Controlling isolated attosecond pulse generation in MoSe₂ using polarization gating and TDDFT simulations

Abstract

The dependence of high harmonic generation spectra in solids on the driving wavelength remains an open question, especially regarding its impact on the generation of isolated attosecond pulses. Achieving active control over isolated attosecond pulse generation remains challenging due to the complex dynamics underlying solid-state high harmonic generation. In this work, we investigate high harmonic generation and the corresponding attosecond pulses generated from monolayer molybdenum diselenide using a real-time, ab initio time-dependent density functional theory approach. Our results reveal a linear increase in harmonic cutoff energy with driving wavelength, both parallel and perpendicular to the laser polarization direction. Using a polarization gating technique with symmetric amplitude components, we evaluate attosecond pulse isolation across multiple wavelengths and identify 2.5 $\:{\upmu\:}\text{m}$ as the most favorable for generating clean isolated attosecond pulses. Momentum-resolved electron dynamics reveal conditions that favor isolated pulse formation at 2.5 $\:{\upmu\:}\text{m}$. Furthermore, by applying an asymmetric polarization gating at this optimal wavelength, isolated attosecond pulses with favorable temporal confinement are achieved. This study provides insight into wavelength-dependent control of high harmonic generation and efficient isolated attosecond pulse generation in solids.

Introduction

High harmonic generation (HHG), an extreme nonlinear optical phenomenon that occurs beyond the perturbative regime, has been studied extensively over the past several decades^1,2. As a tabletop coherent x-ray source, HHG has enabled attosecond $\:(1\:\text{a}\text{s}=\:{10}^{-18}\text{s})\:$pulse generation³ for probing ultrafast electron dynamics. HHG has been reported from various systems such as atomic and molecular gases^4,5, and more recently, in solids⁶. Compared to gaseous systems, solid-state HHG can be driven with significantly lower peak intensities $\:(\sim{10}^{12}\:\text{W}/{\text{c}\text{m}}^{2})$. Moreover, due to the higher electron density in solids, HHG in these systems can result in more compact and brighter extreme-ultraviolet sources⁷.

The HHG process in atomic gas is well described by a three-step semi-classical model: (1) tunnel ionization, an intense femtosecond laser field induces ionization in the system, releasing a valence electron into the continuum. (2) acceleration, the free electron is accelerated by the time-dependent driving laser field. and (3) recombination, when the electron is driven back to the parent ion due to the reversal of the field’s sign, it recombines, emitting high-energy photons with energies corresponding to multiples of the fundamental laser frequency⁸. As a coherent process, HHG occurs at successive half-cycles of the driving laser pulse, leading to the creation of an HHG spectrum that is characterized by a plateau that extends to a cutoff region. Compared to the atomic gases, describing of laser–solid interactions in the strong field regime is more complex due to the intricate electronic structures of solids. HHG in solids can originate from two mechanisms: (1) interband dynamics, an electron in the valence band is excited to the conduction band under the influence of the driving laser field, leaving behind a hole. The resulting electron-hole pairs oscillate under the time-dependent field and, upon recombination, emit high-energy photons⁹. (2) intraband current, this arises from the acceleration of electrons within a single band under the laser field. These electrons undergo nonlinear motion due to the crystal lattice, leading to harmonic emission. The cycle-averaged intraband energy (ponderomotive energy) in this case is momentum-dependent^10,11. In the strong-field regime, the interplay between interband and intraband dynamics becomes significant and cannot be neglected. Moreover, multiple ionization and recombination sites, combined with the high density and periodic structure of solids, further complicate HHG process and the generation of isolated attosecond pulses (IAPs).

Since the first observation of HHG from a periodic crystal (ZnO) in 2011⁶, HHG in solids has attracted considerable attention. HHG has been investigated in a wide range of solid-state systems including bulk semiconductors¹², rare-gas solids¹³, strongly correlated materials¹⁴, nanostructured materials¹⁵, and two-dimensional (2D) materials¹⁶. Among these, 2D materials such as monolayer transition metal dichalcogenides (TMDs) exhibit unique symmetry and electronic properties¹⁷. For example, their typical binary crystal structure allows breaking inversion symmetry enabling which results in the emission of harmonics perpendicular to the direction of the laser field, even-order harmonics¹⁸, features absent in centrosymmetric materials like graphene¹⁷. Yoshikawa et al., considering four TMDs, showed that when the laser field is linearly polarized along the zigzag direction, the electron-hole dynamics at the K point differ from those at the $\:{\text{K}}^{{\prime\:}}$ point. As a result, using this polarization selection rule for HHG, they were able to explain the polarization directions of even- and odd-order harmonics¹⁹.

Despite recent progress in solid-state HHG, achieving truly IAPs in crystals remains challenging²⁰. Unlike gases-where ellipticity quickly suppresses harmonic yield-many solids, including 2D materials such as MoSe₂, maintain or even enhance HHG efficiency as the driving field becomes elliptical^21,22. Moreover, both interband and intraband contributions respond strongly to the driving wavelength, because carrier dynamics and band dispersion vary across the spectrum²³. In addition, the coherence of interband polarization is limited by the finite dephasing time of excited carriers, which plays an important role in the cleanliness of the HHG spectra. It has therefore been suggested that dephasing of the driven electronic wave packet during its excursion inside the solid may account for this inconsistency^24,25,26,27. In particular, excited electrons undergo elastic and inelastic scattering with phonons, crystal defects, and impurities, or—at higher excitation densities—experience electron–electron collisions. Incorporating these effects into the dephasing time is treated differently across theoretical frameworks such as the time-dependent Schrödinger equation (TDSE)^28,29, the semiconductor Bloch equations (SBE)³⁰, and time-dependent density functional theory (TDDFT)³¹.

