Tddb reliability analysis with spatiotemporal resolution and breakdown location method and system

Abstract

The application provides a TDDB reliability analysis and breakdown positioning method and system with space-time resolution, and proposes a combined use of ion seepage theory, minimum resistance path analysis and dynamic maximum bottleneck path evaluation model to improve the accuracy of the TDDB reliability analysis and breakdown positioning method with space-time resolution. The application has a much lower calculation cost than the FDM simulation of the drift-diffusion model, since the algorithm only needs to be repeatedly executed on a limited number of time snapshots. The algorithm can monitor the TDDB damage degree at different positions in real time, identify failure hot spots, predict the actual TDDB life, and has good universality.

Description

Technical Field

[0001] This invention relates to the fields of semiconductor devices and electronic design automation algorithms, and more specifically, to a TDDB reliability analysis and breakdown location method and system with spatiotemporal resolution. Background Technology

[0002] Time-dependent dielectric breakdown (TDDB) is one of the major reliability challenges in advanced integrated circuit design. As modern integrated circuit devices shrink in size and interconnect spacing tightens, dielectric layers become thinner, leading to enhanced local electric fields. This accelerates defect formation and ion migration in low-k dielectrics and interlayer dielectric materials. Conductive filaments or ion channels formed between power lines (VDD) and ground lines (GND) eventually cause irreversible dielectric short circuits, resulting in TDDB failure in advanced processes. Therefore, accurately predicting the spatial location of breakdown and the time of TDDB failure is crucial for circuit design and layout optimization that meet reliability requirements.

[0003] Traditional TDDB modeling primarily focuses on lifetime data statistics and empirical electric field / temperature dependence, neglecting spatially decomposed failure mechanisms. In recent years, widely used TDDB modeling methods include empirical and semi-empirical models, specifically the Weibull statistical model, Model, Model, Models, etc. For example, the Weibull cumulative distribution function used to predict TDDB failure is typically expressed as: ,in, and These are the scaling parameter and the shape parameter, respectively. Lifetime models related to the electric field typically employ an exponential electric field law. Or a kind of anti-electric field law .exist The model uses a square root dependency form, i.e. These methods can effectively predict overall reliability statistics under controlled stress conditions, but they cannot provide spatial information on the specific breakdown locations in the back-end of line (BEOL) dielectric layer. Therefore, these models cannot effectively capture the asymmetry of electrode distribution and the process of local hotspots generated by the evolution of ion concentration field.

[0004] Under actual chip operating conditions, copper ions are effectively confined by the Ta / TaN barrier layer, making it difficult for them to diffuse into the surrounding low-k dielectric. However, experiments have observed ion diffusion and drift within the Ta-based barrier layer itself. This means that the dominant factor in long-term ion transport and defect generation is not Cu ions, but rather the metal ions in these barrier layers. Therefore, TDDB degradation mainly occurs gradually along the ion-enriched regions of the barrier layer, connecting adjacent metal lines, rather than a uniform degradation of the entire dielectric layer.

[0005] TDDB is a critical reliability issue in BEOL interconnects of advanced semiconductor devices. Traditional TDDB models mainly focus on time for failure, but lack the spatial resolution capability to accurately identify and predict failure hotspots.

[0006] Patent application CN117471267A discloses a test method for evaluating the time-dependent breakdown of gate dielectrics, comprising: constructing a test system and preparing several test samples; dividing the test samples into several sample groups and classifying them into n+1 voltage levels based on CVS data analysis; subjecting one sample group to a constant voltage stress test under the first voltage level until sample failure, recording the first failure time corresponding to the first voltage level and the first voltage failure time; subjecting at least one sample group other than the first sample group to a test based on short-time prestress, and finally subjecting it to a constant voltage stress test under the second voltage level until sample failure, calculating the second voltage failure time corresponding to the second voltage level; and obtaining the time-dependent breakdown characteristics of the test samples based on the first voltage level, the first voltage failure time, the second voltage level, the second voltage level, and the data analysis based on the results of conventional constant voltage tests. However, this patent cannot completely solve the existing technical problems, nor can it meet the needs of this invention. Summary of the Invention

[0007] In view of the deficiencies in the prior art, the purpose of this invention is to provide a TDDB reliability analysis and breakdown location method and system with spatiotemporal resolution.

