Generalized Effective Medium Theory to Extract the Optical Properties

Article pubs.acs.org/JPCC

Generalized Effective Medium Theory to Extract the Optical Properties of Two-Dimensional Nonspherical Metallic Nanoparticle Layers Yann Battie,*,† Aotmane En Naciri,† William Chamorro,‡,§ and David Horwat‡ †

LCP-A2MC, Institut Jean Barriol, Université de Lorraine, 1 Bd Arago, 57070 Metz, France CNRS, Institut Jean Lamour, Université de Lorraine, UMR7198, Nancy F-54011, France § Department Materials Science and Engineering, Saarland University, D-66123 Saarbrücken, Germany ‡

S Supporting Information *

ABSTRACT: A new effective medium theory is introduced to describe the optical properties of a two-dimensional array of metallic nanoislands. This model which takes into account both the nanoisland orientation and their shape distribution is successfully used to interpret the ellipsometric measurements performed on gold nanoislands sputtered on a silicon substrate. By coupling ellipsometry to atomic force microscopy measurements, we show that the growth mechanism involves a Volmer−Weber growth mode. The optical anisotropy of uniaxial films was attributed to in-plane preferential selforientation of gold nanoislands. Finally, we demonstrate that ̈ of nanoisland layers the optical birefringence and dichroism can be tuned during the film growth and are due to the splitting of the plasmon resonance into two modes: the transversal and the longitudinal modes of gold nanoislands.

I. INTRODUCTION

Different models have been previously proposed to describe the optical properties of two-dimensional (2D) arrays of metallic nanoparticles.16−20 The nonlocal response formula1 which correctly describes the optical properties of materials, is too difficult to compute, and approximate models are often needed. Some models such as rigorous coupled-wave analysis21 or discrete dipole approximation4 can be used to investigate the optical properties of a metallic nanostructure with few approximations. However, these methods are generally time consuming and are not compatible with real-time characterization. On the contrary, effective medium theories, which approximate a layer composed of metallic nanoparticles as a homogeneous medium, require less computing resources. However, this homogenization procedure can introduce some errors, and the quality of the approximation must be evaluated by far-field measurements. This approach can be considered since the scattering cross section of the metal nanoparticle is negligible. To respect this condition, the nanoparticle size must be smaller than a fraction (∼1/10) of the wavelength in the matrix or inclusion material (Au, glass). Classical effective medium theories22 such as the Maxwell Garnett theory were developed for three-dimensional (3D) organizations of monodispersed spherical nanoparticles. Several attempts have

Metallic nanoparticles exhibit strong optical absorption due to surface plasmon resonance (SPR) induced by the collective oscillation of their conduction electrons. The characteristics of the plasmon resonance depend on the refractive index of the surrounding medium as well as the nanoparticle size, shape, and organization.1−5 Thus, supported nanoparticles are interesting building blocks for solar cells,6,7 chemical sensors,8,9 or optical filters.10 Spectroscopic ellipsometry is a nondestructive optical characterization technique which is highly sensitive to the optical properties of thin films. It has been recently exploited to investigate the growth mechanism of metallic thin films elaborated by sputtering, a common technique used to produce supported metallic nanoislands.11−13 The growth mechanism involves the diffusion of adatoms, the nucleation of metallic islands, and their coalescence.14,15 Real time ellipsometric measurements performed during the growth of gold film have been reported by Beyene et al.13 As ellipsometry is an indirect characterization tool, a B-spline parametrization coupled with a Kramers−Kronig inversion was used to analyze their ellipsometric spectra.13 However, due to the limited spectral range of measurements, the Kramers−Kronig inversion requires us to extrapolate them. Thus, the development of new effective medium theory is required and highly sought to correlate the optical properties of metallic thin films to their morphology. © 2014 American Chemical Society

