Superconductivity without a hole pocket in electron-doped FeSe: Analysis beyond the Migdal-Eliashberg formalism

High-$T_{\mathrm{c}}$ pairing mechanism absent of hole pockets in heavily electron-doped FeSe is one of the key unsolved problems in Fe-based superconductors. Here, this problem is attacked by focusing on the higher-order many-body effects neglected in conventional Migdal-Eliashberg formalism. We uncover two significant many-body effects for high-$T_{\mathrm{c}}$ superconductivity. (i) Due to the “vertex correction”, the dressed multiorbital Coulomb interaction acquires prominent orbital dependence for low-energy electrons. The dressed Coulomb interaction not only induces the orbital fluctuations, but also magnifies the electron-boson coupling constant. Therefore, moderate orbital fluctuations give strong attractive pairing interaction. (ii) The “multi-fluctuation-exchange pairing process” causes large interpocket attractive force, which is as important as the usual single-fluctuation-exchange process. Due to these two significant effects dropped in the Migdal-Eliashberg formalism, the anisotropic $s_{++}$-wave state in heavily electron-doped FeSe is satisfactorily explained. The proposed “inter-electron-pocket pairing mechanism” will enlarge $T_{\mathrm{c}}$ in other Fe-based superconductors.

Figure 1

(a) Hubbard interaction for the spin (charge) channel ${\widehat{U}}^{0s\left(c\right)}$, denoted as the black circle. (b) The three-point VC for the coupling constant between the electron and the fluctuations $\left(U\text{-VC}\right),{\mathrm{\Lambda }}_{l,l^{\prime };m,m^{\prime }}^x(k,p)$ $\left(x=s,c\right)$. $({\widehat{\mathrm{\Lambda }}}^x=\widehat{1}$ for $U\to 0$.) The $\left(U^0\right)$-linear terms are dropped since they are included in the RPA diagrams. The dressed coupling constant modified by the $U\text{-VC}$ is ${\widehat{U}}^x(k,p)={\widehat{\mathrm{\Lambda }}}^x(k,p){\widehat{U}}^{0x}$. (c) Beyond the RPA: ${\widehat{\mathrm{\Phi }}}^x\left(q\right)$ is the irreducible susceptibility with the VC. (d) Single-fluctuation-exchange pairing interaction beyond the ME approximation. The $U\text{-VC}$ is given by Eq. (4).

Figure 2

(a) FSs and band structure of the 1Fe-UC FeSe model with 15% $e$-doping. Green, red, and blue lines correspond to $xz,yz$, and $xy$ orbitals, respectively. In each FS, the $xy$ orbital has large weight near the $k_x$ or $k_y$ axis. (b) The spin and orbital susceptibilities obtained by the RPA. The latter is very small in the RPA. (c) The spin and orbital susceptibilities obtained by the SC-VC theory. The $U\text{-VCs}$ for the irreducible susceptibilities ${\widehat{\mathrm{\Phi }}}^{s,c}\left(q\right)$ are calculated self-consistently. Moderate ferro-orbital fluctuations are induced by the charge channel $U\text{-VC}$, consistent with the experimental phase diagram of $e$-doped FeSe.

Figure 3

Diagrammatic expression for the $U\text{-VC}$ analyzed in the present theoretical study. Double counting terms are subtracted carefully. The MT (AL) diagram inside the red (blue) dotted rectangle corresponds to the MT (AL) diagram in the pairing interaction in Fig. 1 [Fig. 5].

Figure 4

(a) Schematic fully gapped $s$-wave and $d$-wave states in the 1Fe-UC Brillouin zone. For former (latter) is realized when the interpocket pairing interaction is attractive (repulsive). (b) The $s_{++}$-wave state and $d$-wave state in the 2Fe-UC Brillouin zone. The inner FS and outer FS are formed due to the SOI-induced band hybridization. In the $d$-wave state, the gap structure becomes nodal and $T_{\mathrm{c}}$ is suppressed due to the SOI-induced hybridization [48]. (c) The $s_{\pm }$-wave state discussed in Ref. [59].

