Czochralski silicon single crystal growth process V / G value soft measurement modeling method based on PINN

Abstract

The invention discloses a PINN-based Czochralski silicon single crystal growth process V / G value soft measurement modeling method. The method is specifically implemented according to the following steps: step 1, constructing a physical model containing a crystal reference shape and a heat conduction model; the method comprises the steps of 1, designing a physical information neural network PINN, 2, designing a neural network model of the physical information neural network PINN, 3, designing a loss function of the PINN, performing training by combining the difference between network prediction and actual measurement values, a partial differential equation PDE and a physical constraint formed by boundary conditions, and optimizing the weight of the neural network model; step 4, network training and optimization; and step 5, model prediction. According to the invention, the problem that the V / G value is difficult to directly measure in the growth process of the czochralski silicon single crystal in the prior art is solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of soft measurement modeling of Czochralski silicon single crystal growth, and in particular relates to a soft measurement modeling method of V / G value in a Czochralski silicon single crystal growth process based on PINN. Background Art

[0002] Silicon is the most important semiconductor material. Over 90% of integrated circuits are fabricated on silicon single crystals. In recent years, the continuous development of the large-scale integrated circuit industry has placed higher demands on the purity, uniformity, and structural integrity of silicon wafers. Among these, defects generated during the silicon single crystal growth process have become a key factor affecting the quality of the final product.

[0003] Defects in crystals can be divided into point defects, line defects, surface defects, and volume defects according to their dimensions; and can be divided into primary defects and secondary defects according to the cause of their generation. Most defects evolve from point defects. As the characteristic linewidth of integrated circuits continues to shrink, the impact of smaller primary point defects in crystals on integrated circuit performance becomes increasingly significant. According to the Voronkov theory, during the growth of silicon single crystals, the V / G value at the solid-liquid interface (where V represents the crystal growth rate in mm / min and G represents the axial temperature gradient at the solid-liquid interface in K / mm) directly affects the generation and distribution characteristics of point defects. By precisely controlling the V / G ratio, the point defect concentration can be effectively controlled, thereby growing defect-free, high-quality silicon single crystals, laying the foundation for meeting the needs of advanced processes.

[0004] In actual engineering, the V / G value at the solid-liquid interface is difficult to obtain directly, and soft measurement technology is generally used for online prediction. Currently, the soft measurement modeling methods mainly include mechanism-based soft measurement modeling methods and data-driven soft measurement modeling methods.

[0005] Mechanism-based soft sensor modeling approaches are based on a deep understanding of physical mechanisms and the establishment of soft sensor models with well-defined mathematical expressions. Mechanism models are primarily categorized into lumped parameter models and distributed parameter models. Lumped parameter models, however, fail to fully account for the spatial characteristics of the system, resulting in limitations in accurately depicting complex physical processes. Distributed parameter models, by incorporating spatial distribution characteristics, can more accurately simulate the system's dynamics and improve physical accuracy. However, these models typically consist of a set of partial differential equations and multiple boundary conditions, resulting in complex solution processes and slow computational convergence. Existing methods for solving numerical mechanism models include finite difference methods, finite element methods, finite volume methods, and various numerical simulation software. These methods require discretization of the solution domain, but silicon single crystal growth involves complex geometries and boundary conditions, resulting in high computational and time complexity. Furthermore, they require explicit initial and boundary conditions, and the numerical solution algorithms are highly sensitive to boundary conditions, making them difficult to apply to real-time computations and predictions. In summary, although the mechanism-based soft measurement model has clear physical meaning and strong interpretability, its prediction results may have errors due to the inevitable simplified assumptions in the construction of the lumped parameter model. In addition, the computational complexity and time complexity of the distributed parameter model are high, and its efficiency and real-time performance in actual industrial production still face great challenges.

[0006] Data-driven soft-sensor modeling approaches demonstrate flexibility and adaptability in dealing with complex systems and nonlinear relationships, offering high modeling efficiency and robust processing capabilities for high-dimensional data. However, for the complex characteristics of silicon single crystal growth processes, such as nonlinearity, high coupling, and large time lags, these approaches suffer from poor interpretability, overfitting and underfitting, and poor model generalization. They also place high demands on data quality and quantity, can significantly degrade model performance when data is noisy or missing, and struggle to address issues such as large time lags and strong coupling.

[0007] Based on the above problems, the present invention proposes a method based on PINN to solve the temperature distribution during the growth process of silicon single crystals and predict the V / G value. It not only fully considers the spatiotemporal characteristics of the silicon single crystal growth process, but also avoids the discretization of traditional numerical solution methods. It has high physical accuracy and strong generalization ability, and can achieve real-time prediction. Summary of the Invention

[0008] The purpose of the present invention is to provide a soft measurement modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN, which solves the problem in the prior art that V / G value is difficult to directly measure in the Czochralski silicon single crystal growth process.

