  Understanding the role of atomic-level structures in determining amorphous
material properties has been a long-standing challenge in solid-state physics.
Upon mechanical loading, amorphous materials undergo both simple affine
displacement and spatially inhomogeneous non-affine displacement. These two
types of displacement contribute differently to the elastic modulus, i.e., the
Born (or affine) and non-affine terms. Whether "soft" local structures
characterized by either small Born terms or large non-affine displacements
differ has remained an unanswered question despite the importance in
fundamental and applied physics. To address this question, we combined
molecular dynamics simulations and persistent homology analyses for amorphous
Si. We found that the characteristics of local structures with large non-affine
displacements differed significantly from those with small Born terms. The
local structures surrounding atoms with small Born terms are characterized at
the scale of short-range order (SRO), whereas those surrounding atoms with
large non-affine displacements have hierarchical structures ranging from SRO to
medium-range order. Furthermore, we found that these hierarchical structures
are related to low-energy localized vibrational excitations. The correlation
between the non-affine displacement and hierarchical geometric features
elucidated by persistent homology provides a new viewpoint for understanding
and designing the mechanical properties of amorphous materials based on their
static structures.
