Competing Extended-$s$- and $d$-Wave Pairing from Distinct Spin-Fluctuation Channels in Stoichiometric $\mathrm{FeTe}$
The recent observation of superconductivity in stoichiometric $\mathrm{FeTe}$ raises the question of how pairing develops in this tetragonal 11-type chalcogenide once interstitial Fe is removed. We construct an experimentally constrained five-orbital tight-binding model from first-principles calculations and treat electronic correlations and pairing within the fluctuation-exchange approximation. The linearized Eliashberg equation yields competing extended-$s$- and $d_{x^2-y^2}$-wave spin-singlet pairing instabilities. Near stoichiometric filling, spin fluctuations near $(π,0)$ and $(0,π)$ in the unfolded one-Fe Brillouin zone connect the $Γ/M$ hole pockets with the $X/Y$ electron pockets and favor an extended-$s$ gap that changes sign between the hole and electron sheets. Upon electron doping, depletion of the hole pockets shifts the dominant scattering toward the $X$--$Y$ channel near $(π,π)$, making the $d_{x^2-y^2}$-wave state the leading instability, with nodal lines that avoid most of the Fermi surface. The relative strengths of the two pairing channels vary with filling and interaction strength as the dominant spin-fluctuation channel changes. Under matched interaction strength, temperature, and filling, the extended-$s$ eigenvalue is larger in $\mathrm{FeTe}$ than in $\mathrm{FeSe}$ throughout the range considered, while the $d_{x^2-y^2}$-wave eigenvalue is also generally larger, particularly under electron doping. These results give concrete gap structures against which spectroscopic measurements of stoichiometric $\mathrm{FeTe}$ can be compared.