Method for analyzing silicon crystal swing in Czochralski silicon single crystal pulling system under eccentric excitation
By constructing a silicon crystal oscillation model for the Czochralski silicon single crystal pulling system under eccentric excitation, and using energy analysis and the Lagrange equation to solve the problem of unstable silicon crystal oscillation under eccentric excitation, the stability and yield of silicon crystal growth were improved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- XIAN UNIV OF TECH
- Filing Date
- 2026-03-17
- Publication Date
- 2026-05-26
AI Technical Summary
The existing technology lacks systematic and accurate analysis methods for the oscillation behavior of silicon crystals in the Czochralski single crystal pulling system under eccentric excitation, which leads to unstable silicon crystal growth and may cause crystal defects and fractures.
A silicon crystal oscillation model of the Czochralski silicon single crystal pulling system under eccentric excitation was constructed. Nonlinear equations were established through energy analysis and the second-type Lagrange equation, and the Runge-Kutta method was used to solve the equations to obtain the oscillation angle of the soft shaft and the pulled silicon crystal.
It accurately describes the oscillation of silicon crystals under eccentric excitation, improving the stability and yield of silicon crystal growth, and reducing production costs and time investment.
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Figure CN121859608B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of semiconductor silicon single crystal preparation technology, and in particular to a method for analyzing the oscillation of silicon crystals in a Czochralski silicon single crystal pulling system under eccentric excitation. Background Technology
[0002] As a core substrate material for integrated circuit chips, the fabrication technology of silicon single crystal directly affects the performance and manufacturing cost of semiconductor devices. The Czochralski method is currently the mainstream method for fabricating large-size, high-quality silicon single crystals. During the Czochralski process, the silicon crystal is subjected to various excitations, especially eccentric excitations (such as imbalances in the rotating mechanism or melt flow disturbances), which can cause unpredictable oscillations in the silicon crystal, affecting the stability and uniformity of silicon crystal growth, and even leading to crystal defects and intergranular fracture.
[0003] In related technologies, some research methods have been developed to address the vibration and oscillation issues during the Czochralski (CZ) silicon single crystal growth process. However, most methods focus on analysis under symmetrical or ideal conditions, lacking systematic and accurate analytical means for the oscillation behavior of the soft shaft and silicon crystal in the CZ silicon single crystal pulling system under eccentric excitation. Therefore, it is necessary to propose a solution to improve one or more of the problems existing in the aforementioned related technologies.
[0004] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention
[0005] This application provides a method for analyzing the oscillation of silicon crystals in a Czochralski silicon single crystal pulling system under eccentric excitation. The method includes:
[0006] Construct a silicon crystal oscillation model for a Czochralski silicon single crystal pulling system under eccentric excitation;
[0007] Energy analysis was performed on the Czochralski silicon single crystal pulling system, and the nonlinear equation of the Czochralski silicon single crystal pulling system under eccentric excitation was established by applying the second type of Lagrange equation.
[0008] The Runge-Kutta method was used to solve the nonlinear equations to obtain the swing angles of the flexible shaft and the pulled-out silicon crystal.
[0009] The variation of the swing angle of the flexible shaft and the pulled silicon crystal under different eccentricities was analyzed.
[0010] Furthermore, the steps for constructing a silicon crystal oscillation model for a Czochralski (CZ) single-crystal pulling system under eccentric excitation include:
[0011] Modeling is based on the following assumptions:
[0012] Both the flexible shaft and the pulled silicon crystal are uniformly distributed in space, and their masses are equivalently concentrated at their respective centers of mass.
[0013] The silicon crystal has been pulled out into a rigid body;
[0014] Both the flexible shaft and the pulled-out silicon crystal rotate at a constant angular velocity.
[0015] Furthermore, the silicon crystal oscillation model includes:
[0016] Construct a spatial rectangular coordinate system with the rotation center of the suspension point of the flexible shaft as the origin. The eccentricity of the suspension point of the flexible shaft from the origin is... Both the flexible shaft and the pulled-out silicon crystal move at a constant angular velocity. Rotate clockwise, the first The length of the flexible shaft at time is The mass per unit length of the flexible shaft is , No. The length of the silicon crystal pulled out at that time was The mass per unit length of the silicon crystal that has been pulled out is The lifting speed of the flexible shaft is The growth rate of the pulled silicon crystal is... The flexible axis and the spatial rectangular coordinate system The included angle along the axial direction is The silicon crystal and the spatial rectangular coordinate system have been drawn. The included angle along the axial direction is .
