A motion trajectory generation method for double-side grinding of a single crystal silicon wafer
By establishing a kinematic model of the double-sided polishing machine and optimizing its operating parameters, the motion trajectory of the silicon wafer was generated, solving the problem of surface non-uniformity of the silicon wafer and realizing high-quality double-sided polishing of single-crystal silicon wafers.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- MCL ELECTRONICS MATERIALS
- Filing Date
- 2026-05-06
- Publication Date
- 2026-06-05
AI Technical Summary
In existing double-sided polishing machines, improper setting of the rotation speed parameters of each rotating component leads to repetitive and uneven motion trajectories at various points on the silicon wafer, affecting processing quality and yield.
By establishing a kinematic model of a double-sided grinding machine, the motion trajectory parameter equations of any point on the silicon wafer are generated and discretized to generate a set of motion trajectory points. The coefficient of variation is calculated by combining the density matrix to evaluate the uniformity of the motion trajectory and optimize the operating parameters to achieve uniform grinding.
It achieves uniform grinding of silicon wafer surfaces, reduces total thickness variation and surface roughness, improves yield and consistency, and is suitable for silicon wafers and grinding machines of different specifications.
Smart Images
Figure CN122154243A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of double-sided grinding technology for monocrystalline silicon wafers, specifically a method for generating motion trajectories during double-sided grinding of monocrystalline silicon wafers. Background Technology
[0002] Double-sided grinding is a key process in semiconductor silicon wafer manufacturing, aiming to remove damaged layers from the silicon wafer surface, reduce surface roughness, and improve surface flatness. The uniformity during the grinding process directly determines the final processing quality and yield of the silicon wafer. Furthermore, as the feature size of integrated circuits continues to shrink, higher demands are placed on the surface quality and thickness uniformity of the silicon wafer.
[0003] Currently, double-sided grinding machines generally adopt planetary mechanisms and four-axis independent control methods, including an upper grinding disc, a lower grinding disc, a sun gear, planetary gears, and an internal gear ring. Among them, the planetary gears that carry the silicon wafers revolve around the center of the grinding machine and rotate on their own axes under the joint drive of the sun gear and the internal gear ring. At the same time, the silicon wafers are subjected to grinding action between the upper and lower grinding discs. Meanwhile, the upper grinding disc, lower grinding disc, sun gear, and internal gear ring are each driven by an independent motor, meaning that the speed of each of the four can be adjusted independently.
[0004] However, improper setting of the rotation speed parameters of each rotating component can lead to repetitive and uneven motion trajectories at various points on the silicon wafer, resulting in uneven removal of material from the silicon wafer surface and affecting processing quality and yield. Summary of the Invention
[0005] The purpose of this invention is to provide a method for generating motion trajectories for double-sided grinding of single-crystal silicon wafers, which can accurately simulate the motion trajectory of any point on the silicon wafer, thereby evaluating the uniformity of the motion trajectory and achieving uniform grinding of the silicon wafer surface.
[0006] To achieve the above objectives, the specific solution adopted by the present invention is as follows: a method for generating motion trajectories during double-sided grinding of a single-crystal silicon wafer, comprising the following steps: A kinematic model of the planetary gears in the double-sided grinding machine is established based on the operating parameters of the double-sided grinding machine: , , , in, This represents the angular velocity of the planetary gears' revolution. This represents the angular velocity of the planetary gear's rotation; This indicates the rotational speed of the sun gear in a double-sided grinding machine. This indicates the rotational speed of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the radius of the sun gear in a double-sided grinding machine; Indicates the radius of the planetary gears in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in the double-sided grinding machine, and k represents the slip coefficient. Based on the kinematic model, establish the trajectory parametric equations for any point P on the silicon wafer: , Where L represents the distance from the center of the planetary gear to the center of the double-sided grinding machine; This represents the average angle of motion of the grinding disc in a double-sided grinding machine; This indicates the sliding angle of the grinding disc in a double-sided grinding machine; This indicates the distance between the mounting position of the silicon wafer on the planetary gear and the center of the planetary gear; Discretize the trajectory parametric equations to generate a set of points representing the trajectory of any point P on the silicon wafer within a given time.
[0007] As an optimized scheme for the above-mentioned method of generating motion trajectory for double-sided grinding of single-crystal silicon wafers: k = 0.1~0.3.
[0008] As an alternative optimization of the motion trajectory generation method for the aforementioned double-sided grinding of single-crystal silicon wafers: the average motion angle of the grinding disc in the double-sided grinding machine is: , in, This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in a double-sided grinding machine.
[0009] As an alternative optimization of the motion trajectory generation method for double-sided grinding of single-crystal silicon wafers mentioned above: the distance from the center of the planetary gear to the center of the double-sided grinding machine: , in, This indicates the radius of the sun gear in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine.
[0010] As an alternative optimization scheme for the above-mentioned method of generating motion trajectory for double-sided grinding of single-crystal silicon wafers, the method of discretizing and solving the trajectory parameter equation to generate the point set of motion trajectory of any point P on the silicon wafer within a given time is as follows: set the total simulation time T and time step dt, and input the operating parameters of the double-sided grinding machine; calculate the revolution angular velocity, rotation angular velocity and sliding angular velocity of the planetary gear at each moment according to the time sequence, and substitute them into the trajectory parameter equation to obtain the trajectory parameters of any point P on the silicon wafer at each moment; connect all the obtained trajectory parameters into a line to generate the motion trajectory diagram of the silicon wafer within a given time.
[0011] A method for evaluating the uniformity of the motion trajectory during double-sided grinding of a single-crystal silicon wafer includes the following steps: The above generation method is used to generate a set of motion trajectory points for any point P in the silicon wafer; The working area of the silicon wafer is divided into an N×N uniform grid; Count the number of trajectory points falling into each grid and construct the density matrix D(i,j); The coefficient of variation is calculated based on the density matrix and compared with a preset threshold to evaluate the uniformity of the motion trajectory.
[0012] As an optimization scheme for the above-mentioned method for evaluating the uniformity of the motion trajectory during double-sided grinding of single-crystal silicon wafers, the method for calculating the coefficient of variation based on the density matrix is as follows: Calculate the average number of points per grid cell μ and the standard deviation σ based on the density matrix: Average number of points per grid μ = the average number of trajectory points in all grids. Standard deviation σ = Standard deviation of the number of trajectory points in all grids. The coefficient of variation is calculated based on the average number of points per grid μ and the standard deviation σ: .
[0013] As an alternative optimization of the above-mentioned method for evaluating the uniformity of the motion trajectory during double-sided grinding of single-crystal silicon wafers, the method for evaluating the uniformity of the motion trajectory by comparing the coefficient of variation with a preset threshold is as follows: When CV < 0.3, the judgment result is excellent; When 0.3 ≤ CV ≤ 0.5, the result is judged as good; When CV > 0.5, it is judged as poor.
[0014] A method for optimizing the operating parameters of double-sided grinding of monocrystalline silicon wafers, wherein the operating parameters include the rotational speed of the sun gear, the rotational speed of the internal gear ring of the sun gear, the rotational speed of the upper grinding disc, and the rotational speed of the lower grinding disc in the double-sided grinding machine. The method is used to obtain the coefficient of variation under different combinations of motion parameters, and the set of motion parameters with the smallest coefficient of variation is selected.
[0015] A motion trajectory generation system for double-sided grinding of monocrystalline silicon wafers, the motion trajectory system being used to implement the aforementioned motion trajectory generation method for double-sided grinding of monocrystalline silicon wafers, includes a module for establishing a kinematic model and trajectory parameters.
[0016] Compared with existing technologies, the present invention has the following beneficial effects: The present invention provides a method for generating motion trajectories for double-sided grinding of monocrystalline silicon wafers. A dynamic model is constructed based on the operating parameters of double-sided grinding, and a trajectory parameter equation for any point on the silicon wafer is obtained by synthesizing the equations based on the planetary gear revolution, rotation, and sliding effect caused by the difference in rotational speed between the upper and lower grinding discs. Solving this trajectory parameter equation accurately obtains the trajectory point set for any point on the silicon wafer. Based on the trajectory point set, the operating parameters are adjusted, thereby avoiding repetition and uneven distribution of motion trajectories at various points on the silicon wafer, achieving uniform grinding of the silicon wafer, significantly reducing the total thickness variation (TTV) and surface roughness of the silicon wafer, and improving yield and consistency. Furthermore, this method can be applied to silicon wafers and grinding machines of different specifications, exhibiting wide applicability. Attached Figure Description
[0017] Figure 1 This is the trajectory diagram in the application example. Detailed Implementation
[0018] The technical solution of the present invention will be further described in detail below with reference to specific embodiments. Parts not described or disclosed in detail in the following embodiments of the present invention should be understood as prior art known or should be known by those skilled in the art. Example 1
[0019] A method for generating motion trajectories for double-sided grinding of monocrystalline silicon wafers, wherein the double-sided grinding machine is a four-axis independently controlled double-sided grinding machine, including an upper grinding disc, a lower grinding disc, a sun gear, and an internal gear ring, and the upper grinding disc, lower grinding disc, sun gear, and internal gear ring are each controlled by an independent drive motor to achieve independent speed adjustment; planetary gears mesh with the sun gear and internal gear ring to support the silicon wafer to be processed. The motion trajectory generation method includes the following steps: A kinematic model of the planetary gears in the double-sided grinding machine is established based on the operating parameters, including the rotational speeds of the upper and lower grinding discs, the sun gear, and the internal gear ring. The kinematic model is constructed based on the gear meshing principle, specifically the revolution angular velocity, rotation angular velocity, and sliding angular velocity of the planetary gears. The revolution angular velocity of the planetary gears is determined by both the sun gear and the internal gear ring, and its calculation formula is as follows: .
