Method for constructing unsteady thermal stress analytical model of through silicon via

Abstract

The invention discloses a method for constructing an unsteady thermal stress analytical model of a through silicon via, which belongs to the technical field of microelectronics and comprises the following steps of: establishing a transient thermal conduction equation and a temperature field boundary condition of the through silicon via; solving the transient heat conduction equation to obtain a transient temperature field analytical model, and obtaining transient temperature field distribution according to the transient temperature field analytical model; establishing an equilibrium equation and a mechanical boundary condition based on a working environment of the through silicon via, and determining a displacement field of the through silicon via and a basic form of a stress field analytical model according to the equilibrium equation and the transient temperature field analytical model; coupling the transient temperature field and the transient stress field according to the mechanical boundary condition and the basic form of the displacement field and stress field analytical model to obtain an unsteady thermal stress analytical model; and obtaining the unsteady thermal stress according to the unsteady thermal stress analytical model. According to the method, the problem that temperature field distribution and thermal stress change along with time are not considered in a thermal stress analysis model is solved, and real-time prediction of thermal stress distribution in the silicon through hole is realized.

Description

Technical Field

[0001] The invention belongs to the field of microelectronic technology, and in particular relates to a method for constructing an unsteady thermal stress analytical model of a through silicon via. Background Art

[0002] After integrated circuits entered the 22nm process node, Moore's Law gradually approached its limit, and three-dimensional integrated circuits were regarded as an effective solution to continue Moore's Law. However, while achieving high integration, it also inevitably brought more severe heat dissipation problems and thermal stress problems caused by the mismatch of thermal expansion coefficients between materials, which may cause the failure of local structures or even the entire system in severe cases. The thermal stress problem near silicon vias is particularly significant. Common silicon via structures contain several materials, including silicon substrate, metal filling materials (such as copper) and insulating layers. For more complex silicon via structures, such as coaxial silicon vias and annular silicon vias, the structure is more complex and contains more different materials. Due to the mismatch of thermal expansion coefficients between these materials, interaction forces will be generated at the material interface, thereby causing thermal stress. This stress may cause problems such as interface delamination and cracking, affecting the electrical properties and reliability of the silicon via. Long-term thermal stress may also cause deformation and fracture of metal lines, thereby affecting the normal operation of the chip. Therefore, it is very important to seek a general method that can accurately and quickly calculate the thermal stress around silicon vias.

[0003] At present, many studies have developed computational analysis methods for thermal stress of silicon vias: the first is finite element simulation, which discretizes the continuous structure into a finite number of small units (such as tetrahedrons and hexahedrons), and approximates the behavior of the real system by solving the physical equations of each unit; the second is the analytical model, which is based on the theory of thermoelasticity and obtains a closed-form solution through mathematical derivation, and directly gives an analytical expression for thermal stress; the third is machine learning, which establishes a statistical mapping relationship between input (such as silicon via parameters) and output (such as thermal stress) through data-driven, instead of solving physical equations.

[0004] However, the above thermal stress analysis methods have some limitations. Method 1 requires fine meshing to improve the accuracy of the results, and it takes a long time to perform transient thermal-mechanical coupling simulation analysis, especially when performing structural optimization, which requires parametric scanning and large computing resource requirements; the analytical model of method 2 is currently not perfect in the field of TSV thermal stress analysis. First, it only derives the case of uniform temperature distribution and lacks analysis of the temperature field; second, it only focuses on the steady thermal stress of TSVs and lacks research on unsteady thermal stress; method 3 is highly data-dependent and requires a large amount of high-quality data to train the model. When data is insufficient, the prediction is unreliable, and the model decision-making process is opaque, making it difficult to explain the physical mechanism. Summary of the invention

[0005] To solve the above problems existing in the prior art, the present invention provides a method for constructing an unsteady thermal stress analysis model of a through-silicon via. The technical problems to be solved by the present invention are realized through the following technical solutions: The present invention provides a method for constructing an unsteady thermal stress analysis model of a through-silicon via, and the method includes: According to the structure and working environment of the through-silicon via, establish the transient heat conduction equation and temperature field boundary conditions of the through-silicon via; According to the transient heat conduction equation and temperature field boundary conditions of the through-silicon via, obtain the transient temperature field analysis model of the through-silicon via, and obtain the transient temperature field distribution of the through-silicon via according to the transient temperature field analysis model to verify the accuracy of the transient temperature field analysis model; Based on the working environment of the through-silicon via, establish the equilibrium equation and mechanical boundary conditions of the through-silicon via, and determine the basic forms of the displacement field analysis model and the stress field analysis model of the through-silicon via according to the equilibrium equation and the transient temperature field analysis model; According to the mechanical boundary conditions, the basic form of the displacement field analysis model and the basic form of the stress field analysis model of the through-silicon via, couple the transient temperature field and the transient stress field to obtain the unsteady thermal stress analysis model of the through-silicon via; obtain the unsteady thermal stress of the through-silicon via according to the unsteady thermal stress analysis model.

