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Abstract
High harmonic generation from bilayer h-BN materials with different stacking configurations is theoretically investigated by solving the extended multiband semiconductor Bloch equations in strong laser fields. We find that the harmonic intensity of AA'-stacking bilayer h-BN is one order of magnitude higher than that of AA-stacking bilayer h-BN in high energy region. The theoretical analysis shows that with broken mirror symmetry in AA'-stacking, electrons have much more opportunities to transit between each layer. The enhancement in harmonic efficiency originates from additional transition channels of the carriers. Moreover, the harmonic emission can be dynamically manipulated by controlling the carrier envelope phase of the driving laser and the enhanced harmonics can be utilized to achieve single intense attosecond pulse.
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1. Introduction
In the past decade, the interaction of strong laser with solid materials has provided a new platform for high harmonic generation (HHG) [1–6], making it possible to realize compact, solid-state vacuum ultraviolet and extreme ultraviolet (EUV) attosecond sources [7–10], which has gradually become a key research topic of the strong-field physics [11–15]. Owing to the higher density of solid medium, the conversion efficiency of HHG from solids is naturally higher than that from gaseous atoms/molecules under the same laser condition. In gas medium, HHG process can be described by the semiclassical three-step model [16,17]: ionization of an electron, its acceleration in the laser field and subsequent recombination to the parent ion. Whereas in the case of solid materials, HHG can be decomposed into interband and intraband contributions. Analogous to the three-step model in gases, the generation of interband polarization and intraband oscillation of the HHG in a crystal solid can be described as similar three stages [7,14,18]: (i) electronic tunneling excitation from the valence band (VB) to the conduction band (CB), (ii) movement of electron/hole following the band dispersion and electric field, (iii) electron-hole recombination between VB and CB. In the recombination step, a high-energy photon is emitted, and different conjectures [19–21] provide intuitive pictures of the recombination process.
In 2011, Ghimire et al. [1] experimentally observed non-perturbative HHG in ZnO crystal using mid-infrared few-cycle laser pulses. They suggested Bloch oscillation as the source of HHG in ZnO, which would occur due to the reflection of the electron and hole at the Brillouin zone boundaries. Subsequently, Luu et al. [5] applied a subcycle synthesized field with bandwidths that cover the ultraviolet and near-infrared spectral regions to a silicon dioxide crystal and experimentally generated coherent EUV radiation with photon energy of about 40 eV. In their work, the non-resonant oscillation of electrons under an external field was accounted for high-frequency radiation. Almost at the same time, Vampa et al. [7] investigated the process of HHG in ZnO by adding a weak second-harmonic beam and experimentally proved that the interband current generated by the generalized recollision between an electron and its associated hole was the main mechanism for the HHG in ZnO crystals under the mid-infrared laser field. In addition, You et al. [22] measured strongly anisotropic HHG in MgO crystal experiments, indicating that the all-optical method can be used to extract information such as crystal structure, interatomic potential, valence electron density and even wave function. A series of experimental observation of non-perturbative HHG from two-dimensional materials have also attracted considerable attention because of the contrasting behavior in harmonic yield and polarization, such as graphene [23,24] and transitional metal dichalcogenides [25–28], suggesting distinct generation mechanism. These experiments reveal how crystal symmetry governs the process of solid-state HHG.
In order to understand the mechanism of HHG in solid, researchers have proposed a series of theoretical methods [29–48]. Korbman et al. [29,30], Wu et al. [31] and Guan et al. [32] numerically solved the one-dimensional time-dependent Schrödinger equation (TDSE) to simulate the high harmonic spectra in solids. Then, Jin et al. [33], Li et al. [23] and Li et al. [47] solved the two-dimensional TDSE based on model potentials to study the isotropic characteristics of solid HHG. Besides, Vampa et al. [18,35,36] proposed to simulate high-harmonic radiation in solids by solving the density matrix equations. Remarkably, by solving two-band semiconductor Bloch equations (SBEs) under the external laser field [37–41], parts of experimental phenomena can be successfully reproduced and well explained. Hohenleutner et al. [6] improved this theoretical model to analyze the interaction between intense laser and crystal by solving the multiple-band SBEs. It should be noted that time-dependent density-functional theory (TDDFT) calculation can also study the HHG in solids, although its computational cost remains challenge. For instance, based on TDDFT, Tancogne-Dejean and Rubio [42] explored the atomic-like HHG in monolayer h-BN, and Yu et al. [48] further proposed a backscattering assisted atomic-like HHG in bilayer h-BN. Recently, Dasol et al. [49] investigated the dependence of the HHG on the laser polarization angle, ellipticity of the driving laser and interlayer coupling through SBE simulations in h-BN monolayer and bilayers, revealing that high-order harmonic radiation strongly depends on crystal symmetry and interlayer interaction.
