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Abstract
We describe a novel approach for angle-resolved spectral reflectometry using a digital light processing (DLP) projector. Here, the DLP generates ring patterned images which are projected on the back focal plane of an objective lens. This way, the proposed method quickly changes the angle of incidence with ease based on the relation between the radius of the back focal plane and the angle of incidence. As a result, a detector captures the intensity of the image plane based on the angular and spectral axis. As the proposed method detects the interesting spot of a sample image, it can easily locate the measurement spot with viewing the full field of view, and the spot size is reduced by adopting the fiber. This method is verified by comparing the measurement output of the thin-film samples with a commercial ellipsometer. The result shows that our the proposed method enables the high accuracy of the thin-film inspection.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Continuous deposition and etching for the integration of the thin films play an important role in the semi-conductor and display industry. The measurement of the thickness and optical properties is an essential part of this process. Among numerous measurement researches that determine film thickness and optical properties, the optical measurement methods which detect the change in the optical response of the incident radiation are preferred for their nondestructive way of avoiding detrimental effects on the intended design. As the two of the most common optical measurement methods, spectroscopic ellipsometry and reflectometry have been widely used to investigate thin films. Ellipsometry is capable of measuring the thickness and optical properties with high accuracy at a given wavelength and angle of incidence [1–5]. Variable angle spectroscopic ellipsometry (VASE) is an improved type of scheme to obtain a wide spectral region and a variable angle of incidence. The large amount of data in the two independent variables offer a better optimization condition of the inverse problem. However, this method has been limited to industrial applications due to its off-axis configuration and low spatial resolution caused by its large spot size. Besides, it takes long measurement time to modulate polarization components, and change the incident angle and wavelength.
Reflectometry has been an alternative measurement technique for the investigation of thin films [6–10]. This method is configured as on-axis, and measures the reflectance as a function of wavelength based on the normal incidence assumption. However, this limits the use of high numerical aperture (NA) systems. It is difficult for systems with a low NA lens to measure fine patterns integrated on films due to its low magnification. In order to adopt a high NA lens for high magnification, several approaches have been developed to compensate the NA effect of the objective lens such as the averaged factor [8] and the effective angle of incidence [9]. Nevertheless, these compensation methods are based on the mathematical approximation and do not fundamentally eliminate the error. Furthermore, spectroscopic reflectometry only obtains a function of wavelength, which is insufficient to measure complicated structure associated with multi-parameter fits and thin-film with low experimental sensitivity.
As the film has become thinner and more complex by multi-layered fabrication, another approach called angle-resolved spectroscopic reflectometry has been devised to overcome the drawbacks of conventional measurement techniques. In this approach, obliquely incident light by the numerical aperture from the objective lens is used. After reflecting on a sample, the focus of the light with the same incident angle is set on the same position-the back focal plane of the objective lens, which is imaged by a camera. It has the advantages of the on-axis configuration and the use of a high NA objective lens which offers high spatial resolution. Furthermore, it measures the reflectance of the sample with various angles of the incidence. In this field, the beam profile reflectometry technique was firstly developed to measure angle-resolved reflectance in real time. However, it could only be performed at a single wavelength [11–13]. In order to obtain both the spectral and angular variations of reflectance, angle-resolved spectral reflectometry has been developed through analyzing the part of the back focal plane image using a spectrometer [14,15] and a pixelated polarizer mask [16]. However, the optical systems of these researches view the sample’s back focal plane. For such reason, these techniques measure the entire field of the objective lens view. The average results are obtained over all features within the illumination spot. It is hard to reduce the spot size in the case of a broadband light source and inspect the interesting spot of the sample.
