Giant and controllable nonlinear magneto-optical effects in two-dimensional magnets

The study explores how breaking inversion (P) and time-reversal (T) symmetries in magnetic materials enables strong nonlinear magneto-optical (NLMO) phenomena, especially second-harmonic generation (SHG)-related effects. SHG is sensitive to crystal symmetry (T-even “crystal SHG”) and magnetization (T-odd “MSHG”). Two-dimensional magnets offer exceptional tunability compared with bulk materials and exhibit unusual SHG responses. The research question is how to realize and control giant, device-relevant NLMO effects in 2D magnets through the interference of crystal SHG and MSHG, and how such effects can be harnessed to detect magnetic ordering and manipulate light polarization. The work emphasizes frequency-dependent polarization rotation and intensity asymmetry that are independent of thickness, promising for ultrathin photonic devices.

Prior work demonstrated strong SHG in various 2D and layered materials (e.g., MoS2, NbO2-based crystals, twisted h‑BN), and pure c‑type SHG in PT-symmetric systems and magnetic thin films. Studies on CrI3 bilayers uncovered stacking-dependent magnetism and striking nonreciprocal SHG. SHG has been widely used to probe magnetic symmetries and interfaces. These advances suggest 2D materials are excellent platforms to explore and engineer NLMO effects, motivating a systematic symmetry-and first-principles-based investigation in prototypical 2D magnets.

The authors combined symmetry analysis with first-principles density functional theory calculations to compute SHG susceptibilities and NLMO responses. DFT calculations were performed using VASP with spin–orbit coupling. The exchange–correlation functional was PBE and PAW pseudopotentials were used. A Hubbard U correction (values 3.7 eV and 11.6 eV for relevant magnetic transition-metal states) was applied. van der Waals interactions were included via DFT-D3 without damping. Plane-wave cutoffs were 450 eV for CrI3-based materials and 500 eV for H‑VSe2-based materials. Monkhorst–Pack k-point grids were 13×13×1 for CrI3-related structures and 15×15×1 for H‑VSe2. A vacuum spacing of about 20 Å separated periodic images. After self-consistent calculations, maximally localized Wannier functions (Wannier90) were constructed to build tight-binding Hamiltonians for optical response calculations, carefully accounting for T-symmetry breaking in magnets. They used about 56 MLWFs per layer for CrI3/CrI3Br3 and 22 per layer for H‑VSe2. A Lorentzian broadening of 0.05 eV was adopted. k-mesh sampling for SHG susceptibility convergence used roughly 50×50×1 for CrI3-based and 150×150×1 for H‑VSe2-based systems. For SHG susceptibility calculations in trilayer CrI3, the band gap was adjusted to about 1.5 eV. Analytical expressions were derived for the NLMO polarization rotation angle under linearly polarized light and for intensity asymmetry under circularly polarized light, linking the effects to interference between T-even and T-odd SHG tensor components and their relative magnitudes and phases.

- Giant NLMO effects are predicted in CrI3-based 2D magnets due to interference between crystal SHG (T-even) and magnetization-induced SHG (T-odd): - Near 90° rotation in SHG polarization direction under magnetization reversal at specific frequencies (LPL characteristic frequencies where the rotation becomes ill-defined for a single component but the full pattern rotates maximally). - On/off switching of SHG intensity for specific light helicities upon magnetization reversal, achieving η^T_SHG = ±1 (complete asymmetry). - 100% SHG circular dichroism at certain CPL characteristic frequencies (α_CPL), demonstrating helicity–magnetization equivalence in intensity control. - Analytical expressions relate the NLMO rotation angle to tensor component ratios and phases; maximum NLMO angle reaches π/6 (30°) in pattern rotation units corresponding to a 90° change in selected polarization directions due to pattern symmetry. - Frequency-resolved calculations for representative systems (ABA-stacked CrI3 with AFM order, monolayer CrI3Br3 with FM order, and monolayer H‑VSe2 with FM order) show: - CrI3-based systems exhibit giant effects (rotation near 90°, η^T_SHG reaching ±1 at multiple frequencies), e.g., ω_LPL ≈ 1.71 eV in ML CrI3Br3 for dramatic polarization switching. - H‑VSe2 shows smaller asymmetries (|η^T_SHG| < 0.4), attributable to an imbalanced magnitude ratio of T-even and T-odd SHG tensor components. - The effects are extremely sensitive to subtle magnetic ordering (AFM, uncompensated AFM, mixed configurations), enabling differentiation of spin textures via polarization-resolved SHG and CPL SHG-CD. - Design principles identified: - Comparable magnitudes of T-even and T-odd SHG tensor components (e.g., χ+ and χ−, often χxx- and χxy-like terms) are essential for giant NLMO. - Interlayer interaction (spacing), spin–orbit coupling, and stacking/magnetic order synergy enable tuning of relative magnitude and phase, controlling NLMO strength. - Layer-number engineering (e.g., 7-layer AA′-AFM H‑VSe2) can proportionally enhance specific tensor components and induce inherent NLMO by symmetry.

The findings demonstrate that when crystal SHG and MSHG are both symmetry-allowed and of comparable strength, their interference can be harnessed to strongly modulate SHG polarization and intensity by reversing magnetization or light helicity. This directly addresses the goal of achieving large, controllable NLMO responses in 2D magnets. The predicted near-90° polarization switching and complete on/off SHG under CPL provide clear, device-relevant signatures and pathways for ultrathin optical components (polarization switches, CPL filters) and for non-contact probing of subtle magnetic orders. The equivalence between helicity reversal and magnetization reversal provides operational flexibility. Sensitivity to stacking and interlayer coupling underscores the feasibility of engineering NLMO via van der Waals assembly and strain. The analysis clarifies why CrI3-based magnets outperform H‑VSe2: balanced T-even/T-odd tensor components are crucial, suggesting concrete material-by-design strategies.

The study predicts and explains giant, tunable nonlinear magneto-optical effects in 2D magnets, especially CrI3-based systems, arising from interference between crystal SHG and MSHG. Key outcomes include near-90° SHG polarization rotation, complete on/off SHG switching tied to magnetization or helicity, and 100% SHG circular dichroism at specific frequencies. The authors derive analytical conditions for maximizing NLMO and propose general design principles based on tuning interlayer spacing, SOC, stacking sequence, magnetic order, and layer number to balance T-even and T-odd SHG components. These insights enable ultrathin NLMO devices (polarization switches, CPL filters) and advanced optical detection of complex magnetic orders. Future work could experimentally validate the predictions, extend the search to additional 2D magnets, quantify temperature and disorder effects, and integrate device architectures exploiting the identified characteristic frequencies.

The results are primarily based on first-principles calculations with specific functional choices (PBE+U), parameter selections (U values), and computational adjustments (e.g., band-gap tuning for SHG calculations), which may affect quantitative accuracy. Many-layer and finite-temperature effects, disorder, substrate interactions, and fabrication-induced strains are not fully treated and could impact observed NLMO magnitudes. While symmetry arguments are general, experimental realization depends on precise control of stacking, interlayer distances, and magnetic configurations, and on minimizing sample inhomogeneity.