Anomalous phonon behavior of carbon nanotubes: First-order influence of   external load
  External loads typically have a 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 {\em 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. 
Amin Aghaei, Kaushik Dayal and Ryan S. Elliott
cond-mat.mtrl-sci cond-mat.mes-hall physics.comp-ph