Eliashberg functionα2F(ω)and phonon spectrumF(ω). A simple model for an amorphouss−psuperconductor

A simple expression of the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ for an amorphous $s-p$ superconductor is obtained. One finds that ${\alpha }^{2}F\left(\omega \right)=\left(\frac{1}{\omega }\right){\Sigma }_q\left[L\right(\overrightarrow{\mathrm{q}})+N(\overrightarrow{\mathrm{q}}\left)\right]\delta (\omega -{\omega }_q)$, where $\overrightarrow{\mathrm{q}}$ includes the phonon branch index; $\frac{N\left(\overrightarrow{\mathrm{q}}\right)}{{\omega }_q}$ is the $N$ process which conserves "lattice" momentum; $\frac{N\left(\overrightarrow{\mathrm{q}}\right)}{{\omega }_q}$ replaces the $U$ process in the crystalline phase and it selects "lattice" momentum through the structure factor of the amorphous phase. Both $L\left(\overrightarrow{\mathrm{q}}\right)$ and $N\left(\overrightarrow{\mathrm{q}}\right)$ are evaluated using the Heine-Animalu pseudopotential form factor and experimental structure factor. A qualitative deconvolution of the phonon spectrum $F\left(\omega \right)$ from ${\alpha }^{2}F\left(\omega \right)$ for four alloys is attempted. The main findings are: (i) Due to the nonconservation of "lattice" momentum in the conventional sense, ${\alpha }^{2}F\left(\omega \right)$ depends linearly on $\omega$ in the low-$\omega$ region, similar to Bergmann's results on disordered superconductors. The calculated first derivatives of ${\alpha }^{2}F\left(\omega \right)$ are in good agreement with tunneling data on amorphous superconductors. (ii) The deconvoluted $F\left(\omega \right)$ are in better agreement with theoretical results obtained using the Morse potential than the Lennard-Jones potential. (iii) The Hopfield-McMillan parameter $\eta$ tends to decrease in the amorphous phase. Results (ii) and (iii) are discussed in terms of structural short-range order in the amorphous state. For the purpose of comparison, a similar calculation is made for crystalline Pb; it is found that there is no unambiguous enhanced ${\alpha }^{2}F\left(\omega \right)$ in the low-$\omega$ region.