CN113158518A - Method for solving stress of thin substrate - Google Patents
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Abstract
The invention belongs to the technical field of precision measurement and analysis, and particularly relates to a method for solving stress of a thin substrate, which comprises the following steps: s1: obtaining basic parameters of a thin substrate to be analyzed for stress; s2: establishing a finite element model of the thin substrate; s3: applying unit stress load to each surface in the model in sequence, solving the finite element model, and outputting a solving result; s4: obtaining the coordinates of the central points of the surfaces applying unit stress loads in the finite element model, and further constructing a matrix L containing the distance relationship of the central points among the surfaces; s5: acquiring the surface shape of a real thin substrate, and drawing an accepting and rejecting curve; s6: and solving a linear equation by adopting a regularization method, and calculating to obtain the real stress distribution of the thin substrate. The method solves the problems of discontinuous stress distribution and large fluctuation of the thin substrate caused by directly solving an equation set, and simultaneously does not introduce additional physical damage in the process of obtaining the stress distribution.
Description
Technical Field
The invention belongs to the technical field of precision measurement and analysis, and particularly relates to a method for solving stress of a thin substrate.
Background
The thin substrate is a mechanical thin plate, and the geometric characteristic is a circle, and the thickness dimension of the thin substrate is far smaller than the plane dimension. At present, 300mm large-size silicon wafers with the thickness less than 0.2mm are widely used. In the processing process, the surface of the processed silicon wafer can generate residual stress due to the difference of the temperatures of the upper surface and the lower surface. The residual stress can cause damage to the sub-surface, and the larger the damage, the larger the residual stress, so the damage to the sub-surface of the silicon wafer can be characterized by the residual stress. At present, in practical application, a laser triangulation meter and a laser interferometer can be used for measuring the deformation of the silicon chip, the error is in a micron or nanometer level, and the obtained precision is relatively high.
The residual stress detection method is divided into a damage method and a non-damage method. The most commonly used destructive method is the drilling method, which damages the surface of the silicon wafer. The most used nondestructive method is the X-ray diffraction method, which does not damage the surface of the silicon wafer, but has obvious stress gradient in the X-ray penetration depth range when measuring the silicon wafer, and for the non-planar stress state (three-way stress state), the X-ray diffractometer can only measure the surface stress state, so that the common X-ray instrument is not suitable for detecting the residual stress of the sub-surface of the silicon wafer. Generally, an X-ray diffractometer is only suitable for polycrystalline materials in stress measurement, the lattice spacing and the grain orientation inside single crystal materials are uniform, and when the incident angle changes continuously, only a single angle can meet the bragg diffraction condition to form data, and the stress value cannot be solved through fitting.
Indian physicists C.V.Raman and K.S.Krishnan find the Raman effect based on the principle of inelastic scattering of light, and the Raman spectrometer based on the Raman spectrum can respectively reflect the states of the tensile stress and the compressive stress through the left and right frequency shifts of a Raman peak. However, in actual measurement, the raman frequency shift is still affected by the temperature such as the depth of focus, laser heating effect, etc., and has not very high accuracy in the absence of an effective calibration criterion; and because the penetration depth of the light wave is limited, the stress state of the most surface layer is obtained, the micro-Raman measurement result reflects the microstructure change of the processing damage, and the fluctuation of the measurement values at different positions is relatively large.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method for solving the stress of a thin substrate, which can accurately obtain the residual stress of the thin substrate, and simultaneously does not introduce additional physical damage in the process of obtaining the stress distribution.
The invention is realized by adopting the following technical scheme:
a method of resolving thin substrate stress, the method comprising the steps of:
s1: obtaining basic parameters of a thin substrate to be analyzed for stress; the base parameters include diameter, thickness, modulus of elasticity, shear module, poisson's ratio, and density;
s2: establishing a finite element model with the same basic parameters as the thin substrate to be analyzed; the method for constructing the finite element model comprises the following steps:
s21: defining material properties and unit types;
s22: establishing a finite element model of the thin substrate;
s23: dividing meshes for the established finite element model;
s24: adding constraints to the nodes in the finite element model;
s3: sequentially applying unit stress load to each surface in the finite element model, and then solving the finite element model; defining an output path, and mapping the solving result of each surface applying unit stress load in the model to the path; outputting the solving results of all the surfaces in the model;
s4: according to the solving result of the finite element model, obtaining a surface applying unit stress load in the model and an adjacent surface thereof, and further constructing a matrix L containing the distance relation of central points among the surfaces;
s5: measuring the thin substrate to obtain the surface shape of the real thin substrate, constructing a matrix b by a column vector consisting of displacements of a plurality of real thin substrate measuring points, and recording a matrix W consisting of the displacements of the plurality of measuring points under the action of each surface applying unit stress load;
s6: solving a linear equation set by adopting a regularization method to meet the requirement
And minimum. Punishment item by regularization method when drawing different mu values
As the abscissa, in
An accept-reject curve of a vertical coordinate; mu represents the coefficient of penalty term of the regularization method, x is the vector to be solved and is formed by alphajForming an n-dimensional column vector;
the actual stress distribution of the thin substrate is calculated according to the following formula:
σj＝αjσu
in the above formula, σjRepresenting the actual stress of the j-th surface of the thin substrate; sigmauRepresenting the unit stress load value of each surface in the finite element model;
further, the detailed procedure of step S21 is as follows:
firstly, defining unit numbers and selecting the types of shell units, and then sequentially setting the densities, elastic models, Poisson ratios and shear moduli of the materials of the first material layer and the second material layer; and finally, sequentially defining the thicknesses of the first material layer and the second material layer.
