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

Ultra-strong nonlinear optical processes and trigonal warping in MoS2 layers ARTICLE DOI: 10.1038/s41467-017-00749-4 OPEN Ultra-strong nonlinear optical processes and trigonal warping in MoS layers 1,2 1 3 1 4 Antti Säynätjoki , Lasse Karvonen , Habib Rostami , Anton Autere , Soroush Mehravar , 5 4 5 1,2,4 1 Antonio Lombardo , Robert A. Norwood , Tawfique Hasan , Nasser Peyghambarian , Harri Lipsanen , 4 5 3 1 Khanh Kieu , Andrea C. Ferrari , Marco Polini & Zhipei Sun Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, high-order har- monic 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 non- linearities in monolayer MoS (1L-MoS ): the third harmonic is 30 times stronger than the 2 2 second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1L-MoS is approximately three times higher than that for graphene, which was reported (3) to have a large χ . We explain this by calculating the nonlinear response functions of 1L- MoS with a continuum-model Hamiltonian and quantum mechanical diagrammatic pertur- bation 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. 1 2 Department of Electronics and Nanoengineering, Aalto University, Tietotie 3, FI-02150 Espoo, Finland. Institute of Photonics, University of Eastern Finland, 3 4 Yliopistokatu 7, FI-80100 Joensuu, Finland. Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy. College of Optical Sciences, University of Arizona, 1630 E University Blvd, Tucson, AZ 85721, USA. Cambridge Graphene Centre, University of Cambridge, Cambridge CB3 0FA, UK. Antti Säynätjoki, Lasse Karvonen and Habib Rostami contributed equally to this work. Correspondence and requests for materials should be addressed to Z.S. (email: zhipei.sun@aalto.fi) NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 1 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 onlinear optical phenomena can generate high-energy no second harmonic (SH) signal. On the other hand, MoS with photons by converting n = 2, 3, 4,… low-energy photons odd number of layers (ON) is non-centrosymmetric. SHG from 1L- 14–21 Ninto one high-energy photon. These are usually referred to MoS was reported by several groups . as second-, third-, and fourth-harmonic generation (SHG, THG, Here we present combined experimental and theoretical work 1 1, 2 and FHG) . Due to different selection rules , harmonic processes on nonlinear harmonic generation in 1L and few-layer (FL) MoS . are distinct from optically pumped laser phenomena (e.g., optically We report strong THG and FHG from 1L-MoS . THG is more pumped amplification ), and other typical single-photon processes than one order of magnitude larger than SHG, while FHG has the (e.g., single-photon excited photoluminescence ), in which the same magnitude as SHG. This is surprising, since one normally energy of the generated photons is smaller than the pump photons. expects the intensity of nonlinear optical processes to decrease 1, 2 Multiphoton harmonic processes have been widely exploited for with n , with the SHG intensity much larger than that in THG various applications (e.g., all-optical signal processing in tele- and FHG, although even-order processes only exist in non-cen- 1, 4 5 6 communications ,medicine , and data storage ), as well as to trosymmetric materials. Our results show that this expectation is study various transitions forbidden under low-energy single-photon wrong in the case of 1L-MoS . At sufficiently low photon fre- 5, 6 excitation . The physical origin of these processes is the nonlinear quencies (in our experiments the photon energy of the pump is polarization induced by an electromagnetic field E. This gives rise to 0.8 eV), SHG only probes the low-energy band structure of 1L- 22–29 higher harmonic components, the n-th harmonic component MoS . This is nearly rotationally invariant , but with correc- 1 n amplitude being proportional to |E| . Quantum mechanically, tions due to trigonal warping. It is because of these 23, 26, 27 1 1 higher-harmonic generation involves the annihilation of n pump corrections , fully compatible with the D space group , 3h photons and generation of a photon with n times the pump energy. but reducing the full rotational symmetry of the low-energy bands Because an n-th order nonlinear optical process requires n photons to a three-fold rotational symmetry ,thata finite amplitude of to be present simultaneously, the probability of higher-order pro- nonlinear harmonic processes can exist at low photon energies in cessesislower than that of lowerorder . Thus, higher-order pro- EN-MoS . The lack of spatial inversion symmetry is a necessary 7, 8 cesses are typically weaker and require higher pump intensities . but not sufficient condition for the occurrence of SHG. A purely 30–33 Graphene and related materials are at the center of an ever- isotropic band structure gives a vanishing SHG signal , despite increasing research effort due to their unique and complementary some terms in the Hamiltonian explicitly breaking inversion 27, 34–37 properties, making them appealing for a wide range of photonic and symmetry . Terms proportional to the σ Pauli matrix 9–11 optoelectronic applications . Among these, semiconducting break inversion symmetry. Breaking the continuous rotational transition-metal dichalcogenides (TMDs) are of particular interest symmetry of isotropic models (e.g., by including trigonal warping) due to their direct bandgap when in monolayer (1L) form ,leading is required to obtain a non-zero second-order response in a two- to an increase in luminescence by a few orders of magnitude band system. In hexagonal lattices, trigonal warping is a deviation 12, 13 compared with the bulk material . 