[0017] Furthermore, the coordinates of the suspension point of the flexible shaft are:
[0018] (1)
[0019] in, Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the first The angular displacement of the suspension point of the flexible shaft at any given moment;
[0020] The centroid coordinates of the flexible shaft are:
[0021] (2)
[0022] in, Indicates the centroid of the flexible shaft Axis coordinates Indicates the centroid of the flexible shaft Axis coordinates Indicates the centroid of the flexible shaft Axis coordinates Indicates the first Half the length of the flexible shaft at any given moment;
[0023] The centroid coordinates of the silicon crystal have been determined as follows:
[0024] (3)
[0025] in, This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates Indicates the first The silicon crystal has been pulled out to half its length at this point;
[0026] The expression for the linear velocity of the centroid of the flexible shaft is:
[0027] (4)
[0028] in, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, This indicates the rate of change of the swing angle of the flexible shaft over time.
[0029] The expression for the linear velocity of the centroid of the pulled silicon crystal is:
[0030] (5)
[0031] in, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This represents the rate of change of the swing angle of the pulled-out silicon crystal over time.
[0032] Furthermore, the steps for performing energy analysis on the Czochralski silicon single crystal pulling system and establishing the nonlinear equations for the Czochralski silicon single crystal pulling system under eccentric excitation by applying the second type of Lagrange equation include:
[0033] Calculate the kinetic energy of the Czochralski silicon single crystal pulling system;
[0034] Calculate the potential energy of the Czochralski silicon single crystal pulling system;
[0035] By applying the second type of Lagrange equation, the Lagrange equation system for the Czochralski silicon single crystal pulling system is constructed using the kinetic and potential energies of the Czochralski silicon single crystal pulling system.
[0036] Solving the Lagrange equations yields the nonlinear equations for the Czochralski silicon single crystal pulling system.
[0037] Furthermore, the expression for the kinetic energy of the Czochralski silicon single crystal pulling system is as follows:
[0038] (6)
[0039] in, This represents the kinetic energy of the Czochralski silicon single crystal pulling system. Indicates the mass of the flexible shaft. , The velocity of the center of mass of the flexible shaft is indicated. , Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, This represents the moment of inertia of the flexible shaft. , This represents the rate of change of the swing angle of the flexible shaft over time. This indicates the quality of the silicon crystal that has been pulled out. , This indicates the speed at which the centroid of the silicon crystal has been pulled out. , This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates the moment of inertia of the pulled-out silicon crystal. , This represents the rate of change of the swing angle of the pulled-out silicon crystal over time.
[0040] Furthermore, the expression for the potential energy of the Czochralski silicon single crystal pulling system is as follows:
[0041] (7)
[0042] in, The potential energy of the Czochralski silicon single crystal pulling system is represented by the horizontal plane at the origin of the coordinate system, which is taken as the zero potential energy surface of the Czochralski silicon single crystal pulling system. Represents gravitational acceleration. Indicates the first Half the length of the flexible shaft at that moment. Indicates the first The time has been half the length of the silicon crystal.
[0043] Furthermore, by applying the second type of Lagrange equations, the steps to construct the Lagrange equations for the Czochralski silicon single crystal pulling system using the kinetic and potential energies of the system include:
[0044] The Lagrangian quantity of the Czochralski silicon single crystal pulling system is determined using the kinetic and potential energy of the system.
[0045] The expression for the Lagrangian of the Czochralski silicon single crystal pulling system is:
[0046] (8)
[0047] in, This indicates that the Czochralski silicon single crystal pulling system is in the first... The generalized coordinates of time are Generalized speed is Lagrange quantity at time The generalized coordinates represent the Czochralski (CZ) silicon single crystal pulling system. This represents the generalized velocity of the Czochralski silicon single crystal pulling system. This represents the kinetic energy of the Czochralski silicon single crystal pulling system. This represents the potential energy of the Czochralski silicon single crystal pulling system;
[0048] Take generalized coordinates , The transpose is represented, and combined with the Lagrangian of the Czochralski silicon single crystal pulling system, the Lagrangian equation of the Czochralski silicon single crystal pulling system is constructed;
[0049] The expression for the Lagrange equation of the Czochralski silicon single crystal pulling system is:
[0050] (9)
[0051] in, This represents the partial derivative operation. Generalized coordinates representing the flexible axis. This indicates that the generalized coordinates of the silicon crystal have been extracted. Represents the generalized velocity of the flexible shaft. This represents the generalized speed at which the silicon crystal has been pulled out. The generalized damping force of the flexible shaft. This represents the generalized damping force that has been pulled out of the silicon crystal. for abbreviation;
[0052] Therefore, the expression for the Lagrange equations of the Czochralski silicon single crystal pulling system is as follows:
[0053] (10)
[0054] in, This represents the rate of change of the swing angle of the flexible shaft over time. This represents the rate of change of the swing angle of the pulled silicon crystal over time. , This represents the damping coefficient of the flexible shaft. , This indicates the damping coefficient of the silicon crystal that has been pulled out.
