A platform for zero-field isolated skyrmions: 4$d$/Co atomic bilayers on Re(0001)

TL;DR

This work demonstrates that 4$d$/Co atomic bilayers on Re(0001) can host nanoscale zero-field isolated skyrmions when described by an extended spin model that includes higher-order multi-spin interactions derived from DFT. Atomistic spin simulations reveal spontaneous formation of ISk on FM backgrounds for fcc-Rh/Co/Re(0001) and hcp-Pd/Co/Re(0001), with radii around $6\,\mathrm{nm}$ and $12\,\mathrm{nm}$ respectively, and robust energy barriers of about $150\ \mathrm{meV}$ mainly arising from DMI and HOI terms. The skyrmion stability and collapse pathways are characterized via GNEB, showing radial collapse as the annihilation mechanism and revealing how DMI and 3-site HOI contributions raise the annihilation barrier while exchange and MAE oppose skyrmion stability. The results position 4$d$/Co bilayers on Re(0001) as a viable nanoscale platform for zero-field skyrmions and point to potential integration with superconducting substrates for magnet-superconductor hybrids at cryogenic temperatures.

Abstract

Using first-principles density functional theory (DFT) combined with atomistic spin simulations, we explore the possibility of realizing zero-field isolated skyrmions in three 4$d$/Co atomic bilayers -- Rh/Co, Pd/Co, and Ru/Co -- grown on the Re(0001) surface. Our investigation employs an extended atomistic spin model, which goes beyond the standard model by including the multi-spin higher-order exchange interactions (HOI) in addition to the Heisenberg pairwise exchange interaction, Dzyaloshinskii-Moriya interaction (DMI), and magnetocrystalline anisotropy energy (MAE). All magnetic interactions of the extended spin model are calculated using DFT. The phase diagram obtained from atomistic spin simulations based on this spin model for Rh/Co and Pd/Co on Re(0001) reveals that isolated skyrmions emerge spontaneously on the ferromagnetic background even in the absence of an external magnetic field. The radius of zero-field isolated skyrmions in Rh/Co/Re(0001) is around 6 nm, whereas the radius of those skyrmions in Pd/Co/Re(0001) is about 12 nm. Transition-state theory calculations show that the skyrmions are protected by substantial energy barriers, approximately 150 meV, which predominantly arise from DMI, with a small contribution from the HOI interactions. The height of the barriers suggests that skyrmions could be observed in low-temperature experiments. Based on this work, we propose 4$d$/Co bilayers on Re(0001) as a new platform to realize nanoscale zero-field isolated skyrmions.

Figures (4)

