Influence of nonequilibrium quasiparticles in phonon imaging of superconductors: Consistency with BCS theory in Pb crystals

Abstract

This paper is the second part of a study motivated by the prediction of Overhauser and Daemen [A. W. Overhauser and L. L. Daemen, Phys. Rev. Lett. 61, 1885 (1988)] that the electronic ground state of Pb exhibits a spin-density wave (SDW) that leads to a highly anisotropic superconducting gap. The first part of our study [J. D. Short, T. L. Head, and J. P. Wolfe, Phys. Rev. B 78, 054515 (2008)] extended the theoretical predictions to the actual Fermi surface of Pb and initiated phonon imaging experiments on high purity Pb. The temperature dependence of the phonon absorption by quasiparticles over a 1.45--2.1 K temperature range was found to be much weaker than the $\text{exp}(\ensuremath{-}{\ensuremath{\Delta}}_{o}/{k}_{B}T)$ form using the generally accepted gap parameter ${\ensuremath{\Delta}}_{o}=1.35\text{ }\text{meV}$ for Pb. In addition, the absorption coefficients obtained for two crystals of different thickness were not consistent with phonon absorption by quasiparticles in thermal equilibrium with the lattice. To explain these two anomalies in the context of the BCS theory of superconductivity, we examine the possible effects of nonequilibrium quasiparticles generated by the laser source and bolometer detector. Detailed experiments varying crystal length, source size, excitation power, and pulse duration over wide limits enable us to isolate the phonon absorption by both nonequilibrium and thermal-equilibrium quasiparticles. The wide variety of data yields an equilibrium quasiparticle density that varies as $\text{exp}(\ensuremath{-}\ensuremath{\Delta}/{k}_{B}T)$ with $\ensuremath{\Delta}=1.32\ifmmode\pm\else\textpm\fi{}0.07\text{ }\text{meV}$, in good agreement with BCS theory with ${\ensuremath{\Delta}}_{o}=1.35\text{ }\text{meV}$, and inconsistent with the electronic-specific-heat data of van der Hoeven and Keesom [B. J. C. van der Hoeven, Jr. and P. Keesom, Phys. Rev. 137, 103 (1965)] that motivated the SDW hypothesis in Pb. In summary, the highly anisotropic absorption of ballistic phonons in superconducting Pb provides a unique measure of equilibrium quasiparticle density that agrees with BCS theory extending to $\text{exp}(\ensuremath{-}\ensuremath{\Delta}/{k}_{B}T)=2.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ at 1.45 K.