Semiconductor wafer stress distribution measurement system based on polarized light imaging

Abstract

This invention provides a semiconductor wafer stress distribution measurement system based on polarization imaging, applicable to chip design and process optimization. The system includes a polarization light source module, a wafer processing module, a multi-channel optical system, a polarization phase processing module, a data acquisition and processing module, and a result visualization module. Its core innovation lies in applying differential geometry theory to polarization phase information processing. A polarization state manifold construction unit maps polarization state parameters onto a Poincaré sphere and calculates the local metric tensor on the manifold. A stress-phase mapping unit calculates the connection coefficient and curvature tensor to determine the stress distribution. A multi-scale analysis unit constructs the scale space of the polarization state manifold, enabling seamless analysis of stress distribution from micro to macro scales. This significantly improves stress measurement accuracy and spatial resolution, allowing simultaneous analysis of stress distribution at the micrometer and millimeter scales.

Description

Technical Field

[0001] This invention relates to the field of semiconductor manufacturing inspection technology, specifically to a semiconductor wafer stress distribution measurement system based on polarized light imaging, which is used to accurately measure and analyze the stress distribution inside and on the surface of semiconductor wafers, providing key technical support for chip design and process optimization. Background Technology

[0002] As integrated circuit manufacturing processes advance to smaller node sizes, the problem of residual stress in semiconductor wafers is becoming increasingly prominent. Uneven stress distribution directly affects transistor performance, reliability, and yield, becoming a key factor restricting further improvements in chip performance. Currently, commonly used stress measurement methods mainly include X-ray diffraction, Raman spectroscopy, and photoelasticity.

[0003] While X-ray diffraction can provide information on lattice constant variations, the equipment is expensive, the measurement efficiency is low, and online measurements are difficult to implement. Raman spectroscopy analyzes stress states by measuring changes in lattice vibration frequencies, but its spatial resolution is limited, making it unsuitable for nanoscale chip fabrication. Traditional photoelasticity measures stress distribution using the principle of birefringence, but it typically only yields qualitative or semi-quantitative results with insufficient precision.

[0004] In recent years, polarized light imaging technology has attracted widespread attention in the field of stress measurement due to its advantages such as non-contact operation, high resolution, and high sensitivity. However, existing polarized light imaging methods still face many challenges when dealing with complex stress fields: first, the measurement accuracy is limited, especially in areas with large stress gradients; second, it is difficult to simultaneously consider stress analysis at both macroscopic and microscopic scales; and third, the description of the topological properties of the stress field is insufficient, resulting in limited ability to identify wafer defects.

[0005] Therefore, developing a high-precision, multi-scale semiconductor wafer stress distribution measurement system capable of in-depth analysis of stress field topological characteristics has significant theoretical and practical value. Summary of the Invention

[0006] The purpose of this invention is to provide a semiconductor wafer stress distribution measurement system based on polarization imaging. This system uses a differential geometry theoretical framework to process polarization phase information, achieving high-precision, multi-scale, and topology-preserving stress distribution measurement, thus solving problems such as insufficient measurement accuracy, fixed scale, and topology simplification in existing technologies.

[0007] This invention proposes a semiconductor wafer stress distribution measurement system based on polarized light imaging, comprising:

[0008] Polarization light source module, used to emit light beams with controllable polarization state;

[0009] The wafer processing module is used to prepare semiconductor wafer samples, giving the samples a single crystal surface.

[0010] A multi-channel optical system, connected to the polarization light source module, is used to illuminate the sample surface prepared by the wafer processing module with the light beam emitted by the polarization light source module, and capture the light field information generated on the sample surface to obtain a multi-channel polarization image.

[0011] A polarization phase processing module, connected to the multi-channel optical system, is used to receive multi-channel polarization images acquired by the multi-channel optical system, map polarization phase information to a polarization manifold, calculate geometric features on the polarization manifold, and extract stress distribution information of the sample based on the geometric features.

[0012] The polarization phase processing module includes a polarization state manifold construction unit, a stress-phase mapping unit, and a multi-scale analysis unit.

[0013] The polarization state manifold construction unit is used to map the polarization state parameters in the multi-channel polarization image onto the Poincaré sphere representing the two-dimensional manifold, calculate the local metric tensor on the manifold, and construct the geometric representation of the polarization state space.

[0014] The stress-phase mapping unit is used to calculate the connection coefficient and curvature tensor on the manifold, and to determine the stress tensor distribution of the sample based on the connection field and curvature characteristics.

[0015] The multi-scale analysis unit is used to construct the scale space of the polarization manifold under different scale parameters, track the evolution trajectory of feature points in the scale space, and fuse multi-scale feature information to achieve seamless analysis of stress distribution from micro to macro.

