Wafer bonding displacement field prediction method based on PINN neural network

Abstract

The invention relates to the technical field of bonding control, in particular to a wafer bonding displacement field prediction method based on a PINN neural network. The method comprises the steps of establishing a coupling control equation of wafer bonding, constructing a hard constraint neural network structure, designing a joint loss function, setting an adaptive weight adjustment and sampling strategy, and jointly solving a displacement field and a pressure field through a PINN to realize dynamic response prediction in a wafer bonding process. According to the method, dynamic contact constraint is introduced, physical behaviors of a contact area are constrained through a mask function, and it is ensured that a prediction result conforms to physical reality. Compared with a traditional numerical solution, the method does not need grid subdivision, is high in calculation efficiency, is high in physical consistency, is suitable for the multi-scale and multi-physical field collaborative wafer bonding modeling and optimization process, and has good engineering adaptability and popularization value.

Description

Technical Field

[0001] This invention relates to the technical field of bonding control, and specifically to a wafer bonding displacement field prediction method based on a PINN neural network. Background Technology

[0002] Wafer bonding, a core process in advanced manufacturing technologies such as 3D IC, MEMS, and heterogeneous integrated packaging, enables the vertical stacking of chips made of different materials or with different functions, significantly improving integration density, device performance, and packaging efficiency. During bonding, factors such as temperature gradients, mismatched coefficients of thermal expansion of materials, and changes in process stress can easily induce complex displacement and stress fields at the interface or in local structures. Non-uniform displacement not only affects the reliability of the bonding interface but can also lead to problems such as microcracks, warpage, and bonding failure, thereby limiting the packaging accuracy and yield of high-performance chips.

[0003] Traditional numerical methods (such as the finite element method) face the following challenges when simulating such problems: (1) Mesh dependency problem. Dynamic contact leads to mesh distortion, requiring cumbersome meshing and incurring high computational costs; (2) Complexity of multiphysics coupling. The coupling of air resistance and elastic deformation requires solving the Navier-Stokes equations and solid mechanics equations simultaneously, making iterative convergence difficult; (3) High resolution requirement. Micro- and nanoscale bonding requires local mesh refinement, significantly increasing computational resource consumption; (4) Insufficient accuracy in capturing contact interfaces. Traditional contact algorithms are prone to describing penetration or oscillation phenomena during dynamic bonding.

[0004] In recent years, Physical-Informed Neural Networks (PINNs), as a numerical method that integrates physical constraints and neural network training, have shown broad application prospects in areas such as complex physical field modeling, inverse problem solving, and parameter identification. However, existing research based on PINNs mostly focuses on single-physics problems, with little research on multi-coupling, dynamic contact constraints, and complex geometric boundary handling, especially in wafer bonding scenarios.

[0005] Therefore, there is an urgent need to propose a method that integrates PINN neural network structure with wafer thermo-mechanical coupling modeling, which can significantly improve the efficiency and generalization ability of displacement field simulation while ensuring prediction accuracy, thereby providing efficient simulation support for parameter optimization, failure prediction and structural reliability design of wafer bonding process. Summary of the Invention

[0006] To address the above problems, the present invention aims to propose a wafer bonding displacement field prediction method based on a PINN neural network, comprising the following steps:

[0007] S1. Establish the governing equations for the wafer bonding process;

[0008] S2. Design a neural network based on a predetermined function and set constraints;

[0009] S3. Construct the joint loss function;

[0010] S4. Determine the weight adjustment strategy;

[0011] S5. Determine a sampling strategy, wherein the sampling strategy has a different training point density in at least one region than in other regions;

[0012] S6. Solve the governing equations for the wafer bonding process to obtain the prediction equations for the wafer bonding displacement field.

[0013] Furthermore, in step S1, the governing equations of the wafer bonding process are expressed in the form of partial differential equations.

[0014] Furthermore, in step S1, the governing equations for the wafer bonding process include the wafer elastic vibration equation and the NS transient equation for the air gap between the wafers.

[0015] Furthermore, in step S2, the predetermined function for designing the neural network is a multiplier function.

[0016] Furthermore, in step S2, the constraint condition is that the coordinate parameters of the wafer center are set to be forcibly fixed.