Further, the detailed procedure of step S22 is as follows:
firstly, setting a coordinate system in a model as a cylindrical surface coordinate system; generating key points in a fan-shaped area with the length equal to 1/3 circumference in the model, connecting the generated internal key points by straight lines, and connecting the external key points by circular arcs, so that the fan-shaped area is divided into a plurality of quadrangles; then selecting all lines in the circumference, generating equally divided key points on each line, and further dividing the quadrangle into smaller quadrangles; then generating key points for the rest 2/3 parts in the model according to the same method, connecting the key points, and combining repeated key points; and finally, selecting an arc line of the edge in the model to generate a circular surface.
Further, the detailed procedure of step S23 is as follows:
firstly, selecting the grid type of a model as quadrilateral grid division, and designating a grid division method as mapping grid division; then selecting all lines except the circumference in the model, forming the selected lines into a line group, dividing the circular surface generated in the previous step by using the line group, and bonding the divided surfaces together; deleting the previous line group, selecting all lines in the model, and recombining the selected lines into a new line group; and finally, controlling the grid density through the new line group, and performing grid division and generating units and nodes.
Further, the detailed procedure of step S24 is as follows:
selecting three nodes to impose constraints, limiting X, Y, Z directional movement for the first node; limiting movement in the X, Z directions for a second node; the movement in the Z direction is restricted for the third node.
Further, the specific process of step S3 is as follows:
s31: acquiring the total number of surfaces of the model;
s32: setting an initial coordinate system as a cylindrical surface coordinate system, applying unit stress load on the first material layer, selecting shell units related to the surface with the minimum number, and defining the numerical value of the unit stress load on the shell units;
s33: solving the model;
s34: reading a solving result, creating a cylindrical coordinate system, creating a circular path according to 2 points in the cylindrical coordinate system, setting mapping items, and phasing a fractional segment between two adjacent points;
s35: defining an array, mapping the displacement in the Z direction in the analysis and calculation result to the created path, acquiring data in the path and storing the data in the array;
s36: sequentially establishing a plurality of paths from inside to outside in the model so as to ensure that the sampling points are uniformly distributed; meanwhile, assigning the array corresponding to each path to a new array, so that the new array contains the position and displacement information of all previous sampling points;
s37: defining a new array, respectively assigning the X coordinate, the Y coordinate and the displacement in the Z direction of the sampling point in the array in the step S36 to the newly defined array, and outputting the data in the array;
s38: repeating the steps S31-S37 to load the unit stress load on all the surfaces in the model in sequence, wherein the unit stress load is loaded on only one surface of the model when the load is loaded, the unit stress load loaded on the previous surface is deleted before the load is loaded, and the output result of the array in each loaded surface is obtained in sequence.
Further, the specific process of step S4 is as follows:
s41: defining a new empty array, selecting all entities in all models, and acquiring the total number of the models;
s42: creating a new two-dimensional array, setting the row number of the two-dimensional array as the total number of the surfaces of the model obtained in the step S41, setting the column number as 4, and setting the first column of the two-dimensional array as 1 to the total number of the surfaces;
s43, selecting the surface with the smallest number in the model, calculating the geometric data of the surface with the smallest number, obtaining the X, Y, Z coordinate of the central point of the surface with the smallest number, and assigning the coordinate values to the 2 nd, 3 rd and 4 th columns of the two-dimensional arrays; assigning X, Y, Z coordinates of the central points of all the surfaces applying the load to each row of the two-dimensional array in sequence; finally, outputting data in the two-dimensional array;
s44: performing numerical analysis on the output result of the previous step, forming a matrix containing the serial number of the surface and the central point coordinate of the surface based on the analysis result, and acquiring the number of rows of the matrix;
s45: creating a full 0 matrix, wherein the row number of the matrix is the total surface number of the model, and the column number is 5;
s46: assigning the first column element of the matrix in step S44 to the first column of the matrix created in step S45; sequentially calculating the distances from the center point of the first surface of the model corresponding to the matrix to the center points of all the surfaces through the matrix in the step S44, and assigning the calculation result to the fourth column of the matrix in the step S44;
s47: arranging the assigned matrixes in the step S46 in ascending order of a fourth column, and then assigning temporary variables to the arranged matrixes to form a new matrix;
s48: assigning the elements of the first column of the 2 nd, 3 rd, 4 th and 5 th rows of the new matrix created in the step S47 to the 2 nd, 3 rd, 4 th and 5 th columns of the first row of the matrix in the step S45, respectively; sequentially calculating the distances from the center points of the remaining surfaces to the center points of all the surfaces according to the method in the S46, and finishing the assignment operation of all rows of the matrix in the step S45;
s49: creating a square matrix or a unit matrix L, wherein the dimension of the matrix L is the total area number of the model; defining elements on a main diagonal in a matrix L to be 4; assigning values to the 1 st row of the matrix L, and assigning values to-1 to the elements in the 1 st row of the matrix L corresponding to the sequence numbers of the columns of the matrix L and the element values of the 2, 3, 4 and 5 columns of the 1 st row of the matrix processed in the step S48; and sequentially finishing the assignment operation of all the rows in the matrix L according to the same mode.