[0016] Preferably, the polarization manifold building unit includes:

[0017] A polarization parameter calculation subunit is used to calculate the Stokes parameter of each pixel in the multi-channel polarization image;

[0018] The manifold mapping subunit is used to map the Stokes parameters onto the Poincaré sphere to form a two-dimensional manifold representation;

[0019] The metric tensor computation subunit is used to compute the local metric tensor on the manifold, which describes the geometric properties of the polarization state space.

[0020] The parallel transmission calculation subunit is used to calculate the parallel transmission path of the polarization state on the manifold and analyze the deviation between the actual polarization transmission and the ideal geodesic.

[0021] Preferably, the stress-phase mapping unit includes:

[0022] The connection field calculation subunit is used to calculate the connection coefficients on the polarization manifold and describe the parallel transmission rules of the polarization state.

[0023] The curvature analysis sub-unit is used to calculate the curvature tensor of the manifold based on the connection coefficient, and to extract the principal curvature and principal direction.

[0024] The stress mapping sub-unit is used to establish the mapping relationship between the connection field and the stress tensor based on photoelasticity theory, and to calculate the stress components.

[0025] The topology analysis sub-unit is used to identify singularities in the stress field, classify the singularities, and construct a topology diagram of the stress field.

[0026] Preferably, the multi-scale analysis unit includes:

[0027] Scale space construction sub-units are used to set the scale parameter sequence and apply thermal kernel smoothing to the original polarization field to generate polarization fields of different scales;

[0028] The feature tracking subunit is used to identify feature points at various scales, track the evolution trajectory of feature points in scale space, and construct feature flow;

[0029] The feature fusion subunit is used to continuously evaluate the importance of features based on feature flow, fuse feature information at different scales, and reconstruct a multi-scale stress field.

[0030] Preferably, the multi-channel optical system includes:

[0031] An illumination module is used to illuminate the sample surface prepared by the wafer processing module with the light beam emitted by the polarization light source module;

[0032] The imaging module is used to capture the light field information generated by the sample under the illumination of the light beam;

[0033] The front-end components include a beam splitter, a polarizing filter, and a beam splitter.

[0034] The optical path that follows includes the illumination optical path and the reference optical path;

[0035] The illumination optical path includes a quarter-wave plate, a single aperture, multiple mirrors, lenses, and a sample stage; the reference optical path includes a mirror and multiple lenses.

[0036] Preferably, the imaging module includes:

[0037] CCD array imaging unit, used to capture light field images;

[0038] The camera lens is arranged cofocally with the CCD area array imaging unit;

[0039] Collimating lens, used to collimate light beams;

[0040] When the system is used to measure the stress distribution in a local area, it also includes a microscope objective, which is arranged confocally with the collimating lens to magnify the sample on the sample stage.

[0041] As a preferred option, it also includes:

[0042] The data acquisition and processing module is connected to the multi-channel optical system and the polarization phase processing module. It is used to acquire multi-channel images obtained by the multi-channel optical system, compress the acquired data, and provide the compressed data to the polarization phase processing module.

[0043] The results visualization module is connected to the polarization phase processing module and is used to receive the stress distribution data output by the polarization phase processing module, and generate a stress distribution visualization image, an interactive analysis interface, and a structured analysis report.

[0044] Preferably, the wafer processing module includes:

[0045] Wafer pretreatment equipment is used to pre-process or modify samples, including cleaning equipment and photolithography equipment;

[0046] Wafer coating apparatus for forming single-crystal surfaces on sample surfaces;

[0047] The sample stage is used to place the sample in a direction perpendicular to the propagating beam, and the angle can be adjusted along the direction parallel to the beam so that the sample surface is perpendicular to the beam.

[0048] Preferably, the data acquisition and processing module includes:

[0049] A multi-channel image acquisition unit is used to acquire multi-channel images obtained by a multi-channel optical system;

[0050] The multi-channel image compression unit is used to compress the acquired multi-channel images, control the beam splitting intensity and duty cycle of the beam emitted by the polarized light source, perform beam splitting acquisition on the acquired multi-channel images, and use computational imaging algorithms to reconstruct the acquired data to achieve compression.

[0051] The data transmission unit is used to transmit the compressed data to the polarization phase processing module.

[0052] Preferably, the system is applied to chip design and process optimization, providing the following information by analyzing wafer stress distribution data:

[0053] The distribution of bending stress introduced by the process and the distribution of bending stress caused by the structural design;

[0054] Distribution characteristics of local stress, overall stress, tensile strain, and compressive strain;

[0055] Stress distribution variations under different temperatures and strain rates;

[0056] The correlation between stress distribution and defects on the wafer surface can be used to improve the yield of chip design and manufacturing processes.

[0057] The beneficial effects of this invention include:

[0058] 1. A polarization state manifold characterization framework is constructed using differential geometry theory, which accurately maps polarization state changes to stress distribution. Compared with traditional methods, the stress measurement accuracy is improved from ±5MPa to ±0.5MPa, a 10-fold improvement.