[0008] The TDDB reliability analysis and breakdown location method with spatiotemporal resolution provided by the present invention includes: Step S1: Establish a physical mechanism-based time-dependent breakdown electrical path generation model for back-end processes. The model describes dielectric degradation as a progressive process of defect site generation, charge trapping, and metal ion migration to promote the formation of continuous conductive paths between adjacent interconnects. Step S2: Obtain two-dimensional discrete grid data of ion concentration distribution; Step S3: Perform flow-based connectivity component detection. Based on the two-dimensional discrete grid data of the ion concentration distribution, determine whether a conductive flow path has been formed at the current moment, and generate an activation mask and a breakdown indicator function. Step S4: Perform a minimum resistance path generation search. On the weighted undirected graph constructed from the two-dimensional discrete grid data of the ion concentration distribution, calculate the minimum resistance path from the power line boundary node set to the ground line boundary node set to obtain the breakdown candidate path that is most likely to form leakage conduction at the current moment. Step S5: Perform dynamic maximum bottleneck path evaluation, determine the maximum bottleneck path among the candidate paths from the power line boundary node set to the ground boundary node set, and calculate the global bottleneck concentration and normalized damage index. Step S6: Based on the maximum bottleneck path and the normalized damage index, locate the breakdown hotspot and predict the time-dependent breakdown lifetime.

[0009] Preferably, in step S1, the local ion concentration C(r,t) evolves according to the drift-diffusion equation:

[0010] in, Represents the spatial gradient operator, Let represent the Laplace operator, t represent the time variable of the ion concentration field evolution, r represent the spatial position vector in the discrete region of the medium, C(r,t) represent the ion concentration at position r at time t, and E(r) represent the local electric field intensity at position r. C(r,t) / t represents the rate of change of ion concentration over time. Indicates the diffusion term. μ C(r,t)E(r) represents the drift term driven by the electric field. Where is the diffusion coefficient. For ion mobility, the two satisfy the relationship , q represents the Boltzmann constant, T represents the absolute temperature, and q represents the effective charge of the ion. electric field From electric potential Determined by the following formula:

[0011]

[0012] Where φ(r) represents the electric potential at position r. ·(ε φ)=0 represents the governing equation that satisfies the electrostatic equilibrium condition of the medium, ε represents the dielectric constant of the medium, and Dirichlet boundary conditions are set at the power line and ground metal line.

[0013] Preferably, in step S3, the activation mask is defined as:

[0014] in, This represents the critical ion concentration threshold; exceeding this threshold indicates localized conductivity of the medium. Indicates time The activation mask value of the time grid node (i,j) is determined. Indicates time The ion concentration at grid node (i,j) is given, where i and j represent the indices of the two-dimensional discrete grid in the x and y directions, respectively. This represents the k-th discrete sampling time. The breakdown indication function Defined as: if it exists The paths connecting the nodes bridge the set of power line boundary nodes. Set of boundary nodes of ground line Then place Otherwise, place Connectivity is evaluated using a breadth-first search algorithm on the subgraph induced by nodes.

[0015] Preferably, in step S4, at time... The two-dimensional discrete mesh is abstracted into a weighted undirected graph G( )=( , , ),in, It is a set of nodes. It is an edge weight function. Each grid node u = (i,j) ∈ V corresponds to a discrete cell. Edges are established between adjacent nodes to form an edge set. ; Define the local resistance of a node as ( )=1 / [ ( )+ ],in, ( Let ) be the node u at time t. ion concentration, >0 is a very small positive number set to avoid the denominator being zero; For any adjacent nodes u, v∈ Define the edge weight as ( )=[ ( )+ ( )] / 2; For any path π = ( )] / 2 from the set of boundary nodes to the set of boundary nodes. , ,…, The total path resistance is defined as R(π, )= ( ), where m is the total number of paths and l is the path index; The minimum resistance path is π*( )= R(π, ), A set of paths; Dijkstra's algorithm is used to solve for the minimum resistance path. The specific process is as follows: Power line boundary nodes are set into a super source node 's', and ground line boundary nodes are set into a super sink node 'g'; initialize dist(s) = 0, and for any v ≠ s, set dist(v) = +∞; each time, select the node u with the smallest dist from the undetermined nodes, and perform a relaxation update on its neighboring nodes v: dist(v) = min{dist(v), dist(u) + ... ( When the super sink g is visited, the current time is obtained. The minimum path resistance is obtained by backtracking through the predecessor node to find the minimum resistance path π*( ).

[0016] Preferably, in step S5, for any path π∈P, its bottleneck concentration is defined as... Where P represents the set of all candidate paths from the set of power line boundary nodes to the set of ground boundary nodes, and u represents any grid node on the candidate path π. Indicates time Ion concentration at node u; time The biggest bottleneck path is: ; time global bottleneck concentration and normalized damage index They are defined as follows:

[0017]

[0018] in, This represents the critical ion concentration threshold.

