Characterization of iron ferromagnetism by the local atomic volume: from three-dimensional structures to isolated...

We present a comprehensive study of the relationship between the ferromagnetism and the structural properties of Fe systems from three-dimensional ones to isolated atoms based on the spin-density functional theory. We have found a relation between the magnetic moment and the volume of the Voronoi polyhedron, determining, in most cases, the value of the total magnetic moment as a function of this volume with an average accuracy of ±0.28 μ(B) and of the 3d magnetic moment with an average accuracy of ±0.07 μ(B) when the atomic volume is larger than 22 ų. It is demonstrated that this approach is applicable for many three-dimensional systems, including high-symmetry structures of perfect body-centered cubic (bcc), face-centered cubic (fcc), hexagonal close-packed (hcp), double hexagonal close-packed (dhcp), and simple cubic (sc) crystals, as well as for lower-symmetry ones, for example atoms near a grain boundary (GB) or a surface, around a vacancy or in a linear chain (for low-dimensional cases, we provide a generalized definition of the Voronoi polyhedron). Also, we extend the validity of the Stoner model to low-dimensional structures, such as atomic chains, free-standing monolayers and surfaces, determining the Stoner parameter for these systems. The ratio of the 3d-exchange splitting to the magnetic moment, corresponding to the Stoner parameter, is found to be I(3d) = (0.998 ± 0.006) eV /μ(B) for magnetic moments up to 3.0 μ(B). Further, the 3d exchange splitting changes nearly linearly in the region of higher magnetic moments (3.0-4.0 μ(B)) and the corresponding Stoner exchange parameter equals I(h)(3d) = (0.272 ± 0.006) eV /μ(B). The existence of these two regions reflects the fact that, with increasing Voronoi volume, the 3d bands separate first and, consequently, the 3d magnetic moment increases. When the Voronoi volume is sufficiently large (≥22 ų), the separation of the 3d bands is complete and the magnetic moment reaches a value of 3.0 μ(B). Then, when the volume further increases, the 4s bands start to separate, increasing thus the 4s magnetic moment. Surprisingly, in the region of higher magnetic moments (≥3.0 μ(B)), there is also a linear relationship between the 4s exchange splitting and the total magnetic moment with a slope of I(h)(4s) = (1.053 ± 0.016) eV /μ(B), which is nearly identical to I(3d) for magnetic moments up to 3.0 μB. These linear relations can be considered as an extension of the Stoner model for low-dimensional systems.