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We present a study of the optical properties of three-armed square nanospirals made of silver and realized as nanostructured thin films with Glancing Angle Deposition. Calculation of current flows in the nanospirals show excited resonant modes resembling those observed in U-shaped resonators. Four principal resonances were determined: near 200 THz and 480 THz for one polarization and 250 THz and 650 THz for the polarization orthogonal to the first one. In particular, a mode with anti-parallel current flow in opposite arms, associated with the observed resonance near 650 THz, indicates the existence of a magnetic-like resonance in the square nanospiral arrays. The robustness of the resonances against variations in the structural parameters of the nanospirals was investigated. This study revealed that the main parameter driving the position of the resonances was the overall dimension of the nanospiral, directly related to the length of their arms. Optical properties of a sample were measured by generalized spectroscopic ellipsometry at near-normal incidence, and evidence conversion between polarization states even for light polarized in the plane containing one of the arms in agreement with the numerical study. The measurements compared favorably to the results of the numerical simulations taking into account the disorder in the sample.
©2010 Optical Society of America
1. Introduction
Metamaterials are artificial materials engineered to exhibit negative real components of effective permittivity and permeability over some range of excitation frequencies. The first structures proposed consisted of periodic arrays of thin metallic wires [1] associated with split-ring resonators [2] which exhibited negative refraction at microwave frequencies [3]. Since then, there has been a considerable amount of effort to push the frequency of resonance to the optical domain (400-750 THz). However, the requirement of homogenization imposes the constraint that the elementary structures must be much smaller than the wavelength of the incident optical wave (at least below λ/3), necessitating three-dimensional fabrication at the scale of tens of nanometers. Realizations of metamaterials in the visible range have so far relied primarily on electron lithography to synthesize the elementary resonators [4–7]. One realization of parallel rods uses an electro-chemical synthesis route [8]. Consequently, the structures proposed are essentially 2D and, in the case of flat U-shaped resonators, present magnetic resonances only at oblique incidence [4]. Three-dimensional resonators are predicted to exhibit interesting properties. For instance, three arm nanospirals may be seen, in first approximation, as U-shaped resonators adiabatically pulled out of their plane by one end and should present resonances similar to those of flat U-shaped resonators illuminated at oblique incidence. In a recent article, we numerically studied the optical response of 3D gold square nanospirals and showed that they could exhibit magnetic-like resonances in the visible [9]. Recently, gold nanospirals have been realized by laser photo-inscription in a photoresist followed by electrochemical deposition of gold in the resulting template [10]. The fabricated structure presented strong circular dichroism in the near infrared. However, the dimensions of the structures realized by this method are limited by the finite focal volume of the laser light used to inscribe the photoresist. Alternatively, Glancing Angle Deposition (GLAD) has been used to shape many materials at the nanoscale on large surfaces as columns, spirals, chevrons, etc [11,12]. and, in the case of silver columns realized by GLAD, evidence of a negative refractive index has been observed in the visible [13]. GLAD, as many other growth processes based on self-organization introduces some degree of disorder in the material synthesized. First, nucleation of the nanostructures with GLAD is mostly a random process and, although the average first neighbor distance between nanospirals is probably fixed, no long range order is expected. Second, structural parameters as the diameter of the columns, their angle with respect to the surface… should present some variations from one nanostructure to another. We have first performed a numerical parametric study in order to investigate the influence of disorder on the optical properties of the nanospirals. This study will be presented in part A. Then, we have synthesized three-armed silver nanospirals deposited onto a silicon substrate with GLAD. The optical properties have been measured by generalized spectroscopic ellipsometry in the near infrared - visible range. The synthesis, measurements and comparison with calculations will be presented in Part B. Finally, Part C will discuss and compare the results of Parts A and B.
