer thickness increases is related to the decay of the wavelength distance Δλm between the two Fabry−Pérot modes m and m + 1 which are the closest to the LSPR wavelength, as it obeys Δλm = |λm+1 − λm| ≈ λp2/2d. In the following, we will call λeff the asymptotic value of λ̅max(d), which matches the average oscillation wavelength of λmax(d) (Figure 2a). In fact, λeff corresponds to the maximum absorption wavelength of the same NS placed inside an homogeneous medium of effective refractive index equal to the average refractive index felt by the nanostructure in its near-field, for d > 2h. We will refer a few times in the following to the LSPR mode of this effective system as the effective LSPR mode. Influence of the Volume and the Particle Number. Actually, the size and shape of the particle makes it difficult to describe only with one polarizable cell. Figure 3a shows the evolution of λmax(d) and λ̅max(d) for NS of cross section described in Figure 1. The background structure is the same as previously. Compared to the single cell study, the average oscillation wavelength λeff is red-shifted because of the particle volume increase and its more elongated shape. Moreover, contrary to the simple dipolar model, the position of the normalized absorption maximum still presents oscillations because the unperturbed

Figure 2. (a) Evolution with the silica layer thickness d of the position λmax of the absorption (Aabs) maximum (black) and of the position λ̅max of the normalized absorption (A̅ abs) maximum (red, see definition below) for an ITO/Au NS/SiOx/air system. The NS has a squareshape cross section, of 4 nm side, and is described by a single polarizable element. (b) Absorption curves computed for several silica layer thicknesses d < 100 nm. (c) Same as part b but for several values of d > 100 nm.

shifts to the red when the silica layer is made thicker. Hence, the jumps observed in Figure 2a are simply the result of one peak being too far from the average plasmon wavelength (here about 500 nm) compared to the next peak (at shorter wavelength), which becomes the new maximum. We will see that these multiple peaks result from the modulation of the particle plasmon by the interference pattern due to the multiple reflections of the incoming plane wave inside the silica overcoating layer. Thin Layer Behavior (Figure 2a, Inset). The rapid shift of the maximum wavelength for small layer thicknesses is explained in the current geometry by an increase of the average refractive index around the particle. Indeed, it is well-known that the plasmon mode of a single particle in a homogeneous environment shifts to longer wavelength when the refractive index increases. As the electric field distribution of the plasmon mode is localized in the particle near-field area, its position 17821



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transmission spectrum is normalized to the power spectrum of the incident wave. We finally compute 1 − T(ω), where T(ω) is the normalized transmission spectrum. In the following, the geometry of the gold NSs is fixed to l = 25 nm, h = 15 nm, and a = 70 nm. Figure 4a shows several absorption spectra for a typical ITO/ Au NSs/SiOx/air structure of varying layer thickness d, where

Figure 4. (a) Absorption spectra of an ITO/Au NSs/SiOx/air structure, for increasing SiOx layer thicknesses d. (b) Evolution of the LSPR position λmax as a function of d.

refractive indexes are n1 = 2.00 for ITO, n2 = 1.45 for SiOx, and n3 = 1.00 for air. Similarly to the previous section, we observe that the LSPR resonance position λmax is not the same when d changes. The curve displaying the evolution of λmax as a function of d in Figure 4b exhibits the same features as in the single NS section: • an anharmonic oscillation of the maximum wavelength around an average value of approximately 610 nm • the λmax shift is more abrupt on the decreasing slope than on the increasing slope • the oscillation amplitude increases during the three first periods and then starts to slowly decrease • the oscillation period does not change with d and is approximately equal to P = 210 nm.

Figure 3. (a) Evolution of the position of the absorption maximum and of the normalized absorption maximum, computed for an ITO/Au NS/SiOx/air system as a function of d. The NS has a rectangular cross section of dimensions l = 25 nm and h = 15 nm. (b) Absorption (per volume unit) curves computed for an increasing number of NSs, regularly spaced every a = 70 nm.

field is not homogeneous inside the particle. The function λ̅max(d) oscillates with an amplitude of about 3 nm. Finally, we can expect the array property to be somewhat different than the single particle, given the short period a = 70 nm considered here. Figure 3b shows the evolution of the absorption spectrum when the number of particles separated by 70 nm increases from 1 to 6. We can see that a small red shift of about 7 nm is induced by the collective coupling, but the maximum position remains unchanged for N > 4.