Further, the step of step S5 is as follows:
s51: creating a new matrix, wherein the number of rows of the new matrix is the number of measuring points in the thin substrate surface shape obtaining process; the column number is the total number of the surfaces applying the load in the finite element model, and the 3 rd column data in the result output in the step S38 is sequentially imported into the created new matrix according to columns to complete the construction of the W matrix;
s52: obtaining the surface shape of a real thin substrate by using a laser triangular displacement measuring instrument, coinciding a measuring coordinate system with a finite element coordinate system to enable the Z-direction displacement component of a point in the measuring coordinate system and a corresponding constraint point of the finite element to be 0, obtaining a measuring point of the real thin substrate matched with a displacement extraction point of the finite element model through surface shape interpolation, and obtaining X, Y and Z coordinates of the measuring point to obtain a b matrix;
further, the step of step S6 is as follows:
s61: defining a variable mu as a coefficient of a penalty term of the regularization method, wherein the variable variation range is larger than 0; given a μ, x satisfies the following equation:
x＝(WTW+μL)-1WTb，
in the above formula, the first and second carbon atoms are,Trepresents a transpose of a matrix; w represents the matrix constructed in step S51; b represents the matrix obtained in step S52;
then in order to correspond to mu
Is shown as the abscissa of the graph,
is a vertical coordinate; drawing a rounding curve fitted by scattered points with mu larger than 0;
s62: and selecting mu corresponding to the point at the turning point of the alternative curve, calculating x, and selecting the final mu value according to the difference value between the surface shape calculated by x and the measured surface shape and the stress distribution.
The method provided by the invention adopts the following technical ideas and utilized principles:
in elastic mechanics, the assumption of small deformation is that after an object is stressed, the displacement of each point in the object is far smaller than the original size of the object, and the strain and the rotation angle are far smaller than 1. Thus, the geometry before deformation can be used to replace the dimensions after deformation to establish the equilibrium equation. Furthermore, the second order trace of strain is negligible, thereby linearizing the geometric equation.
The rigidity of the material of the thin substrate is unchanged under the condition of small deformation, and the displacement of each unit stress load at a measuring point meets a linear superposition relation. The laser triangular displacement sensor can be used for obtaining the surface shape of a real silicon chip, and the Z-direction displacement of each measuring point is interpolated and recorded as biAnd the Z-direction displacement value calculated at the corresponding measuring point by the model established by the finite element is recorded as WijWhere, subscript i 1 to m represents the number of measurement points, and j 1 to n represents a plane to which a unit stress load is applied.
Under the condition of linear deformation, the silicon wafer deformation meets Hooke's law, and the displacement is linearly changed along with the force. The condition that the thickness of the thin substrate is consistent in the real thin substrate and the finite element modelUnder the condition that the real rigidity of each measuring point is kept consistent with the rigidity of the finite element data point, the experimentally measured displacement of the ith point of the thin substrate and the displacement of the measuring point at the corresponding position under the finite element model have the following relation: bi＝αjWijIn which α isj＝σj/σu。σuRepresenting the unit stress load, σ, in the finite element modeljRepresenting the actual stress on the jth face of the thin substrate corresponding to the finite element model.
Sequentially applying unit stress load to all n surfaces in the finite element model to achieve the same effect as the stress condition of the real silicon wafer, and measuring the displacement b of the points by using m real silicon wafersiThe column vector formed is denoted b, the displacements of the m measurement points under the action of the n surfaces to which the unit stress loads are applied form an m x n matrix denoted W, alphajAn n-dimensional column vector x is formed and is a quantity to be solved, and the b vector and the W array can be obtained by experimental measurement and a finite element model respectively. The two columns of the W matrix are approximately proportional when the two planes applying the initial stress are very close together, and the W matrix is a ill-conditioned matrix. In the practical application process, the experimental measured value biUnavoidable errors, W matrix being ill-conditioned matrix will cause alphajAnd a stable solution cannot be obtained due to great disturbance.
