1. Introduction

Fast nondestructive characterization of micro- and nanostructures on substrates is key in the design, production, and quality control of modern functional nanomaterials. A number of forward scattering models for particles on substrates were developed [1–5], and various optical characterization techniques were described and validated experimentally [6–10]. In Bidirectional Reflectance Distribution Function (BRDF) measurements, the considered structure is illuminated monochromatically, and its size, shape, or material composition is estimated from the measured angular resolved reflected field intensity. This paper concerns the cross-section estimation of submicron wires (SW) on rough substrates, based on noisy in-plane BRDF measurement data. The presented method is applicable to dielectric as well as to highly conductive 2D particles, and it is not limited to canonical cross-section geometries. For fast and simple data acquisition, it is of interest to develop inversion methods that rely on limited measurement data. Therefore, the SW cross-section estimation is here based on a single BRDF measurement over a relatively narrow, 40° aperture.

In Karamehmedović et al. [5,11] we described an efficient forward scattering model applicable in the numerical reconstruction of submicron particles on perfectly planar (‘smooth’) substrates, based on a single measured angular-resolved scattering pattern. In [11] we used this model in conjunction with the decomposition method, described in general in Colton and Kress [12], Section 7.3, to estimate the cross-section radius of perfectly conducting and silver (Ag) submicron wires on smooth Si substrate from simulated scattering data. In this paper, we extend the scattering model of [5,11] to handle particles on rough substrates, and we validate the model against experimental data. The substrate roughness is represented by a heuristic, denoised surface transfer function computed from the measured bare-substrate BRDF. We apply the scattering model in the estimation of the cross section of a Pt submicron wire on a rough Si substrate, based on the BRDF measured in the cross-section plane, in the angular range ± 20° from specular reflection. The experimental data do not include phase information for the scattered far field, and the inverse problem is thus highly ill-posed. We regularize the problem by considering a limited dynamical range of SW cross sections, and we demonstrate that the scattering model can be used as a means of physically justified interpolation within a table-based inversion. In contrast to [11], the inversion here is not based on estimating the near field that corresponds to the measured far-field pattern, although good near-field reconstructions are achieved.

All measurements were performed by Danish Fundamental Metrology. Section 2 presents the experimental setup. Section 3 describes the forward scattering model, including the computation and use of the surface transfer function. Numerical results are shown and discussed in Section 4, and the conclusion and outlook are stated in Section 5.

2. Experimental setup

A Pt submicron wire with an approximately elliptical cross-section rests on a 2-inch polished Si substrate. The wire was formed bottom-up in a FIB-SEM, by depositing three stripes of Pt with nominal length 100 µm on top of each other. BRDF measurements, described shortly, are made at the free-space wavelength λ₀ = 325 nm. Using reference Scanning Electron Microscopy (SEM) and Atomic Force Microscopy (AFM) measurements, shown in Figs. 1a )–1c), the length, width and height of the SW are estimated to λ = 98.86 μm ≈304λ₀, 𝓌: = 2a_SEM = 537.5 nm ≈1.65λ₀ and 𝒽: = 2b_AFM = 562.5 nm ≈1.73λ₀, respectively.

 

Fig. 1 Pt submicron wire on Si substrate: a)-b) SEM. c) AFM. d) Cross-section schematic.

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The thickness of the Si substrate is 20 μm ≈61.5λ₀, and the minimal distance of the SW to the substrate edge is greater than 1 cm ≈30769λ₀. The SW can therefore be modeled accurately as a 2D cylindrical scatterer placed on an infinite half-plane substrate, see Fig. 1d). In the following, as indicated in Fig. 1d), the BRDF setup has the incidence and the measurement plane coincide with the plane of the SW cross-section σ. The angle of incidence θ₀ is fixed at 60° from normal, and the measured BRDF is hence a function of just one angle, namely the angle of observation in the SW cross-section plane. Such in-plane BRDF is defined, e.g., in Stover [13, Section 1.5].

