A method for quantitatively characterizing in-plane stress of a substrate defect cluster region, in-plane stress of a substrate in an epitaxial wafer, and in-plane stress of an epitaxial layer

Abstract

The application discloses a method for quantitatively characterizing stress in a substrate defect aggregation area, stress in a substrate in an epitaxial wafer and stress in an epitaxial layer, and belongs to the technical field of semiconductor detection. The method comprises the following steps: first, testing the substrate to obtain peak displacement values and stress values of the substrate at different test points, and calculating a peak displacement-stress conversion coefficient mu; second, internally scanning and testing the substrate defect aggregation area by using a Raman tester to obtain a peak value, and calculating a peak displacement value, and calculating a stress value of the substrate defect aggregation area according to the peak displacement value and the peak displacement-stress conversion coefficient mu in the first step. The application can realize quantitative testing of stress in a substrate defect aggregation area, stress in a substrate in an epitaxial wafer and stress in an epitaxial layer, and can play a feedback role on each process parameter in the growth of a crystal or the growth of a substrate epitaxial layer according to the tested stress information, so that a substrate or an epitaxial wafer with low stress is obtained.

Description

TECHNICAL FIELD

[0001] The application relates to a method for quantitatively characterizing in-situ stress of a defect aggregation area of a substrate, in-situ stress of a substrate in an epitaxial wafer and in-situ stress of an epitaxial layer, and belongs to the technical field of semiconductor detection. BACKGROUND

[0002] SiC in-plane will cause defects such as microtubules, dislocations and the like due to lattice misalignment, lattice distortion or inclusion foreign matter. Taking microtubules as an example, stress birefringence patterns thereof can be observed under polarized light due to stress relaxation. At present, microtubules or stress lines can be observed by polarized light, but it is difficult to quantitatively observe the difference between the stress of the defect aggregation area and the stress of the normal area. Moreover, the stress in the substrate is complex, and the residual stress in the in-situ defect of the silicon carbide causes the silicon carbide crystal growth thermal stress and the in-situ stress, which is mixed with the surface stress introduced in the processing process of the silicon carbide substrate, and it is difficult to distinguish the surface stress and the in-situ stress, so there is a lack of a quantitative testing method for the in-situ stress of the defect aggregation area of the silicon carbide substrate.

[0003] SiC in-plane will cause defects such as microtubules, dislocations and the like due to lattice misalignment, lattice distortion or inclusion foreign matter. Taking microtubules as an example, stress birefringence patterns thereof can be observed under polarized light due to stress relaxation. At present, microtubules or stress lines can be observed by polarized light, but it is difficult to quantitatively observe the difference between the stress of the defect aggregation area and the stress of the normal area. Moreover, the stress in the substrate is complex, and the residual stress in the in-situ defect of the silicon carbide causes the silicon carbide crystal growth thermal stress and the in-situ stress, which is mixed with the surface stress introduced in the processing process of the silicon carbide substrate, and it is difficult to distinguish the surface stress and the in-situ stress, so there is a lack of a quantitative testing method for the in-situ stress of the defect aggregation area of the silicon carbide substrate.

[0004] The silicon carbide substrate can be further divided into semi-insulating and conductive substrates, according to their respective physical characteristics, semi-insulating silicon carbide is mainly used in high-power devices and radio frequency chips such as 5G base stations, while conductive silicon carbide is mainly used in new energy industries. Based on the differences in their respective physical characteristics and application directions, semi-insulating SiC is used as a heteroepitaxial substrate to epitaxially grow GaN on the Si surface of the SiC substrate, while conductive SiC is used as a homoepitaxial substrate to epitaxially grow SiC on the Si surface of the SiC substrate.

[0005] Due to the CVD method based epitaxial process, epitaxial thin films are further grown on the substrate end, so that the epitaxial layer and the substrate have lattice mismatch during the growth process, thereby introducing stress into the substrate and the epitaxial layer. The lattice mismatch of heteroepitaxy is more serious than that of homoepitaxy. Quantitative characterization of the in-situ stress of the substrate in the epitaxial wafer and the in-situ stress of the epitaxial layer can provide feedback for the epitaxial process. However, there is still a lack of a quantitative testing method for the in-situ stress of the substrate in the epitaxial wafer and the in-situ stress of the epitaxial layer. SUMMARY

[0006] In order to solve the above problems, a method for quantitatively characterizing the in-situ stress of a substrate defect aggregation area, the in-situ stress of a substrate in an epitaxial wafer and the in-situ stress of an epitaxial layer is provided. The method can quantitatively test the in-situ stress of a substrate defect aggregation area, the in-situ stress of a substrate in an epitaxial wafer and the in-situ stress of an epitaxial layer, and can obtain complete stress information of a substrate or an epitaxial layer. According to the stress information, feedback can be provided to the process parameters in the growth of a crystal or the growth of an epitaxial layer of a substrate, so as to optimize the process parameters in the growth process of a crystal or the growth process of an epitaxial layer of a substrate, reduce the generation of internal defects of a substrate or reduce the in-situ stress introduced in the growth of an epitaxial layer, and thus obtain a substrate or an epitaxial wafer with low stress.

[0007] According to an aspect of the present application, a method for quantitatively characterizing the in-situ stress of a substrate defect aggregation area is provided, comprising the following steps:

[0008] First step: testing a substrate to obtain actual peak position values and stress values of the substrate at different test points, calculating the difference between the actual peak values and the standard substrate peak position, i.e. the peak position displacement values of different test points, and calculating the peak position displacement and stress conversion coefficient μ according to the peak position displacement values and the stress values;

[0009] Second step: using a Raman tester to perform internal scanning test on a substrate defect aggregation area, the diameter of the substrate is more than 150 mm, the defects in the defect aggregation area include at least one of microtubules, dislocations, polytypes and carbon inclusions, the internal region is a region extending from the surface of the substrate to 30%-60% of the thickness of the substrate, peak position values are obtained, peak position displacement values are calculated, and the in-situ stress value of the substrate defect aggregation area is calculated according to the peak position displacement values and the peak position displacement and stress conversion coefficient μ in the first step.

[0010] The substrate is cut from a crystal, and the thickness of the surface damage layer of the substrate will be different due to different cutting processes. However, even with the worst cutting process, the thickness of the introduced surface damage layer does not exceed 30% of the thickness of the substrate. Therefore, the region extending from the surface of the substrate to 30%-60% of the thickness of the substrate is the in-situ stress region of the substrate, which is not affected by later cutting and other factors.