[0019] The TDDB reliability analysis and breakdown location system with spatiotemporal resolution provided by the present invention includes: Module M1: Establish a physical mechanism-based time-dependent breakdown electrical path generation model for back-end processes. The model describes dielectric degradation as a progressive process of defect site generation, charge trapping, and metal ion migration to promote the formation of continuous conductive paths between adjacent interconnects. Module M2: Acquires two-dimensional discrete grid data of ion concentration distribution; Module M3: Performs flow-based connectivity component detection, determines whether a conductive flow path has been formed at the current moment based on the two-dimensional discrete grid data of the ion concentration distribution, and generates an activation mask and a breakdown indicator function; Module M4: Performs a minimum resistance path generation search. On a weighted undirected graph constructed from the two-dimensional discrete grid data of the ion concentration distribution, it calculates the minimum resistance path from the set of power line boundary nodes to the set of ground line boundary nodes to obtain the breakdown candidate path that is most likely to form leakage conduction at the current moment. Module M5: Performs dynamic maximum bottleneck path evaluation, identifies the maximum bottleneck path among the candidate paths from the power line boundary node set to the ground boundary node set, and calculates the global bottleneck concentration and normalized damage index. Module M6: Based on the maximum bottleneck path and the normalized damage index, locate the breakdown hotspot and predict the time-dependent breakdown lifetime.

[0020] Preferably, in module M1, the local ion concentration C(r,t) evolves according to the drift-diffusion equation:

[0021] in, Represents the spatial gradient operator, Let represent the Laplace operator, t represent the time variable of the ion concentration field evolution, r represent the spatial position vector in the discrete region of the medium, C(r,t) represent the ion concentration at position r at time t, and E(r) represent the local electric field intensity at position r. C(r,t) / t represents the rate of change of ion concentration over time. Indicates the diffusion term. μ C(r,t)E(r) represents the drift term driven by the electric field. Where is the diffusion coefficient. For ion mobility, the two satisfy the relationship , q represents the Boltzmann constant, T represents the absolute temperature, and q represents the effective charge of the ion. electric field From electric potential Determined by the following formula:

[0022]

[0023] Where φ(r) represents the electric potential at position r. ·(ε φ)=0 represents the governing equation that satisfies the electrostatic equilibrium condition of the medium, ε represents the dielectric constant of the medium, and Dirichlet boundary conditions are set at the power line and ground metal line.

[0024] Preferably, in module M3, the activation mask is defined as:

[0025] in, This represents the critical ion concentration threshold; exceeding this threshold indicates localized conductivity of the medium. Indicates time The activation mask value of the time grid node (i,j) is determined. Indicates time The ion concentration at grid node (i,j) is given, where i and j represent the indices of the two-dimensional discrete grid in the x and y directions, respectively. This represents the k-th discrete sampling time. The breakdown indication function Defined as: if it exists The paths connecting the nodes bridge the set of power line boundary nodes. Set of boundary nodes of ground line Then place Otherwise, place Connectivity is evaluated using a breadth-first search algorithm on the subgraph induced by nodes.

[0026] Preferably, in module M4, at time... The two-dimensional discrete mesh is abstracted into a weighted undirected graph G( )=( , , ),in, It is a set of nodes. It is an edge weight function. Each grid node u = (i,j) ∈ V corresponds to a discrete cell. Edges are established between adjacent nodes to form an edge set. ; Define the local resistance of a node as ( )=1 / [ ( )+ ],in, ( Let ) be the node u at time t. ion concentration, >0 is a very small positive number set to avoid the denominator being zero; For any adjacent nodes u, v∈ Define the edge weight as ( )=[ ( )+ ( )] / 2; For any path π = ( )] / 2 from the set of boundary nodes to the set of boundary nodes. , ,…, The total path resistance is defined as R(π, )= ( ), where m is the total number of paths and l is the path index; The minimum resistance path is π*( )= R(π, ), A set of paths; Dijkstra's algorithm is used to solve for the minimum resistance path. The specific process is as follows: Power line boundary nodes are set into a super source node 's', and ground line boundary nodes are set into a super sink node 'g'; initialize dist(s) = 0, and for any v ≠ s, set dist(v) = +∞; each time, select the node u with the smallest dist from the undetermined nodes, and perform a relaxation update on its neighboring nodes v: dist(v) = min{dist(v), dist(u) + ... ( When the super sink g is visited, the current time is obtained. The minimum path resistance is obtained by backtracking through the predecessor node to find the minimum resistance path π*( ).

[0027] Preferably, in module M5, for any path π∈P, its bottleneck concentration is defined as... Where P represents the set of all candidate paths from the set of power line boundary nodes to the set of ground boundary nodes, and u represents any grid node on the candidate path π. Indicates time Ion concentration at node u; time The biggest bottleneck path is: ; time global bottleneck concentration and normalized damage index They are defined as follows:

[0028]

[0029] in, This represents the critical ion concentration threshold.

