Near-field ellipsometry for thin film characterization

Zhuang Liu; Ying Zhang; Shaw Wei Kok; Boon Ping Ng; Yeng Chai Soh

doi:10.1364/OE.18.003298

1. Introduction

Nowadays thin films have been widely used in various applications such as sensors [1, 2] and optical waveguides [3,4]. This is due to the unique optical and electric features of thin film structures. However, imperfections such as small crackers, isolated islands, and layer unevenness are inevitable in practical fabrication process of thin films [5]. The size of the imperfections can be as small as in nano-scale level [6]. But they do degrade the performance of thin film structure. Hence, characterization of thin film with high spatial resolution becomes critical to ensuring the performance of designed thin film devices [7]. Optical characterization methods are preferred due to their non-destructiveness. The commonly used approaches include interferometry methods [8], surface plasmon resonance methods [9], and ellipsometry methods [10, 11]. However, all these optical methods can only attain the best lateral resolution of half a wavelength due to the diffraction limit. Moreover, characterization of thin film needs to solve ellipsometric equations that are transcendental with respect to the unknowns such as thickness and dielectric constant of all thin film layers. Since the transcendental equations are not convex with respect to the unknowns, their numerical solutions are vulnerable to initial values, measurement noises, and numerical errors. This contributes to artifacts or even faulty measurements. Moreover, the calculation is also very time-consuming, especially for the multi-layer thin film. With advance of thin film technology, it is imperative to have high resolution, fast, and reliable thin film characterization methods.

In this paper, we propose a near-field ellipsometry method for thin film characterization. The idea of using apertureless SNOM in ellipsometry was reported [12] and demonstrated nano-scale resolution. Like existing ellipsometry methods, the proposed method still uses light polarization to establish the ellipsometric equations of the refractive index and the thickness of all thin-film layers. However, unlike the existing ellipsometry methods, the proposed method uses some unique features of aperture SNOM. The first feature is that only the transverse polarized field can be detected by a SNOM and it carries high spatial frequency components beyond the diffraction limit. The second feature of SNOM is that it can measure simultaneously the optical and the topographical characteristics at the top layer of the thin-film. This contributes an extra constraint of the unknowns to be characterized. From the information fusion perspective, the extra constraint helps reduce the artifacts caused by uncertainties such as measurement noises and numerical errors. On the other hand, the constraint helps decouple the complex ellipsometric equations and speed up the calculation of the ellipsometric equations. In the following sections, we first derive the theoretical formulations of the near-field ellipsometry for single layer thin-film. Then, the near-field ellipsometry is extended to a general case for multi-layer thin films. Simulation and experiment studies are carried out to verify the performance of the proposed near-field ellipsometry. It is shown that the proposed near-field ellipsometry can achieve nano-scale characterization of thin film. The fusion of the topographic and ellipsometric signal not only helps reduce artifacts in thin film characterization, it also improves the signal-to-noise (SNR) performance and reduces the computational load of the near-field ellipsometry for thin film characterization.

2. Theoretical formulation of near-field ellipsometry

For sake of clear illustration, the near-field ellipsometry for a single layer thin film on a prism is first derived, as shown in Fig. 1. The central medium is one layer of thin film. The medium below the thin film is the prism from where the light enters. The layer on top of the thin film can be air, water or some other solutions. These two media are considered to extend to semi-infinite regions. The general case of multi-layer thin film is considered in a later section. To make the derivation extendable to multi-layer thin film, we regard the bottom medium and the top medium respectively as 1-st and 3-rd layer for a three-layer structure. We use ε_i = n_i ² (i = 1,2,3) to denote the dielectric constants of the prism, the thin film, and the top medium, respectively. n_i is the refractive index of i-th layer. E_i and H_i denote respectively the electric field and the magnetic field in i-th layer. The task of thin film characterization is to find the thickness and the dielectric constant of the thin film. For this case, it is to find the thickness and dielectric constant of the 2-nd layer, assuming that ε ₁ and ε ₃ are known.

Fig. 1. A single layer thin film on a prism.