The technical scheme adopted by the invention is to adopt a regularization method on the basis of a least square method and introduce a penalty term
The problem of W array collinearity is eliminated, and the influence of a ill-conditioned matrix is improved; namely, it is
Wherein each value W in the W matrixijNamely, the displacement value obtained by the finite element is accurate, namely W is an accurate value, when mu is 0, x is the solution of the least square method, but the matrix W has the problem of collinearity, the condition number of the W array is very large, the fluctuation of the solved value x is large, and when mu is too large, the difference value between the solved surface shape and the real surface shape is large.
L is an n multiplied by n matrix, the first access of the L matrix is a unit matrix, and the second access is as follows: and sequentially setting the elements in the u-th row and the i-th column as 4, wherein the elements in the i-th row correspond to the number of the adjacent surfaces of the load-applying surface with the unit stress, the elements in the i-th row corresponding to the number of the adjacent surfaces of the load-applying surface with the unit stress are-1, and the other elements are 0.
Will be provided with
The vector x is derived to yield:
the preceding formula is equivalent to x ═ WTW+μL)-1WTb, the stress of the j-th surface of the thin substrate can be obtained by using the calculated x vector: sigmaj＝αjσuAnd further obtaining the stress distribution of the real measurement silicon wafer.
The method for solving the stress of the thin substrate has the following beneficial effects:
1. the invention combines experimental measurement data with finite element simulation data, and solves the residual stress on the surface of the thin substrate by introducing physical constraint. The invention is a non-contact nondestructive residual stress acquisition method, does not introduce additional physical damage in the whole process, can accurately evaluate the processing technology quality of the thin substrate and optimize the processing technology parameters.
2. In the measuring method provided by the invention, the problem of W matrix collinearity is eliminated by adopting a regularization method to construct an L matrix, the stress continuously distributed in the thin substrate is obtained, the problem of discontinuous stress distribution of the thin substrate caused by directly solving an equation set by a least square method is solved, the continuity of stress distribution of a silicon wafer is improved, the problem of large stress fluctuation caused by directly solving the equation set is obviously reduced, and the continuous stress distribution in the thin substrate can be accurately obtained.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a simplified flowchart of a method for solving stress of a thin substrate in example 1 of the present invention;
FIG. 2 is a view showing a deformation in the Z direction of a finite element model in which a unit stress load is applied to a single surface of the finite element model in embodiment 1 of the present invention;
FIG. 3 is a Z-direction deformation diagram obtained by interpolation of the silicon wafer surface shape measured in example 1 of the present invention;
FIG. 4 is a rounding curve drawn by the continuous constraint regularization method in embodiment 1 of the present invention;
FIG. 5 is a Z-direction deformation diagram of a silicon wafer solved by using an L matrix of a continuous constraint structure with μ being 300 in example 1 of the present invention;
FIG. 6 is a diagram illustrating a difference between Z-direction deformation of a silicon wafer and an actually measured surface shape of the silicon wafer, where μ is 300, and a L matrix of a continuous constraint structure is used to solve in example 1 of the present invention;
FIG. 7 is a silicon wafer stress distribution diagram obtained by constructing an L-matrix solution linear equation set by using a continuous constraint method with μ being 300 in embodiment 1 of the present invention;
FIG. 8 is a Z-direction deformation diagram of a silicon wafer obtained by solving equations by the least square method in example 1 of the present invention;
FIG. 9 is a graph showing the difference between the silicon wafer surface shape obtained by the least square method and the measured silicon wafer surface shape in example 1 of the present invention;
FIG. 10 is a stress distribution diagram of a silicon wafer obtained by the least square method before the normalization in example 1 of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
The embodiment provides a method for solving stress of a thin substrate, as shown in fig. 1, the technical solution can be summarized as follows: finite element simulation and regularization methods solve two processes of a linear equation set.
Wherein the finite element simulation comprises: defining material property and unit type, building geometric model, dividing grid, adding constraint, applying load to solve, defining path and outputting solving result. The regularization method solves a system of linear equations comprising: constructing an L matrix, selecting a regularization method to punish a term coefficient mu, drawing a rounding curve, performing matrix operation to solve an x vector, and calculating the stress of the thin substrate.
In detail, the method provided by the embodiment includes the following steps:
s1: obtaining basic parameters of a thin substrate to be analyzed for stress; the basic parameters comprise a diameter of 200mm, a thickness of 0.327mm, and three coordinate axes x, y and z of 100],[010],[001]Modulus of elasticity Ex＝Ey＝Ez130GPa, shear modulus Gyz＝Gzx＝Gxy79.6GPa and the Poisson's ratio vyz＝νzx＝νxy0.28 and a density of 2329kg/m3；
S2: establishing a finite element model with the same basic parameters as the thin substrate of the stress to be analyzed in finite element analysis software; the method for constructing the finite element model comprises the following steps:
s21: defining material properties and unit types; the detailed process is as follows:
firstly defining the number of a unit as 1, selecting the type of a shell unit as shell181, and then sequentially setting the density, the elastic modulus, the Poisson ratio and the shear modulus of the materials of a first material layer and a second material layer to be the same as those of a thin substrate to be analyzed; finally, the thickness of the first material layer is 1 μm and the thickness of the second material layer is 0.326 mm.
