Giant orbital magnetization in two-dimensional materials

Orbital magnetization typically plays a minor role in compounds where the magnetic properties are governed by transition metal elements. However, in some cases, the orbital magnetization may be fully unquenched, which can have dramatic consequences for magnetic anisotropy and various magnetic response properties. In the present work, we start by summarizing how unquenched orbital moments arise from particular combinations of crystal field splitting and orbital filling. We exemplify this for the cases of two-dimensional (2D) VI$_3$ and FePS$3$, and show that Hubbard corrections as well as self-consistent spin-orbit coupling are crucial ingredients for predicting correct orbital moments from first principles calculations. We then search the Computational 2D Materials Database (C2DB) for monolayers having tetrahedral or octahedral crystal field splitting of transition metal $d$-states and orbital occupancy that is expected to lead to large orbital moments. We identify 112 monolayers with octahedral crystal field splitting and 62 monolayers with tetrahedral crystal field splitting and for materials with partially filled $t{2g}$ bands, we verify that inclusion of Hubbard corrections as well as self-consistent spin-orbit coupling typically increases the magnitude of predicted orbital moments by an order of magnitude.

💡 Research Summary

The manuscript “Giant orbital magnetization in two‑dimensional materials” investigates why orbital magnetization—normally a minor correction to the total magnetic moment in transition‑metal compounds—can become unusually large (unquenched) in certain 2D crystals, and how this phenomenon can be reliably predicted from first‑principles calculations. The authors first review the microscopic origin of unquenched orbital moments: a specific combination of crystal‑field splitting, partial filling of the t₂g manifold, and spin‑orbit coupling (SOC) can generate a finite expectation value of the orbital angular momentum operator. In octahedral or tetrahedral ligand environments the d‑orbitals split into a three‑fold t₂g set and a two‑fold e_g set; only the t₂g states can carry orbital angular momentum because the diagonal matrix elements of L̂ vanish for e_g. When the t₂g band is partially occupied, SOC can lift the remaining degeneracy and align the orbital moment either parallel or perpendicular to the atomic plane, depending on symmetry.

A central methodological point is that standard density‑functional approximations (LSDA, PBE) fail to capture this effect because they underestimate electronic correlations, leaving the partially filled t₂g band metallic and the orbital moment essentially zero. Adding a Hubbard‑U term (DFT+U) localizes the d‑electrons and opens a gap in the t₂g manifold, but if SOC is applied only as a non‑self‑consistent post‑processing step the resulting wavefunctions remain arbitrary linear combinations of t₂g orbitals; consequently the orbital polarization does not develop. The authors demonstrate that only a fully self‑consistent, non‑collinear DFT+U calculation with SOC included during the self‑consistent field cycle can correctly re‑mix the t₂g states and produce a large orbital moment.

Two prototypical 2D magnets, VI₃ and FePS₃, are used as test cases. Both possess an octahedral crystal field and six (VI₃) or five (FePS₃) d‑electrons, leading to a partially filled t₂g band. Pure LSDA predicts a metallic ground state; LSDA+U opens a gap but yields orbital moments of only ~0.1 μB. When self‑consistent SOC is added, the orbital moment rises to ~1 μB, in line with experimental reports of unusually high magnetic anisotropy in these materials. The analysis shows that the orbital moment aligns out‑of‑plane (perpendicular to the 2D lattice), preserving in‑plane rotational symmetry while generating a strong uniaxial anisotropy.

To assess the broader relevance, the authors query the Computational 2D Materials Database (C2DB) for all monolayers in which a transition‑metal d‑shell experiences either octahedral or tetrahedral coordination. They identify 112 octahedral and 62 tetrahedral candidates (174 total). For each material they perform four calculations: (i) plain LSDA/PBE, (ii) LSDA+U, (iii) LSDA+U with non‑self‑consistent SOC, and (iv) LSDA+U with self‑consistent SOC. The results reveal a systematic trend: materials with partially filled t₂g bands show an order‑of‑magnitude increase in the calculated orbital moment only in case (iv). In many cases the orbital moment exceeds 0.5 μB, and several reach the 1 μB threshold, indicating that large orbital contributions are far more common in 2D transition‑metal compounds than previously thought.

The paper discusses the implications of these findings. Large orbital moments directly enhance magnetic anisotropy, which is crucial for stabilizing long‑range order in two dimensions (circumventing the Mermin‑Wagner theorem). They also affect spin‑wave spectra, magnon gaps, and the efficiency of spin‑orbit torque mechanisms, opening new avenues for 2D spintronic and quantum‑information devices. Experimentally, the predicted orbital moments can be probed by X‑ray magnetic circular dichroism (XMCD), electron spin resonance, or torque magnetometry.

In conclusion, the authors establish a clear three‑step recipe for achieving giant orbital magnetization in 2D materials: (1) a crystal‑field environment that yields a partially filled t₂g manifold, (2) a Hubbard‑U correction that opens a gap and localizes the d‑electrons, and (3) a fully self‑consistent inclusion of SOC during the DFT+U calculation. Their high‑throughput screening demonstrates that over a hundred existing 2D monolayers satisfy these criteria, providing a rich playground for future experimental exploration and device engineering. The work underscores the necessity of treating electron correlation and spin‑orbit effects on equal footing when modeling magnetism in low‑dimensional systems.