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Interband resonant high-harmonic generation by valley polarized electron–hole pairs

Interband resonant high-harmonic generation by valley polarized electron–hole pairs ARTICLE https://doi.org/10.1038/s41467-019-11697-6 OPEN Interband resonant high-harmonic generation by valley polarized electron–hole pairs 1,2 1 1 3 3 3 Naotaka Yoshikawa , Kohei Nagai , Kento Uchida , Yuhei Takaguchi , Shogo Sasaki , Yasumitsu Miyata & 1,2 Koichiro Tanaka High-harmonic generation in solids is a unique tool to investigate the electron dynamics in strong light fields. The systematic study in monolayer materials is required to deepen the insight into the fundamental mechanism of high-harmonic generation. Here we demonstrated nonperturbative high harmonics up to 18th order in monolayer transition metal dichalco- genides. We found the enhancement in the even-order high harmonics which is attributed to the resonance to the band nesting energy. The symmetry analysis shows that the valley polarization and anisotropic band structure lead to polarization of the high-harmonic radia- tion. The calculation based on the three-step model in solids revealed that the electron–hole polarization driven to the band nesting region should contribute to the high harmonic radiation, where the electrons and holes generated at neighboring lattice sites are taken into account. Our findings open the way for attosecond science with monolayer materials having widely tunable electronic structures. 1 2 Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Institute for Integrated Cell-Material Sciences (WPI-iCeMS), Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan. Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan. Correspondence and requests for materials should be addressed to N.Y. (email: yoshikawa@thz.phys.s.u-tokyo.ac.jp) or to K.T. (email: kochan@scphys.kyoto-u.ac.jp) NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 1 1234567890():,; ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 esearch on generation of high harmonics (HHG) from Results atomic or molecular gases has produced coherent High harmonic spectra from monolayer TMDs. We excited the Rbroadband radiation in the extreme ultraviolet region monolayer TMDs by using mid-infrared pulses (0.26eV photon 1,2 −2 and attosecond pulses . Moreover, HHG has recently been energy, 1.7TWcm peak intensity) and detected HHG in the 3–12 reported from several solid-state materials .Non- near-infrared to ultraviolet energy region (see the Methods and perturbative HHG in solids can be achieved with a driving laser Supplementary Figure 1). Figure 1 shows the high harmonic 12 −2 with a peak intensity as low as 10 Wcm , which is much spectra from fifth to sixteenth order of (a) MoSe , (b) WSe , (c) 2 2 smaller than that required in the gas phase, suggesting MoS , and (d) WS monolayers (see the Methods and Supple- 2 2 the potential for a compact, highly efficient attosecond mentary Figure 2 for the full spectrum up to 18th harmonics). We light source. In addition, solid-state HHG can capture the set the polarization of the mid-infrared pulses in parallel with the electronic properties of materials as tomographic imaging of zigzag direction (M–M(M=Mo, W) direction) (see Supplemen- molecular gases has been demonstrated by high-harmonic tary Note 1). The polarization of the even-order harmonic 13–15 spectroscopy . High harmonic spectroscopy in solids shows radiation was perpendicular to that of the incident laser light, the possibility of probing the electronic band structure as it while the polarization of the odd-order harmonics was parallel 3,4,7,10,11 23 relates to crystal symmetry and interatomic bonding .A (Supplementary Figure 3a) . The excitation power dependence unified understanding of the mechanism of HHG in solids is shows the nonperturbative nature of HHG from monolayer necessary for the application of high harmonic spectroscopy. TMDs (see Supplementary Note 2). Although several theoretical models have been proposed for solid-state HHG in terms of interband polarization and intra- Interband resonant enhancement of HHG. Figure 2a–d shows 3–11,16–19 band electron dynamics as well as their interplay ,the the polarization-unresolved emission intensities of the even-order underlying mechanism is still under debate. It will be important harmonics, and Fig. 2e–h the optical absorption spectra. In the to gain an understanding of HHG in single-atomic-layer solids, absorption spectra, the peaks labeled A and B are attributed to so- because propagation effects such as the phase matching con- called A and B excitons, consisting of electrons and holes loca- dition, which would otherwise obscure intrinsic features of lized in the K valleys in momentum space. In addition, the band HHG in bulk crystals, do not affect ideal two-dimensional structures of the monolayer TMDs have a band nesting region, systems. 30,31 causing van Hove singularities in the joint density of states . 20–22 HHG from monolayer graphene was recently reported , and its observed anomalous ellipticity dependence was able to be reproduced by a fully quantum mechanical theory in which both intraband and interband contributions are included . MoSe 4 2 HHG was also observed from monolayer MoS ,which is one of the monolayer transition metal dichalcogenides (TMDs) . 2 11 Monolayer TMDs, which have a finite bandgap in contrast to 10 8 13 12 15 graphene, have attracted much attention for their exotic properties such as enhancement of luminescence derived from 0 their indirect-to-direct bandgap transition from bulk to 24,25 26 monolayer form , extremely large exciton binding energy , 10 WSe b 7 2 and valley pseudospin physics arising from inversion symmetry 27,28 breaking and a large spin-orbit interaction .Liu et al. found 10 12 that the even-order high harmonics are predominantly polar- 10 ized perpendicular to the pump laser and attributed this to the intraband anomalous transverse current arising from the material’s Berry curvature . On the other hand, a recent experiment with an optical pump showed that the interband MoS 5 c 10 2 contribution dominates the intraband one in bulk ZnO .Liu et al. also showed that the even orders of HHG are enhanced at 2 12 much higher energies compared with the lowest exciton 10 or bandgap energy, while the odd orders are monotonically suppressed at higher energies. The enhancement at higher 0 orders and its relation to the perpendicular polarization of even-order HHG is not yet understood. 