S22: establishing a geometric model of the thin substrate; the detailed process is as follows:
firstly, setting a coordinate system in a model as a cylindrical surface coordinate system; generating key points in a fan-shaped area with the length equal to 1/3 circumference in the model, connecting the generated internal key points by straight lines, and connecting the external key points by circular arcs, so that the fan-shaped area is divided into 7 quadrangles; then all lines in the circumference are selected, 4 equally-divided key points are produced on each line, and the quadrangle is further divided into smaller quadrangles; then generating key points for the rest 2/3 parts in the model according to the same method, connecting the key points, and combining repeated key points; and finally, selecting an arc line of an edge in the model to generate a circular surface, and selecting an internal line to divide the circular surface into 336 quadrilateral surfaces.
S23: dividing meshes for the established finite element model; the detailed process is as follows:
firstly, selecting the grid type of a model as quadrilateral grid division, and designating a grid division method as mapping grid division; then selecting all lines except the circumference in the model, forming the selected lines into a line group, dividing the circular surface generated in the previous step by using the line group, and bonding the divided surfaces together; deleting the previous line group, selecting all lines in the model, and recombining the selected lines into a new line group; and finally, controlling the grid density to divide each line into 6 equal parts through a new line group, and carrying out grid division.
S24: adding constraints to the nodes in the finite element model; the detailed process is as follows:
firstly, selecting a node with the radial size of 2/3 radiuses and the coincidence of the circumferential direction and the positive direction of an X axis in a coordinate mode under a cylindrical surface coordinate system, applying constraint on the node and limiting the movement in the X, Y, Z direction; then, selecting a node with the radial size of 2/3 radiuses and the circumferential direction of 120 degrees with the positive direction of the X axis in a coordinate mode under a cylindrical surface coordinate system, and applying constraint to the node to limit the movement in the X and Z directions; and finally, selecting a node with the radial size of 2/3 radiuses and the circumferential direction of 240 degrees with the positive direction of the X axis in a coordinate mode under a cylindrical surface coordinate system, and applying constraint to the node to limit the movement in the Z direction.
S3: in finite element analysis software, sequentially applying unit stress load to each surface in a finite element model, and then solving the finite element model; defining an output path, and mapping a solving result of unit stress load applied to each surface in the model to the path; outputting the solving results of all the surfaces in the model; the specific process is as follows:
s31: obtaining 336 total faces of the model;
s32: setting an initial coordinate system as a cylindrical coordinate system, applying unit stress load on the second material layer, selecting related units of the first surface, and defining the numerical value of the unit stress on the units as 100 MP;
s33: solving the model;
s34: reading a solving result, creating a cylindrical coordinate system, and creating 37 circular paths according to 2 points in the cylindrical coordinate system;
s35: defining an array, mapping the displacement in the Z direction in the analysis and calculation result to the created path, acquiring data in the path and storing the data in the array;
s36: defining a new array, respectively assigning the X coordinate, the Y coordinate and the displacement in the Z direction of the sampling point in the array in the step S36 to the newly defined array, and outputting the data in the array; the output of the two-dimensional array is a text file;
s38: using a loop, with an initial value of 1, a final value of 336, a step size of 1, repeating steps S31-S36 to sequentially load the unit stress loads on all the faces in the model, wherein each time a load is loaded, only one of the faces in the model is loaded, the load loaded on the previous face is deleted before loading, and the output results of the arrays in each loaded face are sequentially obtained. The output result is output in a text file mode, 336 text files are output in total, and 1 column, 2 columns and 3 columns in each text file are respectively the X coordinate, the Y coordinate and the displacement value in the Z direction of the point on the path. The Z-direction deformation diagram of the finite element model with unit stress load applied to a single surface is shown in FIG. 2.
