Thin Film Thickness Analysis Using a Deep Learning Algorithm with a Consideration of Reflectance Fluctuation

Joonyoung Lee; Jonghan Jin

doi:10.57062/ijpem-st.2024.00164

Int. J. Precis. Eng. Manuf.-Smart Tech. > Volume 3(1); 2025 > Article

Lee and Jin: Thin Film Thickness Analysis Using a Deep Learning Algorithm with a Consideration of Reflectance Fluctuation

Regular Paper

International Journal of Precision Engineering and Manufacturing-Smart Technology 2025;3(1):31-38.

Published online: January 1, 2025

DOI: https://doi.org/10.57062/ijpem-st.2024.00164

Thin Film Thickness Analysis Using a Deep Learning Algorithm with a Consideration of Reflectance Fluctuation

Joonyoung Lee^1.² · Jonghan Jin^1.^2.³

¹Meter-Lab. Inc., 114 Gyeryong-ro, Yuseong-gu, Daejeon, 34188, Republic of Korea

²Major of Precision Measurement, Korea National University of Science and Technology (UST), 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic of Korea

³Division of Physical Metrology, Korea Research Institute of Standards and Science (KRISS), 267 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic of Korea

Jonghan Jin: jonghan@kriss.re.kr, jonghan@ust.ac.kr, jonghan@meter-lab.com

Received September 30, 2024 Revised November 7, 2024 Accepted November 9, 2024

Abstract

A deep learning algorithm for thin film thickness analysis based on spectral reflectometry, using a dataset that reflects experimental conditions, has been proposed and implemented. This study extends our previous research, in which we designed an artificial neural network (ANN) algorithm using theoretical reflectance spectrum datasets and quantitatively evaluated it according to the international standard traceability system. The evaluation results indicated that one of the major sources of uncertainty was the offset between the outputs of the ANN algorithm and the certified values of certified reference materials (CRMs). In this study, we focused on how much the uncertainty factor related to the offset is affected by using a dataset that reflects experimental conditions instead of theoretical reflectance spectrum datasets. By applying the fluctuations in reflectance obtained from experiments to the theoretical reflectance spectrum, we created a dataset to train the ANN algorithm under the same conditions as in our previous studies for comparison. As a result, the major uncertainty factor related to the offset improved by about 30%. This study demonstrates the importance of having datasets that accurately reflect real-world conditions for training ANN algorithms.

Keywords: Thickness measurement · Thin-film · Artificial neural network · Uncertainty evaluation

1 Introduction

Recently, deep learning techniques have been widely used across various fields [1–16]. In the field of measurement and inspection, pattern recognition and classification of images acquired through cameras are representative applications [11–16]. For training algorithms in deep learning, it is essential to secure a large number of images with exactly known true labels. For example, to improve the recognition rate of a deep learning model in handwriting recognition, a vast number of handwritten images written in various styles and by different individuals are required for each character. Thus, securing accurately labeled training data is the most fundamental and essential element for training a deep learning model.

In a more rigorous sense, measurement is a process that provides rational quantitative values, differing from traditional pattern recognition and classification. Recently, various cases have been reported where deep learning has been applied in precision measurement fields, including thin film thickness measurement, three-dimensional profile measurement, surface roughness measurement, temperature, and pressure measurements [17–30]. However, considering the characteristics of measurements that provide accurate values, it can be challenging to fully trust the outputs of the algorithms. In other words, it is difficult to verify whether a deep learning algorithm provides accurate values. This difficulty arises from the challenge of securing a large amount of training data with exactly known true labels, as previously mentioned. For instance, in a spectral reflectometer, while interference signals from thin film samples can be obtained, determining the exact thickness of the measured thin film is always problematic, which serves as a true label. Additionally, the interference signals can vary due to factors such as the material, uniformity, purity, and oxidation of the thin film, rather than just changes in thickness [31–36]. Therefore, even abundant interference signals from thin film samples whose exact thickness and material information are unknown may significantly decrease the learning rate of machine learning or even become ineffective. To address this issue, a study was conducted in 2023 to evaluate deep learning algorithms for thin film thickness analysis in spectral reflectometers based on an international standard system [37,38].

