Finite difference method for electromigration analysis of multi-branch interconnects

Electromigration (EM) becomes one of the most challenging reliability issues for current and future ICs in 10-nm technology and below. In this article, a novel method is proposed for the EM hydrostatic stress analysis on 2-D multibranch interconnect trees, which is the foundation of the EM reliability assessment for large-scale on-chip interconnect networks, such as on-chip power grid networks. The proposed method, which is based on an eigenfunction technique, could efficiently calculate the hydrostatic stress evolution for multibranch interconnect trees stressed with different current densities and nonuniformly distributed thermal effects. The proposed method solves the partial differential equations of transient EM stress more efficiently since it does not require any discretization either spatially or temporally, which is in contrast to numerical methods, such as the finite difference method and finite element method. The accuracy of the proposed transient analysis approach is validated against the analytical solution and commercial tools. The convergence of the proposed method is demonstrated by numerical experiments on practical power/ground networks, showing that only a small number of eigenfunction terms are necessary for the accurate solution. Thanks to its analytical nature, the proposed method is also utilized in efficient EM analysis techniques, such as searching for the void nucleation time by a modified bisection algorithm. The numerical results show that the proposed method is 10X–100X faster than the finite difference method and scales better for larger interconnect trees.

Dynamic electromigration modeling for transient stress evolution and recovery under time-dependent current and temperature stressing

Voltage drop (IR drop) and Electromigration (EM) reliability are two key related aspects for on-chip power grid design considerations. Good EM reliability ensures no EM induced void formation in the interconnect to cause a fatal resistance increase during product lifetime. It is important to understand the difference of EM characteristics between a power grid and traditional single link EM structures. In this paper, we report the experimental results of EM characteristics of power grid like structures, to demonstrate the power grid design environment, and physical and statistical redundancy benefits for EM reliability. These results can serve as fundamental references for power grid EM limit determination.

Electromigration (EM) is a major concern for very large-scale integration (VLSI) interconnect reliability, particularly for interconnect trees with multibranch metal wires representing continuously connected metal (Cu) lines terminated at diffusion barriers. For EM modeling and assessment, one important problem is to perform fast EM time to failure analysis for practical VLSI chips. Compact modeling for EM effects for the interconnect tree has been studied recently to better EM signoff analysis. But the existing method cannot consider wires stressed under time-varying temperature, which is very typical for practical chip working conditions. In this paper, we develop the exact analytic model for the stress evolution of interconnect trees under different current density and varying segment length from the first principle. Due to the difficulty in obtaining the exact analytical solution, we focus on three-terminal wire in this paper. The stress evolution is modeled by two Korhonen’s equations coupled through boundary conditions which are solved with the Laplace transformation technique. The new analytical EM model is further extended to consider the time-varying temperature stressing condition and initial non-zero residual stress. The proposed method is compared with the finite-element method (FEM) tool COMSOL, the recently proposed eigenfunction-based method, and the published EM simulator XSim. The comparison shows that the analytical solution agrees well with the results from the FEM numerical analysis. It uses much fewer terms compared to the eigenfunction method for the same accuracy. It also agrees very well with XSim, which is consistent with the previously reported measured results.

For practical testing and detection of electromigration (EM) induced failures in dual damascene copper interconnects in today's and future sub-10nm ICs, one critical issue is how to create stressing conditions so that the chip will fail exclusively under EM in a very short period of time so that EM signoff and validation can be carried out efficiently. In this work, we propose novel EM wearout-acceleration techniques for practical VLSI chips. We will first review the recently proposed three-phase physics-based EM models and discuss the important factors contributing to the EM aging process. Then we propose a new formula for fast estimation of the void's saturation volume for general multi-segment interconnect wires, which is important for EM mortality check. We then investigate two strategies to accelerate the EM failure process: reservoir-enhanced acceleration and temperature-based acceleration. First we show that multi-segment interconnects with reservoir structures and their stressing currents can be exploited to significantly speedup the EM wearout process. Such configurable reservoir-based wires are very flexible and can achieve various EM accelerations at the costs of some routing resources. Additionally, we show that further acceleration can be achieved by increasing temperature. On average, 10% increase in temperature yields about 10X wearout acceleration. However, purely temperature based acceleration is not possible since practical VLSI chips have temperature limitations which must be strictly enforced to ensure the chip only fails under EM, and not due to other reliability effects. In this study, we show that it is possible to achieve significantly high acceleration while staying within the feasible operating zones by combining the two acceleration techniques. Experimental results show that by combining temperature and reservoir accelerations, we can reduce the EM lifetime of a chip from 10 years down to a few hours (about 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> acceleration) under the 150°C temperature limit, which is sufficient for practical EM testing of typical nanometer CMOS ICs.