These observations are in agreement with what has been previously observed for a similar structure (glass substrate (n1 = 1.51), ITO layer (ITO, n2 = 2.0), and water as the detection medium (n3 = 1.33).9,14 We show in the next part that, following the observations of the previous section, we can build a simple analytical model allowing one to qualitatively describe the response of such a system. Simplified Analytical Description. As a reference system, we show in Figure 5a the absorption spectrum of an ITO/Au NSs/SiOx structure (no air above the silica), which exhibits a LSPR resonance at λeff = 613 nm. The LSPR wavelength depends on the NS material (gold in this article), shape (l, h, a), and environment (the refractive indexes of the substrate and the overcoating layer (n1, n2)). In particular, the effect of an increase of either n1 or n2 is to red-shift the LSPR. No oscillation of the LSPR wavelength occurs in this case, as there is no silica/air interface. However, when the gold NSs are placed inside a silica layer of finite thickness, the LSPR is shifted, as shown in Figure 4. This oscillation results from the interaction between the NSs and the system of stationary waves established inside the linear Fabry−Pérot cavity (FP-cavity) formed by the two ITO/silica and silica/air interfaces (see Figure 5b). In other words, when the SiOx thickness d increases, the wavelength for which a maximum of the interference pattern

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PERIODIC SYSTEM OF GOLD NANOSTRUCTURES Numerical simulations for the periodic system were performed using the bidimensionnal FDTD method.26,27 We generate a discrete grid along the x- and y-axis using a mesh interval equal to Δx = Δy = 1 nm. Along the z-direction, the structure is supposed to be infinite and its properties are unchanged. We use periodic boundary conditions (PBC) to repeat periodically the unit cell along the x-axis. An incoming, TM-polarized pulse is generated inside the substrate (dielectric 1) by a current source parallel to the x-axis. The emitted electromagnetic wave propagates along the y-axis and excites localized surface plasmon modes inside each nanostructure. In order to avoid reflections of outgoing waves, perfect matching layers (PML) are applied at y-boundaries of the unit cell.28 The signal transmitted by the structure is first recorded in the air as a function of time, and finally Fourier-transformed in order to obtain the transmission coefficient as a function of frequency. Every 17822



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with L (λ ) =

K 2H (λ − λeff )2 + K 2

where L(λ) is an ad hoc fitting function of the Au-NS plasmon response inside the ITO/SiOx matrix (without air) as a Lorentzian function centered on λeff = 608 nm, of arbitrary amplitude H and full width at half-maximum (fwhm) equal to 2K = 200 nm (λeff and K were extracted from FDTD comparison, see Figure 5a). Figure 6a shows a good agreement between the λmax evolution extracted from the analytical model and from the FDTD

Figure 5. (a) Absorption spectra of an ITO/Au NSs/SiOx (n2 = n3 = nSiOx) structure. LSPR signal is probed and recorded in SiOx. (b) Fabry− Pérot cavity of length d delimited by ITO/SiOx and SiOx/air interfaces.

inside the layer occurs at the particle location is blue-shifted due to its FP-cavity mode nature. As a consequence, the resonance wavelength λmax oscillates around the “average value” λeff, determined by the system of Figure 5a, inset. These observations are consistent with the fact that the λmax oscillation period obeys the following equation: P=