To effectively utilize deep learning in precision measurement fields, two key elements are important. The first is securing training data that accurately reflects real experimental conditions, and the second is the objective evaluation of the trained deep learning algorithms. Previous studies have primarily relied on theoretical data for training due to the difficulty in obtaining sufficient training data with accurately known true labels and have evaluated the trained algorithms using multiple certified reference materials. In this research, we aim to generate training data that is more similar to experimental data and utilize it for training. The training data used in this study was generated by considering the wavelength-dependent reflectivity changes caused by various experimental variables, specifically by adding experimentally obtained reflectivity changes to theoretical data. This approach was taken because it is one of the most noticeable key factors when comparing the reflectance spectra obtained from experiments with the theoretical reflectance spectra. Subsequently, based on previous research, we train an artificial neural network (ANN) algorithm and perform quantitative evaluations in terms of measurement uncertainty. The measurement uncertainty factors are calculated, and the combined uncertainty is derived. Through this, we aim to investigate the impact of training data that reflects experimental conditions in deep learning algorithms for measurement.

2 Basic Principles

2.1 Spectral Reflectometer

The spectral reflectometer utilizes the physical phenomenon where the spectrum of reflected light from a transparent thin film changes according to its thickness due to multiple interference. Since this spectral reflectometer employs light for measurement, it allows for non-contact and non-destructive measurement, offering advantages such as simple configuration and real-time measurement. Fig. 1 illustrates (a) the optical layout of a spectral reflectometer, consisting of a broadband light source and a spectrometer, and (b) the physical phenomenon of light reflection and transmission in a thin film on a substrate. Light emitted from the broadband light source is incident perpendicularly onto the thin film through a circulator, then reflects back into the spectrometer. The light that is incident on the thin film undergoes two optical phenomena: first, it is reflected and transmitted at the air-thin film interface and the thin film-substrate interface. Most of the light enters the spectrometer after undergoing multiple transmission and reflection processes at each interface. The transmission and reflection coefficients at an interface of material i and j as functions of the light’s wavelength are expressed by the Fresnel equations in Eqs. (1) and (2), respectively [37]. These coefficients are determined by the wavelength of the incident light and the complex refractive indices of the incident medium (N_i(λ)) and the transmitting medium (N_j(λ)).

(1)

tij(λ)=2·Ni(λ)Ni(λ)+Nj(λ)

(2)

rij(λ)=Ni(λ)-Nj(λ)Ni(λ)+Nj(λ)

Second, the light incident on the thin film experiences a phase delay proportional to the path it travels within the film. For example, starting from the air-thin film interface, the light reflects once at the thin film-substrate interface and returns to the air-thin film interface, resulting in a phase delay of 2TN_j(λ), where T is the thickness of the thin film. The amount of phase delay varies depending on how many times the light reflects within the thin film. Consequently, the spectrometer detects multiple light waves with different phases that interfere with each other, depending on the number of reflections occurring inside the thin film. Here, the electric field of the interfered light (E′) can be summarized as a function of the reflection coefficient, transmission coefficient, and phase delay, as shown in equation (3). The phase delay can be expressed in terms of the refractive index and thickness of the thin film, as represented in Eq. (4). Thus, the reflection coefficient of the thin film is defined as the ratio of the electric field of the interfered light to that of the incident light (E₀), as given in Eq. (5). Finally, the theoretical reflectance spectrum of the thin film can be calculated as the complex square of its reflection coefficient, as shown in Eq. (6).