[0055] Furthermore, the process of solving the Lagrange equations to obtain the nonlinear equations for the Czochralski silicon single crystal pulling system includes:
[0056] ;
[0057] ;
[0058] in, Indicates the first The silicon crystal has been pulled out to half its length at this point. Indicates the first Half the length of the flexible shaft at any given moment.
[0059] Furthermore, the steps of solving the nonlinear equations using the Runge-Kutta method to obtain the swing angle of the flexible shaft and the pulled-out silicon crystal include:
[0060] exist The Runge-Kutta method is used in the simulation software to numerically solve the nonlinear equations;
[0061] Numerical solutions of the generalized coordinates of the Czochralski silicon single crystal pulling system at each time point are obtained. All numerical solutions represent the changes in the oscillation angle of the flexible shaft and the pulled silicon crystal over time.
[0062] This application provides a method for analyzing the oscillation of silicon crystals in a Czochralski (Czochralski) single-crystal pulling system under eccentric excitation, which has at least the following beneficial effects:
[0063] This application constructs a silicon crystal oscillation model for a Czochralski (CZ) single-crystal pulling system under eccentric excitation. Through energy analysis of the CZ single-crystal pulling system and applying the second-type Lagrange equation, nonlinear equations for the system under eccentric excitation are established. Finally, the Runge-Kutta method is used to solve the nonlinear equations, obtaining the oscillation angles of the flexible shaft and the pulled silicon crystal. This application considers the time-varying characteristics of the length and mass of the flexible shaft and the pulled silicon crystal, establishing a more accurate physical model that can accurately determine the oscillation behavior of the flexible shaft and the pulled silicon crystal under eccentric excitation. Attached Figure Description
[0064] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.
[0065] Figure 1 This illustration shows a schematic diagram of the steps in a silicon crystal oscillation analysis method in a Czochralski silicon single crystal pulling system under eccentric excitation according to an exemplary embodiment of this application;
[0066] Figure 2 This diagram illustrates a silicon crystal oscillation model of the Czochralski silicon single crystal pulling system under eccentric excitation in an exemplary embodiment of this application.
[0067] Figure 3 The diagram shows the curves of the swing angle of the flexible shaft changing with time under different eccentricities in the simulation experiment of this application.
[0068] Figure 4 The diagram shows the curves of the swing angle of the pulled silicon crystal changing over time under different eccentricities in the simulation experiment of this application. Detailed Implementation
[0069] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided to make this application more comprehensive and complete, and to fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.
[0070] Furthermore, the accompanying drawings are merely illustrative of this application and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.
[0071] In recent years, in order to increase production capacity and reduce unit costs, the diameter of silicon crystals has been reduced from its original size. Towards As silicon single crystal growth equipment evolved, the amount of material fed into the furnace increased significantly. This led to a continuous increase in the furnace height, and consequently, a corresponding increase in the length of the flexible lifting shaft (e.g., flexible tungsten wire rope), while the stiffness decreased significantly. Furthermore, due to installation errors or slight deformations in the structure of the silicon single crystal growth equipment during operation, the suspension point of the flexible shaft deviated from the rotation center, resulting in eccentricity. In engineering practice, it was found that during the rotation-lifting coupled motion of the Czochralski silicon single crystal pulling system, the seed crystal exhibited unpredictable oscillations. The amplitude and frequency of these oscillations suddenly became unstable under certain operating conditions, exhibiting aperiodic, large-amplitude drift. This phenomenon not only disrupted the stability of the solid-liquid interface during silicon crystal growth, inducing crystal defects such as dislocations and voids, but in severe cases, it could even cause breakage, rendering the entire silicon single crystal rod unusable.
[0072] However, the oscillation angle of the silicon crystal in the Czochralski silicon single crystal pulling system under eccentric excitation is difficult to measure in actual operating equipment, and conducting related experiments in actual production will require a large investment of material and time resources.
[0073] Therefore, in order to solve the silicon crystal oscillation problem caused by eccentric excitation in practical engineering, this application proposes a method for analyzing silicon crystal oscillation in a Czochralski silicon single crystal pulling system under eccentric excitation, such as... Figure 1 As shown, the method may include the following steps:
[0074] In this application embodiment, a set of reasonable modeling assumptions are first proposed to simplify the description of the Czochralski silicon single crystal pulling system and highlight its core physical mechanism.
