Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model

Pattern Formation under Deep Supercooling by Classical Density Functional-Based Approach

Solidification patterns during nonequilibrium crystallization are among the most important microstructures in the natural and technical realms. In this work, we investigate the crystal growth in deeply supercooled liquid using the classical density functional-based approaches. Our result shows that the complex amplitude expanded phase-field crystal (APFC) model containing the vacancy nonequilibrium effects proposed by us could naturally reproduce the growth front nucleation (GFN) and various nonequilibrium patterns, including the faceted growth, spherulite, symmetric and nonsymmetric dendrites among others, at the atom level. Moreover, an extraordinary microscopic columnar-to-equiaxed transition is uncovered, which is found to depend on the seed spacing and distribution. Such a phenomenon could be attributed to the combined effects of the long-wave and short-wave elastic interactions. Particularly, the columnar growth could also be predicted by an APFC model containing inertia effects, but the lattice defect type in the growing crystal is different due to the different types of short-wave interactions. Two stages are identified during the crystal growth under different undercooling, corresponding to diffusion-controlled growth and GFN-dominated growth, respectively. However, compared with the second stage, the first stage becomes too short to be noticed under the high undercooling. The distinct feature of the second stage is the dramatic increments of lattice defects, which explains the amorphous nucleation precursor in the supercooled liquid. The transition time between the two stages at different undercooling is investigated. Crystal growth of BCC structure further confirms our conclusions.

Simulation of TSV Protrusion in 3DIC Integration by Directly Loading on Coarse-Grained Phase-Field Crystal Model

As thermal management in 3DIC integration becomes increasingly important in advanced semiconductor node processes, novel experimental and modeling approaches are in great demand to reveal the critical material issues involving multiscale microstructures that govern the behavior of through-silicon-via (TSV) protrusion. Here, a coarse-grained phase-field crystal model properly coupled with mechanics through the atomic density field is used to simulate the formation of polycrystalline structures and protrusion of nano-TSVs from the atomic scale. TSVs with different grain structures are directly loaded, and protrusion/intrusion profiles are obtained along with displacement, stress, and strain fields. Thermodynamic driving forces from external loadings and the mismatch of Young’s modulus between adjoining grains as well as detailed displacement and strain distributions are ascribed to control the complex deformation in TSVs. TSVs with sizes up to around 30 nm and an aspect ratio of 4 are successfully investigated, and a further increase in the size and aspect ratio to cover the micrometer range is feasible, which lays down a solid basis toward a multiscale material database for simulation inputs to the design of TSV-based 3DIC integration and relevant electronic design automation (EDA) tools.

Mesoscale Defect Motion in Binary Systems: Effects of Compositional Strain and Cottrell Atmospheres

The velocity of dislocations is derived analytically to incorporate and predict the intriguing effects induced by the preferential solute segregation and Cottrell atmospheres in both two-dimensional and three-dimensional binary systems of various crystalline symmetries. The corresponding mesoscopic description of defect dynamics is constructed through the amplitude formulation of the phase-field crystal model, which has been shown to accurately capture elasticity and plasticity in a wide variety of systems. Modifications of the Peach-Koehler force as a result of solute concentration variations and compositional stresses are presented, leading to interesting new predictions of defect motion due to effects of Cottrell atmospheres. These include the deflection of dislocation glide paths, the variation of climb speed and direction, and the change or prevention of defect annihilation, all of which play an important role in determining the fundamental behaviors of complex defect network and dynamics. The analytic results are verified by numerical simulations.

Traveling waves of the solidification and melting of cubic crystal lattices

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

Enabling simulations of grains within a full rotation range in amplitude expansion of the phase-field crystal model

This paper introduces improvements to the amplitude expansion of the phase-field crystal model that enable the simulation of grains within a full range of orientations. The unphysical grain boundary between grains, rotated by a crystal's symmetry rotation, is removed using a combination of the auxiliary rotation field described in our previous work and an algorithm that correctly matches the complex amplitudes according to the differences in local rotation.