Dependence Of Effective Properties Research Articles

(Invited, Digital Presentation) Advanced Methodology for Assessing Chip Package Interaction Induced Stress Effects on Chip Performance and Reliability

In IC industry, the use of multiple die stack packaging has emerged to meet the increasing demand in miniaturization and improved functionality of mobile devices. During chip operation, transistor power dissipation raises temperature unevenly across a die. The generated thermal hotspots negatively impact reliability and degrade performance. In mechanical aspects, dies become thinner, and bumps and pitch become smaller, which makes heat dissipation more difficult, and lead to increase in mechanical stress. Such stress may cause carrier mobility degradation for transistors and could lead to parametric circuit failure. In the back-end-of-line (BEoL) interconnects, the employed ultra-low k materials prone to damage interconnects when mechanical stress is present, due to its brittle nature and poor adhesion to the barrier materials. These stresses originated at the die packaging step due to thermal mismatch between die and package materials, which is termed as chip package interaction (CPI). We call mechanical CPI (mCPI) when such stress affects reliability of the whole chip, i.e., BEoL, RDL (redistribution layer), bump, or TSVs (through silicon vias). When such stress affects device performance, we call electrical CPI (eCPI).To analyze CPI effects on a feature scale, i.e., in transistor channel or in the individual metal line or ILD (inter layer dielectric) /IMD (inter metal dielectric) gap, an analysis tool must generate accurate feature-scale stress variation across a die. Finite element analysis (FEA) is widely used for analyzing CPI induced problems. However, the traditional FEA cannot effectively handle feature-scale geometries due to huge memory consumption, and instead, treats a die as a uniform material block. Therefore, this approach cannot describe stress distribution caused by local non-uniformity of metal line distribution and fail to provide the needed accuracy for feature-scale analysis. [1]Here, we present an advanced physics-based EDA tool that overcomes the above-mentioned problems by introducing the novel methodology of extracting effective anisotropic thermal-mechanical properties (EMP), as well as employing FEA-based multi-scale simulation procedure. Prior to running FEA, the tool extracts EMP that accurately represent non-uniformities at different scales within a simulation domain. Here, each metal layer in a die is considered a binary system that consists of metal inclusions embedded in an insulator matrix. By dividing the die area into bins, metal density dependent effective properties for each bin are calculated according to theory of anisotropic composite materials. Anisotropy of properties can be obtained by taking routing direction of metal lines into account [2, 3]. EMP can adjust to multi-scale by varying bin size as shown in Fig.1. Here, Young’s modulus is extracted globally with coarse grid, and on sub-modeling region with very fine grid, which shows the corresponding property variation with much finer scale. Since EMP constructs no actual geometrical objects, the methodology can efficiently handle feature-scale objects on a large layout region. When a user selects a circuit block, or a region to be analyzed in detail, the automated tool flow enables two step stress simulation procedure, which is schematically shown in Fig. 2. First, the global-scale stress simulation is performed with coarse both the simulation mesh and EMP bin and extracts the boundary displacements for the circuit block. These boundary displacements are employed in the sub-modeling, with employed fine mesh and EMP bin.Figure 3 demonstrates the importance of EMP for accurate resolution of stress field. The 2D color maps show the x-component of stress distributions in a circuit block as a result of sub-modeling. Here, die BEoL is represented by EMP in (a), while in (b), the entire die including BEoL is represented by silicon, which is employed in traditional FEA. The stress pattern due to interconnect layout details are visible only when EMP is employed. The difference is even more pronounced when 1D stress profile is compared.By back annotating the obtained stress components in a SPICE netlist, the tool enables a user to perform accurate circuit simulation with accounted CPI effects. In eCPI analysis, the tool has been validated by employing measurements of different types of devices [4].The additional tool capabilities that will be presented are mCPI analysis and thermomechanical stress analysis during chip operating conditions.[1] R. Radojcic, More-than-Moore 2.5D and 3D SiP Integration, Springer, 2017.[2] V. Sukharev et al. J. Electron Test, vol. 28, pp. 63-72, 2012[3] V. Sukharev et al., Proc. Int. 3D Systems Integration Conference, 2019[4]. A. Kteyan, et al. Proc. ISPD 2022 Figure 1

