Interlayer Breathing and Shear Modes in Few-Layer Black Phosphorus
  The interlayer breathing and shear modes in few-layer black phosphorus are investigated for their symmetry and lattice dynamical properties. The symmetry groups for the even-layer and odd-layer few-layer black phosphorus are utilized to determine the irreducible representation and the infrared and Raman activity for the interlayer modes. The valence force field model is applied to calculate the eigenvectorw and frequencies for the interlayer breathing and shear modes, which are explained using the atomic chain model. The anisotropic puckered configuration for black phosphorus leads to a highly anisotropic frequency for the two interlayer shear modes. More specifically, the frequency for the shear mode in the direction perpendicular to the pucker is less than half of the shear mode in the direction parallel with the pucker. We also report a set of interlayer modes having the same frequency for all few-layer black phosphorus with layer number N=3i with integer i, because of their collective vibrational displacements. The optical activity of the collective modes supports possible experimental identification for these modes. 
Jin-Wu Jiang, Bing-Shen Wang, Harold S. Park
cond-mat.mtrl-sci