Research

Semiconductor Research Corporation (SRC) Center for Heterogeneous Integration Research on Packaging (CHIRP)

Prof. Subbarayan Co-Directs, along with Prof. B.G. Sammakia of SUNY Binghamton,  the $4.5 Million Purdue-Binghamton SRC Center for Heterogeneous Integration Research in Packaging. There are currently 15 projects related to heterogeneous integration ongoing at the center. More details below.

Summary

Prof. Subbarayan’s research broadly spans the continuum thermodynamic theory for phase interfaces, computational techniques for moving boundary problems such as crack initiation/propagation or phase nucleation/evolution, and their practical applications to modeling and experimentally characterizing the failure of microelectronic devices and assemblies. 

2017-3-6 A Pursuit of Geometry-Centric Solutions to Engineering Problems -  Purdue University Celebration of Faculty Careers Talk

At its core, his research utilizes principles of continuum mechanics, computational and experimental solid mechanics, geometrical modeling, and numerical analysis. Selected examples from this research are described below:

1. Computational Mechanics: Isogeometric Analysis (IGA)

We were among the first to use geometric models, based on Non-Uniform Rational B-Splines or NURBS, directly for analysis eliminating the meshing step needed in finite element analysis [1,2]. The use of geometric models for analysis is commonly referred to, as Isogeometric Analysis (IGA, [3]), and is a popular strategy for directly integrating CAD and CAE without needing an intervening mesh.

A challenging problem in IGA is stitching together, parametrically described spline patches. We developed a parametric stitching, or p-stitching, procedure for coupling patches with assured, arbitrary smoothness at the interface between the subdomains, including problems with sharp changes in gradient, as at dissimilar material interfaces [4]. The coupling procedure relies on Enriched Isogeometric Analysis (EIGA) previously developed by our group in 2012 [5].

Figure 1: Isoparametric descriptions of the NURBS geometrical surface and behavior for predictions of optimal shape of a droplet constrained between a cylindrical surface and a circular pad [1].
 

Automated crack initiation and propagation along arbitrary paths across interfaces in multilayer materials was first demonstrated by our group in 2012 [6]. In the figure below, simulations of arbitrary crack initiation and propagation in micron-scale semiconductor chip dielectric stacks using enriched isogeometric analysis is illustrated [7]. We later used EIGA for singular as well as discontinuous enrichments [8] based on asymptotic analysis of singularities in multi-material wedge corners and crack tips [9]. 

Figure 2: Simulations of crack initiation and propagation in chip-dielectric stacks during assembly [7]. The inset SEM image shows a representative stack. The plot on the left shows the results of the asymptotic analysis [9] that was used to identify the critical corner in the stack where crack might originate. The contour plot on the right shows the numerically simulated initiation site and crack path assuming a cohesive damage description. The stack is approximately a few microns in size and width. The asymptotic analysis correctly identified the potential critical corners for crack initiation.

2. CAD: Algebraic Level Sets and Implicit Boolean Compositions

EIGA, mentioned earlier, utilizes measures of distance (algebraic level sets) constructed from the parametric enrichments; it explicitly preserves the CAD geometry of the interface allowing ease in calculating the curvature and normal that drive the interface motion under physical forces. The algebraic level sets are constructed from parametric boundaries using the theory of resultants of algebraic equations [10,11]. The algebraic level sets enable one to avoid the vexing CAD problem of needing to numerically calculate the intersection between complex three-dimensional parametric surfaces and stitching the surfaces together to form the common B-rep solid model (Figure 3).

Furthermore, a rigorous technique was developed by our group to recover implicitizations of parametric surfaces with trivially singular Dixon resultants and to carryout behavioral analysis of complex intersecting closed regions bounded by parametric surfaces without surface-surface intersection calculations [12].

Figure 3: Algebraic level sets constructed from low-degree parametric surfaces of primitive geometric entities (left) composed using R-functions to solve the point-containment problem of complex B-rep CAD models (right) [11]. The purely algebraic procedure enables one to make inside/outside decisions of a given point relative to the solid model. Furthermore, the procedure provides a robust estimate of the proximity (distance fields) to the boundary, which is very useful for behavioral enrichment. 

3. Continuum Thermodynamics of Interfaces

Interfacial phenomena are of critical importance in many fields, however the conditions governing phase evolution at the interface are often unclear. We derived the thermodynamic configurational force associated with a moving interface with multiple diffusing species and arbitrary surface stress [13], inspired by the work of Truesdell and Toupin [14] and Gurtin [15]. The mass, momentum, energy balance as well as the second law condition were derived on the evolving phase interface subject to mechanical loads, heat and multiple diffusing species. The derived second law condition naturally extends the Eshelby energy–momentum tensor associated with moving cracks to include species diffusion terms (Eq. (43) in [13]), and naturally yields the interface equivalents of Fick’s (Eq. (41)) and Fourier’s laws (Eq. (42)). The second law restriction was then used to derive the condition for the growth and nucleation of new phases in a body undergoing finite deformation subject to inhomogeneous as well as anisotropic interface stress, and multiple diffusing species.

Figure 4: Explicit interface EIGA simulations of void and Cu6Sn5 intermetallic compound (IMC) growth in solder microbumps [17] (a) experimental observations of void growth after 1000 hours at 150 oC (b) initial mass concentrations of Sn in the model (c) mass concentrations of Sn after 200 hours at 150 oC (d) electric potential in the domain and void evolution after 200 hours at 150 oC under a current density of 𝟑 × 𝟏𝟎𝟓 𝑨/𝒄𝒎𝟐. The model accurately captures diffusion dominated IMC growth as well as the greater IMC growth at anode with addition of current.

 

We also posed a configurational optimization problem and derived the sensitivity of an arbitrary objective to arbitrary motions of one or more finite-sized heterogeneities inserted into a homogeneous domain [18]. In the derivation, they naturally obtained the definition of a generalized Eshelby energy-momentum tensor for arbitrary objectives and expressed the sensitivities as surface integrals with jump terms across the heterogeneity boundaries that vanished on homogeneous domains yielding generalized conservation laws for arbitrary objectives. They then showed that the specific path- independent forms of the sensitivity of the objective to arbitrary translation, rotation or scaling of the inserted heterogeneities naturally yielded the J-, L- and M-integrals of fracture mechanics when the objective is strain energy. The theory was computationally implemented using EIGA to optimally identify best/worst-case locations for line cracks that are inserted into the domain as well as to optimally mitigate the risk of fracture due to a crack at its worst-case location by sequentially inserting and optimizing the configurations of circular/elliptical stiff/soft inclusions (Figure 5). The simulations required singular as well as discontinuous enrichments mentioned earlier for modeling the cracks [8].

Figure 5: Optimal location obtained by translating a line crack to minimize the strain energy of the plate while holding the configuration of the voids and inclusions fixed (left). Optimal configuration of four heterogeneities to minimize the strain energy of the plate with a line crack at its worst location (right) [18]. Simulations required singular as well as discontinuous isogeometric enrichments [8].

4. Multiphysics Modeling and Experimental Characterization of Failure in Microelectronic Devices and Assemblies

Solder joints are ubiquitous in microelectronic assemblies. They operate at high homologous temperatures, experience significant rate-dependent plastic deformation, and develop fatigue cracks that are large relative to the size of the joint. A significant effort in the Subbarayan group is to characterize the constitutive and fatigue behavior of a wide variety of solder alloys under a wide range of strain rates (Figure 6, [19,20,21]).