Metal interconnect structure electromigration failure life prediction method, prediction system and computer readable storage medium

Abstract

The application discloses a metal interconnection structure electromigration failure life prediction method, a prediction system and a computer readable storage medium, and belongs to the technical field of semiconductor package reliability prediction. The prediction method comprises the following steps: S1, a simulation model of a metal interconnection structure is established; S2, an electromigration condition parameter is applied to the simulation model; S3, multi-physical field coupling simulation is performed on the simulation model in a time domain, and a change of a joule heat power density in a metal bump in the metal interconnection structure with time is obtained; S4, volume integration is performed on the joule heat power density in the metal bump, and a change of a total joule heat power of the metal bump with time is obtained; S5, according to the total joule heat power and an electric current flowing through the metal bump, a change of an equivalent resistance of the metal bump with time is calculated; S6, based on the change of the equivalent resistance with time, a change of a resistance growth rate of the metal bump with time is determined; and the electromigration failure time of the metal interconnection structure is determined by taking the resistance growth rate reaching a failure threshold as an electromigration failure criterion.

Description

Technical Field

[0001] This invention relates to the field of semiconductor packaging reliability technology, and in particular to a method, system and computer-readable storage medium for predicting electromigration failure lifetime of metal interconnect structures. Background Technology

[0002] With the rapid development of advanced packaging technologies, semiconductor packaging structures are trending towards high integration, small size, and high power density. Metal interconnect structures (interconnect structures with metal bumps), as key interconnect structures for electrical connection and heat conduction between the chip and the substrate, are experiencing continuously increasing current densities. Under the combined effects of high current density and high temperature, atoms within the metal interconnect structure undergo directional migration along the electron flow direction, a phenomenon known as electromigration. As atoms continuously migrate from the cathode side to the anode side, voids or cracks are easily generated on the cathode side, while hump-like protrusions or localized short circuits may occur on the anode side. As voids continue to grow and alter the conductive path, the resistance of the interconnect structure gradually increases, potentially leading to open-circuit failure. Electromigration failure has become a significant factor affecting the reliability of advanced packaging. Accurately predicting the electromigration failure lifetime of metal bumps is crucial for the reliability design and process optimization of packaging structures.

[0003] In the prior art, the methods for obtaining the electromigration failure lifetime of metal interconnect structures mainly include the following two categories.

[0004] One type is accelerated life testing methods. Please refer to reference 1 (Chang et al., Sci. Rep. 2017, 7, 17950). This method applies current density and ambient temperature higher than actual operating conditions to metal bumps to accelerate aging. Electromigration failure is defined as the point at which the resistance of the metal bumps increases by 20%, and the failure lifetime is statistically calculated. This type of method is a destructive test, with a testing cycle of thousands of hours. It requires specialized experimental equipment and samples, and the number of samples available during the R&D phase is limited, making it impossible to quickly assess the lifetime of package structures that are in the design iteration process.

[0005] Another type of criterion is based on the proportion of void area. This method observes the morphology of voids within metal bumps through simulation or experiments, and determines failure when the proportion of the void's projected area to the cross-sectional area of ​​the metal bump reaches a preset value. For example, when the proportion of the void's projected area reaches 16.7%, the corresponding resistance increases by 20%. However, because the shape of voids is irregular and their distribution is uneven during electromigration failure, the resistance distribution at different locations on the metal bump is also uneven. The correlation between the proportion of void area and the increase in resistance changes with the location and shape of the void, leading to significant deviations in failure determination based on the proportion of void area and insufficient accuracy in lifetime prediction.

[0006] However, existing technologies struggle to predict the electromigration failure lifetime of metal bumps quickly and cost-effectively while maintaining accuracy. There is an urgent need in the industry for a prediction method that can address the problems of existing technologies, such as reliance on destructive accelerated experiments leading to high time and cost, or insufficient accuracy based on void area ratio criteria for predicting electromigration failure lifetime. Summary of the Invention

[0007] The purpose of this invention is to provide a method, system, and computer-readable storage medium for predicting the electromigration failure lifetime of metal interconnect structures. This method enables multi-physics coupling simulation of the electromigration process and calculates the change of equivalent resistance over time based on the volume integral of Joule thermal power density. The method uses the equivalent resistance growth rate reaching a threshold as a failure criterion, thereby accurately predicting the electromigration failure lifetime of metal bumps.

[0008] In a first aspect, the present invention provides a method for predicting the electromigration failure lifetime of a metal interconnect structure, comprising: S1 establishing a simulation model of a metal interconnect structure, the metal interconnect structure including metal traces, a pair of metal bumps disposed on the metal traces, and pads stacked on the metal bumps; the simulation model including interconnect structure parameters; S2 applying electromigration condition parameters to the simulation model, the electromigration condition parameters including the current density flowing through the metal interconnect structure and the ambient temperature; S3 performing multiphysics coupling simulation on the simulation model in the time domain to obtain the change of Joule thermal power density within the metal bumps over time; S4 performing volume integral on the Joule thermal power density within the metal bumps to obtain the change of total Joule thermal power of the metal bumps over time; S5 calculating the change of equivalent resistance of the metal bumps over time based on the total Joule thermal power and the current flowing through the metal interconnect structure; S6 determining the change of resistance growth rate of the metal bumps over time based on the change of equivalent resistance over time; and determining the electromigration failure time of the metal interconnect structure by using the failure threshold reached by the resistance growth rate as an electromigration failure criterion.

[0009] In some embodiments, the interconnect structure parameters include the geometry and material properties of the metal trace, the metal bump, and the pad; and / or, the metal bump includes a metal pillar, a first interface layer disposed between the metal pillar and the pad, and a second interface layer disposed between the metal pillar and the metal trace; the first interface layer is a UBM layer, and the second interface layer includes a Ni layer in contact with the metal pillar and an Au layer located between the Ni layer and the metal trace.

[0010] In some implementations, the multiphysics coupling simulation includes the coupling of the following physical fields: solid heat transfer physical field, solid mechanical physical field, solid transport physical field, current physical field, and domain ordinary differential and differential algebraic equation physical field.

[0011] In some implementations, the multiphysics coupling simulation also couples the thermal expansion multiphysics field and the electromagnetic thermal multiphysics field for a coupled solution.

[0012] In some implementations, the multiphysics coupling simulation establishes an atomic migration model based on the mass conservation equation. This atomic migration model couples the atomic migration flux caused by four migration mechanisms: electron wind, stress gradient, thermal gradient, and atomic concentration gradient. In the atomic migration model, the electromigration process within the metal bump satisfies the following mass conservation equation: Where c is the normalized atomic concentration and t is time. ·J represents the divergence of the total diffusion flux; The atomic migration flux It can be expressed by the following formula: ;in, The atomic diffusion coefficient, Let be Boltzmann's constant, T be absolute temperature, and e be the elementary charge. Where ρ is the effective charge number, ρ is the resistivity, j is the current density, and Ω is the atomic volume. For hydrostatic stress, This is the activation energy for thermal migration.

[0013] In some implementations, during the multiphysics coupled simulation, the atomic concentration distribution within the metal bump is calculated based on an atomic migration model, and the distribution of the normalized atomic concentration c within the metal bump over time is solved; in response to the minimum value of the normalized atomic concentration c being lower than the void nucleation threshold, the current simulation time is determined as the void nucleation time; wherein, the normalized atomic concentration c = C v / C v0 C v C represents the atomic concentration at a local location within the metal bump. v0 The initial atomic concentration within the metal bump.

