A method for electromigration stress analysis of multi-segment metal interconnect lines and related apparatus
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- GUANGDONG UNIV OF TECH
- Filing Date
- 2026-03-12
- Publication Date
- 2026-05-29
Smart Images
Figure CN122113650A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of integrated circuit technology, and in particular to a method and related equipment for analyzing electromigration stress in multi-segment metal interconnects. Background Technology
[0002] Electromigration in metal interconnects refers to the phenomenon of metal ions migrating within metal interconnects under the influence of an electric field. Electromigration has become a critical reliability challenge in integrated circuit design, exacerbated by miniaturization, high current densities, and narrow interconnects. Driven by "electron wind," atomic migration leads to cathode voids and anode buildup, reducing the effective cross-section and increasing resistance. This phenomenon is particularly pronounced in power distribution networks. Therefore, developing efficient and accurate electromigration assessment methods is crucial for ensuring the long-term reliability of chips.
[0003] Traditional electromigration modeling relies heavily on empirical methods, which poses a significant challenge in characterizing complex multi-branch structures. While the Black equation and Blech criterion can estimate failure time and identify immortal segments by multiplying current density by length, these methods are based on the core assumption that each segment is treated as an independent, isolated unit. This simplification ignores the coupling of the internal physical mechanisms of the interconnect tree, leading to significant biases in assessing the reliability of complex interconnect architectures.
[0004] Building upon previous research, the Korhonen equation, as a physical alternative to the Black equation, can comprehensively consider the current-driven "electron wind" force and the reverse stress driven by the diffusion gradient, thus effectively characterizing the electromigration stress evolution of tree-level interconnects rather than being limited to single wires. In terms of computational methods, numerical methods such as the finite difference method and the finite element method typically require fine mesh generation and time-step discretization during stress analysis. Due to the involvement of solving a massive number of unknown variables, these methods generally suffer from low computational efficiency. In contrast, analytical methods, by constructing basis functions through Laplace transforms, can obtain closed-form solutions to the hydrostatic stress of simple interconnect trees, providing a new approach for electromigration reliability analysis. For example, existing techniques have improved the separation of variables method, achieving computational acceleration for specific interconnect structures. However, although the above research has enabled tree-level stress analysis, solving the Korhonen equation still faces the severe challenges of long computation time and limited accuracy when dealing with interconnect trees of ever-expanding scale.
[0005] In recent years, machine learning has demonstrated great potential as an efficient tool in simulating electromigration stress analysis dominated by complex partial differential equations. Based on this, the academic community has proposed a series of purely data-driven and semi-analytical solutions. Some researchers have introduced hierarchical physical information neural networks to achieve rapid evaluation of electromigration stress in multi-branch interconnected structures. However, this method relies on supervised learning to predict stress gradients under different boundary conditions and geometric configurations, which not only requires massive amounts of labeled data but also limits the algorithm's generalization ability. Summary of the Invention
[0006] This invention provides a method and related equipment for electromigration stress analysis of multi-segment metal interconnects, aiming to solve the problem of poor neural network fitting effect caused by the increase of simulation segments in large-scale interconnect tree stress analysis.
[0007] To achieve the above objectives, the present invention provides a method for electromigration stress analysis of multi-segment metal interconnects, comprising: Step 1: Obtain the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node, and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; Step 2: Calculate the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters to identify the stress change trend of each metal interconnect. Then, calculate the stress trend change at both ends based on the stress change trend, geometric parameters, local spatial position of each interconnect node and simulation time point to determine the priority of each metal interconnect, and obtain the priority segment set and the secondary segment set. Step 3: Based on the calculated equivalent length threshold of secondary line segments, the secondary line segments in the secondary line segment set are segmented to obtain auxiliary line segments. The auxiliary line segments are added to the priority line segment set, and all metal interconnects in the secondary line segment set except for the auxiliary line segments are equivalent to a secondary line segment. Step 4: Input the first data into the trained spatiotemporal convolutional neural network. Input the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each priority line segment, and simulation time point of each priority line segment into the trained spatiotemporal convolutional neural network as the second data. Use the spatiotemporal convolutional neural network to simulate the interconnection tree to be processed and obtain the derivative of the line segment atomic flux over time. Step 5: Calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
[0008] Furthermore, the formula for calculating the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters is as follows: ; in, Represents line segment At the endpoint The atomic flux at that point is zero. This represents the electromagnetic driving force at the endpoint. The numbers 1, 2, 3, and 4 indicate the directions of the endpoints within the line segment, with 1 representing the left, up, right, and down directions, respectively. Indicates the sequence number of the currently simulated line segment. Represents line segment Length, , , , Representing line segments respectively The line segment numbers connecting the four directions: left, top, right, and bottom. Represents line segment The current density.
