A semiconductor wafer temperature, warpage and stress field combined prediction method and system

Abstract

The application is suitable for the technical field of multi-physical field monitoring of semiconductor manufacturing, and relates to a semiconductor wafer temperature, warping and stress field joint prediction method and system, which comprises the following steps: constructing a thermal-mechanical coupling finite element model to generate a multi-working condition data set, constructing a double-branch neural operator network model based on a Fourier operator neural network, extracting global spatial correlation through frequency domain feature transformation and synchronously outputting a temperature field and a warping displacement field, and then calculating a stress field through a thermoelastic constitutive relationship; in the training, a physical residual loss is constructed by introducing a heat conduction and an elastic mechanics partial differential equation, and is combined with a data fitting loss to be optimized. The application realizes millisecond-level output of multi-physical field distribution in a single forward reasoning, has high reconstruction accuracy and strong generalization ability under extreme sparse temperature measurement conditions, and can be widely applied to online monitoring and digital twin systems of semiconductor equipment.

Description

Technical Field

[0001] This invention belongs to the field of multi-physics field monitoring technology in semiconductor manufacturing, and particularly relates to a method and system for jointly predicting semiconductor wafer temperature, warpage and stress field. Background Technology

[0002] In semiconductor manufacturing processes, particularly critical steps such as chemical vapor deposition (CVD), physical vapor deposition (PVD), etching, and rapid thermal processing, the thermal uniformity of the wafer under high-temperature environments directly impacts thin film deposition rates, stress distribution, and subsequent pattern transfer accuracy. Non-uniform temperature fields not only lead to film thickness variations but also cause thermal stress concentration and wafer warping, thus affecting overlay accuracy and chip yield. Therefore, accurate monitoring and prediction of the wafer's thermal field are crucial for achieving stable control of advanced processes and upgrading digital equipment. Due to the complex environmental conditions within the process chamber, including high temperatures, vacuum, plasma, or reactive atmospheres, it is impossible to deploy sensors on a large scale on the wafer surface. In practical engineering, thermocouples are typically used to measure the wafer temperature, or a limited number of temperature measurement points are placed at the bottom and edges of the heating pad, resulting in sparsely distributed temperature data. Under these conditions, how to accurately reconstruct the overall temperature distribution of the wafer based on limited measurement point information becomes a key technical challenge in process monitoring and control.

[0003] Existing methods for thermal field reconstruction using sparse measuring points mainly include numerical simulation methods based on physical models and data-driven regression prediction methods. Numerical simulation methods typically use finite element analysis to solve the heat conduction equation and combine it with parameters such as material thermal conductivity, boundary conditions, and contact thermal resistance for modeling. These methods rely on precise geometric structures and thermophysical parameter settings, resulting in complex model construction, high computational cost, and an inability to predict multiphysics fields using realistic process parameters, making them unsuitable for real-time monitoring and control equipment. Data-driven methods collect historical simulation data or process data and use neural networks, Gaussian process regression, or interpolation algorithms to establish a mapping relationship between sparse temperature measuring points and the overall field temperature, achieving rapid prediction. Although these methods have high computational efficiency, most employ a global fitting structure based on grid point regression. When facing complex thermal distribution scenarios such as multi-zone heating structures, edge heat dissipation effects, and abrupt changes in local thermal gradients, they often struggle to simultaneously maintain the accuracy of the overall thermal trend and the precise prediction of local high-gradient regions, easily leading to over-smoothing or local prediction bias. Patent CN120895491B discloses a method and system for intelligent wafer packaging and testing in semiconductor chips. The method includes initializing magnetoelectric field coordination parameters, optimizing these parameters using a multi-objective genetic algorithm, and driving core-shell nanoparticles to metallize and fill high-precision through-hole wafers, generating wafers with metallized through-hole structures. Based on the metallized through-hole wafers, bonding parameters are generated using laser Doppler frequency spectroscopy analysis, and flip-chip interconnection is achieved using thermo-ultrasonic virtual technology, generating low-defect bonded wafers. A finite element inversion algorithm is used to establish a stress distribution map of the bonding interface, and a thermal stress-resistant encapsulation layer is constructed using an intelligent stress matching algorithm to encapsulate the low-defect bonded wafers. This patent also uses the finite element method to analyze and test semiconductor wafers, which involves a large computational load and suffers from the same drawbacks as existing technologies.

[0004] Therefore, how to construct a multiphysics field reconstruction method that combines physical consistency, structural generalization ability and computational efficiency under limited temperature measurement information, and realize the coordinated prediction of wafer temperature field, stress field and warpage field, has become an urgent technical problem to be solved in the field of semiconductor manufacturing process monitoring and intelligent modeling. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the purpose of this invention is to provide a method for jointly predicting the temperature, warpage, and stress field of semiconductor wafers. This method solves the technical problems in semiconductor wafer manufacturing processes where the extremely sparse temperature measurement points prevent traditional numerical simulations from meeting real-time requirements, and where purely data-driven models struggle to synchronously and physically decouple the prediction of the multi-physics distribution of temperature, warpage, and stress. Furthermore, this invention also provides a system for jointly predicting the temperature, warpage, and stress field of semiconductor wafers.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] In a first aspect, the present invention provides a method for jointly predicting semiconductor wafer temperature, warpage, and stress field, comprising the following steps:

[0008] S10. Construct a wafer thermo-mechanical coupled finite element model and generate a multi-condition training dataset containing sparse temperature measurement data and corresponding full-field temperature and stress distributions.

