1. Introduction

In the semiconductor and flat panel display industry, fabrication processes are composed of continuous deposition and etching to integrate dielectric, semi-conducting, and metallic films on a Si substrate or transparent glass. The metrology of thickness and optical constants for these films are essential to increase yields and product qualities [1]. Among the many optical metrology tools, ellipsometry is a well established and powerful tool for characterizing the optical properties with astonishing accuracy and precision [2]. However, its lateral resolution of the most versatile technique is poor due to its larger beam size than the pattern size of sample. Thus, a number of different research groups have developed a spatially resolved imaging ellipsometer (IE) to study a two-dimensional thickness profile along with optical properties of thin film [3–5]. In the IE, the reflected lights from a sample surface are detected with CCD (charge coupled device) or CMOS (complementary metal oxide semiconductor) instead of a PD (photodiode) or PMT (photomultiplier tube) measuring an averaged light beam intensity. Also an expanded light beam of 10–25 mm in diameter is used to illuminate a wide area of sample. Although the two-dimensional thickness distribution of thin film structures could be determined by analyzing the reflected lights with null and off-null method, for more quantitative information one has to perform additional null and off-null procedures and the spatial resolution of this optical system is limited by a pixel size of two-dimensional array detector [5]. Thus, to obtain more spatially resolved data, an imaging microellipsometer (IME) with microscope objective lens placed in the reflected light beam path of the analyzer arm has been suggested [6–8]. However, if one prefers to obtain the focused image of the overall field of view, one must move the sample in its plane because image plane is not perpendicular to the direction of the reflected light. If one chooses a high NA objective to obtain a higher spatial resolution, collision between sample and objective lens is occurred as the focal length of the objective becomes short. However in spite of these limitations, IE and IME are suitable for metrology of two-dimensional thickness distribution and visualization of biomolecular interactions [5, 8].

In the viewpoint of industrial application, patterned structure size for metrology recently down to 45 µm² is deposited on the dummy region, which is placed between chip structures [1]. As the dummy gap size becomes small, smaller probe beam size is required. It also needs to be compact size attachable within fabrication processes for real-time metrology, moreover, to be novel ellipsometric data acquisition method for monitoring optical characteristics of sample. However traditional ellipsometer employing a goniometer to adjust an incident angle is too bulky to install in the conveyor chamber and it is hard to move at fast for multi-point measurement with high scanning speed. Thus, other researchers have introduced a microellipsometer (ME) with an objective lens placed in the normal direction to the sample surface [9–12]. Microellipsometer with radial symmetry has been suggested to measure thin film thickness within sub-µm region of sample [10]. A high spatial resolution imaging ellipsometry with 0.5 nm thickness sensitivity was also reported by using rigorous coupled wave analysis (RCWA) algorithm [11]. There are some approaches to obtain ellipsometric data from an intensity image at the exit pupil plane of objective which is termed with a beam profile reflectometer or ellipsometer [13, 14].

In this work, we adopt aME type for satisfying the industrial application needs and introduce a novel peripheral data acquisition method for aME hardware scheme. This measurement technique enables one to determine ellipsometric parameters within an extremely narrow region of sample and to monitor thin film thickness in real-time due to no moving optical components.

2. Microellipsometer

2.1. Theoretical background

To minimize a probe beam size into sub-µm and to achieve ellipsometric data at multiple angles of incidence, we use a high NA objective lens and arrange it in the normal direction to the sample surface. After linearly polarized lights passing through an objective, they strike a sample surface with various angles of incidence as shown in Fig. 1(a). A maximum angle of incidence (θ _max) is determined by the inverse sine of the NA of an objective. The focal length (f) becomes shorter as the NA of objective increases to achieve a high spatial resolution. However the collision between sample surface and objective is not occurred because they are perpendicular to each other. We can obtain larger maximum incident angle by the relationship of NA=sin(θ _max). If using an objective lens with NA=0.9, one can achieve a maximum angle of incidence of 64.16°. A dielectric material such as glass with a refractive index of 1.5 has a Brewster angle of 56.3° and a gold with a complex refractive index of 0.35-j2.45 has a pseudo-Brewster angle of 67.9° at the wavelength of 546.1 nm. Therefore a high NA objective enables to measure ellipsometric data with better sensitivity, because there are no background Fresnel contribution from the substrate at this angle [4, 5].