[0009] Furthermore, based on the stress variation trend, geometric parameters, the local spatial location of each interconnecting node, and the simulation time point, the calculation expression for the stress variation at both ends is as follows: ; in, This represents the upper limit of the order of the summation. This indicates the order of the summation. Indicates the length of the line segment. , Let represent the first derivatives of the stress at the left and right endpoints of the line segment with respect to the local coordinates of the line segment, respectively. Indicates the stress diffusion coefficient of a line segment. This represents an auxiliary function.
[0010] Furthermore, the calculation process for the equivalent length threshold of secondary line segments includes: Establish the constraint relationship between equivalent line segment data and numerical stability; Based on the stress diffusion coefficient, under the premise of satisfying the constraint relationship between equivalent line segment data and numerical stability, an approximate correlation relationship between the local spatial position of each interconnection node and the simulation time point is established, and the approximate kernel function value is obtained as the numerical truncation threshold. The spatial boundary threshold equation is derived using numerical stage thresholds. The equivalent length threshold of minor line segments is calculated using the spatial boundary threshold equation.
[0011] Furthermore, the approximate correlation between the local spatial location of each interconnect node and the simulation time point is established as follows: ; in, Represents the kernel function. This represents the local spatial location of each interconnected node. Indicates the simulation time point, This represents the stress diffusion coefficient.
[0012] Furthermore, the simulation of the interconnection tree to be processed is performed using a spatiotemporal convolutional neural network, including: Stack the first data to obtain a high-dimensional input tensor; The high-dimensional input tensor is input into the feature extraction module for feature extraction, resulting in a feature tensor. The feature tensor is input into the linear interpolation module for smooth mapping to obtain the dimension-reduced feature map. The reduced feature map is input into a multilayer perceptron for processing to obtain the derivative of the atomic flux of the line segment over time.
[0013] The present invention also provides an electromigration stress analysis device for multi-segment metal interconnects, comprising: The acquisition module is used to acquire the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node, and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; The determination module is used to calculate the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters to identify the stress change trend of each metal interconnect. Based on the stress change trend, geometric parameters, local spatial position of each interconnect node and simulation time point, the change in stress trend at both ends is calculated to determine the priority of each metal interconnect, resulting in a priority segment set and a secondary segment set. The segmentation module is used to segment the secondary segments in the secondary segment set based on the calculated equivalent length threshold of the secondary segment to obtain auxiliary segments, add the auxiliary segments to the priority segment set, and treat all metal interconnects in the secondary segment set except for the auxiliary segments as equivalent to a secondary segment. The simulation module is used to input the first data into the trained spatiotemporal convolutional neural network, and input the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each priority line segment, and simulation time point of each priority line segment into the trained spatiotemporal convolutional neural network as the second data. The spatiotemporal convolutional neural network is then used to simulate the interconnection tree to be processed, and the derivative of the line segment atomic flux with time is obtained. The calculation module is used to calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
[0014] The present invention also provides a terminal device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement a method for analyzing the electromigration stress of multiple metal interconnects.
[0015] The present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements a method for analyzing the electromigration stress of multiple metal interconnects.
[0016] The above-described solution of the present invention has the following beneficial effects: Compared with existing technologies, this invention determines the priority of each metal interconnect by analyzing the stress variation trend and the change in stress at both ends of each metal interconnect. Based on the calculated equivalent length threshold of secondary segments, secondary segments in the secondary segment set are segmented to obtain auxiliary segments. These auxiliary segments are added to the priority segment set, and all metal interconnects in the secondary segment set except for auxiliary segments are treated as equivalent to a secondary segment. This combination significantly reduces computational complexity while strictly preserving the physical boundary conditions of key nodes. A spatiotemporal convolutional neural network is used to simulate the interconnect tree to be processed. Key areas prone to hole formation are targeted for high-precision simulation, while smaller areas are processed with lower precision, achieving a synergy between high and low precision. This solves the problem of poor neural network fitting performance caused by the increase in simulated segments in large-scale interconnect tree stress analysis.