Yet no study has systematically explored how wavelength governs burst isolation in monolayer MoSe₂ (1 L-MoSe₂) or other TMDs. Understanding this link is crucial, because residual bursts that appear after the polarization gate closes destroy the temporal confinement required for isolated attosecond emission. 1 L-MoSe₂ exhibits strong light-matter interaction due to its reduced dimensionality and direct bandgap in the visible-near-infrared range, which enables efficient coupling with long-wavelength driving fields³². The non centrosymmetric crystal structure allows the generation of even-order harmonics, providing additional control over the harmonic spectrum—a feature absent in many centrosymmetric materials¹⁹. Moreover, MoSe₂ has relatively high damage thresholds and robust structural stability under intense mid-infrared excitation³³, which allows for higher HHG yield and enhances IAP intensity.

In this work, we employ an ab initio approach based on real-time TDDFT^34,35 as implemented in the Octopus package³⁶, to investigate attosecond pulse generation from 1 L-MoSe₂ (see the Method and simulation setup section for more details). Which is a powerful tool for investigating electron dynamics in such a complex system under interaction with an intense and femtosecond laser pulse. We focus on the generation of IAPs using the polarization gating technique, a widely used method in atomic gases³⁷ and condensed matter systems³⁸ to temporally confine high-harmonic emission. Attosecond pulses are synthesized from the superposition of nonperturbative emission of harmonics arising from the interplay between intraband and interband contributions. We show that the maximum photon energy (cutoff energy) emitted from 1 L-MoSe₂ is linearly dependent on the driving wavelength, when the laser is linearly polarized along the zigzag direction. Within the polarization gating method formed by vector potential, we identify that at a driving wavelength of 2.5 $\:{\upmu\:}\text{m}$, the HHG spectrum exhibits enhanced temporal coherence, leading to more effective isolation of attosecond pulse trains. To further explore this behavior, we examine the momentum-resolved distribution of excited electrons at key times during the interaction for different driving wavelengths. Finally, we show that at the optimal wavelength of 2.5 $\:{\upmu\:}\text{m}$ by reducing the amplitude of the trailing edge of the vector potential in the polarization gating framework-asymmetric polarization gating-It is possible to generate IAPs with favorable full width at half maximum (FWHM) across a range of driving intensities and minimize residual bursts after the polarization gate is closed. The Keldysh parameter in this study is equal to 0.61¹⁰, considering a wavelength of 2.5 $\:{\upmu\:}\text{m}$ and $\:{\text{I}}_{0}=0.5\:\text{T}\text{W}/{\text{c}\text{m}}^{2}$; therefore, the system operates in the tunneling regime.

Result and discussion

Dependence of cutoff energy on driving wavelength

Since this study focuses on the effects of wavelength on the high harmonic spectrum, we first investigate the response of the high harmonic spectra to variations in the driving wavelength of the laser pulse. We therefore begin our discussion by analyzing the dependence of cutoff energy on driving wavelength. In this section, the driving laser field is linearly polarized along the Zigzag direction of the 1 L-MoSe₂ crystal. A schematic of the hexagonal structure of MoSe₂ with Zigzag and Armchair directions from the top view is shown in Fig. 1(a). Monolayer MoSe₂ exhibits a direct bandgap, with the valence band maximum and conduction band minimum located at the k-point. The DFT-calculated band structure, with subtracting the Fermi energy, is displayed in Fig. 1(b), yielding a bandgap of 1.57 eV³⁹. The peak intensity of the single-pulse linear driving laser is fixed at $\:{\text{I}}_{0}=0.5\:\text{T}\text{W}/{\text{c}\text{m}}^{2}$ and is kept below the damage threshold of 1 L-MoSe₂¹⁸. The laser pulse has a total duration of 5 optical cycles. The HHG spectrum for two driving wavelengths of 2.0 $\:{\upmu\:}\text{m}$ and 2.5 $\:{\upmu\:}\text{m}$ is shown in Figs. 1 (c)-(f).

The influence of the driving wavelength on the HHG cutoff energy can be observed by comparing the plateau regions – highlighted by pale yellow shading – in Figs. 1(c)–1(f). For the 2.0 $\:{\upmu\:}\text{m}$ case [Figures 1(c) and 1(d)], the cutoff energy is approximately 6.4 eV. In contrast, for the 2.5 $\:{\upmu\:}\text{m}$ case [Figures 1(e) and 1(f)], the plateau extends further, with a higher cutoff energy around 6.7 eV and a greater number of both odd and even harmonics. The cutoff energy values ​​are in good agreement with the approximate relation E_cutoff = E_w + 3.17 U_p, which is valid under the condition that the laser pulse is linearly polarized⁷. The effect of increasing the cutoff energy and consequently increasing the number of harmonics due to increasing the driving wavelength can be observed in the full width at half maximum (FWHM) of the generated attosecond pulse train. As Fig. 1 shows, the FWHM for the attosecond pulse train at 2.5 $\:{\upmu\:}\text{m}$ wavelength is much smaller than that of the 2.0 $\:{\upmu\:}\text{m}$ case, so that the FWHM for the attosecond pulse train resulting from the superposition of odd harmonics in the HHG spectrum of 2.5 $\:{\upmu\:}\text{m}$, Fig. 1(e) corresponds to 648 as. Therefore, considering wavelengths of 2.5 $\:{\upmu\:}\text{m}$ and longer for the following purposes, namely generating attosecond pulses with minimum pulse width, would be appropriate. In addition, odd harmonics also have better potential for use in generating attosecond pulses for gaining minimum FWHM.