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2.1. Transmission coefficients for ellipsometry

We first investigate the optical transmission characteristics of the thin film. In the proposed ellipsometry, polarized light is incident on the thin film bottom through the prism to generate evanescent wave on the thin film surface. Evanescent wave is detected by a SNOM probe above the thin film surface. The thickness and dielectric constants are recovered from the ellipsometric equations obtained by the p and s-polarized components of the evanescent wave. Here p- and s-polarized light represent the electric fields parallel and perpendicular to the incident plane respectively.

For the s-polarized light, the amplitudes of electric fields in two adjacent layers can be linked by a transfer matrix as [11]:

(\begin{matrix} A_{i + 1}^{s} \\ B_{i + 1}^{s} \end{matrix}) = M_{i}^{s} (\begin{matrix} A_{i}^{s} \\ B_{i}^{s} \end{matrix}) = \frac{1}{2 k_{z_{i + 1}}} (\begin{matrix} (k_{z_{i + 1}} + k_{z_{i}}) \exp (j k_{z_{i}} d_{i}) & (k_{z_{i + 1}} - k_{z_{i}}) \exp (- j k_{z_{i}} d_{i}) \\ (k_{z_{i + 1}} - k_{z_{i}}) \exp (j k_{z_{i}} d_{i}) & (k_{z_{i + 1}} + k_{z_{i}}) \exp (- j k_{z_{i}} d_{i}) \end{matrix}) (\begin{matrix} A_{i}^{s} \\ B_{i}^{s} \end{matrix}),

where A^s_i and B^s_i are complex amplitudes of the incident and the reflected light in the i-th layer respectively, and d_i is the thickness of the i-th layer. k_x = k ₀√ε ₁ sinθ and $k_{z_{i}} = k_{0} \sqrt{ε_{i} - ε_{1} \sin^{2} θ}$ , where k ₀ is the wavenumber in free space and θ is the incident angle on the bottom of the thin film. It should be noticed that in our configuration, if k_{z_i} is a real number, the light is a propagating wave in this layer, and if k_{z_i} is imaginary, the light is an evanescent wave.

The amplitude of electric field in the 1-st layer and the field in the 3-rd layer have following relationship:

(\begin{matrix} A_{3}^{s} \\ B_{3}^{s} \end{matrix}) = M_{1}^{s} M_{2}^{s} (\begin{matrix} A_{1}^{s} \\ B_{1}^{s} \end{matrix}) .

Since no reflected light exists in the top medium for the case as in Fig. 1, it yields B^s ₃ = 0. Using Eq. (2), the transmission coefficient of s-polarized light is obtained by:

T^{s} = \frac{A_{3}^{s}}{A_{1}^{s}} = \frac{4 k_{z_{1}} k_{z_{2}} \exp (j k_{z_{2}} d)}{(k_{z_{1}} + k_{z_{2}}) (k_{z_{2}} + k_{z_{3}}) + (k_{z_{1}} - k_{z_{2}}) (k_{z_{2}} - k_{z_{3}}) \exp (2 j k_{z_{2}} d)},

where d is the thickness of the thin film sample.

Similarly, the transmission coefficient for the p-polarized incident light can be obtained. For p-polarized incident light, the transfer matrix is given by:

M_{i}^{p} = \frac{1}{2 ε_{i} k_{z_{i + 1}}} (\begin{matrix} (ε_{i} k_{z_{i + 1}} + ε_{i + 1} k_{z_{i}}) \exp (j k_{z_{i}} d_{i}) & (ε_{i} k_{z_{i + 1}} - ε_{i + 1} k_{z_{i}}) \exp (- j k_{z_{i}} d_{i}) \\ (ε_{i} k_{z_{i + 1}} - ε_{i + 1} k_{z_{i}}) \exp (j k_{z_{i}} d_{i}) & (ε_{i} k_{z_{i + 1}} + ε_{i + 1} k_{z_{i}}) \exp (- j k_{z_{i}} d_{i}) \end{matrix}) .

But for p-polarized light, the electric field has two components: one along x direction and the other along z direction. Using the conclusion of [13], only the transverse field (i.e., the x component) can be coupled into the SNOM fiber probe and measured by SNOM. Therefore, it is appropriate to express the transmission coefficient of x component. It is obtained by:

T^{x} = \frac{k_{z_{3}} ε_{1} A_{3}^{p}}{k_{z_{1}} ε_{3} A_{1}^{p}}

Then, we can define the transmission coefficient ratio between y component and x component. Using Eq. (3) and Eq. (5), the ratio is obtained as follows:

ρ = \frac{T^{y}}{T^{x}}

where T^y = T^s, because s-polarized light contains only y-component.