In this paper, we propose an improved configuration of angle-resolved spectral reflectometry. We use a digital light processing (DLP) projector to illuminate the light with different angles of incidence. Generally, DLP projector has been used in structured illumination microscopy which realizes the three-dimensional surface topography [17–23]. In this field, DLP generates a sinusoidal pattern projected on the focal plane of the imaging objective. Here, it is the back focal plane of the objective lens on which the DLP-generated images are projected. The illumination patterns are designed as a ring. The ring patterned light is sequentially projected while varying the radius. The incident angle is changed in accordance with the radius of the ring pattern, because the spatial location of light at the back focal plane determines the incident angle. Thus, the digitally controlled illumination system easily changes the angle of incidence with high accuracy in comparison to the previous approaches for angle-resolved illumination [24–27]. Moreover, it has the advantages of high spatial resolution and short imaging time. Though the focus of the incident light is set on the back focal plane, the detector of the proposed system is set on the image plane. The proposed method can easily locate the interesting spot viewing the full field of view and the small spot size is achieved by adopting the fiber. Therefore, our proposed method can obtain the reflectance of the sample in a broad spectral range and a wide incident angle while locating the interesting spot with viewing the image plane of the sample.
2. Method
2.1 Hardware configuration
The schematic diagram of the proposed angle-resolved spectral reflectometry is shown in Fig. 1. The DLP projector is composed of a computer-controlled digital micromirror device (DMD) and a light source. The DMD is a precise light switch that digitally modulates the light by an array of microscopic mirrors. Each mirror corresponds to a pixel of the projected image. In order to generate a patterned image, a digital signal activates the electrode below DMD to tilt either towards or away from the light source. The pixel turns ON or OFF according to the tilted direction of the mirror. In this system, the DMD is utilized to flexibly produce a series of illuminated patterns. The DMD modulates the light emitted from the light source in the form of a ring pattern while changing the radius. The relay lens delivers the modulated light onto the back focal plane of the objective lens. The light passes onto a sample at a specific angle of incidence which is determined by the radius of the ring pattern. After it is reflected by the sample, the light is not only imaged on a review camera by a zoom lens but also detected by a spectrometer. As described in Fig. 1, the patterned illumination is projected on the back focal plane and the camera works to view the sample plane. This feature is different from the back focal plane imaging method. The digitally controlled illumination system easily, quickly and accurately changes the angle of incidence. The spectrometer detected the specific spot on the sample surface with the specific angle of incidence described with red lines in Fig. 1, which is same as the marked spot on the sample plane image. The ring patterned image generated in accordance with the change of radius is projected on the back focal plane as shown in Fig. 2(a). The spectrometer sequentially obtains the spectral intensity during the radius change of the ring. Figure 2(b) shows the relationship between the radius of the ring $(r)$ and the incident angle $(\theta )$. The light passing through the same point of the back focal plane proceeds with the same incident angle on the sample plane. The detailed equation between $r$ and $\theta$ is expressed as
Fig. 2. Explanation of projection onto the back focal plane: (a) projection of the ring images with varying the radius, (b) relation between the radius and the angle of incidence, (c) the example of projected images for different angle of incidence.