Due to the high local current densities, Through Silicon Vias (TSVs) are susceptible to electromigration (EM) degradation, which reduces the reliability of integrated circuits. Unlike traditional methods for TSV modeling and simulation, this paper introduces a unified hard constraint physics-informed learning neural network approach, called HCPINN, for the transient analysis of electromigration in TSVs by solving the conventional mass balance equation. The proposed method allows simultaneous consideration of atomic depletion and accumulation, effective resistance degradation, electric current evolution, and stress distribution. Specifically, we propose a hard constraint method for solving partial differential equations (PDEs) with general boundary conditions (BCs) for transient TSV electromigration analysis. By using the extra fields derived from the mixed finite element method, we reconstruct the corresponding PDEs by transforming general BCs into linear forms. Based on this derivation, we embed general BCs of mass balance equation into the proposed ansatz and employ sub-networks for the approximation on general BCs. The main neural network is responsible for training the internal part of the problem domain without adding loss terms with BCs, overcoming the convergence issue due to unbalanced gradients among different loss terms. Besides, we theoretically demonstrate that this reformulation of general BCs can stabilize the training process. Experimental results indicate that the proposed HCPINN exhibits superior performance and reduces boundary error in TSV electromigration analysis. Compared to the finite element method, the proposed network achieves approximately more than 70 times faster inference with a minimal mean squared error increase of less than 0.1% under different TSV geometries and varying current density profiles.

A more accurate finite difference approach to the pricing of contingent claims

Keys to high wiring density semiconductor packages include flip-chip bonding and build-up substrate technologies. The current issues are the establishment of a fine pitch flip-chip bonding technology and a low coefficient of thermal expansion (CTE) substrate technology. In particular, electromigration and thermomigration in fine pitch flipchip joints have been recognized as a major reliability issue. In this paper, electromigration and thermomigration in Cu/Sn-3Ag-0.5Cu (SAC305)/Cu flip-chip joints and electromigration in Cu/In/Cu flip chip joints are investigated. In the electromigration test, a large electromigration void nucleation at the cathode, large growth of intermetallic compounds (IMCs) at the anode, a unique solder bump deformation towards the cathode, and the significantly prolonged electromigration lifetime with the underfill were observed in both types of joints. In addition, the effects of crystallographic orientation of Sn on electromigration were observed in the Cu/SAC305/Cu joints. In the thermomigration test, Cu dissolution was accelerated on the hot side, and formation of IMCs was enhanced on the cold side at a thermal gradient of about /cm, which was lower than previously reported. The rate of Cu atom migration was found comparable to that of electromigration under current conditions.

The population density approach has been used for modeling the dynamics of large-scale neuronal networks to consider the stochastic nature of the signal processing in the brain. From the conservation principle, the time evolution of population density can be described by a nonlinear partial differential equation (PDE). The finite difference method (FDM) has been proposed for solving this PDE. However, FDM is sensitive to the density gradient of the solution and is confined to problems that have a regular state space. An irregular state space is always obtained if realistic neuronal models are considered. In this study, the finite element method (FEM) is formulated to solve this POE and apply it to solve the orientation tuning problem. The results show that when the state space discretization is coarse, FEM retains high accuracy whereas FDM does not. Furthermore, FEM reduces computation time by 90% compared to that required for FDM. In addition, FEM can easily handle the existence of a point source without any modifications. Due to its superior accuracy, efficiency, and consistency, FEM may be a better numerical technique for applying the population density approach to large-scale neuronal networks.

