Dependence between the Refractive-Index Sensitivity of Metallic

Article pubs.acs.org/JPCC

Dependence between the Refractive-Index Sensitivity of Metallic Nanoparticles and the Spectral Position of Their Localized Surface Plasmon Band: A Numerical and Analytical Study Ophélie Saison-Francioso, Gaeẗ an Lévêque, Rabah Boukherroub, Sabine Szunerits, and Abdellatif Akjouj* Institut d’Electronique, de Microélectronique et de Nanotechnologie, UMR CNRS 8520, Université Lille 1, Sciences et Technologies, Avenue Poincaré BP 60069, 59652 Villeneuve d’Ascq, France ABSTRACT: The refractive-index sensitivity of metallic nanoparticles was investigated numerically using the FDTD method as well as analytically. The obtained results show that the sensitivity of nanoparticles situated in a homogeneous host matrix is independent of their shape. Moreover, it exclusively depends on the LSPR band location, the dielectric function of the metal constituting the nanoparticles, and the refractive index of the host matrix. In the case of nanoparticles deposited on a substrate or in close interaction with it, the sensitivity is however dependent on the shape. Additionally, a loss of sensitivity is observed. Different theoretical trails are examined in order to better understand those phenomena. The study emphasizes that the shape-dependent sensitivity can be related to the modification of the depolarization factors of the nanoparticles in the presence of a substrate.

1. INTRODUCTION In the past decade, the plasmonic properties of noble metal nanostructures have attracted much attention for potential use in optical spectroscopy, photonic devices, and sensors.1−9 The localized surface plasmon resonances (LSPRs) supported by these nanostructures play an important role in such applications. LSPRs are collective electron charge oscillations confined in metallic nanostructures that exhibit enhanced near-field amplitudes at the resonance frequency. The position of the LSPR band and the profile of the electromagnetic field depend largely on the morphology of the nanostructures such as size10−12 shape,13−15 but also on their composition16−18 as well as on the dielectric environment. The dependence of the optical properties of metal nanostructures on their local environment is indeed the basic principle for the use of LSPR structures for labelfree molecular sensing.19−25 That dependency is typically revealed through a shift of the LSPR wavelength induced by a change in the interfacial refractive index (RI) when target molecules are in close contact or immobilized to the plasmonic structures. A surface-supported array of noble nanoparticles constitutes one of the most versatile platforms for this kind of RI sensing. However, surface-supported particles suffer from an intrinsic drawback compared to particles in solution, as the fixed RI of the supporting material will reduce the overall RI sensitivity of the particles.10 The effect of the underlying substrate on the sensing capabilities has therefore been a recurrent subject of studies.26−28 Next to these parameters, the interaction of plasmons on adjacent nanoparticles will have a strong influence on the final plasmon response.29−31 Due to the large number of parameters influencing the LSPRs carried by nanostructures, it is desirable to have accurate theoretical models that predict the properties of plasmonic nanostructures.32−36 Particularly, © 2015 American Chemical Society

numerical methods for solving Maxwell’s equations for light scattering from nanostructures of arbitrary shape in a complex environment are precious tools because they allow optimization of plasmonic nanostructures prior to experiment.37,38 In this paper, we use the finite difference time domain (FDTD) method39,40 to investigate numerically the bulk RI sensitivity of the lowest energy resonance mode of different types of plasmonic nanowires as schematically outlined in Figure 1. Two situations where the structures are embedded in a

Figure 1. Schematic representation of the three LSPR structures studied in this work. Arrays of infinite gold nanowires are deposited on a substrate of refractive index n1. The nanowires are excited by an input ⎯ ⎯→ → source placed in the glass substrate ( E0 , k 0 ). The electromagnetic wave propagates along the z axis, and its electric field is polarized along the x axis. The signal transmitted by the LSPR structures is recorded in the detection medium, which is only represented for the rectangular nanowires on the figure and whose refractive index is n2. Both substrate and detection medium are dielectrics. Received: August 27, 2015 Revised: November 23, 2015 Published: November 30, 2015 28551

DOI: 10.1021/acs.jpcc.5b08357 J. Phys. Chem. C 2015, 119, 28551−28559

Article

The Journal of Physical Chemistry C homogeneous matrix and when surface supported will be investigated. The numerical results for nanowires in a homogeneous matrix will be compared with those obtained by an analytical model for the RI sensitivity, available in the quasistatic approximation and developed by Miller and Lazarides.41,42 Our goal is to explore which parameters control the RI sensitivity of plasmon resonances and if general tendencies could be found. Finally, we show that using the method of image charges43 to complete the model of Miller and Lazarideswhich is only available in the case of a homogeneous matrixgives new perspectives in understanding the decrease of RI sensitivity when plasmonic nanostructures lie on a substrate.10,28

