Thin film thickness rapid measurement method based on Chirp-Z transformation

Abstract

The invention discloses a method for rapidly measuring the thickness of a thin film based on Chirp-Z transformation. The method comprises the following steps: 1) collecting spectral data; 2) data preprocessing; 3) CZT (linear frequency modulation Z transform) parameter configuration and CZT frequency spectrum calculation; according to the method, after the processed data collected by the spectrograph are received, data preprocessing is carried out, spectral analysis is carried out through Chirp-Z transformation, the spectrum peak value is obtained for accurate positioning, the film thickness is calculated through the effective refractive index, and therefore efficient and high-precision film thickness measurement is achieved.

Description

Technical Field

[0001] This invention relates to the field of thin film thickness measurement technology, and more specifically to a rapid method for measuring thin film thickness based on Chirp-Z transform. Background Technology

[0002] Optical methods for measuring thin film thickness, as a non-contact and highly efficient non-destructive testing method, hold an irreplaceable position in materials science and the semiconductor industry. Their core advantage lies in avoiding sample damage caused by mechanical contact, while enabling rapid measurements with nanometer to micrometer-level precision. This makes them particularly suitable for applications with stringent surface integrity requirements, such as wafer coatings, optical thin films, and flexible electronic devices.

[0003] Currently, reflectance spectroscopy is commonly used as the basis for detection. Conventional measurement methods typically rely on iterative optimization algorithms (such as the Levenberg-Marquardt algorithm or the conjugate gradient method) to repeatedly calculate and fit the data collected by the actual spectrometer, thereby retrieving the thickness parameters. However, these iterative algorithms have significant limitations: firstly, their convergence effect heavily depends on the selection of initial parameters. If the initial values ​​deviate significantly from the true solution, they are prone to getting trapped in local optima, leading to distorted output results; secondly, the iterative process itself has a high computational load, especially when dealing with complex film systems or low signal-to-noise ratio spectra, which is particularly time-consuming and may even result in an incorrect solution after a significant amount of time has been spent.

[0004] Another detection method is based on Fast Fourier Transform (FFT) measurement. The reflection spectrum is affected by the interference between the thin film layers, resulting in periodic fluctuations in the form of cosines. The formula for its spectral intensity is related to the thickness. Taking a single-layer thin film as an example, the reflection spectrum is proportional to the cosine of the wavelength. .in Let λ be the wavelength and n be the wavelength-dependent refractive index function of the material. The Fast Fourier Transform (FFT) can quickly extract thickness information by analyzing the transform domain of the reflection spectrum. While this method overcomes the dependence on initial values ​​to some extent, significant technical bottlenecks remain in measurement efficiency, accuracy, and adaptability. First, the FFT method requires extending and interpolating the spectral data to integer powers of 2, adding unnecessary redundant processing and computation. Second, FFT requires spectrum calculation across the entire frequency band, while thin film thickness measurement only relies on peak signals within a specific frequency range. Calculating across the entire frequency band wastes computational resources and cannot meet the rapid online measurement needs of industrial scenarios. Third, the frequency resolution of FFT is fixed by the data length. When the thin film reaches the nanometer scale or approaches the theoretical resolution, the fixed resolution of FFT cannot accurately capture the peak signal; FFT only provides good measurement results for thin films with thicknesses on the micrometer scale. Fourth, to prevent spectral leakage, FFT typically performs windowing, increasing preprocessing time and causing broadening of the main lobe of the spectrum, further reducing peak localization accuracy.

[0005] Therefore, providing a rapid method for measuring thin film thickness based on Chirp-Z transform is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0006] In view of this, the present invention provides a rapid thin film thickness measurement method based on Chirp-Z transform to solve the problems of initial value dependence, low computational efficiency and insufficient measurement accuracy of thin films in existing thin film thickness measurement methods.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A rapid method for measuring thin film thickness based on Chirp-Z transform includes the following steps:

[0009] 1) Spectral data acquisition;

[0010] 2) Data preprocessing: The collected spectral data is converted into a uniform wavenumber correlation spectrum, and then some boundary interference is removed by truncation.

