The roots of pd can be traced back to the early works of gabrio piola according to dellisola et al. The equation of motion in the peridynamic theory is where ismassdensity, isthedisplacementvector. Researcharticle peridynamic model for the numerical simulation of anchor bolt pullout in concrete jiezhilu,1 yaotingzhang,1 habibmuhammad,1,2 andzhijunchen1. The main process is a diffusion of irons that is driven by their. Peridynamic pd theory is a new continuum mechanics formulation introduced to overcome the limitations of classical continuum mechanics such as predicting crack initiation and propagation, and. Numerical simulation of stress evolution during electromigration in ic interconnect lines hua ye, cemal basaran, and douglas c. Reformulating hyperelastic materials with peridynamic modeling simple elastic models, e. The framework allows four coupled physicalprocessestobemodeledsimultaneously. Peridynamic simulation of crack propagation in bulk. A modified druckerprager constitutive model was adopted to model ice and was implemented in the statebased peridynamic equations to analyze the elasticplastic deformation of ice.
Siam journal on numerical analysis siam society for. Hopkins, senior member, ieee abstract a finite element simulation of stress evolution in thin metal film during electromigration is reported in this paper. Jan 01, 2008 a theoretical framework, based upon the peridynamic model, is presented for analytical and computational simulation of electromigration. Gerardin 1950s first systematic studies of electromigration by w. Essentially, a finite difference method is used in the bulk for wave propagation modeling, while peridynamics is automatically inserted in high strain areas to model crack initiation and growth. It occurs when the current density is high enough to cause the drift of metal ions in the direction of the electron flow, and is characterized by the ion flux density. Abstract pdf 564 kb 2015 selecting the kernel in a peridynamic formulation. A thesis submitted in partial fulfillment of the requirements of the university of. Electromigration em is one of the major reliability concerns in advanced ic technology 1. International journal of computational physics, vol. Description of finite element fe model for eis simulation 2.
Electromigration is a critical problem, especially as the ic feature such as width and pitch of thinfilm interconnects have been reduced to submicronlevels. Thermal issues with the pcb design are largely locked in during the component i. In simulations, we use the fracture toughness failure. Simulating damage mechanics of electromigration and.
The effect is important in applications where high direct current densities are used, such as in microelectronics and related structures. Timing issues are also affected by temperature differences between components. It refers to the transfer of metal atoms due to the electron current, and is becoming more challenging as feature size shrinks. Peridynamic modeling of the kalthoffwinkler experiment submission for the 2001 sandia prize in computational science stewart a. Mesh density increased from 189k elements to 1,507k elements. The electromigration process is modeled by a coupled diffusion. Predictive simulation of anchor pullout from concrete structures is not only a serious problem in structural mechanics but also very important in structural design safety. Peridynamics is a nonlocal extension of classical continuum mechanics. Wever correlation between the direction of the current flow and the material transport. An introduction to electromigrationaware physical design. Multiphysics computer simulation ofthe electromigration phenomenon xiaoxin zhui, hiren kotadia3, sha xu2, hua lui, samjid h. Artificial boundary conditions for nonlocal heat equations on. Some remarks on finite element modeling of electromigration. Electromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms.
Electromigration analysis at advanced nodes mentor graphics. Modeling of electromigration induced resistance change in threedimensional interconnects with through silicon vias r. School ofcomputing and mathematical sciences, university ofgreenwich. This paper studies the numerical simulation method for electromigration in ic device and solder joint in a package under the combination of high current density, thermal load and mechanical load. Modeling electromigration for microelectronics design. We then propose a form for a multiphysics constitutive model, and a method for calibrating the peridynamic parameters. Peridynamics as an upscaling of molecular dynamics. Peridynamic multiscale finite element methods timothy costa, stephen bond, david littlewood, stan moore prepared by sandia national laboratories albuquerque, new mexico 87185 and livermore, california 94550 sandia national laboratories is a multiprogram laboratory managed and operated by sandia corporation. Atomic diffusion due to high current density also depends on temperature and stress can generate voids, hillocks 4 metal atom movement etsv void m1 void 2 d. Examples of applications of the peridynamic theory to the solution of static equilibrium problems volume 119 issue 1216 m. If the current density is high enough, there can be a momentum transfer from moving electrons to the metal ions that make up the lattice of the interconnect material. Implementation of peridynamic beam and plate formulations in. First, we present the solutions of the dispersion relations, group velocities, and phase velocities of longitudinal and transverse waves, and.
