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We investigate the chiral magnetic order in free-standing planar 3$d$-5$d$bi-atomic metallic chains (3$d$: Fe, Co; 5$d$: Ir, Pt, Au) usingfirst-principles calculations based on density functional theory. We found thatthe antisymmetric exchange interaction, commonly known as Dzyaloshinskii-Moriyainteraction (DMI), contributes significantly to the energetics of the magneticstructure. We used the full-potential linearized augmented plane wave methodand performed self-consistent calculations of homogeneous spin spirals,calculating the DMI by treating the effect of spin-orbit interaction (SOI) inthe basis of the spin-spiral states in first-order perturbation theory. To gaininsight into the DMI results of our ab initio calculations, we develop aminimal tight-binding model of three atoms and 4 orbitals that contains allessential features: the spin-canting between the magnetic $3d$ atoms, thespin-orbit interaction at the $5d$ atoms, and the structure inversion asymmetryfacilitated by the triangular geometry. We found that spin-canting can lead tospin-orbit active eigenstates that split in energy due to the spin-orbitinteraction at the $5d$ atom. We show that, the sign and strength of thehybridization, the bonding or antibonding character between $d$-orbitals of themagnetic and non-magnetic sites, the bandwidth and the energy differencebetween states occupied and unoccupied states of different spin projectiondetermine the sign and strength of the DMI. The key features observed in thetrimer model are also found in the first-principles results.

Dzyaloshinskii-Moriya interaction and chiral magnetism in 3$d$-5$d$ zig-zag chains: Tight-binding model and ab initio calculations