[0009] The technical solution adopted by the present invention is a soft-sensing modeling method for V / G values in the Czochralski silicon single crystal growth process based on PINN, which is specifically implemented in the following steps:

[0010] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0011] Step 2: Design the neural network model of the physical information neural network PINN.

[0012] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0013] Step 4: Network training and optimization;

[0014] Step 5: Model prediction.

[0015] The present invention is also characterized in that:

[0016] The crystal growth in step 1 includes four stages: shoulder release, shoulder rotation, equal diameter and tailing. The growth conditions in each stage, including growth rate and temperature gradient, are different. The deformation of the solid-liquid interface is also different, and the V / G value will also show dynamic changes.

[0017] Step 1 is implemented as follows:

[0018] Set the crystal reference shape during crystal growth, the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions, expressed as Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

[0019] Step 2 is implemented as follows:

[0020] A fully connected neural network is used for function approximation. The input layer contains three neurons, the output layer is one neuron, and Tanh is used as the activation function after the input layer and each hidden layer.

[0021] The loss function in step 3 is designed as follows:

[0022] Consider a general nonlinear PDE of the following form:

[0023]

[0024] Where x and t represent space and time, and the subscripts represent partial differentials. represents the nonlinear partial differential operator, is the boundary of Ω, u(x,t) represents the solution of the PDE;

[0025] Assume a feedforward fully connected neural network with a depth of M. The input of the network is x, and the output of the mth layer is represented as A neural network is defined as:

[0026]

[0027] Among them, σ is the activation function, W [m] and b [m] Represents the weight and bias of the mth layer, and defines the physical information neural network f(t,x) as:

[0028] f:=u t (x,t;θ)+N x [u(x,t;θ)] (3)

[0029] The PDE residual neural network of Equation (3) is derived by using automatic differentiation and then applying the chain rule, where θ is the set of all parameters in the network, and is learned by minimizing the loss function:

[0030] L(θ)=L f (θ)+L b (θ)+L0(θ) (4)

[0031] Among them, L f (θ) is the loss function of PDE, L b (θ) is the loss function of the boundary condition, L0(θ) is the loss function of the initial condition, and the mean square error MSE is used as the loss function:

[0032]

[0033]

[0034] in, is the starting point, is the boundary point, is a set of points randomly distributed in Ω, N f , N b and N0 represent the total number of configuration points, boundary points, and initial points, respectively. The network parameters are adjusted by minimizing the total loss L(θ) through the gradient descent process.

[0035] In step 3, during the growth of the Czochralski silicon single crystal, the heat conduction model inside the crystal is expressed as:

[0036]

[0037] In formula (8), is the Peclet constant, x(r,z,t) is the temperature distribution in the crystal region, represents the spatial gradient operator in cylindrical coordinates, v0, R cruc 、k s 、k r are the pulling speed, crucible radius, heat conduction coefficient and thermal conductivity respectively, V(r,z,t) is the velocity of the crystal region, and the boundary velocity along the radial direction is ignored. The crystal heat conduction PDE is expressed as:

[0038] (4)

[0039] In formula (9), k0=k r / Pe,V z (t) is the crystal's axial velocity. The variables in the equation are normalized, and the temperature T is relative to the crystal melting temperature T. f Normalized, r and z are normalized to the crucible radius, t is normalized to R cruc. / v0 is normalized, and the normalization formula is as follows:

[0040] x=(TT f ) / T f (5)

[0041] r=(rR cruc. ) / R cruc. (6)

[0042] z=(zR cruc. ) / R cruc. (7)

[0043]

[0044] At the solid-liquid interface, the silicon melt solidifies into silicon crystals. Therefore, the first type of boundary condition is set at the solid-liquid interface, and the temperature is set to be the melting point of silicon, T. f :

[0045] x| z＝0 =0 (9)

[0046] The top temperature of the crystal is T top As the crystal grows, the first type of boundary conditions are set as follows:

[0047] x| z＝l(t) =x top (10)

[0048] Among them, l(t) is the growth height of the crystal, xtop is the normalized top temperature of the crystal;

[0049] The heater provides radial heat flux within a certain height range of the crystal, and the corresponding boundary conditions are as follows:

[0050]

[0051] Where Q(t) is the heat flux provided by the local heater, R c (z) is the radius of the crystal at height z, z1(t) and z2(t) represent the spatial intervals where the heaters are placed. Due to the axial symmetry of the crystal, the heat flux of the crystal is zero at the axis of symmetry, and at other boundaries, the heat flux is assumed to be zero. The second type of boundary conditions are set as follows:

[0052]

[0053] In the crystal growth process model established above, the network parameters are learned by minimizing the mean square error loss. The loss function is expressed as:

[0054] L(θ)=K1·L1(θ)+K2·L2(θ)+K3·L PDE (θ) (15)

[0055] Where K1 represents the equilibrium parameter of the first type of boundary condition, K2 represents the equilibrium parameter of the second type of boundary condition, and K3 represents the equilibrium parameter of the partial differential equation, which is used to balance the losses of the boundary conditions and PDE.