[0017] Furthermore, in step S3, the joint loss function includes at least one of the following: partial differential equation loss, dynamic contact constraint loss, boundary condition loss, and initial condition loss.

[0018] Furthermore, in step S3, the joint loss function includes at least two of the following: partial differential equation loss, dynamic contact constraint loss, boundary condition loss, and initial condition loss.

[0019] Furthermore, in step S3, the joint loss function includes at least three of the following: partial differential equation loss, dynamic contact constraint loss, boundary condition loss, and initial condition loss.

[0020] Furthermore, in step S3, the joint loss function includes at least partial differential equation loss, dynamic contact constraint loss, and initial condition loss of boundary conditions.

[0021] Furthermore, in step S4, the weight adjustment strategy adopts an adaptive weight adjustment strategy, which dynamically adjusts the weights of various losses during the training process.

[0022] Furthermore, in step S5, an adaptive sampling strategy is adopted.

[0023] Furthermore, the adaptive sampling strategy increases the density of training points in the vicinity of the already fitted area.

[0024] Furthermore, in step S6, solving the control equations for the wafer bonding process specifically refers to solving the displacement field and pressure field jointly through PINN.

[0025] Furthermore, the prediction equation for the wafer bonding displacement field refers to the equation for the spatiotemporal evolution of the dynamic bonding process.

[0026] The elastic vibration equation for the wafer is:

[0027]

[0028] in, For the Heaviside function, Let be the elastic vibration response of the wafer at position 𝑥 and time 𝑡, i.e., the displacement response function; and These are the material density and cross-sectional area of ​​the wafer, respectively. This is the acceleration term of the vibration, reflecting the inertial response of the wafer; The distributed load function applied to the external environment is a function of 𝑥 and 𝑡; This is the deflection curvature of the wafer, i.e., the fourth spatial derivative of the wafer; For the bending stiffness of the wafer, the following condition must be met:

[0029]

[0030] Where 𝐸 is Young's modulus, B is wafer thickness, and 𝜈 is Poisson's ratio.

[0031] The Navier-Stokes transient equation for inter-wafer air is:

[0032]

[0033] in, These are the position coordinates of the wafer. The gas gap thickness of the wafer (as of time) and location change), Gas pressure distribution in the gas gap of the wafer (as of time) and location change), This represents the dynamic viscosity coefficient of the air in the air gap. The rate of change of the air gap thickness over time (its positive or negative sign reflects whether the wafer is moving closer or further away). The pressure gradient along the wafer plane causes gas flow;

[0034] The left side of the equation This represents the flow of gas in the x-direction driven by the pressure gradient, taking into account flow resistance; the right side of the equation... This indicates the rate at which gas is compressed or extracted due to the proximity or separation of the upper and lower wafers.

[0035] The hard-constrained neural network is designed as follows:

[0036]

[0037] in, To predict the displacement field, This represents the output portion of the neural network before boundary conditions are set. This indicates hard-coded boundary conditions.

[0038] For the center of the circle, the following condition is satisfied:

[0039]

[0040]

[0041] For the boundary, the following condition is satisfied:

[0042]

[0043]

[0044] in, Denotes the wafer radius, and satisfies... .

[0045] pressure field This refers to the output of a neural network without boundary constraints or implicit constraints.

[0046] Furthermore, dynamic contact constraints are set, meaning that during wafer bonding, the surfaces of the two wafers cannot "non-physically overlap." In other words, once contact or bonding occurs, the following two constraints must be met:

[0047] T1: The relative displacement of the contact area is zero, that is: ;

[0048] T2: The surface cannot continue to sink after contact, that is: .

[0049] The dynamic contact constraint is implemented using a mask function:

[0050]

[0051] in, For contact loss; For relative displacement constraints, the following conditions must be met within the forced contact area: ; For relative velocity constraints, the following conditions must be met within the mandatory contact area: .

[0052] Furthermore, the masking function It is a hard binary function that satisfies:

[0053]

[0054]

[0055] in, For masking functions, This is the contact tolerance threshold; The initial air gap thickness is the thickness under static conditions. For displacement, if A negative value indicates that the wafer is folded downwards, reducing the gap.

