Force-constant model for the vibrational modes in black-phosphorene and phosphorene nanoribbons (PNRs)

Abstract We present in this paper, force constant model developed for Black phosphorene in order to reproduce the vibrationnal properties calculated from density functional theory. The results of this model are compared with the experimental data available from Raman spectroscopy measurements. Excellent agreement is obtained between calculation and Raman experiment. On the basis of the resulting force model, the non-resonant Raman spectra of a large number of armchair and zigzag phosphorene nanoribbons (PNRs) are calculated using the bond polarizability model in the framework of spectral moments method. We have found a good agreement with group theory concerning the number of the Raman-active modes of black phosphorene. We report the effect of the edge and width on the vibrational properties of PNRs by increasing the width, the main characteristic feature is dictated by A g 2 Raman active mode. It exhibits different characteristic for armchair and zigzag edges. The mode is upshifted under armchair edge and downshifted in the zigzag one. Moreover, we observe additional Raman modes as a function of the ribbon width and we propose an equation, ωmin = A/L (A=98 cm−1 nm), to estimate the PNRs width L from the knwoledge of the lowest Raman frequency mode ωmin.