2. Numerical parametric study
The shape of the three-armed nanospirals investigated here and the definition of the (x,y,z) referential are presented in Fig. 1 . The view of each nanospiral along the z axis would yield a structure very similar to U-shaped resonators with sides along axes x and y (Fig. 1a and 1b). The nanospirals were located on a hexagonal lattice and periodic conditions were applied at the boundaries of the unit cell along x and y (Fig. 1c). We varied independently the diameter of the arms between 49 nm and 71 nm, their length between 148 and 256 nm and their angle with the (x,y) plane between 20.1° and 30°. The variations were performed as to keep the packing density around 25%. These are typical values which would be obtained with GLAD samples.
Fig. 1 Schematic of: (a) one nanospiral with the geometrical parameters used in side and (b) top views and (c) one unit cell of the dense array of nanospirals where one nanospiral has been highlighted. In (c) periodic boundaries were applied along x and y.
We used HFSS by Ansoft to evaluate the optical properties of the nanospirals. Calculations were performed in the range 50 THz to 900 THz. The nanospirals were illuminated at normal incidence (i.e. along the z axis) and incident light was polarized either along x or y. Calculations revealed systematic conversion of polarization from x-polarization (resp. y) to y-polarization (resp. x) which prevented any effective permittivity and permeability retrieval from calculated values of the S-matrix. With that aim, we would have needed to define the effective permittivity and permeability as tensor, but this would have also required knowledge of the optic axes of the structure. Because of the chiral shape of the nanospirals the eigenmodes of the nanospirals may be partly circularly polarized modes [10,14]. Consequently, to study the optical properties of the nanospirals, we have calculated the absorption A in like-polarized conditions of a free standing film for each polarization case. Absorption in like-polarized conditions was defined as A = 1-R-T with R and T the reflection and transmission coefficient from x-polarized (resp. y-) to x-polarized (resp. y-) light. It must be noted that owing to the conversion of polarization observed in the nanospirals (see below) the sum A + R + T as defined above will be actually slightly different from unity. The optical constants of silver in the arms were obtained by keeping the optical constants of bulk silver above the onset of the interband transitions and using a damped Drude term (plasma frequency ωp = 1.3 1016 rad.s−1, collision frequency γ = 2.7 1015 rad.s−1) to describe the free electrons response modified by scattering at defects in the arms and/or at the surface of the arms [15,16]. This part will be organized as follow: in a first step we will identify the resonant modes at the origin of the structures in absorption A from the inspection of the instantaneous current distribution near the maxima of A as a function of polarization ; in a second step the influence of the different structural parameters and of positional disorder on the magnitudes and positions of the resonances will be investigated as a function of polarization.
2.1. Identification of the resonances
In this part we will use average structural parameters to identify the resonances in the nanospirals. The diameter of the arms will be 60 nm, their length will be 200 nm their angle with respect to the (x,y) plane will be 26°, as depicted in Fig. 1a.
Figure 2a presents the absorption spectrum for light polarized along x. A broad band centered near 480 THz was observed with a shoulder near 200 THz. The instantaneous current distributions in the arms are inserted in the graphic. The resonance near 200 THz was associated with current flowing in the second arm and the resonance near 480 THz was associated with excitation of the same arm but with a current node in the middle of the arm.
Fig. 2 Absorption calculated for (a) x-polarized light and (b) y-polarized light and. The insets present snapshots of the instantaneous current distribution in one nanospiral at the positions of the resonances in absorption. The current intensities are encoded on a gray scale ranging from black to white for current intensity increasing. The arrows indicate the current directions.
Figure 2b presents the absorption spectrum obtained for light polarized along y. Two well separated resonances were observed near 250 THz and 650 THz. The absorption band near 250 THz resulted from the excitation of the first and third arms with current flowing in parallel directions (see instantaneous current distribution inserted) and the resonance near 650 THz revealed excitation of the first and third arms but with current flowing in opposite directions.
2.2. Influence of structural parameters
We compared the absorption of the array of nanospirals for different values of the parameters. In the following we will present the influence of the variation of each parameter independently.
