Variable-angle and Mueller matrix spectroscopic ellipsometry are used to determine the effective dielectric tensors of random and aligned silver nanoparticles and nanorods thin films. Randomly arranged particles are uniaxially anisotropic while aligned particles are biaxially anisotropic, with the anisotropy predominantly at the plasmonic resonances. The strong resonances in nanorod arrays result in the real part of the effective in-plane permittivities being opposite in sign over a significant range in the visible, suggesting the potential to design materials that display tunable negative-refraction. A structural tilt in the particle arrays results in monoclinic dielectric properties.
©2011 Optical Society of America
Negative refraction has been demonstrated in highly anisotropic materials . Unlike negative index materials composed of split ring resonators  or cut wire pairs  that are described by an effective magnetic resonance, highly anisotropic materials offer a purely dielectric route to negative refraction with lower losses and simpler fabrication . In these materials – termed indefinite materials – one orthogonal effective permittivity differs in sign. The dispersion relation thus describes a hyperbola and the Poynting vector is negatively refracted for all incident angles . Arrays of aligned metallic nanowires fulfill the requirements of indefinite materials and have been used to demonstrate negative refraction [1, 6, 7].
Arrays of metallic nanoparticles, on the other hand, are non-conductive and therefore do not have any direction where the real part of the permittivity is negative in the classical Drude sense. They do however show regions of anomalous dispersion near to a plasmon resonance, especially for noble metals. If the strength of the oscillation is large enough there will be a region on the high frequency side of the resonance where the real part of the effective permittivity is negative. We shall show here that in nanoparticle arrays with a well defined structural anisotropy the range of negative permittivity may not coincide in all directions and one may predict a negative refraction over a limited spectral range. Even for materials where the resonance is weak, materials with an anisotropic plasmonic response are still an interesting occurrence . Such materials may arise from an anisotropic arrangement of spherical particles, or from an isotropic ordered arrangement of non-spherical particles. Alternatively, an anisotropic or disordered arrangement of non-spherical particles may display an isotropic response.
In this paper we aim to use spectroscopic ellipsometry to determine the effective dielectric functions (DF) of patterned and un-patterned silver nanoparticle films, and relate the spectral features to the nanostructure determined using electron microscopy. Ellipsometry has the advantage over reflection and transmission measurements of directly measuring the phase shift of the reflected light, thereby avoiding the need to use the Kramers-Kronig relations to determine the dielectric functions. We are interested in determining the anisotropic properties of the films and how the plasmon resonances relate to the structural anisotropy. Additionally we aim to investigate the properties of the Mueller matrix elements and their dependence on the plasmon resonances in anisotropic films. The potential of using stacked layers of these materials as indefinite metamaterials will be discussed. Finally, a structural tilt in the nanoparticles, as observed by electron microscopy, is included in the optical model as a monoclinic tilt in the dielectric tensor axes, improving the model fit to the measured data.
2. Materials and methods
Silver sphere and rod arrays were deposited by evaporation onto silicon substrates. Unstructured substrates (with ca. 100nm thermal SiO2) resulted in randomly nucleated silver island films (sample A). Silicon substrates (native oxide) with periodically rippled surfaces formed by low-energy ion sputtering were used to template silver island arrays with defined periodicity (35 nm). Short deposition times result in nanosphere arrays (sample B) while longer deposition times results in nanorod arrays (sample C). Details of the method have been reported previously [9–12].The films and substrates were structurally analyzed by SEM and TEM.
SEM images of the three samples used for the optical analysis are shown in Fig. 1 . Sample A is a typical randomly nucleated silver island film with particle diameters around 20 nm. In the templated samples B and C the particles are also randomly nucleated parallel to the ripple direction (from here on denoted the x direction) however they are aligned along the ripple ridges in the perpendicular plane (y direction). Sample C shows aligned nanorods with lengths up to 50 nm.
Optical measurements were performed using spectroscopic ellipsometry. For films on the unstructured substrates we used a J.A.Woollam Co., Inc. rotating analyzer VASE with measurement from 1 to 4.7 eV at 10° interval angle of incidence (AOI) from 15° to 75°. For the Mueller matrix measurements of the patterned samples we used a J.A.Woollam Co., Inc. RC2 spectroscopic ellipsometer in the polarizer-rotating compensator-sample-rotating compensator-analyzer arrangement. This instrument allows the measurement of the 16 normalized MM elements . Generalized ellipsometric data were simultaneously recorded . Data were collected from 0.7 to 5.2 eV at 10° interval AOI from 45° to 75°, and full ϕ rotation around the surface normal (z) axis at 30° intervals. An optical model with a silicon substrate, an oxide layer (native or thermally grown) and a nanoparticle layer (isotropic, uniaxial or biaxial) was employed for all samples. Available optical data in the WVASE32 software (J. A. Woollam Co., Inc.) for silicon, native oxide and thermally grown oxides were used.
