A density functional theory-based finite element method to study the vibrational characteristics of zigzag phosphorene nanotubes

In this paper, the vibrational characteristics of zigzag phosphorene nanotubes are investigated by using a three-dimensional finite element model. The beam elements are used to simulate the P–P bonds in the structure of the phosphorene nanotubes. The elastic properties of the beam elements are computed from the similarity of energy terms in the molecular and structural mechanics. Besides, mass elements are located at the place of the atoms. Considering the zigzag phosphorene nanotubes with different diameters, it is shown that the effect of the diameter on the first natural frequencies of the nanotubes can be neglected. However, this effect increases for higher modes. Besides, at the same diameter, the zigzag phosphorene nanotubes with larger aspect ratios (length/diameter) have smaller frequencies.