[0075] Step S101 of this embodiment: Constructing a silicon crystal oscillation model for the Czochralski (CZ) single-crystal pulling system under eccentric excitation. In step S101 of this embodiment, modeling is based on the following assumptions:
[0076] First, it is assumed that both the flexible shaft and the pulled silicon crystal are uniformly distributed in space, and that their masses are always equivalently concentrated at their respective centers of mass. This is beneficial for subsequent calculations of the kinetic and potential energy of the Czochralski silicon single crystal pulling system. This assumption significantly reduces the complexity of the silicon crystal oscillation model while ensuring a reasonable expression of the inertia and gravitational effects of the Czochralski silicon single crystal pulling system, laying the foundation for the subsequent establishment of the Lagrange equation.
[0077] Secondly, we assume the pulled-out silicon crystal is a rigid body, neglecting its elastic deformation. Here, we assume the pulled-out silicon crystal is a rigid body, not a point mass, because the nature of the silicon crystal's oscillation involves not only the movement of its center of mass, but more importantly, the rotation of its own orientation. The rigid body model acknowledges the finite size and shape of the silicon crystal and explicitly introduces rotational degrees of freedom and moment of inertia, thus enabling the subsequent deriving of more accurate and complete dynamic equations that include rotational kinetic energy.
[0078] Finally, it is assumed that both the flexible shaft and the pulled silicon crystal rotate at a constant angular velocity. This assumption simplifies the rotational excitation to a steady-state excitation, facilitating the analysis of the continuous influence of eccentricity on the silicon crystal oscillation in the Czochralski silicon single crystal pulling system, and improving the resolvability of the silicon crystal oscillation model.
[0079] In one embodiment, such as Figure 2 As shown, the silicon crystal oscillation model includes:
[0080] Construct a spatial rectangular coordinate system with the rotation center of the suspension point of the flexible shaft as the origin. The eccentricity of the suspension point of the flexible shaft from the origin is... Both the shaft and the pulled-out silicon crystal are moving at a constant angular velocity. Rotate clockwise, the first The length of the flexible shaft at time is The mass per unit length of the flexible shaft is , No. The length of the silicon crystal pulled out at that time was The mass per unit length of the silicon crystal that has been pulled out is The lifting speed of the flexible shaft is The growth rate of the pulled silicon crystal is... The flexible axis and the spatial rectangular coordinate system The included angle along the axial direction is The silicon crystal and the spatial rectangular coordinate system have been drawn. The included angle along the axial direction is .
[0081] By establishing the position coordinates of the suspension point, the coordinates of the centroid, and the expression for its linear velocity, a complete mathematical description of the state of the Czochralski silicon single crystal pulling system is realized. This facilitates the silicon crystal oscillation model to accurately reflect the spatial motion relationship between the flexible shaft and the pulled silicon crystal under eccentric excitation, providing a geometric basis for subsequent energy analysis and dynamic modeling.
[0082] Furthermore, the coordinates of the suspension point of the flexible shaft are:
[0083] (1)
[0084] in, Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the first The angular displacement of the suspension point of the flexible shaft at any given time.
[0085] The centroid coordinates of the flexible shaft are:
[0086] (2)
[0087] in, Indicates the centroid of the flexible shaft Axis coordinates Indicates the centroid of the flexible shaft Axis coordinates Indicates the centroid of the flexible shaft Axis coordinates Indicates the first Half the length of the flexible shaft at any given moment.
[0088] The centroid coordinates of the silicon crystal have been determined as follows:
[0089] (3)
[0090] in, This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates Indicates the first The time has been half the length of the silicon crystal.
[0091] The expression for the linear velocity of the centroid of the flexible shaft is:
[0092] (4)
[0093] in, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, This represents the rate of change of the swing angle of the flexible shaft over time.
[0094] The expression for the linear velocity of the centroid of the pulled silicon crystal is:
[0095] (5)
[0096] in, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This represents the rate of change of the swing angle of the pulled-out silicon crystal over time.
[0097] Step S102 of this embodiment: Perform energy analysis on the Czochralski silicon single crystal pulling system and apply the second type of Lagrange equation to establish the nonlinear equation of the Czochralski silicon single crystal pulling system under eccentric excitation. Step S102 of this embodiment may include the following sub-steps:
[0098] Sub-step S1021: Calculate the kinetic energy of the Czochralski silicon single crystal pulling system.