1L-MoS has a single layer of from purely isotropic bands that emerges as one moves away from 23, 26, 27, 36–38 Mo atoms sandwiched between two layers of S atoms in a trigonal the corners K and K′ of the Brillouin zone . Since the prismatic lattice. Therefore, in contrast to graphene, it is non-cen- lattice has a honeycomb structure, this distortion displays a three- 1 14 23, 26, 27, 36–38 trosymmetric and belongs to the space group D .The lack of fold rotational symmetry . We demonstrate that the 3h spatial inversion symmetry makes 1L-MoS an interesting material observed THG/SHG intensity ratio can be explained by quantum for nonlinear optics, since second-order nonlinear processes are mechanical calculations based on finite-temperature many-body 1 39 present only in non-centrosymmetric materials . However, when diagrammatic perturbation theory and low-energy continuum- stacked, MoS layers are arranged mirrored with respect to one model Hamiltonians that include trigonal warping . We conclude 14 14–18 another , therefore MoS with an even number of layers (EN) is that, similar to SHG , the THG process is sensitive to the 3 14 centrosymmetric and belongs to the D space group ,producing number of layers, their symmetry, relative orientation, as well as 3d ab 1g 2g 5L-MoS 1L 2L 2L 2L-MoS 1L 2L 1L 1L-MoS 360 380 400 420 440 20 µm 5L –1 Raman shift (cm ) Fig. 1 Optical image and Raman spectra of the MoS flakes. a Optical micrograph with single-layer, bilayer, and five-layer areas marked by 1L, 2L, and 5L. b Raman spectra of the same sample 2 NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications | | | Intensity (a.u.) PMT NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 ARTICLE Collimator ab QWP VA SHG THG Galvo scanner Scans lens MLL Power (pW) Power (nW) 0 4.5 0 3.4 Dichroic mirror (562 nm) THG Tube lens Collection lens 520 nm BP filter 0.1 SHG 0.01 Dichroic mirror FHG PMT (870 nm) 1E–3 780 nm BP filter 1E–4 20X 1E–5 400 500 600 700 800 Wavelength (nm) Sample Fig. 3 Multiphoton images of MoS flakes. a SHG and b THG map of the flakes in Fig. 1a. c Optical spectrum of the nonlinear signal from 1L-MoS −2 with a peak irradiance ~30 GW cm Fig. 2 Schematic diagram of multiphoton microscope. MLL, linearly polarized mode-locked fiber laser. VA, variable attenuator. QWP, quarter- wave plate used to study the dependence of SHG and THG on the elliptical polarization of the pump light. BP filter, bandpass filter. PMT, and enables high signal-to-noise-ratio, even with acquisition time photomultiplier tube per pixel in the μs range. SHG and THG images of the MoS sample are shown in Fig. 3a, b. The SH photon energy is ~1.6 eV, lower than the 12, 13 bandgap of 1L-MoS . This is not unexpected, as harmonic the elliptical polarization of the excitation light. Similar effects are generation can occur when the harmonic energy is below expected for all other TMDs. This paves the way for the assembly 1, 47, 48 the bandgap . The SHG signal is generated in 1L-MoS , of heterostructures with tailored nonlinear optical properties. while 2L-MoS appears dark. As discussed above, the second- order nonlinear response is present in 1L-MoS , which is non- Results centrosymmetric. However, when stacked to form 2L-MoS , 14, 15 Samples. MoS flakes are produced by micromechanical cleavage MoS layers are mirrored . Therefore, EN-MoS is centro- 2 2 14, 15 3 14, 15 (MC) of bulk MoS onto Si + 285 nm SiO . 1L-MoS and bilayer 2 2 2 symmetric , and belongs to the D space group , 3d (2L-MoS ) flakes are identified by a combination of optical 2 producing no SHG signal. On the other hand, ON-MoS flakes 41, 42 43 14, 15 contrast and Raman spectroscopy . Raman spectra are are non-centrosymmetric . acquired by a Renishaw micro-Raman spectrometer equipped We note that strong THG is detected compared with SHG, even with a 600 line/mm grating and coupled with an Ar ion laser at for 1L-MoS ,Fig. 3b. THG was previously reported for a 7L-MoS 2 2 18 49 514.5 nm. Figure 1a shows the MoS flakes studied in this work 2 flake , but here we see it down to 1L-MoS . Reference followed and their Raman signatures. A reference MC graphene sample is our work and reported THG and SHG from 1L-MoS , giving ð2Þ also prepared and placed on a similar substrate. effective bulk-like second- and third-order susceptibilities χ eff ð3Þ −11 −1 −19 2 −2 and χ of 2.9 × 10 mV and 2.4 × 10 m V , respectively. eff However, ref. did not provide a detailed explanation of the large SHG and THG charcterization. Nonlinear optical measurements THG signal compared to the SHG. Instead it assigned the large 44, 45 are carried out with the set-up shown in Fig. 2 . As excitation THG/SHG ratio to a possible enhancement of THG by the edge of source, we use an erbium-doped mode-locked fiber laser with a 51, 52 the B exciton. However, refs. demonstrated that SHG is 50 MHz repetition rate, maximum average power 60 mW, and enhanced only when the SHG wavelength overlaps the A or B pulse duration 150 fs, which yields an estimated pulse peak power excitons. A similar behavior is expected for THG. Thus, the of ~8 kW . The laser beam is scanned with a galvo mirror 49 53 explanationinref. may not be correct. Reference reported and focused on the sample using a microscope objective. The high-harmonic (>6th-order) generation in the non-perturbative back-scattered second and third harmonic signals are split into regime with mid-infrared (IR) excitation (0.3 eV), unlike our THG different branches using a dichroic mirror and then detected and FHG results with near-IR excitation (0.8 eV). We do not detect using photomultiplier tubes (PMTs). For two-channel detection, THG from the thickest areas of our flake, with N > 30, as in ref. . the light is split into two PMTs using a dichroic mirror with The output spectrum in Fig. 3c further confirms that we observe 562 nm cutoff. After the dichroic mirror, the detected wavelength both SHG and THG. Peaks for THG and SHG at 520 and 780 nm range can be further refined using bandpass filters. The light can can be seen, as well as at 390 nm, corresponding to a four-photon also be directed to a spectrometer (OceanOptics QE Pro-FL). The process. This is detected only in 1L-MoS . Its intensity is ~5.5 times average power on the sample is kept between 10 and 28 mW with lower than SHG, and two orders of magnitude smaller than THG. a typical measurement time ~5 μs, which prevents sample damage NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 3 | | | Intensity (a.u.) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 (780 nm), below the band gap of 1L-MoS , therefore are not a 0.4 15 adsorbed, unlike the SHG signal in ref. . SHG Second- and third-order nonlinear susceptibilities. Based on 0.3 the measured SHG and THG intensities, we can estimate the (2) (3) (2) 10 nonlinear susceptibilities χ and χ . χ can be calculated from the measured average powers of the fundamental and SH signals 0.2 as follows : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 5 ϵ cλ P RτðÞ n þ 1ðÞ n þ 1 0 2ω 2 1 ð2Þ ð1Þ χ ¼ ; 0.1 s 32N τ P ϕ 2 pump where τ is the pulse width, P is the average power of the pump incident fundamental (pump) beam, and P stands for 0 0 2ω 1 234 567 the generated SH beam power, R is the repetition rate, N = 0.5 Number of layers is the numerical aperture, λ = 780 nm is the SH wavelength, τ = τ = 150 fs are the pulse durations at fundamental and R pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 b 1 2 SH wavelengths, ϕ ¼ 8π jcos ρ  ρ 1  ρ j ρ dρ ¼ 3:56 from ref. , and n ¼ n  1:45 are the refractive indexes