The third principal direction besides armchair and zigzag in single-layer black phosphorus

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 ${\varphi }_{tp}=0.268\pi$, 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. (paper)