Mechanical properties characterization and zero Poisson’s ratio design for perforated auxetic metamaterial by computational homogenized method

Metamaterial with elliptical perforations is characterized by a distinctive computational homogenized method. Two kinds of periodic displacement constraints representing tension and shear deformations, respectively, are applied to the unit cell. Then, the relationship of the undetermined displacement constraints in the micromechanical models and the assumed stress conditions in the equivalent orthotropic medium is derived to determine the effective properties. The method is validated by comparing to the experimental and FEM results. Next, the dependence of effective properties on microstructure parameters and base material composition is investigated. Correspondingly, the zero Poisson’s ratio design criterion is formulated to guide the metamaterial design.

Global sensitivity analysis of uncertain parameters based on 2D modeling of solid oxide fuel cell

In this study, we propose an enhanced quasi–two-dimensional and nonisothermal model for solid oxide fuel cell (SOFC) parametric simulation and optimization. The dependence of effective properties on microstructural parameters is fully considered in this model. Besides, an elementary effect (EE) approach based on Monte Carlo experiments is adopted to comprehensively evaluate the sensitivity of totally 24 parameters. A two sample Kolmogorov-Smirnov (K-S) test is carried out to evaluate the ability of EE method for robust and accurate sensitivity analysis. The investigation focuses on the important microstructural parameters of the composite anode/cathode function layers (AFL/CFL). With this research, relative volume fractions of conducting materials in the AFL/CFL are the most sensitive factors among all input parameters while particle radius is found to be the least sensitive microstructural parameters. The particle size ratio of electronic particles to ionic particles is found to be much more sensitive than particle size due to its significant effect on effective conductivity. The cathodic electrochemical parameters reflect cell performance more significantly than the anodic ones. To further elucidate the role of input factors, this study provides a principle for parametric sensitivity classification as well. Besides, impacts of current density variation on parametric sensitivity are comprehensively considered. Negative or positive effects of parameters on cell performance and their influencing mechanisms are also discussed. Furthermore, global sensitivity analysis for single parameters at different positions is performed to assess the probability of structural optimization along the channel length. Then, a feasible nonuniform distribution method in allusion to function layers is proposed for further improvement of cell performance according to global SA results of single parameters along the channel direction.

Computational homogenisation of acoustic metafoams

Acoustic metafoams are novel materials recently proposed for low frequency sound attenuation. The design of their microstructure is based on the combination of standard acoustic foams with locally resonant acoustic metamaterials. This results in improved sound attenuation properties due to the interaction between viscothermal dissipation effects and the local resonance effects at the pore level. In this paper, the non-standard behaviour of such a metafoam with a complex two-phase microstructure is analysed through a multiscale approach. The macroscopic problem is described by general balance equations and at the microscopic scale a detailed representation of the microstructure is considered. The frequency dependent effective properties are used to explain the extraordinary acoustic performance. The homogenisation approach is also validated using direct numerical simulations, showing that the homogenisation technique is adequate in modelling both viscothermal dissipation and the local resonance effect within the metafoam microstructure.

Predicting elastic anisotropy of dual-phase steels based on crystal mechanics and microstructure

