Continuum approach for long-wavelength acoustic phonons in quasi-two-dimensional structures

As an alternative to atomistic calculations of long-wavelength acoustic modes of atomically thin layers, which are known to converge very slowly, we propose a quantitatively predictive and physically intuitive approach based on continuum elasticity theory. We describe a layer, independent of its thickness, by a membrane and characterize its elastic behavior by a $(3\ifmmode\times\else\texttimes\fi{}3)$ elastic matrix as well as the flexural rigidity. We present simple quantitative expressions for frequencies of long-wavelength acoustic modes, which we determine using two-dimensional elastic constants calculated by ab initio density functional theory. The calculated spectra accurately reproduce observed and calculated long-wavelength phonon spectra of graphene and phosphorene, the monolayer of black phosphorus. Our approach also correctly describes the observed dependence of the radial breathing mode frequency on the diameters of carbon fullerenes and nanotubes.