Quantized phonon spectrum of single-wall carbon nanotubes

The electronic spectra of carbon nanotubes and other nanoscale systems are quantized because of their small radii. Similar quantization in the phonon spectra has been difficult to observe because of the far smaller energy sca le. We probed this regime by measuring the temperature-dependent specific h eat of purified single-wall nanotubes. The data show direct evidence of one -dimensional quantized phonon subbands. Above 4 kelvin, they are in excelle nt agreement with model calculations of individual nanotubes and differ mar kedly from the specific heat of two-dimensional graphene or three-dimension al graphite. Detailed modeling yields an energy of 4.3 millielectron volts for the Lowest quantized phonon subband and a tube-tube (or "lattice") Deby e energy of 1.1 millielectron volts, implying a small intertube coupling in bundles.