[0006] Advantages of the present invention: In the solution provided by the present invention, the equilibrium equation and mechanical boundary conditions of the through-silicon via established based on the working environment of the through-silicon via are used to couple the transient temperature field and the transient stress field of the through-silicon via, and an unsteady thermal stress analysis model of the through-silicon via is obtained, realizing the calculation of the transient temperature and unsteady thermal stress at any point and any time in the through-silicon via; it is more convenient in parametric analysis and can quickly evaluate the influence of different design variables. Compared with the prior art that does not consider the temperature field distribution and unsteady thermal stress in the thermal stress analysis model of the through-silicon via, the results calculated by using the unsteady thermal stress analysis model of the through-silicon via in the present invention fully consider the distribution of the transient temperature field and the unsteady thermal stress. Compared with finite element simulation, especially in parametric analysis, the calculation time is greatly shortened and the calculation resources are saved; compared with machine learning, it provides a reliable theoretical basis for the thermal stress analysis of the through-silicon via. Description of the drawings

[0007] Figure 1 It is a schematic flowchart of a method for constructing an unsteady thermal stress analysis model of a through-silicon via provided by an embodiment of the present invention; Figure 2 It is a schematic diagram of the structure and boundary conditions of a through-silicon via provided by an embodiment of the present invention; Figure 3Schematic diagram of transient temperature field distribution of a through - silicon via provided by an embodiment of the present invention; Figure 4 Schematic diagram of relative error of transient temperature field of a through - silicon via provided by an embodiment of the present invention; Figure 5 Schematic diagram of unsteady thermal stress distribution of a through - silicon via provided by an embodiment of the present invention; Figure 6 Schematic diagram of relative error of unsteady thermal stress of a through - silicon via provided by an embodiment of the present invention. Detailed implementation manners

[0008] The following further describes the present invention in detail with specific embodiments, but the implementation manners of the present invention are not limited thereto.

[0009] An embodiment of the present invention provides a method for constructing an unsteady thermal stress analysis model of a through - silicon via, as Figure 1 shown, which may include: S1. According to the structure and working environment of the through - silicon via, establish the transient heat conduction equation and temperature field boundary conditions of the through - silicon via, which may include: S11. According to the structure and working environment of the through - silicon via, obtain the structural parameters, material parameters and thermo - mechanical coupling boundary conditions of the through - silicon via.

[0010] For S11, in this embodiment, a common through - silicon via may be selected. The through - silicon via may be a three - layer axisymmetric cylindrical structure, which may include: a metal layer, an insulating layer and a substrate layer.

[0011] In a specific example, the structure and boundary conditions of the through - silicon via are as Figure 2 shown. The material of the metal layer of the through - silicon via is copper (Cu), the material of the insulating layer is silicon dioxide (SiO2), and the material of the substrate layer is silicon (Si). 、 、 respectively represent the distances from the central axis of the through - silicon via to the outermost axes of copper, silicon dioxide and silicon, which may be 15 microns, 16 microns and 66 microns respectively. The thickness of copper is , that is, 15 microns; the thickness of silicon dioxide is , that is, 1 micron; the thickness of silicon is , that is, 50 microns; the height of the through - silicon via is , that is, 200 microns. It is assumed that the top of the through - silicon via is connected to a heat sink device with good heat dissipation effect, and the heat dissipation effects of the side and bottom surfaces can be ignored. Therefore, a constant - temperature boundary condition is set at the top, and adiabatic boundary conditions are set for the side and bottom surfaces. The temperature and heat flux are continuous at all interfaces of the through - silicon via.

[0012] Exemplarily, takingFigure 2 Taking the through-silicon via provided as an example, its structural parameters and material parameters are shown in Table 1.

[0013] Table 1 Structural and Material Parameters of Through-Silicon Vias

[0014] Among them, represents the thickness of each layer of material, represents the thermal conductivity, represents the constant-pressure heat capacity, represents the density, represents the thermal diffusivity.

[0015] S12. According to the structural parameters, material parameters of the through-silicon via and the thermo-mechanical coupling boundary conditions, establish the transient heat conduction equation of the through-silicon via and the corresponding temperature field boundary conditions in the cylindrical coordinate system.