[0011] Optionally, before the internal scanning test of the substrate defect aggregation area by the Raman tester in the second step, an axial test of the substrate defect aggregation area by the Raman tester is performed first, and the depth of the surface damage layer and the damage stress extension layer of the substrate defect aggregation area is determined according to the stress curve fed back by the axial test.

[0012] Optionally, before the second step of scanning the interior of the defect aggregation area of the substrate using the Raman tester, the surface of the substrate is first tested for curvature, and the curvature of the substrate at the defect aggregation area is obtained. Based on the curvature, in the second step of scanning, an equal-interval focusing mode and / or an equal-focusing plane mode is selected, wherein in the equal-interval focusing mode, the distance between the Raman laser spot at each test point and the surface of the substrate is equal, and in the equal-focusing plane mode, the Raman laser spot at each test point is in the same horizontal plane.

[0013] According to another aspect of the present application, a method for quantitatively characterizing the in-situ stress of a substrate in an epitaxial wafer is provided, comprising the following steps:

[0014] First step: testing the substrate to obtain the actual peak position value and stress value of the substrate at different test points, calculating the difference between the actual peak value and the standard substrate peak position, i.e., the peak position displacement value at different test points, and calculating the peak position displacement-stress conversion coefficient μ based on the peak position displacement value and the stress value.

[0015] Second step: scanning the interior of the substrate in the epitaxial wafer using a Raman tester, wherein the interior of the substrate is the area extending inward from the surface of the substrate to 30%-60% of the thickness of the substrate, obtaining the peak position value, calculating the peak position displacement value, and calculating the in-situ stress value of the substrate in the epitaxial wafer based on the peak position displacement value and the peak position displacement-stress conversion coefficient μ in the first step.

[0016] Optionally, before the second step of scanning the interior of the defect aggregation area of the substrate using the Raman tester, the epitaxial wafer is first subjected to an axial test using the Raman tester, and the depth of the epitaxial layer, the surface damage layer, and the damage stress extension layer in the epitaxial wafer is determined based on the stress curve fed back by the axial test.

[0017] Optionally, before the second step of scanning the interior of the defect aggregation area of the substrate using the Raman tester, the surface of the epitaxial wafer is first tested for curvature, and the curvature of the surface of the epitaxial wafer is obtained. Based on the curvature, in the second step of scanning, an equal-interval focusing mode and / or an equal-focusing plane mode is selected, wherein in the equal-interval focusing mode, the distance between the Raman laser spot at each test point and the surface of the substrate is equal, and in the equal-focusing plane mode, the Raman laser spot at each test point is in the same horizontal plane.

[0018] According to another aspect of the present application, a method for quantitatively characterizing the in-situ stress of a substrate in an epitaxial wafer is provided, comprising the following steps:

[0019] First step: testing the substrate to obtain the actual peak position value and stress value of the substrate at different test points, calculating the difference between the actual peak value and the standard substrate peak position, i.e., the peak position displacement value at different test points, and calculating the peak position displacement-stress conversion coefficient μ based on the peak position displacement value and the stress value.

[0020] Second step: using a Raman tester to scan and test the internal part of the epitaxial layer, which is the area extending from the surface of the epitaxial wafer inward to 30% to 60% of the thickness of the epitaxial layer, to obtain the peak value and calculate the peak displacement value, and then calculate the in-situ stress value of the epitaxial layer according to the peak displacement value and the peak displacement-stress conversion coefficient μ in the first step.

[0021] Before the second step of scanning and testing the internal part of the epitaxial layer using a Raman tester, the surface of the epitaxial wafer is first tested for bending degree, and the bending degree of the surface of the epitaxial wafer is obtained. In the second step of scanning and testing, an equal-interval focusing mode is selected, in which the distance between the Raman laser spot at each test point and the surface of the substrate is equal.

[0022] In the above-mentioned substrate defect aggregation area in-situ stress test, epitaxial wafer in-situ stress test and epitaxial layer in-situ stress test, the calculation step of the peak displacement-stress conversion coefficient μ is as follows:

[0023] Firstly, the interplanar spacing dx in the substrate or epitaxial layer is tested by a tester, and the difference between the interplanar spacing dx and the theoretical interplanar spacing d0 is obtained to obtain the interplanar spacing difference △d at different test points. The stress value of the substrate at different test points is obtained by using the Hook's law σ / S=E×(△d / d0) or the stress-strain formula σ=E×△d.

[0024] Then, the substrate is scanned and tested by using a Raman tester to obtain the actual peak value at different test points, and the difference between the actual peak value and the standard peak value is calculated, i.e. the peak displacement value at different test points, and the ratio of the peak displacement value to the stress value is the peak displacement-stress conversion coefficient μ.

[0025] Preferably, the tester for testing the interplanar spacing dx in the substrate is selected from any one or more of an XRD diffractometer, a synchrotron tester, a white light tester, a neutron diffractometer, a scanning electron microscope and a transmission electron microscope.

[0026] When an XRD diffractometer is used to test the substrate or epitaxial layer, the interplanar spacing dx at different positions of the substrate is calculated by the Bragg diffraction formula.

[0027] In the above-mentioned substrate defect aggregation area in-situ stress test, epitaxial wafer in-situ stress test and epitaxial layer in-situ stress test, the surface of the substrate is taken as the zero point, and the test range of the axial test is from -150 μm to -30 μm inside the substrate to above the surface of the substrate or epitaxial layer.

[0028] Preferably, the surface of the substrate is taken as the zero point, and the test range of the axial test is from -100 μm to -30 μm inside the substrate to above the surface of the substrate or epitaxial layer.

[0029] In the substrate defect aggregation area in-situ stress test, the epitaxial wafer in-situ stress test and the epitaxial layer in-situ stress test, the focusing depth (d) and the spot diameter (D) of the Raman spot are determined according to the thicknesses of the epitaxial layer, the surface damage layer and the stress extension layer, so that the Raman spot is located inside the substrate or the epitaxial layer, and the calculation formula for selecting the focusing depth (d) is:

[0030] wherein Dp is the penetration depth of the Raman laser in the sample, a is the absorption coefficient, λ is the laser wavelength, and k is the extinction coefficient, the penetration depth (Dp) obtained by calculation is used to select the appropriate focusing depth (d), and the requirement to be met is that the penetration depth (Dp) is greater than the focusing depth (d), in the substrate defect aggregation area in-situ stress test and the epitaxial wafer in-situ stress test, the Raman laser spot is located inside the substrate, and in the epitaxial layer in-situ stress test, the Raman laser spot is located inside the epitaxial layer.