A power-stabilized He-Cd laser delivers a Gaussian output beam. The beam is collimated and s-polarized, and it is chopped to reduce both electrical and optical noise. The output power is regulated by a set of neutral density filters placed in front of a polarizer. On the detector side the beam passes through an analyzer before it is detected by a linear silicon detector connected to a lock-in amplifier. The distance, width and height of the detector slit are 30 cm, 0.2 mm and 3 mm, respectively. The Si substrate with the SW is mounted on a sample holder system with 6 degrees of freedom. We first measure the incident power P_i by moving the sample holder system out of the light path. After moving the sample holder back, the laser light is placed in the focus of a microscope equipped with a 50 × magnifying long working distance objective (Mitutoyo). Then, we scan the Si sample until we see a microscope image of the SW. The alignment of the laser beam on SW is checked before the microscope is removed and the BRDF scan is started. During the BRDF scan we measure the scattered power per unit solid angle, dP_s / dΩ_s, for a given scattering angle θ_s, and we thereby obtain the BRDF signal (see Eq. (1).9) in [13], Section 1.5),

The receiver step size equals the width of the detector slit (0.2 mm). The measurements are performed in the angular range θ_s ∈ [–20°,20°] from specular reflection, and varying the incident power using the filters while always maintaining the same setting on the lock-in amplifier. Since only a discrete set of filters are available, there are gaps in the resulting scattering spectra.

3. Forward scattering and roughness model

The forward scattering model is illustrated in Fig. 2 . The time dependence factor used in the following is exp(jωt). The angle of incidence and of specular reflection is θ₀ = π/3 (: 60°) from normal. As shown in Fig. 2b), r = (x,y) is the vector specifying the observation point, with length |r| and angle θ from the direction of specular reflection. The laser illumination is approximated by an incident TE-polarized plane wave with Gaussian amplitude modulation,

 

Fig. 2 The forward scattering model. a) Interior sources with complex images. b) Exterior sources.

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The plane wave reflected off the Si substrate, in the absence of the SW, is written (E^ref,H^ref); here, E^ref = ẑE^ref = ẑΓ_refE^inc, where Γ_ref = (n₀ – n_Si) / (n₀ + n_Si) ≈–0.742 + j0.137 is the Fresnel reflection coefficient for TE waves incident upon the air-Si interface at λ₀ = 325 nm (the refractive index n_Si is here taken from Palik [14]). The total field in the exterior of the SW cross-section σ in the upper half-plane, that is, in ϒ²₊ \ σ, is the sum of the incident wave, the reflected wave, and the field (E^sca,H^sca) scattered by the SW, E^tot = E^inc + E^ref + E^sca. The field scattered by the SW is approximated in ϒ²₊ \ σ by a linear combination of fields emitted by discrete z-directed electric line currents located within the cross-section σ and radiating in the presence of the substrate,

Here Φ_1/2,Si(⋅,r’) is the half-plane Green’s function for the Helmholtz operator in the air-substrate medium, with singularity at r’. Since |n_Si² / n₀²| ≈35.8 is large, we use the ‘image at a complex depth’ approximation derived, e.g., in Lindell and Alanen [15],

Here Φ_Pt(⋅,r”) is the Green’s function for the Helmholtz operator in the Pt-filled plane, with singularity at r”, Φ_Pt(r,r_ν”) = H₀⁽²⁾(k_Pt|r–r”) / 4j. The wave number k_Pt = n_Ptk₀, with the refractive index for Pt at 325 nm taken from [14]. Approximating the fields in terms of discrete sources instead of in terms of surface current densities simplifies the numerical implementation and accelerates the scattering computation, since the numerical integration of the densities is avoided; the classical radiation integrals are replaced with readily computable finite sums in Eq. (3) and Eq. (5). The complex source images account for the particle-surface interaction in the field scattered by the SW. The discrete sources are uniformly distributed along ellipses with semi-diameters 0.86a and 0.86b (interior sources and their images) and a/0.86, b/0.86 (exterior sources). These semi-diameters were found well-suited through numerical experimentation. The complex amplitudes C_ν, D_ν of the discrete sources are computed by imposing the transmission conditions (continuity of the total tangential electric and magnetic field) at a number of discrete testing points t_μ distributed uniformly along the circumference of σ,

The subscripts + and – signify limit values from the exterior and from the interior of σ, respectively. Having found C_ν, D_ν, the BRDF and the near field can be computed using Eqs. (3) and (5). For the numerical results of Section 4, we use 20 interior and 20 exterior discrete sources (M = N = 20), as well as 20 testing points.