[0014] In some implementations, when the atomic concentration in a local region is lower than the void nucleation threshold, the location is identified as a void region, the resistivity is updated to N times the initial resistivity, the region where no voids have formed retains the initial resistivity, and the spatial distribution of Joule heat power density is recalculated based on the updated resistivity; the void nucleation threshold is 0.85.

[0015] In some embodiments, obtaining the spatial distribution of the Joule thermal power density within the metal bump over time includes: expressing the Joule thermal power density as q = E·j according to the differential form of Joule's thermal law, where q is the local Joule thermal power density, E is the local electric field strength, and j is the local current density.

[0016] In some embodiments, the total Joule heat power of the metal bump at any time t can be obtained by taking the volume integral of the Joule heat power density within the metal bump without voids: Where t is time, Let be the total Joule heat power at time t. Let be the local point flow field intensity at time t. Let t be the local current density at time t, and V be the volume of the metal bump.

[0017] In some implementations, the change of the equivalent resistance over time is expressed as R. eff (t)=P(t) / I 2 (t); where R eff P(t) is the equivalent resistance of the metal bump at time t, P(t) is the total Joule heat power at time t, and I(t) is the current flowing through the metal bump at time t.

[0018] In some implementations, the current I(t) flowing through the metal bump is a constant current.

[0019] In some implementations, the current I(t) flowing through the metal bump is a varying current value.

[0020] In some implementations, the change in resistance growth rate over time is: ΔR(t) = (R eff (t)-R0) / R0; where ΔR(t) is the resistance growth rate at time t, and R0 is the initial resistance.

[0021] In some implementations, the failure threshold is 20%; using the resistance growth rate reaching the failure threshold as the electromigration failure criterion includes: determining whether the resistance growth rate reaches the failure threshold; if so, the metal bump has undergone electromigration failure, and the current simulation time is determined as the electromigration failure lifetime of the metal bump; if not, return to steps S3 to S6 for iteration.

[0022] Secondly, the present invention also provides a system for predicting the electromigration failure lifetime of a metal interconnect structure, comprising: a model building module configured to build a simulation model of a metal interconnect structure, the metal interconnect structure including metal traces, a pair of metal bumps disposed on the metal traces, and pads stacked on the metal bumps; the simulation model including interconnect structure parameters, the interconnect structure parameters including the geometric dimensions and material properties of the metal interconnect structure; a condition application module configured to apply electromigration condition parameters to the simulation model, the electromigration condition parameters including the current density flowing through the metal bumps and the ambient temperature; and a multiphysics coupling simulation module configured to perform time-domain simulation on the simulation model. A multiphysics coupled simulation is performed to obtain the change of Joule thermal power density within the metal bump over time. A power integration module is configured to perform volume integration on the Joule thermal power density within the metal bump to obtain the change of the total Joule thermal power of the metal bump over time. An equivalent resistance calculation module is configured to calculate the change of the equivalent resistance of the metal bump over time based on the total Joule thermal power and the current flowing through the metal bump. A failure determination module is configured to determine the change of the resistance growth rate of the metal bump over time based on the change of the equivalent resistance over time. The electromigration failure time of the metal bump is determined by using the failure threshold reached by the resistance growth rate as the electromigration failure criterion.

[0023] In some implementations, the multiphysics coupling simulation includes the coupling of the following physical fields: solid heat transfer physical field, solid mechanical physical field, solid transport physical field, current physical field, domain ordinary differential and differential algebraic equation physical field, thermal expansion multiphysics field and electromagnetic thermal multiphysics field.

[0024] In some embodiments, the electromigration failure lifetime prediction system for metal interconnect structures further includes: an atomic concentration monitoring module configured to: during the multiphysics coupled simulation, solve for the time distribution of the normalized atomic concentration c within the metal bumps; and, in response to the minimum value of the normalized atomic concentration c being lower than the void nucleation threshold, determine the current simulation time as the void nucleation time; wherein, the normalized atomic concentration c = C v / C v0 C v C represents the atomic concentration at a local location within the metal bump. v0 The initial atomic concentration within the metal bump is 0.85; the void nucleation threshold is 0.85.

[0025] Thirdly, the present invention also provides a method for manufacturing a semiconductor device, comprising the following steps: evaluating metal bumps using the aforementioned method for predicting electromigration failure lifetime of metal interconnect structures; adjusting the parameters of the metal bumps according to the prediction results; and manufacturing the metal bumps according to the adjusted parameters.

[0026] Fourthly, the present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the method for predicting the electromigration failure lifetime of the metal interconnect structure.

[0027] The beneficial effects of this invention are as follows: Compared with traditional empirical models (such as the Black equation), this prediction method, based on multiphysics coupling simulation and dynamic monitoring of equivalent resistance, and using the resistance growth rate as the failure criterion, can significantly improve prediction accuracy. By directly tracking the physical chain of spatial distribution of Joule heat power density, total Joule heat power, equivalent resistance, and resistance growth rate through time-domain simulation, and using the failure threshold of the resistance growth rate as the failure criterion, the time and economic costs generated by a large number of lifetime and failure analysis tests can be greatly reduced or avoided, and electromigration reliability assessment can be completed during the product design stage. Attached Figure Description

[0028] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings involved in the description of the embodiments are briefly introduced below. The drawings in this application are all descriptive schematic diagrams and do not limit the specific morphology of the actual device.

[0029] Figure 1 This is a schematic diagram of the electromigration void evolution analysis process in an embodiment of the present invention.

[0030] Figure 2 This is a schematic diagram of the electromigration simulation model of the metal interconnect structure in an embodiment of the present invention.

[0031] Figure 3 This is a graph showing the change of minimum normalized atomic concentration over time in Example 1 of the present invention.

[0032] Figure 4 This is a graph showing the change in resistance growth rate over time in Embodiment 1 of the present invention.

[0033] Figure 5 This is a schematic diagram of the temperature gradient distribution within the interconnect structure at the time of failure in Embodiment 1 of the present invention.

[0034] Figure 6 This is a schematic diagram of the current density distribution within the interconnect structure at the time of failure in Embodiment 1 of the present invention.

[0035] Figure 7 This is a schematic diagram of the normalized atomic concentration distribution within the interconnect structure at the time of failure in Embodiment 1 of the present invention.

[0036] Figure 8 This is a graph showing the change of minimum normalized atomic concentration over time in Example 2 of the present invention.

[0037] Figure 9 This is a graph showing the change in resistance growth rate over time in Embodiment 2 of the present invention.

[0038] Figure 10 This is a schematic diagram of the temperature gradient distribution within the interconnect structure at the time of failure in Embodiment 2 of the present invention.

[0039] Figure 11 This is a schematic diagram of the current density distribution within the interconnect structure at the time of failure in Embodiment 2 of the present invention.

[0040] Figure 12 This is a schematic diagram of the normalized atom concentration distribution within the interconnect structure at the failure time of Embodiment 2 of the present invention.

[0041] Figure 13 This is a schematic diagram of the three-dimensional response of failure time, current density, and temperature when using the Black equation for prediction in the embodiment.

