  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) using
first-principles calculations based on density functional theory. We found that
the antisymmetric exchange interaction, commonly known as Dzyaloshinskii-Moriya
interaction (DMI), contributes significantly to the energetics of the magnetic
structure. We used the full-potential linearized augmented plane wave method
and performed self-consistent calculations of homogeneous spin spirals,
calculating the DMI by treating the effect of spin-orbit interaction (SOI) 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 4 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 found 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 non-magnetic sites, the bandwidth and the energy difference
between states 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.
