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Topological and spin-orbit effects on orbital moments in ultra-thin magnetic films

Using first-principles electronic structure theory, this study demonstrates that topological orbital moments (TOMs) can be distinguished from spin-orbit induced contributions in ultra-thin magnetic films by comparing non-trivial spin textures like the triple-Q state with trivial counterparts, revealing that TOMs provide a significant, detectable signal in spin-compensated systems where spin-orbit effects are nearly canceled.

Original authors: Felix Nickel, Soumyajyoti Haldar, Mara Gutzeit, Stefan Heinze

Published 2026-09-01

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CC BY 4.0

Original authors: Felix Nickel, Soumyajyoti Haldar, Mara Gutzeit, Stefan Heinze

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Magnetism is a familiar force, the invisible pull that guides a compass needle or holds a refrigerator note in place. In most everyday magnets, this pull comes from the tiny, intrinsic spins of electrons, which act like miniature bar magnets all pointing in the same direction. However, in the microscopic world of ultra-thin films—layers of material just a few atoms thick—magnetism can become far more complex. Here, electrons can arrange themselves in intricate, swirling patterns where their spins point in different directions, sometimes canceling each other out so that the total magnetic pull of the entire group is zero. For decades, scientists believed that if the spins canceled out, the magnetic influence of the material would vanish. Yet, a deeper layer of physics suggests that even when the spins are neutralized, the electrons' orbital motion—the way they circle the atomic nucleus—can still generate a magnetic moment. A new study by researchers at Kiel University in Germany investigates how these orbital motions behave in complex magnetic textures, specifically looking for a unique type of magnetism that arises from the shape of the electron paths themselves, rather than just the strength of the spin.

The researchers focused on a specific challenge: distinguishing between two different sources of this orbital magnetism. One source is a standard effect caused by the interaction between an electron's spin and its motion, known as spin-orbit coupling. This effect exists in almost all magnetic materials. The other source is a more exotic phenomenon called a topological orbital moment. This type of magnetism does not depend on the strength of the spins but rather on the global shape or "topology" of the magnetic pattern. Imagine a magnetic texture that twists in a way that creates a specific, non-repeating loop; this twist generates a topological orbital moment. The difficulty lies in the fact that both effects happen at the same time, making it hard to tell which one is responsible for the magnetic signal a scientist might measure. The team set out to untangle these two contributions to see if the topological effect could be isolated and measured, even in materials where the total spin is zero.

To solve this puzzle, the team used powerful computer simulations based on the laws of quantum mechanics to model ultra-thin films of manganese and palladium on a rhenium surface. They compared two different magnetic arrangements. The first was a simple, orderly pattern where spins pointed in alternating rows, a configuration with a straightforward shape that produces no topological orbital moments. The second was a complex, three-dimensional twist where the spins pointed in different directions, forming a structure with a non-trivial topology. By running simulations on both, the researchers could calculate exactly how much magnetism came from the standard spin-orbit interaction and how much came from the topological shape. They found that in the complex, twisted structure, the topological orbital moments were a significant force, creating a net magnetic effect even though the spins themselves canceled out.

The study revealed a crucial insight about how these forces interact. In the complex magnetic state, the standard spin-orbit induced magnetism and the topological magnetism often point in opposite directions, partially canceling each other out at the atomic level. However, when the researchers looked at the entire magnetic unit cell, the topological contribution remained robust. Because the standard spin-orbit effects were nearly balanced out by the opposing spins, the remaining magnetic signal was dominated by the topological orbital moments. This means that in materials where the spins are compensated, the topological effect is not just a minor detail; it is the primary source of the orbital magnetism. The researchers also confirmed that this topological signal is stable and does not change significantly even when the standard spin-orbit effects are included in the calculation, suggesting that it is a reliable feature of these twisted magnetic structures.

The findings extend beyond the specific manganese films to other materials, including iron layers on different metal surfaces that form tiny magnetic vortices known as skyrmions. In these systems, the researchers observed the same trend: when the spins cancel out, the topological orbital moments stand out as the decisive factor. This is a significant discovery because it offers a potential way to detect and manipulate these magnetic textures. Since the topological orbital moments point in a specific direction perpendicular to the surface, they could be influenced by an external magnetic field, even if the spins themselves are too weak to respond. This opens the door to using these subtle orbital effects to control magnetic domains in future technologies. The study provides a clear theoretical roadmap for experimentalists, suggesting that by looking at the local electronic structure with high-resolution tools, they can distinguish the topological signal from the background noise of standard magnetic effects, finally allowing for the direct observation of these elusive topological orbital moments.

Technical Summary: Topological and Spin-Orbit Effects on Orbital Moments in Ultra-Thin Magnetic Films

Problem Statement
Topological orbital moments (TOMs) are a distinct signature of magnetic textures with non-trivial spin topology, arising from a non-vanishing Berry curvature. While TOMs offer a potential mechanism to manipulate magnetic structures with compensated total spin moments, their experimental identification is complicated by the concurrent presence of orbital moments induced by spin-orbit coupling (SOC). In systems with compensated spin magnetization, SOC-induced orbital moments ( $L_{SOC}$ ) are often expected to cancel out, yet their interplay with TOMs ( $L_{TOM}$ ) remains poorly understood. Specifically, it is unclear whether $L_{SOC}$ and $L_{TOM}$ can be disentangled experimentally or theoretically, and whether the additive model $L_{total} = L_{SOC} + L_{TOM}$ holds for complex, non-coplanar spin textures. Furthermore, direct observation of TOMs remains elusive, as previous evidence has been limited to indirect transport measurements in van-der-Waals materials.

