In recent years, with the development of mid-infrared laser technology, experimental measurements of solid-state high-order harmonics have been conducted in various materials, such as crystalline materials, two-dimensional materials, and topological materials. Compared to gases, solid-state high-order harmonics exhibit unique properties, including even-order harmonic generation, linear cutoff law, anomalous ellipticity dependence, and multi-plateau structures. Among these materials, monolayer MoS₂ as a direct bandgap two-dimensional material possesses large exciton binding energy and strong spin-orbit coupling effects. Due to the breaking of inversion symmetry, its Brillouin zone contains two inequivalent energy valleys of K and K'. The rich characteristics of monolayer MoS₂ have brought extensive research in recent years. Therefore, we employ tight-binding approximation and the density matrix equation under the velocity gauge to theoretically calculate the high-order harmonics generated in monolayer MoS₂ under a bichromatic field. By adjusting the bichromatic field delay, we observe the intensity of the generated high-order harmonics as a function of the delay. We find that under different driving field polarization directions, the oscillation period of the high-order harmonics varies. Additionally, we propose an intuitive model to explain this phenomenon. Furthermore, we extract the amplitude and phase information of even-order harmonic oscillations parallel and perpendicular to the polarization direction of the driving field. We discover that the intensity ratio of the second harmonic with respect to the fundamental field exerts different effects on the phase of the even-order harmonic in parallel and perpendicular directions. Our theoretical calculations indicate that adopting a bichromatic field as the driving field for interacting with monolayer MoS₂ not only introduces the competition between the incident field and the medium’s inversion symmetry but also encompasses the contribution of Berry curvature in the perpendicular direction to the even-order harmonic. Finally, this approach aids in further understanding the dynamic processes in solid materials.

The atomic structure of monolayer MoS₂ in a top-down view is depicted in Fig. 1(a), where Mo atoms and S atoms are arranged alternately in a two-dimensional hexagonal honeycomb lattice structure. Fig. 1(b) illustrates the first Brillouin zone of monolayer MoS₂, which contains two high-symmetry points K and K'. The band structure of monolayer MoS₂ is obtained from a three-band tight-binding model that includes third-nearest-neighbor Mo-Mo hopping, as shown in Fig. 2. This model fits the band structure calculated throughout the Brillouin zone using the local density approximation (LDA) and generalized gradient approximation (GGA) methods. Under the independent-particle approximation, we utilize the density matrix equation in the velocity gauge to simulate the process of monolayer MoS₂ interacting with a bichromatic field to generate high-order harmonics.

