Package interconnection electromigration behavior prediction method based on strain field
By constructing a multi-physics coupled finite element model of thermal-electromechanical stress, calculating the overall strain field and predicting the electromigration behavior of package interconnects, the low prediction accuracy of the electromigration behavior of package interconnects and the impact of multi-physics in the prior art is solved. The sufficient problem is to achieve high-precision prediction at small size and high current density.
Patent Information
- Application Number
- CN202411195526.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-08-29
- Publication Date
- 2024-12-17
AI Technical Summary
The prior art package interconnection electromigration behavior prediction accuracy in small sizes and high current density is low, and the combined effects of temperature and mechanical stress are not fully considered, making it difficult to meet the needs of integrated circuit design.
Using the package interconnect electromigration behavior prediction method based on strain field, the temperature field, current density field and mechanical stress field are solved, and the overall strain field is calculated by constructing geometric model and thermal-electromechanical stress multi-physical field coupled finite element model. And combine strain mode and yield strain to predict electromigration behavior.
It realizes accurate prediction of the electromigration behavior of package interconnects, improves the accuracy of prediction results, and can effectively consider the impact of multi-physics in complex scenarios to meet the needs of integrated circuit design.
Smart Images
Figure CN119150781A_ABST
Abstract
Description
Technical Field
[0001] The invention provides a method for predicting electromigration behavior of package interconnection based on strain field, belonging to the field of electronic packaging. Background Art
[0002] In the field of electronic packaging, with the rapid development of integrated circuit technology, the size of components continues to decrease, and the current density continues to increase, which puts higher requirements on the reliability of package interconnection. Electromigration, as a physical phenomenon caused by the migration of metal atoms due to current, has become an important factor affecting the reliability of package interconnection. Accurate prediction of the electromigration behavior of package interconnection is the key to achieving the optimized design of package interconnection structure and materials. Traditional electromigration behavior prediction methods are mainly based on Black's empirical formula. This method is suitable for application scenarios with larger size and low current density, but it is not effective under small size and high current density. In addition, Black's empirical formula mainly uses the length and current density of package interconnection for prediction and analysis, and lacks comprehensive consideration of temperature and mechanical stress, so the prediction accuracy in complex scenarios is limited, which is difficult to meet the needs of integrated circuit design. Therefore, for package interconnection, there is an urgent need for an electromigration behavior prediction method that integrates the effects of multiple physical fields to improve the accuracy of prediction results. Summary of the invention
[0003] In order to solve the problems of low prediction accuracy of electromigration behavior of package interconnection and insufficient consideration of the influence of multiple physical fields in the prior art, the present invention proposes a method for predicting electromigration behavior of package interconnection based on strain field, which can integrate the effects of multiple physical fields and accurately predict the electromigration behavior of package interconnection.
[0004] A method for predicting electromigration behavior of package interconnects based on strain fields specifically comprises the following steps:
[0005] Step 1: Build the geometry model of the package interconnect and add material parameters.
[0006] Step 2: Based on the geometric model constructed in the first step, a thermal-electric-mechanical stress multi-physics field coupling finite element model is constructed to solve the temperature field, current density field and mechanical stress field of the model under given ambient temperature and current conditions; wherein, (1) the thermal field of the package interconnection is modeled for solid heat transfer and natural convection; (2) the electric field of the package interconnection is modeled based on charge conservation and Ohm's law; (3) the mechanical stress field of the package interconnection is modeled based on the theory of linear elastic mechanics, and the equivalent body load of the current stress is calculated using formula (1); (4) the above-mentioned electric-thermal and thermal-mechanical stress physical fields are coupled respectively for electromagnetic heat generation and thermal expansion;
[0007]
[0008] Where f is the volume load of the current stress, which is a column vector containing components in the three directions of x, y, and z. V0 is the average atomic volume, Z is the effective charge number of the material, e is the unit charge, ρ is the resistivity of the material, J is the current density obtained by solution, Γ is the cross-scale correction factor related to the processing technology, which is a 3×3 matrix. The operator diag(·) represents extracting the diagonal elements of the matrix to form a column vector.
[0009] Step 3: Based on the temperature field obtained in step 2, use formula (2) to calculate the thermally induced strain component of the overall strain field of the model:
[0010] ε t =α(TT ref )I, (2)
[0011] Among them, ε t is the thermal strain matrix, α is the thermal expansion coefficient of the material, T and T ref are the solved temperature and reference temperature respectively, and I is the third-order unit matrix.