[0031] The calculation formula of the spot diameter (D) is:

[0032] wherein λ is the laser wavelength, and Na is the numerical aperture of the device.

[0033] In the substrate defect aggregation area in-situ stress test, the epitaxial wafer in-situ stress test and the epitaxial layer in-situ stress test, the diameter of the substrate is greater than 200 mm.

[0034] In the substrate defect aggregation area in-situ stress test, the epitaxial wafer in-situ stress test and the epitaxial layer in-situ stress test, the specific method of the bending test is:

[0035] (1) Ensure that the test platform is horizontal, and select a fixed origin (0, 0, 0) on the test platform;

[0036] (2) Use the Raman to automatically focus to the position with the strongest test signal on the surface of the substrate or the epitaxial wafer, mark this position as (a, b, c), and then adjust the Raman laser to move downward by a distance H, and focus the position to (a, b, c-H);

[0037] (3) Adjust the Raman laser test position to (0, 0, c-H) to perform an equal focusing plane test, obtain a peak intensity scan graph of the surface of the substrate or the epitaxial wafer, calculate the distance from the focusing plane at different positions to the surface of the substrate or the epitaxial wafer, and thus determine the bending degree of the surface of the substrate or the epitaxial wafer.

[0038] Optionally, the value of H is in the range of 5-150 μm, and preferably in the range of 5-100 μm.

[0039] In the substrate defect aggregation area in-situ stress test, the epitaxial wafer in-situ stress test and the epitaxial layer in-situ stress test, the peak position value and the peak position displacement value in the second step are obtained by fitting, and the function used in the fitting includes Gaussian, Lorentz, GaussLor, AGauss, Aloren and AgaussLor.

[0040] In the substrate defect aggregation area in-situ stress test and the epitaxial wafer in-situ stress test, S1 is the in-situ stress value calculated by the equal-interval focusing test mode, S2 is the in-situ stress value calculated by the equal-focusing plane mode, and -0.2MPa≤S1-S2≤0.2MPa; or

[0041] V1 is the peak position value obtained by fitting the equal-interval focusing mode, V2 is the peak position value obtained by fitting the equal-focusing plane mode, and -0.0005cm -1 V1-V2<0.0005cm -1 .

[0042] Preferably, -0.05MPa≤S1-S2≤0.05MPa; or

[0043] -0.0001cm -1 V1-V2<0.0001cm -1 .

[0044] According to the bending degree of the substrate surface, the influence of the bending degree of the substrate surface on the in-situ stress of the substrate can be eliminated in the substrate defect aggregation area in-situ stress test and the epitaxial wafer in-situ stress test, and according to the bending degree of the epitaxial wafer surface, the influence of the bending degree of the epitaxial wafer surface on the in-situ stress of the epitaxial layer can be eliminated in the epitaxial layer in-situ stress test.

[0045] Optionally, the substrate is selected from a silicon carbide substrate.

[0046] Preferably, the crystal form of the silicon carbide substrate is one of 2H-SiC, 4H-SiC, 6H-SiC, 3C-SiC and 15R-SiC.

[0047] In the substrate defect aggregation area in-situ stress test, the epitaxial wafer in-situ stress test and the epitaxial layer in-situ stress test, the in-situ stress value calculated in the second step includes in-situ absolute stress and in-situ relative stress.

[0048] In the calculation of the in-situ absolute stress, the peak position displacement value obtained by calculation is the difference between the actual peak position of the test point and the standard peak position; in the calculation of the in-situ relative stress, the peak position displacement value obtained by calculation is the difference between the actual peak position of the test point and the actual peak position of the relative point.

[0049] The peak value of the relative point can be a reference value obtained in the whole test plane, and the peak value of the relative point is obtained by performing a relevant numerical operation on the value of each test point and the reference value, which includes but is not limited to any one of the average value, the median, the mode and other statistical function calculation results of all test values in the whole test plane.

[0050] The absolute stress can reflect the difference between the substrate and a perfect SiC crystal without defects, and is used to determine the stress level of the SiC substrate. However, it is difficult to realize a perfect SiC substrate without defects by current technology, and therefore the relative stress can be used to determine the in-plane relative stress distribution of the substrate.

[0051] For example, when the absolute stress of the substrate is large and the relative stress is small, it indicates that there are many in-plane defects, but the stress distribution is uniform, the crystal quality is poor, and the uniformity of the quality in each region of the substrate is poor; if the absolute stress is small and the relative stress is large, the overall substrate quality is good, but there is an abnormal quality deviation in a local area.

[0052] The beneficial effects of the present application include but are not limited to:

[0053] 1. The present application calculates the stress values at different positions of the substrate by calculating the crystal face spacing and the crystal face spacing difference at different positions of the substrate, and calculates the peak displacement and the stress conversion coefficient μ according to the peak displacement value obtained by Raman testing, thereby providing a calculation basis for quantitative testing of the in-situ stress of the substrate.

[0054] 2. The present application determines the thickness of the surface damage layer and the damage stress extension layer of the substrate by axial testing of the substrate, and tests the inside of the substrate by Raman laser according to the calculation and selection of the Raman laser penetration depth and the spot diameter, and realizes quantitative testing of the in-situ stress of the defect aggregation area of the substrate and the in-situ stress of the substrate in the epitaxial wafer according to the calculation of the peak displacement value and the peak displacement and stress conversion coefficient μ.

[0055] 3. The present application tests the bending degree of the substrate, and can eliminate the influence of the bending and warping of the substrate surface when using equal-interval focusing testing or equal-focusing plane testing according to the calculated bending degree, thereby realizing quantitative testing of the in-situ stress of the defect aggregation area of the substrate or the in-situ stress of the substrate in the epitaxial layer and improving the testing accuracy of the testing method.

[0056] 4. The present application provides a method for quantitatively characterizing the in-situ stress of the defect aggregation area of the substrate, which can quantitatively test the difference between the stress of the defect aggregation area and the stress of the normal area, and can quantitatively test the stress existing in the lattice distortion of the substrate which has not been released by defects, thereby improving the accuracy of the stress detection of the defect aggregation area of the substrate.