[0099] Furthermore, the expression for the kinetic energy of the Czochralski silicon single crystal pulling system is as follows:
[0100] (6)
[0101] in, This represents the kinetic energy of the Czochralski silicon single crystal pulling system. Indicates the mass of the flexible shaft. , The velocity of the center of mass of the flexible shaft is indicated. , Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, Indicates the centroid of the flexible shaft The linear velocity of the shaft, This represents the moment of inertia of the flexible shaft. , This represents the rate of change of the swing angle of the flexible shaft over time. This indicates the quality of the silicon crystal that has been pulled out. , This indicates the speed at which the centroid of the silicon crystal has been pulled out. , This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates the moment of inertia of the pulled-out silicon crystal. , This represents the rate of change of the swing angle of the pulled-out silicon crystal over time.
[0102] This expression fully considers the kinetic energy term of the Czochralski silicon single crystal pulling system. In particular, the introduction of rotational inertia makes it easier for the silicon crystal oscillation model to accurately describe the dynamic behavior of the coupling between rotation and oscillation.
[0103] Sub-step S1022: Calculate the potential energy of the Czochralski silicon single crystal pulling system.
[0104] Furthermore, here, taking the horizontal plane at the origin of the coordinate system as the zero potential energy surface of the Czochralski silicon single crystal pulling system, an expression for the potential energy of the Czochralski silicon single crystal pulling system is established. This expression reflects the positional changes of the flexible shaft and the silicon crystal in the gravitational field.
[0105] The expression for the potential energy of the Czochralski silicon single crystal pulling system is:
[0106] (7)
[0107] in, This represents the potential energy of the Czochralski silicon single crystal pulling system. Represents gravitational acceleration. Indicates the first Half the length of the flexible shaft at that moment. Indicates the first The time has been half the length of the silicon crystal.
[0108] The introduction of the aforementioned potential energy term enables the silicon crystal oscillation model to reflect the influence of the gravitational restoring torque on the oscillation, providing a basis for the stability analysis of the Czochralski silicon single crystal pulling system.
[0109] Sub-step S1023: Applying the second type of Lagrange equation, construct the Lagrange equation system for the Czochralski silicon single crystal pulling system using the kinetic and potential energies. The process of sub-step S1023 is as follows:
[0110] The Lagrangian quantity of the Czochralski silicon single crystal pulling system is determined using the kinetic and potential energy of the system.
[0111] Furthermore, the expression for the Lagrangian of the Czochralski silicon single crystal pulling system is:
[0112] (8)
[0113] in, This indicates that the Czochralski silicon single crystal pulling system is in the first... The generalized coordinates of time are Generalized speed is Lagrange quantity at time The generalized coordinates represent the Czochralski (CZ) silicon single crystal pulling system. This represents the generalized velocity of the Czochralski silicon single crystal pulling system. This represents the kinetic energy of the Czochralski silicon single crystal pulling system. This represents the potential energy of the Czochralski silicon single crystal pulling system;
[0114] Take generalized coordinates , The transpose is represented, and combined with the Lagrangian of the Czochralski silicon single crystal pulling system, the Lagrangian equation of the Czochralski silicon single crystal pulling system is constructed;
[0115] Furthermore, the expression for the Lagrange equation of the Czochralski silicon single crystal pulling system is as follows:
[0116] (9)
[0117] in, This represents the partial derivative operation. Generalized coordinates representing the flexible axis. This indicates that the generalized coordinates of the silicon crystal have been extracted. Represents the generalized velocity of the flexible shaft. This represents the generalized speed at which the silicon crystal has been pulled out. The generalized damping force of the flexible shaft. This represents the generalized damping force that has been pulled out of the silicon crystal. for abbreviation;
[0118] Therefore, the expression for the Lagrange equations of the Czochralski silicon single crystal pulling system is as follows:
[0119] (10)
[0120] in, This represents the rate of change of the swing angle of the flexible shaft over time. This represents the rate of change of the swing angle of the pulled silicon crystal over time. , This represents the damping coefficient of the flexible shaft. , This indicates the damping coefficient of the silicon crystal that has been pulled out.
[0121] By using the Lagrange equations, the dynamics problem of complex multi-system systems is transformed into an analytical set of differential equations. At the same time, the introduction of the damping term makes the silicon crystal oscillation model closer to the energy dissipation characteristics of the actual Czochralski silicon single crystal pulling system.
[0122] Sub-step S1024: Solve the Lagrange equations to obtain the nonlinear equations for the Czochralski silicon single crystal pulling system. This process includes:
[0123] ;
[0124] ;
[0125] in, Indicates the first The silicon crystal has been pulled out to half its length at this point. Indicates the first Half the length of the flexible shaft at any given moment.
[0126] Understandably, by differentiating and rearranging the Lagrange equations, a set of coupled nonlinear second-order differential equations is obtained. This set of equations fully describes... , The dynamic behavior that evolves over time involves a variety of physical mechanisms, including geometric nonlinearity, rotational excitation, gravitational restoration, and damping effects.