[0042] Figure 14 To verify the prediction of electromigration failure lifetime of the metal interconnect structure of the present invention, the curve of the resistance growth rate over time is shown.

[0043] Among them, 1-metal trace, 2-metal bump, 21-metal pillar, 22-first interface layer, 23-second interface layer, 231-Ni layer, 232-Au layer, 3-pad, 4-printed circuit board. Detailed Implementation

[0044] The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0045] It should be noted that, unless otherwise explicitly specified and limited, the term "connection" should be interpreted broadly. For example, a connection can be a direct connection or an indirect connection through an intermediate medium; it can be a fixed connection, a movable connection, a detachable connection, or an integral connection. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0046] In the process of realizing this invention, it was discovered that the essence of electromigration failure is the formation and expansion of voids caused by atomic migration. The electromigration void evolution analysis process is as follows: Figure 1 As shown, the formation and expansion of voids lead to a decrease in the conductive cross-section of the metal bump 2 within the metal interconnect structure, a redistribution of current density, an increase in Joule heat power density, and a continuous increase in equivalent resistance. Accurate prediction of void evolution, resistance changes, and failure time during electromigration is of great significance for the design, material selection, and reliability assessment of the metal bump 2.

[0047] This invention aims to provide a method for predicting the electromigration failure lifetime of a metal interconnect structure, comprising the following steps: S1 establishing a simulation model of the metal interconnect structure, the metal interconnect structure including a metal trace 1, a pair of metal bumps 2 disposed on the metal trace 1, and pads stacked on the metal bumps 2; the simulation model includes interconnect structure parameters; S2 applying electromigration condition parameters to the simulation model, the electromigration condition parameters including the current density flowing through the metal bumps 2 and the ambient temperature; S3 performing multiphysics coupling simulation on the simulation model in the time domain to obtain the change of Joule heat power density in the metal bumps 2 over time; S4 performing volume integral on the Joule heat power density in the metal bumps 2 to obtain the change of total Joule heat power of the metal bumps 2 over time; S5 calculating the change of equivalent resistance of the metal bumps 2 over time based on the total Joule heat power and the current flowing through the metal bumps 2; S6 determining the change of resistance growth rate of the metal bumps 2 over time based on the change of equivalent resistance over time; using the failure threshold reached by the resistance growth rate as the electromigration failure criterion to determine the electromigration failure time of the metal bumps 2.

[0048] This prediction method is based on the Joule heat power density theory within the metal bump 2 in the metal interconnect structure. By simulating the change in resistance over time, it can more accurately predict the electromigration failure lifetime of the metal interconnect structure.

[0049] Temperature gradient, hydrostatic stress gradient, atomic concentration gradient, and electron wind are primarily driving forces for atomic migration, rather than terminating failure criteria. This prediction method couples the electric field, temperature field, stress field, and atomic concentration field. These factors influence current congestion, Joule heating, void evolution, and atomic flux, ultimately reflected in changes in equivalent resistance. Therefore, a 20% increase in resistance is used as the overall criterion.

[0050] Compared to traditional empirical models (such as the Black equation), this prediction method, based on multiphysics coupled simulation and dynamic monitoring of equivalent resistance, and using the resistance growth rate as the failure criterion, can significantly improve prediction accuracy. By directly tracking the physical chain of spatial distribution of Joule heat power density, total Joule heat power, equivalent resistance, and resistance growth rate through time-domain simulation, and using the failure threshold reached by the resistance growth rate as the failure criterion, the time and economic costs incurred by numerous lifetime and failure analysis tests can be greatly reduced or avoided, allowing electromigration reliability assessment to be completed during the product design phase.

[0051] The technical solution of the present invention will be described in detail below from aspects such as method flow, simulation model, criterion setting, and verification results. In each embodiment, the same terms have the same meaning. Those skilled in the art should understand that the following embodiments are used to illustrate the technical solution of the present invention and are not intended to limit the scope of protection of the present invention.

[0052] In step S1, the parameterized simulation model can cover metal interconnect structures with different material systems and different geometric dimensions, so that the prediction method can be applied to different process nodes and packaging forms, and has good versatility.

[0053] Interconnect structure parameters include the geometric dimensions and material properties of each layer (metal trace 1, metal bump 2, pad) in the metal interconnect structure. Geometric dimensions include the lateral dimensions (dimensions of each layer within the extension plane) and thickness of each layer. Material properties include material composition, etc.

[0054] In some embodiments, the metal trace 1 is located on the printed circuit board 4 (PCB), and the metal trace 1 is preferably a copper trace (Cu trace). The system includes a metal pillar 21, a first interface layer 22 disposed between the metal pillar 21 and the pad, and a second interface layer 23 disposed between the metal pillar 21 and the metal trace 1. The first interface layer 22 is an under bump metallurgy layer (UBM layer); the UBM layer can be any of those used in the prior art, such as a superimposed film layer of titanium and copper. The second interface layer 23 includes a Ni layer 231 in contact with the metal pillar 21 and an Au layer 232 located between the Ni layer 231 and the metal trace 1.

[0055] In one embodiment, the UBM layer and the metal pillar 21 have the same lateral dimensions, but the thickness of the UBM layer is less than the thickness of the metal pillar 21. The lateral dimensions of the pads are greater than the lateral dimensions of the UBM layer.

[0056] In step S2, by flexibly combining current density and ambient temperature, the entire stress space from normal working conditions to accelerated aging test conditions can be covered, supporting various current loading methods such as constant current and variable current. This can simulate the dynamic working conditions (such as current fluctuations and temperature fluctuations) of metal interconnect structures in actual use, which is closer to the real use scenario than traditional constant temperature and constant current accelerated test.

[0057] In step S3, the coupled evolution of multiple physical fields is analyzed simultaneously under a unified framework, accurately capturing the spatial distribution of temperature gradient, current density, normalized atomic concentration, and Joule thermal power density within the metal bump 2 over time. This allows for the identification of characteristics at each stage of the electromigration failure process. Furthermore, it can identify the non-uniform distribution of current density and temperature within the metal bump 2, reducing electromigration lifetime prediction errors and improving prediction accuracy.

[0058] To clearly illustrate multiphysics coupling simulation, the electromigration mechanism will be explained in detail first.

[0059] Electromigration (EM) generally refers to the directional migration of metal atoms within a conductor under high current density, driven by electron wind. As atoms migrate continuously from the cathode side to the anode side, voids or cracks are easily generated on the cathode side, while mound-like protrusions or local short circuits may appear on the anode side, ultimately leading to open-circuit failure of the interconnect. Numerous studies have found that void failure occurs on the cathode side much earlier than on the anode side; therefore, the electromigration failure lifetime prediction method for metal interconnect structures in this invention focuses on predicting the electromigration failure lifetime on the cathode side.

[0060] Two metal bumps 2 were used as the anode and cathode, respectively, for testing. Figure 2 The metal bump 2 on the left is denoted as the left bump, and the metal bump 2 on the right is denoted as the right bump. The left bump and the right bump are calculated separately.

[0061] This invention uses an atomic migration model based on mass conservation to describe the electromigration process in the metal bump 2.

