Giant unquenched orbital moments in two-dimensional magnets: robust first-principles prediction and materials discovery beyond octahedral coordination, up to the $L_z=2\hbar$ quantum

In two-dimensional (2D) magnets whose transition-metal $d$ shells have a partially filled, degenerate crystal-field manifold, the orbital moment can be fully unquenched, contributing 20–50% of the magnetization and dominating the magnetic anisotropy that stabilizes magnetic order in an isolated monolayer. Ovesen and Olsen (2D Materials 13, 015001 (2025)) showed with LSDA+ $U$ plus self-consistent spin–orbit coupling that 44 monolayers from the Computational 2D Materials Database carry atomic orbital moments above 0.5 $\mu_\mathrm{B}$ , but left the framework's quantitative reliability open: the Hubbard $U=3$ eV was heuristic, no cheap substitute for self-consistent spin–orbit coupling exists, ferromagnetic versus antiferromagnetic order was compared only for one prototype, ordering temperatures were not predicted for the candidates, and no material with a non-octahedral/tetrahedral ligand geometry — trigonal prismatic in particular, where the degenerate $(d_{xy}, d_{x^2-y^2})$ pair admits an $L_z=2\hbar$ circulating state — was found. This problem asks to establish the design principle as a quantitatively reliable predictive framework and to identify real 2D materials realizing the larger $L_z=2\hbar$ orbital quantum, together with their anisotropies.

Background

Orbital magnetization is the part of a magnet's magnetic moment generated by the orbital motion of electrons, as opposed to their spin. In most transition-metal compounds the electric field of the surrounding ligands quenches it: the five $d$ orbitals split into real-valued eigenstates that carry no angular-momentum expectation value. In an octahedral environment (six ligands at the vertices of an octahedron) the $d$ shell splits into a lower $t_{2g}$ triplet and an upper $e_g$ doublet; the $t_{2g}$ states can form complex (circulating) superpositions carrying angular momentum $\pm\hbar$ about a trigonal axis, so a magnetically split, partially filled $t_{2g}$ band can host a fully unquenched orbital moment. In a trigonal prismatic environment (six ligands at the vertices of a trigonal prism, point group $D_{3h}$ ) the $d$ shell splits instead into three levels — a singlet and two doublets, one doublet, conventionally $e'$ , spanned by $d_{xy}$ and $d_{x^2-y^2}$ ; complex combinations of that pair carry angular momentum $\pm 2\hbar$ about the prism axis, the basis for the sought $L_z=2\hbar$ state. Magnetic anisotropy (the dependence of a magnet's energy on the direction of its magnetization) matters especially in two dimensions: the Mermin–Wagner theorem forbids spontaneous breaking of continuous spin-rotation symmetry in a 2D system at finite temperature, so monolayer magnetic order survives only through anisotropy — as in the Ising antiferromagnet FePS $_3$ , ordered below about 118 K. Orbital moments are a principal microscopic source of anisotropy: for monolayer FePS $_3$ , Kim and Park (Nano Letters 21, 10114 (2021)) showed that the predicted anisotropy increases by an order of magnitude once the orbital magnetization is captured by treating spin–orbit coupling (SOC) self-consistently. Two methodological ingredients proved essential in density-functional theory (DFT) studies of these states: a Hubbard correction (a penalty term $U$ that penalizes partial $d$ -shell occupation, splitting partially filled $t_{2g}$ bands and opening a gap) and self-consistent SOC in a noncollinear calculation (SCF-SOC), in which the spin–orbit interaction is included while the charge density is relaxed. The common cheap alternative — adding SOC non-self-consistently (NSCF-SOC) after a collinear calculation — yields nearly identical band structures but only small orbital moments, because the collinear ground state occupies an arbitrary real combination of the degenerate states; the correct unquenched orbital polarization is obtained only when SOC is treated self-consistently. Ovesen and Olsen systematically screened monolayers from the Computational 2D Materials Database (C2DB), a public database of hypothetically exfoliable 2D crystals with computed properties (Haastrup et al., 2D Materials 5, 042002 (2018)): starting from 112 monolayers with octahedral and 62 with tetrahedral field splitting, LSDA+ $U$ +SCF-SOC identified 44 with local orbital moments above 0.5 $\mu_\mathrm{B}$ (moments evaluated with the atom-centered projector-augmented-wave approximation), whereas a previous PBE+NSCF-SOC screening of 822 magnetic C2DB monolayers by the same authors had found only one. Their prototypes were VI $_3$ (partially filled majority $t_{2g}$ ) and FePS $_3$ (partially filled minority $t_{2g}$ ). The work stops short, however, of a quantitatively reliable and general framework: $U=3$ eV was chosen heuristically, all candidate calculations were initialized as ferromagnets (the ferromagnetic-versus-antiferromagnetic comparison was made only for FePS $_3$ ), no reduced-cost surrogate for SCF-SOC was validated, ordering temperatures of the candidates were not computed, and the search was restricted to octahedral and tetrahedral coordination — no trigonal-prismatic (or other-geometry) material stabilizing an $L_z=2\hbar$ moment could be found in the C2DB, even though such a state is symmetry-allowed and would roughly double the orbital moment and its anisotropy leverage.

