Algebraic theory of crystal vibrations: Singularities and zeros in vibrations of 1D and 2D lattices

TL;DR

This paper introduces a new algebraic method for calculating the energy dispersion relation and density of states in 1D and 2D lattices, analyzing singularities and zeros, with applications to graphene-like materials and comparisons to experimental data.

Contribution

A novel algebraic approach for computing EDR and DOS in 1D and 2D lattices, including analysis of singularities and zeros, applied to various lattice types.

Findings

Identified van Hove singularities and Dirac zeros in DOS.

Validated the method with experimental data from microwave photonic crystals.

Provided insights into vibrational properties of 2D materials like graphene.

Abstract

A novel method for the calculation of the energy dispersion relation (EDR) and density of states (DOS) in one (1D) and two (2D) dimensions is introduced and applied to linear lattices (1D) and square and hexagonal lattices (2D). The (van Hove) singularities and (Dirac) zeros of the DOS are discussed. Results for the 2D hexagonal lattice (graphene-like materials) are compared with experimental data in microwave photonic crystals.