Fig. 3 depicts the high-order harmonic spectra of the fundamental and second harmonics along the Γ—K direction with different intensity ratios of the second harmonic and fundamental fields. The intensities of each harmonic are normalized, and the delay between the second harmonic and fundamental fields is given in units of the fundamental field period. Additionally, we observe oscillations of each harmonic along the parallel and perpendicular directions with the delay between the second harmonic and fundamental fields oscillating four times within one fundamental field period. This can be explained by an intuitive model. As shown in Fig. 4, the fundamental and second harmonic fields propagate together in monolayer MoS₂. Since both are polarized along the Γ—K direction in monolayer MoS₂, which possesses inversion symmetry in the Γ—K direction, the coherent combination field reaches its maximum either upward or downward a quarter of the fundamental field period after a change in delay between the two, which results in four oscillations within each fundamental field period. By employing a least squares fitting, we can obtain the amplitudes and phases of the oscillations of even-order harmonics, as shown in Fig. 5. Fig. 5(c) indicates that in the parallel direction, the intensity ratio changes of the second harmonic to the fundamental field (increasing from 0.001 to 0.01) alter the dominant phase of the even-order harmonic oscillations. This is because when the second harmonic and fundamental fields are polarized parallelly and along the Γ—K direction, the even-order harmonics in the parallel direction are generated due to the breaking of the inversion symmetry of the driving field by the competitive relationship between the second harmonic and fundamental fields. Thus, as the intensity ratio of the second harmonic to the fundamental field increases, the disturbance of the fundamental field by the second harmonic becomes stronger, thus changing the dominant phase of the even-order harmonic oscillations. As shown in Fig. 5(d), we can observe that in the perpendicular direction, the dominant phase is ?2. The intensity ratio variation of the second harmonic to the fundamental field only causes minor changes in the oscillation of ?2 with the harmonic order, which is different from the parallel direction. This is because when the second harmonic and fundamental fields are polarized parallelly and along the Γ—K direction, the even-order harmonics in the perpendicular direction are generated by the anomalous currents. The anomalous current is equal to the product of the net Berry curvature and the electric field intensity, where the Berry curvature determines the dominant phase in the even-order harmonic oscillations. As the intensity ratio of the second harmonic to the fundamental fields rises, the net Berry curvature remains unchanged, and the dominant phase in the even-order harmonic oscillations consistently remains at ?2. Fig. 6 depicts the high-order harmonic spectra in the parallel direction of the fundamental and second harmonic fields with polarization along the Γ—M direction under different intensity ratios of the second harmonic and fundamental fields. As shown in Fig. 6(a), when the intensity ratio is small (0.0001, 0.001), the even-order harmonics oscillate four times within one fundamental field period. However, Fig. 6(b) reveals that when the intensity ratio increases (0.01, 0.05), the even-order harmonics oscillate twice within one fundamental field period, which indicates that the intensity ratio change can alter the oscillation period of the even-order harmonics. When the intensity ratio is small (0.0001, 0.001), we can still explain this by adopting the intuitive model in Fig. 4. The fundamental and second harmonic fields propagate together in monolayer MoS₂. Due to the polarization of both along the Γ—M direction, and the lack of inversion symmetry in the Γ—M direction of the monolayer MoS₂, the interaction between the fundamental field and the medium itself has generated an asymmetric electric field. We can consider this as a competition between two broken inversion symmetries (the medium and the electric field), or a coherent effect generated by the asymmetric electric field and the incident second harmonic field. Fig. 4 demonstrates that since the amplitude of the asymmetric electric field in the upward direction is greater than that in the downward direction, the maximum field strength of the coherent combined field can only be achieved by delaying the second harmonic by half of the fundamental laser period. Therefore, after half of the fundamental field period, the coherent combined field reaches its maximum in the upward direction, resulting in two times of oscillation within each fundamental field period. When the intensity ratio increases to 0.01 and 0.05, the model in Fig. 4 is no longer applicable. To explore the reasons behind this, we calculate the contribution of the second harmonic to the harmonic generation rate at different intensity ratios by considering only the second harmonic, as shown in Fig. 7. In the case of a single-color field (only the second harmonic), it can be observed that the intensity of even-order harmonics is much lower than that of the bichromatic field (with an intensity ratio of 0.001). By taking the sixth harmonic as an example, we find its intensity in the single-color field is five orders of magnitude lower than that in the bichromatic field, as shown in Fig. 7(a). However, when the intensity ratio is 0.01, the contribution of the second harmonic field to the even-order harmonics in the bichromatic field increases. For the sixth harmonic, its intensity in the single-color field is only three orders of magnitude lower than that in the bichromatic field, as shown in Fig. 7(b). Therefore, as the intensity ratio increases from 0.001 to 0.01, the role of the second harmonic in even-order harmonics increases to change the oscillation period of even-order harmonics from two times of oscillation within one fundamental frequency field period to four times of oscillation. Finally, we extract the amplitudes and phases of the even-order harmonics in parallel directions as shown in Fig. 8. It can be observed that in the parallel direction, the change in the intensity ratio of the second harmonic to the fundamental field (from 0.001 to 0.01) leads to a shift in the dominant phase of the even-order harmonic oscillations from ?1 to ?2. This is because when the second harmonic and fundamental fields are polarized in parallel directions along the Γ—M direction, the even-order harmonics in the parallel direction are generated due to the breaking of inversion symmetry of the driving field, which brings a competitive relationship between the second harmonic and fundamental fields. Therefore, as the intensity ratio of the second harmonic to the fundamental field increases, the disturbance of the fundamental field by the second harmonic becomes stronger, leading to a shift in the dominant phase of the even-order harmonic oscillations.

We explore the influence of a bichromatic field on the high-order harmonic generation in monolayer MoS₂ through theoretical calculations. We find that the oscillation period of high-order harmonics is related to the polarization direction of the bichromatic driving field (the Γ—K or the Γ—M directions), which is also associated with the inversion symmetry of the material itself. Specifically, when the driving field is polarized along the Γ—M direction, the oscillation period of the generated high-order harmonics is affected by the intensity ratio of the second harmonic and fundamental fields to shift from two times of oscillation within one fundamental field period to four times of oscillation. This is due to the enhancement of the second harmonic, which shifts from disturbing the fundamental field to becoming a controlling factor in the harmonic generation. Additionally, we investigate the phase of the even-order harmonics oscillating in parallel and perpendicular directions to the polarization of the driving field, and find that the phase of these two directions of even-order harmonics is differently affected by the intensity ratio of the second harmonic and fundamental fields, which is related to their generation mechanisms. Therefore, we not only contribute to understanding the structural information related to the inversion symmetry breaking in materials, but also deepens the understanding of the strong-field interaction mechanisms in solid materials. Finally, a new approach is provided for utilizing bichromatic fields to control ultra-fast electron dynamics in solids.