Electron-phonon interaction and superconductivity in Tl-Pb-Bi alloys from first principles: Importance of spin-orbit coupling

The importance of spin-orbit coupling for lattice dynamics, electron-phonon coupling, and superconducting properties of face-centered cubic Tl-Pb-Bi alloys is investigated from first principles based on density-functional perturbation theory and a mixed-basis pseudopotential method. The alloys are modeled by the virtual crystal approximation. Electron-phonon coupling and superconducting properties are analyzed within the framework of the strong-coupling Eliashberg theory. A large increase of the coupling constant $\lambda$ for Bi doping can be traced back to a progressive softening of the phonon spectrum, signaling an incipient lattice instability. Performing calculations with and without spin-orbit interaction, we found a profound influence of this relativistic correction on the coupling strength, enhancing in most cases the coupling constant $\lambda$ by as much as 50$\%$, but reducing it for Tl-rich alloys. For the whole alloy series, spin-orbit interaction impressively improves the agreement with both experimental phonon spectra as well as with superconducting quantities obtained from tunneling spectroscopy measurements, which at the same time demonstrates the applicability of the virtual crystal approximation.

Figure 1

Evolution of the structural parameters $a$ and $B$ for the Tl-Pb-Bi alloys as a function of content $x$ for non-SOC and SOC schemes. Experimental data at low[16, 44] and room temperatures[45, 46] are included for comparison.

Figure 2

(a) Electronic band structures for selected Pb${}_{1-x}$Tl${}_x$ and Pb${}_{1-x}$Bi${}_x$ alloys, both without (black dashed lines) and with SOC (red full lines). (b) Density of states at the Fermi energy as a function of doping.

Figure 3

Phonon dispersion of Pb${}_{1-x}$Tl${}_x$ and Pb${}_{1-x}$Bi${}_x$ alloys for several doping concentrations. Shown are theoretical results obtained with (red full lines) and without SOC (black dashed lines). They are compared with experimental data by Ng et al. (Ref. 23) and Aynajian (Ref. 24) available for selected dopings.

Figure 4

Eliashberg functions for several Pb${}_{1-x}$Tl${}_x$ and Pb${}_{1-x}$Bi${}_x$ alloys as calculated with SOC (full red lines) and without SOC (black dashed lines). Open (blue) symbols represent spectra derived from tunneling spectroscopy measurements.[10]

Figure 5

(a) Average electron-phonon coupling constant $\lambda$ as a function of doping for Pb-Tl and Pb-Bi alloys. Results are given for both scalar-relativistic calculations (non-SOC, black dashed lines) and those including SOC (red full lines). The (green) line with diamonds represents a hypothetical $\lambda$, which is obtained by replacing the true doping-dependent phonon spectra by the one from pure Pb. Shown are also experimental data taken from Dynes et al.[10] (b) Doping dependence of the ratio of $\lambda$ and density of states at $E_F$ for the three calculations of (a).

Figure 6

(a) Superconducting critical temperature, $T_c\left(x\right)$, and (b) superconducting gap at $T=0$ K, $\Delta \left(x\right)$, as a function of doping concentration $x$ for Pb-Tl and Pb-Bi alloys calculated with (red full lines) and without SOC (black dashed lines). Filled (blue) squares denote experimental data taken from Dynes et al.[10]