λeff 2n2

(5)

which corresponds to the resonance condition of linear FP cavity modes. In our case, dp = 613.0/(2 × 1.45) ≈ 211.4 nm. Through FDTD simulations (see Figure 4b), we found P ≈ 210 nm, which is in good agreement with the theoretical value. From these preliminary observations, we can build a simple analytical model. Let us consider a plane wave in normal incidence on the silica layer, sent from the ITO substrate. The electrical field modulus at the ITO/SiOx interface is continuous and reads without NSs: ⎛ 1 + 2r cos(ϕ) + (r )2 ⎞1/2 23 23 ⎟ E0(λ , d) = Eit12⎜ 2 ⎝ 1 − 2r23r21 cos(ϕ) + (r23r21) ⎠

Figure 6. Evolution of λmax with the SiOx thickness (d) for ITO/Au NSs/SiOx/air structure. The comparison between FDTD (red line) and analytical results (blue dashed line) is shown in two cases where interband transitions are neglected (a) or added (b) in the plasmon response of the gold NSs L(λ).

(6)

simulation results: the global shape, the red-shift amplitude, and the oscillation period are all correctly described. However, the analytical model underestimates the wavelength decay around values where the LSPR position rapidly shifts to the blue. Indeed, as gold is a noble metal, its optical response depends both on free electrons (described by the Lorentzian function in L(λ)) and core electrons (which experience interband transition for wavelengths smaller than 500 nm). Hence, the absorption spectrum of Au-NSs in ITO/SiOx matrix must be modified in order to include the core electron response as

where Ei is the amplitude of the incoming planewave electric field and with 2n1 t12 = n1 + n2 (7) rij =

ϕ=

ni − nj ni + nj

(8)

4πn2d λ

(9)

Lc(λ) =

where t12 is the transmission coefficient at the ITO/SiOx interface, r21 and r23 are, respectively, the reflection coefficients at SiOx/ITO and SiOx/air interfaces, and ϕ is the phase difference after one back and forth of the light inside the silica layer. Hence, we can use the following expression to model the absorption spectrum of the gold NSs (th meaning theory): A th (λ , d) = E0 2(λ , d) × L(λ)

K12H1 (λ − λ1)2 + K12

+

K 2 2H2 (λ − λ 2)2 + K 2 2

(11)

The new fitting parameters are then K1 = 78.1 nm, H1 = 0.5 au, λ1 = 613.0 nm, K2 = 400.0 nm, H2 = 0.5 au, and λ2 = 5.0 nm. Figure 6b shows a better agreement between analytical and numerical results, particularly concerning the blue-shift amplitude. The main difference occurs now for thin SiOx layers, as the λmax starting values are not the same: λmax(d = 0 nm) = 593 nm for FDTD, whereas λmax(d = 0 nm) = 613 nm for the analytical

(10) 17823



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model. In fact, in the analytical model, we only consider the effect of FP cavity modes on the plasmonic response, which was modeled as the response of the particle placed inside an infinitely thick layer. For values of d comparable and smaller than the particle height, the NS feels an average refractive index which changes progressively from n3 = 1.0 (d = 0 nm, no silica) to n2 = 1.5 (d = ∞): it results in a red-shift of the LSPR wavelength and a modification of the function Lc(λ). As a conclusion, and similarly to the results obtained with the single particle calculations, there are two physical mechanisms explaining the plasmon wavelength shift with the thickness of the silica layer: a first fast red-shift due to the increase of the average refractive index around the particle (for d up to a few times the NS height), followed by the oscillating behavior due to the interaction of the effective LSPR mode (ITO/silica) with the FP mode inside the silica layer (larger thicknesses). Influence of Refractive Indexes (n1, n2, n3) on the Oscillation of the LSPR Position. Following the first qualitative explanations given with the analytical model, the curve representing the evolution of the LSPR position λmax with the silica layer thickness d can be characterized by four parameters: the oscillation period P, the oscillation amplitude on the first half-period A = |λmax(first extremum) − λmax(second extremum)|, the “average value” of the LSPR wavelength λeff, and the LSPR wavelength starting shift Δλi = λmax(first extremum) − λmax(d = 0 nm) (see Figure 7). The purpose of this section is to investigate the

Figure 8. Influence of refractive indexes on λeff for the following structures: (a) n1/Au NSs/SiOx/air; (b) ITO/Au NSs/n2/air; (c) ITO/Au NSs/SiOx/n3.