Turning to the symmetry properties of the harmonics, those emitted along the same direction as the laser polarization (Zigzag direction) correspond to odd harmonics [Figures 1(c) and 1(e)], while those emitted perpendicular to it (Armchair direction) correspond entirely to even harmonics [Figures 1(d) and 1(f)]. This polarization-selective generation of harmonics has been reported in other TMDs as well¹⁸. Unlike in atomic gases, where temporally symmetric driving fields lead solely to odd harmonics, certain solid-state systems such as 1 L-MoSe₂ can support both odd and even harmonics due to the lack of inversion symmetry. This intrinsic broken inversion permits even-order HHG alongside the usual odd-orders¹⁸. In crystals with broken inversion symmetry Berry curvature is nonzero⁴⁰. The effect of this nonvanishing Berry curvature is to introduce an anomalous current that is perpendicular to the polarization of the driving laser pulse and therefore manifests itself through the generation of even harmonics alongside with the Armchair direction.

As Figs. 1(c)–1(f) show, the yields of below-bandgap odd and even harmonics behave differently. Below the bandgap, photons do not have enough energy to drive real interband excitations. As a result, the interband current decay rapidly in strength as the harmonic frequency moves further away from resonance. Without this resonant enhancement, even-order harmonics lose their primary amplification channel. Although the Berry-curvature-driven intraband mechanism can still produce even harmonics below the bandgap, its efficiency is generally much lower. The Berry curvature is largest near the valleys, but the electron trajectories in below-bandgap dynamics often do not sample these regions strongly enough to generate a high yield. In addition, interference between the intraband and the weak residual interband contributions can further suppress the spectra. Consequently, odd harmonics remain strong in the sub-bandgap regime, while even harmonics tend to be very weak.

Fig. 1

(a) Schematic illustration of the 1 L-MoSe₂ crystal structure, highlighting the Zigzag and Armchair directions. (b) DFT-calculated electronic band structure of 1 L-MoSe₂, with energies referenced to the Fermi level (shifted by 0.20 a.u.). (c) and (d) HHG spectra emitted along the Zigzag and Armchair directions, respectively, under excitation with a linearly polarized driving field of 2.0 $\:{\upmu\:}\text{m}$ wavelength. (e) and (f) HHG spectra for the same directions under a 2.5 $\:{\upmu\:}\text{m}$ driving field. The inset panels show the corresponding attosecond pulse trains obtained by superposing harmonics within the plateau region. For driving wavelengths below 2.5 $\:{\upmu\:}\text{m}$, the resulting FWHM values are in the femtosecond regime. The pale-yellow shaded area in marks the plateau bounded by the 1 L-MoSe₂ DFT calculated bandgap energy, E_g​ = 1.57 eV and the predicted cutoff energy, E_cutoff = E_w + 3.17 U_p, as estimated by the three-step model. This approximation holds only for linearly polarized laser pulses.

In addition to the fact that the cutoff energy in solids remains an active research topic⁴¹, establishing its behavior is essential for interpreting both the spectral features and the temporal structure of the resulting IAPs. We next examine the dependence of the cutoff energy in harmonic spectra on the driving wavelength. Figures 2(a) and 2(b) show this dependence for harmonics polarized along the Zigzag and Armchair directions of 1 L-MoSe₂. The driving wavelength is varied from 1.4 μm to 3.2 μm and the laser polarization is fixed along the Zigzag direction. The laser peak intensity is fixed at 0.5 TW/cm², which corresponds to an electric field amplitude of 0.38$\:\:\times\:\:$10^-2 a.u. (see Supplementary Fig. S1 online). As can be seen from Figs. 2(a) and 2(b), with increasing driving wavelength, the cutoff energy increases linearly for both components of the harmonic spectrum, which subsequently increases the plateau area (see Supplementary Fig. S2 online). Therefore, there is a linear dependence between driving wavelength and cutoff energy. The observed linear dependence of the cutoff energy on the driving wavelength originates from interband recombination processes, where the increased vector potential at longer wavelengths enables electrons to reach higher conduction bands near the BZ edge, resulting in the emission of higher-energy photons upon recombination. These results are in good agreement with previous studies reporting a linear dependence of the cutoff energy on the driving wavelength^42,43.

Fig. 2

HHG spectra as a function of driving wavelength for (a) Zigzag and (b) Armchair components of the harmonic spectra generated from 1 L-MoSe₂ under linearly polarized vector potential along the zigzag direction. The peak intensity of the driving field is fixed at 0.5 TW/cm². The white dashed lines indicate the cutoff energy at different driving laser wavelengths.

Generating isolated attosecond pulse using the polarization gating method

In the following, we focus on generating IAPs from 1 L-MoSe₂ using the polarization gating method. Figure 3 shows the results for four different driving wavelengths: 2.0 $\:{\upmu\:}\text{m}$, 2.5 $\:{\upmu\:}\text{m}$, 3.0 $\:{\upmu\:}\text{m}$, and 3.5 $\:{\upmu\:}\text{m}$. In all cases, the peak laser intensity is kept constant at I₀ = 0.5 TW/cm² ($\:{\text{I}}_{0}={\text{I}}_{1}={\text{I}}_{2}$). For gaining insight into the subcycle field evolution of bursts and analyzing the generated pulses in both time and frequency domain, the Gabor transform is displayed in the left panel of Fig. 3. The width of the time window $\:\tau\:$ for Gabor transform calculation is set to $\:\sim\:1\:\text{f}\text{s}$, where T denotes one optical cycle. The black solid and dashed lines represent the scaled x and y components of the time-dependent vector potential, respectively. The vertical white dashed line represents the gate opening time t_g​ (middle of the gate window), where the polarization becomes linear. This is also the point where A_y reaches zero, aligning the linear polarization along the Zigzag direction of the crystal structure.

