Method for measuring and calculating thickness of SiC epitaxial layer
By using the Fresnel reflectivity formula and multi-angle data processing, combined with Kalman filtering and the Nelder-Mead method, the accuracy and speed issues of SiC epitaxial layer thickness measurement were solved, achieving fast, non-destructive, and accurate thickness detection.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-19
- Publication Date
- 2026-03-13
AI Technical Summary
Existing methods for measuring the thickness of SiC epitaxial layers are insensitive to low-resistivity substrates, and their accuracy depends on the refractive index. Furthermore, existing technologies and equipment are complex or destructive, making it difficult to meet the needs of production lines for rapid, non-destructive, and accurate testing.
The Fresnel reflectivity formula is used in conjunction with multi-angle reflectivity data processing. Peaks and valleys are identified by Kalman filtering and sliding window averaging. The refractive index and thickness are fitted by the Nelder-Mead method, and the L-BFGS-B optimization method is used for rapid convergence to achieve thickness calculation.
It enables rapid, non-destructive, and accurate measurement of SiC epitaxial layer thickness, improving the robustness and computational efficiency of the inspection, and is suitable for online inspection on production lines.
Smart Images
Figure CN121658747A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of semiconductor testing technology, and more specifically, to a method for measuring the thickness of SiC epitaxial layers. Background Technology
[0002] Silicon carbide (SiC), as the core of third-generation wide-bandgap semiconductor materials, is an ideal substrate for manufacturing high-voltage, high-frequency, and high-temperature power devices. In the fabrication of SiC-based devices (such as MOSFETs, JFETs, and SBDs), a high-quality, homogeneous epitaxial layer with specific thickness and doping concentration needs to be grown on a conductive SiC substrate. The thickness of the epitaxial layer is one of the core indicators determining key performance parameters such as device breakdown voltage and on-resistance. Therefore, developing a fast, non-destructive, accurate, and online-detectable thickness measurement method is crucial.
[0003] Currently, the following methods are used for measuring the thickness of SiC epitaxial layers: Fourier transform infrared spectroscopy (FTIR) utilizes the interference of reflected light at the interfaces between the air / epitaxial layer and the epitaxial layer / substrate. The thickness of the epitaxial layer is calculated by analyzing the period of the interference fringes in the reflection spectrum or by using Fast Fourier Transform (FFT) analysis. This method is non-destructive to the sample, and the measurement process takes only tens of seconds, making it ideal for 100% in-line inspection on production lines. However, the measurement results require precise knowledge of the epitaxial layer's refractive index, which is affected by doping concentration and temperature, introducing systematic errors. For low-resistivity (highly doped) substrates, infrared light cannot penetrate, resulting in extremely weak or no interference signals, making measurement difficult or impossible.
[0004] Spectral ellipsometer (SE) measures the changes in polarization state (amplitude and phase) of polarized light after reflection from a sample surface. It then fits this result to an optical model (including surface roughness, interface layers, and epitaxial layer optical constants) to simultaneously obtain the epitaxial layer's thickness and optical constants (such as refractive index n and extinction coefficient k). For ultrathin epitaxial layers below the sub-micron level, this method offers significantly higher measurement accuracy than FTIR, reaching the angstrom level. However, it requires a complex and accurate optical model, and the modeling process and fitting algorithm are intricate, demanding highly skilled operators. Single-point measurements and fitting times are much longer than with FTIR, making it less suitable for rapid full-wafer inspection, and the equipment is also expensive.
[0005] Scanning electron microscopy (SEM) creates a clear cross-section of the sample through dissociation or focused ion beam (FIB) imaging, allowing direct observation and measurement of the epitaxial layer thickness under an electron microscope. However, this is a destructive method, typically only allowing measurement of a few points on the wafer, making it difficult to assess overall uniformity. In actual production, it is impossible to inspect all products; only random sampling is possible.
[0006] Given the problems with existing methods for measuring the thickness of SiC epitaxial layers, especially the insensitivity of FTIR to low-resistivity substrates and the dependence of accuracy on refractive index, as well as the core pain points of complex and slow SE modeling, those skilled in the art urgently need to develop an innovative and integrated detection scheme. Summary of the Invention
[0007] In view of this, the present invention proposes a method for measuring the thickness of SiC epitaxial layers, aiming to solve the problem that the accuracy of the FTIR detection method in the current technology depends on the refractive index.
