Electromigration stress modeling method and device for embedded power rail

Abstract

The invention discloses an electromigration stress modeling method and device for an embedded power supply rail, relates to the technical field of embedded power supply rails, and aims to solve the problem of low fidelity during modeling simulation of the embedded power supply rail in the prior art. The modeling method comprises the following steps: constructing a two-dimensional grain boundary structure based on grain boundary line segment coordinate data of the embedded power supply rail; determining an included angle parameter between the grain boundary direction of each grain boundary in the two-dimensional grain boundary structure and the current direction; constructing a target electromigration stress model based on the included angle parameters; and solving the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail. The method is used for improving the fidelity of modeling simulation of the embedded power rail.

Description

Technical Field

[0001] This invention relates to the field of embedded power rail technology, and more particularly to a method and apparatus for modeling electromigration stress of embedded power rails. Background Technology

[0002] As chip interconnect widths continue to shrink and current densities significantly increase, electromigration (EM) has become a critical reliability bottleneck in advanced integrated circuit back-end packaging. To continuously drive the evolution of technology nodes to 2nm and below, the industry is actively deploying Backside Power Delivery Network (BSPDN) architectures. This architecture effectively separates signal and power routing by migrating the power network to the back of the transistors, significantly alleviating front-end routing congestion, effectively reducing voltage drop, and improving overall power-performance-area (PPA) metrics.

[0003] Buried Power Rails (BPRs), a core component of the BSPDN architecture, play a crucial role in transferring power from the back metal layer to the front-end active devices. This structure is buried beneath the transistors in a shallow trench isolation (STI) region, achieving low-resistance power supply through a high aspect ratio metal channel. However, due to its extremely narrow linewidth, limited heat dissipation channels, and complex process constraints, BPR structures are more prone to electromigration failures (EM failures) under high current density conditions. Therefore, there is an urgent need to develop high-fidelity physical modeling and simulation methods for BPR structure characteristics to assess its electromigration reliability. Summary of the Invention

[0004] The purpose of this invention is to provide a method and apparatus for modeling electromigration stress of embedded power rails, so as to solve the low-fidelity problem of existing methods for modeling electromigration stress of embedded power rails.

[0005] To achieve the above objectives, the present invention provides the following technical solution: In a first aspect, the present invention provides a method for modeling electromigration stress in embedded power rails, comprising: A two-dimensional grain boundary structure is constructed based on the coordinate data of the grain boundary line segments of the embedded power rail; Determine the angle parameters between the grain boundary direction and the current direction of each grain boundary in a two-dimensional grain boundary structure; A target electromigration stress model is constructed based on the included angle parameter; Solve the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail.

[0006] Optionally, a target electromigration stress model is constructed based on the included angle parameter, including: Substituting the cosine value of the included angle parameter into the electron wind term of atomic flux in the pre-constructed initial electromigration stress model, the target electromigration stress model is obtained.

[0007] Optionally, before constructing the two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail, the method further includes: constructing an initial electromigration stress model; Constructing an initial electromigration stress model includes: The Korhonen equation, applicable to grain boundary structures: ; This was determined to be the initial electromigration stress model; in, For stress; For time; Bulk modulus; It is atomic flux; ; For electronic wind power; This is the stress gradient term; The atomic diffusion coefficient; Initial electronic wind force; Atomic volume; Boltzmann's constant; For temperature; For stress gradient; Substituting the cosine value of the included angle parameter into the electron wind term of atomic flux in the pre-constructed initial electromigration stress model, including: Substituting the cosine of the included angle into the initial electronic wind force yields the corrected electronic wind force: ; Substituting the corrected electron wind into the electron wind term in the atomic flux yields the target electromigration stress model; where... This is the corrected electronic wind speed; Effective charge number; It is the elementary charge; Resistivity; Current density; The included angle is the included angle parameter. The value is the cosine of the included angle.

[0008] Optionally, the angle parameter between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure is determined, including: A first automated script, written using numerical calculation software, extracts characteristic line segments from each grain boundary. The direction of the characteristic line segment is taken as the grain boundary direction of the corresponding grain boundary, and the angle between the characteristic line segment and the current direction is calculated. The included angle of each grain boundary is associated with the corresponding grain boundary number and solidified into included angle parameters that can be recognized by the physical field simulation platform; the grain boundary number is used to locate the grain boundary region corresponding to the cosine value of the included angle.

[0009] Optionally, before constructing the target electromigration stress model based on the included angle parameter, the method further includes: The second automated script is used to bind each included angle parameter to the corresponding solid grain boundary in the physical field simulation platform through the linkage interface between the numerical calculation software and the physical field simulation platform, thus forming a mapping relationship between the included angle parameter and the solid grain boundary. The two-dimensional grain boundary structure, included angle parameters, and mapping relationships are collectively solidified into a model file; the model file is used by the physics simulation platform to provide readable included angle parameters and mapping relationships when constructing the target electromigration stress model.

[0010] Optionally, characteristic line segments in each grain boundary are extracted, including: Read the grain boundary polygon data corresponding to the two-dimensional grain boundary structure; the two-dimensional grain boundary structure is composed of multiple polygons, each polygon corresponds to a grain boundary, and the grain boundary polygon data contains the coordinates of all vertices of the polygon corresponding to each grain boundary; From all the vertex coordinates corresponding to each grain boundary, select two target vertices that represent the dominant extension direction of the corresponding grain boundary; Connect the two target vertices and use the resulting line segment as the characteristic line segment of the corresponding grain boundary.

[0011] Optionally, a two-dimensional grain boundary structure is constructed based on the coordinate data of the grain boundary segments of the embedded power rail, including: Import the grain boundary line segment coordinate data into numerical calculation software, and use the second automated script written in the numerical calculation software to set the physical thickness. Calculate the length of each grain boundary segment, filter out invalid short segments with a length less than a preset threshold, and calculate the offset perpendicular to the normal direction to generate multiple polygons with thickness. Eliminate overlapping areas between different polygons in multiple polygons with thickness, and filter valid areas with an area larger than a preset area; Collect all effective regions to form a two-dimensional grain boundary structure.

[0012] Optionally, before solving the target electromigration stress model, the method further includes performing a preprocessing step; Perform preprocessing steps, including: In the physics simulation platform, an automatic triangular mesh is used to perform mesh generation on the simulation domain of the target electromigration stress model and set the mesh parameters. After the mesh is generated, the material parameters and electrical parameters are set in the parameter module of the physics simulation platform. The material parameters include the initial atomic diffusion coefficient, and the initial atomic diffusion coefficient of the internal grain boundaries is greater than that of the grain boundaries at the interface. The electrical parameters include the initial current density and the initial temperature. Based on the physical properties of the target electromigration stress model, the model boundary conditions and initial conditions are set; among them, the atomic flux of all boundaries is set to zero, and the initial stress is uniformly set to 0. To meet the time-series analysis requirements of stress evolution, the step time and total simulation duration of the simulation solution parameters are set.

[0013] Optionally, before solving the target electromigration stress model, the method further includes: model loading and parameter configuration; Perform model loading and parameter configuration, including: Load the model file into the physics simulation platform; The target electromigration stress model is set in the partial differential equation module of the physical field simulation platform, and the partial differential equation module is used as the core carrier for stress evolution solution. Based on the mapping relationship, each included angle parameter is associated with the variable library in the partial differential equation module so that each grain boundary domain can call the corresponding included angle parameter during simulation; Based on the expression for atomic flux in the target electromigration stress model, flux source parameters are set in the partial differential equation module; the flux source parameters include flux sources in the x-axis direction and flux sources in the y-axis direction.