[0127] The established nonlinear equations enable the Czochralski silicon single crystal pulling system to accurately characterize the complex dynamic response of silicon crystal oscillation under eccentric excitation, especially to reflect the nonlinear characteristics of silicon crystal oscillation amplitude and frequency as a function of eccentricity, providing a mathematical model basis for subsequent numerical simulations.
[0128] Step S103 of this embodiment: Solve the nonlinear equation using the Runge-Kutta method to obtain the swing angle of the flexible shaft and the pulled-out silicon crystal. Step S103 of this embodiment may include the following sub-steps:
[0129] Sub-step S1031: In The Runge-Kutta method is used in the simulation software to numerically solve the nonlinear equations.
[0130] Sub-step S1032: Obtain the numerical solutions of the generalized coordinates of the Czochralski silicon single crystal pulling system at each time point. All numerical solutions are the changes in the swing angle of the flexible shaft and the pulled silicon crystal over time.
[0131] Specifically, it can be found Write a solver program in the simulation environment and set the initial conditions and parameters of the Czochralski silicon single crystal pulling system (such as...). , , , , , Then, by step-by-step integration using the Runge-Kutta method, the numerical solutions of the generalized coordinates of the Czochralski silicon single crystal pulling system at each time point are obtained. All numerical solutions represent the changes in the oscillation angle of the flexible shaft and the pulled silicon crystal over time.
[0132] This numerical solution method overcomes the difficulties of analytically solving nonlinear equations and can efficiently and stably obtain the dynamic response of the Czochralski silicon single crystal pulling system, providing an operable simulation tool for engineering practice.
[0133] Step S104 of this embodiment: Analyze the changes in the swing angle of the flexible shaft and the pulled-out silicon crystal under different eccentricities.
[0134] This section is divided into the application and analysis phases of the method, which involves changing the eccentricity. By taking the value of , repeat the simulation calculation of step S103 in this embodiment to obtain the curves of the swing angle of the flexible shaft and silicon crystal changing with time under different eccentric excitations.
[0135] This analysis can intuitively demonstrate the impact of different eccentricities on the oscillation amplitude, frequency, and stability of silicon crystals in the Czochralski silicon single crystal pulling system. It provides a quantitative reference for eccentricity control and system optimization in actual production, which helps to reduce the oscillation risk during silicon crystal growth and improve the quality and yield of silicon crystals.
[0136] To verify the superior performance of the silicon crystal oscillation analysis method in the Czochralski silicon single crystal pulling system proposed in this application, the following simulation experiments were conducted.
[0137] Based on relevant literature and the process parameters of the actual Czochralski silicon single crystal pulling system, the linear density of the flexible shaft was determined. Linear density of silicon crystal The initial length of the silicon crystal has been pulled out. The initial length of the flexible shaft The growth rate of the pulled silicon crystal is equal to the pulling rate of the flexible shaft, that is... Constant angular velocity of silicon crystal and flexible shaft Gravitational acceleration Initial generalized coordinates Linear density is the mass per unit length. Here, linear density refers to the mass per unit length.
[0138] The results are as follows Figure 3 and Figure 4 It can be seen that when the eccentricity of the Czochralski silicon single crystal pulling system is 0, neither the flexible shaft nor the pulled silicon crystal will wobble, indicating that the Czochralski silicon single crystal pulling system is in a stable state without eccentric excitation. However, as the eccentricity increases... With the increase of eccentricity, the swing amplitude of the flexible shaft and the pulled silicon crystal increases significantly, indicating that the eccentric force has a significant impact on the dynamic stability of the Czochralski silicon single crystal pulling system. Although the swing angle of the flexible shaft and the pulled silicon crystal oscillates in the initial stage of the Czochralski silicon single crystal pulling system, various damping mechanisms, including internal material damping, frictional damping at the connection points, and damping at the solid-liquid interface, help reduce energy accumulation and dissipation, thereby promoting the stability of the Czochralski silicon single crystal pulling system and enabling it to eventually stabilize at a certain angle.
[0139] contrast Figure 3 and Figure 4 As can be seen, under the same eccentricity, the swing angle of the flexible shaft and the swing angle of the pulled silicon crystal are not the same, and the swing angle of the flexible shaft is always smaller than that of the pulled silicon crystal. This further proves the rationality of modeling the pulled silicon crystal as a rigid body, rather than simply treating it as a point mass. Furthermore, this application considers the time-varying characteristics of the length and mass of the flexible shaft and the pulled silicon crystal. This modeling method can more accurately capture the dynamic behavior of the silicon crystal during the pulling process, thus providing an important theoretical basis for optimizing the pulling process.