[0056] Step 4 is implemented as follows:

[0057] The optimizer selected is a combination of the Adam optimizer and the LBFGS optimizer. First, the Adam optimizer is used for preliminary training and parameter adjustment to ensure rapid initial convergence of the PINN model designed in step 3. Then, the finite memory quasi-Newton method LBFGS optimizer is switched to perform fine tuning to improve the accuracy of the model. The back propagation algorithm is used to optimize the neural network weights and minimize the loss function, thereby learning the temperature distribution law during the crystal growth process.

[0058] Step 5 is implemented as follows:

[0059] The trained neural network model was used to predict the temperature distribution during crystal growth and calculate the V / G value, where V represents the crystal growth rate in mm / min and G represents the axial temperature gradient at the solid-liquid interface in K / mm.

[0060] The beneficial effect of the present invention is that the method combines the heat conduction mechanism model in the crystal growth process with deep learning technology, and constructs and predicts the crystal temperature distribution based on known physical laws when experimental data is limited.

[0061] The present invention is based on a PDE model of heat conduction inside a solid crystal with time-varying boundaries during crystal growth, takes into account the changes in crystal shape during seeding, shouldering, and equal diameter processes of crystal growth, and takes into account the deformation of the solid-liquid interface during crystal growth. Based on PINN and automatic differentiation of deep learning, PDE and boundary conditions are introduced as physical constraints into the loss function of the neural network, and the neural network is trained to approximate the solution of the PDE, so that the neural network can learn complex physical relationships and meet the constraints of the PDE and boundary conditions. The method of the present invention combines the advantages of mechanism-based modeling methods and data-driven modeling methods, and can effectively overcome the shortcomings of traditional numerical simulation methods with high computational complexity, high time complexity, and difficulty in real-time prediction. At the same time, it also solves the problems of strong data dependence and poor generalization ability of data-driven modeling alone, effectively improving prediction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 It is a distribution diagram of crystal shape and boundary conditions in the soft measurement modeling method of V / G value in the Czochralski silicon single crystal growth process based on PINN of the present invention;

[0063] Figure 2 It is a simplified crystal shape and boundary condition distribution diagram in the V / G value soft measurement modeling method of the PINN-based Czochralski silicon single crystal growth process of the present invention;

[0064] Figure 3 This is the crystal temperature distribution diagram predicted by Example 5 of the PINN-based soft-sensing modeling method for V / G values in the Czochralski silicon single crystal growth process of the present invention;

[0065] Figure 4 This is the V / G value change curve predicted by Example 5 in the PINN-based V / G value soft measurement modeling method for the Czochralski silicon single crystal growth process of the present invention;

[0066] Figure 5 This is the crystal temperature distribution diagram predicted by Example 6 of the PINN-based soft-sensing modeling method for V / G values in the Czochralski silicon single crystal growth process of the present invention;

[0067] Figure 6 This is the V / G value variation curve predicted by Example 6 of the V / G value soft measurement modeling method for the PINN-based Czochralski silicon single crystal growth process of the present invention. DETAILED DESCRIPTION

[0068] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0069] The present invention is based on the soft-sensor modeling method of V / G value in the Czochralski silicon single crystal growth process of PINN, which is specifically implemented according to the following steps:

[0070] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0071] The crystal growth in step 1 consists of four stages: shoulder release, shoulder rotation, equal diameter, and tailing. Each stage has different growth conditions, including growth rate and temperature gradient, resulting in different solid-liquid interface deformation and dynamic changes in the V / G value. Therefore, in the subsequent Example 6, the PINN prediction of the V / G value at different growth stages is considered.

[0072] Step 1 is implemented as follows:

[0073] Set the crystal reference shape during crystal growth, taking into account the solid-liquid interface deformation of the crystal, and the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions and is represented by Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

[0074] Step 2: Design the neural network model of the physical information neural network PINN.

[0075] Step 2 is implemented as follows:

[0076] A fully connected neural network is used for function approximation. The input layer contains three neurons, the output layer is one neuron, and the number of hidden layers and the number of neurons in each hidden layer are determined according to the specific instance. Tanh is used as the activation function after the input layer and each hidden layer.