[0014] Compared with existing technologies, the electromigration stress modeling method for embedded power rails provided by this invention, through the technical process of constructing a two-dimensional grain boundary structure → determining the grain boundary-current angle parameters → constructing a target model based on the angle → solving the model to obtain the stress distribution, directly addresses the core defects of traditional electromigration modeling methods and achieves multiple beneficial effects with strong correlation between steps: First, the first step of constructing a two-dimensional grain boundary structure based on grain boundary line segment coordinate data breaks the limitation of traditional modeling that only relies on macroscopic statistical data and ignores the actual grain boundary morphology of BPR. By transforming discrete grain boundary line segment coordinates into a two-dimensional solid grain boundary structure, it can truly restore the number, orientation, and spatial distribution relationship of grain boundaries in BPR, avoiding the modeling deviation caused by the traditional model using macroscopic average results to replace the microscopic real structure, laying a structural fidelity foundation for subsequent accurate analysis of electromigration stress; Second, the second step of determining the angle parameters between the grain boundary direction and the current direction addresses the defect of traditional models that cannot quantify the influence of grain boundary orientation on electromigration by extracting the angle parameters of each grain boundary direction and the current direction. The angle between the grain boundary direction and the current direction is used as a key parameter, and for the first time, the difference in grain boundary direction is incorporated into the modeling process. Compared with the traditional model's indiscriminate treatment of all grain boundaries, this model can identify the directional characteristics of different grain boundaries, providing core microscopic parameters for the subsequent construction of a direction-sensitive electromigration stress model, and avoiding stress prediction deviations caused by ignoring grain boundary direction. Finally, the third and fourth steps construct the target model based on the angle and solve for the stress distribution, connecting the microscopic information of the previous steps. This solves the problem that the traditional model outputs a macroscopic average stress distribution due to the lack of microscopic parameter input, and cannot reflect the stress differences at different grain boundary locations within the BPR. By incorporating the angle parameter into the target model, the solved model can output a stress distribution strongly correlated with the microscopic grain boundary structure, which can intuitively present the stress magnitude and concentration location in different grain boundary regions. This provides a more accurate quantitative basis for the risk assessment of electromigration failure of BPR structures (such as locating weak areas with stress concentration), and avoids the problem of the stress results of the traditional model being disconnected from the microscopic structure.

[0015] In a second aspect, the present invention also provides an apparatus for modeling electromigration stress of embedded power rails, comprising: The first construction module is used to construct a two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail; The determination module is used to determine the angle parameter between the grain boundary direction and the current direction of each grain boundary in a two-dimensional grain boundary structure. The second building module is used to construct a target electromigration stress model based on the included angle parameter; The solver module is used to solve the target electromigration stress model and obtain the electromigration stress distribution of the embedded power rail. Attached Figure Description

[0016] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings: Figure 1 A flowchart illustrating a method for modeling electromigration stress in an embedded power rail, as provided in an embodiment of the present invention. Figure 2 Schematic diagrams of two-dimensional grain boundary structures of different sizes provided for a specific embodiment of the present invention; Figure 3 A schematic diagram showing the angle between the grain boundary direction of an internal grain boundary and the current direction on the interface, provided for a specific embodiment of the present invention; Figure 4 for Figure 3 A schematic diagram of a two-dimensional grain boundary structure of one of the sizes; Figure 5 for Figure 4 A schematic diagram of the simulation distribution results; Figure 6 A schematic diagram of the structure of an electromigration stress modeling device for embedded power rails provided in one embodiment of the present invention. Detailed Implementation

[0017] To facilitate a clear description of the technical solutions in the embodiments of the present invention, the terms "first" and "second" are used to distinguish identical or similar items with essentially the same function and effect. For example, the first threshold and the second threshold are merely used to distinguish different thresholds and do not limit their order. Those skilled in the art will understand that the terms "first" and "second" do not limit the quantity or execution order, and the terms "first" and "second" do not necessarily imply that they are different. It should be noted that in the present invention, the words "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design scheme described as "exemplary" or "for example" in the present invention should not be construed as being more preferred or advantageous than other embodiments or design schemes. Specifically, the use of the words "exemplary" or "for example" is intended to present the relevant concepts in a specific manner. In the present invention, "at least one" refers to one or more, and "more than one" refers to two or more. "And / or" describes the relationship between related objects, indicating that three relationships may exist.

[0018] Electromigration is the movement of atoms under the influence of electron wind. It can cause atomic accumulation in some areas and voids in others, eventually leading to open circuits or short circuits in the interconnect structure of integrated circuits. It is a key reliability bottleneck in the back-end packaging of advanced integrated circuits. Especially in narrow linewidth and high current density structures such as BPR, it can significantly increase the risk of failure and affect the lifespan and stability of the chip.

[0019] Electromigration stress refers to the internal stress generated during the electromigration process, caused by the migration of atoms driven by both electron wind and stress gradient, resulting in atomic accumulation or depletion within the material. Specifically, regions with atomic accumulation will generate compressive stress due to excessive atomic density; regions with atomic depletion will form voids due to atomic loss, causing tensile stress in the surrounding material.

[0020] To achieve high-fidelity simulation of BPR structures, such as Figure 1 As shown, this embodiment of the invention provides a method for modeling electromigration stress in embedded power rails, which may include: Step 100: Construct a two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail; Specifically, existing open-source grain boundary generation algorithms can be used to generate coordinate data representing all grain boundary segments in BPR. This data is usually presented in the form of coordinate arrays, with each array element corresponding to key location information of a grain boundary segment, such as the starting coordinates and the ending coordinates.

[0021] Next, a two-dimensional structure reconstruction is performed based on the aforementioned coordinate data: the coordinates of the discrete grain boundary segments are integrated according to the spatial positional relationship of the actual physical structure of BPR, forming a two-dimensional structural model consistent with the actual BPR grain boundary distribution. This two-dimensional structural model is actually a collection of grain boundary polygons; in other words, the two-dimensional grain boundary structure is composed of multiple polygons, each polygon corresponding to a grain boundary.

[0022] The constructed two-dimensional grain boundary structure corresponds to its own grain boundary polygon data, which contains the coordinates of all vertices of the polygon corresponding to each grain boundary. All vertices are arranged in clockwise or counterclockwise order to ensure that the complete geometric morphology of the grain boundary is represented.

[0023] Two-dimensional grain boundary structures can reflect the number, orientation, and interconnection of grain boundaries in BPR.

[0024] Step 200: Determine the angle parameters between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure; Specifically, the grain boundary direction is the representative direction of each grain boundary in a two-dimensional grain boundary structure.

[0025] The current direction, i.e. the direction of current flow in the interconnect structure when the BPR is actually working, needs to be determined in combination with the circuit connection logic and working state of the BPR in the integrated circuit. It is usually a preset fixed direction or a specific path direction determined according to the circuit design, such as the horizontal direction.

[0026] The angle parameter is directly related to the driving strength of electron wind on atomic migration at grain boundaries. For example, under different angle parameters, the force components of electron wind on atoms are different, which in turn affect the rate and location of atomic accumulation / depletion.

[0027] Step 300: Construct a target electromigration stress model based on the included angle parameter; Step 300 may specifically include: Substituting the cosine value of the included angle parameter into the electron wind term of atomic flux in the pre-constructed initial electromigration stress model, the target electromigration stress model is obtained.

[0028] Step 400: Solve the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail.

