Orientation-dependent High Harmonic Generation from h-BN （Invited）

Wenyang ZHENG; Candong LIU; Ya BAI; Peng LIU; Ruxin LI

doi:10.3788/gzxb20245306.0653210

0　Introduction

High Harmonic Generation （HHG） presents an extremely nonlinear optical radiation resulting from the interaction of a strong laser field and matter. For many years， HHG in gaseous materials has been extensively studied and regarded as a vital tool for advancements in ultrafast science^{［1， 2］}. Recently， there have been reports of HHG occurring in various solid-phase materials， which holds promise as a compact ultrafast light source^［3-5］. Additionally， it has shown potential applications such as reconstructing crystal band structures^［6］， measuring Berry curvature^{［7， 8］}， and investigating topological phase transitions^［9-13］. The fundamentals of gaseous HHG can be explained by the semi-classical three-step model ^{［14， 15］}， and solid HHG is currently described mainly by the Semiconductor Bloch Equations （SBEs）^［16-18］.

Interference effects play a significant role in the solid-state HHG process. Various forms of interference have been observed and studied， including those between inter-band and intra-band processes^{［16， 19］}， those involving multiple energy bands^［20-23］， those in different momentum^{［24， 25］}， different half- periods^{［26， 27］}， and different electron trajectories^［28］. In a study by WANG X Q et al.^［19］， time-frequency analysis was used to investigate the interference effect between inter-band and intra-band currents in HHG from ZnO crystal. It was discovered that when the inter-band and intra-band currents are well synchronized， destructive interference dominates the harmonic spectrum. WU M X et al.^［21］ employed the Time-Dependent Schrödinger Equation （TDSE） and found that strong coupling between conduction bands leads to a weaker second plateau in the HHG spectrum. BIELKE L et al.^［24］ demonstrated that the dependence of cut-off energy on laser intensity is attributed to the interference of harmonics emitted by electrons in different initial states within the Brillouin zone. CAO J Y et al.^［26］ showed that the anomalous intensity variations of even-order harmonics in monolayer MoS₂ originate from the interference between the HHG burst from adjacent half-cycles of the driving field. KIM Y W et al.^［28］ reported the spectral splitting and boarding of a certain harmonic order that produced by quantum path interference between long and short electron （hole） trajectories.

In this paper， we investigate symmetry of HHG from monolayer h-BN using the two-band SBEs. We show that the yield of odd-order harmonics displays a periodicity of 60° as the azimuthal angle between the h-BN and driving field are varied. Specifically， an intriguing pattern in the orientation-dependent HHG is observed. The parallel component of the odd-order harmonics in the cut-off region exhibits an angular shift of 30° compared to the other orders. Analyse indicates that the angular shift of the modulation arises from the interference of different momenta and that of different polarity half-periods. This phenomenon holds significant potential for developing techniques to probe the energy band structure of solids through HHG.

1　Theoretical model

There is a growing interest in extending HHG into two-dimensional （2D） materials^{［7， 29， 30］}，due to the unique electronic and optical properties of monolayer crystals. The h-BN is a transparent insulator with a large indirect bandgap of about 4~6 eV^{［31， 32］}. Since its high damage threshold， h-BN is also an ideal material for studying solid HHG. Fig. 1（a） shows the hexagonal honeycomb structure of a single-layer h-BN crystal in real space. We study the polarization properties of HHG resulting from the interaction between a strong laser field and monolayer h-BN by solving the two-band SBEs^{［17， 33］}

\frac{\partial}{\partial t} p_{c, v} = (ε_{c} (k) + ε_{v} (k) - i \frac{ℏ}{T_{2}}) p_{c, v} - (1 - n_{c} - n_{v}) d_{k} E (t) + i e E (t) \nabla_{k} p_{c, v}

（1）

\frac{\partial}{\partial t} n_{c (v)} = - 2 I m [d_{k} E (t) p_{c, v}^{*}] + e E (t) ∙ \nabla_{k} n_{c (v)}