[0077] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0078] The loss function in step 3 is designed as follows:

[0079] Consider a general nonlinear PDE of the following form:

[0080]

[0081] Where x and t represent space and time, and the subscripts represent partial differentials. represents the nonlinear partial differential operator, is the boundary of Ω, u(x,t) represents the solution of the PDE;

[0082] Assume a feedforward fully connected neural network with a depth of M. The input of the network is x, and the output of the mth layer is represented as A neural network is defined as:

[0083]

[0084] Among them, σ is the activation function, W [m] and b [m] Represents the weight and bias of the mth layer, and defines the physical information neural network f(t,x) as:

[0085] f:=u t (x,t;θ)+N x [u(x,t;θ)] (3)

[0086] The PDE residual neural network of Equation (3) is derived by using automatic differentiation and then applying the chain rule, where θ is the set of all parameters in the network, and is learned by minimizing the loss function:

[0087] L(θ)=L f (θ)+L b (θ)+L0(θ) (4)

[0088] Among them, L f (θ) is the loss function of PDE, L b (θ) is the loss function of the boundary condition, L0(θ) is the loss function of the initial condition, and the mean square error MSE is used as the loss function:

[0089]

[0090] in, is the starting point, is the boundary point, is a set of points randomly distributed in Ω, N f , N b and N0 represent the total number of configuration points, boundary points, and initial points, respectively. The network parameters are adjusted by minimizing the total loss L(θ) through the gradient descent process.

[0091] In step 3, during the growth of the Czochralski silicon single crystal, the heat conduction model inside the crystal is expressed as:

[0092]

[0093] In formula (8), is the Peclet constant, x(r,z,t) is the temperature distribution of the crystal region, ▽ represents the spatial gradient operator in the cylindrical coordinate system, v0, R cruc 、k s 、k r are the pulling speed, crucible radius, heat conduction coefficient and thermal conductivity respectively, V(r,z,t) is the velocity of the crystal region, and the boundary velocity along the radial direction is ignored. The crystal heat conduction PDE is expressed as:

[0094]

[0095] In formula (9), k0=k r / Pe,V z (t) is the crystal's axial velocity. The variables in the equation are normalized, and the temperature T is relative to the crystal melting temperature T. f Normalized, r and z are normalized to the crucible radius, t is normalized to R cruc. / v0 is normalized, and the normalization formula is as follows:

[0096] x=(TT f ) / T f (10)

[0097] r=(rR cruc. ) / R cruc. (11)

[0098] z=(zR cruc. ) / R cruc. (12)

[0099]

[0100] The thermal boundary conditions during crystal heat conduction are given by Figure 1 As shown. At the solid-liquid interface, the silicon melt solidifies into silicon crystals. Therefore, the first type of boundary condition is set at the solid-liquid interface, and the temperature is set to be the melting point of silicon T. f :

[0101] x| z＝0 =0 (14)

[0102] The top temperature of the crystal is T top As the crystal grows, the first type of boundary conditions are set as follows:

[0103] x| z＝l(t) =x top (15)

[0104] Among them, l(t) is the growth height of the crystal, x top is the normalized top temperature of the crystal;

[0105] The heater provides radial heat flux within a certain height range of the crystal, and the corresponding boundary conditions are as follows:

[0106]

[0107] Where Q(t) is the heat flux provided by the local heater, R c (z) is the radius of the crystal at height z, z1(t) and z2(t) represent the spatial intervals where the heaters are placed. Due to the axial symmetry of the crystal, the heat flux of the crystal is zero at the axis of symmetry, and at other boundaries, the heat flux is assumed to be zero. The second type of boundary conditions are set as follows:

[0108]

[0109]

[0110] In the crystal growth process model established above, the network parameters are learned by minimizing the mean square error loss. The loss function is expressed as:

[0111] L(θ)=K1·L1(θ)+K2·L2(θ)+K3·L PDE (θ) (20)

[0112] Where K1 represents the equilibrium parameter of the first type of boundary condition, K2 represents the equilibrium parameter of the second type of boundary condition, and K3 represents the equilibrium parameter of the partial differential equation, which is used to balance the losses of the boundary conditions and PDE.

[0113] Step 4: Network training and optimization;

[0114] Step 4 is implemented as follows:

[0115] The optimizer selected is a combination of the Adam optimizer and the LBFGS optimizer. First, the Adam optimizer is used for preliminary training and parameter adjustment to ensure rapid initial convergence of the PINN model designed in step 3. Then, the finite memory quasi-Newton method LBFGS optimizer is switched to perform fine tuning to improve the accuracy of the model. The back propagation algorithm is used to optimize the neural network weights and minimize the loss function, thereby learning the temperature distribution law during the crystal growth process.

[0116] Step 5: Model prediction.

[0117] Step 5 is implemented as follows:

[0118] The trained neural network model was used to predict the temperature distribution during crystal growth and calculate the V / G value, where V represents the crystal growth rate in mm / min and G represents the axial temperature gradient at the solid-liquid interface in K / mm.

[0119] Example 1

[0120] The present invention is based on the soft-sensor modeling method of V / G value in the Czochralski silicon single crystal growth process of PINN, which is specifically implemented according to the following steps:

[0121] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0122] Step 2: Design the neural network model of the physical information neural network PINN.