[0008] This invention proposes a method for calculating the thickness of SiC epitaxial layers, comprising the following steps: 1) According to the Fresnel reflectance formula / perpendicular incident reflectance R s The expression for and parallel incident reflectivity R p The expression yields R ij The expression for ': Fresnel reflectance formula: Vertical incident reflectivity: Parallel incident reflectivity: ; The objective function can be obtained as follows: according to Solve for e ij achievable Right now ; Where e: film thickness, e ji : The film thickness, m, obtained from the i-th incident angle at the j-th incident angle. ij : The interference order of the wavenumber of the j-th peak at i incident angles, k ji : Wavenumber of the i-th peak at the j-th incident angle, θ ji : The angle of refraction of the wavenumber of the i-th peak at the j-th incident angle, α j : The j-th incident angle, α1=10°, α2=15°; 2) Perform Karman filtering and sliding window averaging on the reflectivity-wavelength curves obtained at incident angles of 10° and 15° respectively, and identify and record the peak and trough values. The Nelder-Mead method was used to fit the most accurate n values for each peak within the constraints. ij ; according to The derivation yields ; 3) Substitute the corresponding wavenumber series and refractive index into the formula. The thickness of the epitaxial layer is obtained from this.
[0009] Compared with the prior art, the beneficial effects of the present invention are as follows: This application introduces Kalman filtering and sliding window averaging in the data processing stage of the modeling process, effectively suppressing noise while preserving peak characteristics, making the input data smoother and more reliable. Secondly, in peak identification and determination of the interference order m, multiple parallel schemes and closed-loop verification are employed to avoid biases caused by a single assumption, improving the robustness of thickness and refractive index calculations. Thirdly, the model incorporates optimization methods such as L-BFGS-B and Nelder-Mead, enabling rapid convergence under constraints and ensuring computational efficiency and accuracy. This invention enhances the stability and reliability of the solution through multi-angle data verification, thickness assumption traversal, and standard deviation curve analysis. Attached Figure Description
[0010] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings: Figure 1 The spectral data for silicon carbide detection using a spectrometer at incident angles of 10° and 15° are as follows. Figure 1 In the image, A represents the spectrum at an incident angle of 10°. Figure 1 In the image, B represents the spectrum at an incident angle of 15°. Figure 2 The spectral data are obtained after Karman filtering and sliding window averaging, with an incident angle of 10°. Figure 3 The spectral data are obtained after Karman filtering and sliding window averaging, with an incident angle of 15°. Figure 4 The middle image shows the spectrum after smoothing at an incident angle of 10°. Figure 5 The middle image shows the spectrum after smoothing at an incident angle of 15°. Figure 6The refractive index distribution is shown when the incident angle is 10°. Figure 7 The refractive index distribution is shown when the incident angle is 15°. Figure 8 The distribution of inversion thickness with wavenumber is shown when the incident angle is 10°. Figure 9 This is a graph showing the distribution of inversion thickness as a function of wavenumber when the incident angle is 15°. Figure 10 This is a comparison chart of the measured reflectance and the predicted reflectance when the incident angle is 10°. Figure 11 This is a comparison chart of the measured reflectance and the predicted reflectance when the incident angle is 15°. Detailed Implementation
[0011] Various exemplary embodiments of the present invention will now be described in detail. This detailed description should not be considered as a limitation of the present invention, but rather as a more detailed description of certain aspects, features, and embodiments of the present invention. It should be understood that the terminology used in this invention is merely for describing particular embodiments and is not intended to limit the present invention.
[0012] Furthermore, regarding the numerical ranges in this invention, it should be understood that each intermediate value between the upper and lower limits of the range is also specifically disclosed. Every smaller range between any stated value or intermediate value within a stated range, and any other stated value or intermediate value within said range, is also included within this invention. The upper and lower limits of these smaller ranges may be independently included or excluded from the range.
[0013] Unless otherwise stated, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. While only preferred methods and materials have been described herein, any methods and materials similar or equivalent to those described herein may be used in the implementation or testing of this invention. All references to this specification are incorporated by way of citation to disclose and describe methods and / or materials associated with those references. In the event of any conflict with any incorporated reference, the content of this specification shall prevail.
