Effective thickness and mechanical properties of β-phases of two-dimensional pnictogen nanosheets

Thickness is not a well-defined quantity for two-dimensional (2D) materials. Hence, there are several choices adopted widely to assign the thickness of a 2D nanosheet: (a) the interlayer distance in layered bulk crystals; (b) the vertical distance between outmost atoms; (c) the value of (b) plus atomic van der Waals radius. Except these definitions of thickness, we determine the effective thickness of 2D pnictogen nanosheets based on continuum elastic theory combined with first principle calculations, from which the tension rigidities are extracted by fitting the strain energies of biaxially stretched samples with a quadratic relation on the lateral and transversal strains for blue phosphorene, arsenene, and antimonene. Whereas the bending rigidities are computed from the pnictogen nanotubes rolled from flat 2D nanosheets. Given the tension and bending rigidities, the effective thickness could be determined from the ratio between these two quantities. Our results indicate that the effective thickness, being much smaller than the widely used interlayer distance in the corresponding bulk material, is suited to describe mechanical properties of 2D nanosheets in a consistent way.