[0016] In S12, based on the fact that the through-silicon via has a multi-layer cylindrical structure introduced in S11, it can be known that: the heat conduction problem of the through-silicon via can be described as a heat conduction problem in the cylindrical coordinate system, which can include the heat conduction equation and the corresponding temperature field boundary conditions in the axisymmetric cylindrical coordinate system.

[0017] Establishing the heat conduction equation and the temperature field boundary conditions of the through-silicon via in the cylindrical coordinate system can include: the heat conduction equation, the radial boundary condition, the axial boundary condition, and the initial condition.

[0018] The heat conduction equation is as follows: ; Among them, represents the temperature difference between the -th layer in the through-silicon via and the ambient temperature, , respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents time, represents the thermal diffusivity of the -th layer in the through-silicon via, represents the second-order partial derivative in the radial direction, represents the partial derivative in the radial direction, represents the second-order partial derivative in the axial direction, represents the partial derivative with respect to . For the convenience of representation, subsequent is abbreviated as .

[0019] The radial boundary conditions include the bounded condition of the central axis, the continuity conditions of temperature and heat flux at the interfaces between copper and silica, between silica and silicon, and the adiabatic boundary condition on the side surface. The bounded condition of the central axis is expressed as follows: Bounded, .

[0020] The continuity conditions of temperature and heat flux at the interface between copper and silica are expressed as follows: ; The continuity conditions of temperature and heat flux at the interface between silica and silicon are expressed as follows: ; Wherein, , and respectively represent the temperature differences between the metal layer, the insulating layer, and the substrate layer and the ambient temperature. , and respectively represent the thermal conductivities of the metal layer, the insulating layer, and the substrate layer.

[0021] The adiabatic boundary condition on the side surface is expressed as follows: .

[0022] The axial boundary conditions include the adiabatic boundary condition at the bottom of the silicon via hole and the constant temperature boundary condition at the top.

[0023] The adiabatic boundary condition at the bottom is expressed as follows: ; The constant temperature boundary condition at the top is expressed as follows: ; The initial conditions are as follows: ; Wherein, the subscript respectively represents the metal layer, the insulating layer, and the substrate layer, represents the temperature difference between the ambient temperature and the temperature of the th layer in the silicon via hole, represents the temperature of any point at any time in the th layer in the silicon via hole, represents the external ambient temperature; and respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents time, represents the temperature difference between the ambient temperature and the temperature of the silicon via hole at the initial time, represents the temperature of any point in the silicon via hole at the initial time.

[0024] S2. According to the transient heat conduction equation of the through-silicon via and the temperature field boundary conditions, an analytical model of the transient temperature field of the through-silicon via is obtained, and the distribution of the transient temperature field of the through-silicon via is obtained according to the analytical model of the transient temperature field to verify the accuracy of the analytical model of the transient temperature field.

[0025] Regarding S2, in S21, according to the transient heat conduction equation of the through-silicon via and the temperature field boundary conditions, obtaining the analytical model of the transient temperature field of the through-silicon via may include: S211. Determine the basic form of the analytical model of the transient temperature field according to the transient heat conduction equation and the temperature field boundary conditions.

[0026] Specifically, in S211, applying the method of separation of variables to the heat conduction equation in S12, the basic form of the analytical model of the transient temperature field can be obtained as follows: ; Among them, represents the serial number of the eigenvalue corresponding to the axial eigenfunction, represents the serial number of the eigenvalue corresponding to the radial eigenfunction, represents the superposition coefficient, 、 and are all eigenfunctions obtained by separation of variables, represents the radial eigenfunction of the th layer in the through-silicon via, that is, the direction eigenfunction, represents the axial eigenfunction, that is, the direction eigenfunction. It can be understood that the axial eigenfunctions of each layer are the same, represents the time function of the th layer in the through-silicon via. Among them, the temperature difference between the temperature of each layer in the through-silicon via and the ambient temperature is calculated using the basic form of the analytical model of the transient temperature field.

[0027] can be expressed as: ; Among them, represents the coefficient of the first kind of Bessel function in the radial eigenfunction of the th layer in the through-silicon via, represents the coefficient of the second kind of Bessel function in the radial eigenfunction of the th layer in the through-silicon via, represents the first kind of zero-order Bessel function, represents the second kind of zero-order Bessel function, represents the eigenvalue of the radial eigenfunction of the th layer in the through-silicon via.

[0028] It can be expressed as: ; Among them, , respectively represent the coefficients of the axial eigenfunction, represents the sine function, represents the cosine function, represents the serial number The eigenvalue of the corresponding axial eigenfunction.

[0029] In this embodiment, , and It can be expressed as: ; ; It can be expressed as: ; Among them, represents the attenuation coefficient of the time function of the th layer in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via. Since the heat flux is continuous at the through-silicon via interface, the following relational expression exists: .