[0017] Other beneficial effects of the present invention will be described in detail in the following detailed description section. Attached Figure Description
[0018] Figure 1 This is a flowchart illustrating an embodiment of the present invention; Figure 2 This is a schematic diagram of an embodiment of the present invention; Figure 3 This is a schematic diagram of the electromigration stress analysis device in an embodiment of the present invention; Figure 4 This is a schematic diagram of the structure of the terminal device in an embodiment of the present invention. Detailed Implementation
[0019] To make the technical problems, solutions, and advantages of this invention clearer, a detailed description will be provided below with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.
[0020] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.
[0021] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a locking connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.
[0022] Furthermore, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0023] This invention addresses existing problems by providing a method and related equipment for analyzing electromigration stress in multi-segment metal interconnects.
[0024] like Figure 1 , 2 As shown, an embodiment of the present invention provides a method for electromigration stress analysis of multi-segment metal interconnects, comprising: Step 1: Obtain the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node, and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; Step 2: Calculate the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters to identify the stress change trend of each metal interconnect. Then, calculate the stress trend change at both ends based on the stress change trend, geometric parameters, local spatial position of each interconnect node and simulation time point to determine the priority of each metal interconnect, and obtain the priority segment set and the secondary segment set. Step 3: Based on the calculated equivalent length threshold of secondary line segments, the secondary line segments in the secondary line segment set are segmented to obtain auxiliary line segments. The auxiliary line segments are added to the priority line segment set, and all metal interconnects in the secondary line segment set except for the auxiliary line segments are equivalent to a secondary line segment. Step 4: Input the first data into the trained spatiotemporal convolutional neural network. Input the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each priority line segment, and simulation time point of each priority line segment into the trained spatiotemporal convolutional neural network as the second data. Use the spatiotemporal convolutional neural network to simulate the interconnection tree to be processed and obtain the derivative of the line segment atomic flux over time. Step 5: Calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
[0025] In this embodiment of the invention, the geometric parameters of each metal interconnect in the interconnect tree to be processed include the width of the line segment. and the length of the line segment The electromagnetic driving force parameter is the current density. The local spatial location of interconnected nodes is used Representation, simulation time points express.
[0026] Since electromagnetic stress simulation of interconnect trees is mainly used to predict the lifespan of metal segments in a chip, and the key factor determining lifespan is the metal segment with the highest electromigration stress, this embodiment of the invention performs stress simulation on all segments, prioritizing the metal segment with the highest electromigration stress. The stress of a metal segment is closely related to the current density and atomic flux at both ends of the segment, thus allowing for further analysis of stress variations with local spatial location. The partial derivatives are: ; in, This indicates the order of the summation. Indicates the length of the line segment. Indicates stress on a line segment. This represents the upper limit of the order of the summation. , It is also an auxiliary function; in: ; ; ; ; , Let represent the first derivatives of the stress at the left and right endpoints of the line segment with respect to the local coordinates of the line segment, respectively. , Let represent the second derivatives of the stress at the left and right endpoints of the line segment with respect to the local coordinates of the line segment, respectively.
[0027] Specifically, step 2 includes: The formula for calculating the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters is as follows: ; in, Represents line segment At the endpoint The atomic flux at that point is zero. This represents the electromagnetic driving force at the endpoint. The numbers 1, 2, 3, and 4 indicate the directions of the endpoints within the line segment, with 1 representing the left, up, right, and down directions, respectively. Indicates the sequence number of the currently simulated line segment. Represents line segment Length, , , , Representing line segments respectively The line segment numbers connecting the four directions: left, top, right, and bottom. Represents line segment The current density; The stress variation trend of each metal interconnect is identified based on the zero atomic flux value of each metal interconnect segment, as shown in Table 1 below: Table 1 Stress Variation Trend Table
[0028] Based on the stress variation trend, geometric parameters, local spatial location of each interconnect node, and simulation time point, the stress trend change at both ends is calculated to determine the priority of each metal interconnect line, thus obtaining the priority segment set and the secondary segment set.