Received: December 5, 2013 Revised: February 13, 2014 Published: February 17, 2014 4899

dx.doi.org/10.1021/jp4119343 | J. Phys. Chem. C 2014, 118, 4899−4905

The Journal of Physical Chemistry C

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or 120 s sputtering time, respectively. The number of evaporated atoms varied linearly from 1.2 × 1021 to 9.5 × 1021 cm−2 as the deposition time increased from 15 s (S1) to 2 min (S4). The surface topography of each film was investigated by atomic force microscopy (Pacific nanotechnology, NanoR2). An n-type Si tip with a 10 nm radius is used in tapping mode. Ellipsometric measurements were recorded with an UVISEL Horiba Jobin Yvon ellipsometer in the 0.6−4.97 eV spectral range. Four angles of incidence are considered: 60°, 65°, 70°, and 75°. The configuration of our ellipsometer measures two parameters Is and Ic which are linked to the ellipsometric angles Ψ and Δ by the following equations

been made to date to introduce the effects of nanoparticle size and shape distributions. Goncharenko et al.23 have extended these theories by taking into account the shape distribution of spheroidal nanoparticles. However, the generalization of the Goncharenko effective medium theory (GEMT) to the 2D medium remains a challenge. By considering nanoparticles as interacting point dipoles, Yamaguchi et al.18,19 have proposed a model to describe the optical properties of an array of monodispersed spherical nanoparticles distributed at the nodes of a square lattice. However, a collection of nanoparticles is always dispersed in size and shape. An extension of the Yamaguchi model was developed by Toudert et al.16,17 to take into account the nanoparticle shape distribution. This model was applied to 2D arrays of nanoparticles embedded in a dielectric medium.17 However, this model fails to describe the nanoparticle−substrate interactions.3 Moreover, it requires a preliminary estimation of the pair correlation function of nanoparticles by transmission electron microscopy. A more realistic approach was proposed by Bedeaux and Vlieger20 by considering the nanoparticle layer as an excess of susceptibility. This model was successfully applied to monodisperse truncated spheroidal nanoparticles24,25 by taking into account multipolar expansion. However, due to its complex parametrization, this model was rarely applied to polydispersed nanoparticle spheroids.26 In this paper, an alternative effective medium theory is introduced to describe the optical properties of a 2D array of metallic nanoislands. This two-dimensional effective medium theory (2DEMT), which is based on a mean field approach, generalized the GEMT23 by taking into account both the optical anisotropy induced by nanoisland orientation and their shape distribution. 2DEMT is successfully used to interpret ellipsometric measurements performed on 2D arrays of the gold nanoislands sputtered on a silicon substrate which are unconventional samples for ellipsometry. We show that 2DEMT gives a better description of the optical properties of the nanoisland films than conventional effective medium theories. By comparing ellipsometric to atomic force microscopy (AFM) measurements, we unambiguously demonstrate that 2DEMT is useful to study the first step of nanoisland nucleation and to understand the origin of the uniaxal anisotropy of metallic nanoisland thin films.

Is = sin 2Ψ·sin Δ

(1)

Ic = sin 2Ψ· cos Δ

(2)

III. RESULTS AND DISCUSSION Details of the nanoisland morphology were obtained from AFM observations of the film surface (Figure 1). Isolated

Figure 1. AFM images of the surfaces of (a) S1, (b) S2, (c) S3, and (d) S4 gold films sputtered on silicon substrates. Inset: the same films deposited on glass substrates.

II. EXPERIMENTAL METHODS Gold nanoislands have been synthesized on monocrystalline 100-oriented silicon substrates using direct current magnetron sputtering of a gold target (2 in. diameter, 3 mm thickness, and purities over 99.9%) in argon. Note that glass substrates are also used to give some insight into the film visual aspect. The deposition chamber (approximately 40 L) was pumped down via a combination of a mechanical and a turbomolecular pump, allowing a base vacuum of 10−4 Pa before deposition. The argon gas flow rate was fixed to 50 standard cubic centimeters per minute, which resulted in a deposition pressure of 0.3 Pa. The target−substrate distance was fixed at 55 mm, and the target was placed off-axis relative to the rotating substrate− holder axis enabling a good lateral homogeneity of the gold loading. The surface of the silicon substrates was etched in situ before deposition using radiofrequency polarization at 100 W for 2 min, allowing an optimal adhesion of the gold nanoislands to the silicon substrates. A target current of 11 mA was applied to the gold target using an MDX 500 advanced energy supply. Four films, S1, S2, S3, and S4, were elaborated with a 15, 30, 60,

metallic islands, randomly distributed in the substrates plane, i.e., the (X, Y) plane, are clearly observed. In other words, the nanoislands are randomly distributed on the top flat surface of the SiO2 native layer. In a first approximation and despite their truncated shape, these metallic nanoislands can be considered as nearly oblate. Their minor axis is predominantly along the normal direction of the substrate. This orientation is typical of the sputtering technique used to produce gold nanoislands and is closely related to the gold wettability.12,27 The projected surface density of nanoislands monotonously increases from 1% to 15% as the sputtering time varies from 15 s to 2 min. Figure 2 depicts the average and standard deviation of the inplane (dxy) and out-of-plane (dz) diameters estimated from a statistical analysis of more than 50 nanoislands. dz can also be assimilated to the film thickness. Considering the convolution of the AFM tip with the nanoisland profile, dxy is slightly overestimated. dz is 10 to 17 times smaller than dxy, confirming the oblate shape and the orientation of nanoislands. The in4900



