Open Access
Add to BibSonomy
Abstract
High-order harmonic generation in solid state has attracted a lot of attentions. The Berry curvature (BC), a geometrical property of the Bloch energy band, plays an important role for the harmonic generation in crystal. As we all know, the influence of BC on the harmonic emission has been investigated before and BC is simplified as a 1D structure. However, many other materials including MoS2 are 2D materials. In this work, we extend the investigation for BC to 2D structure and get a generalized equation, which not only gives a new method to control the harmonic emission with BC, but also gives a deeper understanding for the influence of the BC. We show the ability to control the harmonic emission related to the BC using the orthogonal two-color (OTC) laser field. By tuning the delay of OTC laser field, one can steer the trajectory of electrons and modulate the emission of harmonics. This study can provide us a deeper insight into the role of the BC which is difficult to be measured experimentally.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High harmonic generation (HHG), a strong field nonlinear non-perturbative phenomenon, has been researched extensively in gas phase [1–5]. It is viewed as a practical technique to generate bright and tabletop extreme ultraviolet sources [1,2,5–8] and attosecond pulses [9–12]. Recently with the development of mid-infrared laser technology study of HHG has been extended to a variety of solid state materials such as bulk crystalline solids [13–17], two-dimensional (2D) layer materials [18–22] and topological materials [23–26], etc. HHG in solid state materials exhibits distinctive properties such as linear cutoff law [13], abnormal ellipticity dependence [17,27,28] and generation of even order harmonics [19]. Studying solid state high harmonics provides a way for exploring the electron dynamics in the presence of laser field. It can serve as a unique tool to investigate multiband dynamics [29–31] as well as ultrafast dynamical quantum interference [32–35] in solid state materials. The mapping between HHG spectra and electron dynamics in Brillouin zone has made it possible to reconstruct band structures [36,37], measure Berry curvature [38] and probe topological phase transitions [24,39].
HHG from gases can be well explained by the three-step model including ionization, acceleration and recollision [40,41]. However, mechanism of HHG in solid materials is qualitatively different from that of atomic phase due to the condensed state and periodic structure. HHG in solids are generally explained in terms of electron dynamics in the reciprocal space. The movement of electrons in momentum space can be divided into two parts [42–45]: interband and intraband dynamics. Interband dynamic means the polarization of electron-hole between conduction band and valence band and intraband dynamic is referred to be as dynamical Bloch oscillation. The overall harmonic yield is determined by the coupling of interband and intraband dynamics. Even order harmonics in solid materials can be ascribed to nonzero Berry curvature [19], quantum interference of direct and indirect excitation path [46], transition dipole phase [47] or interband polarization [48]. However, which physical mechanism dominating the even order harmonics is still under debate and needs more investigation. In this work, we mainly investigate the even order harmonics from BC.
HHG in monolayer MoS2 has been investigated before. In Ref. [19], harmonics in MoS2 can be divided into two components with respect to the polarization of laser: parallel and perpendicular components. Odd harmonics are mainly parallel with laser polarization. Even harmonics are mainly polarized perpendicular to the laser orientation. The difference is considered to come from that even harmonics are generated by BC, while odd harmonics are from band group velocity.
In Ref. [19] and Ref. [38], BC is simplified as a 1D structure and expanded as a series of sine function. However, many other materials including MoS2 are 2D materials. In this work, we extend the 1D BC model to 2D BC model, get a generalized equation and explore the anomalous electron dynamics induced by BC in monolayer MoS2 in 2D Brillouin zone. The equation is derived to illustrate the influence of BC in 2D plane and it can also be extended to 3D for bulk materials. To further manifest the properties of BC, we use an OTC laser field to drive the electron move in 2D Brillouin zone and manipulate the trajectory of electron to control the generation of even order harmonics. Orthogonally polarized two-color field has been widely used in controlling the HHG dynamics in atomic phase [49,50]. Changing temporal delay between OTC field can be used to select long- and short-trajectory [51,52]. Harmonic yield can be enhanced up to one order of magnitude compared with SC field [53–55]. Manipulation of free electron’s trajectory based on OTC field can also be used to probe atomic wavefunctions [56]. Recently, OTC investigation has been extended to solid state HHG. Y. Sanari et al. found that application of OTC laser field can modify the selection rules for frequency-mixing processes of HHG [57]. Tang et al. investigated the HHG from ZnO driven by OTC laser fields and found the HHG yield is highly sensitive to the relative phase between the two fields [58]. Here, the OTC laser field is only used to show the ability to control BC with complex laser field. For 3D material, the result can be extended to 3D.