[0057] 5. The method for quantitatively characterizing the in-situ stress of the substrate and the in-situ stress of the epitaxial layer according to the present application, which can quantitatively test the in-situ stress of the substrate and the in-situ stress of the epitaxial layer during the epitaxial process, thereby calculating the stress value introduced in the substrate by the lattice matching of the epitaxial layer and the substrate, and playing a feedback role on the process parameters during the growth of the epitaxial layer of the substrate. BRIEF DESCRIPTION OF DRAWINGS

[0058] The accompanying drawings, which are included to provide a further understanding of the present application and are incorporated in and constitute a part of this application, illustrate embodiments of the present application and together with the description serve to explain the present application. In the drawings:

[0059] Figure 1 The axial peak intensity distribution graph of the Si substrate involved in the embodiment 2 of the present application and the 4H-SiC substrate.

[0060] Figure 2 The surface type principle graph of the SiC substrate for the Raman test involved in the embodiment 2 of the present application.

[0061] Figure 3 The principle graph of the Raman axial test of the SiC substrate and the stress axial distribution graph involved in the embodiment 3 of the present application.

[0062] Figure 4 The schematic graph of the Raman equal-interval focusing mode and the equal-focusing plane mode for testing the in-situ stress of the substrate involved in the embodiment 4 of the present application.

[0063] Figure 5 The substrate defect aggregation area test graph involved in the embodiment 5 of the present application.

[0064] Figure 6 The in-situ stress test graph of the substrate before and after the growth of the epitaxial layer involved in the embodiment 6 of the present application.

[0065] Figure 7 The in-situ stress test graph of the epitaxial layer in the embodiment 7. DETAILED DESCRIPTION

[0066] The present application will be described in detail below with reference to the embodiments, but the present application is not limited to these embodiments. The quantitative test method of the present application can be used to detect the surface stress and the in-situ stress of the substrate, and the following embodiments are described by taking the 4H-SiC single crystal as an example, which does not constitute a limitation on the test method.

[0067] The stress principle of the Raman test substrate in the following examples is as follows: a monochromatic laser of a certain wavelength is irradiated onto the surface of the substrate by converging through a series of light paths by an objective lens, and the interaction of the laser photons and the substrate lattice causes photon scattering. The occurrence of Raman scattering is related to the lattice vibration of the substrate itself. When residual stress exists in the sample, tensile stress or compressive stress will cause the atomic bond length to be elongated or shortened accordingly, so that the vibration frequency of the atom will be reduced or increased, which is reflected in the Raman spectrum as the characteristic peak of the substrate moving to low frequency or high frequency, i.e. peak shift.

[0068] Quantitative test method of substrate stress in Example 1

[0069] (1) Peak shift and calculation of stress conversion coefficient μ

[0070] Firstly, the single crystal XRD diffractometer is used to test the silicon carbide substrate and analyze the substrate according to the inverse lattice space. The parameters of the standard sample 4H-SiC substrate are compared with the standard parameters. Then, according to the Bragg diffraction equation: 2dsinθ = nλ, where d is the interplanar spacing, θ is the diffraction half angle, n is the diffraction order, and λ is the wavelength, the θ and d of different substrates are tested, and it is known that the tensile stress or compressive stress in the crystal causes the interplanar spacing d to change correspondingly. According to the different interplanar spacings d0 and d x , d x are divided into d1 and d2, where d0 is the theoretical interplanar spacing, d1 is the interplanar spacing after being subjected to tensile stress, d1 is greater than d0, d2 is the interplanar spacing after being subjected to compressive stress, d2 is less than d0, the difference between the interplanar spacing dx and the theoretical interplanar spacing d0 is obtained, and the difference between the interplanar spacing dx and the theoretical interplanar spacing d0 is obtained. The difference △d at different test points is △d = d1-d0 or △d = d0-d2. Then, the stress value σ of the silicon carbide substrate at different test points is calculated by combining Hooke's law or the stress-strain formula:

[0071] The calculation formula of Hooke's law is σ / S = E × (△d / d0), where S is the area of σ, which is a fixed value. When Hooke's law is applicable to microcosmic calculation, S is the micro area, which can be counted as 1, E is the Young's modulus of the substrate, which is a fixed parameter of the substrate, △d is the interplanar spacing difference at different positions, d0 is the theoretical interplanar spacing, and σ is the stress value to be calculated.

[0072] The calculation formula of stress-strain is σ = E × △d, where E is the Young's modulus of the substrate, which is a fixed parameter of the substrate, △d is the interplanar spacing difference at different positions, and σ is the stress value to be calculated.

[0073] According to the Raman tester, the actual peak value of the substrate at different test points is obtained, the difference between the actual peak value and the standard peak value is calculated, that is, the peak displacement value Δv of different test points; and the ratio of the peak displacement value at different test points to the stress value is the peak displacement and stress conversion coefficient μ, which ranges from -125 to -2500.

[0074] (2) Quantitative test method of substrate stress

[0075] First, the Raman test point is set. Single-point test can be performed, or Mapping scanning can be performed on the entire substrate. Therefore, the single-point test is set for sporadic coordinate points, and the Mapping scanning is set for 10x10-50x50 and more points, which are not equal. The appropriate test method can be selected according to the properties of the substrate to determine the Raman test coordinates of the silicon carbide substrate.

[0076] Then, the test results are outputted, which include but are not limited to editable peak, peak intensity data and Mapping.

[0077] Subsequently, the output data is fitted with a peak, and the fitting function includes but is not limited to Gaussian, Lorentz, GaussLor, AGauss, Aloren and AGaussLor, to obtain the accurate peak value and the peak displacement value, and the peak is accurate to more than one decimal place.

[0078] Finally, according to the peak displacement and stress conversion coefficient μ calculated in (1) and the peak displacement value Δv obtained in the Raman test, the stress value of the silicon carbide is calculated, and the calculation formula is: σ (MPa) = μ × Δv (cm -1 ), and finally the stress test image of the substrate is obtained.

[0079] Example 2: Substrate surface or epitaxial wafer surface bending test

[0080] Because SiC is prepared from a crystal through a series of processing to obtain a wafer, a large amount of external mechanical stress is applied, and it is difficult to ensure that the surface is completely horizontal, that is, there is a certain bending degree. In order to eliminate the influence of the bending degree of the substrate surface on the substrate surface stress and the body stress test, the bending degree of the substrate needs to be tested first. According to the bending degree, the substrate surface stress is selected for equal-interval focusing test, and the substrate body stress is selected for equal-interval focusing test or equal-focusing plane test.

