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Abstract
Sub-optical-cycle electron dynamics in materials driven by intense laser fields can be investigated by high harmonic generation. We observed frequency shift of high harmonic spectrum near the band gap of monolayer MoS2 experimentally. Through semi-classical quantum trajectory analysis, we demonstrated that the phase of transition dipole moment varies according to the recombination timing and momentum of tunneled electrons. It results in either blue- or red-shift of harmonic frequencies, determined by the modulated energy gap by transition dipole phases (TDPs) and Berry connections. Our finding reveals the effect of TDPs on high harmonic frequency in non-central symmetric materials.
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1. Introduction
High harmonic generation (HHG) in gaseous atoms and molecules has been studied for a couple of decades and founded the attosecond science for generating isolated attosecond pulses [1], probing transient processes of electron excitation and tunneling [2], and mapping molecular orbitals [3,4]. HHG in gas phase can be grasped with a semi-classical three-step model: tunnel ionization, free acceleration and recombination [5,6]. Recently, HHG in a variety of solid materials has been reported as a promising and compact ultrashort light source [7–9], as well as a potential tool to reconstruct crystal band structures [10], probe Berry curvature [11,12], and detect topological phase transitions [13–16]. Currently, it is generally accepted that two major contributions constitute HHG: intra-band current and inter-band polarization [17,18]. The intra-band mechanism comes from nonlinear gradient of potential curve leading to motion of electrons within [19,20], and the inter-band one has a re-collision-like dynamics as those in laser driven gaseous atoms or molecules [21,22]. In the generalized re-collision image an electron is excited to conduction band and leaving a hole in the valence band, then they are accelerated in their respective bands, and finally they recombine to emit a high-energy photon with the energy of instantaneous band gap [23].
Interesting phenomena were found for the HHG in systems with a broken symmetry such as MoS2, GaSe and ZnO et al. [11,16,24–32]. For the generation of even-order harmonics, interference of multiple transition paths and TDPs have been considered essential to describe the non-centrosymmetric system. O. Schubert et al. [30] proposed that even-order harmonics are produced as a result of interference of multiple transition paths. Jiang et al. [25,29] identified the role of non-zero TDPs on the generation of even-order harmonics and its orientation dependence. Recently, HHG from two-dimensional (2D) materials has been observed and offers rare opportunity to investigate the carrier dynamics upon the Dirac cone potential [11,33–36]. Compared with graphene, monolayer transition metal dichalcogenides (TMDs) attract broad attention owing to limited band gap and non-central symmetric structure. It has been proposed that the Berry curvature of MoS2 causes even-order harmonics in the direction of polarization perpendicular to the pump laser [11]. Yoshikawa et al. [37] revealed that the electron-hole polarization driven to the band nesting region significantly contribute to the even-order harmonic in resonance with the optical transition. This comes from the asymmetric nature of the dynamics in the K and K’ valleys. Cao et al. [35] demonstrated that spectral interference turns to be constructive for even-order harmonics above a certain photon energy and such interference is controlled by Berry connections. Thus, among the HHG spectra observed in MoS2, even-order harmonics appear an abnormal enhancement [11,35,37].
However, quantum trajectory analysis indicates both TDPs and Berry connections modulate the energy and momentum of the released photon [26,27,38], but no experimental observation has confirmed it so far. In this work, we revisited the HHG in monolayer of MoS2, and we found that frequencies of even-order harmonics near bandgap region shifts significantly, i.e. the 6th harmonic (H6) is red-shifted while the 8th harmonic (H8) is blue-shifted. The analysis using semiconductor Bloch equations (SBEs) reproduces the frequency shifts by considering the dominant inter-band currents. Saddle point analysis revealed that the TDPs and Berry connections jointly modulate the timing of electron-hole recombination, leading to the varied frequency shifts in adjacent half optical cycles. Our work reveals the role of TDPs and Berry connections on harmonic frequency shifts of non-central symmetric materials.
