Simulations of diffusion driven phase evolution in heterogenous solids

With reduction in size, ever greater operational demands are placed on electronics components at all levels of the device, starting from the transistor level to the level of the package and the solder interconnects. Concurrently, there has been a move to more complicated materials systems in order to meet health and environmental guidelines. These trends of reducing size, increasing loads have increased the necessity to understand the mechanisms of the failure. ^ As the length scales are reduced, it becomes increasingly important to consider interfacial and micro-structural effects that can be safely ignored at larger length scales owing to the randomness. It has become important to model the effect of interfacial motion and micro-structural evolution due to diffusion on the reliability of micro-electronics components. Examples of interfacial motion phenomena in solids include crack propagation, grain boundary motion, diffusion driven void motion through sur- face and bulk diffusion. The presence and evolution of these over the life-cycle of electronics components such as metal lines and solder joints presents a significant reliability challenge. The mathematical models that describe the evolution of these interfaces are usually formulated as systems of non-linear equations and hence, numerical methods provide an important method to study and understand them. The primary challenge in the study of these moving boundary problems is the tracking of the moving boundary and the application of appropriate boundary conditions on the moving boundary. ^ The phase field method tracks through smooth approximations of the Heaviside step and Dirac δ functions, which are maintained through the solution of a system of nonlinear differential equations. In this work, phase field approaches are developed for the study of diffusion driven phase evolution problems. First a phase field model for the evolution of voids in solder joints owing to electromigration and stress-migration both at the interface due to the surface gradients of the electric potential, temperature, curvature and strain energy, as well as self diffusion in the bulk on account of the chemical potential gradients as well as the electromigration force. This is modeled using a vacancy diffusion mechanism, while the growth of the voids is assumed to be due to the absorption of voids at the interface of pre-existing voids. A formal asymptotic analysis is performed to show the equivalence of the diffuse interface model to its sharp interface equivalents. Several numerical examples are presented. ^ Finally, an n-phase system of Cahn-Hilliard equations is developed to allow for the simulation of void evolution and growth in a multi-phase system. This is derived through a micro-force balance in order to eliminate the use of Lagrange multipliers that are commonly seen in such methods. A limited formal asymptotic analysis is performed to show the equivalence of the model to the standard surface diffusion model in regions where only two phase are present. This is numerically implemented and various numerical examples of phase evolution under simple surface diffusion, as well as surface diffusion with electromigration are demonstrated

Numerical analysis of lead-free solder joints: effects of thermal cycling and electromigration

To meet the requirements of miniaturization and multifunction in microelectronics, understanding of their reliability and performance has become an important research subject in order to characterise electronics served under various loadings. Along with the demands of the increasing miniaturization of electronic devices, various properties and the relevant thermo-mechanical-electrical response of the lead-free solder joints to thermal cycling and electro-migration become the critical factors, which affect the service life of microelectronics in different applications. However, due to the size and structure of solder interconnects in microelectronics, traditional methods based on experiments are not applicable in the evaluation of their reliability under complex joint loadings. This thesis presents an investigation, which is based on finite-element method, into the performance of lead-free solder interconnects under thermal fatigue and electro-migration, specifically in the areas as follows: (1) the investigation of thermal-mechanical performance and fatigue-life prediction of flip-chip package under different sizes to achieve a further understanding of IMC layer and size effects of a flip chip package under thermal cycling; (2) the establishment of a numerical method, simulating void-formation/crack-propagation based on the results of finite-element analysis, to allow the prediction of crack evolution and failure time for electro-migration reliability of solder bumps; (3) the establishment of a flow-based algorithm for combination effects of thermal-mechanical and electro-migration that was subsequent implemented in to an FE model to evaluate the reliability assessment of service lives associated with a flip chip package

Phase-field modeling on the diffusion-driven processes in metallic conductors and lithium-ion batteries

