Topological orbital magnetization and emergent Hall effect of an atomic-scale spin lattice at a surface

We predict the occurrence of a novel type of atomic-scale spin lattice in an Fe monolayer on the Ir(001) surface. Based on density functional theory calculations we parametrize a spin Hamiltonian and solve it numerically using Monte Carlo simulations. We find the stabilization of a three-dimensional spin structure arranged on a $(3\times 3)$ lattice. Despite an almost vanishing total spin magnetization we predict the emergence of orbital magnetization and large anomalous Hall effect, to which there is a significant topological contribution purely due to the real space spin texture at the surface.

Figure 1

Energy dispersion of homogeneous flat spin spirals for Fe/Ir(001). The energies $E\left(\mathbf{q}\right)$ (filled circles) are calculated via DFT along the high-symmetry lines of the two-dimensional Brillouin zone and given with respect to the $c(2\times 2)$ antiferromagnetic state. The solid lines are fits to the Heisenberg model with up to sixth-nearest neighbors [44]. The inset shows the energy dispersion close to the $\overline{\mathrm{M}}$ point for left- and right-rotating spirals including SOC, i.e., the effect of the DM interaction.

Figure 2

Spin structures of lowest energy obtained via Monte Carlo simulations as a function of the strength of the four-spin interaction. The biquadratic interaction is changed according to the condition $2K_{4\mathrm{spin}}+B=0.7$ meV from the DFT calculations. The sketches only display a small section of the actually simulated spin lattice. Units in the figure are in meV.

Figure 3

(a) Side view and (b) top view of the proposed atomic-scale $\left(3\times 3\right)$ spin lattice. Two unit cells are shown.

Figure 4

Top: BZ distribution of the Berry curvature without (left) and with (right) SOC for Fe monolayer in $\left(3\times 3\right)$ state with one layer of Ir substrate, superimposed with the real-space distribution of the spins (blue arrows). Bottom: Calculated values of ${\sigma }_{3\times 3}^{\mathrm{AH}}$ as function of the Ir substrate thickness.