• Figure 1: (a) Two-dimensional Brillouin zone (2DBZ) displaying two high-symmetry directions $\overline{\Gamma \mathrm{KM}}$ (blue) and $\overline{\Gamma \mathrm{M}}$ (red). The reciprocal vectors ($\mathbf{b}_1$ and $\mathbf{b}_2$) are also shown. Energy dispersion $E$(q) of homogeneous flat spin spirals along the $\overline{\Gamma \mathrm{KM}}$ and $\overline{\Gamma \mathrm{M}}$ directions of the 2DBZ without (light blue) and with (dark blue) SOC for (b) fcc-Rh/Co/Re(0001), (c) hcp-Pd/Co/Re(0001) and (d) fcc-Ru/Co/Re(0001). Filled circles present DFT data and solid lines are fits to the pairwise Heisenberg exchange (Eq. (1)) (light blue) and pairwise Heisenberg exchange plus the DMI (Eq. (2)) (dark blue) spin models. Filled diamonds, highlighted by solid red border, present the DFT total energies of multi-$Q$ states ($uudd$ and 3$Q$), evaluated without SOC, and are shown at the $\textbf{q}$ points of the corresponding single-$Q$ (spin spiral) states. Insets show $E$(q) in the vicinity of the $\overline{\Gamma}$ point ($\mid$$\textbf{q}$$\mid \leq 0.1\times\frac{2\pi}{a}$) without and with SOC. Note that the dispersions with SOC contain the effect of DMI and MAE. The MAE shifts the energy dispersion ($E(\textbf{q})$) by $K_{\textrm{MAE}}$/2 along the positive energy axis with respect to the FM state ($\overline{\Gamma}$ point). Total energy contribution due to SOC (green) and individual contributions from 4$d$ overlayers (pink), i.e., (e) fcc-Rh, (f) hcp-Pd, and (g) fcc-Ru, Co (yellow) and Re substrate (brown) along the two high-symmetry directions $\overline{\Gamma \mathrm{KM}}$ and $\overline{\Gamma \mathrm{M}}$ of the 2DBZ. Solid circles denote DFT data and solid green lines are fits to the DMI Hamiltonian (Eq. (2)). The full energy dispersion of fcc-Rh/Co/Re(0001) without SOC (light blue) in (b) is taken from Ref. paul2024
• Figure 2: Variation of the magnetic moments in (a) fcc-Rh/Co/Re(0001), (b) hcp-Pd/Co/Re(0001) and (c) fcc-Ru/Co/Re(0001) for spin spirals along the two high-symmetry directions $\overline{\Gamma \mathrm{KM}}$ and $\overline{\Gamma \mathrm{M}}$ of the 2DBZ shown in Figs. \ref{['fig:f1']}(b-d), respectively. Top, middle, and bottom panels of (a) show the magnetic moment of the Rh, Co, and the first Re layers, respectively; those of (b) show the Pd, Co, and the first Re layers, respectively, and those of (c) show of the Ru, Co, and the first Re layers, respectively.
• Figure 3: Zero-temperature magnetic phase diagram of (a) fcc-Rh/Co/Re(0001), and (b) hcp-Pd/Co/Re(0001). Energies of the FM state (green), skyrmion lattice (SkX, orange), and isolated skyrmions (ISk, blue) are shown as a function of the external magnetic field. (c) Spin structure of the skyrmion lattice (SkX), an isolated skyrmion, and the FM state for fcc-Rh/Co/Re(0001) at zero magnetic field. Only a small portion of the $150 \times 150$ simulation box is shown. Annihilation (yellow) and creation (green) barriers of isolated skyrmion for (d) fcc-Rh/Co/Re(0001) and (e) hcp-Pd/Co/Re(0001) as a function of applied magnetic field obtained via the GNEB method. The solid circles are obtained from the atomistic spin model (Eq. (13)), the annihilation barriers ($E^{a}_b$) are fitted with $E^{a}_b(B)= ae^{bB} + c_1$ and the creation barriers ($E^{c}_b$) with $E^{c}_b(B)= mB + c_2$, where $a$, $b$, $c_1$, $c_2$, and $m$ are fitting parameters and $B$ denotes the magnetic field. The inset shows variation in the radius of isolated skyrmions for (d) fcc-Rh/Co/Re(0001) and (e) hcp-Pd/Co/Re(0001) with the magnetic field. The solid circles are obtained from the atomistic spin model (Eq. (13)) and solid lines are fit to $R(B)= ae^{bB} + c$, where $R$ denotes the radius, $B$ denotes the magnetic field and $a$, $b$, and $c$ are the fitting parameters.
• Figure 4: Total and individual energy contributions with respect to the initial (skyrmion) state along the minimum energy path (MEP) for the radial collapse of isolated skyrmions into the FM (final) state at zero magnetic field for (a) fcc-Rh/Co/Re(0001) and (b) hcp-Pd/Co/Re(0001). Energy of the saddle point with respect to the initial (skyrmion) and final (FM) states defines the annihilation and creation barriers, respectively. Total and individual energy decomposition of the annihilation and creation barriers for (c) hcp-Rh/Co/Re(0001) and (d) hcp-Pd/Co/Re(0001). Three-site four spin and four-site four spin interactions are abbreviated as 3-Spin and 4-spin, respectively. Spin structures before (SP-3, SP-2, and SP-1), after (SP+1 and SP+2), and at the saddle point (SP) for (e) fcc-Rh/Co/Re(0001) and (f) fcc-Pd/Co/Re(0001) along the MEP shown in (a) and (b), respectively, indicating radial collapse. Note that only a small portion around the skyrmions in the $150 \times 150$ simulation box is shown.