[0011] 3) CZT parameter configuration and CZT spectrum calculation, where CZT is linear frequency modulated Z-transform;

[0012] 4) Peak selection: For monolayer films, there is only one peak, which can be directly calculated by extracting the maximum value, or selected after fitting with a Gaussian function; for multilayer films, set the amplitude threshold T=0.6×max(F(f)), or use the judgment criterion that the first derivative is zero and the second derivative is negative, or perform multi-peak fitting by fitting with a Gaussian function; where F(f) is the spectral function of the spectrum;

[0013] 5) Calculate the film thickness based on the peak value and equivalent refractive index: Convert the horizontal axis coordinates to thickness using the equivalent refractive index. The equivalent refractive index is calculated using the endpoint simplified method, multi-point linear weighted method, or integral method. The corresponding formula for the horizontal axis value is:

[0014]

[0015] in It is the equivalent refractive index. It is the minimum thickness when the algorithm is configured. N is the maximum thickness configured, N is the point index, and m is the number of sampling points.

[0016] By adopting the above technical solutions, the beneficial effects of the present invention are as follows:

[0017] After receiving the processed spectrometer data, the data is preprocessed, and Chirp-Z transform (CZT) is used for spectral analysis to obtain the precise location of the spectral peaks. The film thickness is then calculated using the equivalent refractive index, thereby achieving efficient and high-precision film thickness measurement.

[0018] Further, step 1) specifically includes using a spectrometer to obtain the reflection or transmission spectral intensity of the thin film, and calculating the reflectance of the thin film sample by comparing the theoretical reflectance and actual reflection intensity of the standard sheet.

[0019] Furthermore, the spectrometer is a dispersive spectrometer or an interferometric spectrometer. If an interferometric spectrometer is used, the acquired spectral data is directly a wavenumber-correlated spectrum, thus eliminating the need for step 2) to convert the acquired spectral data into a uniform wavenumber-correlated spectrum.

[0020] Furthermore, step 2) also includes smoothing the spectral data and removing DC signals.

[0021] Furthermore, the smoothing method can be any one of low-pass filtering, windowing filtering, and mean filtering; the DC signal removal method is to use spectral filtering or subtract the mean.

[0022] Furthermore, step 3) specifically includes the following steps:

[0023] Step 1: Calculate the core scaling factor and spiral parameters based on the input parameters:

[0024] Calculate the frequency scaling factor: The formula is ((fu-fl)*m) / (fs*(m-1)), where fs is the wavenumber sampling frequency of the spectral data and m is the number of output points;

[0025] Calculate the starting point factor A: a complex number exp(2.0*I*PI*fl / fs), which corresponds to the starting phase of the target frequency range, where I is the imaginary unit, PI is pi, and exp is the natural exponent.

[0026] Calculate the spiral factor W: a complex number exp(-2.0*I*PI / m*scale), used to control the spiral sampling trajectory on the Z plane;

[0027] Step 2: Pre-calculate the quadratic phase factor vector wk2:

[0028] The quadratic phase factor of each element is calculated iteratively, with the formula wk2(i) = exp(-(I * PI * scale * i²) / m). This vector is used for phase correction of the subsequent signal and construction of the convolution kernel.

[0029] Step 3: Calculate the input sequence correction coefficient Awk2:

[0030] Construct the vector ak, where each element is exp(-2.0*I_*PI_*fl / fs*i).

[0031] To calculate Awk2, multiply ak by the first n elements of wk2 point by point, where n is the length of the input data;

[0032] Step 4: Construct the convolution kernel and pre-compute its FFT:

[0033] Construct the convolution kernel temphn: take the elements from index 1 to n-1 in wk2, and concatenate them in reverse order with the elements from index 0 to m-1 in wk2;

[0034] The pointwise inverse of temphn is saved to Fwk2 for subsequent fast calculation of convolution via FFT;

[0035] Step 5: Preprocess the input signal:

[0036] Multiply the input signal by Awk2 to perform phase preprocessing on the original signal, and the result is denoted as gn;

[0037] Step 6: Accelerate convolution calculation using FFT transformation:

[0038] Calculate the Fourier transform Gn of gn;

[0039] Multiply Gn by the convolution kernel Fwk2 point by point;

[0040] Perform an inverse FFT on the product to obtain the convolution result y;

[0041] Step 7: Phase Correction

[0042] By extracting effective segments from y using the starting index and target length, the effective parts of the convolution result that are related to the target frequency range are selected, and redundant edge data is excluded.

[0043] The extracted valid results are multiplied point by point with the pre-calculated wk2 to obtain the spectrum of the target range.

