Ultrasonic Studies of the Superconductivity of Doped Tin

Detailed information concerning the anisotropy of the effective limiting superconducting energy gap $\Delta \mathrm{}\left(0\right)\mathrm{}$ of doped Sn has been obtained from measurements of the temperature dependence of the attenuation of compressional ultrasonic waves. Earlier results for the In-doped Sn system have been confirmed for other impurities, and analysis of the results has been made in terms of recent developments in the theory of dirty superconductivity. Measurements have been made for the sound-propagation vector q parallel to the [001], [110], and [100] directions in single crystals of In-, Cd-, Sb-, and Bi-doped Sn with impurity concentrations ranging from ∼1 to 5 000 parts per million (ppm). The measured anisotropy between the [110] and [001] directions for pure Sn is ∼20% of the average energy gap. It is found that this anisotropy is reduced to ∼2% for values of the inverse resistivity ratio $\rho \gtrsim {10}^{-2\mathrm{}}$ and for $\Delta T_c>15\mathrm{}$ mdegK. BCS theory, which was developed for isotropic, weak-coupling superconductors, is applicable to the most heavily doped samples; it is found that the energy gap ($\sim 1.76k_B{T}_c$) and the temperature dependence of ultrasonic attenuation are in agreement with the predictions of the theory. By comparing the energy-gap measurements from tunneling data with the energy gap obtained from ultrasonic-attenuation data, it is possible to demonstrate that the temperature dependence of the attenuation gives a measure of more than one energy gap. In addition to the gap obtained from the attenuation data at low temperatures, a larger gap can be observed near $T_c$. From tunneling data, the larger gap for $\mathbf{q}\parallel \mathrm{}\left[001\right]\mathrm{}$ is $(1.9\mathrm{}\pm 0.05)k_B{T}_c$; the ultrasonic determination yields $(1.92\mathrm{}\pm 0.02)k_B{T}_c$. In addition, the temperature dependence of the larger gap is obtained near $T_c$ and is compared with BCS theory. For [$1-\frac{T}{T_c}$] ranging from ∼4×${10}^{-2\mathrm{}}$ to ∼4×${10}^{-1\mathrm{}}$, it is found that the BCS-predicted approximation, $\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{\Delta \mathrm{}\left(0\right)\mathrm{}}=1.74\mathrm{}{[1-\frac{T}{T_c}]}^{\frac{1}{2}}$, is accurate to ∼2%.