Peridynamic modeling of localization in ductile metals. Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks y liao, l liu, q liu, x lai, m assefa, j liu journal of thermal stresses 40 12, 14841501, 2017. The framework allows four coupled physical processes to be. Computer simulation of electromigration in microelectronics. Electromigration induced interconnect aging and its repercussions on the performance of nanometerscale vlsi circuits a dissertation submitted to the faculty of the graduate school of the university of minnesota by vivek mishra in partial fulfillment of the requirements for the degree of doctor of philosophy sachin s. I want to do the emir analysis as it is described in spectreuser. Hence, this study presents the peridynamic diffusion modeling and implementation. To overcome this limitation, an improved bpd model is proposed to analyze the deformation and crack propagation of microelastic brittle materials with emphasis on varying poissons ratios. The term peridynamic, an adjective, was proposed in the year 2000 and comes from the prefix peri, which means all around, near, or surrounding. After this we provide a section describing the discretized peridynamic electromigrationmodel of a onedimensionalexample problem. The peridynamic theory is a nonlocal extension of continuum mechanics that is compatible with the physical nature of cracks as discontinuities. Index termsperidynamics, finite element, diffusion, model.
Simulation of electromigration is a field under development and it is noticeable that new promising approaches are emerging. Modeling of electromigration phenomena sciencedirect. A percolative simulation of electromigration phenomena. Electromigration em refers to the unwanted movement of materials in a semiconductor. Simulation of electromigration induced stress of solder. Implementation of peridynamic beam and plate formulations in finite.
Discontinuities require special treatment methods of lefm, for example. Electromigration what ic designers need to know industry. The peridynamic formulation for transient heat conduction in bodies with discontinties. Electromigration em refers to the undesired migration of metal atoms along a metal interconnect. Peridynamic simulation of electromigration semantic scholar. Electromigration study for multiscale powerground vias in. Modeling electromigration using the peridynamics approach nist. Computer simulation of electromigration in microelectronics interconnect by xiaoxin zhu centre of numerical modelling and process analysis, school of computing and mathematical sciences,the university of greenwich, london, u.
Threedimensional 3d coupled electricalthermalmechanical finite element analysis is performed. Peridynamic simulation 0 500 1500 2000 2500 3000 0 0. Introduction electromigration em is a mass migration phenomenon that occurs in conductors under the influence of strong electric current, temperature, temperature gradient, and mechanical stress ref. The peridynamic theory has been used to model and simulate numerically various kinds of mechanical behavior of solids. This handbook covers the peridynamic modeling of failure and damage. In the peridynamic constitutive model, concrete material was described as a series of interacting particles, and the shortrange repulsive force and anisotropic behavior of concrete were taken into account in the expression of the interactive bonding force, which was given in terms of classical. Simulation laboratory csl, established in 1997, addresses both research and educational aspects of diffusion electromigration, thermal grooving, mechanical spectroscopy. Comparison between our numerical simulation results and the experimental results from previous work has shown good agreement. Research article damage and failure process of concrete. This work is devoted to analytical solutions of the elastodynamic behavior of linearized isotropic statebased peridynamic materials. Peridynamic model for the numerical simulation of anchor. Methodology for electromigration signoff in the presence of. Previous works the effect of electromigration and its relation to the mean time to failure mttf of an electronic circuit has.
The discrete peridynamic model has the same computational structure as a molecular dynamics model. Peridynamic modeling and simulation of ice craters by impact. Black, electromigration a brief survey and some recent results, ieee transactions on electron devices volume 16, issue 4, apr 1969 pages. One is the indenter scratching parallel to the interface. Simulation of electromigration through peridynamics. Peridynamic simulation of electromigration walter gerstle1, stewart silling2,davidread3,vinodtewary4 and richard lehoucq5 abstract. Peridynamic theory of solid mechanics sandia national. Nonlocal peridynamic modeling and simulation on crack.