2. Results
Firstly, we simulate harmonic generation from monolayer MoS2 based on the density-matrix equation in the velocity gauge. The feasibility of the model has been verified in Ref [59]. The energy bands of monolayer MoS2 are obtained from a tight-binding model involving the third-nearest-neighbor Mo-Mo hopping, which can well fit the ones from first-principle calculation in the local-density approximation (LDA) across the whole Brillouin zone [60]. Figure 1(a) is the schematic of first Brillouin zone of monolayer MoS2, which is a hexagonal structure characterized by two unequal high symmetry point K and K’. The three energy bands used in the simulation contains one valence band and two conduction bands as is shown in Fig. 1(b).
Fig. 1. (a) Schematic of first Brillouin zone of monolayer MoS2. (b) Energy bands used in the density-matrix equation contain one valence band and two conduction bands. (c) Typical high harmonic spectra generated from monolayer MoS2 with linear laser polarization along Γ-Κ direction. (d) BC Ω(k) of first conduction band across the whole Brillouin zone extracted from third-nearest tight-binding model.
The interaction of monolayer MoS2 with laser pulse is simulated by propagating density-matrix elements. The relaxation processes are included by phenomenologically introducing the transverse and longitudinal dephasing time T2 and T1. Dephasing time T1 represents the relaxation of population in energy band. Dephasing time T2 denotes the decoherence of interband polarization. Generally, T1 is much longer than the pulse duration and T2, so it is usually omitted in many theoretical simulations. Dephasing time T2 and T1 are set to be 20fs and 500fs respectively. Temporal evolution of density-matrix elements is expressed by:
where $\rho $ is the density matrix, density matrix element ${\rho _{mn}}(k,t) = \left\langle {m,k} \right|\rho |{n,k} \rangle$, $|{n,k} \rangle$ and $|{m,k} \rangle$ is the eigenstate of band n and m at a particular momentum k, A(t) is the vector potential of external field, p(k) is the momentum matrix, Ei(k) (i = m, n) is the band energy and $f_m^0(k)$ is the initial distribution of the m-band. The time-dependent current density at a particular k is obtained by:
Then the total current density can be acquired by integrating the jk(t) over the first BZ:
Finally the harmonic spectrum can be obtained by making a Fourier transformation of the total current:
The vector potential of laser electric field is expressed by:
In monolayer MoS2, the dynamics of carrier within a single band in the presence of laser field can be expressed by following equations:
where the $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} $ is electric field, ${\varepsilon _g}$ is the energy difference between first conduction band and valence band, $\Omega $ is the Berry curvature difference between first conduction band and valence band. Equation (8) indicates that the intraband harmonics generated in the wake of electron's movement come from two parts. The first term of Eq. (8) corresponds to band group velocity and the latter anomalous velocity induced by BC, which acts like a magnetic field. Figure 1(d) displays the BC of first conduction band extracted from the third-nearest tight-binding model using the Kubo formula [61,62].