（2）

Figure 1.The structure and the energy dispersion of a monolayer h-BN crystal

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where $p_{c, v}$ is the inter-band polarization between the conduction and valence band， $n_{c (v)}$ the occupation of electrons （holes）， $ε_{c}$ and $ε_{v}$ are the energies of the conduction and valence bands， respectively； $E (t)$ is the laser electric field， and $d_{k}$ is the transition dipole matrix elements. We consider the nearest neighbor tight binding model， whose Hamiltonian for h-BN is^{［34， 35］}

H_{0} = (\begin{matrix} \frac{Δ}{2} & γ f (k) \\ γ f^{*} (k) & \frac{- Δ}{2} \end{matrix})

（3）

f (k) = e x p (- i \frac{a k_{y}}{\sqrt[]{3}}) + 2 e x p (- i \frac{a k_{y}}{2 \sqrt[]{3}}) c o s (\frac{a k_{x}}{2})

（4）

where， $Δ$ is the minimum bandgap of h-BN， $γ = - 2.33 e V$ is the is the hopping integral between the nearest adjacent atoms， and $a = 0.251 n m$ is the lattice constant^［35］. The characteristic energy of the tight-binding Hamiltonian $H_{0}$ can be obtained by the following formula

E_{\pm} (k) = \pm \sqrt[]{γ^{2} {|f (k)|}^{2} + \frac{Δ^{2}}{4}}

（5）

where $+$ and $-$ represent Conduction Band （CB） and Valence Band （VB）， respectively. Fig. 1（b） shows $Δ = 5 e V$ calculated energy dispersion of the bandgap.

The current generated by ultrafast field driving is $J (t) = \{J_{x}, J_{y}\}$ ，and there are inter-band and intra-band contributions $J (t) = J^{i n t e r} (t) + J^{i n t r a} (t)$ . The inter-band polarization and intra-band current respectively

J^{i n t e r} (t) = \int d_{k} p_{c, v} d^{2} k + c . c

（6）

J^{i n t r a} (t) = \int [n_{c} \nabla_{k} E_{+} + n_{v} \nabla_{k} E_{-}] d^{2} k

（7）

$\nabla_{k} E_{λ}$ is the group velocity， which is proportional to the derivative of the energy band， $λ = \pm$ . In the simulation， the center wavelength of driving light is 2.9 μm （ $ω_{0} = 0.43 e V$ ）， the Full Width at Half Maximum （FWHM） of the pulse is 60 fs， the peak power is 1 TW/cm²， and the dephasing time T₂=2 fs.

2　Results and discussion

We calculated the polarization resolved high harmonics as a function of the crystal orientation， by solving the two-band SBEs. As shown in Fig. 2（a）， the inter-band HHG polarized parallel to the laser polarization（HHG parallel components） are modulated with a 60° periodicity， and this modulation is determined by the crystal symmetry. The orientation angle of 0° is defined as the pump pulse polarized along the zigzag direction of the crystal in real spaces. The colour bar represents the harmonic yield on a log scale. For the parallel components， we find that the intensity modulation of the even-order harmonics is synchronized. Whereas， an asynchronous orientation dependence of odd-order harmonics appears at the cut-off region. The intensity modulation has a 30° angular shift between the 17th harmonic（H17） and other odd-orders， i.e.， the other odd-order harmonics intensity is enhanced when the driving field is polarized along the zigzag direction， vice versa for the H17.

Figure 2.The high harmonics as a function of the crystal orientation

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The inter-band HHG polarized perpendicular to the laser polarization（HHG perpendicular components） of even-order harmonics relative to the odd-order harmonic dominates in Fig. 2（b）， and exhibits the intensity modulation with a period of 60°. Moreover， there is a 30° angular shift between the modulation of the perpendicular and parallel components of the even-order harmonics， the breaking of the inversion symmetry of the phase difference of the momentum matrix elements along the two orthogonal directions leads to the generation of even harmonics perpendicular components， and the result is consistent with these symmetry restrictions on the HHG response in non-centrosymmetric materials^{［7， 12， 30］}. Another obvious feature in Fig. 2（b） is that the perpendicular component of odd-order harmonics rather weak and their intensity is modulated with a period of 60°， and the single peak is split into two sub-peaks. The location of the split where the emission is suppressed occurs at the orientation angle ξ=60°×n. The angular shift phenomenon of the parallel polarized odd-order harmonic intensity modulation is also been found in the experiments and calculations of Bi₂Se₃ and MoS₂^{［36， 37］}. However， the effect of multi-band interactions on the symmetry of the HHG has been mainly considered in past studies. We found that there is also a significant angular shift phenomenon in the results of two-band SBEs.