[0029] The beneficial effects of this embodiment: 1) Address the shortcomings of relying on empirical models and insufficient physical mechanism characterization, and improve the physical reliability of the model.

[0030] Existing technologies largely rely on empirical lifetime formulas (such as the Black equation) or equivalent diffusion models. These methods are highly dependent on empirical parameters and cannot accurately reflect the physical control mechanisms of material microstructure, such as grain boundary orientation, on electromigration behavior, resulting in a lack of physical support for the actual migration process. This embodiment takes a physics-based modeling logic as its core, relying on the underlying principle of atomic flux conservation. By constructing a two-dimensional grain boundary structure to restore the microstructure and integrating physical driving factors to build the model, it breaks away from the excessive reliance on empirical parameters and fully depicts the physical chain of "microstructure-atomic migration-stress evolution." This allows the model to truly reflect the essential process of electromigration and solves the problem of the lack of physical mechanisms in empirical models.

[0031] (ii) To address the shortcomings of neglecting the influence of grain boundary geometry and orientation and poor adaptability, and to improve the applicability of the model to nanoscale structures.

[0032] As process nodes shrink to 2nm and below, interconnect structure dimensions are miniaturized to the point where grain boundary size is close to linewidth. Grain boundaries no longer exhibit a statistically average distribution; their geometric morphology and orientation become key factors influencing atomic flux paths and local stress evolution. Existing models do not consider this crucial variable and cannot adapt to the characteristics of nanoscale structures. This embodiment determines the angle parameter between the grain boundary direction and the current direction in step 200 and incorporates it into the model in step 300 to match the regulation law of the driving force by the cosine relationship between the EM term and the angle. This accurately captures the influence of grain boundary orientation on the magnitude of the EM driving force and stress distribution characteristics, enabling the model to adapt to the grain boundary-dominated electromigration behavior at the nanoscale and solving the deficiency of insufficient adaptability of existing models to miniaturized structures.

[0033] (iii) To address the shortcomings of weak time-varying process and cross-scale modeling capabilities and improve the reliability of stress prediction.

[0034] Existing empirical or equivalent diffusion models are insufficient in modeling time-varying processes (such as the dynamic evolution of stress over time) and in terms of cross-scale adaptability (especially at the nanoscale), making it difficult to provide reliable stress predictions in nanoscale interconnect structures. This embodiment is based on a physics-driven target electromigration stress model, combined with the numerical solution process in step 400. It can characterize the dynamic evolution of stress through time-varying parameters in the model, and adapt to the nanoscale through microscopic grain boundary structure modeling in step 100. The final output stress distribution can accurately reflect the stress state at different times and microscopic locations, providing a reliable basis for the electromigration reliability assessment of nanoscale interconnect structures and solving the problem of weak time-varying and cross-scale modeling capabilities of existing models.

[0035] In an optional embodiment, before constructing the two-dimensional grain boundary structure based on the grain boundary segment coordinate data of the embedded power rail, the method further includes: constructing an initial electromigration stress model. Constructing an initial electromigration stress model includes: The Korhonen equation, applicable to grain boundary structures: (1); This was determined to be the initial electromigration stress model; in, For stress; For time; Bulk modulus; It is atomic flux; ; For electronic wind power; This is the stress gradient term; The atomic diffusion coefficient; Initial electronic wind force; Atomic volume; Boltzmann's constant; For temperature; For stress gradient; Substituting the cosine value of the included angle parameter into the electron wind term of atomic flux in the pre-constructed initial electromigration stress model, including: Substituting the cosine of the included angle into the initial electronic wind force yields the corrected electronic wind force: (2); Substituting the corrected electron wind into the electron wind term in the atomic flux yields the target electromigration stress model; where... This is the corrected electronic wind speed; Effective charge number; It is the elementary charge; Resistivity; Current density; The included angle is the included angle parameter. The value is the cosine of the included angle.

[0036] In summary, the final target electromigration stress model is as follows: (3); Beneficial effects of this embodiment: 1) Anchoring to physical mechanisms, completely eliminating reliance on empirical models and improving the core reliability of the model. Existing empirical models such as the Black equation heavily rely on empirical parameters and cannot reflect the microstructure's control mechanism on electromigration, making them only suitable for macroscopic statistical scenarios. This embodiment explicitly selects the Korhonen equation, applicable to grain boundary structures, as the initial electromigration stress model. This equation, based on the conservation of atomic flux, naturally possesses the ability to characterize the complete physical chain of "current density-stress gradient-atomic migration-stress evolution." By directly adopting this mature physical model framework, subjective biases caused by empirical model fitting are avoided. At the same time, relying on the built-in electron wind and stress gradient terms of the equation, the underlying physical logic of electromigration stress generation is fully preserved, giving the model output results solid physical support and solving the core defects of empirical models, namely "weak physical correlation and limited reliability."

[0037] 2) Precise coupling of angle information improves the characterization of microscopic influencing factors and adapts to the structural characteristics of nanoscale BPR. As the process node evolves to 2nm and below, BPR, as a core component of BEOL, experiences continuous linewidth reduction, leading to grain boundary sizes approaching the linewidth. Grain boundary orientation becomes a key factor affecting atomic flux paths and stress evolution, a variable not considered in existing models. This embodiment derives a corrected electron wind by substituting the cosine of the angle into the initial electron wind term, precisely matching the physical law that "the angle between the EM term and the grain boundary migration path direction and the current direction has a cosine relationship." This correction makes the calculation of electron wind no longer a macroscopic average value, but strongly correlated with the specific orientation of each grain boundary, thus allowing the target model to accurately capture the differences in EM driving force at different grain boundary locations. This design perfectly adapts to the structural characteristics of grain boundary main conductive migration behavior at the nanoscale, solving the problem that traditional models cannot adapt to microscale structures due to neglecting the influence of grain boundary orientation.

[0038] 3) Enhance the model's dynamic characterization capabilities to match the core requirements of BEOL (Back End of Line) structural reliability assessment. Electromigration stress evolves dynamically over time, and its spatial distribution is a key indicator for assessing the failure risk of interconnect structures, such as fracture and short circuit. However, existing empirical models are insufficient in modeling time-varying processes and adaptability across scales. The Korhonen equation, upon which this embodiment is based, inherently possesses excellent time-varying process modeling capabilities. The expression directly relates to the rate of change of stress over time. Combined with the corrected electron wind term, the target model can dynamically output the stress distribution at various locations in the BPR grain boundary structure at different time points. Furthermore, this model is adapted to advanced BEOL structures and nanoscale interconnect scenarios, meeting the urgent need for high-precision stress evolution simulation due to the increased risk of BPR electromigration failure under conditions of reduced linewidth and increased current density. This provides accurate time-varying stress data support for the reliability design of BEOL structures.

[0039] In an optional embodiment, step 100: constructing a two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail may include: Import the grain boundary line segment coordinate data into numerical calculation software, and use the second automated script written in the numerical calculation software to set the physical thickness. Calculate the length of each grain boundary segment, filter out invalid short segments with a length less than a preset threshold, and calculate the offset perpendicular to the normal direction to generate multiple polygons with thickness. Eliminate overlapping areas between different polygons in multiple polygons with thickness, and filter valid areas with an area larger than a preset area; Collect all effective regions to form a two-dimensional grain boundary structure.

[0040] For example, MATLAB is the preferred choice for numerical computation software. Of course, other types of commercial software can also be used, as long as they can perform numerical computation and automate scripting; there are no restrictions on the specific software type.

[0041] For example, the preset threshold is 0.5nm.

[0042] For example, the preset area is 1e-8nm².

