CN-115935812-B - Silicon melt flow rate prediction method in Czochralski silicon single crystal growth process

CN115935812BCN 115935812 BCN115935812 BCN 115935812BCN-115935812-B

Abstract

The invention discloses a silicon melt flow rate prediction method in the growth process of a Czochralski silicon single crystal, which comprises the steps of firstly establishing a two-dimensional axisymmetric rotation model of the silicon melt in the growth process of the Czochralski silicon single crystal, and setting boundary conditions of the fluid model; and finally, solving a two-dimensional axisymmetric rotation model of the silicon melt in the growth process of the Czochralski silicon single crystal by using the space factor physical information neural network, and finally obtaining a prediction result of the flow velocity of the silicon melt. The method solves the problem that the flow velocity of the silicon melt cannot be predicted in real time in the growth process of the silicon single crystal in the prior art.

Inventors

LIU DING
MU LINGXIA
SHI SHUYAN
HUANG WEICHAO
Huo Zhiran

Assignees

西安理工大学

Dates

Publication Date: 20260508
Application Date: 20221128

Claims (6)

1. A method for predicting the flow rate of a silicon melt in the growth process of a Czochralski silicon single crystal is characterized by comprising the following steps: step1, establishing a two-dimensional axisymmetric rotation model of a silicon melt in the growth process of a Czochralski silicon single crystal, and setting boundary conditions of the fluid model; The step1 specifically comprises the following steps: step 1.1, constructing a silicon melt fluid model comprising a continuity equation and a momentum equation; step1.2, establishing a silicon melt fluid model under a cylindrical coordinate; step 1.3, constructing boundary conditions of a two-dimensional axisymmetric rotation model of the silicon melt; step 2, constructing a space factor physical information neural network; The step 2 comprises the following steps: step 2.1, constructing a physical information neural network: loss function of physical information neural network Comprising two parts, boundary point loss Loss of configuration point The boundary points are sampling points of the analog domain boundary, which need to meet the boundary conditions, the configuration points are sampling points in the analog domain, which need to meet partial differential equations corresponding to the model, and the method is as follows: (14) (15) (16) in the formula, The number of boundary points is indicated, Indicating the number of configuration points and, Representing network pair number Boundary points Is used to determine the output value of the (c), Represent the first Boundary points The corresponding true value of the true, Representing network pair number Each configuration point Is used to determine the output value of the (c), Representing the first of the partial differential equations Error of the individual equations; Step 2.2, space factor physical information neural network: Loss function of space factor physical information neural network is lost by boundary points Loss of configuration point In the training process, in order to relatively balance the boundary point loss and the configuration point loss, when the boundary point loss reaches a set target value, the optimization of the configuration point loss is enhanced, so that the loss function formulas (14) to (16) of the spatial parameter physical information neural network are improved to be that (22) (23) (24) In the middle of As a threshold value for the boundary loss, As a switching function, satisfy: (25); and 3, solving a two-dimensional axisymmetric rotation model of the silicon melt in the growth process of the Czochralski silicon single crystal in the step 1 by using the space factor physical information neural network in the step2, and finally obtaining a prediction result of the flow velocity of the silicon melt.
2. The method for predicting the silicon melt flow rate during the growth of a czochralski silicon single crystal of claim 1, wherein the step 1.1 comprises: (1) (2) In the middle of Indicating the degree of dispersion of the fluid, Is a hamiltonian, which is a hamiltonian, Is the velocity vector in the flow field, The term of convection is indicated as such, Is the density of the fluid and, Is the pressure in the flow field and, Is the viscosity coefficient of the silicon melt, Is a laplace operator of the device, Is the acceleration of gravity; Expanding equation (1) (2) yields: (3) (4) In the middle of 、、 Respectively represent A shaft(s), A shaft(s), A speed component of the shaft.
3. The method for predicting the silicon melt flow rate during the growth of a czochralski silicon single crystal according to claim 2, wherein the step 1.2 is specifically as follows: Taking a silicon melt as Newton, incompressible and axisymmetric rotary fluid, taking the axes of a crystal and a crucible as symmetry axes, converting equations (3) (4) into a cylindrical coordinate according to the characteristics, and establishing a silicon melt fluid model under the cylindrical coordinate: (5) In the middle of 、、 Respectively representing the radial, axial and azimuthal velocity components, 、、 Is the coordinates of radial, axial and azimuth; Since the model is symmetrical about the rotation axis, any variable is represented by equation (5) (6) Satisfy the following requirements The two-dimensional axisymmetric rotation model of the silicon melt is expressed as follows: (7) (8) and (7) and (8) are two-dimensional axisymmetric rotation models of the silicon melt under the cylindrical coordinates finally.
