A method for constructing an analytical model of unsteady thermal stress in through-silicon vias

Abstract

The present invention discloses a method for constructing an unsteady thermal stress analytical model for a through-silicon via (TSV), belonging to the field of microelectronics technology. The method comprises: establishing a transient heat conduction equation and temperature field boundary conditions for the TSV; solving the transient heat conduction equation to obtain a transient temperature field analytical model, and obtaining a transient temperature field distribution based on the transient temperature field analytical model; establishing an equilibrium equation and mechanical boundary conditions based on the TSV's working environment, and determining the basic form of the TSV's displacement field and stress field analytical model based on the equilibrium equation and the transient temperature field analytical model; coupling the transient temperature field with the transient stress field based on the mechanical boundary conditions, the displacement field, and the basic form of the stress field analytical model to obtain an unsteady thermal stress analytical model; and obtaining unsteady thermal stress based on the unsteady thermal stress analytical model. The present invention solves the problem that the thermal stress analytical model does not consider the time-varying temperature field distribution and thermal stress, and realizes real-time prediction of the thermal stress distribution in the TSV.

Description

Technical Field

[0001] The present invention belongs to the field of microelectronics 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. Three-dimensional integrated circuits were seen 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. In severe cases, it may cause failure of local structures or even the entire system. The thermal stress problem near silicon vias is particularly significant. Common silicon via structures contain several materials, including silicon substrate, metal filling material (such as copper), and insulation layer. 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 inducing 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 find 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 TSV thermal stress: 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 analytical model, which is based on the theory of thermoelasticity, obtains closed-form solutions 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 TSV parameters) and output (such as thermal stress) through data-driven, instead of solving physical equations.

[0004] However, these thermal stress analysis methods have some limitations. Method 1 requires fine meshing to improve the accuracy of the results, and transient thermal-mechanical coupled simulation analysis is time-consuming, especially when performing structural optimization, requiring parametric sweeps and requiring high computational resources. The analytical model used in Method 2 is currently imperfect for TSV thermal stress analysis. First, it only derives for uniform temperature distribution and lacks analysis of the temperature field. Second, it focuses only 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. Predictions are unreliable when data is insufficient, and the model's decision-making process is opaque, making it difficult to explain the physical mechanisms. Summary of the Invention

[0005] To address the aforementioned issues in the prior art, the present invention provides a method for constructing an analytical model of unsteady thermal stress in through-silicon vias. The technical issues addressed by the present invention are achieved through the following technical solutions:

[0006] The present invention provides a method for constructing an unsteady thermal stress analytical model of a through silicon via, the method comprising:

[0007] According to the structure of TSV and its working environment, the transient heat conduction equation and temperature field boundary conditions of TSV are established;

[0008] According to the transient heat conduction equation and the temperature field boundary conditions of the through silicon via, a transient temperature field analytical model of the through silicon via is obtained, and 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;

[0009] Based on the working environment of TSV, the equilibrium equation and mechanical boundary conditions of TSV are established. Based on the equilibrium equation and transient temperature field analytical model, the basic form of the TSV displacement field analytical model and the basic form of the stress field analytical model are determined.

[0010] 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 the unsteady thermal stress analytical model of the through-silicon via; the unsteady thermal stress of the through-silicon via is obtained based on the unsteady thermal stress analytical model.

[0011] Beneficial effects of the present invention:

[0012] In the solution provided by the present invention, the equilibrium equations and mechanical boundary conditions of the TSV established based on the working environment of the TSV are used to couple the transient temperature field and transient stress field of the TSV, thereby obtaining an analytical model of the unsteady thermal stress of the TSV, and realizing the calculation of the transient temperature and unsteady thermal stress at any point and at any time in the TSV; it is relatively convenient when performing parametric analysis, and the influence of different design variables can be quickly evaluated. Compared with the current prior art, which does not consider the problem of temperature field distribution and unsteady thermal stress in the thermal stress analytical model of the TSV, the results obtained by the present invention using the analytical model of the unsteady thermal stress of the TSV fully consider the distribution of transient temperature field and the distribution of unsteady thermal stress. Compared with finite element simulation, especially when performing parametric analysis, the calculation time is greatly shortened and computing resources are saved; compared with machine learning, it provides a reliable theoretical basis for the thermal stress analysis of TSV. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 A schematic flow chart of a method for constructing an unsteady thermal stress analytical model of a through-silicon via provided by an embodiment of the present invention;