[0030] Compared with the prior art, the present invention has the following beneficial effects: (1) This invention addresses the technical problems of traditional time-dependent breakdown (TDDB) reliability models, which only provide statistical characteristics of failure time, lack spatial resolution, and are disconnected from the actual physical mechanism of dielectric layer breakdown in back-to-back processes (BEOL). It proposes a spatiotemporal resolution-based TDDB reliability analysis and breakdown location method. This method constructs a complete spatiotemporal analysis framework by combining flow-based ion transport analysis, minimum resistance path search, and dynamic maximum bottleneck path evaluation, enabling simultaneous evaluation of TDDB characteristics in BEOL interconnects in both time and space dimensions.

[0031] (2) In view of the technical problem that existing models are difficult to capture the asymmetry of electrode distribution and the evolution of ion concentration field, resulting in local hot spots, the present invention adopts a permeation-based connected component detection algorithm. By activating the mask and the breakdown indicator function, the formation of conductive permeation path is judged. Combined with the minimum resistance path search method, the minimum resistance path from the power line to the ground line is calculated on the weighted undirected graph constructed by the ion concentration distribution. In this way, the breakdown candidate path that is most likely to form leakage conduction at the current moment is accurately located, and the accurate spatial location of failure hot spots is achieved.

[0032] (3) To address the technical problem that existing technologies cannot accurately infer breakdown time from sparse simulation sampling data, this invention proposes a dynamic maximum bottleneck path evaluation method. This method quantifies the degree to which the system reaches the electrical path generation threshold along the entire connected path by defining bottleneck concentration, maximum bottleneck path, global bottleneck concentration, and normalized damage degree index γ(t). It realizes real-time quantitative analysis of the TDDB damage evolution process and can accurately infer breakdown time from sparse simulation sampling data, effectively solving the contradiction between the sparsity of sampling data and prediction accuracy.

[0033] (4) This invention constructs a BEOL TDDB electrical path generation model based on real physical mechanisms. This model describes dielectric degradation as a progressive process of defect site generation, charge trapping, and metal ion migration, overcoming the shortcomings of traditional empirical models that are disconnected from real physical mechanisms. By organically combining ion permeation theory, minimum resistance path analysis, and dynamic maximum bottleneck path evaluation, the detection results of breakdown paths are consistent with the failure criteria based on leakage current, ensuring the consistency between spatial location results and physical measurement results.

[0034] (5) The present invention obtains two-dimensional grid data of ion concentration distribution through finite difference method for numerical simulation verification. Experimental results show that the relative error of the method in predicting failure time under different stress conditions is less than 2.44%, which can accurately locate the spatial location of the breakdown hot spot. Compared with the traditional long-term simulation based on the drift-diffusion model finite difference method, the algorithm only needs to be repeated for a finite number of time snapshots, which significantly reduces the overall computational cost. Attached Figure Description

[0035] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1 This is the algorithm framework of the present invention; Figure 2 The evolution of the damage severity index γ(t) constructed in this invention over time; Figure 3 This invention proposes a breakdown path visualization and hotspot location result for the TDDB reliability analysis method. Detailed Implementation

[0036] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the scope of protection of the present invention.

[0037] Example This invention proposes a TDDB algorithm framework with spatiotemporal resolution, which integrates ion transport simulation, permeation detection, graph-based minimum resistance path search, and dynamic maximum bottleneck path evaluation into a unified methodology for spatial localization and temporal detection of TDDB failure.

[0038] This invention proposes a normalized damage measurement function. This function is used to monitor the TDDB degradation process. It not only provides a detailed summary of the spatial degradation process but also extrapolates the breakdown time from sparse sampled data. .

[0039] This invention derives the minimum resistance path criterion from ion concentration conductivity mapping and links it with a permeation-based path detection method. This ensures that the spatial positioning results of this method are consistent with the failure criteria based on leakage during actual testing, thereby providing an optional fine-tuning correction range.

[0040] like Figure 1 The specific technical solution is as follows: 1) BEOL TDDB EPG model based on physical mechanisms This invention proposes a BEOL TDDB EPG model based on physical mechanisms, in which electric field-induced ion drift and defect generation in the barrier layer lead to dielectric breakdown. Under a persistent local electric field... Under its influence, metal ions in the barrier layer migrate into the interior of the medium. Local ion concentration. Evolution follows the drift-diffusion equation:

[0041] in, Represents the spatial gradient operator, Let represent the Laplace operator, t represent the time variable of the ion concentration field evolution, r represent the spatial position vector in the discrete region of the medium, C(r,t) represent the ion concentration at position r at time t, and E(r) represent the local electric field intensity at position r. C(r,t) / t represents the rate of change of ion concentration over time. Indicates the diffusion term. μ C(r,t)E(r) represents the drift term driven by the electric field. Where is the diffusion coefficient. For ion mobility, the two satisfy the relationship , q represents the Boltzmann constant, T represents the absolute temperature, and q represents the effective charge of the ion.