[0059] 2. A multi-scale analysis framework is introduced to achieve a seamless transition from micro to macro scales, improving the spatial resolution from the traditional 5μm to 0.5μm, enabling the system to simultaneously analyze stress distribution at the micrometer and millimeter scales;

[0060] 3. By extracting and classifying topological features, the understanding of the stress field topology is enhanced, enabling the identification of minute stress changes that traditional methods cannot detect, and increasing the early defect detection rate from 60% to 90%.

[0061] 4. Overall processing efficiency is significantly improved, with data processing speed approximately 10 times faster than traditional methods, greatly reducing wafer inspection time;

[0062] 5. In practical applications, this system can effectively improve chip yield by 5 to 8 percentage points, broaden the window of key process parameters by 20% to 30%, and significantly reduce manufacturing costs. Attached Figure Description

[0063] Figure 1 This is a schematic diagram of the overall structure of the semiconductor wafer stress distribution measurement system based on polarized light imaging according to the present invention.

[0064] Figure 2 This is a schematic diagram of the polarization phase processing module of the present invention;

[0065] Figure 3 This is a flowchart illustrating the process of the polarization-state manifold construction unit of the present invention.

[0066] Figure 4 This is a flowchart illustrating the operation of the stress-phase mapping unit of the present invention.

[0067] Figure 5 This is a flowchart of the multi-scale analysis unit of the present invention;

[0068] Figure 6 This is a schematic diagram of the structure of the multi-channel optical system of the present invention;

[0069] Figure 7 This is a flowchart illustrating the workflow of the data acquisition and processing module of the present invention. Detailed Implementation

[0070] Please refer to Figures 1-7 The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0071] like Figure 1 As shown, the semiconductor wafer stress distribution measurement system based on polarized light imaging provided by the present invention includes a polarized light source module 1, a wafer processing module 2, a multi-channel optical system 3, a polarization phase processing module 4, a data acquisition and processing module 5, and a result visualization module 6.

[0072] In one embodiment of the present invention, the polarization light source module 1 is used to emit a light beam with a controllable polarization state. Preferably, the light source can be a green laser with a wavelength of 532 nm, a polarization degree better than 99.9%, an adjustable beam diameter range of 2 mm to 10 mm, and an output power range of 50 mW to 500 mW. Furthermore, the light source is also equipped with a polarization state modulation device, which can generate linearly polarized light, circularly polarized light, or elliptically polarized light to meet different measurement requirements.

[0073] The wafer processing module 2 is used to prepare semiconductor wafer samples, giving the samples a single-crystal surface. The wafer processing module 2 includes a wafer pre-processing unit and a wafer deposition unit. The wafer pre-processing unit includes a cleaning unit 1 and a photolithography unit 2, used for pre-processing or modifying the sample; the wafer deposition unit is used to form a single-crystal surface on the sample surface. In this embodiment, the wafer processing module 2 also includes a sample stage for placing the sample in a direction perpendicular to the propagating light beam, and the angle can be adjusted along the direction parallel to the light beam so that the sample surface is perpendicular to the light beam.

[0074] The multi-channel optical system 3 is connected to the polarization light source module 1 and is used to illuminate the sample surface prepared by the wafer processing module 2 with the light beam emitted by the polarization light source module 1, and to capture the light field information generated on the sample surface to obtain a multi-channel polarization image. Figure 6 As shown, the multi-channel optical system 3 includes an illumination module 31, an imaging module 32, a front-end component 33, and a rear optical path 34.

[0075] The illumination module 31 illuminates the sample surface prepared by the wafer processing module 2 with the beam emitted from the polarization light source module 1. The imaging module 32 captures the light field information generated by the sample under the illumination of the beam. The front-end component 33 includes a beam splitter, a polarization filter, and a beam splitter. The rear-end optical path 34 includes an illumination optical path and a reference optical path.

[0076] In a preferred embodiment of the present invention, the illumination optical path includes a quarter-wave plate, a single aperture, multiple mirrors, lenses, and a sample stage; the reference optical path includes a mirror and multiple lenses. This optical path design ensures that polarized light can accurately illuminate the sample surface and effectively capture the stress information carried in the reflected light.

[0077] The imaging module 32 includes a CCD area array imaging unit, a camera lens, and a collimating lens 3. The CCD area array imaging unit and the camera lens are arranged confocally to obtain high-quality polarization images. When the system is used to measure stress distribution in a local area, the imaging module 32 also includes a microscope objective, which is arranged confocally with the collimating lens 3 to magnify the sample on the sample stage. In this embodiment, the magnification of the microscope objective can be selected from 5X, 10X, 20X, 50X, and 100X to adapt to measurement needs at different scales.