[0056] Dynamic air gap thickness Areas less than or equal to the contact tolerance threshold are considered contact areas. Areas with a dynamic air gap thickness greater than the contact tolerance threshold are also considered contact areas. Contact areas are masked with 1 and constraints are triggered. Non-contact areas are masked with 0, do not generate gradients, and do not trigger constraints.

[0057] In another preferred embodiment, the mask function A smooth sigmoid approximation is used, satisfying:

[0058]

[0059] in, For the sigmoid function, The steepness control parameter, This is the contact tolerance threshold.

[0060] Beneficial effects

[0061] This invention provides a wafer bonding displacement field prediction method based on PINN neural network, which can achieve high-precision modeling and prediction of displacement and pressure fields during wafer bonding without requiring a large amount of experimental data. By constructing a neural network structure with the control equations as the core, this method effectively integrates the wafer elastic vibration equation and the gap gas flow control equation under the premise of satisfying physical constraints, significantly improving the model's ability to learn complex physical coupling behaviors.

[0062] Among these methods, the multiplier function is used to construct the neural network structure, and the boundary conditions at the wafer center are forcibly applied through hard constraints. This ensures prediction accuracy while avoiding the instability problem of conventional "soft penalty" methods during training. Furthermore, by introducing a dynamic contact constraint mechanism, combined with a mask function to dynamically identify the bonding region and apply physical constraints of zero displacement and zero velocity, non-physical overlap in the wafer bonding region can be effectively avoided, enhancing the physical consistency of the modeling and the reliability of the simulation.

[0063] Furthermore, this invention introduces a joint loss function construction method, incorporating partial differential equation residuals, boundary conditions, initial conditions, and contact loss into a unified framework. An adaptive weighting strategy dynamically adjusts the contribution of each loss term during training, effectively improving the network's convergence efficiency and robustness. Simultaneously, a sampling strategy enhances the training point density based on the fitting region, ensuring numerical accuracy in key physical process regions.

[0064] By jointly solving the displacement and pressure fields using the PINN structure, this method can output predictive equations that evolve with time and space, fully reflecting the dynamic changes in the wafer bonding process. This helps improve the simulation efficiency of process modeling, the sensitivity of parameter optimization, and the accuracy of subsequent process control, and has good engineering applicability and promotion value. Attached Figure Description

[0065] Figure 1 This is a schematic diagram of a method flow according to an embodiment of the present invention;

[0066] Figure 2 This is a comparison of the actual boundary conditions and the predicted boundary conditions in one embodiment of the present invention;

[0067] Figure 3 This is a spatiotemporal diagram of the bonding process according to an embodiment of the present invention. Detailed Implementation

[0068] To enhance understanding of the present invention, the present invention will be further described in detail below with reference to embodiments. These embodiments are only used to explain the present invention and do not constitute a limitation on the scope of protection of the present invention.

[0069] Example 1

[0070] according to Figure 1 As shown, this embodiment provides a wafer bonding displacement field prediction method based on PINN neural network, including the following steps:

[0071] S1. Establish the governing equations for the wafer bonding process; the governing equations for the wafer bonding process are expressed in the form of partial differential equations; the governing equations for the wafer bonding process include the wafer elastic vibration equation and the Navier-Stokes transient equation for the air in the wafer gap.

[0072] S2. Design a neural network based on the multiplier function method and set constraints; the constraints are that the coordinate parameters of the wafer center are set to be fixed.

[0073] S3. Construct a joint loss function; the joint loss function shall include at least one of the following: partial differential equation loss, dynamic contact constraint loss, boundary condition loss, and initial condition loss.

[0074] S4. Determine the weight adjustment strategy; the weight adjustment strategy adopts an adaptive weight adjustment strategy, which dynamically adjusts the weights of various losses during the training process.

[0075] S5. Determine an adaptive sampling strategy, wherein the adaptive sampling strategy has a training point density different from other regions in at least one region; the adaptive sampling strategy enhances the training point density in adjacent already-fitted regions; including:

[0076] Basic sampling: 60% of the sampling points are uniformly distributed in the spatiotemporal domain;

[0077] Fine sampling in the adjacent bonding area: 40% of the sampling points are distributed in the adjacent bonding area.

