Stability and magnetic properties of Fe double layers on Ir (111)

We investigate the interplay between the structural reconstruction and the magnetic properties of Fe double layers on Ir (111) substrate using first-principles calculations based on density functional theory and mapping of the total energies on an atomistic spin model. We show that if a second Fe monolayer is deposited on Fe/Ir (111), the stacking may change from hexagonal-close-packed to bcc (110)-like accompanied by a reduction of symmetry from trigonal to centered rectangular. Although the bcc-like surface has a lower coordination, we find that this is the structural ground state. This reconstruction has a major impact on the magnetic structure. We investigate in detail the changes in the magnetic exchange interaction, the magnetocrystalline anisotropy, and the Dzyaloshinskii-Moriya interaction depending on the stacking sequence of the Fe double layer. Based on our findings, we suggest a technique to engineer Dzyaloshinskii-Moriya interactions in multilayer systems employing symmetry considerations. The resulting anisotropic Dzyaloshinskii-Moriya interactions may stabilize higher-order skyrmions or antiskyrmions.

Figure 1

Given are the stacking sequences for the bulk case for structures (a) fcc, (b) hcp, and (c) pseudomorphically strained bcc. All layers are close packed in this scenario.

Figure 2

Top views of the atomic configurations of films with stackings ${\mathrm{fb}}^{*}$ and ff and symmetry elements of their respective plane groups $\mathrm{c}m$ and $\mathrm{p}3m1$. Shown are the atoms of the three outmost layers, only.

Figure 3

Brillouin zones of the double-layer stackings. In (a) the relationships between real space (gray) and reciprocal (blue) unit cells including the first Brillouin zone (yellow) for the hexagonal reference structure is given. For simplicity, the vectors of the real space and reciprocal space were chosen to have the same length. The dotted-lined black triangle indicates the irreducible part of the first Brillouin zone. Also shown is the Cartesian coordinate system within the first Brillouin zone (yellow), which is used for the calculation of magnetic interaction energies. The high-symmetry $q$ points $\overline{\mathrm{\Gamma }}$, $\overline{\mathrm{K}}$, and $\overline{\mathrm{M}}$ are indicated as well. In the double layers, the hexagonal symmetry is reduced to trigonal and centered rectangular. In (b) the trigonal setting is shown where half of the points $\overline{\mathrm{K}}$ are lost, indicated by the additional ${\overline{\mathrm{K}}}^{\prime }$ and a doubling of the size of the irreducible Brillouin zone. In the centered rectangular systems, given in (c), the Brillouin zone changes its shape and size (the hexagonal one is shown for comparison) and new special $k$ points $\overline{\mathrm{X}}$, $\overline{\mathrm{Y}}$, and $\overline{\mathrm{M}}$ result.

Figure 4

Schematic pictures of flat spin spirals on the hexagonal lattice in real space corresponding to $q=\frac{1}{6}\frac{2\pi }{a}$ propagating along (a) $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{K}}$ direction with the wave vector $(q,0,0)$ and (b) $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{M}}$ direction of reciprocal space with the wave vector $(0,q,0)$.

Figure 5

Asymmetric film geometry as it was used in the calculations of the magnetic properties. Shown is the ff stacking of the Fe double layer.

Figure 6

Total energies of the different stackings relative to the energy of the ff stacking. Decisive for the stability of the double layer is the top-layer configuration.

Figure 7

Total energies along the transformation paths from ff and fh to ${\mathrm{fb}}^{*}$ stacking. The line is a guide to the eye.

Figure 8

Spin spiral energy dispersions in the stackings ff, ${\mathrm{fb}}^{*}$, hf, and fh. In (a) the spin spiral propagates in both the Fe@vac layer and the Fe@Ir. Exchange stabilized spin spirals are found in ff and fh. In (b), the spin spiral propagates only in the Fe@vac layer exposed to the vacuum, but not in the interfacial Fe@Ir layer. In this spin configuration, the spin spiral is only stable in stacking ff; all other stackings favor the ferromagnetic state. In (c), the spin spiral propagates in the interfacial Fe@Ir layer, but not in the Fe@vac layer at the surface. In this case, we find a deep energy minimum for the spin spiral state in the hf stacking, while the others remain ferromagnetic.

