Role of Dzyaloshinskii-Moriya interaction for magnetism in transition-metal chains at Pt step edges

We explore the emergence of chiralmagnetism in one-dimensional monatomic Mn, Fe, and Co chains deposited at the Pt(664) step edge carrying out an ab initio study based on density functional theory (DFT). The results are analyzed employing several models: (i) a micromagnetic model, which takes into account the Dzyaloshinskii-Moriya interaction (DMI) besides the spin stiffness and the magnetic anisotropy energy, and (ii) the Fert-Levy model of the DMI for diluted magnetic impurities in metals. Due to the step-edge geometry, the direction of the Dzyaloshinskii vector (D vector) is not predetermined by symmetry and points in an off-symmetry direction. For the Mn chain we predict a long-period cycloidal spin-spiral ground state of unique rotational sense on top of an otherwise atomic-scale antiferromagnetic phase. The spins rotate in a plane that is tilted relative to the Pt surface by 62. towards the upper step of the surface. The Fe and Co chains show a ferromagnetic ground state since the DMI is too weak to overcome their respective magnetic anisotropy barriers. An analysis of domain walls within the latter two systems reveals a preference for a Bloch wall for the Fe chain and a Neel wall of unique rotational sense for the Co chain in a plane tilted by 29. towards the lower step. Although the atomic structure is the same for all three systems, not only the size but also the direction of their effective D vectors differ from system to system. The latter is in contradiction to the Fert-Levy model. Due to the considered step-edge structure, this work provides also insight into the effect of roughness on DMI at surfaces and interfaces of magnets. Beyond the discussion of the monatomic chains we provide general expressions relating ab initio results to realistic model parameters that occur in a spin-lattice or in a micromagnetic model. We prove that a planar homogeneous spiral of classical spins with a given wave vector rotating in a plane whose normal is parallel to the D vector is an exact stationary state solution of a spin-lattice model for a periodic solid that includes Heisenberg exchange and DMI. In the vicinity of a collinear magnetic state, assuming that the DMI is much smaller than the exchange interaction, the curvature and slope of the stationary energy curve of the spiral as a function of the wave vector provide directly the values of the spin stiffness and the spiralization required in micromagnetic models. The validity of the Fert-Levy model for the evaluation of micromagnetic DMI parameters and for the analysis of ab initio calculations is explored for chains. The results suggest that some care has to be taken when applying the model to infinite periodic one-dimensional systems.