1. Introduction

There have been great demands on precise and high-speed thin film and nanopattern measurements for numerous nanoscale manufacturing processes in the display and semiconductor fields. As an alternating nanometrology tool to conventional non-optical nanoscopy such as scanning electron microscopy (SEM) and atomic force microscopy (AFM), optical metrologies have been playing a crucial role in thin-film thickness and 3-D nanopattern profile measurement [1–3]. Spectroscopic ellipsometry (SE) is among the most impactful optical nanometrology. SE determines optical and structural characteristics through an assessment of ellipsometric parameters (Ψ, Δ) signified by the polarization state of the light reflected off a specimen [4–6]. These parameters are typically acquired using an active polarization modulation incorporating a mechanical rotating unit or an electrical photo-elastic modulator. Such active configuration provides a highly precise measurement capability but needs to be applied over time. For real-time measurements necessary for in-line monitoring, various snapshot SE configurations that operate with a passive polarization modulation have been proposed, including a thick retarder [7–11], multi-channel sensing modules [12–16], and a monolithic polarizing interferometer [17–20].

Despite such numerous studies on SE for high precision and real-time in-line monitoring, conventional SE systems are mainly designed for measuring a single spot only and have limited spatial resolution. To overcome the spatial resolution limitation of non-imaging SE, imaging ellipsometry (IE) that typically employs high-magnification objectives and a 2-D imaging sensor has been studied to explore ellipsometric microscopic image characterizations in biochemical and material science fields [6,21,22]. Typical imaging ellipsometry employs a null-type configuration in which a complex rotating mechanism of a polarizer or analyzer is required to find null positions, resulting in a relatively long measurement time although currently available commercial null-type IE can provide a high spatial resolution of less than 1µm. Recently, rotating compensator-based IE has become a standard configuration for IE instead of finding the null positions due to its benefit in measurement speed [23,24]. In addition, a rotatable Offner system has been proposed with a large field of view (FOV) to overcome the defocus and distortion in the edge part caused by the oblique incidence configuration of the conventional IE [25]. To obtain imaging SE data for a certain 2-D area, eventually, an additional wavelength scanning process (i.e., grating or band-pass filter, etc.) is required, and it also increases the total imaging SE measurement time. As another alternative approach to implementing an imaging SE system, a hyperspectral sensing scheme employing a spectrograph and a 2-D array sensor for spatio-spectral ellipsometric imaging has been studied although those attempts still have limitations in the lack of either measurement speed or rigorous system accuracy [26,27]. Recently, our group has proposed a dynamic spectroscopic imaging ellipsometer (DSIE) employing a monolithic polarizing interferometer [28,29]. We have successfully demonstrated that it can provide high-throughput large-scale Δ(k,X,Y) mapping capability of patterned semiconductor wafer with a spatial resolution of around 50 × 50µm². To meet the stricter measurement system reliability required for measurement and inspection (MI) tools in semiconductor manufacturing, however, the previous single-channel DSIE system needs to employ an additional compensation channel to overcome its inherent shortcomings in the long-term stability due to the imperfectness of the practical monolithic polarizing interferometer module [20].

In this paper, a robust dynamic spectroscopic imaging ellipsometer system based on a monolithic Linnik-type polarizing interferometer is described. We claim the importance of employing a Linnik-type monolithic interferometer module combined with a compensation channel by which the long-term stability of a DSIE can be attained in the general environment with various disturbances. Detail compensation procedures including theoretical derivations and experiments for implementing the robustness of DSIE are provided in comparison with the previously suggested Michelson-type single-channel DSIE scheme. Moreover, we address a global mapping phase error compensation method capable of accurate SE mapping for large-scale applications. To demonstrate the performance of the proposed dual-channel Linnik-based DSIE system, we have conducted a whole wafer mapping experiment of a 12-inch thin film wafer.

2. Methodology

2.1 Instrumentation

The proposed robust DSIE system based on a monolithic polarizing Linnik interferometer is depicted in Fig. 1. For an object measurement channel, a 2-D hyperspectral imaging sensor is employed to obtain a spatio-spectral interference fringe from a line-shaped illumination incident on a measured specimen. In contrast to the previous single sensing channel scheme, we have added a 1-D single spectrometer as a compensation channel to compensate for the unstable phase drift caused by ambient temperature change. A 600µm diameter optical fiber connected to a 20W white LED source generates a diverging beam, and it passes through a lens (L_in) to create a collimating wave, which enters a linear polarizer (P_in) aligned at 45 degrees. A linearly polarized light (E_in) is incident on the mechanically fixed monolithic interferometer, which consists of a non-polarizing beam splitter (BS₁), two linear polarizers (P_1,2) precisely aligned at 0 and 90 degrees, two focusing lenses (L_1,2), and two plane mirrors (M_1,2). The two beams separated by the non-polarizing beam splitter (BS₁), respectively pass through the linear polarizers P₁(0°) and P₂(90°) and then are focused to the center of the plane mirrors (M_1,2) through the lenses (L_1,2). The light reflected from each plane mirror again passes through the focusing lenses, and travels back maintaining the collimation. The distances between the mirror and the lens in both paths are aligned with the same focal length (F = 50 mm), and both beams are precisely aligned on-axis by a 3-axis kinematic mount coupled to the M₁.

 

Fig. 1. Schematic of the proposed robust dynamic spectroscopic imaging ellipsometer based on a monolithic polarizing Linnik interferometer (for more clear understanding about the optical system, optical components have been enlarged in size in comparison with that of the XY stage).