So far, we have established the ellipsometric equation of the two unknowns including the thin film thickness d and the thin film dielectric constant ε ₂. They are based on the near-field optical signals.

2.2. Near-field elipsometry

Now, we explore the extra equation of the two unknowns, by virtual of the topographic information that is obtained simultaneously with the near-field optical signals in the SNOM. A SNOM is usually based on an atomic force microscopy (AFM) system to achieve the probe-sample distance control as sketched in Fig. 2. When the SNOM is operated in a constant separation mode, the sum of the topography reading and the sample thickness should be a constant.

We can assume a virtual straight line A along the probe’s scanning direction and the sample has a flat bottom line. The distance between these two lines is kept constant which is denoted by d ₀. Denote the topography reading as d_A and the sample thickness d. They are related by:

d_{0} = d + d_{A},

Fig. 2. Schematic of the AFM and ellipsometric SNOM relationship.

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Inserting Eq. (7) into Eq. (6), we obtain an equation with only one unknown ε ₂:

\frac{X_{1} + X_{2} \exp (j 2 Δ)}{Y_{1} + Y_{2} \exp (j 2 Δ)} - ρ = 0,

where

X_{1} = k_{z_{1}} (ε_{2} k_{z_{1}} + ε_{1} k_{0} \sqrt{ε_{2} - ε_{1} \sin^{2} θ}) (ε_{3} k_{0} \sqrt{ε_{2} - ε_{1} \sin^{2} θ} + ε_{2} k_{z_{3}}),

and ρ is defined in Eq. (6).

Equation (8) establishes the basic equations of near-field ellipsometry. But it is difficult to have a close analytical solution to ε ₂ from Eq. (8). Instead, numerical methods are used to solve this transcendental equation. For example, using the well-known Newton-Raphson algorithm [14], we can calculate the unknown dielectric constant by:

ε_{2}^{n + 1} = ε_{2}^{n} - \frac{H (ε_{2}^{n})}{H' (ε_{2}^{n})}, n = 0,1,2, \dots,

where

H (ε_{2}) ≜ \frac{X_{1} + X_{2} \exp (j 2 Δ)}{Y_{1} + Y_{2} \exp (j 2 Δ)} - ρ = 0 .

As a remark, the near-field ellipsometry can also be implemented in another form that finds the two unknowns ε ₂ and d by only solving Eq. (6). It does not need the topography measurements. A Newton-Raphson algorithm for this second implementation of the near-field ellipsometry is given by the following iterations:

(\begin{matrix} ε_{2}^{m + 1} \\ d^{m + 1} \end{matrix}) = (\begin{matrix} ε_{2}^{m} \\ d^{m} \end{matrix}) - J_{F}^{- 1} (ε_{2}^{m}, d^{m}) F (ε_{2}^{m}, d^{m}), m = 0,1,2, \dots,

where

F (ε_{2}, d) ≜ (\begin{matrix} Re [f (ε_{2}, d) - ρ] \\ Im [f (ε_{2}, d) - ρ] \end{matrix})

and J_F(ε ₂,d) is Jacobian matrix of F(ε ₂,d).

Comparing the two implementations of near-field ellipsometry, the latter one needs to conduct complicated matrix computation especially for the Jacobian matrix J_F(ε ₂,d). This takes much more time than the former one which only calculates a scalar function. The subsequent simulation studies also show that the first implementation of near-field ellipsometry can outperform the second one in removal of artifacts and reduction of measurement noise effects.

3. Extension to multi-layer thin film

The theoretical formulation of the near-field ellipsometry can be extended to multi-layer thin film structures.

We use the same denotations for dielectric constants, electric fields, and thicknesses as used before, and i takes values greater than 3. The overall transfer matrix of multi-layer thin film, denoted as M^s for s-polarized light, can be derived by cascading Eq. (1):

M^{s} = M_{1}^{s} M_{2}^{s} \dots M_{n - 1}^{s}

and

(\begin{matrix} A_{n}^{s} \\ 0 \end{matrix}) = M^{s} (\begin{matrix} A_{1}^{s} \\ B_{1}^{s} \end{matrix}) .