Figures 3a and 3b present the influence on absorption of arms’ diameter variations for each polarisation case. The diameter of the arms was varied between 71 nm and 49 nm. The length of the arms and their angle with respect to the (x,y) plane were maintained at 200 nm and 24°, respectively. The diameter values used induced variations from 2.8 to 4.1 in the aspect ratio (length/diameter) of each arm. The positions of the resonances presented a slight shift toward lower frequencies when the aspect ratio of the arms increased. Figure 3c presents the variation of the coefficient of conversion of polarization. It was defined as ratio of transmitted intensity in y-polarized light to incident intensity in x-polarized light. Similar results would be obtained for the reverse case. Two bands of similar magnitudes were observed near 200 THz and 450 THz. The positions of the maxima did not vary much, however, the magnitude of conversion increased largely with diameter of arms.
Fig. 3 Absorption spectra (a) for x-polarized light and (b) y-polarized light calculated for different diameter of the arms. (c) Coefficients of conversions of polarization. The length of the arms and their angle with the (x,y) plane were set to 200 nm and 24°, respectively.
Figures 4a and 4b present the influence of arms’ length variations on absorption for each polarization case. The length of the arms was varied between 248 nm and 148 nm. The diameter of the arms and their angle with respect to the (x,y) plane were maintained at 64 nm and 24°, respectively. The aspect ratio of each arm varied from 2.3 to 3.9 in these conditions. A large shift of all resonances toward low frequencies was observed when the aspect ratio ofthe arms increased. Figure 4c presents the variation of the coefficient of conversion of polarization. Similarly, the positions of the maxima shifted to lower energy with increasing length of arms. Simultaneously, the magnitude coefficient of conversion decreased.
Fig. 4 Absorption spectra (a) for x-polarized light and (b) y-polarized light calculated for different length of the arms. (c) Coefficients of conversions of polarization. The diameter of the arms and their angle with the (x,y) plane were set to 64 nm and 24°, respectively.
Figures 5a and 5b presents the influence of the arms’ angle with respect to the (x,y) plane on absorption. The angle was varied between 20.1° and 30°. The length of the arms and their diameter were maintained to 203 nm and 64 nm, respectively. The positions of the resonances did not vary much upon angle variations. In contrast the magnitude of absorption at the position of the resonances varied. Figure 5c presents the variation of the coefficient of conversion of polarization. The same effects of the angle of the arms on the magnitude of the coefficient of conversion were observed.
Fig. 5 Absorption spectra (a) for x-polarized light and (b) y-polarized light calculated for different angle of the arms with the (x,y) plane. (c) Coefficients of conversions of polarization. The length of the arms and their diameter were set to 64 nm and 203 nm, respectively.
We have recently shown that aperiodicity may also modify the resonances [17]. We briefly recall our results here as they evidenced a completely different effect induced by aperiodicity as compared to the influence of variations of the structural parameters.
Figures 6a and 6b present the effect of aperiodicity on absorption for each polarization case. The diameter of the arms, their length and the angle with respect to the (x,y) plane were set to 60 nm, 200 nm and 26°, respectively. Figure 6c illustrates the effect of aperiodicity on the coefficient of conversion of polarization. Aperiodicity was obtained by changing randomly the initial positions of each nanospiral by ± 10% at most along x and y in a supercell containing 16 nanospirals located on a square lattice with packing density of 20%. These conditions (square lattice, packing density of 20%) were chosen as to allow for the modifications of the positions without overlap of the nanospirals. Although the calculations performed in the case of aperiodicity cannot be used quantitatively here, the effects can be compared qualitatively. Indeed, it can be seen in Figs. 6 that the lineshape of absorptions were very similar to those calculated in Figs. 2 to 5. Figures 6 were obtained by averaging tendifferent random realizations. It can be seen in Figs. 6a and 6b that the lineshape of absorption remained mostly unaffected by aperiodicity. Concerning the coefficient of conversion (Fig. 6c), the effect of aperiodicity was to modify the relative magnitudes of both bands.
