Oscillatory behavior of interlayer Dzyaloshinskii-Moriya interaction by spacer thickness variation

The interlayer Dzyaloshinskii-Moriya interaction (IL-DMI) has recently emerged as an ingredient promoting chiral orthogonal coupling between adjacent magnetic layers in multilayered systems. IL-DMI offers an additional tuning knob to engineer the magnetic behavior in spintronic devices, which could be useful for nonvolatile logic and memory technologies. Here, we systematically study, via first-principles calculations and the three-site Fert-Lévy model, the spacer thickness dependence of the IL-DMI between an out-of-plane ferrimagnet TbCo and an in-plane ferromagnet Co through Pt, Ir, Pd, and Ru. We observed a damped oscillatory behavior with increasing spacer thickness in all cases with characteristic amplitude and periodicity. Furthermore, we established a direct correlation between the IL-DMI and density of states of bottom and top Co atoms, dominated by the spacer thickness, which is attributed to a hybridization of electronic orbitals. Based on this compelling evidence, we propose that the electronic orbital hybridization contributes to the microscopic origin of the IL-DMI metallic magnetic multilayers. We anticipate that our results will provide insights into the understanding and precise control of IL-DMI in a wide range of materials and spintronic device concepts.

Figure 1

Schematics of the model TbCo (3 ML)/spacer/Co (4 ML) structure where the spacer thickness is varied from 1 to 6 ML. The direction of the magnetic moments of the bottom TbCo layer is fixed in the $z$ direction (${\mathbf{M}}_{\text{TbCo}}$ || $\mathbf{z}$), while that of the top Co layer is fixed sequentially in the $x$ (${\mathbf{M}}_{\text{Co}}$ || $\mathbf{x}$) and $-x$ (${\mathbf{M}}_{\text{Co}}$ || $-\mathbf{x}$) directions to construct an orthogonal magnetic configuration and determine the preferred chirality for each spacer thickness value.

Figure 2

Perpendicular magnetic anisotropy (PMA) of TbCo. (a) The dependence of the magnetic anisotropy energy (MAE) of the TbCo layer on the ratio of lattice parameters $c/a$. The background gradient shows the formation energy relative to the optimized geometry. The maximum relative formation energy indicated by blue region at the left end of the figure is 0.56 eV. A positive value of MAE indicates a PMA for TbCo. (b)and (c) Top and side views of TbCo crystal.

Figure 3

Interlayer Dzyaloshinskii-Moriya interaction (IL-DMI) in the TbCo/Pt/Co multilayer structure. The variation of IL-DMI between TbCo and Co layers is given with respect to Pt spacer thickness. Positive IL-DMI values with red background indicate that the favorable configurations are (${\mathbf{M}}_{\text{TbCo}}$ || $\mathbf{z}, {\mathbf{M}}_{\text{Co}}$ || $-\mathbf{x}$) and (${\mathbf{M}}_{\text{TbCo}}$ || $-\mathbf{z}, {\mathbf{M}}_{\text{Co}}$ || $\mathbf{x}$), while negative IL-DMI values with blue background denote that the favorable configurations are (${\mathbf{M}}_{\text{TbCo}}$ || $\mathbf{z}, {\mathbf{M}}_{\text{Co}}$ || $\mathbf{x}$) and (${\mathbf{M}}_{\text{TbCo}}$ || $-\mathbf{z}, {\mathbf{M}}_{\text{Co}}$ || $-\mathbf{x}$).

Figure 4

Relation between interlayer Dzyaloshinskii-Moriya interaction (IL-DMI) and density of states(DOS). (a)–(c) Orbital-resolved partial DOS and (d)–(f) layer-resolved partial DOS of the structure with Pt spacer of 1, 2, and 6 ML thickness, respectively. (g) A qualitative analysis on the link between IL-DMI strength and DOS. The normalized IL-DMI strength and the integrated DOS display a strong correlation. Here, we integrated the DOS in the energy interval of $-0.5$ to 0.5 eV relative to the Fermi energy; however, that of $-1$ to 1 eV and $-1.5$ and 1.5 eV showed a similar trend, which can be seen in the Supplemental Material [31].

Figure 5

Relation between interlayer Dzyaloshinskii-Moriya interaction (IL-DMI) and spacer element. (a) The spacer thickness dependence of the IL-DMI for different spacer elements. Here, IL-DMI of the Pt spacer is reproduced by the same data set as in Fig. 3, for comparison. (b)–(e) Element-resolved partial density of states for different spacers. The dashed lines indicate the region corresponding to near the Fermi level between $-2$ and 2 eV.