🔬 materials science

Large spin splitting metallic altermagnets from machine-learned design rules

This study employs interpretable machine learning to identify design rules for metallic altermagnets, leading to the discovery of a stable family of tetragonal $A_2XY$ Heusler compounds with large spin splittings that surpass existing benchmarks and exhibit promising tunneling magnetoresistance.

Original authors: Ali Sufyan, Brahim Marfoua, J. Andreas Larsson, Rickard Armiento, Erik van Loon

Published 2026-09-15

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CC BY 4.0

Original authors: Ali Sufyan, Brahim Marfoua, J. Andreas Larsson, Rickard Armiento, Erik van Loon

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

In the world of electronics, controlling the flow of electrons based on their spin is the holy grail for building faster, more efficient devices. For decades, scientists have relied on two main types of magnetic materials to do this: ferromagnets, which act like tiny bar magnets with a strong net pull, and antiferromagnets, where the internal magnetic forces cancel each other out perfectly, leaving no external pull. Ferromagnets are easy to use but create stray magnetic fields that interfere with neighboring components, while antiferromagnets are quiet and dense but difficult to manipulate because they lack that net magnetic field. A third category, recently named altermagnets, has emerged to bridge this gap. These materials possess the quiet, cancelled-out magnetic order of antiferromagnets, yet they still manage to split their electron energy levels based on spin direction, a feature usually reserved for ferromagnets. This unique combination allows them to carry spin-polarized currents without creating the messy magnetic fields that plague traditional devices, making them a promising candidate for the next generation of spintronic technology.

Despite their theoretical appeal, finding real-world materials that fit this description has been a challenge. While researchers have identified a few examples, most are either insulators that block electricity or semiconductors that are difficult to integrate into standard circuits. The most sought-after prize has been a metallic altermagnet that exhibits a large, clear separation of spin states, specifically one with a "d-wave" pattern, which is the most efficient shape for generating useful electrical currents. Until now, such materials have been scarce, and the search for them has often been a slow process of trial and error, testing one chemical compound after another without a clear map of what to look for.

A new study by researchers at Lund University and Linköping University in Sweden has changed the approach to this search by combining artificial intelligence with high-speed computer simulations. Instead of blindly testing thousands of random chemicals, the team first taught a machine-learning model to recognize the hidden patterns that lead to large spin splitting. They fed the computer data from 180 previously studied magnetic materials, allowing the algorithm to learn which structural and chemical features mattered most. The model quickly identified two simple rules for success: the crystal structure needs to be compact, with a small number of atoms packed tightly together, and the magnetic atoms within that structure must be capable of holding strong local magnetic moments. These findings provided a clear set of instructions for the next phase of the hunt.

Armed with these rules, the researchers turned their attention to a specific family of chemical compounds known as Heusler alloys, which are made of three different elements arranged in a tetragonal crystal shape. They screened 307 different variations of these compounds using advanced density-functional theory, a rigorous method for calculating the behavior of electrons in solids. The computer simulations acted as a filter, first checking if the symmetry of the crystal allowed for the special altermagnetic state, and then verifying if the material remained metallic. Out of the hundreds of candidates, the simulations confirmed that 157 were indeed metallic altermagnets. The team then narrowed this list down by checking for stability, ensuring the materials would not fall apart or change shape under normal conditions. This process left them with a shortlist of sixteen promising candidates that were not only stable but also possessed the desired magnetic properties.

The results were striking. Several of these new materials exhibited spin splitting far larger than any previously known metallic altermagnet. The top performer, a compound made of cobalt, aluminum, and scandium, showed a spin splitting of 2.24 electronvolts, while a second candidate made of iron, aluminum, and germanium reached 2.07 electronvolts. To put this in perspective, the previous benchmark for metallic altermagnets was a material called chromium antimonide, which split at roughly 1.43 electronvolts under the same testing conditions. The new materials more than doubled the performance of the best previous metallic examples. The researchers also calculated how these materials would behave in a real device, estimating that the cobalt-aluminum-scandium compound could generate a tunneling magnetoresistance of up to 203 percent. This is a measure of how effectively the material can switch electrical resistance on and off using spin, a critical function for memory storage and logic gates.