[0123] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0124] Step 4: Network training and optimization;

[0125] Step 5: Model prediction.

[0126] Example 2

[0127] The present invention is based on the soft-sensor modeling method of V / G value in the Czochralski silicon single crystal growth process of PINN, which is specifically implemented according to the following steps:

[0128] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0129] The crystal growth in step 1 includes four stages: shoulder release, shoulder rotation, equal diameter and tailing. The growth conditions in each stage, including growth rate and temperature gradient, are different. The deformation of the solid-liquid interface is also different, and the V / G value will also show dynamic changes.

[0130] Step 1 is implemented as follows:

[0131] Set the crystal reference shape during crystal growth, taking into account the solid-liquid interface deformation of the crystal, and the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions, expressed as Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

[0132] Step 2: Design the neural network model of the physical information neural network PINN.

[0133] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0134] Step 4: Network training and optimization;

[0135] Step 5: Model prediction.

[0136] Example 3

[0137] The present invention is based on the soft-sensor modeling method of V / G value in the Czochralski silicon single crystal growth process of PINN, which is specifically implemented according to the following steps:

[0138] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0139] In step 1, crystal growth involves four stages: shoulder release, shoulder rotation, equal diameter, and finalization. Each stage exhibits different growth conditions, including growth rate and temperature gradient, leading to varying solid-liquid interface deformation and dynamic changes in the V / G value. Therefore, in the subsequent Example 2, PINN predictions of the V / G values at different growth stages are considered.

[0140] Step 1 is implemented as follows:

[0141] Set the crystal reference shape during crystal growth, taking into account the solid-liquid interface deformation of the crystal, and the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions, expressed as Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

[0142] Step 2: Design the neural network model of the physical information neural network PINN.

[0143] Step 2 is implemented as follows:

[0144] A fully connected neural network is used for function approximation. The input layer contains three neurons, the output layer is one neuron, and the number of hidden layers and the number of neurons in each hidden layer are determined according to the specific instance. Tanh is used as the activation function after the input layer and each hidden layer.

[0145] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0146] Step 4: Network training and optimization;

[0147] Step 5: Model prediction.

[0148] Example 4

[0149] The present invention is based on the soft-sensor modeling method of V / G value in the Czochralski silicon single crystal growth process of PINN, which is specifically implemented according to the following steps:

[0150] Step 1: Construct a physical model including a crystal reference shape and a heat conduction model;

[0151] In step 1, crystal growth involves four stages: shoulder release, shoulder rotation, equal diameter, and finalization. Each stage exhibits different growth conditions, including growth rate and temperature gradient, leading to varying solid-liquid interface deformation and dynamic changes in the V / G value. Therefore, in the subsequent Example 2, PINN predictions of the V / G values at different growth stages are considered.

[0152] Step 1 is implemented as follows:

[0153] Set the crystal reference shape during crystal growth, taking into account the solid-liquid interface deformation of the crystal, and the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions, expressed as Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

[0154] Step 2: Design the neural network model of the physical information neural network PINN.

[0155] Step 2 is implemented as follows:

[0156] A fully connected neural network is used for function approximation. The input layer contains three neurons, the output layer is one neuron, and the number of hidden layers and the number of neurons in each hidden layer are determined according to the specific instance. Tanh is used as the activation function after the input layer and each hidden layer.

[0157] Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model;

[0158] The loss function in step 3 is designed as follows:

[0159] Consider a general nonlinear PDE of the following form:

[0160]

[0161] Where x and t represent space and time, and the subscripts represent partial differentials. represents the nonlinear partial differential operator, is the boundary of Ω, u(x,t) represents the solution of the PDE;

[0162] Assume a feedforward fully connected neural network with a depth of M. The input of the network is x, and the output of the mth layer is represented as A neural network is defined as:

[0163]

[0164] Among them, σ is the activation function, W [m] and b [m] Represents the weight and bias of the mth layer, and defines the physical information neural network f(t,x) as:

[0165] f:=u t (x,t;θ)+N x [u(x,t;θ)] (3)

[0166] The PDE residual neural network of Equation (3) is derived by using automatic differentiation and then applying the chain rule, where θ is the set of all parameters in the network, and is learned by minimizing the loss function:

[0167] L(θ)=L f (θ)+L b (θ)+L0(θ) (4)

[0168] Among them, L f (θ) is the loss function of PDE, L b (θ) is the loss function of the boundary condition, L0(θ) is the loss function of the initial condition, and the mean square error MSE is used as the loss function:

[0169]

[0170] in, is the starting point, is the boundary point, is a set of points randomly distributed in Ω, N f , N b and N0 represent the total number of configuration points, boundary points, and initial points, respectively. The network parameters are adjusted by minimizing the total loss L(θ) through the gradient descent process.