[0009] S20. Arrange several discrete temperature sensing units on the semiconductor process cavity wafer to acquire sparse temperature measurement data.

[0010] S30. Construct a two-branch neural network model based on Fourier operator neural network, and reconstruct the continuous distribution of the entire wafer field by learning the operator mapping between the input space and the output function space.

[0011] S40. Supervise the training of the dual-branch neural operator network model using the multi-condition training dataset, so that it can synchronously output the wafer's full-field temperature distribution, warpage displacement distribution and stress distribution.

[0012] S50. Input the sparse temperature measurement data collected in real time during the actual semiconductor process into the trained dual-branch neural operator network model, obtain the full-field temperature distribution and displacement distribution of the wafer through forward inference, and obtain the full-field stress distribution of the wafer in real time through thermoelastic constitutive relation.

[0013] S60, the output results are used for dynamic adjustment of power distribution in the heating zone of semiconductor equipment.

[0014] Furthermore, the specific steps of S10 are as follows:

[0015] S101. Construct a three-dimensional geometric model that includes the wafer, heater, and partitioned heating structure;

[0016] S102. Set the temperature-related thermal and mechanical parameters of the material;

[0017] S103. By changing the heating power, boundary heat dissipation conditions and environmental parameters, multiple sets of simulation samples under different working conditions are generated. Each set of simulation samples includes sparse temperature measurement point temperature data, corresponding full-field temperature field, stress field and displacement field true data.

[0018] Furthermore, the specific steps for constructing the two-branch neural operator network model based on the Fourier operator neural network in S30 are as follows:

[0019] S301. Map the sparse temperature measurement data into a discrete two-dimensional input field, wherein the input field includes a temperature interpolation field, a measurement point mask matrix, and a spatial coordinate encoding.

[0020] S302. The discrete two-dimensional input field is mapped to the frequency domain space through a fast Fourier transform;

[0021] S303. Perform a linear transformation on the low-frequency modes in the frequency domain space to extract global spatial correlation features;

[0022] S304. The spatial domain is recovered through inverse Fourier transform, and the field features are iteratively updated through a multi-layer operator network structure, as shown in the following formula:

[0023] ;

[0024] in, and The first Layer and First The characteristic field of the layer; The weight matrix is ​​a local linear transformation matrix; It is a frequency-domain learnable complex weight tensor; It is a non-linear activation function.

[0025] Furthermore, in step S40, a thermophysical residual loss based on the heat conduction control equation, a mechanical residual loss based on the mechanical equilibrium equation, and a data fitting loss function are constructed, and a total loss function is formed by weighting them to achieve synergistic optimization of the temperature field and the stress field. The expression of the total loss function is as follows:

[0026] ;

[0027] in, For data fitting loss, For the residuals of the heat conduction equation, The residuals of the mechanical equilibrium equations, To reduce data loss weights, The weighting coefficients for physical consistency constraints. This is the scaling factor for the physical residual.

[0028] Furthermore, in S50, the real-time calculation of the full-field stress distribution of the wafer through thermoelastic constitutive relations specifically includes:

[0029] The spatial derivative of the continuous displacement field output by the model is obtained by using automatic differentiation to obtain the total strain tensor;

[0030] Based on the plane stress assumption, the stress tensor is calculated by subtracting thermal strain using the generalized Hooke's law. .

[0031] Furthermore, the stress tensor The constitutive equation components are expressed as:

[0032] ;

[0033] Where E is Young's modulus. Poisson's ratio, The coefficient of thermal expansion varies with temperature. Where T is temperature. This is the stress-free reference temperature.

[0034] Furthermore, the sparse temperature measurement data originates from a limited number of temperature sensors arranged on electrostatic chucks, thermocouple structures, or equivalent temperature measurement units, and the number of temperature sensors is less than the number of grid nodes used to obtain the overall temperature distribution.

[0035] Furthermore, in S40, supervised training employs a dynamic annealing strategy:

[0036] In the purely data-driven era: This allows the model to quickly fit the approximate outline of the label;

[0037] During the period of introduction of physical constraints: from The index rose to 1.0, forcing the model to correct for non-physical temperature differences and stress concentrations in areas without measurement points;

[0038] During the multi-physics fine-tuning period: a gradient-based adaptive weighting algorithm is adopted to automatically balance the gradient magnitudes of thermal and mechanical residuals during backpropagation, ensuring synchronous convergence of the multiphysics fields.

[0039] Furthermore, the dual-branch neural operator network model achieves implicit learning of coupled features between the temperature field and the warped displacement field by sharing a deep feature extraction backbone. The dual-branch projection head includes a temperature branch and a warped branch, which are respectively projected in a dimension reduction manner through independent multilayer perceptron networks.

[0040] Secondly, the present invention also provides a joint prediction system for semiconductor wafer temperature, warpage, and stress field, comprising:

[0041] The monitoring module is used to reconstruct the full-field temperature distribution, warpage, and stress distribution of the wafer in real time;

[0042] The decision module is used to compare the reconstructed real-time distribution with preset process thresholds to identify local overheating areas or stress concentration points.

[0043] The control module is used to dynamically adjust the power distribution of the heating element or the load-bearing parameters of the mechanical support structure in the semiconductor device according to the identification results of the decision module, so as to achieve closed-loop uniform control of the wafer thermal field and stress field.

