Electromigration Model Research Articles

Analysis of Electromigration in Dual-Damascene Interconnect Structures

The effects of grain boundaries on electromigration failure are presented. The electromigration model incorporates the grain boundary as a separate medium which acts not only as fast diffusivity path for material transport but also absorbs and releases vacancies. Moreover, we analyze the void nucleation condition and its implication on electromigration failure is discussed. Our results show that high stress and a high vacancy concentration develop at the triple points formed on the interface between copper grains and the capping layer, where a void nucleation is likely to take place. This scenario of weak triple points in combination with a stress threshold also supports the mechanism of multiple void nucleation as it has been experimentally observed.

Avoidance vs. repair: New approaches to increasing electromigration robustness in VLSI routing

Studies on further IC development mutually predict that the reliability of future integrated circuits (ICs) will be severely endangered by the occurrence of electromigration (EM). The reason for the increasing number of EM damages are the ongoing structural reductions in the IC. Digital circuits are particularly at risk because they have been neglected in the consideration of EM, resulting in a lack of suitable EM measures. For this reason, a paradigm shift in physical design must be accomplished, complementing the traditional EM verification step after layout creation with a proactive EM-robust physical synthesis. This work presents the necessary adaptations and new approaches by modifying the routing step of digital circuits, resulting in an EM-robust routing result. Our contribution includes the development of EM models, the derivation of EM-suppressing measures, and finally the consideration of these countermeasures in an EM-robust routing process. In summary, our work is an important contribution to increase the EM robustness in digital layouts, thereby ensuring the reliability of future ICs.

Mixing and heat transfer in micro/nano-channel due to charged corrugated surfaces

Fluid transport and mixing analysis is performed numerically in micro/nano-channels, where the channel walls are fabricated with separated charged blocks. Two different streams of miscible fluids with different concentrations are supplied through the inlet port with an external applied electric field. The fluid is flowing under low Reynolds numbers experiences convection and diffusion effects which promote the desired mixing. In this paper theory of dynamic equations are used for the time scales with the coefficient as zero linear growth. Both Joule heating effects and pressure variation is considered for the fluid transport using a lubrication theory approximation, considering the combined formulation of the Navier-Stokes and Nernst-Planck model for electromigration. It is observed that axial pressure variation are maximum above the heterogeneous regions providing a zero gradient along the fully-developed regions. The diffusive transport of thermal energy for an unaltered wall temperature can be enhanced due to the augmentation of the local fluid temperature with the applied electric field. It is observed that wall heterogeneity creates a high convective zone in the diffuse layers resulting complex flow structures of fluids and ions. The improvement of ion transport in the species layers at different horizontal positions significantly enhance the mixing rate.

Comprehensive model of electromigrative transport in microfluidic paper based analytical devices.

A complete mathematical model for electromigration in paper-based analytical devices is derived, based on differential equations describing the motion of fluids by pressure sources and EOF, the transport of charged chemical species, and the electric potential distribution. The porous medium created by the cellulose fibers is considered like a network of tortuous capillaries and represented by macroscopic parameters following an effective medium approach. The equations are obtained starting from their open-channel counterparts, applying scaling laws and, where necessary, including additional terms. With this approach, effective parameters are derived, describing diffusion, mobility, and conductivity for porous media. While the foundations of these phenomena can be found in previous reports, here, all the contributions are analyzed systematically and provided in a comprehensive way. Moreover, a novel electrophoretically driven dispersive transport mechanism in porous materials is proposed. Results of the numerical implementation of the mathematical model are compared with experimental data, showing good agreement and supporting the validity of the proposed model. Finally, the model succeeds in simulating a challenging case of free-flow electrophoresis in paper, involving capillary flow and electrophoretic transport developed in a 2Dgeometry.