[0029] Specifically, based on the stress variation trend, geometric parameters, the local spatial location of each interconnect node, and the simulation time point, the calculation expression for the stress variation at both ends is as follows: ; in, This represents the upper limit of the order of the summation. This indicates the order of the summation. Indicates the length of the line segment. , Let represent the first derivatives of the stress at the left and right endpoints of the line segment with respect to the local coordinates of the line segment, respectively. Indicates the stress diffusion coefficient of a line segment. This represents an auxiliary function, and its specific expression is: .
[0030] This invention, through analyzing the relationship between floating-point precision limitations and kernel function decay characteristics, derives the effective spatial boundary of equivalent line segments, thereby calculating the equivalent length threshold of secondary line segments. The calculation process of the equivalent length threshold of secondary line segments specifically includes: Establish the constraint relationship between equivalent line segment data and numerical stability; Based on the stress diffusion coefficient, under the premise of satisfying the constraint relationship between equivalent line segment data and numerical stability, an approximate correlation relationship between the local spatial position of each interconnection node and the simulation time point is established, and the approximate kernel function value is obtained as the numerical truncation threshold. The spatial boundary threshold equation is derived using numerical stage thresholds. The equivalent length threshold of minor line segments is calculated using the spatial boundary threshold equation.
[0031] In constructing the neural network model for stress simulation, this invention first establishes a constraint relationship between equivalent line segment data and numerical stability. While theoretically increasing the number of equivalent line segments can significantly reduce simulation parameters and improve acceleration, the system identifies a critical point for numerical stability: when the number of equivalent line segments exceeds this critical value and the equivalent line segment length is too large, the kernel function value becomes extremely small. Based on the default single-precision floating-point format used by deep learning frameworks (such as PyTorch mps), when the kernel function value is below the lower limit of floating-point precision, gradient vanishing occurs, preventing the gradient of the atomic flux at both ends of the line segment from effectively backpropagating in the neural network, thus significantly reducing the approximate ability of solving the PDE equation. Therefore, a numerical truncation threshold is introduced as a constraint.
[0032] Specifically, the approximate correlation between the local spatial location of each interconnect node and the simulation time point is established as follows: ; in, Represents the kernel function. This represents the local spatial location of each interconnected node. Indicates the simulation time point, This represents the stress diffusion coefficient.
[0033] In order to determine the boundary of the effective computational domain, this embodiment of the invention uses an approximate kernel function as a numerical truncation threshold, thereby solving for the maximum spatial distance that guarantees gradient propagation. Through mathematical derivation, substituting the numerical truncation threshold into the above formula and eliminating the exponential term using logarithmic operations, the spatial boundary threshold function of spatial location with respect to time is obtained as follows: ; To maximize the number of equivalent line segments while ensuring their numerical value remains non-zero, it is necessary to determine the maximum effective value of the simulation time t in the calculation formula. The simulation time t is set as the sum of the preset maximum simulation time and the maximum zeros of the Legendre polynomials. This simulation time is then substituted into the spatial boundary threshold function to calculate... The value is the threshold for the equivalent length of the secondary line segment; Then, depth-first search is used to locate the secondary segments in the priority segment set, and the secondary segments in the secondary segment set are segmented according to the equivalent length threshold of the secondary segments to obtain auxiliary segments, which are then added to the priority segment set.
[0034] To efficiently simulate the stress evolution of secondary line segments, this invention proposes two equivalent simplification algorithms: the first algorithm is for series-connected multi-segment straight line structures, the core of which is to model the stress difference at the nodes at both ends of the equivalent line segments as a time-varying function related to the electromigration driving force; the second algorithm is for cross-interconnected structures.
[0035] Theorem 1: For any given condition, ... A structure composed of consecutive straight segments has an equivalent relationship. and Under these conditions, the stress difference between the left and right nodes of the equivalent segment is different.
[0036] Specifically, the stress difference at the left node is given by the following formula: ; The stress difference at the right node is given by the following equation: ; in, ; ; ; ; For the first Current density of each line segment For the first Current density of each line segment This is the set of indices for the internal nodes. , , This is an auxiliary function.