2. Experiment
We excited a monolayer MoS2 crystal using s-polarized mid-infrared laser pulses (3.8 μm, 60 fs, 25 µJ) delivered from a home-built laser system [39]. Power and polarization direction of the laser pulses were adjusted by two wire grid polarizers with the extinction ratio of 300:1. The pulses were focused onto the sample surface using a parabolic mirror (f = 101.6 mm), and the focused spot size is ∼ 20 µm in full width at half maximum (FWHM). HHG signal was collected by a pair fused silica lens and directed into a spectrometer equipped with a thermoelectrically cooled Si charge-coupled device (CCD) camera (Andor Shamrock 303i with Newton 920). The monolayer MoS2 single crystal was grown by chemical vapor deposition (CVD) on a barium fluoride (BaF2) substrate [40]. Monitored by a microscope that can switch into the experimental optics to replace HHG detection, the MoS2 crystal is in a triangular shape and with the side length of 40 µm [34,35,41]. Figure 1 shows a measured harmonic spectrum as the pump polarization along armchair direction (I = 2.3 TW/cm2). The spectrum shows that the odd order HHG yields larger than even orders, and H10 is abnormally stronger than H8, which is consistent with previous investigations [35,37]. However, one may notice that the spectra of H6 and H8 both shift away from their center positions.
Fig. 1. High harmonic spectrum measured from monolayer MoS2 (I = 2.3 TW/cm2). The driving pulse is polarized along armchair direction. The structure of MoS2 is shown in the inset in Fig. 1, where black-S atoms and blue-Mo atoms are alternately arranged in a two-dimensional hexagonal honeycomb lattice possess trigonal prismatic structure.
In order to identify the observation, we calculate the weighted frequency shift for each harmonic order by $\Delta {\mathrm{\omega }_s} = \frac{{\mathop \smallint \nolimits_{s - 0.5}^{s + 0.5} \omega \cdot {I_\omega }d\omega }}{{\mathop \smallint \nolimits_{s - 0.5}^{s + 0.5} {I_\omega }d\omega }} - {\mathrm{\omega }_\textrm{s}}$, in which ${I_\omega }$ is the measured intensity profile of the $sth$ order harmonic. Figure 2(a) shows the result of frequency shift of the high harmonics. Red-shift of H5 and H6, and blue-shift of H7 - H9 are clearly seen, with H6 having the largest red-shift of 9 meV, and H8 the largest blue-shift of 11 meV. Then we adjust the pump intensity in the range of 1.9 ∼ 2.7 TW/cm2, one can find that H6 stays in red shifted and H8 keeps in blue, as shown in Fig. 2(b), where an overall red-shift of all harmonics can be distinguished with a linear slope. This observation confirms the significant spectral shift for even-order harmonics of H6 and H8.
Fig. 2. (a) Measured frequency shift of each harmonic at the pump intensity of 2.3 TW/cm2, and (b) intensity-dependence of H6 and H8.
3. Results and discussion
It is noted that the significant frequency shift may relate to the abnormal enhancement of the same harmonic orders for the same monolayer material [35,37]. In order to explore the different frequency shift between H8 and H6, we solve the semiconductor-Bloch-equations (SBEs) [25–27] to analyze the electron dynamic in the monolayer material irradiated by intense laser fields. According to the selection rules for HHG in solids, when the crystal is excited along the armchair direction, all harmonics are polarized in parallel with the incident light [34,42,43]. Thus the SBEs can be solved in one-dimension to the parallel direction of K-K (or K'-K’) in reciprocal space. Since MoS2 has the smallest band gap along K (or K’) coordinate in reciprocal space, $k = 0$ is set to K (or K’), and the momentum space is $\left[ { - \frac{\pi }{{\sqrt 3 a}},\frac{\pi }{{\sqrt 3 a}}} \right]$, in which a is the lattice constant. The band structure and the transition dipole moment (TDM) are derived from the tight binding approximation model [44,45].
In the calculation, we consider only the lowest conduction band c1 and the highest valence band v. The driving laser field used in the simulation is a Gaussian pulse, and its full width at half maximum (FWHM) duration τ = 60 fs, laser wavelength ${\lambda _0} = 3.8\; \mu m$, which is consistent with the experimental conditions. In the SBEs model, a phenomenological dephasing time is often introduced to account for the loss of coherence due to the interaction of the excited electron with the crystal medium. The parameter T2 will modify the relative strength and the “coherence” of the harmonics generated [25]. To obtain narrow harmonic peaks as observed in the experiment, the phenomenal dephasing time in SBEs is set to 1 fs [35]. We also find that the inter-band current dominantly contributes to the HHG emission, similar with the previous study of Ref. [21,35,46]. The calculated inter-band spectrum is shown in Fig. 3 in comparison with the experimental result.