Solid-state harmonics are emerging as a potential approach to compact, table-top attosecond light sources. How to further increase the conversion efficiency of HHG has always been the focus in this research field. Li et al. [50] theoretically optimized the efficiency of the second plateau of solid high harmonic spectra through the two-color laser technology. They presented a promising method to generate isolated attosecond pulses from solids by selecting the appropriate parameters of two-color lasers. Also, Liu et al. [51] effectively enhanced the harmonic emission by about one order of magnitude by combining nanofabrication technology and ultrafast strong-field laser technology. Recently, Franz et al. [52] found that nanostructured semiconducting waveguides can greatly improve the intensity of HHG in solid.
In this paper, taking bilayer h-BN as an example, the characteristics of high-order harmonics radiated in different stacking structures are studied. Here, two typical stacking configurations (AA and AA’ as named in Ref [53].) were considered. Numerical simulations show that the spatial symmetry of bilayer will be broken by interlayer atomic dislocation, which makes part of interband transitions that are forbidden originally become allowed and gives a rise to extra interband excitation and recombination channels. Thus, the carrier transition probability and the high-order harmonic efficiency are significantly improved. Furthermore, we purpose to utilize these enhanced harmonics to generate single intense attosecond pulse by dynamically controlling the carrier envelope phase (CEP) of the driving laser.
2. Theoretical model
Our analyses of the laser-crystal interaction are based on the solving of the extended multiband SBEs [54]. We consider a linearly polarized laser field propagating through crystal along the optical axis. In the independent-particle approximation, the multiband SBEs read (atomic units are used throughout the paper, unless stated otherwise) [4,6,54]
Here, $f_{\boldsymbol k}^e$ and $f_{\boldsymbol k}^h$ are the populations of electrons and holes, respectively. We assume that electrons are initially filled in valence band. $p_{\boldsymbol k}^{\lambda \lambda ^{\prime}}$ is the microscopic interband polarization between CBs and VBs, where λ = e, h is the index that specifies either an electron or a hole. T1 represents the phenomenological damping of the antisymmetric part of the carriers. T2 is the dephasing time. $\mathrm{{\cal E}}_{\boldsymbol k}^e$= $\mathrm{{\cal E}}_{\boldsymbol k}^c$ and $\mathrm{{\cal E}}_{\boldsymbol k}^h$= $- \mathrm{{\cal E}}_{\boldsymbol k}^v$ are k-dependent energy of the corresponding carriers in CBs and VBs. The energy bands and transition dipole elements ${\boldsymbol d}_{\boldsymbol k}^{\lambda \lambda ^{\prime}}$ were calculated by using the Vienna Ab-initio Simulation Package code [55–57]. Geometric optimizations of bilayer h-BN crystals in AA-stacking and AA'-stacking were done within generalized gradient approximation in the parametrization of Perdew-Burke-Ernzerhof [56]. The energy cutoff was set to be 500 eV. A Monkhorst-Pack mesh of 25 × 25 × 1 k-points was used in Brillouin zone for geometry optimizations and electronic structure calculations. Geometry structures were fully relaxed until the convergence criteria of energy and force are less than 10−5 eV and 0.01 eV/Å, respectively. The accurate band dispersions and transition dipole elements were calculated using the HSE06 hybrid functional. Then, the energy bands were fitted by a function ${\mathrm{{\cal E}}_n}(k )= \mathop \sum \limits_{i = 0}^\infty {\mathrm{\alpha }_n}(i )cos({ika} )$, where a is the lattice constant along the laser polarization direction.
The total time-dependent macroscopic interband polarization P(t) and intraband electric current J(t) are given by
3. Results and discussion
The schematic configurations of bilayer h-BN in AA-stacking (eclipsed with N over N and B over B) and AA'-stacking (eclipsed with B over N) are shown in Figs. 1(a) and 1(b), respectively. After geometric optimizations by high level first-principles calculations, both lattice constants of AA-stacking and AA'-stacking are found to be near 2.51 Å. Moreover, the interlayer spacing of AA-stacking is found to be 3.41 Å, which is larger than 3.11 Å for the most stable AA'-stacking bilayer h-BN. Besides, the energy bands in Γ–M direction of AA- and AA'-stacking bilayers are shown in Figs. 1(d) and 1(e), respectively. We note that the harmonic efficiency of HHG in Γ–K direction is much lower than that in Γ–M direction, especially in high energy region [54]. Therefore, only the energy bands in Γ–M direction were considered in this work. Here, both band dispersion and band gap are similar for these two h-BN bilayers, where AA-stacking (AA'-stacking) shows a band gap of 6.18 eV (6.32 eV). We set the Fermi energy levels of both AA-stacking and AA'-stacking materials as 0 eV (dashed lines in Figs. 1(d) and (e)).