Diffusionsgetriebene Prozesse sind wichtige Phänomene der Materialwissenschaft im Bereich der Energieumwandlung und -übertragung. Während der Umwandlung von chemischer Energie in elektrische Energie ist die Speziesdiffusion im Allgemeinen mit der Austauschrate und folglich mit der Leistung der Umwandlungsvorrichtung verbunden. Alternativ diffundiert die Übertragung des elektrischen Feldes durch die Spezies, wenn sie durch irgendein Medium verläuft. Die Konsequenzen dieses Effekts können reguliert werden, um Oberflächen-Nanomuster abzustimmen. Andernfalls können die unkontrollierten Morphologien zu einer dauerhaften Verschlechterung der metallischen Leiter führen. Daher ist das Verständnis des materiellen Verhaltens bei Vorhandensein der treibenden Kräfte von Diffusionsspezies von wissenschaftlichem Interesse. Die vorgestellte Dissertation schlägt eine Untersuchung von jeweils einem Beispiel der Speziesdiffusion während der Energieumwandlung und -übertragung vor. Ziel der Studie ist es insbesondere, sowohl die Lithiumeinfügung von Lithium-Ionen-Batterien in die Kathodenelektrode als auch die morphologische Entwicklung von Einschlüssen zu untersuchen, während sie sich unter der Elektromigration in den metallischen Leitern ausbreiten. Lithium-Manganoxid-Spinell, ein Kathodenelektrodenmaterial von Lithium-Ionen-Batterien, zeigt während des Einfügens eine Koexistenz von Li-reichen und Li-armen Phasen. Für ein besseres Verständnis des Mechanismus einer zweiphasigen Koexistenz wird ein mathematisches Modell der Phasentrennung abgeleitet, das auf der Cahn-Hilliard-Gleichung basiert. Zunächst wird die geometrische Formpolydispersität eines isolierten Partikels betrachtet, um den mesoskopischen Effekt der Oberflächenkrümmung zu untersuchen. Die Simulationsergebnisse zeigen, dass der Beginn der Phasentrennung bevorzugt in Bereichen auftritt, in denen das Partikel eine starke Krümmung aufweist. Weiterhin wird das elliptische Teilchen mit einem höheren Querschnittsverhältnis dem Einsetzen der Phasentrennung vor den Teilchen mit einem niedrigeren Querschnittsverhältnis ausgesetzt. Abschlißend wird der Einfluss der Variation der Parameter auf die Ladungsdynamik diskutiert. Die Studie wird weiter auf mehrere Partikelsysteme ausgedehnt, um den Einfluss verschiedener mikrostruktureller Deskriptoren wie Partikelgröße, Porosität und Tortuosität auf den Transportmechanismus zu verstehen. Bei Tortuosität wird eine lineare Abhängigkeit der Transportrate beobachtet. Die Steigung dieser linearen Beziehung ist unabhängig von der Partikelgröße, zeigt jedoch eine gewisse Abhängigkeit von der Porosität. Darüber hinaus legen die vorgestellten Ergebnisse nahe, dass Systeme, die aus kleineren Partikeln bestehen, der durch Oberflächenreaktionen begrenzten Theorie genau folgen, während größere Partikel zu der durch Massentransporte begrenzten Theorie tendieren, die für planare Elektroden abgeleitet wurde. Um die hierarchisch strukturierten Elektroden zu identifizieren, könnten die vorgestellten Simulationsergebnisse verwendet werden, um den experimentellen Aufwand zu optimieren. Die durch Elektromigration induzierte morphologische Entwicklung von Einschlüssen (Hohlräume, Ausfällungen und Inseln) wurde kürzlich im Hinblick auf die effiziente Auslegung der Verbindungen und Oberflächen-Nanomuster untersucht. Um die morphologischen Entwicklungen zu verstehen, wird ein Phasenfeldmodell abgeleitet, um Einschlüsse zu berücksichtigen, die unter dem externen elektrischen Feld wandern. Die Erkenntnisse aus den numerischen Ergebnissen zu isotropen Einschlüssen bestätigen zunächst die Ergebnisse der linearen Stabilitätsanalyse. Zusätzlich können die numerischen Ergebnisse den Übergang eines kreisförmigen Einschlusses zu einem fingerartigen Schlitz elegant erläutern. Die nachfolgende Drift des Schlitzes ist durch eine Forminvarianz zusammen mit einer stationären Schlitzbreite und -geschwindigkeit gekennzeichnet, die mit dem angelegten elektrischen Feld jeweils als $E_{\infty}^{-1/2}$ und $E_{\infty}^{3/2}$ skaliert werden. Die Ergebnisse aus Phasenfeldsimulationen werden kritisch mit der Lösung mit scharfen Grenzflächen verglichen. Die Auswirkungen der Studie auf die Vorhersage einer Hohlraumwanderung in Flip-Chip-Sn-Ag-Cu-Lötperlen und die Herstellung von Kanälen mit gewünschten Mikro- / Nanodimensionen werden diskutiert. Die Studie wird weiter auf anisotrope Einschlüsse ausgedehnt, die in $\{110\}$, $\{100\}$ und $\{111\}$ kristallografischen Ebenen von flächenzentrierten kubischen Kristallen wandern. Basierend auf numerischen Ergebnissen werden morphologische Karten in der Ebene des Fehlorientierungswinkels und des Leitfähigkeitskontrasts zwischen dem Einschluss und der Matrix erstellt. Die Simulationen sagen eine Vielzahl von Morphologien voraus, darunter stationäre und zeitperiodische Morphologien sowie Zick-Zack-Oszillationen und eine Einschlussauflösung. Darüber hinaus wird beobachtet, dass der Einfluss der Variation des Leitfähigkeitskontrasts und der Fehlorientierung Einfluss auf die morphologische Entwicklung der zeitperiodischen Schwingungen, der stationären Formen und der Art und Weise hat, wie Einschlüsse auseinander brechen. Schließlich werden die numerischen Ergebnisse der stationären Dynamik, die für anisotrope Einschlüsse erzielt wurden, kritisch mit isotropen analytischen und numerischen Ergebnissen verglichen. Die vorgestellte Dissertation zeigt, dass die Phasenfeldmethoden die wesentliche Physik der oben diskutierten diffusionsgetriebenen Phänomene elegant erfassen können