[0171] In step 3, during the growth of the Czochralski silicon single crystal, the heat conduction model inside the crystal is expressed as:

[0172]

[0173] In formula (8), is the Peclet constant, x(r,z,t) is the temperature distribution of the crystal region, ▽ represents the spatial gradient operator in the cylindrical coordinate system, v0, R cruc 、k s 、k r are the pulling speed, crucible radius, heat conduction coefficient and thermal conductivity respectively, V(r,z,t) is the velocity of the crystal region, and the boundary velocity along the radial direction is ignored. The crystal heat conduction PDE is expressed as:

[0174]

[0175] In formula (9), k0=k r / Pe,V z (t) is the crystal's axial velocity. The variables in the equation are normalized, and the temperature T is relative to the crystal melting temperature T. f Normalized, r and z are normalized to the crucible radius, t is normalized to R cruc. / v0 is normalized, and the normalization formula is as follows:

[0176] x=(TT f ) / T f (10)

[0177] r=(rR cruc. ) / R cruc. (11)

[0178] z=(zR cruc. ) / R cruc. (12)

[0179]

[0180] The thermal boundary conditions during crystal heat conduction are given by Figure 1 As shown. At the solid-liquid interface, the silicon melt solidifies into silicon crystals. Therefore, the first type of boundary condition is set at the solid-liquid interface, and the temperature is set to be the melting point of silicon T. f :

[0181] x| z＝0=0 (14)

[0182] The top temperature of the crystal is T top As the crystal grows, the first type of boundary conditions are set as follows:

[0183] x| z＝l(t) =x top (15)

[0184] Among them, l(t) is the growth height of the crystal, x top is the normalized top temperature of the crystal;

[0185] The heater provides radial heat flux within a certain height range of the crystal, and the corresponding boundary conditions are as follows:

[0186]

[0187] Where Q(t) is the heat flux provided by the local heater, R c (z) is the radius of the crystal at height z, z1(t) and z2(t) represent the spatial intervals where the heaters are placed. Due to the axial symmetry of the crystal, the heat flux of the crystal is zero at the axis of symmetry, and at other boundaries, the heat flux is assumed to be zero. The second type of boundary conditions are set as follows:

[0188]

[0189] In the crystal growth process model established above, the network parameters are learned by minimizing the mean square error loss. The loss function is expressed as:

[0190] L(θ)=K1·L1(θ)+K2·L2(θ)+K3·L PDE (θ) (20)

[0191] Where K1 represents the equilibrium parameter of the first type of boundary condition, K2 represents the equilibrium parameter of the second type of boundary condition, and K3 represents the equilibrium parameter of the partial differential equation, which is used to balance the losses of the boundary conditions and PDE.

[0192] Step 4: Network training and optimization;

[0193] Step 5: Model prediction.

[0194] Example 5

[0195] According to step 1, combined with the crystal growth process, considering the computational complexity and the feasibility of the verification scheme, the reference shape of the crystal is simplified as follows Figure 2As shown, the crystal has a radius of 0.15m and its height changes during growth. Uniform sampling of the crystal's height and radius provides the network input. Heater positions z1 = 0mm and z2 = 26.7mm are used. Considering the heater's effect, the crystal sidewalls are set to second-class boundary conditions, while the top and bottom temperatures are set to first-class boundary conditions. The bottom temperature is the crystal's melting point, 1685K.

[0196] Following step 2, design the PINN neural network architecture, using a fully connected neural network for function approximation. The input layer contains three neurons, the hidden layer consists of 16 layers, each with eight neurons, and the output layer contains one neuron. Tanh is used as the activation function after the input layer and each hidden layer. The network's three input parameters are (r, z, t), including the location and time of the sampling points, and the output is the predicted crystal temperature.

[0197] According to step 3, the loss function of PINN is designed. The loss function of PINN consists of two parts: boundary condition loss and PDE loss. The boundary conditions of the crystal are divided into the first type of boundary conditions and the second type of boundary conditions. The loss function of the first type of boundary conditions is expressed as L b1 (θ), the loss function of the second type of boundary condition is expressed as L b2 (θ), the loss of PDE is expressed as L PDE (θ), the loss function of PINN can be expressed as:

[0198] L(θ)=K PDE ·L PDE (θ)+K b1 ·L b1 (θ)+K b2 ·L b2 (θ) (21)

[0199] Among them, K b1 represents the equilibrium parameter of the first type of boundary condition, K b2 represents the equilibrium parameter of the second type of boundary condition, K PDE Represents the equilibrium parameters of the partial differential equation and learns the network parameters by minimizing the mean square error (MSE) loss.

[0200] Following step 4, training is performed by combining the physical equation loss and network prediction error to optimize the neural network weights. A combination of the Adam optimizer and the LBFGS optimizer is used. Training is first performed using the Adam optimizer. After 40,000 epochs, the LBFGS optimizer is used, and training is terminated when the gradient is less than 1e-7. The initial learning rate for the Adam optimizer is set to 0.001, and the learning rate for the LBFGS optimizer is set to 1.