 

Fig. 1. Basic principle of microellipsometer: (a) Multiple rays are refracted at different angles of incidence by an objective lens (b) Polarization states of input and output beam are decribed in the upper and lower semicircle, respectively

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If the transmittance axis of linear polarizer is aligned at x axis, the polarization states of light beam incoming into an objective and reflected off from a sample are described in the upper and lower semicircle of Fig. 1(b), respectively. In the upper semicircle, it is noted that the x axis contains p-polarized lights in contrast to y axis containing s-polarized lights due to the definition of incident plane. Moreover, the rays incident into the region between x and y axis have different p-/s-wave component ratios because of the rotating effect of an incident plane according to the azimuth angle (φ) in Fig. 1(b). In polarization microscopy, this phenomenon is referred to as a Fresnel effect and is treated as a nuisance because it causes four-corners problem [15]. In microellipsometer system, however, incident plane rotating effect plays a role of rotating polarizer just like typical rotating polarizer ellipsometer. This is a key algorithm of real-time ellipsometric data acquisition method without moving parts. Thus, if the light intensity profile is measured as a function of an azimuth angle (φ) at the same radius, one can calculate ellipsometric parameters of the sample structures. In addition, by changing the radius, one can obtain various ellipsometric data at multiple angles of incidence.

2.2. Jones matrix calculation

To explore a proposed principle in this paper, Jones formalism can be utilized to calculate a light intensity in optical detector. Optical components and coordinates system are described in Fig. 2.

 

Fig. 2. Schematic of optical components and coordinates for microellipsometer

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A beam from a white light source is typically randomly polarized. It can be expressed as follows,

This ray becomes a linearly polarized light after passing through a linear polarizer. The Jones matrix of polarizer is given as Eq. (2),

Here, P is an azimuth angle of polarizer’s transmittance axis with respect to the reference x axis and R(P) is rotating matrix with rotation angle P as follows,

Because a ray has a different plane of incidence as an azimuth angle changes, another rotation operation R(P) is taken to consider the rotating effect of plane of incidence for each ray. A ray reflected off a sample surface is recollected by the objective. In this optical configuration, a pure geometry effect has to be included in the Jones matrix. This effect can be expressed as R[-(φ+π/2)]T_M, where

is a reflection operation matrix to reorientate into the reference x axis [11]. Finally, this ray arrives at the detector after passing the analyzer with a Jones matrix of

Here, A is the azimuth angle between the transmittance axis of the analyzer and the reference x axis. The electric vector of this ray having an angle of incidence of θ and passing a plane of incidence of the azimuth angle φ at the detector can be expressed as

Here, T_S(θ) is a Jones matrix for isotropic sample and it can be given as

where r_p(θ) and r_s(θ) are the Fresnel reflection coefficients for p- and s-wave, respectively. As mentioned in section 2.1, if the transmittance axes of polarizer and analyzer are parallel to the reference x axis, i.e., choosing P=0 and A=0, the intensity I(θ,φ) detected by a CCD detector can be obtain with the relationship of I(θ,φ)=|E_out(θ,φ)·E*_out(θ,φ)| as follows.

where I ₀ is the average intensity,

and {α₂(θ),α₄(θ)} are the second- and fourth-order normalized Fourier coefficients,

Ellipsometric parameters {Ψ,Δ} are defined by the Fresnel reflection coefficients {r_p,r_s} as shown in Eq. (11),

In these equations, parameters {Ψ,Δ,r_p,r_s} are the physical quantities related to the angle of incidence, however, index(θ) is omitted throughout this article for simplicity. From Eqs. (8)–(11), it is clear that if one calculates normalized Fourier coefficients from intensity profile in the same radius according to azimuth angle(φ), ellipsometric parameters {Ψ,Δ} can be estimated by using the relation of Eqs. (10a,b).