About half a cycle after t_g, high intensity attosecond pulse resulting from the driving vector potential interaction is observed. The right panels show the corresponding time-domain profiles of the attosecond pulses. Attosecond pulses for wavelengths of 2.0 $\:{\upmu\:}\text{m}$, 2.5 $\:{\upmu\:}\text{m}$, 3.0 $\:{\upmu\:}\text{m}$, and 3.5 $\:{\upmu\:}\text{m}$ were obtained from the superposition of the 5th to 13th, 3th to 20th, 6th to 24th, and 6th to 28th harmonics, respectively (see Supplementary Fig. S3 online). For 2.0 $\:{\upmu\:}\text{m}$ and 2.5 $\:{\upmu\:}\text{m}$, the pulse is well isolated with FWHMs of 934 as and 420 as, respectively. While the larger harmonic bandwidth at 2.5 $\:{\upmu\:}\text{m}$ contributes to the shorter duration, this factor alone cannot account for the observed reduction. In addition to bandwidth, the spectral phase of the harmonics plays an important role in shaping the temporal profile, and differences in phase lead to a significantly shorter pulse for 2.5 $\:{\upmu\:}\text{m}$. As the wavelength increases to 3.0 $\:{\upmu\:}\text{m}$ and 3.5 $\:{\upmu\:}\text{m}$, the weaker side pulses begin to appear after the gate closes, when the polarization of the vector potential becomes circular again. These extra emissions suggest the isolation is less effective at longer wavelengths. We also investigated the 4.0 μm wavelength in this study and similar behavior to the 3.0 and 3.5 μm cases is still observed (see Supplementary Fig. S4 online). These results confirm that while isolation can, in principle, be achieved below 2.5 μm, only the 2.5 μm case yields an IAP with both effective confinement and sub-femtosecond duration.

Fig. 3

Generation of IAPs from 1 L-MoSe₂ using the polarization gating method for different wavelengths. The IAPs are obtained by coherently superposing harmonics above the band gap. Gabor time-frequency analysis of HHG spectra at (a1) 2.0 $\:{\upmu\:}\text{m}$, (b1) 2.5 $\:{\upmu\:}\text{m}$, (c1) 3.0 $\:{\upmu\:}\text{m}$ and (d1) 3.5 $\:{\upmu\:}\text{m}$. Light-yellow solid and dashed curves represent the scaled x- and y-components of the time-dependent vector potential. The dashed white vertical line marks the gate opening time t_g (middle of the gate window). Correspondingly in the (a2), (b2), (c2) and (d2) they show their related IAPs. The measured FWHM durations are 934 as, 420 as, 489 as, and 513 as, respectively. While the 2.5 $\:{\upmu\:}\text{m}$ pulse is well isolated, longer wavelengths lead to weaker secondary emissions following the main peak.

Momentum-resolved distribution of excited electrons

In solid-state systems, HHG and the resulting isolated attosecond pulses are strongly influenced by the material’s band structure. To gain deeper insight, it is effective to examine the momentum-resolved distribution of excited electrons. Especially in our system, where IAPs generated by 3.0 $\:{\upmu\:}\text{m}$ and 3.5 $\:{\upmu\:}\text{m}$ driving wavelengths show persistent emission even after the gate closes, indicating emission events continue after the temporal window of polarization gating has passed, contrary to the expected suppression of HHG beyond the gating region. In order to demonstrate the driving of the electron wavepacket in momentum space by the driving vector potential in 1 L-MoSe₂, we computed the time-resolved electron excitation dynamics using a real-time TDDFT approach. At each time step, the number of excited electrons was calculated as the total number of electrons $\:{\text{N}}_{\text{e}}$ minus the poplulation remaining in the valence bands. The valence band population at time t was obtained by projecting the time-evolved Kohn-Sham states $\:{|{\uppsi\:}}_{\text{n},\text{k}}\left(\text{t}\right)\rangle$ onto the basis of the ground-state wavefunctions $\:{|{\uppsi\:}}_{\text{n}\:,\mathbf{k}}^{\text{G}\text{S}}\rangle$, and summing the squared overlaps over all valence bands n’, all initial occupied states n, and all k-points using the expression:

$$\:{\text{N}}_{\text{e}\text{x}}\left(\text{t}\right)={\text{N}}_{\text{e}}-\frac{1}{{\text{N}}_{\text{k}}}\sum_{\text{n},{\text{n}}^{{\prime\:}}}^{\text{o}\text{c}\text{c}.}\sum_{\text{k}}{\left|\langle{{\uppsi\:}}_{{\text{n}}^{{\prime\:}},\mathbf{k}}^{\text{G}\text{S}}|{{\uppsi\:}}_{\text{n},\text{k}}\left(\text{t}\right)\rangle\right|}^{2}$$

(1)

Our results shown in Fig. 4 indicate that in the BZ region the electrons undergo motion dictated by the instantaneous value of the time-dependent vector potential. Moreover, the various snapshots of the excited electrons show a complex modulation at a sub-cycle timescale, due to the complex band structure of 1 L-MoSe₂, which results in many conduction bands being involved in dynamics.

Figure 4 presents the time-resolved excitation dynamics calculated from TDDFT approach. In Fig. 4(a), The first Brillouin zone and the k-path containing high-symmetry points of 1 L-MoSe₂ are shown. The driving laser pulse is illustrated in Fig. 4(b), composed of right- and left-handed circularly polarized pulses combined to generate a polarization gate field. The distributions of excited electrons in momentum space at three key time steps – one optical cycle before the middle of the gate window (t_g – T), at the middle of the gate window (t_g), and one optical cycle after the middle of the gate window (t_g + T) are displayed in Figs. 4(c), 4(d), 4(e) and 4(f), corresponding to driving wavelengths of 2.0 $\:{\upmu\:}\text{m}$, 2.5 $\:{\upmu\:}\text{m}$, 3.0 $\:{\upmu\:}\text{m}$, and 3.5 $\:{\upmu\:}\text{m}$, respectively. All results are calculated for a fixed laser peak intensity of I₀ = 0.5 TW/cm² ($\:{\text{I}}_{0}={\text{I}}_{1}={\text{I}}_{2}$). The fulltime evolution of the electron dynamics is provided in Supplementary Videos S1–S3, which visualize the momentum-resolved distributions for the three driving wavelengths. When comparing the excited electron distribution at the same time across different driving wavelengths, it is observed that: excitation becomes noticeably stronger as the wavelength increases (see Supplementary Fig. S6 online). This behavior can be understood by considering two main factors. First, the amplitude of the time-dependent vector potential increases with the driving wavelength. As a result, the field becomes strong enough to excite electrons beyond the lowest conduction band. Each additional conduction band then provides a new quantum path for recombination, producing harmonics via many recombination paths⁴⁴. Since increasing the number of recombination paths enhances quantum interference between them, temporal confinement decreases, leading to the formation of multiple bursts even after the polarization gate is closed, as shown in Fig. 3(c1, c2) and (d1, d2). Second, longer wavelengths are associated with longer optical cycles, meaning that the electron’s excursion in the conduction band becomes more extended. This prolonged intraband motion increases the likelihood of electrons reaching the edges of the first Brillouin zone, thereby opening up additional harmonic emission channels⁴⁵. This explains not only the relatively high intensity bursts observed after gate closure at longer driving wavelengths, but also the increase in FWHM. In fact, it was initially expected that the broader plateau in the HHG spectrum at longer wavelengths would result in a shorter FWHM (given the superposition of the 6th to 28th harmonics for a 3.5 $\:{\upmu\:}\text{m}$ driving wavelength Fig. 3 (d1, d2)). However, the opposite occurred due to increased phase mismatch and loss of coherence among the contributing harmonics.

A closer inspection of the excitation profile at t_g + T further confirms this behavior: as the wavelength increases, the distribution of excited electrons becomes broader in momentum space, signaling greater participation of higher conduction bands. Electrons that reach these bands may not recombine quickly and may remain across multiple optical cycles. This residual population contributes to the delayed bursts observed after the main attosecond pulse, particularly for 3.0 $\:{\upmu\:}\text{m}$ and 3.5 $\:{\upmu\:}\text{m}$ driving wavelengths, as shown in Figs. 3(c1, c2) and 3(d1, d2).

Additional insight into this behavior can be obtained by examining the magnitude of the dipole matrix elements (see Supplementary Note 3 and Supplementary Fig. S5 online). Because the dipole strength is strongly dependent on the band structure at different points in the Brillouin zone, regions of high dipole magnitude correspond to enhanced transition probabilities. As the electron’s excursion in k-space increases with longer driving wavelengths, the likelihood of transitions into higher conduction bands is consequently amplified. This explains the broader excited-electron distributions and delayed emissions observed at longer wavelengths. Another important observation from the dipole matrix elements is that the transition strengths between different bands are comparable and the band gaps are relatively small. As a result, the multi-plateau phenomenon typically reported in HHG spectra of solids is not observed here; instead, the spectra become denser with increasing harmonic order.

Overall, these findings highlight an important balance in generating isolated attosecond pulses from solids. Longer driving wavelengths increase the cutoff energy and spectral extension, offering clear advantages. But on the other hand, they also make the electron dynamics more complex. As excitation spreads more broadly in momentum space and more conduction bands become involved, the chances of phase mismatch and decoherence between harmonics rise. The results showed that in MoSe₂, using longer driving wavelengths extends the harmonic cutoff but also creates additional phase mismatches between harmonics. That phase mismatches in the HHG spectrum lead to a weakening of the temporal confinement required for effective isolation of the attosecond pulse train. Thus, identifying the optimal driving wavelength is crucial for stable isolated attosecond pulse generation.

Fig. 4

TDDFT simulations of excited electron dynamics in 1 L-MoSe₂. (a) First BZ showing the high-symmetry path. (b) Vector potential of the polarization-gated laser field composed of right- and left-handed circularly polarized components. (c)-(f) Excited electron distributions in momentum space at times t_g - T (left), t_g (center), and t_g + T (right) for driving wavelengths of 2.0 $\:{\upmu\:}\text{m}$, 2.5 $\:{\upmu\:}\text{m}$, 3.0 $\:{\upmu\:}\text{m}$, and 3.5 $\:{\upmu\:}\text{m}$, respectively. Longer wavelengths result in both a higher excitation level and broader distribution in the BZ, indicating stronger access to upper conduction bands.

Generating IAPs using the polarization gating method with asymmetric amplitude ratios

Considering the optimal wavelength of 2.5 $\:{\upmu\:}\text{m}$ as the driving wavelength, we reduced the intensity of the trailing circularly polarized pulse to half that of the leading pulse, aiming to suppress HHG after the closing gate time. In this approach, the driving field amplitude at the leading edge of the pulse is reduced by a certain ratio compared to that at the trailing edge. As a result, ground-state depletion and emissions occurring after the main attosecond burst are suppressed, which could enhance the isolation of the primary emission, as was previously demonstrated in high-harmonic generation in gases⁴⁶. The vector potential corresponding to such asymmetric amplitude ratios is shown in Fig. 5(a). The results of the vector potential interaction within this framework, for peak intensities ranging from $\:0.50\text{T}\text{W}/{\text{c}\text{m}}^{2}$ to $\:1.50\text{T}\text{W}/{\text{c}\text{m}}^{2}$, are shown in Fig. 5(b). As seen, IAPs with FWHMs of 440–541 as are generated from 1 L-MoSe₂. These results confirm that by reducing the amplitude of the trailing edge of the driving pulse, we can effectively reduce the harmonic radiation after the main attosecond burst in the gate window, thus resulting in the generation of a single attosecond pulse with a proper isolation. Among the intensities, which all have a ratio of $\:{\text{I}}_{1}/{\text{I}}_{2}=2$, the generated attosecond pulse FWHMs remained relatively constant, suggesting that the gating efficiency is maintained over a broad intensity range.

Fig. 5

(a) Time-dependent vector potential in the framework of the polarization gating method with asymmetric amplitude ratio ($\:{\text{I}}_{1}=0.50\text{T}\text{W}/{\text{c}\text{m}}^{2},\:{\text{I}}_{2}=0.25\:\text{T}\text{W}/{\text{c}\text{m}}^{2}$). (b) It shows attosecond pulses for a driving laser pulse intensity range of $\:{\text{I}}_{1}=0.50\text{T}\text{W}/{\text{c}\text{m}}^{2}$ to $\:{\text{I}}_{1}=1.50\text{T}\text{W}/{\text{c}\text{m}}^{2}$ with a ratio of $\:{\text{I}}_{1}/{\text{I}}_{2}=2$. The driving wavelength is fixed at 2.5 $\:{\upmu\:}\text{m}$. The FWHM of each IAP is shown in the right panel. The intensity of the attosecond pulses is normalized. All IAPs obtained from different driving laser intensities are obtained from the superposition of the 3th to 20th harmonics. The delay factor $\:\gamma\:$ is taken to be 1.74.

Conclusion

In conclusion, we explored the generation of attosecond pulses from 1 L-MoSe₂ using the polarization gating method via real-time, ab initio TDDFT approach. Our analysis of the HHG cutoff energies revealed a linear dependence on the driving laser wavelength, a trend that appeared in both the Zigzag and Armchair components of harmonic spectra. Focusing on the polarization gating method, we demonstrated the generation of IAPs using driving wavelengths of 2.0 $\:{\upmu\:}\text{m}$, 2.5 $\:{\upmu\:}\text{m}$, 3.0 $\:{\upmu\:}\text{m}$, and 3.5 $\:{\upmu\:}\text{m}$, with the 2.5 $\:{\upmu\:}\text{m}$ case producing the cleanest, most isolated pulse. Through time-frequency Gabor analysis and momentum-resolved excited electron distributions, we found that the increase in excited electron populations at longer wavelengths – due to increased vector potential amplitudes and extended optical cycles – leads to residual post-gate emission, thus compromising pulse isolation. To further improve pulse isolation, we implemented an asymmetric amplitude scheme in the polarization gate field by reducing the intensity of the trailing circularly polarized component. These results demonstrate the effectiveness of using the polarization gating method with unequal amplitude ratio to achieve better pulse isolation. The results of our study introduce 1 L-MoSe₂ as a good potential candidate for generating IAP. Moreover, the strong dependence of the pulse isolation on the driving laser wavelength emphasizes the importance of carefully controlling excitation conditions to optimize the generation of isolated attosecond pulses while minimizing post-gate emission.

Method and simulation setup

The dynamics of the laser-matter interaction are simulated using time-dependent density functional theory (TDDFT). This approach involves solving the time-dependent Kohn-Sham equations, where the wave functions evolve under the influence of a time-dependent Hamiltonian. The Kohn-Sham equations in atomic⁴⁷ unit are given by ⁵⁴:

$$\:\text{i}\frac{\partial\:}{\partial\:\text{t}}{{\upphi\:}}_{\text{j}}\left(\mathbf{r},\text{t}\right)=\left[-\frac{1}{2}{\nabla\:}^{2}+{\text{v}}_{\text{K}\text{S}}\left[\text{n}\right]\left(\mathbf{r},\text{t}\right)\right]{{\upphi\:}}_{\text{j}}\left(\mathbf{r},\text{t}\right),$$

(2)

where, $\:{\phi\:}_{j}\left(\varvec{r},t\right)$ denotes the orbitals of the auxiliary Kohn-Sham system.