[0012] Step 4: Based on the current density field obtained in the second step, use formula (3) to calculate the electrostrain component of the overall strain field of the model:
[0013] ε e =ZeρJkΓ, (3)
[0014] Among them, ε e is the electrostrain matrix, Z is the effective charge number of the material, e is the unit charge, ρ is the resistivity of the material, J is the current density obtained by solution, k is a parameter that characterizes the flexibility of the material at the microscopic scale, which is defined as the inverse of the product of the average atomic area and the elastic modulus, and Γ is the cross-scale correction factor related to the processing technology, which is a 3×3 matrix.
[0015] Step 5: Based on the mechanical stress field obtained in the second step, use formula (4) to calculate the mechanical strain component of the overall strain field of the model:
[0016] ε m =E -1 σ m , (4)
[0017] Among them, ε m is the mechanical strain matrix, σ m The mechanical stress matrix is obtained by solving, and E is the elastic modulus matrix of the material.
[0018] Step 6: Based on the thermal strain components, electrical strain components and mechanical strain components obtained in steps 3, 4 and 5, the overall strain field of the model is calculated using formula (5):
[0019] εtotal =ε t +ε e +ε m , (5)
[0020] Among them, ε total is the overall strain matrix.
[0021] Step 7: Predict the electromigration behavior of the interconnect based on the overall strain field of the model calculated in step 6.
[0022] First, for the overall strain field obtained in step 6, the strain mode is determined by the positive and negative signs of the diagonal components of the strain matrix, where positive values represent tension and negative values represent compression.
[0023] Next, according to the strain pattern obtained by the above analysis, the pattern of electromigration behavior is predicted, wherein the strain of the tensile mode corresponds to the void-type electromigration phenomenon, and the strain of the compressive mode corresponds to the hillock or whisker-type electromigration phenomenon;
[0024] Then, based on the overall strain field calculated in step 6, the evolution degree of electromigration behavior is predicted according to formula (6):
[0025]
[0026] Among them, the subscripts i = x, y, z represent the x, y and z directions respectively, the subscripts j = +, - represent the tensile and compressive strain modes respectively, and η i It is a quantitative index to measure the evolution degree of electromigration behavior in the i direction. The higher the index is, the easier it is for electromigration to occur and the more obvious the phenomenon is. The lower the index is, the less likely it is for electromigration to occur and the less obvious the phenomenon is. total,ii is the strain component in the i direction in the overall strain field, ε c,j is the yield strain of the material at the microscopic scale, Γ ii is the component in the i-direction of the cross-scale correction factor matrix. BRIEF DESCRIPTION OF THE DRAWINGS
[0027] Figure 1 The present invention is a flowchart of the steps of a method for predicting electromigration behavior of package interconnects based on strain fields according to an embodiment of the present invention.
[0028] Figure 2 Schematic diagram of the geometric model and materials of the package interconnection to be predicted according to an embodiment of the present invention.
[0029] Figure 3 This is a diagram showing the solution results of the temperature field of the package interconnect to be predicted according to an embodiment of the present invention.
[0030] Figure 4 This is a diagram showing the solution results of the electric field density field of the package interconnect to be predicted according to an embodiment of the present invention.
[0031] Figure 5 This is a diagram showing the solution results of the mechanical stress field (in the xx direction) of the package interconnect to be predicted according to an embodiment of the present invention.
[0032] Figure 6 This is a diagram showing the solution results of the strain field (in the xx direction) of the package interconnect to be predicted according to an embodiment of the present invention.
[0033] Figure 7 This is a diagram showing the prediction results of the electromigration behavior of the package interconnect to be predicted according to an embodiment of the present invention. DETAILED DESCRIPTION
[0034] The method for predicting electromigration behavior of package interconnect based on strain field proposed by the present invention is further described below in conjunction with the accompanying drawings and specific implementation examples.
[0035] like Figure 1 As shown, the method for predicting electromigration behavior of package interconnect based on strain field includes the following steps:
[0036] S1. In this embodiment, construct Figure 2 The package interconnect geometry model shown in the figure includes three parts: strips, pads and substrate. The strips and pads are made of copper, and the substrate is made of silicon. The relevant materials use the default material parameters of copper and silicon in the COMSOL Multiphysics multi-physics field simulation software.
