Stacking-dependent magnetic ordering in bilayer ScI$_{2}$

Stacking-dependent magnetism in two-dimensional van der Waals materials offers an effective route for controlling magnetic order without chemical modification. Here, we present a combined first-principles and finite-temperature study of magnetic ordering in bilayer ScI${2}$ with different stacking configurations. Using density functional theory with Hubbard-U corrections, we investigate the structural, electronic, and magnetic properties of monolayer and bilayer ScI${2}$ in $AA$, $AB$, and $BA$ stackings. The electronic structure exhibits a spin-polarized ground state dominated by Sc-$d$ states near the Fermi level. Mapping total energies onto an effective Heisenberg spin Hamiltonian reveals strong intralayer ferromagnetic exchange that is largely insensitive to stacking, while the inter-layer exchange depends strongly on stacking geometry, favoring ferromagnetic coupling for $AA$ and $BA$ stackings and antiferromagnetic coupling for the $AB$ stacking. Spin-orbit coupling calculations show that both monolayer and bilayer ScI${2}$ possess a robust out-of-plane magnetic easy axis. Finite-temperature Monte Carlo simulations indicate that all bilayer configurations sustain magnetic ordering at and above room temperature, with ordering temperatures in the range $360-375$ K, as confirmed by Binder cumulant analysis and finite-size scaling. These results demonstrate that stacking geometry enables control of the magnetic ground state in bilayer ScI${2}$ without significantly affecting its thermal stability.

💡 Research Summary

This paper investigates how the stacking arrangement of bilayer scandium diiodide (ScI₂), a two‑dimensional van der Waals magnetic material, influences its magnetic ordering and thermal stability. Using density‑functional theory (DFT) with a Hubbard‑U correction (U_eff = 1.7 eV) and Grimme D3 dispersion, the authors first establish that the octahedral 1T polymorph is energetically favored over the trigonal‑prismatic 1H form by ~0.11 eV per formula unit, with an optimized lattice constant of a ≈ 3.9 Å and Sc–I bond length of 2.8 Å. Three bilayer stacking configurations are constructed from the 1T monolayer: AA (perfect registry), AB (lateral shift t₁ = (‑1/3,‑2/3)), and BA (t₂ = (1/3,2/3) or a 180° rotation of AB). After full relaxation, AB is the lowest‑energy stacking (‑0.22 eV/f.u.), BA is slightly higher (‑0.21 eV/f.u.), and AA is the least stable (‑0.18 eV/f.u.). The interlayer distances differ: AA ≈ 3.75 Å, AB/BA ≈ 3.45 Å, indicating stronger interlayer orbital overlap for the shifted configurations.

Electronic‑structure calculations reveal that a single Sc²⁺ ion (d¹) yields a spin‑polarized half‑metallic monolayer: the majority‑spin Sc‑d band crosses the Fermi level while the minority‑spin channel is gapped by ~0.5 eV. Iodine p‑states lie ≈ 4 eV below the Fermi level and hybridize weakly with Sc‑d. In the bilayers, the overall band topology remains similar, but subtle stacking‑dependent modifications appear near the Fermi level, especially in the Sc‑d‑derived bands, reflecting changes in interlayer hybridization.

Magnetic exchange interactions are extracted by mapping total‑energy differences of several spin configurations onto a Heisenberg Hamiltonian H = ‑∑_ij J_ij S_i·S_j. The intralayer exchange J_intra is robustly ferromagnetic (≈ 10 meV) and essentially independent of stacking. In contrast, the interlayer exchange J_inter is highly sensitive to registry: AA and BA exhibit ferromagnetic J_inter > 0, whereas AB shows antiferromagnetic J_inter < 0. This sign reversal is rationalized by the Goodenough‑Kanamori super‑exchange rules: the geometry of the Sc–I–Sc super‑exchange pathway changes with lateral shift, altering the overlap symmetry and thus the sign of the exchange integral.

Spin‑orbit coupling (SOC) calculations demonstrate a strong out‑of‑plane magnetic anisotropy for all stackings. The magnetic anisotropy energy (MAE) is positive, confirming that the easy axis lies along the z‑direction. This anisotropy lifts the Mermin‑Wagner restriction, allowing long‑range order at finite temperature in a truly two‑dimensional system.

To assess finite‑temperature behavior, the authors perform classical Monte Carlo (MC) simulations on square lattices with linear sizes L = 30–200, employing the Metropolis algorithm and periodic boundary conditions. Using the extracted J values, they compute the magnetization, susceptibility, and fourth‑order Binder cumulant. Finite‑size scaling of the Binder cumulant crossing points yields precise Curie (or Néel) temperatures: AA and BA stackings order at ≈ 370 K, while the AB stacking orders at ≈ 360 K. All three configurations therefore retain magnetic order well above room temperature, indicating that the stacking‑induced switch between ferromagnetic and antiferromagnetic ground states does not compromise thermal robustness.

The study concludes that (i) stacking provides a non‑chemical, reversible knob to toggle interlayer magnetic coupling between FM and AFM, (ii) strong intralayer ferromagnetism together with robust out‑of‑plane anisotropy ensures high‑temperature stability, and (iii) the combined DFT + U and MC methodology offers a quantitative predictive framework for designing vdW magnetic heterostructures. The authors suggest future work on external‑field or strain‑driven stacking manipulation, as well as experimental verification via scanning probe microscopy or magneto‑optical Kerr effect measurements, to translate these theoretical insights into functional spintronic devices.