Orbital fluctuation mediated superconductivity and structure transition in iron-based superconductors

The main features in Fe-based superconductors are summarized as (i) orthorhombic transition accompanied by remarkable softening of the shear modulus $C_{66}$, (ii) high-$T_{\rm c}$ superconductivity close to the orthorhombic phase, and (iii) stripe-type magnetic order induced by orthorhombicity. To understand them, we analyze the multiorbital Hubbard-Holstein model with Fe-ion optical phonons. In the random-phase-approximation (RPA), a small electron-phonon coupling constant ($\lambda\sim0.2$) is enough to produce large orbital (=charge quadrupole) fluctuations. The most divergent susceptibility is the $O_{xz}$-antiferro-quadrupole (AFQ) susceptibility, which causes the $s$-wave superconductivity without sign reversal ($s_{++}$-wave state). \footnote{H. Kontani and S. Onari, Phys. Rev. Lett. {\bf 104}, 157001 (2010).} The $_{s++}$-wave state is robust against impurities, \footnote{S. Onari and H. Kontani, Phys. Rev. Lett. {\bf 103}, 177001 (2009).} consistently with experimental observations. At the same time, divergent development of $O_{x^2-y^2}$-ferro-quadrupole (FQ) susceptibility is brought by the ``two-orbiton process'' with respect to the AFQ fluctuations.