Interconnect Electromigration Modeling and Analysis for Nanometer ICs: From Physics to Full-Chip

In this article, we will present recent advances in VLSI reliability effects with a focus on electromigration (EM) failure/aging effect on interconnects, which is one of the most important reliability concerns for VLSI systems especially at the nanometer regime. One of the most important advances for EM analysis in recent years is the recognition that EM failure analysis can't depend on single wire segment anymore, as done in the traditional Black and Blech's based methods. New generation of EM modeling and design must consider all the wire segments in an interconnect as the hydrostatic stress in those wire segments affect each other. Such recognition bring both challenges and opportunities. We will start with physics-level stress-oriented characterization of EM failure effects and recently proposed three-phase EM models. Then we present a new EM immortality check at the circuit level considering multi-segment interconnects and void saturation volumes. After this, we will present how to accelerate EM aging effects for fast EM validation at the circuit level under normal working conditions using advanced structure-based techniques. Finally, we will present new EM sign-off analysis tool, called EMspice, at the full-chip power grid level considering the interplays between resistance changes from post-voiding processes and current density changes from power grids over the aging process. A number of other relevant works will be reviewed and compared as well.

Semi In-Situ Observation on Void Formation in Electromigrated Sn-Ag Film and its Prediction by Random-Walk Simulation

In this study a thin metallic film composed of Sn-3.5Ag (wt.%) was subjected to a current density of 7.77×104 A/cm2 at room temperature, in order to test the ability of existing models of electromigration (EM) to predict the nucleation and evolution of voids generated by the resulting atomic migration. The computer simulation used to compute the coupled current distribution, thermal distribution and atomic migration problems relied on a Random Walk (RW) method that has not previously been applied to this problem but is particularly well suited to modelling a domain which is undergoing changes due to the formation of voids. Comparison of the experimental results and computer simulations provide proof of concept that the RW method can be successfully applied to this class of problem, but also that imperfections in the film can lead to deviations from the predicted patterns.

Phase field modeling of pore electromigration in anisotropic conducting polycrystals

Abstract A phase field model is developed to investigate the migration of pores driven by an electric field in polycrystalline materials with anisotropic electrical conductivity. Mass diffusion describing pore migration is coupled with charge conduction by solving microscopic Ohm’s law. The model accounts for grain structure dependent conductivity distribution which affects pore migration velocity and path. As an example, the model is applied to simulate pore migration in tin polycrystals. Significant effects of the grain orientation on pore migration is observed which are analyzed in terms of the orientation-dependent conductivity in individual grains and the mismatch of conductivity across grain boundaries. The effects of conductivity anisotropy on the interactions between pores and pore coalescence are also discussed.

The dynamics of band (peak) shape development in capillary zone electrophoresis in the case of two co-migrating analytes: The displacement and the tag-along effects.

Peak shapes in electrophoresis are often distorted from the ideal Gaussian shape due to disturbing phenomena, of which the most important is electromigration dispersion. For fully dissociated analytes, there is a tight analogy between nonlinear models describing a separation process in chromatography and electrophoresis. When the velocity of the separated analyte depends on the concentration of the co-analyte, the consequence is a mutual influence of the analytes couples, which distorts both analyte zones. In this paper, we introduce a nonlinear model of electromigration for the analysis of two co-migrating fully dissociated analytes. In the initial stages of separation, they influence each other, which causes much more complicated peak shapes. The analysis has revealed that the two most important phenomena-the displacement and the tag-along effects-are common both for nonlinear chromatography and electrophoresis, though their description is partly based on rather different phenomena. The comparison between the nonlinear model of electromigration we describe and the numerical computer solution of the original continuity equations has proven an almost perfect agreement. The predicted features in peak shapes in initial stages of separation have been fully confirmed by the experiments.