[0044] The method and system for jointly predicting semiconductor wafer temperature, warpage, and stress field provided by this invention have at least the following advantages compared with the prior art:

[0045] In semiconductor wafer manufacturing, existing technologies suffer from extremely sparse temperature measurement points, making traditional numerical simulations unable to meet real-time requirements. Furthermore, purely data-driven models struggle to synchronously and physically decouple the predicted temperature, warpage, and stress multi-physics distributions. This invention eliminates the reliance on high-density sensor arrays. By embedding heat conduction and mechanical equilibrium equations into a neural network, it utilizes physical laws to compensate for the information gaps caused by sparse sensors. Even with only a limited number (e.g., 17) of discrete measurement points, it ensures that the reconstructed full-field temperature and stress distributions conform to physical reality, avoiding the non-physical oscillations or overfitting phenomena common in purely data-driven methods. This invention overcomes the limitation of traditional methods that can only monitor temperature. Through physical constraints and automatic differentiation techniques, it can simultaneously calculate macroscopic warpage displacement and the internal thermal stress of the wafer while outputting the temperature field. This is equivalent to constructing a "virtual sensor" across the entire wafer field, providing process engineers with a crucial indicator for directly assessing whether wafer warpage has occurred, leading to uneven coating or breakage risks. Compared to traditional finite element simulation (FEM), which requires hours of mesh generation and iterative solving, the inference speed of the trained model in this invention can reach the millisecond level. Combined with Sim2Real transfer learning technology, this method can quickly adapt to different process cavities and has industrial application value for direct embedding into equipment control systems for real-time closed-loop feedback (run-to-run control). Attached Figure Description

[0046] To more clearly illustrate the solution of the present invention, a brief introduction will be given to the drawings used in the description of the embodiments below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0047] Figure 1 A flowchart of a method for jointly predicting semiconductor wafer temperature, warpage, and stress field provided in an embodiment of the present invention;

[0048] Figure 2 This is a schematic diagram of the semiconductor equipment used in a method for jointly predicting the temperature, warpage, and stress field of a semiconductor wafer, as provided in an embodiment of the present invention.

[0049] Figure 3 A flowchart illustrating the architecture of a neural network in a method for jointly predicting temperature, warpage, and stress field of a semiconductor wafer, provided in an embodiment of the present invention.

[0050] Figure 4 This is a schematic diagram illustrating the effect of a method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to an embodiment of the present invention. Detailed Implementation

[0051] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0052] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0053] This invention provides a method for jointly predicting semiconductor wafer temperature, warpage, and stress field, applicable to advanced semiconductor packaging and manufacturing processes. The method includes the following steps:

[0054] S10. Construct a wafer thermo-mechanical coupling finite element model to generate a multi-condition training dataset containing sparse temperature measurement data and corresponding full-field temperature and stress distributions; S20. Deploy several discrete temperature sensing units on the semiconductor process cavity wafer to acquire sparse temperature measurement data; S30. Construct a two-branch neural network model based on a Fourier operator neural network, and reconstruct the continuous distribution of the wafer's full field by learning the operator mapping between the input space and the output function space; S40. Supervised training of the two-branch neural network model using the multi-condition training dataset to enable it to synchronously output the wafer's full-field temperature distribution, warpage displacement distribution, and stress distribution; S50. Input the sparse temperature measurement data collected in real time during actual semiconductor process operation into the trained two-branch neural network model, obtain the wafer's full-field temperature and displacement distribution results through forward inference, and obtain the wafer's full-field stress distribution in real time through thermoelastic constitutive relations; S60. The output results are used for dynamic adjustment of the power distribution in the heating zone of semiconductor equipment.

[0055] This invention achieves millisecond-level output of multiphysics distribution in a single forward inference, and has both high reconstruction accuracy and strong generalization ability under extremely sparse temperature measurement conditions. It can be widely used in online monitoring of semiconductor equipment and digital twin systems.

[0056] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0057] This invention provides a method for jointly predicting semiconductor wafer temperature, warpage, and stress field, applied in advanced semiconductor packaging and manufacturing processes. In this embodiment, the system architecture of the method includes:

[0058] Multiphysics simulation data generation module: used to build a thermo-mechanical coupling simulation model covering multiple process scenarios, and generate a true dataset containing sparse temperature measurement points, full field temperature, warping displacement and stress;

[0059] The neural operator network pre-training module is based on the Fourier operator network (FNO) architecture. It realizes the basic learning of frequency domain mapping from discrete sparse temperature measurement input field to continuous multi-physics field (temperature, warping) and calculates the stress field.

[0060] Physical consistency constraint calculation module: Utilizes automatic differentiation technology to construct a physical residual loss function based on the heat conduction equation and the thermoelastic equilibrium equation;

[0061] Joint optimization and multi-field decoupling module: By dynamically balancing data fitting loss and physical constraint loss, it achieves high-precision training of neural operator models and decoupling of implicit features of multi-physics fields;

[0062] Online real-time prediction and monitoring module: Receives sparse temperature measurement data from the actual wafer carrier, and outputs a real-time distribution cloud map of temperature, stress and warpage across the entire field in milliseconds through a single forward inference.

[0063] Combination Figures 1 to 4 In this embodiment, the method for jointly predicting semiconductor wafer temperature, warpage, and stress field includes the following steps:

[0064] S10. Construct a wafer thermo-mechanical coupled finite element model and generate a multi-condition training dataset containing sparse temperature measurement data and corresponding full-field temperature and stress distributions.