[0078] S6. By jointly solving the displacement field and pressure field using PINN, the prediction equation for the wafer bonding displacement field is obtained; the prediction equation for the wafer bonding displacement field refers to the spatiotemporal evolution process equation of the dynamic bonding process.

[0079] Example 2

[0080] This embodiment provides a wafer bonding displacement field prediction method based on PINN (Physics-Informed Neural Network), specifically including the following steps:

[0081] S1. Establish the governing equations:

[0082] A governing equation describing the physical behavior during wafer bonding is established. Mathematically, this governing equation is a set of coupled partial differential equations, including the elastic vibration equation of the wafer and the transient Navier-Stokes equation for the air within the wafer gap. The elastic vibration equation is as follows:

[0083]

[0084] in, For the Heaviside function, Let be the elastic vibration response of the wafer at position 𝑥 and time 𝑡, i.e., the displacement response function; and These are the material density and cross-sectional area of ​​the wafer, respectively. This is the acceleration term of the vibration, reflecting the inertial response of the wafer; The distributed load function applied to the external environment is a function of 𝑥 and 𝑡; This is the deflection curvature of the wafer, i.e., the fourth spatial derivative of the wafer; This refers to the bending stiffness of the wafer.

[0085] Meanwhile, the NS transient equation for the air within the air gap is:

[0086]

[0087] in, These are the position coordinates of the wafer. The gas gap thickness of the wafer (as of time) and location change), Gas pressure distribution in the gas gap of the wafer (as of time) and location change), This represents the dynamic viscosity coefficient of the air in the air gap. The rate of change of the air gap thickness over time (its positive or negative sign reflects whether the wafer is moving closer or further away). The pressure gradient along the wafer plane causes gas flow;

[0088] S2. Neural Network Structure Design and Hard Constraint Application:

[0089] Design a PINN neural network based on multiplier functions, where the displacement prediction network satisfies a hard constraint form:

[0090]

[0091] in, To predict the displacement field, This represents the output portion of the neural network before boundary conditions are set. This indicates hard-coded boundary conditions.

[0092] By introducing multiplier functions into the network structure This makes it possible to have the wafer center (i.e. The boundary conditions are satisfied:

[0093]

[0094]

[0095] pressure field By another neural network Output.

[0096] The boundary is defined as follows: the wafer edge satisfies the natural boundary condition.

[0097]

[0098]

[0099] in Where is the wafer radius.

[0100] S3. Construct the joint loss function:

[0101] The loss function includes the following physical constraints:

[0102] Residual loss in partial differential equations;

[0103] Initial and boundary condition losses;

[0104] Dynamic contact constraint loss, which incorporates a masking function:

[0105]

[0106] The mask function is used to identify whether there is contact.

[0107] S4. Set the adaptive weight adjustment strategy:

[0108] During training, an adaptive weighting mechanism is used to dynamically balance the weights of the above loss terms, ensuring that different physical terms can be effectively optimized in the early and late stages of training, and avoiding a certain type of loss from dominating the training process.

[0109] S5. Design sampling strategy:

[0110] To improve training efficiency and accuracy, the sampling area is divided, and the density of training points is increased in the adjacent bonding area (the area where the air gap thickness is close to zero). That is, an adaptive sampling strategy is used to adjust the distribution of the point set.

[0111] S6. Solve the governing equations and output the prediction model:

[0112] The constructed PINN network is input into the above control equations, and multiple rounds of backpropagation iterations are performed through a joint training optimizer to finally obtain the displacement field during wafer bonding. With pressure field The predictive model can output the spatiotemporal evolution behavior during wafer bonding, providing a theoretical basis for bonding path optimization and process control.

[0113] The fitting graph of the simulated and actual values ​​of the spatiotemporal evolution equation of the dynamic bonding process in this embodiment.

[0114] Example 3

[0115] This embodiment further optimizes the mask function form and sampling strategy based on the first embodiment, and introduces a parallel neural network structure to improve prediction accuracy and training efficiency.