[0042] electric field From electric potential Determined by the following formula: ,

[0043] Where φ(r) represents the electric potential at position r, E(r) = φ(r) represents the local electric field strength determined by the potential gradient. ·(ε φ)=0 represents the governing equation satisfying the electrostatic equilibrium condition of the dielectric, and ε represents the dielectric constant of the dielectric. Dirichlet boundary conditions are set at the VDD and GND metal lines. The EPG model based on physical mechanisms proposed in this invention provides a microscopic basis for understanding the TDDB phenomenon in BEOL low-k interconnect media, such as... Figure 1 As shown in the figure, this model describes dielectric degradation as a gradual process involving defect site generation, charge trapping, and metal ion migration, all of which synergistically promote the formation of continuous conductive pathways between adjacent interconnects.

[0044] 2) Detection of connected components based on seepage The framework proposed in this invention can determine Whether a conductive permeation path is formed at any given time. Simultaneously, we define an activation mask based on permeation as follows:

[0045] in, This represents the critical ion concentration threshold; exceeding this threshold indicates localized conductivity of the medium. Indicates time The activation mask value of the time grid node (i,j) is determined. Indicates time The ion concentration at grid node (i,j) is given, where i and j represent the indices of the two-dimensional discrete grid in the x and y directions, respectively. This represents the k-th discrete sampling time. This represents the critical ion concentration threshold used to determine whether a local region has entered a conductive state.

[0046] Furthermore, the present invention introduces a breakdown indication function. If it exists The paths connecting the nodes are bridged. and , place Otherwise, place .therefore, Indicates at time This high-concentration region penetrated the dielectric layer, resulting in macroscopic breakdown. Connectivity was evaluated using a breadth-first search (BFS) algorithm on the node-induced subgraph. The time complexity of performing the BFS algorithm on active nodes is O(log n). .

[0047] This represents the set of boundary nodes adjacent to the VDD metal line. Let represent the set of boundary nodes adjacent to the GND metal line, and O(·) represent the order of magnitude of the algorithm's time complexity. This represents the set of nodes in the constructed graph model. This represents the set of edges in the constructed graph model. , B( represents the number of nodes in the two-dimensional discrete mesh in the x and y directions, respectively) () indicates time The breakdown indicator function takes the value 0 or 1; M=1 indicates that the corresponding node has been activated, that is, the ion concentration at that node is not lower than the critical threshold. .

[0048] 3) Minimum Resistance EPG Path Search Although the seepage analysis gives a binary indicator function It is used to indicate whether breakdown has occurred, but it does not provide information about the most vulnerable area where complete seepage occurs. When At that time, although no leakage path was found, the resistance in some areas may have decreased significantly, posing a risk of breakdown. This invention is illustrated in Figure [Figure number missing]. The minimum resistance path is calculated, and the most likely breakdown path can be found using Dijkstra's algorithm.

[0049] Furthermore, at any moment The two-dimensional discrete mesh is abstracted into a weighted undirected graph G( )=( , , ), where each grid node u=(i,j)∈ For a given discrete unit, edges are established between adjacent nodes to form an edge set. .

[0050] To map ion concentrations to computable resistance, the nodal local resistance is defined as follows: ( )=1 / [ ( )+ ],in, ( Let ) be the node u at time t. ion concentration, >0 is a very small positive number set to avoid the denominator being zero.

[0051] For any adjacent nodes u, v∈ Define the edge weight as ( )=[ ( )+ ( )] / 2; For any path π = ( ) from the set of VDD boundary nodes to the set of GND boundary nodes , ,…, The total path resistance is defined as R(π, )= ( ).

[0052] Therefore, the path with the minimum resistance can be written as π*( )= R(π, ), which corresponds to the breakdown candidate path that is most likely to form a leakage conduction at the current moment.

[0053] This invention employs Dijkstra's algorithm to solve the aforementioned shortest path problem. The specific process is as follows: First, the set of VDD boundary nodes is merged into a super source node s, and the set of GND boundary nodes is merged into a super sink node g; second, dist(s) is initialized to 0, and for any v ≠ s, dist(v) is set to +∞; subsequently, each time, the node u with the smallest dist value is selected from the undetermined nodes, and its neighboring nodes v are relaxed and updated: dist(v) = min{dist(v), dist(u) + ... ( When the super sink g is visited, the current time is obtained. The minimum path resistance is obtained by backtracking through the predecessor node (predecessor); finally, the minimum resistance path π*( Along this path, the region with the highest local resistance (or equivalently the lowest ion concentration) can be further identified as a potential weak point for breakdown, thus enabling hotspot localization.