Fig. 3. Calculated inter-band contributions of harmonic spectrum (I = 0.23 TW/cm2) in comparison with the experimental result. The dephasing time is set as 1 fs.
Based on the simulated spectrum, frequency shift of the inter-band harmonics is determined and plotted in Fig. 4(a). The frequency shift for even-order harmonics is in good agreement with the experimental observation. The large difference between the frequency shift of odd-order harmonics and the experimental results may be due to the fact that our calculations only consider the one-dimensional energy band and do not take into account the motion of electrons in the full Brillouin zone. The results of the two-dimensional calculations are given in Section 5, where the frequency shift of odd-order harmonics is small. Frequency shift at a range of laser intensities is shown in Fig. 4(b) and they show a flat intensity dependence, which is different with the experimental observation. Because the overall red-shifts for all orders have similar slopes, we attribute to the non-adiabatic mechanism [47].
Fig. 4. (a) Calculated frequency shift of each order of the inter-band harmonic spectrum. (b) Intensity-dependence of the inter-band harmonic frequency shift.
4. Semi-classical analysis
The inter-band process of the HHG of solids includes excitation, motion and recombination of the electrons and corresponding holes [21,46]. In the two-band model, the inter-band spectrum is [26,27]
With $E_g^k = E_c^k - E_v^k$ is the band gap, shown in Fig. 5(a), ${\alpha ^k} = \textrm{arg}({d_{c,v}^k} )$ the TDPs, $\Delta {A^k} = A_c^k - A_v^k$, with $A_n^k = iu_n^k\textrm{|}{\nabla _k}\textrm{|}u_n^k$ as Berry connections, $n = c,v$.
Fig. 5. (a) The band gap obtained by tight-binding model. (b) The ${D^k} $ in smooth periodic gauge.
According to the saddle point equation [26,27]
with ${D^k} = \mathrm{\Delta }{A^k} - \nabla {\mathrm{\alpha }^k}$, and $\Delta r = \mathop \smallint \nolimits_{t\mathrm{^{\prime}}}^t v_c^{k(\tau )} - v_v^{k(\tau )}d\tau $ where $v_c^{k(\tau )} = {\nabla _k}E_n^{k(\tau )}$. It is noted that $\textrm{E}(t )\cdot {D^k}$ is the energy dressed by the light field, which is the main factor causing the frequency shift. The value of ${D^k}$ is shown in Fig. 5(b). By constructing a smooth and periodic structure gauge for the Bloch states, the Berry connections $A_c^k$ and $A_v^k$ are independent of k. Therefore, according to Eq. (3a), the Berry connections does not affect the motion of the electron-hole pair, and TDPs does have an observable impact on the recombination time and momentum of the electron-hole pair.According to the Landau–Zener tunneling theory [49,50], tunneling has an exponential dependence on the band gap. Tunneling ionization mainly occurs at the minimum band gap, namely, the K (or $\mathrm{K^{\prime}}$) in the reciprocal space. And it is reasonable to ignore tunneling ionization of other K (or $\mathrm{K^{\prime}}$) channels [51]. Using Eq. (3)(a, b), time and momentum of electron-hole recombination are calculated, as shown in Fig. 6, the blue curve represents electron excitation at the positive half-cycle and the red curve indicates electron excitation at the negative half-cycle. The black curve represents the energy released by the recombination of electron-hole pairs when TDPs and Berry connections are not considered. This shows that the frequency shift caused by TDPs and Berry connections is time dependent.
The frequency shift $\textrm{E}(t )\cdot {D^k}$ in different times is shown in Fig. 7(a). When the frequency shift is greater than 0, harmonic blue shift, and when it is less than 0, harmonic red-shift. We use a filter function to retrieve H6 and H8, and the time domain spectra of H6 and H8 are shown in Fig. 7(b, c), respectively. It is seen that H6 is mainly produced at the harmonic red-shift moment, and H8 is mainly generated at the harmonic blue-shift moment. Different generation time for harmonic orders leads to different harmonic frequency shift.
Fig. 7. (a) Frequency shift of the high harmonics. (b) H6 time domain spectroscopy. (c) H8 time domain spectroscopy.
5. Effect of crystal orientation on high harmonic frequency shift
The TDPs and Berry connections are related to the symmetry of the crystal [11,52], therefore, the frequency shift depends on the crystal orientation is a noteworthy issue. In section 2, we have shown that measured total harmonic spectrum and frequency shift in experiment as the pump polarization along armchair direction. Figure 8 shows a measured total harmonic spectrum and frequency shift in experiment as the pump polarization along zigzag direction (I = 2.3 TW/cm2). When driving pulse is polarized along zigzag direction, H6 having the largest blue-shift, and H8 the largest red-shift, opposite to the frequency shift of armchair direction
Fig. 8. (a) High harmonic spectrum measured from monolayer MoS2 (I = 2.3 TW/cm2). The driving pulse is polarized along zigzag direction. (b) Measured frequency shift of each harmonic.
The orientation-dependence of total harmonic of monolayer MoS2 is simulated by density-matrix equations within the independent-particle approximation [34,43]. The driving mid-infrared laser field is modelled by a Gaussian temporal envelope with 60 fs duration and 20 GW/cm2 peak intensity at a central wavelength of 3800 nm. The dephasing times are set as T2 = 1 fs. Figure 9(a) shows the total harmonic spectrum in simulation as the pump polarization along armchair and zigzag direction. The orientation-dependence of total harmonic frequency shift is shown in Fig. 9(b). The frequency shift of the odd-order harmonics does not vary with the crystal orientation angle, while and the frequency of the even-order harmonics varies periodically. We believe that this periodic change is due to the anisotropy of the MoS2 Berry connections and transition dipole moment. This is additional evidence to prove that the frequency shift of even-order harmonics comes from the influence of Berry connections and TDPs.
Fig. 9. (a)The total harmonic spectrum obtained at armchair and zigzag direction. (b) Orientation-dependence of total harmonic frequency shift.
6. Conclusion
In conclusion, we attribute the frequency shift of the even-order harmonic near the monolayer MoS2 band gap to the modulate of the TDPs and Berry connections on the recombination energy of the electron hole pair. The k dependent TDPs leads to changes in the light field embellishment energy during the sub-cycle as a function of the time and momentum of electron hole pair recombination. H6 is mainly generated at the harmonic red-shift moment, while H8 is mainly generated at the harmonic blue-shift moment, which leads to the sixth harmonic red-shift and the eighth harmonic blue-shift of MoS2. We believe that the effects of TDPs and Berry connections on HHG frequency are universal, such results aid in the fundamental understanding of carriers dynamics in non-centrosymmetric material.
Appendix A: effect of carrier-envelope phase on high harmonic frequency shift
When the duration of ultrashort light pulses approaches the few-cycle regime, the high harmonic spectrum shifts with the vary of the carrier envelope phase (CEP) used is $E = {E_0}exp({ - {t^2}/({2\tau {\;^2}} )} )cos({{\omega_0}t + {\phi_{CEP}}} )$, in which τ = 30 fs. In Fig. 10, we shown intensity spectra while ${\phi _{CEP}} $ varying continuously from 0 to 6π, the frequency shift repeats itself with a phase period of 2π. This variation is due to the fact that the exact positions and amplitudes of the electric-field extrema change periodically with the CEP [30].
Appendix B: effect of dephasing time on high harmonic frequency shift
In Fig. 11, we shown the high-harmonic spectra and frequency shift calculated with dephasing time T2 = 5fs. The spectrum is enhancement in comparison with the T2 = 1fs, and is a significant difference between the harmonic frequency shift and the experimental results. Therefore, we speculate the short dephasing time is more likely to determine even-order harmonic frequency shift.
Fig. 11. (a) Calculated inter-band contributions of harmonic spectrum (I = 0.23 TW/cm2). The dephasing time is set as 5 fs. (b) Calculated frequency shift of each order of the inter-band harmonic spectrum.
In addition, we show time-frequency analysis results for T2 = 1 fs and T2 = 5 fs in Fig. 12. Only the short trajectories are seen in the time-frequency spectrum when dephasing time T2 = 1fs. When set the dephasing time T2 = 5fs, long trajectories can be observed. The long trajectories lead to in the temporal overlap between two adjacent half-cycle trajectories, this makes the interference in the process of HHG more complex. This is consistent with a previous study [23,53].
Funding
National Natural Science Foundation of China (11874373, 12174412, 12174413); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021241); Scientific Instrument Developing Project of the Chinese Academy of Sciences.
Disclosures
The authors declare that they have no competing interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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