(3)

E′=[rij+Σm=0∞tij·rjk·(rji·rjk)m·tji·e-i(m+1)φ(λ)]·E0

(4)

φ(λ)=2πλ·2·Nj(λ)·T

(5)

r(T,λ)=rij+rjk·e-i·4π/λ·T·Nj(λ)1+rij·rjk·e-i·4π/λ·T·Nj(λ)

(6)

R(T;λ)=r(T;λ)·r(T;λ)*

2.2 Deep Learning Algorithm

Deep learning algorithms are an advanced form of machine learning that learns complex patterns in data through no-nlinear transformations within a multi-layered structure. Generally, a deep learning model consists of an input layer, one or more hidden layers, and an output layer; this structure is called an artificial neural network. In the input layer, each element of the data is multiplied by a weight, a bias is added, and then a non-linear active function (e.g., sigmoid, ReLU) is applied to deliver the computed result to the next layer. This process continues from the input layer through the hidden layers to the output layer, and it is referred to as forward propagation. To train a deep learning algorithm, a dataset consisting of input data and corresponding true labels is required. The algorithm takes the input data from the dataset, performs forward propagation, and then compares the output to the true labels to calculate the error. Subsequently, the backpropagation algorithm is used to adjust the model’s weights and biases to minimize this error. This process is repeated multiple times, allowing the deep learning model to gradually improve its ability to analyze data. In this study, the deep learning algorithm for thin film thickness analysis is trained using a dataset where the reflectance spectrum (R(T;λ)) from Eq. (6) serves as the input data and the thin film thickness (T) is set as the true label. The trained deep learning model analyzes the reflectance spectrum obtained from thin film samples in real time, determining the thin film thickness without any iterations or additional processes. In the future, it is expected to be more useful in cases that require extensive computations, such as multilayer thin thickness measurements. However, this study focused solely on the analytical reliability of the deep learning algorithm, limiting the research to single-layer measurements.

3 Design of Ann Algorithm

To generate training data similar to actual experiments, the fluctuation in the reflectance spectrum caused by the measurement environment was analyzed. First, using a spectral reflectometer, the interference spectra of silicon oxide thin film certified reference materials (CRMs) with thicknesses of 10, 30, 50, and 100 nm were measured continuously for 100 cycles each. The measured interference spectra were obtained with 881 samples at intervals of about 0.34 nm across the visible wavelength range from 355 to 657 nm. Fig. 2(a) shows the continuously measured interference spectrum for a 10 nm thick silicon oxide thin film as an example. Although the measurement duration was less than 1 s, significant changes in light intensity were observed, which could not be ignored. This might be coming from factors such as the stability of the light source and environmental variations like temperature and humidity. Since the spectrometer used for the measurements was wavelength-calibrated, changes in light intensity due to wavelength variations can be disregarded. When this data is redrawn as a reflectance spectrum according to Eq. (6), as shown in Fig. 2(b), significant variations in reflectance can still be observed.

To quantitatively represent these reflectance variations, standard uncertainty was employed. For the four thin film samples, the differences between the maximum and minimum light intensity values (Peak-to-Valley, PV) were calculated for each of the 881 wavelengths in the reflectance spectra measured 100 times like Fig. 2(b). The standard uncertainty of the reflectance variations was derived as 0.003 using the maximum value from a total of 3,524 PV values. To express the calculated standard uncertainty as a quantitative value for relative changes, the previously obtained PV value of reflectance variations was divided by each average reflectance. Fig. 3 presents a histogram of the relative standard uncertainty obtained in this manner, with most values falling within 1%. Finally, to simulate the noise that may occur in actual measurement environments, theoretical reflectance spectra with a maximum fluctuation of 1% were generated and utilized as training data.

Fig. 4 illustrates the process of generating training data that reflects a 1% fluctuation in reflectance. First, theoretical reflectance spectra were generated for thin film thicknesses ranging from 1 to 110 nm, in increments of 1 nm, under the conditions of a silicon oxide layer on a silicon substrate, resulting in a total of 110 spectra. Then, for each theoretical reflectance at each thickness, 10 reflectance values were randomly generated based on a normal distribution with a relative standard uncertainty of ±1%. In total, 1,100 training data points were generated.