This work utilizes mean-field self-consistent and full-field fast Fourier transform-based homogenizations to study the effective elastic behavior of several steels: three dual-phase (DP), DP 590, DP 980, and DP 1180, and one martensitic (MS), MS 1700. Crystallographic textures and phase fractions of these steels are characterized using electron microscopy along with electron-backscattered diffraction to initialize the models. A comprehensive set of Young's modulus and Poisson's ratio data, measured at the ambient temperature as a function of orientation with respect to the rolling direction for each steel sheet, is used to calibrate and validate the models. The calibration of the models enabled us to estimate the single crystal elastic constants for both the martensitic phase and ferrite, while calculating the orientation dependent effective properties. Half of the data was used in the calibration. Subsequent predictions of the orientation dependent effective elastic properties for the remaining data verified that the estimated single crystal properties are reliable. As the steels exhibit a different level of anisotropy in their effective behavior, good predictions allowed us to discuss the role of texture, grain structure, phase fraction and distribution on the effective properties. The results of this work represent a significant incentive to introduce elastic anisotropy in numerical tools for simulating metal forming processes of dual-phase steels, in particular those processes involving springback, using the texture informed crystal mechanics-based models to more accurately estimate the effective elastic properties required by such simulations.

Straightforward mathematical modeling of the time-dependent behavior of non-ageing viscoelastic composites with multi-coated inclusions

In this paper a straightforward mathematical approach in time domain is elaborated to predict the effective time behavior of non ageing viscoelastic composites with multi-coated inclusions. The developed modeling stands on the dynamic Green’s tensor, integral equations in Stieltjes space and the interfacial operators. The Mori-Tanaka mean field approach is considered. Straightforward expressions of the effective behavior are obtained in time domain for various coating types. Based on the predicted time dependent effective properties, the overall response of the composite is investigated for uniaxial and multi-axial time dependent loadings. Various numerical results are presented versus volume fraction, shape of inclusions and the coating thickness.

Viscoelectroelastic closed form models for frequency and time dependent effective properties of reinforced viscoelectroelastic composites

In this paper, closed form models of time and frequency dependent effective properties of reinforced viscoelectroelastic composites are elaborated. Based on the correspondence principle of linear viscoelectroelasticity and the Mori-Tanaka micromechanical mean field approach, the effective properties are derived in the Laplace-Carson domain and then inverted numerically to the time domain. New analytical models for decoupled and coupled effective properties of transversely isotropic viscoelectroelastic materials are elaborated using compact matrix formulations. Explicit relationships are presented for different effective viscoelectroelastic moduli of piezoelectric materials with various inclusions shapes. Various viscoelastic time dependent models for the phases can be used. Only few Eshelby tensor coefficients are required to be analytically or numerically computed. Results based on the elaborated effective viscoelectroelastic models are presented for various types of piezoelectric inclusions and compared to the available predictions. These models are suitable for designers to predict composite viscoelectroelastic moduli of heterogeneous materials with optimized active and passive characteristics.

Mathematical modeling of the overall time-dependent behavior of non-ageing viscoelastic reinforced composites

New mathematical and numerical formulations are developed for effective time-dependent non ageing viscoelastic behavior of linear viscoelastic composites. The modeling is based on the dynamic Green’s functions, integral equations, and Volterra product. The time concentration tensor is derived through numerical solution of the integral equation and the Mori–Tanaka micromechanical model. The developed modeling gives explicit expressions of effective properties through convolution products in the Stieltjes space. A numerical procedure is developed allowing the calculation of the tensoriel convolution product and its inverse. Based on Carson–Laplace transform, the frequency dependent effective properties are also obtained and converted to the time domain. For viscoelastic Prony laws and inclusions with various shapes, numerical results are presented and compared with the approach based on the Laplace transform and the correspondence principle.

Finite element modelling on time dependent ferroelectric behaviour of 1–3 piezocomposites

Experiments are conducted to measure the effective properties and also the time dependent behaviour of 1–3 piezocomposites and bulk piezoceramics subjected to cyclic electric field at room temperature under quasi static loading condition in the non-linear regime. Experimental dielectric and butterfly hysteresis show that 1–3 piezocomposite exhibits the time dependent behaviour. Hence a viscoelastic based numerical model has been proposed to predict the time dependent effective properties of 1–3 piezocomposites and bulk piezoceramics. The predicted effective properties are incorporated in the proposed 3D finite element based micromechanical model to predict the time dependent non-linear electromechanical behaviour of 1–3 piezocomposites and compared with experimental observations. The experimental and simulated results show that the presence of epoxy polymer constituent in 1–3 piezocomposite has an influence on the non-linear time dependent electromechanical response of 1–3 piezocomposite.