Given a numerically computed function I_num and the corresponding measured data I_meas at n observation points, the two will here be compared using the following root mean square error measure:

The micro-roughness of the air-substrate interface is modeled using a surface transfer function. Ideally, given an incident field and a substrate, with or without a structure present, the transfer function modifies the Fourier transform of the BRDF that would be measured without roughness to the Fourier transform of the corresponding rough-substrate BRDF. For small observation angles θ, i.e., for observation points close to specular reflection θ₀, it holds that cos θ ≈1, and it is seen from Eq. (1) that the reflected field intensity is approximately proportional to the corresponding BRDF. We therefore approximate the surface transfer function by the ratio of the Fourier transforms of the experimental bare-substrate BRDF I_sca,0 and the incident field intensity |E^inc|². Close to specular reflection, it holds that sin θ ≈θ, so by Eq. (2) we have |E^inc|²(θ) ≈(max |E^inc|²) exp(–(|r| θ)²/σ_inc²) for θ ∈ [–20°,20°], and the Fourier transform of |E^inc|² satisfies ℱ|E^inc|²(ξ) ≈(max |E^inc|²) σ_inc π^1/2 |r|⁻¹ exp(–ξ²σ_inc² / 4|r|²). The bare-substrate BRDF I_sca,0 is fitted by a smooth function,

 

Fig. 3 The heuristic surface roughness model. a) Fitting the bare-substrate BRDF.
b) The effect on the forward model output.

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The fit parameters that minimize the RMSE, as defined in Eq. (6), over the angular range ± 20° are found to be A₁ = 0.8624 sr⁻¹, A₂ = 0.004452 sr⁻¹, A₃ = 8.219⋅10⁻⁶ sr⁻¹, σ₁ = 0.001933 m, σ₂ = 0.009941 m and σ₃ = 0.08051 m. To qualify this fit, the achieved RMSE of 0.4284 is comparable to the RMSE of 0.4276 obtained for the least-squares fit of log₁₀I_sca,0 with a truncated Fourier series up to 11th harmonic,

The interface roughness is included in the scattering model as follows. First, the forward model of Fig. 2 is used to compute the BRDF I_sca,1,SMOOTH for the SW on the smooth substrate. The Fourier transform of I_sca,1,SMOOTH is then modified using the surface transfer function H, ℱI_sca,1,ROUGH: = HℱI_sca,1,SMOOTH, and the final BRDF I_sca,1,ROUGH is computed by taking the inverse Fourier transform, I_sca,1,ROUGH = ℱ⁻¹HℱI_sca,1,SMOOTH. Figure 3b) compares the measured BRDF with the predictions obtained using the scattering model with the measured cross-section semi-diameters a_SEM = 268.75 nm and b_AFM = 281.25 nm. It is seen that including the roughness in the model improves the correspondence between the computed and the measured BRDF; the RMSE w.r.t. the experimental data is reduced from 0.6386 to 0.3750.

Interface roughness is commonly modeled by inserting an additional material layer between air and substrate, and setting the permittivity of the layer according to an effective medium theory [17, sections 5.3.1-5.3.2]. For example, for a substrate with relative permittivity ε₂ and a volume fraction f of air in the rough interface, the effective medium layer relative permittivity ε₁ is given in Eq. (5).43) on p. 179 in [17],

A disadvantage of this approach is that the resulting stratified Green’s function, which describes the fields radiated by the individual discrete sources in the presence of the two-layer substrate/effective medium structure, is more elaborate and numerically expensive than the complex image approximation for the single-interface case. Our roughness model keeps the scattering problem geometry simple, and allows the use of the efficient formulation with discrete sources and complex images.

4. Numerical results and discussion

Figures 4a ) and 4b) show the achieved root mean square error, as defined in Eq. (6), in the computed BRDF relative to the measured BRDF, with and without the roughness included in the forward scattering model (‘rough’ and ‘smooth’ case), respectively. A total of 200 and 801 measurement points in the angular range θ ∈ [–20°,20°] are used for the two figures, respectively. The down-sampling of the data is done in the ‘rough’ case to speed up the computation, but also because this turns out to produce slightly better (inverse) Fourier transform results. The RMSE is evaluated over a 16 × 16 grid within the range a ∈ [200 nm, 350 nm] = [0.74a_SEM,1.30a_SEM], b ∈ [200 nm, 350 nm] = [0.71b_AFM,1.24b_AFM] of cross-section semi-diameters a and b. Figures 4c) and 4d) present the same RMSE in top view, to better show the global minima. These are circled in black, and located at a = 290 nm, b = 320 nm in the rough case and at a = 350 nm, b = 290 nm in the smooth case. The average computation time for a single value of the RMSE is 4.0 sec and 3.8 sec in the two cases, with a single-core MATLAB® implementation on a standard PC. It is seen that including the roughness makes the inverse problem better posed, in that the global minimum, indicating the best estimate for a and b, is better localized. Also, it generally improves the correspondence between the computation and the measurement results, since the RMSE takes on smaller values.