In Fig. 3（a）， the intensity modulation of H17 is shown as the function of pump intensity. The colour bar represents the H17 normalized intensity on a linear scale. We found that this angular shift is independent of intensity of the driving pulse （0.5 to 1.5 TW/cm²）. The total inter-band harmonic spectrum of the driving light along the zigzag and armchair directions is shown Fig. 3（b）. The black dotted line shows the order-dependent intensity of the odd-order harmonics when the driving light along the zigzag direction. It can be seen that the harmonic spectrum has a sharp decrease （cut-off region） for harmonics above H17 when the driving field is along the zigzag direction. The harmonic intensities below the H17 are stronger in the zigzag direction than in the armchair direction， and the sharp decrease in the zigzag direction harmonics at the H17 results in the harmonic intensities being less than those in the armchair direction harmonics. Therefore， we believe that the angular shift of the intensity modulation of odd-order harmonic parallel components are related to the cut-off of the zigzag directional harmonic spectrum.

Figure 3.Intensity-dependence of the H17 and the high harmonics spectrum

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The H17 intensities generated by different momentum in reciprocal space are shown in Fig. 4 when the driving field is along the zigzag and the armchair direction of h-BN， respectively. The colour bar represents the H17 intensity on a linear scale. Moreover， we found that the harmonics are mainly generated in the momentum near $K (K^{'})$ ， which is consistent with the Landau-Zener tunneling theory^［38］ with the maximum probability of excitation of the minimum bandgap. As we can see from Fig. 4（a）， the H17 is generated mainly in the path along the $Γ - K^{'} - K - Γ$ in reciprocal space when the driving light is along the zigzag direction， and when the driving optical is along the armchair direction， the H17 is mainly generated in the path along the $K - K - K$ . The bandgap along paths in reciprocal space marked by the red lines in Fig. 4， we found the Γ-K'-K-Γ path， M momentum bandgap energy （k_x=0， $E_{g_{M}}$ =6.83 eV） is close to the harmonic energy （7.31 V） that the angular shift occurs. Therefore， we believe that the odd-order harmonic energy at which the angular shift appears in the intensity modulation of HHG is related to the bandgap.

Figure 4.The intensities of the H17 generated by different momentum and the bandgaps along paths in reciprocal space marked by the red lines

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Add a Gaussian bandgap with a peak value of $2 ω_{0}$ to the original bandgap near the M momentum channel， $E_{g 1} = 2 ω_{0} ∙ e x p (\frac{- {(k x - k x_{M})}^{2} - {(k y - k y_{M})}^{2}}{2 σ^{2}})$ ，（ $k x_{M}, k y_{M}$ ） are the coordinates of M points in the reciprocal space， and σ is the variance of Gaussian function， here we choose $\frac{π}{3 a}$ ， so that the variation of the energy band of the Γ-K'-K-Γ path occurs between the two minimum bandgaps as much as possible. The bandgap of the path of Γ-K'-K-Γ after the band change is shown in Fig. 5（a）， and parallel components of the high harmonics as a function of the crystal orientation after the increase and decreases of the bandgap is shown in Fig. 5（b），（c）. The colour bar represents the harmonic yield on a log scale. We find that the harmonic energy at which the intensity modulation undergoes an angular shift varies with the change of the bandgap near the bandgap local maximum， so that the angular shift of the odd-order harmonics is closely related to the bandgap of the M momentum.