Then

(\begin{matrix} A_{1}^{s} \\ B_{1}^{s} \end{matrix}) = {(M^{s})}^{- 1} (\begin{matrix} A_{n}^{s} \\ 0 \end{matrix}) = (\begin{matrix} {\tilde{m}}_{11}^{s} & {\tilde{m}}_{12}^{s} \\ {\tilde{m}}_{21}^{s} & {\tilde{m}}_{22}^{s} \end{matrix}) (\begin{matrix} A_{n} \\ 0 \end{matrix}) = (\begin{matrix} {\tilde{m}}_{11}^{s} \\ {\tilde{m}}_{21}^{s} \end{matrix}) A_{n}^{s} .

So we have the transmission coefficient of s-polarized light:

T^{s} = \frac{A_{n}^{s}}{A_{1}^{s}} = \frac{1}{{\tilde{m}}_{11}^{s}} .

Similarly, the transmission relationship of n-layer structure for p-polarized light can be expressed as:

(\begin{matrix} A_{1}^{p} \\ B_{1}^{p} \end{matrix}) = {(M^{p})}^{- 1} (\begin{matrix} A_{n}^{p} \\ 0 \end{matrix}) = (\begin{matrix} {\tilde{m}}_{11}^{p} & {\tilde{m}}_{12}^{p} \\ {\tilde{m}}_{21}^{p} & {\tilde{m}}_{22}^{p} \end{matrix}) (\begin{matrix} A_{n}^{p} \\ 0 \end{matrix}) = (\begin{matrix} {\tilde{m}}_{11}^{p} \\ {\tilde{m}}_{21}^{p} \end{matrix}) A_{n}^{p},

where M^p = ∏M_i^p, and M_i^p is defined as in Eq. (4).

For the case of the multi-layer structure, only the transverse field (i.e., x component) can be observed by SNOM. The transmission coefficient of x component is given by:

T^{x} = \frac{k_{z_{n}} A_{n}^{p}}{k_{z_{1}} A_{1}^{p}} = \frac{k_{z_{n}} ε_{1}}{k_{z_{1}} ε_{n} {\tilde{m}}_{11}^{p}} .

Then the total transmission coefficient ratio between y and x components is expressed by:

ρ_{λ} = \frac{k_{z_{1}} ε_{n} {\tilde{m}}_{11}^{p}}{k_{z_{n}} ε_{1} {\tilde{m}}_{11}^{s}} = f_{λ} (ε_{1} \dots ε_{n} d_{1} \dots d_{n}),

where m̃^s ₁₁ and m̃^p ₁₁ are functions of k_{z_i} and ε_i, and λ represents the wavelength of the incident light. If there is a total of n layers in the system and the central n − 2 layers are thin film sample, n − 2 different wavelengths or n − 2 different incident angles θ are needed to obtain n − 2 complex equations from which thickness and dielectric constant of each thin film layer can be solved.

If combining with AFM, the relation involves all the thicknesses of thin film layers, and Eq. (7) becomes:

d_{0} = d_{s_{2}} + d_{s_{3}} + \dots + d_{s_{n - 1}} + d_{A},

where d_{s_i} is the thickness of the i-th layer of the thin film. Unlike the case for single layer thin film, this relation contributes an extra constraint to multiple unknowns to be solved.

Now the problem can be considered as an optimization problem subject to one equation constraint, as follows:

min {F^{T} (E, D) F (E, D)} s . t . g (d_{s_{i}}) = d_{A} + \sum d_{s_{i}} - d_{0} = 0,

where E = [ε _s₂, ⋯, ε _{s_n-1}]^T, D = [d _s₂, ⋯, d _{s_n-1}]^T, ε_{s_i} is dielectric constant of i-th layer of thin film and

F (E, D) ≜ (\begin{matrix} Re [f_{λ_{1}} - ρ_{λ_{1}}] \\ Im [f_{λ_{1}} - ρ_{λ_{1}}] \\ ⋮ \\ Re [f_{λ_{n - 2}} - ρ_{λ_{n - 2}}] \\ Im [f_{λ_{n - 2}} - ρ_{λ_{n - 2}}] \end{matrix}) .

