Anisotropic Migdal-Eliashberg theory using Wannier functions

We combine the fully anisotropic Migdal-Eliashberg theory with electron-phonon interpolation based on maximally localized Wannier functions, in order to perform reliable and highly accurate calculations of the anisotropic temperature-dependent superconducting gap and critical temperature of conventional superconductors. Compared with the widely used McMillan approximation, our methodology yields a more comprehensive and detailed description of superconducting properties, and is especially relevant for the study of layered or low-dimensional systems as well as systems with complex Fermi surfaces. In order to validate our method we perform calculations on two prototypical superconductors, Pb and MgB${}_2$, and obtain good agreement with previous studies.

Figure 1

Calculated isotropic Eliashberg spectral function ${\alpha }^{2}F$ (black solid line) of Pb, and cumulative contribution to the electron-phonon coupling strength $\lambda$ (red dashed line). The top of the red dashed curve corresponds to $\lambda =1.24$.

Figure 2

Calculated energy-dependent superconducting gap of Pb at $T=0.3$ K. The gap is obtained by solving the isotropic Eliashberg equations with ${\mu }_{\mathrm{c}}^{*}=0.1$. (a) Superconducting gap along the imaginary energy axis (black solid line). (b) Superconducting gap along the real energy axis. We show both the solutions obtained from the approximate analytic continuation using Padé functions (black solid line and red dashed line), and the solutions obtained using the iterative analytic continuation (green dash-dotted line and blue dotted line).

Figure 3

(a) Calculated quasiparticle density of states of Pb at $T=0.3$ K (black solid lines). The superconducting gap is obtained from Fig. 2. (b) Same quantity as in (a), magnified in order to show the structure which is used in tunneling experiments for extracting the Eliashberg spectral function.

Figure 4

(a) Calculated superconducting gap of Pb at the Fermi level as a function of temperature (disks). The Coulomb parameter is set to ${\mu }_{\mathrm{c}}^{*}=0.1$. (b) Calculated superconducting gap of Pb at the Fermi level for $T=0$ K as a function of the Coulomb parameter ${\mu }_{\mathrm{c}}^{*}$ (disks). (c) Calculated critical temperature as a function of the Coulomb parameter ${\mu }_{\mathrm{c}}^{*}$ (disks). In all panels the solid lines are guides to the eye.

Figure 5

(a) Calculated isotropic Eliashberg spectral function ${\alpha }^{2}F$ of MgB${}_2$ (black solid line) and cumulative contribution to the electron-phonon coupling strength $\lambda$ (red dashed line). (b) Distribution of the electron-phonon coupling strength ${\lambda }_{\mathbf{k}}$ for MgB${}_2$ (black solid line).

Figure 6

Calculated energy-dependent superconducting gap of MgB${}_2$ at $T=10$ K. The gap is obtained by solving the fully anisotropic Eliashberg equations with ${\mu }_{\mathrm{c}}^{*}=0.16$. (a) Superconducting gap along the imaginary energy axis (black dots). (b) Superconducting gap along the real energy axis, obtained from the approximate analytic continuation using Padé functions (black dots). Only a representative sample of ${10}^5$ data points out of entire set of calculated gaps (${10}^7$ points) is shown for clarity.

Figure 7

(a) Calculated anisotropic superconducting gaps of MgB${}_2$ on the Fermi surface as a function of temperature. The Coulomb potential is set to ${\mu }_{\mathrm{c}}^{*}=0.16$. (b) Corresponding quasiparticle density of states for a few representative temperatures (15 K, black solid line; 30 K, red dashed line; 45 K, blue dash-dotted line).

Figure 8

Calculated energy distribution of the superconducting gaps on the $\sigma$ sheets of the Fermi surface of MgB${}_2$ at $T=30$ K (black lines). The gaps are obtained for various $\mathbf{k}$- and $\mathbf{q}$-point meshes, as indicated by the labels on the horizontal axis. The Coulomb parameter is set to ${\mu }_{\mathrm{c}}^{*}=0.16$.