The time-dependent electron density of the system $\:n\left(\mathbf{r},\text{t}\right)$ is calculated as:

$$\:\text{n}\left(\mathbf{r},\text{t}\right)=\sum_{\text{j}=1}^{\text{N}}{\left|{{\upphi\:}}_{\text{j}}\left(\mathbf{r},\text{t}\right)\right|}^{2}$$

(3)

The Kohn-Sham potential is expressed as⁴⁷:

$$\:{\text{v}}_{\text{K}\text{S}}\left[\text{n}\right]\left(\mathbf{r},\text{t}\right)={\text{v}}_{\text{e}\text{n}}\left(\mathbf{r}\right)+\int\:{\text{d}}^{3}{\mathbf{r}}^{{\prime\:}}\frac{\text{n}\left({\mathbf{r}}^{{\prime\:}},\text{t}\right)}{\left|\mathbf{r}-{\mathbf{r}}^{{\prime\:}}\right|}+{\text{v}}_{\text{X}\text{C}}\left[\text{n}\right]\left(\mathbf{r},\text{t}\right)+{\text{v}}_{\text{e}\text{x}\text{t}}\left(\mathbf{r},\text{t}\right)$$

(4)

The first and second terms represent the electron-nucleus pseudopotential and the Hartree potentials, respectively. The Hartree potential is computed over the volume Ω of the unit cell. Third term is the exchange-correlation potential, and the last term is the external potential due to the laser. The external potential is expressed in the velocity gauge as $\:{\text{V}}_{\text{e}\text{x}\text{t}}\left(\varvec{r},t\right)=\frac{1}{\text{c}}\mathbf{P}.\mathbf{A}\left(\text{t}\right)$, where $\:\mathbf{P}$ is the momentum operator and $\:\mathbf{A}\left(\text{t}\right)$ is the time-dependent vector potential, assuming the scalar potential $\:{\varphi\:}=0$.

In the polarization gating scheme used for interaction with 1 L-MoSe₂, the left- and right-handed circularly polarized components of the vector potential is described as:

$$\:{\mathbf{A}}_{\text{l}}\left(\text{t}\right)=\frac{\sqrt{{\text{I}}_{1}}\text{c}}{{\upomega\:}}{\text{f}}_{1}\left(\text{t}\right)\left[\frac{1}{\sqrt{1+{{\varepsilon\:}}^{2}}}\text{cos}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right){\widehat{\text{e}}}_{\text{x}}+\frac{{\varepsilon\:}}{\sqrt{1+{{\varepsilon\:}}^{2}}}\text{sin}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right){\widehat{\text{e}}}_{\text{y}}\right],$$

(5)

$$\:{\mathbf{A}}_{\text{r}}\left(\text{t}\right)=\frac{\sqrt{{\text{I}}_{2}}\text{c}}{{\upomega\:}}{\text{f}}_{2}\left(\text{t}\right)\left[\frac{1}{\sqrt{1+{{\varepsilon\:}}^{2}}}\text{cos}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right){\widehat{\text{e}}}_{\text{x}}-\frac{{\varepsilon\:}}{\sqrt{1+{{\varepsilon\:}}^{2}}}\text{sin}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right){\widehat{\text{e}}}_{\text{y}}\right]$$

(6)

Here, $\:{\varepsilon\:}=\pm\:1$ represents the ellipticity parameter for circular polarization. Throughout this study, the carrier envelope phase $\:{{\upphi\:}}_{\text{C}\text{E}}$ is assumed to be zero. The time-dependent envelope functions $\:{\text{f}}_{1}\left(\text{t}\right)$ and $\:{\text{f}}_{2}\left(\text{t}\right)$ are defined by:

$$\:{\text{f}}_{1}\left(\text{t}\right)={\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\text{t}}{{{\uptau\:}}_{\text{p}}}\right),$$

(7)

$$\:{\text{f}}_{2}\left(\text{t}\right)={\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\left(\text{t}-{\text{T}}_{\text{d}}\right)}{{{\uptau\:}}_{\text{p}}}\right),$$

(8)

where, $\:{\text{T}}_{\text{d}}={{\upgamma\:}{\uptau\:}}_{\text{p}}$ is the delay time between the two pulses with $\:{{\uptau\:}}_{\text{p}}$ denoting the pulse duration and the unitless parameter $\:\gamma\:$ represents the delay factor. Considering the direction of laser pulse propagation in the z direction, the in-plane vector potential interacting with 1 L-MoSe₂ becomes:

$$\:\mathbf{A}\left(\text{t}\right)={\mathbf{A}}_{\text{l}}\left(\text{t}\right)+{\mathbf{A}}_{\text{r}}\left(\text{t}\right)={\text{A}}_{\text{d}\text{r}\text{i}\text{v}\text{e}}\left(\text{t}\right){\widehat{\text{e}}}_{\text{x}}+\:{\text{A}}_{\text{g}\text{a}\text{t}\text{e}}\left(\text{t}\right){\widehat{\text{e}}}_{\text{y}},$$

(9)

where, $\:{\text{A}}_{drive}\left(t\right)$ is responsible for generating the attosecond pulses, and $\:{A}_{gate}\left(t\right)$ is responsible for suppressing the generation of attosecond pulses outside the gate by adding a transverse component to the path of charge carriers. These can be defined as:

$$\:{\text{A}}_{\text{d}\text{r}\text{i}\text{v}\text{e}}\left(\text{t}\right)=\frac{\sqrt{{\text{I}}_{0}}\text{c}}{{\upomega\:}\sqrt{1+{{\varepsilon\:}}^{2}}}\left[{\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\text{t}}{{{\uptau\:}}_{\text{p}}}\right)+{\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\text{t}}{{{\uptau\:}}_{\text{p}}}-{\upgamma\:}\right)\right]\text{c}\text{o}\text{s}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right),$$

(10)

$$\:{\text{A}}_{\text{g}\text{a}\text{t}\text{e}}\left(\text{t}\right)=\frac{{\varepsilon\:}\sqrt{{\text{I}}_{0}}\text{c}}{{\upomega\:}\sqrt{1+{{\varepsilon\:}}^{2}}}\left[{\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\text{t}}{{{\uptau\:}}_{\text{p}}}\right)-{\text{s}\text{i}\text{n}}^{2}\left(\frac{{\uppi\:}\text{t}}{{{\uptau\:}}_{\text{p}}}-{\upgamma\:}\right)\right]\text{s}\text{i}\text{n}\left({\upomega\:}\text{t}+{{\upphi\:}}_{\text{C}\text{E}}\right)$$