 

Fig. 4 Objective function: a) with, and b) without the roughness model.
c)-d) Top view of a) and b), respectively.
Best cross-section match: e) reconstructed |(E)^inc + (E)^sca|, f) reconstructed total near field amplitude.

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Figures 4e) and 4f) show the amplitude of the reconstructed field E^inc + E^sca and the reconstructed total electric near field E^inc + E^ref + E^sca, respectively, corresponding to the estimates a = 290 nm, b = 320 nm. We here do not make use of the reconstructed near field in the estimation of the SW cross-section, and Figs. 4e)-4f) are only shown to demonstrate the quality of the reconstructed near field. The contour of the SW cross section is visible as a locus of minima of the field in Fig. 4e); the discrete sources can also be seen in this figure. The SW cross-section itself is visible as the region where the total field vanishes in Fig. 4f). The plotted field amplitude in Fig. 4f) has been limited to 2V/m to enhance the contrast. This imaging of the particle is possible because the Pt SW is highly conductive, and the total field nearly vanishes within the cross-section σ. However, we stress that the forward model is directly applicable to arbitrary penetrable SW, and the imaging step is not necessary in the estimation of the cross-section semi-diameters a and b. The near-field plots in Figs. 4e) and 4f) are of resolution 100 × 100 points, and each took approx. 48 sec to compute. Table 1 summarizes the estimation of the SW cross-section.

The average relative error in the estimates of the SW cross-section parameters is 10.85% with the rough model and 16.67% with the smooth model. In comparison, Germer et al. [6] estimated the radius of 100 nm polystyrene latex spheres on a substrate with 0.3% error. Also, de la Peña and Saiz [7] estimated the radii of metallic, circular cross-section 2D particles on a metallic substrate with a relative error of approx. 0.4%, while Bell and Bickel [10] achieved a relative error of only 0.1% in estimating the cross-section radius of silica fibers on a substrate. We note, however, that our estimation is based on less experimental data: in [6], the BRDF is measured over a 100° aperture, in [7] multiple angles of incidence are used and a full scattering pattern is needed for normal incidence, and [10] measures and fits 16 components of the Mueller matrix describing the scattering by the fiber on the substrate.

To improve the correspondence between the predicted and the measured BRDF in Fig. 3b) and reduce the error in the cross-section estimate, one should use a more elaborate/realistic geometrical representation of the SW cross-section. The forward scattering model presented in Section 3 is, in principle, readily applicable to any cross-section geometry. One could also take into account the material residues near the SW resulting from the fabrication process, and include a native oxide layer on the substrate in the scattering model.

The forward scattering model of Section 3 can be used in conjunction with a table of pre-computed BRDF spectra, as a physically justified means of interpolation between the table entries. In this approach to the solution of the inverse problem, matching of the table entries with the measured BRDF would first be used to define a relatively small dynamical range where the semi-diameters a and b are estimated. (In Karamehmedović et al. [18] we interpolated pre-computed far-field patterns using discrete sources, for defect estimation in nanogratings.) Finally, instead of computing the surface transfer function heuristically as above, one can attempt a statistical description of the rough surface, as done, e.g., in [16,19], and construct a transfer function in terms of the root mean square roughness, surface autocovariance length, surface autocovariance function etc. The statistical roughness parameters can be included as variables of optimization when minimizing the RMSE of Fig. 4.

5. Conclusion and outlook

The forward scattering model of [5,11] for penetrable particles on substrates was extended to include the substrate roughness, and used in the estimation of the profile of a Pt submicron wire on a rough Si substrate. The micro-roughness of the substrate was modeled by a heuristic, denoised surface transfer function. Based on experimental BRDF data given over a 40° aperture, the semi-diameters of the wire were estimated with relative errors of approx. 8% and 14%, respectively. The presented forward model can be used with regularized inversion schemes. For example, it would make a useful supplement to table-based inversion, as a physically justified means of interpolation/refinement of far-field fitting results with pre-computed scattering patterns. We expect the inversion would be better posed if more BRDF spectra were available, that is, if measurements at different wavelengths and angles of incidence were included. Also, it is desirable to include the effect of cross-polarization introduced by the rough interface. Finally, the scattering model is extendible to the 3D case.