10 WS d In this study, we systematically investigated HHG from four kinds of monolayer TMDs with different bandgap energies (MoSe ,WSe ,MoS ,and WS )toexamine theresonance 10 2 2 2 2 8 12 effect and polarization of HHG in monolayer TMDs. By 13 14 comparing the HHG and optical absorption spectra of the monolayer TMDs, we found that HHG is enhanced when it is in resonance with the optical transition due to band nesting, 2.0 3.0 4.0 indicating that the interband polarization mainly contributes Photon energy (eV) to the even-order HHG. A simple calculation that is an extension of the three-step model for electron–hole pairs in Fig. 1 High harmonic generation from monolayer TMDs. High harmonic solids shows that the electron–hole pairs driven to the band spectra from a MoSe , b WSe , c MoS , and d WS monolayers at room 2 2 2 2 nesting region significantly contributes to the resonance temperature induced by mid-infrared pulse excitation (photon energy −2 enhancement, and anisotropic driving of electron–hole polar- 0.26eV, peak intensity 1.7TWcm ). We set the excitation polarization in izations around the K and K’ points determines polarization parallel with the zigzag direction. The harmonics from fifth to sixteenth selection rule of HHG. order are labeled 2 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications Intensity (arb. units) NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE Harmonic order Harmonic order Harmonic order Harmonic order 8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14 ab c d 1 MoSe WSe MoS WS 2 2 2 2 ef g h C C B B A B 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 Photon energy (eV) Photon energy (eV) Photon energy (eV) Photon energy (eV) Harmonic order Harmonic order Harmonic order Harmonic order 5 7 9 11 13 15 5 7 9 11 13 15 5 7 9 11 13 15 5 7 9 11 13 15 ij k l –2 –4 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 Photon energy (eV) Photon energy (eV) Photon energy (eV) Photon energy (eV) Fig. 2 Resonant enhancement of even-order high harmonics. Intensities of even-order harmonics in a MoSe , b WSe , c MoS , and d WS monolayers. 2 2 2 2 Error bars represent the standard deviations. e, f, g, h Corresponding optical absorption spectra of the TMD monolayers. The peaks of the A and B excitons are labeled A and B, respectively. The peaks due to the band nesting effects are labeled C and D. i, j, k, l Corresponding intensities of odd-order harmonics The band nesting results in strong optical absorption at higher regime, therefore, electron–hole pairs are generated coherently by 32–34 photon energies (labeled C and D in the figure) . As shown in Zener tunneling at the K and K’ points in the two-dimensional Fig. 2a, the intensity of the 10th harmonic in monolayer MoSe is hexagonal Brillouin zone. We will assume that electron–hole pairs larger than that of the 8th, and the intensity of the 14th harmonic are created when the electric field of the incident light reaches is larger than that of the 12th. Interestingly, the energies of the positive or negative maxima. Since the generated electrons and 10th and 14th harmonics coincide with the absorption peaks C holes are accelerated differently in the conduction and valence and D. This is not only the case in MoSe . Figure 2b–d also shows bands, their motions are asymmetric in real space (Fig. 3c). When enhancements at the band nesting energies for WSe , MoS , and the electron and hole meet again in real space (the circle in 2 2 WS . The observed enhancement in resonance with the interband Fig. 3d), electron–hole recombination occurs and HHG is emit- optical transition strongly suggests that the interband contribu- ted. The electron–hole pairs created at the positive peak field (t = tion is dominant in even-order HHG. On the other hand, the 0 in Fig. 3d) under excitation with a polarization along the zigzag intensities of the odd orders monotonically decrease with direction (Fig. 3a) have anisotropic driving processes: those cre- increasing order as shown in Fig. 2i–l. However, though it is not ated at the K point are driven in the K – Γ – K′ direction, while so apparent, the shoulder-like structure appears at the C and D those created at the K’ point are driven in the K′ – M – K transitions. This could suggest that odd-order harmonics are direction. This anisotropy in turn causes anisotropic acceleration contributed by both intraband process, which may show no and recombination dynamics in the K and K’ bands (0<t<T=2, resonant enhancement around interband transitions, and inter- where T is the period of the incident light). The dynamics of the band process, which should show the resonances. The experi- electron–hole pairs generated at the negative peak field (t = T/2) mental data imply that, at lower energy region, especially below are the reverse of those generated at the positive peak field the bandgap energy, the intraband contribution is much larger ðÞ T=2<t<T . On the other hand, when the excitation light is than the interband one for the odd-order harmonics, but the polarized along the armchair direction, the electron–hole pairs interband polarization may substantially contributes it at the generated at the positive peak field and those generated at the resonance energies. negative peak field have the same dynamics in the K and K’ bands (Fig. 3e). Next, we show how the anisotropic and alternating nature of Polarization selection rules of HHG. The resonant enhancement the electron–hole dynamics gives rise to a polarization selection of HHG indicates that the interband polarization mainly con- rule for HHG. The electric field of HHG is the sum of the left- tributes to even-order HHG. The question is how the perpendi- and right-hand circular polarized light: cular polarization arises in the even-order harmonics. The HHG HHG þ HHG transverse current due to a material’s Berry curvature might not E ðÞ t ¼ E ðÞ t σ þ E ðÞ t σ ð1Þ contribute to interband HHG. Here, we will discuss the pffiffiffi mechanism