Investigating the electromigration limits of Cu nano-interconnects using a novel hybrid physics-based model

To predict the impact of technological variables such as materials, dimensions, interfaces, and operating conditions on Cu electromigration, in this study a hybrid modeling framework is developed by coupling a global Korhonen-type electromigration modeling module with a cellular automaton-based void dynamics module. The modeling framework is corroborated and benchmarked using experiments on Cu interconnects and is used to predict the impact of scaling on Cu electromigration induced stress evolution. The simulations shed light on the impact of dimensional scaling on stress kinetics, void nucleation, and growth phases, where the nucleation phase is found to become longer than the growth phase and voids are found to grow relatively more rapidly upon nucleation in highly scaled linewidth. In addition, the simulations predict 22% lower median time to failure and a higher variability of time to failure for downstream vs upstream electromigration modes due to the more critical impact of near via voiding in downstream cases. Lending further credence to its predictive merits, the model predicts and explains complex R-shift signatures occurring at high temperature electromigration experiments due to void dynamics.

Saturation-Volume Estimation for Multisegment Copper Interconnect Wires

Recently, many physics-based electromigration (EM) models have been proposed to mitigate the over-conservativeness of existing Black–Blech-based EM models. To assess EM failures, one needs to estimate how the void grows after the nucleation phase. As a result, it is important to estimate the saturation volume of a void, which is important for EM mortality check. The existing saturation void volume model only works for a single wire segment. In this paper, we propose a new model for fast estimation of the void’s saturation volume for general multisegment interconnect wires. The new model is based on the fundamental atom conservation at the steady-state condition of void growth phases. The new formula agrees with the existing saturation void volume formula for the single-segment wire case and is a natural extension of the single-segment case to general multisegment wires. In addition, we consider the impacts of the void volume on final stress distributions of the wire to further improve the accuracy of the proposed formula. Based on the new formula, we propose a new EM immortality check flow, which considers both the recently proposed EM immortality in the void nucleation phase and the void saturation volume in the growth phase. The new flow can further reduce the conservativeness of the existing EM failure effect analysis. The numerical results show that the proposed formula agrees well with a published work for two-segment cases, which are supported by experimental data. The formula is also validated by the recently proposed physics-based 3-D finite-element (FEM) analysis tool for general multisegment interconnect wires. We also demonstrate new EM immortality check flow to quickly identify the new type of immortal wires, which are nucleated but with smaller-than-critical voids.

General coupling model for electromigration and one-dimensional numerical solutions

A three-dimensional (3D) general coupling model for electromigration has been developed with the use of the mass conservation equation. The flux terms that include concentration gradient, electron wind force, stress migration, and thermal migration are considered. The constitutive equation for the electromigration strain has been derived. Then, the governing equations for one-dimensional (1D) metal lines are obtained for both totally fixed and stress-free mechanical boundary conditions. The numerical results reveal that the hydrostatic stress is significantly lower than the predicted results in the existing literature for the totally fixed configuration. Extensive discussions are presented to provide the explanations of such difference. The vacancy concentration gradient plays an important role in formulating electromigration problems. The current-driven flux can be entirely balanced by the concentration gradient that acts as an opposing force during electromigration under a stress-free condition in steady-state. The new solutions of the critical threshold jL, the product of current density, and metal line length are obtained in terms of vacancy concentration. As electromigration is eventually determined by the void growth, the critical vacancy concentration is used to reanalyze Blech's experiment data. The theoretical predictions are consistent with the experimental observations.

Reliability based hardware Trojan design using physics-based electromigration models

In recent years the concern over Hardware Trojans has come to the forefront of hardware security research as these types of attacks pose a real and dangerous threat to both commercial and mission-critical systems. One interesting threat model utilizes semiconductor physics, specifically aging effects such as Electromigration (EM). However, existing methods for EM-based Trojans rely on empirical Black's models can easily lead to performance degradation and less accuracy in Trojan activation time prediction. In this article, we study the EM-based Trojan attacks based on recently developed physics-based EM models. We propose novel EM attack techniques in which the EM-induced hydrostatic stress increase in a wire is caused by wire structure or layer changes without changing the current density of the wires. The proposed techniques consist of sink/reservoir insertion or sizing and layer switching techniques based on the early and late failure modes of EM wear-out effects. As a result, the proposed techniques can have minimal impact on circuit performance, which is in contrast with existing current-density-based EM attacks. The proposed techniques can serve as a trigger for the EM attack on power/ground networks and signal and clock networks. Furthermore, we also present two potential EM attack mitigation techniques, namely, the split fabrication and burn-in testing.

