Possible Superconductivity Induced by Strong Spin-Orbit Coupling in Carrier Doped Iridium Oxides Insulators

$5d$ transition metal oxide Sr$_{2}$IrO$_{4}$ and its relevant Iridium oxides have attracted much interest because of exotic properties arising from highly entangled spin and orbital degrees of freedom due to strong spin-orbit coupling (SOC). Sr$_{2}$IrO$_{4}$ crystalizes in the layered perovskite structure, similar to cuprates. Five $5d$ electrons in Ir occupy its $t_{2g}$ orbitals which are split by strong SOC, locally inducing an effective total angular momentum $J_{\textrm{eff}}=1/2$, analogous to a $S=1/2$ state in cuprates. Because of the similarities to cuprates, the possibility of superconductivity (SC) in Iridium oxides has been expected theoretically once mobile carriers are introduced into the $J_{\textrm{eff}}=1/2$ antiferromagnetic insulator [1]. To study theoretically possible SC in carrier doped Sr$_{2}$IrO$_{4}$, we investigate a three-orbital Hubbard model with SOC. By solving the Eliashberg equation in the random phase approximation, we find that $J_{\textrm{eff}}=1/2$ antiferromagnetic fluctuations favor $d_{x^{2}-y^{2}}$-wave SC with a mixture of singlet and triplet Cooper pairings. We will also discuss the particle-hole asymmetry of the SC induced by electron and hole doping. [1] H. Watanabe, et. al., Phys. Rev. Lett. {\bf 110}, 027002 (2013)