2. MODELING PART: DESCRIPTION OF THE SIMULATION PROGRAM AND THE LSPR STRUCTURES INVESTIGATED Three different nanostructures were investigated and are schematically depicted in Figure 1. They all consist of arrays of infinite gold nanowires of different cross sections leading to rectangular, triangular, and U-shaped nanowires. The electromagnetic properties of all three LSPR structures are invariant along the y axis. Consequently, numerical simulations can be performed using a bidimensional FDTD method,39,40 which allows solving Maxwell’s curl equations by replacing all derivatives by finite time and space differences. A discrete grid was generated along the x- and z-axis using a mesh interval equal to Δx = Δz = 1 nm. Periodic Boundary Conditions (PBC) were used to repeat periodically the unit cell along the x-axis with a being the lattice parameter. An incoming TM-polarized pulse is generated inside the substrate (n1) by a current source parallel to the x-axis. The emitted electromagnetic wave propagates along the z-axis and excites localized surface plasmon modes inside each nanostructure. In order to avoid reflections of outgoing waves, Perfectly Matched Layers (PML)44 were applied at z-boundaries of the unit cell. The signal transmitted by the LSPR structures is recorded as a function of time in the detection medium (n2), situated above and around the nanowires. This signal is Fourier-transformed in order to obtain the transmission coefficient as a function of frequency. The gold dielectric constant in our FDTD code is described by a Lorentz−Drude model M

ε(ω) = εr, ∞ +

∑ m=0

Figure 2. Transmission spectra computed using the FDTD method for an array of rectangular gold nanowires (black dotted line), triangular gold nanowires (red line), and U-shaped gold nanowires (blue dashed line). Refractive indexes of the host matrix are n1 = n2 = 1.5.

(1) The case where n1 = n2, which corresponds to nanowires situated in a homogeneous matrix. (2) The case where nanowires are deposited on a dielectric substrate with refractive index n1 = 1.5. The RI sensitivities of the nanostructures were approximated by ΔλLSPR/Δn2. The straight lines, representing the linear increase of λLSPR with n2 for the homogeneous matrix case and the case of a matrix with a substrate, intersect at one point whose abscissa is n2 = 1.5. Consequently, in order to save computation time, RI sensitivities were determined by using extinction spectra for the following detection media: n2 = 1.0 and n2 = 1.5. 3.1. Numerical Results and Discussion. The nanowires RI sensitivity S as a function of their LSPR wavelength, determined by FDTD calculation, is displayed in Figure 3. The data corresponding to nanowires with a rectangular cross-section are represented by circles. One of these points matches a system

fm ωp2 ωm2 − ω 2 + jω Γm

(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 use M = 5 damped harmonic oscillators in order to take interband transitions of gold into account. The values, taken from Rakic et al.,45 are listed in the Table 3.

3. RESULTS Figure 2 shows transmission spectra of three of the LSPR nanostructures studied in this work. The triangular and U-shaped gold nanowire arrays display a multimodal plasmonic response. Only the RI sensitivity S of the lowest energy resonance mode of all LSPR nanostructures considered was studied in the following (in Figure 2, the mode at λLSPR = 877 nm for the triangular nanowire array and at λLSPR = 974 nm for the U-shaped nanowire array). Two different cases are investigated.

Figure 3. Representation of the RI sensitivity for the different nanowires studied as a function of their LSPR wavelength for n2 = 1.5. Points obtained for nanowires with a rectangular, triangular, and U-shaped cross-section, respectively, correspond to circles, triangles, and squares. Open symbols match the case where nanowires are situated in a homogeneous matrix (n1 = n2). Filled symbols match the case where nanowires are deposited on a substrate with refractive index n1 = 1.5. Geometrical parameters of all nanowire arrays studied are listed in the Table 1a−c. 28552


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The Journal of Physical Chemistry C

Table 1. Geometrical Parameters, LSPR Wavelength for n1 = n2 = 1.5, and RI Sensitivity of Each Type of LSPR Structures Studied by the FDTD Methoda (b) nanowire arrays of triangular cross-section

(a) nanowire arrays of rectangular cross-section geometrical parameters and λLSPR (nm) l

h

100 90 80 70 60 100 100 100 100 100 100 100 100 125

15 15 15 15 15 25 35 45 55 15 15 15 15 15

dip

λLSPR

geometrical parameters and λLSPR (nm)

S (nm/RIU) with substrate

without substrate

25 904.0 270.3 35 819.2 213.6 45 759.9 178.8 55 712.4 149.0 65 672.2 126.5 25 782.5 210.2 25 714.3 159.1 25 650.5 81.7 25 612.5 45.5 15 996.8 339.6 35 856.7 232.5 45 829.5 214.8 55 811.5 200.3 15 1110.8 394.6 (b) nanowire arrays of triangular cross-section

geometrical parameters and λLSPR (nm)

446.2 370.3 317.3 273.8 238.3 328.5 253.0 166.3 128.1 518.9 404.5 380.3 365.8 613.2

h

dip

λLSPR

with substrate

without substrate

126 136 116 106 96 126 126 126 126 126 126

55 55 55 55 55 65 75 85 95 55 55

19 9 29 39 49 19 19 19 19 29 39

876.8 996.8 832.1 753.3 732.2 816.7 799.2 784.9 773.3 842.6 824.4

197.8 263.4 178.9 134.2 126.8 166.8 158.0 148.9 144.7 183.5 176.0

447.7 545.3 424.7 302.7 307.0 354.5 341.1 333.4 318.8 425.0 412.3

h 55 55

dip

λLSPR

with substrate

without substrate

49 814.2 171.9 59 806.6 168.9 (c) nanowire arrays of U-shaped-cross-section

geometrical parameters and λLSPS (nm)