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(εeff). Generally, such an inhomogeneous top layer is modeled by conventional effective medium theories such as Maxwell Garnett (MG) or Bruggeman (BEM).22 Here, we demonstrated that optical responses of 2D nanoisland films are improperly determined by these theories. As shown in Figure 3, MG and BEM22 give the same results. Indeed, experimental data are well adjusted in the 2.5−4.5 eV spectral range where the absorption of the substrate is dominant. However, both models fail to adjust the ellipsometric data in the 0.6−3 eV spectral range, i.e., close to the plasmon resonance of nanoislands. The disagreement is observed for all angles of incidence. Both models are restricted to three-dimensional materials.22 In addition, the Maxwell Garnett approach considers only monodispersed spherical inclusions. Thus, a more realistic effective medium theory must be developed to take into account the nanoisland shape distribution and orientation. This effective medium theory is built in three steps. First, we consider an isotropic threedimensional material composed of nonoriented ellipsoidal nanoparticles distributed in shape and embedded in a matrix. In a second step, all nanoparticles are oriented in the same direction. Finally, we consider only a thin slab of this material. This slab is composed of a single layer of nanoparticles. The continuity of the normal component of the electric displacement field at the interface is not taken into account in this effective theory. Indeed, in our approach the nanoisland layer is considered as a three-dimensional film with a film thickness d and not as an interface. However, the continuity of the normal component of the electric displacement field at the nanoisland film/SiO2 interface is taken into account in another step by modeling the propagation of light through the whole sample structure. In a composite material, the effective dielectric tensor [εeff] is related to the spatial averages of the displacement ⟨D⟩ and electric field ⟨E⟩

Figure 2. Evolution of the mean values of (a) the nanoislands in plane diameter (dxy) and (b) the film thickness (dz) estimated from (a,b) AFM and (b) ellipsometric measurements during the deposition process. The gray areas are the 68% confidence interval of the (a) dxy and (b) dz distributions estimated from AFM measurements.

plane diameter dxy of S1 is slightly smaller than that of S2. Recent works28 reveal that, in the initial stages of gold island nucleation, the nanoisland lattice parameter is larger than the bulk value. In other words, this variation of the in-plane diameter dxy could be attributed to the gold lattice expansion in S1. Beyond 30 s, dxy increases with the sputtering time. A linear dependence between the sputtering time and dz is evident even in the initial stages of film growth. Moreover, the distributions of dz and dxy are broadened as the sputtering time increases. As shown in the insets of Figure 1, the film color depends on the sputtering times and changes from slightly pink to blue. In other words, the optical properties of these films are correlated to both the nanoisland morphology and their concentration. Figure 3 shows the S4 experimental Ic and Is spectra recorded at different angles of incidence. Similar spectra are obtained for other films (not shown). To exploit ellipsometric data, an optical model must be introduced. The model consists of a silicon substrate covered by a native oxide layer and a top nanoisland layer described by an effective dielectric function

Figure 3. Is (red) and Ic (blue) experimental spectra (empty squares) measured on S4 at four angles of incidence: (a) 60°, (b) 65°, (c) 70°, and (d) 75°. The simulated spectra by using MG, BEM theories (dash lines), and 2DEMT (solid lines) are also reported. Note that MG and BEM theories give the same solutions. 4901


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Combining eqs 3, 4, 5, and 6, the effective dielectric tensor [εeff] of a medium which consists of a collection of spheroids distributed in shape and randomly oriented can be expressed by the following equation

(3)

The sign ⟨⟩ tranduces a spatial averaging. By considering only one-particle correlations and the mean field approximation, the averages of the displacement and electric field can be decomposed into two contributions D = (1 − f )εm Em +fεi E i

(4)

E = (1 − f ) Em +f E i

(5)