Figure 5.The bandgap along Γ-K'-K-Γ paths and the parallel components of the high harmonics after the band change

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Time-frequency analysis of the high harmonic current generated by K and K' momentum are shown in Fig. 6（b），（c）， respectively. The colour bar represents the harmonic instantaneous yield on a linear scale. We find that H17 is generated mainly in the half-cycle of a particular polarity of the driving light， i.e.， the higher order harmonics of the K momentum are generated mainly in the positive half-cycle， and the higher order harmonics of the K' momentum are generated mainly in the negative half-cycle. This is due to the fact that along Γ-K'-K-Γ path， the bandgap energy of the M momentum （bandgap local maximum） between the minimum bandgap K and K' is 15 $ω_{0}$ （6.45 eV）< $E_{g_{M}}$ =6.83 eV<17 $ω_{0}$ （7.31 eV）， which leads to a large difference in the process of generating H17 harmonics in the laser half-cycle of different polarities.

Figure 6.Instantaneous electric field of the driving laser and time-frequency analysis of the harmonic current

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When the harmonics generated from two adjacent half-cycles of opposite polarity interfere with each other， the total high harmonic current is given by^［26］

J (s) = J_{p} (s) - J_{n} (s) e^{i s ω_{0} \frac{T_{0}}{2}}

（8）

where $J (s)$ is the high harmonic current of order s， $J_{p} (s)$ and $J_{n} (s)$ is the current generated in positive and negative half-cycle， respectively. $e^{i s ω_{0} \frac{T_{0}}{2}}$ stands for the phase difference results from the time interval between two adjacent half-cycles. The minus sign comes from the change in laser field direction. For odd-order harmonics， we have $- e^{i s ω_{0} \frac{T_{0}}{2}} = 1$ . In addition， for the K and K' momentum， the phase difference between them is shown in Fig. 7， this explains the constructive interference of odd-order harmonics of K and K' momentum. Thus， for odd-order harmonics of energy larger than the bandgap at the M momentum， $J_{1} (s) = J_{p_{K^{'}}} (s) + J_{n_{K}} (s)$ ， and $J_{2} (s) = J_{p_{K}} (s) + J_{p_{K'}} (s) + J_{n_{K}} (s) + J_{n_{K'}} (s)$ for odd-order harmonics of energies lower than the bandgap at the M momentum. Therefore， the energy of M momentum is closely related to the cut-off of harmonics. In conclusion， we believe that when the driving light is polarized along different directions， the momentum that contribute more to the total high harmonics in the reciprocal space are different， and the valley polarization of the energy band will cause the harmonics generated in the zigzag direction to weaken rapidly near the bandgap at M momentum and enter the cut-off region. And the rapid weakening of the harmonic energy in the cut-off region causes the angular shift phenomenon of the orientation-dependent of h-BN. We believe that the observation of the angular shift of the odd-order harmonics in cut-off region can developing techniques to probe the energy band structure of solids using high harmonics.

Figure 7.Phase difference between the harmonics of K and K' momentum channel

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3　Conclusion

In conclusion， we investigate the symmetry of the interaction between a strong laser field and a monolayer h-BN to generate high harmonics using the SBEs model， find the angular shift of odd-order harmonics， and analyze the influence of the interference effect in the process of harmonic generation. Our study demonstrates that the orientation-dependence of the high harmonics is closely related to the energy band structure， the odd-order harmonic intensity modulation will be angular shifted at the order that energy is close to the bandgap of M-momentum. The higher order harmonics being generated only at laser half-cycles of a particular polarity， and interference between the different half-cycles and interference between the different momentum results in an angular shift of the cut-off order intensity modulation. We believe that the interference phenomenon is prevalent in the process of HHG in solids， which has a certain improve to the utilization of high harmonic probing of the energy band structure of solids.

Category: Special Issue for Attosecond Optics

Received: Jan. 31, 2024

Accepted: Mar. 21, 2024

Published Online: Jul. 16, 2024

The Author Email: Candong LIU (cdliu@siom.ac.cn)

DOI:10.3788/gzxb20245306.0653210