Tunneling ${\alpha }^{2}F\left(\omega \right)$ and heat-capacity measurements in high-$T_c$ ${\mathrm{Nb}}_3$Ge

Improved film synthesis has allowed us to prepare ${\mathrm{Nb}}_3\frac{\mathrm{Ge}}{\mathrm{Si}}\frac{{\mathrm{O}}_x}{\mathrm{Pb}}$ tunnel junctions on samples with high $T_c$ (up to 21.2 K) and large energy gap (${\Delta }_{{\mathrm{Nb}}_3\mathrm{Ge}}$ up to 3.85 meV). These junctions have satisfactory features for taking derivative measurements. The data were reduced by the modified McMillan-Rowell proximity gap inversion analysis developed by Arnold and Wolf to generate ${\alpha }^{2}F\left(\omega \right)$ and related microscopic parameters. The trend previously seen of a movement of the lowest phonon branch to lower energies as the $T_c$ and gap increase is continued, resulting in the lowest-energy phonon mode being well defined and enhanced in strength. Theoretical functional derivatives for $T_c$ (by Bergmann and Rainer) and the energy gap (by Mitrovic et al.) qualitatively explain the rise in $T_c$, energy gap, and $\frac{2\Delta }{k_B{T}_c}$. Heat-capacity measurements have been performed on various samples to give bulk ${\mathrm{Nb}}_3$Ge properties, including one sample which was analyzed by tunneling ${\alpha }^{2}F\left(\omega \right)$ and heat capacity. $\gamma =34\mathrm{}\pm 1.5$ mJ/mole ${\mathrm{K}}^2$ and the bare density of states, $N\left(0\right)=1.5\mathrm{}\pm 0.1$ states/eV atom, suggests that a high density of states is inadequate to explain the high $T_c$ in ${\mathrm{Nb}}_3$Ge. Values for $\frac{2\Delta }{k_B{T}_c}=4.2\mathrm{}\pm 0.1$ and $\frac{\Delta C}{\gamma T_c\sim 1.9}$ indicate a strong-coupled superconductor, in agreement with tunneling results on this sample.