4. A method for predicting the silicon melt flow rate during the growth of a czochralski silicon single crystal as defined in claim 3, wherein the step 1.3 is specifically as follows: The two-dimensional axisymmetric rotation model of the silicon melt totally comprises five boundaries, and boundary conditions corresponding to each boundary are as follows in sequence: Axisymmetric boundary, boundary position is 、 The corresponding boundary conditions are 、、 ; The boundary position of the bottom of the crucible is 、 The corresponding boundary conditions are 、、 ; The boundary position of the side wall of the crucible is 、 The corresponding boundary conditions are 、、 ; The solid-liquid interface is defined as the boundary position 、 The corresponding boundary conditions are 、、 ; Free surface boundary position is 、 The corresponding boundary conditions are 、、 ; Wherein is The radius of the crystal is such that, For the radius of the crucible, Is the crucible height.
5. The method for predicting the silicon melt flow rate during the growth of a Czochralski silicon single crystal of claim 4, wherein step 2 comprises: step 2.1, constructing a physical information neural network: first, a general expression of partial differential equations is given: (9) In the middle of As a result of the non-linear operator, Representing a solution that satisfies the partial differential equation, In order to be in the spatial domain, ; The boundary conditions are expressed as: (10) In the middle of Representing boundaries, i.e. solutions to partial differential equations, are required to meet set boundary conditions ; The physical information neural network is input as The physical information neural network is output as In the following For network parameters, including weight matrix Bias vector Physical information neural network output True solution to partial differential equation The dimensions remain the same and act as The expression of the physical information neural network is as follows: (11) (12) (13) In the middle of Representing network No The output of the layer is provided with, In order to activate the function, As the spatial sample points of the input, 、 Represent the first The weight matrix and the bias vector of the layer, Representing the final output of the network, The total layer number of the physical information neural network; Step 2.2, space factor physical information neural network: inputting the original space Each hidden layer of the network is reintroduced to construct a space factor physical information neural network, and the hidden layer (12) is updated as follows: (17) in the formula, As an additional offset vector to be used, Representing vector multiplication; Using two encoders 、 Instead of 、 Hidden layer (17) is updated as: (18) In the middle of 、 , 、 Respectively encoders Is used for the weight matrix and the offset vector of the model (1), 、 Respectively encoders Weight matrix and bias vector of (a); The space factor physical information neural network has the expression: (19) (20) (21)。
6. The method for predicting the silicon melt flow rate during the growth of a Czochralski silicon single crystal of claim 5, wherein the step 3 comprises: Step 3.1, setting specific parameters of a space factor physical information neural network: The input layer of the space factor physical information neural network is 2 neurons, which are used for inputting the radius of the space coordinate And height of The output layer is 4 neurons, and outputs radial velocity respectively Axial velocity Azimuth velocity Pressure and force The hidden layers are 5 layers, 128 neurons are arranged on each layer, the initial learning rate is set to be 0.002, each time of iteration is 1000, the learning rate is attenuated to be 0.9 times of the previous learning rate, and the total time of the whole training process is 30000 times; Step 3.2, constructing a training set: the training set includes a set of boundary points Configuration point set The boundary points in the boundary point set need to be sampled at the boundary of the simulation area, and the boundary point set totally comprises five boundaries, namely an axisymmetric boundary, a crucible bottom, a crucible side wall, a solid-liquid interface and a free surface, and is used for meeting the set boundary conditions, the sampling mode is selected for equidistant sampling, the number of sampling points of each boundary is 400, the configuration points in the configuration point set are sampled in the simulation area and are used for meeting a partial differential equation, the sampling mode is selected for Latin hypercube sampling, and the number of sampling is 6000; Step 3.3, constructing a loss function of a two-dimensional axisymmetric rotation model of the silicon melt: Constructing a space factor physical information neural network loss function according to the established silicon melt two-dimensional axisymmetric rotation model Comprising two parts, boundary point loss Loss of configuration point , Boundary point loss function The following are provided: (26) Configuring a point loss function The following are provided: (27) (28) (29) (30) (31) Total loss of The method comprises the following steps: (32) step 3.4, substituting experimental parameters into: Physical parameters of the silicon melt: 、 ; assuming a crystal radius of The radius of the crucible is The height of the crucible is Rotation speed of crucible Crystal rotation speed ; Step 3.5, training a space factor physical information neural network and predicting: Experiments are carried out on a Python framework Tensorflow-GPU 1.15.2, and the computer is configured as follows, the CPU is AMD Ryzen G, 5600G, the frequency is 3.90 GHz, the running memory is 16G, the display card is RTX2060, and the display memory is 6G; Optimizing network parameters by adopting an ADAM gradient descent method, wherein the exponential decay rate is 0.99, stopping training when the iteration times reach 30000 times or the loss value is lower than 1e-5, and after the network training is finished, carrying out radius adjustment on space coordinates Height of As an input to the network, the flow rate of the analog region is predicted.

