Abstract
Nonlinear optical materials possess wide applications, ranging from terahertz and midinfrared detection to energy harvesting. Recently, the correlations between nonlinear optical responses and certain topological properties, such as the Berry curvature and the quantum metric tensor, have attracted considerable interest. Here, we report giant roomtemperature nonlinearities in noncentrosymmetric twodimensional topological materials—the Janus transition metal dichalcogenides in the 1 T’ phase, synthesized by an advanced atomiclayer substitution method. High harmonic generation, terahertz emission spectroscopy, and second harmonic generation measurements consistently show ordersofthemagnitude enhancement in terahertzfrequency nonlinearities in 1 T’ MoSSe (e.g., > 50 times higher than 2H MoS_{2} for 18^{th} order harmonic generation; > 20 times higher than 2H MoS_{2} for terahertz emission). We link this giant nonlinear optical response to topological band mixing and strong inversion symmetry breaking due to the Janus structure. Our work defines general protocols for designing materials with large nonlinearities and heralds the applications of topological materials in optoelectronics down to the monolayer limit.
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Introduction
Advances in nonlinear optics empower a plethora of applications, such as attosecond light sources based on high harmonic generation (HHG) and photodetectors for sensitive terahertz (THz) detection at elevated temperatures^{1,2,3,4}. Inherently, the nonlinear optical properties of materials are connected with their magnetic structures^{5,6}, crystalline symmetries^{7,8}, and electronic band topologies. In particular, nontrivial band topologies lead to exotic electronic dynamics and enhanced optical responses^{9,10,11,12,13,14}. Notable examples include anomalous HHG in various classes of topological materials^{15,16,17,18}. The observation of enhanced optical responses in topological materials have been found primarily in threedimensional systems until now^{12,13,15,16,17,19}. Designing twodimensional (2D) platforms with strong optical responses is advantageous for optoelectronic applications at the nanoscale with easy controllability and scalability, but so far is limited to topologically trivial materials such as graphene^{20} and 2Hphase transition metal dichalcogenides (TMDs)^{21}. A promising topologically nontrivial candidate are the monolayer Janus TMDs (JTMDs) in the distorted octahedral (1 T’) phase^{3}. Similar to 1 T’ pristine TMDs^{22,23,24}, 1 T’ JTMDs are topologically nontrivial with an inverted bandgap in the THz regime (tens of meV). Generally, a topologically protected band structure and small electronic bandgap result in larger Berry connections, larger electronic interband transition rate, and thus stronger optical response. In addition, by replacing the top layer chalcogen atoms (e.g., sulfur) in the monolayer 1 T’ TMDs with a different type of chalcogen (e.g., selenium), the resulting Janus structure has strong inversion asymmetry and electric polarization^{25,26}, which can further improve the nonlinear optical response.
In this work, we report experimental observations of giant nonlinearities at THz frequencies in monolayer 1 T’ JTMDs, which are synthesized via a roomtemperature atomiclayer substitution (RTALS) method^{27} under ambient conditions. It is revealed that, although the electromagnetic interaction occurs only in a single monolayer flake of 1 T’ MoSSe (~10–20 µm in transverse size), the generation of midinfrared high harmonics, THz emission, and infrared second harmonic generation are all exceptionally efficient. Further comparison with topologically trivial TMDs and theoretical analyses indicate that the keys to such giant THzfrequency nonlinearities are strong inversion symmetry breaking and topological band mixing. Our results suggest that 1 T’ JTMDs is a promising material class that could lead to an era in THz/infrared sensing using atomicallythin materials. Our results also deepen the understanding of the fundamental mechanisms underlying strong nonlinear optical responses, which could have a profound influence in, for example, roomtemperature THz detection and clean energy harvesting via the bulk photovoltaic effect^{2,28}.
