Skyrmions in 2D chiral magnets with noncollinear ground states stabilized by higher-order interactions

Abstract

Magnetic skyrmions are intriguing topological spin textures that have attracted great attention due to their potential for future spintronic devices. Skyrmions have so far been explored in different magnetic materials, such as ferromagnets, antiferromagnets, and ferrimagnets. Here, we propose a new type of unconventional skyrmions stabilized in noncollinear magnets. Using first-principles calculations and atomistic spin simulations, we demonstrate that a noncollinear ground state can be stabilized in Rh/Co and Pd/Co atomic bilayers on the Re(0001) surface by four spin exchange interactions, although Co -- a material often used in applications -- is a prototypical ferromagnet with strong pairwise exchange interaction. We further show that unconventional skyrmion lattices and isolated skyrmions can emerge on this noncollinear magnetic background. Transition-state theory calculations reveal that these metastable skyrmions are protected by large energy barriers, suggesting that they could be observed in experiments. These unconventional types of skyrmions in noncollinear magnets might open new possibilities for topological spin transport or magnet-superconductor hybrid systems.

Figures (5)

• Figure 1: Magnetic phase diagram, spin structures, and Fourier transforms of Rh/Co/Re(0001) including and excluding HOI.a Illustration of an isolated skyrmion in the Co layer of Rh/Co/Re(0001). b,c Zero temperature phase diagram of Rh/Co/Re(0001) obtained by neglecting HOI and including all interactions, i.e., including HOI, respectively. Energies of spin spirals (SS, black circles and line), skyrmion lattice on the FM background (SkX, orange circles and lines), FM state (green circles and lines), excluding HOI, and energies of the spin spirals (SS-2$Q$, yellow circles and line), noncollinear ground state (nc-GS, red circles and lines), skyrmion lattice on the same noncollinear background (nc-SkX, blue circles and lines), including all interactions, are shown with reference to the energies of homogeneous spin spirals (dashed black line) at zero field. d-g Spin structure of SS, FM, SkX and isolated skyrmions on the FM background (ISk), respectively. h-k Spin structure of SS-2$Q$, nc-GS, nc-SkX and isolated skyrmions on the nc-GS (nc-ISk), respectively. Magnetic unit cell of nc-GS, contains three spins, is shown with solid white lines. Two spins inside the unit cell point close to the out-of-plane direction (polar angle $\theta \approx 17^{\circ}$, azimuth angle $\phi \approx 315^{\circ}$), while the third spin at the edge of the unit cell is inclined towards the in-plane direction ($\theta \approx 39^{\circ}, \phi \approx 135^{\circ}$). For better visualization, the spin direction, expect for the FM state and nc-GS, is reversed. Fourier transform of l-n SS, FM and SkX, respectively, excluding HOI, and o-q SS-2$Q$, nc-GS and nc-SkX, respectively, including HOI, in the 2DBZ. r Mapping of an isolated skyrmion on the noncollinear ground state (nc-ISk) onto a unit sphere, $S^2$, via stereographic projection. The color indicates the $z-$component of spins. Transition from spin up ($S_z= 1$) to spin down ($S_z= -1$) is marked by a color change from orange to dark blue via yellow.
• Figure 2: Sublattice decomposition of spin structures in Rh/Co/Re(0001).a The noncollinear ground state (nc-GS) in Fig. \ref{['fig:fig1']}i is decomposed into b-d three sublattices (SL), SL 1-3. All three sublattices display FM alignment of spins, however, the magnetization axis of SL 2 (c) and SL 3 (d) points close to the out-of-plane direction ($\theta \approx 17^{\circ}$, $\phi \approx 315^{\circ}$), whereas the axis of SL 1 (b) is inclined more towards the in-plane direction ($\theta \approx 39^{\circ}$, $\phi \approx 135^{\circ}$). e Spin spiral state (SS-2$Q$) in Fig. \ref{['fig:fig1']}h is decomposed into f-h three sublattices, SL 1-3. Each sublattice supports spin spirals characterized as a single-$Q$ state. i Noncollinear skyrmion lattice (nc-SkX) in Fig. \ref{['fig:fig1']}j is decomposed into j-l three sublattices, SL 1-3. Each sublattice contains a skyrmion lattice on the FM background. However, the magnetization axes of FM background follow the same directions as of the sublattices obtained from nc-GS. The topological charge of each sublattice is same as of the