4D Microstructural Characterization of Electromigration and Thermal Aging Damage in Tin-Rich Solder Joints

abstract: As the microelectronics industry continues to decrease the size of solder joints, each joint will have to carry a greater current density, making atom diffusion due to current flow, electromigration (EM), a problem of ever-increasing severity. The rate of EM damage depends on current density, operating temperature, and the original microstructure of the solder joint, including void volume, grain orientation, and grain size. While numerous studies have investigated the post-mortem effects of EM and have tested a range of current densities and temperatures, none have been able to analyze how the same joint evolves from its initial to final microstructure. This thesis focuses on the study of EM, thermal aging, and thermal cycling in Sn-rich solder joints. Solder joints were either of controlled microstructure and orientation or had trace alloying element additions. Sn grain orientation has been linked to a solder joints’ susceptibility to EM damage, but the precise relationship between orientation and intermetallic (IMC) and void growth has not been deduced. In this research x-ray microtomography was used to nondestructively scan samples and generate 3D reconstructions of both surface and internal features such as interfaces, IMC particles, and voids within a solder joint. Combined with controlled fabrication techniques to create comparable samples and electron backscatter diffraction (EBSD) and energy-dispersive spectroscopy (EDS) analysis for grain orientation and composition analysis, this work shows how grain structure plays a critical role in EM damage and how it differs from damage accrued from thermal effects that occur simultaneously. Unique IMC growth and voiding behaviors are characterized and explained in relation to the solder microstructures that cause their formation and the possible IMC-suppression effects of trace alloying element addition are discussed.Dissertation/ThesisDoctoral Dissertation Materials Science and Engineering 201

On modelling the constitutive and damage behaviour of highly non-linear bio-composites - Mesh sensitivity of the viscoplastic-damage law computations

The large strain fracture of non-linear complex solids concerns a wide range of applications, such as material forming, food oral processing, surgical instrumental penetration as well as more recently, the design of biodegradable composites for packaging and bio-medical use. Although simulations are a powerful tool towards understanding and designing such processes, modelling ductile fracture in materials such as soft natural composites imposes a new challenge, particularly when the fracture patterns cannot be pre-defined. Here we bring to light new information on these aspects of benefit to the multidisciplinary community, by characterising and modelling the deformation and fracture of short cellulose fibre starch extruded composites. Hyperviscoelastic-Mullins damage laws show merits in modelling such complex systems. Yet they are inferior to a viscoplastic-damage law able to capture exactly their highly non-linear, rate dependent and pressure dependent pseudo-plastic stress-strain response. The viscoplastic-damage law also predicts fracture based on experimental toughness values without pre-specifying the crack path in a Finite Element (FE) model, displaying superiority over the conventional cohesive zone approach. Yet, despite using a toughness parameter to drive crack propagation, spurious mesh dependency is still observed while other previously unreported sources of error imposed by the finite element aspect ratio are also highlighted. The latter is rectified by developing a novel numerical strategy for calculating the characteristic element length used in the damage computations. Inherent mesh dependency suggests that non-local damage models may be essential to model this newly investigated class of natural composites