of HHG in monolayer TMDs on the basis of a semi- HHG HHG HHG ± where E ¼ E ±iE = 2 are the amplitudes of the σ ± x y classical three-step model in solids. The discussion is funda- components. Here, we regard the x axis as the zigzag direction mentally the same as the one on the polarization of high-order and suppose mid-infrared excitation with the polarization of the sideband generation in ref. . In our experiment, the incident pffiffiffi MID zigzag direction: E ðÞ t ¼ E ðÞ tðÞ σ þ σ = 2. We only con- light was linearly polarized; i.e., it was a linear combination of x zigzag left- and right-hand circular polarized light. In the strong-field sider the ballistic driving process for HHG since the electrons and NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 3 Harmonic intensity Absorbance Harmonic intensity (arb. units) (arb. units) (arb. units) ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ac b Mo, W K′ x M K S, Se e R K′ y (Armchair) K′ K x (Zigzag) de E: Zigzag E: Armchair e–h creation e–h creation E(t) E(t) K,K′ t , t 3 4 t T/2 t T/2 2 3 t t 0 t t T 0 t , t T 1 4 1 2 K′ K,K′ K′ e–h creation e–h creation Fig. 3 electron–hole dynamics of HHG in monolayer TMDs. a Top view of the crystal of monolayer TMDs. The brown dots are transition-metal atoms, while the yellow dots are chalcogenide atoms. The x (y) axis corresponds to the zigzag (armchair) direction. b Two-dimensional hexagonal Brillouin zone of monolayer TMDs. The high symmetry, Γ, K (K’), M points are labeled. The electric field with a polarization in the x (y) direction in real space drives carriers in the k (k ) direction. c Schematic diagrams of the real space trajectories of electrons (blue curve) and holes (magenta curve) driven by light field. The x y electron–hole pairs generated by Zener tunneling at t = 0 are accelerated by the electric field, and they recombine when they meet againðÞ t ¼ t . d Schematic drawings of electron–hole creation and recombination dynamics under excitation with a polarization along the zigzag direction. The electron–hole pairs generated at t = 0 follow different traces depending on whether they were created at the K or K’ point. Thus, the electron–hole pairs recombine at different times t and t (blue solid and magenta open circles in the dashed orange oval). The electron–hole pairs generated at t = T/2 follow 1 2 reversed dynamics to those created at t = 0 (magenta open and blue solid circles in the dashed turquoise oval). e On the other hand, under excitation with the polarization along the armchair direction, the acceleration and recombination dynamics of the electron–hole pairs do not depend on whether the pairs were created at the K point or at the K’ point holes losing their initial coherence do not contribute to high momentum space. Here, we calculated the electron–hole harmonics. We assume that the relative phase between dynamics in real and momentum space for monolayer MoS on electron–hole pairs generated at K and K’ points is conserved the basis of the three-step model. The valence and conduction in the whole process . Even if their spin states change by driving band were given by a tight-binding model without spin-orbit 35,36 process, the following discussion is valid when the initially couplings (see the Supplementary Note 3). We supposed that generated, relative phase is not lost. The time-reversal symmetry the incident light in the calculation has a sine-like electric field −1 of the system and the periodicity of the incident light with the with a maximum field of 27MVcm and a polarization in the polarization of the zigzag direction follows the relation (see zigzag direction (see the Supplementary Note 5 for the case of the Supplementary Note 4 for the detailed discussion): polarization in the armchair direction). Moreover, we assumed that the electron–hole pairs are created at the K and K’ points HHG HHG when the electric field reaches a peak (at t ¼ 0; T=2; T; ¼ in E ðtÞ¼E t þ ð2Þ Fig. 3d) through the Zener tunneling process. The electron–hole pairs are accelerated by the light field and eventually recombine This directly gives the selection rules: with each other by emitting high-energy photons. When the HHG HHG electron–hole pairs in the K valley are driven in the K – Γ – K′ E ðÞ t ¼ E t þ ð3Þ x x direction, those in the K’ valley are driven in the K′ – M – K direction (Fig. 3b). and Here, we propose an extension of the conventional three-step HHG HHG model that includes quantum mechanical effects for E ðÞ t ¼ E t þ ð4Þ y y electron–hole pairs in solids: the trajectories of electrons and hi P P holes in momentum space are calculated using the equation of HHG inωt n n inωt By considering E ðÞ t ¼ E e ¼ E ^ x þ E ^ y e , n x y _ motion for Bloch electrons: h k ¼ qE, where k is the electron’s n n wave vector, q its charge, and E the electric field. In real space, we equations (3) and (4) tell us that the odd-order HHG has parallel should consider a wave packet of Bloch electrons and holes. The polarization and even-order HHG have perpendicular polariza- tions to the incident light. The polarization selection rules of the equation of motion should be r_ ¼ h ∂εðkÞ=∂k, where εðkÞ is the electron’s energy. It is obvious that the wave packet can occupy HHG obtained here explain the results of our experiment and ref. . We also discuss the polarization selection rule of HHG any lattice site in real space due to the periodicity. Therefore, we considered several trajectories starting from different atomic sites. with the armchair excitation (see Supplementary Note 4). Here, we limited our calculation to one dimension because the transverse motion due to the Berry curvature is relatively small Calculation of dynamics of electrons and holes by semi- and the obtained results do not change qualitatively. Figure 4a, b classical treatment. To understand the resonant effect of inter- shows the real-space trajectories of the carriers generated at t = 0 band HHG in the band nesting energy region, it is worthwhile to at the K (K’) point. The turquoise curves indicate the motion of a know where the recombined electron–hole pairs are in 4 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE a c 0 < t < T/2 1 3 4 2 0 1 –1 e–h e–h creation creation –2 –1 –3 –4 –2 e –h @ K point –5 0 T/4 T/2 K′ K Γ K′ K Time Wave vector b d T/2 < t < T 4 3 1 e–h e–h 0 0 creation creation –1 –1 –2 –2 e –h @ K′ point –3 0 T/4 T/2 K′ K Γ K′ K Time Wave vector Fig. 4 Semi-classical calculation of electron–hole dynamics in the real and momentum space. a Real space dynamics of carriers generated at t = 0 and the K point under excitation with polarization along zigzag direction. The turquoise curve shows the trajectory of the hole generated at x/a= 0, and the black dashed curves show the electrons generated at x=a ¼ 0; 1; 2; 3; 4. The possible recombination paths of the hole and electrons are marked as blue solid circles. b Same as a but for the carriers generated at the K’ point. The possible recombination paths of the hole and electrons are marked as magenta open circles. c Recombined electron–hole pairs in momentum space generated at t = 0. The electron–hole pairs created at the K (K’) point are represented as blue solid (magenta open) circles. The labels from 1 to 4 correspond to those in a and b. The asymmetric features of the electron–hole dynamics in the K and K’ valleys lead to different times and energies of recombination. Only the electron–hole pairs generated at the K point reach the band nesting region. d Same as c but for electron–hole pairs generated at t = T/2. The dynamics in the K and K’ valleys are reversed from those of t = 0 and show the alternating nature. Only electron–hole pairs generated at the K point reach the band nesting region hole generated at x/a = 0. The black dashed curves show the energy distribution than those of the latter, which cause the motion of electrons generated at several neighboring atomic sites resonant enhancement in the band nesting energy. Figure 4d around the hole generation siteðÞ x=a ¼ 0 . HHG emissions can shows the same kind of plot as Fig. 4c for the electron–hole pairs happen when an electron and hole meet again in real space. For a generated at t = T/2. A half period later, the dynamics in hole generated at the K point, there are four ways for it to momentum space reverse and the electron–hole pairs generated recombine with the electrons, marked as circles in Fig. 4a. In this at the K’ point reach the band nesting region. From the viewpoint case, it should be noted that the hole recombines with an electron of the polarization selection rule, the alternating nature of the generated at a different atomic site. This would be a characteristic dynamics in the K and K’ valleys at t = 0 and t = T/2 is feature of HHG in solids. For the hole generated at the K’ point, important. The interband emission process changes every half there are also four recombination possibilities, marked as circles period of the incident light; that is, light with the same photon in Fig. 4b. By comparing the dynamics in real space with those in energy but opposite helicity is emitted in the subsequent half momentum space, one can determine the momentum and energy period. As discussed in the previous section, the asymmetric of electron–hole pairs at the recombination time. The circles in nature of the dynamics in the K and K’ valleys directly lead to Fig. 4c represent the recombined electron–hole pairs generated at polarization selection rules. t = 0 in momentum space. The blue solid (magenta open) circles indicate the electron–hole pairs generated at the K (K’) point, Discussion which result in an σ ðÞ σ emission. The two pairs generated at In summary, we investigated generation of high harmonics in the K points labeled 2 and 3 are in the band nesting region at the monolayer TMDs. By comparing four monolayer TMDs, we time of recombination. The band nesting causes a large joint found resonant enhancement of HHG with the interband optical density of states in the C and D energy region, as shown in the transition due to band nesting effects. We also found that odd absorption spectra (Fig. 2a–d). The large joint density of states in and even orders of HHG have parallel and perpendicular polar- turn causes a resonant enhancement of HHG at those energies izations under excitation with polarization along with the zigzag (Fig. 2e–h). Actual electrons and holes are wave packets with a direction. This comes from the asymmetric nature of the finite uncertainty in momentum, originating from the uncertainty dynamics in the K and K’ valleys. Our findings give the important of the time at which the pair was created via the Zener tunneling indication that the nonlinear interband polarization significantly process. Thus, if we think about the electron–hole pairs in the contributes to the high harmonic generation in solids and opens band nesting region (magenta open circles labeled 2 and 3) and the way for attosecond science with monolayer materials having the others, the former cause light emissions with a narrower widely tunable electronic structures. NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 5 x/a x/a Energy (eV) Energy (eV) ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 Methods 13. Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, Samples. The monolayer TMDs were grown on sapphire substrates by chemical 867–871 (2004). vapor deposition. 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Real-time observation of interfering crystal JP18H01832) from the Ministry of Education, Culture, Sports, Science and Technology electrons in high-harmonic generation. Nature 523, 572–575 (2015). (MEXT), Japan. 9. Ndabashimiye, G. et al. Solid-state harmonics beyond the atomic limit. Nature 534, 520–523 (2016). 10. Langer, F. et al. Lightwave-driven quasiparticle collisions on a subcycle timescale. Nature 533, 225–229 (2016). Author contributions 11. You, Y. S., Reis, D. A. & Ghimire, S. Anisotropic high-harmonic generation in N.Y. and K.T. conceived the experiments. N.Y. and K.U. carried out the experiments and bulk crystals. Nat. Phys. 13, 345–349 (2016). analyses. Y.T., S.S. and Y.M. fabricated the samples. K.N. and K.T. performed the cal- 12. Ghimire, S. & Reis, D. High-harmonic generation from solids. Nat. Phys. 15, culation. N.Y. and K.T. wrote the manuscript. All the authors contributed to the dis- 10–16 (2019). cussion and interpretation of the results. 6 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE Additional information Open Access This article is licensed under a Creative Commons Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- Attribution 4.0 International License, which permits use, sharing, 019-11697-6. adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Competing interests: The authors declare no competing interests. Commons license, and indicate if changes were made. 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Interband resonant high-harmonic generation by valley polarized electron–hole pairs

