Dzyaloshinskii-Moriya interaction at an antiferromagnetic interface: First-principles study of Fe/Ir bilayers on Rh(001)

We study the magnetic interactions in atomic layers of Fe and $5d$ transition metals such as Os, Ir, and Pt on the (001) surface of Rh using first-principles calculations based on density functional theory. For both stackings of the $5d$/Fe bilayer on Rh(001) we observe a transition from an antiferromagnetic to a ferromagnetic nearest-neighbor exchange interaction upon $5d$ band filling. In the sandwich structure $5d$/Fe/Rh(001) the nearest-neighbor exchange is significantly reduced. For Fe/Ir bilayers on Rh(001) we consider spin spiral states in order to determine exchange constants beyond nearest neighbors. By including spin-orbit coupling we obtain the Dzyaloshinskii-Moriya interaction (DMI). The magnetic interactions in Fe/Ir/Rh(001) are similar to those of Fe/Ir(001) for which an atomic scale spin lattice has been predicted. However, small deviations between both systems remain due to the different lattice constants and the Rh vs Ir surface layers. This leads to slightly different exchange constants and DMI and the easy magnetization direction switches from out-of-plane for Fe/Ir(001) to in-plane for Fe/Ir/Rh(001). Therefore a fine tuning of magnetic interactions is possible by using single $5d$ transition-metal layers which may allow to tailor antiferromagnetic skyrmions in this type of ultrathin films. In the sandwich structure Ir/Fe/Rh(001) we find a strong exchange frustration due to strong hybridization of the Fe layer with both Ir and Rh which drastically reduces the nearest-neighbor exchange. The energy contribution from the DMI becomes extremely large and DMI beyond nearest neighbors cannot be neglected. We attribute the large DMI to the low coordination of the Ir layer at the surface. We demonstrate that higher-order exchange interactions are significant in both systems which may be crucial for the magnetic ground state.

Figure 1

Unit cell of $\mathrm{Fe}\text{/}5d$ bilayers on Rh(001). The $5d$ elements are Os, Ir, or Pt. Two different stackings of the bilayer are considered. Left: The Fe layer at the surface. Right: The Fe layer in a sandwich structure between the $5d$ layer and the Rh surface.

Figure 2

Sketch of the Dzyaloshinskii-Moriya vectors for the Fe monolayer on the Rh(001) surface from first to fifth neighbors (first red, second orange, third blue, fourth green, fifth gray) with the directions of the high symmetry lines of the two-dimensional Brillouin zone. The DM vectors are perpendicular to the bond between the black reference Fe atom and the corresponding neighbor. The size of the vectors illustrate the expected decreasing strength of the DMI with distance. The propagation directions of spin spirals for $\mathbf{q}$ along the $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ and the $\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ direction are shown.

Figure 3

Considered spin structure to test the influence of higher-order exchange interactions. The angle $\alpha$ is varied from $0^{\circ }$ to ${45}^{\circ }$, where theses structures correspond to the $p(2\times 1)$ antiferromagnetic state and the $2Q$ state, respectively.

Figure 4

Calculated total energy differences $\mathrm{\Delta }E$ between the FM and the AFM state for $\mathrm{Fe}\text{/}5d$/Rh(001) (dashed line) and $5d$/Fe/Rh(001) (solid line). Positive values indicate that the FM state is preferred, negative values denote a favorable $c(2\times 2)$ AFM structure. All energies are calculated for structurally relaxed films in the AFM state. The green (red) line is the value taken from Ref. [44] (Ref. [46]).

Figure 5

Calculated spin-resolved local density of states (LDOS) of the top three layers of (a) $\mathrm{Fe}\text{/}5d$/Rh(001) and (b) $5d$/Fe/Rh(001) in the $c(2\times 2)$ antiferromagnetic state. Upper (lower) parts of each panel correspond to the majority (minority) spin channel.

