Nonlinear magneto-optical (NLMO) effects, arising from the interplay of light, polarization, and magnetism, are of significant interest. Second-harmonics generation (SHG)-related NLMO effects are particularly advantageous for magnetization detection and light modulation due to their high spatial and temporal resolution and independence from sample thickness. The SHG signal primarily originates from electric dipole transitions, requiring the breaking of inversion symmetry. This can occur through crystal SHG and magnetization-induced SHG (MSHG). In materials with inversion symmetry, only the MSHG survives. Recent experiments on two-dimensional (2D) materials have shown unusual SHG responses, highlighting their potential as a platform for both fundamental research and device applications. This study focuses on exploring and controlling NLMO effects in 2D magnets, aiming to uncover giant NLMO effects and design principles for their manipulation in various materials and systems.

The paper reviews previous studies on NLMO effects, focusing on SHG-related phenomena in magnetic materials and 2D materials. It cites research on giant nonreciprocal second-harmonic generation in CrI₃, switchable magnetic bulk photovoltaic effects, and electrically and magnetically switchable nonlinear photoconductivity in PT-symmetric topological quantum materials. The review highlights the unique advantages of 2D materials for studying and exploiting NLMO effects due to their tunability and atomic-scale control. The authors emphasize the lack of a comprehensive theoretical understanding for controlling the magnitude of NLMO effects, especially in 2D systems, which motivates their current research.

The research employed a computational approach based on first-principles calculations using the Vienna Ab initio Simulation Package (VASP) with spin-orbit coupling (SOC) included. The Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional and projector augmented-wave (PAW) potentials were used. Van der Waals corrections (DFT-D3) were applied for layered materials. The cutoff energy and k-point grid were carefully selected for convergence. Maximally localized Wannier functions were generated using Wannier90 to construct tight-binding Hamiltonians for calculating optical responses. The calculations accounted for the breaking of time-reversal symmetry to accurately evaluate SHG contributions in magnetic systems. The authors considered representative 2D magnets including CrI₃, CrI₃Br₃, and H-VSe₂, analyzing their atomic structures, magnetic symmetries, and SHG tensor components. They investigated the NLMO angle for linearly polarized incident light and NLMO intensity asymmetry for circularly polarized incident light at different frequencies. The effects of interlayer interaction, SOC, and stacking order were also analyzed.

The study uncovered giant NLMO effects in CrI₃-based 2D magnets due to the interference between comparable crystal SHG and MSHG. Key findings include a near 90° polarization rotation of SHG light upon magnetization reversal and an on/off switching effect. A 100% SHG circular dichroism (CD) effect was also observed. These NLMO effects were found to be highly sensitive to subtle changes in magnetic order. The authors analytically derived conditions for achieving these giant effects, showing that comparable magnitudes of T-even and T-odd SHG components are crucial. They also investigated the influence of interlayer spacing, SOC, and stacking order on NLMO effects, providing design principles for manipulating these effects in 2D magnets. The analysis of ABA-stacked CrI₃ and ML CrI₃Br₃ revealed distinct NLMO angle and intensity asymmetry behaviors as a function of frequency. The study also demonstrated the capability of NLMO effects to distinguish subtle magnetic orderings, such as spin fluctuations and spin canting. Finally, the authors demonstrated that the control of the relative and absolute magnitude of the T-even and T-odd SHG components can be achieved by varying the interlayer distance, the strength of SOC, and the synergistic effect of stacking order and magnetic order.

The findings demonstrate the potential of 2D magnets for developing ultra-thin multifunctional NLMO devices, such as optical polarization switches and filters. The significant NLMO effects observed in CrI₃-based materials provide promising candidates for such applications. The sensitivity of these effects to subtle magnetic orderings also opens avenues for exploring novel magnetic sensing techniques. The analytical derivation of conditions for giant NLMO effects provides valuable insights for designing and engineering materials with enhanced NLMO properties. The investigation of the influence of structural parameters and magnetic order on NLMO effects offers a framework for tailoring these effects in different 2D systems. The results significantly advance our understanding of NLMO phenomena in 2D magnets and pave the way for their exploitation in next-generation optical and magneto-optical devices.

This work uncovered giant and controllable NLMO effects in CrI₃-based 2D magnets, highlighting their potential for ultra-thin optical devices and magnetic order detection. The authors analytically derived conditions for achieving these effects and proposed general strategies for manipulating them in 2D magnets. Future research could explore other 2D magnetic materials and investigate the impact of temperature and external fields on these NLMO effects.

The study relies on first-principles calculations, which have inherent limitations regarding the accuracy of approximations used in the computational models. Experimental verification of the predicted NLMO effects is necessary to confirm the theoretical findings. The analysis focused on specific 2D magnet structures; further investigations into a wider range of materials and stacking configurations are needed to generalize the findings.