We can define a Lagrangian:

L (E, D, v) = F^{T} (E, D) F (E, D) - v g (D),

where v is a Lagrange multiplier such that the solution of

L_{ε_{s_{i}}}^{'} = 0, L_{d_{s_{i}}}^{'} = 0, L_{v}^{'} = 0

must be a stationary point of the Lagrangian L(E,D,v).

Using Newton-Raphson algorithm to solve above equations, the iterations can be written as:

(\begin{matrix} E^{(m + 1)} \\ D^{(m + 1)} \\ v^{(m + 1)} \end{matrix}) = (\begin{matrix} E^{(m)} \\ D^{(m)} \\ v^{(m)} \end{matrix}) - J_{\nabla L}^{- 1} (E^{(m)}, D^{(m)}, v^{(m)}) \nabla L (E^{(m)}, D^{(m)}, v^{(m)}), for m = 0,1,2, \dots .

It is noted that the extra constraint here does not reduce the dimension of the characterization problem as it does for the case of single layer thin film characterization. But the subsequent simulation studies show that the extra constraint does improve the performance of thin film characterization.

4. Simulations for thin film characterization

In order to verify the theoretical derivation, numerical simulations are conducted to evaluate the performance of applying the proposed near-field ellipsometry to characterize thin film. The simulation is carried out by using a commercial finite difference time domain (FDTD) software (RSOFT).

4.1. Setup of near-field ellipsometry

The configuration of the near-field ellipsometry setup considered in the simulation is illustrated in Fig. 3. It is quite similar to the setup in [12] where an apertureless SNOM and the reflection light are used, but here, the transmission light is measured by an aperture SNOM so that the feature of aperture SNOM can be exploited. The thin film under characterization is assumed to be coated on the surface of the prism. Without loss of generality, the thin film is only considered to have one-dimensional variation along x direction. Monochromatic polarized light is incident on the thin film from its bottom through the prism. The incident light angle is adjusted to satisfy the total reflection condition so that evanescent waves exist on the top surface of the thin film. The transmitted light is detected with a SNOM probe in the near-field region on top of the thin film surface.

In the simulation, a glass prism (SFL11) is chosen with high refractive index n ₁ = 1.778 at 633 nm [15], and the top layer is chosen as air with refractive index n ₃ = 1. The wavelength of incident light is chosen at 633 nm and the incident angle θ is adjusted to π/4.

Fig. 3. Schematic of the prism-sample configuration.

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4.2. Characterization result of a single layer thin film

Figure 4(a) shows a single layer thin film under characterization with a step variation in topography and dielectric constant. For the left half of the film, the thickness is 50 nm with refractive index n ₂ = 2 and for the right half, the thickness is 70 nm with n ₂ = 3. In the simulation, the evanescent field is detected with the probe-sample separation set at 50 nm above the thin film surface. The numerical calculation simulates scanning the SNOM probe over the thin film from left to right while maintaining the probe separation at 50 nm.

Using Eq. (9) with the given topography, we can calculate the refractive index and thickness for every lateral position of the thin film. The results are displayed in Fig. 4(b) and Fig. 4(c). The topography is supposed to be measured by AFM, but we assume it available here. The results show that the refractive index calculated by the near-field ellipsometry fits very well the original settings given in Fig. 4(a). It is also noticed that there are small discrepancies between the measured values and the settings. They are basically numerical errors resulting from two limitations with FDTD software. One is the limited grid size in FDTD calculations. The smallest grid size for our FDTD simulation software is 10 nm and round-ups have to be done in calculations. The other numerical error source is that the plane waves considered in derivation of the proposed ellipsometry is approximated by Gaussian beam in FDTD software simulation.

To further demonstrate the advantage of using topography in ellipsometry, we perform the characterization by using the second implementation of near-field ellipsometry as given in Eq. (11). In this case, the refractive index and thickness are calculated simultaneously. The results are depicted as curves in Fig. 4(d) and Fig. 4(e). Compared to the results in Fig. 4(b) and Fig. 4(c), the calculated results from the second implementation show much more significant deviation from the true values, in addition to the slightly more fluctuation on the refraction index measurements for the thicker thin film part. It witnesses the advantage of the first implementation Eq. (9) over the second one given by Eq. (11). It is understood that the fusion of topography measurements reduces the uncertainties in calculating the topography through solving Eq. (6). Here the uncertainties are due to numerical errors. Thus, the fusion of topographic and ellipsometric information improves the accuracy of characterization.