In Fig. 2, we scan the laser polarization along different directions θ in Brillouin zone. Harmonic yield of both components are normalized according to perpendicular components respectively. We can see that even harmonic yield is maximized along Γ- Κ direction and even harmonics are mainly perpendicular to laser polarization in agreement with previous work. As has been illustrated in Ref. [19], it can be seen from Eq. (8) that BC Ω(k) can induce a perpendicular current and hence the even harmonics. Figure 3(a) shows the harmonic spectrum calculated using Eq. (8). Red solid line represents the even order harmonics generated by Berry curvature. Blue solid line represents the odd order harmonics from band group velocity. It can be seen that intensity of BC induced even harmonics is close to that of band group velocity induced odd harmonics. This indicates that the BC is one of the origin of perpendicular even order harmonics. Parallel components are considered to originate from interband polarization.
Fig. 2. Comparisons between perpendicular and parallel components of even harmonics along different laser orientations. The subscript // and ⊥ denote the harmonic components parallel and perpendicular to laser polarization respectively.
Fig. 3. (a) HHG spectrum calculated from the Eq. (8). Berry curvature (BC, red) contribute to the generation of even order harmonics and band group velocity(GV, blue) leads to odd order harmonics. (b) Distribution of net BC $\Delta \Omega^{\mathrm{K} 1}$ around K1 point. Two orthogonal dashed lines mark the Γ- M and Γ- K direction.
Monolayer MoS2 have a direct bandgap located at six valley points in Brillouin zone. When laser is irradiated onto the MoS2, electron would tunnel through the direct bandgap into conduction band leaving a hole in valence band and accelerate from the bottom of the valley located at K(K’). Dynamics of electrons at energy valley play a decisive role in the nonlinear optical properties of MoS2. The nonlinear current generated by the BC given in Ref. [19] reads:
In this work, we consider all of six valley points, and the distribution of electron in momentum space can be expressed by:
Considering the symmetry: Ω(K)=Ω(-K), we can have the relationship between different K point: K4 = -K1, K5 = -K2, K6 = -K3. Then the total current of six valley points can be expressed by:
Due to the C3h rotation symmetry, K3, K5 can be obtained by rotating K1 an angle ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \theta } ^ \pm }$ about out-of-plane z axis with ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \theta } ^ \pm } = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \theta } ({ \pm {\raise0.7ex\hbox{${2\pi }$} \!\mathord{\left/ {\vphantom {{2\pi } \textrm{3}}} \right.}\!\lower0.7ex\hbox{$\textrm{3}$}}} )$. By considering the symmetry, we can perform this rotation on the vector potential of external field to get the same result because the BC vector always points to z axis:
In this way, the total contribution of the BC of six valley points reduces to the difference between the two points across the K point in one single valley. $\Delta k=-e A(t)$ is the drift from the valley point in the presence of laser field. If the BC is expanded as Fourier series along some direction as in the previous work [38] and the coefficients can be determined by the experimental data, hence the BC can be measured. Furthermore, from this generalized formula we can get more information. First, the current jB is decided by the inversion symmetry property of BC around K point. When the MoS2 is irradiated by a linearly polarized laser with polarization along Γ-Μ direction, the emission contributed by BC disappears because the BC is symmetry in this direction, not because of zero BC. Second, the emission from the current jB can only measure the inverse symmetry part of BC. Third, this formula can be easily extended to 3D for bulk materials. Figure 3(b) depicts the distribution of net Berry curvature $\Delta \Omega^{\mathrm{K} 1}$ in two dimensions. The abnormal current induced by BC is decided by the inverse symmetry property directly. When laser is polarized along Γ- Κ direction, net BC is nonzero when carrier move away from Κ point. However, When laser is polarized along Γ- M direction, net BC is zero all the time during the laser pulse.