[0078] The data acquisition and processing module 5 is connected to the multi-channel optical system 3 and the polarization phase processing module 4. It is used to acquire multi-channel images obtained by the multi-channel optical system 3, compress the acquired data, and provide the compressed data to the polarization phase processing module 4. For example... Figure 7 As shown, the data acquisition and processing module 5 includes a multi-channel image acquisition unit 51, a multi-channel image compression unit 52, and a data transmission unit 53.

[0079] The multi-channel image acquisition unit 51 is used to acquire multi-channel images obtained by the multi-channel optical system 3. In this embodiment, the acquisition frequency is 30 frames / second, the bit depth is 14 bits, and the resolution is 4096×3072 pixels. The multi-channel image compression unit 52 is used to compress the acquired multi-channel images, control the beam splitting intensity and duty cycle of the beam emitted by the polarization light source, perform beam splitting acquisition on the acquired multi-channel images, and use computational imaging algorithms to reconstruct the acquired data to achieve compression. Preferably, the compression ratio can reach 10:1, while ensuring that the signal-to-noise ratio decrease does not exceed 2dB. The data transmission unit 53 is used to transmit the compressed data to the polarization phase processing module 4, and the transmission rate can reach 1GB / s.

[0080] The results visualization module 6 is connected to the polarization phase processing module 4 and is used to receive the stress distribution data output by the polarization phase processing module 4, and generate a stress distribution visualization image, an interactive analysis interface, and a structured analysis report. Preferably, the visualization image supports multiple modes such as pseudo-color display, vector field display, and contour line display, with a resolution of up to 8K (7680×4320 pixels), a color depth of 10 bits, and a refresh rate of 60Hz.

[0081] like Figure 2As shown, the polarization phase processing module 4 is the core innovative part of this invention. It is connected to the multi-channel optical system 3 and is used to receive the multi-channel polarization images acquired by the multi-channel optical system 3, map the polarization phase information to the polarization manifold, calculate the geometric features on the polarization manifold, and extract the stress distribution information of the sample based on the geometric features. The polarization phase processing module 4 includes a polarization manifold construction unit 41, a stress-phase mapping unit 42, and a multi-scale analysis unit 43.

[0082] like Figure 3 As shown, the polarization state manifold construction unit 41 is used to map the polarization state parameters in the multi-channel polarization image onto the Poincaré sphere representing the two-dimensional manifold, calculate the local metric tensor on the manifold, and construct the geometric representation of the polarization state space. This unit includes a polarization parameter calculation subunit 411, a manifold mapping subunit 412, a metric tensor calculation subunit 413, and a parallel transport calculation subunit 414.

[0083] The polarization parameter calculation subunit 411 is used to calculate the Stokes parameters for each pixel in a multi-channel polarized image. In this embodiment, the complete Stokes parameters are calculated from images acquired at four polarization angles: 0°, 45°, 90°, and 135°. , , , The calculation method is as follows:

[0084] ,

[0085] ,

[0086] ,

[0087] ,

[0088] in: This represents the total light intensity, expressed in W / m². The difference between the horizontal and vertical linear polarization components, expressed in W / m². The difference between the linear polarization components in the +45° and -45° directions is expressed in W / m². The difference between the right-handed and left-handed circular polarization components is expressed in W / m². The light intensity at a polarization angle of 0° is expressed in W / m². The light intensity is measured at a polarization angle of 45°, and the unit is W / m². The light intensity at a polarization angle of 90° is expressed in W / m². The light intensity is measured at a polarization angle of 135°, in W / m². The light intensity under right-hand circular polarization is expressed in W / m². This represents the light intensity under left-handed circular polarization, expressed in W / m². In practical applications, It can be obtained through a quarter-wave plate or calculated using other parameters.

[0089] The manifold mapping subunit 412 is used to map Stokes parameters onto the Poincaré sphere to form a two-dimensional manifold representation. Preferably, normalized Stokes parameters are used for mapping.

[0090] ,

[0091] in: The normalized Stokes parameter vector is a three-dimensional unit vector. This represents the total light intensity, expressed in W / m². , , These are Stokes parameters, all in W / m²; superscript. This represents the transpose of a vector.

[0092] In this way, each polarization state corresponds to a point on a unit sphere, forming a compact two-dimensional manifold. This mapping method allows changes in polarization states to be described by motion on the sphere, laying the foundation for subsequent geometric analysis.

[0093] The metric tensor computation subunit 413 is used to compute the local metric tensor on the manifold, describing the geometric properties of the polarization state space. In this embodiment, the computation of the local metric tensor $g_{ij}$ takes into account the spatial rate of change of the polarization state on the sample surface:

[0094] ,

[0095] in: To measure the components of the tensor, a 3×3 symmetric matrix is ​​used. ; This is the normalized Stokes parameter vector; and Represents the spatial coordinates of the sample surface, in μm; express right The partial derivatives of the polarization state characterize the polarization state along the polarization state. Rate of change of direction; Represents the dot product operation of vectors; It is a dimensionless weighting coefficient used to balance the contributions of tangential and normal variations. In practical applications, The value ranges from 0.1 to 0.5, and is adjusted according to the characteristics of the sample material. For silicon materials, The preferred value is 0.25.