Fig. 6 Absorption spectra (a) for x-polarized light and (b) y-polarized light calculated in the periodic case (dotted line) and aperiodic case (full line). (c) Coefficients of conversions of polarization. The aperiodic case was obtained by averaging ten different random realizations. One realization consisted in random variations by ± 10% at most of the initial locations of all nanospirals.
3. Experimental results
Silver square right handed helixes, with three arms, were deposited using electron beam evaporator with GLAD. The Si substrate was placed so that its normal formed an angle of 85° with respect to the incoming flux of silver atoms leading to the formation of well spaced columns. The first arm was deposited to be a 200 nm long slanted column. The substrate was then rotated by 90° around its normal to form the second arm, linked to the first one but azimuthally rotated by 90°. This process was repeated one additional time to obtain the third arm of the nanospirals. The optical measurements were performed over the 185-900 THz spectral range (i.e. approximately 1660 nm – 330 nm) by generalized ellipsometry in the Jones configuration. The Jones matrix relates the reflected Jones vector to the incident Jones vector of light:
where p-polarization refers to light with electric field parallel to the incidence plane and s-polarization to light with electric field perpendicular to the incidence plane. We define the ellipsometric reflectance ratio , where and are the usual ellipsometric angles. Similarly we define and which quantify the systematic conversions of polarization upon reflection at the surface. The measurements were performed with light incident at 25° from the substrate surface normal with the incidence plane containing the growth direction of the first and third arms (plane (y,z)). We used the smallest incidence angle possible as to reduce as much as possible the contribution of resonant modes excited by light polarized along the z axis and to allow for an easier comparison with the modes of the nanospirals identified in Part A (Fig. 2). For the incidence angles generally used in ellipsometry (typically 70°), the component of the p-polarized electric field along the z-axis would no longer be negligible.Figure 7 presents cross-section image of the film obtained by Scanning Electron Microscopy (SEM). One of the nanospirals has been outlined. On the average, the total thickness of the film was near 270 nm, the length and diameter of the arms of each nanospiral were 200 nm and 60 nm, respectively. The arms were inclined by approximately 25° from the surface of the substrate.
Fig. 7 Scanning Electron Microscopy cross-section view of the 270 nm thick Ag nanospirals film. One nanospiral has been outlined.
Figure 8 presents the measured and calculated generalized ellipsometric angles for light propagating in the (y,z) plane. Because of the different definitions of references of phase between measurements and calculations, the sine of phase shift was plotted. As observed in the calculations, a significant signal was observed for the ellipsometric angles corresponding to polarization conversions. The calculations were performed in conditions similar to the measurements, i.e. a substrate was added with optical constants of bulk crystalline Si [18], the incidence angle was set to 25° and different configurations of arms’ length, diameter and angle were averaged [19]. We neglected aperiodicity, because the parameters used in the calculations differed from the values expected for the sample: this should lead to some error near 600 THz (Fig. 6). Fair qualitative agreement was observed for ΨE and ΔE. Similar agreement was observed for light propagating in the (x,z) plane (not shown here). The calculations reproduced rather well the broad structure on ΨE near 500 THz and the weak structures near 360 THz and 740 THz observed in the measured ellipsometric angles. The low transmission through the nanospirals (below 10%) reduced the contribution of image dipoles in the substrate although the presence of substrate may shift the resonances to low frequency as compared to free standing films. The agreement was not as good for the off-diagonal elements of the Jones matrix although the right trends were observed. It must be noted that the structures appearing on ellipsometric angles, although related to resonant modes of the nanospirals, are not at the energetic position of these resonances.
Fig. 8 Generalized ellipsometric parameters ψ and Δ presented as Jones matrix elements column (a) measured (line + dots) and column (b) calculated (full line) for incidence angle of 25°. The first quadrant of the first column depicts the top view of one nanospiral with the orientation of s-polarized light.
