Third sound and stability of thin ³He-⁴He films

We study third sound in thin ³He-⁴He mixture films from first-principles, microscopic theory and compare these results to the usual film-averaged, hydrodynamic approach. The hydrodynamic approach yields third-sound speeds that depend only on the thickness of the superfluid film and the distribution of impurities—i.e., ³He. In very thin films, this result clearly must be modified to account for the effects of nonuniform ⁴He film density. Utilizing the variational, hypernetted-chain–Euler-Lagrange theory as applied to inhomogeneous boson systems, we calculate accurate chemical potentials for both the ⁴He superfluid film and the physisorbed ³He. Numerical density derivatives of the chemical potentials lead to the sought-after third-sound speeds that clearly reflect a layered structure of at least seven oscillations. We are thus able to gauge the range of applicability of the film-averaged hydrodynamic results as applied to thin quantum liquid films. We study third sound on two model substrates: Nuclepore and glass. We compute the change in third-sound speed as a function of ³He coverage in the linear (low-concentration) regime, which is then studied for the two substrates as a function of ⁴He film thickness and compared to existing experiments.³He density profiles are calculated as a function of ⁴He film thickness, and we show explicitly the smooth transition from Andreev states in the thick-film limit to lateral mixtures in the submonolayer limit. This effect was first seen by Noiray et al. Phys. Rev. Lett. 53 2421 (1984). Our results predict that the addition of a small amount of ³He can increase, as well as decrease, the third-sound speed relative to that of the pure ⁴He film. Further, we show that the addition of a small amount of ³He can destabilize the film and drive a phase separation into lateral regions of ³He-rich and ³He-poor patches. This latter result may help explain the phase transitions reported by Bhattacharyya and Gasparini Phys. Rev. Lett. 49 919 (1982) and Csáthy, Kim and Chan Phys. Rev. Lett. 88 045301 (2002) in thin mixture films.