S (nm/RIU)

b

b 126 126

S (nm/RIU)

lc

e

dip

71 61 51 41 31 71 71 71 71 71 71 71 71

5 5 5 5 5 10 15 20 25 5 5 5 5

30 30 30 30 30 30 30 30 30 20 40 50 60

51 41 41

10 15 20

50 70 80

λLSPR

406.2 405.1

S (nm/RIU) with substrate

l = 100 nm, h = 50 nm 1956.0 1031.2 1899.3 1010.2 1833.2 963.6 1771.5 934.6 1702.1 895.6 1464.7 772.0 1225.1 610.6 1083.1 522.8 974.4 446.4 2184.6 1171.8 1845.9 975.4 1771.5 934.6 1713.1 880.8 l = 80 nm, h = 50 nm 1248.4 634.6 1012.1 477.8 882.6 374.6

without substrate 1263.8 1228.8 1182.0 1149.2 1084.0 905.2 702.8 596.4 504.4 1423.6 1195.0 1149.2 1095.8 731.6 543.2 424.2

a

For the record, the refractive index of the substrate is n1 = 1.5. The parameter dip refers to interparticles distance which is equal to dip = a− l for nanowire arrays of rectangular and U-shaped cross-section, and dip= a−b for nanowire arrays of triangular cross-section. The significance of all geometrical parameters is explained on Figure 1.

characterized by nanowires of height h = 15 nm, width l = 125 nm, and period a = 140 nm. The other points were obtained by changing successively the height, the width, and the period of a reference system characterized by h = 15 nm, l = 100 nm, and a = 125 nm. The height, width and period were varied from 15 to 55 nm, 60 to 100 nm and 115 to 155 nm, respectively. The data corresponding to nanowires with a triangular cross-section are represented by triangles. They were obtained by changing successively the height, the base, and the period of a reference system characterized by h = 55 nm, b = 126 nm, and a = 145 nm. The height, base and period were varied from 55 to 95 nm, 96 to 126 nm and 145 to 185 nm, respectively. Finally, the data corresponding to nanowires with a U-shaped cross section are represented by squares. Three of these points match systems characterized by l = 80 nm, h = 50 nm, a = 130 nm, e = 10 nm, and lc = 51 nm; l = 80 nm, h = 50 nm, a = 150 nm, e = 15 nm, and lc = 41 nm; l = 80 nm, h = 50 nm, a = 160 nm, e = 20 nm, and lc = 41 nm. The other points are obtained by changing successively the period, the base thickness (e), and the gap between the two branches of the U-shaped cross section (lc). The parameters of the reference system are l = 100 nm, h = 50 nm, a = 130 nm, e = 5 nm, and lc = 71 nm. The period, thickness e and lc were varied from 120 to 160 nm, 5 to 25 nm and 31 to 71 nm, respectively. The geometrical parameters of each LSPR structure considered

here with its corresponding sensitivity and its associated resonance wavelength for n1 = n2 = 1.5 are listed in Table 1a−c. Two important findings can be deduced from Figure 3. In the case of a homogeneous embedding of the nanostructures, the evolution of the RI sensitivity with the LSPR wavelength is governed by a law that is not influenced by the shape, size, or period of the nanowires. In the case where the nanostructures are deposited on a dielectric substrate with refractive index n1 = 1.5, the evolution of the RI sensitivity with the resonance wavelength is dependent on the nanowires morphology. Moreover, for a given type of nanowires, no dependence of the RI sensitivity on the geometrical parameters or the coupling strength is observed. Finally, Figure 3 highlights the decrease of the sensitivity when a substrate is introduced below the nanowires. Particularly, a more important decrease can be noticed for larger wavelengths. Expanding the wavelength range of the plasmon resonance for the rectangular or triangular particle is difficult in this set of simulations. Indeed, the plasmon wavelength for particles with these shapes is essentially modified by changing their aspect ratio, which must be decreased (flatter particles) to obtain a red shift. For that we can only decrease the thickness or increase the width of the particle. In the first case, we are limited by the discretization step (a decrease of the step increases a lot the computation time) and in the second case by the finite period of the grating. Let us emphasize that for rectangular or triangular 28553


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Figure 4. (A) Comparison between S numerically predicted by FDTD for gold nanowire arrays considered in this work (n1 = n2 = 1.5) and the RI sensitivity given by the analytical expression (eq 9 with n = 1.5, A = −3.8251 × 10−5 nm−2, B = −0.0074 nm−1, and C = 10.0976). (B) Error between the RI sensitivity evaluated with the analytical expression (Sth) and the RI sensitivity predicted by FDTD (SFDTD). The error is calculated as follows: Error = (Sth − SFDTD)/SFDTD.

particles, the field at the resonance is enhanced at sharp edges of the same particle, while the large resonance wavelength for the Ushaped particles comes from the fact that the field is strongly localized and enhanced in between the two harms of adjacent particles: the wavelength is more in the red if the space in between two harms is shorter. Hence, the origin of the localized plasmon is a bit different in the two cases; this why we cannot really expand the overlap of the λLSPR range for these two types of particles. 3.2. Nanoparticles in a Homogeneous Host Matrix: Analytical Model Describing How S Evolves with the LSPR Wavelength. Miller and Lazarides have developed an analytical expression available in the quasistatic approximation for the RI sensitivity of unique metallic nanoparticles situated in a homogeneous matrix.41,42 They defined the RI sensitivity S as S=

dλLSPR = dn

parameters. The Li parameters depend on the nanoparticle shape and reflect the nanoparticle depolarization in the three directions of space. The localized plasmon resonance occurs when the polarizability reaches a maximum. If the dielectric function imaginary part of the metal constituting the nanoparticle is small or slowly varying then the resonance condition is given by the well-known formula ⎛ 1 − Li ⎞ ′ = − n 2⎜ εLSPR ⎟ ⎝ Li ⎠