S4: acquiring a central point X coordinate, a central point Y coordinate and a central point Z coordinate of a surface applying unit stress load in a model from finite element analysis software, and further constructing a matrix L in a regularization method through Matlab; the specific process is as follows:
s41: defining a new empty array, selecting all entities in all models, and obtaining the total number of faces of the models 336;
s42: creating a new two-dimensional array, setting the row number of the two-dimensional array as the total number of the surfaces of the model obtained in the step S41, setting the column number as 4, and setting the first column of the two-dimensional array as 1 to 336;
s43, selecting the surface with the smallest number in the model, calculating the geometric data of the surface with the smallest number, obtaining the X, Y, Z coordinate of the central point of the surface with the smallest number, and assigning the coordinate values to the 2 nd, 3 rd and 4 th columns of the two-dimensional arrays; assigning X, Y, Z coordinates of the central points of all the surfaces applying the load to each row of the two-dimensional array in sequence; finally, outputting data in the two-dimensional array;
s44: performing numerical analysis on the output result of the previous step, and forming a matrix containing the serial number of the surface and the coordinates of the center point of the surface on the basis of the analysis result, wherein the number of rows is 336 and the number of columns is 4;
s45: creating an all 0 matrix with the number of rows 336 and the number of columns 5;
s46: assigning the first column element of the matrix in step S44 to the first column of the matrix created in step S45; sequentially calculating the distances from the center point of the first surface of the model corresponding to the matrix to the center points of all the surfaces through the matrix in the step S44, and assigning the calculation result to the fourth column of the matrix in the step S44;
s47: arranging the assigned matrixes in the step S46 in ascending order of a fourth column, and then assigning temporary variables to the arranged matrixes to form a new matrix;
s48: assigning the elements of the first column of the 2 nd, 3 rd, 4 th and 5 th rows of the new matrix created in the step S47 to the 2 nd, 3 rd, 4 th and 5 th columns of the first row of the matrix in the step S45, respectively; sequentially calculating the distances from the center points of the remaining surfaces to the center points of all the surfaces according to the method in the S46, and finishing the assignment operation of all rows of the matrix in the step S45;
s49: creating a square matrix L, the dimension 336 of said matrix L; defining elements on a main diagonal in a matrix L to be 4; assigning values to the 1 st row of the matrix L, and assigning values to-1 to elements in the 1 st row of the matrix L corresponding to element values of 2, 3, 4 and 5 columns of the 1 st row of the matrix processed in the sequence number S48; and sequentially finishing the assignment operation of all the rows in the matrix L according to the same mode.
S5: interpolating the measured surface shape of the thin substrate to obtain a column vector consisting of Z-direction displacements of 4218 measuring points, recording the column vector as b, and recording a displacement composition matrix of the measuring points under the action of each surface applying unit stress load as W;
the method comprises the following specific steps:
s51: creating a new matrix, the number of rows 4218 of the new matrix; the number of columns is 336, and the 3 rd column data in the result output in the step S38 is sequentially imported into the created new matrix by columns, so that the structure of the W matrix is completed;
s52: obtaining the surface shape of the real thin substrate by using a laser triangular displacement measuring instrument, and coinciding the measuring coordinate system with the finite element coordinate system to ensure that the Z-direction displacement component of a point in the measuring coordinate system and a corresponding constraint point of the finite element is 0, and connecting
And (3) obtaining measurement points of the real thin substrate matched with the finite element model displacement extraction points through surface shape interpolation, and obtaining X, Y and Z coordinates of the measurement points so as to obtain a b vector, wherein the number of rows is 4218, and the number of columns is 1. The Z-direction distortion of the thin substrate profile by interpolation is shown in fig. 3.
S6: solving a linear equation set by adopting a regularization method to meet the requirement
And minimum. Punishment item by regularization method when drawing different mu values
As the abscissa, in
An accept-reject curve of a vertical coordinate; and selecting mu at the turning point of the rounding curve, calculating a vector x, and finally calculating the residual stress of each surface of the thin substrate.
The actual stress distribution of the thin substrate is calculated according to the following formula:
σj＝αjσu
in the above formula, σjRepresenting the actual stress of the j-th surface of the thin substrate; sigmauRepresenting the unit stress load value of each surface in the finite element model;
the method comprises the following specific steps:
s61: defining a variable mu as a coefficient of a penalty term of the regularization method, wherein the variable variation range is 0-50000;
to correspond to mu
Is shown as the abscissa of the graph,
is a vertical coordinate; drawing a rounding curve with mu scattered within 0-50000 for fitting, as shown in FIG. 4;
s62: and selecting mu corresponding to the point at the turning point of the alternative curve, calculating x, and selecting the final mu value according to the difference value between the surface shape calculated by x and the measured surface shape and the stress distribution. In this example, a value of 300 μ is selected and x is calculated. The stress value of each surface finally obtained is the unit stress load 100MPa multiplied by x.
And guiding the solved stress result into finite element analysis software to extract the Z-direction displacement and the actual stress value of the finite element model, respectively drawing a deformation graph as shown in FIG. 5, a deformation and silicon wafer actual deformation difference graph as shown in FIG. 6, which is solved by the continuous constraint regularization method, wherein the error is within 2 mu m, and a stress distribution graph as shown in FIG. 7, wherein the maximum stress is 19.9MPa, the minimum stress is-35.4 MPa, the stress fluctuation value is 55.3MPa, and the continuity of the stress distribution in the graph is obviously improved.
The Z-direction deformation distribution diagram of the silicon wafer obtained by directly solving the equation by the least square method is shown in fig. 8, the error distribution of the subtraction of the obtained silicon wafer surface shape and the measured silicon wafer surface shape is shown in fig. 9, the surface shape error is within 2 μm, the stress diagram of the silicon wafer is shown in fig. 10, the maximum stress obtained by solving by the least square method is 84.6MPa, the minimum stress is-71.6 MPa, the stress fluctuation value is 156.2MPa, and is about 3 times of the stress fluctuation value obtained by solving by the continuous constraint regularization method.