[0014] Figure 2 A schematic diagram of the structure and boundary conditions of a through silicon via provided by an embodiment of the present invention;

[0015] Figure 3 A schematic diagram of transient temperature field distribution of a through silicon via provided by an embodiment of the present invention;

[0016] Figure 4 A schematic diagram of a relative error of a transient temperature field of a through silicon via provided by an embodiment of the present invention;

[0017] Figure 5 A schematic diagram of unsteady thermal stress distribution of a through silicon via provided by an embodiment of the present invention;

[0018] Figure 6 A schematic diagram of the relative error of unsteady thermal stress of a through silicon via provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0019] The present invention will be further described in detail below with reference to specific examples, but the embodiments of the present invention are not limited thereto.

[0020] The embodiment of the present invention provides a method for constructing an unsteady thermal stress analytical model of a through silicon via. Figure 1 As shown, this may include:

[0021] S1. Based on the structure of the TSV and the working environment, establish the transient heat conduction equation and temperature field boundary conditions of the TSV, which may include:

[0022] S11, according to the structure of the through silicon via and the working environment, obtain the structural parameters, material parameters and thermal-mechanical coupling boundary conditions of the through silicon via.

[0023] With respect to 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.

[0024] In a specific example, the structure and boundary conditions of the through silicon via are as follows: Figure 2 As shown, the metal layer material of the through silicon via is copper (Cu), the insulating layer material is silicon dioxide (SiO2), and the substrate layer material is silicon (Si). 、 、 Represents the distance from the center axis of the silicon via to the outermost axis of copper, silicon dioxide, and silicon, which can be 15 microns, 16 microns, and 66 microns respectively. The thickness of copper is , i.e. 15 microns; the thickness of silicon dioxide is , i.e. 1 micron; the thickness of silicon is , i.e. 50 microns; the height of the through silicon via is , which is 200 microns. Assuming the top of the TSV is connected to a heat sink, providing good heat dissipation, and the heat dissipation effect on the sides and bottom is negligible, a constant temperature boundary condition is set at the top, and adiabatic boundary conditions are set at the sides and bottom. The temperature and heat flow at all TSV interfaces are continuous.

[0025] For example, Figure 2 Taking the provided through silicon via as an example, its structural parameters and material parameters are shown in Table 1.

[0026] Table 1 Structural parameters and material parameters of through silicon vias

[0027]

[0028] in, Indicates the thickness of each layer of material, represents thermal conductivity, represents the constant pressure heat capacity, represents density, represents the thermal diffusivity.

[0029] S12, based on the structural parameters, material parameters and thermal-mechanical coupling boundary conditions of the through-silicon via, a transient heat conduction equation of the through-silicon via and corresponding temperature field boundary conditions are established in a cylindrical coordinate system.

[0030] In S12, based on the multi-layer cylindrical structure of the TSV introduced in S11, it can be seen that the heat conduction problem of the TSV can be described as a heat conduction problem in a cylindrical coordinate system, which can include the heat conduction equation in the axisymmetric cylindrical coordinate system and the corresponding temperature field boundary conditions.

[0031] The heat conduction equation and temperature field boundary conditions of the through silicon via are established in a cylindrical coordinate system, which may include: a heat conduction equation, radial boundary conditions, axial boundary conditions and initial conditions.

[0032] The heat conduction equation is as follows:

[0033] ;

[0034] in, Indicates the first The temperature difference between the layer temperature and the ambient temperature, 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. Indicates time, Indicates the first Thermal diffusivity of the layer, express The second-order partial derivative in the radial direction, express The partial derivative in the radial direction, express The second-order partial derivative in the axial direction, express about For convenience of representation, the following Abbreviated as .

[0035] The radial boundary conditions include the bounded condition of the central axis, the temperature and heat flow continuity conditions at the interface between copper and silicon dioxide, the temperature and heat flow continuity conditions at the interface between silicon dioxide and silicon, and the adiabatic boundary conditions on the side. The bounded condition of the central axis is expressed as follows:

[0036] Bounded, .