Results
Multimodal nonlinearity characterization of 1 T’ MoSSe
The schematic illustration of multimodal characterization methods is shown in Fig. 1a. Our experiments investigated the THzfrequency nonlinearities of monolayer 1 T’ MoSSe with three different techniques, i.e., high harmonic generation (HHG)^{29,30}, THz emission spectroscopy (TES), and second harmonic generation (SHG). These techniques access nonlinear coefficients with different orders (2nd to 18th order) and spectral ranges (THz to infrared). As a comparison, we also studied the responses of monolayer 2H MoSSe, 1 T’ MoS_{2}, and 2H MoS_{2} under the same measurement conditions. Such combined information unequivocally indicates giant THzfrequency nonlinearities for 1 T’ MoSSe. As shown in Fig. 1b, c, 1 T’ MoSSe and MoS_{2} have distorted octahedral structures, with band inversion between metal dorbitals and chalcogen porbitals^{22}. In contrast, the 2H phase is characterized by a trigonal prismatic structure and is topologically trivial. In this work, Janus 1 T’ MoSSe and 2H MoSSe (Fig. 1d) are respectively converted from 1 T’ MoS_{2} and 2H MoS_{2} by the roomtemperature atomiclayer substitution method^{2}. Highly reactive hydrogen radicals produced by a remote plasma were used to strip the toplayer sulfur atoms. Meanwhile, selenium vapor was supplied in the same lowpressure system to replace the missing sulfur, resulting in the asymmetric Janus MoSSe in 1 T’ phase and 2H phase. To confirm the fidelity of material conversion, Raman scattering measurements were performed due to their sensitivity to the crystal lattice structure (Fig. 1e). For Janus 2H MoSSe, the positions of the A_{1g} mode (~288 cm^{−1}) and E_{2g} mode (~355 cm^{−1}) are consistent with literature results^{2}; Meanwhile, the multiple A’ modes of Janus 1 T’ MoSSe located at ~226.2 cm^{−1}, ~298.4 cm^{−1}, ~429.8 cm^{−1} agree well with the theoretical calculations as well, indicating the successful material substitution.
Efficient highharmonic generation
We first show highly efficient HHG from a single monolayer flake of 1 T’ MoSSe. The excitation source for HHG is midinfrared (MIR) pulses with inplane linear polarization at 5µm wavelength, 1kHz repetition rate, and ~ 20 MV/cm peak field strength (setup schematic shown in Supplementary Fig. 1). The HHG image acquired in 1 T’ MoSSe (Fig. 2a) contains at least up to 18th order response, limited by our detection scheme. The evenorder HHG, which is absent in bulk TMDs^{30}, is a direct consequence of the broken spatial symmetry of the monolayer Janus systems. We varied the incident MIR polarization and observed nearly perfect cancellation of HHG intensity at specific angles to one of the crystallographic axes, indicating the HHG signal originates from a single flake instead of an average over many flakes with random orientations (shown in Supplementary Fig. 4). This is consistent with the laser spot size (1/e^{2} size) ~100 µm and the sparse flakeflake spacing (shown in Supplementary Fig. 4). The HHG intensity of single flake 1 T’ MoSSe are further compared with that of millimeterscale 2H MoS_{2} under the same condition. Despite the irradiated flake being generally ~10 times smaller than the laser spot, the HHG of 1 T’ MoSSe is over an order of magnitude stronger than that of the millimeterscale 2H MoS_{2} with 100% coverage (Fig. 2b–d)^{31}. The strong THz nonlinearity of 1 T’ MoSSe is further confirmed by comparing it with other reference samples (2H MoSSe and 1 T’ MoS_{2}). Figure 2e shows the HHG spectrum of 2H MoSSe, which has much weaker evenorder harmonics than those of 1 T’ MoSSe. Meanwhile, the HHG of 1 T’ MoS_{2}, which is also topological nontrivial^{22}, shows relatively strong oddorder harmonics but no evenorder harmonics, due to the inversion symmetry (Fig. 2f). Further semiquantitative HHG efficiency comparison with other literature^{16,30} is summarized in Table 1 showing clear advantages of 1 T’ MoSSe over most solidstate bulk or film samples.