2.2 Analysis method
For a transparent thin film layer shown in Fig. 3, the Fresnel reflection coefficient of the light passing through the boundary of media is expressed as
Even though the DMD has the advantages of high spatial resolution and short imaging time compared to the previous angle-resolved illumination methods, the generated pattern has a line-width as shown in Fig. 4 due to the pixel resolution. For this reason, the radius cannot be specified with a single incidence angle. Thus, it is required to modify the theoretical reflectance with the consideration of the effect caused by the line-width. In the case of the DMD, the effect of the line-width is easily calculated with high accuracy from the known spatial resolution of the DMD when compared to the previous angular illumination approaches. The effective reflectance $\hat {\mathfrak {R}}(r)$ is described as the following equation with $r$ and $\Delta r$ being the radius and the line-width of the ring respectively:
By acquiring data in both wavelengths and incident angles, the proposed angle-resolved spectral system could obtain thickness and the refractive index of the sample with better accuracy compared to the conventional reflectometry. In the nonlinear fitting process of minimizing the cost function, it is better to increase the amount and dimension of the accessible data for better optimization conditions [14,28]. Figure 5 shows the effect of the proposed method in comparison with the conventional spectral analysis through simulations. In the case of the extremely thin film, it is difficult for the spectral reflectometry to accurately measure the thickness due to the monotonicity of reflectance as shown in Fig. 5(a). This monotonicity of the profile means lack of distinguishing features such as the existence of extremes, which could be misguided to express each reflectance with different optical properties with a linear operator of one another in the measurement range. As a matter of fact, it leads to the ill-conditioned optimization in the non-linear fitting. Furthermore, this problem increases the likelihood of converging wrong parameters affected by external noise. On the contrary, the proposed method can obtain reflectance in the two independent dimensional space as shown in Fig. 5(b), and each reflectance is more discrete from one another. Moreover, the proposed method is robust in the case of simultaneously determining thickness and refractive index. Figures 5(c) and 5(d) demonstrate the cost of Eq. (8) below 0.005 with variables of thickness and the real term of the refractive index. The conventional spectral reflectometry presents high correlation between fit parameters. In addition, small experimental errors particularly have a strong effect on the measurement, which makes the the results be subjected to a systematic error for both film thickness and refractive index. However, the proposed method has distinct global minimum that enables rapid and accurate convergence to the unique solution.
Fig. 5. Comparison of simulation results with the conventional spectral reflectometry in two cases: SiO2 film on a Si substrate whose thickness is 10 nm, 20 nm, and 30 nm ((a) spectral reflectance, (b) angle-resolved spectral reflectance) ; cost value of SiO2 film on a Si substrate according to film thickness and the real term of refractive index ((c) spectral reflectance, (d) angle-resolved spectral reflectance).
3. Experiment setup
The experimental equipment of the proposed method was constructed based on the configuration described in Fig. 1. The system for the experiment is made up of a DLP projector, a relay lens, beam splitters, an objective lens, a zoom lens, a spectrometer and a camera. The DLP projector is composed of Single Texas Instruments 0.65" 1080p DMD and the 330W Halogen lamp. The available spectrum is covered from 450 nm to 700 nm in this system. The pixel resolution of the projector on the back focal plane is 7.56 µm, and the radius of the ring is changed at intervals of 1 pixel in radius. Therefore, the proposed system has high spatial resolution in angle-resolved illumination. In the configuration of the projector, it is required that the frame of the detector is synchronized with the projector so that the entire projection is captured [29,30]. The relay lens is utilized to deliver the modulated light from the projector onto the back focal plane of the objective lens. The apochromatic relay lens(Edmund Optics) is used in this experiment, and the objective lens is used to set the focus of a ray on the sample surface. Furthermore, it is necessary to select a high NA lens to acquire a large range of incidence angles, and a Nikon CF IC EPI Plan Apo (100$\times$ and NA: 0.95) is employed. The full pupil size of the objective lens is 3.8 mm in diameter, which means that the maximum radius of the projected image is set to 250 pixels. Obtaining the light reflected from the sample plane is enabled by the 1$\times$ zoom lens. The spectrometer is Ocean Optics Maya2000 Pro that covers wavelengths from 165 nm to 1100 nm by 2048 CCD arrays. The reflected light enters into the spectrometer through the optical fiber. As the core diameter of the fiber is 50 µm, the spatial sampling of the system is numerically calculated as 0.5 µm. Here, the diffraction limited spot size is in the order of 0.71 µm at a central wavelength of 550 nm according to the Rayleigh criterion. The camera is Basler acA1300-200um which is the CMOS camera with 1280$\times$1024 pixel resolution and 4.8 µm $\times$ 4.8 µm pixel sizes. It is implemented to review the measurement position by viewing the sample plane.
