Effects of spin-orbit coupling on the spin structure of deposited transition-metal clusters

The influence of the spin-orbit coupling on the magnetic structure of deposited transition-metal nanostructures has been studied by fully relativistic electronic-structure calculations. The interplay of exchange coupling and magnetic anisotropy was monitored by studying the corresponding magnetic torque calculated within ab initio and model approaches. We find that a spin-orbit-induced Dzyaloshinski-Moriya interaction has a profound effect on the spin structure of such complex magnetic systems and that in combination with magnetic anisotropies and isotropic exchange this can result in peculiar magnetic properties.

Figure 1

(Color online) Magnetic torque results for Fe, Co, and Ni dimers on Pt(111) for $\theta =\pi /4$ as a function of the azimuth angle $\varphi$. (a): magnetic configuration of the transition-metal dimers deposited on a Pt(111) substrate. The large (small) spheres present Pt surface (subsurface) atoms. The medium size spheres represent the dimer atoms with the projection of their magnetic moments onto the surface ($xy$ plane) represented by arrows. The projection of the DM-vector ${\vec{D}}_{12}$ onto the surface is represented by a short arrow. (b): ab initio $T_i^{\text{dir}}$ compared with HM $T_i^{\text{mod}}$ for ${\text{Co}}_2$ on Pt(111). (c): $T_i^{\text{DM}}$ and $T_i^K$ contributions to $T_i^{\text{mod}}$ for ${\text{Co}}_2$ on Pt(111). [(d)–(f)]: ab initio $T_i^{\text{dir}}$ for the two atoms of (d) ${\text{Fe}}_2$, (e) ${\text{Co}}_2$, and (f) ${\text{Ni}}_2$. $T_1^{\text{dir}}$ and $T_2^{\text{dir}}$ are presented by circles and squares, respectively, and their sum by the thick solid line. The thin solid lines give the DM contributions according to Eq. (4).

Figure 2

(Color online) Magnetic structure of a 91-atom FePt $\left(2\times 1\right)$ 2D alloy cluster deposited on a Pt(111) surface determined via MC simulations for $T=0.1\text{K}$. The arrows denote the orientation of the magnetic moments associated with the iron atoms.