Fig. 1. Top views (up) and side views (down) of bilayer h-BN for (a) AA-stacking and (b) AA'-stacking. (c) First Brillouin zone of bilayer h-BN and high symmetric points Г, M, and K in reciprocal space. Energy bands of bilayer h-BN for (d) AA-stacking and (e) AA'-stacking in Γ–M direction.
Figure 2(a) presents the high harmonic spectra from bilayer h-BN in AA-stacking (blue area) and AA'-stacking (red area) with six VBs and ten CBs used in simulation. The driving external field used here is a mid-infrared laser with the wavelength of 1600 nm and the full width at half maximum (FWHM) of 10 optical cycles. The maximum amplitude of the laser field is 0.012 a.u. (i.e., the peak intensity of laser is about I = 5.0 × 1012 W/cm2), which is much lower than the damage threshold of h-BN under such a short pulse irradiation. Surprisingly, the harmonic efficiency of AA'-stacking bilayer h-BN is above one order of magnitude higher than that of AA-stacking bilayer h-BN in high energy region > 30 eV (> the 39th harmonic order). Usually, the high energy harmonics in solids come from the higher CBs [58]. In Figs. 2(b) and 2(c), when two VBs (v3, v4) and four CBs (c1, c2, c3, c4) were included in SBE calculations, we found that the high energy harmonics disappear as expected and the harmonic spectra are very similar for two stacking cases. However, if two extra CBs (c5, c6) were added, the simulation results shown in Fig. 2(c) (red area) are almost close to the total high harmonic spectra in Fig. 2(a) (red area). Therefore, the c5 and c6 CBs dominate harmonic emission in high energy region especially for AA'-stacking bilayer h-BN. In both Figs. 2(b) and 2(c), even though the two VBs (v1,v2) are absent, the spectra (red area) is almost same to the results calculated with four VBs(v1,v2,v3,v4) and six CBs (black line) in high energy region. We could come to a conclusion that the two VBs (v3, v4) and the two CBs (c5, c6) are heavily involved in the generation process of high-frequency composition. Owing to the band dispersions are similar in these two stacking bilayers shown in Figs. 1(d) and 1(e), we expect that the transition dipole moment would play a key role in harmonic enhancement in AA'-stacking case. It should be noted that we also simulated the intraband contributions from all involved bands. However, the total intensity of intraband HHG in high energy region is found very small.
Fig. 2. Simulated high harmonic spectra (a) from AA-stacking bilayer h-BN and AA'-stacking bilayer h-BN with six VBs and ten CBs, (b) from AA-stacking bilayer h-BN and (c) from AA'-stacking bilayer h-BN with four VBs and six CBs (black line), two VBs (v1, v2) and six CBs (blue area), two VBs (v3, v4) and six CBs (red area), two VBs (v3, v4) and four CBs (yellow area). The used laser parameters: wavelength of 1600 nm, FWHM of 10 optical cycles, and intensity of 0.012 a.u. (about I = 5.0 × 1012 W/cm2).
To better understand the physics behind the harmonic enhancement in AA'-stacking bilayer, the transition dipole moments among two VBs (v3 and v4) and two CBs (c5 and c6) are shown in Figs. 3(a) and 3(b) for AA-stacking and AA'-stacking, respectively. For AA-stacking bilayer h-BN, two layers are mirror symmetric with each other. The interaction between the two layers are very weak, so electrons in one layer can hardly transit to the other. As a result, only two transition dipole moments (v3–c6, v4–c5) dominate in Fig. 3(a) which correspond to the transition dipole moments inside each layer. However, with broken mirror symmetry in AA'-stacking, electrons have the opportunity to transit from one layer to the other layer. Hence, other two transition dipole moments v3–c5 and v4–c6 corresponding to the interlayer excitation channels are obviously increased in Γ–M direction. Therefore, compared to AA-stacking bilayer h-BN, we believe that the enhancement of harmonic radiation in AA'-stacking bilayer h-BN originates from the extra transition channels of the carriers.
Fig. 3. The transition dipole moments among two VBs (v3, v4) and two CBs (c5, c6) for (a) AA-stacking (b) AA'-stacking bilayer h-BN.
