A method for quickly calculating a large range and high-precision wafer thickness
By combining adaptive bandpass filtering, window function and fitting method, combined with FFT and Hilbert transformation, the problem of low wafer thickness measurement accuracy in the prior art is solved, and a high-precision, fast and stable thickness solution method is achieved.
Patent Information
- Application Number
- CN202211361994.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-11-02
- Publication Date
- 2025-05-27
- Estimated Expiration
- 2042-11-02
AI Technical Summary
The prior art when measuring thicker wafer thicknesses, the accuracy is lower, and improving the accuracy requires a large amount of FFT zero-compensation, which increases calculation time and reduces measurement stability.
Adaptive bandpass filtering, window function and fitting method are adopted, combined with FFT and Hilbert transformation, to provide a high-precision fast solution method for wafer thickness.
High-precision measurement of thicker wafers is achieved, with a large measurement range, high stability, short calculation time, and easy integration into the online measurement system.
Smart Images
Figure CN115682964B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of optical thickness measurement of wafers, and particularly to a method for rapidly calculating the thickness of wafers with a large measurement range and high precision. Background Art
[0002] As the most widely used substrate material for IC devices, when the back side of a wafer is thinned to meet the requirements of advanced packaging technology, strict geometric accuracy needs to be satisfied.
[0003] The current mainstream on-line measurement method for wafer thickness is infrared interferometry. Infrared light can penetrate single-crystalline silicon, reflect on the upper and lower surfaces to form interference, and then obtain its frequency information through Fourier transform to obtain the silicon wafer thickness. Among them, the two parameters of the measurement range and resolution of the spectrometer affect the thickness range and accuracy of silicon wafer measurement. When measuring a thicker silicon wafer, a high resolution requirement is imposed on the spectrometer. However, due to the band limitation of the current high-resolution spectrometer, the measurement accuracy is poor at this time. The most commonly used method to improve the accuracy currently is to achieve it through zero-padding of FFT. However, a large amount of zero-padding is often required to achieve the target calculation accuracy, which increases the calculation time and the measurement stability is poor at the same time. Summary of the Invention
[0004] In view of the above-mentioned prior art, based on spectral analysis and phase extraction, the present invention provides a method for rapidly calculating the thickness of wafers with a large measurement range and high precision by using the methods of adaptive band-pass filtering, window function, and fitting. The technical solutions are as follows:
[0005] A method for rapidly calculating the thickness of wafers with a large measurement range and high precision, comprising the following steps:
[0006] S1: Determine the refractive index n of the wafer to be measured at the measurement wavelength;
[0007] S2: Establish an optical model of the reflected optical electric field vector of the wafer to be measured;
[0008] S3: Obtain the applicable intervals of the Fourier transform method and the Hilbert transform method, and the method is as follows:
[0009] Select a thickness simulation range. Within this range, for different thickness values, use the optical model in S2 to generate the simulated reflection spectrum at this thickness and perform thickness calculation. When performing thickness calculation, use the Fourier transform method and the Hilbert transform method respectively to perform thickness calculation, and take the difference between the calculated thickness value and the theoretical value as the error. Take the standard deviation of the thickness calculated by the two methods as the repeatability evaluation criterion to obtain the respective error curves of the two methods; according to the respective error curves, find the applicable intervals of the Fourier transform method and the Hilbert transform method to obtain the thickness calculation threshold d;
[0010] S4: For the wafer to be measured, collect its original reflection spectrum, perform median filtering on the original reflection spectrum to filter out the fundamental frequency, and obtain the high-frequency signal;
[0011] S5: Convert the high-frequency signal of the wafer sample to be measured to the wavenumber domain, perform fast Fourier transform, and obtain the initial optical thickness value D according to the abscissa corresponding to the maximum amplitude of the fast Fourier transform curve of the oscillation waveform l ;
[0012] S6: If the initial optical thickness value D l is greater than d*n, obtain the physical thickness D of the wafer to be measured by the Hilbert transform method; otherwise, obtain the physical thickness D of the wafer to be measured by the Fourier transform method.
[0013] Furthermore, the method of S1 is as follows: Collect the original reflection spectrum of the standard sample, calculate its optical thickness by FFT, and divide the optical thickness by the physical thickness of the standard sample to obtain the refractive index n.