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ARTICLE https://doi.org/10.1038/s41467-019-11697-6 OPEN Interband resonant high-harmonic generation by valley polarized electron–hole pairs 1,2 1 1 3 3 3 Naotaka Yoshikawa , Kohei Nagai , Kento Uchida , Yuhei Takaguchi , Shogo Sasaki , Yasumitsu Miyata & 1,2 Koichiro Tanaka High-harmonic generation in solids is a unique tool to investigate the electron dynamics in strong light fields. The systematic study in monolayer materials is required to deepen the insight into the fundamental mechanism of high-harmonic generation. Here we demonstrated nonperturbative high harmonics up to 18th order in monolayer transition metal dichalco- genides. We found the enhancement in the even-order high harmonics which is attributed to the resonance to the band nesting energy. The symmetry analysis shows that the valley polarization and anisotropic band structure lead to polarization of the high-harmonic radia- tion. The calculation based on the three-step model in solids revealed that the electron–hole polarization driven to the band nesting region should contribute to the high harmonic radiation, where the electrons and holes generated at neighboring lattice sites are taken into account. Our findings open the way for attosecond science with monolayer materials having widely tunable electronic structures. 1 2 Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Institute for Integrated Cell-Material Sciences (WPI-iCeMS), Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan. Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan. Correspondence and requests for materials should be addressed to N.Y. (email: yoshikawa@thz.phys.s.u-tokyo.ac.jp) or to K.T. (email: kochan@scphys.kyoto-u.ac.jp) NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 1 1234567890():,; ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 esearch on generation of high harmonics (HHG) from Results atomic or molecular gases has produced coherent High harmonic spectra from monolayer TMDs. We excited the Rbroadband radiation in the extreme ultraviolet region monolayer TMDs by using mid-infrared pulses (0.26eV photon 1,2 −2 and attosecond pulses . Moreover, HHG has recently been energy, 1.7TWcm peak intensity) and detected HHG in the 3–12 reported from several solid-state materials .Non- near-infrared to ultraviolet energy region (see the Methods and perturbative HHG in solids can be achieved with a driving laser Supplementary Figure 1). Figure 1 shows the high harmonic 12 −2 with a peak intensity as low as 10 Wcm , which is much spectra from fifth to sixteenth order of (a) MoSe , (b) WSe , (c) 2 2 smaller than that required in the gas phase, suggesting MoS , and (d) WS monolayers (see the Methods and Supple- 2 2 the potential for a compact, highly efficient attosecond mentary Figure 2 for the full spectrum up to 18th harmonics). We light source. In addition, solid-state HHG can capture the set the polarization of the mid-infrared pulses in parallel with the electronic properties of materials as tomographic imaging of zigzag direction (M–M(M=Mo, W) direction) (see Supplemen- molecular gases has been demonstrated by high-harmonic tary Note 1). The polarization of the even-order harmonic 13–15 spectroscopy . High harmonic spectroscopy in solids shows radiation was perpendicular to that of the incident laser light, the possibility of probing the electronic band structure as it while the polarization of the odd-order harmonics was parallel 3,4,7,10,11 23 relates to crystal symmetry and interatomic bonding .A (Supplementary Figure 3a) . The excitation power dependence unified understanding of the mechanism of HHG in solids is shows the nonperturbative nature of HHG from monolayer necessary for the application of high harmonic spectroscopy. TMDs (see Supplementary Note 2). Although several theoretical models have been proposed for solid-state HHG in terms of interband polarization and intra- Interband resonant enhancement of HHG. Figure 2a–d shows 3–11,16–19 band electron dynamics as well as their interplay ,the the polarization-unresolved emission intensities of the even-order underlying mechanism is still under debate. It will be important harmonics, and Fig. 2e–h the optical absorption spectra. In the to gain an understanding of HHG in single-atomic-layer solids, absorption spectra, the peaks labeled A and B are attributed to so- because propagation effects such as the phase matching con- called A and B excitons, consisting of electrons and holes loca- dition, which would otherwise obscure intrinsic features of lized in the K valleys in momentum space. In addition, the band HHG in bulk crystals, do not affect ideal two-dimensional structures of the monolayer TMDs have a band nesting region, systems. 30,31 causing van Hove singularities in the joint density of states . 20–22 HHG from monolayer graphene was recently reported , and its observed anomalous ellipticity dependence was able to be reproduced by a fully quantum mechanical theory in which both intraband and interband contributions are included . MoSe 4 2 HHG was also observed from monolayer MoS ,which is one of the monolayer transition metal dichalcogenides (TMDs) . 