[0030] S212. According to the basic form of the transient temperature field analysis model, the radial eigenfunction is obtained.

[0031] In S212, the control equation and boundary conditions of the radial eigenfunction are as follows: The control equation of the radial eigenfunction is expressed as follows: ; Among them, represents the second-order partial derivative of the radial eigenfunction of the th layer in the through-silicon via in the direction (radial direction), represents the partial derivative of the radial eigenfunction of the th layer in the through-silicon via in the direction.

[0032] The boundary conditions of the radial eigenfunction correspond to the boundary conditions of the heat conduction problem and are expressed as follows: Bounded, ; ; ; ; Among them, , and respectively represent the radial eigenfunctions of the metal layer, the insulating layer, and the substrate layer. , and respectively represent the thermal conductivities of the metal layer, the insulating layer, and the substrate layer.

[0033] The control equation of the radial eigenfunction is a Bessel equation of order zero, and the form of its general solution is as follows: ; Among them, , and satisfy the following relational expression: ; can be obtained through the following formula: ; Among them, is the determinant operation symbol, , , are respectively , , abbreviations.

[0034] , can be expressed as: ; ; ; ; ; Among them, , need to be obtained through the eigenvalues of the known radial eigenfunction, while , need to be obtained on the premise of knowing , , represents the Bessel function of the first kind of order one, represents the Bessel function of the second kind of order one.

[0035] S213, according to the orthogonality relation of the radial eigenfunction, the superposition coefficients are obtained.

[0036] The superposition coefficient in the temperature field analysis model can be expressed as: ; wherein, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer, represents the temperature difference between the temperature of the through-silicon via at the initial moment and the ambient temperature; and respectively represent the direction characteristic function and the modulus of the direction characteristic function, and can be obtained through the following orthogonal relationship: ; ; and can be respectively expressed as: ; .

[0037] wherein, represents the th layer corresponding th radial characteristic function in the through-silicon via, represents the th layer corresponding th radial characteristic function in the through-silicon via, and are used here to distinguish two different radial characteristic functions, represents the th layer corresponding th radial eigenvalue in the through-silicon via calculated through , represents the th layer corresponding th radial eigenvalue in the through-silicon via calculated through , and are used here to distinguish two different radial eigenvalues.

[0038] S214, based on the radial characteristic function and the superposition coefficient, the transient temperature field analysis model of the through-silicon via is obtained.

[0039] The analytical model of the transient temperature field of the through-silicon via can be expressed as: ; ; ; Wherein, 、 、 respectively represent the temperature differences from the ambient temperature of the metal layer, the insulating layer, and the silicon substrate, 、 respectively represent the integration variables of the radial coordinate and the axial coordinate, and represent the radial coordinate and the axial coordinate in the cylindrical coordinate system respectively.

[0040] The expression of the analytical model of the transient temperature field of the through-silicon via is as follows: ; Wherein, represents the temperature difference between the temperature of the th layer in the through-silicon via and the ambient temperature, represents the number of layers of the through-silicon via structure, represents the thermal conductivity of the th layer in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer. Specifically, when , it represents the position of the central axis of the through-silicon via, , represents the height of the through-silicon via, represents the serial number of the eigenvalue corresponding to the axial eigenfunction, represents the serial number of the eigenvalue corresponding to the radial eigenfunction, 、 respectively represent the integration variables of the radial coordinate and the axial coordinate, The integration range of ~ , The integration range of is 0~ 、 respectively represent the integration differential element of the radial coordinate and the integration differential element of the axial coordinate, 、 respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, Represents time, Represents the serial number The eigenvalue of the corresponding axial eigenfunction, Represents the layer in the through - silicon via Corresponding radial eigenvalue, Represents the layer in the through - silicon via Corresponding radial eigenvalue, Represents the attenuation coefficient of the time function of the layer in the through - silicon via, Represents the modulus of the axial eigenfunction, Represents the modulus of the radial eigenfunction, Represents the coefficient of the Bessel function of the first kind in the radial eigenfunction of the layer in the through - silicon via, Represents the coefficient of the Bessel function of the second kind in the radial eigenfunction of the layer in the through - silicon via, Represents the Bessel function of the first kind of order zero, Represents the Bessel function of the second kind of order zero, Represents the cosine function, Represents the exponential function, Represents the temperature difference between the initial temperature of the through - silicon via and the ambient temperature.

[0041] S22, The transient temperature field distribution of the through - silicon via is obtained according to the transient temperature field analytical model to verify the accuracy of the transient temperature field analytical model.

