# Possible superconductivity induced by a large spin-orbit coupling in carrier doped iridium oxide insulators: A weak coupling approach

## TL;DR

This study investigates the potential for superconductivity in doped iridium oxide Sr$_{2}$IrO$_{4}$, revealing that large spin-orbit coupling favors $d_{x^{2}-y^{2}}$-wave singlet pairing in electron-doped cases and $s_{ ext{±}}$-wave pairing in hole-doped cases, with implications for strongly correlated materials.

## Contribution

It provides a systematic analysis of pairing symmetries in doped iridium oxides considering strong spin-orbit coupling using a three-orbital Hubbard model and Eliashberg equations.

## Key findings

- Large SOC favors $d_{x^{2}-y^{2}}$-wave pairing in electron doping.
- Small SOC and large Hund's coupling favor $s_{ ext{±}}$-wave pairing in hole doping.
- Pseudospin triplet pairing is unfavorable across studied parameters.

## Abstract

We study possible superconductivity in a carrier-doped iridium oxide insulator Sr$_{2}$IrO$_{4}$ based on an effective $t_{2g}$ three-orbital Hubbard model on the square lattice with a large spin-orbit coupling (SOC). Numerically solving the linearized Eliashberg equation for the superconducting (SC) gap function with the random phase approximation, we systematically examine both singlet and triplet SC gap functions with possible pairing symmetries and the parameter dependence of the superconductivity. For the realistic SOC $\lambda$ and Hund's coupling $J/U$ relevant to Sr$_{2}$IrO$_{4}$, namely, for a large $\lambda$ and small $J/U$ region, we find that the intra-band antiferromagnetic (AF) pseudospin $\bm{j}_{\text{eff}} = -\bm{l}+\bm{s}$ fluctuations favor a $d_{x^{2}-y^{2}}$-wave pseudospin $j_{\text{eff}}=1/2$ singlet pairing in the electron-doping. We also find that the $d_{x^{2}-y^{2}}$-wave pairing is more stabilized with increasing the SOC and decreasing the Hund's coupling. Furthermore, we show for a small $\lambda$ and large $J/U$ region that an $s_{\pm}$-wave singlet pairing is favored in the hole-doped region. The origin of the $s_{\pm}$-wave pairing is due to the inter-band pair scattering arising from the intra-orbital AF spin $\bm s$ fluctuations. Although the possibility of a pseudospin triplet pairing is considered, we find it always unfavorable for all parameters studied here. The experimental consequences for other strongly correlated materials with a large SOC are also discussed.