2.3. Experimental setup

The optical system layout of microellipsometer with a high NA objective lens to minimize probe beam size is shown in Fig. 3. White light source is transmitted through an optical fiber and becomes the plane wave by collimating lens. A laser line filter(FL632.8-1, Thorlabs) is employed to make monochromatic light source. This filter has a optical performance such as center wavelength of 632.8 nm±0.2 nm and a FWHM (full width half maximum) of 1 nm±0.2 nm. A neutral density filter is used to adjust a light intensity for preventing a CCD detector from saturation. After passing through a Glan taylor polarizing cube (CGTP-20, Lambda Research Optics), a linearly polarized lights are focused on the sample surface by a Nikon CFI LU Plan Apo Epi (100×and a NA of 0.9). After reflection from the sample, the light is recollected by the objective and steered to the analyzer arm by the beam splitter. To minimize a reflection difference between p- and s-polarized light in the beam splitter, laser line non-polarizing cube beam splitter (10BC16NP.4, Newport) is used. Finally, a CCD camera (12bit, Uniqvision) captures an image on the exit pupil plane of objective.

 

Fig. 3. Schematic of optical layout in microellipsometer

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The exit pupil plane imaging has been typically used in polarization microscope and adopted in other microellipsometer [11, 13]. Traditionally, observation with a polarization microscope have been categorized under orthoscopic and conoscopic. Conoscopic observations, especially, involve illuminating a crystalline surface with a cone of light then imaging the exit pupil of the objective lens. This mode of observation is used for characterizing the crystal’s ellipsoid of birefringence and for identifying its optical axes [15]. Thus, pupil plane imaging provides for maximum light collection efficiency and it is suitable for detecting polarization state changes after reflection on the sample.

3. Application demonstration

3.1. Thin film measurement and results

SiO₂(N=1.460) thin film sample on the Si(N=3.875-j0.0156) substrate is utilized to demonstrate a proposed ellipsometric data acquisition method in this paper. Figure 4(a) shows two-dimensional intensity distribution based on theoretical calculation of Eq. (8) and Fig. 4(b) shows exit pupil plane image of SiO₂ thin film sample of 32.9 nm thickness on the CCD detector. It is noted that x axis direction contains a p-polarized lights, y axis containing s-polarized lights, because the transmittance axis of polarizer and analyzer is aligned to x axis. Thus light intensities on the y axis are slightly brighter than that of x axis. By obtaining the light intensity according to azimuth angle (φ) as shown in Fig. 4(b), one can calculate ellipsometric parameters {Ψ,Δ} from normalized Fourier coefficients of measured light intensities. Furthermore, by changing a radius of circle, various {Ψ,Δ} at the multiple angles of incidence can be achieved.

For the SiO₂ thin film sample of 32.9 and 53.6 nm thickness, theoretical data {α ^Theo ₂,α^Theo ₄} obtained in Eq. (10) and measured normalized Fourier coefficients {α^Exp ₂,α^Exp ₄} are depicted in Fig. 5. Here, solid- and dotted-line mean theoretically calculated normalized Fourier coefficients; circle and rectangle points represent experimental data for 32.9 and 53.6 nm thickness sample, respectively. Each angle of incidence θ_i (measured from the center of circle) is determined with the relationship as follows,

where ρ_max is the maximum radius of the exit pupil image and sin(θ _max) is the NA of the objective. In the theoretical and experimental results, two different thickness sample data become apart from each other as an angle of incidence is increased. It means that this microellipsometer has higher sensitivities near the maximum incident angle, because the pseudo-Brewster angle of Si(N=3.875-j0.0156) substrate is about 75.5° for 632.8 nm wavelength light. It is reason that a large NA(=0.9) objective lens is adopted in this system. There are some discrepancies between theoretical and experimental data. The error factors and calibration method will be discussed in next section.

 

Fig. 4. Two-dimemsional intensity distribution on the exit pupil plane of an objective lens: (a) theoretical results (b) experimental results with SiO2 thin film of 32.9 nm thickness

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Fig. 5. Normalized Fourier coefficients {α₂,α₄}: solid- and dotted-line mean theoretically calculated normalized Fourier coefficients {α^Theo ₂,α^Theo ₄}, circle and rectangle points represent experimental data {α^Exp ₂,α^Exp ₄} for 32.9 and 53.6 nm thickness SiO₂ sample.

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3.2. Error analysis and calibration

These errors mainly come from non-normal incidence on the optical surfaces and birefringence effect of the optical components [10, 12]. The former is caused from different transmittance properties of p- and s-polarized lights passing through an objective lens. The latter is induced from physical stress to mount the components such as imaging lens and polarization parts into a barrel or housing.