Figure 9

Shown are the preferred (in color and bold) and disfavored spin orientations (in gray and fine) for the stackings hf, ${\mathrm{fb}}^{*}$, fh, and ff. The preferred orientation of spins changes from an easy plane in hf, ${\mathrm{fb}}^{*}$, and fh stacking to an out-of-plane easy axis in the ff stacking. The anisotropy coefficients $K_1$ increase at the same time from 0.1 meV in ${\mathrm{fb}}^{*}$ to 1.2 meV in ff.

Figure 10

Spin-orbit coupling contribution to energy dispersion of cycloidal flat spin spirals related to the Dzyaloshinskii-Moriya interaction along high-symmetry lines for stackings (a) ff and (b) ${\mathrm{fb}}^{*}$. Given are the contributions of the different atoms as well as the fit for the total energy considering DMI up to the third-nearest-neighbor shell (see Table 5). The DMI is considerably suppressed in the $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{Y}}$ direction of the ${\mathrm{fb}}^{*}$ stacking.

Figure 11

Total energy contributions to spin spiral energy in the ${\mathrm{fb}}^{*}$ stacking dispersions due to SOC resulting from spin spirals in only one of the Fe layers. Given are the atomic contributions and the total SOC energies. In (a), the spin spiral propagates in the interfacial Fe@Ir layer, but not in the Fe@vac layer at the surface. The magnitudes along both propagation directions are the same. Thus, the SOC possesses 3-fold symmetry in this spin configuration. The corresponding DMI coefficient is $D_1(\mathrm{Fe}@\mathrm{Ir})=0.40\mathrm{meV}$. In (b), the spin spiral propagates in the Fe@vac layer of the surface. The magnitudes along the two propagation directions are completely different. The SOC reflects the reduced symmetry of the bcc-like stacking. The obtained DMI coefficient is $D_1(\mathrm{Fe}@\mathrm{vac})=0.34\mathrm{meV}$. In (c), we show how the Ir-Fe DMI (yellow arrows) and Fe-Fe DMI (red arrows) arise from different 3-atom scattering events possible in the two different spin spiral configurations. Shown are the centered rectangular unit cells (gray solid line) in top view, along with the rhombic unit cell (dotted line) for orientation. The quantization axis of the SOC is always perpendicular to the spin spiral propagation direction which leads to cycloidal spin spirals along $q$. The scattering partners are connected via dashed lines, in gray for Ir-Fe and in turquoise/dark blue for Fe-Fe interactions. Only for the spin spiral in Fe@vac along the $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{M}}$ direction, there is no Ir-Fe DMI possible, which gives rise to the strong asymmetry in (b).

Figure 12

Orbital and angular momentum resolved partial densities of states (PDOSs) for the $3d$ states of (a) Fe@vac, (b) Fe@Ir and for the $5d$ states of (c) Ir@Fe in the stackings ff (lines) and ${\mathrm{fb}}^{*}$ (filled). Both majority spins (positive PDOS) and minority spins (negative PDOS) are given. The stacking of the Fe@vac layer changes the hybridization in the Ir@Fe layer as indicated by the arrows. Please note the different PDOS scales for Fe and Ir atoms.

Figure 13

Full energy dispersions including SOC of flat cycloidal spin spirals from DFT relative to the FM state of stackings (a) ff, (b) ${\mathrm{fb}}^{*}$, (c) fh, and (d) hf along high-symmetry directions $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{K}}$ and $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathrm{M}}$. Given are values with (filled circles, colored solid lines) and without SOC (open triangles, gray dashed lines). Positive (negative) values of $q$ indicate clockwise (anticlockwise) rotating spin spirals. In all stackings clockwise-rotating spin spirals are favored. The contribution of the magnetocrystalline anisotropy to spin spirals amounts to $\frac{1}{2}{K}_1$.

Figure 14

Total energies of the different stackings relative to the energy of the ff stacking with different epitaxial strains. Epitaxial strain does not alter the relative stabilities of the different double-layer stackings.

Figure 15

Energy difference between FM states when SOC is applied along the $x$ and $z$ direction (dashed green triangle), along the $y$ and $z$ direction (dashed blue square), and along the $x$ and $z$ direction (dashed red dot) as a function of the the number of Ir layers in which SOC is applied in the calculation for the ff (a), ${\mathrm{fb}}^{*}$ (b), fh (c), and hf (d).