[0065] Specifically, in this embodiment, a temperature and stress field dataset covering different process scenarios is generated through finite element multiphysics simulation. In wafer structures with a thickness-to-diameter ratio much less than 1, based on the thin-plate assumption and plane stress conditions, the temperature field is considered as a two-dimensional distribution, driving three-dimensional thermoelastic bending behavior. This dataset includes sparse temperature measurement point inputs and corresponding full-field temperature, stress tensor distribution, and warpage displacement field true values ​​output. The simulation process is based on the constructed thermo-mechanical coupled finite element model, establishing geometric models of the wafer, heater, and partitioned heating structure, configuring nonlinear physical property parameters such as temperature-related thermal conductivity, elastic modulus, and coefficient of thermal expansion of the material, and setting different spatial distributions of heating power, boundary heat dissipation conditions, and environmental parameters. Multiple sets of supervised samples under various operating conditions are obtained by solving the heat conduction equation and the linear elasticity equation. To improve the model's generalization ability, the dataset can be expanded by perturbing the heating power, material parameters, and boundary conditions. Under steady-state or quasi-steady-state heating conditions, due to the thin wafer thickness and high thermal conductivity of silicon, the temperature gradient in the thickness direction is negligible compared to the in-plane temperature gradient. Therefore, the temperature field can be approximately represented as a two-dimensional function:

[0066] ;

[0067] This means that the temperature is assumed to be uniformly distributed along the thickness direction.

[0068] The steady-state heat conduction equation is:

[0069] ;

[0070] Thermoelastic equilibrium equation:

[0071] ;

[0072] in, For temperature, Thermal conductivity as a function of temperature For heat source items, This is a physical activity. Let be the stress tensor.

[0073] S20. Arrange several discrete temperature sensing units on the semiconductor process cavity wafer to obtain sparse temperature measurement data.

[0074] S30. Construct a two-branch neural network model based on Fourier operator neural network, and reconstruct the continuous distribution of the entire wafer field by learning the operator mapping between the input space and the output function space.

[0075] Specifically, in this embodiment, this step aims to enable the neural operator model to learn the mapping relationship from a discrete sparse temperature observation field to a continuous temperature field and stress warping field. The neural operator model adopts a Fourier Neural Operator (FNO) structure. The Fourier transform-based neural operator network extracts and maps features through the following steps:

[0076] S301. Constructing the input feature field: First, the sparse temperature measurement point data is concatenated into a two-dimensional input feature field. The input field contains the initial interpolated full-field temperature distribution, a mask matrix marking the actual temperature measurement point locations, and spatial coordinate information. Subsequently, a simple linear mapping is used to... Convert to initial feature field , and then fed into the operator network.

[0077] S302, Dual-branch feature extraction: The network consists of multiple stacked Fourier operator layers. For the first... The feature update process of this layer is divided into two parallel paths:

[0078] Frequency domain path (learning global trends): using Fast Fourier Transform (FFT) The current feature field Transform to the frequency domain. In the frequency domain, discard high-frequency components and retain only the low-frequency core modes, multiplying them by a learnable complex weight tensor. This is used to extract global spatial correlation features; finally, inverse fast Fourier transform (IFFT) is applied. Restore it to the spatial domain.

[0079] S303, Spatial Path (Preserving Local Details): Directly Using the Weight Matrix For the current feature field A local linear transformation is performed to preserve local features near the wafer edge or heating zone.

[0080] S304, Feature Update Formula: Add the processing results of the above two paths, process them through a non-linear activation function, and output them to the next layer. The feature update formula for the layer operator is:

[0081] ;

[0082] in, and The first Layer and First The characteristic field of the layer; The weight matrix is ​​a local linear transformation matrix; It is a frequency-domain learnable complex weight tensor; It is a non-linear activation function.

[0083] S40. Supervised training of the dual-branch neural operator network model is performed using a multi-condition training dataset, enabling it to synchronously output the wafer's full-field temperature distribution, warpage displacement distribution, and stress distribution.

[0084] Specifically, in this embodiment, the network achieves implicit learning of coupled features between temperature, stress, and warpage by sharing a deep feature extraction backbone, thereby avoiding the error accumulation problem caused by calculating temperature first and then calculating stress and warpage in traditional methods.

[0085] Furthermore, in this embodiment, a multi-physics joint supervised loss function is constructed to optimize and train the model. The total loss function includes the temperature field physical fitting loss, the stress field physical fitting loss, and the sparse temperature data fitting loss, which are balanced by weighting coefficients. Preferably, a physical consistency constraint loss is introduced, incorporating the residuals of the heat conduction control equation and the mechanical equilibrium equation into the loss function. The specific calculation process includes: selecting sampling points within the wafer region, using automatic differentiation to calculate the spatial derivatives of the predicted temperature and displacement fields; calculating the residuals based on the heat conduction equation and the linear elastic mechanical equilibrium equation; and using the squared mean of the residuals as the physical consistency loss. The total loss function is composed of a weighted average of the data loss and the physical loss, and the neural operator model parameters are jointly optimized through backpropagation.