of 1 2 the substrate at the wavelengths of the fundamental and SHG, respectively. The effective bulk-like second-order susceptibility of ð2Þ ð2Þ ð2Þ MoS ðχ Þ can be obtained from Eq. (1) with χ ¼ , where eff eff t MoS 10, 24 t ¼ 0:65 nm is the 1L-MoS thickness . We obtain the MoS 2 ð2Þ −1 effective second-order susceptibility χ  2:2 pmV for 1L- eff MoS . Reference reported a bulk-like second-order suscept- −1 ibility 29 pmV , which is ~10 times larger than here. However, −1 20, 52, 55 0.1 several other studies reported ~5 pmV for 1560 nm . ð2Þ Thus, our measured χ agrees well with earlier values measured eff with similar excitation wavelength. ð3Þ The third-order susceptibility χ can be estimated eff 0.5 1 1.5 2 by comparing the measured THG signal from MoS to that of Pump peak power on sample (kW) 1L-graphene (SLG): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t THG ð3Þ SLG MoS ð3Þ k a = 0.85 c 0 ð2Þ χ  χ : SLG eff 64 t THG k a = 0.9 MoS SLG c 0 k a = 0.95 c 0 with t = 0.33 nm the SLG thickness, and THG and SLG SLG k a = 1.0 c 0 THG the measured signals from SLG and MoS , respectively. MoS 2 16 2 EXP. Our results show that THG from 1L-MoS is around three times (3) larger than THG , which indicates that χ of 1L-MoS is SLG 2 comparable to that of SLG, in the frequency range of our 4 ð3Þ 49, 56 experiments. Previous reports indicate that χ is SLG −17 −19 2 −2 (3) ~10 –10 m V . Thus, based on Eq. (2), χ of 1L-MoS is in the same range. This is remarkable, as SLG is known to have (3)56–59 18 (3) a large χ . Reference reported χ of 7L-MoS to be ð3Þ approximately three orders of magnitude smaller than χ of SLG 200 400 800 1800 ð3Þ 56 56 58 ref. . χ from ref. is much higher than other theoretical SLG P (W) 49 pump and experimental values. We believe that our measured ratio between 1L-MoS and SLG is more accurate, since we measured Fig. 4 Experimental and theoretical nonlinear optical processes in MoS . both materials at the same time under the same conditions. a SHG and THG intensities as functions of N. b Power dependence of We note that large discrepancies can be found in earlier reported SHG and THG in 1L-MoS . c Experimental and theoretical THG/SHG effective susceptibilities for layered materials (LM). For example, irradiance ratio as a function of P . Different theoretical curves refer to pump (3) pffiffiffi there is a approximately four orders of magnitude difference in χ different values of the ultra-violet cutoff k (in units of 1=a ¼ 3=a; where c 0 −15 2 −2 57 −19 2 −2 49 for SLG (~10 m V in ref. ;~10 m V in ref. ). There is a = 3.16 Å is the lattice constant of 1L-MoS ). Black dashed lines in panels (2) an approximately three orders of magnitude difference in χ a and b are a guide to the eye. The error bars in c account for experimental −7 −1 15 reported for 1L-MoS at 800 nm (e.g., ~10 mV in ref. ;and uncertainties −10 −1 17 ~10 mV in ref. ). Effective susceptibilities are well defined only in three-dimensional materials, since their definition involves a polarization per unit volume . Therefore, given the large discre- SHG signals on areas with N = 3, 5, 7 have nearly the same pancies in literature, it is better to describe the nonlinear processes intensity as 1L-MoS , Fig. 4a. This contrasts ref. , where a pump in LMs using the ratio between the harmonic signal power and the laser at 810 nm was used. We attribute this difference to the fact incident pump power (i.e., harmonic conversion efficiency). In this that photons generated in the second-order nonlinear process in case, when comparing the efficiencies in our measurements with 49 −10 our set-up with a 1560 nm pump have an energy ~1.6 eV those in ref. , our THG efficiency (~4.76 × 10 )is~1.4times 4 NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications | | | THG THG SHG Detected peak power (µW) THG SHG THG peak power (µW) I / I SHG peak power (µW) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 ARTICLE resonance effects are not observed. In our experiments, the energy THG (pW) of 3ω photons is above the A exciton but does not directly SHG x 37 (pW) 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 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 270 90 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 . 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 Fig. 5 SHG and THG dependence on the pump light polarization. The polar calculations based on the k·p theory (see Supplementary plot angle corresponds to linearly polarized light when θ = 0° + m·90°, and Note 1) and finite-temperature diagrammatic perturbation to circularly polarized pump light when θ = 45° + m·90°. The SHG power is theory (see Supplementary Notes 3 and 4). The calculations are multiplied by a factor of 37 to fit in the same scale as THG a factor 2 smaller than the experiments. Considering the com- plexity of the nonlinear optical processes and that our calculations ignore high-energy band structure effects and −10 49 many-body renormalizations , we believe this to be a satisfactory larger than that (~3.38 × 10 )inref. ,while our SHG efficiency −11 49 agreement, indicating the importance of trigonal warping in (~6.47 × 10 )istwice that of ref. . Since the effective harmonic generation. susceptibilities are not well defined for LMs and also depend on FHG generally derives from cascades of lower-order nonlinear the calculation method, we believe that the conversion efficiency is a multiphoton processes . With an excitation wavelength of better figure of merit for LMs. 1560 nm, this could be, e.g., a cascade of two SHG processes, where 780 nm photons are first generated through SHG Discussion (ω + ω ⇒ ω ) and then undergo another SHG 1560 nm 1560 nm 780 nm Our measurements show that the nonlinear response of 1L-MoS process (ω + ω ⇒ ω ). To yield a FHG at 390 nm 780 nm 780 nm 390 nm and SLG are comparable in magnitude, both revealing stronger of the same intensity as SHG at 780 nm in this cascaded process, nonlinear efficiency than three-dimensional nonlinear materials, 1 one would need a conversion efficiency (defined as P /P ) 2ω pump 1 59 such as diamond and quartz . This can be explained by for the second SHG process (i.e., ω + ω ⇒ ω )to 780 nm 780 nm 390 nm 27, 34–37, 60 considering their effective Hamiltonians . The main be close to unity. However, we observe a conversion efficiency contribution to THG is paramagnetic. This is described by the −10 ~10 for SHG. Therefore, we conclude that our FHG does not square diagram in Supplementary Notes 1–4 (Supplementary arise from cascaded SHGs. Another possible cascade process is Figs. 1–6). This paramagnetic contribution is mainly related to based on THG (ω + ω + ω ⇒ ω ) and 1560 nm 1560 nm 1560 nm 520nm the strong inter-band coupling in the effective Hamiltonian, sum-frequency generation (ω + ω ⇒ ω ). We 520 nm 1560 nm 390 nm c t a 0 0 controlled by large velocity scales, v  and v ¼ 300 h find that THG strongly increases up to N = 5, as for Fig. 4a. 0:65 ´ for SLG and 1L-MoS , with c the speed of light. The 300 Therefore, we expect this cascaded process to have a similar trend SLG paramagnetic third-harmonic efficiency (PTHE) is propor- with N. However, we only observe FHG in 1L-MoS . Thus, we tional to the square of third-order conductivity. Since also exclude this cascade process, and conclude that this is a ð3Þ 2 4 σ / v , we get an overall prefactor v , which explains the (4) yyyy F F direct χ process. strong nonlinear SLG response. Similarly, for 1L-MoS , the We now consider the dependence of our results on the elliptical square diagram contains four paramagnetic current vertices, polarization of the incident light. We consider an incident which gives an overall prefactor v , and an integral over laser beam with arbitrary polarization, i.e., E ¼jj E ε with the dummy momentum variables, which gives a prefactor v ^ε ¼ b x cosðθÞ ± ib y sinðθÞ. Using the crystal symmetries of (see Supplementary Note 3). Therefore, the third-order response 1L-MoS , we derive (see Supplementary Note 2) the following ð3Þ (2) function, Π , is proportional to v , which implies a scaling expressions for the second- and third-order polarizations P yyyy 4 (3) of PTHE as v . Exciton physics is not considered because our and P : experimental conditions only capture off-resonance transitions. ð2Þ ð2Þ 2 1L-MoS is transparent at this wavelength due to its ~1.9 eV P ¼ ϵ χ jj E½ i sinð2θÞb x  b y ð3Þ yyy 12 61 gap , while SLG absorbs 2.3% of the light . Therefore, 1L-MoS and other TMDs are promising for integration with waveguides and or fibers for all-optical nonlinear devices, such as all-optical modulators and signal processing devices, where materials with ð3Þ ð3Þ 3 P ¼ ϵ χ jj E ^ε cosð2θÞ: ð4Þ 0 ± yyyy nonlinear properties are essential . TheSHG andTHG powerdependencefollows quadraticand cubic trends, Fig. 4b. At our power levels, THG is up to 30 times Note that θ = 0° corresponds to a linearly polarized laser along stronger than SHG. 1L-TMDs have strongly bound excitons that the b x direction, perpendicular to the D mirror symmetry plane, 3h 62–64 can modify their optical properties . The exciton resonances while θ = 45° corresponds to a circularly polarized laser. From 17, 65, 66 51, 55 also affect their nonlinear optical responses . References Eq. (3), we expect the intensity of SHG in response to a circularly reported that when the SHG energy is above the A and B excitons, polarized pump laser to be twice that of a linearly polarized laser. NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 5 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 a bc SHG THG 1L 7L 2L 6L 3L 4L 5L 20 µm Fig. 6 Optical and multiphoton images of few-layer MoS flake. a Optical micrograph, b SHG, and c THG images of flake with few-layer areas under 1560 nm excitation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) ð1Þ χ (−3ω;ω, ω, ω) is the third-order optical susceptibility, n ¼ ϵ ðjωÞ, with j¼1;3 Equation (4) implies vanishing THG in response to a circularly ð1Þ ϵ the TMD linear dielectric function. Δkt is the phase mismatch between the polarized pump laser. fundamental and third harmonic generated waves. We measure the dependence of SHG and THG on elliptical For Δkt ≈ 0, THG adds up quadratically with light propagation length (i.e., t ∝ polarization using a linearly polarized laser and a rotating QWP. N). The signal starts to saturate for N = 6. The possible reasons for subquadratic signal build-up can be either phase mismatch, or absorption . For THG, Δk = Depending on the angle θ between the QWP axes and the laser 3k ± k , where k and k are the wavevectors of the incident and THG signals, in 3ω in 3ω polarization, the excitation light is linearly (θ = 0° + m·90°) or respectively, where the plus sign indicates THG generated in the backward circularly (θ = 45° + m·90°) polarized. Figure 5 shows that the direction, while minus identifies forward generated THG. Even for backward experimental data are in agreement with Eqs. (3) and (4). generated THG, Δkt ≈ 0 for 6L-MoS (~4.3 nm ). This rules out phase mismatch as the origin of the signal saturation when N ≤ 6. Therefore, we assume that the The THG signal is maximum for a linearly polarized excitation signal saturation is due to absorption of the third harmonic light. laser, while it vanishes for circularly polarized light. SHG is always visible, but its intensity is maximum for circularly polarized light. Diagrammatic nonlinear response theory. To quantify theoretically the strength Given that harmonic generation is strongly dependent on the of nonlinear harmonic generation processes, we generalize the diagrammatic perturbation theory approach to the case of TMDs. We combine this technique symmetry and stacking of layers and that different 1L-TMDs 11, 14, 15, 21 with a low-energy k·p model Hamiltonian HðÞ k for 1L-MoS . In such low-energy (e.g., WSe , MoSe ) all have similar nonlinear response , 2 2 35, 39 model, light-matter interactions are treated by employing minimal coupling , one could use heterostructures (e.g., MoS /WSe ) to engineer 2 2 k → k + eA(t)/ħ, where A(t) is a time-dependent uniform vector potential. SHG and other nonlinear processes for high photon-conversion Nonlinear response functions are calculated via the multi-legged Feynman diagrams depicted in Supplementary Figs. 2 and 3. efficiency for a wide range of applications requiring the genera- tion of higher frequencies. This may lead to the use of LMs and heterostructures for applications utilizing optical nonlinearities Data availability. The data that support the findings of this study are available (e.g., all-optical devices, frequency combs, high-order harmonic from the corresponding author on request. generation, multiphoton microscopy, and therapy etc.). Received: 8 November 2016 Accepted: 26 July 2017 Methods Determination of MoS thickness from SHG and THG signals. SHG and THG for FL-MoS (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 . Therefore, References areas with intensity between the 3, 5, and 7L regions in Fig. 6c, but dark in SHG, 1. Boyd, R. W. Nonlinear Optics (Academic Press, 2003). are 4 and 6 L. The dependence of the intensities of THG and SHG on N is plotted 2. Pavone, F. S. & Campagnola, P. J. Second Harmonic Generation Imaging in Fig. 4a. The combination of SHG and THG can be used to identify N at least up (Taylor & Francis, 2013). to 7. The THG signal develops as a function of N. Using Maxwell’s equations for a 3. Saleh, B. E. A. & Teich, M. C. Fundamentals of Photonics (Academic Press, nonlinear medium with thickness t and considering the slowly varying amplitude 2003). 1, 69 approximation , we obtain: 4. Willner, A. E., Khaleghi, S., Chitgarha, M. R. & Yilmaz, O. F. All-optical signal processing. J. Lightwave Technol 32, 660–680 (2014). 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Abstract

