Third sound and stability of thinHe3−He4films

Abstract

We study third sound in thin $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ 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., $^{3}\mathrm{He}$. In very thin films, this result clearly must be modified to account for the effects of nonuniform $^{4}\mathrm{He}$ film density. Utilizing the variational, hypernetted-chain--Euler-Lagrange theory as applied to inhomogeneous boson systems, we calculate accurate chemical potentials for both the $^{4}\mathrm{He}$ superfluid film and the physisorbed $^{3}\mathrm{He}$. 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 $^{3}\mathrm{He}$ coverage in the linear (low-concentration) regime, which is then studied for the two substrates as a function of $^{4}\mathrm{He}$ film thickness and compared to existing experiments.$^{3}\mathrm{He}$ density profiles are calculated as a function of $^{4}\mathrm{He}$ 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 $^{3}\mathrm{He}$ can increase, as well as decrease, the third-sound speed relative to that of the pure $^{4}\mathrm{He}$ film. Further, we show that the addition of a small amount of $^{3}\mathrm{He}$ can destabilize the film and drive a phase separation into lateral regions of $^{3}\mathrm{He}$-rich and $^{3}\mathrm{He}$-poor patches. This latter result may help explain the phase transitions reported by Bhattacharyya and Gasparini [Phys. Rev. Lett. 49, 919 (1982)] and Cs\'athy, Kim, and Chan [Phys. Rev. Lett. 88, 045301 (2002)] in thin mixture films.