[0081] After the bending test, a single crystal silicon is used to calibrate the tester. First, the single crystal Si is tested by Z-axis Mapping, and it is confirmed that it is a Gaussian distribution. The surface of the single crystal Si substrate is taken as zero point, and the axial Mapping test is performed from -100 μm inside the single crystal Si substrate to +20 μm above the single crystal Si substrate. The change of the Raman signal intensity is analyzed, and the result is as follows:Figure 1 As shown, Figure 1 (a) is the axial peak intensity distribution diagram of a single crystal Si substrate, the single crystal Si substrate is a non-transparent material, so its peak intensity shows a Gaussian distribution, with the surface zero position as the symmetry axis; then the single crystal Si is subjected to surface focusing single-point testing, and the single crystal Si Raman peak position is corrected to 520.70 cm -1 At this time, it indicates that the instrument optical path has been calibrated.

[0082] The following takes a 4H-SiC substrate as an example to illustrate the testing process of surface curvature, and the corrected tester is used to test the 4H-SiC substrate in the Z direction, the Raman laser is from-100 μm inside the 4H-SiC substrate to +20 μm above the 4H-SiC substrate, Figure 1 (b) is the axial peak intensity distribution diagram of a 4H-SiC substrate, the 4H-SiC substrate and other silicon carbide have the characteristics of transparent material, so the Raman laser can be shot into the inside of the silicon carbide substrate, and as the depth of the Raman laser increases, the feedback signal gradually weakens. Other silicon carbide includes semi-insulating, conductive and 3C, 4H, 15R, 6H and other different crystal types of silicon carbide substrates, so the Raman peak intensity of the silicon carbide substrate gradually decreases as the depth of the Raman laser increases.

[0083] Through the automatic focusing function of the Raman device, the strongest peak of the test signal on the substrate surface can be automatically set to find, so that the function can be used to measure the surface curvature of the substrate. The specific method is as follows:

[0084] (1) First, ensure the level of the test platform, select a fixed origin (0, 0) on the test platform, and mark the point as a three-dimensional coordinate (0, 0, 0);

[0085] (2) According to the substrate thickness, the objective lens is raised to a height not in contact with the substrate, and an arbitrary point is automatically focused to the strongest position of the substrate surface test signal, which is marked as (a, b, c). Then adjust the Raman laser to move down H distance, and the focusing position is (a, b, c-H);

[0086] (3) Adjust the Raman laser test position to (0, 0, c-H) to perform equal focusing plane testing, and obtain the two-dimensional peak intensity scanning diagram of the substrate surface. According to the peak intensity distribution relationship of the silicon carbide substrate in Figure 1 and the principle in Figure 2 , the distance between the focusing plane of the Raman laser and the surface of the silicon carbide substrate can be obtained, so that the in-plane curvature of the substrate can be obtained, and the in-plane curvature distribution diagram of the substrate can be obtained.

[0087] The same steps and operation methods can be used to test the surface curvature of the epitaxial wafer.

[0088] Example 3 Axial test to distinguish surface stress and bulk stress of substrate

[0089] Since the substrate is affected by the temperature field and the introduction of the inclusion during the crystal growth process, there are thermal stress and residual stress in the crystal. After the crystal is cut, ground and polished to obtain the substrate, the above-mentioned processing of the crystal forming the substrate will cause surface lattice distortion, thereby generating surface stress. In order to further distinguish the surface stress and bulk stress of the substrate, according to the light transmission characteristics of silicon carbide, the substrate is subjected to axial test, i.e. Z direction Mapping, taking the surface of the substrate as zero point, and the test range of the axial test is from-150 μm to-30 μm in the substrate to above the surface of the substrate.

[0090] The test principle diagram is shown in Figure 3 (a), that is, the laser spot is gradually focused from the bulk of the substrate to the damage stress extension layer, further to the surface damage layer, and then continues to be focused to the air, so as to determine the thickness of the surface damage layer and the damage stress extension layer. Taking the surface of the substrate as zero point, the test range of the axial test is that the Raman laser spot is gradually moved from-100 μm in the substrate to 30 μm above the surface of the substrate, Figure 3 (b) is a stress axial distribution diagram of 5-point test on the upper and lower left and right of the substrate. The 5 points on the substrate show the same trend, and only the negative stress maximum value at the surface damage layer is different.

[0091] Figure 3 In (b), the laser spot is gradually moved from-100 μm in the substrate to-30 μm, at which time the stress extension layer fluctuates up and down, representing the uniform state of the bulk internal stress. When the laser spot is focused to the damage stress extension layer, the stress curve starts to show a downward trend at-30 μm, and the downward trend of the stress gradually increases. When it approaches the zero point of the substrate surface, the negative stress maximum value is reached, which represents the stress value of the surface damage layer at this time. Finally, the laser spot continues to move upward and gradually separates from the substrate surface to the air, and the stress shows a rapid upward trend to a chaotic fluctuation. Therefore, the thickness of the surface damage layer and the damage stress extension layer of the substrate can be determined according to the axial test results of the substrate.

[0092] Example 4 Quantitative test of bulk stress of substrate

[0093] Silicon carbide in the PVT method of crystal growth by temperature gradient and inclusion of the resulting residual stress in the crystal. And the difference between the surface processing stress, in the body stress located at a certain depth from the surface, the reaction of the crystal growth thermal stress and residual stress. And silicon carbide from the crystal through a series of processing to prepare the process of the substrate by a large number of external mechanical stress, it is difficult to ensure that the surface is completely horizontal, that is, the substrate surface exists a certain degree of bending warping. According to the measurement method of the substrate surface bending degree obtained by Raman test in example 2, this embodiment combines the physical properties of silicon carbide, relates to a method for quantitatively characterizing the internal stress of the substrate without the influence of the surface bending warping.

[0094] According to the physical properties of silicon carbide, the method for quantitatively characterizing the internal stress of the substrate by Raman is divided into equal interval focusing test and equal focusing plane test, as follows:

[0095] 1. Equal interval focusing mode

[0096] The test principle of equal interval focusing mode is shown in Figure 4 (a), in which the focusing depth (d) of the Raman laser focused inside the substrate at different positions remains unchanged, the penetration depth (Dp) and spot diameter (D) of the Raman laser are calculated according to the formula, and the appropriate focusing depth (d) is selected. The requirements to be met are that the penetration depth (Dp) is greater than the focusing depth (d), and since the test is for internal stress, the focusing depth d needs to be greater than the thickness of the surface damage layer and the damage stress extension layer obtained by test in example 3.