[εeff ] = [εm(1 − f ) + fεi

∫ ∫ P(Lx , Ly) /[(1 − f ) + f

The subscripts m and i refer to the matrix and the inclusions, respectively. f is the volume fraction of the inclusions, while εm and εi are the dielectric function of the surrounding medium and nanoparticles, respectively. By assuming a spheroidal nanoparticle shape and the quasistatic limit, the electric field vector in the inclusions ⟨Ei⟩ is proportional to the electric field vector in the surrounding medium ⟨Em⟩29 ⎛ λ(Lx) 0 0 ⎞ ⎜ ⎟ ⎜ 0 λ (Ly ) 0 ⎟ ⎜ ⎟ ⎜ ⎟ λ(Lz)⎠ 0 ⎝ 0

⟨E i⟩ =

εm , εm + Lj(εi − εm)

dLx dLy]

∫ ∫ P(Lx , Ly) 3

dLx dLy]1

(9)

where P(Lx, Ly) is the normalized distribution of the spheroid depolarization factors and 1 is the 3 × 3 identity matrix. In other words, eq 9 takes into account the shape distribution of the nanoparticles provided the volume fraction of nanoislands is sufficiently small to neglect multipolar interaction between nanoparticles. The optical properties of a large number of nanoparticle shapes can be modeled by eq 9 such as prolate or oblate nanoparticles, nanowires, nanorods, and nanospheres. For monodispersed nanoparticle shape this equation is similar to the quasistatic Gans theory,29 while in the particular case of monodispersed spherical nanoislands, this equation is equivalent to the Maxwell Garnett theory. In other words, eq 9 is a general formulation of standard effective medium theories. Due to the nearly oblate shape and the two-dimensional organization of the sputtered nanoislands, an anisotropic behavior is expected. Thus, eq 9 must be modified to introduce the nanoisland shape and orientation. By considering the symmetry of nanoisland films (Figure 1), the effective dielectric tensor can be defined as

⟨Em⟩ (6)

j = x, y, z (7)

Lx, Ly, and Lz are the spheroid depolarization factors which vary from 0 to 1. Moreover, the sum rule condition involves Lx + Ly + Lz = 1

3

λ(Lx) + λ(Ly) + λ(1 − Lx − Ly)

where λ(Lj) =

λ(Lx) + λ(Ly) + λ(1 − Lx − Ly)

(8)

0.5 ⎛ ⎞ ⎜ εm(1 − f ) + fεi ∫0 P(Lxy)λ(Lxy)dLxy ⎟ 0 0 ⎜ ⎟ 0.5 − + (1 ) ( ) ( )d λ f f P L L L ∫ ⎜ ⎟ xy xy xy 0 ⎜ ⎟ 0.5 ⎜ ⎟ εm(1 − f ) + fεi ∫ P(Lxy)λ(Lxy)dLxy 0 ⎟ [εeff ] = ⎜ 0 0 0.5 ⎜ ⎟ (1 − f ) + f ∫ P(Lxy)λ(Lxy)dLxy ⎜ ⎟ 0 ⎜ ⎟ 0.5 ⎜ εm(1 − f ) + fεi ∫ P(Lxy)λ(1 − 2Lxy)dLxy ⎟ 0 ⎜ ⎟ 0 0 0.5 ⎜ (1 − f ) + f ∫ P(Lxy)λ(1 − 2Lxy)dLxy ⎟⎠ ⎝ 0

where Lx = Ly = Lxy. In other words, this two-dimensional effective medium theory (2DEMT) considers the nanoisland film as a uniaxial layer with an optical axis perpendicular to the substrate surface. The nanoparticle image effects are neglected.1,3 Note that this formalism is also well suited for biaxial materials. Dielectric tensor details are given in the Supporting Information. The ellipsometric data have been adjusted by coupling eq 10 to the Berreman transfer matrix formalism.30 The transfer matrix technique is useful to describe the change of the polarization state into a stratified medium. This formalism is based on the continuity of the electric field and displacement field in the direction parallel and perpendicular to each interface, respectively. It also takes into account the phase change and absorption induced by the light propagation through a medium. Four matrices done by M. Schubert30 are used to describe the light propagation in the air/nanoisland medium/native silicon oxide/silicon substrate structure. In first approximation, the influence of the nanoislands/substrate