[0037] Theorem 2: Consider a line with a total length of of Segment interconnects, in any global coordinates Among them, Equivalent homogeneous model With precise piecewise model Stress difference between Given by the following formula: ; in ; as well as ,exist The first item in the table represents the node. The first two terms reflect the local error caused by the internal flux divergence, while the latter two terms reflect the error caused by the flux divergence within the local flux divergence. and The propagation error caused by the mismatch of boundary conditions at the point of origin. It is asymptotically stable in the time domain and bounded in the spatial domain.
[0038] Theorem 3: For segments connected to internal nodes, except for one segment where both nodes have a degree greater than 1, all other connected segments have the following structure: one node has a degree of 1. A segment connected to an endpoint can be equivalently represented as a single segment, with the following equivalence rules: ; ; in, , , , , These represent the stress gradient information of the internal nodes in the left, right, down, and up directions, respectively.
[0039] Before equivalence, the original N segments required prediction of 2N parameters. Under ideal conditions, after equivalence, only 2 parameters need to be predicted, which greatly reduces the difficulty of stress simulation.
[0040] It should be noted that the equivalent algorithm described in the embodiments of the present invention specifically refers to port equivalence. For the preferred line segment, the stress boundary conditions at the interconnecting nodes remain consistent before and after the equivalent transformation. However, for the observation points inside the equivalent structure, the stress values are not physically completely equivalent.
[0041] Specifically, the simulation of the interconnection tree to be processed is performed using a spatiotemporal convolutional neural network, including: The first set of data is stacked, and this first set of data covers the simulation time. Local spatial location And the six-dimensional eigenvectors of the electromigration driving force components in the four directions, resulting in a shape of The high-dimensional input tensor, where, Specifically refers to the total number of original internal interconnect nodes without equivalent simplification, thus ensuring the integrity of physical topology information; The high-dimensional input tensor is input into the feature extraction module for feature extraction, resulting in a feature tensor. The feature tensor is input into the linear interpolation module for smooth mapping to obtain the dimension-reduced feature map. The reduced feature map is input into a multilayer perceptron for processing to obtain the derivative of the atomic flux of the line segment over time.
[0042] In this embodiment of the invention, the feature extraction module includes a one-dimensional temporal convolutional layer and a one-dimensional spatial convolutional layer. The one-dimensional temporal convolutional layer has a kernel size of 3 and padding of 1, used to capture the dynamic evolution features of stress over time, and its output shape is as follows. The temporal feature tensor, after dimensionality transformation, enters a one-dimensional spatial convolutional layer. The node dimension extracts the flux gradient and coupling relationship between adjacent nodes. At this point, the shape of the output feature tensor transforms into... .
[0043] This invention introduces a linear interpolation module to replace the traditional pooling layer. This structural innovation enables the model to smoothly map and reduce the dense feature space containing the original physical information to a sparse equivalent node space, achieving a precise feature transition from the physical real structure to the computationally equivalent model. This effectively avoids the loss of key topological information caused by traditional downsampling, which is the biggest highlight that distinguishes this model from conventional convolutional neural networks.
[0044] It should be noted that the spatiotemporal convolutional neural network provided in this embodiment of the invention uses the Adam optimizer for optimization training, and this embodiment of the invention does not limit its specific training method and process.
[0045] This invention compares the performance of the proposed method with that of an MLP-based model in terms of stress prediction accuracy and computational speedup. As a benchmark, the MLP employs a recommended structure of 5 hidden layers with 50 neurons per layer, and is optimized using the LBFGS algorithm.
[0046] All methods used in the experiments were implemented using Python 3.9.5 and PyTorch 1.12. As a baseline, the COMSOL simulation stress solution was obtained by solving the Korhonen equation under a given geometry and current density configuration, and exported and generated using the MATLAB Live Link for COMSOL Multiphysics 6.2 interface. All experiments were run on a computing platform equipped with an Apple M4 processor and 16GB of memory.
[0047] A consistent data scaling factor was used in all experiments: , , Regarding model configuration, the MLP model uses the Xavier scheme for weight initialization and the LBFGS optimizer for training; while the model provided in this embodiment of the invention uses the Adam optimizer for training, and the learning rate of both is uniformly set to 0.001.