4. Discussion
The resonant modes observed in the nanospirals (Figs. 2) were multipolar modes and similar modes have been observed in U-shape resonators [5,6,20–22]. For x-polarized light, the lowest resonant mode, corresponding to the excitation of the whole nanospiral, was not observed in the spectral range investigated here: it should be obtained near 150 THz in isolated nanospirals [17]. It disappeared in strongly coupled nanospirals either because it was shifted outside the spectral range investigated or because it was screened by the mode corresponding to the excitation of the second arm. In this polarization case, only the resonant modes corresponding to excitation of the second arm were observed near 200 THz and 480 THz (Fig. 2a). These modes would be referred as to electric in metamaterials.
For y-polarized light, the parallel current flow associated with the resonance near 250 THz (Fig. 2(b)) has already been predicted and observed in the case of U-shaped resonators [5,6,20–22]. This mode would be referred as to electric in metamaterials. In contrast, the anti-parallel current flow associated with the resonances near 650 THz (Fig. 2b) has been observed only on U-shaped resonators illuminated at oblique incidence [5,6] and on asymmetric rods [23]. The presence of the two modes would originate in the excitation of a mode of parallel rods, silent in the symmetric case (i.e. normal incidence or symmetric rods), a consequence of the asymmetry of the very shape of our resonators, in which the first and third arms are not in the same plane [23,24]. Depending on the particular asymmetry and coupling of the arms in one nanospiral, and among many nanospirals, the relative energetic position of these modes with respect to each other may be changed. It is then difficult to compare our results to those of Ref. 23. Nevertheless, the mode corresponding to currents flowing in opposite directions, near 650 THz in our work, would be associated with a resonance generally referred as to magnetic [9,21,23,24].
The variations in the position of the resonances in absorption were larger in the case of variations of the length of the arms (Fig. 4) than in the case of variations in the diameter of the arms (Fig. 3), although the aspect ratio variations for each arm were of similar magnitude. In contrast, the positions of the resonances in absorption were mostly insensitive to variations in the angles of the arms without change in their length (Fig. 5). These observations should be attributed to the differences in the overall dimensions of the nanospirals induced by the variations in diameter and length of the arms. In the former case, the dimensions of the nanospirals (seen for example as their footprint in the (x,y) plane, Fig. 1b) did not change much while in the later case the dimensions of the nanospirals scaled with the length of the arms. Concerning the case of variation of the angles of the arms, the result was mainly that the magnitude of the component of the electric field of light parallel to the arms changed. To summarize, the variations in absorption observed in Figs. 3 to 5 would reflect the same scaling effect as observed in resonators for metamaterials [1,2,25,26]. The relative insentivity of the modes to the amount of aperiodicity used here has been observed in metamaterials in the microwave regime [27,28] and more recently in the THZ domain too [29]. To conclude this parametric study, from the growth prospect, we have shown that the optical properties of the nanospirals were rather robust against variations in the structural parameters aimed at. The optical properties were mainly related to the overall dimension of the nanospirals which is the easiest parameter to control since it is mainly related to the growth duration of each arm. The variations in diameter of columns with deposited thickness, often observed in samples synthesized with GLAD, would induce only minor variations in the resonances. Without pre-patterning of the substrate, the random nucleation of the nanospirals would not affect much the resonances. Consequently, a numerical study on an ideal array of nanospirals should provide a first good insight in the optical response of nanospirals synthesized with GLAD.
