Phonon imaging in superconducting Pb crystals: Absence of large gap anisotropy and spin-density waves

Abstract

We show that anisotropies in the electronic structure of a superconducting metal can be probed with ballistic acoustic phonons at low temperatures. We were motivated by the possibility that Pb has narrow valleys of lowered gap, possibly caused by the spin-density wave (SDW) postulated by Overhauser and Daemen [Phys. Rev. Lett. 61, 1885 (1988)]. Our experiments and analysis on crystals of different thickness, plus results with a phonon-frequency filter, do not support their idea of deep valleys in the superconducting gap parameter. For a spherical Fermi surface, phonon scattering would be isotropic but for the highly anisotropic Fermi surface of Pb, the scattering rate depends on the direction of the phonon wave vector, $\mathbf{q}$. To observe this anisotropy, we perform phonon-imaging experiments on high-purity Pb crystals cooled between 1.45 and 2.1 K. At this low temperature the mean-free path of a phonon can extend to millimeter distances if its energy is less than the superconducting energy gap, $2{\ensuremath{\Delta}}_{o}$. In this regime, phonons are absorbed by thermally excited quasiparticles on the Fermi surface. Phonon images show a striking pattern of phonon absorption that becomes stronger as the temperature is raised. The anisotropy is explained by a calculation of phonon scattering on the actual Fermi surface of Pb without recourse to SDWs. The preliminary temperature dependences reported in this paper, which assumes that the quasiparticles doing the scattering are in equilibrium at the lattice temperature, are much weaker than the low-temperature form $\text{exp}(\ensuremath{-}{\ensuremath{\Delta}}_{o}/{k}_{B}T)$ predicted by the BCS theory of superconductivity. Even accounting for the inherent scattering of phonons from isotopes in the crystal, we find a path-length dependence of phonon-absorption coefficient inconsistent with deep SDW valleys. A phonon filter experiment designed to attenuate phonons that would eject quasiparticles from such valleys clearly supports this conclusion. A resolution of the anomalous temperature and sample-length dependences, plus a measurement of the superconducting gap by phonon-imaging, is reported in the following paper.