[0014] Furthermore, the method of S2 is as follows: Take the upper surface of the wafer to be measured as the 0 optical path reference plane, establish an upper surface reflection optical electric field vector model based on Snell's law, and obtain the optical model of the reflected optical electric field vector of the wafer to be measured.
[0015] Furthermore, the selected thickness simulation range in S3 is 100um - 770um; within this range, take a thickness value every 10um.
[0016] Furthermore, the thickness calculation threshold d = 300.
[0017] Furthermore, the method of obtaining the physical thickness D of the wafer to be measured by the Hilbert transform method in S6 is as follows:
[0018] S61: Extract the high-frequency signal of the original reflection spectrum and solve the phase by the Hilbert transform;
[0019] S62: Solve the physical thickness of the wafer to be measured according to the phase and refractive index:
[0020]
[0021] where phase_slope is the slope of the phase curve.
[0022] Furthermore, the method of obtaining the physical thickness D of the wafer to be measured by the Fourier transform method in S6 is as follows:
[0023] S61: Use the initial optical thickness value D lConstruct a digital band - pass filter with the corresponding spectral signal frequency as the center frequency; select a Butterworth filter as the digital band - pass filter to perform adaptive band - pass filtering on the high - frequency signal of the original reflection spectrum, and obtain the oscillating waveform after the second filtering.
[0024] S62: Apply a Hamming window to the filtered signal, zero - pad it, and perform the second fast Fourier transform; perform Gaussian spectral interpolation to obtain the exact peak position Peak. index :
[0025]
[0026] where k m is the abscissa index of the FFT peak of the second fast Fourier transform, and S is the mapping from the abscissa index to the FFT amplitude.
[0027] S63: Calculate the physical thickness of the wafer to be measured:
[0028]
[0029] where Peak index is the exact peak position index obtained by Gaussian spectral interpolation, λ min is the lower limit of the spectral data wavelength, and λ max is the upper limit of the spectral data wavelength.
[0030] Compared with the prior art, the beneficial effects of the present invention are:
[0031] (1) The measurement range is large, and it can stably measure the thickness of sub - millimeter - level wafers;
[0032] (2) The measurement system is simple, the measurement is fast, and it is easy to be integrated into the wafer thinning process to realize on - line measurement;
[0033] (3) The measurement accuracy is high. Brief Description of the Drawings
[0034] Figure 1 is the process flow block diagram of a method for high - range and high - precision rapid calculation of wafer thickness according to the present invention;
[0035] Figure 2 is the schematic diagram of a measurement system according to an embodiment of the present invention
[0036] The meanings of the reference numerals are as follows:
[0037] 1 - light source 2 - spectrometer 3 - fiber optic coupler 4 - lens group 5 - wafer to be measured
[0038] Figure 3 Original spectral signal
[0039] Figure 4High-frequency signals in the spectrum
[0040] Figure 5 FFT spectrum of the high-frequency spectral signal
[0041] Figure 6 Error curve
[0042] Figure 7 Phase curve
[0043] Figure 8 Filter response curve
[0044] Figure 9 Spectral signal passing through the band-pass filter
[0045] Figure 10 FFT spectrum of the filtered signal Detailed implementation manners
[0046] Based on spectral analysis and phase extraction, the present invention provides a method for rapid calculation with a large measurement range and high precision of the wafer thickness through adaptive filtering, window function, Gaussian spectral interpolation, etc. The detailed implementation manners of the present invention will be further described in detail below with reference to the accompanying drawings.
[0047] For the interference generated by light on the upper and lower surfaces of the wafer, its interference model can be expressed as:
[0048]
[0049] Where A(r) is the electric vector of the reflected light, and A(i) is the electric vector of the incident light. Here, λ is the wavelength, n is the refractive index of single-crystalline silicon, R is the reflectivity of the upper surface of single-crystalline silicon, and h is the thickness of single-crystalline silicon.
[0050] The oscillating waveform seen in the spectrometer is due to the presence of a term similar to sine e iσ in this model. The present invention aims to accurately, stably, and rapidly extract the frequency information of the term similar to sine in this signal.
[0051] As Figure 2 shown, the light emitted by the light source 1 is converged on the surface of the sample 5 through the optical fiber 3 and the lens 4. The light reflected from the upper and lower surfaces of the sample interferes, and the spectrometer 2 acquires the interference signal. The signal collected by the spectrometer is processed.
