Group theory for structural analysis and lattice vibrations in phosphorene systems

TL;DR

This paper applies group theory to analyze the symmetry and lattice vibrations of phosphorene-related two-dimensional materials, including graphene and its allotropes, considering layer number and stacking, aiding in understanding their physical properties.

Contribution

It introduces a group theory framework for analyzing symmetry, vibrational modes, and structural features of phosphorene systems and related 2D materials.

Findings

Identifies symmetry relations among different 2D allotropes.

Distinguishes materials based on vibrational mode analysis.

Provides tools for studying strain effects and phase transitions.

Abstract

Group theory analysis for two-dimensional elemental systems related to phosphorene is presented, including (i) graphene, silicene, germanene and stanene, (ii) dependence on the number of layers and (iii) two stacking arrangements. Departing from the most symmetric $D_{6h}^{1}$ graphene space group, the structures are found to have a group-subgroup relation, and analysis of the irreducible representations of their lattice vibrations makes it possible to distinguish between the different allotropes. The analysis can be used to study the effect of strain, to understand structural phase transitions, to characterize the number of layers, crystallographic orientation and nonlinear phenomena.