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Two perpendicularly linear-polarized lights after the monolithic Linnik interferometer are separated into two paths by another non-polarizing beam splitter (BS₂). One beam (E_out1) is incident to a compensation channel employing a spectrometer (SP) that has a spectral range of 530-680 nm with a spectral resolution of 2.6 nm. The compensation channel is applied for real-time compensation of the spectroscopic ellipsometric phase drift in the monolithic Linnik interferometer which is caused by external disturbances including temperature variation and vibration. The other collimated wave (E_out2) is changed to a line-shaped beam by a cylindrical lens (CL), with a focal length of 100 mm. Then, after the reflective measured specimen, the reflected beam enters the object measurement channel employing a hyperspectral imaging (HSI) sensor module coupled with a telecentric lens (L_out). The hyperspectral imaging module is comprised of a spectrograph with a slit size of 14.6mm × 30µm, a monochrome complementary metal-oxide semiconductor (CMOS) camera with a 6.5 × 6.5µm pixel size, and 2160 × 2560 pixels. Continuous spatio-spectral intensity data are dynamically acquired by the hyperspectral imaging module with a spatio-spectral field of view of 11.5 mm (spatial range) × 170 nm (spectral range from 500 nm to 670 nm). Note that the effect of the AOI variation due to the employment of the cylindrical lens is ignorable since the AOI spread incident on the slit of the spectrograph is within 0.1 degrees.

2.2 Theoretical background

An inlet wave before the Linnik-based monolithic polarizing interferometer E_in(k) and outlet waves E_out1(k) and E_out2(k) separated by the non-polarizing beam splitter (BS₂) are defined as follows, respectively [17,18].

And,

Here, k is a wavenumber defined as 2π/λ, and i represents a complex operator meaning i²= -1. u, v and ξ, η denote amplitudes and phases of the inlet wave along the x- and y-axis, respectively. E₁(k) and E₂(k) correspond to the perpendicularly linear-polarized beam traveling to each arm of the monolithic interferometer module with the p- and s-polarization and are represented by the Jones matrix as follows.

Here, J_BS1,2, J_M1,2 and J_P denote the Jones matrix of the non-polarizing beam splitters, the plane mirrors, and the linear polarizers, respectively. z₁ and z₂ represent optical path lengths between the separation plane of the non-polarizing beam splitter (BS₁) and the plane mirrors M₁ and M₂. All optical components in both arms of the monolithic interferometer are mechanically fixed like a rigid optical device to maintain a constant optical path difference (OPD) z₀= z₁ − z₂. In this paper, we set the OPD to be around 40µm. Note that, we newly define unknown amplitude and phase terms of E₁(k) and E₂(k) as u′, v′ and ξ′, η′, respectively.

After traveling through the monolithic interferometer, the two linearly polarized complex waves that are perpendicular to each other are separated into E_out1(k) and E_out2(k) by the non-polarizing beam splitter (BS₂) as denoted in Eq. (2). E_out1(k) is incident to the spectrometer sensor in the compensation channel as described in Eq. (5) before passing through an outlet linear polarizer aligned at 45 degrees.

Here, $E_{SP}^{1,2}(k)$ represent scalar forms of the complex waves with p- and s-polarization, respectively, which are generated independently by the monolithic polarizing interferometer. Note that E_out1(k) entering the compensation channel only without experiencing the measured object allows us to monitor instability occurring in the monolithic polarizing interferometer due to external disturbances. The line-shaped beam generated after E_out2(k) passes through the cylindrical lens (CL) is incident on the measured object at an incidence angle of θ_AOI as depicted in Fig. 1. The reflected wave entering the object measurement channel before passing through another outlet linear polarizer aligned at 45 degrees can be represented as follows.

Here, $E_{HSI}^{1,2}(k,x)$ indicate scalar forms of the complex waves with p- and s-polarization containing the spectroscopic ellipsometric information of measured specimen, respectively. The complex wave measured by the hyperspectral imaging module can be represented as a function of both spectral(k) and spatial(x) variables. x represents the spatial coordinate along the line-shaped illumination direction on the measured object. Note that we need to define new optical path length z₁′(x) and z₂′(x) with the spatial coordinate x to analyze the optical path difference in the monolithic polarizing interferometer spatially. |r_p,s(k,x)| and |δ_p,s(k,x)| are the amplitude and phase changes for the p- and s- waves caused by the spectroscopic ellipsometric characteristics of the measured object, respectively. As the waves entering the object measurement channel experience different polarization circumstances, we define new unknown amplitude and phase terms of $E_{HSI}^1(k,x)$ and $E_{HSI}^2(k,x)$ as u′′, v′′ and ξ′′, η′′, respectively. Importantly, in addition, ε_p(k,X,Y) and ε_s(k,X,Y) representing a global mapping phase error Ε(k,X,Y) = ε_p(k,X,Y) − ε_s(k,X,Y) need to be added in the phase terms to describe the real system correctly. They denote the unknown phase terms that are caused by the 3-D pose of the reflective surface of the measured object at a specific point of (X,Y) which represents the coordinates of the scanning and spatial axis as illustrated in Fig. 1. Note that the global mapping error compensation is critical in attaining the high accuracy of the proposed robust DSIE system as the incidence angle of light entering the inlet slit of the hyperspectral imaging module can be varied by the object 3-D pose change. In this study, we show that the unwanted 3-D pose of the measured object mainly comes from the inherent runout error of the XY linear stage used for the SE mapping. To overcome this inherent error of the mapping stage that affects the final accuracy of the proposed ultrafast SE mapping system, the global mapping phase error compensation should be performed as a significant step. The detailed compensation process to remove the global mapping phase error Ε(k,X,Y) is described in section 2.4. Note again to avoid confusion that the uppercase X and Y defined in Fig. 1 indicate the coordinate axes of the XY-linear stage, while the lowercase x corresponds to the spatial-axis of the spatio-spectral image.