An extended peridynamic approach for crack propagation analysis in concrete structures was proposed. Electromigration, computer simulation laboratory, internal. It is convenient to assume that vanishes outside some horizon. Transient modeling of tsvwire electromigration and lifetime analysis of power distribution network for 3d ics xin zhao1, yang wan 2, michael scheuermann3, and sung kyu lim 1ibm systems and technology group, hopewell junction, ny, usa. Peridynamic simulation of electromigration, comput mater continua. Electromigrationinduced interconnect aging and its. Peridynamics pd is a novel continuum mechanics theory established by stewart silling in 2000. A p l i e d m e c h an e ournal of applied j mechanical. This density depends on the magnitude of forces that tend to hold the ions in place, i. Beginning from an example nonordinary statebased model, lower dimensional peridynamic models of plates, beams, and shells are developed and validated against classical models. Two classes of artificial boundary conditions abcs are designed, namely, nonlocal analog dirichlettoneumanntype abcs global in time and highorder pade approximate abcs local in time. Pdf implementation of peridynamic beam and plate formulations.
Request pdf peridynamic simulation of electromigration a theoretical framework, based upon the peridynamic model, is presented for analytical and computational simulation of electromigration. This document provides a brief overview of the peridynamic model of a continuum, then discusses how the peridynamic model is discretized within lammps. Pdf peridynamic pd theory is a new continuum mechanics formulation. Electromigration is the gradual displacement of metal atoms in a semiconductor. Simulating damage mechanics of electromigration and thermomigration shidong li mohd f. Abstract peridynamic pd theory is a new continuum mechanics. Request pdf electromigration in thin films and electronic devices. Physicsbased predictive monte carlo simulation is able to model void incubation, its coalescence and void growth, 84 a physical mtf formulation being derived. Calculation and analysis of em test structure with excel and simulation tool 5. Classical continuum mechanics is, how ever, not efficacious for analysis of fields that are, or may become, manifestly discontinuous, as shown for example in fig.
A robust, scalable computational framework based on these extensions to the peridynamic theory is developed, and numerical examples are provided which demonstrate the ability to capture experimentally observed ballistic limit curves for ductile materials, as well as realistic fracture patterns in brittle materials subjected to projectile impact loadings. The mixed blessing of locality origins of nonlocality in a modellongrange forces coarsening a finescale material system smoothing of a heterogeneous material system nonlocality at the macroscale the mixed blessing of nonlocalityintroduction to peridynamics equilibrium in terms of integral equations material modeling bond based. For em, local vias in between metal layers of ic have been regarded as emprone structures, and have been actively studied. Electromigrationir drop emir custom ic design cadence. Peridynamics is a reformulation of continuum mechanics based on integration of interactions rather than spatial differentiation of displacements. A finite differenceaugmented peridynamics method for.
T1 simulation of electromigration through peridynamics. The term peridynamics, a noun, is a shortened form of the phrase peridynamic model of solid mechanics. From the simulation results, it is observed that the damage and failure of the column is just spontaneous, and no special failure criteria are required in order to predict the pathalongwhichthecrackdevelops. Electromigration protection requires accurate interconnect. Electromigration em is a key reliability challenge for the economically very important electronics industry. The ice forces obtained from peridynamic simulation are in the same order as experimental data. Peridynamic modeling of the kalthoffwinkler experiment. A theoretical framework, based upon the peridynamic model, is presented for analytical and computational simulation of electromigration. Force state or deformation state has infinite degrees of freedom.
Atomistic monte carlo simulation is suitable to deal with interatomic interaction under electromigration 85 or atomic driftdiffusion in the grain boundary. In this study, the peridynamic pd framework is applied to model the electromigration process by coupling physical fields of mechanical deformation, heat transfer, electrical potential distribution, and vacancy diffusion simultaneously. Extension of the peridynamic theory of solids for the. N2 thin film metallic conductors or interconnects are subjected to increasingly high current densities as the feature sizes decrease that can lead to failure of interconnects in moderately short. Modeling electromigration using the peridynamics approach. Peridynamic simulation of crack propagation in bulk superconductors with an electromagneticthermal model yanyun ru, huadong yong, youhe zhou department of mechanics and engineering sciences, college of civil engineering and mechanics, lanzhou university, lanzhou, gansu 730000, p. Peridynamic simulation of electromigration tech science press. Various problems are simulated based upon the present peridynamic heat. Madenci, simulation of electromigration through peridy.
