Reciprocal Space Constraints Create Real-Space Anomalies in Doped Carbon Nanotubes

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When graphite is doped with electrons, carbon-carbon bonds lengthen and Raman-active phonons soften as antibonding states fill. However, in semiconducting carbon nanotubes, one Raman-active G-band mode increases in frequency at low doping levels. We show how phase constraints on the conduction-band wave function expose a latent bonding character in the conduction band of certain nanotubes. In these tubes, filling the lowest conduction band shortens the axial bonds even as it lengthens the circumferential bonds. The A{1}{LO} phonon, which preferentially stretches the axial bonds, then hardens even as the other phonons soften. Quantum confinement eliminates the angular averaging taken for granted in higher-dimensional systems and develops a new class of states, neither bonding nor antibonding, whose character depends on the angular orientation of the bonds in question.