Symmetry control of high-harmonic generation in bilayer hexagonal boron nitride

High-order harmonic generation (HHG) from atoms and molecules has provided effective light sources and sparked significant interest in both experimental and theoretical domains of attosecond science.^1–3 This technology facilitates the extension of techniques used in attosecond science to condensed matter systems.^4–6 The emergence of HHG from solids in attosecond science and materials physics presents a substantial opportunity for developing compact attosecond light sources and has been utilized in ultrafast spectroscopy to detect rapid electronic dynamics. Given that different solids possess unique band structures, the study of HHG in solids unveils diverse characteristics of interband and intraband electronic dynamics, which are critical for understanding solid-state HHG.⁷ However, comprehending solid-state HHG still remains challenging, as the atomic three-step model does not fully encapsulate the complex interband and intraband responses of solids.

The mechanism of solid-state HHG is primarily governed by the electronic dynamics associated with interband and intraband processes. The interplay between these two types of dynamics is complex, typically exhibiting either competitive or cooperative behaviors. Thus, a thorough investigation of their collective contributions to HHG is essential. Notably, two-dimensional (2D) materials have emerged as promising candidates for exploring HHG mechanisms. Their atomic thickness, nonlinear optical properties, and exceptional performance under ultrashort intense laser pulses make them particularly suitable. These characteristics offer significant potential for effective HHG modulation.

Among these materials, hexagonal boron nitride (hBN) has become a highly sought-after 2D transparent insulator, favored for its wide indirect bandgap, high damage threshold, and robust thermal, chemical, and mechanical stability. These characteristics establish hBN as an ideal material for applications in the mid- and far-ultraviolet regions. Over the past few years, numerous studies on HHG in hBN have been conducted, addressing phenomena such as interlayer coupling,⁸ rotational anisotropy,⁹ time–frequency behavior,¹⁰ and wavelength scaling of harmonics.¹¹ Recent research on monolayer hBN has revealed that its phonon classical modes are sensitive to the carrier-envelope phase (CEP), opening up new possibilities for multi-dimensional spectroscopy measurements of phonon motion. Moreover, the atomic thickness of monolayer hBN provides a unique opportunity to explore both bulk and atomic HHG by adjusting the angle of incidence, which serves as a control parameter to toggle between the two extremes of in-plane and out-of-plane electric fields.

Furthermore, recent discoveries have shown that optimizing double- or multilayer nanostructures can simultaneously control and enhance the harmonic generation in 2D hBN materials, while also managing the emission of atomic-like harmonics.^12,13 For instance, a notable correlation has been found between the cutoff energy of higher harmonics and the spacing between material layers.¹⁴ Du et al. demonstrated that employing two-color lasers can significantly enhance the plateau of the harmonic spectrum, increasing it by two to three orders of magnitude.¹⁵ In addition, the efficiency of harmonic conversion in multilayer hBN can be significantly enhanced due to the effect of the coupling mechanism.¹⁶ Simulations indicate that in both monolayer and bilayer hBN, harmonic emission is intricately linked to the crystal inversion symmetry, rotational symmetry, and interlayer interactions. Given hBN’s high damage threshold, mechanical strain provides another way to control and optimize harmonic outputs. Typically, stretching strain improves harmonic yields because higher excitations can be achieved with a lower bandgap under strain. Novel experimental techniques have also been developed, enabling hBN to be stacked in various configurations, including the notable magic angle hBN, which has attracted considerable attention. A recent study has shown that the AB-stacking order is the most stable among the five highly symmetric stacking configurations, applicable to both bilayer and bulk systems.¹⁷ 

In this study, utilizing monolayer and AB bilayer hBN, we investigate the impact of laser field symmetry on the selection rules for high harmonics by carefully adjusting key laser parameters, such as frequency and amplitude ratio.^18,19 In particular, we numerically solve the semiconductor Bloch equations (SBEs) for laser–matter interactions within the dipole approximation.^20,21 Our systematic investigation encompasses the selection rules, ellipticity dependence, and polarization properties of the high harmonics generated under a variety of laser fields.^22–24 

This paper is organized as follows: In Sec. II, we briefly present the framework of the SBEs and the numerical details. In Sec. III, we discuss the selection rules of HHG and its dependence on the laser parameters for monolayer and bilayer hBN. Finally, the conclusion is drawn in Sec. IV.

