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

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.

💡 Research Summary

In this work the authors combine first‑principles density‑functional theory (DFT) with large‑scale atomistic spin simulations to investigate whether isolated magnetic skyrmions can exist without an external magnetic field in ultrathin 4d/Co bilayers grown on a Re(0001) substrate. Three systems are studied: Rh/Co, Pd/Co and Ru/Co, each with a specific stacking (fcc‑Rh, hcp‑Pd, fcc‑Ru) on the Co monolayer.

The magnetic interactions are extracted from DFT in a comprehensive way. By calculating flat spin‑spiral dispersions (without spin‑orbit coupling) the Heisenberg exchange constants Jij are obtained up to the 10th nearest neighbour, revealing strong ferromagnetic nearest‑neighbour coupling together with competing antiferromagnetic longer‑range terms that generate exchange frustration. Spin‑orbit coupling is treated within first‑order perturbation theory to obtain the Dzyaloshinskii‑Moriya interaction (DMI) vectors Dij and the magnetocrystalline anisotropy energy (MAE) constant KMAE. All three systems display a positive MAE, i.e. an out‑of‑plane easy axis.

Beyond the conventional pairwise terms, the authors evaluate higher‑order exchange (HOI) – biquadratic (B1), three‑site four‑spin (Y1) and four‑site four‑spin (K1) interactions – by comparing the energies of special multi‑Q states (up‑up‑down‑down, double‑row AFM, and the 3Q state) with single‑Q spin‑spiral energies. For Rh/Co and Pd/Co the biquadratic term is positive (≈0.2 meV) while Y1 and K1 are negative (≈‑0.05 meV), indicating a modest but non‑negligible contribution to the total energy landscape. In Ru/Co the HOI are essentially negligible.

All extracted parameters are inserted into an extended atomistic spin Hamiltonian
H = Hex + HDMI + HMAE + HZeeman + Hbiquad + H3‑site + H4‑site,
which is then solved on a 150 × 150 hexagonal lattice using a damped Landau‑Lifshitz dynamics to obtain relaxed spin configurations. Phase diagrams are constructed by varying an external magnetic field B from 0 to several tesla. The simulations reveal that both Rh/Co and Pd/Co host spontaneous isolated skyrmions on a ferromagnetic background at B = 0 T. The skyrmion radius is ≈6 nm for Rh/Co and ≈12 nm for Pd/Co, the difference being traced back to the stronger DMI and larger exchange frustration in the Rh system. Ru/Co fails to stabilise skyrmions because its DMI is too weak (≈0.2 meV·Å).

To assess thermal stability, the authors employ the geodesic nudged elastic band (GNEB) method to compute the minimum‑energy path for skyrmion collapse into the ferromagnetic state. Energy barriers of ~150 meV (Rh/Co) and ~140 meV (Pd/Co) are obtained. A detailed decomposition shows that roughly 85 % of the barrier originates from the DMI, while the remaining ~15 % is contributed by the three‑site four‑spin term, confirming that HOI act as a secondary stabilising factor. Such barriers are an order of magnitude larger than kBT at temperatures below 10 K, indicating that the skyrmions should be observable in low‑temperature spin‑polarized STM experiments.

The paper also discusses the broader relevance of the platform. Re(0001) is a well‑studied substrate that becomes superconducting below 1.7 K, opening the possibility of integrating these magnetic skyrmions with superconductivity to explore topological superconducting phases in magnet‑superconductor hybrids. Moreover, the ability to obtain zero‑field skyrmions eliminates the need for on‑chip magnetic field generators, simplifying device architectures for skyrmion‑based memory or logic.

In summary, the study provides a rigorous, fully parameterised theoretical demonstration that Rh/Co and Pd/Co atomic bilayers on Re(0001) constitute a new material platform for nanoscale, zero‑field isolated skyrmions. By incorporating higher‑order exchange into the spin model, the authors achieve quantitative predictions of skyrmion size, stability, and collapse mechanisms, thereby offering concrete guidance for future experimental verification and for the design of skyrmion‑based spintronic devices.