[0042] Specifically, in this embodiment, starting from the initial moment, the temperature of the through - silicon via gradually decreases with time, and the temperature distribution of the through - silicon via reaches a steady state in about 1 millisecond. The transient temperature field distribution of the through - silicon via is as Figure 3 shown, Figure 3 where the abscissa and respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, with the unit of μm, and the ordinate represents the temperature, with the unit of K. It can be seen from Figure 3 that: for the global temperature distribution of the through - silicon via at 0.1 millisecond, the temperature of the through - silicon via shows an axisymmetric distribution, which is related to its axisymmetric geometric structure and boundary conditions; a large temperature gradient is generated at the interface of different materials, which is related to the low thermal conductivity of silicon dioxide; at the top surface of the through - silicon via, that is, the position in the figure , the temperature is constantly 300K, which corresponds to the constant - temperature boundary condition of the top surface. The relative error of the transient temperature field of the through - silicon via is as Figure 4 shown, Figure 4 where the abscissa and respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, with the unit of μm, and the vertical coordinate represents the relative error with the unit of %. From Figure 4 it can be seen that: the relative error distribution of the global temperature of the TSV at 0.1 ms, and this error is the result of comparison with the finite element simulation. It can be seen that the maximum relative error is below 0.6%, and the average relative error calculated is 0.24%, which fully proves the accuracy of the analytical model of the transient temperature field of the TSV. Among them, the calculation formula for the average relative error of the temperature is as follows: ; where represents the relative error, represents the temperature difference of the TSV in the finite element simulation, represents the temperature difference of the TSV calculated by the analytical model of the transient temperature field.

[0043] S3. Based on the working environment of the TSV, establish the equilibrium equation and mechanical boundary conditions of the TSV, and determine the basic forms of the displacement field and stress field analytical models of the TSV according to the equilibrium equation and the analytical model of the transient temperature field.

[0044] Specifically, the expression of the equilibrium equation of the TSV is as follows: ; where represents the radial normal stress, represents the circumferential normal stress, represents the axial normal stress, represents the radial-axial shear stress, , respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents the partial derivative of the radial normal stress in the radial direction ( direction), represents the partial derivative of the radial-axial shear stress in the axial direction ( direction), represents the partial derivative of the radial-axial shear stress in the radial direction, represents the partial derivative of the axial normal stress in the axial direction. The former is the radial equilibrium equation, and the latter is the axial equilibrium equation.

[0045] The mechanical boundary conditions can include the displacement continuity condition at the interface, the stress continuity condition at the interface, and the free boundary condition at the boundary.

[0046] The displacement continuity condition at the interface includes the radial displacement continuity condition and the axial displacement continuity condition, which are expressed as follows: ; ; ; ; Among them, , , are the radial displacements of the metal layer, the insulating layer, and the substrate layer, respectively, , , are the axial displacements of the metal layer, the insulating layer, and the substrate layer, respectively.

[0047] The stress continuity conditions at the interface include the radial normal stress continuity condition and the radial-axial shear stress continuity condition, which are expressed as follows: ; ; ; ; Among them, , , are the radial normal stresses of the metal layer, the insulating layer, and the silicon substrate, respectively, , , are the radial-axial shear stresses of the metal layer, the insulating layer, and the silicon substrate, respectively.

[0048] The free boundary conditions at the boundary: ; .

[0049] In addition, to avoid meaningless displacements of the vias, the boundary conditions of fixed constraints are satisfied at .

[0050] The expressions of the basic forms of the analytical models of the displacement field and stress field of the vias are as follows: ; ; ; ; ; ; Among them, represents the radial displacement of the th layer in the via, represents the thermoelastic displacement potential of the th layer in the via, Denote the Love displacement function of the th layer in the through-silicon via, and represent the radial coordinate and the axial coordinate in the cylindrical coordinate system respectively, denote the partial derivative with respect to in the radial direction, denote and the second-order mixed partial derivative with respect to Denote the axial displacement of the th layer in the through-silicon via, Denote the Poisson's ratio of the th layer in the through-silicon via, denote the Laplace operator, denote the operation of applying the Laplace operator to , denote the partial derivative with respect to in the axial direction, denote the second-order partial derivative with respect to in the axial direction of the radial normal stress of the th layer in the through-silicon via, denote the axial normal stress of the th layer in the through-silicon via, denote the shear modulus of the th layer in the through-silicon via, denote the coefficient of thermal expansion of the th layer in the through-silicon via, denote the temperature difference between the temperature of the th layer in the through-silicon via and the ambient temperature, denote the second-order partial derivative with respect to in the radial direction, denote the circumferential normal stress of the th layer in the through-silicon via, denote the partial derivative with respect to in the radial direction, denote the radial-axial shear stress of the th layer in the through-silicon via, denote and the second-order mixed partial derivative with respect to . The mechanical material parameters of the through-silicon via are shown in Table 2.

