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.

Efficient Fully Discrete Finite-Element Numerical Scheme with Second-Order Temporal Accuracy for the Phase-Field Crystal Model

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically.

Multiscale analysis of crystalline defect formation in rapid solidification of pure aluminium and aluminium–copper alloys

Rapid solidification leads to unique microstructural features, where a less studied topic is the formation of various crystalline defects, including high dislocation densities, as well as gradients and splitting of the crystalline orientation. As these defects critically affect the material’s mechanical properties and performance features, it is important to understand the defect formation mechanisms, and how they depend on the solidification conditions and alloying. To illuminate the formation mechanisms of the rapid solidification induced crystalline defects, we conduct a multiscale modelling analysis consisting of bond-order potential-based molecular dynamics (MD), phase field crystal-based amplitude expansion simulations, and sequentially coupled phase field–crystal plasticity simulations. The resulting dislocation densities are quantified and compared to past experiments. The atomistic approaches (MD, PFC) can be used to calibrate continuum level crystal plasticity models, and the framework adds mechanistic insights arising from the multiscale analysis. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model

The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called ‘triangle phase’) is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Energy quadratization Runge-Kutta method for the modified phase field crystal equation

In this paper, we propose high order and unconditionally energy stable methods for a modified phase field crystal equation by applying the strategy of the energy quadratization Runge–Kutta methods. We transform the original model into an equivalent system with auxiliary variables and quadratic free energy. The modified system preserves the laws of mass conservation and energy dissipation with the associated energy functional. We present rigorous proofs of the mass conservation and energy dissipation properties of the proposed numerical methods and present numerical experiments conducted to demonstrate their accuracy and energy stability. Finally, we compare long-term simulations using an indicator function to characterize the pattern formation.

Evaluation of grain boundary energy, structure and stiffness from phase field crystal simulations

A two-mode phase field crystal (PFC) model is employed to investigate the equilibrium configurations of a range of grain boundaries in fcc-structured materials. A total of 80 different symmetrical tilt grain boundaries are evaluated by PFC simulations in 3D and the results are shown to agree well with data taken from the literature, both regarding the variation of grain boundary energy and also in terms of the resulting grain boundary structures. This verification complements existing PFC studies which are almost exclusively focused either on grain boundaries found in 2D systems or in bcc lattices in 3D. The present work facilitates application of PFC in the analysis of grain boundary mechanics in an extended range of materials, in particular such mechanics that take place at extended time scales not tractable for molecular dynamics (MD) simulations. In addition to the verification of predicted grain boundary energies and structures, wavelet transforms of the density field are used in the present work to obtain phase fields from which it is possible to identify grain boundary fluctuations that provide the means to evaluate grain boundary stiffness based on the capillarity fluctuation method. It is discussed how PFC provides benefits compared to alternative methods, such as MD simulations, for this type of investigations.