[0048] The key physical parameters involved in the simulation are as follows: Boltzmann constant elementary charge Effective charge number Activation energy Effective bulk modulus diffusion coefficient resistivity and atomic volume For the training dataset, we set the time interval ( Uniform sampling was performed at 100 points.
[0049] Table 2 Relative Error Results
[0050] Table 3
[0051] The experimental results show that when the number of interconnecting segments is small, the proposed method does not exhibit a significant advantage over COMSOL in terms of accuracy and computational speed. Compared to MLP-based methods, in scenarios with a very small number of segments, the proposed model's speed advantage is not significant because it needs to use a depth-first search (DFS) algorithm to determine priority segments; however, in terms of accuracy, thanks to the simplification of the simulated segment parameters, the model can more fully fit the physical evolution of segment stress, resulting in significantly higher accuracy than the MLP model. It should be noted that for equivalent segments, because they are integrated into the overall segment, the internal test points and the current density information of the original segment are missing, leading to relatively low prediction accuracy in this part.
[0052] When dealing with large-scale interconnect trees, the proposed method achieves a speedup of up to 469 times compared to COMSOL on the test dataset. Compared to MLP models, the proposed method not only significantly outperforms the prediction accuracy of both overall and priority segments, but also demonstrates a significant advantage in computational speed. Under optimal conditions, the prediction accuracy of stress evolution for priority segments is improved by 91.9270%, while the computational speed is improved by 27.8730 times.
[0053] Compared with existing technologies, this invention determines the priority of each metal interconnect by analyzing the stress variation trend and the change in stress at both ends of each metal interconnect. Based on the calculated equivalent length threshold of secondary segments, secondary segments in the secondary segment set are segmented to obtain auxiliary segments. These auxiliary segments are added to the priority segment set, and all metal interconnects in the secondary segment set except for auxiliary segments are treated as equivalent to a secondary segment. This combination significantly reduces computational complexity while strictly preserving the physical boundary conditions of key nodes. A spatiotemporal convolutional neural network is used to simulate the interconnect tree to be processed. Key areas prone to hole formation are used as targets for high-precision simulation, while smaller areas are processed with lower precision, achieving a synergy between high and low precision. This solves the problem of poor neural network fitting effect caused by the increase in the number of simulated segments in large-scale interconnect tree stress analysis.
[0054] Corresponding to the electromigration stress analysis method for multi-segment metal interconnects described in the above embodiments, such as Figure 3 As shown, the present invention also provides an electromigration stress analysis device 100 for multi-segment metal interconnects, the electromigration stress analysis device 100 comprising: The acquisition module 101 is used to acquire the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; The determination module 102 is used to calculate the zero atomic flux of each metal interconnect segment based on geometric parameters and electromagnetic driving force parameters to identify the stress change trend of each metal interconnect. Based on the stress change trend, geometric parameters, local spatial position of each interconnect node and simulation time point, the change in stress trend at both ends is calculated to determine the priority of each metal interconnect, thus obtaining the priority segment set and the secondary segment set. The segmentation module 103 is used to segment the secondary segments in the secondary segment set based on the calculated equivalent length threshold of the secondary segment to obtain auxiliary segments, add the auxiliary segments to the priority segment set, and treat the metal interconnects in the secondary segment set other than the auxiliary segments as equivalent to a secondary segment. The simulation module 104 is used to input the first data into the trained spatiotemporal convolutional neural network, and input the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each priority line segment, and simulation time point of each priority line segment into the trained spatiotemporal convolutional neural network as the second data. The spatiotemporal convolutional neural network is used to simulate the interconnection tree to be processed to obtain the derivative of the line segment atomic flux over time. The calculation module 105 is used to calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
[0055] It should be noted that the information interaction and execution process between the above-mentioned devices / units are based on the same concept as the method embodiments of this application. For details on their specific functions and technical effects, please refer to the method embodiments section, and they will not be repeated here.
[0056] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the above-described division of functional units and modules is merely an example. In practical applications, the above functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiments can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit. Furthermore, the specific names of the functional units and modules are only for easy differentiation and are not intended to limit the scope of protection of this application. The specific working process of the units and modules in the above system can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.
