Dzyaloshinskii-Moriya interaction and chiral magnetism in 3d − 5d zigzag chains: Tight-binding model and ab initio calculations

We investigate the chiral magnetic order in freestanding planar 3d-5d biatomic metallic chains (3d :F e, Co; 5d: Ir, Pt, Au) using first-principles calculations based on density functional theory. We find that the antisymmetric exchange interaction, commonly known as the Dzyaloshinskii-Moriya interaction (DMI), contributes significantly to the energetics of the magnetic structure. For the Fe-Pt and Co-Pt chains, the DMI can compete with the isotropic Heisenberg-type exchange interaction and the magnetocrystalline anisotropy energy, and for both cases a homogeneous left-rotating cycloidal chiral spin-spiral with a wavelength of 51 u A and 36 u A, respectively, was found. The sign of the DMI, which determines the handedness of the magnetic structure, changes in the sequence of the 5d atoms Ir(+), Pt(−), Au(+). We use the full-potential linearized augmented plane wave method and perform self-consistent calculations of homogeneous spin spirals, calculating the DMI by treating the effect of spin-orbit interaction in the basis of the spin-spiral states in first-order perturbation theory. To gain insight into the DMI results of our ab initio calculations, we develop a minimal tight-binding model of three atoms and four orbitals that contains all essential features: the spin canting between the magnetic 3d atoms, the spin-orbit interaction at the 5d atoms, and the structure inversion asymmetry facilitated by the triangular geometry. We find that spin canting can lead to spin-orbit active eigenstates that split in energy due to the spin-orbit interaction at the 5d atom. We show that the sign and strength of the hybridization, the bonding or antibonding character between d orbitals of the magnetic and nonmagnetic sites, the bandwidth, and the energy difference between occupied and unoccupied states of different spin projection determine the sign and strength of the DMI. The key features observed in the trimer model are also found in the first-principles results.