[0086] Thermal conductivity residual:

[0087] ;

[0088] Mechanical equilibrium residuals:

[0089] ;

[0090] The mean squared residuals are used as the physical consistency loss:

[0091] ;

[0092] Total loss function It consists of a weighted average of data loss and physical loss:

[0093] ;

[0094] in, This is the scaling factor for the physical residual. To reduce data loss weights, These are the weighting coefficients for physical consistency constraints. The parameters of the neural operator model are jointly optimized through backpropagation.

[0095] S50. Input the sparse temperature measurement data collected in real time during the actual semiconductor process into the trained dual-branch neural operator network model, obtain the full-field temperature distribution and displacement distribution of the wafer through forward inference, and obtain the full-field stress distribution of the wafer in real time through thermoelastic constitutive relations.

[0096] Specifically, in this embodiment, the input data is preprocessed (outlier removal, normalization, and mask construction) to form a discrete input field, which is then input into the trained neural operator model. Through a single forward inference, a high-resolution full-field temperature distribution and warping displacement field can be output simultaneously.

[0097] To ensure physical consistency between temperature, displacement, and stress field, this embodiment does not directly use a neural network to predict stress. Instead, it uses a built-in analytical calculation layer to calculate the stress based on the temperature field output by the network. and displacement field Real-time stress field export The specific calculation logic is as follows:

[0098] Continuous displacement field output by the network using automatic differentiation Calculate the spatial derivative. Based on the Kirchhoff-Love thin-plate assumption, the total strain tensor... The components are expressed as:

[0099] ;

[0100] in, For in-plane displacement, This is an out-of-plane displacement (curvature).

[0101] Note: The formula includes the first derivative of the displacement (in-plane strain) and the second derivative (bending strain caused by warping).

[0102] Under the plane stress assumption Under, stress tensor The components of the physical constitutive equation are expressed as:

[0103] ;

[0104] Where E is Young's modulus (it should be specified whether it varies with temperature). change), Poisson's ratio, The coefficient of thermal expansion as a function of temperature (a key parameter in semiconductor manufacturing). For one of them This is the stress-free reference temperature.

[0105] S60, the output results are used for dynamic adjustment of power distribution in the heating zone of semiconductor equipment.

[0106] Specifically, in this embodiment, the neural operator network branches directly output the displacement distribution across the entire wafer. Subsequently, using the built-in analytical operator layer, the first and second spatial derivatives of the displacement field are obtained through automatic differentiation (Autograd). Combined with the aforementioned geometric equations and thermoelastic constitutive equations, the stress distribution is calculated analytically in real time. This process does not involve training network parameters, thus forcing the displacement field and stress field to meet physical consistency in structural design.

[0107] This invention also provides a joint prediction system for semiconductor wafer temperature, warpage, and stress field. In this embodiment, the joint prediction system for semiconductor wafer temperature, warpage, and stress field includes:

[0108] The monitoring module is used to reconstruct the full-field temperature distribution, warpage, and stress distribution of the wafer in real time;

[0109] The decision module is used to compare the reconstructed real-time distribution with preset process thresholds to identify local overheating areas or stress concentration points.

[0110] The control module is used to dynamically adjust the power distribution of the heating elements or the load-bearing parameters of the mechanical support structure in the semiconductor device based on the identification results of the decision module, so as to achieve closed-loop uniform control of the wafer thermal field and stress field.

[0111] Example 1

[0112] This invention provides a method for jointly predicting semiconductor wafer temperature, warpage, and stress field, comprising the following steps:

[0113] After system initialization, a temperature-stress-warping dataset covering different heating scenarios is first generated using multiphysics simulation software to provide high-quality supervised samples for operator learning.

[0114] Next, a Fourier Operator Network (FNO) model is constructed. Sparse temperature measurement points are mapped to discrete input fields containing masks and coordinate codes. The FNO model is trained using the generated simulation dataset to learn the mapping relationship from the sparse temperature observation space to the continuous multiphysics function space.

[0115] During model training, physical consistency constraints are introduced. Automatic differentiation is used to calculate the spatial derivative of the continuous field output by the model. This derivative is then substituted into the heat conduction equation and the elasticity equation to calculate the residuals. The physical residuals are weighted with the data fitting error to form a joint loss function, which is then used to jointly optimize the network parameters through backpropagation.

[0116] After the model training converges, it is deployed on semiconductor equipment. In actual process operation, it receives sparse temperature data from thermocouple-sensored wafers in real time, corresponding to the sparse temperature data on the heater surface.

[0117] After normalization and masking, the input data is fed into the Fourier Neural Operator (FNO) model. The model outputs high-resolution full-field temperature distribution, in-plane stress cloud map and warpage displacement field of the wafer in milliseconds through fast frequency domain convolution and inverse transform.

[0118] Finally, the multiphysics prediction results are visualized and safety thresholds are set to provide real-time decision-making basis for the dynamic adjustment of process recipes and the run-to-run closed-loop control of equipment.

[0119] In this embodiment, the finite element multiphysics simulation dataset is constructed as follows:

[0120] Geometric model definition: Based on the actual cavity structure of physical vapor deposition (PVD) or chemical vapor deposition (CVD) processes, a three-dimensional thermo-mechanical coupled geometric model is constructed using finite element software such as COMSOL Multiphysics.

[0121] Wafer parameters: 300mm (12 inches) in diameter, 0.775mm in thickness, made of monocrystalline silicon (Si).