By introducing a new parameter χi = (1 − Li)/2Li, which depends on the nanoparticle shape, the resonance condition can be rewritten as

′ εLSPR = −2χi n2

′ dεLSPR

(2)

′ dεLSPR 2 ′ = εLSPR dn n

where n is the refractive index of the host matrix and ε′ is the real part of the dielectric function of the metal constituting the nanoparticle. Equation 2 shows that the RI sensitivity is related to the same quantities that determine LSPR wavelength for a given nanoparticle, namely, the resonance condition (ε′LSPR) which depends on the nanoparticle’s shape and on the refractive index of the host matrix and the way the real part of the metal dielectric function evolves with the wavelength (ε′(λ)). Moreover, eq 2 shows that the RI sensitivity is equal to the rate of change with n of ε′LSPR divided by the rate of change with the wavelength of ε′, taken at the resonance wavelength. For particles much smaller than the excitation light wavelength (quasi-static approximation), the principal components of their polarizability tensor46 can be written as εNP(λ) − n2 2

(5)

We deduce from eq 5 that the first derivative of the resonance condition with respect to n is equal to

dn dε ′ (λ) dλ λ

LSPR

αi = V

(4)

2

n + Li(εNP(λ) − n )

(6)

Concerning the evolution of the real part of the metal dielectric constant ε′(λ), we did not opt for using a linear function like Miller and Lazarides did. Indeed, we worked on a wider range of wavelengths (500−2200 nm), and consequently, a quadratic function to accurately fit ε′(λ) was needed ε′(λ) = Aλ 2 + Bλ + C

(7)

From eq 7, the first derivative of ε′(λ) with respect to the wavelength at the resonance is dε′(λ) dλ

= 2AλLSPR + B λLSPR

(8)

Finally, eqs 6, 7, and 8 gave the analytical expression for the RI sensitivity

, i = x, y, z (3)

where V, εNP(λ) = ε′ (λ) + iε″ (λ), and Li, are respectively, the volume, the nanoparticle’s dielectric function, and geometrical

S= 28554

2 2(AλLSPR + BλLSPR + C) n(2AλLSPR + B)

(9) DOI: 10.1021/acs.jpcc.5b08357 J. Phys. Chem. C 2015, 119, 28551−28559

Article

The Journal of Physical Chemistry C This analytical expression does not depend anymore on the nanoparticles shape (Li). The RI sensitivity is now only related to the LSPR wavelength, the refractive index of the host matrix, and the wavelength dispersion of the real part of the dielectric function of the nanoparticle. Figure 4A depicts graphically the analytical expression for the RI sensitivity as a function of the resonance wavelength for a gold nanoparticle situated in a host matrix of refractive index n = 1.5. In order to compare this curve with our FDTD results, a quadratic fit of the real part of the gold dielectric function given by the Lorentz−Drude model used in our FDTD code has been realized between 500 and 2200 nm, which gives A = −3.8251 × 10−5 ± 2.5340 × 10−7nm−2, B = −0.0074 ± 0.0007 nm−1, and C = 10.0976 ± 0.4308. Figure 4A shows a good agreement between the evolution of the RI sensitivity with LSPR wavelength given by the analytical expression (eq 9) and FDTD results. Nonetheless, we observe a slight overestimation of the sensitivity compared to the results predicted by the numerical simulation. This overestimation increases when the resonance wavelength decreases. Figure 4B shows that eq 9 allows approximating more accurately the sensitivity of the plasmonic structures studied for large wavelengths. Two points present a significant error in Figure 4A: one at λLSPR = 613 nm with an error of 68.5%. This corresponds to a nanowire array with a rectangular cross-section whose width is l = 100 nm and height h = 55 nm. The second point at λLSPR = 651 nm reaches an error of 52.9%. It corresponds to a nanowire array with rectangular cross-section with l = 100 nm as well, but its height is slightly smaller and equal to 45 nm. In fact, the two points are related to the largest nanowires studied. Consequently, it is highly probable that the significant discrepancy between theoretical and numerical values of sensitivity is due to the nanowires size (and especially to their height), which is too large to satisfy the quasi-static approximation. Moreover, the plasmonic resonance of those nanowires is situated in a wavelength range where the real part of the gold dielectric function is low and its imaginary part is increasing because of interband transitions appearance below λ = 667 nm. The approximation which allowed establishing the resonance condition (eq 4) is not as good as in the case of nanowire arrays with a LSPR wavelength in the near-infrared. Every other point shows an error lower than 20%, except the point at λLSPR = 714.3 nm with an error of 24.9%. This point corresponds to a nanowire array of rectangular cross-section with l = 100 nm and h = 35 nm. In principle, the analytical expression for S (eq 9) is only available for plasmonic systems composed by a single nanoparticle. Every nanowire studied in this work is organized in arrays. Figure 5 is the same as Figure 4A with more information. It includes in addition the change in S, when a single gold nanowire situated in the same host matrix is studied. These points were obtained by using a bidimensional FDTD code where the computational domain is entirely bounded by PML44 instead of having periodic boundary conditions along one direction. In the case of a single gold nanowire with a U-shaped cross-section, the LSPR wavelength is λLSPR = 1542 nm and S = 970 nm/RIU. The resonance wavelength of nanowires organized in an array with a period a = 130 nm is λLSPR = 1956 nm and their RI sensitivity is S = 1264 nm/RIU. Having a system composed by a single nanowire instead of an array produced a decrease of the resonance wavelength and a loss of sensitivity. Moreover, we see that the combined displacement of the resonance wavelength and the sensitivity occurred along the pink dashed curve when the nanostructure changes from an array of U-shaped nanowires