The performance of the proposed method is evaluated through using single layer thin film and multi- layered film samples. All of the samples were fabricated by PECVD (PlasmaPro System100, Oxford instruments). The single-layered sample is a uniform SiO2 thin film layer deposited on a Si substrate. The single-layered sample set consists of 5 different thickness. Multi-layered films are sequentially stacked on a silicon wafer in the order of SiNx-SiO2 and SiNx-SiO2-SiNx. The measurement results of the proposed method were compared with reference thickness values obtained from a commercial spectroscopic ellipsometer (Horiba UVISEL).
4. Results
The experiment was conducted to validate the accuracy of the proposed method by measuring various thin-film samples. The sample set consists of not only a single-layer deposited SiO2 films on a silicon substrate, but also multi-layer films sequentially stacked on a silicon wafer in the order of SiNx-SiO2 and SiNx-SiO2-SiNx.
4.1 Single-layer film
The single-layer films were measured for the verification. Figures 6(a) and (b) respectively shows the results from the measurement of Sample 3 and 4 in Table 1. In this figure, the angle-resolved spectral reflectance measured by the proposed method is compared with the theoretical reflectance obtained from Eq. (4) after the optimization. The experimental reflectance in the domain of wavelengths and incidence angles is consistent with the theoretical one. It confirmed that the theoretical model was well established. In addition, we demonstrated the reflectance at a specific wavelength of 555.17 nm and at a specific angle of incidence of 52.47$^{\circ }$ with the theoretical model to show the detailed result of the optimization. Even though there is a slight discrepancy, the results show that the experimental results are well fitted with the theoretical model based on the spectral and angle-resolved analysis. For the result of the optimization, we have obtained 123.74 nm and 492.56 nm thickness values in Table 1. The deviations were less than 2 nm, and the results are close to the reference results from the commercial ellipsometer.
4.2 Thin single-layer film
The samples of the thin single-layer film were measured to verify the accuracy of the measurement in very thin films. As mentioned in Section 2.2, it is difficult for the spectral reflectometry to accurately measure the thickness due to the lack of distinguishing features. Figures 7(a) and 7(b) respectively shows the results from the measurement of Sample 1 and 2 in Table 1. The experimental reflectance well-matches with the theoretical model obtained from the optimization process. The reflectance at the specific wavelength and angle of incidence is well-fitted with the theoretical model. Through the comparison between Figs. 7(a) and 7(b), the monotonicity of the reflectance is shown in the range of very thin thickness. It reveals that the analysis with a single variable is hard to obtain an accurate thickness of very thin films. For the result of the optimization, we have obtained 13.38 nm and 35.44 nm thickness values. These can be considered close to the reference results from commercial ellipsometer, because the difference is less than the sub-nanometer range. The result of the sample 1 and 2 verifies that the proposed method could accurately measure very thin film which is a challenging task for the conventional spectral reflectometry.
4.3 Multi-layered film
The samples with the multi-layered films on the substrate were also measured to verify the performance of our method. Such a multi-layer film is hard to be measured because it requires a large amount of data to determine the thickness of each layer through the multi-variable non-linear fitting process. Figure 8(a) shows the measurement result of SiNx-SiO2 on the Si substrate, and Fig. 8(b) exhibits the result of films stacked on a silicon wafer in the order of SiNx-SiO2-SiNx. For the result of the multi-layered films, the angle-resolved spectral reflectance was obtained with a significant appearance in two-dimensional planes in comparison with the spectral reflectance. The results in only spectral domain show that the conventional spectral analysis has the risk of falling into the local minimum. On the contrary, by gathering an abundant amount of data with varying wavelength and incident angle, the proposed method avoids the convergence to the local minimum unlike the conventional spectral analysis. The results in Table 1 indicate that the proposed method is capable of an accurate measurement of thin film layers in the nanometer range.