[0122] Multi-zone Heater: 315mm in diameter, 25mm thick, made of aluminum nitride (AlN) ceramic. It is divided into three independently controlled annular heating zones: a central zone (Radius 0-50mm), a middle zone (50-100mm), and an edge zone (100-150mm).

[0123] Material property configuration: To ensure the accuracy of thermoelasticity simulation, nonlinear material property parameters that vary with temperature T (unit: K) are configured.

[0124] Single-crystal silicon (Si):

[0125] Thermal conductivity:

[0126] ;

[0127] Coefficient of linear expansion: ;

[0128] Elastic modulus:

[0129] Pa;

[0130] Poisson's ratio:

[0131] Aluminum nitride (AlN): Thermal conductivity is configured as an empirical polynomial that decreases slightly with temperature, and the coefficient of linear expansion is constant. .

[0132] In this embodiment, the boundary conditions and solution settings are as follows:

[0133] Thermal conduction boundary: The bottom surface of the wafer is in contact with the upper surface of the heater, which is set as a contact thermal conduction boundary condition.

[0134] To reflect the actual mechanical bonding state, an equivalent thermal contact conductivity is applied between the wafer bottom surface and the heater. The value is set according to the surface roughness and contact pressure, with a typical range of 1000-3000 W / (m²·K).

[0135] Surface-to-surface radiation is set between the front side of the wafer and the cavity dome, with the surface emissivity set to 0.7.

[0136] Solid mechanical boundary: Bottom support constraint: To simulate the mechanical support effect of the wafer on the heating platform, while allowing in-plane thermal expansion slip, a normal elastic support is applied to the bottom surface of the wafer:

[0137] ;

[0138] in: Normal support stiffness coefficient; typical order of magnitude: ;In-plane direction ( No constraints are imposed to allow free thermal expansion.

[0139] Rigid body displacement elimination condition: To avoid overall rigid body translation or rotation during the solution process, a minimum constraint is applied to the wafer's geometric center: fix the center point. Displacement;

[0140] A steady-state two-way thermo-mechanical coupled solver is used. First, the nonlinear steady-state heat conduction equation is solved iteratively. After the temperature field converges (residuals) The thermal strain load is then passed into the linear elasticity module to solve for displacement and stress.

[0141] In this embodiment, the dataset generation and multi-scenario generalization expansion are as follows:

[0142] Basic operating condition sampling: Latin hypercube sampling was used to randomly combine the power of the three inner, middle and outer heating zones (0-3000W / zone) to generate 5000 basic samples.

[0143] Data perturbation augmentation: Temperature measurement noise injection: Random white noise with a signal-to-noise ratio (SNR) of 10dB is superimposed on the extracted sparse temperature labels.

[0144] Contact thermal resistance fluctuation: This involves analyzing contact parameters. The uniform disturbance simulates the local heat transfer changes caused by gas fluctuations in actual processes.

[0145] Through the above enhancement strategies, a total of 10,000 training samples were finally obtained. For each sample, the coordinates and temperature of M=17 discrete points were extracted as input. Full field temperature at resolution VonMises equivalent stress And Z-axis displacement (warping) As a truth label.

[0146] In this embodiment, the model based on the Fourier neural operator is constructed as follows:

[0147] Construction and mapping of discrete input fields: Transforming a thermocouple wafer with 17 regularly arranged temperature measurement points into a regular grid input field that FNO can process:

[0148] Coordinate Encoding and Mask Generation: Construction A blank background mesh. Generate a measurement point mask matrix. A value of 1 is assigned to the grid pixels corresponding to 17 measurement points, and 0 is assigned to the rest. A coordinate matrix is ​​generated. Includes normalized physical coordinates.

[0149] Discrete-point interpolation initialization: Using radial basis function (RBF) interpolation or bicubic interpolation, the temperature at 17 points is expanded into a coarse full-field initial temperature field. .

[0150] Feature concatenation: The above channels are concatenated to form a dimension of Input feature tensor .

[0151] Lifting maps 4-channel features to a higher-dimensional latent space (e.g., using a fully connected network (MLP) layer) through a single layer. (channel), to obtain initial hidden variables .

[0152] In this embodiment, the frequency domain global convolution operation (FourierLayers) is as follows:

[0153] Stacking The Fourier neural operator layer of the first layer. Layer, feature updates follow these steps:

[0154] Fast Fourier Transform (FFT): transforms latent variables Transform to the frequency domain.

[0155] Low-frequency mode cutoff: Set the cutoff wavenumber By retaining only low-frequency information in the spatial frequency range and directly filtering out high-frequency disturbances that may be introduced by interpolation or sensor noise, this closely matches the physical smoothness of the heat diffusion process.

[0156] Complex linear transformation: transforming the truncated frequency domain tensor into a learnable complex weight tensor. Perform multiplication operations to extract global spatial thermal correlations.

[0157] Inverse Fourier Transform (IFFT): Maps the transformed features back to the spatial domain.

[0158] Residual addition and activation: The output of the IFFT is summed with the output of a local convolution kernel in the spatial domain. The processed local features are summed, then passed through the GELU activation function to output. .

[0159] In this embodiment, the decoupling design (projection) of the multiphysics output head is as follows:

[0160] For the deepest hidden variables Dimensionality reduction projection is performed. To avoid gradient conflicts between physical quantities of different magnitudes (temperature hundreds of K, stress tens of MPa, displacement tens of micrometers) during feature extraction, a dual-branch independent projection network is used:

[0161] Temperature branch: ;

[0162] Warped branches: ;

[0163] Calculated from constitutive relations ;

[0164] The above structure ensures that the model can decouple macroscopic multiphysics fields from a set of underlying physical characteristics (such as thermal expansion trends) through different nonlinear mappings.