The deep learning algorithm can be generated under various conditions; however, this study focuses solely on utilizing training data that simulates the experimental environment to compare with prior research findings. Therefore, an ANN algorithm having one hidden layer with 150 nodes, which was chosen in previous studies, was used. The objective function was defined as the mean absolute error, as shown in Fig. 5(a), which calculates the average absolute difference between the thicknesses predicted by the deep learning model and the true label values in the training data for each iteration. Fig. 5(a) shows the learning trend based on the number of iterations. The iterations repeated until the error of the objective function fell below 10% of the largest uncertainty of the CRMs used in this study, which was 0.2 nm in thickness. In this training, the analysis reliability was validated for only four thicknesses within the range of 1 to 110 nm, making it necessary to confirm the analysis reliability for other thickness values. To achieve this, linearity was verified using both theoretical data and CRM data, as shown in Fig. 5(b). The coefficient of determination (R²) was 0.999, indicating that the analysis reliability can be valid for the thickness measurement range of the thin film.

4 Uncertainty Evaluation

The trained ANN algorithm performed uncertainty evaluation based on the methods proposed in our previous work. The analytical uncertainty of the artificial neural network algorithm for thin film thickness measurement is largely composed of three factors: the offset between the certified values of 4 thin film CRMs and its outputs(u(t_offset)), the standard uncertainty of the thin film thickness CRM(u(t_CRM)), and the repeatability of thickness measurements(u(t_rep)), as described in Eq. (7). First, the uncertainty of the offset was defined as the maximum difference between the output values obtained by inputting the reflectance spectra from measuring four thin film thickness CRMs into the trained artificial neural network algorithm and the certified values of each thin film thickness CRM. In previous studies, it was the largest source of uncertainty, which was evaluated as the largest offset of 1.0 for a 10 nm thin film thickness CRM. In this study, by using training data that reflects the experimental conditions, the uncertainty was evaluated to be improved by 30% compared to previous research, resulting in a value of 0.7 for a 50 nm thin film thickness CRM. Here, this study is a simple comparison between considering and not considering the intensity fluctuation of actual measurements [37]. This does not imply that data with more noise is more accurate than theoretical data. To show how uncertainty varies depending on the degree of fluctuation in the actual measurements, future research will be necessary to focus on uncertainty related to the degree of fluctuation in measurement values.

(7)

u(d)=u2(toffset)+u2(tCRM)+u2(trep)

The second uncertainty factor, the uncertainty related to the CRMs, was defined as the largest standard uncertainty among the values of the four CRMs. Since this is determined by standard institutes, it is independent of the measurement method and has a constant value. The CRMs used in this study were calibrated by the Korea Research Institute of Standards and Science. The expanded uncertainty for the certified values of the 50 and 100 nm thin film CRMs was 2.2 nm (k = 2), which is greater than ones of the other thin film CRMs; thus, the standard uncertainty for the CRMs can be evaluated to be 1.1 nm. Lastly, the repeatability of thickness measurements was determined by calculating the standard deviation of the thickness values obtained from analyzing the reflectance spectra of the four thin film thickness CRMs using the trained artificial neural network algorithm. The uncertainty for repeatability of the thickness values obtained from all thin film thickness CRMs was very small, at less than 0.1 nm. The combined standard uncertainty was calculated to be 1.3 nm through Eq. (7), and Table 1 represents the uncertainty budget summarizing the uncertainty factors.

In summary, when using data that reflects the reflectance fluctuations, the uncertainty related to the offset of the output values, previously reported as the largest uncertainty factor in our prior research, was improved by approximately 30%, resulting in a value of 0.7 nm. This improvement occurred because the differences between the analyzed thickness outputs from the deep learning algorithm and the certified thickness values of the reference materials were dramatically reduced. According to the results of this study, the largest factor affecting the combined standard uncertainty was the uncertainty of the CRM, which is not directly related to the measurements or analyses conducted. Hence, this demonstrates that utilizing theoretical reflectance spectrum data that reflects the fluctuations can directly enhance the performance of the artificial neural network algorithm. It underscores the importance of how well the training data reflects the experimental conditions. One important point to note is that the deep learning algorithm in this study was trained based on reflectance variations observed under specific measurement equipment and environmental conditions. When different measurement equipment is used or environmental conditions change, reflectance variations under the new conditions must be analyzed and then a new deep learning model must be generated. This can be a practical limitation of this study.