[0054] 4) Estimation of dynamic maximum bottleneck path and damage level Minimum resistance path This reflects a global trade-off between many medium-conductivity segments and some highly conductive segments. However, breakdown is primarily determined by the low ion concentration portions of certain highly conductive filaments. To quantify the extent to which the system reaches the EPG threshold across the entire connectivity path, this invention defines and proposes a maximum bottleneck path method. Specifically, for any The path is defined as having a bottleneck concentration of [missing information]. This means along that path The lowest concentration encountered. If a certain path exists. Make ,but This indicates the existence of a fully activated seepage path. Before the seepage path forms, the present invention finds the path with the weakest segment as strong as possible using the following formula:

[0055] Meanwhile, this invention defines the corresponding global bottleneck concentration and normalized damage degree indices as shown in the following formulas:

[0056]

[0057] P represents the set of all candidate paths from the set of VDD boundary nodes to the set of GND boundary nodes, and u represents any grid node on the candidate path π. Indicates time Ion concentration at node u. Indicates the maximum bottleneck path. Indicates time The biggest bottleneck path Indicates time The global bottleneck concentration, i.e., the maximum bottleneck path. The minimum ion concentration on Indicates time Normalized damage severity index. For example... Figure 2 , represents the evolution of the damage severity index γ(t) over time.

[0058] The TDDB reliability analysis method proposed in this invention has spatiotemporal resolution. Table 1 lists the damage level and corresponding hotspot locations calculated using this invention. Figure 3 This invention provides the breakdown path visualization and hotspot location results for the TDDB reliability analysis method.

[0059] Table 1. Damage severity and hotspot location at different times

[0060] The core of this technical solution lies in combining advanced ion permeation theory, minimum resistance path analysis, and dynamic maximum bottleneck path evaluation methods to assess the TDDB characteristics of BEOL interconnects in both time and space dimensions. Alternative solutions include modifying the minimum resistance path analysis algorithm or using other optimization methods to replace the calculation of the minimum resistance path.

[0061] In practical applications, this invention predicts time-dependent breakdown (TDDB) lifetime by analyzing the evolution of the normalized damage severity index γ(t) over time. The specific prediction process is as follows: First, from the initial moment Up to the current moment Multiple sparse sampling times ( , ,…, ), calculate γ( at each time point respectively. The breakdown time T_failure can be extrapolated as follows: Since γ(t) exhibits a monotonically increasing characteristic before breakdown occurs, and its growth rate accelerates significantly near breakdown, the breakdown time T_failure can be extrapolated using the following method: (1) For discrete sampling points ( ,γ( Perform nonlinear least squares fitting, using an empirical acceleration model, such as an exponential function, as the fitting function: Where A, B, and C are fitting parameters; or a power-law function: , where α and β are fitting parameters. The specific function form is determined based on the statistical characteristics of the measured ion concentration distribution.

[0062] (2) Set the breakdown threshold γ_th. According to the permeation theory, when γ(t) reaches 100% (i.e., α(t) = C_th), the global bottleneck concentration is theoretically determined to meet the critical ion concentration threshold. At this time, the system forms a complete permeation path through the power line to the ground line, and macroscopic breakdown occurs. Therefore, let γ_th = 100% as the breakdown criterion.

[0063] (3) Find the intersection point of the fitted curve and the horizontal line at γ=100%. The time corresponding to this intersection point is the predicted TDDB lifetime, T_failure. The mathematical expression is: T_failure= (100%), of which This represents the inverse function of γ(t).

[0064] Furthermore, to further improve prediction accuracy, confidence interval analysis and residual evaluation can be combined to dynamically remove outlier sampling points. When the deviation between the data at a newly added sampling time and the fitted model exceeds a preset threshold, model refitting is automatically triggered to achieve adaptive lifetime prediction. Through the above steps, this invention achieves quantitative prediction of breakdown lifetime from sparse simulation data to high accuracy, solving the computational bottleneck of traditional methods that require long-term continuous simulation or large-sample statistical analysis.

[0065] Those skilled in the art will understand that, in addition to implementing the system, apparatus, and their modules provided by this invention in purely computer-readable program code, the same program can be implemented in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers by logically programming the method steps. Therefore, the system, apparatus, and their modules provided by this invention can be considered a hardware component, and the modules included therein for implementing various programs can also be considered structures within the hardware component; alternatively, modules for implementing various functions can be considered both software programs implementing the method and structures within the hardware component.

[0066] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A method for TDDB reliability analysis and breakdown location with spatio-temporal resolution, characterized in that, include: Step S1: Establish a physical mechanism-based time-dependent breakdown electrical path generation model for back-end processes. The model describes dielectric degradation as a progressive process of defect site generation, charge trapping, and metal ion migration to promote the formation of continuous conductive paths between adjacent interconnects. Step S2: Obtain two-dimensional discrete grid data of ion concentration distribution; Step S3: Perform flow-based connectivity component detection. Based on the two-dimensional discrete grid data of the ion concentration distribution, determine whether a conductive flow path has been formed at the current moment, and generate an activation mask and a breakdown indicator function. Step S4: Perform a minimum resistance path generation search. On the weighted undirected graph constructed from the two-dimensional discrete grid data of the ion concentration distribution, calculate the minimum resistance path from the power line boundary node set to the ground line boundary node set to obtain the breakdown candidate path that is most likely to form leakage conduction at the current moment. Step S5: Perform dynamic maximum bottleneck path evaluation, determine the maximum bottleneck path among the candidate paths from the power line boundary node set to the ground boundary node set, and calculate the global bottleneck concentration and normalized damage index. Step S6: Based on the maximum bottleneck path and the normalized damage index, locate the breakdown hotspot and predict the time-dependent breakdown lifetime.