Electromigration of Cell Surface Macromolecules during Galvanotaxis

Weak, DC electric fields (EFs) direct cellular polarity, migration and outgrowth. They are used to induce polarity during tissue engineering and direct migration to promote epidermal wound repair. Fields as weak as 10 mV/mm, are sufficient to direct migration, although the mechanisms used by cells to sense such weak fields are still debated as weaknesses exist in the competing models. We hypothesize that cellular polarization is achieved by the redistribution of a cell surface, electric field receptor (EFR) in the presence of an applied EF. We are investigating the electromigration model that depends on electrically generated forces in the plane of the plasma membrane and parallel to the electric field vector, specifically, electrophoretic and electro-osmotic forces. Electrophoretic force drives net negatively charged macromolecules to the anode while electro-osmotic flow of water, in the boundary layer, drives macromolecules with large extracellular domains to the cathode. We have derived a model based on these opposing forces, to predict the relative concentration of surface proteins over space and time, allowing us to test the redistribution of known plasma membrane surface macromolecules in applied EFs and under controlled conditions. The model closely describes accumulation of a net negatively charged, GPI-anchored, fluorescent protein to the cathode under different field strengths, showing that redistribution reaches steady-state within minutes, significantly faster than cathodally migrating cells turn toward the cathode. The model is useful for determining the most likely candidates for the EFR on many different cell types.

The dynamics of band (peak) shape development in capillary zone electrophoresis in light of the linear theory of electromigration.

The continuity equations that describe the movement of ions in liquid solutions under the influence of an external stationary electric field, as it is utilized in electrophoresis, were introduced a long time ago starting with Kohlrausch in 1897. From that time on, there have been many attempts to solve the equations and to discuss the results. In electrophoresis, special attention has always been devoted to the peak shapes obtained by the detector since the shapes have a tight connection with the phenomena taking place during electromigration and influence the efficiency and selectivity of the separation. Among these phenomena, the most important is electromigration dispersion. In this commented review paper, we compare various models of electromigration, try to find points that connect them, and discuss the range of their validity in light of the linear and nonlinear theory of electromigration.

The electrothermal impact on a contact metal-semiconductor: applications to the germanium–silver system

PurposeThis study aims to examine the electromigration processes resulting from thermal overloads of semiconductor devices. While in operation, parts of such devices can heat up to 330°C for a short period, resulting in the emergence of molten zones and the devices’ inevitable degradation. Therefore, this study examines the mechanisms behind the formation and migration of silver-based molten zones in bulk germanium and on its surface.Design/methodology/approachExperimental data concerning the correlation between the migration velocities of the inclusions and their sizes are obtained.FindingsBy comparing these experimental data with known electromigration models, it is concluded that inclusions move through the mechanism of melting and crystallization. The dynamics of Ge–Ag zones in the volume of a germanium crystal are compared to those on its surface and accelerated electromigration on the surface of the crystal is observed. This increased migration velocity is shown to be associated with additional contributions of the electrocapillary component.Originality/valueThe results of this study can be used to calculate the operating modes of semiconductor power devices under intense heat loading.