[0014] Various modifications and variations can be made to the specific embodiments described in this specification without departing from the scope or spirit of the invention, as will be apparent to those skilled in the art. Other embodiments derived from this specification will also be apparent to those skilled in the art. This specification and embodiments are merely exemplary.
[0015] The terms “include,” “including,” “have,” “contain,” etc., used in this article are all open-ended terms, meaning that they include but are not limited to.
[0016] This invention provides a method for calculating the thickness of a SiC epitaxial layer, comprising the following steps: 1) According to the Fresnel reflectance formula / perpendicular incident reflectance R s The expression for and parallel incident reflectivity R p The expression yields R ij ’ The expression: Fresnel reflectance formula: Vertical incident reflectivity: Parallel incident reflectivity: ; The objective function can be obtained as follows: according to Solve for e ij achievable Right now ; Where e: film thickness, e ji : The film thickness, m, obtained from the i-th incident angle at the j-th incident angle. ij : The interference order of the wavenumber of the j-th peak at i incident angles, k ji : Wavenumber of the i-th peak at the j-th incident angle, θ ji : The angle of refraction of the wavenumber of the i-th peak at the j-th incident angle, α j : The j-th incident angle, α1=10°, α2=15°; 2) Perform Karman filtering and sliding window averaging on the reflectivity-wavelength curves obtained at incident angles of 10° and 15° respectively, and identify and record the peak and trough values. The Nelder-Mead method was used to fit the most accurate n values for each peak within the constraints. ij ; according to The derivation yields ; 3) Substitute the corresponding wavenumber series and refractive index into the formula. The thickness of the epitaxial layer is obtained from this.
[0017] Example 1 (1) According to the Fresnel reflectance formula / perpendicular incident reflectance R s The expression for and parallel incident reflectivity R p The expression yields R ij ’ The expression: Fresnel reflectance formula: Vertical incident reflectivity: Parallel incident reflectivity: ; The objective function can be obtained as follows: according to Solve for e ij achievable Right now ; Where e: film thickness, e ji : The film thickness, m, obtained from the i-th incident angle at the j-th incident angle. ij : The interference order of the wavenumber of the j-th peak at i incident angles, k ji : Wavenumber of the i-th peak at the j-th incident angle, θ ji : The angle of refraction of the wavenumber of the i-th peak at the j-th incident angle, α j : The j-th incident angle, α1=10°, α2=15°; (2) Silicon carbide was detected using a spectrometer at incident angles of 10° and 15° respectively, and spectral data were obtained, such as... Figure 1 As shown, Figure 1 In the image, A represents the spectrum at an incident angle of 10°. Figure 1 B in the image represents the spectrum at an incident angle of 15°.
[0018] The spectral data were processed by Karman filtering and sliding window averaging, resulting in the following data: Figure 2 and Figure 3 As shown, Figure 2 The data represents the spectral data with an incident angle of 10° after Karman filtering and sliding window averaging. Figure 3 The graph shows spectral data with an incident angle of 15° after Karman filtering and sliding window averaging. The blue line represents the original data, the orange line represents the data after Karman filtering, and the red line represents the data after sliding window averaging. It can be observed that the final curve is smoother and does not exhibit significant fluctuations after each processing step.
[0019] After smoothing, we identified and labeled the peaks and troughs of the processed spectral data, and then plotted the spectrum using these data. Figure 4 and Figure 5 . Figure 4 The middle image shows the spectrum at an incident angle of 10°. Figure 5 This is the spectrum at an incident angle of 15°.
[0020] (3) Use the Nelder-Mead method (simplex method) to fit the most accurate n for each peak under the constraints. ij : 3.1 Sorting Vertices First, calculate the objective function value Error(Nj) for each vertex and sort them in ascending order: Error(N0)≤Error(N1)≤……≤Error(N n-1 )≤Error(N n ) N0: Optimal point (minimum objective function value) N n Worst case (maximum objective function value) N n-1 Secondary near miss 3.2 Calculate the center point Excluding the worst-case vertex Nn, calculate the geometric centroids of the remaining vertices: ; 3.3 Reflection Through the worst reflection point N n To the other side of the center of gravity, a reflection point N is generated. n : A: Reflection coefficient (value 1) 3.4 Extension When the reflection point N r When N0 is better than the optimal point, it extends further in the reflection direction, generating an extension point N. e : γ: Expansion coefficient (value = 2) 3.5 contraction When the reflection point Nr is poor (Error(N) r Error(N) n-1 When contracting towards the center of gravity, a contraction is generated. Point N c.