[0062] In the atomic migration model, the rate of change of atomic concentration at a local location within the metal bump 2 is equal to the divergence of the atomic diffusion flux flowing into that location, and the electromigration process satisfies the following mass conservation equation (Equation 1):

[0063] Where c is the normalized atomic concentration; t is time; J It is the total diffusion flux of atoms (abbreviated as "atomic migration flux"). ·J represents the divergence of the total diffusion flux.

[0064] Atom migration flux Equation 2 represents:

[0065] J E It is the atomic migration flux caused by electron wind, determined by atomic migration driven by an electric field, and is called electromigration flux. J E Equation 3 represents:

[0066] J S It is the atomic migration flux caused by the stress gradient, determined by atomic migration driven by the stress gradient, and is called the stress migration flux. J S Equation 4 represents:

[0067] J TIt is the atomic migration flux caused by the thermal gradient, which is determined by atomic migration driven by the temperature gradient and is called thermal migration flux. J T Equation 5 represents:

[0068] J C It is the atomic migration flux caused by the atomic concentration gradient, determined by atomic diffusion driven by the concentration gradient, and is called the concentration gradient diffusion flux. J C Equation 6 represents:

[0069] In Formula 3~Formula 6, The atomic diffusion coefficient, Let be Boltzmann's constant, T be absolute temperature, and e be the elementary charge. Where ρ is the effective charge number, ρ is the resistivity, j is the current density, and Ω is the atomic volume. For hydrostatic stress, This is the activation energy for thermal migration.

[0070] Combining the four migration mechanisms mentioned above, atomic migration flux It can be expressed by the following formula:

[0071] It can be seen that the electromigration process in the metal bump 2 is not determined by the current density alone, but is the result of the combined effect of the current density, temperature field, stress field and concentration field.

[0072] Under high current density operating conditions, electron wind promotes atomic migration, Joule heating increases local temperature and enhances diffusion capacity, and temperature gradient further induces electromigration, while stress and concentration gradients compensate for or regulate the atomic migration process. By solving the above mass conservation equation and total migration flux equation, the change law of atomic concentration inside metal bump 2 over time can be obtained, which can be further used to determine the time of void formation, resistance increase, and electromigration failure. Specifically, based on the interconnect structure parameters and electromigration condition parameters, a multiphysics coupling simulation model of metal bump 2 is established, and the multiphysics coupling simulation model is solved to obtain the current field, temperature field, stress field, and atomic concentration field inside metal bump 2.

[0073] Multiphysics coupling simulation includes the coupling of the following physical fields: current physical field, solid heat transfer physical field, solid mechanics physical field, solid transport physical field, and domain ordinary differential and differential algebraic equation physical field. The synergistic coupling of these five physical fields constructs the complete physical framework for electromigration simulation.

[0074] By solving the heat conduction equation using the solid heat transfer physics field, the temperature field distribution in metal bump 2 was obtained. Temperature affects the atomic diffusion coefficient and the resistivity of the material, and is a key accelerating factor for electromigration rate.

[0075] By solving the current continuity equation using the physical field of the current, the current density distribution in metal bump 2 can be obtained. The current density distribution is a prerequisite for calculating the Joule heat power density and is also the driving force for electromigration (the source of electron wind).

[0076] By solving the mechanical equilibrium equations using solid mechanics, the stress field distribution in metal bump 2 is obtained. Vacancies accumulated due to electromigration cause volume changes, which in turn alter the stress state in local areas; stress gradients can also cause stress migration due to electron migration.

[0077] By solving the vacancy diffusion equation using the solid-state transport physics field, the evolution of the vacancy concentration in metal bump 2 over time can be obtained. The change in vacancy concentration can reflect the microscopic evolution process of electron migration damage.

[0078] Domain ordinary differential equations and differential-algebraic equations are used in physics to define and solve evolution equations for global and state variables that are independent of spatial coordinates. For example, they can manage the updating of material parameters, enable variable transfer and coupled control across physics fields, and ensure information synchronization between different physics fields.

[0079] There are complex coupling relationships among the aforementioned physical fields. For example, the electric current field generates heat based on the Joule heating effect, causing temperature changes and altering the temperature field; it also drives electron migration based on the electron wind mechanism. Temperature changes affect the resistivity of materials and also cause thermal stress based on thermal expansion, altering the stress field. Changes in the stress field, in turn, cause vacancy migration, and the accumulation of vacancy leads to volume deformation, which also alters the stress field.

[0080] Therefore, multiphysics coupling simulation also couples thermal expansion multiphysics and electromagnetic-thermal multiphysics for coupled solution. Thermal expansion coupling couples the temperature field with the solid mechanical field through the thermal strain relationship, enabling the simulation to simultaneously consider both electromigration and thermal migration failure mechanisms, and to analyze the influence of migration force generated by thermal gradient on electromigration force.

[0081] Electromagnetic thermal coupling bidirectionally couples the physical field of electric current with the physical field of heat transfer in a solid. During electromigration, the formation and growth of voids leads to a reduction in the conductive cross-sectional area and a further increase in local current density, generating stronger Joule heating and creating positive feedback acceleration for void growth. Electromagnetic thermal coupling can capture this accelerated failure mechanism and will not underestimate this positive feedback effect.

[0082] Compared to the weak coupling method that solves each physical field and then superimposes them, the multiple physical field coupling model of this invention uses a strong coupling method to solve the problem within a unified framework. This method can more accurately capture the real-time feedback and positive feedback acceleration mechanism between physical fields, and analyze the interaction mechanism between stress evolution, void growth, resistance change and Joule heating effect during electromigration.

[0083] By coupling the above-mentioned multiple physical fields, the interaction of the four migration mechanisms can be considered simultaneously in a unified simulation environment, providing a relatively comprehensive physical simulation basis for the prediction of electromigration failure lifetime of metal bump 2.

[0084] It should be noted that the multiphysics coupled simulation uses a time-domain fully coupled solver. Within each time step, electric potential, temperature, displacement, and atomic concentration are treated as joint unknowns and assembled into a unified set of nonlinear equations, which are then updated synchronously through Newton iteration. The next time step is only initiated after the residuals of all governing equations meet the convergence tolerance.

[0085] The specific coupling relationships of the various physical fields are as follows: the current density generates Joule heating, which changes the temperature field; temperature affects the electric field, atomic migration, and hydrostatic stress through conductivity, diffusion coefficient, and thermal expansion; the current density, temperature gradient, hydrostatic stress gradient, and atomic concentration gradient jointly drive atomic migration; the atomic concentration is fed back to the electric field and Joule heating through concentration-dependent resistivity, thus forming a closed-loop coupling.

[0086] The Joule heat power density q refers to the Joule heat generated per unit volume per unit time within the metal bump 2. Obtaining the spatial distribution of the Joule heat power density within the metal bump 2 over time involves: expressing the Joule heat power density as the dot product of the electric field strength and current density within the metal bump 2, according to the differential form of Joule's law, i.e., q = E·j. Here, q is the local Joule heat power density, E is the local electric field strength, and j is the local current density.

[0087] In metal bump 2, the current density is not uniformly distributed. The local Joule thermal power density can be calculated independently on each grid cell using a differential form, rather than directly obtaining the Joule thermal power density of the entire metal bump 2, thus providing spatial resolution.

[0088] In the process of realizing this invention, it was also discovered that the electromigration failure process can include void nucleation, void growth, and interconnect structure breakage. Significant changes in resistance only occur after the voids have grown to a certain size; therefore, relying solely on resistance changes for judgment can only capture signals from the later stages of void growth or the interconnect structure breakage stage.