[0097] According to the characteristics of silicon carbide material, the peak intensity reaches the strongest when the laser is focused on the surface of the substrate, so the automatic focusing function of the Raman tester can be used to focus the laser on the surface of the substrate, and then the focusing depth (d) is determined according to the above method, so that the equal interval focusing test of the substrate at different positions can be realized, that is, the distance between the focusing plane and the surface of the substrate is always equal. This method is not affected by the bending warping of the substrate surface, so that the obtained Raman peak intensity always remains at the maximum value of the test point axis direction. According to the stress quantitative test method of example 1, the internal stress map of the substrate is obtained.

[0098] 2. Equal focusing plane mode

[0099] The test principle of equal focusing plane mode is shown in Figure 4 (b), in which the focusing depth (d) of the Raman laser spot focused inside the substrate at different positions changes, and the laser spot at different positions is focused on the same plane inside the substrate. The penetration depth (Dp) of the Raman laser is calculated according to the same formula as the equal interval focusing test, and the appropriate focusing depth (d) is selected in the same way.

[0100] Since the focusing planes are equal in the present test method, the characteristics of silicon carbide, such as Figure 1 (b) are combined, as shown in the figure, the peak intensity reaches the strongest when the laser is focused on the substrate surface, and the signal gradually decays with the increase of the focusing depth. And according to the effective signal-to-noise ratio of the Raman tester, it is found that the effective signal-to-noise ratio is inversely related to the Raman laser focusing depth (d), so in the present test method, the focusing depth (d) at different positions needs to be combined with the bending degree of the substrate to obtain the effective peak position displacement value by the stress quantitative test method of embodiment 1, so as to reduce the error of the body stress test, and the body stress test diagram of the substrate is obtained by the test method.

[0101] Through the above two kinds of test methods of equal interval focusing mode and equal focusing plane mode, the peak position displacement values in the substrate are obtained respectively, and the body stress distribution of the substrate is calculated, S1 is the body stress value calculated by the equal interval focusing mode, S2 is the body stress value calculated by the equal focusing plane mode, -0.2MPa≤S1-S2≤0.2MPa, V1 is the peak position value obtained after fitting by the equal interval focusing mode, V2 is the peak position value obtained after fitting by the equal focusing plane mode, -0.0005cm -1 V1-V2<0.0005cm -1 .

[0102] The body stress includes absolute body stress and relative body stress. Taking 4H-SiC as an example for illustration:

[0103] The absolute body stress is determined by the difference between the actual 4H-SiC peak position and the standard peak position, such as FTO(2 / 4), and the calculation formula of the absolute body stress is:

[0104] σabsolute=μ×(v 实际 -V FTO(2 / 4) ), σ is in MPa, v 实际 and V FTO(2 / 4) are in cm -1 , wherein v 实际 is the actual FTO(2 / 4) peak position of the substrate test point, and V FTO(2 / 4) is the FTO(2 / 4) standard peak position of the substrate;

[0105] The relative body stress is the difference between the actual 4H-SiC peak position of the SiC substrate and the peak position of the relative point, and the calculation formula of the relative body stress is:

[0106] σrelative=μ×(v (x1,y1) -V (x2,y2) ), σ is in MPa, v (x1,y1) and V (x2,y2) are in cm -1 , wherein v (x1,y1)V (x2,y2) The actual FTO (2 / 4) peak position of the substrate relative point, the relative stress value of the test point is calculated according to the above formula, if the relative value of σ is positive, it indicates that the stress of the test point relative to the relative point is tensile stress, if the relative value of σ is negative, it indicates that the stress of the test point relative to the relative point is compressive stress.

[0107] Example 5 quantitatively tests the stress in the defect aggregation area of the substrate

[0108] According to the axial test of example 3, the thickness of the surface damage layer and the damage stress extension layer of the substrate is determined. According to the measurement method of the surface curvature of the substrate obtained by Raman test in example 2, this embodiment combines the physical characteristics of silicon carbide, and relates to a method for quantitatively characterizing the stress in the defect aggregation area of the substrate which is not affected by the surface bending and warping.

[0109] See Figure 5 (b) is the polarized stress scanning diagram of the substrate, Figure 5 (c) is the third quadrant high-density microtube aggregation area of the substrate, two test methods of equal interval focusing mode and equal focusing plane mode in example 4 are used to quantitatively test the defect aggregation area of the substrate. Since the error of the two methods is small, the mapping result is reserved to 0.1 MPa, so there is no difference in the results. The test results are shown in Figure 5 (a), it can be seen that there is a difference between the stress in the defect aggregation area of the substrate and the stress in the normal area of the substrate, and the stress value in the defect aggregation area of the substrate can be quantitatively tested, which proves that the test method of this embodiment can quantitatively test the stress in the defect aggregation area of the substrate.

[0110] Example 6 quantitatively tests the stress in the substrate of the epitaxial wafer

[0111] According to the axial test of example 3, the thickness of the surface damage layer and the damage stress extension layer of the substrate is determined. According to the measurement method of the surface curvature of the substrate obtained by Raman test in example 2, this embodiment combines the physical characteristics of silicon carbide and the obtained epitaxial wafer, and relates to a method for quantitatively characterizing the stress in the substrate of the epitaxial wafer which is not affected by the surface bending and warping. It is used to detect the stress change of the substrate in the growth of the epitaxial layer.

[0112] Before the growth of the epitaxial layer, the test method of example 4 is used to test the stress in the substrate of the normal silicon carbide substrate from Si and C surfaces. It is found through multiple tests that the stress test results of the same silicon carbide substrate Si and C surfaces are basically the same, such as Figure 6 (a) is the stress distribution diagram of the substrate tested from the Si surface; Figure 6(b) is the stress distribution map of the substrate from the C face test, so for a piece of silicon carbide substrate, the axial in-situ residual stress distribution can be considered uniform, and it can be regarded as a stress uniform body, as shown in Figure 6 (c) shown.