[0043] In the first specific implementation method, the implementation logic of the second automation script is described as follows.

[0044] I. Initial variable definition: preparing basic parameters and storage containers for grain boundary processing.

[0045] Define an empty array edges to receive the coordinate data of the grain boundary line segments output by the open-source grain boundary generation algorithm. The data to be filled in afterward is in the format of 4 columns per row, corresponding to the starting coordinates (x1, y1) and ending coordinates (x2, y2) of a grain boundary line segment respectively. Reserve the storage space for the original grain boundary line segment data and establish the connection with the upstream algorithm. Set the grain boundary physical thickness parameter grain_boundary_thickness=1, in nm. Since grain boundaries are essentially solid regions with thickness rather than geometric lines, they need to be given thickness to match the actual BPR structure and meet COMSOL's modeling requirements for solid domains. Create an empty MATLAB geometry object all_polyshapes of type polyshape to store the processed non-overlapping grain boundary polygons. This will serve as the benchmark for judging the overlap of newly generated grain boundary polygons and avoid overlapping regions generated by different grain boundary segments. Define an empty cell array all_polygons to store the final, non-overlapping, valid grain boundary polygon vertex coordinates, which will serve as the core input data for the subsequent first automatic script to import COMSOL.

[0046] II. Loop processing.

[0047] The loop iterates through the total number of grain boundary segments, with the loop condition being `for i=1:size(edges,1)`, where `size(edges,1)` is the row number of the `edges` array, representing the total number of grain boundary segments to be processed. Each grain boundary segment is processed one by one to ensure no omissions. The specific steps are as follows: 1. Extract the endpoint coordinates of the current grain boundary segment.

[0048] Extract data from the first and second columns of the i-th row of the edges array to obtain the starting coordinates p1 of the i-th grain boundary segment, corresponding to x1 and y1, and determine the starting position of the current grain boundary segment, providing a basic positioning for subsequent thickness generation; Extract data from the 3rd-4th column of the i-th row of the edges array to obtain the endpoint coordinates p2 of the i-th grain boundary segment, corresponding to x2 and y2, determine the end position of the current grain boundary segment, and clarify the direction and length characteristics of the segment.

[0049] 2. Filter out invalid short line segments to avoid algorithm errors interfering with subsequent modeling.

[0050] Calculate the direction vector vec from the starting point p1 to the ending point p2, with the x-direction component being x2-x1 and the y-direction component being y2-y1, which simultaneously represents the direction and length of the line segment; Calculate the magnitude len of the direction vector vec, which is the actual length of the line segment, in nm. The calculation formula is simplified to quantify the line segment length to determine whether it is an invalid short line. If the line segment length len is less than 0.5nm, a preset threshold is set, and a skip operation is performed without further processing. This filters out extremely short and invalid line segments caused by errors in the open-source grain boundary generation algorithm, and avoids generating excessively small polygons that could affect the accuracy of COMSOL mesh generation and simulation efficiency.

[0051] 3. Calculate the offset direction of grain boundary thickness, perform key geometric calculations, and ensure uniform thickness.

[0052] Divide the direction vector vec by its modulus len to obtain the unit direction vector dir_vec, which has a length of 1. Only the direction information of the line segment is retained, eliminating the influence of the line segment length on the "thickness offset" and ensuring that the physical thickness of all grain boundaries is uniform, which is the preset 1nm. The normal direction is calculated by taking the y component of dir_vec as the new x component and the negative of the x component of dir_vec as the new y component. Essentially, it is a simplified calculation of rotating the vector by 90°. Since the grain boundary thickness needs to be extended to both sides along the direction perpendicular to the line segment, rather than along the line segment direction, this vertical vector is the one-sided offset direction. Calculate the offset on one side: Since the total thickness of the grain boundary is 1nm, it is offset by 0.5nm to each side. By multiplying the offset distance of 0.5nm on one side by the vertical vector normal_vec, the specific offset coordinates are obtained, which determines the distance and direction of the grain boundary line segment extending to both sides, in order to prepare for the generation of a polygon with thickness.

[0053] 4. Generate grain boundary polygons with thickness, and transform line segments without thickness into solid geometry.

[0054] Based on the starting point p1, the ending point p2, and the single-sided offset offset, four vertex coordinates are generated and arranged in clockwise order to form a quadrilateral vertex set poly_pts: ①p1+offset, the point after the starting point is offset outwards, ②p2+offset, the point after the ending point is offset outwards, ③p2-offset, the point after the ending point is offset inwards, ④p1-offset, the point after the starting point is offset inwards. In this way, the line segment with no thickness is transformed into a quadrilateral solid with thickness, which matches COMSOL's modeling requirements for geometric domains. COMSOL cannot directly define the simulation domain using line segments. The MATLAB polyshape function is called to convert the poly_pts composed of four vertex coordinates into a polygon geometry object poly_new. This object supports subsequent geometric operations, such as merging and subtraction, and encapsulates discrete vertex coordinates into an operable geometric entity, which facilitates subsequent handling of overlap problems.

[0055] 5. Eliminate overlapping regions, ensure the uniqueness of grain boundary geometry, and avoid COMSOL geometry conflicts.

[0056] Perform geometric subtraction: Subtract the processed set of non-overlapping polygons all_polyshapes from the newly generated polygon poly_new to obtain the non-overlapping remaining region poly_trimmed. This avoids the polygons generated by different grain boundary segments from overlapping each other, such as the overlapping parts of intersecting grain boundaries, and prevents geometric conflicts of multiple domains at the same location when importing into COMSOL later. Determine if the trimmed polygon poly_trimmed has any valid vertices. If the vertices are empty, it means that the newly generated polygon poly_new completely overlaps with the set of non-overlapping polygons all_polyshapes, so skip the subsequent processing; if the vertices exist, it means that there are valid non-overlapping regions, so proceed to the next step of filtering.

[0057] 6. Filter valid regions and update the cumulative set to ensure the validity and consistency of the output data.

[0058] The MATLAB regions function is called to split the trimmed polygon poly_trimmed into a connected region array regions_array: if poly_trimmed contains multiple non-connected small regions, it will be automatically split into multiple independent sub-regions to handle the multi-region problem that may occur after trimming and ensure that each connected region can be judged separately. The loop iterates through each connected region one by one, with the loop condition being `for r=1:length(regions_array)`. The inner loop then performs the trimming of individual grain boundary segments. Extract the r-th connected region this_region, and focus on the current sub-region for subsequent validity judgment; Calculate the area of ​​this_region. If the area is greater than 1e-8nm², it is determined to be a valid region. If the area is less than or equal to the threshold, it is considered an invalid corner and is skipped in subsequent processing. This is to filter out the tiny fragmented regions caused by geometric subtraction, which may be calculation errors, and to avoid such regions affecting the quality of the COMSOL mesh and simulation efficiency. Store the vertex coordinates of the effective region this_region into the end of the cell array all_polygons, that is, all_polygons{end+1}=this_region.Vertices, collect all available grain boundary geometry data, and provide input for the algorithm corresponding to the first automated script below to import COMSOL; The effective region this_region is merged into the processed set of non-overlapping polygons all_polyshapes, all_polyshapes=union(all_polyshapes,this_region). All_polyshapes is updated as the latest benchmark for the next grain boundary segment overlap judgment, ensuring that the subsequently processed grain boundary segments will not overlap with the current effective region, thus maintaining the uniqueness of the grain boundary geometry.

[0059] The algorithm steps involved in the second automation script in Specific Implementation Method 1 can be implemented in MATLAB using the corresponding full code, which will not be repeated in this embodiment.

