Exchange Interactions in D5 Kitaev Materials: from Na2IrO3 to Α−

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 $\mathrm{\ensuremath{\Delta}}$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction $K$ within a wide range of $\mathrm{\ensuremath{\Delta}}$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\mathrm{\ensuremath{\Delta}}$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic $K$ and positive nondiagonal $\mathrm{\ensuremath{\Gamma}}$ values. However, their Heisenberg $J$ terms have opposite signs: AFM $J>0$ in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and FM $J<0$ in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. 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 $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. The differences between the exchange couplings of these two materials originate from the opposite $\mathrm{\ensuremath{\Delta}}$ 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.