[0037] S2. Based on the geometric model constructed in S1, a multi-physics coupling finite element model of thermal-electrical-mechanical stress is constructed with the help of the solid heat transfer module, current module and solid mechanics module in COMSOL Multiphysics multi-physics simulation software, as well as the electromagnetic heat and thermal expansion multi-physics coupling interface. In the solid heat transfer module, the ambient temperature and reference temperature are set to 25°C, and the heat flux type on the package interconnect surface is set to natural convection; in the current module, the current of the positive side pad of the package interconnect is set to a given working current of 2.5A, and the potential of the negative side pad is set to ground; in the solid mechanics module, the body load f of the package interconnect is calculated according to formula (7):
[0038]
[0039] Among them, the average atomic volume V0 is 4.73E-29m, the effective charge number Z is 5.00, the unit charge e is 1.60E-19C, the resistivity ρ is 7.64E-11×T / K-5.11E-9Ωm, which is a function of temperature T, J is the current density obtained by solving the current module, and the cross-scale correction factor matrix Γ is [1.12, 0.36, 0.14; 0.36, 1.08, 0.07; 0.14, 0.07, 0.95]. By solving the above multi-physics simulation model, the steady-state temperature field of the package interconnection is obtained (such as Figure 3 As shown), electric field density field (as shown Figure 4 As shown) and mechanical stress field (taking the xx direction as an example, Figure 5 shown).
[0040] S3. Based on the temperature field obtained by solving S2, the thermally induced strain component of the overall strain field of the model is calculated using formula (8):
[0041] ε t =α(TT ref )I, (8)
[0042] Among them, the thermal expansion coefficient α is 1.812E-12×(T / K) 3 +1.461E-9×(T / K) 2 +1.530E-5×T / K-4.739E-3, reference temperature T ref is 25℃.
[0043] S4. Based on the current density field obtained by solving S2, the electrostrain component of the overall strain field of the model is calculated using formula (9):
[0044] ε e =ZeρJkΓ, (9)
[0045] Among them, the flexibility parameter k is 4.887E16 GPa -1 m -2 , the rest of the parameter values are the same as above.
[0046] S5. According to the mechanical stress field obtained by solving S2, the mechanical strain component of the overall strain field of the model is calculated using formula (10):
[0047] ε m =E -1 σ m , (10)
[0048] Among them, the elastic modulus matrix E of the material is that the diagonal elements are 1.396E11-5.078E6×T / K-1.911E5×(T / K) 2 +290.7×(T / K) 3-0.2058×(T / K) 4 +5.385E-5×(T / K) 5 The third-order diagonal matrix of Pa.
[0049] S6. According to the thermal strain components, electrical strain components and mechanical strain components obtained by solving S3, S4 and S5, the overall strain field of the model is calculated using formula (11):
[0050] ε total =ε t +ε e +ε m . (11)
[0051] Taking the xx direction as an example, the overall strain of the model, as well as its thermal strain component, electrical strain component and mechanical strain component are as follows: Figure 6 shown.
[0052] S7. Predict the electromigration behavior of the package interconnect based on the overall strain field of the model calculated in S6.
[0053] First, for the overall strain field obtained by S6, the strain mode is determined by the positive and negative signs of the diagonal components of its strain matrix, where positive values represent tension and negative values represent compression.
[0054] Next, according to the strain pattern obtained by the above analysis, the pattern of electromigration behavior is predicted, wherein the strain of the tensile mode corresponds to the void-type electromigration phenomenon, and the strain of the compressive mode corresponds to the hillock or whisker-type electromigration phenomenon;
[0055] Then, based on the overall strain field calculated in S6, the evolution degree of electromigration behavior is predicted according to formula (12):
[0056]
[0057] Among them, the subscripts i = x, y, z represent the three directions of x, y and z respectively, and the subscripts j = +, - represent the two strain modes of tension and compression respectively. The tensile mode yield strain ε of the material at the microscale is c,+ The compression mode yield strain ε is 0.21%. c,- The value of other parameters is consistent with the previous one.
[0058] Finally, taking the xx direction as an example, the prediction results of the electromigration behavior of the package interconnect are as follows: Figure 7 shown.
Claims
1. A method for predicting electromigration behavior of package interconnects based on strain field, characterized in that: The following steps are involved: Step 1: Build the geometric model of package interconnection and add material parameters; Step 2: Based on the geometric model in step 1, a thermal-electrical-mechanical stress multi-physics field coupling finite element model is constructed to solve the temperature field, current density field and mechanical stress field of the model under given ambient temperature and current conditions; Step 3: Calculate the thermally induced strain component of the overall strain field of the model based on the temperature field solved in step 2; Step 4: Calculate the electrostrain component of the overall strain field of the model based on the current density field obtained in step 2; Step 5: Calculate the mechanical strain component of the overall strain field of the model based on the mechanical stress field obtained in step 2; Step 6: Calculate the overall strain field of the model according to the thermally induced strain components, electrically induced strain components and mechanical strain components obtained in steps 3, 4 and 5; Step 7: Based on the overall strain field of the model calculated in step 6, predict the electromigration behavior of the package interconnect.
2. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The "constructing a thermal-electrical-mechanical stress multi-physics field coupling finite element model" in step 2 further includes the following contents: (1) Modeling the thermal field of package interconnects based on solid heat transfer and natural convection; (2) Modeling the electric field of package interconnects based on charge conservation and Ohm’s law; (3) Modeling the mechanical stress field of package interconnects based on linear elasticity theory; (4) In view of electromagnetic heat generation and thermal expansion, the above-mentioned electric-thermal and thermal-mechanical stress physical fields are coupled respectively.
3. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 2, characterized in that: The mechanical stress field modeling of the package interconnect further includes the following: The equivalent body load of current stress is calculated using formula (1): Where f is the volume load of the current stress, which is a column vector containing components in the three directions of x, y, and z. V0 is the average atomic volume, Z is the effective charge number of the material, e is the unit charge, ρ is the resistivity of the material, J is the current density obtained by solution, Γ is the cross-scale correction factor related to the processing technology, which is a 3×3 matrix. The operator diag(·) represents extracting the diagonal elements of the matrix to form a column vector.
4. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The step 3 of "calculating the thermally induced strain component of the overall strain field of the model according to the temperature field solved in step 2" further includes the following contents: According to the temperature field obtained in step 2, the thermally induced strain component of the overall strain field of the model is calculated using formula (2): e t =α(TT ref )I, (2) Among them, ε t is the thermal strain matrix, α is the thermal expansion coefficient of the material, T and T ref are the solved temperature and reference temperature respectively, and I is the third-order unit matrix.
5. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The step 4 of "calculating the electrostrain component of the overall strain field of the model according to the current density field solved in step 2" further includes the following contents: According to the current density field obtained in step 2, the electrostrain component of the overall strain field of the model is calculated using formula (3): e e =ZeρJkΓ, (3) Among them, ε e is the electrostrain matrix, Z is the effective charge number of the material, e is the unit charge, ρ is the resistivity of the material, J is the current density obtained by solution, k is a parameter that characterizes the flexibility of the material at the microscopic scale, which is defined as the inverse of the product of the average atomic area and the elastic modulus, and Γ is the cross-scale correction factor related to the processing technology, which is a 3×3 matrix.
6. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The step 5 of "calculating the mechanical strain component of the overall strain field of the model according to the mechanical stress field solved in step 2" further includes the following contents: According to the mechanical stress field obtained in step 2, the mechanical strain component of the overall strain field of the model is calculated using formula (4): e m =E -1 s m , (4) Among them, ε m is the mechanical strain matrix, σ m The mechanical stress matrix is obtained by solving, and E is the elastic modulus matrix of the material.
7. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The step 6 of "calculating the overall strain field of the model according to the thermally induced strain components, electrically induced strain components and mechanical strain components obtained in steps 3, 4 and 5" further includes the following contents: According to the thermal strain components, electrical strain components and mechanical strain components obtained in steps 3, 4 and 5, the overall strain field of the model is calculated using formula (5): e total =e t +e e +e m , (5) Among them, ε total is the overall strain matrix.
8. The method for predicting electromigration behavior of package interconnect based on strain field according to claim 1, characterized in that: The step 7 of "predicting the electromigration behavior of the package interconnection according to the overall strain field of the model calculated in step 6" further includes the following contents: Step 71: for the overall strain field obtained in step 6, the strain mode is determined by using the positive and negative signs of the diagonal components of the strain matrix, wherein for the diagonal components, positive values represent stretching and negative values represent compression; Step 72: predicting the mode of electromigration behavior according to the strain mode obtained by the analysis in step 71, wherein the strain in the tensile mode corresponds to the void-type electromigration phenomenon, and the strain in the compressive mode corresponds to the hillock or whisker-type electromigration phenomenon; Step 73: Based on the overall strain field calculated in step 6, the evolution degree of the electromigration behavior is predicted according to formula (6): Among them, the subscripts i = x, y, z represent the x, y and z directions respectively, the subscripts j = +, - represent the tensile and compressive strain modes respectively, and η i It is a quantitative index to measure the evolution degree of electromigration behavior in the i direction. The higher the index is, the easier it is for electromigration to occur and the more obvious the phenomenon is. The lower the index is, the less likely it is for electromigration to occur and the less obvious the phenomenon is. total,ii is the strain component in the i direction in the overall strain field, ε c,j is the yield strain of the material at the microscopic scale, Γ ii is the component in the i-direction of the cross-scale correction factor matrix.