First-principles thermodynamics simulations provide a powerful framework for understanding the growth mechanisms of two-dimensional materials at the atomic scale. These simulations rely on density functional theory and ab initio calculations to predict the energetics of growth processes, including edge configurations, kink formation, and adatom kinetics. By coupling these calculations with thermodynamic models, researchers can simulate the equilibrium and non-equilibrium conditions that govern material synthesis, particularly for transition metal dichalcogenides (TMDs) and hexagonal boron nitride (hBN) grown via chemical vapor deposition (CVD) or molecular beam epitaxy (MBE).

The growth of 2D materials begins with the formation of nucleation sites, where the edge energetics play a critical role. Ab initio calculations reveal that the edge energies of TMDs, such as MoS2 or WS2, depend on the chalcogen termination and the metal-to-chalcogen ratio. For instance, sulfur-terminated edges in MoS2 exhibit lower formation energies compared to molybdenum-terminated edges under sulfur-rich conditions. The stability of different edge configurations directly influences the shape of the growing islands, with zigzag edges often dominating under equilibrium conditions due to their lower energy. Kink sites, which are deviations from straight edges, also contribute to growth kinetics. Simulations show that kink formation energies range between 0.5 and 1.5 eV, depending on the local stoichiometry and edge type. These kinks serve as preferential attachment sites for adatoms, facilitating lateral expansion.

Adatom mobility is another key factor in 2D material growth. First-principles molecular dynamics simulations predict that metal adatoms, such as Mo or W, exhibit higher diffusion barriers on chalcogen-terminated surfaces compared to chalcogen adatoms. For example, the diffusion barrier for a molybdenum adatom on a sulfur-covered surface can exceed 0.8 eV, while sulfur adatoms may diffuse with barriers as low as 0.3 eV. This difference in mobility leads to distinct growth modes, where chalcogen adatoms contribute more readily to edge propagation. The presence of substrates further modifies these energetics. Calculations demonstrate that sapphire or graphene substrates can reduce adatom diffusion barriers by up to 30%, promoting larger domain sizes.

Chemical potential diagrams are essential tools for predicting the thermodynamic stability of 2D materials under varying growth conditions. These diagrams map the phase space as a function of metal and chalcogen chemical potentials, revealing the conditions under which stoichiometric or chalcogen-deficient phases form. For TMD growth, the chemical potential of the chalcogen precursor, such as H2S or Se vapor, must be carefully controlled to avoid metal-rich or excessively chalcogen-rich compositions. In the case of hBN, the ratio of boron to nitrogen precursors determines the prevalence of boron- or nitrogen-terminated edges. Phase diagrams derived from ab initio thermodynamics show that nitrogen-rich conditions favor nitrogen-terminated edges, while boron-rich conditions lead to boron-terminated edges, each with distinct electronic properties.

The role of precursors in CVD and MBE growth is also elucidated through simulations. For TMDs, metal-organic precursors like Mo(CO)6 or W(CO)6 decompose to release metal atoms, while chalcogen precursors such as H2S or H2Se provide the chalcogen component. First-principles studies reveal that the decomposition pathways of these precursors are highly sensitive to temperature and pressure. For instance, Mo(CO)6 decomposes at temperatures above 500°C, releasing Mo atoms that subsequently react with sulfur species. The partial pressures of these precursors influence the adatom concentrations on the growth substrate, directly affecting nucleation density and domain size. In MBE growth, the flux rates of metal and chalcogen sources must be precisely balanced to achieve stoichiometric films. Simulations indicate that excess chalcogen flux can lead to multilayer growth or chalcogen adlayer formation, while insufficient flux results in metal-rich defects.

Substrate interactions are equally critical. Ab initio calculations show that epitaxial strain and interfacial bonding can alter the growth kinetics of 2D materials. For example, MoS2 grown on sapphire experiences a slight compressive strain due to lattice mismatch, which can enhance nucleation density but also introduce defects. Graphene substrates, with their weak van der Waals interactions, allow for more relaxed growth but may lead to rotational misalignment of domains. The substrate temperature also plays a role, as higher temperatures increase adatom mobility but may also promote desorption or decomposition of precursors.

Experimental validation of these simulations comes from high-resolution scanning transmission electron microscopy (STEM) observations. Predicted morphologies, such as triangular or hexagonal domains in TMDs, align closely with experimental findings. STEM images reveal that sulfur-rich conditions produce predominantly triangular MoS2 domains with sulfur-terminated edges, consistent with thermodynamic predictions. In contrast, selenium-rich conditions lead to hexagonal domains due to the lower kink formation energies in selenide-based TMDs. Similarly, hBN grown under nitrogen-rich conditions exhibits smooth edges, while boron-rich conditions result in more jagged morphologies, corroborating theoretical edge energy calculations.

The chalcogen ratio during growth further influences defect formation. Simulations demonstrate that sulfur-deficient conditions in MoS2 growth promote the formation of sulfur vacancies, which act as nucleation sites for additional layers or grain boundaries. Conversely, excess sulfur can passivate these vacancies but may also lead to adsorbed sulfur clusters that disrupt crystalline order. Experimental STEM studies confirm these trends, showing that optimized chalcogen ratios yield the most uniform and defect-free monolayers.

First-principles thermodynamics simulations thus provide a comprehensive understanding of 2D material growth, bridging atomic-scale energetics with macroscopic synthesis conditions. By quantifying edge energies, kink dynamics, and adatom behavior, these simulations guide the rational design of growth protocols for TMDs, hBN, and related materials. The agreement between predicted morphologies and experimental observations underscores the predictive power of these methods, enabling advances in the controlled synthesis of 2D materials for electronic, optoelectronic, and quantum applications.