The complex waves acquired by the two sensors (SP and HIS) after passing through the outlet linear polarizers P_out1 and P_out2 for the dual-channel scheme are expressed as Eq. (7) and Eq. (8), respectively.

Eventually, the spectral interference signals $I_{SP}^{comp}(k)$ and $I_{HSI}^{Object\,(X,\,Y)}(k,x)$ acquired by the dual channels are represented as follows, respectively.

Here, α₁(k) and β₁(k) correspond to the square roots of the DC terms of the p- and s- polarized complex waves traveling to compensation channel. γ₁(k) represents a spectral coherence function of the compensation channel. The variables α₂(k,x), β₂(k,x) and γ₂(k,x) are corresponding definitions for the object measurement channel. $\Phi _{HSI}^{Object\,(X,\,Y)}(k,x)$ and $\Phi _{SP}^{comp}(k)$ represent the spatio-spectral and spectral phase function extracted by the hyperspectral imaging and the 1-D spectrometer sensor, and they can be described as follows.

Here, Δ(k,x)=δ_p(k,x) − δ_s(k,x) represents the phase difference between the p- and s-polarization created by the measured reflective thin film object, and a reliable and accurate Δ(k,x) is what we need to extract from the proposed system eventually. As denoted, z₀′(x) is the optical path difference of the monolithic interferometric module meaning z₁′(x) − z₂′(x). Note here again that x is the spatial coordinate used to describe the spatial variation in analyzing the monolithic interferometric module, while Y represents a spatial coordinate indicating a specific point of the measured object along the x-direction. X denotes a spatial coordinate indicating the scanning direction of the XY stage as illustrated in Fig. 1.

For a normal operation of the proposed dual-channel scheme, to begin with, the two spectral sensors (spectrometer and hyperspectral imaging sensor) need to be calibrated spectrally by employing a regression algorithm with the peak wavelengths where constructive interference occurs. Figure 2(a) is a spatio-spectral interference fringe observed by the 2-D hyperspectral imaging module for a bare Si wafer at an incidence angle of 45 degrees. After the spectral domain calibration, the peak wavelengths between the spectral interference signals from the 1-D spectrometer used for the compensation channel and the spectral line profile of the spatio-spectral interference fringe at the center coordinate of the spatial axis in Fig. 2(a) get well-matched as depicted in Fig. 2(b).

 

Fig. 2. Wavelength calibration between HIS and SP: (a) a spatio-spectral interference fringe of the measured bare Si acquired by the HSI in the object measurement channel, and (b) spectral interference signal comparison between the SP data in the compensation channel and the line profile (x = 6 mm) of the HSI data in Fig. 2(a) after wavelength calibration process.

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The spectral phase functions $\Phi _{HSI}^{Object (X,\,Y)}(k,x)$ and $\Phi _{SP}^{comp}(k)$ described in Eq. (11) and Eq. (12) are extracted as illustrated in Fig. 3(c) and 3(d) from the spatio-spectral interference fringe and the spectral interference signal by applying the Fourier transform method as illustrated in Fig. 3 [30,31]. Figure 3(a) and Fig. 3(b) illustrate the spatio-spectral frequency domain and the spectral frequency domain data obtained after applying a 2-D and a 1-D Fourier transform to the measured raw data depicted in Fig. 2, respectively. Then, $\Phi _{HSI}^{Object (X,\,Y)}(k,x)$ and $\Phi _{SP}^{Air}(k)$ are extracted by applying an inverse Fourier transform to the selected AC term as illustrated in Fig. 3(a) and 3(b), and eventually by unwrapping the wrapped phase functions, as depicted in Fig. 3(c) and Fig. 3(d). As illustrated in the blue solid lines of Fig. 3(b) and Fig. 3(d), the spectral line profile of Fig. 3(a) and Fig. 3(c) at the central spatial position are well-matched with the data (blue dotted line) extracted by the spectrometer sensor in the compensation channel.

 

Fig. 3. Phase function extraction results: (a) spatio-spectral frequency domain data after 2D FFT, and (c) spatio-spectral ellipsometric phase map after applying an inverse 2D FFT, which is extracted from the HSI-based object measurement channel. (b) spectral frequency domain data after 1D FFT, and (d) spectral ellipsometric phase function after an inverse 1D FFT of the SP-based compensation channel (dotted lines represent line profiles depicted in Fig. 3(a) and 3(c)).

