The study of two-dimensional (2D) materials and their stacking configurations has opened new avenues for exploring emergent properties. Recent advances in controlling strong light fields have also enabled manipulation of coherent electron transport in these materials on ultrafast timescales. This research builds upon these advancements by introducing a light-wave-driven analogue to twisted layer stacking. The core research question is whether the spatial symmetry of a strong light field can be precisely controlled and used to manipulate the electronic properties of a 2D material, specifically breaking time-reversal symmetry and inducing a Haldane model behavior. The importance lies in achieving ultrafast and precise control over topological properties in a material, opening possibilities for novel electronic devices and technologies based on valleytronics. The purpose is to demonstrate this control experimentally and to characterize the resulting effects using time-resolved optical harmonic polarimetry. This is a significant step forward in ultrafast light-wave electronics and 2D material engineering. The study is important because it presents a versatile and robust technique for manipulating electronic band structures in 2D materials, potentially leading to novel applications in areas such as ultrafast optoelectronics and valleytronics.

The electronic and topological properties of materials are profoundly influenced by discrete symmetries like spatial inversion and time reversal. In materials like graphene, where both symmetries are preserved, Dirac-like dispersion leads to semimetallic behavior. Breaking spatial inversion symmetry, as in hBN, creates a bandgap and insulating behavior. Breaking time-reversal symmetry enables distinguishing carrier dynamics at the K and K' valleys, which are time-reversal partners. The Haldane model, a theoretical model of a Chern insulator, demonstrates time-reversal symmetry breaking through a staggered magnetic field, inducing complex next-nearest neighbor (CNNN) hoppings. Previous work has also explored valley-selective excitation using circularly polarized light in materials like transition metal dichalcogenides, giving rise to the field of valleytronics. Controlling these symmetries enables engineering materials with properties not found naturally, such as through vertical stacking of 2D materials or the application of time-dependent electric and magnetic fields. Floquet theory has shown that the characteristics of electric fields (frequency, intensity, helicity) can modify a crystal's Hamiltonian parameters. Light-wave electronics leverage the temporal control of strong light fields to manipulate electron transport on sub-laser-cycle timescales. This research builds upon this body of work by using a novel approach involving structured light fields to control and modify the band structure of hBN.

The researchers employed a tailored light-wave, resembling a trefoil structure, to interact with the hBN monolayer. This trefoil waveform, generated by interferometrically combining counter-rotating circularly polarized 2 µm (ω) and 1 µm (2ω) light waves with a 2:1 intensity ratio, matches the threefold symmetry of the hBN lattice. The orientation of the trefoil waveform relative to the hBN lattice was precisely controlled by adjusting the subcycle time delay between the ω and 2ω pulses. This manipulation of the light waveform directly modifies the CNNN hoppings in the hBN lattice, analogous to the Haldane model's staggered magnetic field, breaking time-reversal symmetry. The effective Haldane-type Hamiltonian parameters were controlled by rotating the light waveform, leading to tunable bandgap modification at the K and K' valleys. Time-dependent simulations based on a two-band model of gapped graphene, using hBN parameters, were performed to predict the valley-specific electron populations. Experimentally, the valley polarization was measured using all-optical time-resolved harmonic polarimetry. A time-delayed linearly polarized 2 µm probe pulse measured changes in the polarization of the third harmonic generation. The asymmetry in the s- and p-polarized components of this third harmonic was used to quantitatively determine the degree of valley polarization. The experimental setup involved focusing the pump trefoil waveform and the probe pulse onto the hBN monolayer. The pump intensity was kept in the 4-7 TW cm⁻² range to avoid damaging the sample. The subcycle-precise control over the waveform's orientation allowed for systematic exploration of the induced changes in the band structure and valley polarization. The data was analyzed by Fourier transformation to identify dominant periodicities and asymmetries, allowing for comparison with the theoretical predictions.

The experiments successfully demonstrated subcycle-controlled time-resolved symmetry breaking and valley bandgap control in hBN using a tailored light waveform. Rotating the trefoil-shaped light wave resulted in a periodic oscillation of the effective bandgap at the K and K' valleys, with a dominant 120° periodicity. This oscillation reflects the modification of the band structure topology due to the controlled time-reversal symmetry breaking. Time-dependent simulations confirmed the experimental observations. The simulations and experimental measurements both showed a clear asymmetry in the electron populations between the K and K' valleys as the trefoil waveform was rotated. The resulting valley polarization could be directly mapped onto the asymmetry of the helicity components of the third-harmonic probe signal. The valley polarization exhibited a strong 120° periodicity, consistent with the threefold symmetry of the hBN lattice and the induced CNNN hoppings. A secondary 60° periodicity was also observed and attributed to the incomplete symmetry switching when only the light waveform's orientation, but not its helicity, was changed. The experiments showed no such periodicity when the laser irradiated only the fused silica substrate. By Fourier filtering of the experimental data around the 120° periodicity, the asymmetry parameter α reflecting the valley polarization was extracted and compared to the simulated results, showing excellent agreement. An estimate of the valley lifetime in hBN under these strong-field conditions was approximately 60 fs, corresponding to around nine optical cycles of the driving field.

The findings directly address the research question by demonstrating the ability to control the electronic properties of hBN using a precisely tailored light waveform. The observed 120° periodic oscillation in valley polarization provides clear evidence of the light-wave-induced modification of the band structure, analogous to the Haldane model. The good agreement between theoretical simulations and experimental results validates the interpretation of the light-wave interaction in terms of a light-driven Haldane model. The ability to control the magnitude, location, and curvature of the bandgap opens up exciting possibilities for non-resonant valleytronics and ultrafast switching in insulating 2D materials. The method's robustness and universality make it applicable to a wider range of materials, beyond hBN, and pave the way for the creation of ultrafast quantum switches. The results extend valleytronics to the few-femtosecond timescale and insulating systems, surpassing limitations of previous approaches. The study also suggests that similar techniques could be used to create micron-sized domains with different electronic properties within a single material by spatially varying the light waveform's trefoil rotation.

This research successfully demonstrated a light-wave-driven analogue to twisted layer stacking, achieving ultrafast and precise control over the band structure of hBN. The results showcase the creation of a light-driven Haldane model in an insulating material, enabling subcycle control over valley polarization and bandgap engineering. The methodology offers a robust and universal platform for manipulating electronic properties in 2D materials. Future research could explore the application of this technique to other 2D materials, investigate the effects of different light waveforms, and develop novel devices based on ultrafast light-driven valleytronics.

While the study provides strong evidence for light-wave control of the Haldane model in hBN, some limitations exist. The experiments were performed at room temperature, and the influence of lower temperatures on valley lifetimes and polarization dynamics remains unexplored. Further investigations could analyze the precise mechanism of energy relaxation and decoherence processes in this regime. The study primarily focused on a specific range of laser intensities, and future experiments could explore a wider range to further understand the scaling of valley polarization with field strength. The current experiments did not explicitly measure the Hall current directly. While the results strongly suggest the presence of a valley Hall current based on the polarization of the generated third harmonic, future studies could confirm this using more direct measurements.