composite nc-SkX state. m Noncollinear isolated skyrmions (nc-ISk) in Fig. \ref{['fig:fig1']}k is decomposed into n-p three sublattices, SL 1-3. Each sublattice contains a skyrmion on the FM background. However, the magnetization axes of FM background follow the same directions as of the sublattices obtained from nc-GS. The topological charge of nc-ISk and the three sublattices is equal to 1. For better visualization, the spins of the three sublattices are rotated around an axis in the $x$-$y$ plane to align the FM spins in the out-of-plane direction. It is important to note that the rotated spin structure is nearly 6 meV per spin higher in energy than the unrotated nc-ISk spin structure.
• Figure 3: Simulated SP-STM images of spin structures in Rh/Co/Re(0001).a-d SP-STM image of the spin spiral (SS-2$Q$, Fig. \ref{['fig:fig1']}h), the nc-GS (Fig. \ref{['fig:fig1']}i), the noncollinear skyrmion lattice (nc-SkX, Fig. \ref{['fig:fig1']}j), and the noncollinear isolated skyrmions (nc-ISk, Fig. \ref{['fig:fig1']}k), respectively, simulated for an out-of-plane direction of the tip magnetization. The 2D magnetic unit cell of the noncollinear ground state (nc-GS) is shown with solid red lines. The tip magnetization is indicated above the panels by a circled dot for the out-of-plane direction.
• Figure 4: Radius, minimum energy path, and energy barriers for skyrmions in Rh/Co/Re(0001).a Radius of noncollinear isolated skyrmions (nc-ISk) in the original lattice (Fig. \ref{['fig:fig2']}m) and three sublattices (Fig. \ref{['fig:fig2']}n-p) as a function of the applied magnetic field. The filled circles represent data from the atomistic spin simulations, the solid lines are fits to the function $y(x)= a \left( x+b \right)^{-1} + c$, where $y$ is the skyrmions radius, $x$ is the magnetic field, and $a$, $b$ and $c$ are fitting constants. b Creation and annihilation energy barriers of the noncollinear isolated skyrmions (nc-ISk) as a function of applied magnetic field. The atomistic spin simulation data (filled circles) of the annihilation and creation barriers are fitted with $y(x)= a \left( x+b \right)^{-1} + c$ and $y(x)= mx + c$, respectively, where $y$ is the energy barriers, $x$ is the magnetic field, and $a$, $b$, $c$, and $m$ are fitting constants. c Total and individual energy contributions along the minimum energy path (MEP) corresponding to the radial collapse of the noncollinear isolated skyrmions (nc-ISk) at 3.0 T including all interactions. The energy of the saddle point relative to the initial state (nc-ISk) represents the annihilation barriers, whereas its energy relative to the final state (nc-GS) represents the creation barriers. Topological charge of noncollinear isolated skyrmions (nc-ISk) and isolated skyrmions on the FM background in three sublattices along MEP is also shown. d Energy decomposition of the saddle point with respect to the initial (nc-ISk) and final states (nc-GS). e Spin structures of noncollinear isolated skyrmions (nc-ISk) at the saddle point (SP), before the saddle point (SP-2 and SP-1) and after the saddle point (SP+1 and SP+2) corresponding to the minimum energy path in c. The three-site ($Y_1$) four spin interaction and four-site four spin interactions ($K_1$) are abbreviated as 3-spin and 4-spin, respectively. The values of the fitting constants are listed in Supplementary Table 5.
• Figure 5: Ground states from a competition of HOI and pair-wise exchange.a Phase diagram displaying the variation of three higher-order exchange interactions, i.e., three-site four spin ($Y_1$), four-site four spin ($K_1$) and biquadratic ($B_1$) interactions, as a function of effective exchange interaction ($J_{\mathrm{eff}}$). The light red indicates the FM phase, while the cyan denotes the noncollinear phase as shown in Fig. \ref{['fig:fig1']}i, on which the isolated skyrmions are formed. The black surface marks the boundary of these two phases. The filled circles present previously studied TM ultrathin films malottki2017ameyer19paul2020bpaulhoiromming18gutzeit2021meyer2020 and the filled diamonds indicate the two films studied here. The magnetic interactions of the additional TM ultrathin films used here are summarized in Supplementary Table 6. b Zoomed view around fcc-Pd/Co/Re(0001) at $B_1/J_{\mathrm{eff}}=$ 0.053. The effect of DMI ($D_{\mathrm{eff}}$) and MAE ($K$) on the phase diagram is negligible (Supplementary Fig. 16).