[0165] In this embodiment, the joint training strategy incorporating physical constraints is as follows:

[0166] Automatic Differentiation and Physical Residual Calculation: To overcome overfitting under sparse input in purely data-driven scenarios, physical partial differential equations (PDEs) are introduced as a penalty term.

[0167] Selection within the wafer domain via Latin hypercube sampling Configuration points Since FNO is an approximation of continuous functions, we can use PyTorch's autograd function to obtain the exact derivative of the model's output field with respect to spatial coordinates.

[0168] residuals of heat conduction equation :calculate Thermal conductivity Calculate in real time based on the aforementioned physical property formulas.

[0169] Mechanical equilibrium equation residuals : Through the predicted continuous displacement field Calculate the total strain tensor, subtracting thermal strain. Then, the stress is calculated using the elasticity matrix. The divergence is then calculated. It serves as a force balance residual.

[0170] Joint Loss Function and Dynamic Weight Allocation: The total loss function consists of the data fit term (MSE) and the physical residual term (PDELoss).

[0171] ;

[0172] Dimensional normalization: Due to the relationship between stress (MPa) and displacement ( The numerical differences are extremely large. Before calculating the MSE, all physical quantities are normalized to a distribution with a mean of 0 and a variance of 1 by StandardScaler.

[0173] Dynamic annealing strategy:

[0174] Epoch 0-500 (Purely Data-Driven Period): This allows FNO to quickly fit the approximate outline of the label.

[0175] Epoch 500-1000 (Physical Constraint Introduction Period): from The index rose to 1.0. The forced model correction addresses non-physical temperature differences and stress concentrations occurring in areas without measurement points (blind zones).

[0176] Epoch 1000-3000 (Multi-physics fine-tuning period): A gradient-based adaptive weighting algorithm (such as GradNorm) is used to automatically balance the gradient magnitudes of thermal and mechanical residuals during backpropagation, ensuring synchronous convergence of the multiphysics fields.

[0177] In this embodiment, the actual application of online monitoring of semiconductor processes is as follows:

[0178] After the model is trained and the weights are fixed, it is deployed in the control computer of the PVD or CVD machine.

[0179] Online reasoning process:

[0180] Data acquisition: The machine collects the real-time temperature of 17 thermocouples on the surface of the heating plate at a frequency of 10Hz.

[0181] Forward propagation: The 17 temperature values ​​are encapsulated as input tensors and fed into the FNO model. Benefiting from the extremely high forward computation efficiency of FNO, the single inference time is <15ms on a single RTX 3090 graphics card.

[0182] Result output and alarm logic: Synchronous output Temperature cloud map, stress cloud map and warping deformation map.

[0183] Security check: The system extracts the maximum temperature difference across the entire venue in real time. Maximum local thermal stress and center-edge warping range .

[0184] Closed-loop feedback: When the predicted maximum stress approaches 80% of the silicon wafer's fracture yield strength, or when the predicted warpage exceeds the lithography machine's depth of focus (DOF) tolerance limit, the system triggers an alarm and dynamically fine-tunes the power ratio of the four heating zones through a PID algorithm, thereby achieving feedforward control of the wafer's thermomechanical state.

[0185] The semiconductor wafer temperature, warpage, and stress field joint prediction method and system described in the above embodiments, compared with the prior art, addresses the limitations of existing technologies in semiconductor wafer manufacturing processes. Existing technologies suffer from extremely sparse temperature measurement points, leading to limitations in real-time performance of traditional numerical simulations. Furthermore, purely data-driven models struggle to synchronously and physically decouple the prediction of temperature, warpage, and stress multi-physics field distributions. This invention eliminates the reliance on high-density sensor arrays. By embedding heat conduction and mechanical equilibrium equations into a neural network, it utilizes physical laws to compensate for the information loss caused by sparse sensors. Even with only a limited number (e.g., 17) of discrete measurement points, it ensures that the reconstructed full-field temperature and stress distribution conforms to physical reality, avoiding the non-physical oscillations or overfitting phenomena common in purely data-driven methods. This invention overcomes the limitation of traditional methods that can only monitor temperature. Through physical constraints and automatic differentiation techniques, it can simultaneously calculate macroscopic warpage displacement and internal wafer thermal stress while outputting the temperature field. This is equivalent to constructing a "virtual sensor" across the entire wafer field, providing process engineers with key indicators to directly determine whether wafer warpage has occurred, leading to uneven coating or breakage risks. Compared to traditional finite element simulation (FEM), which requires hours of mesh generation and iterative solving, the inference speed of the trained model in this invention can reach the millisecond level. Combined with Sim2Real transfer learning technology, this method can quickly adapt to different process cavities and has industrial application value for direct embedding into equipment control systems for real-time closed-loop feedback (run-to-run control).

[0186] Obviously, the embodiments described above are merely preferred embodiments of the present invention, and not all embodiments. The accompanying drawings illustrate preferred embodiments of the present invention, but do not limit the scope of the patent. The present invention can be implemented in many different forms; rather, these embodiments are provided to provide a more thorough and complete understanding of the disclosure of the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing specific embodiments, or make equivalent substitutions for some of the technical features. Any equivalent structures made using the content of this specification and drawings, directly or indirectly applied to other related technical fields, are similarly within the scope of patent protection of this invention.