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2.3 Phase compensation (1): long-term stability (LTS) problem

The dynamic spectroscopic imaging ellipsometer (DSIE) system employing a monolithic polarizing interferometer can provide high-throughput large-scale Δ(k,X,Y) mapping capability with a high spatial resolution of around 50 × 50µm² [28,29]. As reported recently, a monolithic polarizing interferometric module is inherently affected by ambient temperature change [20], and the previous DSIE scheme with a single sensing channel has the same long-term stability problem in ambient conditions. To overcome the instability problem, in this study, we add a compensation channel in the same way as reported in Ref. [20]. Due to the high complexity of temperature dependency of the spatial OPD z₀′(x) causing an unwanted shift in the measured Δ(k,x), however, we claim that such a simple addition of the compensation channel employing a 1-D spectrometer sensor cannot be a compensatory solution for solving the long-term stability problem of the DSIE system as a Michelson-type monolithic scheme can cause unpredictable and unreproducible angular deflection, resulting in the unwanted spatial variation of the optical path difference z₀′(x) between the two arms. Such micro-level tilt variation of the mirrors in the Michelson-type monolithic interferometer module can occur in a real system. We emphasize that we can solve the long-term stability problem of the DSIE system by employing a simple palm-size spectrometer sensor rather than another complicated bulky hyperspectral imaging sensor in the compensation channel. In this section, in addition, we show that a Linnik-based monolithic polarizing interferometer should be employed for a highly robust and reliable DSIE system through a quantitative analysis on the effect of the spatial optical path difference z₀′(x) variation for high robustness of DSIE system. For this, we derive a rigorous formula by which we can compare numerically how much degradation can take place in the long-term stability between the Michelson- and the Linnik-type monolithic interferometric scheme.

To measure the spatio-spectral phase difference between the p- and s-polarization Δ(k,x) containing the object spectroscopic ellipsometric information from the spatio-spectral phase function $\Phi _{HSI}^{Object (X,\,Y)}(k,x)$ in Eq. (11), the unknown phase terms 2kz₀′(x) + (ξ′′ − η′′) and E(k,X,Y) should be removed. For this, in this study, we measure a well-known and easily-gettable specimen like a bare Si as a reference. Air reference also can be employed for compensation as reported in the previous study [20]. However, we employ a bare Si wafer as the reference since entire bare Si wafer mapping data are needed to compensate for the global XY stage mapping error E(k,X,Y) that is addressed in section 2.4 as well as the bare Si reference is even easier to apply for the phase compensation of 2kz₀′(x) + (ξ′′ − η′′) than the air reference.

As the first phase compensation step described in this section, by measuring a bare Si wafer at an incidence angle of θ_AOI, we extract the spectral phase function $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)$ in Eq. (13) from a spatio-spectral interference signal $I_{HSI}^{Ref (X,\,Y)}(k,x)$ measured at a specific position of the XY-linear stage [28,29]. The compensation for E(k,X,Y) is addressed in section 2.4 in detail. As the next step, the main compensation for the long-term instability caused by the spatial dependency of the optical path difference z₀′(x) variation is performed by extracting the spectral phase function $\Phi _{SP}^{Ref}(k)$ from the spectral interference signal $I_{SP}^{comp}(k)$ indicated in Eq. (14), which is acquired by the spectrometer sensor in the compensation channel.

As the OPD z₀′(x)|_t at time t is varied not only temporally but also spatially in contrast to the dynamic spectroscopic ellipsometer system where the spatial OPD variation is not significant [20]. The OPD z₀′(x)|_t of the DSIE system can be changed both temporally and spatially due to the imperfect rigidity of the mechanical mounts and the refractive index temperature dependency of the cemented anisotropic optical layer of the non-polarizing beam splitter (BS₁) used for the monolithic interferometric module. To overcome such a complicated phase compensation problem, the employment of another hyperspectral imaging module in the compensation channel can be considered. However, the total system gets too bulky and costly. In this study, we suggest a compact and inexpensive approach to solve the long-term stability problem of the DSIE system by combining a Linnik-type monolithic interferometer with a compact palm-size spectrometer sensor in the compensation channel.

Let us assume first that the OPD z₀′(x)|_t is varied just only temporally by external temperature variation, and it is constant spatially along the x-coordinate. The OPD z₀′(x)|_t at time t is not the same as that of when the reference bare Si is measured at time t = t₀ mainly due to the temperature dependency of the anisotropic optical layer of the non-polarizing beam splitter used for the monolithic interferometric module. In this assumption, as z₀′(x)|_t is regarded as a constant spatially, the temperature-dependent z₀′(x)|_t can be compensated in the same way as reported in the previous study [20] by duplicating the 1-D spectral phase function $\Phi _{SP}^{Ref}(k)\,{|_{t}}$ to the x-axis direction to create new 2-D spatio-spectral phase references $\Phi _{SP}^{Air}(k,x)\,{|_{t}}$ and $\Phi _{SP}^{Ref}(k,x)\,{|_{{t0}}}$ as denoted in Eq. (15). Eventually, the compensated $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$ for the object measured at time t can be obtained as described in Eq. (15). Note that the global mapping phase error term Ε(k,X,Y)=ε_p(k,X,Y) − ε_s(k,X,Y) denoted in Eq. (11) has not been removed yet in this stage as described in Eq. (15). In this section, we assume additionally that the XY-stage movements in the X- and Y-direction have translational motions only with no runout for all the coordinates of (X,Y), resulting in Ε(k,X,Y)=ε_p(k,X,Y) − ε_s(k,X,Y) in Eq. (15) becomes zero for both object and reference measurements.