Electromigration behavior and reliability of bamboo Al(Cu) interconnects for integrated circuits

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 1999.Includes bibliographical references (leaves 103-108).by V.T. Srikar.Ph.D

Micro-mechanical characteristics and dimensional change of Cu-Sn interconnects due to growth of interfacial intermetallic compounds

Sn-based solder alloys are extensively used in electronic devices to form interconnects between different components to provide mechanical support and electrical path. The formation of a reliable solder interconnects fundamentally relies on the metallurgic reaction between the molten solder and solid pad metallization in reflowing. The resultant IMC layer at the solder/pad metallization interface can grow continuously during service or aging at an elevated temperature, uplifting the proportion of IMCs in the entire solder joint. However, the essential mechanical properties of interfacial IMC (i.e. Cu6Sn5, Cu3Sn) layers, such as Young s modulus and hardness, are drastically different in comparison with Sn-based solder and substrate. Therefore, the increasing fraction of interfacial IMCs in the solder joint can lead to significant deformation incompatibility under exterior load, which becomes an important reliability concern in the uses of solder joints for electronic interconnects. In the past decades, extensive research works were implemented and reported regarding the growth of interfacial IMC layers and its effect on the mechanical integrity of solder joints. But, the following fundamental issues in terms of mechanical and microstructural evolution in the uses of solder joints still remain unclear, demanding further research to elaborate: (1) The protrusion of IMCs: Though the growth of interfacial IMC layers along the diffusion direction in solder joints were studied extensively, the growth of IMCs perpendicular to the diffusion direction were reported in only a few papers without any further detailed investigation. This phenomena can crucially govern the long-term reliability of solder interconnects, in particular, in the applications that require a robust microstructural integrity from a solder joint. (2) Fracture behaviour of interfacial IMC layers: The fracture behaviour of interfacial IMC layers is a vital factor in determining the failure mechanism of solder joints, but this was scarcely investigated due to numerous challenges to enable a potential in-situ micro-scale tests. It is therefore highly imperative to carry out such study in order to reveal the fracture behaviour of interfacial IMC layers which can eventually provide better understanding of the influence of interfacial IMC layers on the mechanical integrity of solder joints. (3) Volume shrinkage: The volume shrinkage (or solder joint collapse) induced by the growth of interfacial IMC layers was frequently ascribed as one of the main causes of the degradation of mechanical reliability during aging due to the potentially resulted voids and residual stress at the solder/substrate interface. However, very few experimental works on the characterisation of such type of volume shrinkage can be found in literatures, primarily due to the difficulties of observing the small dimensional changes that can be encountered in the course of IMCs growth. (4) Residual stress: The residual stress within solder joints is another key factor that contributes to the failure of solder joints under external loads. However, the stress evolution in solder joints as aging progresses and the potential correlation between the residual stress and the growth of interfacial IMC layers is yet to be fully understood, as stress/strain status can fundamentally alter the course of total failure of a solder joint. (5) Crack initiation and propagation in solder joints: Modelling on the mechanical behaviour of solder joints is often undertaken primarily on the stress distribution within solder joints, for instance, under a given external loading. But there is lack of utilising numerical analysis to simulate the crack initiation and propagation within solder joints, thus the effect of interfacial IMC layers on the fracture behaviour of the solder joints can be elaborated in further details. In this thesis, the growth of interfacial IMCs in parallel and perpendicular to the interdiffusion direction in the Sn99Cu1/Cu solder joints after aging was investigated and followed by observation with SEM, with an intention of correlating the growth of IMCs along these two directions with aging durations based on the measured thickness of IMC layer and height of perpendicular IMCs. The mechanism of the protrusion of IMCs and the mutual effect between the growth of IMCs along these two directions was also discussed. The tensile fracture behaviour of interfacial Cu6Sn5 and Cu3Sn layers at the Sn99Cu1/Cu interface was characterised by implementing cantilever bending tests on micro Cu6Sn5 and Cu3Sn pillars prepared by focused ion beam (FIB). The fracture stress and strain were evaluated by finite element modelling using Abaqus. The tensile fracture mechanism of both Cu6Sn5 and Cu3Sn can then be proposed and discussed based on the observed fracture surface of the micro IMC pillars. The volume shrinkage of solder joints induced by the growth of interfacial IMC layers in parallel to the interdiffusion direction in solder joint was also studied by specifically designed specimens, to enable the collapse of the solder joint to be estimated by surface profiling with Zygo Newview after increased durations of aging. Finite element modelling was also carried out to understand the residual stress potentially induced due to the volume shrinkage. The volume shrinkage in solder joints is likely to be subjected to the constraint from both the attached solder and substrate, which can lead to the build-up of residual stress at the solder/Cu interface. Depth-controlled nanoindentation tests were therefore carried out in the Sn99Cu1 solder, interfacial Cu6Sn5 layer, Cu3Sn layer and Cu with Vickers indenter after aging. The residual stress was then evaluated in the correlation with aging durations, different interlayers and the locations in the solder joint. Finally, finite element models incorporated with factors that may contribute to the failure of solder joints, including microstructure of solder joints, residual stress and the fracture of interfacial IMC, were built using Abaqus to reveal the effect of these factors on the fracture behaviour of solder joints under applied load. The effect of growth of IMC layer during aging on the fracture behaviour was then discussed to provide a better understanding of the degradation of mechanical integrity of solder joints due to aging. The results from this thesis can facilitate the understanding of the influence of interfacial IMC layers on the mechanical behaviour of solder joints due to long-term exposure to high temperatures