It can be seen from the analysis above that the harmonics result from the disparity of BC between K1 and –K1. The emission of abnormal current jB is decided by the symmetry of BC and the electron trajectory. This raises an intriguing question: can we control the yield of a specific harmonic by designing the subcycle electron movement? As an illustration we use the electric fields of OTC lasers at frequencies ω0 and 2ω0 to simulate the harmonic generation induced by BC. The two color field is expressed by:
The polarization of fundamental field is along Γ- K direction. Laser parameters of fundamental laser field is the same as stated above. We compare the harmonic spectrum generated by two-color laser field at different phase delay in Fig. 4. The field ratio Ey: Ex is 1:1.
Fig. 4. (a) HHG spectrum generated by BC at phase delay A and C, as illustrated in (b). (b) Harmonic yield dependence on the phase delay. Each order harmonic is normalized according to the intensity of the counterpart generated in single-color field. A, B and C mark three phase delay. The dot-dashed curves in (c), (d) and (e) present the electron trajectory of net BC in a laser cycle corresponding to delay A, B and C, respectively. The two color field ratio Ey: Ex is 1:1. Laser parameters of fundamental laser field is the same as stated above.
Figure 4(a) shows harmonic spectrum originating from the abnormal current at two different phase delay. Phase delay A and C correspond to π/2 and π respectively. We have compared the intensity of HHG spectrum generated from single- and two-color field (figure is not showed here). Even order harmonics get enhanced most obviously and odd order harmonics increase less than even order. This can also be seen from Fig. 4(a). It can be seen that intensity of some even order harmonics can surpass that of neighboring odd order harmonics at both phase delay. In addition, we can see that lower order harmonics are stronger at phase A while higher order harmonics are stronger at phase C.
To see more clearly the phase dependence, we plot the harmonic yield of H6, H10 and H30 in Fig. 4(b). The intensity of H6, H10 and H30 is normalized respectively according to their counterpart generated in SC field along Γ- K direction. We can see that the optimum phase is different for different order harmonic. The optimum phase delay transits from π/2 to π while harmonic order changes from lower to higher order. In addition, harmonic yield gets remarkably enhanced. Harmonic yield of H10 is 60 times stronger than that in SC field, and H30 can be enhanced near three orders of magnitude. In Fig. 4(c), (d) and (e) we present the trajectory of net BC $\Delta \Omega^{\mathrm{K} 1}$ at delay A, B and C respectively. With excitation of the OTC laser field, electron will move in a 2D plane and presents totally different trajectory of net BC at different time delay. During one optical cycle, the net BC changes its sign multiple times, which means that net BC oscillates with multiple frequency of the laser field to produce the harmonics. Another important thing to be noted is that overall amplitude of net BC is biggest at delay A and smallest at delay C, which can be seen from their trajectories. At delay A, electron would travel across the zone with biggest $\Delta \Omega^{\mathrm{K} 1}$, while it would travel in the zone with small $\Delta \Omega^{\mathrm{K} 1}$ at delay C. In the meantime, the weight of different order harmonics will change with the variation of trajectory as can be seen in Fig. 4(a). Finally, yield of different order harmonics can display a different dependence on time delay.
We continue to change the relative intensity of the two-color field. The intensity of fundamental field keep fixed, while the field ratio between second harmonic field and fundamental field varies from 0 to 3. The harmonic yield at different field ratio and phase delay is displayed in Fig. 5. Each order harmonic is normalized according to their counterpart generated by SC field. As the dependence on field ratio and phase delay is different for different order harmonics, we only present here four orders H6, H8, H10 and H12 as examples. From Fig. 5 we can see dependence on phase delay will change with the increase of field ratio. This can be intuitively understood by electron’s trajectory. If we keep the phase delay unchanged, electron can explore the farer region in the Brillouin zone while the field ratio is increasing. If we keep the field ratio fixed and vary the phase delay, electron can still explore the different region as illustrated in Fig. 4(c-e). And we can see from Fig. 1(d) and Fig. 3(b) that the distribution of BC in Brillouin zone is very complex and BC at different region can vary dramatically. From Eq. (13), we can see variation of the anomalous current comes from both the variation of laser electric field and BC. And net BC $\Delta \Omega^{\mathrm{K} 1}$(-eA(t)) is determined by laser vector potential. Laser vector potential and electric field has a phase difference of π/2. So even order harmonics induced by anomalous current will show a very complex dependence on field ratio and phase delay. And it is hard to find a unified pattern. However, this OTC investigation can be helpful in the following aspects. First, we can drive the electron into different region by adjusting field ratio and phase delay. One maybe measure the Berry curvature in a 2D plane using two-color field. Second, the harmonic yield of even order harmonics gets enhanced by adjusting the delay between two color laser field. Finally, this scheme provides us a deeper insight into the electron’s dynamics in 2D space.