Here, $\Phi _{SP}^{Ref}(k,x)\,{|_{{t = t0}}}$ and $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)\,{|_{{t = t0}}}$ represent spatio-spectral reference phase functions extracted from the compensation and the object measurement channel, respectively. The compensation channel measures air all the measurement time to monitor the instability of the monolithic polarizing interferometric module. Note that the retardance of the non-polarizing beam splitter (BS₂) does not cause any measurement error as the retardation effect is corrected through the proposed compensation steps [20]. However, the OPD z₀′(x)|_t of the DSIE system can be varied not only temporally, but also spatially due to the imperfect rigidity of mechanical mirror mounts used for the Michelson-type monolithic interferometric module although they are firmly tightened in alignment process. Unreproducible 3-D angular deflection can occur unpredictably as the kinematic mounts coupled to two mirrors M₁ and M₂ are not free from temperature-dependency [32]. Thus, the OPD z₀′(x)|_t in Eq. (15) can be changed spatially as well by ambient temperature change in the Michelson-type DSIE configuration. As a result, the 1-D spectrometer sensor that measures spectral information only is unable to compensate for the spatially varying OPD changes of z₀′(x)|_t in $\Phi _{HSI}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$. Note that, $\Delta _{meas}^{Ref}(k,x)$ indicates Δ(k,x) measured by a commercial SE system that can provide a reliable data of a reference specimen such as a bare Si [18].

To minimize the spatial variation of z₀′(x)|_t, as illustrated in Fig. 4, a Linnik configuration is applied for implementing the proposed robust DSIE system. As depicted in Fig. 4(a), l₁(x) and l₂(x) indicate optical path lengths (OPL) corresponding to the distance between the split plane of the non-polarizing beam splitter (BS₁) and the 1_st surface of each focusing lens (L_1,2), respectively. As marked in Fig. 4(a), the optical path differences between the p- and s-polarizing modulation arms (A_1,2∼E_1,2) can be expressed as a function of x as in Eq. (16) [33].

 

Fig. 4. (a) A ray diagram of a monolithic polarizing Linnik interferometer and (b) simulated comparison of the spatially-varying OPD variation due to slight angular deflection of the mirror (M₁) between a Michelson- and a Linnik-type monolithic interferometer scheme.

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Note also that both focusing lenses should be of the same material to exclude any unwanted OPD z₀′(x)|_t variation caused by the mismatch of the refractive index change due to temperature variation. For simplicity in equation representations, we assume the mirror M₂ is firmly fixed and the normal plane of the mirror M₂ is perfectly parallel to the propagating wave direction while the mirror M₁ is tilted slightly.

Here, OPL_L1,2 represent optical path lengths of waves passing between the L_1,2 and the M_1,2. f(x) denotes a hyperbolic function of the convex surface of the lens, and we define z = f(x) < 0. n_l(k), t_c and f_b denote the refractive index, center thickness, and back focal length of the lens, respectively. θ_f= tan^-1x/f_b indicates the angle of the focused light on the z-axis, and ε_M1 means the deflection angle (tilted angle) of the kinematic mirror mount for M₁. Rays passing through the L_1,2 are incident at (f_b, 0). However, the coordinates on the mirror surface move by (a(x), b(x)) when angular deflection occurs. We have simulated the OPD change δz₀′(x)|_t=z₀′(x)|_t − z₀′(x)|_t = t0 when the kinetic mount has a deflection of 10µrad by using Eq. (16). The red solid line in Fig. 4(b) depicts δz₀′(x)|_t for a Michelson-based DSIE where no lenses are employed and collimated beams are reflected from plane mirrors. It varies a lot spatially while the blue solid line in Fig. 4(b) illustrating δz₀′(x)|_t for a Linnik-type DSIE has a little variance in the OPD change. This simulation results explains how the Linnik-type scheme can compensate for the complicated instability of the DSIE system in the spatial domain to obtain a reliable 2-D spatio-spectral map $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$ described in Eq. (15). Note that the proposed monolithic polarizing Linnik interferometer employs focusing lenses with a focal length of 50 mm (NA≈0.1), so it has a relatively high depth of focus (∼10um) unlike a commonly adopted high-magnification (NA > 0.5) Linnik interferometer. As a result, it is relatively insensitive to changes in focus position.

2.4 Phase compensation (2): Global mapping phase error

In the phase compensation described in section 2.3, a bare Si wafer has been measured at a specific coordinate (X₀,Y₀) of the XY-linear stage to calibrate the unknown phase terms in the assumption that the XY-stage movements in the X- and Y-direction have translational motions only with no runout for all (X,Y) coordinates. As diffraction angles from a grating employed for a spectral system can be deviated by the incidence angle change to the inlet slit of the spectral system, the measurement accuracy of the hyperspectral imaging system can be affected by the 3-D pose variation of the reflective surface of the measured object. For a large-scale full wafer measurement, the global mapping phase error Ε(k,X,Y) = ε_p(k,X,Y) − ε_s(k,X,Y) denoted in Eq. (11) should be calibrated for accurate measurements.

This section describes why the remaining global mapping phase error terms in Eq. (15) should be compensated for correctly. The global mapping phase error terms cannot be calibrated just by the $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)\,{|_{{t = t0}}}$ obtained at a specific position of the reference bare Si wafer since the global mapping phase error Ε(k,X,Y) is not constant, but is varied for different (X,Y) coordinates due to the XY-stage runout variation over the XY translational position. Thus, Ε(k,X,Y) should be compensated for by saving the measurement data of the entire reference bare Si wafer for all coordinates (X,Y) in advance as a pre-preparation step, by which the XY-stage runout error factor can be removed. Experimental details on how to compensate for the global mapping phase error term for a 12-inch full wafer measurement are described in section 3.2.