Figure 5. Comparison between the sensitivity numerically predicted by FDTD for the single gold nanowires and gold nanowire arrays considered in this work (n1 = n2 = 1.5) and the sensitivity given by the analytical expression (eq 9 with n = 1.5, A = −3.8251 × 10−5 nm−2, B = −0.0074 nm−1, and C = 10.0976). Blue symbols correspond to three systems of single nanowires characterized by the geometrical parameters l = 100 nm and h = 15 nm (blue circle), b = 126 nm, and h = 55 nm (blue triangle) and l = 100 nm, h = 50 nm, lc = 71 nm, and e = 5 nm (blue square). Yellow symbols correspond to the same systems as those represented by blue symbols with nanowires organized in arrays of period a = 125 nm (yellow circle), a = 145 nm (yellow triangle), and a = 130 nm (yellow square).

to a single nanowire. This fact is indicated by an orange arrow on the figure. We observe the same behavior when we have a single nanowire with a triangular cross-section (blue triangle) or a single nanowire with a rectangular cross-section (blue circle) instead of having arrays (yellow triangle and circle). We consequently deduce that for a given type of nanoparticle, plasmonic systems organized in arrays have better RI sensitivities because their resonance wavelengths are increased due to the interparticle coupling. Additionally, it is interesting to note that the points (λLSPR, S) for the systems organized in arrays and for the systems composed by a single nanoparticle align along the same curve. 3.3. Nanoparticles Deposited on a Substrate: Analytical Ways To Understand the Decrease of Sensitivity. Introducing a substrate below the nanoparticles leads to a modification of the above resonance condition (eq 4) and in particular of the influence of the geometrical parameters Li. Yamaguchi et al.47 developed an interesting theoretical approach, recently taken up again by Pinchuk et. al48 and available in the quasistatic approximation. This approach allows calculating absorption spectra of ellipsoidal and spheroidal nanoparticles which are closely interacting with their substrate. It uses the method of the image charges43 in order to obtain the local electric field acting on the nanoparticles (see Figure 6A). Considering an ellipsoidal metallic nanoparticle in the vicinity of ⎯→ the substrate (εm) (Figure 6B), the applied electric field E0 induces an oscillation of the charges at the nanoparticle surface. In the quasistatic approximation, this can be represented by a ⎯p = ql ⃗ located at the center of the ellipsoid (l is the dipole → 1 28555


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The Journal of Physical Chemistry C εm + Lxeff, [εNP(λ) − εm] = 0

(13)

with εm = n22, where n2 is the refractive index of the detection medium. If the imaginary part of the dielectric function of the metal constituting the nanoparticle is small or slowly varying then the resonance condition is given by ⎛ 1 − L eff x, ′ = −n2 2⎜⎜ εLSPR eff L ⎝ x,

⎞ ⎟ ⎟ ⎠

(14)

eff After introducing a new geometrical parameter χeff X,∥ = (1 − LX,∥)/ eff LX,∥, which depends on the nanoparticle shape, the resonance

condition can be rewritten as Figure 6. (A) Schematic representation of a point charge q situated in a medium of dielectric function εm and in the vicinity of a substrate whose dielectric function is εs. This charge q creates an image charge in the substrate equal to q′ = q(εm − εs)/(εm + εs). (B) Schematic representation of an ellipsoidal metallic nanoparticle situated in a medium of dielectric function εm = (n2)2 and in the vicinity of a substrate whose dielectric function is εs = (n1)2. The ellipsoid in the substrate is the image of the nanoparticle.

′ εLSPR = −n2 2χxeff,

In the case where nanoparticles are deposited on or placed close to a substrate (n1), their RI sensitivity can be defined as dλ S = LSPR = dn 2

distance between the charges constituting the dipole). The induced charges in the substrate can be described by an image ⎯p = −q′l ⃗ located at the center of the image ellipsoid, dipole → 2 where q′ = q(εm − εs)/(εm + εs). In fact, the direction and the ⎯p and → ⎯p depend on the applied electric field orientation of → 1 2 polarization. Here, we consider an electric field polarized along the x axis, like in our FDTD code, in order to compare our theoretical and numerical results. To obtain an analytical expression for the sensitivity of this ellipsoidal nanoparticle interacting with the substrate, we need its effective polarizability tensor ⎡ α eff 0 0 ⎤ ⎢ x, ⎥ ⎢ ⎥ eff [αeff ] = ⎢ 0 αy , 0 ⎥ ⎢ ⎥ 0 α⊥eff ⎦ ⎣ 0

αxeff,

axayaz ⎛ εs − εm ⎞ ⎟ ⎜ 24d3 ⎝ εs + εm ⎠

dε ′ (λ) dλ λLSPR

(16)

⎛ n ⎞2 ∂L eff ′ ′ dεLSPR εLSPR x, 2 ⎟ =2 + ⎜⎜ eff ⎟ ∂n dn 2 n2 L 2 ⎝ x, ⎠

(17)

Unlike what happens in the homogeneous host matrix case, we see that the resonance condition’s rate of change with n2 depends on the nanoparticles’ shape via the parameter Leff x,∥. After a few calculations, we obtain the following formula for the RI sensitivity