[0021] If Error(N n-1 )≤Error(N r ) <Error(N n This is an external contraction: If Error(N0) > Error(N0), then it is an inward contraction: β is the shrinkage coefficient, usually taken as β=0.5 δ is the shrinkage coefficient, usually taken as δ=0.5.
[0022] (4) Model solution results According to the quantitative expression , can be obtained .
[0023] By substituting the wavenumber order and refractive index obtained in the above process into the formula, the thickness of the epitaxial layer can be calculated. Through research, we preset reasonable ranges for the epitaxial layer parameters, excluding results that significantly exceeded these ranges. Specifically, we limited the refractive index to between 1 and 4, and controlled the reflectivity to between 0 and 1.
[0024] The reliability of refractive index and thickness is determined by the degree of agreement between measured reflectance and theoretically predicted reflectance (i.e., using R...). 2 The coefficient of determination is the core indicator. After the above steps, the graph was plotted. Figures 6-11 ,in Figure 6 This is a diagram showing the refractive index distribution at an incident angle of 10°. Figure 7 The refractive index distribution is shown when the incident angle is 15°. Figure 6 as well as Figure 7 The blue dot represents the refractive index corresponding to the horizontal wavenumber, and the two red dashed lines represent the reasonable range of refractive index values obtained from the query. This chart not only helps us eliminate unreasonable values, but also allows us to intuitively observe the number of unreasonable values. If there are too many unreasonable values, it can prompt us to check whether there are any errors in the model. Figure 8 The graph shows the distribution of inversion thickness as a function of wavenumber when the incident angle is 10°. Figure 9 This is a distribution of inversion thickness as a function of wavenumber when the incident angle is 15°; through... Figure 8 and Figure 9 It can more quickly locate abnormal data points, which facilitates subsequent data cleaning, and can also analyze whether the thickness of the epitaxial layer is uniform. Figure 10 This is a comparison chart of measured and predicted reflectance when the incident angle is 10°. Figure 11 This is a comparison chart of the measured and predicted reflectance when the incident angle is 15°. The fitting effect is visually compared by observing the degree of overlap between the blue measured points and the orange predicted points. Finally, R is obtained. 2 Used to quantify goodness of fit.
[0025] Based on the optimized overlap of the images and the density of the measured points, the model solutions for refraction angles of 10 degrees and 15 degrees were calculated: The refraction angle is 10°; the average thickness is 9.2317 μm; and the refractive index fit at each peak is 1.0000.
[0026] The refraction angle is 15°; the average thickness is 9.0908 μm; and the refractive index fit of each peak is 0.9982.
[0027] Based on the data obtained above, the thickness of the SiC sample is e = 9.16125 μm.
[0028] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.
Claims
1. A method for calculating the thickness of a SiC epitaxial layer, characterized in that, Includes the following steps: 1) According to the Fresnel reflectance formula / perpendicular incident reflectance R s The expression for and parallel incident reflectivity R p The expression yields R ij The expression for ': Fresnel reflectance formula: Vertical incident reflectivity: Parallel incident reflectivity: ; The objective function can be obtained as follows: according to Solve for e ij achievable Right now Where e: film thickness, e ji : The film thickness, m, obtained from the i-th incident angle at the j-th incident angle. ij : The interference order of the wavenumber of the j-th peak at i incident angles, k ji : Wavenumber of the i-th peak at the j-th incident angle, θ ji : The angle of refraction of the wavenumber of the i-th peak at the j-th incident angle, α j : The j-th incident angle, α1=10°, α2=15°; 2) Perform Karman filtering and sliding window averaging on the reflectivity-wavelength curves obtained at incident angles of 10° and 15° respectively, and identify and record the peak and trough values. The Nelder-Mead method was used to fit the most accurate n values for each peak within the constraints. ij ; according to The derivation yields ; 3) Substitute the corresponding wavenumber series and refractive index into the formula. The thickness of the epitaxial layer is obtained from this.