Ion implantation has been the dominant doping technique for silicon integrated circuits (ICs) and most other semiconductors for the past 45 years. It is expected to retain this position of dominance for the foreseeable future. In this process, dopant ions are accelerated to 0.1–1000 keV of energy and smashed into a crystalline semiconductor substrate, creating a cascade of damage that may displace hundreds or thousands of lattice atoms for each implanted ion. In this chapter, we will seek to understand how such an energetic and violent technique has become the dominant and preferred method of doping semiconductor wafers in manufacturing. At first glance, it seems that the technique would not be of much use in the precise art of fabricating integrated circuits. Indeed, although the original patent for ion implantation was issued to William Shockley in 1954, it was not until the late 1970s that ion implantation was used in manufacturing.

If workers from one of today’s multi-billion-dollar integrated circuit (IC) manufacturing plants were suddenly transported to a 1960s semiconductor plant, they would likely be amazed that chips could be successfully manufactured in such a place. Such factories were “dirty” by today’s standards, and wafer cleaning procedures were poorly understood. Of course, chips were manufacturable even in those days, but they were very small and contained very few components by today’s standards. Since defects on a chip tend to reduce yields (fraction of good chips on a wafer) exponentially as chip size increases, small chips can be manufactured with a yield greater than zero even in quite dirty environments. However, all of the progress that has been made in the past six decades in shrinking device sizes and designing very complex chips would have been for naught if similar advances had not been made in manufacturing capability, especially in defect density.

In this chapter, 2D Computational Grains (CGs) with elastic inclusions or voids and 3D CGs with spherical/ellipsoidal inclusions/voids or without inclusions/voids are developed for micromechanical modeling of composite and porous materials. A compatible displacement field is assumed along the outer boundary of each CG. Independent displacement fields in the CG are assumed as characteristic-length-scaled T-Trefftz trial functions. Muskhelishvili’s complex functions are used for 2D CGs, and Papkovich-Neuber solutions are used for 3D CGs to construct the T-Trefftz trial displacement fields. The Papkovich-Neuber potentials are linear combinations of spherical/ellipsoidal harmonics. To develop CG stiffness matrices, multi-field boundary variational principles are used to enforce all the conditions in a variational sense. Through numerical examples, we demonstrate that the CGs developed in this chapter can estimate the overall material properties of heterogeneous materials, and compute the microscopic stress distributions quite accurately, and the time needed for computing each SERVE is far less than that for the finite element method.

The preprocessing of Computational Grains (CGs) is introduced in Chapter 3, and several types of CGs have been developed for the micromechanical modeling of different kinds of composites with particulates, fibers, and so on in Chapters 5–11. A multi-scale analysis framework of composite structures by using the CGs and the standard FEM is developed in this chapter, based on the homogenization of composite materials at the microlevel, and slender or shell structures at the meso- and macro-levels. The specific process of the multi-scale algorithm is illustrated with an example of a stiffened composite panel. The results show the multi-scale analysis method is an accurate and efficient tool for large composite structures, not only simulating the overall structural responses in a bottom-up fashion, but also obtaining the detailed stresses at multiple scales in the dehomogenization process.

In this chapter, a new type of Computational Grains is proposed to study the micro-electro-mechanical behavior of composite piezoelectric materials. This method is based on a new hybrid variational principle, and independently assumed displacements and electric potentials in each CG. Each CG can efficiently model a single physical grain of the composite material, thus saving a significant time of generating complex FEM mesh. Computational Grains can also model porous and composite piezoelectric materials even if the distribution of voids/inclusions is not symmetrical (which is assumption used with all unit cell models). Because the trial solutions are complete but do not satisfy the governing differential equations a priori, the formulation is very simple, and can accurately account for the local field concentrations efficiently and accurately. This is illustrated using different examples where the fields along the void/inclusion periphery are calculated, and the effective material properties of porous and composite materials are predicted, and compared with other analytical and computational models. The proposed CG in this chapter is expected to become a very powerful tool of direct numerical simulations of the micro/meso mechanics of composite piezoelectric materials, and can possibly lead to efficient multi-scale modeling of piezoelectric devices.