Squa =

2 2(AλLSPR + BλLSPR + C) + n2(2AλLSPR + B)

⎛ n2 ⎞2 axayaz n2n12 ⎜ ⎟ ⎝ Lxeff, ⎠ 6d3 (n12 + n2 2)2 2AλLSPR + B (18)

(10)

The wavelength dispersion of the real part of the metal dielectric function is still defined as a second-degree polynomial. Using a linear fit of ε′ (λ) (ε′ (λ) = Dλ + E), the RI sensitivity is equal to

S lin =

2(DλLSPR + E) + Dn2

⎛ n2 ⎞2 axayaz n2n12 ⎜ ⎟ ⎝ Lxeff, ⎠ 6d3 (n12 + n2 2)2 D

(19)

The first part of eqs 18 and 19 corresponds to the RI sensitivity of nanoparticles situated in a homogeneous host matrix of refractive index n2. This part does not explicitly depend on the shape of the nanoparticles or on the nanoparticles’ coupling because all of the information concerning the coupling is included in the parameter λLSPR. The black part of eqs 18 and 19 is a negative term which is related to the decrease in sensitivity in the presence of a substrate. This term explicitly depends on the nanoparticles’ shape via the parameters ax, ay, az, and Leff x,∥. In the case of flat nanoparticles on a substrate (d = az), the quantity axayaz/d3 becomes large and leads to a huge overestimation of the interaction between the nanoparticle and the substrate. That is the reason why Valamanesh et al.,49 based on the work of Bobbert and Vlieger,50 proposed to use another effective depolarization factor which is in our case49

(11)

associated with a new geometrical factor Lxeff, = Lx −

′ dεLSPR dn 2

which gives

The symbols ∥ and ⊥ are related to the applied electric field polarization. They, respectively, correspond to a polarization direction parallel or perpendicular to the substrate. Thus, in this work we are only interested in the principal value αeff x,∥. The calculation by Pinchuk et al.48 gives εNP(λ) − εm 4 = πaxayaz 3 εm + Lxeff, [εNP(λ) − εm]

(15)

(12)

where ax, ay, and az are the semimajor axis of the ellipsoid, d is the distance between the nanoparticle and the substrate (see Figure 6B), εNP (λ) is the complex permittivity of the metal constituting the nanoparticle, and Lx is the depolarization factor of the nanoparticle along the x axis. Equations 11 and 12 show that nanoparticles in the presence of a substrate do not have the same polarizability tensor as nanoparticles in a homogeneous matrix because of their geometrical factors. The new resonance condition is 28556


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The Journal of Physical Chemistry C 2 2 ⎧ 3 ⎞ 1 ⎛ n − n2 ⎞ 2 ⎛ ⎜ ⎟ L͠ xeff, = Lx + ⎜ 12 (1 ) + ξ + ξ0 2⎟ξ0 2 0 ⎨⎝ 2 ⎠ 2 ⎝ n1 + n2 ⎠ ⎩ 2

computation time. Its parameter values, taken from Vial et al.,51 are listed in the Table 3. The position of the lowest energy resonance mode is numerically determined for two different detection media (n2 = 1.000 and n2 = 1.333). The RI sensitivity of the system is approximated by ΔλLSPR/Δn2. We considered two cases: the case where the nanoparticles are directly in contact with the substrate (d = az) and the case where the nanoparticles closely interact with the substrate (d−az = 5 nm). Table 2 brings together the RI sensitivities obtained by FDTD and calculated using the analytical expressions in eqs 18, 19, 23, and 24.

⎪

⎪

⎫ ⎛ ⎛1⎞ 1 ⎞ ln⎜1 + 2 ⎟ −ξ0 arctan⎜ ⎟ − ξ0 2⎬ ξ0 ⎠ ⎝ ξ0 ⎠ ⎝ ⎭

⎪

⎪

(20)

with 1

ξ0 =

2

,r=

r −1

ax az

(21)

⎧ 1 ξ2 ⎛ 1 ⎞⎫ 1 0 + ξ0 arctan⎜ ⎟⎬ Lx = (1 + ξ0 2)⎨− 2 2 ⎝ ξ0 ⎠⎭ ⎩ 2 1 + ξ0 ⎪

⎪

⎪

⎪

Table 2. Comparison between FDTD Results and Analytical Resultsa (22)

FDTD results

This new geometrical factor is only available in the case of spheroidal nanoparticles whose revolution axis is perpendicular to the interface substrate/detection medium. Furthermore, it leads to new formula for the RI sensitivity which are, respectively, for a quadratic and a linear fit of ε′ (λ)

S͠ qua =

2 2(AλLSPR + BλLSPR + C) − n2(2AλLSPR + B)

⎛ ⎜ ⎝

n2 L͠ eff x,

analytical results

⎞2 2Ψn2n12 ⎟ (n 2 + n 2)2 ⎠ 1 2

2AλLSPR + B (23)

̆

S lin =

2(DλLSPR + E) − Dn2

⎛ ⎜ ⎝

n2 L͠ xeff,

⎞ ⎟ ⎠

2

(24)

where ⎧ ⎛ ⎪⎛ 3 1 ⎞ 2⎞ 2 ⎜ ⎟ξ ⎟ ⎜ Ψ = (1 + ξ0 2)⎨ + ξ + ln 1 0 0 ⎪⎝ ⎠ ξ0 2 ⎠ ⎝ ⎩ 2 ⎫ ⎛1⎞ −ξ0 arctan⎜ ⎟ − ξ0 2⎬ ⎝ ξ0 ⎠ ⎭