Figure 7. Summary of the parameters characterizing the λmax(d) oscillation, in the case of an ITO/Au NSs/SiOx/air structure: the oscillation period P, the oscillation amplitude on the first half-period A = |λmax(first extremum) − λmax(second extremum)|, the “average value” of LSPR wavelength λeff, and the LSPR wavelength starting shift Δλi = λmax(first extremum) − λmax(d = 0 nm) (positive (red-shift) or negative (blue-shift).

Figure 9. Influence of refractive indexes on the oscillation period. Black triangles correspond to FDTD results, whereas color dots correspond to P-values obtained with the formula P = λeff/2n2: (a) effect of n1 (n1/Au NSs/SiOx/air structure); (b) effect of n2 (ITO/Au NSs/n2/air structure); (c) effect of n3 (ITO/Au NSs/SiOx/n3 structure).

influence of the refractive indexes n1, n2, and n3 on every parameter, starting from the same structure as previously: ITO/Au NSs/SiOx/air. Influence on the Average Oscillation Wavelength λeff. We find as expected that the average LSPR wavelength λeff is redshifted when n1 or n2 increases (Figure 8a and b). However, we can notice that λeff is more sensitive to a change of n2 than a change of n1, probably because only one face of the particle is in contact with the substrate (n1). Finally, Figure 8c confirms that λeff is not contingent upon n3. Influence on the Oscillation Period P. Figure 9 shows a small discrepancy between periods obtained from FDTD and from analytical results P = λeff/2n2. This can be attributed to the fact that the period has been measured between the two first extrema of the λmax(d) curve, in order to save computation time. It would be required to work out more periods for better

accuracy. However, we can verify that the period has the same trend for both FDTD and analytical results. Effect of n1 (Figure 9a). The oscillation period P increases with the refractive index n1 of the substrate. This is consistent with the analytical expression for P, as increasing n1 induces an increase of the effective LSPR mode wavelength λeff. Consequently, the period increases as n2 is kept constant. Effect of n2 (Figure 9b). Two phenomena come up against when n2 increases. Similarly to what happens for n1, increasing n2 red-shifts the LSPR and thus λeff. However, as the period P is proportional to the inverse of n2, the final effect is not obvious. However, Figure 9b shows that P decreases when n2 increases.When n1 = n2 = nITO = 2.0, the oscillation of the LSPR wavelength is due to the interaction with a system of partially stationary waves created by reflection of the incoming wave on the n2/air interface. An additional effect could come from the 17824



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n3. It appears that the amplitude and sign of the initial wavelength shift depend a lot on the value of n3 compared to n2: a red-shift occurs for n3 < n2 = nSiOx = 1.45, whereas a blue-shift occurs for n3 > n2. The shift direction depends on whether the materials around the gold NSs are replaced by a material of larger or smaller index than the detection medium during the layer growth. For n3 < n2, the average refractive index neff(d) increases, which induces a red-shift of the LSPR, whereas, for n3 > n2, the average refractive index around NSs decreases and the LSPR peak is blue-shifted. We can as well notice in Figure 11 that a phase difference of π in the oscillation exists between n3 > n2 and n2 > n3. Finally, the larger the difference between n2 and n3, the larger the initial shift Δλi. The corresponding values are summarized in Figure 10c. Influence on the Oscillation Amplitude A. In principle, there are no simple rules to predict the amplitude of the wavelength oscillation when the different refractive indexes change. However, it can be qualitatively understood using simple considerations about the behavior of the electric field inside the Fabry−Pérot cavity and the plasmon response of the particle. Figure 12a represents the absorption curves computed from

back reflection of light by the NS array, but this contribution has not been taken into account in our analytical model. Effect of n3 (Figure 9c). We see in Figure 9c that the oscillation period does not depend on n3. The oscillation period is not defined when n3 = n2 = nSiOx = 1.45 (green dot) as the reflection of light on the upper interface vanishes: the plasmonic resonance wavelength does not depend anymore on d in that particular case. Influence on the Initial Wavelength Shift Δλi. Effect of n1. When the refractive index n1 increases from 1.45 to 1.8, Δλi decreases and reaches a minimum. For larger values, Δλi becomes n1-independent and is fixed to about 23 nm (Figure 10a).

