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

Abstract

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, characterize its elastic behavior by a (3×3) elastic matrix as well as the flexural rigidity. We present simple quantitative expressions for frequencies of long-wavelength acoustic modes, which we determine using 2D 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 diameter of carbon fullerenes and nanotubes.

Long-wavelength deformations and vibrational modes in empty and liquid-filled microtubules and nanotubes: A theoretical study

“…The elastic behavior of carbon nanotubes can be described using quantities previously obtained using DFT calculations for graphene [10]. The calculated elements of the elastic stiffness matrix (1) are c 11 = c 22 = 352.6 N/m, c 12 = 59.6 N/m, and c 66 = 146.5N/m, all in very good agreement with experimental results [20].…”

“…Finally, the radial breathing mode (RBM) of the nanotube, depicted Fig. 1(d), has a nearly k−independent frequency given by [10]…”

“…As shown earlier [10], elastic in-plane deformations of a plate of indefinite thickness may be described by the (3×3) 2D elastic stiffness matrix, which is given in Voigt notation by (1)…”

Towards topological quasifreestanding stanene via substrate engineering

“…In such a configuration, all the frequencies of phonon modes associated to stanene are positive. The small still imaginary frequencies in the long-wavelength q → 0 flexural acoustic ZA mode is a well known artifact that is not related to a structural instability [61][62][63].…”

Origin of Unusually High Rigidity in Selected Helical Coil Structures

“…In the following we explore the elastic behavior of this structure using continuum elasticity theory in order to identify the reason for its rigidity [2][3][4] . Since continuum elasticity theory applies from nanometer-sized fullerenes and nanotubes [5][6][7] to the macro-scale, we expect our approach to be useful to explore the rigidity of helical structures on the microand nanometer scale.…”