λLSPR (n2 = 1.333) = 570 nm SFDTD = 111 nm/RIU Slin ̆ (eq 19) = 83 nm/RIU error: −25% Slin̆ (eq 24) = 126 nm/RIU error: 14% Squa (eq 18) = 95 nm/RIU error: −14% Squă (eq 23) = 142 nm/RIU error: 28%

λLSPR (n2 = 1.333) = 567 nm SFDTD = 126 nm/RIU Slin ̆ (eq 19) 121 nm/RIU error: −4% Slin̆ (eq 24) = 121 nm/RIU error: −4% Squa (eq 18) = 138 nm/RIU error: 10% Squă (eq 23) = 138 nm/RIU error: 10%

The RI sensitivity of this nanoparticles array is estimated to be 148 nm/RIU by FDTD and 160 nm/RIU by the analytical model (eq 9) in the case of a homogeneous matrix. A better RI sensitivity than in the case of nanoparticles deposited on a substrate was observed in both cases. We can see that analytical expressions, despite their simplicity, predict RI sensitivities with a surprising accuracy. The analytical expressions are more precise in the case of the plasmonic system where d−az = 5nm, in other words when nanoparticles are slightly above the substrate. Moreover, the analytical expressions modified to better describe flat nanoparticles (eqs 23 and 24) give the same results as nonmodified expressions (eqs 18 and 19). A very good accuracy between the sensitivities predicted by FDTD and theoretical results in the case of a linear fit of ε′(λ) since the error is equal to −4% can be observed. In the case of a quadratic fit of ε′(λ), the error is slightly higher and reaches 10%. Concerning the nanoparticles in contact with the substrate (d = az), using modified analytical expressions for flat nanoparticles (eqs 23 and 24) improves the agreement between the sensitivity predicted by FDTD and the sensitivity analytically calculated. This is however only true for the linear fit of ε′(λ). Indeed, when ε′(λ) is fitted by a second-degree polynomial, we observe an increase of the error from −14% to 28%.

⎪

⎪

d-az = 5 nm

a The linear and quadratic fits of the real part of the gold dielectric function have been performed on the Lorentz−Drude model used in our FDTD code51 and on a wavelength range between 500 and 900 nm. We obtained A= −3.0653 × 10−5 ± 1.4339 × 10−6 nm−2, B = −0.0315 ± 0.0020 nm−1, and C = 20.6022 ± 0.6901 for the quadratic fit and D = −0.0744 ± 0.0009 nm−1 and E = 35.1625 ± 0.6222 for the linear fit. The error is calculated as follows: Error = (Sth − SFDTD)/ SFDTD, where Sth is the RI sensitivity evaluated with the analytical expressions.

2Ψn2n12 (n12 + n2 2)2

D

d = az

(25)

We used a three-dimensional FDTD code in order to verify the accuracy of eqs 18, 19, 23, and 24. Figure 7 shows a schematic representation of the investigated plasmonic system. It is a biperiodic array of gold oblate spheroids. The gold dielectric constant of the nanoparticles is described by a Lorentz−Drude model with M = 1 and not 5 like previously in order to save

Figure 7. Schematic representations of the systems studied [(xOz) plan cross-section]. Gold nanoparticles are oblate spheroids (ax = ay = 25 nm and az = 12.5 nm) and organized in a biperiodic array characterized by the periods Ax = Ay = 120 nm along the x axis and the y axis, respectively. Two cases are considered: (A) the nanoparticles are deposited on the substrate (n1 = 1.5) and (B) the nanoparticles closely interact with the substrate (d = 17.5 nm).

4. CONCLUSION In this work, the refractive-index sensitivity of gold nanowire arrays and of single oblate spheroids was investigated numerically using the FDTD method. The results show that the refractive28557


Article


Table 3. Values of the Parameters of the Lorentz−Drude Models Describing the Gold Dielectric Constant in our FDTD Codes values of the Lorentz−Drude model parameters taken from Rakic et al.45

fm ωp2

M

ε(ω) = εr, ∞ +

∑

ω2 m=0 m

ε r ,∞ = 1

f 0 = 0.760 f1 = 0.024 f 2 = 0.010 f 3 = 0.071 f4 = 0.601 f5 = 4.384

∑ m=0

ωp = 1.37188 × 1016 rad/s

ω0 = 0.0 rad/s ω1 = 6.30488 × 1014 rad/s ω2 = 1.26098 × 1015 rad/s ω3 = 4.51065 × 1015 rad/s ω4 = 6.53885 × 1015 rad/s ω5 = 2.02364 × 1016rad/s values of the Lorentz−Drude model parameters taken from Vial et al.51

ωm2 − ω2 + jω Γm

εr,∞ = 5.9673

f0 = 1 f1 = 1.09

Ω0 = 1.3280 × 1016 rad/s Ω1 = 4.0845 × 1015 rad/s

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS A.A. and O.S.F. thank the Centre de Ressources Informatiques (CRI) de Lille 1. R.B. and S.S. gratefully acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS), the University Lille 1, and Nord Pas de Calais region. S.S. thanks the Institut Universitaire de France (IUF).