Viscoelastic modeling and experimental characterization of thermo-electromechanical response of 1–3 piezocomposites

The viscoelastic behaviour of 1–3 piezocomposites with different volume fractions and bulk piezoceramics is studied under thermal environment subjected to electrical loading. Experiments are conducted to measure the temperature dependent effective properties based on IEEE standards and time dependent thermo-electromechanical response of 1–3 piezocomposites and piezoceramic subjected to high cyclic electrical loading under elevated thermal environment. The temperature dependent effective properties are predicted using the proposed numerical model based on unit cell approach and implemented into the viscoelastic model to predict the time dependent thermo-electro-elastic effective properties. The predicted effective properties are incorporated in a finite element based 3-D micromechanical model to predict the time dependent thermo-electro-mechanical performance behaviour of 1–3 piezocomposites. Comparison between the experiments and simulations shows that this model can reproduce the characteristics of time dependent non-linear electromechanical response under thermal environment. It is observed that the variation in fiber volume fraction and elevated operating temperature has a significant influence on the response of the 1–3 piezocomposites.

On the limit and applicability of dynamic homogenization

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. We also study the effect of the interface location on the applicability of homogenization. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction.

Modeling and experimental characterization on temperature-dependent ferroelastic switching of 1–3 type piezocomposites

Under high mechanical loading, 1–3 piezocomposites experience ferroelastic switching that causes depolarization. The combination of thermal and mechanical loads may affect the performance of 1–3 piezocomposites. A theoretical and experimental analysis has been carried out to study the ferroelastic behaviour of 1–3 piezocomposites under thermal load. Temperature dependent effective properties are determined analytically using equivalent layered approach and compared with the experimental values measured based on resonance technique. Experiments are conducted on poled piezocomposite samples subjected to large compressive stress, parallel to the fiber (poling) direction at elevated temperatures. A thermodynamically consistent uni-axial model is proposed to simulate the macroscopic behaviour of ferroelastic switching on piezocomposites. Volume fractions of three distinct uni-axial variants (instead of six variants) are used as internal variables, which describes the distribution of crystallographic orientation and polarization state of the crystal. Non-linear hardening parameter, based on gaussian function, is introduced to incorporate the grain boundary effects. The predicted effective properties are used in the uni-axial model and the classical stress versus strain and stress versus electrical displacement curves are simulated. Comparison between the experiments and simulations show that the proposed model can reproduce the characteristics of non-linear response. It is observed that the fiber volume fraction and temperature has a strong influence on the ferroelastic response of 1–3 piezocomposites.

Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites

ABSTRACTCentral to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Effective properties of terahertz double split-ring resonators at oblique incidence

By combining experimental and theoretical efforts we study the optical properties of single-layer, double split-ring resonators at oblique incidences for terahertz frequencies and reveal their angular dependent effective properties. Contrary to what is usually anticipated, for incidence angles deviating from normal we do not observe any significant dispersion in the effective permeability. Effective properties are dominated instead by a strong dispersion in the effective permittivity, causing also the wavenumber in the medium to be large. The structure is shown to possess almost invariant effective properties for oblique incidences.

Generalization of the multiparticle effective field method to random structure matrix composites

In this paper linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the generalization of the ``multiparticle effective field'' method (MEFM, see [7] for references), previously proposed for the estimation of stress field averages in the phases. The refined version of the MEFM takes into account both the variation of the effective fields acting on each pair of fibers and inhomogeneity of statistical average of stresses inside the inclusions. One considers in detail the connection of the method proposed with numerous related methods. The explicit representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to the homogeneous loading at infinity. Just with some additional assumptions (such as an effective field hypothesis) the involved tensors can be expressed through the Green's function, Eshelby tensor and external Eshelby tensor. The dependence of effective properties and stress concentrator factors on the radial distribution function of the inclusion locations is analyzed.