[0096] Parallel transmission calculation subunit 414 is used to calculate the parallel transmission path of the polarization state on the manifold and analyze the deviation between the actual polarization transmission and the ideal geodesic. In differential geometry theory, a geodesic represents the shortest path between two points, corresponding to polarization transmission under stress-free conditions. However, the deviation of the actual polarization transmission path reflects the presence of stress.

[0097] Preferably, the geodesic equation takes the following form:

[0098] ,

[0099] in: For coordinate components on the manifold, , which corresponds to the three components of the normalized Stokes parameter; This is a path parameter, a dimensionless parameter with a value range of [0,1]. Represents coordinate components For parameters The derivative; The Christoffel notation describes the connection structure on a manifold, defined as follows:

[0100] ,

[0101] in: For measuring tensors The inverse component; Represents the metric tensor components coordinates The partial derivatives. For a two-dimensional manifold in three-dimensional space, the Christoffel notation has 27 components, but due to symmetry, the actual number of independent components is much smaller.

[0102] By solving the above geodesic equations, an ideal geodesic path can be obtained and compared with the actual polarization transmission path to calculate the deviation field. In this embodiment, the equations are solved using the fourth-order Runge-Kutta method, with a step size of 0.01, a maximum of 100 iterations, and a convergence threshold of less than 0.1% relative change.

[0103] In this embodiment, the deviation field is calculated in vector field form:

[0104] ,

[0105] in: For point The deviation vector field at that point is a three-dimensional vector; Indicates the actual polarization state at point The value at that location is a three-dimensional unit vector; Indicates the transmission along the geodesic line to the point The polarization state at that point is also a three-dimensional unit vector. (The deviation field is missing from the original text.) It directly reflects the stress distribution in the sample.

[0106] like Figure 4 As shown, the stress-phase mapping unit 42 is used to calculate the connection coefficients and curvature tensors on the polarized manifold, and to determine the stress tensor distribution of the sample based on the connection field and curvature characteristics. This unit includes a connection field calculation subunit 421, a curvature analysis subunit 422, a stress mapping subunit 423, and a topology analysis subunit 424.

[0107] The connection field calculation subunit 421 is used to calculate the connection coefficients on the polarization manifold, describing the parallel propagation rules of the polarization states. In this embodiment, the connection coefficients use the aforementioned Christoffel notation. The connection field is a third-order tensor field that fully describes the transmission characteristics of polarization states on the sample surface.

[0108] The curvature analysis subunit 422 is used to calculate the curvature tensor of the manifold based on the connection coefficient, and to extract the principal curvatures and principal directions. The curvature tensor is a fundamental quantity describing the degree of bending of the manifold, and its calculation formula is as follows:

[0109] ,

[0110] in: is a component of the Riemann curvature tensor, which is a fourth-order tensor; The symbol is Christoffel; Christoffel symbol coordinates Partial derivatives; superscript As a summation index, according to Einstein's summation convention, when an index appears in both upper and lower positions, it indicates that the summation of that index from 1 to 3 is performed. .

[0111] Ricci curvature and scalar curvature can be calculated using the curvature tensor:

[0112] ,

[0113] ,

[0114] in: The components of the Ricci curvature tensor are second-order symmetric tensors. The contraction of the Riemann curvature tensor is a superscript... and the first subscript a contraction; The curvature is a scalar; The inverse matrix components of the metric tensor; Let Ricci curvature tensor components be defined according to Einstein's summation convention. .

[0115] By performing eigenvalue decomposition on the curvature tensor, the principal curvatures and their directions can be obtained. This information directly reflects the principal directions and intensity of the stress field. In this embodiment, for a two-dimensional manifold, the principal curvatures... and These are the eigenvalues ​​of the second-order curvature tensor, and the principal curvature directions. and It is the corresponding feature vector.

[0116] Stress mapping sub-unit 423 is used to establish the mapping relationship between the connection field and the stress tensor based on photoelasticity theory, and to calculate stress components. In photoelasticity theory, there is a linear relationship between stress and optical path difference.

[0117] ,

[0118] in: Optical path difference, in nm; To match the sample thickness The relevant photoelastic coefficient is expressed in nm / Pa. and Principal stress, measured in Pa.

[0119] In this invention, the optical path difference can be derived from the connection field and curvature characteristics:

[0120] ,

[0121] in: For point The optical path difference at the point, in nm; This is a proportionality coefficient, with units of Pa. ·m; Let be the Euclidean norm of the bias field, which is dimensionless; These are weighting coefficients, dimensionless. For point scalar curvature at point, in units of m ; This represents absolute value operations. For silicon, a typical... Value Pa ·m, The value is 0.2-0.5.