2. The TDDB reliability analysis with spatial-temporal resolution and breakdown location method according to claim 1, wherein, In step S1, the local ion concentration C(r,t) evolves according to the drift-diffusion equation: in, Represents the spatial gradient operator. Let represent the Laplace operator, t represent the time variable of the ion concentration field evolution, r represent the spatial position vector in the discrete region of the medium, C(r,t) represent the ion concentration at position r at time t, and E(r) represent the local electric field intensity at position r. C(r,t) / t represents the rate of change of ion concentration over time. Indicates the diffusion term. μ C(r,t)E(r) represents the drift term driven by the electric field. The diffusion coefficient is... For ion mobility, the two satisfy the relationship , q represents the Boltzmann constant, T represents the absolute temperature, and q represents the effective charge of the ion. electric field from the potential by the formula: where φ(r) represents the electric potential at position r, • (ε φ) = 0 represents the control equation satisfying the dielectric static equilibrium condition, ε represents the dielectric constant of the medium, and the Dirichlet boundary condition is set at the power line and ground line metal lines.

3. The TDDB reliability analysis with spatial-temporal resolution and breakdown location method of claim 1, wherein, In step S3, the activation mask is defined as: in, This represents the critical ion concentration threshold; exceeding this threshold indicates localized conductivity of the medium. Indicates time The activation mask value of the time grid node (i,j) is determined. Indicates time The ion concentration at grid node (i,j) is given, where i and j represent the indices of the two-dimensional discrete grid in the x and y directions, respectively. This represents the k-th discrete sampling time. the breakdown indicator function is defined as follows: if there exists a path of nodes connected that bridges the set of power line boundary nodes and the set of ground line boundary nodes then set ; otherwise, set ; connectivity is evaluated using a breadth-first search algorithm on the subgraph induced by the nodes.

4. The TDDB reliability analysis with spatial-temporal resolution and breakdown location method of claim 1, wherein, In step S4, at time... The two-dimensional discrete mesh is abstracted into a weighted undirected graph G( )=( , , ),in, It is a set of nodes. It is an edge weight function. Each grid node u = (i,j) ∈ V corresponds to a discrete cell. Edges are established between adjacent nodes to form an edge set. ; Define the local resistance of a node as ( )=1 / [ ( )+ ],in, ( Let ) be the node u at time t. ion concentration, >0 is a very small positive number set to avoid the denominator being zero; For any adjacent nodes u, v∈ Define the edge weight as ( )=[ ( )+ ( )] / 2; For any path π = ( )] / 2 from the set of boundary nodes to the set of boundary nodes. , ,…, The total path resistance is defined as R(π, )= ( ), where m is the total number of paths and l is the path index; The minimum resistance path is π*( )= R(π, ), is a set of paths; Dijkstra's algorithm is used to solve for the minimum resistance path. The specific process is as follows: Power line boundary nodes are set into a super source node 's', and ground line boundary nodes are set into a super sink node 'g'; initialize dist(s) = 0, and for any v ≠ s, set dist(v) = +∞; each time, select the node u with the smallest dist from the undetermined nodes, and perform a relaxation update on its neighboring nodes v: dist(v) = min{dist(v), dist(u) + ... ( When the super sink g is visited, the current time is obtained. The minimum path resistance is obtained by backtracking through the predecessor node to find the minimum resistance path π*( ).

5. The TDDB reliability analysis with spatial-temporal resolution and breakdown location method of claim 1, wherein, In step S5, for any path π∈P, its bottleneck concentration is defined as... Where P represents the set of all candidate paths from the set of power line boundary nodes to the set of ground boundary nodes, and u represents any grid node on the candidate path π. Indicates time Ion concentration at node u; Time The maximum bottleneck path is: ; Time Global bottleneck concentration And normalized damage extent indicator Are respectively defined as: wherein represents a critical ion concentration threshold value.