[0051] Table 2 Mechanical Material Parameter Table

[0052] In the table, represents the Young's modulus. The relationship between the Young's modulus and the shear modulus is as follows: ; where represents the Poisson's ratio of the th layer in the through-silicon via, which can be obtained based on the transient temperature field analysis model of the through-silicon via and is expressed as: .

[0053] It can be expressed as: ; where , , , are the coefficients to be solved. To make the thermal stress of the through-silicon via meaningful at , , and the remaining coefficients to be solved can be obtained through the above mechanical boundary conditions; represents the zero-order Bessel function of the first kind, represents the zero-order Bessel function of the second kind, represents the th layer corresponding to the th radial eigenvalue in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via, represents time, represents the decay coefficient of the time function of the th layer in the through-silicon via, represents the serial number corresponding to the eigenvalue of the axial eigenfunction, represents the modified zero-order Bessel function of the first kind; represents the modified zero-order Bessel function of the second kind; represents the modified first-order Bessel function of the first kind; represents the modified first-order Bessel function of the second kind.

[0054] S4. According to the mechanical boundary conditions, the basic forms of the displacement field and stress field analysis models of the through-silicon via, couple the transient temperature field and the transient stress field to obtain the unsteady thermal stress analysis model of the through-silicon via; obtain the unsteady thermal stress of the through-silicon via according to the unsteady thermal stress analysis model.

[0055] In S4, specifically in S41, according to the mechanical boundary conditions, the basic form of the displacement field analytical model of the through-silicon via, and the basic form of the stress field analytical model, the transient temperature field and the transient stress field are coupled to obtain an unsteady thermal stress analytical model of the through-silicon via, which may include: S411, according to the mechanical boundary conditions and the basic form of the displacement field analytical model of the through-silicon via, obtain the displacement field analytical model of the through-silicon via.

[0056] Specifically, the displacement field analytical model of the through-silicon via can be obtained according to the thermoelastic displacement potential and the Love displacement function expression in S3, and the displacement field analytical model can be expressed as: ; ; where 、 、 、 are the coefficients to be solved in step S3, represents the first kind of Bessel function of the first order, represents the second kind of Bessel function of the first order.

[0057] S412, according to the displacement field analytical model of the through-silicon via and the basic form of the stress field analytical model of the through-silicon via, couple the transient temperature field and the transient stress field to obtain the unsteady thermal stress analytical model of the through-silicon via.

[0058] Specifically, the unsteady thermal stress analytical model of the through-silicon via is as follows: ; ; ; ; where, in order to simplify the formula, both ends of the stress field analytical model expression of the through-silicon via in S3 are divided by , then the left ends of the equal sign become , , , .

[0059] After obtaining the unsteady thermal stress analytical model of the through-silicon via, the unsteady thermal stress of the through-silicon via can be calculated according to the unsteady thermal stress analytical model of the through-silicon via. The unsteady thermal stress can be characterized by the Von Mises equivalent stress, and the calculation formula of the Von Mises equivalent stress is as follows: ; where represents the Von Mises equivalent stress.

[0060] Exemplarily, the unsteady thermal stress distribution of the through-silicon via is as Figure 5 shown, Figure 5 where the abscissa in the figure represents the radial coordinate in the cylindrical coordinate system, with the unit of μm, and the ordinate represents the von Mises equivalent stress, with the unit of MPa. Figure 5 The calculation results of the von Mises equivalent stress of the through-silicon via on the cross-section at different moments are given. Five moments are selected, which are 0.01 ms, 0.05 ms, 0.10 ms, 0.18 ms, and 0.30 ms. Among them, FEM in the figure represents the result of finite element simulation, and Analytical represents the calculation result of the analytical model of the unsteady thermal stress of the through-silicon via proposed in the present invention. The interfaces between copper and silicon dioxide and between silicon dioxide and the silicon substrate are marked in the figure. From Figure 5 the figure, it can be seen that: at different moments, the calculation results of the analytical model of the through-silicon via are basically in agreement with the results of finite element simulation. The maximum thermal stress appears at the interface between silicon dioxide and the silicon substrate. In addition, the thermal stress at the interface between copper and silicon dioxide is also relatively large. Therefore, the risk of failure at the interfaces of different materials is relatively high. The closer to the free boundary, the smaller the equivalent stress. Figure 5