Problem Statement

The problem is to turn the crystal-field design principle for giant unquenched orbital moments in two-dimensional magnets — a partially filled, degenerate crystal-field $d$ manifold plus Hubbard splitting and self-consistent spin–orbit coupling yields a fully unquenched orbital moment, which in turn drives the magnetic anisotropy that sustains monolayer magnetic order — into a quantitatively reliable, materials-level predictive framework, and to extend it beyond the octahedral and tetrahedral coordinations in which it was demonstrated. Concretely, the formulation established by Ovesen and Olsen (2D Materials 13, 015001 (2025)) leaves two linked open directions toward this objective, both of which a complete resolution must address, a validated route from crystal-field filling rules to predicted high-anisotropy two-dimensional magnets: (1) robustness of the predicted atomic orbital moments and anisotropy energies across the 44 monolayers identified by Ovesen and Olsen with atomic orbital moments above 0.5 $\mu_\mathrm{B}$ (the materials sought under direction (2) below are assessed within that direction), assessed on all three of the following axes: their dependence on a material-specific or self-consistently determined Hubbard $U$ (rather than the heuristic $U=3$ eV), their dependence on the long-range magnetic order (ferromagnetic versus antiferromagnetic configurations, checked beyond the single FePS $_3$ prototype), and their behavior under cheaper spin–orbit-coupling treatments (i.e. a controlled assessment of when, if ever, non-self-consistent SOC yields reliable orbital moments and anisotropies); and (2) extension of the design principle to other ligand geometries: a search, within and beyond the C2DB, for specific two-dimensional materials — trigonal prismatic coordination in particular, where partial filling of the degenerate $(d_{xy}, d_{x^2-y^2})$ manifold can stabilize a circulating $L_z=2\hbar$ state — that realize such larger orbital quanta, together with the computed orbital moments and magnetocrystalline anisotropies of any identified candidates; a negative finding — that no such monolayer exists — resolves this direction only in a bounded, verifiable form: an exhaustive survey of the C2DB together with a survey of at least one further, named public database of two-dimensional materials beyond the C2DB (a survey confined to the C2DB alone cannot support the negative, since this direction demands a search beyond it), together with a symmetry- or electron-counting argument excluding a partially filled $(d_{xy}, d_{x^2-y^2})$ manifold (the $e'$ doublet) in a stable $D_{3h}$ monolayer; an unsupported or unbounded claim of absence does not resolve it. Predicting magnetic ordering temperatures is not part of the stated objective; the source work raises them only as motivation for the design principle.

References

Martin Ovesen and Thomas Olsen, Giant orbital magnetization in two-dimensional materials, 2D Materials 13, 015001 (online November 2025; print issue 2026). DOI: 10.1088/2053-1583/ae1513; arXiv:2506.18563; https://doi.org/10.1088/2053-1583/ae1513
Martin Ovesen and Thomas Olsen, Orbital magnetization in two-dimensional materials from high-throughput computational screening, 2D Materials 11, 045010 (2024). DOI: 10.1088/2053-1583/ad6ba3; arXiv:2405.04239; https://doi.org/10.1088/2053-1583/ad6ba3
Shuhang Chen, Wenjing Xu, Yueyue Ning, and Ke Yang, Exploration of the Two-Dimensional Ising Magnetic Materials in the Triangular Prismatic Crystal Field, The Journal of Physical Chemistry C 128, 556–562 (2024). DOI: 10.1021/acs.jpcc.3c06603; arXiv:2312.15625; https://doi.org/10.1021/acs.jpcc.3c06603
Varun Rajeev Pavizhakumari and Thomas Olsen, Critical temperatures of two dimensional magnets beyond linear spin wave theory: application to CrI3, MPS3 (M=Ni, Mn, Fe) and CrSBr, 2D Materials (2026). DOI: 10.1088/2053-1583/ae9779; arXiv:2511.08426; https://doi.org/10.1088/2053-1583/ae9779
Tae Yun Kim and Cheol-Hwan Park, Magnetic Anisotropy and Magnetic Ordering of Transition-Metal Phosphorus Trisulfides, Nano Letters 21, 10114–10121 (2021). DOI: 10.1021/acs.nanolett.1c03992; https://doi.org/10.1021/acs.nanolett.1c03992
Sten Haastrup, Mikkel Strange, Mohnish Pandey, Thorsten Deilmann, Per S. Schmidt, Nicki F. Hinsche, Morten N. Gjerding, Daniele Torelli, Peter M. Larsen, Anders C. Riis-Jensen, Jakob Gath, Karsten W. Jacobsen, Jens Jørgen Mortensen, Thomas Olsen, and Kristian S. Thygesen, The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals, 2D Materials 5, 042002 (2018). DOI: 10.1088/2053-1583/aacfc1; https://doi.org/10.1088/2053-1583/aacfc1
Ruilin Mao and Zhongyao Li, 5d transition metal atom substituting N doped MoSi2N4 and WSi2N4: Electronic structure, magnetic anisotropy, and strain effect, Physics Letters A 534, 130260 (2025). DOI: 10.1016/j.physleta.2025.130260; https://doi.org/10.1016/j.physleta.2025.130260
Kunal Dutta, Anupam Mondal, Sayantika Bhowal, Subhradip Ghosh, and Indra Dasgupta, Interplay of Valley, Orbital, Spin, and Layer Degrees of Freedom in Ta2CS2 MXene, arXiv:2605.01271 (2026); https://arxiv.org/abs/2605.01271