Beyond just finding a single winner, the study demonstrated that the machine-learning approach works. The two simple criteria identified by the algorithm—compact unit cells and strong local magnetic moments—proved to be reliable predictors for finding large spin splitting in a completely new chemical family. The researchers confirmed that these materials are not just theoretical curiosities; they are dynamically and mechanically stable, and many of them sit at a low energy state that suggests they could be synthesized in a laboratory. The study also mapped out how the spin of the electrons moves through these materials, revealing that the splitting is not uniform but follows a specific four-lobed pattern that changes direction depending on the angle of travel. This directional behavior is a hallmark of the d-wave order and is essential for creating the complex spin currents needed for advanced electronics.

The work does not stop at identification. The researchers have provided a detailed roadmap for experimentalists, highlighting specific compounds like the cobalt-aluminum-scandium and iron-aluminum-germanium variants as the most viable targets for immediate synthesis. They have also calculated the intrinsic spin Hall conductivity, a measure of how well the material can convert an electric current into a spin current, finding values that are substantial and comparable to the best materials currently used in the industry. By combining interpretable machine learning with targeted first-principles calculations, the team has not only discovered a new family of high-performance materials but also established a new method for finding them. This approach suggests that the search for the perfect magnetic material is no longer a game of chance, but a guided exploration where the path forward is illuminated by the patterns the data itself reveals.

Technical Summary: Large Spin Splitting Metallic Altermagnets from Machine-Learned Design Rules

Problem Statement
Altermagnets represent a distinct class of collinear magnetism characterized by compensated magnetic order (zero net magnetization) combined with momentum-dependent spin splitting of nonrelativistic origin. While theoretically promising for spintronics due to their ability to support spin-polarized currents without stray fields, experimentally validated bulk realizations of metallic altermagnets with large, directly resolved spin splittings remain scarce. Existing candidates are often insulating, semiconducting, or possess splitting magnitudes insufficient for robust transport applications. Furthermore, while computational surveys have identified symmetry-allowed altermagnets, there is a lack of general design rules to quantitatively predict the magnitude of spin splitting across broad chemical spaces, hindering the discovery of materials with exceptionally large splittings.

Methodology
The authors employ a two-stage approach combining interpretable machine learning (ML) with targeted high-throughput density-functional theory (DFT):

Machine Learning Model Development:
- Training Data: A dataset of 180 DFT-labeled altermagnetic structures from the authors' previous MAGNDATA survey was used.
- Descriptors: 223 features were constructed, including compositional statistics (Magpie framework), structural parameters (lattice constants, space group), and DFT-derived magnetic quantities (local moments, Hubbard $U$ , band gaps).
- Model Selection: A gradient-boosted regressor (GBR) was selected after comparing four architectures. The model was trained on the logarithm of the maximum spin splitting ( $\Delta_{max}$ ) to handle the skewed distribution.
- Interpretability: SHapley Additive exPlanations (SHAP) were used to identify the most influential descriptors, translating statistical correlations into physical design principles.
Targeted High-Throughput Screening:
- Target Family: Guided by ML insights, the authors screened ternary $A_2XY$ Heusler compounds in the tetragonal space group $P4/mmm$ (No. 123). This family was chosen for its compact four-atom primitive cells and the potential for large local moments on transition-metal $A$ sites.
- Workflow:
  - Symmetry Filtering: From 307 relaxed structures, symmetry analysis of the magnetic $A$ -atom Wyckoff positions distinguished altermagnetic arrangements (2e and 2f sites) from conventional antiferromagnetic ones (2g and 2h sites).
  - DFT Validation: 169 symmetry-allowed candidates underwent full DFT relaxation and static calculations to confirm metallicity and altermagnetic ground states.
  - Stability Assessment: Candidates were evaluated for dynamical stability (phonon spectra), mechanical stability (Born criteria), and thermodynamic stability (convex hull distance relative to competing phases from Materials Project and Alexandria databases).
- Transport Properties: For stable candidates, spin-resolved band structures, Fermi surfaces, intrinsic spin Hall conductivity, and Julliere-model tunneling magnetoresistance (TMR) estimates were calculated.