Fig. 4. Simulations for the AFM and near-field ellipsometry measurement with line scanning of the sample. (a) Schematic view of sample. (b) Calculated sample refractive index using the first implementation of near-field ellipsometry Eq. (9). (c) Calculated sample thickness using the first implementation of near-field ellipsometry Eq. (9). (d) Calculated sample refractive index using the second implementation Eq. (11). (e) Calculated sample thickness using the second implementation Eq. (11).

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4.3. Performance of near-field ellipsometry in the presence of noises

Now we consider more realistic situations by taking into account of the noise effect. The noises of a SNOM signal come from several sources. The noises disturbing the topography measurements are mainly due to the electric detection circuits, the mechanical vibration of the probe, and the probe scanning processes. The noise affecting the optical measurements are various optical noises.

In the simulation, characterization of a single layer thin film with thickness of 50 nm is considered. We compare the performance of the two implementations of the near-field ellipsometry given respectively in Eq. (9) and Eq. (11). For the first implementation, we consider also the different cases depending upon the noises on the optical signal and/or on the topographical signal. The noise-free near-field signals are generated by using FDTD as illustrated in the last section. The noise-polluted signals are simulated by adding Gaussian noises onto the polarization ratio ρ of SNOM and/or the topography readings from the AFM function of a SNOM. All the noise levels are chosen the same as 14 dB.

Table 1 summarizes the refractive index measurement results that are obtained for the different noise situations. A clear observation is that the first implementation of the near-field ellipsometry outperforms the second implementation even when both the near-field optical signal and the topographical signal are subject to noises. This is attributed to the fusion of topographic signals in the recovery of refractive index in the near-field ellipsometry. The information fusion of the two signals do improve the performance of near-field ellipsometry in the presence of noises.

It is also observed from Table 1 that the noise on topography measurements would affect more the characterization results than the optical noises do for the first implementation of the near-field ellipsometry. For a practical SNOM, the topographic measurement usually have much finer resolution and sensitivity than its optical counterparts. The noise level on the topographic measurement is much lower. Thus, the fusion of the topographic measurement in the ellipsometry characterization still delivers better results than the one without the fusion.

Table 1. Variances of refractive index and thickness calculated under different noise situations using the 1st and 2nd nano ellipsometry implementation.

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4.4. Characterization of multi-layer thin film

The capability of the near-field ellipsometry for multi-layer thin film characterization is simulated by applying Eq. (25) on a two-layer thin film as specified in Fig. 5(a). The lower layer of the film is assumed to have even distribution in thickness and refractive index. The thickness is chosen as d ₂ = 50 nm and the refractive index as n ₂ = 2. The upper layer has a step-like distribution in thickness and refractive index. The thickness and the refractive index of the left part are chosen respectively as d ₃ = 60 nm and n ₃ = 2.5. For the corresponding right part, d ₃ = 80 nm and n ₃ = 3. We use Eq. (25) to establish the ellipsometric equations at the wavelengths of 633 nm and 532 nm. The calculated thickness and refractive index are recorded as curves in Fig. 5(b) and Fig. 5(c). The results show that the measurements are quite close to the true settings. This verifies the effectiveness of the proposed near-field ellisometry for multi-layer thin film characterization.

Fig. 5. Simulation for line scan of multi-layer sample. (a) Schematic view of multi-layer sample. (b) Calculated thickness of multi-layer sample. (c) Calculated refractive index of multi-layer sample.

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5. Experiment results

The performance of the proposed near-field ellipsometry is further investigated in experiments by using near-field ellipsometry to characterize a real thin film on a prism. The experimental setup is the same as in Fig. 3. The thin film sample is prepared by coating 50 nm-thick gold on the surface of a glass plate, and the plate is placed on a BK7 glass prism with refractive index matching oil filled in between the glass plate and the prism. The light is generated by a He-Ne laser with wavelength of 632.8 nm, and the polarization of the light is controlled by a polarizer and quarter-wave plate.