All the used lasers have ten fundamental cycles, and the basic intensity is B₀ = 1.2 × 10¹² W/cm². f(t) is a dimensionless envelope function with the cos² form. In total, the laser fields used in this work can be divided into three categories, according to the laser symmetry defined by Floquet group theory in Ref. 29. To properly address the amplitude ratio of each fundamental laser component, we introduce dimensionless parameters, A1 and A2, for convenience.

1. π-rotation laser fields with different frequency ratios and phases. In this work, π-rotation laser fields contain two representative types, i.e., the G and C₂ point groups, with equal amplitude ratio,

   $EG(t)=B0f(t)[(sin(2ωt)+cos(3ωt))x⃗+cos(ωt)y⃗]$

   (9)

   and

   $EC2(t)=B0f(t)[sin(ωt)x⃗+sin(3ωt+π/7)y⃗].$

   (10)
2. Counter-rotating bicircular (CRB) laser fields with different frequency ratios (ω₁/ω₂) or amplitude ratios (A₁/A₂) can be given by

   $ECRB(t)=B0f(t)(A1⁡cos(ω1t)+A2⁡cos(ω2t))x⃗+(A1⁡sin(ω1t)−A2⁡sin(ω2t))y⃗.$

   (11)

   In this work, we consider typical frequency ratios, including ω₂ = ω₁, 2ω₁, 3ω₁, 4ω₁, and 5ω₁, with the same amplitude ratio (A₁ = 1 and A₂ = A₁). Meanwhile, we consider typical amplitude ratios, including $A2=5A1$⁠, 5A₁, and 5$5A1$ with the same frequency ratio (ω₁ = ω₂). Using the laser pulse of ω₂ = 5ω₁ and $A2=55A1$⁠, we further discuss the effect of the rotational symmetry axis on the HHG selection rules.

3. We study the selection rules under two typical symmetrical laser fields, which belong to the C_5,2 and C₅ groups. These two selected kinds of groups are related to the symmetry group of the hBN, so different selection rules are expected. The following two typical lasers of the C_5,2 and C₅ groups are used for comparison:

   $EC5(t)=B0f(t)(2⁡cos(ωt)+cos(4ωt))x⃗+(2⁡sin(ωt)−sin(4ωt))y⃗$

   (12)

   and

   $EC5,2(t)=B0f(t)(cos(2ωt)+cos(3ωt))x⃗+(sin(2ωt)−sin(3ωt))y⃗.$

   (13)

In this study, we present the energy-band dispersion relations and the ground-state geometries for monolayer and bilayer hBN, as depicted in Fig. 1. Observing the top view of monolayer hBN, it is classified under the D_3h point group, exhibiting a threefold rotational symmetry and lacking inversion symmetry. The band structure of the monolayer is illustrated in Fig. 1(a). Figures 1(b) and 1(d) depict the band structure and geometry for the AB-stacked bilayer hBN, respectively. We adopt the AB stacking notation consistent with that used by Ribeiro and Peres³⁰ For the ground-state properties of both monolayer and bilayer hBN, the band structures are in agreement with those presented in Ref. 17. It is noteworthy that the in-plane symmetry of the AB bilayer closely mirrors that of the monolayer hBN, yet the AB bilayer lacks a mirror plane in the out-of-plane direction. Consequently, even without the application of out-of-plane lasers, it is anticipated that both monolayer and AB bilayer hBN will exhibit identical symmetry configurations in our analysis.