[0060] The beneficial effects of this embodiment are as follows: This embodiment achieves standardized processing of grain boundary line segment coordinate data through MATLAB automated scripts. First, it assigns physical thickness to the grain boundary, filters out invalid short segments and extremely small fragment regions, and then eliminates polygon overlap and filters effective regions through geometric operations. This transforms discrete grain boundary lines into conflict-free, high-fidelity two-dimensional solid grain boundary structures. This not only accurately matches the solid domain modeling and mesh generation requirements of the simulation platform used subsequently, but also avoids errors and inefficiencies caused by manual processing through automated processes. At the same time, it ensures the accurate restoration of the nanoscale BPR grain boundary microstructure, laying a high-quality geometric foundation for the accurate solution of the subsequent target electromigration stress model.

[0061] In an optional embodiment, step 200: determining the angle parameter between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure, including: A first automated script, written using numerical calculation software, extracts characteristic line segments from each grain boundary. The direction of the characteristic line segment is taken as the grain boundary direction of the corresponding grain boundary, and the angle between the characteristic line segment and the current direction is calculated. The included angle of each grain boundary is associated with the corresponding grain boundary number and solidified into included angle parameters that can be recognized by the physical field simulation platform; the grain boundary number is used to locate the grain boundary region corresponding to the cosine value of the included angle.

[0062] Understandably, the method also includes the following steps before constructing the target electromigration stress model based on the included angle parameter: The second automated script is used to bind each included angle parameter to the corresponding solid grain boundary in the physical field simulation platform through the linkage interface between the numerical calculation software and the physical field simulation platform, forming a mapping relationship between the included angle parameter and the solid grain boundary; the solid grain boundary is the grain boundary polygon located in the physical field simulation platform.

[0063] The two-dimensional grain boundary structure, included angle parameters, and mapping relationships are collectively solidified into a model file; the model file is used by the physics simulation platform to provide readable included angle parameters and mapping relationships when constructing the target electromigration stress model.

[0064] Extracting characteristic line segments from each grain boundary can specifically include: Read the grain boundary polygon data corresponding to the two-dimensional grain boundary structure; as mentioned above and it needs to be emphasized again here, the two-dimensional grain boundary structure is composed of multiple polygons, each polygon corresponds to a grain boundary, and the grain boundary polygon data contains the coordinates of all vertices of the polygon corresponding to each grain boundary. All vertices are arranged in clockwise or counterclockwise order to ensure the complete geometric shape of the grain boundary is represented. From all the vertex coordinates corresponding to each grain boundary, select two target vertices that represent the dominant extension direction of the corresponding grain boundary; Connect the two target vertices and use the resulting line segment as the characteristic line segment of the corresponding grain boundary.

[0065] For example, the two target vertices could be the first two vertices in label order from all the vertices.

[0066] For example, if COMSOL is preferred as the physics simulation platform, then the linkage interface could be Livelink for MATLAB.

[0067] It should be noted that the physics simulation platform is not limited to COMSOL simulation software. Other industrial or academic simulation software with multiphysics simulation capabilities and support for solving partial differential equations can also be used. The present invention prioritizes COMSOL based on its industrial versatility and the flexibility of the PDE module, rather than any exclusive limitation.

[0068] It should be noted that different software has different API interfaces, geometric modeling logic, and solver configurations. The implementation methods of grain boundary import and parameter mapping need to be adjusted accordingly. However, the core ideas of this invention (angle extraction, Korhonen equation coupling, and automated process) can be transferred to other compatible software.

[0069] In the second specific implementation method, the logic that the first automation script can implement is described as follows by example.

[0070] 1. Initialize the COMSOL model environment and build the basic framework of the model.

[0071] Import the Java class libraries required for COMSOL modeling and This provides a dependency environment for subsequent calls to COMSOLAPI to create models and manipulate geometry and parameters; Call the COMSOL model utility class ModelUtil to create a new blank model object; Set the model's storage path to the current working directory pwd to ensure that subsequent model files can be read and written in the current path; Set the model label to “GrainDomainModel” to easily identify the model’s purpose, which is to model grain boundary domains. Create a new physical component in the model, identify it as comp1, and enable the component as true, as a container for grain boundary geometry and parameters; Create a two-dimensional geometric object geom1 in the comp1 component and set the length unit of the geometric object to nm to match the nanoscale of the BPR structure, providing a unit reference for the subsequent import of grain boundary polygon coordinates.

[0072] 2. Import the grain boundary polygons generated by the second automated script to construct the basic geometric structure.

[0073] The loop condition is Process each grain boundary polygon one by one: extract the vertex coordinate data of the i-th grain boundary polygon from the all_polygons array and assign it to the variable poly; Generate a unique identifier tag for the polygon in COMSOL, in the format of "pg + serial number", such as "pg1" for the first polygon, to avoid duplicate geometric feature names; Create a "Polygon" feature in the 2D geometric object geom1 and associate it with the unique identifier tag mentioned above; Set vertex coordinates for this polygon feature: use the x-coordinate of the first column of the poly array. y-coordinate of column 2 The x and y parameters assigned to the polygon must be transposed into a row vector to match the COMSOL input format. Set the polygon type to solid to ensure that it is recognized in COMSOL as a geometric domain that can participate in the simulation, rather than just a boundary line; Execute the command to generate the 2D geometric object geom1 Complete the geometric rendering of all grain boundary polygons to form the basic grain boundary geometry of BPR.

[0074] 3. Initialize key parameters and temporary variables to prepare for angle calculation and mapping.

[0075] Define the total number of grain boundaries parameter gb, whose value is equal to "the total number of grain boundary polygon sets output by the second automated script", i.e., n=length(all_polygons). In subsequent steps, the total number of grain boundaries parameter gb is used to control the number of loops to ensure that all grain boundaries are covered. Define a variable n_polygons and assign it the length of all_polygons to specify the total number of grain boundary polygons to be processed, which corresponds to the parameter gb of the total number of grain boundaries. Attempt to create a temporary variable var1 in the comp1 component using a try-catch block: If the temporary variable var1 does not already exist in the comp1 component, then execute... Create a temporary variable for use as a transition in subsequent parameter operations; If the temporary variable var1 already exists, the catch branch is triggered, skipping the creation operation to avoid code errors caused by duplicate variable creation.

[0076] IV. Calculate the angle θ between the grain boundary and the current direction segment by segment, and extract the core parameters.

[0077] Based on "total number of grain boundary polygons" " represents the number of loops, Calculate the included angle θ of each grain boundary one by one: Extract the vertex coordinates poly of the i-th grain boundary polygon. First, determine the number of vertices of the polygon: if the number of vertices is less than 2 and size(poly,1) is less than 2, then a valid line segment cannot be formed, and continue is executed to skip the angle calculation of the current polygon. If the number of vertices meets the requirement, calculate the difference between the coordinates of the first two vertices of the polygon: x-direction difference dx = x-coordinate of the second vertex - x-coordinate of the first vertex, y-direction difference dy = y-coordinate of the second vertex - y-coordinate of the first vertex; The angle between the grain boundary segment and the horizontal direction and the direction of the main current is calculated by the arctangent function atan2(dy,dx), and the result is assigned the value theta_val. This function can determine the quadrant of the angle by the sign of the difference between x and y, so as to ensure the accuracy of the direction. Convert theta_val to the string format expr that retains 10 decimal places to avoid loss of numerical precision; Generate a unique name var_name for this included angle parameter in COMSOL, in the format "theta_ + serial number", such as "theta_1" for the included angle parameter of the first grain boundary; Call the COMSOL parameter module model.param to store the expr and θ value strings into the parameter with the corresponding name var_name, thus completing the COMSOL storage of the included angle parameter.