[0113] The silicon carbide substrate is subjected to epitaxial layer growth. Different epitaxial layers are obtained on the Si face of the silicon carbide substrate by homoepitaxy and heteroepitaxy of conductive and semi-insulating silicon carbide, that is, epitaxial wafers are obtained. After the epitaxial layer growth, there is a lattice mismatch between the epitaxial layer and the substrate, thereby introducing stress, and the substrate needs to be heat treated during the epitaxy process, so the in-situ stress in the substrate changes accordingly. The epitaxial wafer obtained after the epitaxial layer growth is tested by the test method of Example 4, and the in-situ stress of the substrate from the Si and C faces is tested. Figure 6 (e) is the stress distribution map of the substrate from the Si face test, Figure 6 (f) is the stress distribution map of the substrate from the C face test. It can be seen that during the epitaxy of the epitaxial layer on the Si face of the substrate, the Si face substrate and the epitaxial layer exhibit lattice mismatch, which can cause the in-situ stress of the Si face of the substrate to increase, and during the epitaxy process, the C face is heat treated, so the stress is released to a certain extent, and therefore the in-situ stress of the C face is reduced to a certain extent, resulting in that the axial in-situ stress distribution of the substrate increases more and more close to the Si face, as shown in Figure 6 (d) shown.

[0114] Example 7: Quantitative test of in-situ stress of epitaxial layer

[0115] According to the measurement method of the surface curvature of the epitaxial layer obtained by Raman test in Example 2, this embodiment combines the physical properties of silicon carbide and the epitaxial wafer obtained therefrom, and relates to a method for quantitatively characterizing the in-situ stress of the epitaxial layer without being affected by the surface curvature and warping, so as to detect the in-situ stress of the epitaxial layer.

[0116] The epitaxial layer is quantitatively tested by the test method of equal interval focusing in Example 4. The thickness of the epitaxial layer is limited, so only the test method of equal interval focusing can be used. The focal point of the Raman laser is inside the epitaxial layer, that is, it extends from the surface of the epitaxial wafer to the region of 30%-60% of the thickness of the epitaxial layer. The test results are shown in Figure 7 The overall morphology of the epitaxial layer is the same as that of the substrate, but the stress value is increased compared with the substrate.

[0117] According to the characteristics of the SiC crystal obtained by the PVT method, the crystal growth stress is simulated and analyzed, and the in-plane radial relative stress actually measured by the above quantitative test method and the simulated radial relative stress are both symmetrically distributed with high edge stress and low center stress, the simulation result is consistent with the actually measured result; the simulation axial absolute stress and the actually measured in-plane axial absolute stress are both distributed with positive front end stress and negative rear end stress, the simulation result is also consistent with the actually measured result. And the absolute stress and relative stress values calculated by the calculation formula of the embodiment are consistent with the order of magnitude of the simulation result, both are 10~10 2 MPa. Since the SiC crystal has a relatively thick thickness, the penetration depth of the Raman laser is limited, the SiC crystal is cut into multiple substrates, the actually measured axial absolute stress is obtained by testing the in-situ absolute stress of different substrates on the same axis.

[0118] In the embodiment 1 of the present application, the peak position displacement of the substrate and the stress conversion coefficient μ are obtained according to the different characteristics of the substrate, which provides a basis for the quantitative test of the substrate stress; according to the axial test method of the embodiment 3, the thickness of the surface damage layer and the damage stress extension layer of the substrate can also be calculated, and the thickness of the epitaxial layer, the surface damage layer and the damage stress extension layer in the epitaxial wafer can also be calculated; and according to the above embodiment 2, the curvature of the substrate is determined, the focusing depth (d) and the spot diameter (D) of the Raman laser spot are calculated, the influence of the surface curvature of the substrate on the in-situ stress test of the substrate is eliminated, and the influence of the surface curvature of the epitaxial wafer on the in-situ stress test of the substrate and the in-situ stress test of the epitaxial layer is also eliminated, which provides a unified standard for the quantitative test of the in-situ stress of the substrate defect aggregation area, the in-situ stress of the substrate in the epitaxial wafer and the in-situ stress of the epitaxial layer.

[0119] The above is only an embodiment of the present application, and the protection scope of the present application is not limited by these specific embodiments, but is determined by the claims of the present application. The present application can have various changes and variations for those skilled in the art. Any modification, equivalent replacement, improvement, etc. within the technical thought and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for quantitatively characterizing the bulk stress in a substrate defect aggregation region, characterized in that, The method comprises the following steps: The first step is to test the substrate to obtain the actual peak position value and stress value of the substrate at different test points, calculate the difference between the actual peak value and the standard substrate peak position, i.e. the peak position displacement value at different test points, and calculate the peak position displacement and stress conversion coefficient μ according to the peak position displacement value and the stress value. The second step is to use a Raman tester to perform internal scanning test on the defect aggregation area of the substrate, the diameter of the substrate is greater than or equal to 150 mm, the defects in the defect aggregation area include at least one of microtubules, dislocations, polytypes, and carbon inclusions, the internal part is a region extending from the surface of the substrate to 30%-60% of the thickness of the substrate, the peak position value is obtained, the peak position displacement value is calculated, and the in-situ stress value of the defect aggregation area of the substrate is calculated according to the peak position displacement value and the peak position displacement and stress conversion coefficient μ in the first step. The calculation step of the peak position displacement and stress conversion coefficient μ is as follows: First, the test instrument is used to test the crystal face spacing dx in the substrate, the difference between the crystal face spacing dx and the theoretical crystal face spacing d0 is obtained, and the stress value of the substrate at different test points is obtained by using the Hook's law σ / S=E×(△d / d0) or the stress and strain formula σ=E×△d. Then, the Raman tester is used to perform scanning test on the substrate to obtain the actual peak position value at different test points, calculate the difference between the actual peak value and the standard peak position, i.e. the peak position displacement value at different test points, and the ratio of the peak position displacement value to the stress value is the peak position displacement and stress conversion coefficient μ.

2. The method of quantitatively characterizing in- situ stress of a defect cluster region of a substrate according to claim 1, wherein, Before the second step of using the Raman tester to perform internal scanning test on the defect aggregation area of the substrate, the Raman tester is used to perform axial test on the defect aggregation area of the substrate, and the depth of the surface damage layer and the damage stress extension layer of the defect aggregation area of the substrate is determined according to the stress curve fed back by the axial test.