Fig. 5. Harmonic yield from Berry curvature at different field ratio and phase delay. Field ratio Ey: Ex varies from 0 to 3. Harmonic yield is normalized according to their counterpart generated by SC field. The colorbar shows the degree of enhancement compared with SC field.
3. Conclusion
In this work, we extend the investigation of BC from 1D to 2D field and explore role of BC in the harmonic generation with abnormal current in monolayer MoS2. Density-matrix elements propagation in the velocity gauge can well reproduce polarization properties of even order harmonics. The semiclassical analysis in 2D Brillouin zone shows that anomalous current comes from the disparity of BC. The anisotropic structure of BC makes driving laser capable of driving the electron to specific band position which leads to a fast oscillating subcycle BC dynamics. With the case of OTC field, one can get yield of specific order harmonic enhanced by tuning the delay time between two color field. The yield of even order harmonics can be enhanced greatly, which is unavailable in SC field. Even order harmonics from BC display different dependence on phase delay and field ratio due to the complex structure of BC and electric field. This maybe lead to the measurement of BC in a 2D plane based on two-color field. This can advance the perception of strong field interaction mechanism in solid materials and provide us a new route to manipulate the subcycle electron dynamics with BC field.
Funding
National Natural Science Foundation of China (No. 11774363, No. 11874374, No. 61690223, No. 91950203); the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB16); Chinese Academy Sciences (CAS) Youth Innovation Promotion Association.
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.
References
1. T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef]
2. C. Jin, G. J. Stein, K. H. Hong, and C. D. Lin, “Generation of Bright, Spatially Coherent Soft X-Ray High Harmonics in a Hollow Waveguide Using Two-Color Synthesized Laser Pulses,” Phys. Rev. Lett. 115(4), 043901 (2015). [CrossRef]
3. P. Ye, X. He, H. Teng, M. Zhan, S. Zhong, W. Zhang, L. Wang, and Z. Wei, “Full quantum trajectories resolved high-order harmonic generation,” Phys. Rev. Lett. 113(7), 073601 (2014). [CrossRef]
4. F. Calegari, M. Lucchini, K. S. Kim, F. Ferrari, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “Quantum path control in harmonic generation by temporal shaping of few-optical-cycle pulses in ionizing media,” Phys. Rev. A 84(4), 041802 (2011). [CrossRef]
5. M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science 291(5510), 1923–1927 (2001). [CrossRef]
6. A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, “Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases,” J. Opt. Soc. Am. B 4(4), 595–601 (1987). [CrossRef]
7. T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. G. Tisch, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996). [CrossRef]
8. L. Zhang, Y. Zheng, G. Li, Z. Jia, Y. Li, Y. Xu, Y. Leng, Z. Zeng, R. Li, and Z. Xu, “Bright High-Order Harmonic Generation around 30 nm Using Hundred-Terawatt-Level Laser System for Seeding Full Coherent XFEL,” Appl. Sci. 8(9), 1446 (2018). [CrossRef]
9. G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009). [CrossRef]