[0078] V. Create cumulative selection of grain boundaries to establish a bridge between geometry and parameters.

[0079] Using the "total number of grain boundaries parameter gb" as the number of iterations, For each grain boundary polygon, a "cumulative selection" is created to associate the polygon with subsequent θ values: Generate a unique name for the cumulative selection, selName, in the format "csel + serial number", such as "csel1" for the cumulative selection of the first grain boundary; For each cumulative selection, determine its associated polygon feature name pgName, which is "pg + serial number" generated by the first automated script, such as "pg1"; In the selection module of the 2D geometry object geom1, create a selection object of type CumulativeSelection and associate it with selName; Set the label for this cumulative selection to the current grain boundary number num2str(i) to facilitate identification of its corresponding grain boundary; Set the "contributeto" of the polygon feature pgName to the current cumulative selection selName to ensure that the geometry of the polygon is included in the range of the corresponding cumulative selection. Execute the command geom1.run to generate the 2D geometric object geom1 again, update the geometric selection information, and make the cumulative selection associated with the polygon effective.

[0080] 6. Extract and organize the geometric domain IDs to locate the specific domain bound to the θ value.

[0081] Initialize an empty array allDomainIDs to store the COMSOL geometry domain IDs corresponding to all grain boundary polygons; Using "total number of grain boundaries (gb)" as the loop count, extract the cumulative selection of the corresponding domain IDs one by one: Generate the field selection path selPath corresponding to the cumulative selection, in the format "geom1_csel+serial_dom", such as the path of the first cumulative selection is "geom1_csel1_dom". This path is the default format for identifying "geometry + cumulative selection + field" in COMSOL. The COMSOL selection tool mphgetselection is invoked to extract the geometric domain entity sel.entities corresponding to the cumulative selection via selPath and append it to the allDomainIDs array; The allDomainIDs array is deduplicated using the unique(allDomainIDs, 'stable') method to ensure that only one ID for each geometry domain is retained, thus avoiding duplicate bindings. Reshape the deduplicated allDomainIDs array into a row vector reshape(allDomainIDs,1,[]) to unify the data format; The sorted allDomainIDs array is assigned to the MATLAB base workspace using assignin('base','allDomainIDs',allDomainIDs) for easy access and verification in subsequent steps.

[0082] 7. Establish a unique mapping between the θ value and the geometric domain, and bind the core parameters.

[0083] Delete the created temporary variable var1 to clean up redundant variables and avoid interfering with subsequent parameter operations; Define the total number of domains n_domains, and assign it the length of the allDomainIDs array after sorting it out, thus specifying the number of geometric domains that need to be bound to θ values; Using "total number of domains n_domains" as the loop count, bind each θ value to its corresponding geometric domain one by one: Extract the ID domain_id of the i-th geometry from the allDomainIDs array to determine the binding target of the θ value; Generate the θ parameter expression thetaExpr corresponding to the field, in the format "theta_+ordinal number", such as "theta_1" for the first field, and associate it with the stored θ parameter name; Generate the variable name varName corresponding to this field, in the format of "var + serial number", such as "var1" for the first field, which will serve as the variable identifier in COMSOL that carries the value of θ; Create the variable varName in the variable module of the comp1 component; Set the geometric association of this variable: specify the geometric object associated with the variable as geom1, with a dimension of 2, to adapt to two-dimensional geometry; Set the domain selection of variables: limit the scope of variables to the current geometry domain ID domain_id, ensuring that variables are only effective within the corresponding domain; Assigning values ​​to variables: The value of the variable "theta" is taken by thetaExpr, the name of the θ parameter, to achieve a unique mapping of "θ parameter → variable → geometric domain", so that COMSOL simulation can call the θ value of the corresponding domain to correct the EM term.

[0084] 8. Save the COMSOL model to complete the process loop.

[0085] Call the COMSOL model save function "mphsave" to save the model after all the above operations, including geometry, parameter and variable mapping, as a file "BPR.mph". This file can be opened directly in COMSOL and used for subsequent electromigration stress simulation, such as PDE module modeling, mesh generation and solving.

[0086] In the second specific implementation method, all the algorithm steps involved in the first automated script can be implemented in MATLAB using the corresponding full code, and will not be described again in this embodiment.

[0087] The beneficial effects of this embodiment are as follows: 1) It achieves accurate extraction of the included angle parameter and unique association with the grain boundary, solving the problem of ambiguity in the correspondence between the parameter and the grain boundary. By reading the vertex coordinates of the grain boundary polygon, the characteristic line segment is selected to determine the grain boundary direction. The included angle is calculated using the atan2 function and 10 decimal places are retained. Then, the included angle is bound to the corresponding grain boundary through the grain boundary number, ensuring that the included angle parameter of each grain boundary truly reflects its actual relationship with the current direction, avoiding simulation errors caused by parameter confusion. 2) It constructs a seamless linkage mechanism between the automated script and the physical field simulation platform, breaking through the inefficiency and error-prone bottleneck of manual parameter import. By leveraging a linkage interface, the initialization of the physical field simulation platform model, import of grain boundary polygons, storage of included angle parameters, and binding of geometric domains are automated through scripts, forming a closed-loop process of "parameter extraction - model import - domain binding," significantly reducing manual operations. Simultaneously, through cumulative selection and geometric domain ID deduplication, the uniqueness of the mapping between included angle parameters and solid grain boundaries is ensured, avoiding parameter call errors during simulation. 3) Balancing the versatility of the simulation platform with the portability of its core logic, breaking the limitations of single software compatibility. While prioritizing COMSOL as the physical field simulation platform, it is not exclusive. For differences in API interfaces and modeling logic among different simulation software, only the implementation methods of grain boundary import and parameter mapping need to be adjusted. The core included angle extraction, Korhonen equation coupling, and automation processes can be directly transferred, adapting to various industrial or academic simulation tools, thus expanding the applicability of the technical solution.

[0088] In an optional embodiment, the method further includes performing a preprocessing step before solving the target electromigration stress model; Perform preprocessing steps, including: In the physics simulation platform, an automatic triangular mesh is used to perform mesh generation on the simulation domain of the target electromigration stress model, and mesh parameters are set; for example, the maximum cell size is no more than 70.5 nm, the minimum cell size is no less than 10.7 nm, the maximum cell growth rate is 2, the curvature factor is 1, and the resolution of narrow regions is 0.9. After the mesh is generated, the material parameters and electrical parameters are set in the parameter module of the physics simulation platform. The material parameters include the initial atomic diffusion coefficient, and the initial atomic diffusion coefficient of the internal grain boundaries is greater than that of the grain boundaries at the interface. The electrical parameters include the initial current density and the initial temperature. Based on the physical properties of the target electromigration stress model, the model boundary conditions and initial conditions are set; among them, the atomic flux of all boundaries is set to zero, and the initial stress is uniformly set to 0. To meet the time-series analysis requirements of stress evolution, the step time and total simulation duration of the simulation solution parameters are set.

