Ab initio theory of plasmonic superconductivity within the Eliashberg and density-functional formalisms

We extend the two leading methods for the ab initio computational description of phonon-mediated superconductors, namely Eliashberg theory and density-functional theory for superconductors (SCDFT), to include plasmonic effects. Furthermore, we introduce a hybrid formalism in which the Eliashberg approximation for the electron-phonon coupling is combined with the SCDFT treatment of the dynamically screened Coulomb interaction. The methods have been tested on a set of well-known conventional superconductors by studying how the plasmon contribution affects the phononic mechanism in determining the critical temperature $\left(T_C\right)$. Our simulations show that plasmonic SCDFT leads to a good agreement between predicted and measured $T_C$'s, whereas Eliashberg theory considerably overestimates the plasmon-mediated pairing and, therefore, $T_C$. The hybrid approach, on the other hand, gives results close to SCDFT and overall in excellent agreement with experiments.

Figure 1

Left: Eliashberg superconducting gap function $\varphi$ for bulk Nb in the full dynamical (blue), weak-coupling dynamical (red), and static (black) Coulomb approach. Orange ticks show the positions of the Matsubara points at $T=2.8K$. Right (same color code): Eliashberg mass renormalization function $Z$ and its decomposition, for the dynamical case, into Coulomb $Z^c$ and phononic $Z^{\mathrm{ph}}$ components.

Figure 2

Left: SCDFT gap function ${\mathrm{\Delta }}_s$ for bulk Nb in the dynamical (blue), weak-coupling dynamical (red), and static (black) Coulomb approach. Right: SCDFT $\mathcal{Z}$ kernel and its decomposition (for the dynamical case) into Coulomb ${\mathcal{Z}}^c$ and phononic ${\mathcal{Z}}^{\mathrm{ph}}$ components. All the quantities are computed in the low-temperature limit and using the SPG phononic functional.

Figure 3

Electron energy loss spectra in the low-$\mathbf{q}$ limit, showing the position of the main plasmonic peaks for the chosen set of materials.

Figure 4

Bottom: Eliashberg critical temperatures computed within the full dynamical (violet), weak-coupling dynamical (green), and static (blue) approach for the screened Coulomb interaction. Note the use of a logarithmic scale on the ordinate axis. Top: Percentage error (computed with respect to the experimental critical temperature). Errors are highlighted by a color scale from green (accurate) to yellow and red (large deviation).

Figure 5

Bottom: SCDFT critical temperatures computed using the LM (left) and SPG (right) phononic functional, within the full dynamical (violet), weak-coupling dynamical (green), and static (blue) approach for the screened Coulomb interaction. Note the use of a logarithmic scale on the ordinate axis. Top: Percentage error (computed with respect to the experimental critical temperature). Errors are highlighted by a color scale from green (accurate) to yellow and red (large deviation).

Figure 6

Bottom: Hybrid Eliashberg-SCDFT critical temperatures computed within the full dynamical (violet), weak-coupling dynamical (green), and static (blue) approach for the screened Coulomb interaction. Note the use of a logarithmic scale on the ordinate axis. Top: Percentage error (computed with respect to the experimental critical temperature). Errors are highlighted by a color scale from green (accurate) to yellow and red (large deviation).