Figure 5(a) and 5(b) depict experimentally-obtained spatio-spectral 2-D ellipsometric $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)$ maps extracted from a specific position of the reference bare Si wafer before and after compensating for the global mapping phase error terms, respectively. To obtain the global mapping phase error compensation result in Fig. 5(b), $\Phi _{HSI}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$ and $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)\,{|_{{t = t0}}}$ in Eq. (15) should be extracted from the same location of the bare Si wafer. When $\Phi _{HSI}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$ and $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)\,{|_{{t = t0}}}$ are measured at different locations of the measured bare Si, we obtain Fig. 5(a). Figure 5(c) represents spatial line profiles selected from the extracted $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)$ in Fig. 5(a) and 5(b) at λ=560.4 nm, respectively. As illustrated in Fig. 5(a), unwanted residual unknown spectral ellipsometric phase term caused by the global mapping phase error is observed mainly along the spatial axis. In contrast, as in Fig. 5(b), we obtain highly accurate spatio-spectral spectroscopic ellipsometric phase map $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)$ after removing the global mapping phase error terms. Note that the unwanted 3-D pose of the measured object mainly comes from the runout error of the XY-linear stage, and it is crucial to compensate for the unwanted but inevitable XY-stage runout for reliable SE mapping.

 

Fig. 5. Spatio-spectral 2-D ellipsometric maps of a reference bare Si wafer (a) before and (b) after compensating for the global mapping phase error term. (c) Line profile comparison between the vertical dotted line (at λ=560.4 nm) in Fig. 5(a) and 5(b).

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3. Experimental results

3.1 Long-term stability (LTS) compensation

We have conducted an experiment to evaluate the temperature-dependent OPD instability compensation method first described in section 2.3. In this section, we show experimentally that a reliable long-term stability of the DSIE system can be achieved by employing a Linnik-type monolithic interferometric scheme combined with a 1-D spectrometer sensor in the compensation channel. To demonstrate the long-term stability evaluation of the proposed DSIE system, a transmissive measurement mode has been used where the measured object is air by setting the angle of incidence θ_AOI =90°.

Figure 6 illustrates the long-term stability measurement result (for 20 hours) obtained by using a DSIE system employing a Michelson-type monolithic interferometric module. Figure 6(b) shows that the Michelson-type interferometric scheme can cause unwanted spatial instability in measuring the ellipsometric phase map $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ although the compensation channel is applied. Figure 6(a) depicts an ambient temperature variation plot during the LTS experiment. While the ambient temperature is decreased by 1.7℃ over 20 hours, the measured $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ is varied by more than 300 degrees temporally with the non-uniform spatial distribution of around 70 degrees. Figure 6(c) reflects OPD changes δz₀′(x)|_t at t = 1hr (blue dotted line), 10hrs (red dotted line), and 20hrs (black dotted line) extracted from the slope of spectral phase functions. The star points in Fig. 6(c) indicate δz₀(x)|_t at t = 1hr (blue star), 10hrs (red star), and 20hrs (black star) extracted by the spectrometer sensor in the compensation channel. As shown in Fig. 6(c), the observed δz₀(x)|_t in the central spatial area is matched well with that observed by the 1-D spectrometer sensor in the compensation channel, and the δz₀′(x)|_t observed in the object measurement channel varies spatially up to 140 nm for 20 hours. From this result, we infer that the extracted $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ cannot be compensated for by only employing the 1-D spectrometer-based compensation channel only. Note that this OPD change comes mainly from the unwanted tilt motion of mirrors used in the monolithic polarizing interferometric module due to ambient temperature change.

 

Fig. 6. Long-term stability (LTS) experiment results of a Michelson-type DSIE (for 20 hours): (a) ambient temperature variation recorded by a thermometer for 20 hours, (b) measured spatio-temporal ellipsometric phase map $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ at λ=600.4 nm, and (c) corresponding OPD changes as a function of x occurred in the monolithic interferometric module.

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To overcome such limitations of the Michelson-type monolithic approach, we have employed a Linnik-type monolithic interferometer scheme as illustrated in Fig. 1. Figure 7 depicts the long-term stability result for 60 hours at θ_AOI=90° by using the Linnik-type DSIE system. Figure 7(a) and 7(b) show the long-term stability measurement results of $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ and $\Delta _{Compensated}^{Object}(k,x)\,{|_{t}}$ described in Eq. (15) for a specific wavelength of 600.4 nm, respectively. In comparison with Fig. 6(b), the spatio-temporal long-term instability of Δ(k₀,x)|_t depicted in Fig. 7(a) is almost perfectly uniform along the spatial axis, which means that the long-term stability problem of the DSIE system can be compensated for as illustrated in Fig. 7(b) just by employing the 1-D spectrometer sensor in the compensation channel. Figure 7(c) and 7(d) illustrate the line profiles of Fig. 7(a) and 7(b) at x = 1, 5.5, and 10.5 [mm], respectively. The black solid line in Fig. 7(c) represents the $\Delta _{SP}^{Air}(k)\,{|_{t}}$ extracted from the 1-D single spectrometer of the compensation channel. Also, the gray solid line indicates the ambient temperature and the scale is same as in Fig. 6 for clear comparison. As illustrated in Fig. 7(b) and 7(d), the maximum deviations of the long-term stability are stably maintained within 2 degrees along the entire spatial axis for 60 hours.