[0020] The angular velocity of a planetary gear's rotation around its own center is determined by the difference in rotational speed between the sun gear and the internal ring gear, and its calculation formula is as follows: .
[0021] Considering the relative motion between the upper and lower grinding discs and the silicon wafer, when there is a difference in rotational speed between the upper and lower grinding discs, relative sliding will occur between the silicon wafer and the grinding discs. Therefore, the sliding angular velocity is introduced: .
[0022] in, This represents the angular velocity of the planetary gears' revolution. This represents the angular velocity of the planetary gear's rotation; This indicates the rotational speed of the sun gear in a double-sided grinding machine. This indicates the rotational speed of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the radius of the sun gear in a double-sided grinding machine; Indicates the radius of the planetary gears in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in the double-sided grinding machine. k represents the sliding coefficient, and the value of k ranges from 0.1 to 0.3. The specific value is determined based on the grinding disc material, the characteristics of the grinding fluid, and the workpiece.
[0023] The motion of any point P on the silicon wafer is decomposed into the synthesis of multiple motions: the revolution of the planetary gear center around the equipment center, the rotation of the silicon wafer with the planetary gear, and the sliding caused by the average rotation of the upper and lower grinding discs. Based on the kinematic model, the trajectory parametric equation of any point P on the silicon wafer is established: , Where L represents the distance from the center of the planetary gear to the center of the double-sided grinding machine; This represents the average angle of motion of the grinding disc in a double-sided grinding machine; This indicates the sliding angle of the grinding disc in a double-sided grinding machine; This indicates the distance between the mounting position of the silicon wafer on the planetary gear and the center of the planetary gear.
[0024] The average angle of motion of the grinding disc in the double-sided grinding machine is: , in, This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in a double-sided grinding machine.
[0025] Distance from the center of the planetary gear to the center of the double-sided grinder: , in, This indicates the radius of the sun gear in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine.
[0026] The trajectory parametric equations are discretized and solved to generate a set of points representing the trajectory of any point P on the silicon wafer within a given time period. Numerical computation software, such as MATLAB, Python, or Excel VBA, is used to discretize and solve the trajectory parametric equations. Specifically: the total simulation time T and time step dt are set to ensure trajectory continuity and control computational load, and the operating parameters of the double-sided grinding machine are input; the revolution angular velocity, rotation angular velocity, and sliding angular velocity of the planetary gears are calculated sequentially according to the time sequence, and substituted into the trajectory parametric equations to obtain the trajectory parameters X and Y of any point P on the silicon wafer at each time point. All the obtained trajectory parameters are connected into a line to generate a motion trajectory diagram of the silicon wafer within a given time period. The trajectory shape can be visually observed through the motion trajectory diagram. Example 2
[0027] A method for evaluating the uniformity of the motion trajectory during double-sided grinding of a single-crystal silicon wafer includes the following steps: The motion trajectory point set of any point P in the silicon wafer is generated using the generation method described in the embodiments; The working area of the silicon wafer is divided into an N×N uniform grid, where N is set according to the required calculation accuracy, for example, 10, 15 or 20. Grids that fall outside the circular boundary of the working area are ignored in subsequent statistics.