[0052] The specific embodiments are described as follows:
[0053] First, determine the refractive index of the sample to be measured at the measurement wavelength: Collect the original spectrum of the standard sample with the measurement system, calculate its optical thickness through FFT, and divide the optical thickness by the physical thickness of the standard sample to obtain the refractive index n. The calculation formula is as follows:
[0054] Formula 1:
[0055]
[0056] where N is the abscissa index of the peak value of the FFT result (the index starts from 0); λ min is the lower limit of the wavelength of the spectral data, and λ max is the upper limit of the wavelength of the spectral data; d is the physical thickness of the standard sample.
[0057] An optical model is established based on the sample, with the upper surface of the wafer to be measured as the 0 optical path reference plane. Based on Snell's law, an optical model of the reflected optical electric field vector on the upper surface is established to obtain the optical model of the reflected optical electric field vector of the wafer to be measured; the incident light is perpendicularly incident on the wafer to be measured to obtain the original light intensity signal of the light intensity distribution with respect to the wavelength, as Figure 3 shown.
[0058] The original reflected spectrum is median-filtered to filter out the fundamental frequency. A high-frequency signal is obtained as Figure 4 .
[0059] The high-frequency signal is converted from the wavelength to the wavenumber domain, and a fast Fourier transform is performed. According to the abscissa corresponding to the maximum amplitude of the fast Fourier transform curve of the oscillating waveform, the FFT spectrum is shown in Figure 5 , and the initial value D of its optical thickness is obtained from the thickness calculation formula 2 l .
[0060] Formula 2:
[0061]
[0062] where N is the abscissa index of the peak value of the FFT result (the index starts from 0); λ min is the lower limit of the wavelength of the spectral data, and λ max is the upper limit of the wavelength of the spectral data.
[0063] In order to more accurately obtain the applicable intervals of the Fourier transform method and the Hilbert transform method, it is first necessary to obtain the error curves of the corresponding methods. In the range of 100 μm - 770 μm, a thickness value is taken every 10 μm. The simulation reflected spectrum at this thickness is generated using the previous model and the thickness is solved. This process is repeated 100 times for each thickness, and the difference between the mean value of the solved thickness and the theoretical value is taken as the error. The standard deviation of the solved thickness is used as the repeatability evaluation criterion. Through the above method, an error curve as Figure 6 shown can be obtained.
[0064] The error curve shows that when using the simulation reflected spectrum, the Fourier transform method can achieve a measurement accuracy of 0.5% in the entire thickness range; the Hilbert transform method has a large error at low thicknesses due to multi-beam interference and frequency extraction problems, but performs extremely well when d > 300 μm.
[0065] According to the test performances of the Hilbert and FFT methods in terms of different thicknesses, a thickness calculation threshold of 300 um is set. When the initial value of the optical thickness is greater than 300 um, the Hilbert transform method is selected; when the thickness is less than 300 um, the Fourier transform method is selected.
[0066] If the initial value of the optical thickness is greater than the set thickness calculation threshold multiplied by the refractive index, the high-frequency signal of the original spectrum is extracted, and the phase is solved through the Hilbert transform ( Figure 7 ); the physical thickness of the sample is solved according to the phase and the refractive index (Formula 3);
[0067] Formula 3:
[0068]
[0069] where phase_slope is the slope of the phase curve and n is the refractive index
[0070] If the initial value of the optical thickness is less than the set thickness calculation threshold multiplied by the refractive index, a digital band-pass filter is constructed with this initial value as the center frequency; the Butterworth filter is selected for the digital band-pass filter, and the maximum attenuation Ap of the passband is set to 3; the minimum attenuation As to be achieved in the stopband is 20; see the filter response curve in Figure 8 , and the original spectrum is adaptively band-pass filtered to obtain the oscillatory waveform after the second filtering as shown in Figure 9 ;
[0071] Then, a Hamming window is added, zero-padding is performed, and the second fast Fourier transform is carried out. See the FFT spectrum in Figure 10 . According to the abscissa index k of the FFT peak m The Gaussian spectrum interpolation is used with Formula 4 to improve the FFT frequency measurement resolution to obtain the accurate peak position Peak index , and the accurate thickness D is calculated using the refractive index in combination with Formula 5 and Formula 1. S is the mapping from the abscissa index to the FFT amplitude.