[0140] In summary, the silicon crystal oscillation analysis method provided in this application for the Czochralski silicon single crystal pulling system under eccentric excitation establishes a physical model that reflects the time-varying characteristics and nonlinear effects of the Czochralski silicon single crystal pulling system through reasonable assumptions and a complete kinematic description. The dynamic model constructed based on the Lagrange equation systematically expresses various physical mechanisms such as rotational excitation, gravity recovery, and damping dissipation. The numerical method achieves efficient solution and is suitable for parameter analysis and optimization design in engineering practice. By analyzing the oscillation response under different eccentricities, it provides a theoretical basis for equipment installation calibration, process parameter selection, and stability improvement of the Czochralski silicon single crystal pulling system.
[0141] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of the embodiments of this application, "multiple" means two or more, unless otherwise explicitly specified.
[0142] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. In addition, those skilled in the art can combine and integrate the different embodiments or examples described in this specification.
[0143] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in this application, and these modifications or substitutions should all be covered within the scope of protection of this application.
[0144] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein.
Claims
1. A method for analyzing the oscillation of silicon crystals in a Czochralski silicon single crystal pulling system under eccentric excitation, characterized in that, The method includes the following steps: Construct a silicon crystal oscillation model for a Czochralski silicon single crystal pulling system under eccentric excitation; Energy analysis was performed on the Czochralski silicon single crystal pulling system, and the nonlinear equations of the second kind of Lagrange equation were applied to establish the Czochralski silicon single crystal pulling system under eccentric excitation, including: Calculate the kinetic energy of the Czochralski silicon single crystal pulling system; The expression for the kinetic energy of the Czochralski silicon single crystal pulling system is: (6) in, This represents the kinetic energy of the Czochralski silicon single crystal pulling system. Indicates the mass of the flexible shaft. , This indicates the mass per unit length of the flexible shaft. Indicates the first The length of the flexible shaft at any given time. The velocity of the center of mass of the flexible shaft is indicated. , Indicating the centroid of the flexible shaft The linear velocity of the shaft, Indicating the centroid of the flexible shaft The linear velocity of the shaft, Indicating the centroid of the flexible shaft The linear velocity of the shaft, This represents the moment of inertia of the flexible shaft. , This represents the rate of change of the swing angle of the flexible shaft over time. This indicates the quality of the silicon crystal that has been pulled out. , This indicates the mass per unit length of the silicon crystal that has been pulled out. Indicates the first The time has now stretched the length of the silicon crystal. This indicates the speed at which the centroid of the silicon crystal has been pulled out. , This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates the moment of inertia of the pulled-out silicon crystal. , This represents the rate of change of the swing angle of the pulled-out silicon crystal over time. Calculate the potential energy of the Czochralski silicon single crystal pulling system; The expression for the potential energy of the Czochralski silicon single crystal pulling system is: (7) in, The potential energy of the Czochralski silicon single crystal pulling system is represented by the horizontal plane at the origin of the coordinate system, which is taken as the zero potential energy surface of the Czochralski silicon single crystal pulling system. Represents gravitational acceleration. Indicates the first Half the length of the flexible shaft at any given moment. Indicates the first The silicon crystal has been pulled to half its length at this point. Represents the relationship between the flexible axis and the spatial rectangular coordinate system. The angle along the axial direction, This indicates that the silicon crystal has been pulled out into a Cartesian coordinate system. Angle along the axial direction; By applying the second type of Lagrange equations, the Lagrange equations of the Czochralski silicon single crystal pulling system are constructed using the kinetic and potential energy of the Czochralski silicon single crystal pulling system. Solving the Lagrange equations yields the nonlinear equations of the Czochralski silicon single crystal pulling system. The Runge-Kutta method was used to solve the nonlinear equations to obtain the swing angles of the flexible shaft and the pulled-out silicon crystal. The changes in the swing angle of the flexible shaft and the pulled-out silicon crystal under different eccentricities were analyzed.
2. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 1, characterized in that, The steps for constructing the silicon crystal oscillation model of the Czochralski silicon single crystal pulling system under eccentric excitation include: Modeling is based on the following assumptions: Both the flexible shaft and the pulled silicon crystal are uniformly distributed in space, and their masses are equivalently concentrated at their respective centers of mass. The pulled-out silicon crystal is a rigid body; Both the flexible shaft and the pulled-out silicon crystal rotate at a constant angular velocity.
3. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 2, characterized in that, The silicon crystal oscillation model includes: A spatial rectangular coordinate system is constructed with the rotation center of the suspension point of the flexible shaft as the origin. The eccentricity of the suspension point of the flexible shaft from the origin is: Both the flexible shaft and the pulled-out silicon crystal move at a constant angular velocity. Rotate clockwise, the first The length of the flexible shaft at time is The mass per unit length of the flexible shaft is , No. The length of the silicon crystal pulled out at the specified time is The mass per unit length of the pulled silicon crystal is The lifting speed of the flexible shaft is The growth rate of the pulled silicon crystal is The flexible shaft and the spatial rectangular coordinate system The included angle along the axial direction is The pulled-out silicon crystal and the spatial rectangular coordinate system The included angle along the axial direction is .
4. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 3, characterized in that, The coordinates of the suspension point of the flexible shaft are: (1) in, Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the suspension point of the flexible shaft Axis coordinates Indicates the first The angular displacement of the suspension point of the flexible shaft at any given moment; The centroid coordinates of the flexible shaft are: (2) in, Indicating the centroid of the flexible shaft Axis coordinates Indicating the centroid of the flexible shaft Axis coordinates Indicating the centroid of the flexible shaft Axis coordinates Indicates the first Half the length of the flexible shaft at any given moment; The centroid coordinates of the pulled-out silicon crystal are: (3) in, This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates This indicates that the centroid of the silicon crystal has been pulled out. Axis coordinates Indicates the first The silicon crystal has been pulled out to half its length at this point; The expression for the linear velocity of the centroid of the flexible shaft is: (4) in, Indicating the centroid of the flexible shaft The linear velocity of the shaft, Indicating the centroid of the flexible shaft The linear velocity of the shaft, Indicating the centroid of the flexible shaft The linear velocity of the shaft, This indicates the rate of change of the swing angle of the flexible shaft over time. The expression for the linear velocity of the centroid of the pulled-out silicon crystal is: (5) in, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This indicates that the centroid of the silicon crystal has been pulled out. The linear velocity of the shaft, This represents the rate of change of the swing angle of the pulled-out silicon crystal over time.
5. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 3, characterized in that, The step of applying the second type of Lagrange equation to construct the Lagrange equation system of the Czochralski silicon single crystal pulling system using the kinetic and potential energies of the Czochralski silicon single crystal pulling system includes: The Lagrange quantity of the Czochralski silicon single crystal pulling system is determined using the kinetic and potential energy of the Czochralski silicon single crystal pulling system. The expression for the Lagrange quantity of the Czochralski silicon single crystal pulling system is: (8) in, This indicates that the Czochralski silicon single crystal pulling system is in the first... The generalized coordinates of time are Generalized speed is Lagrange quantity at time The generalized coordinates represent the Czochralski (CZ) silicon single crystal pulling system. This represents the generalized velocity of the Czochralski silicon single crystal pulling system. This represents the kinetic energy of the Czochralski silicon single crystal pulling system. This represents the potential energy of the Czochralski silicon single crystal pulling system; Take generalized coordinates , The transpose is represented, and combined with the Lagrangian quantity of the Czochralski silicon single crystal pulling system, the Lagrangian equation of the Czochralski silicon single crystal pulling system is constructed. The expression for the Lagrange equation of the Czochralski silicon single crystal pulling system is as follows: (9) in, This represents the partial derivative operation. Generalized coordinates representing the flexible axis. This indicates that the generalized coordinates of the silicon crystal have been extracted. Represents the generalized velocity of the flexible shaft. This represents the generalized speed at which the silicon crystal has been pulled out. The generalized damping force of the flexible shaft. This represents the generalized damping force that has been pulled out of the silicon crystal. for abbreviation; Therefore, the expression for the Lagrange equations of the Czochralski silicon single crystal pulling system is: (10) in, This represents the rate of change of the swing angle of the flexible shaft over time. This represents the rate of change of the swing angle of the pulled silicon crystal over time. , This represents the damping coefficient of the flexible shaft. , This indicates the damping coefficient of the silicon crystal that has been pulled out.
6. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 5, characterized in that, The process of solving the Lagrange equations to obtain the nonlinear equations of the Czochralski silicon single crystal pulling system includes: ； ； in, Indicates the first The silicon crystal has been pulled to half its length at this point. Indicates the first Half the length of the flexible shaft at any given moment.
7. The method for analyzing the oscillation of silicon crystals in the Czochralski silicon single crystal pulling system under eccentric excitation as described in claim 6, characterized in that, The step of solving the nonlinear equation using the Runge-Kutta method to obtain the swing angle of the flexible shaft and the pulled-out silicon crystal includes: exist The Runge-Kutta method is used in the simulation software to numerically solve the nonlinear equations. Numerical solutions of the generalized coordinates of the Czochralski silicon single crystal pulling system at each time point are obtained. All of the numerical solutions are the changes in the swing angle of the flexible shaft and the pulled silicon crystal over time.