Figure 10. Influence of n1, n2, and n3 on the initial wavelength shift at the beginning of the λmax(d) curve for the following structures: (a) n1/ Au NSs/SiOx/air; (b) ITO/Au NSs/n2/air; (c) ITO/Au NSs/SiOx/n3.

Effect of n2. Figure 10b shows a large increase of Δλi with n2, where it evolves from Δλi = 22.3 nm for n2 = 1.45 to Δλi = 140.0 nm for n2 = 2.63. As the starting value of λmax does not depend on n2 (d = 0 nm), the increase of Δλi is only due to the red-shift of the LSPR with n2 for a layer thickness corresponding to the first extremum of the λmax(d) curve. Effect of n3. Figure 11 shows several λmax(d) curves for increasing value of the refractive index of the detection medium

Figure 12. (a) Normalized absorption computed with the previous analytical model, for n1 = n2, n3 = 1.5, and d = 770/n2 nm. Blue lines represent E(λ, d), the thicker black curve represents Lc(λ), and the red curves correspond to Ath(λ, d) = E(λ, d) × Lc(λ). The gray arrows indicate the position of the absorption maximum. (b) Evolution of PF-cavity contrast with refractive indexes n1, n2, and n3.

Figure 11. Evolution of λmax with SiOx layer thickness d in an ITO/Au NSs/SiOx/n3 structure, for several values of n3. 17825



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the analytical model in a simple situation where n1 = n2 and n3 = 1.5. In this case, the excitation field at the NS location is a partially stationary wave arising from the reflection of the incident plane wave on the n2/n3 interface. When the refractive index of the substrate increases, the contrast of the stationary wave increases as well. It clearly appears that the larger the contrast, the larger the shift of the LSPR wavelength (indicated by a gray arrow). The shift is maximal when the contrast of the interference pattern under the interface is equal to 1. Hence, a first parameter allowing to predict the evolution of the wavelength shift amplitude is the contrast of the interference pattern inside the cavity. Figure 12b shows the evolution of this contrast as a function of n3/n2 and n1/n2. Figure 13 shows the FDTD results of the amplitude evolution when either n1, n2, or n3 is changed. We can see that

Figure 14. (a) Absorption spectra for a ITO/Au NSs/SiOx structure and increasing refractive index n1. (b) Absorption spectra for an ITO/ Au NSs/SiOx structure and increasing refractive index n2.

the LSPR mode width with n1, which tends to increase the oscillation amplitude. Effect of n2. When n2 increases, the oscillation amplitude first increases (for n2 < 2.4) and then decreases (Figure 13b). The increase of the oscillation amplitude is in agreement with the contrast evolution of the field inside the layer. Again, Figure 14b shows several absorption spectra for systems where n2 = n3 = nSiOx in order to see the evolution of the plasmonic response with n2. It appears, similarly to the previous paragraph, that the decrease of the oscillation amplitude coincides with the multimodal plasmon response of the gold NSs, as the number of LSPR modes increases with n2. Effect of n3. The evolution of the oscillation amplitude with n3 is consistent with the contrast evolution presented above. The oscillation amplitude decreases down to 0 for 1.0 < n3 < n2 = nSiOx, whereas it increases for n3 > n2 (Figure 13c). Indeed, when n3 increases from 1.0, the reflection coefficient at the SiOx/n3 interface and consequently the contrast both start to decrease. They cancel for n2 = n3 and finally increase again (with opposite sign) when n3 becomes larger than n2.

Figure 13. Influence of refractive indexes on the oscillation amplitude. Oscillation amplitudes reported are the first half-period amplitudes for the following structures: (a) n1/Au NSs/SiOx/air; (b) ITO/Au NSs/ n2/air; (c) ITO/Au NSs/SiOx/n3.