Methodology
The authors employ first-principles electronic structure theory based on Density Functional Theory (DFT) to investigate the origin of orbital moments in ultra-thin magnetic films. The study utilizes two primary computational frameworks:

FLEUR Code: Based on the full-potential linearized augmented plane wave (FLAPW) method, used to analyze Pd/Mn/Re(0001) film systems and freestanding Pd/Mn/Re trilayers. Calculations were performed in both the Row-wise Antiferromagnetic (RW-AFM) state and the non-coplanar Triple-Q (3Q) state.
VASP Code: Based on the projector augmented wave (PAW) method, used to investigate atomic-scale skyrmion lattices (SkX) in Fe monolayers on various substrates (Ir(111), Rh/Ir(111), and Ir-3/Re(0001)).

To isolate the topological contribution from the SOC contribution, the authors perform calculations in two distinct regimes:

Scalar-relativistic approximation: SOC is neglected, yielding purely topological orbital moments ( $L_{TOM}$ ).
Full SOC inclusion: Yields the total orbital moment ( $L_{total}$ ).
By comparing these results, the SOC-induced contribution ( $L_{SOC}$ ) is derived. The study focuses on systems where the net spin moment vanishes (spin-compensated) to minimize the dominance of $L_{SOC}$ , allowing for a clearer analysis of $L_{TOM}$ .

Key Contributions and Results

Disentanglement of $L_{SOC}$ and $L_{TOM}$ : The authors use the RW-AFM state as a reference. Since this state has trivial topology, it exhibits no $L_{TOM}$ . By rotating the spin moments in the RW-AFM state, they demonstrate that $L_{SOC}$ depends strongly on the global orientation of the spins relative to the atomic lattice (both polar and azimuthal angles). The magnitude of $L_{SOC}$ scales with the polar angle, consistent with the Bruno formula, and varies with the azimuthal angle due to anisotropic symmetric exchange (ASE) interactions.
Validation of the Additive Model: For the non-coplanar 3Q state (a superposition of three RW-AFM states), the study confirms that the total orbital moment is well-approximated by the sum of SOC-induced and topological contributions. Crucially, the inclusion of SOC causes only negligible changes to the magnitude of $L_{TOM}$ itself, suggesting that $L_{TOM}$ is robust against small SOC-induced modifications to the electronic structure.
Compensation in Spin-Compensated Systems: In the 3Q state and spin-compensated skyrmion lattices (e.g., SkX in Fe/Ir-3/Re(0001)), the net spin moment vanishes. Consequently, the $L_{SOC}$ contributions, which are roughly aligned with local spin moments, also nearly cancel out over the unit cell. As a result, the net $L_{total}$ is dominated by the uniform $L_{TOM}$ .
Local vs. Global Behavior: While the net orbital moment in spin-compensated systems is largely determined by $L_{TOM}$ , the local orbital moments at individual lattice sites differ significantly. The vector sum of $L_{SOC}$ and $L_{TOM}$ at specific sites can either enhance or reduce the local moment depending on the relative orientation of the spin moment and the surface normal. This local variation is absent in the scalar-relativistic limit.
Material Dependence: The study compares Mn-based (Pd/Mn/Re) and Fe-based systems. Fe-based systems exhibit a topological orbital susceptibility ( $\kappa_{TO}$ ) roughly four times larger in magnitude (and opposite in sign) compared to Mn-based systems.
Skyrmion Lattices: The analysis extends to complex nanoskyrmion lattices. In spin-compensated skyrmion states, the net orbital moment remains close to the topological value. However, in non-compensated skyrmion lattices (e.g., SkX-12, SkX-19, SkX-27), the uncompensated spin moment leads to a significant uncompensated $L_{SOC}$ , creating a large discrepancy between the topological and total orbital moments.

Significance and Claims
The paper claims that for spin-compensated magnetic structures with non-trivial topology, the topological orbital moment is the decisive factor for the net orbital magnetization, as SOC-induced contributions are nearly compensated. This finding supports the feasibility of detecting TOMs in such systems, potentially via local probes like spin-polarized scanning tunneling microscopy (SP-STM) which can resolve the local variations in orbital moments caused by the interplay of $L_{SOC}$ and $L_{TOM}$ .

The authors assert that the simple picture of additive contributions ( $L_{total} = L_{SOC} + L_{TOM}$ ) holds very well for these systems. They conclude that spin-compensated states are superior candidates for experimental studies of TOMs compared to spin-uncompensated states, where the large net SOC contribution would obscure the topological signal. The work provides a theoretical foundation for interpreting future experiments, suggesting that the orientation of magnetic domains in 3Q states could potentially be manipulated by external magnetic fields acting on the net orbital moment, even in the absence of a net spin moment.

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