[0057] This invention also provides a terminal device, such as... Figure 4 As shown, the terminal device D10 of this embodiment includes: at least one processor D100 ( Figure 4 The diagram shows only one processor, a memory D101, and a computer program D102 stored in the memory D101 and executable on the at least one processor D100. When the processor D100 executes the computer program D102, it implements the above-described method for analyzing the electromigration stress of multi-segment metal interconnects.
[0058] The terminal device D10 can be a desktop computer, laptop, handheld computer, server, server cluster, or cloud server, etc. This terminal device may include, but is not limited to, a processor D100 and a memory D101. Those skilled in the art will understand that... Figure 4 This is merely an example of terminal device D10 and does not constitute a limitation on terminal device D10. It may include more or fewer components than shown in the figure, or combine certain components, or different components, such as input / output devices, network access devices, etc.
[0059] The processor D100 can be a central processing unit (CPU), or it can be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor can be a microprocessor or any conventional processor.
[0060] In some embodiments, the memory D101 may be an internal storage unit of the terminal device D10, such as a hard disk or memory of the terminal device D10. In other embodiments, the memory D101 may be an external storage device of the terminal device D10, such as a plug-in hard disk, smart media card (SMC), secure digital card (SD), flash card, etc., equipped on the terminal device D10. Furthermore, the memory D101 may include both internal and external storage units of the terminal device D10. The memory D101 is used to store the operating system, applications, bootloader, data, and other programs, such as the program code of the computer program. The memory D101 can also be used to temporarily store data that has been output or will be output.
[0061] It should be noted that the information interaction and execution process between the above-mentioned devices / units are based on the same concept as the method embodiments of this application. For details on their specific functions and technical effects, please refer to the method embodiments section, and they will not be repeated here.
[0062] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the above-described division of functional units and modules is merely an example. In practical applications, the above functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiments can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit. Furthermore, the specific names of the functional units and modules are only for easy differentiation and are not intended to limit the scope of protection of this application. The specific working process of the units and modules in the above system can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.
[0063] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements a method for analyzing the electromigration stress of multiple metal interconnects.
[0064] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments of this application can be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or certain intermediate forms. The computer-readable medium can include at least: any entity or device capable of carrying the computer program code to a building device / terminal device, a recording medium, a computer memory, a read-only memory (ROM), a random access memory (RAM), an electrical carrier signal, a telecommunication signal, and a software distribution medium. Examples include USB flash drives, portable hard drives, magnetic disks, or optical disks.
[0065] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.
Claims
1. A method for analyzing electromigration stress in multi-segment metal interconnects, characterized in that, include: Step 1: Obtain the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node, and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; Step 2: Calculate the zero atomic flux value of each metal interconnect segment based on the geometric parameters and the electromagnetic driving force parameters to identify the stress change trend of each metal interconnect. Calculate the stress change trend at both ends based on the stress change trend, the geometric parameters, the local spatial position of each interconnect node, and the simulation time point to determine the priority of each metal interconnect, thus obtaining a priority segment set and a secondary segment set. Step 3: Based on the calculated equivalent length threshold of the secondary line segment, the secondary line segments in the secondary line segment set are segmented to obtain auxiliary line segments. The auxiliary line segments are added to the priority line segment set, and all metal interconnects in the secondary line segment set except for the auxiliary line segments are equivalent to a secondary line segment. Step 4: Input the first data into the trained spatiotemporal convolutional neural network, and input the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each priority line segment, simulation time point, geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnection node in each secondary line segment, and simulation time point of the secondary line segment as the second data into the trained spatiotemporal convolutional neural network. Use the spatiotemporal convolutional neural network to simulate the interconnection tree to be processed to obtain the derivative of the line segment atomic flux over time. Step 5: Calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
2. The electromigration stress analysis method for multi-segment metal interconnects according to claim 1, characterized in that, The calculation expression for the zero atomic flux value of each metal interconnect segment based on the geometric parameters and the electromagnetic driving force parameters is as follows: ； in, Represents line segment At the endpoint The atomic flux at that point is zero. This represents the electromagnetic driving force at the endpoint. The numbers 1, 2, 3, and 4 indicate the directions of the endpoints within the line segment, with 1 representing the left, up, right, and down directions, respectively. Indicates the sequence number of the currently simulated line segment. Represents line segment Length, , , , Representing line segments respectively The line segment numbers connecting the four directions: left, top, right, and bottom. Represents line segment The current density.