2 11 Monolayer TMDs, which have a finite bandgap in contrast to 10 8 13 12 15 graphene, have attracted much attention for their exotic properties such as enhancement of luminescence derived from 0 their indirect-to-direct bandgap transition from bulk to 24,25 26 monolayer form , extremely large exciton binding energy , 10 WSe b 7 2 and valley pseudospin physics arising from inversion symmetry 27,28 breaking and a large spin-orbit interaction .Liu et al. found 10 12 that the even-order high harmonics are predominantly polar- 10 ized perpendicular to the pump laser and attributed this to the intraband anomalous transverse current arising from the material’s Berry curvature . On the other hand, a recent experiment with an optical pump showed that the interband MoS 5 c 10 2 contribution dominates the intraband one in bulk ZnO .Liu et al. also showed that the even orders of HHG are enhanced at 2 12 much higher energies compared with the lowest exciton 10 or bandgap energy, while the odd orders are monotonically suppressed at higher energies. The enhancement at higher 0 orders and its relation to the perpendicular polarization of even-order HHG is not yet understood. 10 WS d In this study, we systematically investigated HHG from four kinds of monolayer TMDs with different bandgap energies (MoSe ,WSe ,MoS ,and WS )toexamine theresonance 10 2 2 2 2 8 12 effect and polarization of HHG in monolayer TMDs. By 13 14 comparing the HHG and optical absorption spectra of the monolayer TMDs, we found that HHG is enhanced when it is in resonance with the optical transition due to band nesting, 2.0 3.0 4.0 indicating that the interband polarization mainly contributes Photon energy (eV) to the even-order HHG. A simple calculation that is an extension of the three-step model for electron–hole pairs in Fig. 1 High harmonic generation from monolayer TMDs. High harmonic solids shows that the electron–hole pairs driven to the band spectra from a MoSe , b WSe , c MoS , and d WS monolayers at room 2 2 2 2 nesting region significantly contributes to the resonance temperature induced by mid-infrared pulse excitation (photon energy −2 enhancement, and anisotropic driving of electron–hole polar- 0.26eV, peak intensity 1.7TWcm ). We set the excitation polarization in izations around the K and K’ points determines polarization parallel with the zigzag direction. The harmonics from fifth to sixteenth selection rule of HHG. order are labeled 2 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications Intensity (arb. units) NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE Harmonic order Harmonic order Harmonic order Harmonic order 8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14 ab c d 1 MoSe WSe MoS WS 2 2 2 2 ef g h C C B B A B 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 Photon energy (eV) Photon energy (eV) Photon energy (eV) Photon energy (eV) Harmonic order Harmonic order Harmonic order Harmonic order 5 7 9 11 13 15 5 7 9 11 13 15 5 7 9 11 13 15 5 7 9 11 13 15 ij k l –2 –4 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 2.0 3.0 4.0 Photon energy (eV) Photon energy (eV) Photon energy (eV) Photon energy (eV) Fig. 2 Resonant enhancement of even-order high harmonics. Intensities of even-order harmonics in a MoSe , b WSe , c MoS , and d WS monolayers. 2 2 2 2 Error bars represent the standard deviations. e, f, g, h Corresponding optical absorption spectra of the TMD monolayers. The peaks of the A and B excitons are labeled A and B, respectively. The peaks due to the band nesting effects are labeled C and D. i, j, k, l Corresponding intensities of odd-order harmonics The band nesting results in strong optical absorption at higher regime, therefore, electron–hole pairs are generated coherently by 32–34 photon energies (labeled C and D in the figure) . As shown in Zener tunneling at the K and K’ points in the two-dimensional Fig. 2a, the intensity of the 10th harmonic in monolayer MoSe is hexagonal Brillouin zone. We will assume that electron–hole pairs larger than that of the 8th, and the intensity of the 14th harmonic are created when the electric field of the incident light reaches is larger than that of the 12th. Interestingly, the energies of the positive or negative maxima. Since the generated electrons and 10th and 14th harmonics coincide with the absorption peaks C holes are accelerated differently in the conduction and valence and D. This is not only the case in MoSe . Figure 2b–d also shows bands, their motions are asymmetric in real space (Fig. 3c). When enhancements at the band nesting energies for WSe , MoS , and the electron and hole meet again in real space (the circle in 2 2 WS . The observed enhancement in resonance with the interband Fig. 3d), electron–hole recombination occurs and HHG is emit- optical transition strongly suggests that the interband contribu- ted. The electron–hole pairs created at the positive peak field (t = tion is dominant in even-order HHG. On the other hand, the 0 in Fig. 3d) under excitation with a polarization along the zigzag intensities of the odd orders monotonically decrease with direction (Fig. 3a) have anisotropic driving processes: those cre- increasing order as shown in Fig. 2i–l. However, though it is not ated at the K point are driven in the K – Γ – K′ direction, while so apparent, the shoulder-like structure appears at the C and D those created at the K’ point are driven in the K′ – M – K transitions. This could suggest that odd-order harmonics are direction. This anisotropy in turn causes anisotropic acceleration contributed by both intraband process, which may show no and recombination dynamics in the K and K’ bands (0<t<T=2, resonant enhancement around interband transitions, and inter- where T is the period of the incident light). The dynamics of the band process, which should show the resonances. The experi- electron–hole pairs generated at the negative peak field (t = T/2) mental data imply that, at lower energy region, especially below are the reverse of those generated at the positive peak field the bandgap energy, the intraband contribution is much larger ðÞ T=2<t<T . On the other hand, when the excitation light is than the interband one for the odd-order harmonics, but the polarized along the armchair direction, the electron–hole pairs interband polarization may substantially contributes it at the generated at the positive peak field and those generated at the resonance energies. negative peak field have the same dynamics in the K and K’ bands (Fig. 3e). Next, we show how the anisotropic and alternating nature of Polarization selection rules of HHG. The resonant enhancement the electron–hole dynamics gives rise to a polarization selection of HHG indicates that the interband polarization mainly con- rule for HHG. The electric field of HHG is the sum of the