Enhanced terahertz emission and secondharmonic generation
Dramatic enhancements in the TES and SHG measurements further validate giant nonlinearities in 1 T’ MoSSe. Figure 3a shows the TES measurements under 800nm laser excitation (details in Supplementary Fig. 2) on four kinds of chemical vapor deposition (CVD) grown samples (1 T’ MoSSe, 1 T’ MoS_{2}, 2H MoSSe, and 2H MoS_{2}), among which 1 T’ MoSSe shows distinctly higher THz emission efficiency. We do not observe a detectable signal in 1 T’ MoS_{2} with the same excitation fluence, consistent with its centrosymmetric structure, which forbids secondorder nonlinear response. The weak TES signal in 2H MoS_{2} has been attributed to an inefficient surface photocurrent^{32,33}. The augmented TES in 1 T’ MoSSe aligns with the theory that 1 T’ TMDs exhibit giant nonlinearities at THz frequencies^{3}. The polarization analysis of the THz emission (Fig. 3b) reveals the emitted radiation is mainly polarized in the labframe xdirection and contains a slightly weaker ydirection component (axis definition shown in Fig. 3b). Based on our experimental configuration, the xdirection emission has contributions from both inplane and outofplane photoresponses, while the ydirection emission originates only from inplane photoresponses. Thus, the observation of emission in the ydirection indicates the existence of an inplane current contributing to the TES signal, but the stronger emission in the xdirection indicates there are likely significant contributions as well from outofplane currents. Further experiments are needed to disentangle the inplane and outofplane contributions. For fixed excitation fluence, the peak THz field as a function of the pump polarization exhibits a sinusoidal modulation with a periodicity of approximately \(\pi\) (Fig. 3c), reflecting the ranktwo tensor nature of the photoresponses, which are second order in electric fields. Detailed analysis is included in Supplementary Figs. 5–10. Finally, the TES signal shows a linear dependence on the excitation fluence (Fig. 3d) at low fluences and continues to increase at higher fluences exceeding 60 µJ/cm^{2}, albeit with a smaller slope. Such phenomena are likely due to the combined effects of photocurrent saturation due to carrier generation^{32} and nonlinearly increasing photocurrents (detailed discussion in Supplementary Note 5).
Figure 3e shows the SHG measurements on the four kinds of CVDgrown samples excited with 800nm pulses (details in Supplementary Fig. 3). The SHG of several different flakes from each sample was measured to estimate the average intensity and flaketoflake deviation. 2H MoS_{2}, 1 T’ MoSSe, and 2H MoSSe show high SHG efficiency, and no SHG signal is detected in 1 T’ MoS_{2}. In 1 T’ MoSSe and 2H MoSSe, SHG is further enhanced by a factor of 4 and 3 compared to monolayer 2H MoS_{2} respectively, for which high SHG efficiency has been extensively reported^{21,34,35,36}. This highlights the importance of augmented inversion symmetry breaking in Janus structures, which improves evenorder nonlinearities. The SHG efficiency in Janustype samples is further amplified in an angleresolved SHG measurement (Fig. 3f) that is particularly sensitive to outofplane dipoles^{25}. In this experiment, the incident angle of the 800nm fundamental beam deviates from the normal incidence so that the tilted incident beam provides a vertical electric field and interacts with the outofplane dipoles effectively. To exclude other geometric factors, an spolarized SHG I_{s} induced by an inplane dipole with the same collection efficiency is measured and used to normalize ppolarized SHG I_{p} that contains outofplane dipole contribution at nonnormal incidence. For 1 T’ MoSSe and 2H MoSSe, I_{p}/I_{s} symmetrically increases as a function of the incident angle, while 2H MoS_{2} shows much smaller angledependent changes. This confirms the presence of outofplane dipoles in Janustype samples.
