1. Introduction

The development of functionalized nanostructures has recently attracted a great deal of research interests [1,2]. Among a variety of nanostructures, a myriad of studies related to noble metal nanoparticles have been conducted in hope of utilizing their strong interaction with light for physical, chemical, and biological applications [3–5]. Structural metrology of nanoparticles or the understanding of their geometry and distribution traditionally relies upon different microscopies, such as scanning electron microscopy (SEM), transmission electron microscopy, atomic force microscopy,...etc. However, the above techniques require additional sample preparations, and/or exert energetic electrons (> 1 keV) or mechanical forces upon samples, which may incur annoying artifacts or even cause irreversible damages to samples.

In contrast, optical techniques can use relatively low-energy photons (< 10 eV) to probe the samples nondestructively although direct imaging of nanoparticles is very challenging due to the diffraction limit. Spectroscopic ellipsometry (SE) has been widely applied for the characterization of thin films and bulk materials. In recent years, the characterizations of two-dimensional gratings [6], heterogeneous layers [7], and various nanostructures including nanocolumns [8,9] and nanoparticles [10] have also been reported. It has been demonstrated that SE can find applications in process control and obtaining information about nanostructures’ geometry, anisotropy, as well as quantum size effects [11].

In this study, SE measurements coupled with efficient theoretical modeling were used to metrologically characterize the nanostructure consisting of randomly distributed gold nanoparticles on a substrate. The polarization-dependent reflection coefficients were calculated by a finite-element Green’s function (FEGF) method [12], and the size, shape, and distribution of nanoparticles are to be determined.

2. Experimental

The Au nanoparticles samples under investigation were prepared by the following procedures. The surfaces of microscopic slides were first modified through a silanization treatment using the APTES (3-aminopropyl triethoxysilane, Fluka) [13]. After being rinsed with deionized water and baked in an oven at 120°C for 30 minutes, the substrates were immersed into an aqueous solution of commercially available Au colloids (BBInternational) with mean particle size ranging from 20 to 80 nm for overnight so that the solvent can evaporate completely. Samples immobilized with Au nanoparticles were then rinsed with deionized water again to remove unattached nanoparticles and gently dried with N₂ purge prior to SE measurements. Variable-angle spectroscopic ellipsometry measurements were done using a rotating-analyzer ellipsometer (RAE) system with an adjustable retarder (VUV-VASE, J. A. Woollam Co.) in the ultraviolet to near infrared region (≈1-9 eV) at incident angles of 55°, 60°, and 65°. To obtain the optical constants of the multilayered substrate for the model calculations, SE measurements of bare APTES coated on a microscopic slide were carried out. We found that optical contrast between the APTES coating and the glass slide is negligibly small; the discrepancy between SE spectra of the slides with and without an APTES coating is within the measurement error bars. This left us without choices to use the peusodielectric constant of APTES on glass in lieu of the real optical constants of the multilayered substrate.

After all the SE measurements were completed, we sent the Au nanoparticles samples for a field-emission SEM inspection in order to understand their geometry and distribution. As shown in Fig. 1 , most of the nanoparticles are of spherical shape viewing from the top, and their sizes are relatively uniform and close to the nominal sizes specified by the supplier. However, the distribution density and pattern do vary significantly among the samples. For the sample of the smallest particle size (20 nm), the distribution is denser and somewhat of a periodic pattern. But the distributions become more dilute and random for samples of larger particle sizes. Based on those SEM micrographs shown in Fig. 1 and other similar ones (not shown), the surface coverage was estimated to be around 22-25% (for 20-nm nanoparticles), 12-15% (for 40 nm), 16-19% (for 60 nm), and 17-18% (for 80 nm), respectively.

 

Fig. 1 Top-view SEM images of Au nanoparticles samples with nominal particle sizes of (a) 20, (b) 40, (c) 60, and (d) 80 nm, respectively. Scale bar: 300 nm.

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3. Modeling

Optical properties of aggregated islands or small particles on a substrate have been extensively investigated [14–16], and the SE characterization of Au nanocolloids deposited on Si substrates has been reported [7] by applying the theory developed for island films [16]. Alternatively, one may employ the effective medium approximation (EMA) to model metal nanoparticles by treating them as a uniform thin film with ambience as the host material and numerous spherical or ellipsoidal inclusions of the metal. Unfortunately, the EMA theory can only work well when the finite-wavelength effects are negligible [11,17], which becomes invalid for measurements at short wavelengths. Therefore, it is desirable to have another approach which can handle light scattering from nanostructures in a broader spectral range.

The rigorous coupled-wave analysis (RCWA) [18] could be the most developed and widely used method for the analysis of diffraction by periodic structures. To alleviate the RCWA’s calculation inefficiency in analyzing complex three-dimensional periodic structures, a more efficient numerical method based on the FEGF approach [12] has been developed and was employed in our theoretical calculation of ellipsometry parameters Ψ [ = tan⁻¹(|R_p/R_s|)] and Δ [ = arg(R_p/R_s)], where R_s(R_p) is the s (p)-polarized complex reflection coefficient. The discrete dipole approximation [19] and GranFilm code [20] based on multipole expansion can also be used for modeling the random distribution of spheroidal nanoparticles on a substrate. These approximated methods work very well when the size of nanoparticles is much smaller than the wavelength considered. In comparison, the FEGF method adopted here is valid even when the feature size is comparable or larger than the wavelength, and it can be extended to objects of arbitrary shape and more complicated distribution patterns.

Based on the findings from SEM (Fig. 1), Au nanoparticles were assumed to be spheroids with equatorial diameters d along the x and y axes. The height t along the z axis could be different from d. The spheroidal shape was modeled by 15 (for a 20-nm particle size) to 25 (for 80 nm) slices of circular discs with slowly varying diameters, as shown in Fig. 2(a) .