(10)

contacts on the effective dielectric function is neglected. Thus, the surrounding medium was assimilated to air, and εi is the gold dielectric function extracted from Palik.31 The intrinsic confinement effects,1 which change the dielectric function of gold inclusion, have been neglected. Moreover, a Gaussian distribution was used for the depolarization distribution. The mean value and standard deviation of Lxy, the volume fraction, and the film thickness assimilated to the mean value dz have been simultaneously fitted by using the Levenberg−Marquard algorithm. In other words, only four parameters are fitted. Note that the Lz distribution is directly deduced from the Lxy distribution by using eq 8. As shown in Figure 3, a good agreement is obtained between the experimental spectra and the calculated ones. The root-meansquare errors between the experimental data and the simulated ones do not exceed 0.01 for all films (spectra not shown), confirming the correctness of this model. Thus, the 2DEMT model allows describing the measurements performed at different angles of incidences. This condition is necessary to 4902



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homogeneous thin film as the deposition time increases. Moreover, in accordance with AFM measurements, the distribution is wider for longer deposition time. By comparing AFM measurements to the ellipsometric analysis, a Volmer− Weber growth mechanism13,15,32 can be involved (Figure 4c). In the first nucleation step (sample S1), gold adatoms diffuse on the surface until they encounter other adatoms to form flattened nanoislands with a narrow distribution in shape. For longer sputtering times (samples S2 to S4), gold adatoms can be trapped by nanoislands. As a consequence, the in-plane and out-of-plane diameters are both enlarged. Adatoms can also aggregate together to produce other small islands. Thus, upon film growth, the number of nanoislands and their volume fraction increase, while the size and shape distributions are broadened. The nanoisland size and shape distributions have a profound effect on the optical properties of nanoisland layers. The evolution of the ordinary (εxy) and extraordinary (εz) effective complex dielectric functions of the films is shown in Figure 5. Their imaginary part exhibits a strong asymmetric absorption band in the 1.5−2.7 eV spectral range attributed to the SPR of gold nanoislands. Compared to spherical nanoparticles, the SPR energy of gold nanoislands is red-shifted by more than 0.45 eV. This behavior has also been observed by Gaspar et al.33 who found similar SPR energies to ours. The same trends are also reported in the case of evaporated silver islands.34 Nevertheless, no analytical model was previously proposed to explain these discrepancies, so 2DEMT can be very useful for the scientific community working in plasmonics. In accordance with the Kramers−Kronig relations,35 a large variation of their real parts is expected close to the SPR. Compared to the SPR of εxy, the SPR of εz is blueshifted by 0.3 eV. This splitting is a consequence of the flattening of gold nanoislands along the optical axis. In other words, SPRs of εz

validate the use of the effective medium approach to describe the optical properties of such a film. In other words, the scattering and diffraction effects by nanoislands can be neglected. Moreover, the Lxy mean value can be extracted from AFM measurement after correcting the in-plane diameter by the AFM tip radius. The error between the mean value of Lxy estimated by ellipsometry and calculated from AFM measurements does not exceed 5% for all samples (not shown). The film thickness and the volume fraction estimated from ellipsometric measurements are shown in Figure 2b and Figure 4a, respectively. In agreement with AFM results, both

Figure 4. Evolution of (a) the volume fraction and (b) the Lxy distribution of nanoislands during the film deposition estimated from ellipsometry. (c) Schematic description of the nanoisland nucleation and growth.

parameters increase with the sputtering time. The Gaussian distributions of the in-plane depolarization (Figure 4b) are centered at a value less than 0.33, the typical value for spherical nanoislands. This confirms that nanoislands are oblate, and their major axes are located in the film plane. Except for S1, the Lxy mean value decreases as the sputtering time increases, in line with the preferential in-plane expansion of gold islands. In other words, as Lxy approaches to zero, the film becomes more and more dense. This suggests that nanoislands tend to form a

Figure 5. (a) Real and (b) imaginary parts of the extraordinary effective dielectric function (εz) of the nanoisland films. (c) Real and (d) imaginary parts of the ordinary effective dielectric function (εxy) of the nanoisland films. 4903