 

Fig. 7. LTS experiment results of the proposed Linnik-based DSIE (for 60 hours): measured spatio-temporal ellipsometric phase map $\Delta _{HSI}^{Air}(k,x)\,{|_{t}}$ at λ=600.4 nm (a) before and (b) after employing the 1-D spectrometer-based compensation channel. (c and d) Line profiles for x = 1, 5.5 and 10.5 mm of Fig. 7(a) and 7(b), respectively.

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3.2 Global mapping phase error compensation

Although highly reliable 2-D spatio-spectral ellipsometric phase map $\Delta _{Compensated}^{Object(X,Y)}(k,x)\,{|_{t}}$ in Eq. (15) can be extracted at a static condition for LTS evaluation as described in section 3.1, the global SE mapping accuracy can be degraded since the XY-stage runout error is not uniform spatially, and it creates unknown global mapping phase error terms. As addressed in section 2.4, however, the global mapping phase error Ε(k,X,Y) = ε_p(k,X,Y) − ε_s(k,X,Y) denoted in Eq. (11) can be calibrated by measuring the $\Phi _{HSI}^{Ref\,(X,\,Y)}(k,x)$ indicated in Eq. (15) for all coordinates (X,Y) in-advance. For more clear representation for the global SE mapping performed by the XY-stage, we have defined the compensated 2-D spatio-spectral ellipsometric phase map $\Delta _{Compensated}^{Object\,(X,\,Y)}(k,x)\,{|_{t}}$ in Eq. (15) newly as a 3-D cubic ellipsometric phase map $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ representing the final SE mapping data extracted over the entire (X,Y) coordinates.

For the global mapping phase error compensation, we measure a 12-inch bare Si wafer placed on the XY-stage to extract $\Delta _{Compensated}^{Bare\,Si(X,\,Y)}(k,X,Y) = {\varepsilon _p}(k,X,Y) - {\varepsilon _s}(k,X,Y)$, by which the XY-stage runout error factor can be compensated. Note that this process for extracting $\Delta _{Compensated}^{Bare\,Si(X,\,Y)}(k,X,Y)$ for all coordinates (X,Y) is carried out only once. Thus, consecutive SE mapping of multiple thin-film wafer objects can be performed without being interrupted by the global mapping phase error compensation step. Figure 8(a) and 8(b) depict $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ map for the bare Si wafer object before and after the global mapping phase error compensation at a selected wavelength of 600.4 nm, respectively. These 2-D spatial ellipsometric phase maps have been measured over the large area of 15 cm × 23 cm by scanning the XY-stage horizontally (along the X-axis) with a stage velocity of 5(mm/s) by varying the Y-coordinate for the consecutive 20 lines. The difference between Fig. 8(a) and 8(b) represents the global mapping phase error term Ε(k,X,Y) = ε_p(k,X,Y) − ε_s(k,X,Y) caused by the runout errors of the XY stage. After removing the global mapping phase error over the entire measured area, the corrected continuous SE mapping result of the bare Si wafer is attained as illustrated in Fig. 8(b). Note that the global mapping phase error Ε(k,X,Y) depends significantly on the condition of alignment of the object mount and the repeatability of the XY-linear stage. Figure 8(c) shows the line profile for the black dotted lines of Fig. 8(a) and Fig. 8(b). As indicated by the blue solid line in Fig. 8(c), the proposed global mapping phase error compensation method removes the discontinuity of the large-scale object dramatically with a mean error of around 0.47 degrees for the entire $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ map.

 

Fig. 8. $\Delta _{Compensated}^{Object(X,Y)}(k,X,Y)$ mapping result of a bare Si wafer: (a) before and (b) after the global mapping phase error compensation method. (c) line profiles of Fig. 8(a) and 8(b) for X = 0.

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3.3 Thin-film uniformity measurement: Tetraethyl orthosilicate (TEOS) thin film

In this section, we demonstrate a thin film thickness uniformity mapping capability of the proposed robust DSIE system by using a 12-inch Tetraethyl orthosilicate (TEOS) thin-film deposited on a bare Si substrate. TEOS (Si(OC2H5)4) is a commonly used precursor in the synthesis of silicon dioxide (SiO2) films using PECVD (Plasma-enhanced chemical vapor deposition). The refractive index of TEOS is around 1.449 (@λ=632.8 nm), while the extinction coefficient is relatively low. The 12-inch TEOS thin film object is measured with an incidence angle of 45 degrees by scanning horizontally to the X-direction by a total of 27 lines over the entire 30cm × 30 cm mapping area with a scanning speed of 2 mm/s. The total SE mapping takes approximately 1 hour and 20 minutes with the current system. However, we estimate that the total measurement time with the same level of spatial resolution can be reduced to around 16 minutes just by replacing the current light source with higher power source as the integration time of the 2-D imaging sensor can be decreased around 5 times more.

Figure 9(a)–9(c) illustrate how the proposed compensation steps affect the 3-D cubic ellipsometric phase map $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ for the 12-inch TEOS thin film object. Figure 9(a) represents $\Delta _{HSI}^{Object(X,\,Y)}(k,X,Y)$ extracted without employing the compensation channel. Figure 9(b) illustrates $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ calibrated only at the center of the 12-inch bare Si wafer (X = 0, Y = 0), so still containing the global mapping phase error terms in Eq. (15). After applying the proposed holistic compensation method described in section 2.3 and 2.4, eventually, we obtain an accurate 3-D cubic ellipsometric phase map $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ of the 12-inch TEOS thin film object for the entire spectral range from 500 to 670 nm [Visualization 1].