 

Fig. 2 (a) The stack of slices used to model the spheroidal shape of nanoparticles (side view); (b) a schematic drawing of randomly distributed nanoparticles with variable distances between particles R_j (top view).

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The nanoparticles were assumed to be placed randomly with an average inter-particle distance R, as shown in Fig. 2(b). The multiple scatterings from this random distribution were taken in account by introducing a “structure factor”, S(k) = 1 + f ∑_j≠0 exp{i(k-k ₀)∙R _j}, where k is the in-plane wavevector to be integrated in the scattering integral equation [12,21]. k ₀ is the in-plane component of the wavevector for the incident light. R _j denote the positions of nanoparticles on the surface, relative to a given one (j = 0) under consideration. f describes the uniformity factor. f = 1 if the nanoparticles are evenly distributed [as in Fig. 1(a)], but f < 1 if the distribution is not uniform [as in Fig. 1(b)-(d)]. For a random distribution, it can be shown that S(k)≈1 + f [Nδ(k-k ₀)-∫_cell dR exp{i(k-k ₀)∙R}/A_cell] [21], where we have replaced the discrete sum by a continuum integral. A _cell denotes the average cell area for the region where the distribution is considered uniform (A_cell = πR ²). N is the total number of nanoparticles under consideration.

To describe the optical properties of materials under investigation, the dielectric constants of bulk Au [22] were adopted for the nanoparticles. For the multilayered substrate, as discussed above, the optical contrast between the APTES coating and the glass slide was found to be extremely small. Therefore, it is possible for us to treat the substrate plus APTES as a single semi-infinite layer and use its pseudodielectric constants in the theoretical calculations. In the following fittings, four parameters describing nanoparticles’ shape and size (lateral diameter d and height t) as well as distribution (the average distance between nanoparticles R and a “uniformity factor” f) were the only fitting parameters for this study.

4. Results and discussion

The comparisons of the SE measurements at three angles of incidence and the model calculations of the best fits are depicted in Fig. 3 . The four fitting parameters and the number of slices (a fixed value for each size) are summarized in Table 1 . These best fits were determined by varying the fitting parameters to minimize the mean-squared errors (MSEs) of measurements and calculations at 55° and 60° angles of incidence. As can be seen from the comparisons, the calculated spectra can fit experimental data fairly well simultaneously at all three angles of incidence. This also verifies that the validity of our model.

 

Fig. 3 SE measurements and model calculations of Au nanoparticles samples with nominal sizes of (a) 20, (b) 40, (c) 60, and (d) 80 nm at incident angles of 55°, 60°, and 65°.

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Table 1. A summary of parameters used in the theoretical modelings for Au nanoparticles.

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Based on these results, the Au nanoparticle sizes used for the best fits are still close to the values specified by the supplier although the shapes after immobilization on the substrate seem to become somewhat oblate instead of spherical. This demonstrates SE’s capability and sensitivity of distinguishing a minute difference in nanoparticles’ geometry. Moreover, distribution properties found from the fittings are in good agreement with what SEM micrographs revealed. For example, the average distance is close to the estimation calculated by counting the particle number N in a square of area A:$R_{\text{est}\text{.}}=\sqrt{A/N}$.

Figure 4 displays side-by-side a 1 × 1 μm representative SEM micrograph and its particle distribution pattern for the 20-nm Au nanoparticles sample. The locations and particle number (N ≈566 for Fig. 4) were determined by using a homemade MATLAB procedure to analyze the SEM micrograph. The average distance between two particles was found to be about 40 to 42 nm (analyzed from several representative SEM micrographs), which is consistent with the value for the best fit (42 nm). For samples with larger particle sizes, the average distances estimated using the same approach are around 135-140, 156-175, and 210-249 nm for 40-, 60-, and 80-nm nanoparticles, respectively. If we compare the model fitting results with the previous work assuming perfectly periodic distribution of nanoparticles [10], as shown in Table 2 , it is clear that random-distribution models work much better because the average distances obtained here are much closer to estimations obtained from SEM micrographs in comparison with periodicity for periodic distribution. The MSE becomes much smaller if we only consider the range 2-9 eV. The poor fit in the 1-2 eV range is attributed to the clustering of nanoparticles, which is apparent in the SEM micrographs for sizes 40-80 nm

 

Fig. 4 A representative SEM micrograph for 20-nm Au nanoparticles and its particle distribution pattern. The panel size is 1 × 1 μm and contains around 566 particles.

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Table 2. A comparison of theoretical modelings based on assumptions of random and periodic distributions.

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Although the above theoretical calculations were done by using extracted pseudodielectric constants of APTES on a glass slide to represent the real layered structure, it should be noted that our modeling approach is actually applicable to arbitrarily multiple layers and not limited to a single-layer substrate. Moreover, the above analyses could be further improved by several ways, such as averaging over particle size variation and taking into account the presence of clustered particles. A more complicated description of the sample by considering all the contributions from isolated particles, clustered particles, and the vacant area (without any particles) should be able to model the system more accurately.

5. Conclusion

In this report, we applied SE to characterize the size, shape, and distribution of randomly distributed Au nanoparticles on a multilayer film. The calculations based on the FEGF method which takes into account contributions from multiple scatterings and an improved model over the one used in our previous studies [10] agree with the SE measurements fairly well when suitable geometry parameters are used in the modeling. These results demonstrate that this technique could provide key information about immobilized nanoparticles, such as the particle size, shape, and distribution. And it suggests that the SE measurements coupled with efficient and sophisticated theoretical modeling and some SEM analysis can be a very useful tool for nondestructive metrology of nanostructures.