Description

Silicon melt flow rate prediction method in Czochralski silicon single crystal growth process Technical Field The invention belongs to the technical field of preparation of semiconductor silicon single crystal materials, and particularly relates to a silicon melt flow rate prediction method in a Czochralski silicon single crystal growth process. Background Integrated circuit chips play a vital role in the development of social technology, and silicon is one of the most important semiconductor materials as a base material for manufacturing the chips. The Czochralski method, also called CZ method, is a main method for producing silicon single crystals, and is widely used for growth and production of large-diameter, high-quality semiconductor silicon crystals. The flow field, the temperature field, the magnetic field and the like are main physical fields for silicon single crystal growth research, wherein the flow field is used as a starting point for other physical field researches, and various complex convection such as natural convection, forced convection, marangoni convection and the like are contained inside the flow field. The interaction between the complex convection currents affects the temperature distribution, concentration distribution, impurity distribution, etc. of the silicon melt during the growth of the silicon single crystal. The flow velocity of the silicon melt is predicted, so that the analysis of the internal convection and temperature distribution of the melt is facilitated, and the optimizer adopts a corresponding control strategy, thereby having very important significance for optimizing the crystal growth process and improving the crystal quality. However, in actual engineering, the flow rate of the silicon melt cannot be directly measured due to the complicated environment in the furnace, the excessively high temperature of the silicon melt and the like. The silicon melt flow rate is usually predicted by adopting a computational fluid dynamics method, the solution is needed to be solved after the discretization of a fluid model, and the obtained solution is in a discrete form and has limited application scenes such as real-time prediction, model optimization control and the like. The invention provides a silicon melt flow velocity prediction method based on a space-based factor physical information neural network, which meets the strong form of partial differential equation in a fluid model. After the network training is finished, the flow velocity of the silicon melt at the position can be rapidly predicted by only inputting the space coordinates of the target point, and the method can be used for application scenes such as silicon single crystal growth model control, crystal quality optimization and the like. Disclosure of Invention The invention aims to provide a silicon melt flow rate prediction method in the growth process of a Czochralski silicon single crystal, which solves the problem that the silicon melt flow rate in the growth process of the silicon single crystal cannot be predicted in real time in the prior art. The technical scheme adopted by the invention is that the silicon melt flow rate prediction method in the growth process of the Czochralski silicon single crystal is implemented according to the following steps: step1, establishing a two-dimensional axisymmetric rotation model of a silicon melt in the growth process of a Czochralski silicon single crystal, and setting boundary conditions of the fluid model; step 2, constructing a space factor physical information neural network; and 3, solving a two-dimensional axisymmetric rotation model of the silicon melt in the growth process of the Czochralski silicon single crystal in the step 1 by using the space factor physical information neural network in the step2, and finally obtaining a prediction result of the flow velocity of the silicon melt. The present invention is also characterized in that, The step 1 is specifically as follows: step 1.1, constructing a silicon melt fluid model comprising a continuity equation and a momentum equation; step1.2, establishing a silicon melt fluid model under a cylindrical coordinate; And 1.3, constructing boundary conditions of a two-dimensional axisymmetric rotation model of the silicon melt. Step 1.1 is specifically as follows: In the middle of Indicating the degree of dispersion of the fluid,Is a hamiltonian, which is a hamiltonian,Is the velocity vector in the flow field,Representing the convection term, ρ is the density of the fluid, P is the pressure in the flow field, μ is the viscosity coefficient of the silicon melt,Is the Laplacian, g is the gravitational acceleration; Expanding equation (1) (2) yields: where u, v, w represent the velocity components of the x-axis, y-axis, and z-axis, respectively. Step 1.2 is specifically as follows: Taking a silicon melt as Newton, incompressible and axisymmetric rotary fluid, taking the axes of a crystal and a crucible as symmetry a