[0072] Formula 4:
[0073]
[0074] where, k m is the abscissa index of the FFT peak (the index starts from 0), and S is the mapping from the abscissa index to the FFT amplitude.
[0075] Formula 5:
[0076]
[0077] where, Peak index is the accurate peak position index obtained by Gaussian spectrum interpolation, and λmin is the lower wavelength limit of spectral data, λ max is the upper wavelength limit of spectral data, and n is the refractive index.
[0078] Table 1 Measurement data
[0079]
[0080] By applying the above solution method, high-stability and high-precision measurement results have been obtained in the measurement of wafers with different thicknesses.
Claims
1. A method for rapidly calculating the thickness of a wafer with a large measurement range and high precision, comprising the following steps: S1: Determine the refractive index n of the wafer to be measured at the measurement wavelength; S2: Establish an optical model of the reflected optical electric field vector of the wafer to be measured; S3: Obtain the applicable ranges of the Fourier transform method and the Hilbert transform method as follows: Select a thickness simulation range. Within this range, for different thickness values, use the optical model in S2 to generate the simulated reflection spectrum at this thickness and perform thickness calculation. When performing thickness calculation, use the Fourier transform method and the Hilbert transform method respectively to calculate the thickness, and take the difference between the calculated thickness value and the theoretical value as the error. Take the standard deviation of the thickness calculated by the two methods as the repeatability evaluation criterion to obtain the error curves of the two methods respectively; According to their respective error curves, find the applicable ranges of the Fourier transform method and the Hilbert transform method to obtain the thickness calculation threshold d; S4: For the wafer to be measured, collect its original reflection spectrum, perform median filtering on the original reflection spectrum to filter out the fundamental frequency and obtain the high-frequency signal; S5: Convert the high-frequency signal of the wafer sample to be measured into the wavenumber domain, perform a fast Fourier transform, and obtain the initial value D of its optical thickness according to the abscissa corresponding to the maximum amplitude of the fast Fourier transform curve of the oscillation waveform l ; S6: If the initial value D of the optical thickness l is greater than d*n, the physical thickness D of the wafer to be measured is obtained by the Hilbert transform method; otherwise, the physical thickness D of the wafer to be measured is obtained by the Fourier transform method.
2. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The method of S1 is as follows: Collect the original reflection spectrum of the standard sample, calculate its optical thickness by FFT, and divide the optical thickness by the physical thickness of the standard sample to obtain the refractive index n.
3. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The method of S2 is as follows: Take the upper surface of the wafer to be measured as the 0 optical path reference plane, and establish an upper surface reflection optical electric field vector model based on Snell's law to obtain the optical model of the reflected optical electric field vector of the wafer to be measured.
4. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The thickness simulation range selected in S3 is 100um - 770um; within this range, take a thickness value every 10um.
5. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The thickness calculation threshold d = 300.
6. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The method for obtaining the physical thickness D of the wafer to be measured by the Hilbert transform method in S6 is as follows: S61: Extract the high-frequency signal of the original reflection spectrum and solve the phase by the Hilbert transform; S62: Solve the physical thickness of the wafer to be measured according to the phase and the refractive index: where phase_slope is the slope of the phase curve.
7. The method for rapidly calculating the thickness of a wafer with a large measurement range and high precision according to claim 1, characterized in that, The method for obtaining the physical thickness D of the wafer to be measured by the Fourier transform method in S6 is as follows: S61: Using the initial value D of the optical thickness l as the center frequency of the corresponding spectral signal frequency, a digital band-pass filter is constructed; The digital band-pass filter selects a Butterworth filter to perform adaptive band-pass filtering on the high-frequency signal of the original reflection spectrum to obtain the oscillating waveform after the second filtering; S62: Apply a Hamming window to the filtered signal, zero-pad it, and perform a second fast Fourier transform; perform Gaussian spectral interpolation to obtain the exact peak position Peak index : where k m is the abscissa index of the FFT peak of the second fast Fourier transform, and S is the mapping from the abscissa index to the FFT amplitude; S63: Calculate the physical thickness of the wafer to be measured: Among them, Peak index is the accurate peak position index obtained by Gaussian spectrum interpolation, λ min is the lower wavelength limit of the spectral data, λ max is the upper wavelength limit of the spectral data.