[0089] In the multiphysics coupled simulation, the atomic concentration distribution within the metal bump 2 is calculated based on the atomic migration model, and the distribution of the normalized atomic concentration c within the metal bump 2 over time is solved. In response to the minimum value of the normalized atomic concentration c falling below the void nucleation threshold, the current simulation time is determined as the void nucleus formation time. That is, the normalized atomic concentration c at each location within the metal bump 2 is monitored in real time during the simulation; when the normalized atomic concentration c at any location within the metal bump 2 decreases to the void nucleation threshold, it is determined that a void nucleus has formed at that location on the metal bump 2.

[0090] By introducing a microscopic failure criterion, precise detection of void nucleation is achieved. A failure threshold (e.g., 20%) is used to determine overall failure, and a void nucleation threshold (e.g., 0.85%) is used to determine void nucleation. Together, they constitute a complete electromigration failure criterion system.

[0091] The time from the start of the test to the formation of a void is recorded as the void nucleation time, and the time between the void nucleation time and the failure time is recorded as the void growth time. Phased evaluation is of great significance for engineering practice. For example, if the void nucleation time accounts for a large proportion of the total lifetime, the optimization focus of metal bump 2 can be placed on suppressing void nucleation; if void growth accounts for a large proportion of the total lifetime, the optimization focus of metal bump 2 can be placed on its resistance to void propagation.

[0092] The normalized atomic concentration *c* represents the ratio of the atomic concentration at a local location within metal bump 2 to the initial atomic concentration, i.e. c = C v / C v0 C v C represents the atomic concentration at a local location within metal bump 2. v0 c represents the initial atomic concentration within metal bump 2. A c of 1 indicates that the atomic concentration at this location has not changed, c less than 1 indicates that atoms have been lost at this location, and c greater than 1 indicates that atoms have accumulated at this location.

[0093] Specifically, when c > 0.9, the resistivity of the local region remains at its initial value. When 0.85 < c < 0.9, the resistivity of the local region increases linearly and is recoverable. When c ≤ 0.85, the local region is considered to have formed a void and permanently locked in high resistivity.

[0094] When the normalized atomic concentration c decreases to the void nucleation threshold, a void nucleus is determined to form at that location; when the normalized atomic concentration c falls below the void nucleation threshold, the region is a void region. The resistivity immediately becomes N times (e.g., 32 times) of the initial value, while the region where no voids have formed retains its initial resistivity.

[0095] The spatial distribution of Joule thermal power density is re-solved based on the updated resistivity. This step introduces the core physical feedback mechanism of "updating the resistivity to N times the initial resistivity after the formation of voids, thereby causing the electric / current field to be re-solved" into the multiphysics coupling simulation. The resulting Joule thermal power density is not static or only changes slightly with temperature, but dynamically changes with the metal bump 2. Therefore, it can simulate the significant nonlinear growth trend of resistance.

[0096] Please refer to reference 2 (Yang et al., ACM Trans. Des. Autom. Electron. Syst. 2025, 30, 1), where the specific definition of resistivity as a function of atomic concentration is as follows: Where ρ0 is the base resistivity (initial resistivity) considering the effect of temperature, and Dam is the concentration damage variable.

[0097] Multiphysics simulation software (such as COMSOL Multiphysics) recalculates local resistivity and conductivity based on the currently obtained atomic concentration and temperature in each nonlinear iteration, without modifying each mesh element individually. This model only establishes the feedback relationship between atomic concentration and resistivity. Parameters such as thermal conductivity, coefficient of thermal expansion, and Young's modulus are expressed using temperature-dependent expressions from the material library of the multiphysics simulation software, and are not set as functions of the normalized atomic concentration c.

[0098] After resolving the current field based on the updated conductivity properties, the current bypasses the void region and forms current congestion at the void edge, increasing the local current density and Joule heat power density, which in turn increases the local temperature. The temperature increase further accelerates atomic migration at the void edge region, causing the void to continue to expand, and the cross-sectional area of ​​the region in metal bump 2 where no void has been formed continues to decrease.

[0099] Through the iteration between updating conductivity properties and resolving the current field, the current congestion and Joule heating enhancement effects caused by void evolution are quantitatively included in the simulation results, thus accurately reflecting the physical mechanism of continuous resistance growth during electromigration failure.

[0100] In some implementations, the void nucleation threshold is 0.85. That is, void nuclei begin to form at this location when the atomic concentration decreases to 85% of the initial atomic concentration. This threshold corresponds to the void nucleation time observed by electron microscopy and is earlier than the moment when the resistance growth rate increases significantly, providing phased information about the failure process, such as the void nucleation time and void growth time.

[0101] In step S4, the overall electrothermal state characterization of the metal bump 2 is obtained by volume integral of the Joule thermal power density of the metal bump 2, eliminating the risk of misjudgment caused by local fluctuations. A volume integral operator is established in the software's "Define—Component Coupling—Integrate" function, and its source selection range is set to the entire metal bump 2 that carries the current transmission, thus obtaining the boundary of the volume integral of the metal bump 2 and accurately defining the target integration domain for calculating the equivalent resistance.

[0102] By dividing the Joule heat power density q within the metal bump 2 by volume, the total Joule heat power of the metal bump 2 at any time t can be obtained:

[0103] Where t is time, Let be the total Joule heat power at time t. Let be the local point flow field intensity at time t. Let t be the local current density at time t, and V be the volume of metal bump 2.

[0104] The volume integral is a weighted sum of all local thermal effects and conforms to the laws of physical conservation. During electromigration, the formation of voids causes the conductivity of local regions to approach zero, and the volume integral can accurately reflect the actual change in the heating power of void regions.

[0105] S5, based on the total Joule heat power and the current flowing through the metal bump 2, uses P=I 2 The equivalent resistance of metal bump 2 is calculated over time using R. The equivalent resistance is calculated using the Joule thermal power density volume integral, allowing the change in equivalent resistance to directly derive from the physical field simulation results. This eliminates the need to introduce an additional resistance network model into the simulation, ensuring consistency between the equivalent resistance calculation and the electromigration damage evolution, and improving the accuracy of failure determination.

[0106] The evolution of equivalent resistance reflects the manifestation of microstructural changes caused by electromigration in electrical performance. It helps to distinguish the contribution of failure mechanisms (such as failure caused by voids) to the change in resistance. The whole process is driven by physical field simulation, which can improve the credibility of the prediction method.

[0107] The equivalent resistance refers to the resistance value when the entire metal bump 2 is considered as a resistive element. The change of equivalent resistance over time is expressed as: R eff (t)=P(t) / I 2 (t). Wherein, R eff P(t) is the equivalent resistance of metal bump 2 at time t, P(t) is the total Joule heat power at time t, and I(t) is the current flowing through metal bump 2 at time t.

[0108] The physical basis of this formula is that Joule heating originates from the resistive loss within metal bump 2. The simulation model includes the geometric dimensions and material properties of each structure in the metal interconnect structure, and the equivalent resistance calculated by this formula is the true equivalent resistance of metal bump 2.

[0109] In some implementations, the current I(t) flowing through the metal interconnect structure (metal bump 2) is a constant current, that is, a constant current is applied to the metal interconnect structure. This avoids calculating the current value at any given time, reducing one variable dimension.