[0201] According to step 5, the temperature distribution during the crystal growth process is predicted, and the obtained temperature distribution during the crystal growth process is as follows Figure 3 As shown, the change of V / G value during crystal growth is calculated as follows Figure 4 shown.

[0202] Example 6

[0203] According to step 1, combined with the crystal growth process, the crystal reference shape is set as the solution domain of the differential equation, such as Figure 1 As shown, the radius of the crystal is 0.15m, the crystal height changes with the growth process, and the maximum height is 0.7m. The solid-liquid interface deformation of the crystal is expressed as Here, h0 determines the direction and magnitude of the deformation at the solid-liquid interface. Its value is derived from actual experimental results and is uniformly sampled within the solution domain to form the network input. The heater positions are set to z1 = 0 mm and z2 = 26.7 mm. Considering the heater's effect, the temperatures at the top of the crystal and at the solid-liquid interface are set as first-class boundary conditions, while the heat flux at the crystal sidewalls and axis of symmetry is set as second-class boundary conditions. The temperature at the top of the crystal varies with the height of the crystal growth, as determined by fitting a nonlinear equation based on actual experimental data. The bottom temperature is the crystal's melting point, 1685 K.

[0204] Following step 2, design the PINN neural network architecture, using a fully connected neural network for function approximation. The input layer contains three neurons, the hidden layer consists of 16 layers, each with eight neurons, and the output layer contains one neuron. Tanh is used as the activation function after the input layer and each hidden layer. The network's three input parameters are (r, z, t), including the location and time of the sampling points, and the output is the predicted crystal temperature.

[0205] According to step 3, the loss function of PINN is designed. The loss function of PINN consists of two parts: boundary condition loss and PDE loss. The thermal boundary conditions of the crystal are divided into the first type of boundary conditions and the second type of boundary conditions. In the first type of boundary conditions, the top loss function is expressed as L b11 (θ), the loss function at the solid-liquid interface is expressed as L b12 (θ), the loss function of the second type of boundary condition is expressed as L b2 (θ), the loss of PDE is expressed as L PDE (θ), the loss function of PINN can be expressed as:

[0206] L(θ)=K PDE ·L PDE (θ)+K b11 ·L b11 (θ)+K b12 ·L b12 (θ)+K b2 ·Lb2 (θ) (22)

[0207] Among them, K b11 , K b12 , K b2 They represent the equilibrium parameters of the boundary conditions, K PDE Represents the equilibrium parameters of the partial differential equation and learns the network parameters by minimizing the mean square error (MSE) loss.

[0208] Following step 4, training is performed by combining the physical equation loss and network prediction error to optimize the neural network weights. A combination of the Adam optimizer and the LBFGS optimizer is used. Training is first performed using the Adam optimizer. After initial error convergence, the LBFGS optimizer is used to train until the gradient is less than 1e-7. The initial learning rate for the Adam optimizer is set to 0.001, and the learning rate for the LBFGS optimizer is set to 1.

[0209] According to step 5, the temperature distribution during the crystal growth process is predicted, and the obtained temperature distribution during the crystal growth process is as follows Figure 5 As shown, the change of V / G value during crystal growth is calculated as follows Figure 6 shown.

Claims

1. A soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN is characterized by: Please follow the steps below to implement: Step 1: Construct a physical model including a crystal reference shape and a heat conduction model; Step 2: Design the neural network model of the physical information neural network PINN. Step 3: Design the PINN loss function, combine the difference between the network prediction and the actual measurement value, the physical constraints formed by the partial differential equation (PDE) and the boundary conditions to train and optimize the weights of the neural network model; Step 4: Network training and optimization; Step 5: Model prediction.

2. The soft-sensing modeling method for V / G value of Czochralski silicon single crystal growth process based on PINN according to claim 1, characterized in that: The crystal growth in step 1 includes four stages: shoulder release, shoulder rotation, equal diameter and tailing. The growth conditions including growth rate and temperature gradient are different in each stage, the solid-liquid interface deformation is different, and the V / G value will also show dynamic change characteristics.

3. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 2, characterized in that: The step 1 is specifically implemented according to the following steps: Set the crystal reference shape during crystal growth, the actual crystal solid-liquid interface position z IF (r) has different shapes depending on the growth conditions, expressed as Where z represents the direction of the growth axis, r represents the radial direction of the crystal, R represents the radius of the crystal, the position at z = 0 is defined as the position of the solid-liquid interface outside the crystal, and the parameter h0 determines the concave and convexity and size of the solid-liquid interface deformation. top Represents the top temperature of the crystal. The reference shape of the crystal is the solution domain of the differential equation. Sampling is performed within the solution domain to obtain the input coordinate points of the network.

4. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 3, characterized in that: The step 2 is specifically implemented according to the following steps: A fully connected neural network is used for function approximation. The input layer contains three neurons, the output layer is one neuron, and Tanh is used as the activation function after the input layer and each hidden layer.

5. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 4, characterized in that: The loss function in step 3 is designed as follows: Consider a general nonlinear PDE of the following form: Where x and t represent space and time, and the subscripts represent partial differentials. represents the nonlinear partial differential operator, is the boundary of Ω, u(x,t) represents the solution of the PDE; Assume a feedforward fully connected neural network with a depth of M. The input of the network is x, and the output of the mth layer is represented as A neural network is defined as: Among them, σ is the activation function, W [m] and b [m] Represents the weight and bias of the mth layer, and defines the physical information neural network f(t,x) as: f:=u t (x,t;θ)+N x [u(x,t;θ)] (3) The PDE residual neural network of Equation (3) is derived by using automatic differentiation and then applying the chain rule, where θ is the set of all parameters in the network, and is learned by minimizing the loss function: L(θ)=L f (θ)+L b (θ)+L0(θ) (4) Among them, L f (θ) is the loss function of PDE, L b (θ) is the loss function of the boundary condition, L0(θ) is the loss function of the initial condition, and the mean square error MSE is used as the loss function: in, is the starting point, is the boundary point, is a set of points randomly distributed in Ω, N f , N b and N0 represent the total number of configuration points, boundary points, and initial points, respectively. The network parameters are adjusted by minimizing the total loss L(θ) through the gradient descent process.

6. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 5, characterized in that: In step 3, during the growth of the Czochralski silicon single crystal, the heat conduction model inside the crystal is expressed as: In formula (8), is the Peclet constant, x(r,z,t) is the temperature distribution of the crystal region, ▽ represents the spatial gradient operator in the cylindrical coordinate system, v0, R cruc 、k s 、k r are the pulling speed, crucible radius, heat conduction coefficient and thermal conductivity respectively, V(r,z,t) is the velocity of the crystal region, and the boundary velocity along the radial direction is ignored. The crystal heat conduction PDE is expressed as: In formula (9), k0=k r / Pe,V z (t) is the crystal's axial velocity. The variables in the equation are normalized, and the temperature T is relative to the crystal melting temperature T. f Normalized, r and z are normalized to the crucible radius, t is normalized to R cruc. / v0 is normalized, and the normalization formula is as follows: x=(T-T f ) / T f (10) r=(r-R cruc. ) / R cruc. (11) z=(z-R cruc. ) / R cruc. (12) At the solid-liquid interface, the silicon melt solidifies into silicon crystals. Therefore, the first type of boundary condition is set at the solid-liquid interface, and the temperature is set to be the melting point of silicon, T. f : x| z＝0 ＝0 (14) The top temperature of the crystal is T top As the crystal grows, the first type of boundary conditions are set as follows: x| z＝l(t) =x top (15) Among them, l(t) is the growth height of the crystal, x top is the normalized top temperature of the crystal; The heater provides radial heat flux within a certain height range of the crystal, and the corresponding boundary conditions are as follows: Where Q(t) is the heat flux provided by the local heater, R c (z) is the radius of the crystal at height z, z1(t) and z2(t) represent the spatial intervals where the heaters are placed. Due to the axial symmetry of the crystal, the heat flux of the crystal is zero at the axis of symmetry, and at other boundaries, the heat flux is assumed to be zero. The second type of boundary conditions are set as follows: In the crystal growth process model established above, the network parameters are learned by minimizing the mean square error loss. The loss function is expressed as: L(θ)=K1·L1(θ)+K2·L2(θ)+K3·L PDE (i) (20) Where K1 represents the equilibrium parameter of the first type of boundary condition, K2 represents the equilibrium parameter of the second type of boundary condition, and K3 represents the equilibrium parameter of the partial differential equation, which is used to balance the losses of the boundary conditions and PDE.

7. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 6, characterized in that: The step 4 is specifically implemented according to the following steps: The optimizer selected is a combination of the Adam optimizer and the LBFGS optimizer. First, the Adam optimizer is used for preliminary training and parameter adjustment to ensure rapid initial convergence of the PINN model designed in step 3. Then, the finite memory quasi-Newton method LBFGS optimizer is switched to perform fine tuning to improve the accuracy of the model. The back propagation algorithm is used to optimize the neural network weights and minimize the loss function, thereby learning the temperature distribution law during the crystal growth process.

8. The soft-sensing modeling method for V / G value in the Czochralski silicon single crystal growth process based on PINN according to claim 7, characterized in that: The step 5 is specifically implemented according to the following steps: The trained neural network model was used to predict the temperature distribution during crystal growth and calculate the V / G value, where V represents the crystal growth rate in mm / min and G represents the axial temperature gradient at the solid-liquid interface in K / mm.

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