10. P. Antoine, A. Lhuillier, and M. Lewenstein, “Attosecond pulse trains using high-order harmonics,” Phys. Rev. Lett. 77(7), 1234–1237 (1996). [CrossRef]
11. P. Lan, E. J. Takahashi, and K. Midorikawa, “Isolated-attosecond-pulse generation with infrared double optical gating,” Phys. Rev. A 83(6), 063839 (2011). [CrossRef]
12. P. K. Maroju, C. Grazioli, M. Di Fraia, M. Moioli, D. Ertel, H. Ahmadi, O. Plekan, P. Finetti, E. Allaria, L. Giannessi, G. De Ninno, C. Spezzani, G. Penco, S. Spampinati, A. Demidovich, M. B. Danailov, R. Borghes, G. Kourousias, C. E. Sanches Dos Reis, F. Bille, A. A. Lutman, R. J. Squibb, R. Feifel, P. Carpeggiani, M. Reduzzi, T. Mazza, M. Meyer, S. Bengtsson, N. Ibrakovic, E. R. Simpson, J. Mauritsson, T. Csizmadia, M. Dumergue, S. Kuhn, H. Nandiga Gopalakrishna, D. You, K. Ueda, M. Labeye, J. E. Baekhoj, K. J. Schafer, E. V. Gryzlova, A. N. Grum-Grzhimailo, K. C. Prince, C. Callegari, and G. Sansone, “Attosecond pulse shaping using a seeded free-electron laser,” Nature 578(7795), 386–391 (2020). [CrossRef]
13. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nature Phys 7(2), 138–141 (2011). [CrossRef]
14. S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110(6), 061101 (2017). [CrossRef]
15. S. C. Jiang, S. Gholam-Mirzaei, E. Crites, J. E. Beetar, M. Singh, R. F. Lu, M. Chini, and C. D. Lin, “Crystal symmetry and polarization of high-order harmonics in ZnO,” J. Phys. B: At. Mol. Opt. Phys. 52(22), 225601 (2019). [CrossRef]
16. K. Kaneshima, Y. Shinohara, K. Takeuchi, N. Ishii, K. Imasaka, T. Kaji, S. Ashihara, K. L. Ishikawa, and J. Itatani, “Polarization-Resolved Study of High Harmonics from Bulk Semiconductors,” Phys. Rev. Lett. 120(24), 243903 (2018). [CrossRef]
17. Y. S. You, David A. Reis, and S. Ghimire, “Anisotropic high-harmonic generation in bulk crystals,” Nature Phys 13(4), 345–349 (2017). [CrossRef]
18. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation,” Nano Lett. 13(7), 3329–3333 (2013). [CrossRef]
19. H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nature Phys 13(3), 262–265 (2017). [CrossRef]
20. K. Liu, L. Zhang, T. Cao, C. Jin, D. Qiu, Q. Zhou, A. Zettl, P. Yang, S. G. Louie, and F. Wang, “Evolution of interlayer coupling in twisted molybdenum disulfide bilayers,” Nat Commun 5(1), 4966 (2014). [CrossRef]
21. F. Langer, C. P. Schmid, S. Schlauderer, M. Gmitra, J. Fabian, P. Nagler, C. Schuller, T. Korn, P. G. Hawkins, J. T. Steiner, U. Huttner, S. W. Koch, M. Kira, and R. Huber, “Lightwave valleytronics in a monolayer of tungsten diselenide,” Nature 557(7703), 76–80 (2018). [CrossRef]
22. F. Langer, M. Hohenleutner, C. P. Schmid, C. Poellmann, P. Nagler, T. Korn, C. Schuller, M. S. Sherwin, U. Huttner, J. T. Steiner, S. W. Koch, M. Kira, and R. Huber, “Lightwave-driven quasiparticle collisions on a subcycle timescale,” Nature 533(7602), 225–229 (2016). [CrossRef]
23. Y. Bai, F. Fei, S. Wang, N. Li, X. Li, F. Song, R. Li, Z. Xu, and P. Liu, “High-harmonic generation from topological surface states,” Nat. Phys. 17(3), 311–315 (2021). [CrossRef]
24. D. Bauer and K. K. Hansen, “High-Harmonic Generation in Solids with and without Topological Edge States,” Phys. Rev. Lett. 120(17), 177401 (2018). [CrossRef]
25. L. Jia, Z. Zhang, D. Z. Yang, M. S. Si, G. P. Zhang, and Y. S. Liu, “High harmonic generation in magnetically-doped topological insulators,” Phys. Rev. B 100(12), 125144 (2019). [CrossRef]