Theoretical origin of giant terahertzfrequency nonlinearity
The experimental results above indicate that the optical nonlinearity of 1 T’ MoSSe can be ordersofmagnitude (e.g., >50 times higher for 18th order HHG; >20 times higher for TES) stronger than those of 2H MoSSe. To understand this effect, we examine the microscopic mechanism underlying the strong THzfrequency nonlinear responses in 1 T’ MoSSe. The band structures of 1 T’ MoSSe is shown in Fig. 4a. The band inversion of 1 T’ MoSSe happens around the \(\varGamma\)point. Due to spinorbit interaction, there is a band reopening at the \(\pm \varLambda\)points (marked in Fig. 4a). When the Fermi level is inside the bandgap, the interband transition dipole (Berry connection) \({r}_{{mn}}(k)\equiv \langle {mk}\leftr\right{nk}\rangle\) plays an essential role in optical processes^{37}, because it determines the strength of the dipole interaction between electrons and the electric fields. Here \(r\) is the position operator, while \({mk}\rangle\) is the electron wavefunction at band \(m\) and wavevector \(k\). In Fig. 4b, c, we plot \(\left{r}_{{vc}}\left(k\right)\right\) of 2H and 1 T’ MoSSe, where \(v\) (\(c\)) denotes the highest valence (lowest conduction) band. For 2H MoSSe, the maximum value of \(\left{r}_{{vc}}\left(k\right)\right\) is around \(\sim 2\,\mathring{\rm A}\) near the bandedge (\(\pm {K}\) points), while for 1 T’ MoSSe, \(\left{r}_{{vc}}\left(k\right)\right\) can reach \(\sim 50\,\mathring{\rm A}\) near the bandedge (\(\pm \varLambda\) points). Consequently, electrons in 1 T’ MoSSe would have stronger dipole interaction and hence faster interband transitions under light illumination. This is attributed to the topological enhancement, that is, band inversions in topological 1 T’ MoSSe lead to wavefunction hybridization and hence larger wavefunction overlap between valence and conduction bands near the band edge, which accelerates the interband transitions^{3,38,39}. The calculated first, second, and thirdorder nonlinear susceptibilities of 2H and 1 T’ MoSSe are shown in Fig. 4d–f. For \(\omega \, \lesssim \, 0.5\,{{{{{\rm{{eV}}}}}}}\), the responses of 1 T’ MoSSe are significantly stronger than those of 2H MoSSe. For \(\omega \, \gtrsim \, 1\,{{{{{\rm{{eV}}}}}}}\), the responses of 1 T’ and 2H MoSSe are relatively close, consistent with experimental HHG, TES, and SHG measurements at different wavelengths. In the insets of Fig. 4d–f, we plot the \(k\)resolved contributions \({I}^{(i)}(k)\) to the optical susceptibility at \(\omega=1\,{{{{{\rm{{eV}}}}}}}\) (see “Methods” for details). Notably, the maximum value of \({I}^{(i)}(k)\) of 1 T’ MoSSe, located around the \(\pm \varLambda\) points, is larger by ordersofmagnitude than that of 2H MoSSe. This indicates that \(k\)points near the \(\pm \varLambda\) points, which are influenced by topological enhancement, make major contributions to the total susceptibility even at \(\omega=1\,{{{{{\rm{{eV}}}}}}}\). Note that the bandgap of 1 T’ MoSSe is on the order of \(10\,{{{{{\rm{{meV}}}}}}}\), and thus interband transitions of electrons near the \(\pm \varLambda\) points are far offresonance with \(\omega=1\,{{{{{\rm{{eV}}}}}}}\) photons. However, the contributions near the \(\pm \varLambda\) points still dominate those at other \(k\)points where resonant interband transitions could happen. This again suggests the importance of the topological enhancement and the large interband transition dipoles near the \(\pm \varLambda\) points. The topological enhancement gradually decays at large \(\omega\). Consequently, the optical responses could be stronger in 2H MoSSe with \(\omega \, \gtrsim \, 1\,{{{{{\rm{{eV}}}}}}}\). Other inplane elements of the SHG tensor are shown in Supplementary Fig. 14. We also note that the theoretical calculations here should only be considered as qualitative estimations, and several issues can lead to inaccuracies. For example, the density functional theory calculations suffer from intrinsic errors regarding some electronic properties, including bandgaps. Some manybody interactions are also ignored in the calculations here. Besides, the theoretical calculations deal with ideal materials, which should be distinguished from the real samples used in experimental that are influenced by doping levels, etc. Future theoretical and experimental developments could yield more accurate information for quantitative theoryexperiment comparison.