[0061] The relative error of the unsteady thermal stress of the through-silicon via is as Figure 6 shown, Figure 6 where the abscissa in the figure represents the radial coordinate in the cylindrical coordinate system, with the unit of μm, and the ordinate represents the relative error, with the unit of %. Figure 6 The relative errors of the von Mises equivalent stress of the through-silicon via on the cross-section at different moments are given. It can be seen that the maximum relative error is below 2.5%. The error is relatively large in the silicon dioxide layer, which may be related to the sudden change of thermal stress at the material interface. However, for the maximum thermal stress at five different moments, that is, the relative errors of the thermal stress at the interface between silicon dioxide and the silicon substrate are: 0.025%, 0.154%, 0.220%, 0.227%, and 0.226% respectively. The average relative errors of the thermal stress at five different moments are: 0.235%, 0.503%, 0.640%, 0.656%, and 0.654% respectively. Overall, the average relative error is at a relatively low level; locally, the relative error in the silicon dioxide layer has a certain increase, but it is within an acceptable range; from the position where the risk of failure is relatively high, the analytical model predicts the maximum thermal stress relatively accurately. This also fully proves the reliability of the analytical model for calculating the unsteady thermal stress of the through-silicon via. In this embodiment, the calculation time of this analytical model is about 2 seconds, while the time for finite element simulation using COMSOL simulation software is about 149 seconds, and the calculation speed is greatly improved.

[0062] ​​A method for constructing an unsteady thermal stress analysis model of a through-silicon via provided by an embodiment of the present invention uses the equilibrium equation and mechanical boundary conditions of the through-silicon via established based on the working environment of the through-silicon via to couple the transient temperature field and transient stress field of the through-silicon via, obtaining an unsteady thermal stress analysis model of the through-silicon via, and realizing the calculation of the transient temperature and unsteady thermal stress at any point and at any time in the through-silicon via; it is more convenient during parametric analysis and can quickly evaluate the influence of different design variables. Compared with the current existing technology where the temperature field distribution and unsteady thermal stress are not considered in the thermal stress analysis model of the through-silicon via, the results calculated using the unsteady thermal stress analysis model of the through-silicon via of the present invention fully consider the distribution of the transient temperature field and the distribution of the unsteady thermal stress. Compared with finite element simulation, especially during parametric analysis, it significantly shortens the calculation time and saves computing resources; compared with machine learning, it provides a reliable theoretical basis for the thermal stress analysis of the through-silicon via.

[0063] It should be noted that in the description of the present invention, it should be understood that the terms "first" and "second" are only used for descriptive purposes and cannot be construed as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include one or more of such features. In the description of the present invention, "a plurality" means two or more unless otherwise specifically defined.

[0064] The above are only the preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.

Claims

1. A method for constructing an unsteady thermal stress analysis model of a through-silicon via, characterized in that Including: Establish the transient heat conduction equation and temperature field boundary conditions of the through-silicon via according to the structure and working environment of the through-silicon via; Obtain the analytical model of the transient temperature field of the through-silicon via according to the transient heat conduction equation and temperature field boundary conditions of the through-silicon via, and obtain the distribution of the transient temperature field of the through-silicon via according to the analytical model of the transient temperature field to verify the accuracy of the analytical model of the transient temperature field; Based on the working environment of the through-silicon via, establish the equilibrium equation and mechanical boundary conditions of the through-silicon via, and determine the basic forms of the analytical model of the displacement field and the analytical model of the stress field of the through-silicon via according to the equilibrium equation and the analytical model of the transient temperature field; Couple the transient temperature field and the transient stress field according to the mechanical boundary conditions, the basic form of the analytical model of the displacement field of the through-silicon via and the basic form of the analytical model of the stress field of the through-silicon via to obtain the analytical model of the unsteady thermal stress of the through-silicon via; Obtain the unsteady thermal stress of the through-silicon via according to the analytical model of the unsteady thermal stress.

2. The method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, characterized in that, The step of establishing the transient heat conduction equation and temperature field boundary conditions of the through-silicon via according to the structure and working environment of the through-silicon via includes: Obtain the structural parameters, material parameters and thermo-mechanical coupling boundary conditions of the through-silicon via according to the structure and working environment of the through-silicon via; Establish the transient heat conduction equation and the corresponding temperature field boundary conditions of the through-silicon via in the cylindrical coordinate system according to the structural parameters, material parameters and thermo-mechanical coupling boundary conditions of the through-silicon via.

3. A method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, characterized in that The step of obtaining the analytical model of the transient temperature field of the through-silicon via according to the transient heat conduction equation and temperature field boundary conditions of the through-silicon via includes: Determine the basic form of the analytical model of the transient temperature field according to the transient heat conduction equation and temperature field boundary conditions; Obtain the radial eigenfunction according to the basic form of the analytical model of the transient temperature field; Obtain the superposition coefficient according to the orthogonality relation of the radial eigenfunction; Obtain the analytical model of the transient temperature field of the through-silicon via according to the radial eigenfunction and the superposition coefficient.