[0028] Count the number of trajectory points falling into each grid and construct the density matrix D(i,j); The coefficient of variation is calculated based on the density matrix and compared with a preset threshold to assess the uniformity of the motion trajectory. This index eliminates the influence of average density and can independently reflect the dispersion of the trajectory distribution, making it a core quantitative indicator for evaluating uniformity. Specifically: Calculate the average number of points per grid cell μ and the standard deviation σ based on the density matrix: Average number of points per grid μ = the average number of trajectory points in all grids. Standard deviation σ = Standard deviation of the number of trajectory points in all grids. The coefficient of variation is calculated based on the average number of points per grid μ and the standard deviation σ: .
[0029] The method for evaluating the uniformity of the motion trajectory by comparing the coefficient of variation with a preset threshold is as follows: When CV < 0.3, the judgment result is excellent; When 0.3 ≤ CV ≤ 0.5, the result is judged as good; When CV > 0.5, it is judged as poor. Example 3
[0030] A method for optimizing the operating parameters of double-sided grinding of monocrystalline silicon wafers, wherein the operating parameters include the rotational speed of the sun gear, the rotational speed of the internal gear ring of the sun gear, the rotational speed of the upper grinding disc, and the rotational speed of the lower grinding disc in the double-sided grinding machine. The operating parameters are optimized by a grid search method. Specifically, the method described in Example 2 is used to obtain the coefficient of variation under different combinations of motion parameters, and the set of motion parameters with the smallest coefficient of variation is selected.
[0031] In other embodiments, operating parameters are optimized based on irrational speed ratios. Specifically, when the ratio of the planetary gear's revolution angular velocity to its rotation angular velocity, and the ratio of the grinding disc's average rotational speed to its revolution angular velocity, are close to irrational numbers, the silicon wafer's motion trajectory is less prone to repetition, resulting in better uniformity. Specifically: Two key speed ratios are defined: Revolution-rotation speed ratio γ1= / , Grinding disc-revolution speed ratio γ2= / , Adjusting the speed of the sun gear with γ1 as the primary target. and the speed of the internal gear ring This makes γ1 approach the target irrational number. Then, fine-tune the rotation speeds of the upper and lower grinding discs. , This makes γ2 approach an irrational number.
[0032] Verification: Extend the simulation time and observe whether the trajectory repeats itself. If there is no obvious periodicity, the adjustment is complete. Example 4
[0033] A sensitivity analysis method for double-sided polishing of monocrystalline silicon wafers is disclosed. The operating parameters include the rotational speed of the sun gear, the rotational speed of the internal gear ring of the sun gear, the rotational speed of the upper polishing disc, and the rotational speed of the lower polishing disc in the double-sided polishing machine. With the other three rotational speed parameters fixed, one of the rotational speed parameters is changed within a set range, and the trend of the coefficient of variation (CV) is observed. For example, by fixing the rotational speeds of the sun gear, the internal gear ring of the sun gear, and the upper polishing disc in the double-sided polishing machine, and changing the rotational speed of the lower polishing disc within a set range, the CV value is obtained using the method described in the examples.
[0034] Application examples A four-axis independently controlled double-sided grinding machine is used, with a sun gear radius of... The radius of the sun gear internal gear ring is 503mm. The radius of the planetary gear is 1453mm. The silicon wafer mounting position is 557mm. The diameter is 312mm, and the sliding coefficient k is 0.14; initial operating parameters are set as follows: rotational speed of the upper grinding disc. The speed of the lower grinding disc is 14.5 rpm. The sun gear speed is 30 rpm. The speed of the internal gear ring of the sun gear is 7 rpm. The angular velocity of the planetary gears is 15 rpm. Discretization is performed using MATLAB, with a simulation time T of 60 s and a step size dt of 0.1 s, to obtain the revolution angular velocity of the planetary gears. The rotational angular velocity of the planetary gears is 1.35 rad / s. The value is -0.38 rad / s; the trajectory plot is displayed as a complex petal shape.
[0035] Using a 10×10 grid for counting, the coefficient of variation (CV) was 0.28, indicating a good uniformity rating. The actual TTV of the silicon wafer after polishing was less than 1 μm.
[0036] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims
1. A method for generating the motion trajectory of double-sided grinding of a single-crystal silicon wafer, characterized in that, Includes the following steps: A kinematic model of the planetary gears in the double-sided grinding machine is established based on the operating parameters of the double-sided grinding machine: ， ， ， in, This represents the angular velocity of the planetary gears' revolution. This represents the angular velocity of the planetary gear's rotation; This indicates the rotational speed of the sun gear in a double-sided grinding machine. This indicates the rotational speed of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the radius of the sun gear in a double-sided grinding machine; Indicates the radius of the planetary gears in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine; This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in the double-sided grinding machine, and k represents the slip coefficient. Based on the kinematic model, establish the trajectory parametric equations for any point P on the silicon wafer: ， Where L represents the distance from the center of the planetary gear to the center of the double-sided grinding machine; This represents the average angle of motion of the grinding disc in a double-sided grinding machine; This indicates the sliding angle of the grinding disc in a double-sided grinding machine; This indicates the distance between the mounting position of the silicon wafer on the planetary gear and the center of the planetary gear; Discretize the trajectory parametric equations to generate a set of points representing the trajectory of any point P on the silicon wafer within a given time.