[0116] S1. Establish the governing equations:

[0117] Similar to the first embodiment, the wafer elastic vibration equation and the Navier-Stokes transient equation are used to describe the dynamic coupling behavior of wafer displacement and air pressure field. The governing equation is expressed as a set of coupled partial differential equations, which includes material parameters E, B, ρ, ν and dynamic viscosity coefficient μ.

[0118] S2. Construct a multi-network PINN structure:

[0119] Unlike the first embodiment, and Using a separate network output, this embodiment splits the neural network structure into two nested layers. The first layer predicts intermediate variables. Then, through analytical transformation, the displacement and pressure are obtained, i.e.:

[0120] Define intermediate functions

[0121] Calculated using function mapping relationships:

[0122]

[0123] This structure implicitly introduces physical continuity constraints, which improves the consistency between predicted fields.

[0124] S3. Construct the joint loss function:

[0125] The loss function still includes the partial differential equation residuals, boundary conditions, initial conditions, and dynamic contact loss. The dynamic contact loss uses a sigmoid smoothing mask function.

[0126]

[0127] in, For the sigmoid function, This represents the air gap thickness.

[0128] The final contact loss term is expressed as:

[0129]

[0130] S4. Optimization of Adaptive Weight Adjustment Strategy:

[0131] During training, a gradient norm-based method is used to automatically assign weights to each loss term, ensuring that the network has a more sensitive response capability to highly dynamic regions (such as oscillating boundaries and contact boundaries).

[0132] S5. Information-driven sampling strategy based on physical gradients:

[0133] In this embodiment, the sampling method is based on the gradient distribution of the network prediction results to perform dynamic resampling. In areas where the gradient of the pressure field p(x,t) changes drastically, denser training points are automatically generated to enhance the local learning ability of key areas and improve the analytical accuracy near the contact boundary.

[0134] S6. Solve the governing equations together and output the prediction results:

[0135] By using the aforementioned parallel PINN structure, combined with multi-round iterative training and loss convergence control, a unified prediction model containing wafer displacement field and air gap pressure field is finally output.

[0136] The prediction results include: displacement field Evolution over time; pressure field Distribution map; Spatial distribution evolution map of contact area;

[0137] The equations in this embodiment have good visualization capabilities and are suitable for simulation of wafer bonding behavior and failure mode analysis under different processes.

[0138] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for predicting wafer bonding displacement field based on PINN neural network, characterized in that, Includes the following steps: S1. Establish the governing equations for the wafer bonding process; S2. Design a neural network based on a predetermined function and set constraints; S3. Construct the joint loss function; S4. Determine the weight adjustment strategy; S5. Determine a sampling strategy, wherein the sampling strategy has a different training point density in at least one region than in other regions; S6. Solve the governing equations for the wafer bonding process to obtain the prediction equations for the wafer bonding displacement field.

2. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S1, the governing equations of the wafer bonding process are expressed in the form of partial differential equations.

3. The wafer bonding displacement field prediction method based on PINN neural network according to claim 2, characterized in that: In step S1, the governing equations for the wafer bonding process include the wafer elastic vibration equation and the NS transient equation for the inter-wafer air.

4. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S2, the predetermined function for designing the neural network is a multiplier function.

5. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S2, the constraint condition is that the coordinate parameters of the wafer center are set to be fixed.

6. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S3, the joint loss function includes at least one of partial differential equation loss, dynamic contact constraint loss, boundary condition loss, and initial condition loss.

7. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S4, the weight adjustment strategy adopts an adaptive weight adjustment strategy, which dynamically adjusts the weights of various losses during the training process.

8. The wafer bonding displacement field prediction method based on PINN neural network according to claim 1, characterized in that: In step S5, an adaptive sampling strategy is adopted.

9. The wafer bonding displacement field prediction method based on PINN neural network according to claim 8, characterized in that: The adaptive sampling strategy increases the density of training points in the vicinity of the already fitted area.

10. The wafer bonding displacement field prediction method based on PINN neural network according to claim 8, characterized in that: In step S6, solving the control equations for the wafer bonding process specifically refers to solving the displacement field and pressure field together using PINN.

Images