Ultrasonic Studies of the Superconductivity of Doped Tin. II

Abstract

Recent ultrasonic-attenuation studies have led to conflicting interpretations concerning the reduced anisotropy of the superconducting energy gap characteristic of impure Sn. Perz and Dobbs have concluded that the reduced anisotropy is only due to an averaging of the electron-sound-wave interaction over the Fermi surface, while we have concluded that under the specified conditions the reduced anisotropy is indeed a real measure of the superconducting properties. The two conclusions are discussed here in detail, and additional ultrasonic data are presented. Finally, comparison is made with recent thermal-conductivity measurements in impure Sn. I N a recent paper Perz and Dobbs 1 have reported the observation of a dependence on ql of the anisotropy of the superconducting energy gap as measured by ultrasonic attenuation, where q is the propagation vector of the sound wave and lis the "average" electron mean free path as determined from resistivity-ratio measurements. This result is in conflict with (1) the observations of Thomas 2 for ql values as low as ^0.25, and (2) the observation by the authors 3,4 of no frequency dependence of the anisotropy for ql values as low as 0.5. The authors 3,4 found a reduced anisotropy of the superconducting energy gap in impure Sn; it was ascertained that the anisotropy was a function of the transition temperature and not a function of ql Consequently, the reduced anisotropy was interpreted as a real change in the superconducting properties of the doped Sn system. The resolution of these differing results and interpretations are important to the application of the ultrasonic technique in the elucidation of the properties of real superconductors. This paper is intended as a comment on Ref. 4 in the light of the questions raised by Perz and Dobbs and also in consideration of recent thermal-conductivity measurements. 5 The BCS 6 theory predicts that for an isotropic superconductor the ratio of the compressional wave attenuation in the superconducting state to that in the normal state is given by a s /a n =2f(A), where/ is the Fermi function and A(T) the temperature-dependent superconducting energy gap. This result has been shown by Tsuneto 7 to be independent of sound frequency and electron mean free path for impurity-1 J. M. Perz and E. R. Dobbs, Proc. Roy. Soc. (London) A279, 408 (1967).166 limited scattering. Morse 8 and Leibowitz 9 have assumed that the above form is correct for anisotropic superconductors, but that for a given crystal direction the value of the limiting energy gap A(0) can assume a value different from the BCS value of 1.76 k B T c , where T c is the transition temperature. The A(0) for a given crystallographic direction was obtained in early studies by measuring the slope of a plot of ln(a s /a n ) versus T c /T over the low-temperature range, for which /(A)-exp[-A(0)/^r] and A(r)~A(0). In more recent studies the authors 3,4 have taken into account the temperature dependence of the energy gap and have used nonlinear least-squares fitting techniques to analyze t...