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Spin-orbit coupling (SOC), which is the core of many condensed-matter phenomena such as nontrivial band gap and magnetocrystalline anisotropy, is generally considered appreciable only in heavy elements. This is detrimental to the synthesis and application of functional materials. Therefore, amplifying the SOC effect in light elements is crucial. Herein, focusing on 3d and 4d systems, we demonstrate that the interplay between crystal symmetry and electron correlation can significantly enhance the SOC effect in certain partially occupied orbital multiplets through the self-consistently reinforced orbital polarization as a pivot. Thereafter, we provide design principles and comprehensive databases, where we list all the Wyckoff positions and site symmetries in all two-dimensional (2D) and three-dimensional crystals that could have enhanced SOC effect. Additionally, we predict nine material candidates from our selected 2D material pool as high-temperature quantum anomalous Hall insulators with large nontrivial band gaps of hundreds of meV. Our study provides an efficient and straightforward way for predicting promising SOC-active materials, relieving the use of heavy elements for next-generation spin-orbitronic materials and devices.

However, designing materials with strong SOC in realistic materials has been quite straightforward because it simply links to the atomic mass of the constituent elements. Therefore, the candidates for spin-orbit active materials have mostly been limited to solids with heavy elements such as Bi, Sb, Te, Hg, Pt, and Pb23,24,25,26. Unfortunately, compounds containing heavy atoms usually have weaker chemical bonding; thus, they accommodate more native defects27, leading to poor stability for performing exotic functionalities. A famous example is the topological insulator Bi2Te3 with bulk conductivity owing to the heavily n-type self-doping28,29.

Therefore, it is crucial to thoroughly explore the potential of the SOC effect in materials with lighter elements. However, it is generally believed that SOC does not play an essential role in 3d transition metal materials, which are ideal for studying the interplay between symmetry, electronic occupation, and electron correlation30. On the other hand, the SOC effect was found comparable to the correlation in 4d and 5d series, leading to emergent quantum phases such as Weyl semimetal31, topological Mott insulator32,33, and quantum spin liquid34,35. The SOC effect of these systems was found to be more prominent in the presence of electron correlation, attributed to the electron localization induced by Coulomb repulsion that reduces the kinetic energy33,34. Recently, the cooperative effect between SOC and correlation was considered to explain the Fermi surface puzzle of the paramagnetic Fermi liquid Sr2RhO436, Sr2RuO437,38, as well as relatively large band splitting in other 4d, 5d, and 5f compounds39,40. These studies revealed the essential role of total angular momentum for the cooperative effects between SOC and correlation.

In this study, we aim to theoretically design materials with light elements but large effective SOC strength based on orbital symmetry, electron occupation, and the cooperative effect with correlation. The focus is on transition-metal magnetic materials, especially the 3d series, where the SOC strength is significantly smaller than the typical spin-exchange splitting. We propose that the cooperative effect of the electron correlation can significantly enhance the effective SOC through orbital polarization when there are partially occupied orbital multiplets around the Fermi level. Thereafter, we provide design principles and comprehensive databases, where we list all the Wyckoff positions and site symmetries that allow orbital multiplets in periodic crystals. The results indicate that 32 out of 80 layer groups and 125 out of 230 space groups can support large SOC effect. Therefore, for materials no matter recorded in existing databases or designed artificially, one can easily resort to our symmetry principles to predict promising candidates with strong effective SOC.

2D materials, specifically 2D magnets, have attracted significant attention because of their engineerable and integrable nature for future devices. Particularly, the high-temperature QAH effect has been investigated for the potential application of dissipationless electronics; however, it is challenging to realize41. Hence, we applied our procedure to Computational 2D materials database (C2DB)42,43 and screened out 71 2D materials (from ~1600 candidates) with an orbital multiplet near the Fermi energy, enhancing the SOC effect. As opposed to the previous case-by-case search approach, we systematically obtained nine high-temperature QAH insulators with large nontrivial band gaps of hundreds of meV. Additionally, our symmetry principles and material candidates for enhanced SOC effect are valid for searching materials with strong magnetocrystalline anisotropy, which has a significant influence on industrial ferromagnetic materials with ultrahigh coercive fields. Our study paves a new avenue for realizing light-element materials with strong effective SOC for next-generation functional materials and devices in various fields.

Based on the mechanism described above, we extract the crucial principles to facilitate the design of light-element materials with large effective SOC. The 3d transition-metal elements of the materials should reside at the well-chosen Wyckoff positions, of which the site symmetries should permit the existence of orbital multiplets E1, E2, or T. Doublets E1 and E2 are allowed in both tetragonal, trigonal, and hexagonal point groups, whereas the triplet T only exists in the cubic point groups, leading to 24 single point groups (Supplementary Table 1). The design principles lead to a comprehensive database with all the Wyckoff positions and site symmetries, allowing orbital multiplets in 2D and 3D periodic crystals. The results indicate that 32 out of 80 layer groups and 125 out of 230 space groups can support the correlation-enhanced SOC effect, as listed in Supplementary Tables 2 and 3, respectively. Therefore, for materials no matter recorded in existing databases such as C2DB42 and ICSD46 or artificially designed, one can easily predict candidates with potentially strong SOC effect using our database (Supplementary Tables 2 and 3): (i) if the compound belongs to the required layer/space groups, and ii) if the transition-metal ion sits on the required Wyckoff positions.

Notably, the Dirac cones of Fe2X2 can also be gapped in a trivial way by an out-of-plane external electric field owing to the onsite energy splitting of different d electrons at two FeX sublayers. Because the electric field could not effectively separate the orbital degeneracy of \(\{{d}_{{xz}},{d}_{{yz}}\}\), the correlation effect can hardly enlarge the gap. Therefore, the electric-field-induced gap is more insensitive with increasing U (Fig. 3e). In addition, the competition between the SOC-induced nontrivial gap and the electric field-induced trivial gap in monolayer Fe2X2 is analogous to the Haldane phase diagram64 (Fig. 3f), which is quite rare in realistic materials. More detailed calculation results for Fe2X2 are presented in Supplementary Note 4.

In addition to the large-gap QAH insulators, the paradigm we used to design materials with large SOC effects could be applied to various scenarios. For instance, the long-range magnetism of 2D magnetic materials can only be stabilized by uniaxial anisotropy1. The single-ion magnetocrystalline anisotropy, which is one of the main components of uniaxial anisotropy, originates from the SOC effect65. With the reinforcement of the orbital polarization and SOC effect, the magnetocrystalline anisotropic energy, EMAE, for 2D magnetic materials with \(\{{d}_{{xz}},{d}_{{yz}}\}\) frontier orbital doublet can significantly increase as66

To summarize, we provide design principles for large SOC effects with the help of orbital degeneracy, electron occupation, and correlation, eliminating the need for heavy elements. To activate the correlation-enhanced SOC effect, we combined symmetry analysis of the transition-metal sites residing at specific Wyckoff positions and first-principles calculations to examine the partially occupied orbital multiplets around the Fermi level. Applying the guiding principles to the C2DB database, we found 71 2D material candidates supporting the correlation-enhanced SOC effect and nine compounds as potential candidates for high-temperature QAH insulators. The procedure can be easily extended for designing and searching 3D light-element materials with strong effective SOC in various fields of condensed matter physics, such as spintronics, spin-orbitronics, and topological phases of matter. ff782bc1db