Figure 6

(a) Calculated energy dispersion $E\left(\mathbf{q}\right)$ of flat, cycloidal spin spirals for a freestanding Fe/Ir bilayer without (black dots) and with spin-orbit coupling (red dots) in $\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ direction with both senses of rotation. The dispersion is fitted to the Heisenberg model (black line) and includes the DMI and magnetocrystalline anisotropy (red line). The green diamonds indicate the values of the supercell calculations (see text for details). (b) Layer resolved contribution of $\mathrm{\Delta }{E}_{\text{SOC}}\left(\mathbf{q}\right)$. The black curve is the fit of the DMI including five nearest neighbors. (c) Layer resolved magnetic moments.

Figure 7

One-dimensional sketch to illustrate the effect of DMI beyond nearest neighbors for clockwise-rotating spin spirals close to the FM state (upper panel) and close to the AFM state (lower panel). The cross product ${\mathbf{m}}_i\times {\mathbf{m}}_j$ is shown for the first three neighbors.

Figure 8

(a) Calculated total energy dispersion $E\left(\mathbf{q}\right)$ of flat, cycloidal spin spirals for Fe/Ir/Rh(001) without (black dots) and with spin-orbit coupling (red dots) in $\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ direction for both rotational senses. The dispersion is fitted to the Heisenberg model (black line) and includes the DMI and magnetocrystalline anisotropy (red line). (b) Layer resolved contribution to $\mathrm{\Delta }{E}_{\text{SOC}}\left(\mathbf{q}\right)$. The black curve is the fit of the DMI for three nearest neighbors. (c) Sketch of the spin spiral state according to the energy minimum of the red curve of (a).

Figure 9

Calculated energy of superposition states of spin spirals for Fe/Ir/Rh(001) with respect to the $p(2\times 1)$ AFM state. The considered spin structure is shown in the inset and $\alpha$ is varied from $0^{\circ }$ to ${45}^{\circ }$. The red line is a fit to the energy contribution for nearest-neighbor biquadratic and four-spin interaction (cf. Sec. 2f).

Figure 10

(a) Calculated energy dispersion $E\left(\mathbf{q}\right)$ of flat, cycloidal spin spirals for Ir/Fe/Rh(001) without (black dots) and with spin-orbit coupling (red dots) in $\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ direction for both senses of rotation. The dispersion is fitted to the Heisenberg model (black line) and includes the DMI and magnetocrystalline anisotropy energy (red line). (b) Layer resolved contributions to $\mathrm{\Delta }{E}_{\text{SOC}}$. The black curve is the fit of the DMI for three nearest neighbors. (c) Sketch of the spin spiral state according to the minimum of the red curve of (a).

Figure 11

Calculated energy of superposition states of spin spirals for Ir/Fe/Rh(001) with respect to the $p(2\times 1)$ AFM state. The considered spin structure is shown in the inset and $\alpha$ is varied from $0^{\circ }$ to ${45}^{\circ }$. The red line is a fit to the energy contribution for nearest-neighbor biquadratic and four-spin interaction (cf. Sec. 2f).

Figure 12

(a) Calculated energy dispersion $E\left(\mathbf{q}\right)$ of flat, cycloidal spin spirals for Ir/Ir/Fe/Rh(001) without (black dots) and with spin-orbit coupling (red dots) in $\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ direction for both senses of rotation. The dispersion is fitted to the Heisenberg model (black line) and includes the DMI and magnetocrystalline anisotropy (red line). (b) Layer resolved contribution to $\mathrm{\Delta }{E}_{\mathrm{SOC}}\left(\mathbf{q}\right)$. The black curve is the fit of the DMI for four nearest neighbors.

Figure 13

Energy dispersion of spin spirals along the $\overline{X}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ direction for Ir/Fe/Rh(001). (a) Energy dispersion $E\left(\mathbf{q}\right)$ without spin-orbit coupling. (b) Energy contribution due to SOC, $\mathrm{\Delta }{E}_{\text{SOC}}\left(\mathbf{q}\right)$ and (c) magnetic moments of the topmost three layers. The black points are the values including the induced magnetic moments in the Ir layer with the fit to the Heisenberg model and the DMI. The green points are values if the moments in the Ir layer are suppressed in the calculation.