Directional-dependent thickness and bending rigidity of phosphorene

The strong mechanical anisotropy of phosphorene combined with the atomic-scale thickness challenges the commonly employed elastic continuum idealizations. Using objective boundary conditions and a density functional based potential, we directly uncover the flexibility of individual α, β, and γ phosphorene allotrope layers along an arbitrary bending direction. A correlation analysis with the in-plane elasticity finds that although a monolayer thickness cannot be defined in the classical continuum sense, an unusual orthotropic plate with a directional-dependent thickness can unambiguously describe the out-of-plane deformation of α and γ allotropes. Such decoupling of the in-plane and out-of-plane nanomechanics might be generic for two-dimensional materials beyond graphene.