3. The method of quantitatively characterizing in- stress of a defect cluster region of a substrate according to claim 2, wherein, Before the second step of using the Raman tester to perform internal scanning test on the defect aggregation area of the substrate, the surface of the substrate is tested for bending degree to obtain the bending degree of the substrate at the defect aggregation area, and the bending degree is used as a reference to select an equal-interval focusing mode and / or an equal-focusing plane mode in the second step of scanning test, wherein the distance between the Raman laser spot at each test point and the surface of the substrate is equal in the equal-interval focusing mode, and the Raman laser spot at each test point is in the same horizontal plane in the equal-focusing plane mode.

4. A method of quantitatively characterizing in- bulk stress in a substrate of an epitaxial wafer, characterized by, The method comprises the following steps: The first step is to test the substrate to obtain the actual peak position value and stress value of the substrate at different test points, calculate the difference between the actual peak value and the standard substrate peak position, i.e. the peak position displacement value at different test points, and calculate the peak position displacement and stress conversion coefficient μ according to the peak position displacement value and the stress value. The second step is to use a Raman tester to perform internal scanning test on the substrate of the epitaxial wafer, the internal part of the substrate is a region extending from the surface of the substrate to 30%-60% of the thickness of the substrate, the peak position value is obtained, the peak position displacement value is calculated, and the in-situ stress value of the substrate of the epitaxial wafer is calculated according to the peak position displacement value and the peak position displacement and stress conversion coefficient μ in the first step. The calculation step of the peak position displacement and stress conversion coefficient μ is as follows: Firstly, the interplanar spacing dx in the substrate is tested by a testing instrument, the interplanar spacing dx and the theoretical interplanar spacing d0 are subtracted to obtain the interplanar spacing difference △d at different test points, and the stress values of the substrate at different test points are obtained by using the Hook's law σ / S=E×(△d / d0) or the stress-strain formula σ=E×△d; Then, the substrate is scanned and tested by using a Raman tester to obtain the actual peak value at different test points, the difference between the actual peak value and the standard peak position is calculated, that is, the peak displacement value at different test points is obtained, and the ratio of the peak displacement value to the stress value is the peak displacement and stress conversion coefficient μ.

5. The method of quantitatively characterizing in situ stresses in a substrate in an epitaxial sheet according to claim 4, wherein Before the second step of scanning and testing the internal part of the defect aggregation area of the substrate by using the Raman tester, the axial test of the epitaxial wafer is performed by using the Raman tester, and the depth of the epitaxial layer, the surface damage layer and the damage stress extension layer in the epitaxial wafer is determined according to the stress curve fed back by the axial test.

6. The method of quantitatively characterizing in situ stresses in a substrate in an epitaxial sheet according to claim 4, wherein Before the second step of scanning and testing the internal part of the defect aggregation area of the substrate by using the Raman tester, the surface of the epitaxial wafer is tested for the bending degree, the bending degree of the surface of the epitaxial wafer is obtained, and the equal-interval focusing mode and / or the equal-focusing plane mode are selected in the second step of scanning and testing, wherein in the equal-interval focusing mode, the distance between the Raman laser spot at each test point and the surface of the substrate is equal, and in the equal-focusing plane mode, the Raman laser spot at each test point is in the same horizontal plane.

7. The method of quantitatively characterizing in- plane stress of a defect cluster region of a substrate according to any one of claims 1 to 3 or the method of quantitatively characterizing in- plane stress of a substrate of an epiwafer according to any one of claims 4 to 6, characterized in that, The testing instrument for testing the interplanar spacing dx in the substrate is selected from any one or more of an XRD diffractometer, a synchrotron radiation tester, a white light tester, a neutron diffractometer, a scanning electron microscope and a transmission electron microscope.

8. The method of quantitatively characterizing in- plane stress of a defect cluster region of a substrate according to any one of claims 1 to 3 or the method of quantitatively characterizing in- plane stress of a substrate of an epiwafer according to any one of claims 4 to 6, characterized in that, S1 is the in-vivo stress value calculated by the equal-interval focusing mode, S2 is the in-vivo stress value calculated by the equal-focusing plane mode, and -0.2 MPa≤S1-S2≤0.2 MPa; or V1 is the peak position value obtained after fitting with the equal-interval focusing mode, V2 is the peak position value obtained after fitting with the equal-focusing plane mode, -0.0005 cm -1 < V1-V2 <0.0005cm -1 .

9. The method of quantitatively characterizing in- plane stress of a defect cluster region of a substrate according to any one of claims 1 to 3 or the method of quantitatively characterizing in- plane stress of a substrate of an epiwafer according to any one of claims 4 to 6, characterized in that, The in-vivo stress value calculated in the second step includes an in-vivo absolute stress and an in-vivo relative stress; In the calculation of the in-vivo absolute stress, the peak displacement value obtained is the difference between the actual peak position and the standard peak position at the test point; in the calculation of the in-vivo relative stress, the peak displacement value obtained is the difference between the actual peak position at the test point and the actual peak position at the relative point; and / or The diameter of the substrate is more than 200 mm.

10. A method for quantitatively characterizing the internal stress of an epitaxial layer, characterized in that, The method comprises the following steps: First step: test the epitaxial layer to obtain the actual peak value and stress value of the epitaxial layer at different test points, calculate the difference between the actual peak value and the standard substrate peak position, that is, the peak displacement value at different test points, and calculate the peak displacement and stress conversion coefficient μ according to the peak displacement value and the stress value; Second step: scan and test the internal part of the epitaxial layer by using a Raman tester, the internal part of the epitaxial layer is a region extending from the surface of the epitaxial wafer inwardly to 30%-60% of the thickness of the epitaxial layer, obtain the peak value, calculate the peak displacement value, and calculate the in-vivo stress value of the epitaxial layer according to the peak displacement value and the peak displacement and stress conversion coefficient μ in the first step; The calculation steps of the peak displacement and stress conversion coefficient μ are: First, the interplanar spacing dx in the substrate is tested by a testing instrument, the interplanar spacing difference △d at different testing points is obtained by subtracting the theoretical interplanar spacing d0 from the interplanar spacing dx, and the stress values of the substrate at different testing points are obtained by using the Hook's law σ / S=E×(△d / d0) or the stress-strain formula σ=E×△d; Then, the substrate is scanned and tested by using a Raman testing instrument, the actual peak value at different testing points is obtained, the difference between the actual peak value and the standard peak position is calculated, i.e. the peak position displacement value at different testing points is obtained, and the ratio of the peak position displacement value to the stress value is the peak position displacement and stress conversion coefficient μ.

Images