Valence-force model and nanomechanics of single-layer phosphorene

Abstract

In order to understand the relation of strain and material properties, both a microscopic model connecting a given strain to the displacement of atoms, and a macroscopic model relating applied stress to induced strain, are required. Starting from a valence-force model for black phosphorous (phosphorene) [Kaneta et al., Solid State Communications, 1982, 44, 613] we use recent experimental and computational results to obtain an improved set of valence-force parameters. From the model we calculate the phonon dispersion and the elastic properties of single-layer phosphorene. Finally, we use these results to derive a complete continuum model, including the bending rigidities, valid for long-wavelength deformations of phosphorene. This continuum model is then used to study the properties of pressurized suspended phosphorene sheets.

Multi-scale approach for strain-engineering of phosphorene

“…Introducing a local band-gap, one then finds a spatially changing strain-induced contribution, which is essential for the so-called inverse funneling effect [12]. We obtain a very good agreement between fully numerical results (TB and VFM) with analytical estimates resulting from continuum elasticity [16] combined with the derived strain-induced modification of the band-gap given by Eq. (15).…”

“…Its macroscopic elastic properties are characterized by two bending rigidities and four stiffness constants. We find the shape of the deformed drum and the strain distribution by solving the equations corresponding to the out-of-plane displacement-field w(r) and for the Airy stress function χ(r) [16]. In following we focus our discussion on approximate analytic solutions (details are given in Appendix B) and compare them with numerical results.…”

“…If the drumhead is subjected to uniform pressure, it will be statically deformed, which leads to a nonuniform strain distribution. This system is conveniently described by continuum elasticity [16]. The phosphorene monolayer is effectively modeled as a thin anisotropic plate.…”

“…16 to numerically calculate the shape of a pressurized drum. To validate our approach in the continuum limit while maintaining computational convenience we simulate a drum of radius 103.16Å.…”

“…In this article we develop a theoretical framework that consistently treats both the elastic and electronic degrees of freedom of phosphorene [16][17][18] and is suitable for large-scale modeling of phosphorene devices. To this end, we put forward a minimal TB model with that accurately describes the electronic band-structure of phosphorene in the vicinity of the Γ-point.…”