Persistent homology elucidates hierarchical structures responsible for mechanical properties in covalent amorphous solids.

Understanding how atomic-level structures govern the mechanical properties of amorphous materials remains a fundamental challenge in solid-state physics. Under mechanical loading, amorphous materials exhibit simple affine and spatially inhomogeneous nonaffine displacements that contribute to the elastic modulus through the Born (affine) and nonaffine terms, respectively. The differences between soft local structures characterized by small Born terms or large nonaffine displacements have yet to be elucidated. This challenge is particularly complex in covalent amorphous materials such as silicon, where the medium-range order (MRO) plays a crucial role in the network structure. To address these issues, we combined molecular dynamics simulations with persistent homology analysis. Our results reveal that local structures with small Born terms are governed by short-range characteristics, whereas those with large nonaffine displacements exhibit hierarchical structures in which short-range disorder is embedded within the MRO. These hierarchical structures are also strongly correlated with low-energy localized vibrational excitations. Our findings demonstrate that the mechanical responses and dynamic properties of covalent amorphous materials are intrinsically linked to the MRO, providing a framework for understanding and tailoring their properties.