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

Aluminum-copper alloy (AlCu) is commonly utilized as interconnect material in low-power devices. However, as the size of electronic devices continues to decrease and current density increases, electromigration (EM) has emerged as a significant reliability concern for AlCu interconnects in the microelectronics industry. In this study, two-level AlCu interconnect structures with a Ti/TiN barrier layer were fabricated using 0.18 μm technology to perform accelerated EM tests. The test samples were subjected to three current levels (1.45 mA, 2.40 mA and 5.30 mA) at three ambient temperatures (200 °C, 225 °C and 250 °C) to investigate the nucleation and evolution of voids during EM degradation and then obtain the mean time to failure (MTTF). The numerical simulation method of atomic density integral (ADI) was used to simulate experimental observations based on the ANSYS platform, considering the coupled effects of electron wind force, stress gradient, temperature gradient, and atomic density gradient. A comparison of the experimental results and numerical simulations proves that the ADI method can be applied successfully to EM failure prediction of AlCu interconnects. Finite element models with different reservoir lengths were built to demonstrate the mechanism of the reservoir effect. The results show that the reservoir can improve the EM lifetime of the interconnect, but there is a critical extension length beyond which increasing extension sizes have no effect on EM lifetime.

In this paper, an adaptive algorithm is proposed to perform electromigration (EM) assessment for full-chip power/ground networks. Based on the eigenfunction solutions, the proposed method improves the efficiency by properly selecting the eigenfunction terms and utilizing the closed-form eigenfunctions for commonly seen interconnect wires such as T-shaped or cross-shaped wires. It is demonstrated that the proposed method can trad-off well among the accuracy, efficiency and applicability of the eigenfunction based methods. The experimental results show that the proposed method is about three times faster than the finite difference method and other eigenfunction based methods.

Bellows compensators are critical components in pipeline systems, designed to absorb thermal expansions, vibrations, and pressure reflections. Ensuring their operational reliability requires accurate prediction of the stress–strain state (SSS) and stability under internal pressure. This study presents a comprehensive mathematical model for analyzing corrugated bellows compensators, formulated as a boundary value problem for a system of partial differential equations (PDEs) within the Kirchhoff–Love shell theory framework. Two numerical approaches are developed and compared: a finite difference method (FDM) applied to a reduced axisymmetric formulation to ordinary differential equations (ODEs) and a finite element method (FEM) for the full variational formulation. The FDM scheme utilizes a second-order implicit symmetric approximation, ensuring stability and efficiency for axisymmetric geometries. The FEM model, implemented in Ansys 2020 R2, provides high fidelity for complex geometries and boundary conditions. Convergence analysis confirms second-order spatial accuracy for both methods. Numerical experiments determine critical pressures based on the von Mises yield criterion and linearized buckling analysis, revealing the influence of geometric parameters (wall thickness, number of convolutions) on failure mechanisms. The results demonstrate that local buckling can occur at lower pressures than that of global buckling for thin-walled bellows with multiple convolutions, which is critical for structural reliability assessment. The proposed combined approach (FDM for rapid preliminary design and FEM for final verification) offers a robust and efficient methodology for bellows design, enhancing reliability and reducing development time. The work highlights the importance of integrating rigorous PDE-based modeling with modern numerical techniques for solving complex engineering problems with a focus on structural integrity and long-term performance.

Electromigration (EM) in power distribution networks (PDNs) is a major reliability issue in 3-D ICs. While the EM issues of local vias and through-silicon-vias (TSV) have been studied separately, the interplay of TSVs and conventional local vias in 3-D ICs has not been well investigated. This co-design is necessary when the die-to-die vertical power delivery is done using both TSVs and local interconnects. In this paper, we model EM for PDNs of 3-D ICs with a focus on multiscale via (MSV) structure, i.e., TSVs and local vias used together for vertical power delivery. We study the impact of structure, material, and preexisting void conditions on the EM-related lifetime of our MSV structures. We also investigate the transient IR-voltage change of full-chip level 3-D PDNs with MSVs with our model. The experimental results demonstrate that our EM modeling can effectively capture the EM reliability of the full-chip level 3-D PDNs with MSVs, which can be hard to achieve by the traditional EM analysis based on the individual local via or the TSV.

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow