Ultra-strong nonlinear optical processes and trigonal warping in MoS2 layers

Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, high-order harmonic conversion is typically much less efficient than low-order, due to the weak intrinsic response of the higher-order nonlinear processes. Here we report ultra-strong optical nonlinearities in monolayer MoS2 (1L-MoS2): the third harmonic is 30 times stronger than the second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1L-MoS2 is approximately three times higher than that for graphene, which was reported to have a large χ (3). We explain this by calculating the nonlinear response functions of 1L-MoS2 with a continuum-model Hamiltonian and quantum mechanical diagrammatic perturbation theory, highlighting the role of trigonal warping. A similar effect is expected in all other transition-metal dichalcogenides. Our results pave the way for efficient harmonic generation based on layered materials for applications such as microscopy and imaging.

N onlinear optical phenomena can generate high-energy photons by converting n = 2, 3, 4,… low-energy photons into one high-energy photon. These are usually referred to as second-, third-, and fourth-harmonic generation (SHG, THG, and FHG) 1 . Due to different selection rules 1, 2 , harmonic processes are distinct from optically pumped laser phenomena (e.g., optically pumped amplification 3 ), and other typical single-photon processes (e.g., single-photon excited photoluminescence 1 ), in which the energy of the generated photons is smaller than the pump photons. Multiphoton harmonic processes have been widely exploited for various applications (e.g., all-optical signal processing in telecommunications 1, 4 , medicine 5 , and data storage 6 ), as well as to study various transitions forbidden under low-energy single-photon excitation 5,6 . The physical origin of these processes is the nonlinear polarization induced by an electromagnetic field E. This gives rise to higher harmonic components, the n-th harmonic component amplitude being proportional 1 to |E| n . Quantum mechanically, higher-harmonic generation involves the annihilation of n pump photons and generation of a photon with n times the pump energy. Because an n-th order nonlinear optical process requires n photons to be present simultaneously, the probability of higher-order processes is lower than that of lower order 1 . Thus, higher-order processes are typically weaker and require higher pump intensities 7,8 .
Graphene and related materials are at the center of an everincreasing research effort due to their unique and complementary properties, making them appealing for a wide range of photonic and optoelectronic applications [9][10][11] . Among these, semiconducting transition-metal dichalcogenides (TMDs) are of particular interest due to their direct bandgap when in monolayer (1L) form 12 , leading to an increase in luminescence by a few orders of magnitude compared with the bulk material 12,13 . 1L-MoS 2 has a single layer of Mo atoms sandwiched between two layers of S atoms in a trigonal prismatic lattice. Therefore, in contrast to graphene, it is non-centrosymmetric and belongs to the space group D 1 3h 14 . The lack of spatial inversion symmetry makes 1L-MoS 2 an interesting material for nonlinear optics, since second-order nonlinear processes are present only in non-centrosymmetric materials 1 . However, when stacked, MoS 2 layers are arranged mirrored with respect to one another 14 , therefore MoS 2 with an even number of layers (EN) is centrosymmetric and belongs to the D 3 3d space group 14 , producing no second harmonic (SH) signal. On the other hand, MoS 2 with odd number of layers (ON) is non-centrosymmetric. SHG from 1L-MoS 2 was reported by several groups [14][15][16][17][18][19][20][21] .