3. The electromigration stress analysis method for multi-segment metal interconnects according to claim 1, characterized in that, Based on the stress variation trend, the geometric parameters, the local spatial location of each interconnect node, and the simulation time point, the calculation expression for the stress variation at both ends is as follows: ； in, This represents the upper limit of the order of the summation. This indicates the order of the summation. Indicates the length of the line segment. , Let represent the first derivatives of the stress at the left and right endpoints of the line segment with respect to the local coordinates of the line segment, respectively. Indicates the stress diffusion coefficient of a line segment. This represents an auxiliary function.
4. The electromigration stress analysis method for multi-segment metal interconnects according to claim 1, characterized in that, The calculation process for the equivalent length threshold of the secondary line segment includes: Establish the constraint relationship between equivalent line segment data and numerical stability; Based on the stress diffusion coefficient, under the premise of satisfying the constraint relationship between equivalent line segment data and numerical stability, an approximate correlation relationship between the local spatial position of each interconnection node and the simulation time point is established, and the approximate kernel function value is obtained as the numerical truncation threshold. The spatial boundary threshold equation is derived using the numerical stage threshold. The equivalent length threshold of the secondary line segment is calculated using the aforementioned spatial boundary threshold equation.
5. The electromigration stress analysis method for multi-segment metal interconnects according to claim 4, characterized in that, The approximate correlation between the local spatial location of each interconnect node and the simulation time point is established as follows: ； in, Represents the kernel function. This represents the local spatial location of each interconnected node. Indicates the simulation time point, This represents the stress diffusion coefficient.
6. The electromigration stress analysis method for multi-segment metal interconnects according to claim 4, comprising simulating the interconnect tree to be processed using a spatiotemporal convolutional neural network, including: Stack the first data to obtain a high-dimensional input tensor; The high-dimensional input tensor is input into the feature extraction module for feature extraction to obtain the feature tensor; The feature tensor is input into a linear interpolation module for smooth mapping to obtain a dimension-reduced feature map. The reduced feature map is input into a multilayer perceptron for processing to obtain the derivative of the line segment atomic flux over time.
7. An apparatus for analyzing electromigration stress in multi-segment metal interconnects, characterized in that, include: The acquisition module is used to acquire the geometric parameters, electromagnetic driving force parameters, local spatial position of each interconnect node, and simulation time point of each metal interconnect in the interconnect tree to be processed as the first data; The determination module is used to calculate the atomic flux zero value of each metal interconnect segment based on the geometric parameters and the electromagnetic driving force parameters to identify the stress change trend of each metal interconnect, and to calculate the stress trend change at both ends based on the stress change trend, the geometric parameters, the local spatial position of each interconnect node and the simulation time point to determine the priority of each metal interconnect, thereby obtaining a priority segment set and a secondary segment set. The segmentation module is used to segment the secondary segments in the secondary segment set based on the calculated equivalent length threshold of the secondary segment to obtain auxiliary segments, add the auxiliary segments to the priority segment set, and treat the metal interconnects in the secondary segment set other than the auxiliary segments as equivalent to a secondary segment. The simulation module is used to input the first data into the trained spatiotemporal convolutional neural network, and to input the geometric parameters, electromagnetic driving force parameters, local spatial positions of each interconnection node in each priority line segment, and simulation time points of the secondary line segments as the second data into the trained spatiotemporal convolutional neural network. The spatiotemporal convolutional neural network is then used to simulate the interconnection tree to be processed to obtain the derivative of the line segment atomic flux over time. The calculation module is used to calculate the electromigration stress analysis results of the interconnect tree to be processed based on the derivative of the atomic flux of the line segment over time.
8. A terminal device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the electromigration stress analysis method for multi-segment metal interconnects as described in any one of claims 1 to 6.
9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the electromigration stress analysis method for multi-segment metal interconnects as described in any one of claims 1 to 6.