• there is no oscillation of the plasmonic response for n2 = n3 • the oscillation amplitude increases when n2 increases and when n3 increases beyond n2 Effect of n1. The oscillation amplitude is strongly affected by the value of the substrate refractive index (n1). Figure 13a shows that, when n1 increases, the oscillation amplitude first decreases (for n1 < 1.35), then increases (for 1.35 < n1 < 2.00), and finally decreases (for n1 > 2.00). The first part of the curve is in agreement with the fact that the contrast decays when n1 increases. For larger values, we have to take into account the modification of the LSPR mode of the gold NS with n1. Figure 14 shows the absorption spectra for several systems where n2 = n3 = nSiOx. In Figure 14a, we can verify that λeff is red-shifted when n1 increases, consistent with our previous observations. Moreover, the LSPR becomes wider and multimodal with n1. Hence, if the origin of the oscillation amplitude evolution with n1 is still not clear, it seems to result from a competition between several phenomena. Two of them strive for a decrease of the oscillation amplitude when n1 increases: the decrease of the contrast of the stationary waves system inside the PF-cavity and multimodal plasmonic response. The last phenomenon is the increase of

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CONCLUSION We have presented in this paper an extensive study of the displacement of the LSPR wavelength in a system of gold NSs coated with a dielectric layer of varying thickness. A physical phenomenon leading to this shift has been enlightened using a combination of Green’s tensor method (single NS problem) and FDTD (NS grating problem) numerical simulations. It appears that this system can be described in a first approximation by a simple analytical model, in which the plasmon response of the metallic particle is triggered by the electric field 17826



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resulting from the coupling of the incident wave with the Fabry−Pérot cavity formed by the intermediate layer, without the NSs. For a given layer thickness, the LSPR mode is blue- or red-shifted depending at which wavelength the electric field amplitude is maximum or minimum at the NS position. When the layer thickness increases, the wavelength at which an antinode of the interference pattern inside the layer coincides with the particle is red-shifted. This red-shift corresponds to the rapidly growing branches on the λmax(d) curves, as shown in Figure 6. When the antinode is too far from the particle plasmon, it is replaced by the next one, of lower wavelength. Together with this qualitative description of the physical phenomenon, we have presented a quantitative analysis of the evolution of different parameters characterizing the LSPR shift with the layer thickness d, when the refractive indexes of the background structure are modified. It appears that • The period P of the oscillation obeys the relation P = λeff/2n2 and increases with n1 (red-shift of the LSPR) but decreases with n2. • The average wavelength λeff increases with both n1 and n2 but does not depend on n3. • The initial wavelength shift Δλi for small thickness has a complex behavior with n1, as changing n1 shifts both the starting and the average value of λmax. When n2 or n3 is changed, neither the initial value (n3) nor the average value (n2) of the LSPR wavelength is modified, making the evolution of Δλi simpler to predict, taking into account the fact that increasing n2 or n3 red-shifts the plasmon wavelength. • The oscillation amplitude A evolution is governed by several competing factors. First, the contrast of the interference pattern of the unperturbed field inside the dielectric layer: the larger the contrast, the larger the oscillation amplitude; second, the width of the plasmon mode, which increases as well with the refractive index surrounding the particle; and finally, the multimodal character of the plasmon resonance. The evolution is clear with n3, which, through the modification of the reflection coefficient on the upper interface, changes the contrast and consequently the oscillation amplitude. When either n1 or n2 is changed, no general rule could be found at this point. Hence, this structure is an interesting starting point to design highly sensitive sensors. To optimize the response of such a device, either the multilayer structure or the shape of the particle can be tuned, as both allow one to influence the shift of the LSPR wavelength.

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Article

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported in part by the Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council, the EU-FEDER and Interreg IV, project “Plasmobio”, and FEDER through the “Contrat de Projets Etat Region CPER 2007-2013.” 17827


Plasmonic Nanoparticles Array for High-Sensitivity Sensing: A

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