Dynamical matrix of monolayer graphene including sixth-nearest-neighbor interactions: Phonon dispersion, elastic properties and the Kohn anomaly at the K point

We develop a force-constant model for monolayer graphene including interatomic interactions up to the sixth-nearest neighbors and derive the complete analytical expressions of the 6 × 6 dynamical matrix. The sixth-nearest-neighbor force constants are determined from the long-wavelength elastic limit constrained by the known elastic properties of graphene. Phonon dispersion relations for the six branches (LA, TA, ZA, LO, TO, ZO) are calculated throughout the first Brillouin zone along the Γ − K − M − Γ path and compared with previous analytical models, ab initio calculations, and experimental data, with quantitative agreement varying among the considered references. Convergence analysis reveals that sixth-nearest-neighbor interactions have little effect at the Γ and M points but produce measurable shifts at K , highlighting the wave-vector-dependent sensitivity of lattice dynamics to long-range interactions near the Kohn anomaly. The model also yields elastic and acoustic properties consistent with theoretical and experimental data.