4. A method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 3, characterized in that, The expression of the analytical model of the transient temperature field of the through-silicon via is as follows: ； Among them, represents the temperature difference between the temperature of the th layer in the through-silicon via and the ambient temperature, represents the number of layers of the through-silicon via structure, represents the thermal conductivity of the th layer in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via, represents the thermal diffusivity of the th layer in the through-silicon via, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer, represents the distance from the central axis of the through-silicon via to the outermost axis of the th layer. Specifically, when it is , it represents the position of the central axis of the through-silicon via, represents the height of the through-silicon via, represents the serial number of the eigenvalue corresponding to the axial eigenfunction, represents the serial number of the eigenvalue corresponding to the radial eigenfunction, , respectively represent the integration variable of the radial coordinate and the integration variable of the axial coordinate, The integration range of is from to The integration range of is from 0 to , respectively represent the integration differential element of the radial coordinate and the integration differential element of the axial coordinate, , respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents time, represents the serial number corresponding to the eigenvalue of the axial eigenfunction, represents the th layer in the through-silicon via corresponding to the th radial eigenvalue, represents the th layer in the through-silicon via corresponding to the th radial eigenvalue, represents the attenuation coefficient of the time function of the th layer in the through-silicon via, represents the modulus of the axial eigenfunction, represents the modulus of the radial eigenfunction, represents the coefficient of the Bessel function of the first kind in the radial eigenfunction of the th layer in the through-silicon via, Denote the coefficient of the second kind of Bessel function in the radial eigenfunction of the n-th layer in the through-silicon via, denote the Bessel function of the first kind of order zero, denote the Bessel function of the second kind of order zero, denote the cosine function, denote the exponential function, denote the temperature difference between the initial temperature of the through-silicon via and the ambient temperature.

5. A method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, characterized in that The expression of the equilibrium equation of the through-silicon via is as follows: ； Among them, represents the radial normal stress, represents the circumferential normal stress, represents the axial normal stress, represents the radial-axial shear stress, 、 respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents the partial derivative of the radial normal stress in the radial direction, represents the partial derivative of the radial-axial shear stress in the axial direction, represents the partial derivative of the radial-axial shear stress in the radial direction, represents the partial derivative of the axial normal stress in the axial direction.

6. The method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, wherein The mechanical boundary conditions include: The displacement continuity condition at the interface, the stress continuity condition at the interface and the free boundary condition at the boundary.

7. A method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, characterized in that, The expressions of the basic forms of the analytical models of the displacement field and stress field of the through-silicon via are as follows: ； ； ； ； ； ； Among them, represents the radial displacement of the th layer in the through-silicon via, represents the thermoelastic displacement potential of the th layer in the through-silicon via, represents the Love displacement function of the th layer in the through-silicon via, , respectively represent the radial coordinate and the axial coordinate in the cylindrical coordinate system, represents the partial derivative of with respect to in the radial direction, and the second-order mixed partial derivative of represents the axial displacement of the th layer in the through-silicon via, represents the Poisson's ratio of the th layer in the through-silicon via, represents the Laplace operator, represents the operation of applying the Laplace operator to , represents the partial derivative of with respect to in the axial direction, represents the radial normal stress of the th layer in the through-silicon via, represents the axial normal stress of the th layer in the through-silicon via, represents the shear modulus of the th layer in the through-silicon via, represents the thermal expansion coefficient of the th layer in the through-silicon via, represents the temperature difference between the temperature of the th layer in the through-silicon via and the ambient temperature, represents the second-order partial derivative of with respect to in the radial direction, represents the second-order partial derivative of with respect to in the radial direction, represents the circumferential normal stress of the th layer in the through-silicon via, represents the second-order partial derivative of with respect to in the axial direction, represents the radial-axial shear stress of the and The second-order mixed partial derivative.

8. A method for constructing an unsteady thermal stress analysis model of a through-silicon via according to claim 1, characterized in that, The step of coupling the transient temperature field and the transient stress field according to the mechanical boundary conditions, the basic form of the analytical model of the displacement field of the through-silicon via and the basic form of the analytical model of the stress field of the through-silicon via to obtain the analytical model of the unsteady thermal stress of the through-silicon via includes: Obtain the analytical model of the displacement field of the through-silicon via according to the mechanical boundary conditions and the basic form of the analytical model of the displacement field of the through-silicon via; Couple the transient temperature field and the transient stress field according to the analytical model of the displacement field of the through-silicon via and the basic form of the analytical model of the stress field of the through-silicon via to obtain the analytical model of the unsteady thermal stress of the through-silicon via.

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