■

ω0 = 0.0 rad/s ω1 = 4.0845 × 1015 rad/s

Γ0 = 0.1000283 × 1015 rad/s Γ1 = 0.6588548 × 1015 rad/s

(5) Février, M.; Gogol, P.; Aassime, A.; Mégy, R.; Delacour, C.; Chelnokov, A.; Apuzzo, A.; Blaize, S.; Lourtioz, J.-M.; Dagens, B. Giant Coupling Effect between Metal Nanoparticle Chain and Optical Waveguide. Nano Lett. 2012, 12, 1032−1037. (6) Pillai, S.; Catchpole, K. R.; Trupke, T.; Green, M. A. Surface Plasmon Enhanced Silicon Solar Cells. J. Appl. Phys. 2007, 101, 093105. (7) Catchpole, K. R.; Polman, A. Design Principles for Particle Plasmon Enhanced Solar Cells. Appl. Phys. Lett. 2008, 93, 191113. (8) Mayer, K. M.; Hafner, J. H. Localized Surface Plasmon Resonance Sensors. Chem. Rev. 2011, 111, 3828−3857. (9) Szunerits, S.; Boukherroub, R. Sensing Using Localized Surface Plasmon Resonance Sensors. Chem. Commun. 2012, 48, 8999−9010. (10) Larsson, E. M.; Alegret, J.; Käll, M.; Sutherland, D. S. Characteristics of NIR Localized Surface Plasmon Resonances in Gold Nanorings for Application as Ultrasensitive Biosensors. Nano Lett. 2007, 7, 1256−1263. (11) Scaffardi, L. B.; Tocho, J. O. Size Dependence of Refractive Index of Gold Nanoparticles. Nanotechnology 2006, 17, 1309−1315. (12) Hanarp, P.; Käll, M.; Sutherland, D. S. Optical Properties of Short ̈ Range Ordered Arrays of Nanometer Gold Disks Prepared by Colloidal Lithography. J. Phys. Chem. B 2003, 107, 5768−5772. (13) Chen, H.; Kou, X.; Yang, Z.; Ni, W.; Wang, J. Shape- and SizeDependent Refractive Index Sensitivity of Gold Nanoparticles. Langmuir 2008, 24, 5233−5237. (14) Bukasov, R.; Shumaker-Parry, J. S. Highly Tunable Infrared Extinction Properties of Gold Nanocrescents. Nano Lett. 2007, 7, 1113− 1118. (15) Aizpurua, J.; Hanarp, P.; Sutherland, D. S.; Käll, M.; Bryant, G. W.; Garcia de Abajo, F. J. Optical Properties of Gold Nanorings. Phys. Rev. Lett. 2003, 90, 057401. (16) Langhammer, C.; Schwind, M.; Kasemo, B.; Zoric, I. Localized Surface Plasmon Resonances in Aluminum Nanodisks. Nano Lett. 2008, 8, 1461−1471. (17) Ye, J.; Van Dorpe, P.; Lagae, L.; Maes, G.; Borghs, G. Observation of Plasmonic Dipolar Anti-Bonding Mode in Silver Nanoring Structures. Nanotechnology 2009, 20, 465203. (18) Zoric, I.; Zäch, M.; Kasemo, B.; Langhammer, C. Gold, Platinum, and Aluminum Nanodisks Plasmons: Material Independence, Subradiance, and Damping Mechanisms. ACS Nano 2011, 5, 2535−2546. (19) Anker, J. N.; Hall, W. P.; Lyandres, O.; Shah, N. C.; Zhao, J.; Van Duyne, R. P. Biosensing with Plasmonic Nanosensors. Nat. Mater. 2008, 7, 442−453. (20) Stewart, M. E.; Anderton, C. R.; Thompson, L. B.; Maria, J.; Gray, S. K.; Rogers, J. A.; Nuzzo, R. G. Nanostructured Plasmonic Sensors. Chem. Rev. 2008, 108, 494−521. (21) Zhao, J.; Jensen, L.; Sung, J.; Zou, S.; Schatz, G. C.; Van Duyne, R. P. Interaction of Plasmon and Molecular Resonances for Rhodamine 6G Adsorbed on Silver Nanoparticles. J. Am. Chem. Soc. 2007, 129, 7647− 7656.

index sensitivity of nanoparticles situated in a homogeneous host matrix is independent of their shape. An analytical model for the RI sensitivity developed by Miller and Lazarides41,42 was further presented. From this model it emerged that the refractive-index sensitivity exclusively depends on the LSPR band location, the dielectric constant of the metal constituting the nanoparticles, and the refractive index of the host matrix. On the other hand, when nanoparticles are deposited on a substrate or are in close interaction with it, the sensitivity is dependent on the shape. Different theoretical trails have been examined in order to better understand this phenomenon. From an analytical approach based on the image dipole concept developed by Pinchuk et al.,48 we brought a correction to the model of Miller and Lazarides. This correction gave new insights into the dependence to the shape and the decrease of the refractive-index sensitivity in the presence of a substrate. Our model especially shows that the shape dependence of the sensitivity can be related to the modification of the depolarization factors of the nanoparticles in the presence of a substrate.

■

Γ0 = 8.05202 × 1013 rad/s Γ1 = 3.66139 × 1014 rad/s Γ2 = 5.24141 × 1014 rad/s Γ3 = 1.32175 × 1015 rad/s Γ4 = 3.78901 × 1015 rad/s Γ5 = 3.36362 × 1015 rad/s

fm Ωm2

M

ε(ω) = εr, ∞ +

− ω2 + jω Γm

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Dependence between the Refractive-Index Sensitivity of Metallic

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