[0110] The current can be a variable value, and it needs to be calculated according to the formula for variable current.

[0111] In step S6, the electromigration failure time of metal bump 2 is determined by using the resistance growth rate reaching the failure threshold as the electromigration failure criterion. The resistance growth rate is a more sensitive failure indicator than the absolute resistance value and can be used to identify the inflection point of electromigration from slow degradation to accelerated failure, thus enabling earlier failure warning.

[0112] ΔR(t) is the resistance growth rate at time t, that is, ΔR(t) is the ratio of the increase in equivalent resistance at time t relative to the initial equivalent resistance R0, ΔR(t) = (R0 / T) * R0 / T. eff (t)-R0) / R0. Where R0 is the initial equivalent resistance, that is, the equivalent resistance of metal bump 2 before electromigration occurs.

[0113] The first threshold is the resistance growth rate threshold used to determine whether the metal bump 2 has undergone electromigration failure. In this invention, the failure threshold is 20%, which is consistent with the industry practice of recognizing electromigration failure as a 20% increase in resistance in accelerated life testing. By adopting a unified 20% failure threshold, the failure time determined by simulation and the failure time determined by experiment are comparable, and the simulation results can be directly compared with the accelerated test data.

[0114] Using the resistance growth rate reaching the failure threshold as the criterion for electromigration failure includes: determining whether the resistance growth rate has reached the failure threshold; if yes, then the metal bump 2 has undergone electromigration failure, and the current simulation time is determined as the electromigration failure lifetime of the metal bump 2 (the simulation time from the start of the simulation to the determination of failure); if no, then return to steps S3 to S6 for iteration.

[0115] Specifically, when the resistance growth rate has not reached the failure threshold, the simulation time step is advanced, and the steps of updating the conductivity characteristics and multiphysics coupling simulation are iteratively executed until the resistance growth rate reaches the failure threshold.

[0116] The above process allows the acquisition of electromigration failure lifetime to be completed mainly in the simulation stage, without the need to manufacture samples or conduct destructive accelerated experiments, thereby shortening the lifetime assessment cycle and reducing assessment costs. The equivalent resistance is directly calculated by the volume integral of Joule thermal power density, avoiding the deviations caused by non-uniform resistance measurement and non-uniform void distribution, thus improving the accuracy of lifetime prediction.

[0117] The present invention also uses the Black equation to verify the above prediction method.

[0118] The Black equation is an empirical equation describing the relationship between electromigration failure lifetime and current density and temperature in interconnect structures. Using current density j and temperature T as two independent variables, the Black equation is used to fit the failure time (TTF) bivariately. The fitted equation is as follows:

[0119] Where TTF is the electromigration failure lifetime, A is an empirical constant, j is the current density, and n is the current density exponent. To activate energy, is Boltzmann's constant, and T is the absolute temperature.

[0120] Ea is a preset known value, and A and n are obtained by fitting the electromigration failure lifetime obtained under multiple simulation conditions.

[0121] The simulation of electromigration failure lifetime using the above prediction method was compared with the results verified by the Black equation. The results showed that the error of the simulation results of the prediction method was no greater than -5.00%.

[0122] This invention also provides a system for predicting the electromigration failure lifetime of metal interconnect structures, including a model building module, a condition application module, a multiphysics coupling simulation module, a power integration module, an equivalent resistance calculation module, and a failure determination module. These modules work collaboratively to implement any of the aforementioned methods for predicting the electromigration failure lifetime of metal interconnect structures.

[0123] The model building module is configured to build a simulation model of a metal interconnect structure, which includes a metal trace 1, a pair of metal bumps 2 disposed on the metal trace 1, and pads stacked on the metal bumps 2. The simulation model includes interconnect structure parameters, which include the geometric dimensions and material properties of the metal interconnect structure. The model building module is further configured to perform any of the steps S1 described above.

[0124] The condition application module is configured to apply electromigration condition parameters to the simulation model, including the current density flowing through the metal bump 2 and the ambient temperature. The condition application module is further configured to perform any of the steps S2 described above.

[0125] The multiphysics coupling simulation module is configured to perform multiphysics coupling simulation on the simulation model in the time domain to obtain the change of Joule heat power density in the metal bump 2 in the metal interconnect structure over time.

[0126] The multiphysics coupling simulation module is further configured to perform any of the steps S3 described above. For example, the multiphysics coupling simulation includes the coupling of the following physical fields: solid heat transfer physical field, solid mechanical physical field, solid transport physical field, current physical field, domain ordinary differential and differential algebraic equation physical field, thermal expansion multiphysics field, and electromagnetic thermal multiphysics field.

[0127] The power integration module is configured to perform volume integration on the Joule heat power density within the metal bump 2 to obtain the change in the total Joule heat power of the metal bump 2 over time. The power integration module is further configured to perform any of the steps S4 described above.

[0128] The equivalent resistance calculation module is configured to calculate the change in the equivalent resistance of the metal bump 2 over time based on the total Joule heat power and the current flowing through the metal bump 2. The equivalent resistance calculation module is further configured to perform any of the steps S5 described above.

[0129] The failure determination module is configured to determine the change in the resistance growth rate of the metal bump 2 over time based on the change in equivalent resistance over time; and to determine the electromigration failure time of the metal bump 2 by using the resistance growth rate reaching a failure threshold as the electromigration failure criterion. The failure determination module is further configured to implement any of the steps S6 described above.

[0130] The electromigration failure lifetime prediction system for metal interconnect structures also includes an atomic concentration monitoring module. This module is configured to solve for the distribution of the normalized atomic concentration c within the metal bump 2 over time during multiphysics coupled simulations; in response to the minimum value of the normalized atomic concentration being lower than the void nucleation threshold, the current simulation time is determined as the void nucleation time.

[0131] Wherein, the normalized atomic concentration c=C v / C v0 C v C represents the atomic concentration at a local location within metal bump 2. v0 The initial atomic concentration within metal bump 2 is given. The void nucleation threshold is 0.85.

[0132] The present invention also provides a method for manufacturing a semiconductor device, comprising the following steps: evaluating a metal bump 2 using a metal interconnect structure electromigration failure lifetime prediction method; adjusting the parameters of the metal bump 2 according to the prediction results; and manufacturing the metal bump 2 according to the adjusted parameters.

[0133] The present invention also provides a computer-readable storage medium having a computer program stored thereon. When executed by a processor, the computer program implements the electromigration failure lifetime prediction method for metal interconnect structures described above.

[0134] The prediction method and prediction system of the present invention will be described in detail below with specific embodiments.

[0135] Example 1 Multiphysics coupling simulation is achieved using multiphysics simulation software such as COMSOL Multiphysics.

[0136] like Figure 2 As shown, the simulation model represents the metal interconnect structure between the chip and the substrate, including aluminum pads (Al Pad), metal bumps 2, and copper traces (Cu Trace). Metal bumps 2 consist of a UBM layer, Cu pillars, Ni layer 231, and Au layer 232. The aluminum pads are 1 μm thick and have a lateral dimension of 28 μm × 28 μm. The UBM layer consists of a 150 nm thick titanium layer and a 300 nm thick copper layer. The copper-nickel-gold (CuNiAu) bumps have a diameter of 18 μm, a height of 12 μm, and a copper, nickel, and gold thickness ratio of 8:0.25:1.

