Thermal Expansion and Superfluid Onset for 4 He Films Adsorbed on Carbon Nanotubes

Third sound measurements of superfluid $^4$He thin films adsorbed on 10 nm diameter multiwall carbon nanotubes are used to make the first measurement of the thermal expansion coefficient normal to the film surface, and to probe the superfluid onset temperature as a function of the film thickness. The nanotubes provide a highly ordered carbon surface, with layer-by-layer growth of the adsorbed film as shown by oscillation peaks in the third sound velocity at the completion of the third, fourth, and fifth atomic layers, arising from oscillations in the film compressibility. The high sensitivity of the third sound to film thickness allows a measurement of the thermal expansion coefficient, which is found to display a large-amplitude oscillation between positive and negative values, precisely out of phase with the compressibility oscillation. The linear slope of the superfluid onset temperature with thickness is found to be anomalous, a factor of three smaller than the universal value predicted by the Kosterlitz-Thouless theory.