Elastic Bending Modulus for Single-Layer Black Phosphorus

TL;DR

This paper derives an analytical formula for the elastic bending modulus of single-layer black phosphorus, revealing its dependence on intrinsic thickness and directional differences, with implications for 2D material mechanics.

Contribution

It provides the first analytical derivation of the elastic bending modulus for SLBP, highlighting its larger value compared to graphene and its proportionality to thickness squared.

Findings

Bending modulus is 4.8028 eV (armchair) and 7.9905 eV (zigzag).

Elastic bending modulus is larger than in graphene due to finite thickness.

Modulus is proportional to the square of the intrinsic thickness.

Abstract

We derive an analytic formula for the elastic bending modulus of single-layer black phosphorus (SLBP) based on the valence force field model. The obtained elastic bending modulus is 4.8028 eV and 7.9905 eV along the armchair and zigzag directions in the SLBP, respectively. These values are obviously larger than the bending modulus of 1.4 eV in graphene due to the intrinsic finite thickness for SLBP. Our derivation analytically illustrates that the elastic bending modulus of the SLBP is proportional to the square of the intrinsic thickness of the SLBP.