2. The method for generating motion trajectories during double-sided grinding of a single-crystal silicon wafer as described in claim 1, characterized in that: The value of k is 0.1 to 0.
3.
3. The method for generating motion trajectories during double-sided grinding of a single-crystal silicon wafer as described in claim 1, characterized in that: The average angle of motion of the grinding disc in the double-sided grinding machine is: ， in, This indicates the rotational speed of the upper grinding disc in a double-sided grinding machine. This indicates the rotational speed of the lower grinding disc in a double-sided grinding machine.
4. The method for generating motion trajectories during double-sided grinding of a single-crystal silicon wafer as described in claim 1, characterized in that: Distance from the center of the planetary gear to the center of the double-sided grinder: ， in, This indicates the radius of the sun gear in a double-sided grinding machine; This indicates the radius of the internal gear ring of the sun gear in a double-sided grinding machine.
5. The method for generating motion trajectories during double-sided grinding of a single-crystal silicon wafer as described in claim 1, characterized in that: The method for discretizing and solving the trajectory parametric equation to generate the point set of the motion trajectory of any point P on the silicon wafer within a given time is as follows: Set the total simulation time T and time step dt, and input the operating parameters of the double-sided polishing machine; calculate the revolution angular velocity, rotation angular velocity and sliding angular velocity of the planetary gears at each moment according to the time sequence, and substitute them into the trajectory parametric equation to obtain the trajectory parameters of any point P on the silicon wafer at each moment; connect all the obtained trajectory parameters into a line to generate the motion trajectory diagram of the silicon wafer within a given time.
6. A method for evaluating the uniformity of the motion trajectory during double-sided grinding of a single-crystal silicon wafer, characterized in that, Includes the following steps: The motion trajectory point set of any point P in a silicon wafer is generated using the generation method described in any one of claims 1-5; The working area of the silicon wafer is divided into an N×N uniform grid; Count the number of trajectory points falling into each grid and construct the density matrix D(i,j); The coefficient of variation is calculated based on the density matrix and compared with a preset threshold to evaluate the uniformity of the motion trajectory.
7. The method for evaluating the uniformity of the motion trajectory during double-sided grinding of a single-crystal silicon wafer as described in claim 6, characterized in that, The method for calculating the coefficient of variation based on the density matrix is as follows: Calculate the average number of points per grid cell μ and the standard deviation σ based on the density matrix: Average number of points per grid μ = the average number of trajectory points in all grids. Standard deviation σ = Standard deviation of the number of trajectory points in all grids. The coefficient of variation is calculated based on the average number of points per grid μ and the standard deviation σ: 。 8. A method for evaluating the uniformity of the motion trajectory during double-sided grinding of a single-crystal silicon wafer as described in claim 6, characterized in that, The method for evaluating the uniformity of the motion trajectory by comparing the coefficient of variation with a preset threshold is as follows: When CV < 0.3, the judgment result is excellent; When 0.3 ≤ CV ≤ 0.5, the result is judged as good; When CV > 0.5, it is judged as poor.
9. A method for optimizing the operating parameters of double-sided grinding of monocrystalline silicon wafers, wherein the operating parameters include the rotational speed of the sun gear in the double-sided grinding machine, the rotational speed of the internal gear ring of the sun gear in the double-sided grinding machine, the rotational speed of the upper grinding disc in the double-sided grinding machine, and the rotational speed of the lower grinding disc in the double-sided grinding machine, characterized in that: The coefficient of variation for different combinations of motion parameters is obtained using the method described in claim 6, and the set of motion parameters with the smallest coefficient of variation is selected.
10. A motion trajectory generation system for double-sided grinding of a single-crystal silicon wafer, the motion trajectory system being used to implement the motion trajectory generation method for double-sided grinding of a single-crystal silicon wafer according to any one of claims 1-5, characterized in that: It includes modules for building kinematic models and trajectory parameters.