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

Technical Summary: Platform for Zero-Field Isolated Skyrmions in 4d/Co Atomic Bilayers on Re(0001)

Problem Statement
Magnetic skyrmions, topologically protected spin textures, are promising candidates for next-generation spintronic devices, neuromorphic computing, and quantum technologies due to their nanoscale size and stability. While isolated skyrmions have been experimentally realized in various transition-metal ultrathin films and multilayers, they typically require an external magnetic field for stabilization. Field-free stabilization is highly desirable for device integration to eliminate field-generating components and reduce energy consumption. To date, within the transition-metal ultrathin film family, only Rh/Co atomic bilayers on Ir(111) have been observed to host sub-10 nm isolated skyrmions at zero magnetic field, stabilized by exchange frustration. Theoretical predictions suggest that 4d/Co bilayers on the Re(0001) surface could also host such states, but previous studies relied on effective spin models that neglected higher-order multi-spin interactions (HOI) and lacked atomistic spin simulations to confirm stability.

Methodology
This study employs a combined approach of first-principles density functional theory (DFT) and atomistic spin simulations to investigate three specific systems: fcc-Rh/Co, hcp-Pd/Co, and fcc-Ru/Co atomic bilayers grown on the Re(0001) surface.

1. DFT Calculations: Electronic and magnetic structures were calculated using the fleur code based on the full-potential linearized augmented plane wave (FP-LAPW) method. The study modeled 4d/Co bilayers on nine layers of Re(0001).

   • Energy Dispersion: The energy dispersion of homogeneous spin spirals was calculated as a function of the propagation vector $q$ along high-symmetry directions ($\Gamma K M$ and $\Gamma M$), both with and without spin-orbit coupling (SOC).
   • Interaction Extraction: Pairwise exchange ($J_{ij}$) and Dzyaloshinskii-Moriya interaction (DMI, $D_{ij}$) constants were extracted by mapping DFT energies to Heisenberg and DMI Hamiltonians.
   • Higher-Order Interactions (HOI): To go beyond standard models, the study calculated three HOI constants: biquadratic ($B_1$), three-site four-spin ($Y_1$), and four-site four-spin ($K_1$). These were derived from the energy differences between specific multi-Q states (uudd and 3Q states) and corresponding single-Q spin spiral states.
   • Magnetocrystalline Anisotropy (MAE): Calculated self-consistently to determine the easy magnetization axis.

2. Atomistic Spin Simulations: An extended atomistic spin model was constructed incorporating all DFT-derived parameters (Heisenberg exchange, DMI, MAE, and HOI).

   • Relaxation: Spin structures were relaxed using the Landau-Lifshitz equation on a $150 \times 150$ hexagonal lattice to determine the zero-temperature magnetic phase diagram.
   • Stability Analysis: The Geodesic Nudged Elastic Band (GNEB) method was used to calculate the minimum energy path (MEP) for the collapse of isolated skyrmions into the ferromagnetic (FM) ground state, determining annihilation and creation energy barriers.

Key Contributions and Results

• Identification of Zero-Field Skyrmion Platforms: The extended spin model simulations reveal that isolated skyrmions emerge spontaneously on the ferromagnetic background at zero magnetic field for fcc-Rh/Co/Re(0001) and hcp-Pd/Co/Re(0001). In contrast, isolated skyrmions could not be stabilized in fcc-Ru/Co/Re(0001) at zero or finite fields due to insufficient DMI strength.
• Skyrmion Dimensions: The radius of the zero-field isolated skyrmions is approximately 6 nm for fcc-Rh/Co/Re(0001) and 12 nm for hcp-Pd/Co/Re(0001). These radii decrease as the external magnetic field increases.
• Role of Higher-Order Interactions: The study highlights that beyond nearest-neighbor exchange and DMI interactions induce frustration in these films. Crucially, the inclusion of HOI (specifically the three-site four-spin interaction) is necessary for an accurate description of the system, although the effective spin model (neglecting HOI) correctly predicted the ground state.
• Energy Barriers and Stability: Transition-state theory calculations show that the isolated skyrmions in the stable systems are protected by substantial energy barriers of approximately 150 meV (annihilation barrier) at zero field.
  • Barrier Composition: The DMI provides the dominant positive contribution to the energy barrier, with a smaller positive contribution from the three-site four-spin interaction. The pairwise exchange and MAE terms act to reduce the barrier (favoring the FM state). The four-site four-spin interaction provides a negligible contribution.
  • Observability: The magnitude of these barriers suggests that these skyrmions could be observed in low-temperature experiments.
• Material Properties:
  • All three films exhibit an out-of-plane easy magnetization axis.
  • The DMI in fcc-Rh/Co and hcp-Pd/Co is primarily driven by the Re substrate, whereas in fcc-Ru/Co, the Ru and Co layers contribute differently, resulting in a significantly lower total DMI.
  • Magnetic moments of the Co and 4d layers show significant variation with spin spiral wave vectors, a feature not captured by simple Heisenberg models with fixed moments, though the moments remain fairly constant near the FM state.

Significance and Claims
The authors propose 4d/Co bilayers on the Re(0001) surface as a new, promising platform for realizing nanoscale, zero-field isolated skyrmions. The work demonstrates that specific combinations of 4d transition metals (Rh, Pd) with Co on Re can stabilize these topological textures without external fields, driven by a complex interplay of exchange frustration, DMI, and higher-order spin interactions.

The paper claims that the calculated energy barriers are sufficiently high to allow experimental observation via spin-polarized scanning tunneling microscopy (SP-STM) at low temperatures. Furthermore, the authors note that since the Re substrate becomes superconducting below 1.7 K, these systems could serve as model magnet-superconductor hybrid systems, potentially relevant for emergent topological superconductivity. The study emphasizes that while the effective spin model predicted the existence of these states, the extended model including HOI is essential for accurately assessing skyrmion stability and properties.