ARTICLE DOI: 10.1038/s41467-017-00749-4 OPEN Ultra-strong nonlinear optical processes and trigonal warping in MoS layers 1,2 1 3 1 4 Antti Säynätjoki , Lasse Karvonen , Habib Rostami , Anton Autere , Soroush Mehravar , 5 4 5 1,2,4 1 Antonio Lombardo , Robert A. Norwood , Tawfique Hasan , Nasser Peyghambarian , Harri Lipsanen , 4 5 3 1 Khanh Kieu , Andrea C. Ferrari , Marco Polini & Zhipei Sun Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, high-order har- monic 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 non- linearities in monolayer MoS (1L-MoS ): the third harmonic is 30 times stronger than the 2 2 second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1L-MoS is approximately three times higher than that for graphene, which was reported (3) to have a large χ . We explain this by calculating the nonlinear response functions of 1L- MoS with a continuum-model Hamiltonian and quantum mechanical diagrammatic pertur- bation 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. 1 2 Department of Electronics and Nanoengineering, Aalto University, Tietotie 3, FI-02150 Espoo, Finland. Institute of Photonics, University of Eastern Finland, 3 4 Yliopistokatu 7, FI-80100 Joensuu, Finland. Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy. College of Optical Sciences, University of Arizona, 1630 E University Blvd, Tucson, AZ 85721, USA. Cambridge Graphene Centre, University of Cambridge, Cambridge CB3 0FA, UK. Antti Säynätjoki, Lasse Karvonen and Habib Rostami contributed equally to this work. Correspondence and requests for materials should be addressed to Z.S. (email: zhipei.sun@aalto.fi) NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 1 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 onlinear optical phenomena can generate high-energy no second harmonic (SH) signal. On the other hand, MoS with photons by converting n = 2, 3, 4,… low-energy photons odd number of layers (ON) is non-centrosymmetric. SHG from 1L- 14–21 Ninto one high-energy photon. These are usually referred to MoS was reported by several groups . as second-, third-, and fourth-harmonic generation (SHG, THG, Here we present combined experimental and theoretical work 1 1, 2 and FHG) . Due to different selection rules , harmonic processes on nonlinear harmonic generation in 1L and few-layer (FL) MoS . are distinct from optically pumped laser phenomena (e.g., optically We report strong THG and FHG from 1L-MoS . THG is more pumped amplification ), and other typical single-photon processes than one order of magnitude larger than SHG, while FHG has the (e.g., single-photon excited photoluminescence ), in which the same magnitude as SHG. This is surprising, since one normally energy of the generated photons is smaller than the pump photons. expects the intensity of nonlinear optical processes to decrease 1, 2 Multiphoton harmonic processes have been widely exploited for with n , with the SHG intensity much larger than that in THG various applications (e.g., all-optical signal processing in tele- and FHG, although even-order processes only exist in non-cen- 1, 4 5 6 communications ,medicine , and data storage ), as well as to trosymmetric materials. Our results show that this expectation is study various transitions forbidden under low-energy single-photon wrong in the case of 1L-MoS . At sufficiently low photon fre- 5, 6 excitation . The physical origin of these processes is the nonlinear quencies (in our experiments the photon energy of the pump is polarization induced by an electromagnetic field E. This gives rise to 0.8 eV), SHG only probes the low-energy band structure of 1L- 22–29 higher harmonic components, the n-th harmonic component MoS . This is nearly rotationally invariant , but with correc- 1 n amplitude being proportional to |E| . Quantum mechanically, tions due to trigonal warping. It is because of these 23, 26, 27 1 1 higher-harmonic generation involves the annihilation of n pump corrections , fully compatible with the D space group , 3h photons and generation of a photon with n times the pump energy. but reducing the full rotational symmetry of the low-energy bands Because an n-th order nonlinear optical process requires n photons to a three-fold rotational symmetry ,thata finite amplitude of to be present simultaneously, the probability of higher-order pro- nonlinear harmonic processes can exist at low photon energies in cessesislower than that of lowerorder . Thus, higher-order pro- EN-MoS . The lack of spatial inversion symmetry is a necessary 7, 8 cesses are typically weaker and require higher pump intensities . but not sufficient condition for the occurrence of SHG. A purely 30–33 Graphene and related materials are at the center of an ever- isotropic band structure gives a vanishing SHG signal , despite increasing research effort due to their unique and complementary some terms in the Hamiltonian explicitly breaking inversion 27, 34–37 properties, making them appealing for a wide range of photonic and symmetry . Terms proportional to the σ Pauli matrix 9–11 optoelectronic applications . Among these, semiconducting break inversion symmetry. Breaking the continuous rotational transition-metal dichalcogenides (TMDs) are of particular interest symmetry of isotropic models (e.g., by including trigonal warping) due to their direct bandgap when in monolayer (1L) form ,leading is required to obtain a non-zero second-order response in a two- to an increase in luminescence by a few orders of magnitude band system. In hexagonal lattices, trigonal warping is a deviation 12, 13 compared with the bulk material . 