Exchange interactions in $d^{5}$ Kitaev materials: From Na$_2$IrO$_3$ to $\alpha$-RuCl$_3$

We present an analytical study of the exchange interactions between pseudospin one-half $d^{5}$ ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard U, charge-transfer excitations, and cyclic exchange are considered. Hoppings within $t_{2g}$ orbitals as well as between $t_{2g}$ and $e_g$ orbitals are included. Special attention is paid to the trigonal crystal field $\Delta$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction K within a wide range of $\Delta$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\Delta$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic K and positive non-diagonal $\Gamma$ values. However, their Heisenberg J terms have opposite signs: AFM J>0 in Na$_2$IrO$_3$ and FM J<0 in $\alpha$-RuCl$_3$. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM J is emphasized in $\alpha$-RuCl$_3$. The differences between the exchange couplings of these two materials originate from the opposite $\Delta$ values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of $d^{5}$ spin-orbit Mott insulators.