Effective theory of superconductivity in strongly coupled amorphous materials

A theory of phonon-mediated superconductivity in strong-coupling amorphous materials is developed based on an effective description of structural disorder and its effect on the vibrational spectrum. The theory accounts for the diffusivelike transport of vibrational excitations due to disorder-induced scattering within the Eliashberg theory of strong-coupling superconductivity. The theory provides a good analytical description of the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ in comparison with experiments and allows one to disentangle the effects of transverse and longitudinal excitations on the Eliashberg function. In particular, it shows that the transverse excitations play a crucial role in driving an increase or excess in the Eliashberg function at low energy, which is related to the boson peak phenomenon in vibrational spectra of glasses. This low-energy excess, on the one hand, drives an enhancement of the electron-phonon coupling but at the same time reduces the characteristic energy scale ${\omega }_{\log }$ in the Allen-Dynes formula. As a consequence, the nonmonotonicity of $T_c$ as a function of alloying (disorder) in Pb-based systems can be rationalized. The case of Al-based systems, where disorder increases $T_c$ from the start, is also analyzed. General material-design principles for enhancing $T_c$ in amorphous superconductors are presented.

Figure 1

One-branch calculation of the electron-phonon spectral function ${\alpha }^{2}F\left(\omega \right)$. The $y$ axis is shown in arbitrary units. The diffusion constant goes from $D=0.1$ (lighter line) to $D=2$ (darker line). The values of the other parameters are kept fixed. The inset shows the low-frequency scaling $\sim \omega$ for the red line data. The scaling is independent of the value of the diffusion constant $D$.

Figure 2

Electron-phonon coupling constant $\lambda$ as a function of the diffusivity $D$ of transverse vibrational excitations for various values of the transverse speed of sound. The units for both axes are arbitrary. The inset shows how the position of the maximum varies as a function of the speed of sound; the curve is fitted very well by a linear relation (solid line). The relative amplitude of the peak between the line at large $v$ (bottom purple line) and the one at small $v$ (top blue line) where the peak is maximum is about a factor of 5.

Figure 3

Theoretical model fittings of the electron-phonon spectral function for Pb-based materials. The data are taken from Ref. [7] and refer to pure crystalline Pb (blue), granular microcrystalline Pb (red), the amorphous alloy ${\text{Pb}}_{0.9}{\text{Cu}}_{0.1}$ (black), and the amorphous alloy ${\text{Pb}}_{0.75}{\text{Bi}}_{0.25}$ (orange), with the latter taken from Ref. [37].

Figure 4

Behavior of characteristic physical parameters describing superconductivity for the four different Pb-based systems with varying disorder as a function of the transverse diffusivity parameter $D_T$. Symbols refer to the same systems of Fig. 4, with disorder in the experimental systems increasing from left to right in all plots. Top: the characteristic energy scale ${\omega }_{\log }$ (a prefactor in the Allen formula for $T_c$). Middle: the electron-phonon coupling constant $\lambda$ for the four systems. Bottom: the critical temperature for the onset of superconductivity $T_c$ calculated from the Allen formula (12) in the text. For reference, the experimentally measured $T_c$ values are (in order of increasing $D_T$) 7.2 K (pure Pb), 7.19 K) (granular microcrystalline Pb), 6.5 K (${\text{Pb}}_{0.9}{\text{Cu}}_{0.1}$), and 6.9 K (${\text{Pb}}_{0.75}{\text{Bi}}_{0.25}$). These data are explicitly tabulated in [36, 37], where also the original experimental references are quoted.

Figure 5

Theoretical modeling of Eliashberg function for Al-based materials. Symbols: experimental data for the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ measured by electron tunneling from Ref. [38]. Solid lines: model fitting for crystalline Al (solid line) and granular Al (dashed line). The inset shows the different scalings in the low-energy limit for crystalline and granular Al.