Multiband full-bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and $\mathbf{high}\mathbf{\text{−}}{\mathit{T}}_c$ superconductivity in $\mathbf{FeSe}/{\mathbf{SrTiO}}_{\mathbf{3}}$

We examine the impact of interfacial phonons on the superconducting state of $\mathrm{FeSe}/{\mathrm{SrTiO}}_3$ developing a material's specific multiband, full bandwidth, and anisotropic Eliashberg theory for this system. Our self-consistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak-coupling constant ${\lambda }_m=0.4$, in mediating the high $T_c$, and explain other puzzling experimental observations, such as the $s$-wave symmetry and replica bands. We discover that the formation of replica bands has a $T_c$ decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a $T_c$ enhancing effect. We predict a strong-coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.

Figure 1

Results of self-consistent multiband Eliashberg theory calculations. (a) Calculated full bandwidth spectral function at 10 K, relevant to ARPES measurements, plotted along the $M-\mathrm{\Gamma }-M$ symmetry line of the BZ. The bare electron dispersions, used as input, are depicted by the white dashed lines. Several shake-off effects are visible in the quasiparticle band structure which are caused by interfacial EPI. (b) Spectral function for small energies near the $M$ point $[=(\pi ,\pi \left)\right]$ of the BZ [the red rectangle in (a)]. The white dotted lines are guides to the eye. The formation of replica bands 110 meV deeper than the main electron bands is clearly resolved [6]. (c) The spectral function (the red line) at $M$ exhibits a main peak at $-50\mathrm{meV}$ and an additional weaker peak near $-160\mathrm{meV}$. The former peak is due to the main electron bands that form the Fermi surface in the normal state whereas the latter is their replica band contribution. The integrated experimental intensity at the $M$ point [6] is shown by the black line. (d) Calculated temperature dependence of the superconducting gap edge maxima, giving $\mathrm{\Delta }/k_{\mathrm{B}}{T}_c\approx 2.1$.

Figure 2

Calculated momentum dependence of the superconducting gap and renormalization functions. (a) The gap edge $\mathrm{\Delta }\left(\mathbf{k}\right)$ is anisotropic ($\sim 25\%$) over the Fermi surface with the average value of $\sim 10\mathrm{meV}$. (b) The renormalized chemical potential $\chi \left(\mathbf{k}\right)$ exhibits sign changes across the Fermi surface. (c) The mass renormalization function $Z\left(\mathbf{k}\right)$ at $T<T_c$ and (d) the same quantity at $T>T_c$. The insets show the distributions of the respective quantities over the Fermi surface.

Figure 3

Influence of included band dispersions on the superconductivity in monolayer FeSe/STO. (a) The calculated $T_c$, starting from electrons restricted only to the Fermi surface (blue) and subsequently including their full bandwidth (purple) and electron and hole (red) band contributions until the full ten-band calculation is recovered (green). Spectral weight transfer to the replica bands weakens ${\lambda }_m$ and therefore reduces $T_c$ (purple). The interfacial phonon mode also couples weakly to bands away from the Fermi level, thus providing an additional channel to compensate this loss and even increase $T_c$ (red and green). (b) and (c) Cooper-pair binding energy $\mathrm{\Delta }(\mathbf{k},E)$, projected on each band of the renormalized electronic band structure due to the interfacial EPI. In (b) the red (blue) color depicts electron attraction (repulsion) and thus Cooper-pair formation (breaking). Apart from the dominant contribution of electrons in bands forming the Fermi surface ($M$ point), deep Fermi-sea pairing takes place near the $\mathrm{\Gamma }$ point and along the $X-M$ high-symmetry line of the folded Brillouin zone.

Figure 4

Calculated tunneling spectra and predicted dip-hump signatures. (a) The differential conductance $dI/dV$ in the energy regime relevant to superconductivity normalized by the normal-state value. A full gap with main coherence peaks at $\mathrm{\Delta }\approx \pm 11\mathrm{meV}$ is in agreement with STS observations of plain $s$-wave superconductivity [13]. The kinks near $\pm (\mathrm{\Delta }+\mathrm{\Omega })\approx \pm 91\mathrm{meV}$ are signatures of the underlying electron-boson interaction, caused by the interfacial phonon mode. Additional features with the form of a dip-hump structure appear at higher energies, marked with $1{(}^{\prime }1)$ and 2, respectively. (b) $dI/dV$ in the low-energy regime for direct comparison with STS data [13]. (c) A zoom in of the dip-hump region of the normalized $dI/dV$ reveals a dip with minima near 150 meV and a hump with maxima near 220 meV. Within standard isotropic Eliashberg theory (the green dotted line), this dip-hump signature corresponds to a ratio of $\mathrm{\Delta }/k_{\mathrm{B}}{T}_c\approx 2.1$ but with a strong-coupling value of ${\lambda }_{\mathrm{iso}}=1.6$.