In Fig. 4, the wavelength dependent harmonic yields are shown, which were obtained by integrating the harmonics in high energy region from 26 eV to 50 eV. With the increase of the laser wavelength, we find that the harmonic yield is gradually decreased, apparently for both AA-stacking and AA'-stacking bilayer h-BN. This is mainly due to that the longer laser wavelength corresponds to a smaller photon energy, and therefore, the electron needs to absorb more photons for transition from VBs to CBs with much lower transition probability. The results in Fig. 4 also indicate that the harmonic yield in AA'-stacking bilayer h-BN keeps significantly higher than that in AA-stacking bilayer h-BN as the laser wavelength less than 4500 nm. However, when the wavelength of driving laser is larger than 4500 nm, the harmonic yields of these two stacking bilayers tend to be close in magnitude, and the intraband current dominates the HHG for such long wavelengths instead of interband polarization.
To investigate the ultrafast dynamics of electrons in bilayer h-BN crystal, we restrict the FWHM of the laser field to one optical cycle. The dynamic control of harmonic emission by CEP manipulation of the driving laser for AA-stacking and AA'-stacking bilayer h-BN is shown in Fig. 5. The simulated harmonic spectra exhibit a strong sensitivity to the CEP of the laser such that the harmonic shift both in AA-stacking and AA'-stacking bilayer h-BN when the CEP is changed, which originates in the atto-chirp of the harmonic radiation [59]. Besides, the enhancement of harmonic radiation in AA'-stacking bilayer h-BN is still reserved in high energy region above 25 eV (> the 35th harmonic order) and a supercontinuum harmonic spectrum can be found with appropriate CEP.
At last, we intend to use the harmonics in high energy region in Fig. 5 to generate attosecond pulses by directly filtering out the harmonics below 25 eV. As shown in Fig. 6(a), only weak attosecond pulse train with low intensity can be obtained from AA-stacking bilayer h-BN. However, the strongest isolated 180-as pulse can be achieved by controlling the CEP of laser around 0.65 π from AA'-stacking bilayer h-BN shown in Fig. 6(b). To investigate the temporal structure of HHG for AA'-stacking bilayer h-BN with this CEP, we perform time-frequency analysis by using the wavelet transformation of the total intraband current and interband polarization [60,61], $A({t,{\omega }} )= \smallint [{J({t^{\prime}} )+ P({t^{\prime}} )} ]\sqrt {\omega } W[{{\omega }({t^{\prime} - t} )} ]dt^{\prime}$ where $W[{{\omega }({t^{\prime} - t} )} ]$ is the mother wavelet with the formula $W(x )= \frac{1}{{\sqrt {\tau } }}{e^{ix}}{e^{ - {x^2}/2{{\tau }^2}}}$, and τ = 6 in our calculations. The energy values of the cutoffs in the time–frequency distributions in Fig. 6(b) match with the harmonic cutoffs in Fig. 5(b). In particular, only single dominant emission can be found and the emission time of the photon also agrees with the time of attosecond pulse emission in Fig. 6(a).
Fig. 6. CEP dependent attosecond pulses from (a) AA-stacking and (b) AA'-stacking h-BN bilayers by directly filtering out the harmonics below 25 eV. (c) Time-frequency analysis of HHG in AA'-stacking bilayer h-BN with CEP = 0.65 π.
4. Conclusion
In summary, we theoretically studied the effect of stacking configuration on HHG from bilayer hexagonal boron nitride and found that the intensity of high-energy harmonics generated from AA'-stacking bilayer h-BN is much higher than that of AA-stacking bilayer h-BN. Due to the broken mirror symmetry in AA'-stacking, electrons are easier to transit from one layer to the other layer, which increases the excitation/recombination channels of the carriers and enhances the efficiency of HHG. In addition, we investigated the laser wavelength dependent harmonic yields and notice gradually close harmonic yields of two stacking structures in longer wavelength because of the dominant intraband contribution to the HHG. For application, we used the enhanced harmonics from AA'-stacking bilayer h-BN and got an intense isolated 180-as pulse by controlling the CEP of laser. Our results indicate the potential of sub-cycle control of electron dynamics in harmonic emission from bilayer crystals.
Funding
National Key Research and Development Program of China (2022YFA1604301); National Natural Science Foundation of China (12174195, 11974185, 11704187, 11834004); Fundamental Research Funds for the Central Universities (30920021153); China Postdoctoral Science Foundation (2019M661841).
Disclosures
The authors declare no conflicts of interest.
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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