Dzyaloshinskii-Moriya interaction accounting for the orientation of magnetic domains in ultrathin films: Fe/W(110)

Combining relativistic first-principles calculations with a micromagnetic model, we establish the Dzyaloshinskii-Moriya interaction as an important mechanism in thin-film magnetism, determining the orientation of magnetic domains relative to the lattice, the type of domain wall, and the rotational direction of the magnetization in the wall. Applying the analysis to two monolayers Fe on W(110), we provide an explanation for puzzling experimental data obtained by spin-polarized scanning tunneling microscopy.

Figure 1

(Cf. Ref. 13) Magnetic structure of 2 ML Fe/W(110) imaged with SP-STM. The bright and dark areas correspond to domains that are aligned out of plane and parallel or antiparallel to the magnetization of the STM tip. The thin vertical lines originate from step edges and other geometrical irregularities. The domain walls are oriented normal to the [001] direction.

Figure 2

Schematic representation of domain walls with different spin rotation axes and rotational directions. (a) Two Bloch walls with opposite rotational directions. The magnetization rotates around the $x$ axis, i.e., the rotation axis is in the surface plane and parallel to the propagation direction. (b) Two Néel-type walls with opposite rotational direction. The magnetization rotates around the $y$ axis, i.e., the rotation axis is again in the surface plane but perpendicular to the propagation direction. We denote these two rotational directions as left (l) and right (r).

Figure 3

Planar rotation paths of the magnetization and corresponding model parameters for the domain walls in 2 ML Fe/W(110). The magnetization in the domains points out of plane and is indicated by circles and the magnetization direction in the middle of the wall is indicated by arrows.

Figure 4

Energies of the homogeneous spin spirals. The energies are plotted against the inverse of the period length $\lambda$. ${\lambda }^{-1}=0$ corresponds to the ferromagnetic state. The results shown in (a) are obtained within a scalar-relativistic approach (neglecting SOC). Quadratic fits to these symmetric curves yield the spin stiffness $A$. (b) shows the odd part of $\overline{E}\left({\lambda }^{-1}\right)$ obtained by including SOC. The slope of these curves at ${\lambda }^{-1}=0$ determines $D$ (Ref. 28).

Figure 5

Locality of the magnetostatic interactions. The data points show the sum of dipolar interactions evaluated up to a certain interatomic distance $x$ in the [001] direction. In the $\left[1\overline{1}0\right]$ direction, the sums are truncated at 2000 unit cells. The asymptotic values (dashed lines) are obtained by summing over $\left(2000\times 2000\right)$ unit cells.