Tuning Dzyaloshinskii-Moriya interaction in ferrimagnetic GdCo: A first-principles approach

We present a systematic analysis of our ability to tune chiral Dzyaloshinskii-Moriya interaction (DMI) in compensated ferrimagnetic $\mathrm{Pt}/\mathrm{GdCo}/{\text{Pt}}_{1-x}{\text{W}}_x$ trilayers by cap layer composition. Using first-principles calculations, we show that the DMI increases rapidly for only $\sim 10$% W and saturates thereafter, in agreement with experiments. The calculated DMI shows a spread in values around the experimental mean, depending on the atomic configuration of the cap layer interface. The saturation is attributed to the vanishing of spin-orbit coupling energy at the cap layer and the simultaneous constancy at the bottom interface. Additionally, we predict the DMI in $\mathrm{Pt}/\mathrm{GdCo}/X$ $(X=\text{Ta},\text{W},\text{Ir})$ and find that W in the cap layer favors a higher DMI than Ta and Ir that can be attributed to the difference in $d$-band overlap around the Fermi level. Our results open up exciting combinatorial possibilities for controlling the DMI in ferrimagnets towards nucleating and manipulating ultrasmall high-speed skyrmions.

Figure 1

Schematic of the $\text{Pt}\left(2\right)/\text{Gd}\text{Co}\left(2\right)/{\text{Pt}}_{1-x}{\text{W}}_x\left(2\right)$ structure (the number in parentheses denotes the number of monolayers) corresponding to $x=12.5\%$ for (a) CW and (b) CCW spin configurations. The red arrows show the spin orientations for noncollinear calculations. Even though we select the FM alignment between atoms at sites 1 and 5 (3 and 7) when constructing the spin spiral, we end up subtracting the energies between CW and CCW spin configurations to obtain the DMI, so this particular interaction cancels out. $\text{L1},\cdots ,\text{L6}$ denote the layer number, while numbers in circles label atomic positions.

Figure 2

The DMI as a function of W composition $x$ in $\text{Pt}/\text{Gd}\text{Co}/{\text{Pt}}_{1-x}{\text{W}}_x$. (a) DMI variation with respect to W positions while the Gd atoms are fixed at (1,7) positions. (b) Total spectrum of the DMI as both Gd and W positions are varied. For a specific W composition, each of the different colors represents the variation of Gd atomic positions, and the scattered points within the same color represent different W positions for that particular Gd arrangement in the structure (Fig. 1). (c) Surface DMI in comparison with experimentally observed DMI [28]. The numbers followed by the symbols Gd and W represent the positions of the respective atoms in the structure shown in Fig. 1.

Figure 3

(a) Change in SOC energy at the interfacial HM layers (L2 and L5) as a result of changing spin chirality of the magnetic layers (L3 and L4) from CW to CCW. All the color bars on the left (right) side represent the SOC energy change at L2 (L5) for different W compositions. (b) Layer-resolved DMI for structures with W composition $x=0\%\text{–}50\%$.

Figure 4

(a) Calculated DMI in Pt/GdCo/$X$, where $X$ = Ta, W, Ir. (b) Layer-resolved DMI.

Figure 5

Projected density of states (DOS) showing the $3d-5d$ band overlap between Co (black) and $X$ (colored) in Pt/GdCo/$X$. (a) $X$ = Ta, (b) $X$ = W, and (c) $X$ = Ir. The red up (down) arrow represents the spin-up (spin-down) channel.