(11)

The equation of motion for the total microscopic current, $\:J\left(\varvec{r},t\right)$, can be written as⁴⁸:

$$\:\frac{\partial\:}{\partial\:\text{t}}\text{J}\left(\mathbf{r},\text{t}\right)=-\text{n}\left(\mathbf{r},\text{t}\right)\nabla\:{\text{v}}_{\text{e}\text{x}\text{t}}\left(\mathbf{r},\text{t}\right)-{\text{F}}^{\text{k}\text{i}\text{n}}\left(\mathbf{r},\text{t}\right)-{\text{F}}^{\text{i}\text{n}\text{t}}\left(\mathbf{r},\text{t}\right),$$

(12)

where $\:{\text{F}}^{\text{k}\text{i}\text{n}}\left(\mathbf{r},\text{t}\right)$ and $\:{\text{F}}^{\text{i}\text{n}\text{t}}\left(\mathbf{r},\text{t}\right)$ are correspond to the internal force densities of the many-particle system due to kinetic and interaction effects, respectively. By integrating both sides of the above relation over the total volume of the system, $\:{\Omega\:}$, we can show:

$$\:\frac{\partial\:}{\partial\:\text{t}}\underset{{\Omega\:}}{\int\:}{\text{d}}^{3}\mathbf{r}\text{J}\left(\mathbf{r},\text{t}\right)=-\underset{{\Omega\:}}{\int\:}{\text{d}}^{3}\mathbf{r}\text{n}\left(\mathbf{r},\text{t}\right)\nabla\:{\text{v}}_{\text{e}\text{x}\text{t}}\left(\mathbf{r},\text{t}\right)$$

(13)

Due to the integration over the entire volume of the interacting system, the contributions of the internal force densities become zero⁴⁷. Therefore, the HHG spectrum calculated as follows:

$$\:\text{H}\text{H}\text{G}\left({\upomega\:}\right)={\left|\text{F}\text{T}\left(\frac{\partial\:}{\partial\:\text{t}}\underset{{\Omega\:}}{\int\:}{\text{d}}^{3}\text{r}\text{J}\left(\mathbf{r},\text{t}\right)\right)\right|}^{2},$$

(14)

where FT represents the Fourier transform. As a result, by superimposing an appropriate range of harmonics between $\:{{\upomega\:}}_{\text{i}}$ and $\:{{\upomega\:}}_{\text{f}}$, the attosecond pulse can be calculated in the time domain by substituting Eq. (10) into the following equation:

$$\:\text{I}\left(\text{t}\right)=\:{\left|{\int\:}_{{{\upomega\:}}_{\text{i}}}^{{{\upomega\:}}_{\text{f}}}{\text{e}}^{\text{i}{\upomega\:}\text{t}}\left(\text{H}\text{H}\text{G}\left({\upomega\:}\right)\right)\right|}^{2}$$

(15)

A good option to analyze the time-frequency characteristics of the emitted radiation with high temporal resolution is to use the Gabor transform applied to the time-dependent current in Eq. (9). To calculate the Gabor transform:

$$\:\text{G}\left({\upomega\:},\text{t}\right)=\int\:d{\text{t}}^{{\prime\:}}\text{J}\left({\text{t}}^{{\prime\:}}\right){\text{e}}^{-\text{i}{\upomega\:}{\text{t}}^{{\prime\:}}}{\text{e}}^{\frac{-{(\text{t}-{\text{t}}^{{\prime\:}})}^{2}}{{2{\uptau\:}}^{2}}},$$

(16)

where $\:\tau\:$ is the width of the time window.

In this study, the exchange-correlation potential is treated within the local density approximation (LDA), and Hartwigsen–Goedecker–Hutter norm-conserving pseudopotentials⁴⁹ are employed. While LDA underestimates the bandgap of semiconductors, it reproduces the valence and conduction band dispersions with sufficient accuracy, and is thus suitable for describing the coupled interband and intraband carrier dynamics relevant to HHG.

The 1 L-MoSe₂ crystal structure is modeled using a hexagonal unit cell with a lattice constant of 3.288 A⁵⁰, corresponding to a cell area of 33.434 bohr². The simulation box is a parallelepiped of dimensions (6.213, 6.213, 20.002) bohr³ with a uniform real-space grid spacing of 0.34 a.u. along all directions. To eliminate interlayer interactions due to periodic boundary conditions, a vacuum spacing of at least 20 bohr is added along the out-of-plane (z) direction, ensuring effective isolation of the monolayer. For 1 L-MoSe₂, the work function E_w was determined to be 5.84 eV, derived from the highest occupied state.

The Brillouin zone is sampled using a 28 × 28 × 1 Monkhorst–Pack k-point mesh. Time propagation of the Kohn–Sham equations is performed using the enforced time-reversal symmetry (AETRS) scheme⁵¹ with a time step of 0.2 a.u. ($\:\sim$ 4.8 as). The simulation propagates nine bands, equal to the number of valence bands in the system. The incident laser field is applied in the plane of the monolayer (x–y plane). The interaction is modeled within the dipole approximation, assuming a spatially uniform field. Effects such as electron-electron and impurity scattering, electron–phonon coupling, surface interactions, and diffraction are beyond the scope of this study and are therefore not included in our analysis.