[0044] Therefore, this invention provides a rapid method for measuring thin film thickness based on Chirp-Z transform. Compared with existing technologies, this invention has the following advantages:

[0045] It enables the focus on specific frequency ranges required for thin film thickness measurement for spectrum calculation, reducing redundant operations in data preprocessing and the waste of computing power in full-band calculation, shortening the overall time from spectral data acquisition to thickness output, reducing the influence of initial thickness values, reducing the error between measured values ​​and true values ​​during rapid measurement, realizing rapid detection of thin films, and meeting the needs of rapid online measurement of thin film thickness in fields such as semiconductors and flexible electronics. Attached Figure Description

[0046] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0047] Figure 1 The attached figure is a flowchart of a rapid film thickness measurement method based on Chirp-Z transform provided by the present invention.

[0048] Figure 2 The attached figure is a schematic diagram of using the FFT algorithm to measure the thickness of 800nm ​​thick SiO2. The upper figure is the spectral data that the spectrometer should theoretically measure, the lower figure is the relationship between thickness and intensity after Fourier transform of the measured data, and the lower figure is the relationship between thickness and intensity after Fourier transform of the spectral data calculated by the algorithm (i.e., the measurement result).

[0049] Figure 3 The attached figure is a schematic diagram of using the CZT algorithm to measure the thickness of 800nm ​​thick SiO2. The upper figure shows the spectral data that the spectrometer should theoretically measure, and the lower figure shows the relationship between thickness and intensity after Chirp-Z transformation of the measured data, as well as the relationship between thickness and intensity after Chirp-Z transformation of the spectral data calculated by the algorithm (i.e., the measurement result). Detailed Implementation

[0050] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0051] like Figures 1-3 As shown in the figure, this invention discloses a rapid method for measuring thin film thickness based on Chirp-Z transform, including the following steps:

[0052] Spectral data acquisition: The reflectance or transmission spectral intensity of the thin film is obtained using dispersive or interferometric spectrometers. The reflectance of the thin film sample is calculated by comparing the theoretical reflectance and the actual reflectance intensity of the standard sheet. This is referred to as spectral data.

[0053] 2) Data preprocessing: The collected spectral data is converted into a uniform wavenumber-correlated spectrum. If the data is obtained using an interferometric spectrometer, the spectral data is directly a wavenumber-correlated spectrum, and this step can be omitted. Furthermore, in order to avoid interference from changes in material properties in the low-band or light source intensity in the edge bands, some boundary interference data should be removed by truncation.

[0054] To further optimize the technical solution of the present invention, low-pass filtering, windowing filtering, mean filtering and other methods can be used to smooth the data; at the same time, spectral filtering or subtraction of the mean can be used to remove DC signals from the data to eliminate the influence of baseline peaks on spectral peaks.

[0055] 3) CZT parameter configuration and CZT spectrum calculation, where CZT is a linear frequency modulated Z-transform:

[0056] In the CZT parameter configuration, the CZT algorithm supports the free selection of analysis points on the unit circle or spiral path in the Z-plane, making it suitable for high-precision spectrum analysis and non-uniform signal processing. The parameters A and W that need to be configured are complex numbers, which together define the sampling path in the Z-plane. In this embodiment, sampling is performed on a segment of the arc of the unit circle, but sampling using a spiral path should also fall within the scope of this patent.

[0057] in

[0058]

[0059] This is the sampling starting point. Subsequently, each point is multiplied by W-1 based on the previous point, controlling the fineness of the frequency resolution. For ease of use, this embodiment only requires inputting the thickness range and the number of sampling points; the algorithm will automatically generate the corresponding parameters A and W. Details are as follows:

[0060] The sample film thickness has a rough guess range. For example, assuming an 800nm ​​SiO2 thin film sample, its range is estimated to be between 0-2000nm before measurement (the lower limit is denoted as fl and the upper limit as fu). The number of output sampling points m determines the accuracy of the measurement, that is, the resolution is (fu-fl) / m. Note that the resolution here cannot be less than the theoretical value of the Nyquist sampling theorem. After inputting the start point, end point, and number of sampling points, the algorithm will automatically generate the sampling path in the Z plane and convert it into parameters A and W.

[0061] The algorithm uses the Bluestein algorithm to accelerate the calculation of CZT and outputs the effective frequency band spectrum. The algorithm complexity is O((N)log(N)), which is comparable to FFT and much higher than the standard Discrete Fourier Transform (DFT).

[0062] The specific steps for CZT spectrum calculation include:

[0063] Step 1: Calculate the core scaling factor and spiral parameters based on the input parameters:

[0064] Calculate the frequency scaling factor: The formula is ((fu-fl)*m) / (fs*(m-1)), where fs is the wavenumber sampling frequency of the spectral data and m is the number of output points;

[0065] Calculate the starting point factor A: a complex number exp(2.0*I*PI*fl / fs), which corresponds to the starting phase of the target frequency range, where I is the imaginary unit, PI is pi, and exp is the natural exponent.