Discussion
The giant nonlinearities of 1 T’ JTMD, corroborated by both experimental and theoretical results above, support the giant THz frequency photocurrent responses of 1 T’ JTMDs predicted by theory^{3} and prelude that 1 T’ JTMD could serve as efficient darkcurrentfree THz detectors via the nonlinear bulk photovoltaic effect^{10}. Our calculations indicate that the intrinsic photoresponsivity and noise equivalent power of the 1 T’ JTMD THz detector can outperform many current roomtemperature THz sensors based on Schottky diodes or silicon fieldeffect transistors^{40,41}, albeit lower than the best pyroelectric detectors and bolometers^{40} (Fig. 4g, see also Supplementary Note 1 and Supplementary Fig. 11). We foresee stacking multiple monolayer 1 T’ JTMDs and using fieldenhancement structures^{42} can further enhance the responsivity^{43} and enable a facile usage of this detector for THz sensing purposes.
In conclusion, we demonstrate giant nonlinear responses in monolayer 1 T’ MoSSe, a prototype Janus topological semiconductor. Comparative experiments with different crystal phases (2H vs. 1 T’) and symmetry types (Janus vs. nonJanus) indicate that 1 T’ MoSSe possesses ordersofmagnitude enhancement in HHG and secondorder THz emission efficiency, and a few times enhancement in infrared SHG. Supported by theoretical calculations, our results elucidate that the remarkable enhancements originate from augmented structural asymmetry in Janustype structures and topological bandmixing in 1 T’ phases. The boosted HHG efficiency and the high fabrication versatility^{27} of 1 T’ JTMDs prelude a plethora of applications in lightwave electronics^{44,45} in the monolayer limit. Meanwhile, the giant THzfrequency nonlinearities observed in this work could enable THz detection^{46,47} with a large photoresponsivity at subA/W level and noise equivalent power down to the \({{{\mathrm{pW}}}}/\sqrt{{{\rm{Hz}}}}\) level.
Methods
Growth of 1 T′ MoS_{2} monolayer flakes
The precursor K_{2}MoS_{4} was prepared according to the previously reported synthesis procedures^{48}. The growth of 1 T’ MoS_{2} monolayer flakes was carried out in a standard CVD furnace with a 1inch quartz tube under atmospheric pressure. A freshcleaved fluorophlogopite mica substrate with K_{2}MoS_{4} powders were placed in the center of the furnace. After the system was purged with Ar for 10 min, the furnace was heated up to 750 °C in 40 min with 100 sccm Ar. When the temperature of the furnace reached 750 °C, 10 sccm H_{2} was introduced and the flow rate of Ar was decreased to 90 sccm. After 5 min, the mica substrate was rapidly pulled out of the furnace heating zone. After cooling down to room temperature, the 1 T’ MoS_{2} monolayer flakes were obtained.
Synthesis of 1 T’ MoSSe monolayer flakes
The synthesis of monolayer 1 T’ MoSSe is realized by a roomtemperature atomic layer substitution method^{27} from 1 T’ MoS_{2}^{49}. A remote commercial inductively coupled plasma (ICP) system was used to substitute the toplayer sulfur atoms of monolayer 1 T’ MoS_{2} with selenium. The potassiumassisted CVDgrown monolayer 1 T’ MoS_{2} was placed in the middle of a quartz tube. The plasma coil placed at the upstream of CVD furnace. The distance between the sample and the plasma coil is around 10 cm, with the selenium powder placed on the other side. At the beginning of the process, the whole system was pumped down to a low mTorr to remove air in the chamber. Then, hydrogen was introduced into the system with 10 sccm and the plasma generator was ignited for 20 min. The hydrogen atoms assist the removal of the sulfur atoms on the top layer of MoS_{2}, at the same time, the vaporized selenium filled in the vacancy of the sulfur atoms, resulting in the asymmetric Janus topological structure of MoSSe. The whole process was performed at room temperature. After the reaction, Ar gas was purged into the system with 100 sccm to remove the residual reaction gas, and the pressure was recovered to atmospheric.