Fig. 6. Experimental characterization results of a single layer thin film.

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In experiment, a home-made SNOM system with tuning fork probe is used. It is equipped with the heterodyne detection as in [16] to measure the near-field polarization for x and y polarized light on the thin film surface. The system is operated under a constant probe-sample separation mode. The topography reading is the feedback voltage denoted by V. It is supposed to be proportional to the elongation distance of the PZT that drives the tuning fork probe tip, i.e., d_A = KV where K is a constant coefficient.

Before we proceed the experiments, we first calibrate K and d ₀ that are used in the near-field ellipsometry. The calibration can be done by using a ‘golden sample’ which has ideally flat single layer thin film. Here, we scanned a patch of relatively flat area on the sample. Topographic signals are obtained by the voltage reading and the optical signals for both s and p-polarized incident light are obtained by the heterodyne detector. The sample thickness d is obtained using Eq. (11). After that, we do curve fitting with the topographic signal and the thickness information to characterize the constant K and d ₀. These calibrated values are used for applying Eq. (9) to characterize the rest areas of the sample.

After calibration, an area of 0.8 μm × 0.8 μm of the thin film is scanned. The topographic image of such an area is shown in Fig. 6(a). It reflects the thickness of the thin film over this area. Since we are using single layer thin film, the refractive index distribution should also follow the same pattern as in the topographic image. We first calculate the thickness and the refractive index distribution by using the second implementation of the proposed near-field ellipsometry given in Eq. (11). The result is shown in Fig. 6(d) and Fig. 6(e). Their countpart results obtained by using the first implementation Eq. (9) are given in Fig. 6(b) and Fig. 6(c). Comparing the refractive index distribution obtained by the two methods, the second implementation clearly gives wrong measurements since the refractive index distribution should follow the topographic pattern as we explained above. The fault measurements are due to the significant topographic discrepancies appearing in top-left corner and the right side of Fig. 6(e). The discrepancies are propagated in the iterative calculations of the algorithm Eq. (11), and lead the algorithm converging locally to a wrong equilibrium point of the transcendental ellipsometry equations. The experiment results show that the fusion of the topographic measurement with optical signals in near-field ellipsometry would be able to correct artifacts in characterization results in practice. This advantage is leveraging on the unique capability of a SNOM in simultaneously obtaining both topographic and optical characteristics of the sample.

Moreover, we compare the calculation time of the two implementations of the near-field ellipsometry. For the above images, the first implementation takes only 0.27 s for calculation, while the second implementation takes 4.19 s. That is, one order of magnitude in computational efficiency has been achieved.

6. Conclusion

In this paper, we proposed a near-field ellipsometry method to characterize dielectric constant and thickness of thin film. The method has demonstrated nano-scale lateral resolution by leveraging the capability of SNOM in overcoming the diffraction limit. It further exploited the capability of SNOM in simultaneous measurements of the topographical and optical characteristics of a sample. By fusing the topographical information with the ellipsometric optical measurements, the characterization is shown capable of correcting artifacts and robust to noises. In addition, the calculation load can also be reduced significantly by using the fusion. Thus, the near-field ellipsometry is expected to be a promising technique for nano-scale thin film characterization.

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Standard derivation	The 2nd implementation with noise	The 1st implementation with noise on SNOM	The 1st implementation with noise on AFM	The 1st implementation with noise on both SNOM and AFM
δ_n ²	0.607	0.0049	0.0359	0.0458
δ_d ²(nm ²)	410.775	0	93.5211	114.7463

Near-field ellipsometry for thin film characterization

Abstract

1. Introduction

2. Theoretical formulation of near-field ellipsometry

2.1. Transmission coefficients for ellipsometry

2.2. Near-field elipsometry

3. Extension to multi-layer thin film

4. Simulations for thin film characterization

4.1. Setup of near-field ellipsometry

4.2. Characterization result of a single layer thin film

4.3. Performance of near-field ellipsometry in the presence of noises

4.4. Characterization of multi-layer thin film

5. Experiment results

6. Conclusion

References and links

Cited By

Figures (6)

Tables (1)

Equations (33)

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