5. Summary

In this paper, we proposed and implemented an artificial neural network algorithm for thin film thickness analysis based on spectral reflectometry using a dataset that reflects experimental conditions. For that, a new dataset that reflects experimental conditions was created by adding the fluctuations in reflectance to the theoretical reflectance spectra. The reflectance fluctuation was ±1%, determined by repeated measurements. For comparison, the ANN algorithm was trained under the same conditions as in our previous studies. The ANN algorithm had a single hidden layer with 150 nodes, and the number of input nodes was 881, each representing the reflectance at a wavelength across the visible wavelength range from 355 to 657 nm. The iterations for training the ANN algorithm continued until the error of the objective function fell below 10% of the largest uncertainty of the CRMs used in this study, which was 0.2 nm in thickness. To verify the performance of the ANN algorithm, four CRMs with nominal thicknesses of 10, 30, 50, 100 nm were utilized. Furthermore, linearity was verified across the full thickness range of 10 to 110 nm, with a coefficient of determination (R²) of 0.999. In the uncertainty budget, there were three factors: the offset between the certified values of the four thin film CRMs and their outputs, the standard uncertainty of the thin film thickness CRM, and the repeatability of thickness measurement.

In this study, by using a dataset that reflects the experimental conditions, the uncertainty related to the offset, one of major uncertainty factors, was estimated to be 0.7 nm. It was dramatically improved by 30% compared to previous research. The uncertainty related to the CRM is a constant value of 1.1 nm, which is independent of this work, as it is determined by the capability of the standard institutes. The repeatability was too small to be ignorable, which was less than 0.1 nm. Finally, the combined standard uncertainty was evaluated to be 1.3 nm (k = 1). Therefore, this study demonstrates the importance of having datasets that accurately reflect real-world conditions for training deep learning algorithms. Furthermore, this is the first practical example of how much the performance of deep learning algorithms can be quantitatively enhanced by incorporating real-world conditions into theoretical datasets from the perspective of measurement uncertainty.

Acknowledgement

This work was jointly supported by Meter-Lab. Inc. and the Korea Research Institute of Standards and Science. It was supported by “1-2-05. Multiscale Length Measurement Team: Development of core technology for absolute length measurement based on optical comb” under “Advancement of length measurement standard technology” associated with grant number 24011043.

Fig. 1

(a) Optical layout of spectral reflectometry (b) Light propagation in a thin film (r_ij: reflection coefficient for light traveling from air to thin film, r_ji : reflection coefficient for light traveling from thin film to air, r_jk : reflection coefficient for light traveling from thin film to substrate, t_ij: transmission coefficient for light traveling from air to thin film, t_ji : transmission coefficient for light traveling from thin film to air, T: thickness of thin film)

Fig. 2

Variation of (a) interference spectrum and (b) reflectance spectrum of a 10 nm thick silicon oxide thin film

Fig. 3

Histogram of reflectance spectrum variation

Fig. 4

Generation of training data that reflects a 1% fluctuation in reflectance

Fig. 5

(a) Error convergence in training and (b) correlation between analysis value and thickness of the trained artificial neural network algorithm

Table 1

Uncertainty budget

Uncertainty factor	Value / nm	Sensitivity	Uncertainty / nm
Uncertainty for the output offset, u(t_offset)	0.7	1	0.7
Uncertainty for the CRMs, u(t_CRM)	1.1	1	1.1
Uncertainty for repeatability, u(t_rep)	< 0.1	1	< 0.1
Combined standard uncertainty, u(t)	1.3

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Biography

Jonghan Jin CTO, Meter-Lab. Inc. / Principal research scientist in the Physical Metrology Division, Korea Research Institute of Standards and Science. / Professor in the Department of Precision Measurement, Korea National University of Science and Technology / His research interest is optical metrology.

Biography

Joonyoung Lee received the Ph.D. degree in Precision Measurement from Korea National University of Science and Technology. / Professional, R&D Team, Meter-Lab. Inc. / His research interest is machine learning algorithm for precision metrology.