left- tributes to even-order HHG. The question is how the perpendi- and right-hand circular polarized light: cular polarization arises in the even-order harmonics. The HHG HHG þ HHG transverse current due to a material’s Berry curvature might not E ðÞ t ¼ E ðÞ t σ þ E ðÞ t σ ð1Þ contribute to interband HHG. Here, we will discuss the pffiffiffi mechanism of HHG in monolayer TMDs on the basis of a semi- HHG HHG HHG ± where E ¼ E ±iE = 2 are the amplitudes of the σ ± x y classical three-step model in solids. The discussion is funda- components. Here, we regard the x axis as the zigzag direction mentally the same as the one on the polarization of high-order and suppose mid-infrared excitation with the polarization of the sideband generation in ref. . In our experiment, the incident pffiffiffi MID zigzag direction: E ðÞ t ¼ E ðÞ tðÞ σ þ σ = 2. We only con- light was linearly polarized; i.e., it was a linear combination of x zigzag left- and right-hand circular polarized light. In the strong-field sider the ballistic driving process for HHG since the electrons and NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 3 Harmonic intensity Absorbance Harmonic intensity (arb. units) (arb. units) (arb. units) ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ac b Mo, W K′ x M K S, Se e R K′ y (Armchair) K′ K x (Zigzag) de E: Zigzag E: Armchair e–h creation e–h creation E(t) E(t) K,K′ t , t 3 4 t T/2 t T/2 2 3 t t 0 t t T 0 t , t T 1 4 1 2 K′ K,K′ K′ e–h creation e–h creation Fig. 3 electron–hole dynamics of HHG in monolayer TMDs. a Top view of the crystal of monolayer TMDs. The brown dots are transition-metal atoms, while the yellow dots are chalcogenide atoms. The x (y) axis corresponds to the zigzag (armchair) direction. b Two-dimensional hexagonal Brillouin zone of monolayer TMDs. The high symmetry, Γ, K (K’), M points are labeled. The electric field with a polarization in the x (y) direction in real space drives carriers in the k (k ) direction. c Schematic diagrams of the real space trajectories of electrons (blue curve) and holes (magenta curve) driven by light field. The x y electron–hole pairs generated by Zener tunneling at t = 0 are accelerated by the electric field, and they recombine when they meet againðÞ t ¼ t . d Schematic drawings of electron–hole creation and recombination dynamics under excitation with a polarization along the zigzag direction. The electron–hole pairs generated at t = 0 follow different traces depending on whether they were created at the K or K’ point. Thus, the electron–hole pairs recombine at different times t and t (blue solid and magenta open circles in the dashed orange oval). The electron–hole pairs generated at t = T/2 follow 1 2 reversed dynamics to those created at t = 0 (magenta open and blue solid circles in the dashed turquoise oval). e On the other hand, under excitation with the polarization along the armchair direction, the acceleration and recombination dynamics of the electron–hole pairs do not depend on whether the pairs were created at the K point or at the K’ point holes losing their initial coherence do not contribute to high momentum space. Here, we calculated the electron–hole harmonics. We assume that the relative phase between dynamics in real and momentum space for monolayer MoS on electron–hole pairs generated at K and K’ points is conserved the basis of the three-step model. The valence and conduction in the whole process . Even if their spin states change by driving band were given by a tight-binding model without spin-orbit 35,36 process, the following discussion is valid when the initially couplings (see the Supplementary Note 3). We supposed that generated, relative phase is not lost. The time-reversal symmetry the incident light in the calculation has a sine-like electric field −1 of the system and the periodicity of the incident light with the with a maximum field of 27MVcm and a polarization in the polarization of the zigzag direction follows the relation (see zigzag direction (see the Supplementary Note 5 for the case of the Supplementary Note 4 for the detailed discussion): polarization in the armchair direction). Moreover, we assumed that the electron–hole pairs are created at the K and K’ points HHG HHG when the electric field reaches a peak (at t ¼ 0; T=2; T; ¼ in E ðtÞ¼E t þ ð2Þ Fig. 3d) through the Zener tunneling process. The electron–hole pairs are accelerated by the light field and eventually recombine This directly gives the selection rules: with each other by emitting high-energy photons. When the HHG HHG electron–hole pairs in the K valley are driven in the K – Γ – K′ E ðÞ t ¼ E t þ ð3Þ x x direction, those in the K’ valley are driven in the K′ – M – K direction (Fig. 3b). and Here, we propose an extension of the conventional three-step HHG HHG model that includes quantum mechanical effects for E ðÞ t ¼ E t þ ð4Þ y y electron–hole pairs in solids: the trajectories of electrons and hi P P holes in momentum space are calculated using the equation of HHG inωt n n inωt By considering E ðÞ t ¼ E e ¼ E ^ x þ E ^ y e , n x y _ motion for Bloch electrons: h k ¼ qE, where k is the electron’s n n wave vector, q its charge, and E the electric field. In real space, we equations (3) and (4) tell us that the odd-order HHG has parallel should consider a wave packet of Bloch electrons and holes. The polarization and even-order HHG have perpendicular polariza- tions to the incident light. The polarization selection rules of the equation of motion should be r_ ¼ h ∂εðkÞ=∂k, where εðkÞ is the electron’s energy. It is obvious that the wave packet can occupy HHG obtained here explain the results of our experiment and ref. . We also discuss the polarization selection rule of HHG any lattice site in real space due to the periodicity. Therefore, we considered several trajectories starting from different atomic sites. with the armchair excitation (see Supplementary Note 4). Here, we limited our calculation to one dimension because the transverse motion due to the Berry curvature is relatively small Calculation of dynamics of electrons and holes by semi- and the obtained results do not change qualitatively. Figure 4a, b classical treatment. To understand the resonant effect of inter- shows the real-space trajectories of