[0037] The temperature and heat flow continuity conditions at the copper and silicon dioxide interface are expressed as follows:

[0038] ;

[0039] The temperature and heat flow continuity conditions at the interface between silicon dioxide and silicon are expressed as follows:

[0040] ;

[0041] in, 、 and Represent the temperature difference between the metal layer, insulation layer and substrate layer and the ambient temperature respectively. 、 and represent the thermal conductivity of the metal layer, insulating layer and substrate layer respectively.

[0042] The adiabatic boundary conditions on the side are expressed as follows:

[0043] .

[0044] The axial boundary conditions include an adiabatic boundary condition at the bottom of the TSV and a constant temperature boundary condition at the top.

[0045] The adiabatic boundary condition at the bottom is expressed as follows:

[0046] ;

[0047] The constant temperature boundary condition at the top is expressed as follows:

[0048] ;

[0049] The initial conditions are as follows:

[0050] ;

[0051] Among them, the subscript Represent the metal layer, insulation layer and substrate layer respectively, Indicates the first The temperature difference between the layer temperature and the ambient temperature, Indicates the first The temperature at any point in the layer at any time, Indicates the external ambient temperature; and denote the radial and axial coordinates in the cylindrical coordinate system, respectively. Indicates time, Indicates the temperature difference between the TSV temperature and the ambient temperature at the initial moment, Represents the temperature of any point in the TSV at the initial moment.

[0052] S2. Based on the transient heat conduction equation and temperature field boundary conditions of the TSV, the transient temperature field analytical model of the TSV is obtained. Based on the transient temperature field analytical model, the transient temperature field distribution of the TSV is obtained to verify the accuracy of the transient temperature field analytical model.

[0053] For S2 and S21, based on the transient heat conduction equation and temperature field boundary conditions of the through silicon via, the transient temperature field analytical model of the through silicon via is obtained, which may include:

[0054] S211. Determine the basic form of the transient temperature field analytical model based on the transient heat conduction equation and the temperature field boundary conditions.

[0055] Specifically, in S211, the separation of variables method is applied to the heat conduction equation in S12, and the basic form of the transient temperature field analytical model can be obtained as follows:

[0056] ;

[0057] in, Indicates the serial number of the eigenvalue corresponding to the axial characteristic function, Indicates the serial number of the eigenvalue corresponding to the radial eigenfunction, represents the superposition coefficient, 、 and They are all characteristic functions obtained by separating variables. Indicates the first The radial characteristic function of the layer, that is, Directional characteristic function, represents the axial characteristic function, that is, Directional characteristic function, it can be understood that the axial characteristic function of each layer is the same, Indicates the first The temperature difference between the temperature of each layer in the TSV and the ambient temperature is calculated using the basic form of the transient temperature field analytical model.

[0058] It can be expressed as:

[0059] ;

[0060] in, Indicates the first The coefficients of the Bessel function of the first kind in the radial characteristic function of the layer, Indicates the first The coefficients of the second kind Bessel function in the radial characteristic function of the layer, represents the zero-order Bessel function of the first kind, represents the zero-order Bessel function of the second kind, Indicates the first Eigenvalues of the radial characteristic function of the layer.

[0061] It can be expressed as:

[0062] ;

[0063] in, 、 denote the coefficients of the axial characteristic function, represents the sine function, represents the cosine function, Indicates the serial number The corresponding eigenvalue of the axial characteristic function.

[0064] In this embodiment, 、 and It can be expressed as:

[0065] ;

[0066] ;

[0067] It can be expressed as:

[0068] ;

[0069] in, Indicates the first The attenuation coefficient of the layer as a function of time, Indicates the first Thermal diffusivity of the layer. Since the heat flow at the TSV interface is continuous, the following relationship is obtained:

[0070] .

[0071] S212, obtaining a radial characteristic function according to a basic form of the transient temperature field analytical model.

[0072] In S212, the governing equations and boundary conditions of the radial characteristic function are as follows:

[0073] The governing equation of the radial characteristic function is expressed as follows:

[0074] ;

[0075] in, Indicates the first The radial characteristic function of the layer is The second-order partial derivative in the direction (radial), Indicates the first The radial characteristic function of the layer is Partial derivatives in the direction.