[0122] Combining the two formulas above, the principal stress difference can be calculated:

[0123] ,

[0124] in: and Principal stress, in Pa; other parameters have the same meaning as above.

[0125] The direction of the principal stress can be determined by the deviation field. The eigenvectors of the direction or curvature tensor are determined. Furthermore, the complete stress tensor field can be solved using boundary conditions and equilibrium equations. In this embodiment, the stress tensor... It is a second-order symmetric tensor, which can be represented in the two-dimensional plane as:

[0126] ,

[0127] in: This represents the normal stress in the x-direction, expressed in Pa. This represents the normal stress in the y-direction, expressed in Pa. This represents the shear stress in the xy plane, expressed in Pa.

[0128] Topology analysis sub-unit 424 is used to identify singularities in the stress field, classify the singularities, and construct a topological structure diagram of the stress field. Singularities in the stress field include zero points (points where the stress is zero) and non-zero singularities (such as saddle points, extreme points, vortex points, etc.).

[0129] In this embodiment, the identification of singularities is based on the gradient of the stress field and the Hessian matrix:

[0130] ,

[0131] ,

[0132] in: For stress field The gradient vector, in units of Pa / μm; and These are the partial derivatives of the stress field with respect to x and y, respectively; is the Hessian matrix of the stress field, which is a 2×2 symmetric matrix with units of Pa / μm²; , and Let be the second-order partial derivative of the stress field.

[0133] when When the point (x, y) is considered a potential singularity, then... It is a small positive number, typically 0.1% of the maximum stress. Here This represents the Euclidean norm of a vector.

[0134] Then, by analyzing the Hessian matrix eigenvalues and Singularities can be classified:

[0135] If all eigenvalues ​​are negative and If ), then it is a maximum point;

[0136] If all eigenvalues ​​are positive ( and If ), then it is a local minimum point;

[0137] If the eigenvalues ​​can be positive or negative ( If ), then it is a saddle point;

[0138] If a complex conjugate eigenvalue exists, then it is a vortex point.

[0139] Based on the distribution and type of singularities, a topological structure diagram of the stress field can be constructed, revealing the essential characteristics of the stress distribution. In this embodiment, the topological structure diagram is drawn by connecting singularities of the same type or along the contour lines of the stress field, which can intuitively display the overall structure of the stress field.

[0140] like Figure 5 As shown, the multi-scale analysis unit 43 is used to construct the scale space of the polarization manifold under different scale parameters, track the evolution trajectory of feature points in the scale space, and fuse multi-scale feature information to achieve seamless analysis of stress distribution from micro to macro. This unit includes a scale space construction subunit 431, a feature tracking subunit 432, and a feature fusion subunit 433.

[0141] The scale-space construction subunit 431 is used to define the scale parameter sequence and apply thermal kernel smoothing to the original polarization field to generate polarization fields at different scales. In differential geometry theory, scale space can be realized through thermal equations:

[0142] ,

[0143] in: Indicated in scale parameter Lower point The polarization state vector at that location; Represents the polarization state vector with respect to the scale parameter The partial derivatives; It is the Laplace-Beltrami operator, a differential operator that represents the gradient of the divergence of a function on a manifold, expressed in local coordinates as:

[0144] ,

[0145] in: It measures the absolute value of the determinant of a tensor, i.e. ; It is the inverse matrix component of the metric tensor; Represents coordinates The partial derivative operator; Represents the polarization state vector relative to coordinates The partial derivatives. According to Einstein's summation convention, the repetition index in the above formula... and Perform summation.

[0146] In practical implementation, scale parameter The value ranges from 0.01 to 10, with a total of 8 scale levels, distributed in an exponential growth manner: This setup can cover various scales from micrometers to millimeters, meeting the needs of different application scenarios. In this embodiment, The unit can be understood as m , which corresponds to the square of the spatial resolution.

[0147] The feature tracking subunit 432 is used to identify feature points at various scales, track the evolution trajectory of feature points in scale space, and construct feature flow. Feature points mainly include singular points of the stress field, such as the extrema, saddle points, and vortex points mentioned above.

[0148] In this embodiment, the feature point tracking employs a nearest neighbor matching algorithm. For two adjacent scales... and Feature point set in and Define matching criteria:

[0149] for and If the following conditions are met:

[0150] spatial distance ,in It is a scale-dependent threshold, typically valued at... Pixel;

[0151] Feature strength similarity ,in The typical value is 0.2;

[0152] The topological types are consistent, meaning that the two feature points belong to the same category;

[0153] Then it is believed and These are the same feature points in scale space. By connecting the corresponding feature points at different scales, a feature flow is formed. The feature flow records the life cycle of feature points in scale space, including generation, evolution, and disappearance.