An introduction to electromigrationaware physical design, ispd 2006, pp. All conductors are susceptible to electromigration, therefore it is important to consider the effects the electrical current resulting from the applied field may have on the conductor. Examples of applications of the peridynamic theory to the. Mesoscale peridynamic simulation of crack deflections in heterogeneous material under thermomechanical environment y. Modeling of electromigration induced resistance change in. Artificial boundary conditions for nonlocal heat equations. The book extends the classical theory of continuum mechanics to allow unguided modeli. Transient modeling of tsvwire electromigration and.
Numerical visualization shows advantages of applying peridynamics on ice damage. Meshless peridynamics method for modeling corrosion crack. However, simulation time, integration algorithms, and the required. Electromigration in thin films and electronic devices. Methodology for electromigration signoff in the presence of adaptive voltage scaling weiting jonas chant, andrew b. The simulation has shown the 3d void generation and ttf for both device and solder joint of an ic package in electromigration. Peridynamic simulation of electromigration request pdf. Electromigration, simulation, shunt, solder joint 1.
The framework allows four coupled physical processes to be modeled simultaneously. Handbook of peridynamic modeling 1st edition florin. So far, how to extend peridynamics to model general hyperelastic materials still remains a challenging problem. Page 2 of 3 ao ghaffarima, xiao s 21 peridynamic modeling and simulation of rolling contact fatigue appl mech ng. It avoids the need to evaluate the partial derivatives of the deformation with respect to the spatial coordinates, instead using an integrodifferential equation for the linear momentum balance. Hu, yin yu, xiaojing zhang school of aeronautics and astronautics, shanghai jiao tong university, shanghai 200240, china abstract peridynamic approach with a new nonuniform discretization is applied to investigate crack.
Peridynamic simulation of brittleice crushed by a vertical. Multiphysics computer simulation ofthe electromigration. Two classes of artificial boundary conditions abcs are designed, namely, nonlocal analog dirichlettoneumanntype abcs global in. Madenci department of aerospace and mechanical engineering the university of arizona tucson, az 85721 abstract prediction of crack propagation path is especially a challenging due to the interaction of different fields such as thermal diffusion and deformation. Reformulating hyperelastic materials with peridynamic. In the finite element method fem, the crack paths or the points of crack initiation usually need to be assumed in advance. Pd theory is based on integrodifferential equations and these equations are generally difficult to be solved by using analytical techniques. Em in copper damascene interconnect structures is an extremely complex phenomenon because of the multitude of possible diffusion paths, including surfaces, grain boundaries, other internal interfaces, and bulk material, and the multitude of driving forces for diffusion, which. The emu simulation of this problem is shown in figure1.
Many aspects of a pcbs performance are determined during detailed design, e. Mesoscale peridynamic simulation of crack deflections in. Meanwhile, voids were initiated at the entry location of the electron flow due to the current crowding effect. A copper post wafer level package is used as a test vehicle for simulation. Experimental data on void drift by electromigration are introduced to provide a specific target for a demonstration of the peridynamical approach. Peridynamic modeling of structural damage and failure. Peridynamic modeling and simulation of ice craters 467 adaptive particle refinement theory are introduced in this ection. The center pioneers a number of applications such as elecromigration, internal friction, grain boundary grooving, and sintering. Influence of temperature gradient on the electromigration. This migration occurs due to the electron wind which is the momentum transfer created by electrons in the current pushing the metal atoms that make up the interconnect. Peridynamic simulation of the failure mode of concrete column under uniaxial compression. In this paper, some issues in finite element modeling of electromigration in solder joints are addressed.
Article pdf available in journal of multiscale modelling 103 november. Pd has been attractive to researchers as it is a nonlocal formulation in an integral form, unlike the local differential form of classical continuum mechanics. The nanoscratch simulations are performed along with different scratching direction. Otherwise, some special crack growth treatment or adaptive remeshing algorithm is normally used. Theory of electromigration electromigration is the transport of material in a conductor under the influence of an applied electric field. In 3d packaging technology, em can be due to drastic changes in geometries between metal wires, throughsiliconvias, and landing pads. Jun 11, 2018 peridynamic pd theory is a new continuum mechanics formulation introduced to overcome the limitations of classical continuum mechanics such as predicting crack initiation and propagation, and capturing nonlocal effects. To study the repetitive nature of ice force, damage zone lengths of crushing failure were computed and conclude that damage zone lengths are 0. A useful special case where is any point in the reference con. Electromigration is also affected by recent changes in. In the present work, a statebased peridynamics with adaptive particle refinement is proposed to simulate water ice crater formation due to impact loads.
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