[0137] When the material composition is determined, the necessary material parameters such as electrical conductivity, thermal conductivity, atomic diffusion coefficient, effective charge number, and activation energy of each layer are also determined in this field.

[0138] At an ambient temperature of 150℃ and an applied current density of 7.71×10⁻⁶, 4 A / cm 2 Taking the simulated working conditions as an example, this paper explains the method for predicting the electromigration failure lifetime of metal interconnect structures. Figure 2 The curves showing the changes in minimum normalized atomic concentration, resistance growth rate, temperature, current density, and normalized atomic concentration distribution over time inside the metal bump 2 (cathode) on the right side are as follows: Figures 3 to 7 As shown.

[0139] Please refer to Figure 3 As shown, at the start of the simulation, the electric field, temperature, and atomic concentration distribution within metal bump 2 are uniform, and the equivalent resistance is the initial equivalent resistance R0. As the simulation progresses, atoms within metal bump 2 undergo directional migration under the influence of electron wind, resulting in a continuous loss of atoms from the cathode side and a gradual decrease in the normalized atomic concentration. At 394 hours of simulation, the normalized atomic concentration at the target location on the cathode side of the metal interconnect structure decreases to 0.85, indicating the formation of a void at this location, which then undergoes nucleation.

[0140] After the formation of voids, the renewal of conductivity properties and the iterative resolving of the current field begin to play a role. The void region becomes insulating, and the current becomes crowded at the edge of the void, resulting in a significant increase in local current density and temperature.

[0141] Please refer to Figure 4 As shown, after 3192 hours of simulation, the equivalent resistance growth rate reached 20%, indicating that metal bump 2 has undergone electromigration failure, with an electromigration failure lifetime of 3192 hours. Please refer to... Figures 5 to 7 As shown, at the moment of failure, the highest temperature inside metal bump 2 was 190.55℃, and the maximum current density was 1.07×10⁻⁶. 7 A / cm 2 The minimum normalized atomic concentration is 0.70.

[0142] This embodiment simultaneously obtained the void nucleation time (394h) and the electromigration failure lifetime (3192h). The failure propagation stage after void nucleation lasted for 2798h, accounting for 87.7% of the entire failure process, indicating that the void growth stage is the main component of the electromigration failure lifetime, and also demonstrating that the two-way coupled closed-loop simulation can completely describe the failure process.

[0143] Example 2 The difference from Implementation 1 is that the ambient temperature is 100℃ and the applied current density is 1×10⁻⁶. 5 A / cm 2 Taking the simulated operating conditions as an example, the method for predicting the electromigration failure lifetime of metal bump 2 is explained. The test results are as follows: Figures 8 to 12 As shown.

[0144] At the start of the simulation, the field quantities within the metal bump 2 are uniformly distributed.

[0145] like Figure 8 As shown, when the simulation reached 1997h, the normalized atomic concentration at the target position on the cathode side of the metal interconnect structure decreased to 0.85, indicating that a void was formed at this position, and the void nucleated.

[0146] Please refer to Figure 9 As shown, the void continued to expand thereafter. When the simulation reached 2330h, the equivalent resistance growth rate reached 20%, and it was determined that the metal bump 2 had undergone electromigration failure. The electromigration failure lifetime was 2330h.

[0147] Please refer to Figures 10 to 12 As shown, at the moment of failure, the highest temperature inside metal bump 2 was 162.76℃, and the maximum current density was 2.43×10⁻⁶. 6 A / cm 2 The minimum normalized atomic concentration is 0.83.

[0148] Comparing Examples 1 and 2, it can be seen that when the ambient temperature decreases by 50°C, due to a 29.7% increase in applied current density, the void nucleation time is extended from 394 hours to 1997 hours, while the failure propagation stage is significantly shortened, and the electromigration failure lifetime is reduced from 3192 hours to 2330 hours. This method allows the electromigration failure lifetime under different combinations of current density and ambient temperature to be obtained in the simulation stage, thus providing a quantitative basis for the reliability design of the packaging structure.

[0149] Verification Example 1 At temperatures ranging from 436 K to 537 K and current densities of 7.70 × 10⁻⁶ K, 4 A / cm 2 Up to 1×10 5 A / cm 2 Multiple simulation conditions were set within the range, and simulations were performed using the methods described above to obtain the electromigration failure lifetime under each condition, as shown in Table 1.

[0150] Table 1. Failure times obtained from simulations at different current densities and temperatures.

[0151] Based on the electromigration failure lifetimes obtained under multiple simulation conditions, a bivariate fitting was performed using the Black equation. This was done at different temperatures (436K~537K) and different current densities (7.70×10⁻⁶). 4 ~1×10 5 A / cm 2 Using current density and temperature as two independent variables, a bivariate fitting was performed on the electromigration failure lifetime to obtain the fitted Black equation:

[0152] Using the fitted Black equation, the temperature at 454.3 K and the current density at 7.9 × 10⁻⁶ K were analyzed. 4 A / cm 2 The verification operating conditions are predicted, and the three-dimensional response of failure time, current density, and temperature is as follows: Figure 13 As shown, the electromigration failure lifetime was obtained as 4130 h. Using the above-mentioned method for predicting the electromigration failure lifetime of metal interconnect structures, multiphysics coupling simulation was performed on this verification condition, and the change in resistance growth rate over time was obtained as follows: Figure 14 As shown, the electromigration failure lifetime is 4342 hours. The error between the two is -4.88%, which is comparable to that in reference 3 (Appl. Phys. Lett. 2006, 88, 072102), verifying the accuracy of the electromigration failure lifetime prediction method provided by this invention.

[0153] In Reference 3, the predicted temperature was 432.55 K and the current density was 1.68 × 10⁻⁶. 4 A / cm 2 Below, Chiang et al. simulated SnAg 3.0 Cu 0.5 The error between the measured lifespan and the actual lifespan is -4.76%.

[0154] By fitting and validating the Black equation using two variables, the prediction results of the simulation method are mutually verified with the industry-recognized empirical rules of the Black equation, and the results show good consistency. This improves the reliability of the lifetime prediction results and provides a quantitative basis for the reliability design and process optimization of the packaging structure.

[0155] The above embodiments illustrate the technical solution of the present invention from the perspectives of method flow, simulation model, criterion setting, and verification results. Examples 1 and 2 provide specific data on void nucleation time and electromigration failure lifetime under different electromigration conditions, and the verification examples cross-verify the simulation method using Black equations, etc.

[0156] The above description, based on the embodiments shown in the figures, details the structure, features, and effects of the present invention. The above description is only a preferred embodiment of the present invention, but the present invention is not limited to the scope of implementation shown in the figures. Any changes made in accordance with the concept of the present invention, or equivalent embodiments modified to have equivalent changes, that do not exceed the spirit covered by the specification and figures, should be within the protection scope of the present invention.