In this chapter, Computational Grains are developed for the direct micromechanical modeling of heterogeneous materials reinforced with coated particulate inclusions. Each CG is treated as a three-phase particle/coating/matrix grain, wherein the exact internal displacement field is assumed in terms of the P-N solutions that are further represented by the spherical harmonics. The Computational Grain program generates accurate homogenized moduli as well as exact local interphase stress distributions, with good agreement to the very fine-mesh FE technique and the CSA (Composite Sphere Assemblage) model. The effects of the material properties as well as the thickness of the coating system on the effective properties and localized stress concentrations are also examined for the CGs, where the former parameters play more important roles than the latter ones in altering the response of composite materials. Finally, a simpler implementation of periodic boundary conditions on the SERVEs is developed through the surface-to-surface constraints of the displacement field on the opposite faces. The developed CGs provide accurate and efficient computational tools in the direct modeling of the micromechanical behavior of the particulate composites reinforced with coatings/interphases, which cannot be easily accomplished by the off-the-shelf FE packages and classical models.

This chapter provides a very brief summary of the types of heterogeneous materials considered in this monograph: fiber-reinforced composites, particulate composites, nanocomposites, porous composites, and so on. A succinct summary is given of analytical homogenization methods to determine the overall properties of particulate composites based on the upper and lower bounds of Hashin and Shtrikman; the Eshelby ellipsoidal inclusion theory and the Self-consistent Method of Eshelby; and the Mori-Tanaka Method and some other semi-analytical methods. Numerical methods such as the finite element method, the boundary element method, XFEM, and so on to model a representative volume element (RVE) of a heterogeneous material are reviewed, and thus the motivation for the Computational Grains method discussed in the rest of this book is presented.

This chapter discusses the role of a representative volume element (RVE) in the computational homogenization of heterogeneous materials. The use of the finite element method in modeling an RVE is discussed. The role of using the Hill-Mandel boundary conditions, and the use of periodic displacement and aperiodic traction boundary conditions on an RVE are discussed. The advantages of using the present Computational Grains method in modeling an RVE, not only to determine the macro properties of a heterogeneous material but also to determine the detailed interfacial stress states which are damage precursors at the micro level are discussed.

In this chapter, a Computational Grain is developed for direct numerical modelling of composites with nanoscale inclusions considering both interface stretching and bending effects, using a large number of CGs in a representative volume element. The CGs developed in this chapter are by far the first and the only numerical tool for direct numerical modelling of nanocomposites with a large number of inclusions with Steigman-Ogden matrix/inclusion interfaces. By using a new boundary-type multi-field variational principle together with Papkovich-Neuber potentials, the stiffness matrices of CGs can be directly evaluated and assembled. Together with the parallel algorithms, it is found that very efficient simulations of nanocomposites can be realized, for example, a SERVE containing 10,000 nano inclusions only takes fifty minutes on a sixteen-core computer. The influence of spatial distributions of the nano inclusions on the overall properties of nanocomposites is also investigated in this chapter. We also study the influence of interface bending resistance parameters on the effective modulus of nanocomposites. Numerical results show that interface bending resistance parameters affect the shear modulus of nanocomposites but their effect on the bulk modulus is negligible.

In this chapter, Trefftz trial functions which satisfy identically all the governing equations of linear elasticity in 2D and 3D problems are summarized. These Trefftz functions are later used in conjunction with boundary variational principles (since all the field equations are satisfied identically inside the Voronoi cell elements), to construct planar and 3D Computational Grains to directly model statistically equivalent representative volume elements (SERVEs) of heterogeneous materials at the microscale. In as much as the Trefftz functions are used as trial solutions, this modeling captures the correct and accurate stress solutions in the matrix, inclusions, and at their interfaces. The presented Trefftz solutions include: (1) Muskhelishvili’s complex functions for 2D problems,(2) Papkovich-Neubar solutions for 3 D problems,and (3) Harmonic functions in spherical coordinates, cylindrical coordinates, and ellipsoidal coordinates.