Theoretical investigation of the magnetic dynamics and superconducting pairing symmetry in $\alpha$-RuCl$_3$

We study the spin-wave excitations in $\alpha$-RuCl$_3$ by the spin-wave theory. Starting from the five-orbital Hubbard model and the perturbation theory, we derive an effective isospin-$1/2$ model in the large Hubbard ($U$) limit. Based on the energy-band structure calculated from the first-principle method, we find that the effective model can be further reduced to the $K-\Gamma$ model containing a ferromagnetic nearest-neighbor (NN) Kitaev interaction ($K$) and a NN off-diagonal exchange interaction ($\Gamma$). With the spin-wave theory, we find that the $K-\Gamma$ model can give magnetic excitations which is consistent with the recent neutron scattering experiments. Furthermore, to investigate various particle-hole excitations and possible superconducting pairing symmetry in the doped systems, we ignore the effects of the $e_g$ orbitals and use the random-phase approximation. Our results show that the $J_{\rm eff}=1/2$ picture is robust in the doped systems, and the $d$-wave pairing is the most favourable superconducting pairing symmetry.