Claims

1. A method for predicting the electromigration failure lifetime of a metal interconnect structure, characterized in that, include: S1 establishes a simulation model of a metal interconnect structure, the metal interconnect structure including metal traces, a pair of metal bumps disposed on the metal traces, and pads stacked on the metal bumps; the simulation model includes interconnect structure parameters; S2 applies electromigration condition parameters to the simulation model, the electromigration condition parameters including the current density flowing through the metal bump and the ambient temperature; S3 performs multi-physics coupling simulation on the simulation model in the time domain to obtain the change of Joule heat power density in the metal bump over time. S4 performs a volume integral on the Joule heat power density within the metal bump to obtain the change of the total Joule heat power of the metal bump over time. S5 calculates the change in the equivalent resistance of the metal bump over time based on the total Joule heat power and the current flowing through the metal bump; S6 determines the change in the resistance growth rate of the metal bump over time based on the change in the equivalent resistance over time; and determines the electromigration failure time of the metal bump by using the failure threshold reached by the resistance growth rate as the electromigration failure criterion.

2. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 1, characterized in that, The interconnect structure parameters include the geometry and material properties of the metal traces, metal bumps, and pads; And / or, the metal bump includes a metal pillar, a first interface layer disposed between the metal pillar and the pad, and a second interface layer disposed between the metal pillar and the metal trace; the first interface layer is a UBM layer, and the second interface layer includes a Ni layer in contact with the metal pillar and an Au layer located between the Ni layer and the metal trace.

3. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 1, characterized in that, The multiphysics coupling simulation includes the coupling of the following physical fields: solid heat transfer physical field, solid mechanical physical field, solid transport physical field, current physical field, and domain ordinary differential and differential algebraic equation physical field.

4. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 3, characterized in that, The multiphysics coupling simulation also couples the thermal expansion multiphysics field and the electromagnetic thermal multiphysics field for coupled solution.

5. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 1, characterized in that, The multiphysics coupling simulation establishes an atomic migration model based on the mass conservation equation. The atomic migration model couples the atomic migration flux caused by four migration mechanisms: electron wind, stress gradient, thermal gradient, and atomic concentration gradient. In the atomic migration model, the electromigration process within the metal bump satisfies the following mass conservation equation: Where c is the normalized atomic concentration and t is time. ·J represents the divergence of the total diffusion flux; The atomic migration flux It can be expressed by the following formula: ; in, The atomic diffusion coefficient, Let be Boltzmann's constant, T be absolute temperature, and e be the elementary charge. Where ρ is the effective charge number, ρ is the resistivity, j is the current density, and Ω is the atomic volume. For hydrostatic stress, This is the activation energy for thermal migration.

6. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 5, characterized in that, In the multiphysics coupling simulation, the atomic concentration distribution within the metal bump is calculated based on the atomic migration model, and the distribution of the normalized atomic concentration c within the metal bump over time is solved. In response to the fact that the minimum value of the normalized atomic concentration c is lower than the void nucleation threshold, the current simulation time is determined as the void nucleus formation time; Wherein, the normalized atomic concentration c=C v / C v0 C v C represents the atomic concentration at a local location within the metal bump. v0 The initial atomic concentration within the metal bump.

7. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 6, characterized in that, When the atomic concentration in a local area is lower than the void nucleation threshold, the location is identified as a void region, and the resistivity is updated to N times the initial resistivity. Regions where no voids have formed retain the initial resistivity, and the spatial distribution of Joule heat power density is recalculated based on the updated resistivity. The void nucleation threshold is 0.

85.

8. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 1, characterized in that, Obtaining the spatial distribution of the Joule thermal power density within the metal bump over time includes: expressing the Joule thermal power density as q = E·j according to the differential form of Joule's thermal law, where q is the local Joule thermal power density, E is the local electric field strength, and j is the local current density.

9. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 8, characterized in that, The total Joule heat power of the metal bump at any time t can be obtained by taking the volume integral of the Joule heat power density within the metal bump that does not form a cavity: Where t is time, Let be the total Joule heat power at time t. Let be the local point flow field intensity at time t. Let t be the local current density at time t, and V be the volume of the metal bump.

10. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 9, characterized in that, The change of the equivalent resistance over time is expressed as R. eff (t)=P(t) / I 2 (t); Among them, R eff P(t) is the equivalent resistance of the metal bump at time t, P(t) is the total Joule heat power at time t, and I(t) is the current flowing through the metal bump at time t.

11. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 10, characterized in that, The current I(t) flowing through the metal bump is a constant current.

12. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 10, characterized in that, The change in resistance growth rate over time is: ΔR(t) = (R eff (t)-R0) / R0; Where ΔR(t) is the resistance growth rate at time t, and R0 is the initial resistance.

13. The method for predicting the electromigration failure lifetime of a metal interconnect structure according to claim 1, characterized in that, The failure threshold is 20%; the step of using the resistance growth rate reaching the failure threshold as the electromigration failure criterion includes: determining whether the resistance growth rate reaches the failure threshold; if so, the metal bump has undergone electromigration failure, and the current simulation time is determined as the electromigration failure lifetime of the metal bump; if not, the process returns to steps S3 to S6 for iteration.

14. A system for predicting the electromigration failure lifetime of a metal interconnect structure, characterized in that, include: The model building module is configured to build a simulation model of a metal interconnect structure, wherein the metal interconnect structure includes metal traces, a pair of metal bumps disposed on the metal traces, and pads stacked on the metal bumps; the simulation model includes interconnect structure parameters, which include the geometric dimensions and material properties of the metal interconnect structure. The condition application module is configured to apply electromigration condition parameters to the simulation model, the electromigration condition parameters including the current density flowing through the metal bump and the ambient temperature; The multiphysics coupling simulation module is configured to perform multiphysics coupling simulation on the simulation model in the time domain to obtain the change of Joule heat power density in the metal bump over time. The power integration module is configured to perform volume integration on the Joule thermal power density within the metal bump to obtain the change of the total Joule thermal power of the metal bump over time. The equivalent resistance calculation module is configured to calculate the change of the equivalent resistance of the metal bump over time based on the total Joule thermal power and the current flowing through the metal bump. The failure determination module is configured to determine the change in the resistance growth rate of the metal bump over time based on the change in the equivalent resistance over time; and to determine the electromigration failure time of the metal bump by using the failure threshold reached by the resistance growth rate as the electromigration failure criterion.

15. The system according to claim 14, characterized in that, The multiphysics coupling simulation includes the coupling of the following physical fields: solid heat transfer physical field, solid mechanical physical field, solid transport physical field, current physical field, domain ordinary differential and differential algebraic equation physical field, thermal expansion multiphysics field and electromagnetic thermal multiphysics field.

16. The system according to claim 14, characterized in that, Also includes: The atomic concentration monitoring module is configured to: solve the distribution of the normalized atomic concentration c within the metal bump over time during the multiphysics coupled simulation; and determine the current simulation time as the void nucleation time in response to the minimum value of the normalized atomic concentration c being lower than the void nucleation threshold. Wherein, the normalized atomic concentration c=C v / C v0 C v C represents the atomic concentration at a local location within the metal bump. v0 The initial atomic concentration within the metal bump; The void nucleation threshold is 0.

85.

17. A method for manufacturing a semiconductor device, characterized in that, Includes the following steps: The method for predicting the electromigration failure lifetime of metal interconnect structures as described in any one of claims 1 to 13 is used to evaluate metal bumps. Adjust the parameters of the metal bumps based on the prediction results, and manufacture the metal bumps according to the adjusted parameters.

18. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the method for predicting the electromigration failure lifetime of metal interconnect structures as described in any one of claims 1 to 13.

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