Claims

1. A method for jointly predicting temperature, warpage, and stress field of a semiconductor wafer, characterized in that, Includes the following steps: S10. Construct a wafer thermo-mechanical coupled finite element model and generate a multi-condition training dataset containing sparse temperature measurement data and corresponding full-field temperature and stress distributions. S20. Arrange several discrete temperature sensing units on the semiconductor process cavity wafer to acquire sparse temperature measurement data. S30. Construct a two-branch neural network model based on Fourier operator neural network, and reconstruct the continuous distribution of the entire wafer field by learning the operator mapping between the input space and the output function space. S40. Supervise the training of the dual-branch neural operator network model using the multi-condition training dataset, so that it can synchronously output the wafer's full-field temperature distribution, warpage displacement distribution and stress distribution. S50. Input the sparse temperature measurement data collected in real time during the actual semiconductor process into the trained dual-branch neural operator network model, obtain the full-field temperature distribution and displacement distribution of the wafer through forward inference, and obtain the full-field stress distribution of the wafer in real time through thermoelastic constitutive relation. S60, the output results are used for dynamic adjustment of power distribution in the heating zone of semiconductor equipment.

2. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, The specific steps of S10 are as follows: S101. Construct a three-dimensional geometric model that includes the wafer, heater, and partitioned heating structure; S102. Set the temperature-related thermal and mechanical parameters of the material; S103. By changing the heating power, boundary heat dissipation conditions and environmental parameters, multiple sets of simulation samples under different working conditions are generated. Each set of simulation samples includes sparse temperature measurement point temperature data, corresponding full-field temperature field, stress field and displacement field true data.

3. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, The specific steps for constructing the two-branch neural operator network model based on the Fourier operator neural network in S30 are as follows: S301. Map the sparse temperature measurement data into a discrete two-dimensional input field, wherein the input field includes a temperature interpolation field, a measurement point mask matrix, and a spatial coordinate encoding. S302. The discrete two-dimensional input field is mapped to the frequency domain space through a fast Fourier transform; S303. Perform a linear transformation on the low-frequency modes in the frequency domain space to extract global spatial correlation features; S304. The spatial domain is recovered through inverse Fourier transform, and the field features are iteratively updated through a multi-layer operator network structure, as shown in the following formula: ； in, and The first Layer and First The characteristic field of the layer; The weight matrix is ​​a local linear transformation matrix; It is a frequency-domain learnable complex weight tensor; It is a non-linear activation function.

4. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, In step S40, a thermophysical residual loss based on the heat conduction control equation, a mechanical residual loss based on the mechanical equilibrium equation, and a data fitting loss function are constructed. These are then weighted to form a total loss function to achieve synergistic optimization of the temperature field and the stress field. The expression for the total loss function is as follows: ； in, For data fitting loss, For the residuals of the heat conduction equation, The residuals of the mechanical equilibrium equations, To reduce data loss weights, The weighting coefficients for physical consistency constraints. This is the scaling factor for the physical residual.

5. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, In S50, the real-time calculation of the wafer's full-field stress distribution through thermoelastic constitutive relations specifically includes: The spatial derivative of the continuous displacement field output by the model is obtained by using automatic differentiation to obtain the total strain tensor; Based on the plane stress assumption, the stress tensor is calculated by subtracting thermal strain using the generalized Hooke's law. .

6. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 5, characterized in that, The stress tensor The constitutive equation components are expressed as: ； Where E is Young's modulus. Poisson's ratio, The coefficient of thermal expansion varies with temperature. Where T is temperature. This is the stress-free reference temperature.

7. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, The sparse temperature measurement data comes from a limited number of temperature sensors arranged on electrostatic chucks, thermocouple structures, or equivalent temperature measurement units. The number of temperature sensors is less than the number of grid nodes used to obtain the temperature distribution across the entire field.

8. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, In step S40, supervised training employs a dynamic annealing strategy: In the purely data-driven era: This allows the model to quickly fit the approximate outline of the label; During the period of introduction of physical constraints: from The index rose to 1.0, forcing the model to correct for non-physical temperature differences and stress concentrations in areas without measurement points; During the multi-physics fine-tuning period: a gradient-based adaptive weighting algorithm is adopted to automatically balance the gradient magnitudes of thermal and mechanical residuals during backpropagation, ensuring synchronous convergence of the multiphysics fields.

9. The method for jointly predicting semiconductor wafer temperature, warpage, and stress field according to claim 1, characterized in that, The dual-branch neural operator network model achieves implicit learning of coupled features between the temperature field and the warped displacement field by sharing a deep feature extraction backbone. The dual-branch projection head includes a temperature branch and a warped branch, which are respectively projected in a dimension reduction manner through independent multilayer perceptron networks.

10. A system employing the method as described in any one of claims 1 to 9, characterized in that, include: The monitoring module is used to reconstruct the full-field temperature distribution, warpage, and stress distribution of the wafer in real time. The decision module is used to compare the reconstructed real-time distribution with preset process thresholds to identify local overheating areas or stress concentration points. The control module is used to dynamically adjust the power distribution of the heating element or the load-bearing parameters of the mechanical support structure in the semiconductor device according to the identification results of the decision module, so as to achieve closed-loop uniform control of the wafer thermal field and stress field.

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