[0076] The boundary conditions of the radial characteristic function correspond to the boundary conditions of the heat conduction problem and are expressed as follows:

[0077] Bounded, ;

[0078] ;

[0079] ;

[0080] ;

[0081] in, 、 and Represent the radial characteristic functions of the metal layer, insulation layer and substrate layer respectively. 、 and represent the thermal conductivity of the metal layer, insulating layer and substrate layer respectively.

[0082] The governing equation of the radial characteristic function is the zero-order Bessel equation, and its general solution is as follows:

[0083] ;

[0084] in, 、 and The following relationship is satisfied:

[0085] ;

[0086] It can be obtained by the following formula:

[0087] ;

[0088] in, is the determinant operator symbol, 、 、 They are 、 、 abbreviation of .

[0089] 、 It can be expressed as:

[0090] ;

[0091] ;

[0092] ;

[0093] ;

[0094] ;

[0095] in, 、 It needs to be obtained through the known characteristic value of the radial characteristic function, and 、 Need to know 、 Under the premise of represents the first-order Bessel function of the first kind, represents the first-order Bessel function of the second kind.

[0096] S213: Obtain a superposition coefficient according to the orthogonal relationship of the radial characteristic functions.

[0097] Superposition coefficient in the analytical model of temperature field It can be expressed as:

[0098] ;

[0099] in, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, Indicates the temperature difference between the TSV temperature and the ambient temperature at the initial moment; 、 Respectively Directional characteristic function and The modulus of the directional characteristic function can be obtained through the following orthogonal relationship:

[0100] ;

[0101] ;

[0102] 、 They can be expressed as:

[0103] ;

[0104] .

[0105] in, Indicates the first layer The corresponding radial characteristic function, Indicates the first layer The corresponding radial characteristic function, and Here it is used to distinguish two different radial characteristic functions, Indicates passing The calculated number of TSVs layer The corresponding radial eigenvalues, Indicates passing The calculated number of TSVs layer The corresponding radial eigenvalues, and It is used here to distinguish two different radial eigenvalues.

[0106] S214 , obtaining a transient temperature field analytical model of the through silicon via according to the radial characteristic function and the superposition coefficient.

[0107] The analytical model of the transient temperature field of TSV can be expressed as:

[0108] ;

[0109] ;

[0110] ;

[0111] in, 、 、 Represent the temperature difference between the metal layer, insulating layer and silicon substrate and the ambient temperature, 、 denote the radial coordinate integral variable and the axial coordinate integral variable, respectively, representing the radial coordinate and axial coordinate in the cylindrical coordinate system.

[0112] The expression of the transient temperature field analytical model of TSV is as follows:

[0113] ;

[0114] in, Indicates the first The temperature difference between the layer temperature and the ambient temperature, represents the number of layers of TSV structure, Indicates the first The thermal conductivity of the layer, Indicates the first Thermal diffusivity of the layer, Indicates the first Thermal diffusivity of the layer, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, in particular, when hour , represents the central axis position of the through silicon via, represents the height of the TSV, Indicates the serial number of the eigenvalue corresponding to the axial characteristic function, Indicates the serial number of the eigenvalue corresponding to the radial eigenfunction, 、 denote the radial coordinate integral variable and the axial coordinate integral variable, respectively. The integral range is ~ , The integral range is 0~ , 、 denote the radial coordinate integral differential and the axial coordinate integral differential, respectively. 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. Indicates time, Indicates the serial number The corresponding eigenvalue of the axial characteristic function, Indicates the first layer The corresponding radial eigenvalues, Indicates the first layer The corresponding radial eigenvalues, Indicates the first The attenuation coefficient of the layer as a function of time, represents the modulus of the axial characteristic function, represents the modulus of the radial characteristic function, Indicates the first The coefficients of the Bessel function of the first kind in the radial characteristic function of the layer, Indicates the first The coefficients of the second kind Bessel function in the radial characteristic function of the layer, represents the zero-order Bessel function of the first kind, represents the zero-order Bessel function of the second kind, represents the cosine function, represents the exponential function, It represents the temperature difference between the TSV temperature and the ambient temperature at the initial moment.