26. C. P. Schmid, L. Weigl, P. Grossing, V. Junk, C. Gorini, S. Schlauderer, S. Ito, M. Meierhofer, N. Hofmann, D. Afanasiev, J. Crewse, K. A. Kokh, O. E. Tereshchenko, J. Gudde, F. Evers, J. Wilhelm, K. Richter, U. Hofer, and R. Huber, “Tunable non-integer high-harmonic generation in a topological insulator,” Nature 593(7859), 385–390 (2021). [CrossRef]
27. C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016). [CrossRef]
28. N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017). [CrossRef]
29. G. Ndabashimiye, S. Ghimire, M. X. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016). [CrossRef]
30. Y. S. You, Y. C. Yin, Y. Wu, A. Chew, X. M. Ren, F. J. Zhuang, S. Gholam-Mirzaei, M. Chini, Z. H. Chang, and S. Ghimire, “High-harmonic generation in amorphous solids,” Nat Commun 8(1), 724 (2017). [CrossRef]
31. Y. S. You, M. X. Wu, Y. C. Yin, A. Chew, X. M. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. H. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017). [CrossRef]
32. T. Y. Du, D. Tang, and X. B. Bian, “Subcycle interference in high-order harmonic generation from solids,” Phys. Rev. A 98(6), 063416 (2018). [CrossRef]
33. L. J. Lu and X. B. Bian, “Multielectron interference of intraband harmonics in solids,” Phys. Rev. B 100(21), 214312 (2019). [CrossRef]
34. Y.-L. He, J. Guo, F.-Y. Gao, Z.-J. Yang, S.-Q. Zhang, and X.-S. Liu, “Interference between harmonics of different crystal momentum channels in solid high-order harmonic generation,” Phys. Rev. A 104(1), 013104 (2021). [CrossRef]
35. Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral Interference in High Harmonic Generation from Solids,” ACS Photonics 6(4), 851–857 (2019). [CrossRef]
36. A. A. Lanin, E. A. Stepanov, A. B. Fedotov, and A. M. Zheltikov, “Mapping the electron band structure by intraband high-harmonic generation in solids,” Optica 4(5), 516 (2017). [CrossRef]
37. G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-Optical Reconstruction of Crystal Band Structure,” Phys. Rev. Lett. 115(19), 193603 (2015). [CrossRef]
38. T. T. Luu and H. J. Worner, “Measurement of the Berry curvature of solids using high-harmonic spectroscopy,” Nat Commun 9(1), 916 (2018). [CrossRef]
39. R. E. F. Silva, A. Jimenez-Galan, B. Amorim, O. Smirnova, and M. Ivanov, “Topological strong-field physics on sub-laser-cycle timescale,” Nat. Photonics 13(12), 849–854 (2019). [CrossRef]
40. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]
41. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. Lhuillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]
42. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical Analysis of High-Harmonic Generation in Solids,” Phys. Rev. Lett. 113(7), 073901 (2014). [CrossRef]
43. P. Jürgens, B. Liewehr, B. Kruse, C. Peltz, D. Engel, A. Husakou, T. Witting, M. Ivanov, M. J. J. Vrakking, T. Fennel, and A. Mermillod-Blondin, “Origin of strong-field-induced low-order harmonic generation in amorphous quartz,” Nat. Phys. 16(10), 1035–1039 (2020). [CrossRef]
44. G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015). [CrossRef]
45. N. Tancogne-Dejean, O. D. Mucke, F. X. Kartner, and A. Rubio, “Ellipticity dependence of high-harmonic generation in solids originating from coupled intraband and interband dynamics,” Nat Commun 8(1), 745 (2017). [CrossRef]
46. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nature Photon 8(2), 119–123 (2014). [CrossRef]
47. S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the Transition Dipole Amplitude and Phase on the Generation of Odd and Even High-Order Harmonics in Crystals,” Phys. Rev. Lett. 120(25), 253201 (2018). [CrossRef]
48. N. Yoshikawa, K. Nagai, K. Uchida, Y. Takaguchi, S. Sasaki, Y. Miyata, and K. Tanaka, “Interband resonant high-harmonic generation by valley polarized electron-hole pairs,” Nat Commun 10(1), 3709 (2019). [CrossRef]
49. M. D. Perry and J. K. Crane, “High-order harmonic emission from mixed fields,” Phys. Rev. A 48(6), R4051–R4054 (1993). [CrossRef]
50. H. Eichmann, A. Egbert, S. Nolte, C. Momma, B. Wellegehausen, W. Becker, S. Long, and J. K. McIver, “Polarization-dependent high-order two-color mixing,” Phys. Rev. A 51(5), R3414–R3417 (1995). [CrossRef]
51. L. Brugnera, D. J. Hoffmann, T. Siegel, F. Frank, A. Zaïr, J. W. G. Tisch, and J. P. Marangos, “Trajectory Selection in High Harmonic Generation by Controlling the Phase between Orthogonal Two-Color Fields,” Phys. Rev. Lett. 107(15), 153902 (2011). [CrossRef]
52. D. J. Hoffmann, C. Hutchison, A. Zaïr, and J. P. Marangos, “Control of temporal mapping and harmonic intensity modulation using two-color orthogonally polarized fields,” Phys. Rev. A 89(2), 023423 (2014). [CrossRef]
53. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly Efficient High-Harmonic Generation in an Orthogonally Polarized Two-Color Laser Field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
54. T. T. Liu, T. Kanai, T. Sekikawa, and S. Watanabe, “Significant enhancement of high-order harmonics below 10 nm in a two-color laser field,” Phys. Rev. A 73(6), 063823 (2006). [CrossRef]
55. L. Brugnera, F. Frank, D. J. Hoffmann, R. Torres, T. Siegel, J. G. Underwood, E. Springate, C. Froud, E. I. C. Turcu, J. W. G. Tisch, and J. P. Marangos, “Enhancement of high harmonics generated by field steering of electrons in a two-color orthogonally polarized laser field,” Opt. Lett. 35(23), 3994–3996 (2010). [CrossRef]
56. D. Shafir, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and N. Dudovich, “Atomic wavefunctions probed through strong-field light–matter interaction,” Nature Phys 5(6), 412–416 (2009). [CrossRef]
57. Y. Sanari, T. Otobe, Y. Kanemitsu, and H. Hirori, “Modifying angular and polarization selection rules of high-order harmonics by controlling electron trajectories in k-space,” Nat Commun 11(1), 3069 (2020). [CrossRef]
58. D. Tang and X. B. Bian, “Control of high-order harmonic generation in solids by orthogonally polarized laser fields,” Phys. Rev. B 104, 104302 (2021). [CrossRef]
59. C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Polarization-resolved analysis of high-order harmonic generation in monolayer MoS2,” New J. Phys. 22(7), 073046 (2020). [CrossRef]
60. G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, “Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides,” Phys. Rev. B 88(8), 085433 (2013). [CrossRef]
61. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett. 49(6), 405–408 (1982). [CrossRef]
62. Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, “First Principles Calculation of Anomalous Hall Conductivity in Ferromagnetic bcc Fe,” Phys. Rev. Lett. 92(3), 037204 (2004). [CrossRef]