The selection rules and yield of the HHG are proven to be related to the material’s symmetry as well as the spatiotemporally symmetrical laser fields.¹⁷ With both linear and circular polarizations, we find that the HHG yields of the monolayer and AB bilayer hBN show different maximum intensities in different directions. The y component of the even harmonics has a maximum in the Γ–K direction and the x component has a maximum in the Γ–M direction, which hold for both even and odd harmonics. Due to the mirror symmetry, both monolayer and AB bilayer hBN show vanished even harmonics in the Γ–M direction. For circularly polarized lasers, the selection rule can be seen as the combination of the symmetry of the lasers and the materials, i.e., in this case, it contains 3k ± 1 selection rules. To study the selection rules, we start with the π rotation rules,^14,31,32 which usually contain the symmetry of C₂, G, Q, etc. In this work, we pick up two typical symmetries of the incident laser for comparison. One is the C₂ symmetry, and the other is the G symmetry.

For convenience, we set the rotation axis of the laser fields of C₂ and G symmetries along the z axis, perpendicular to the xy plane. Note that the two in-plane lasers are the same with ten cycles of 1.2 × 10¹² W/cm² at 1.9 μm fundamental wavelength. We proved that, for left and right rotation lasers, the HHG is the same, indicating the symmetry of the rotation symmetry. Figures 2(a) and 2(b) show the HHG of the monolayer and bilayer, respectively. For both G and C₂ symmetries, frequency combs are observed. One can clearly identify even and odd harmonics, manifesting no symmetry effect as a function of layer stacking. For the bilayers with laser fields of G and C₂ symmetries, the time–frequency analysis of the total harmonic emission is shown in Figs. 2(c) and 2(d). According to the Lissajous curves of lasers, the HHG shows correlations with the maximum value and profile of the lasers. For a more comprehensive analysis, we decompose the total harmonics into the x and y components (not shown). For the G symmetry, we find that the HHG yield in the x direction is more pronounced than that in the y direction, while for the C₂ symmetry, the y component is larger than the x component. This finding is expected, because the laser polarization of the C₂ symmetry is along the Γ–M direction, which leads to the disappearance of even harmonics and then makes the y component more pronounced. Moreover, as shown in Figs. 2(a) and 2(b), the laser fields of the C₂ symmetry are more obvious in the selection of odd harmonics. This also belongs to the π rotation rule. For the G symmetry, the vector potential in the x direction becomes dominant and stronger in the Γ–K direction so that near-uniform even harmonics appear with selection rules of k ± 1. The broken symmetry in the z-direction for the AB bilayer does not impact the selection rules of G and C₂ symmetries, while the bilayer enhances the HHG yields.

The differences of the selection rules for the C₂ and G symmetries are partly attributed to the different laser orientations and inversion symmetries. In the following, we get more insights into these effects in two aspects. With fixed inversion symmetry of the laser fields, we tune the laser orientations by changing different ratios between ω₁ and ω₂ in Eq. (11). To this end, for the CRB lasers, we pick up three typical parameters, i.e., ω₂ = ω₁, 2ω₁, and 5ω₁. In Figs. 3(a)–3(d), we present the HHG for the monolayer and bilayer hBN, with the frequency ratios of ω₂ = ω₁, 2ω₁, and 5ω₁. With the increase in ω₂ from ω₁ to 5ω₁, the HHG intensity gets larger. Observe that the selected HHG orders for the AB bilayer are the same as those of the monolayer and the HHG intensities for the AB bilayer are larger than those of the monolayer. The reason is that the layer stacking does not affect the HHG order when the z axis is perpendicular to the materials and laser polarization.³³ However, detailed information, for example, more splits, can be observed in the bilayer. Further comparisons are made in Figs. 3(e) and 3(f). One can see that the electronic excitation in the bilayer is more intense than that in the monolayer (e.g., more obvious hole patterns in the bilayer case). In Fig. 3(g), we plot the ellipticity of the third-order HHG to clarify the effect of the frequency ratios. It shows the similar tendency for the cases of ω₂ = ω₁, 2ω₁, and 5ω₁, while for the case of ω₂ = 4ω₁, the ellipticity monotonically increases.