[0115] S22, obtaining a transient temperature field distribution of the through silicon via according to the transient temperature field analytical model to verify the accuracy of the transient temperature field analytical model.

[0116] Specifically, in this embodiment, the temperature of the TSV gradually decreases over time from the initial moment, and the temperature distribution of the TSV reaches a steady state in about 1 millisecond. Figure 3 As shown, Figure 3 Middle horizontal coordinate and They represent the radial and axial coordinates in the cylindrical coordinate system, respectively, in μm, and the ordinate represents the temperature, in K. Figure 3 It can be seen that the global temperature distribution of the silicon via at 0.1 milliseconds is axisymmetric, which is related to its axisymmetric geometry and boundary conditions; a large temperature gradient is generated at the interface between different materials, which is related to the lower thermal conductivity of silicon dioxide; on the top surface of the silicon via, that is, The temperature at the location is constant at 300K, which corresponds to the constant temperature boundary condition of the top surface. The relative error of the transient temperature field of the silicon via is as follows: Figure 4 As shown, Figure 4 Middle horizontal coordinate and They represent the radial coordinate and axial coordinate in the cylindrical coordinate system, respectively, in μm, and the ordinate represents the relative error, in %. Figure 4 The relative error distribution of the global temperature of the TSV over a 0.1 millisecond period is shown, compared to the finite element simulation results. The maximum relative error is below 0.6%, and the calculated average relative error is 0.24%, fully demonstrating the accuracy of the TSV transient temperature field analytical model. The average relative error in temperature is calculated as follows:

[0117] ;

[0118] in, Represents the relative error, represents the TSV temperature difference from finite element simulation, represents the TSV temperature difference calculated by the transient temperature field analytical model.

[0119] S3, based on the working environment of silicon via, establish the equilibrium equation and mechanical boundary conditions of silicon via, and determine the basic form of the displacement field and stress field analytical model of silicon via according to the equilibrium equation and transient temperature field analytical model.

[0120] Specifically, the expression of the balance equation of the through silicon via is as follows:

[0121] ;

[0122] in, represents the radial normal stress, represents the hoop normal stress, represents the axial normal stress, represents the radial-axial shear stress, 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. Indicates the radial normal stress in the radial direction ( direction), Indicates the radial-axial shear stress in the axial direction ( direction), represents the radial partial derivative of the radial-axial shear stress, 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.

[0123] Mechanical boundary conditions can include displacement continuity conditions at the interface, stress continuity conditions at the interface, and free boundary conditions at the boundary.

[0124] The displacement continuity conditions at the interface include radial displacement continuity conditions and axial displacement continuity conditions, which are expressed as follows:

[0125] ;

[0126] ;

[0127] ;

[0128] ;

[0129] in, 、 、 are the radial displacements of the metal layer, insulation layer and substrate layer, respectively. 、 、 are the axial displacements of the metal layer, insulation layer, and substrate layer, respectively.

[0130] 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:

[0131] ;

[0132] ;

[0133] ;

[0134] ;

[0135] in, 、 、 are the radial normal stresses of the metal layer, insulating layer and silicon substrate respectively, 、 、 are the radial-axial shear stresses of the metal layer, insulating layer and silicon substrate, respectively.

[0136] Free boundary conditions at the boundaries:

[0137] ;

[0138] .

[0139] In addition, in order to avoid meaningless displacement of TSV, The boundary conditions of the fixed constraints are met.

[0140] The basic expressions of the analytical model of displacement and stress fields of TSV are as follows:

[0141] ;

[0142] ;

[0143] ;

[0144] ;

[0145] ;

[0146] ;

[0147] in, Indicates the first Radial displacement of the layer, Indicates the first Thermoelastic displacement potential of the layer, Indicates the first The Love shift function of the layer, 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. express The partial derivative in the radial direction, express about and The second-order mixed partial derivatives of Indicates the first Axial displacement of the layer, Indicates the first Poisson's ratio of the layer, represents the Laplace operator, Express Apply the Laplace operator to perform the operation, express The partial derivative in the axial direction, express The second-order partial derivative in the axial direction, Indicates the first The radial normal stress of the layer, Indicates the first The axial normal stress of the layer, Indicates the first Shear modulus of the layer, Indicates the first Thermal expansion coefficient of the layer, Indicates the first The temperature difference between the layer temperature and the ambient temperature, express The second-order partial derivative in the radial direction, express The second-order partial derivative in the radial direction, Indicates the first The hoop normal stress of the layer, express The partial derivative in the radial direction, express The second-order partial derivative in the axial direction, Indicates the first Radial-axial shear stress of the layer, express about and The mechanical material parameters of TSV are shown in Table 2.