the carriers generated at t = 0 band HHG in the band nesting energy region, it is worthwhile to at the K (K’) point. The turquoise curves indicate the motion of a know where the recombined electron–hole pairs are in 4 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE a c 0 < t < T/2 1 3 4 2 0 1 –1 e–h e–h creation creation –2 –1 –3 –4 –2 e –h @ K point –5 0 T/4 T/2 K′ K Γ K′ K Time Wave vector b d T/2 < t < T 4 3 1 e–h e–h 0 0 creation creation –1 –1 –2 –2 e –h @ K′ point –3 0 T/4 T/2 K′ K Γ K′ K Time Wave vector Fig. 4 Semi-classical calculation of electron–hole dynamics in the real and momentum space. a Real space dynamics of carriers generated at t = 0 and the K point under excitation with polarization along zigzag direction. The turquoise curve shows the trajectory of the hole generated at x/a= 0, and the black dashed curves show the electrons generated at x=a ¼ 0; 1; 2; 3; 4. The possible recombination paths of the hole and electrons are marked as blue solid circles. b Same as a but for the carriers generated at the K’ point. The possible recombination paths of the hole and electrons are marked as magenta open circles. c Recombined electron–hole pairs in momentum space generated at t = 0. The electron–hole pairs created at the K (K’) point are represented as blue solid (magenta open) circles. The labels from 1 to 4 correspond to those in a and b. The asymmetric features of the electron–hole dynamics in the K and K’ valleys lead to different times and energies of recombination. Only the electron–hole pairs generated at the K point reach the band nesting region. d Same as c but for electron–hole pairs generated at t = T/2. The dynamics in the K and K’ valleys are reversed from those of t = 0 and show the alternating nature. Only electron–hole pairs generated at the K point reach the band nesting region hole generated at x/a = 0. The black dashed curves show the energy distribution than those of the latter, which cause the motion of electrons generated at several neighboring atomic sites resonant enhancement in the band nesting energy. Figure 4d around the hole generation siteðÞ x=a ¼ 0 . HHG emissions can shows the same kind of plot as Fig. 4c for the electron–hole pairs happen when an electron and hole meet again in real space. For a generated at t = T/2. A half period later, the dynamics in hole generated at the K point, there are four ways for it to momentum space reverse and the electron–hole pairs generated recombine with the electrons, marked as circles in Fig. 4a. In this at the K’ point reach the band nesting region. From the viewpoint case, it should be noted that the hole recombines with an electron of the polarization selection rule, the alternating nature of the generated at a different atomic site. This would be a characteristic dynamics in the K and K’ valleys at t = 0 and t = T/2 is feature of HHG in solids. For the hole generated at the K’ point, important. The interband emission process changes every half there are also four recombination possibilities, marked as circles period of the incident light; that is, light with the same photon in Fig. 4b. By comparing the dynamics in real space with those in energy but opposite helicity is emitted in the subsequent half momentum space, one can determine the momentum and energy period. As discussed in the previous section, the asymmetric of electron–hole pairs at the recombination time. The circles in nature of the dynamics in the K and K’ valleys directly lead to Fig. 4c represent the recombined electron–hole pairs generated at polarization selection rules. t = 0 in momentum space. The blue solid (magenta open) circles indicate the electron–hole pairs generated at the K (K’) point, Discussion which result in an σ ðÞ σ emission. The two pairs generated at In summary, we investigated generation of high harmonics in the K points labeled 2 and 3 are in the band nesting region at the monolayer TMDs. By comparing four monolayer TMDs, we time of recombination. The band nesting causes a large joint found resonant enhancement of HHG with the interband optical density of states in the C and D energy region, as shown in the transition due to band nesting effects. We also found that odd absorption spectra (Fig. 2a–d). The large joint density of states in and even orders of HHG have parallel and perpendicular polar- turn causes a resonant enhancement of HHG at those energies izations under excitation with polarization along with the zigzag (Fig. 2e–h). Actual electrons and holes are wave packets with a direction. This comes from the asymmetric nature of the finite uncertainty in momentum, originating from the uncertainty dynamics in the K and K’ valleys. Our findings give the important of the time at which the pair was created via the Zener tunneling indication that the nonlinear interband polarization significantly process. Thus, if we think about the electron–hole pairs in the contributes to the high harmonic generation in solids and opens band nesting region (magenta open circles labeled 2 and 3) and the way for attosecond science with monolayer materials having the others, the former cause light emissions with a narrower widely tunable electronic structures. NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications 5 x/a x/a Energy (eV) Energy (eV) ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 Methods 13. Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, Samples. The monolayer TMDs were grown on sapphire substrates by chemical 867–871 (2004). vapor deposition. 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All the authors contributed to the dis- 10–16 (2019). cussion and interpretation of the results. 6 NATURE COMMUNICATIONS | (2019) 10:3709 | https://doi.org/10.1038/s41467-019-11697-6 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-11697-6 ARTICLE Additional information Open Access This article is licensed under a Creative Commons Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- Attribution 4.0 International License, which permits use, sharing, 019-11697-6. adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Competing interests: The authors declare no competing interests. Commons license, and indicate if changes were made. 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