Key Contributions

Derivation of Design Rules: The study identifies two primary physical descriptors strongly associated with large spin splitting: (1) compact unit cells (low number of atoms per cell) and (2) magnetic sublattices capable of sustaining sizable local moments.
Discovery of a New Material Platform: The authors identify the tetragonal $A_2XY$ Heusler family as a chemically tunable platform for metallic $d$ -wave altermagnetism.
Identification of Record-Breaking Candidates: Six compounds are identified with spin splittings exceeding that of the benchmark metallic altermagnet CrSb (calculated at 1.43 eV under the same protocol). The top candidates are Co $_2$ AlSc ( $\Delta_{max} = 2.24$ eV) and Fe $_2$ AlGe ( $\Delta_{max} = 2.07$ eV).
Comprehensive Stability and Response Analysis: The work provides a rigorous stability assessment, confirming that 15 of the 16 top candidates are dynamically and mechanically stable, with 10 lying on or below the thermodynamic convex hull. It also quantifies their directional Fermi-surface polarization and spin Hall conductivity.

Results

Model Performance: The ML model achieved an overall cross-validated $R^2 = 0.70$ and correctly ranked all 14 "large-splitter" materials in the training set above the median, despite class imbalance.
Screening Outcomes:
- Of 307 initial candidates, 169 were symmetry-allowed altermagnets.
- 157 were confirmed as metallic altermagnets by DFT.
- 16 candidates satisfied the energy tolerance ( $\Delta E_{AM-GS} \le 0.1$ meV/f.u.) for having the altermagnetic state as the ground state.
- 15 of these 16 were found to be both dynamically and mechanically stable.
Splitting Magnitudes: The maximum splittings reached 2.24 eV (Co $_2$ AlSc) and 2.07 eV (Fe $_2$ AlGe), significantly surpassing the 1.43 eV benchmark for CrSb.
Transport Properties:
- Fermi Surface: The materials exhibit characteristic $d$ -wave ( $d_{x^2-y^2}$ ) momentum dependence, with spin splitting reversing sign under 90° rotation and vanishing along nodal planes.
- TMR: Using the Julliere model, Co $_2$ AlSc is estimated to have a tunneling magnetoresistance of up to 203% at the Fermi level along the [100] direction.
- Spin Hall Conductivity: The intrinsic spin Hall conductivity ( $\sigma^z_{xy}$ ) reaches magnitudes up to $\sim 486 (\hbar/e)(\Omega \text{cm})^{-1}$ , approximately one-quarter of that of Pt. The anomalous Hall conductivity is symmetry-forbidden for the $c$ -axis Néel vector.

Significance and Claims
The paper claims to demonstrate how interpretable machine learning can guide targeted first-principles searches by extracting physically meaningful design rules from limited data. Specifically, it validates that the combination of compact unit cells and large local moments is a transferable predictor for strong altermagnetic splitting, successfully applied to a chemical family ( $A_2XY$ Heuslers) distinct from the training set.

The authors identify a chemically tunable family of metallic $d$ -wave altermagnets that overcome the scarcity of bulk materials combining metallicity, unambiguous $d$ -wave order, and large spin splitting. They highlight Co $_2$ AlSc and Fe $_2$ AlGe as particularly promising candidates for spin-transport studies. The work concludes by suggesting that the next logical steps involve the epitaxial growth of the most stable candidates (Fe $_2$ AlGe, Mn $_2$ AlTa, Co $_2$ AlSc) and experimental verification of their $d$ -wave splitting and predicted multi-axis magnetoresistance. The study does not claim these materials are immediately ready for device integration but positions them as a validated platform for future experimental exploration.

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