Here we present combined experimental and theoretical work on nonlinear harmonic generation in 1L and few-layer (FL) MoS 2 . We report strong THG and FHG from 1L-MoS 2 . THG is more than one order of magnitude larger than SHG, while FHG has the same magnitude as SHG. This is surprising, since one normally expects the intensity of nonlinear optical processes to decrease with n 1, 2 , with the SHG intensity much larger than that in THG and FHG, although even-order processes only exist in non-centrosymmetric materials. Our results show that this expectation is wrong in the case of 1L-MoS 2 . At sufficiently low photon frequencies (in our experiments the photon energy of the pump is 0.8 eV), SHG only probes the low-energy band structure of 1L-MoS 2 . This is nearly rotationally invariant [22][23][24][25][26][27][28][29] , but with corrections due to trigonal warping. It is because of these corrections 23,26,27 , fully compatible with the D 1 3h space group 1 , but reducing the full rotational symmetry of the low-energy bands to a three-fold rotational symmetry 1 , that a finite amplitude of nonlinear harmonic processes can exist at low photon energies in EN-MoS 2 . The lack of spatial inversion symmetry is a necessary but not sufficient condition for the occurrence of SHG. A purely isotropic band structure gives a vanishing SHG signal [30][31][32][33] , despite some terms in the Hamiltonian explicitly breaking inversion symmetry 27,[34][35][36][37] . Terms proportional to the σ z Pauli matrix break inversion symmetry. Breaking the continuous rotational symmetry of isotropic models (e.g., by including trigonal warping) is required to obtain a non-zero second-order response in a twoband system. In hexagonal lattices, trigonal warping is a deviation from purely isotropic bands that emerges as one moves away from the corners K and K′ of the Brillouin zone 23,26,27,[36][37][38] . Since the lattice has a honeycomb structure, this distortion displays a threefold rotational symmetry 23,26,27,[36][37][38] . We demonstrate that the observed THG/SHG intensity ratio can be explained by quantum mechanical calculations based on finite-temperature many-body diagrammatic perturbation theory 39 and low-energy continuummodel Hamiltonians that include trigonal warping 35 . We conclude that, similar to SHG 14-18 , the THG process is sensitive to the number of layers, their symmetry, relative orientation, as well as

Results
Samples. MoS 2 flakes are produced by micromechanical cleavage (MC) of bulk MoS 2 40 onto Si + 285 nm SiO 2 . 1L-MoS 2 and bilayer (2L-MoS 2 ) flakes are identified by a combination of optical contrast 41,42 and Raman spectroscopy 43 . Raman spectra are acquired by a Renishaw micro-Raman spectrometer equipped with a 600 line/mm grating and coupled with an Ar + ion laser at 514.5 nm. Figure 1a shows the MoS 2 flakes studied in this work and their Raman signatures. A reference MC graphene sample is also prepared and placed on a similar substrate.
SHG and THG charcterization. Nonlinear optical measurements are carried out with the set-up shown in Fig. 2 44,45 . As excitation source, we use an erbium-doped mode-locked fiber laser with a 50 MHz repetition rate, maximum average power 60 mW, and pulse duration 150 fs, which yields an estimated pulse peak power of~8 kW 46 . The laser beam is scanned with a galvo mirror and focused on the sample using a microscope objective. The back-scattered second and third harmonic signals are split into different branches using a dichroic mirror and then detected using photomultiplier tubes (PMTs). For two-channel detection, the light is split into two PMTs using a dichroic mirror with 562 nm cutoff. After the dichroic mirror, the detected wavelength range can be further refined using bandpass filters. The light can also be directed to a spectrometer (OceanOptics QE Pro-FL). The average power on the sample is kept between 10 and 28 mW with a typical measurement time~5 μs, which prevents sample damage and enables high signal-to-noise-ratio, even with acquisition time per pixel in the μs range.
SHG and THG images of the MoS 2 sample are shown in Fig. 3a, b. The SH photon energy is~1.6 eV, lower than the bandgap of 1L-MoS 2 12, 13 . This is not unexpected, as harmonic generation can occur when the harmonic energy is below the bandgap 1, 47, 48 . The SHG signal is generated in 1L-MoS 2 , while 2L-MoS 2 appears dark. As discussed above, the secondorder nonlinear response is present in 1L-MoS 2 , which is noncentrosymmetric. However, when stacked to form 2L-MoS 2 , MoS 2 layers are mirrored 14,15 . Therefore, EN-MoS 2 is centrosymmetric 14,15 , and belongs to the D 3 3d space group 14, 15 , producing no SHG signal. On the other hand, ON-MoS 2 flakes are non-centrosymmetric 14,15 .