[0089] Before solving the target electromigration stress model, the method also includes: model loading and parameter configuration; model loading and parameter configuration includes: Load the model file into the physics simulation platform; The target electromigration stress model is set in the partial differential equation module of the physical field simulation platform, and the partial differential equation module is used as the core carrier for stress evolution solution. Based on the mapping relationship, each included angle parameter is associated with the variable library in the partial differential equation module so that each grain boundary domain can call the corresponding included angle parameter during simulation; in particular, the mapping relationship can ensure that the electronic wind term can automatically call the cosθ value of the corresponding grain boundary domain; Based on the expression for atomic flux in the target electromigration stress model, flux source parameters are set in the partial differential equation (PDE) module; these parameters include flux sources in the x-axis and y-axis directions. For example, the x-axis flux source is set as follows: The flux source in the y-axis direction is set to .

[0090] In the third specific implementation method, an exemplary description is given of the electromigration stress modeling method for embedded power rails.

[0091] I. Construction of two-dimensional grain boundary structures.

[0092] An open-source grain boundary generation algorithm was used to generate coordinate data, or coordinate array, representing all grain boundary segments of the embedded power rail. Each row of data in this array corresponds to the start and end coordinates of a grain boundary segment. This coordinate array was imported into the numerical computation software MATLAB and processed by its second automated script to ultimately form a two-dimensional grain boundary structure composed of multiple polygons, with each polygon corresponding to a grain boundary. Figure 2 The example generated two-dimensional grain boundary structures with average grain sizes of 5 nm, 15 nm and 40 nm, respectively. Subsequent simulations used the structure with an average grain size of 5 nm as the demonstration object.

[0093] II. Extraction and mapping of included angle parameters.

[0094] This document describes the development of the first automated script for the MATLAB-CMOSLive link interface, which extracts and calculates the angles between characteristic segments of grain boundaries and current directions in a two-dimensional grain boundary structure. (See also...) Figure 3 The direction of the current is horizontal.

[0095] The first automated script reads the vertex coordinates of the polygon corresponding to each grain boundary, selects the first two vertices to form a feature segment representing the dominant extension direction of the grain boundary, calculates the angle parameter between the feature segment and the current direction, and assigns each angle parameter to the corresponding theta_i (where i represents the i-th grain boundary). Running this script imports the i grain boundary polygons with angle information into COMSOL. Specifically, it creates a unique cumulative selection for each imported grain boundary polygon, and then assigns a corresponding theta value to the field corresponding to each cumulative selection through the platform's built-in API. This achieves a precise mapping between each grain boundary and the angle parameter, providing data support for subsequent correction of the electronic wind term.

[0096] III. Construction of the target electromigration stress model.

[0097] Open the model file (e.g., .mph file) generated after the above steps, and construct the target electromigration stress model in the PDE module of COMSOL. This model is based on the Korhonen equation applicable to grain boundary structures, i.e., formula (3).

[0098] IV. Simulation Domain Mesh Generation.

[0099] In COMSOL, an automatic triangular meshing method is used to mesh the simulation domain of the target electromigration stress model to ensure the accuracy and efficiency of the simulation calculation. The mesh parameters are set as follows: maximum element size 70.5 nm, minimum element size 10.7 nm, maximum element growth rate 2, curvature factor 1, and narrow region resolution 0.9. Figure 4 As shown.

[0100] V. Material and Electrical Parameter Configuration.

[0101] In the COMSOL parameter module, input the required material and electrical parameters for the model. For the material parameters, the initial atomic diffusion coefficient of the internal grain boundaries is set to be much higher than that of the grain boundaries at the interface. Regarding the electrical parameters, the example sets the initial current density to 40 mA / cm² and the initial temperature to 373 K. These parameter settings are consistent with the high current density operating characteristics of the BPR structure.

[0102] VI. PDE Module Setup and Simulation Solution.

[0103] Import the PDE equations corresponding to the completed target electromigration stress model into the COMSOL PDE module, and set the flux source parameters according to the atomic flux expression, where the flux source in the x-axis direction is set as follows: The flux source in the y-axis direction is set to .

[0104] All model boundary conditions were set to zero flux, and the initial stress was uniformly set to 0. To meet the requirements of time-series stress evolution analysis, the simulation step time was set to 10000 s, and the total simulation duration was 40000 s. Solving the partial differential equations of stress evolution in the simulation domain corresponding to the BPR structure yielded a two-dimensional BPR electromigration stress distribution map that varied over time. The stress distribution results at a simulation duration of 40000 s are shown below. Figure 5 As shown. Figure 5 The range of the medium color bar is: (Pascal) to The stress range of Pa clearly shows the stress distribution differences of the BPR structure under a simulation duration of 40,000 s. It allows direct observation of stress concentration areas (such as high-value areas) and low-stress areas, which echoes the core physical logic that "grain boundary orientation modulates the EM term and thus affects stress evolution." If the cosine value of the angle θ between the grain boundary and the current direction is large in a certain region, the electron wind term is stronger and the atomic accumulation / depletion is more significant, corresponding to the stress extremum region in the figure. This verifies the accuracy of the Korhonen model, which incorporates grain boundary angle information, compared with the traditional empirical model.

[0105] VII. Model Analysis and Engineering Applications.

[0106] The aforementioned modeling and simulation process exhibits good reusability and adaptability, and can be repeatedly executed under conditions of different average grain boundary sizes, different BPR structures, different material parameters, and different electrical parameters. By comparing and analyzing the stress evolution patterns in multiple sets of simulation results, it is possible to assess the electromigration failure trend of BEOL advanced interconnect structures, such as BPR, in integrated circuit back-end processes. This example process is suitable for BPR electromigration reliability studies considering the influence of grain boundary structures, and can provide accurate simulation data support for structural optimization design.

[0107] As can be seen from the above, this invention is the first to realize the systematic application of the grain boundary directionality model in BPR structures: This invention is the first to combine the model with two-dimensional BPR structures, and constructs a highly adaptable and scalable electromigration stress modeling and simulation process based on its physical characteristics such as nanometer linewidth, structural size, and high current density, providing a more physically interpretable tool basis for BPR electromigration reliability modeling.

[0108] An automated and modular modeling process was built on the multiphysics simulation platform: the automatic extraction and transfer of grain boundary angle parameters are realized through the linkage interface, which significantly simplifies the mapping process of the original grain boundary structure to the finite element platform, improves modeling efficiency and engineering adaptability, and avoids the uncertainty and complexity brought about by traditional manual modeling.

[0109] This invention explicitly introduces the angle information between the grain boundary and the current direction, enabling the electron wind term, or EM term, in the Korhonen equation to be physically controlled according to the grain boundary orientation. This allows for precise modeling of the atomic flux distribution and stress evolution process of the BPR structure, significantly improving the consistency between simulation results and real microscopic behavior.

[0110] It can be adapted to various materials, process conditions and structural dimensions, and is suitable for advanced node design optimization: This simulation process can be reused under multiple sets of different grain boundary sizes, material parameters and current densities, and is suitable for high-precision reliability analysis and design improvement of BPR or similar BEOL structures in advanced process nodes of 2nm and below, with broad engineering application prospects.

[0111] This study helps to reveal the dominant role of micrograin boundary structure in the electromigration failure mechanism of BPR: by simulating the influence of grain boundary orientation on stress concentration and evolution path, it helps to understand and predict the spatial distribution of void nucleation and propagation behavior in BPR, providing a theoretical basis and quantitative tools for reliability assessment and process optimization.

[0112] In summary, this invention not only effectively connects the grain boundary electromigration model with the physical field simulation platform at the engineering implementation level, but also fills the technical gap of this type of modeling method in BPR application scenarios, and has significant innovative value and industrial application potential.

[0113] The apparatus provided by the present invention is described below, and the apparatus described below can be referred to in correspondence with the method described above.