3. Results and analysis
3.1. Random arrays - uniaxial anisotropy
A general description of non-depolarizing materials is provided by the Jones formalism. With polarization states defined vectorially by orthogonal electric field components, Ep and Es, where the subscripts p and s define the directions parallel and perpendicular to the plane of incidence, respectively, reflection of light from a surface is expressed in matrix form as 
Ψ and Δ data for sample A is shown in Fig. 2 at 55° AOI. Also shown are fitted model data, assuming the silver particles form a layer with either an isotropic dielectric function, where the diagonal Cartesian tensor elements are equal, or a uniaxial dielectric tensor, where , (z is normal to the substrate surface). The isotropic model is a simple Lorentzian oscillator to account for the localized plasmon resonance. This is equivalent to a Maxwell-Garnett effective medium approximation (MG-EMA) with the metal inclusion modeled by a Drude model . The 4d-5s interband transition around 3.8 eV is modeled by a Tauc-Lorentz (TL) expression  fit to the bulk DF of silver from . The Lorentzian amplitude, broadening and resonance frequency, and the TL amplitude, were used as fit parameters in the final fitting procedure. An additional offset to the real part of the DF, was also fit. In the uniaxially anisotropic model the in-plane DF () is fit in the same way as the isotropic case described above. The out-of-plane DF was approximated by a Drude-TL model (as for the bulk data) with the Drude amplitude and broadening and the TL amplitude set as fitting parameters. We note that the particle size and periodicity of all the samples is around 1/10 or less of the wavelength over most of the measurement range so we assume that we are in the long wavelength limit and the wave vector dependence on the dielectric function may be ignored. The fitting parameters are presented in Table 1 .
It is clear from Fig. 2(a) and 2(b) that the isotropic model does not fit the measurement effectively at energies above 3 eV. The uniaxial model gives a much better fit to the measured data, accounting for the features above 3 eV. The well-documented optical anisotropy of silver island films  arises from different in-plane and out-of-plane interactions between the electric dipoles in the metal particles , and interaction with image charges in the substrate . Methods to account for this anisotropy include introducing a depolarization factor into the MG-EMA  or using the Thin Island Film theory . However, it is difficult in these approaches to separate the individual contributions due to particle size, shape and image charges, and it is not attempted here.
In the case that the optical axes are parallel to the plane of incidence (as is the case here) standard ellipsometric measurements may be performed, however a unique fit for εz is not possible at a single AOI. Measurements using variable angle spectroscopic ellipsometry (VASE) allows the unique determination of εz  if the fitting parameters for each angle are coupled. Figure 2(c) and 2(d) shows the results of fitting the data for sample A at multiple AOIs with the uniaxial anisotropic model. At large AOI the data shows the most contrast since the difference between the Fresnel reflection coefficients is maximum around the Brewster angle of the silicon substrate. Interestingly the fits at high AOI show greater deviation from the measured data around the in-plane plasmon resonance at 2.7 eV. Since the particles are likely to be truncated spheres, as observed in SEM images of such films , the shape deviation from spherical symmetry may introduce higher order resonance modes which are not accounted for by the single Lorentzian oscillator .
The DFs of sample A determined from the VASE fit are shown in Fig. 3 . The in-plane DF shows the characteristic plasmon resonance around 2.7 eV and the interband transition at 3.8 eV. The out-of-plane DF is similar to the bulk DF but the broadening of the Drude component is increased to account for surface scattering while the Drude amplitude is decreased to account for the reduced effective electron density. Note that in the out-of plane direction, although not explicit in the dielectric function, a volume plasmon resonance is still observed in transmission and reflection, as observed in the Ψ data in Fig. 2 at around 3.5 eV. The thin film model thus predicts the volume plasmon resonance in the z direction. This is logical since the Mie theory, which predicts a localized plasmon resonance in metallic spheres using the bulk dielectric function, is the spherical analogue of the Airy planar thin film equation derived from the Fresnel equations. Note that we observe an absorption peak in the z direction at the effective plasma frequency (ε1 = 0) in thin silver island films. In a solid film this resonance is known as a Ferrell mode , the plasmonic analogue of the phononic Berreman mode , and is extensively discussed in basic texts . We should expect that as the metal volume content of the film increases the out-of-plane resonance will blue-shift to the bulk plasma resonance frequency and the in-plane resonance will red-shift to zero (i.e. Drude behavior) .