[0066] Calculate the spiral factor W: a complex number exp(-2.0*I*PI / m*scale), used to control the spiral sampling trajectory on the Z plane;

[0067] Step 2: Pre-calculate the quadratic phase factor vector wk2:

[0068] The quadratic phase factor of each element is calculated iteratively, with the formula wk2(i) = exp(-(I * PI * scale * i²) / m). This vector is used for phase correction of the subsequent signal and construction of the convolution kernel.

[0069] Step 3: Calculate the input sequence correction coefficient Awk2:

[0070] Construct the vector ak, where each element is exp(-2.0*I_*PI_*fl / fs*i).

[0071] To calculate Awk2, multiply ak by the first n elements of wk2 point by point, where n is the length of the input data;

[0072] Step 4: Construct the convolution kernel and pre-compute its FFT:

[0073] Construct the convolution kernel temphn: take the elements from index 1 to n-1 in wk2, and concatenate them in reverse order with the elements from index 0 to m-1 in wk2;

[0074] The pointwise inverse of temphn is saved to Fwk2 for subsequent fast calculation of convolution via FFT;

[0075] The above parameter pre-calculation process is not directly related to the input spectral data; for application scenarios with fixed thickness measurement range and fixed data length, this pre-calculation process can be performed in advance to shorten the calculation time of a single measurement;

[0076] Step 5: Preprocess the input signal:

[0077] Multiply the input signal by Awk2 to perform phase preprocessing on the original signal, and the result is denoted as gn;

[0078] Step 6: Accelerate convolution calculation using FFT transformation:

[0079] Calculate the Fourier transform Gn of gn;

[0080] Multiply Gn by the convolution kernel Fwk2 point by point;

[0081] Perform an inverse FFT on the product to obtain the convolution result y;

[0082] Step 7: Phase Correction

[0083] By extracting effective segments from y using the starting index and target length, the effective parts of the convolution result that are related to the target frequency range are selected, and redundant edge data is excluded.

[0084] The extracted valid results are multiplied point by point with the pre-calculated wk2 to obtain the spectrum of the target range.

[0085] 4) Peak selection: For monolayer films, there is only one peak, which can be directly calculated by extracting the maximum value, or selected after fitting with a Gaussian function; for multilayer films, set the amplitude threshold T=0.6×max(F(f)), or use the judgment criterion that the first derivative is zero and the second derivative is negative, or perform multi-peak fitting by fitting with a Gaussian function; where F(f) is the spectral function of the spectrum;

[0086] 5) Calculate the film thickness based on the peak value and equivalent refractive index: Convert the horizontal axis coordinate points into thickness using the equivalent refractive index. The equivalent refractive index is calculated using the endpoint simplified method (averaging the refractive indices of the maximum and minimum wavelengths), the multi-point linear weighted method (weighting and summing multiple points at equal intervals), or the integral method (integrating and averaging the refractive indices within the wavelength range). The corresponding formula for the horizontal axis value is:

[0087]

[0088] in It is the equivalent refractive index. It is the minimum thickness (fl) when the algorithm is configured. Where is the maximum configured thickness (fu), N is the point index, and m is the number of output sampling points.

[0089] After receiving the processed spectrometer data, this invention performs data preprocessing, uses Chirp-Z transform (CZT) for spectral analysis to obtain precise location of spectral peaks, and calculates the film thickness using the equivalent refractive index, thereby achieving efficient and high-precision film thickness measurement.

[0090] This invention uses an 800nm ​​SiO2 on Si wafer for algorithm testing:

[0091] In the FFT algorithm, the resolution of low-frequency signals is insufficient, making it impossible to identify 800nm ​​thin layers; while CZT focuses on the wavenumber range corresponding to 0-2000nm, ensuring resolution while the CZT algorithm runs at a speed comparable to the FFT algorithm, and the measurement results are very close to the theoretical values.

[0092] The actual value for the current use case is 800nm.

[0093] The FFT algorithm returned a value of 214.142798 nm and took 0.0409 seconds (Python verification code); the CZT algorithm returned a value of 804.201050 nm and took 0.0384 seconds (Python verification code).

