Phase-field crystal (PFC) modeling provides a powerful computational framework for investigating the growth of graphene on metal substrates at atomic resolution. This approach bridges the gap between atomistic simulations and continuum models, enabling the study of long-time dynamics while retaining atomic-scale details. The PFC method describes the evolution of a periodic density field, capturing crystalline order, defects, and grain boundaries in a computationally efficient manner.
The PFC model for graphene growth incorporates a free energy functional that accounts for both the crystalline symmetry of graphene and its interaction with the metal substrate. The density field minima correspond to atomic positions, allowing the simulation of nucleation, domain growth, and defect formation. Key parameters in the model include the undercooling driving force, substrate interaction strength, and lattice mismatch between graphene and the substrate.
One of the primary advantages of PFC modeling is its ability to simulate defect structures at experimentally relevant scales. Dislocations, grain boundaries, and stacking faults emerge naturally during the growth process. For example, the model captures the formation of pentagon-heptagon pairs (Stone-Wales defects) and the misorientation angles between adjacent graphene domains. These defects significantly influence the electronic and mechanical properties of the grown material.
Nucleation density and domain orientation are critical factors determining graphene quality. PFC simulations reveal that nucleation events are highly sensitive to substrate temperature and carbon supersaturation. Higher temperatures and lower supersaturation lead to fewer nucleation sites, promoting larger single-crystalline domains. The substrate steps and terraces play a crucial role in guiding graphene orientation. On copper substrates, simulations show that step edges act as preferential nucleation sites, aligning graphene domains along low-energy crystallographic directions.
The role of substrate steps is particularly important in controlling graphene growth. PFC models demonstrate that step edges can either promote epitaxial alignment or induce rotational disorder, depending on the step height and interaction strength. For instance, monoatomic steps on Cu(111) surfaces tend to align graphene domains, while multiatomic steps may lead to misoriented nucleation. These findings align with experimental observations of graphene grown via chemical vapor deposition (CVD).
Comparisons between PFC simulations and CVD-grown graphene characterization data show strong agreement in defect distributions and domain sizes. Experimental measurements using scanning tunneling microscopy (STM) and transmission electron microscopy (TEM) reveal similar grain boundary configurations and defect densities as those predicted by PFC models. For example, both simulations and experiments observe grain boundaries with misorientation angles between 5° and 30°, consistent with the energetics of graphene on metal substrates.
Coupling PFC models with carbon diffusion dynamics enhances the predictive capability of the simulations. Carbon adatom diffusion on the metal surface is a rate-limiting step in graphene growth. By integrating a diffusion equation with the PFC formalism, the model captures the interplay between carbon supply and graphene domain expansion. The diffusion-limited regime leads to dendritic growth morphologies, while reaction-limited conditions produce compact hexagonal domains. These predictions match experimental observations of graphene growth under varying CVD conditions.
The PFC approach also enables the study of strain effects during graphene growth. Lattice mismatch between graphene and the substrate induces strain, which can lead to wrinkle formation or partial relaxation through defect generation. Simulations show that strain distribution depends on the substrate’s thermal expansion coefficient and the cooling rate post-growth. These insights help explain experimental observations of wrinkle patterns in transferred graphene films.
Further refinements of PFC models include incorporating anisotropic interfacial energies and substrate reconstructions. For example, nickel substrates exhibit stronger graphene-substrate interactions than copper, leading to different growth kinetics and defect distributions. PFC simulations adjusted for these interactions reproduce the experimentally observed differences in graphene quality between these substrates.
The scalability of PFC modeling allows for simulations spanning micrometers in lateral dimensions while resolving atomic-scale features. This capability is crucial for studying polycrystalline graphene films with realistic domain sizes. Statistical analysis of simulated microstructures provides grain size distributions and defect densities that can be directly compared with experimental data.
In summary, phase-field crystal modeling offers a robust framework for simulating graphene growth on metal substrates with atomic resolution. It captures nucleation dynamics, defect formation, and substrate interactions in agreement with experimental observations. Coupling with carbon diffusion models further enhances its predictive power, providing insights into growth kinetics and morphology evolution. The method’s ability to bridge atomic-scale details with large-scale dynamics makes it a valuable tool for optimizing graphene synthesis processes.
Future developments may extend PFC models to include additional physical effects, such as hydrogen interactions during CVD or the role of impurities in defect formation. Such refinements will further improve the accuracy of simulations and their utility in guiding experimental graphene growth strategies.