Based on the results of the above simulation test, it can be found that: in the embodiment, physical constraint is introduced to solve residual stress, the L matrix is constructed by adopting a regularization method to eliminate the collinearity problem of the W matrix, and compared with the stress obtained by directly solving an equation set by a least square method, the stress fluctuation value obtained by adopting the L matrix regularization method of the continuous constraint structure is reduced by about 2/3, the stress continuously distributed in the thin substrate is really obtained, and the problem of accurately obtaining the continuous stress distribution in the thin substrate is solved.
The present invention is not limited to the above preferred embodiments, and any modifications, equivalent substitutions and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (10)
1. A method of resolving stress in a thin substrate, the method comprising the steps of:
s1: obtaining basic parameters of a thin substrate to be analyzed for stress; the base parameters include diameter, thickness, modulus of elasticity, shear module, poisson's ratio, and density;
s2: establishing a finite element model with the same basic parameters as the thin substrate to be analyzed; the method for constructing the finite element model comprises the following steps:
s21: defining material properties and unit types;
s22: establishing a finite element model of the thin substrate;
s23: dividing meshes for the established finite element model;
s24: adding constraints to the nodes in the finite element model;
s3: applying a unit stress load sigma to each surface in the finite element model in sequenceuThen solving the finite element model; defining an output path, and mapping the solving result of each surface applying unit stress load in the model to the path; outputting the solving results of all the surfaces in the model;
s4: according to the solving result of the finite element model, obtaining a surface applying unit stress load in the model and an adjacent surface thereof, and further constructing a matrix L containing the distance relation of central points among the surfaces;
s5: measuring the thin substrate to obtain the surface shape of the real thin substrate, constructing a matrix b by a column vector consisting of displacements of a plurality of real thin substrate measuring points, and recording a matrix W consisting of the displacements of the plurality of measuring points under the action of each surface applying unit stress load;
s6: solving a linear equation set by adopting a regularization method to meet the requirement
And minimum. Punishment item by regularization method when drawing different mu values
As the abscissa, in
An accept-reject curve of a vertical coordinate; mu represents the coefficient of penalty term of the regularization method, x is the vector to be solved and is formed by alphajForming an n-dimensional column vector;
the actual stress distribution of the thin substrate is calculated according to the following formula:
σj＝αjσu,
in the above formula, σjRepresenting the actual stress of the j-th surface of the thin substrate; sigmauThe unit stress load values of the respective faces in the finite element model are shown.
2. The method of resolving thin substrate stress of claim 1, wherein: the detailed procedure of step S21 is as follows:
firstly, defining unit numbers and selecting the types of shell units, and then sequentially setting the densities, elastic models, Poisson ratios and shear moduli of the materials of the first material layer and the second material layer; and finally, sequentially defining the thicknesses of the first material layer and the second material layer.
3. The method of resolving thin substrate stress of claim 1, wherein: the detailed procedure of step S22 is as follows:
firstly, setting a coordinate system in a model as a cylindrical surface coordinate system; generating key points in a fan-shaped area with the length equal to 1/3 circumference in the model, connecting the generated internal key points by straight lines, and connecting the external key points by circular arcs, so that the fan-shaped area is divided into a plurality of quadrangles; then selecting all lines in the circumference, generating equally divided key points on each line, and further dividing the quadrangle into smaller quadrangles; then generating key points for the rest 2/3 parts in the model according to the same method, connecting the key points, and combining repeated key points; and finally, selecting an arc line of the edge in the model to generate a circular surface.
4. The method of resolving thin substrate stress of claim 1, wherein: the detailed procedure of step S23 is as follows:
firstly, selecting the grid type of a model as quadrilateral grid division, and designating a grid division method as mapping grid division; then selecting all lines except the circumference in the model, forming the selected lines into a line group, dividing the circular surface generated in the previous step by using the line group, and bonding the divided surfaces together; deleting the previous line group, selecting all lines in the model, and recombining the selected lines into a new line group; and finally, controlling the grid density through the new line group, and performing grid division and generating units and nodes.
5. The method of resolving thin substrate stress of claim 1, wherein: the detailed procedure of step S24 is as follows:
selecting three nodes to impose constraints, limiting X, Y, Z directional movement for the first node; limiting movement in the X, Z directions for a second node; the movement in the Z direction is restricted for the third node.
6. The method of resolving thin substrate stress of claim 1, wherein: the specific process of step S3 is as follows:
s31: acquiring the total number of surfaces of the model;
s32: applying unit stress load on the first material layer, selecting the shell unit with the smallest serial number and related to the surface, and defining the numerical value of the unit stress load on the shell unit;
s33: solving the model;
s34: reading a solving result, creating a cylindrical coordinate system, creating a circular path under the cylindrical coordinate system, setting a mapping item, and phasing a fractional segment between two adjacent points;
s35: defining an array, mapping the displacement in the Z direction in the analysis and calculation result to the created path, acquiring data in the path and storing the data in the array;
s36: sequentially establishing a plurality of paths from inside to outside in the model so as to ensure that the sampling points are uniformly distributed; meanwhile, assigning the array corresponding to each path to a new array, so that the new array contains the position and displacement information of all previous sampling points;
s37: defining a new array, respectively assigning the X coordinate, the Y coordinate and the displacement in the Z direction of the sampling point of the array in the step S36 to the newly defined array, and outputting the data in the array;
s38: repeating the steps S31-S37 to load the unit stress load on all the surfaces in the model in sequence, wherein the unit stress load is loaded on only one surface of the model when the load is loaded, the unit stress load loaded on the previous surface is deleted before the load is loaded, and the output result of the array in each loaded surface is obtained in sequence.
