Anisotropic bending behaviors and bending induced buckling in single-layered black phosphorus

Abstract

The bending behavior of single-layered black phosphorus was systematically studied using density functional theory. Graphene, containing a single layer of molybdenum disulfide, was studied by considering changes in the bending strain energies as a function of the radius of the curvature. Bending-induced buckling in single-layered black phosphorus and its bent states were found along a zigzag direction as shown by the critical radius of the curvature. Combing the bending strain energies from first-principle calculations of the corresponding single-wall nanotubes of grapheme, single-layered black phosphorus and single-layered molybdenum disulfide with the strain energy densities using pure bending plate theory and by the introduction of a simple equivalent atomic area, the bending stiffnesses of three kinds of two-dimensional crystals were calculated and found to be in good agreement with experimental and empirical results. We also show that the bending stiffness is 5.336 eV under zigzag bending and 1.251 eV under armchair bending in single-layered black phosphorus. Apparent anisotropic bending behavior was found where high bending stiffness under zigzag bending leads to a buckling phenomenon. With insight into the physical mechanism of the anisotropic bending behavior, electron density distributions in the single-layered black phosphorus and the corresponding armchair (11,0) and zigzag (0,15) black phosphorus nanotubes with a similar curvature radius were further analyzed. We show that the anisotropic bending behavior can be attributed to repulsive phosphorus lone pair interactions, which are stronger under zigzag bending compared with armchair bending. When a critical curvature radius is attained, the hybridized sp 3 orbitals in single-layered black phosphorus will be affected by the lone pairs according to Pauli's exclusion principle. The local equilibrium conditions of the electron clouds are thus disrupted and bending-induced buckling finally occurs.