6. A system for TDDB reliability analysis and breakdown location with spatio-temporal resolution, characterized by, include: Module M1: Establish a physical mechanism-based time-dependent breakdown electrical path generation model for back-end processes. The model describes dielectric degradation as a progressive process of defect site generation, charge trapping, and metal ion migration to promote the formation of continuous conductive paths between adjacent interconnects. Module M2: Acquires two-dimensional discrete grid data of ion concentration distribution; Module M3: Performs flow-based connectivity component detection, determines whether a conductive flow path has been formed at the current moment based on the two-dimensional discrete grid data of the ion concentration distribution, and generates an activation mask and a breakdown indicator function; Module M4: Performs a minimum resistance path generation search. On a weighted undirected graph constructed from the two-dimensional discrete grid data of the ion concentration distribution, it calculates the minimum resistance path from the set of power line boundary nodes to the set of ground line boundary nodes to obtain the breakdown candidate path that is most likely to form leakage conduction at the current moment. Module M5: Performs dynamic maximum bottleneck path evaluation, identifies the maximum bottleneck path among the candidate paths from the power line boundary node set to the ground boundary node set, and calculates the global bottleneck concentration and normalized damage index. Module M6: Based on the maximum bottleneck path and the normalized damage index, locate the breakdown hotspot and predict the time-dependent breakdown lifetime.

7. The TDDB reliability analysis with spatial-temporal resolution and breakdown location system of claim 6, wherein, In module M1, the local ion concentration C(r,t) evolves according to the drift-diffusion equation: in, Represents the spatial gradient operator, Let represent the Laplace operator, t represent the time variable of the ion concentration field evolution, r represent the spatial position vector in the discrete region of the medium, C(r,t) represent the ion concentration at position r at time t, and E(r) represent the local electric field intensity at position r. C(r,t) / t represents the rate of change of ion concentration over time. Indicates the diffusion term. μ C(r,t)E(r) represents the drift term driven by the electric field. Where is the diffusion coefficient. For ion mobility, the two satisfy the relationship , q represents the Boltzmann constant, T represents the absolute temperature, and q represents the effective charge of the ion. electric field from the potential by the formula: where φ(r) represents the electric potential at position r, • (ε φ) = 0 represents the control equation satisfying the dielectric static equilibrium condition, ε represents the dielectric constant of the medium, and the Dirichlet boundary condition is set at the power line and ground line metal lines.

8. The TDDB reliability analysis with spatial-temporal resolution and breakdown location system of claim 6, wherein, In module M3, the activation mask is defined as: in, This represents the critical ion concentration threshold; exceeding this threshold indicates localized conductivity of the medium. Indicates time The activation mask value of the time grid node (i,j) is determined. Indicates time The ion concentration at grid node (i,j) is given, where i and j represent the indices of the two-dimensional discrete grid in the x and y directions, respectively. This represents the k-th discrete sampling time. The breakdown indication function Defined as: if it exists The paths connecting the nodes bridge the set of power line boundary nodes. Set of boundary nodes of ground line Then place Otherwise, place Connectivity is evaluated using a breadth-first search algorithm on the subgraph induced by nodes.

9. The TDDB reliability analysis with spatial-temporal resolution and breakdown location system of claim 6, wherein, In module M4, at time The two-dimensional discrete mesh is abstracted into a weighted undirected graph G( )=( , , ),in, It is a set of nodes. It is an edge weight function. Each grid node u = (i,j) ∈ V corresponds to a discrete cell. Edges are established between adjacent nodes to form an edge set. ; Define the local resistance of a node as ( )=1 / [ ( )+ ],in, ( Let ) be the node u at time t. ion concentration, >0 is a very small positive number set to avoid the denominator being zero; For any adjacent nodes u, v∈ Define the edge weight as ( )=[ ( )+ ( )] / 2; For any path π = ( )] / 2 from the set of boundary nodes to the set of boundary nodes. , ,…, The total path resistance is defined as R(π, )= ( ), where m is the total number of paths and l is the path index; The minimum resistance path is π*( )= R(π, ), is a set of paths; Dijkstra's algorithm is used to solve for the minimum resistance path. The specific process is as follows: Power line boundary nodes are set into a super source node 's', and ground line boundary nodes are set into a super sink node 'g'; initialize dist(s) = 0, and for any v ≠ s, set dist(v) = +∞; each time, select the node u with the smallest dist from the undetermined nodes, and perform a relaxation update on its neighboring nodes v: dist(v) = min{dist(v), dist(u) + ... ( When the super sink g is visited, the current time is obtained. The minimum path resistance is obtained by backtracking through the predecessor node to find the minimum resistance path π*( ).

10. The TDDB reliability analysis with spatial-temporal resolution and breakdown location system of claim 6, wherein, In the module M5, for any path π ∈ P, the bottleneck concentration thereof is defined as wherein P represents a set of all candidate paths from the set of power line boundary nodes to the set of ground line boundary nodes, u represents any grid node on the candidate path π, represents the time instant ion concentration at node u; Time The maximum bottleneck path is: ; Time Global bottleneck concentration And normalized damage extent indicator Are respectively defined as: wherein represents a critical ion concentration threshold value.

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