 

Fig. 9. $\Delta _{Compensated}^{Object(X,Y)}(k,X,Y)$ mapping results of 12-inch TEOS thin-film wafer: (a) before and (b) after applying a compensation channel in Michelson-type DSIE. $\Delta _{Compensated}^{Object(X,Y)}(k,X,Y)$ obtained by employing the compensation channel combined with a Linnik-type monolithic interferometer at (c) λ=560.5 nm and (d) λ=660.7 nm. (e) Line profiles of Fig. 9(c) and 9(d) for X = 0, and (f)$\Delta _{Compensated}^{Object(X,Y)}(k,X,Y)$ for X = 0, Y = 0 (For the entire wavelength range, See Visualization 1).

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Figure 9(c) and 9(d) show the 2-D ellipsometric map $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ for 560.5 nm and 660.7 nm, respectively. To verify the reliability of DSIE, a commercial dual-rotating compensator type spectroscopic ellipsometer (RC2, J.A. Woollam Co., Inc.) has been used to measure the TEOS thin film object with a spot size of around 3 mm at the same incidence of angle of 45 degrees. By using a commercial SE, a line of Δ(k)s have been measured along the vertical line of the 12-inch TEOS thin film object with an interval of 5 mm as illustrated in Fig. 9(c)-9(f). The total measurement time of the commercial system for 60 measurement points is around 15 minutes. The grey and black solid line of Fig. 9(e) indicate the line profile for X = 0 of Fig. 9(c) and 9(d), respectively. The blue circle and red diamond marker represent Δ(k) results measured by the commercial SE for the same wavelengths of 560.5 nm and 660.7 nm as those selected for DSIE system, respectively. The estimated spatial accuracy of $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$ for the vertical line of X = 0 depicted in Fig. 9(e) is 99.4 ± 0.4% and 99.5 ± 0.3% at the selected wavelengths of 560.5 nm and 660.7 nm, respectively. The black dashed and dotted lines in Fig. 9(f) indicate Δ(λ)s measured by the proposed DSIE system for the specific spatial points of X = 0, Y = 0 and X = 0, Y = -14, respectively. The red and blue solid line of Fig. 9(f) reflect the measured results of the commercial SE at the same locations. The spectral accuracy of Δ(k)s are also estimated to be 99.6 ± 0.24% and 99.5 ± 0.25% at (0,0) and (0,-14), respectively. Through this system accuracy evaluation, we claim that the proposed SE mapping system can provide a highly reliable and accurate 3-D cubic spectroscopic ellipsometric phase map $\Delta _{Compensated}^{Object(X,\,Y)}(k,X,Y)$.

As the final task of the proposed ultrafast high spatial resolution SE mapping system, a thin film thickness uniformity mapping of the 12-inch TEOS thin-film with 1 μm nominal thickness has been carried out as illustrated in Fig. 10. For the thin film thickness extraction, a look-up table based minimum MSE (Mean Square Error) searching algorithm with a thin film thickness resolution of 0.05 nm is applied. The 12-inch whole thin-film thickness (T) uniformity has been estimated to be 2.55% by using the calculation formula of 100 × (T_max-T_min)/(2×T_avg). The number of total thickness data point measured in this experiment is over 20,000 by 500 pixels, which is over-sampled data for thin film uniformity mapping. Note that, however, the proposed SE mapping system can provide an unprecedent high-quality 3-D thickness profile information in a given period of time depending upon a required spatial resolution. For instance, the high-quality massive information about the volumetric thin film thickness can evaluate the TEOS film deposition process performed by a commercial PECVD (Plasma-enhanced Chemical Vapor Deposition) system in more detail by analyzing that radial lines of the TEOS thickness map indicate that radicals for wafer cleaning are sprayed through holes arranged along the lines of shower head distributor, and the shape of the distributor is reflected in the film quality.

 

Fig. 10. 3-D thickness profile map of the 12-inch TEOS thin-film wafer with a nominal thickness of 1 µm.

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4. Conclusion

In this study, we have described a holistic compensation method by which the temperature-dependency of a monolithic polarization interferometer and global mapping phase errors of a DSIE system can be removed. The proposed dual-channel DSIE system based on a Linnik-type monolithic polarizing interferometer solves the long-term stability problem of a single-channel DSIE system. We have shown that the spatial OPD variation occurs in a Michelson-type monolithic interferometer scheme due to thermal angular deflection in the kinetic mount used for mirrors in a monolithic interferometer module, while the spatial OPD variation becomes almost zero when a Linnik-type interferometer scheme is employed in combination with a 1-D spectrometer-based compensation channel. Details on a global mapping phase error compensation method also have been demonstrated by analyzing the accuracy of a 12-inch thin film wafer mapping results. The current whole 12-inch SE mapping time is approximately 1 hour and 20 minutes. To apply the proposed system for an inline process control tool for HVM (high volume manufacturing), however, the measurement speed needs to be improved further. Furthermore, a high throughput MM-SIE (Mueller matrix spectroscopic imaging ellipsometer) based on our DSIE configuration needs to be explored to measure asymmetry of 3D nanostructures [34,35]. We anticipate that the proposed quantitative robust DSIE system has a significant potential of opening a new in-line SE measurement and inspection field for high-yield control of the large-scale thin film and nano-pattern manufacturing processes.