Quantized Phonon Spectrum of Single-Wall Carbon Nanotubes

The electronic spectra of carbon nanotubes and other nanoscale systems are quantized due to their small radii. Similar quantization in the phonon spectra has been difficult to observe due to the far smaller energy scale. We probed this regime by measuring the temperature-dependent specific heat of purified single-wall nanotubes. The data show direct evidence of 1D quantized phonon subbands. Above 4K, they are in excellent agreement with model calculations of individual nanotubes, and differ markedly from the specific heat of two-dimensional graphene or three-dimensional graphite. Detailed modeling yields an energy of 4.3 meV for the lowest quantized phonon subband, and a tube-tube (or lattice) Debye energy of 1.1 meV, implying a small inter-tube coupling in bundles.