Phonon stiffening in semiconducting single-walled carbon nanotubes under n-type doping

The doping dependence of the high-frequency Raman-active modes in single-walled semiconducting carbon nanotubes is studied by density functional theory. We find that the $A_{1g}$ longitudinal mode in $(3*n+1,0)$ zigzag tubes shows a small anomalous upshift, followed by a large downshift under electron doping. This doping-induced stiffening of the $A_{1g}$ mode is related to the large anharmonicity of the mode. Connections are made to recent experiments in the group of P. C. Eklund.