We choose growth conditions yielding a sample containing silver nanospirals with structural parameters close to the average ones of our numerical study, as confirmed by the SEM image (Fig. 7). The optical measurements were performed by generalized spectroscopic ellipsometry. They revealed the importance of determining all elements of the Jones matrix since significant values of the off-diagonal elements were measured. The measurements evidenced that x and y axes did not correspond to optic axes of the nanospirals. This observation confirmed the calculations and we can suspect that the eigenmodes of our nanospirals were probably elliptically polarized modes [14,24]. In first approximation, the disorder observed on the SEM image may be seen as the incoherent superposition of all the variations presented above. Consequently, we calculated the reflected intensities for different values of the structural parameters and averaged them to compare calculations to measurements [19]. This would be equivalent to assume that coupling among nanospirals was only a short distance interaction. This assumption was supported by the observation of only multipolar resonant modes in the nanospirals (Figs. 2). Multipolar resonances interact on much shorter distances than dipolar ones and would be less sensitive to long range ordering and, additionally, induce broad resonances which would reduce the apparent sensitivity to positional and structural disorder. Differences in agreement between calculated and measured values were observed between on the one hand diagonal elements of the Jones matrix (ΨE,ΔE) and on the other hand the off-diagonal elements of the Jones matrix (Ψps,Δps,Ψsp,Δsp). We can invoke the same effect of sensitivity of the optical properties to structural changes to explain them: the diagonal elements were not very sensitive to structural disorder (Figs. 3 to 5a and 5b) whereas the off-diagonal elements were more sensitive (Figs. 3 to 5c). Consequently, the choice of a particular combinations of disorder will not influence much the values of (ΨE,ΔE) whereas it will modify strongly the values of (Ψps,Δps,Ψsp,Δsp).
At low frequencies, a different source of discrepency originated most likely in a purely optical effect, in the sense of reflectivity properties of thin films, inducing the calculated values of rpp and rss to vanish near 130 THz (not shown here). We observed that this optical effect was not associated with an absorption band (Figs. 2 to 6) although it yielded the sharp rise observed in Ψps and Ψsp on the low frequency side of the spectra (Fig. 8a). Additional phase shifts were associated with the low values of rpp and rss which added up to the ones associated with the resonances of the nanospirals and yielded the differences observed between calculated and measured phases particularly in Δsp and ΔE (Fig. 8b). A shoulder was actually observed in the measured Ψ near 150-200 THz which was overestimated in the calculations. To reproduce this effect, our model should take into account additional structural parameters which would modify the overall reflectivity of the film, for instance: a rough Ag wetting layer at the surface of the substrate, different nanospirals’ dimensions in each calculated unit cell...
The overall agreement between calculated and measured values showed that excitation of electric-like and magnetic-like modes of the nanospirals would explain the spectral variations observed in all ellipsometric angles above 150 THz. Our results evidenced that the strength and frequencies of the electric-like modes of the nanospirals were rather robust against structural and positional disorder induced by the GLAD synthesis method. An interesting point was the presence of the magnetic-like mode near 650 THz (460 nm). Such high resonance frequency has been predicted in SRR only thanks to introduction of multi-gap structures [25,26] to increase capacitive effects. In our case, the multipolar mode at the origin of the resonance observed held the same role as resonance in a multi-gap structure. Consequently, the use of 3D nanostructures supporting multipolar modes allowed for excitation of a magnetic-like mode at high frequencies.
5. Conclusion
Thanks to a numerical parametric study, we have shown that:
- - three-armed square nanospirals exhibit resonant modes similar to those observed in U-shaped resonators used for metamaterials at optical frequencies. In particular, we have shown that the current distribution in the nanospirals associated with the broad resonance near 650 THz (wavelength of 460 nm) resembled that observed in metamaterials at their magnetic-like resonance.
- - the structural disorder induced by growth processes as GLAD should not be too detrimental to the resonances observed in ideal nanospirals. The position of the resonances was mainly related to the length of the arms constituting the nanospirals and their magnitude was related in a more complex manner to the diameter of the arms, their angle with respect to the substrate. However, within the range of parameters used, disorder did not cancel any of the resonances.
We have synthesized on the cm2 scale silver nanospirals using GLAD and measured their optical properties. The resonances predicted numerically allowed reproducing the measurements. One advantage of the 3D shape permitted by GLAD is that it should allow excitation of magnetic-like modes even for illumination at normal incidence.
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