[0148] Table 2 Mechanical material parameters

[0149]

[0150] In the table, represents Young's modulus. The relationship between Young's modulus and shear modulus satisfies the following equation:

[0151] ;

[0152] in, Indicates the first Poisson's ratio of the layer, It can be obtained based on the transient temperature field analytical model of TSV, which is expressed as:

[0153] .

[0154] It can be expressed as:

[0155] ;

[0156] in, 、 、 、 is the coefficient to be solved, in order to make the thermal stress of the silicon via Place is meaningful, , 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, Indicates the first layer The corresponding radial eigenvalues, Indicates the first Thermal diffusivity of the layer, Indicates time, Indicates the first The attenuation coefficient of the layer as a function of time, Indicates the serial number The corresponding eigenvalue of the axial characteristic function, 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.

[0157] S4, based on the basic form of the mechanical boundary conditions, displacement field and stress field analytical model of the silicon via, the transient temperature field and the transient stress field are coupled to obtain the unsteady thermal stress analytical model of the silicon via; the unsteady thermal stress of the silicon via is obtained based on the unsteady thermal stress analytical model.

[0158] Regarding S4 and S41, based on 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 the unsteady thermal stress analytical model of the through silicon via, which may include:

[0159] S411, obtaining a displacement field analytical model of the through silicon via according to the mechanical boundary conditions and the basic form of the displacement field analytical model of the through silicon via.

[0160] Specifically, the displacement field analytical model of the through-silicon via can be obtained based on the thermoelastic displacement potential and the Love displacement function expression in S3. The displacement field analytical model can be expressed as:

[0161] ;

[0162] ;

[0163] in, 、 、 、 is the coefficient to be solved in step S3, represents the first-order Bessel function of the first kind, represents the first-order Bessel function of the second kind.

[0164] S412 , according to the basic forms of the displacement field analytical model and the stress field analytical model of the through silicon via, coupling the transient temperature field with the transient stress field to obtain an unsteady thermal stress analytical model of the through silicon via.

[0165] Specifically, the analytical model of the unsteady thermal stress of TSV is as follows:

[0166] ;

[0167] ;

[0168] ;

[0169] ;

[0170] In order to simplify the formula, the stress field analytical model expression of the silicon via in S3 is divided by both ends of the equal sign. , then the left side of the equal sign becomes , , , .

[0171] After obtaining the analytical model of the unsteady thermal stress of the TSV, the unsteady thermal stress of the TSV can be calculated based on the analytical model. The unsteady thermal stress can be characterized by the Von Mises equivalent stress. The calculation formula of the Von Mises equivalent stress is as follows:

[0172] ;

[0173] in, Represents Von Mises equivalent stress.

[0174] For example, the unsteady thermal stress distribution of TSV is as follows: Figure 5 As shown, Figure 5 The horizontal axis represents the radial coordinate in the cylindrical coordinate system, the unit is μm, and the vertical axis represents the Von Mises equivalent stress, the unit is MPa. Figure 5 Given different moments The calculation results of the Von Mises equivalent stress of the silicon via on the cross-section line were selected at five times, namely 0.01ms, 0.05ms, 0.10ms, 0.18ms and 0.30ms. Figure 5 FEM represents the result of finite element simulation, and Analytical represents the calculation result of the unsteady thermal stress analytical model of the silicon via proposed in the present invention. The interface between copper and silicon dioxide and the interface between silicon dioxide and silicon substrate are marked in the figure. Figure 5 As can be seen from the figure, at different times, the analytical model calculation results for the TSV are generally consistent with the finite element simulation results. The maximum thermal stress occurs at the interface between silicon dioxide and the silicon substrate. Furthermore, the thermal stress at the interface between copper and silicon dioxide is also high, thus increasing the risk of failure at the interface between different materials. The closer to the free boundary, the smaller the equivalent stress.