7. The method of resolving thin substrate stress of claim 1, wherein: the specific process of step S4 is as follows:
s41: defining a new empty array, selecting all entities in all models, and acquiring the total number of the models;
s42: creating a new two-dimensional array, setting the row number of the two-dimensional array as the total number of the surfaces of the model obtained in the step S41, setting the column number as 4, and setting the first column of the two-dimensional array as 1 to the total number of the surfaces;
s43, selecting the surface with the smallest number in the model, calculating the geometric data of the surface with the smallest number, obtaining the X, Y, Z coordinate of the central point of the surface with the smallest number, and assigning the coordinate values to the 2 nd, 3 rd and 4 th columns of the two-dimensional arrays; assigning X, Y, Z coordinates of the central points of all the surfaces applying the load to each row of the two-dimensional array in sequence; finally, outputting data in the two-dimensional array;
s44: performing numerical analysis on the output result of the previous step, forming a matrix containing the serial number of the surface and the central point coordinate of the surface based on the analysis result, and acquiring the number of rows of the matrix;
s45: creating a full 0 matrix, wherein the row number of the matrix is the total surface number of the model, and the column number is 5;
s46: assigning the first column element of the matrix in step S44 to the first column of the matrix created in step S45; sequentially calculating the distances from the center point of the first surface of the model corresponding to the matrix to the center points of all the surfaces in the step S44, and assigning the calculation result to the fourth column of the matrix in the step S44;
s47: arranging the assigned matrixes in the step S46 in ascending order of a fourth column, and then assigning temporary variables to the arranged matrixes to form a new matrix;
s48: assigning the elements of the first column of the 2 nd, 3 rd, 4 th and 5 th rows of the new matrix created in the step S47 to the 2 nd, 3 rd, 4 th and 5 th columns of the first row of the matrix in the step S45, respectively; sequentially calculating the distances from the center points of the remaining surfaces to the center points of all the surfaces according to the method in the S46, and finishing the assignment operation of all rows of the matrix in the step S45;
s49: creating a square matrix L, wherein the dimension of the matrix L is the total number of surfaces of the model; defining elements on a main diagonal in a matrix L to be 4; assigning a value to the 1 st row of the matrix L, and assigning the elements of the indexes of the columns in the 1 st row of the matrix L corresponding to the element values of the 2, 3, 4 and 5 columns in the 1 st row of the matrix after the processing in the sequence number S48 as-1; and sequentially finishing the assignment operation of all the rows in the matrix L according to the same mode.
8. The method of resolving thin substrate stress of claim 6, wherein: the step of step S5 is as follows:
s51: creating a new matrix, wherein the number of rows of the new matrix is the number of measuring points in the thin substrate surface shape obtaining process; the column number is the total number of the surfaces applying the load in the finite element model, and the 3 rd column data in the result output in the step S38 is sequentially imported into the created new matrix according to columns to complete the construction of the W matrix;
s52: and acquiring the surface shape of the real thin substrate by using a sensor, coinciding the measurement coordinate system with the finite element coordinate system to enable the Z-direction displacement component of a point in the measurement coordinate system and a corresponding constraint point of the finite element to be 0, acquiring the measurement point of the real thin substrate matched with the displacement extraction point of the finite element model through surface shape interpolation, and acquiring the X, Y and Z coordinates of the measurement point to obtain a b matrix.
9. The method of resolving thin substrate stress of claim 8, wherein: the step of step S6 is as follows:
s61: defining a variable mu as a coefficient of a penalty term of the regularization method, wherein the variable variation range is larger than 0; given a μ, x satisfies the following equation:
x＝(WTW+μL)-1WTb，
in the above formula, the first and second carbon atoms are,Trepresents a transpose of a matrix; w represents the matrix constructed in step S51; b represents the matrix obtained in step S52;
then in order to correspond to mu
Is shown as the abscissa of the graph,
is a vertical coordinate; drawing a rounding curve fitted by scattered points with mu larger than 0;
s62: and selecting mu corresponding to the point at the turning point of the alternative curve, calculating x, and selecting the final mu value according to the difference value between the surface shape calculated by x and the measured surface shape and the stress distribution.
10. The method of resolving thin substrate stress of claim 7, wherein: in step S49, the L matrix is a unit matrix whose dimension is equal to the number of faces to which the unit stress load is applied, the main diagonal element is 1, and the remaining elements are 0.
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