We note that strong THG is detected compared with SHG, even for 1L-MoS 2 , Fig. 3b. THG was previously reported for a 7L-MoS 2 flake 18 , but here we see it down to 1L-MoS 2 . Reference 49 followed our work 50 and reported THG and SHG from 1L-MoS 2 , giving effective bulk-like second-and third-order susceptibilities χ ð2Þ eff and χ ð3Þ eff of 2.9 × 10 −11 mV −1 and 2.4 × 10 −19 m 2 V −2 , respectively. However, ref. 49 did not provide a detailed explanation of the large THG signal compared to the SHG. Instead it assigned the large THG/SHG ratio to a possible enhancement of THG by the edge of the B exciton. However, refs. 51, 52 demonstrated that SHG is enhanced only when the SHG wavelength overlaps the A or B excitons. A similar behavior is expected for THG. Thus, the explanation in ref. 49 may not be correct. Reference 53 reported high-harmonic (>6th-order) generation in the non-perturbative regime with mid-infrared (IR) excitation (0.3 eV), unlike our THG and FHG results with near-IR excitation (0.8 eV). We do not detect THG from the thickest areas of our flake, with N > 30, as in ref. 18 . The output spectrum in Fig. 3c further confirms that we observe both SHG and THG. Peaks for THG and SHG at 520 and 780 nm can be seen, as well as at 390 nm, corresponding to a four-photon process. This is detected only in 1L-MoS 2 . Its intensity is~5.5 times lower than SHG, and two orders of magnitude smaller than THG.   Fig. 4a. This contrasts ref. 14 , where a pump laser at 810 nm was used. We attribute this difference to the fact that photons generated in the second-order nonlinear process in our set-up with a 1560 nm pump have an energy~1.6 eV (780 nm), below the band gap of 1L-MoS 2 12 , therefore are not adsorbed, unlike the SHG signal in ref. 14 .
Second-and third-order nonlinear susceptibilities. Based on the measured SHG and THG intensities, we can estimate the nonlinear susceptibilities χ (2) and χ (3) . χ (2) can be calculated from the measured average powers of the fundamental and SH signals as follows 54 : where τ is the pulse width, P pump is the average power of the incident fundamental (pump) beam, and P 2ω stands for the generated SH beam power, R is the repetition rate, N a = 0.5 is the numerical aperture, λ 2 = 780 nm is the SH wavelength, τ = τ 2 = 150 fs are the pulse durations at fundamental and SH wavelengths, ϕ ¼ 8π 54  The third-order susceptibility χ SLG is 10 −17 -10 −19 m 2 V −2 . Thus, based on Eq. (2), χ (3) of 1L-MoS 2 is in the same range. This is remarkable, as SLG is known to have a large χ (3)56-59 . Reference 18 reported χ (3) of 7L-MoS 2 to be approximately three orders of magnitude smaller than χ ð3Þ SLG of ref. 56 . χ ð3Þ SLG from ref. 56 is much higher than other theoretical 58 and experimental 49 values. We believe that our measured ratio between 1L-MoS 2 and SLG is more accurate, since we measured both materials at the same time under the same conditions. We note that large discrepancies can be found in earlier reported effective susceptibilities for layered materials (LM). For example, there is a approximately four orders of magnitude difference in χ (3) for SLG (~10 −15 m 2 V −2 in ref. 57 ;~10 −19 m 2 V −2 in ref. 49 ). There is an approximately three orders of magnitude difference in χ (2) reported for 1L-MoS 2 at 800 nm (e.g.,~10 −7 mV −1 in ref. 15 ; and 10 −10 mV −1 in ref. 17 ). Effective susceptibilities are well defined only in three-dimensional materials, since their definition involves a polarization per unit volume 1 . Therefore, given the large discrepancies in literature, it is better to describe the nonlinear processes in LMs using the ratio between the harmonic signal power and the incident pump power (i.e., harmonic conversion efficiency). In this case, when comparing the efficiencies in our measurements with those in ref. 49 , our THG efficiency (~4.76 × 10 −10 ) is~1.4 times larger than that (~3.38 × 10 −10 ) in ref. 49 , while our SHG efficiency (~6.47 × 10 −11 ) is twice that of ref. 49 . Since the effective susceptibilities are not well defined for LMs and also depend on the calculation method, we believe that the conversion efficiency is a better figure of merit for LMs.

Discussion
Our measurements show that the nonlinear response of 1L-MoS 2 and SLG are comparable in magnitude, both revealing stronger nonlinear efficiency than three-dimensional nonlinear materials, such as diamond 1 and quartz 59 . This can be explained by considering their effective Hamiltonians 27,[34][35][36][37]60 . The main contribution to THG is paramagnetic. This is described by the square diagram in Supplementary Notes 1-4 ( Supplementary  Figs. 1-6). This paramagnetic contribution is mainly related to the strong inter-band coupling in the effective Hamiltonian, controlled by large velocity scales, v F % c 300 and v ¼ t 0 a 0 h % 0:65 c 300 for SLG and 1L-MoS 2 , with c the speed of light. The SLG paramagnetic third-harmonic efficiency (PTHE) is proportional to the square of third-order conductivity. Since 39 σ ð3Þ yyyy / v 2 F , we get an overall prefactor v 4 F , which explains the strong nonlinear SLG response. Similarly, for 1L-MoS 2 , the square diagram contains four paramagnetic current vertices, which gives an overall prefactor v 4 , and an integral over the dummy momentum variables, which gives a prefactor 1 v 2 (see Supplementary Note 3). Therefore, the third-order response function, Π ð3Þ yyyy , is proportional to v 2 , which implies a scaling of PTHE as v 4 . Exciton physics is not considered because our experimental conditions only capture off-resonance transitions.