1L-MoS has a single layer of from purely isotropic bands that emerges as one moves away from 23, 26, 27, 36–38 Mo atoms sandwiched between two layers of S atoms in a trigonal the corners K and K′ of the Brillouin zone . Since the prismatic lattice. Therefore, in contrast to graphene, it is non-cen- lattice has a honeycomb structure, this distortion displays a three- 1 14 23, 26, 27, 36–38 trosymmetric and belongs to the space group D .The lack of fold rotational symmetry . We demonstrate that the 3h spatial inversion symmetry makes 1L-MoS an interesting material observed THG/SHG intensity ratio can be explained by quantum for nonlinear optics, since second-order nonlinear processes are mechanical calculations based on finite-temperature many-body 1 39 present only in non-centrosymmetric materials . However, when diagrammatic perturbation theory and low-energy continuum- stacked, MoS layers are arranged mirrored with respect to one model Hamiltonians that include trigonal warping . We conclude 14 14–18 another , therefore MoS with an even number of layers (EN) is that, similar to SHG , the THG process is sensitive to the 3 14 centrosymmetric and belongs to the D space group ,producing number of layers, their symmetry, relative orientation, as well as 3d ab 1g 2g 5L-MoS 1L 2L 2L 2L-MoS 1L 2L 1L 1L-MoS 360 380 400 420 440 20 µm 5L –1 Raman shift (cm ) Fig. 1 Optical image and Raman spectra of the MoS flakes. a Optical micrograph with single-layer, bilayer, and five-layer areas marked by 1L, 2L, and 5L. b Raman spectra of the same sample 2 NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications | | | Intensity (a.u.) PMT NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 ARTICLE Collimator ab QWP VA SHG THG Galvo scanner Scans lens MLL Power (pW) Power (nW) 0 4.5 0 3.4 Dichroic mirror (562 nm) THG Tube lens Collection lens 520 nm BP filter 0.1 SHG 0.01 Dichroic mirror FHG PMT (870 nm) 1E–3 780 nm BP filter 1E–4 20X 1E–5 400 500 600 700 800 Wavelength (nm) Sample Fig. 3 Multiphoton images of MoS flakes. a SHG and b THG map of the flakes in Fig. 1a. c Optical spectrum of the nonlinear signal from 1L-MoS −2 with a peak irradiance ~30 GW cm Fig. 2 Schematic diagram of multiphoton microscope. MLL, linearly polarized mode-locked fiber laser. VA, variable attenuator. QWP, quarter- wave plate used to study the dependence of SHG and THG on the elliptical polarization of the pump light. BP filter, bandpass filter. PMT, and enables high signal-to-noise-ratio, even with acquisition time photomultiplier tube per pixel in the μs range. SHG and THG images of the MoS sample are shown in Fig. 3a, b. The SH photon energy is ~1.6 eV, lower than the 12, 13 bandgap of 1L-MoS . This is not unexpected, as harmonic the elliptical polarization of the excitation light. Similar effects are generation can occur when the harmonic energy is below expected for all other TMDs. This paves the way for the assembly 1, 47, 48 the bandgap . The SHG signal is generated in 1L-MoS , of heterostructures with tailored nonlinear optical properties. while 2L-MoS appears dark. As discussed above, the second- order nonlinear response is present in 1L-MoS , which is non- Results centrosymmetric. However, when stacked to form 2L-MoS , 14, 15 Samples. MoS flakes are produced by micromechanical cleavage MoS layers are mirrored . Therefore, EN-MoS is centro- 2 2 14, 15 3 14, 15 (MC) of bulk MoS onto Si + 285 nm SiO . 1L-MoS and bilayer 2 2 2 symmetric , and belongs to the D space group , 3d (2L-MoS ) flakes are identified by a combination of optical 2 producing no SHG signal. On the other hand, ON-MoS flakes 41, 42 43 14, 15 contrast and Raman spectroscopy . Raman spectra are are non-centrosymmetric . acquired by a Renishaw micro-Raman spectrometer equipped We note that strong THG is detected compared with SHG, even with a 600 line/mm grating and coupled with an Ar ion laser at for 1L-MoS ,Fig. 3b. THG was previously reported for a 7L-MoS 2 2 18 49 514.5 nm. Figure 1a shows the MoS flakes studied in this work 2 flake , but here we see it down to 1L-MoS . Reference followed and their Raman signatures. A reference MC graphene sample is our work and reported THG and SHG from 1L-MoS , giving ð2Þ also prepared and placed on a similar substrate. effective bulk-like second- and third-order susceptibilities χ eff ð3Þ −11 −1 −19 2 −2 and χ of 2.9 × 10 mV and 2.4 × 10 m V , respectively. eff However, ref. did not provide a detailed explanation of the large SHG and THG charcterization. Nonlinear optical measurements THG signal compared to the SHG. Instead it assigned the large 44, 45 are carried out with the set-up shown in Fig. 2 . As excitation THG/SHG ratio to a possible enhancement of THG by the edge of source, we use an erbium-doped mode-locked fiber laser with a 51, 52 the B exciton. However, refs. demonstrated that SHG is 50 MHz repetition rate, maximum average power 60 mW, and enhanced only when the SHG wavelength overlaps the A or B pulse duration 150 fs, which yields an estimated pulse peak power excitons. A similar behavior is expected for THG. Thus, the of ~8 kW . The laser beam is scanned with a galvo mirror 49 53 explanationinref. may not be correct. Reference reported and focused on the sample using a microscope objective. The high-harmonic (>6th-order) generation in the non-perturbative back-scattered second and third harmonic signals are split into regime with mid-infrared (IR) excitation (0.3 eV), unlike our THG different branches using a dichroic mirror and then detected and FHG results with near-IR excitation (0.8 eV). We do not detect using photomultiplier tubes (PMTs). For two-channel detection, THG from the thickest areas of our flake, with N > 30, as in ref. . the light is split into two PMTs using a dichroic mirror with The output spectrum in Fig. 3c further confirms that we observe 562 nm cutoff. After the dichroic mirror, the detected wavelength both SHG and THG. Peaks for THG and SHG at 520 and 780 nm range can be further refined using bandpass filters. The light can can be seen, as well as at 390 nm, corresponding to a four-photon also be directed to a spectrometer (OceanOptics QE Pro-FL). The process. This is detected only in 1L-MoS . Its intensity is ~5.5 times average power on the sample is kept between 10 and 28 mW with lower than SHG, and two orders of magnitude smaller than THG. a typical measurement time ~5 μs, which prevents sample damage NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 3 | | | Intensity (a.u.) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 (780 nm), below the band gap of 1L-MoS , therefore are not a 0.4 15 adsorbed, unlike the SHG signal in ref. . SHG Second- and third-order nonlinear susceptibilities. Based on 0.3 the measured SHG and THG intensities, we can estimate the (2) (3) (2) 10 nonlinear susceptibilities χ and χ . χ can be calculated from the measured average powers of the fundamental and SH signals 0.2 as follows : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 5 ϵ cλ P RτðÞ n þ 1ðÞ n þ 1 0 2ω 2 1 ð2Þ ð1Þ χ ¼ ; 0.1 s 32N τ P ϕ 2 pump where τ is the pulse width, P is the average power of the pump incident fundamental (pump) beam, and P stands for 0 0 2ω 1 234 567 the generated SH beam power, R is the repetition rate, N = 0.5 Number of layers is the numerical aperture, λ = 780 nm is the SH wavelength, τ = τ = 150 fs are the pulse durations at fundamental and R pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 b 1 2 SH wavelengths, ϕ ¼ 8π jcos ρ  ρ 1  ρ j ρ dρ ¼ 3:56 from ref. , and n ¼ n  1:45 are the refractive indexes of 1 2 the substrate at the wavelengths of the fundamental and SHG, respectively. The effective bulk-like second-order susceptibility of ð2Þ ð2Þ ð2Þ MoS ðχ Þ can be obtained from Eq. (1) with χ ¼ , where eff eff t MoS 10, 24 t ¼ 0:65 nm is the 1L-MoS thickness . We obtain the MoS 2 ð2Þ −1 effective second-order susceptibility χ  2:2 pmV for 1L- eff MoS . Reference reported a bulk-like second-order suscept- −1 ibility 29 pmV , which is ~10 times larger than here. However, −1 20, 52, 55 0.1 several other studies reported ~5 pmV for 1560 nm . ð2Þ Thus, our measured χ agrees well with earlier values measured eff with similar excitation wavelength. ð3Þ The third-order susceptibility χ can be estimated eff 0.5 1 1.5 2 by comparing the measured THG signal from MoS to that of Pump peak power on sample (kW) 1L-graphene (SLG): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t THG ð3Þ SLG MoS ð3Þ k a = 0.85 c 0 ð2Þ χ  χ : SLG eff 64 t THG k a = 0.9 MoS SLG c 0 k a = 0.95 c 0 with t = 0.33 nm the SLG thickness, and THG and SLG SLG k a = 1.0 c 0 THG the measured signals from SLG and MoS , respectively. MoS 2 16 2 EXP. Our results show that THG from 1L-MoS is around three times (3) larger than THG , which indicates that χ of 1L-MoS is SLG 2 comparable to that of SLG, in the frequency range of our 4 ð3Þ 49, 56 experiments. Previous reports indicate that χ is SLG −17 −19 2 −2 (3) ~10 –10 m V . Thus, based on Eq. (2), χ of 1L-MoS is in the same range. This is remarkable, as SLG is known to have (3)56–59 18 (3) a large χ . Reference reported χ of 7L-MoS to be ð3Þ approximately three orders of magnitude smaller than χ of SLG 200 400 800 1800 ð3Þ 56 56 58 ref. . χ from ref. is much higher than other theoretical SLG P (W) 49 pump and experimental values. We believe that our measured ratio between 1L-MoS and SLG is more accurate, since we measured Fig. 4 Experimental and theoretical nonlinear optical processes in MoS . both materials at the same time under the same conditions. a SHG and THG intensities as functions of N. b Power dependence of We note that large discrepancies can be found in earlier reported SHG and THG in 1L-MoS . c Experimental and theoretical THG/SHG effective susceptibilities for layered materials (LM). For example, irradiance ratio as a function of P . Different theoretical curves refer to pump (3) pffiffiffi there is a approximately four orders of magnitude difference in χ different values of the ultra-violet cutoff k (in units of 1=a ¼ 3=a; where c 0 −15 2 −2 57 −19 2 −2 49 for SLG (~10 m V in ref. ;~10 m V in ref. ). There is a = 3.16 Å is the lattice constant of 1L-MoS ). Black dashed lines in panels (2) an approximately three orders of magnitude difference in χ a and b are a guide to the eye. The error bars in c account for experimental −7 −1 15 reported for 1L-MoS at 800 nm (e.g., ~10 mV in ref. ;and uncertainties −10 −1 17 ~10 mV in ref. ). Effective susceptibilities are well defined only in three-dimensional materials, since their definition involves a polarization per unit volume . Therefore, given the large discre- SHG signals on areas with N = 3, 5, 7 have nearly the same pancies in literature, it is better to describe the nonlinear processes intensity as 1L-MoS , Fig. 4a. This contrasts ref. , where a pump in LMs using the ratio between the harmonic signal power and the laser at 810 nm was used. We attribute this difference to the fact incident pump power (i.e., harmonic conversion efficiency). In this that photons generated in the second-order nonlinear process in case, when comparing the efficiencies in our measurements with 49 −10 our set-up with a 1560 nm pump have an energy ~1.6 eV those in ref. , our THG efficiency (~4.76 × 10 )is~1.4times 4 NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications | | | THG THG SHG Detected peak power (µW) THG SHG THG peak power (µW) I / I SHG peak power (µW) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 ARTICLE resonance effects are not observed. In our experiments, the energy THG (pW) of 3ω photons is above the A exciton but does not directly SHG x 37 (pW) 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 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 270 90 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 . 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 Fig. 5 SHG and THG dependence on the pump light polarization. The polar calculations based on the k·p theory (see Supplementary plot angle corresponds to linearly polarized light when θ = 0° + m·90°, and Note 1) and finite-temperature diagrammatic perturbation to circularly polarized pump light when θ = 45° + m·90°. The SHG power is theory (see Supplementary Notes 3 and 4). The calculations are multiplied by a factor of 37 to fit in the same scale as THG a factor 2 smaller than the experiments. Considering the com- plexity of the nonlinear optical processes and that our calculations ignore high-energy band structure effects and −10 49 many-body renormalizations , we believe this to be a satisfactory larger than that (~3.38 × 10 )inref. ,while our SHG efficiency −11 49 agreement, indicating the importance of trigonal warping in (~6.47 × 10 )istwice that of ref. . Since the effective harmonic generation. susceptibilities are not well defined for LMs and also depend on FHG generally derives from cascades of lower-order nonlinear the calculation method, we believe that the conversion efficiency is a multiphoton processes . With an excitation wavelength of better figure of merit for LMs. 1560 nm, this could be, e.g., a cascade of two SHG processes, where 780 nm photons are first generated through SHG Discussion (ω + ω ⇒ ω ) and then undergo another SHG 1560 nm 1560 nm 780 nm Our measurements show that the nonlinear response of 1L-MoS process (ω + ω ⇒ ω ). To yield a FHG at 390 nm 780 nm 780 nm 390 nm and SLG are comparable in magnitude, both revealing stronger of the same intensity as SHG at 780 nm in this cascaded process, nonlinear efficiency than three-dimensional nonlinear materials, 1 one would need a conversion efficiency (defined as P /P ) 2ω pump 1 59 such as diamond and quartz . This can be explained by for the second SHG process (i.e., ω + ω ⇒ ω )to 780 nm 780 nm 390 nm 27, 34–37, 