Based on the above results, we also clarify that other parameters, such as laser amplitude or orientation, are able to impact the harmonic emission. For the CRB lasers with the same amplitude ratio (i.e., A₁ = A₂), we verified that the laser amplitude does not affect the emergence of the harmonic order, while it impacts the HHG intensity (not shown), which is in consistent with other Refs. 34 and 35. For both left- and right-hand polarizations, with the increase in the laser amplitude from A₁ to $55A1$⁠, the HHG intensity enhances. Then, using the CRB with the fixed frequency ratio of ω₂ = 5ω₁ and fixed amplitude ratio of $A2=55A1$⁠, we study the orientation effect. The rotation angle and direction are illustrated in the insets of Fig. 4(b). As shown in Figs. 4(a) and 4(b), by rotating an angle of 90°, an inverse laser intensity can be found in the x and y directions, and accordingly, the harmonic intensity and selection rules have changed. Before the rotation, the incident laser intensity in the x direction increases and the HHG from the Γ–K direction is dominant so that the even harmonic orders appear. On the contrary, after the rotation, the incident laser intensity in the y direction increases and the HHG from the Γ–M direction is dominant. Therefore, the odd harmonic orders emerge. In addition, by tuning the left-rotation to the right, one can find a one-order harmonic shifting, manifesting the change of the selection rules.

We then focus on the inversion symmetry of the folded and rotational lasers. The rotation angle of the rotary axis remains constant and only changes the inversion symmetry of the laser field itself. To achieve this, we tune the ratio of ω₂ with respect to ω₁ to form C₅ and C_5,2 symmetries of the lasers. Note that the C₅ symmetry contains only fifth-order cyclic operations, with an angle of inversion symmetry of 72°. The C_5,2 symmetry has additional operations, such as the 72° and 144° inversion symmetries. As shown in Figs. 5(a) and 5(b), one can observe that the selection rules of the C_5,2 symmetry are 5k ± 2, while the C₅ selection rule is 5k ± 1, which also shows one-order difference. This suggests that the axis rotational angle and the inversion symmetry can manipulate the HHG orders. The HHG selection rules of the C_m,n symmetry can be manipulated by adjusting the spatial-symmetry term m and the time-symmetry term n. The term m controls the maximum intensity distribution in space, and therefore, it affects the order selection rule. The term n changes the HHG intensity in time, which actually decides the phase relations. Unlike the space term m, n has a significant impact on the polarization angle and the ellipticity of the HHG spectra, leading to the strong dependence of ellipticity and polarization.^36,37 In addition, polarization angle is defined as the intersection angle between the laser and the x axis, as shown in the insets of Fig. 5(b). For the bilayer hBN, Figs. 5(c) and 5(d) illustrate an obvious change in the polarization angle, which shows different angles of the maximum intensity.

In our study, we analyzed the impact of the dynamical symmetry on the HHG by solving the SBEs for both monolayer and AB bilayer hBN. We manipulated key parameters of the incident laser, such as polarization angle, ellipticity, and amplitude, to demonstrate and validate the selection rules. We discovered that the selection rules for the AB bilayer do not significantly differ from those of the monolayer. In most scenarios involving in-plane lasers, the harmonics generated in the bilayer closely resemble a superposition of harmonics from two independent monolayers. In addition, by examining the inversion symmetry of light fields at various rotation angles, we confirmed the selection rules, noting that certain harmonic orders are either suppressed or enhanced. In addition, we find that the shifts in harmonic orders and the selection rules under various lasers follow the laws of Floquet group theory as described in Ref. 29. In particular, the symmetry of the HHG profile is primarily determined by the symmetry of the Hamiltonian. Our findings could influence techniques for manipulating HHG. Furthermore, we anticipate that various standard optical fields could be employed in optical devices and multilayer systems, potentially useful for measuring lattice deformations.

We thank Xiu-Lan Liu in Peking University for fruitful discussions and manuscript revision. W.-D.Y. acknowledges the financial support from the National Natural Science Foundation of China under Grant No. 12104019 and the Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications (Grant No. NY223203).

The authors have no conflicts to disclose.

Zi-Yuan Gao: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal). Chen-Ni Wang: Formal analysis (equal); Investigation (equal); Writing – original draft (equal). Zhi-Han Huan: Formal analysis (equal); Investigation (equal). Wan-Dong Yu: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhi-Hong Yang: Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal). Yun-Hui Wang: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.