1L-MoS 2 is transparent at this wavelength due to its~1.9 eV gap 12 , while SLG absorbs 2.3% of the light 61 . Therefore, 1L-MoS 2 and other TMDs are promising for integration with waveguides or fibers for all-optical nonlinear devices, such as all-optical modulators and signal processing devices, where materials with nonlinear properties are essential 11 .
The SHG and THG power dependence follows quadratic and cubic trends, Fig. 4b. At our power levels, THG is up to 30 times stronger than SHG. 1L-TMDs have strongly bound excitons that can modify their optical properties [62][63][64] . The exciton resonances also affect their nonlinear optical responses 17, 65, 66 . References 51,55 reported that when the SHG energy is above the A and B excitons, resonance effects are not observed. In our experiments, the energy of 3ω photons is above the A exciton but does not directly overlap with the A or B excitons. Thus, we do not assign the large THG/SHG intensity ratio to an excitonic enhancement, but to the approximate rotational invariance of the 1L-MoS 2 band structure at low energies, which is broken by trigonal warping. SHG is weaker than expected for a non-centrosymmetric material, due to near-isotropic bands contributing to the SHG signal for our low incident photon energies (0.8 eV). Even in the presence of a weak trigonal warping, SHG and THG might be comparable above the threshold for two-and three-photon absorption edges. However, this is not a resonant effect. Resonances only emerge when the laser matches a single level (like an excitonic level) rather than a continuum of states 67 . In our analysis, SHG would be absent without trigonal warping. But, trigonal warping alone cannot explain the magnitude of the FHG signal compared to SHG and THG. Figure 4c compares the THG/SHG ratio from experiments and calculations based on the k·p theory 35 (see Supplementary Note 1) and finite-temperature diagrammatic perturbation theory 39 (see Supplementary Notes 3 and 4). The calculations are a factor 2 smaller than the experiments. Considering the complexity of the nonlinear optical processes and that our calculations ignore high-energy band structure effects 29 and many-body renormalizations 65 , we believe this to be a satisfactory agreement, indicating the importance of trigonal warping in harmonic generation.
FHG generally derives from cascades of lower-order nonlinear multiphoton processes 68 . With an excitation wavelength of 1560 nm, this could be, e.g., a cascade of two SHG processes, where 780 nm photons are first generated through SHG (ω 1560 nm + ω 1560 nm ⇒ ω 780 nm ) and then undergo another SHG process (ω 780 nm + ω 780 nm ⇒ ω 390 nm ). To yield a FHG at 390 nm of the same intensity as SHG at 780 nm in this cascaded process, one would need a conversion efficiency (defined as P 2ω /P pump 1 ) for the second SHG process (i.e., ω 780 nm + ω 780 nm ⇒ ω 390 nm ) to be close to unity. However, we observe a conversion efficiencỹ 10 −10 for SHG. Therefore, we conclude that our FHG does not arise from cascaded SHGs. Another possible cascade process is based on THG (ω 1560 nm + ω 1560 nm + ω 1560 nm ⇒ ω 520nm ) and sum-frequency generation (ω 520 nm + ω 1560 nm ⇒ ω 390 nm ). We find that THG strongly increases up to N = 5, as for Fig. 4a. Therefore, we expect this cascaded process to have a similar trend with N. However, we only observe FHG in 1L-MoS 2 . Thus, we also exclude this cascade process, and conclude that this is a direct χ (4) process.