[0094] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0095] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A rapid method for measuring thin film thickness based on Chirp-Z transform, characterized in that, Includes the following steps: 1) Spectral data acquisition; 2) Data preprocessing: The collected spectral data is converted into a uniform wavenumber correlation spectrum, and then some boundary interference is removed by truncation. 3) CZT parameter configuration and CZT spectrum calculation, where CZT is linear frequency modulated Z-transform; 4) Peak selection: For monolayer films, there is only one peak, which can be directly calculated by extracting the maximum value, or selected after fitting with a Gaussian function; for multilayer films, set the amplitude threshold T=0.6×max(F(f)), or use the judgment criterion that the first derivative is zero and the second derivative is negative, or perform multi-peak fitting by fitting with a Gaussian function; where F(f) is the spectral function of the spectrum; 5) Calculate the film thickness based on the peak value and equivalent refractive index: Convert the horizontal axis coordinates to thickness using the equivalent refractive index. The equivalent refractive index is calculated using the endpoint simplified method, multi-point linear weighted method, or integral method. The corresponding formula for the horizontal axis value is: ； in It is the equivalent refractive index. It is the minimum thickness when the algorithm is configured. N is the maximum thickness configured, N is the point index, and m is the number of sampling points.

2. The method for rapid measurement of thin film thickness based on Chirp-Z transform according to claim 1, characterized in that, Step 1) The specific steps include using a spectrometer to obtain the reflection or transmission spectral intensity of the thin film, and calculating the reflectance of the thin film sample by comparing the theoretical reflectance and the actual reflection intensity of the standard sheet.

3. The method for rapid measurement of thin film thickness based on Chirp-Z transform according to claim 2, characterized in that, The spectrometer is either a dispersive spectrometer or an interferometric spectrometer. If an interferometric spectrometer is used, the acquired spectral data is directly a wavenumber-correlated spectrum, thus eliminating the need for step 2) to convert the acquired spectral data into a uniform wavenumber-correlated spectrum.

4. The method for rapid measurement of thin film thickness based on Chirp-Z transform according to claim 1, characterized in that, Step 2) also includes smoothing the spectral data and removing DC signals.

5. The method for rapid measurement of thin film thickness based on Chirp-Z transform according to claim 4, characterized in that, The smoothing method is any one of low-pass filtering, windowing filtering, and mean filtering; the DC signal removal method is to use spectral filtering or subtract the mean.

6. The method for rapid measurement of thin film thickness based on Chirp-Z transform according to claim 1, characterized in that, Step 3) Specific steps include: Step 1: Calculate the core scaling factor and spiral parameters based on the input parameters: Calculate the frequency scaling factor: The formula is ((fu-fl)*m) / (fs*(m-1)), where fs is the wavenumber sampling frequency of the spectral data and m is the number of output points; Calculate the starting point factor A: a complex number exp(2.0*I*PI*fl / fs), which corresponds to the starting phase of the target frequency range, where I is the imaginary unit, PI is pi, and exp is the natural exponent. Calculate the spiral factor W: a complex number exp(-2.0*I*PI / m*scale), used to control the spiral sampling trajectory on the Z plane; Step 2: Pre-calculate the quadratic phase factor vector wk2: The quadratic phase factor of each element is calculated iteratively, with the formula wk2(i) = exp(-(I * PI * scale * i²) / m). This vector is used for phase correction of the subsequent signal and construction of the convolution kernel. Step 3: Calculate the input sequence correction coefficient Awk2: Construct the vector ak, where each element is exp(-2.0*I_*PI_*fl / fs*i). To calculate Awk2, multiply ak by the first n elements of wk2 point by point, where n is the length of the input data; Step 4: Construct the convolution kernel and pre-compute its FFT: Construct the convolution kernel temphn: take the elements from index 1 to n-1 in wk2, and concatenate them in reverse order with the elements from index 0 to m-1 in wk2; The pointwise inverse of temphn is saved to Fwk2 for subsequent fast calculation of convolution via FFT; Step 5: Preprocess the input signal: Multiply the input signal by Awk2 to perform phase preprocessing on the original signal, and the result is denoted as gn; Step 6: Accelerate convolution calculation using FFT transformation: Calculate the Fourier transform Gn of gn; Multiply Gn by the convolution kernel Fwk2 point by point; Perform an inverse FFT on the product to obtain the convolution result y; Step 7: Phase Correction By extracting effective segments from y using the starting index and target length, the effective parts of the convolution result that are related to the target frequency range are selected, and redundant edge data is excluded. The extracted valid results are multiplied point by point with the pre-calculated wk2 to obtain the spectrum of the target range.

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