Abstract
External loads typically have an indirect influence on phonon curves, i.e., they influence the phonon curves by changing the state about which linearization is performed. In this paper, we show that in nanotubes, the axial load has a direct first-order influence on the long-wavelength behavior of the transverse acoustic (TA) mode. In particular, when the tube is force-free, the TA mode frequencies vary quadratically with wave number and have curvature (second derivative) proportional to the square-root of the nanotube's bending stiffness. When the tube has non-zero external force, the TA mode frequencies vary linearly with wave number and have slope proportional to the square-root of the axial force. Therefore, the TA phonon curves—and associated transport properties—are not material properties but rather can be directly tuned by external loads. In addition, we show that the out-of-plane shear deformation does not contribute to this mode and the unusual properties of the TA mode are exclusively due to bending. Our calculations consist of 3 parts: First, we use a linear chain of atoms as an illustrative example that can be solved in close-form; second, we use our recently developed symmetry-adapted phonon analysis method to present direct numerical evidence; and finally, we present a simple mechanical model that captures the essential physics of the geometric nonlinearity in slender nanotubes that couples the axial load directly to the phonon curves. We also compute the density of states and show the significant effect of the external load.
Highlights
The low-dimensionality of nanotubes causes unusual features in the phonon curves due to geometric effects
We examine the transverse acoustic (TA) phonon mode and its slope and curvature in the long wavelength limit
We show that the out-of-plane shear deformation does not contribute to this mode and the unusual properties of the TA mode are exclusively due to bending
Summary
The low-dimensionality of nanotubes causes unusual features in the phonon curves due to geometric effects. These authors do not analytically show that the TA modes should be linear, but argue that nanotubes do not have modes analogous to the out-of-plane “bending” mode of graphene, and nanotubes must have linear long wavelength acoustic dispersion behavior Among these works, two recent papers state that quadratic dependence is found in some other papers, but argue that this may be attributed to numerical errors. The paper develops a spring model and performs lengthy calculations for zig-zag and armchair nanotubes.37 Among these papers, the works mention that others have observed linear dependence of the TA dispersion relations. We develop a simple mechanical model based on classical beam theory that captures the essential physics of the geometric nonlinearity in slender nanotubes that couple the axial load directly to the phonon curves. V, we show that a simple model based on classical beam theory captures the key nonlinearity that governs the TA mode behavior
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