[0114] like Figure 6 As shown, embodiments of the present invention also provide an electromigration stress modeling device for embedded power rails, used to implement the electromigration stress modeling method for embedded power rails in any of the above embodiments. The device may include: The first construction module 610 is used to construct a two-dimensional grain boundary structure based on the coordinate data of the grain boundary line segments of the embedded power rail; The module 620 is used to determine the angle parameter between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure. The second building module 630 is used to build a target electromigration stress model based on the included angle parameter; Solver module 640 is used to solve the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail.

[0115] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art will understand and implement other variations of the disclosed embodiments by reviewing the accompanying drawings, the disclosure, and the appended claims in carrying out the claimed invention. In the claims, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude a plurality.

[0116] Although the invention has been described in conjunction with specific features and embodiments, it is obvious that various modifications and combinations can be made therein without departing from the spirit and scope of the invention. Accordingly, this specification and drawings are merely exemplary descriptions of the invention as defined by the appended claims, and are considered to cover any and all modifications, variations, combinations, or equivalents within the scope of the invention. Clearly, those skilled in the art can make various alterations and modifications to the invention without departing from its spirit and scope. Thus, if such modifications and modifications of the invention fall within the scope of the claims and their equivalents, the invention is also intended to include such modifications and modifications.

Claims

1. A method for modeling electromigration stress in embedded power rails, characterized in that, include: A two-dimensional grain boundary structure is constructed based on the coordinate data of the grain boundary line segments of the embedded power rail; Determine the angle parameter between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure; A target electromigration stress model is constructed based on the included angle parameters; Solve the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail.

2. The electromigration stress modeling method for embedded power rails according to claim 1, characterized in that, Based on the included angle parameter, a target electromigration stress model is constructed, including: Substituting the cosine value of the included angle parameter into the electron wind term of atomic flux in the pre-constructed initial electromigration stress model, the target electromigration stress model is obtained.

3. The electromigration stress modeling method for embedded power rails according to claim 2, characterized in that, Before constructing the two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail, the method further includes: constructing the initial electromigration stress model; Constructing the initial electromigration stress model includes: The Korhonen equation, applicable to grain boundary structures: This was determined to be the initial electromigration stress model; in, For stress; For time; Bulk modulus; It is atomic flux; ; For electronic wind power; This is the stress gradient term; The atomic diffusion coefficient; Initial electronic wind force; Atomic volume; Boltzmann's constant; For temperature; For stress gradient; Substituting the cosine value of the included angle parameter into the electron wind term of the atomic flux in the pre-constructed initial electromigration stress model includes: Substituting the cosine value of the included angle into the initial electronic wind force yields the corrected electronic wind force: ； Substituting the corrected electron wind into the electron wind term in the atomic flux yields the target electromigration stress model; wherein... This is the corrected electronic wind speed; Effective charge number; It is the elementary charge; Resistivity; Current density; The included angle is the included angle parameter. The value is the cosine of the included angle.

4. The electromigration stress modeling method for embedded power rails according to claim 1, characterized in that, Determining the angle parameters between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure includes: A first automated script, written using numerical calculation software, extracts characteristic line segments from each grain boundary. The direction of the feature line segment is taken as the grain boundary direction of the corresponding grain boundary, and the angle between the feature line segment and the current direction is calculated. The included angle of each grain boundary is associated with the corresponding grain boundary number and solidified into the included angle parameter that can be recognized by the physical field simulation platform; the grain boundary number is used to locate the grain boundary region corresponding to the cosine value of the included angle.

5. The electromigration stress modeling method for embedded power rails according to claim 4, characterized in that, Before constructing the target electromigration stress model based on the included angle parameter, the method further includes: Using the second automated script, through the linkage interface between the numerical calculation software and the physical field simulation platform, each included angle parameter is bound to the corresponding solid grain boundary in the physical field simulation platform, forming a mapping relationship between the included angle parameter and the solid grain boundary; The two-dimensional grain boundary structure, the included angle parameter, and the mapping relationship are collectively fixed into a model file; the model file is used by the physical field simulation platform to provide readable included angle parameters and mapping relationships when constructing the target electromigration stress model.

6. The electromigration stress modeling method for embedded power rails according to claim 4, characterized in that, Extract characteristic line segments from each grain boundary, including: Read the grain boundary polygon data corresponding to the two-dimensional grain boundary structure; the two-dimensional grain boundary structure is composed of multiple polygons, each polygon corresponds to a grain boundary, and the grain boundary polygon data contains the coordinates of all vertices of the polygon corresponding to each grain boundary; From all the vertex coordinates corresponding to each grain boundary, select two target vertices that represent the dominant extension direction of the corresponding grain boundary; Connect the two target vertices and use the resulting line segment as the characteristic line segment of the corresponding grain boundary.

7. The electromigration stress modeling method for embedded power rails according to claim 4, characterized in that, Based on the coordinate data of grain boundary segments of the embedded power rail, a two-dimensional grain boundary structure is constructed, including: Import the grain boundary line segment coordinate data into the numerical calculation software, and use the second automated script written in the numerical calculation software to set the physical thickness; Calculate the length of each grain boundary segment, filter out invalid short segments with a length less than a preset threshold, and calculate the offset perpendicular to the normal direction to generate multiple polygons with thickness. Eliminate overlapping areas between different polygons among multiple polygons with thickness, and filter out valid areas with an area greater than a preset area; Collect all valid regions to form the two-dimensional grain boundary structure.

8. The electromigration stress modeling method for embedded power rails according to claim 4, characterized in that, Before solving the target electromigration stress model, the method further includes: performing a preprocessing step; Perform preprocessing steps, including: In the physical field simulation platform, an automatic triangular mesh is used to perform mesh generation on the simulation domain of the target electromigration stress model and set the mesh parameters; After the mesh is generated, material parameters and electrical parameters are set in the parameter module of the physical field simulation platform. The material parameters include the initial atomic diffusion coefficient, and the initial atomic diffusion coefficient of the internal grain boundaries is greater than that of the grain boundaries at the interface. The electrical parameters include the initial current density and the initial temperature. Based on the physical properties of the target electromigration stress model, model boundary conditions and initial conditions are set; wherein, the atomic flux of all boundaries is set to zero flux, and the initial stress is uniformly set to 0; To meet the time-series analysis requirements of stress evolution, the step time and total simulation duration of the simulation solution parameters are set.

9. The electromigration stress modeling method for embedded power rails according to claim 5, characterized in that, Before solving the target electromigration stress model, the method further includes: model loading and parameter configuration; Perform model loading and parameter configuration, including: Load the model file into the physical field simulation platform; The target electromigration stress model is set in the partial differential equation module of the physical field simulation platform, and the partial differential equation module is used as the core carrier for stress evolution solution. According to the mapping relationship, each included angle parameter is associated with the variable library in the partial differential equation module so that each grain boundary domain can call the corresponding included angle parameter during simulation; Based on the expression for atomic flux in the target electromigration stress model, flux source parameters are set in the partial differential equation module; the flux source parameters include flux sources in the x-axis direction and flux sources in the y-axis direction.

10. A device for modeling electromigration stress of embedded power rails, characterized in that, include: The first construction module is used to construct a two-dimensional grain boundary structure based on the coordinate data of the grain boundary segments of the embedded power rail; The determination module is used to determine the angle parameter between the grain boundary direction and the current direction of each grain boundary in the two-dimensional grain boundary structure; The second construction module is used to construct a target electromigration stress model based on the included angle parameter; The solver module is used to solve the target electromigration stress model to obtain the electromigration stress distribution of the embedded power rail.

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