Multiphysics phase field modeling of electromigration

With the miniaturization of microelectronic devices, the reliability of solder interconnects is a significant concern. As size is reduced, the current density flowing through an interconnect becomes larger and exacerbates electromigration leading to microstructural changes and failure. Since most solder alloys are required to be lead-free over toxicity concerns, there are additional challenges to interconnect performance. Nearly all solder alloys are comprised with its majority component being tin due to its low melting temperature and economical cost. The typical metallic white tin phase has a body-centered tetragonal crystal structure which exhibits strong anisotropy in its physical properties. In particular, the electrical/thermal conductivity and elastic modulus are highly anisotropic. For this reason, the performance of solder bumps that contain only a few grains are sensitive to the orientation of each individual grain. Quantitative description of electromigration at such scales is required to understand the microstructure behavior impacting performance and degradation of interconnects. Electromigration induced microstructure evolution in solder interconnects involves complicated multiphysical processes. It involves the diffusion of atoms driven by charge conduction which is also strongly affected by concurrent heat conduction and mechanical processes. A multiphysics phase field model is developed to investigate the diffusional processes in tin solder interconnects. The driving forces for electromigration are obtained solving for current density and electric field in microstructures with inhomogeneous and anisotropic electrical conductivity using microscopic Ohm\u27s law. Similarly, the driving forces caused by temperature gradients are obtained solving for heat flux using Fourier\u27s law of conduction that accounts for inhomogeneous and anisotropic thermal conductivity. The model is capable of accounting for the generation of heat through Joule heating. Simulations of conduction driven pore and inclusion migration are discussed in terms of volume and surface diffusion mechanisms. Finally, the contribution of stress and its gradient are obtained through microelasticity modeling. From Hooke\u27s law of elasticity, the modeling allows different external loading conditions to be considered and is capable of solving for internal stress concentrations in microstructures with structural and elastic property mismatches near defects including grain boundaries, voids, and precipitates. These internal stresses generated contribute to the diffusional processes through its gradient

Electrical overstress and electrostatic discharge failure in silicon MOS devices

This thesis presents an experimental and theoretical investigation of electrical failure in MOS structures, with a particular emphasis on short-pulse and ESD failure. It begins with an extensive survey of MOS technology, its failure mechanisms and protection schemes. A program of experimental research on MOS breakdown is then reported, the results of which are used to develop a model of breakdown across a wide spectrum of time scales. This model, in which bulk-oxide electron trapping/emission plays a major role, prohibits the direct use of causal theory over short time-scales, invalidating earlier theories on the subject. The work is extended to ESD stress of both polarities. Negative polarity ESD breakdownis found to be primarily oxide-voltage activated, with no significant dependence on temperature of luminosity. Positive polarity breakdown depends on the rate of surface inversion, dictated by the Si avalanche threshold and/or the generation speed of light-induced carriers. An analytical model, based upon the above theory is developed to predict ESD breakdown over a wide range of conditions. The thesis ends with an experimental and theoretical investigation of the effects of ESD breakdown on device and circuit performance. Breakdown sites are modelled as resistive paths in the oxide, and their distorting effects upon transistor performance are studied. The degradation of a damaged transistor under working stress is observed, giving a deeper insight into the latent hazards of ESD damage