The discovery and classification of topological phases of matter in real materials has been a significant focus of research. Initial efforts concentrated on time-reversal symmetric (non-magnetic) materials, employing band representation theory for classification. However, this approach excludes a vast number of magnetic materials and their unique properties arising from broken time-reversal symmetry. These properties include linear response effects like the anomalous Hall effect (AHE), where a transverse voltage drop results from a longitudinal current, and the anomalous Nernst effect (ANE), its thermal analog involving a temperature gradient. Magnetic Heusler compounds, a versatile and tunable material class, are promising candidates for exhibiting large AHE and ANE due to their inherent magnetism and topological band structures. Previous studies have shown exceptionally high ANE in Co₂MnGa, indicating their potential for applications in data storage, data transfer, and thermoelectric power generation. This work addresses the challenge of identifying the most promising Heusler compounds for achieving enhanced linear response effects by establishing simple guidelines for predicting their magnitudes.

The introduction extensively cites prior research on topological phases of matter (refs 1-4), focusing on the challenges posed by magnetic materials and the absence of established classification methods for these systems. Several studies of selected material classes are cited (refs 5-15), highlighting the lack of a comprehensive approach for magnetic materials. Existing literature on the AHE (refs 16, 17) and ANE (refs 18, 19) is reviewed, establishing their importance and the need for materials with stronger effects. The potential of Heusler compounds (refs 20-25) is introduced, and the remarkable AHE and ANE observed in Co₂MnGa (refs 26-31) are highlighted, emphasizing their importance for applications in magneto-optics (refs 32-34). The significance of Berry curvature (refs 17, 19, 35, 36) in enhancing intrinsic contributions to these effects is also mentioned, along with the potential implications for electrical, thermoelectrical, and magneto-optical properties (ref 37).

The research employed a computational approach using the Heusler database from the University of Alabama as a starting point. The workflow (Figure 1) involved selecting stable, cubic, and magnetic full Heusler structures. Density functional theory (DFT) calculations using VASP with pseudopotentials, plane waves, and the generalized gradient approximation (GGA) were performed to determine electronic structures. The magnetic moments were iteratively adjusted until they matched the database values, keeping them aligned along the (001) direction. Wannier functions were generated using Wannier90, with parameter adjustments (orbital projections and energy windows) to minimize the energy difference between Wannier and DFT wave functions. Tight-binding parameters were extracted to construct Hamiltonians, enabling the calculation of Berry curvature (Ω) using the Kubo formalism. The anomalous Hall conductivity (σ) was then calculated from the Berry curvature, while the anomalous Nernst conductivity (α) was determined using the approach proposed by Xiao et al. (ref 19). Finally, the optical Hall conductivity and Kerr angle (θ) were computed using the Kubo formalism.

The analysis revealed a strong correlation between the material's space group and the magnitude of AHE and ANE. Compounds with space group 225, possessing three mirror planes, exhibited significantly larger AHC, ANC, and Kerr angles than those with space group 216 (Figure 2). This difference is explained by the presence of nodal lines (NLs) in space group 225, which are protected by mirror symmetries. When magnetism breaks these symmetries, the NLs gap out, inducing a strong Berry curvature and leading to enhanced AHE and ANE. A four-band model (equation 1) was used to illustrate this mechanism (Figure 3). The study further explored the relationship between the number of valence electrons and AHC/ANC, discovering a double-peak structure with maxima at 21 and 28 valence electrons (Figure 4). This behavior is attributed to the energetic position of Berry curvature-inducing features relative to the Fermi level. While AHC and ANC are both related to Berry curvature, they are not directly correlated, though a linear connection emerges when considering their maximum values within an energy window around the Fermi level. Several Heusler compounds displayed exceptionally high AHC and ANC values, some even surpassing previously reported values (Table 1). Importantly, both AHC and ANC show strong energy dependence (Figure 5) highlighting the importance of considering Fermi level shifts due to doping in comparing theoretical and experimental results. The dominant source of Berry curvature was determined to be either Weyl points (WPs) or gapped nodal lines (NLs).

The findings underscore the importance of symmetry and the Fermi level position in designing materials with giant AHE and ANE. Mirror symmetries in conjunction with magnetism prove crucial in generating large Berry curvature, enhancing linear response coefficients. The identification of optimal valence electron counts (21 and 28) provides a valuable guideline for material selection. The strong energy dependence of AHC and ANC emphasizes the need to consider doping effects when comparing theoretical predictions with experimental measurements. The results significantly expand the potential of Heusler compounds for applications leveraging AHE and ANE, opening avenues for creating high-performance materials with tailored properties.

This study comprehensively investigated the AHE, ANE, and MOKE in a large set of magnetic cubic full Heusler compounds. The results demonstrate the importance of material symmetry and the Fermi level position for achieving giant AHE and ANE. Several Heusler compounds were identified exhibiting superior transport properties compared to previously reported materials. The insights from this work pave the way for the rational design and synthesis of novel high-performance materials for various technological applications. Future studies could focus on experimental verification of the predicted properties and further investigation of the effects of different magnetization directions on the Berry curvature distribution.

The study is theoretical and relies on DFT calculations. Discrepancies might exist between the calculated and experimentally observed values due to factors such as defects, impurities, and sample preparation techniques. The analysis focuses primarily on intrinsic contributions to AHE and ANE; extrinsic effects could influence the overall response. Furthermore, the study assumes a specific magnetization direction; variations in magnetization direction might affect the results.