3.2. Ordered arrays - biaxial anisotropy
If the film and/or substrate is optically anisotropic, and the plane of incidence is not parallel to the optical axis, a proportion of incident p-polarized light is converted to s-polarized light (and vice versa) and the off-diagonal elements rps and rsp of the reflection Jones matrix J in Eq. (1) are non-zero. In principle they may be determined using generalized ellipsometry (GE) . However, if the sample depolarizes the probe beam (i.e. converts it to partially polarized or unpolarized light) then a full description of the optical properties is not possible using J. In the most general description the polarization state of a light beam is described by the Poincaré sphere or the Stokes formalism. The Stokes method defines parameters which are directly measureable as irradiances. The 4x4 Mueller matrix M which represents the transformation of the incident Stokes vector S i to that reflected from a sample surface S r is defined as31]. In a non-depolarizing sample the Jones matrix may be calculated directly from M, and vice versa. In a depolarizing sample the calculations will not be equivalent. M may be decomposed into J plus a depolarization component, D p, or alternatively into a sum of the specularly reflected and near-specular scattered contributions. One may thus check for depolarization in a sample by comparing the measured M elements with those calculated from the measured J elements using the formula 
This can be simplified to expressions for individual M elements in terms of the J elements. Figure 4 shows a comparison of the measured element m 31 with that calculated from the measured J elements for sample B. The difference is appreciable around the two features near 2.7 and 3.3 eV, which correspond to the in-plane resonance (IPR) and out-of-plane resonance (OPR) discussed above. Note that the elements m 13, m 14, m 23, m 24, m 31, m 32, m 41 and m 42, referred to as the off-diagonal elements, are only non-zero when the sample is anisotropic . Thus we can conclude that the sample is optically anisotropic and that there is appreciable depolarization around the plasmon resonances, justifying the need to measure the sample using MME.
The origin of the depolarization is not immediately clear. Common causes of depolarization include; large sample surface roughness causing scattering of the probe beam; thin film thickness inhomogeneity; backside reflection from a weakly absorbing substrate; and variation in the incident angle of the probe beam (e.g. caused by focusing optics). We do not expect these effects to be significant in our measurements. Depolarization phenomena not only occur for heterogeneous samples but are also induced by strong dispersion. Therefore depolarization effects may be linked to plasmonic resonances. The fact that in our results the largest difference between the two traces is exactly where the IPR and OPR are expected suggest that the plasmonic resonances are indeed the origin of the depolarization.
The measured M elements for sample B are shown in Fig. 5(a) for rotation angles ϕ = 0°, 30°, 60° and 90° at 75° AOI. The non-degeneracy in the diagonal elements between 2.5 and 3.0 eV is additional evidence of optical anisotropy. More telling is the non-zero values of the off-diagonal elements, shown at 20x scale of the diagonal elements. The main resonant features occur near 2.5 and 3.2 eV (i.e. near the positions of the plasmon resonances observed in sample A). The M elements may in principle be decomposed to represent the individual contributions from linear and circular birefringence and dichroism . Such an analysis will be the subject of future work.
The M elements were fit using a similar model to that described above, however we now use a biaxial model () to fit independent in-plane dielectric functions, with the results shown in Fig. 5(b). The diagonal elements are well reproduced by the model, however the off-diagonal elements reproduce only the resonance feature around 2.5 eV. Little evidence of the resonance near 3.2 eV is observed.
Figure 6 shows the measured M elements and the fitted model of sample C. The optical anisotropy is clearly much stronger in this sample, as observed in both the lack of rotational degeneracy in diagonal elements below 3 eV, and also the comparatively strong off-diagonal signals (x5 compared with x20 for sample B). The range of the anisotropy is also increased, with significant values of the off-diagonal elements well into the infrared. Once again the model predicts the observed resonant features, although the observed feature near 3.6 eV is also lacking in the off-diagonal fits. Since this is exactly the energy range of the OPR we naturally speculate that the feature is connected with the properties of εz. This will be discussed further below.
The effective dielectric functions from the above model are shown in Fig. 7 . As observed for the uniaxial case above, the out-of-plane dielectric function εz is similar to that of bulk silver, with the characteristic Drude tail at low energies. The two peaks in the in plane components of the imaginary parts ε2 of the dielectric tensor are due to the particle plasmon resonances, occurring at 2.60 and 2.71 eV for sample B and 1.83 and 2.52 eV for sample C. The latter resonances are considerably stronger. ε1 shows a region of anomalous dispersion around the resonances and drops below zero on the high frequency side of the resonance in sample C. In the region between the resonance peaks the signs of ε1 are opposite in the in-plane directions. This leads us to conjecture as to the possibility of creating an indefinite material from multilayers of elongated particles on rippled templates.
An indefinite material is one whose dielectric tensor elements are of opposite signs . A TM plane wave with wavevector k in the x-y plane will be dispersed according to6]. Since the phase velocity direction is still positive the group and phase velocities are antiparallel, resulting in negative refraction.