[0154] in: and respectively scale and The coordinates of the feature points below, in pixels; This represents the Euclidean distance between two points, in pixels. This is the distance threshold, in pixels. and Representing feature points and The characteristic strength at a point can be either a stress value or a curvature value; This indicates taking the maximum value of the two feature intensities; The similarity threshold is dimensionless.

[0155] Feature fusion subunit 433 is used to persistently evaluate feature importance based on feature flow, fusing feature information from different scales to reconstruct a multi-scale stress field. Feature importance is generally positively correlated with its persistence in the scale space.

[0156] ,

[0157] in: It is a characteristic flow The weights are dimensionless; and These are feature flows The starting and ending scales, units and scale parameters Same, that is m ; It is the minimum scale parameter, usually 1. m ; This represents the natural logarithm function.

[0158] Based on feature weights, stress information at different scales can be fused into a multi-scale representation:

[0159] ,

[0160] in: It is a point The stress at a given location is expressed in Pa. It is a scale Lower point The stress value at the point, in Pa; These are the corresponding weights, dimensionless, and satisfy... ; This refers to the number of scale levels, which is 8 in this embodiment. In this embodiment, the weight allocation follows the principle of fine scale: medium scale: coarse scale = 5:3:2 to balance local details and global structure.

[0161] The polarization-based imaging-based semiconductor wafer stress distribution measurement system of this invention is applied to chip design and process optimization. By analyzing the stress distribution data of the wafer, it provides key information to guide the chip manufacturing process. Specifically, the system can provide the following information:

[0162] 1. Bending stress distribution introduced by the process and bending stress distribution caused by structural design;

[0163] 2. Distribution characteristics of local stress, overall stress, tensile strain, and compressive strain;

[0164] 3. Stress distribution variations under different temperatures and strain rates;

[0165] 4. The correlation between stress distribution and defects on the wafer surface.

[0166] In practical applications, this system has been used in 14nm processes, resulting in a 5-8 percentage point improvement in yield, a 20-30% widening of the key process parameter window, and a reduction in the new process development cycle from 18 months to 14 months, a decrease of 22%. Simultaneously, the system's early defect detection rate has increased from 60% to 90%, defect location accuracy from ±50μm to ±5μm, and root cause analysis success rate from 75% to 95%.

[0167] In addition, this system demonstrates outstanding economic benefits, with an equipment investment payback period of no more than 12 months, operating costs reduced by 30% compared to traditional systems, and an average annual added value of over US$5 million per production line.

[0168] In a typical workflow, the system first prepares and loads the wafer sample, followed by parameter settings and system calibration. Next, the system pre-scans the sample, allowing the operator to select regions of interest for detailed analysis. Subsequently, the system acquires multi-channel polarization images, performs real-time data processing, and displays the results. Finally, the system stores the data, generates an analysis report, and unloads the sample. The entire process is highly automated, with a single wafer processing time of no more than 5 minutes in standard mode and no more than 15 minutes in high-precision mode. The system supports automatic loading and unloading and continuous processing of up to 25 samples, significantly improving testing efficiency.

[0169] In summary, the semiconductor wafer stress distribution measurement system based on polarization imaging provided by this invention processes polarization phase information through the framework of differential geometry theory, achieving high-precision, multi-scale, and topology-preserving stress distribution measurement. This provides strong technical support for quality control and process optimization in semiconductor manufacturing, and has significant technical and economic benefits.

[0170] The embodiments described above are merely illustrative of specific implementations of the present invention, and while the descriptions are detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A semiconductor wafer stress distribution measurement system based on polarized light imaging, characterized in that, include: Polarization light source module, used to emit light beams with controllable polarization state; The wafer processing module is used to prepare semiconductor wafer samples, giving the samples a single crystal surface. A multi-channel optical system, connected to the polarization light source module, is used to illuminate the sample surface prepared by the wafer processing module with the light beam emitted by the polarization light source module, and capture the light field information generated on the sample surface to obtain a multi-channel polarization image. The polarization phase processing module is connected to the multi-channel optical system and is used to receive the multi-channel polarization image acquired by the multi-channel optical system, map the polarization phase information to the polarization manifold, calculate the geometric features on the polarization manifold, and extract the stress distribution information of the sample based on the geometric features. The polarization phase processing module includes a polarization state manifold construction unit, a stress-phase mapping unit, and a multi-scale analysis unit. The polarization state manifold construction unit is used to map the polarization state parameters in the multi-channel polarization image onto the Poincaré sphere representing the two-dimensional manifold, calculate the local metric tensor on the manifold, and construct the geometric representation of the polarization state space. The stress-phase mapping unit is used to calculate the connection coefficient and curvature tensor on the manifold, and to determine the stress tensor distribution of the sample based on the connection field and curvature characteristics. The multi-scale analysis unit is used to construct the scale space of the polarization manifold under different scale parameters, track the evolution trajectory of feature points in the scale space, and fuse multi-scale feature information to achieve seamless analysis of stress distribution from micro to macro.

2. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, The polarization-state manifold building unit includes: A polarization parameter calculation subunit is used to calculate the Stokes parameter of each pixel in the multi-channel polarization image; The manifold mapping subunit is used to map the Stokes parameters onto the Poincaré sphere to form a two-dimensional manifold representation. The metric tensor computation subunit is used to compute the local metric tensor on the manifold, which describes the geometric properties of the polarization state space. The parallel transmission calculation subunit is used to calculate the parallel transmission path of the polarization state on the manifold and analyze the deviation between the actual polarization transmission and the ideal geodesic.

3. The semiconductor wafer stress distribution measurement system based on polarized light imaging according to claim 1, characterized in that, The stress-phase mapping unit includes: The connection field calculation subunit is used to calculate the connection coefficients on the polarization manifold and describe the parallel transmission rules of the polarization state. The curvature analysis sub-unit is used to calculate the curvature tensor of the manifold based on the connection coefficient, and to extract the principal curvature and principal direction. The stress mapping sub-unit is used to establish the mapping relationship between the connection field and the stress tensor based on photoelasticity theory, and to calculate the stress components. The topology analysis sub-unit is used to identify singularities in the stress field, classify the singularities, and construct a topology diagram of the stress field.

4. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, The multi-scale analysis unit includes: Scale space construction sub-units are used to set the scale parameter sequence and apply thermal kernel smoothing to the original polarization field to generate polarization fields of different scales; The feature tracking subunit is used to identify feature points at various scales, track the evolution trajectory of feature points in scale space, and construct feature flow; The feature fusion subunit is used to continuously evaluate the importance of features based on feature flow, fuse feature information at different scales, and reconstruct a multi-scale stress field.

5. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, The multi-channel optical system includes: An illumination module is used to illuminate the sample surface prepared by the wafer processing module with the light beam emitted by the polarization light source module; The imaging module is used to capture the light field information generated by the sample under the illumination of the light beam; The front-end components include a beam splitter, a polarizing filter, and a beam splitter. The optical path that follows includes the illumination optical path and the reference optical path; The illumination optical path includes a quarter-wave plate, a single aperture, multiple mirrors, lenses, and a sample stage; the reference optical path includes a mirror and multiple lenses.

6. The semiconductor wafer stress distribution measurement system based on polarized light imaging according to claim 5, characterized in that, The imaging module includes: CCD array imaging unit, used to capture light field images; The camera lens is arranged cofocally with the CCD area array imaging unit; Collimating lens, used to collimate light beams; When the system is used to measure the stress distribution in a local area, it also includes a microscope objective, which is arranged confocally with the collimating lens to magnify the sample on the sample stage.

7. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, Also includes: The data acquisition and processing module is connected to the multi-channel optical system and the polarization phase processing module. It is used to acquire multi-channel images obtained by the multi-channel optical system, compress the acquired data, and provide the compressed data to the polarization phase processing module. The results visualization module is connected to the polarization phase processing module and is used to receive the stress distribution data output by the polarization phase processing module, and generate a stress distribution visualization image, an interactive analysis interface, and a structured analysis report.

8. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, The wafer processing module includes: Wafer pretreatment equipment is used to pre-process or modify samples, including cleaning equipment and photolithography equipment; Wafer coating apparatus for forming single-crystal surfaces on sample surfaces; The sample stage is used to place the sample in a direction perpendicular to the propagating beam, and the angle can be adjusted along the direction parallel to the beam so that the sample surface is perpendicular to the beam.

9. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 7, characterized in that, The data acquisition and processing module includes: A multi-channel image acquisition unit is used to acquire multi-channel images obtained by a multi-channel optical system; The multi-channel image compression unit is used to compress the acquired multi-channel images, control the beam splitting intensity and duty cycle of the beam emitted by the polarized light source, perform beam splitting acquisition on the acquired multi-channel images, and use computational imaging algorithms to reconstruct the acquired data to achieve compression. The data transmission unit is used to transmit the compressed data to the polarization phase processing module.

10. The semiconductor wafer stress distribution measurement system based on polarization imaging according to claim 1, characterized in that, The system is applied to chip design and process optimization, and provides the following information by analyzing wafer stress distribution data: The distribution of bending stress introduced by the process and the distribution of bending stress caused by the structural design; Distribution characteristics of local stress, overall stress, tensile strain, and compressive strain; Stress distribution variations under different temperatures and strain rates; The correlation between stress distribution and defects on the wafer surface can be used to improve the yield of chip design and manufacturing processes.

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