Strain engineering of the lattice vibration modes in monolayer black phosphorus

Strain engineering can be served as an efficient tool for tuning physical properties of two-dimentional (2D) materials. However, the quantifying strain and characterizing its spatially inhomogeneous distribution are always challenging. Here, we report the strain-induced lattice vibration mode variations in monolayer black phosphorus (BP) and clarify the underlying mechanism resorting to the bond relaxation theory based on the combination of mass-spring model and Lagrangian dynamics. We reveal that the Ag2 and B-2g modes exhibit blue (red) shifts while the Ag1 mode shows a red (blue) shift through applying an armchair (zigzag) strain, which can be ascribed to the orientation-dependent lattice vibration mode variations resulting from the different responses of force constants. Moreover, it is found that a negative Poisson's ratio along the out-of-plane direction emerges when the strain is applied in the zigzag direction. Our results offer intriguing insights into the underlying physics of vibration mode variations in strained BP-like 2D layered materials.