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This is a reply to the comment by Midtvedt and Croy on our article 2015 Nanotechnology26 315706. [ABSTRACT FROM AUTHOR]

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 SLBP is proportional to the square of its intrinsic thickness. [ABSTRACT FROM AUTHOR]

Armchair and zigzag are two popular principal directions in many quasi-two-dimensional nanostructures such as graphene, hexagonal boron nitride, and black phosphorus. To date, nearly all published works discuss the anisotropy for physical quantities based on the comparison between quantity values in these two principal directions in single-layer black phosphorus. However, in the present work, by analyzing the directional dependence for the Young’s modulus with an analytic formula, we explore the third principal direction with a direction angle , besides the armchair and zigzag directions. This new principal direction is determined by the puckered configuration of black phosphorus. The largest Young’s modulus value is neither in the armchair, nor in the zigzag direction. Instead it is the third principle direction along which the Young’s modulus has the maximum value. This work reveals that armchair and zigzag directions are not enough for the discussion of anisotropic properties in the single-layer black phosphorus, and it is crucial to include this new principal direction for anisotropy discussions. [ABSTRACT FROM AUTHOR]

We propose parametrizing the Stillinger–Weber potential for covalent materials starting from the valence force-field model. All geometrical parameters in the Stillinger–Weber potential are determined analytically according to the equilibrium condition for each individual potential term, while the energy parameters are derived from the valence force-field model. This parametrization approach transfers the accuracy of the valence force field model to the Stillinger–Weber potential. Furthermore, the resulting Stilliinger–Weber potential supports stable molecular dynamics simulations, as each potential term is at an energy-minimum state separately at the equilibrium configuration. We employ this procedure to parametrize Stillinger–Weber potentials for single-layer MoS2 and black phosphorous. The obtained Stillinger–Weber potentials predict an accurate phonon spectrum and mechanical behaviors. We also provide input scripts of these Stillinger–Weber potentials used by publicly available simulation packages including GULP and LAMMPS. [ABSTRACT FROM AUTHOR]

The single-layer black phosphorus is characteristic for its puckered structure, which has led to distinct anisotropy in its optical, electronic, and mechanical properties. We use the non-equilibrium Greenʼs function approach and the first-principles method to investigate the thermal conductance for single-layer black phosphorus in the ballistic transport regime, in which the phonon–phonon scattering is neglected. We find that the anisotropy in the thermal conduction is very weak for the single-layer black phosphorus—the difference between two in-plane directions is less than 4%. Our phonon calculations disclose that the out-of-plane acoustic phonon branch has lower group velocities in the direction perpendicular to the pucker, as the black phosphorus is softer in this direction, leading to a weakening effect for the thermal conductance in the perpendicular direction. However, the longitudinal acoustic phonon branch behaves abnormally; i.e., the group velocity of this phonon branch is higher in the perpendicular direction, although the single-layer black phosphorus is softer in this direction. The abnormal behavior of the longitudinal acoustic phonon branch is closely related to the highly anisotropic Poissonʼs ratio in the single-layer black phosphorus. As a result of the counteraction between the out-of-plane phonon mode and the in-plane phonon modes, the thermal conductance in the perpendicular direction is weaker than the parallel direction, but the anisotropy is pretty small. [ABSTRACT FROM AUTHOR]

The mechanical properties of single-layer black phosphorus under uniaxial deformation are investigated using first-principles calculations. Both the Young's modulus and ultimate strain are found to be highly anisotropic and nonlinear as a result of its quasi-two-dimensional puckered structure. Specifically, the in-plane Young's modulus is 41.3 GPa in the direction perpendicular to the pucker and 106.4 GPa in the parallel direction. The ideal strain is 0.48 and 0.11 in the perpendicular and parallel directions, respectively. [ABSTRACT FROM AUTHOR]