Electromigration has been studied extensively for simple interconnect segments with blocking boundary conditions at both ends. Stress evolution, void nucleation and failure conditions are well established for such structures. The compact model for simple structures is known as Blech length model or Blech effect. In [20], the steady state solution for vacancy/atom
Figure 2.5: Current and stress distribution in two back-to-back connected segments with
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continuity equation was studied by Blech. The result is a compact EM validation criterion which determines the critical product of current density and wire length (jL) for void nucleation [20]. However, these studies were performed for simple via-to-via interconnects and could not be used for multi-segment structures. While extending Blech criterion to complex structures was studied in some papers [36, 38], they have limitation in terms of either capturing all effects or in accuracy. The model proposed in this Chapter computes the stress distribution and captures the electrical and non-electrical effects of all segments connected in a net. The presented compact model can be easily used in CAD tools to replace inaccurate EM failure prediction. This model is also capable of capturing the effect of passive extensions. Application of this model offers design guidance. This model is justified by finite element implementation of vacancy continuity equation and matched with similar experimental results.
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Chapter 3
3
Models Considering Thermal Effects
In this Chapter, we investigate the effect of electrically induced thermal effects on interconnect reliability and aging. We provide an overview of the material transport underlying physics and study all aging processes including electromigration, stress-migration, and thermomigration. We discuss the impact of technology scaling on electro-thermo-mechanical reliability with an emphasis on Joule heating. We propose new models for uniform and non-uniform temperature evolution and its steady state distribution in interconnects. We demonstrate that neglecting thermal effects in modern technologies may lead to incorrect conclusions about interconnect mortality. We introduce new models for reliability assessment capturing thermal, electrical and mechanical requirements simultaneously leading to criteria for accurate temperature-aware mortality assessment of interconnects. The proposed models are employed in many experiments to demonstrate how various temperature profiles affect voiding and hillock formation. These models are verified by comparing the results against finite element experiments. We also provide a detailed explanation of aging in advanced-technology- manufactured local and global wires.
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3.1. Introduction
Device reliability is often nonnegotiable, and is crucial for sensitive applications such as medical, autonomous vehicles, and space craft. Interconnect aging and failures induced by mass transfer has become a key reliability issue due to the scaling of interconnects in modern ICs.
In accurate interconnect reliability and lifetime assessment models, all thermal, electrical, and mechanical aspects of mass transfer need to be simultaneously taken into account. Electric field applied to a wire causes gradual movement of ions due to momentum transfer between conducting electrons and diffusing metal atoms. This phenomenon is called electromigration (EM). EM has been linked to mechanical effects via hydrostatic stress buildup by Blech [25, 20]. Thermal aspects of mass transport induced aging, due to complexity, have not been fully coupled with electro-mechanical models. It has been experimentally observed that aging and failure of interconnects in VLSI chips are temperature-dependent [42, 12, 43, 44] but the existing reliability models do not capture thermal effects correctly. Common approach is to add a non-varying temperature rise Δ𝑇 due to Joule heating obtained from thermal analysis to models used for EM analysis without considering the true mutual relationships of thermal and electro-mechanical effects [45, 5]. This simplistic treatment ignores thermal gradient and thermal migration issues leading to incorrect reliability assessment and catastrophic failures. This is particularly worrying in stressed scenarios such as scaled interconnect in 3D IC devices.
Temperature has three main effects on mass transfer induced aging: (1) during the annealing process, residual stress builds up due to differences in thermal expansion coefficients of materials adjacent to the interconnect line [5, 28, 46]; (2) electromigration is enhanced at
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high temperatures due to faster atomic diffusion [42, 12, 43, 44, 10]; (3) heat transfer creates temperature gradients which themselves cause atomic motion from hot to cool places due to thermomigration (TM) [12, 13, 14, 15].
The first problem (initial built-in stress) occurs prior to device use. Residual stress may cause atomic motion before EM and TM during circuit operation. Initial stress is generated
during the system cooling process from the stress free annealing temperature 𝑇 down to
the use temperature 𝑇. This problem is relatively straight forward and has been investigated by many researchers [47]. A primary source of this thermally induced residual stress, 𝜎 , is the difference in the coefficients of thermal expansion of the metal, 𝛼 , and confinement 𝛼 ,
modeled by 𝜎 = 𝐾(𝛼 − 𝛼 )(𝑇 − 𝑇) where 𝐾 is the modulus of elasticity. The atomic
motion due to stress gradients is called stress-migration (SM) [5, 28].
The second problem (thermally enhanced EM) is significant since atomic diffusion is very sensitive to temperature. Temperature-varying diffusion lifetime analysis is required for including the thermal effects in defect nucleation or growth [48].
The third problem (thermomigration) relates heat flow to atomic flow. Temperature gradients created by Joule heating cause atoms to move from hot to cool places. Complete description of mass flow requires considering Joule heating, electromigration, and stress migration, together [49].
This Chapter investigates thermomigration under steady state conditions to obtain temperature-dependent mortality criteria. The main contributions are: Section 2: (a) reviews existing methodologies combining thermal effects and electro-mechanical effects, (b) reviews Blech’s model and shows the inaccuracy of simplistic methods. Section 3: (a) briefly revisits
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material transport underlying physics, (b) presents new models for reliability assessment including thermal effects and (c) introduces new criteria capturing both thermal requirements and EM. Section 4: (a) discusses the effect of technology scaling on electro-thermo-mechanical reliability including thermal effects and (b) provides a detailed explanation of aging processes in local and global wires manufactured in advanced technologies. Section 5: (a) presents experimental results for various benchmarks to validate the proposed models, (b) presents a discussion how various temperature profiles affect voiding and hillock formation. Chapter 8 provides a detailed derivation of the new models for temperature evolution and its non-uniform distributions.
To our best knowledge, these are the first compact models to analyze the aging of interconnects including EM, SM and TM effects. The new models explain several experimental observations (e.g. unexpected mortality or immortality) that were inexplicable before.