Bernal Stacking and Symmetry-Inequivalent Antiferromagnetism in MSi$_2$N$_4$ Heterobilayers

Layered MA$_2$Z$_4$ compounds, structural relatives of MoS$_2$ discovered in 2020, exhibit rich magnetic behavior arising from reduced dimensionality, noncentrosymmetric lattice symmetries, and stacking-dependent exchange interactions. Here, we investigate Bernal-like stackings in H-phase MA$_2$Z$_4$ (M = Mn and Fe; A = Si; Z = N) monolayers and bilayers by combining first-principles spin-dependent relaxation energies with a localized-spin Heisenberg description. From density-functional calculations, we extract the dominant intralayer exchange couplings up to third-nearest neighbors and the leading interlayer exchanges up to second-nearest neighbors, enabling construction of an effective bilayer spin Hamiltonian. We first analyze interface-driven proximity effects within a ferromagnetic reference configuration, demonstrating how recovery of AB-type stacking and spin alignment–while varying only the transition-metal species–provides a route for selectively tuning magnetic order and symmetry breaking within the P$\bar{6}$m2 space group. Building on this microscopic understanding of the bonding environment, we then examine antiferromagnetic ordering tendencies in the coupled layers. Exact diagonalization of the resulting bilayer Hamiltonian reveals the magnetic ground state and low-lying excitation spectrum, showing that the interlayer exchange is not merely perturbative but competes directly with intralayer interactions in stabilizing the observed spin configurations. These results establish Bernal-stacked MA$_2$Z$_4$ bilayers as a platform in which stacking geometry and exchange hierarchy jointly govern magnetic reconstruction, offering a controlled pathway toward domain selection and spin-texture engineering in low-dimensional van der Waals materials.

💡 Research Summary

This work investigates the magnetic properties of Bernal‑stacked bilayers composed of the newly discovered MA₂Z₄ family, specifically MnSi₂N₄ (MSN) and FeSi₂N₄ (FSN). Using density‑functional theory with Hubbard‑U corrections (U_eff = 4.6 eV for Mn, 6.0 eV for Fe) and Grimme D3(BJ) dispersion, the authors first obtain fully relaxed monolayer structures in the non‑centrosymmetric P ¯6 m2 space group. Lattice constants are a ≈ b ≈ 2.85 Å, and the transition‑metal–nitrogen bond lengths lie between 1.99 and 3.50 Å. Spin‑orbit coupling (SOC) calculations reveal that FeSi₂N₄ prefers an out‑of‑plane easy axis (MAE ≈ ‑0.67 meV) while MnSi₂N₄ favors an in‑plane orientation (MAE ≈ +0.78 meV).

Three high‑symmetry registries—H3 (Bernal‑like), T4, and Top—are examined for the bilayer. Potential‑energy scans show that the H3 configuration is universally the lowest‑energy stacking, with an equilibrium interlayer distance of ~3.1 Å and negligible strain (<0.3 %). Consequently, the H3 registry is adopted for all subsequent magnetic analyses.

The magnetic exchange interactions are extracted by mapping spin‑polarized DFT total‑energy differences onto a localized‑spin Heisenberg Hamiltonian. Intralyer couplings up to third nearest neighbours (J₁, J₂, J₃) and interlayer couplings up to second nearest neighbours (J⊥₁, J⊥₂) are obtained. For MnSi₂N₄, the dominant intralayer J₁ is strongly antiferromagnetic (≈ ‑5 meV), while the nearest‑interlayer J⊥₁ is ferromagnetic and comparable in magnitude (≈ +4.8 meV). In FeSi₂N₄, J₁ is ferromagnetic (≈ +3 meV) and J⊥₁ is also ferromagnetic but weaker (≈ +2.5 meV). These values demonstrate that interlayer exchange is not a perturbative correction; it competes directly with intralayer exchange and can even dominate the magnetic ground state.

Using the extracted parameters, the authors construct a 2 × 2 supercell Heisenberg model and solve it by exact diagonalization. The low‑energy spectrum reveals that the ground state for both materials is a mixed configuration: one layer adopts a stripelike antiferromagnetic order while the opposite layer aligns ferromagnetically. Energy differences among the three possible interlayer alignments (↑↑, ↑↓, ↓↑) are only 0.1–0.3 meV, indicating a delicate balance that can be tipped by modest external stimuli. The excitation spectrum contains both intralayer spin‑wave modes (≈ 5 meV) and interlayer optical‑like modes (≈ 8 meV), suggesting that neutron scattering or Raman spectroscopy could detect characteristic multi‑branch magnon features.

Symmetry analysis shows that Bernal stacking preserves the P ¯6 m2 point group but the magnetic ordering breaks time‑reversal symmetry in a layer‑dependent manner, leading to effective symmetry lowering that could enable exotic phenomena such as spin‑quantum‑Hall‑like responses. The authors also explore the influence of charge doping and biaxial strain. Electron or hole doping reduces the in‑plane MAE of MnSi₂N₄, potentially driving a transition to an out‑of‑plane easy axis. Small (<1 %) biaxial strain modifies the ratio J₁/J⊥₁, offering a route to electrically or mechanically switch between ferromagnetic and antiferromagnetic interlayer coupling.

In conclusion, the study establishes that (i) Bernal‑type AB stacking is the energetically preferred configuration for MA₂Z₄ bilayers, (ii) interlayer exchange can be as strong as intralayer exchange, fundamentally shaping the magnetic ground state, (iii) the choice of transition metal (Mn vs Fe) dictates both the hierarchy of exchange interactions and the magnetic anisotropy, and (iv) external knobs such as gating, strain, or layer‑specific functionalization provide practical pathways to engineer spin textures and domain selection. These insights position Bernal‑stacked MA₂Z₄ heterobilayers as a versatile platform for spin‑tronic devices, including tunable FM/AFM tunnel junctions, spin‑valves, and magnonic circuits in the emerging family of two‑dimensional van‑der‑Waals materials.