Multiscale modeling of polycrystalline graphene: A comparison of structure and defect energies of realistic samples from phase field crystal models

Abstract

We extend the phase field crystal (PFC) framework to quantitative modeling of polycrystalline graphene. PFC modeling is a powerful multiscale method for finding the ground state configurations of large realistic samples that can be further used to study their mechanical, thermal or electronic properties. By fitting to quantum-mechanical density functional theory (DFT) calculations, we show that the PFC approach is able to predict realistic formation energies and defect structures of grain boundaries. We provide an in-depth comparison of the formation energies between PFC, DFT and molecular dynamics (MD) calculations. The DFT and MD calculations are initialized using atomic configurations extracted from PFC ground states. Finally, we use the PFC approach to explicitly construct large realistic polycrystalline samples and characterize their properties using MD relaxation to demonstrate their quality.

Energetics and structure of grain boundary triple junctions in graphene

“…In the whole of the PFC1, PFC3, MD(2D) and MD(3D) data, R 2  > 0.999 for all the cases except for a few PFC and MD cases where R 2  > 0.996. The total formation energy increases with length, approximately 5 eV per nanometre of grain boundary, which is consistent with our previous results for the grain boundary energy of symmetric boundaries (<8 eV/nm) 24 and with those of earlier works referenced therein. In the insets to the panels we show the extrapolated limit of L  → 0 magnified.…”

“…Coarse graining has been done by filtering out the atomic level structure and by mapping the local crystallographic orientation to different hues; see comment [60] in ref. 24 for details. The chains of dark dots are individual dislocations along small-angle armchair grain boundaries, and the slightly undulating solid black lines are large-angle armchair-zigzag boundaries. …”

Phase-field crystal model for heterostructures

“…In addition, for each value of β 2 , we observed C 12 ≈ C 11 /3, whereby ν ≈ 1/3. This is a feature common to many simple PFC models 33 .…”

“…The following upper bounds for spatial and temporal discretizations were used for all the calculations: ∆x = 0.75, ∆y = 0.75, and ∆t = 0.25. Finally, we also used a model system size optimization algorithm 33 to eliminate strain in our bicrystalline model systems of heterostructures. We did not apply the method to polycrystalline systems, since we did not attempt to extract equilibrium densities from them or to analyze them quantitatively here.…”

Phase ordering, transformation, and grain growth of two-dimensional binary colloidal crystals: A phase field crystal modeling

“…This method thus bridges the gap between continuum modeling that describes the long wavelength behavior of the system but not crystalline details, and atomistic modeling that captures the microscopic details but is computationally challenging for large systems. It has been successfully applied to the study of a broad range of phenomena such as quantum dot growth during epitaxy [24], grain boundaries of 2D materials [25,26], graphene Moiré patterns [27], colloidal solidification and growth [28,29], structural phase transformation [30,31], glass formation [32], and quasicrystal growth [33], among many others. For the application of PFC method to colloidal systems, most existing studies are limited to single-component crystallization process [28,29], while the study of binary colloidal structures is still lacking.…”