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and εxy can be assimilated to the transversal and longitudinal SPR modes of nanoislands. The 5d → 6sp interband transitions1 located in the 2.7−4.9 eV range are also clearly observed. During sputtering, the real part of the interband transitions remains constant, while their imaginary part increases by 50% and 68% for εxy and εz, respectively. As shown in Figure 5, the SPR is more sensitive to the nanoisland volume fraction and shape distribution than interband transitions. Due to the decrease of the volume fraction from S4 (f = 33%) to S1 ( f = 11%), the SPR amplitudes of εz and εxy decline by half. Moreover, the SPRs of εz and εxy are red-shifted by 0.2 eV when the sputtering time evolves from 15 to 120 s. This behavior is attributed to the increase of nanoisland size and anisotropy. Indeed, an incident electric field induces a charge distribution around nanoislands. The Coulomb attraction between positive and negative charges results in restoring forces, also called depolarization field, characterized by an oscillation frequency assimilated to the SPR energy. The restoring force decreases as the nanoisland size increases and the SPR is red-shifted.1 In addition, due to the depolarization effect, the SPR amplitude of εz is two times smaller than the εxy one. As the depolarization field is higher when the incident field is directed along the minor axis of the nanoislands, the incident electric field is more compensated and the oscillator strength of the SPR is drastically reduced in this direction. Except for S1, the real part of the ordinary effective dielectric function is negative in the 1.8−2.5 eV spectral range, i.e., close to the plasmon resonance (Figure 5). This spectral range becomes wider as the sputtering time increases. As suggested by Oates et al.,36,37 the sign of the real part of the effective dielectric function gives some insights into the percolation degree of metallic islands. In other words the percolation threshold corresponds to the nominal thickness at which the real part of the effective dielectric function falls below zero in the near-infrared/visible spectral range.15 As shown from AFM measurements and considering the Oates criteria,36,37 the surface density of nanoislands is smaller than the percolation thresholds, where metallic islands are known to have a pseudodentritical shape.15 Note that the optical properties of percolated nanoislands cannot be described by 2DEMT. As the sputtering time increases, the real part of the in-plane effective dielectric function becomes more and more negative suggesting that the films tend toward metallic materials. Indeed, by increasing the nanoisland size and volume fraction, the interparticle distance is reduced, and the nanoisland dipole coupling is enhanced.4 On the other hand, the real part of the extraordinary effective dielectric function remains positive for all films, suggesting an insulating behavior of the films along the optical axis. Indeed, as dz is smaller than the electron mean free path estimated by Kreibig et al.1 at 40 nm, electrons are always confined along the z direction. This behavior is another indicator of the preferential in-plane growth of metallic islands. ̈ of nanoThe effective birefringence and optical dichroism island films, reported in Figure 6, are independent of the sputtering time close to the interband transitions, while large variations are observed close to the plasmon resonance. This behavior highlights the high sensitivity of the SPR mode to the nanoisland shape and orientation. In the interband transition range, the effective birefringence is negligible. As a consequence, the effective birefringence is only due to the mode degeneracy of the plasmon resonance of the nanoislands. ̈ Despite that the dichroism remains negative in the whole spectra, the effective birefringence changes in sign close to the

̈ of nanoisland films for Figure 6. (a) Birefringence and (b) dichroism different sputtering times.

εz plasmon resonance. A positive effective birefringence is observed when εrz is negative, while below 1.8 eV the effective birefringence becomes positive. On the other hand, the ̈ is maximized at the SPR. amplitude of the effective dichroism ̈ and birefringence become more and The effective dichroism more pronounced as the sputtering time increases. Thus, the optical properties of metallic nanoisland films can be tuned by changing the volume fraction. This result supports the use of metallic nanoisland films as a future building block of birefringent or dichroic̈ devices.

IV. CONCLUSIONS In summary, an effective medium theory based on a mean field and quasistatic approach is introduced to describe the optical properties of orientated spheroids distributed in shape. The optical properties of a large number of nanoparticle shapes such as prolate or oblate nanoparticles, nanowires, nanorods, and nanospheres can be obtained using this effective medium theory. This model is used to explain the optical properties of two-dimensional gold nanoisland films elaborated by sputtering. This model takes into account the shape distribution and the orientation of nanoislands. By coupling ellipsometric to AFM measurements, the growth mechanism which involves a Volmer−Weber growth mode was investigated. The optical anisotropy of uniaxial films was correlated to in-plane preferential self-orientation of oblate gold nanoislands. We ̈ have demonstrated that the optical anisotropy and dichroism are due to the spatial heterogeneity of the nanoisland plasmon resonance. The extraordinary effective dielectric effective function is reduced by a depolarization effect, and the SPR is split into two modes: the transversal and the longitudinal modes of gold nanoislands.

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ASSOCIATED CONTENT

S Supporting Information *

Effective dielectric tensor of isotropic, uniaxial, or biaxial medium. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Jean-Luc Pierrot, from LCP-A2MC, for AFM measurements. W.C. thanks the European Commission for “Erasmus Mundus” PhD fellowship within the DocMase project. 4904



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Generalized Effective Medium Theory to Extract the Optical Properties

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