Interlayer Dzyaloshinskii-Moriya Interactions

The interfacial Dzyaloshinkii-Moriya interaction defines a rotational sense for the spin structure in two-dimensional magnetic films and can be used to create chiral magnetic structures like spin spirals and skyrmions in those films. Here, we show by means of atomistic calculations that in heterostructures an interlayer coupling of the Dzyaloshinskii-Moriya type across a spacer can emerge. We quantify this interaction in the framework of the Lévy-Fert model for trilayers consisting of two ferromagnets separated by a nonmagnetic spacer and show that such an interlayer Dzyaloshinkii-Moriya interaction yields nontrivial three-dimensional spin textures across the entire trilayer, which evolve within as well as between the planes and, hence, combine intraplane and interplane chiralities. This analysis opens new perspectives for three-dimensional tailoring of magnetic chirality in multilayers.

Figure 1

Microscopic interlayer DM vectors in a trilayer crossection. (a),(b) Contributions to the ${\overrightarrow{D}}_{ij}$ (arrows) due to mediating NM sites to the right ${\overrightarrow{D}}_{ij}^{\text{right}}$ and to the left ${\overrightarrow{D}}_{ij}^{\mathrm{left}}$ from the bond (green triangles). In (a), magnetic (blue spheres) and nonmagnetic (gray spheres) atoms lie in the same plane and ${\overrightarrow{D}}_{ij}^{\mathrm{left}}={-\overrightarrow{D}}_{ij}^{\text{right}}$ leading to $D_{ij}^{\mathrm{eff}}=0$. In (b), the NM atoms are shifted along the $y$ axis corresponding to a cross section of a hcp stacking shown in (d). Because of the shift, $D_{ij}^{\mathrm{eff}}\ne 0$. (c) Effective ${\overrightarrow{D}}_{ij}^{\mathrm{eff}}$ for a chain with geometry (b) dependent on the number $n$ of mediating NM atoms ($r_{\mathrm{c}}=an$). The inset displays the dependency of $D_{ij}^{\mathrm{eff}}$ as a function of the NM atom chain location along the $z$ axis ($d_{\mathrm{NM}}$) for a given $d_{\mathrm{FM}}$ and $r_{\mathrm{c}}=15a$. (d) Cross section of a hcp stacking scenario.

Figure 2

Interlayer DM energy in a FM-NM-FM trilayer. Calculated orientations of DM vectors in a hcp trilayer for nearest-neighboring mediating atoms: (a) side view and (b) top view. Triangles define the $ijl$ planes, dark spheres show magnetic atoms, and light spheres correspond to mediating atoms. (c),(d) Blue arrows represent magnetic moments. Energy minimized azimuthal rotations [$d\phi =+\pi /2$ in (c) and $d\phi =-\pi /2$ in (d)] of the top FM layer with respect to the bottom FM layer for in-phase (c) and antiphase (d) $\delta \theta$ magnetization oscillations in the top layer (with respect to the bottom layer oscillations). (e),(f) Energy landscape in the $d\phi ,\delta \theta$ phase space. The exemplary curve for $\delta \theta =+\pi /4$ in (e) defines the color code of (f).

Figure 3

Ground states of trilayers with interlayer DM interaction from MC simulations. (a) Unit cell of stable MC low-temperature ($k_{B}T=0.001J_{ij}^{\text{intra}}$) configuration for dominating interlayer DM ($|{\overrightarrow{D}}_{ij}^{\mathrm{eff}}|>J_{ij}^{\text{intra}}$). Cycloidal spin spirals across the trilayer (dark dashed line) and along the $z$ axis are formed. (b) Top view of the magnetic structure for $D_{ij}^{\mathrm{eff}}/J_{ij}^{\text{intra}}\approx 1.4$ ($V_1/J_{ij}^{\text{intra}}=9.7$) with the same color scale (blue and red correspond to opposite out-of-plane magnetization, while green and orange correspond to opposite in-plane magnetization) and realistic case $D_{ij}^{\mathrm{eff}}/J_{ij}^{\text{intra}}\approx 0.15$. In the last case both layers are in-plane magnetized (cyan and orange spins have orthogonal orientation). (c) Mean interlayer DM energy achieved in MC simulations for net in-plane ferromagnetic states in each layer making an angle $d\phi$ with respect to one another. Large arrows schematize MC states at the energy minimum. Direction gives net in-plane orientation, color gives an out-of-plane contrast. (d) Stable MC low-temperature ($k_{B}T=0.001J_{ij}^{\text{intra}}$) configurations of the top layer for $D_{ij}^{\text{eff}}/J_{ij}^{\text{intra}}\approx 0.2$, and $K_{xy}^{\mathrm{top}}=0.15J_{ij}^{\text{intra}}$ for $d\phi =+\pi /2$. Arrows show net in-plane orientation, while colors show the out-of-plane magnetic contrast.