We now consider the dependence of our results on the elliptical polarization of the incident light. We consider an incident laser beam with arbitrary polarization, i.e., E ¼ E j jε ± witĥ ε ± ¼ b x cosðθÞ ± ib y sinðθÞ. Using the crystal symmetries of 1L-MoS 2 , we derive (see Supplementary Note 2) the following expressions for the second-and third-order polarizations P (2) and P (3) : and Note that θ = 0°corresponds to a linearly polarized laser along the b x direction, perpendicular to the D 1 3h mirror symmetry plane, while θ = 45°corresponds to a circularly polarized laser. From Eq. (3), we expect the intensity of SHG in response to a circularly polarized pump laser to be twice that of a linearly polarized laser. Equation (4) implies vanishing THG in response to a circularly polarized pump laser. We measure the dependence of SHG and THG on elliptical polarization using a linearly polarized laser and a rotating QWP. Depending on the angle θ between the QWP axes and the laser polarization, the excitation light is linearly (θ = 0°+ m·90°) or circularly (θ = 45°+ m·90°) polarized. Figure 5 shows that the experimental data are in agreement with Eqs. (3) and (4). The THG signal is maximum for a linearly polarized excitation laser, while it vanishes for circularly polarized light. SHG is always visible, but its intensity is maximum for circularly polarized light.
Given that harmonic generation is strongly dependent on the symmetry and stacking of layers and that different 1L-TMDs (e.g., WSe 2 , MoSe 2 ) all have similar nonlinear response 11,14,15,21 , one could use heterostructures (e.g., MoS 2 /WSe 2 ) to engineer SHG and other nonlinear processes for high photon-conversion efficiency for a wide range of applications requiring the generation of higher frequencies. This may lead to the use of LMs and heterostructures for applications utilizing optical nonlinearities (e.g., all-optical devices, frequency combs, high-order harmonic generation, multiphoton microscopy, and therapy etc.).

Methods
Determination of MoS 2 thickness from SHG and THG signals. SHG and THG for FL-MoS 2 (N = 1…7) are studied on the flakes in Fig. 6a. SHG and THG images are shown in Fig. 6b, c. At 1560 nm, the contrast between 1 and 3L areas is small, as well as the contrast between 3, 5, and 7 L regions (Fig. 6b).
The THG signal increases up to N = 7, Figs. 4a and 6c. On the other hand, the SHG signal (Fig. 6b) is only generated in ON flakes, due to symmetry 14 . Therefore, areas with intensity between the 3, 5, and 7L regions in Fig. 6c, but dark in SHG, are 4 and 6 L. The dependence of the intensities of THG and SHG on N is plotted in Fig. 4a. The combination of SHG and THG can be used to identify N at least up to 7. The THG signal develops as a function of N. Using Maxwell's equations for a nonlinear medium with thickness t and considering the slowly varying amplitude approximation 1, 69 , we obtain: where I in and I 3ω are the intensities of the incident and THG light, respectively, and χ (3) (−3ω;ω, ω, ω) is the third-order optical susceptibility, n j¼1;3 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ ð1Þ ðjωÞ p , with ϵ ð1Þ the TMD linear dielectric function. Δkt is the phase mismatch between the fundamental and third harmonic generated waves.
For Δkt ≈ 0, THG adds up quadratically with light propagation length (i.e., t ∝ N). The signal starts to saturate for N = 6. The possible reasons for subquadratic signal build-up can be either phase mismatch, or absorption 13 . For THG, Δk = 3k in ± k 3ω , where k in and k 3ω are the wavevectors of the incident and THG signals, respectively, where the plus sign indicates THG generated in the backward direction, while minus identifies forward generated THG. Even for backward generated THG, Δkt ≈ 0 for 6L-MoS 2 (~4.3 nm 70 ). This rules out phase mismatch as the origin of the signal saturation when N ≤ 6. Therefore, we assume that the signal saturation is due to absorption of the third harmonic light.
Diagrammatic nonlinear response theory. To quantify theoretically the strength of nonlinear harmonic generation processes, we generalize the diagrammatic perturbation theory approach 39 to the case of TMDs. We combine this technique with a low-energy k·p model Hamiltonian H k ð Þ for 1L-MoS 2 35 . In such low-energy model, light-matter interactions are treated by employing minimal coupling 35,39 , k → k + eA(t)/ħ, where A(t) is a time-dependent uniform vector potential. Nonlinear response functions are calculated via the multi-legged Feynman diagrams depicted in Supplementary Figs. 2 and 3.
Data availability. The data that support the findings of this study are available from the corresponding author on request.