A Descriptor-Driven Thermodynamic Framework for Achieving Unidirectional Nucleation in 2D Material Epitaxy

Unidirectional nucleation is crucial for achieving wafer-sized single-crystal epitaxy of two-dimensional (2D) materials, yet it is fundamentally hindered by multidirectional nucleations resulting from symmetry mismatch-induced energy equivalence at the epilayer-substrate interfaces. Here, we propose a universal, hierarchical framework that integrates thermodynamic modeling, epitaxial descriptor construction, and substrate-step engineering to enable unidirectional nucleation and epitaxy across diverse 2D materials. Our thermodynamic model classifies 2D nucleation into edge-dominated and surface-dominated regimes, pinpointing that the latter can be controlled only when terrace steps are precisely aligned with the preferred epitaxial axis. A quantitative epitaxial descriptor based on lattice mismatch and interfacial atomic spacing is developed for screening optimal growth orientations for arbitrary 2D/substrate systems, eliminating the need for first-principles calculations. Applied to benchmark systems, this framework predicts that graphene and hexagonal boron nitride (h-BN) tolerate broad step directions, whereas molybdenum disulfide (MoS₂) requires strict orientation engineering, in full agreement with experiments. This work establishes a general protocol for substrate-step engineering that promotes unidirectional nucleation and provides guidelines for wafer-scale single-crystal epitaxy of diverse 2D materials.