The equifrequency curves for sample C at = 1.7 and 2.3 eV are shown in Fig. 8 for a wavevector in the x-y plane. At 2.3 eV the group and phase velocities are antiparallel, with the x-component of the group velocity being negative. At 1.7 eV the material is positive isotropic in the x-y plane and is positively refracting (parallel group and phase velocities). Such negative refraction has been observed in nanowire arrays , however the difference with our material is that the range of negative refraction is limited to a region between the plasmon resonances, which may be desirable in designing photonic devices. Being the ultra-thin version of a nanowire array oriented in the z-direction, we should also expect negative refraction in sample A. However to observe negative refraction requires one to build a wedge of material, which is difficult for a 10 nm film. Conversely, we assume that by the addition of more identical layers, separated by a thin dielectric spacer layer we could construct a multilayer material composed of layers of sample C which would allow light coupling into a waveguide structure. Thus one could design a material where for a certain energy range ε1 is positive for εy and εz and negative for εx. Our simple two-step self-organized patterning procedure is conducive to large scale fabrication. Optical components such as wedges and prisms may then be stamped or etched into such a multilayer.
3.3. Particle tilting
The inability of our optical model to accurately reproduce the out-of-plane resonance in the anisotropic materials deserves further analysis. We note that this is not due to the lack of explicit resonance in the dielectric function; an absorption resonance is predicted from the dielectric functions as for Sample A. A closer look at the rotational symmetry of the off-diagonal elements is telling. Figure 9(a) shows the rotational dependence of the element m31 for sample C at 2.7 and 3.6 eV. Clearly the IPR shows 2-fold symmetry (although not exact) while the higher energy OPR projects back onto itself after a full rotation period. This gives an insight into the origin of the asymmetry – it is likely due to a tilt in the particles on the substrate. Due to the oblique incidence of the evaporated atoms during the deposition process the particles grow preferentially on one side of the ripples  and are thus tilted toward the evaporation source. The particle tilt is clear in the TEM image shown in Fig. 9(b), showing two oblate-spheroid silver particles from a sample similar to that of sample B. . The long axis of the spheroid is tilted by around 15 degrees from the substrate plane (x-y plane). The particles are separated from the crystalline silicon substrate by a silicon oxide layer and are embedded in a protective amorphous carbon coating for TEM preparation.
We investigate whether accounting for the structural tilt in the sample by introducing a tilt in the dielectric tensor will predict the features near the OPR. There are two methods to introduce tilt into the dielectric tensor. The first is to rotate all the tensor unit vectors around the Euler angle θ (i.e. apply a co-ordinate rotation to the dielectric tensor). The second is to introduce an effect analogous to a monoclinic crystal by tilting only a single axis of the tensor. For this case one may use the transfer function T as defined in 
A comparison of the two analysis methods for a single off-diagonal M element m13 is shown in Fig. 10 for both anisotropic samples. Compared to the orthorhombic Cartesian model presented above both co-ordinate rotation around the Euler angle θ and tilting of the monoclinic angle β gave improved results. In both cases we observe a tilt of around 5°. From the TEM images we might expect a larger tilt however it is not necessary that the optical anisotropy is directly aligned with the structural anisotropy . It is clear from Fig. 10 that the feature above 3 eV is better predicted by the monoclinic model than the Euler rotation, however there is still room for improvement.
Silver nanoparticle arrays show significant optical anisotropy due to the spatial distribution of the particles. Randomly ordered silver island films show uniaxial anisotropy with an in-plane plasmon resonance well described by an effective medium model such as the Maxwell Garnett EMA, and an out-of-plane plasmon resonance, similar to a Ferrell mode, which may be described by a Drude model with an effective plasma frequency to account for the reduced electron density. If in-plane anisotropy is introduced into the particle arrays one observes a splitting of the plasmon resonance. The dielectric function may be determined by fitting a dielectric tensor to generalized or Mueller matrix ellipsometry measurements. Since we observe depolarization near the plasmon resonances the use of MME is recommended. In our materials a monoclinic model improved the fit to the measured data although we still could not completely account for the feature near the OPR in the off-diagonal M elements. In the strongly anisotropic material we observe a region between the in-plane resonances where the signs of ε1 are opposite, suggesting the potential to fabricate multilayered indefinite materials which display negative refraction in a tunable range in the visible region.
Thanks to D. Schmidt and H. Wormeester for helpful discussions and Arndt Mücklich for the TEM image. This work has been supported by the Deutsche Forschungsgemeinschaft (Grant No. FA 314/6-1 and Forschergruppe 845). The Knut and Alice Wallenberg foundation is acknowledged for financial support to the ellipsometers.
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