HHG, THz emission, and SHG setups
For HHG, the fundamental laser beam has a wavelength of 5.0 μm with a repetition rate of 1 kHz and a pulse duration of ~70 fs. It is generated by the difference frequency of the signal and idler beam from an optical parametric amplifier pumped with a Ti: Sapphire chirpedpulse amplifier (6 mJ, 1 kHz). The fundamental MIR beam with a 10 µJ pulse energy was focused on the sample with a ZnSe lens with a focal length of 15 cm. The measurements are performed in a transmission geometry at normal incidence. Generated HHG signals transmitted through the sample are collected by a CaF_{2} lens and directed and dispersed in a spectrometer (Princeton Instruments HRS300) and detected by a chargecoupled device (CCD) camera (Princeton Instruments PIXIS 400B). We note that the distorted shapes of highorder harmonics are due to chromatic aberration when focusing and imaging the HHG from the entrance slit of the spectrometer to the CCD camera.
For THz emission measurement, the ultrafast laser excitation was provided by a modelocked Ti:Sapphire laser with pulses of 40fs (FWHM) duration and 5.12MHz repetition rate. After focusing the laser beam on the sample, the refocused THz radiation from the sample was detected using the electrooptic (EO) effect in a noncentrosymmetric crystal (1mmthick ZnTe or 258µmthick GaP). The induced birefringence in the EO crystal was recorded at different delay times by a laser probe pulse passing through a polarizing beamsplitter (Wollaston prism) and impinging on a balanced photodetector. The power imbalance was fed into a lockin amplifier synchronized with modulation of the excitation beam at 320 kHz by an acoustooptic modulator. By scanning the time delay between the excitation and probe pulses, the temporal profile of the transient THz electric field could be mapped. A pair of wiregrid polarizers were used to determine the THz emission polarization.
For SHG measurement, the fundamental pulses are provided by a modelocked Ti:Sapphire oscillator at 800nm wavelength, 5.12MHz repetition rate, and 40fs pulse duration. They are focused on the sample with a ×20 objective, and the generated SHG light from a single flake is filtered and detected by a photomultiplier tube. For outofplane measurement, a collimated ppolarized pump beam with a 1 mm spot size is guided to the objective back aperture (D = 7.6 mm). The beam was focused at the sample with a tilted angle and generated an oscillating vertical electrical field to drive the outofplane dipole for SHG. The SHG (green) is collected with the same objective and analyzed by a polarizer. The beam position at the objective back aperture can be scanned along the xdirection with a motorized stage, which tunes the incident angle accordingly.
Ab initio calculations
The ab initio density functional theory (DFT)^{50,51} calculations are performed using the Vienna ab initio simulation package (VASP)^{52,53}. The exchangecorrelation interactions are included using generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE)^{54}. Core and valence electrons are respectively treated by projector augmented wave (PAW) method^{55} and planewave basis functions. The first Brillouin zone is sampled by a \(13\times 17\times 1\) and \(17\times 17\times 1\) \(k\)mesh for 1 T’ and 2H structures, respectively.
Nonlinear optical susceptibility calculations
After the DFT results are obtained, a tightbinding (TB) Hamiltonian in the Wannier basis is built using the Wannier90 package^{56}. The TB Hamiltonian is then used to interpolate the relevant properties on a denser \(k\)mesh.
The firstorder susceptibility is calculated within the velocity gauge
Here \(m,n\) label the electron states, while \({v}_{{mn}}^{i}\equiv \left\langle m\left{v}^{i}\rightn\right\rangle\) is the velocity operator. \({E}_{{mn}}\) and \({f}_{{mn}}\) are respectively the difference in energy and occupation number between \(\leftm\right\rangle\) and \(\leftn\right\rangle\). \({\varepsilon }_{0}\) is the vacuum permittivity. The secondorder susceptibility is calculated within the length gauge^{57}
where
Here the interband position matrix is \({r}_{{mn}}^{i}=\frac{{{\hslash }}{v}_{{mn}}^{i}}{{{iE}}_{{mn}}}\) for \(m\, \ne \, n\) and \({r}_{{mn}}^{i}=0\) for \(m=n\). \({\Delta }_{{mn}}^{i}\equiv {v}_{{mm}}^{i}{v}_{{nn}}^{i}\) is the difference in band velocities. Meanwhile, for two numbers \(A\) and \(B\) one has \(\left\{{AB}\right\}\equiv \frac{1}{2}({AB}+{BA})\).