Dynamic reliability management based on resource-based EM modeling for multi-core microprocessors

This article presents a new approach for system-level reliability management for multi/many core microprocessors. In the new approach, the electro-migration (EM) induced time to failure (TTF) at the system level is modeled as a resource, which is abstracted, from a recently proposed physics-based EM model, at the chip level. In this model, a single core can spend the TTF resources at different rates specified by the temperature and the related power consumption. As a result, the new resource-based EM model allows more flexible EM-reliability management for multi/many-core systems. As an application of the new model, we propose a novel task migration method to explicitly balance consumption of EM resources for all the cores. The new method aims at equalizing the probability of failure of each core, which will maximize the lifetime of the whole multi/many core system. To more efficiently regulate the lifetime of the multi/many core system, dynamic voltage and frequency scaling (by using different performance states (p-states), which can represent different operating voltages and frequencies) is further employed to compensate for the excessively consumed lifetime of all the cores when the chip is loaded with heavy tasks for a certain period of time. In this way, the TTF of all the cores could be compensated to meet the lifetime requirement, giving the multi/many core system more flexibility to handle heavy task assignment on demand. Experimental results on a 36-core processor platform show that the proposed task migration scheme could balance the lifetime consumption of all the cores, and maintain even-distributed TTF slacks across different cores, while the existing temperature-based task migration schemes lead to diverged TTF of cores. The results also show that the TTF consumption could be easily compensated for by switching to a low p-state, which could not be achieved if TTF consumption diverged across different cores.

Analytical Modeling of Electromigration Failure for VLSI Interconnect Tree Considering Temperature and Segment Length Effects

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.

A model for electromigration induced flow in liquid metals

Electromigration in liquid metals can be used for mass transport over a considerable length with micro- and nano-scale flow features. Exploitation of this phenomenon, however, requires a sound understanding of the liquid metal flow under an applied electric field. Depending on the sign of the effective charge number, Z*, liquid metals flow along a set direction under the applied electric field. A few liquid metals, e.g. Ga, Sn, etc, flow in the direction of the electric field, while a few others, e.g. Pb, flow in the opposite direction. Here, we propose a new model for predicting the direction of the aforementioned flow for a given liquid metal. Our model incorporates Lennard-Jones potential into the cell model of liquids in order to calculate the value of Z* as a function of temperature. We then carry out experiments on a few metals to validate the model and show that it indeed correctly predicts the ensuing flow.

Probabilistic Wire Resistance Degradation Due to Electromigration in Power Grids

Electromigration (EM) is a growing concern in on-chip interconnects, particularly in the computing and automotive domains. EM can cause wire resistances in a circuit to increase, which may result in circuit performance failure within the lifetime of a product. Classical circuit-level EM models have two drawbacks: first, they do not accurately capture the physics of degradation in modern copper dual-damascene (Cu DD) metallization and second, they fail to model the inherent resilience in a circuit that may allow it to continue to function even after some wires fail. This paper overcomes both limitations, and develops a method to analyze the effect of EM on the resistance of wires in a circuit. For a single wire, our probabilistic analysis encapsulates known realities about Cu DD wires, e.g., that some regions of these wires are more susceptible to EM than others, and that void evolution shows statistical behavior. We develop a new criterion for identifying mortal wires based on this analysis. One part of the criterion incorporates the achievement of steady state and provides a result that is slightly tighter than the traditional Blech criterion, while another part is tied to the lifetime of the system. We apply these ideas to the analysis of on-chip power grids and demonstrate the inherent robustness of these grids that maintains supply integrity under some EM failures.

Fast and Automated Electromigration Analysis for CMOS RF PA Design

This paper presents fast and automated electromigration (EM) reliability modeling by using automated modeling generation (AMG) algorithm. The AMG converts human based EM modeling into an automated modeling and simulation process with the help of ANSYS parametric design language (APDL) program. For automating the neural model training process, training-driven adaptive sampling is applied to integrate data generation, data distributions determination, model structure adaptation, training and testing into a unified framework. Fully automated reliability model construction and simulation is achieved for the first time. This method effectively shortens the period of EM modeling by using dynamic sampling method. Furthermore, the heat generation from active devices has been considered to describe the heat effect on the interconnect reliability. Through the proposed technique, the allowable sizes, temperature and output power of a CMOS radio frequency power amplifier (RF PA) are derived to give reliability criteria for PA designer.