[0175] The relative error of the unsteady thermal stress of TSV is as follows: Figure 6 As shown, Figure 6 The horizontal axis represents the radial coordinate in the cylindrical coordinate system, in μm, and the vertical axis represents the relative error, in %. Figure 6 Given different moments The relative error of the Von Mises equivalent stress of the TSV on the cross-section line is shown. The maximum relative error is below 2.5%, with larger errors in the silicon dioxide layer, likely due to a sudden change in thermal stress at the material interface. However, the relative errors for the maximum thermal stress at the interface between silicon dioxide and the silicon substrate at five different times are 0.025%, 0.154%, 0.220%, 0.227%, and 0.226%, respectively. The average relative errors for the thermal stress at these five times are 0.235%, 0.503%, 0.640%, 0.656%, and 0.654%, respectively. Overall, the average relative error is low. Locally, the relative error in the silicon dioxide layer increases slightly, but remains within an acceptable range. The analytical model accurately predicts the maximum thermal stress at locations with a higher risk of failure. This fully demonstrates the reliability of the analytical model for calculating unsteady thermal stress in TSVs. In this embodiment, the calculation time of the analytical model is about 2 seconds, while the time for finite element simulation using COMSOL simulation software is about 149 seconds, which greatly improves the calculation speed.

[0176] The embodiment of the present invention provides a method for constructing an unsteady thermal stress analytical model for a through-silicon via. The method utilizes the equilibrium equations 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, thereby obtaining an unsteady thermal stress analytical model for the through-silicon via. This method enables the calculation of transient temperature and unsteady thermal stress at any point in the through-silicon via at any time. The method is convenient when performing parametric analysis and can quickly evaluate the impact of different design variables. Compared with the existing technology that does not consider the temperature field distribution and unsteady thermal stress in the thermal stress analytical model of the through-silicon via, the present invention uses the results calculated using the unsteady thermal stress analytical model of the through-silicon via to fully consider the distribution of the transient temperature field and the distribution of unsteady thermal stress. Compared with finite element simulation, especially when performing parametric analysis, the method significantly shortens the calculation time and saves computing resources. Compared with machine learning, the method provides a reliable theoretical basis for thermal stress analysis of through-silicon vias.

[0177] It should be noted that, in the description of the present invention, it should be understood that the terms "first" and "second" are used for descriptive purposes only and should not be understood to indicate or imply relative importance or implicitly specify the number of the technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of the present invention, "plurality" means two or more, unless otherwise specifically defined.

[0178] The above description is only a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention are included in the scope of protection of the present invention.

Claims

1. A method for constructing an unsteady thermal stress analytical model of a through silicon via, characterized in that: include: According to the structure of TSV and its working environment, the transient heat conduction equation and temperature field boundary conditions of TSV are established; According to the transient heat conduction equation and the temperature field boundary conditions of the through silicon via, a transient temperature field analytical model of the through silicon via is obtained, and 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; Based on the working environment of TSV, the equilibrium equation and mechanical boundary conditions of TSV are established. Based on the equilibrium equation and transient temperature field analytical model, the basic form of the TSV displacement field analytical model and the basic form of the stress field analytical model are determined. 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 the unsteady thermal stress analytical model of the through-silicon via; the unsteady thermal stress of the through-silicon via is obtained based on the unsteady thermal stress analytical model.

2. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, characterized in that: The transient heat conduction equation and temperature field boundary conditions of the TSV are established based on the TSV structure and the working environment, including: According to the structure of the TSV and the working environment, the structural parameters, material parameters and thermal-mechanical coupling boundary conditions of the TSV are obtained; According to the structural parameters, material parameters and thermal-mechanical coupling boundary conditions of the TSV, the transient heat conduction equation of the TSV and the corresponding temperature field boundary conditions are established in the cylindrical coordinate system.

3. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, wherein: The transient temperature field analytical model of the through silicon via is obtained according to the transient heat conduction equation of the through silicon via and the temperature field boundary conditions, including: According to the transient heat conduction equation and the temperature field boundary conditions, the basic form of the transient temperature field analytical model is determined; According to the basic form of the transient temperature field analytical model, the radial characteristic function is obtained; According to the orthogonal relationship of radial characteristic functions, the superposition coefficient is obtained; According to the radial characteristic function and superposition coefficient, the analytical model of the transient temperature field of the through-silicon via is obtained.

4. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 3, wherein: The expression of the transient temperature field analytical model of the through silicon via is as follows: ； in, Indicates the first The temperature difference between the layer temperature and the ambient temperature, represents the number of layers of TSV structure, Indicates the first The thermal conductivity of the layer, Indicates the first Thermal diffusivity of the layer, Indicates the first Thermal diffusivity of the layer, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, Indicates the center axis of the TSV to the The distance from the outermost axis of the layer, in particular, when hour , represents the central axis position of the through silicon via, represents the height of the through silicon via, Indicates the sequence number of the eigenvalue corresponding to the axial characteristic function, Indicates the serial number of the eigenvalue corresponding to the radial eigenfunction, 、 denote the radial coordinate integral variable and the axial coordinate integral variable, respectively. The integral range is ~ , The integral range is 0~ , 、 denote the radial coordinate integral differential and the axial coordinate integral differential, respectively. 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. Indicates time, Indicates the serial number The corresponding eigenvalue of the axial characteristic function, Indicates the first layer The corresponding radial eigenvalues, Indicates the first layer The corresponding radial eigenvalues, Indicates the first The decay coefficient of the layer as a function of time, represents the modulus of the axial characteristic function, represents the modulus of the radial characteristic function, Indicates the first The coefficients of the Bessel function of the first kind in the radial characteristic function of the layer, Indicates the first The coefficients of the second kind Bessel function in the radial characteristic function of the layer, represents the zero-order Bessel function of the first kind, represents the zero-order Bessel function of the second kind, represents the cosine function, represents the exponential function, It represents the temperature difference between the TSV temperature and the ambient temperature at the initial moment.

5. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, wherein: The balance equation of the through silicon via is expressed as follows: ； in, represents the radial normal stress, represents the hoop normal stress, represents the axial normal stress, represents the radial-axial shear stress, 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. represents the radial partial derivative of the radial normal stress, represents the partial derivative of the radial-axial shear stress in the axial direction, represents the radial partial derivative of the radial-axial shear stress, represents the partial derivative of the axial normal stress in the axial direction.

6. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, wherein: The mechanical boundary conditions include: Displacement continuity conditions at interfaces, stress continuity conditions at interfaces, and free boundary conditions at boundaries.

7. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, wherein: The basic expressions of the displacement field and stress field analytical model of the through silicon via are as follows: ； ； ； ； ； ； in, Indicates the first Radial displacement of the layer, Indicates the first Thermoelastic displacement potential of the layer, Indicates the first The Love shift function of the layer, 、 denote the radial and axial coordinates in the cylindrical coordinate system, respectively. express The partial derivative in the radial direction, express about and The second-order mixed partial derivatives of Indicates the first Axial displacement of the layer, Indicates the first Poisson's ratio of the layer, represents the Laplace operator, Express Apply the Laplace operator to perform the operation, express The partial derivative in the axial direction, express The second-order partial derivative in the axial direction, Indicates the first The radial normal stress of the layer, Indicates the first The axial normal stress of the layer, Indicates the first Shear modulus of the layer, Indicates the first Thermal expansion coefficient of the layer, Indicates the first The temperature difference between the layer temperature and the ambient temperature, express The second-order partial derivative in the radial direction, express The second-order partial derivative in the radial direction, Indicates the first The hoop normal stress of the layer, express The partial derivative in the radial direction, express The second-order partial derivative in the axial direction, Indicates the first Radial-axial shear stress of the layer, express about and The second-order mixed partial derivatives of .

8. The method for constructing an unsteady thermal stress analytical model of a through silicon via according to claim 1, wherein: 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 the unsteady thermal stress analytical model of the through silicon via, including: According to the mechanical boundary conditions and the basic form of the displacement field analytical model of the through silicon via, the displacement field analytical model of the through silicon via is obtained; According to the basic forms of the displacement field analytical model and the stress field analytical model of the through silicon via, the transient temperature field and the transient stress field are coupled to obtain the unsteady thermal stress analytical model of the through silicon via.

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