Bulk penetration of edge properties in two-dimensional materials

Edges are essential for the mechanical, chemical, electronic, and magnetic properties of two-dimensional (2D) materials. Research has shown that features assigned to edges are not strictly localized but often penetrate the bulk to some degree. However, mechanical edge properties, such as edge energies and stresses, are typically assigned at the system level, with spatial bulk penetrations that remain unknown. Here, we use density-functional tight-binding simulations to study how deep various edge properties spatially penetrate the 2D bulk. We study nine different edges made of four materials: graphene, goldene, boron nitride, and molybdenum disulfide. By investigating edge energies, edge stresses, and edge elastic moduli, we find that although the edge properties typically originate near edges, they still penetrate the bulk to some degree. An utmost example is goldene with a staggered edge, whose edge properties penetrate the bulk nanometer-deep. Our results caution against associating system-level edge properties too strictly with the edge, especially if those properties are further used in continuum models.