Figure 5

(a) $|{\mathrm{\Lambda }}_{3,3;3,3}^c(\theta ,{\theta }^{\prime }){|}^2(\gg 1)$ and (b) $|{\mathrm{\Lambda }}_{4,4;4,4}^s(\theta ,{\theta }^{\prime }){|}^2(\ll 1)$ in $e$-doped FeSe model at the lowest frequency $\left({\epsilon }_n={\epsilon }_n^{\prime }=\pi T\right)$, where $\theta$ and ${\theta }^{\prime }$ represent the Fermi momenta; see Fig. 2. We show the pairing interactions (c) $V_{\left(\mathrm{RPA}\right)}$ and (d) $V^{\mathrm{\Lambda }}$ on the FSs. The summation for the lowest Matsubara frequencies $\left({\epsilon }_n=\pm {\epsilon }_{n^{\prime }}=\pi T\right)$ is taken. The attractive interaction in (d) originates from the orbital fluctuations. (e) The crossing-fluctuation-exchange (=AL process) pairing interaction $V^{\mathrm{cross}}(k,p)$, which represents the multi-fluctuation-exchange processes. (f) $V^{\mathrm{cross}}$ on the FSs, in which strong attractive interpocket interaction gives the $s$-wave state.

Figure 6

Gap functions in the 1Fe-UC FeSe model obtained for the pairing interaction (a) $V_{\left(\mathrm{RPA}\right)}$, (b) $V^{\mathrm{\Lambda }}$ (with $U\text{-VC})$, and (c) $V^{\mathrm{tot}}$ (with $U\text{-VC}$ and crossing term), without the SOI. In (b) and (c), the $s$-wave state is realized due to the orbital fluctuations. In all cases, the gap function has maximum on the region with strong $d_{xy}$-orbital character. (d) FSs in the 2Fe-UC FeSe model for ${\lambda }_{\mathrm{SOI}}=80$ meV. The outer (inner) FS is mainly composed of the $xy$ orbital $(xz,yz$ orbitals). We present the obtained $s_{++}$-wave gap functions for (e) $V^{\mathrm{\Lambda }}$ and (f) $V^{\mathrm{tot}}$. (g) The $s_{++}$-wave state in the presence of impurities, by which the gap anisotropy is smeared. The anisotropic $s_{++}$-wave states in (f) and (g) are consistent with the experimental reports [42, 43, 46, 47].

Figure 7

Eigenvalue for the $s$-wave state ${\lambda }^s$ and that for the $d$-wave state ${\lambda }^d$ obtained for the pairing interaction (a) $V_{\left(\mathrm{RPA}\right)}$, (b) $V^{\mathrm{\Lambda }}$, and (c) $V^{\mathrm{tot}}$. We also show the eigenvalues for (d) $V^{\mathrm{SC}}=V^{\mathrm{\Lambda }=1}+V^{\mathrm{cross}}$.

Figure 8

${\left(U^0\right)}^2$ term for the $U\text{-VC}$, by which the spin channel $U\text{-VC}$ is suppressed. One bubble scales as $\sim O(2N_{\mathrm{orb}}-1)$, where $N_{\mathrm{orb}}$ is the $d$-orbital degrees of freedom.

Figure 9

Stoner factors obtained by the “full SC-VC” method performed in the main text. For comparison, the result given by the “charge-channel SC-VC” and that by the RPA are also shown.

Figure 10

(a) FSs and band structure of the bulk FeSe model $\left({\widehat{H}}_{\mathit{k}}^{0,\mathrm{b}}\right)$ with 15% $e$-doping. Green, red, and blue lines correspond to $xz,yz$, and $xy$ orbitals, respectively. (b) FSs in the 2Fe-UC FeSe model for ${\lambda }_{\mathrm{SOI}}=80$ meV. The outer (inner) FS is mainly composed of the $xy$ orbital $(xz,yz$ orbitals). (c) The obtained $s_{++}$-wave gap function for $V^{\mathrm{tot}}$ in the case of $r=0.36$ and $z_4=1/2$; $({\alpha }_S,{\alpha }_C)=(0.83,0.85)$.