Exchange interactions in d(5) Kitaev materials: From Na2IrO3 to alpha-RuCl3

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 Na2IrO3 and alpha-RuCl3 are discussed based on our theory. We have found that both materials share dominant ferromagnetic K and positive nondiagonal Gamma values. However, their Heisenberg J terms have opposite signs: AFM J > 0 in Na2IrO3 and FM J < 0 in alpha-RuCl3. 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-RuCl3. 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.