5. Discussion
The measurement results of various thin-film samples are used to verify the feasibility of the proposed method. The single layer films, the thin single layer films, and the multi-layered films were measured and the results are compared in Table 1 with the commercial ellipsometer as a reference. The maximum difference with the reference value is 2.16 nm of L1 in Sample 6.
Fig. 6. Experimental results of a single-layered SiO2 film: Measurement and theoretical model of angle-resolved spectral reflectance; Comparisons of measured spectral reflectance (red solid line) with theoretical model (blue dashed line) at an incident angle of 52.47$^{\circ }$; Comparisons of measured angle-resolved reflectance (red solid line) with theoretical model (blue dashed line) at a wavelength of 555.17 nm; (a) Sample 3, (b) Sample 4.
Fig. 7. Experimental results of a thin single-layered SiO2 film: Measurement and theoretical model of angle-resolved spectral reflectance; Comparisons of measured spectral reflectance (red solid line) with theoretical model (blue dashed line) at an incident angle of 52.47$^{\circ }$; Comparisons of measured angle-resolved reflectance (red solid line) with theoretical model (blue dashed line) at a wavelength of 555.17 nm; (a) Sample 1, (b) Sample 2.
Fig. 8. Experimental results of a multi-layered SiO2 film: Measurement and theoretical model of angle-resolved spectral reflectance; Comparisons of measured spectral reflectance (red solid line) with theoretical model (blue dashed line) at an incident angle of 52.47$^{\circ }$; Comparisons of measured angle-resolved reflectance (red solid line) with theoretical model (blue dashed line) at a wavelength of 555.17 nm; (a) Sample 5, (b) Sample 6.
Table 1. Measurement results.
The cause of the difference can be analyzed through the error analysis of the system. Considering the function of reflectance as shown in Eq. (5), error contributions can be divided into the wavelength and the angle of incidence. Here, numerical estimations were performed in the reflectance [15,31]. The differential reflectance in each elementary term is calculated for the range equivalent to the corresponding elementary uncertainty magnitude with respect to the specified test condition parameter. In the spectral acquisition, the uncertainty of the wavelength, $\delta \lambda$, is claimed to be $0.4$nm referring to the spectrometer specification. Then, the differential reflectance is estimated as
Table 2. Error analysis of reflectance $\mathfrak {R}$ and thickness with respect to $\delta \lambda$ and $\delta r$.
6. Conclusion
We proposed a novel optical technique of angle-resolved spectral reflectometry using a computer-controlled DMD projector to measure film thickness and optical constants. To the best of our knowledge, the proposed system has promising characteristic of the usage of DMD projection on the back focal plane to obtain the angle-resovled spectral reflectance. The usage of DMD provides the advantage of high spatial resolution and short changing time in the angle of incidence. This system also easily changes the angle of incidence with high accuracy in comparison with the previous angle-resolved illumination system. Furthermore, it has advantages by viewing the image plane of the sample. The small spot size is achieved by adopting the fiber, and the proposed method can locate the interesting spot viewing the full field of view with ease. Therefore, our proposed method can obtain the reflectance of the sample in a broad spectral range and a wide incident angle while locating the interesting spot with viewing the image plane of the sample. In order to validate the accuracy of the proposed method, various thin-film samples were measured. The sample set consists of not only a single-layer deposited SiO2 films on a silicon substrate, but also multi-layer films sequentially stacked on a silicon wafer in the order of SiNx-SiO2 and SiNx-SiO2-SiN. These results are compared with the commercial ellipsometer. This comparison confirms the capability of the accurate measurement of thin films and the determination of multi-layered films of the proposed method. Considering the mentioned strengths and the experimental results, our proposed method will be widespread as a simple and general metrological tool for angle-resolved spectral reflectometry in thin-film metrology.
Acknowledgments
This work was supported in part by the Brain Korea 21 Plus, the Institute of Engineering Research, and the Institute of Advanced Machines and Design at Seoul National University.
Disclosures
The authors declare no conflicts of interest.
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