To compensate for these error factors, Linke et al [12] inserted additional Jones matrix containing calibration parameters {Ψ_obj,Δ_obj} before and after Jones matrix of sample. Similar to this calibration technique, calibration Jones matrix TC(θ) is introduced into the total Jones matrix of optical system. Therefore Eq. (6) can be replaced with

where T_C(θ) is defined as

The calibration parameter ε represents transmittance coefficient ratio of p-polarized light with respect to s-polarized light and δ is a phase delay caused from birefringence effect. Also these parameters {ε,δ} are functions of an angle of incidence, however index(θ) is omitted for simplicity. Normalized Fourier coefficients {α^Cal ₂,α^Cal ₄} including calibration parameters are given by

To determine each compensating value {ε,δ} in the range of overall incident angles, least square error (LSE) calculations are performed with following equation,

here, parameters {α^Cal ₂,α^Cal ₄} and {α^Exp ₂,α^Exp ₄} are normalized Fourier coefficients based on the modified theoretical model of Eq. (13) and deduced from experimental results, respectively. Index n is the number of reference sample used for calibration. It is noted that additional calibration parameter θ is inserted to compensate the finite pixel size effect of CCD detector, because light intensity signal from one pixel is consisted with rays of different incident angles within a pixel size. It means that calibration parameter θ represents an averaged incident angle of selected pixel. Also, analytically determined incident angle from relationship of Eq. (12) can be applied to the back focal plane of objective lens. That is, the distribution of light in the back focal plane of the objective is the Fourier transform of the distribution in the front focal plane. Therefore, in the back focal plane, the plane-wave components incident upon the sample are spatially separated. Thus another parameter θ is introduced to compensate the above problems comparing with Linke et al.

 

Fig. 6. Calibrated {α^Cal ₂₍₄₎, –+–, –∙–, –×–} and experimental {α^Exp ₂₍₄₎, □, ◦, ∆} normalized Fourier coefficients for 53.6, 32.9, and 2.1 nm thickness SiO₂ sample, respectively

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After calculating {θ,ε,δ} parameters with SiO₂ thin film thickness sample of 32.9 and 53.6 nm, LSE values can be obtained under 1×10^-6~10^-5 in the overall range. Calibrated and measured normalized Fourier coefficients for 32.9 and 53.6 nm thickness SiO ₂ sample are depicted in Fig. 6. From these results, after measuring experimental normalized Fourier coefficients α^Exp ₂₍₄₎, one can deduce an ellipsometric data {Ψ,Δ} from Eq. (15) including calibration parameter {θ,ε,δ}. Additional experimental data for 2.1 nm SiO₂ thickness sample are also shown in Fig. 6. Calibrated data of 2.1 nm thickness sample are compensated by utilizing calibration parameters {ε,δ}, which are obtained from two kinds of SiO ₂ thickness sample in previous compensation process. We found appreciable deviations between the experimental and calibrated data in the large angle of incidence range. In this range of incident angle, LSE values have 1×10^-5. These deviations are caused from more oblique incident rays at these angles than those of small angle of incidence region, that is, outside pixels in the exit pupil plane image detect an amount of ray. Therefore, the representative properties of calibration parameters {θ,ε,δ} decrease in these angle range.

Various SiO₂ thin film measurement results are summarized in Table 1. These experimental results show that a novel peripheral data acquisition method enables microellipsometer to measure ellipsometric parameters {Ψ,Δ} and calibration theory is suitable to this optical system.

Table 1. Measurement results for various SiO2 thin film sample with microellipsometer

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However, it is clear that thickness errors increase at the large incident angle range. Thus these pixel size effects have to be considered, we expect that these errors can be removed by inserting pixel size effect into Jones matrix calculations in the future.

4. Conclusion

We introduced a microellipsometry using a high NA objective for measuring a thin film thickness at the sub-µmregion of sample and proposed a novel ellipsometric data acquisition method while polarization optical parts are not rotating, and then we showed that harmonic intensity profile, which is measured around azimuth angle direction, can be interpreted as ellipsometric parameters at multiple angles of incidence. Thus this technique allows in-line monitoring for the thin film thickness. However, there are some errors between experimental and theoretical data until now. If pixel size effect might be considered, we will get more reasonable results in the near future.