The ab initio theory for calculating the thirdorder susceptibility is not welldeveloped. Here we use the velocity gauge formula^{58}
Here \({g}_{{mn}}^{i}\equiv \frac{{f}_{{nm}}{v}_{{mn}}^{i}}{{E}_{{nm}}\hslash \omega }\). Equation (6) experiences a spurious divergence in the real part of \({\chi }_{{ijkl}}^{\left(3\right)}\). Therefore, we first calculate the imaginary part of \({\chi }_{{ijkl}}^{\left(3\right)}\), and then obtain the real part from the Kramers–Kronig relations^{59}.
The \({{{{{\boldsymbol{k}}}}}}\)resolved contributions to the total susceptibility \({I}^{\left(i\right)}\left({{{{{\boldsymbol{k}}}}}}\right)\), which are shown in the inset of Fig. 4b–d, are defined as the integrand of the Brillouin zone integration in Eqs. (1, 2, 6). The Brillouin zone integrations is performed by a \({{{{{\boldsymbol{k}}}}}}\)mesh sampling, \(\int \frac{{d}^{3}{{{{{\boldsymbol{k}}}}}}}{{\left(2\pi \right)}^{3}}=\frac{1}{V}{\sum }_{{{{{{\boldsymbol{k}}}}}}}{w}_{{{{{{\boldsymbol{k}}}}}}}\), where \(V\) is the volume of the unit cell and \({w}_{{{{{{\boldsymbol{k}}}}}}}\) is the weight factor. Since 2D materials do not have welldefined volume, we use \(V=S{l}_{{{{{{\rm{eff}}}}}}}\), where \(S\) is the area of the 2D unit cell, while \({l}_{{{{{{\rm{eff}}}}}}}\) is taken as \(6\,\mathring{\rm A}\) for all materials. The convergence in the \({{{{{\boldsymbol{k}}}}}}\)mesh is tested.
Data availability
Relevant data supporting the key findings of this study are available within the article and the Supplementary Information file. All raw data generated during the current study are available from the corresponding authors upon request.
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Acknowledgements
This work was primarily funded through the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DEAC0276SF00515. The high harmonic generation experiments were supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the AMOS program. Y.G. acknowledges the financial support from Zhejiang University. H.X. and J.L. were supported by an Office of Naval Research MURI through grant #N000141712661. E.S. and H.S. acknowledge the financial support from Research Center for Industries of the Future at Westlake University, National Natural Science Foundation of China (grant no. 52272164). J.K. and T.Z. acknowledge the financial support from US Department of Energy (DOE), Office of Science, Basic Energy Sciences under Award DE‐SC0020042.
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J.S., H.X., Y.G. and A.L. designed the study; Y.G. performed the Janus material synthesis and Raman characterization; H.X. and J.L. performed the theoretical analyses and ab initio calculations.; C.H. and J.S. performed HHG measurements under the supervision of S.G. and A.L.; C.X. and J.S. performed TES measurements under the supervision of A.L.; C.H.F. synthesized typical transition metal dichalcogenides under the supervision of L.J.; J.S., F.Q. and L.Y. performed SHG measurements under the supervision of A.L. and T.H.; A.J. synthesized waferscale 2H MoS2 under the supervision of F.L.; H.S., T.Z., E.S. and J.K. participated in data analysis; J.S., H.X. and Y.G. wrote the manuscript; All authors read and revised the manuscript.
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Shi, J., Xu, H., Heide, C. et al. Giant roomtemperature nonlinearities in a monolayer Janus topological semiconductor. Nat Commun 14, 4953 (2023). https://doi.org/10.1038/s4146702340373z
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DOI: https://doi.org/10.1038/s4146702340373z
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