60 considering their effective Hamiltonians . The main be close to unity. However, we observe a conversion efficiency contribution to THG is paramagnetic. This is described by the −10 ~10 for SHG. Therefore, we conclude that our FHG does not square diagram in Supplementary Notes 1–4 (Supplementary arise from cascaded SHGs. Another possible cascade process is Figs. 1–6). This paramagnetic contribution is mainly related to based on THG (ω + ω + ω ⇒ ω ) and 1560 nm 1560 nm 1560 nm 520nm the strong inter-band coupling in the effective Hamiltonian, sum-frequency generation (ω + ω ⇒ ω ). We 520 nm 1560 nm 390 nm c t a 0 0 controlled by large velocity scales, v  and v ¼ 300 h find that THG strongly increases up to N = 5, as for Fig. 4a. 0:65 ´ for SLG and 1L-MoS , with c the speed of light. The 300 Therefore, we expect this cascaded process to have a similar trend SLG paramagnetic third-harmonic efficiency (PTHE) is propor- with N. However, we only observe FHG in 1L-MoS . Thus, we tional to the square of third-order conductivity. Since also exclude this cascade process, and conclude that this is a ð3Þ 2 4 σ / v , we get an overall prefactor v , which explains the (4) yyyy F F direct χ process. strong nonlinear SLG response. Similarly, for 1L-MoS , the We now consider the dependence of our results on the elliptical square diagram contains four paramagnetic current vertices, polarization of the incident light. We consider an incident which gives an overall prefactor v , and an integral over laser beam with arbitrary polarization, i.e., E ¼jj E ε with the dummy momentum variables, which gives a prefactor v ^ε ¼ b x cosðθÞ ± ib y sinðθÞ. Using the crystal symmetries of (see Supplementary Note 3). Therefore, the third-order response 1L-MoS , we derive (see Supplementary Note 2) the following ð3Þ (2) function, Π , is proportional to v , which implies a scaling expressions for the second- and third-order polarizations P yyyy 4 (3) of PTHE as v . Exciton physics is not considered because our and P : experimental conditions only capture off-resonance transitions. ð2Þ ð2Þ 2 1L-MoS is transparent at this wavelength due to its ~1.9 eV P ¼ ϵ χ jj E½ i sinð2θÞb x  b y ð3Þ yyy 12 61 gap , while SLG absorbs 2.3% of the light . Therefore, 1L-MoS and other TMDs are promising for integration with waveguides and or fibers for all-optical nonlinear devices, such as all-optical modulators and signal processing devices, where materials with ð3Þ ð3Þ 3 P ¼ ϵ χ jj E ^ε cosð2θÞ: ð4Þ 0 ± yyyy nonlinear properties are essential . TheSHG andTHG powerdependencefollows quadraticand cubic trends, Fig. 4b. At our power levels, THG is up to 30 times Note that θ = 0° corresponds to a linearly polarized laser along stronger than SHG. 1L-TMDs have strongly bound excitons that the b x direction, perpendicular to the D mirror symmetry plane, 3h 62–64 can modify their optical properties . The exciton resonances while θ = 45° corresponds to a circularly polarized laser. From 17, 65, 66 51, 55 also affect their nonlinear optical responses . References Eq. (3), we expect the intensity of SHG in response to a circularly reported that when the SHG energy is above the A and B excitons, polarized pump laser to be twice that of a linearly polarized laser. NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 5 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 a bc SHG THG 1L 7L 2L 6L 3L 4L 5L 20 µm Fig. 6 Optical and multiphoton images of few-layer MoS flake. a Optical micrograph, b SHG, and c THG images of flake with few-layer areas under 1560 nm excitation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) ð1Þ χ (−3ω;ω, ω, ω) is the third-order optical susceptibility, n ¼ ϵ ðjωÞ, with j¼1;3 Equation (4) implies vanishing THG in response to a circularly ð1Þ ϵ the TMD linear dielectric function. Δkt is the phase mismatch between the polarized pump laser. fundamental and third harmonic generated waves. We measure the dependence of SHG and THG on elliptical For Δkt ≈ 0, THG adds up quadratically with light propagation length (i.e., t ∝ polarization using a linearly polarized laser and a rotating QWP. N). The signal starts to saturate for N = 6. The possible reasons for subquadratic signal build-up can be either phase mismatch, or absorption . For THG, Δk = Depending on the angle θ between the QWP axes and the laser 3k ± k , where k and k are the wavevectors of the incident and THG signals, in 3ω in 3ω polarization, the excitation light is linearly (θ = 0° + m·90°) or respectively, where the plus sign indicates THG generated in the backward circularly (θ = 45° + m·90°) polarized. Figure 5 shows that the direction, while minus identifies forward generated THG. Even for backward experimental data are in agreement with Eqs. (3) and (4). generated THG, Δkt ≈ 0 for 6L-MoS (~4.3 nm ). This rules out phase mismatch as the origin of the signal saturation when N ≤ 6. Therefore, we assume that the The THG signal is maximum for a linearly polarized excitation signal saturation is due to absorption of the third harmonic light. laser, while it vanishes for circularly polarized light. SHG is always visible, but its intensity is maximum for circularly polarized light. Diagrammatic nonlinear response theory. To quantify theoretically the strength Given that harmonic generation is strongly dependent on the of nonlinear harmonic generation processes, we generalize the diagrammatic perturbation theory approach to the case of TMDs. We combine this technique symmetry and stacking of layers and that different 1L-TMDs 11, 14, 15, 21 with a low-energy k·p model Hamiltonian HðÞ k for 1L-MoS . In such low-energy (e.g., WSe , MoSe ) all have similar nonlinear response , 2 2 35, 39 model, light-matter interactions are treated by employing minimal coupling , one could use heterostructures (e.g., MoS /WSe ) to engineer 2 2 k → k + eA(t)/ħ, where A(t) is a time-dependent uniform vector potential. SHG and other nonlinear processes for high photon-conversion Nonlinear response functions are calculated via the multi-legged Feynman diagrams depicted in Supplementary Figs. 2 and 3. efficiency for a wide range of applications requiring the genera- tion of higher frequencies. This may lead to the use of LMs and heterostructures for applications utilizing optical nonlinearities Data availability. The data that support the findings of this study are available (e.g., all-optical devices, frequency combs, high-order harmonic from the corresponding author on request. generation, multiphoton microscopy, and therapy etc.). Received: 8 November 2016 Accepted: 26 July 2017 Methods Determination of MoS thickness from SHG and THG signals. SHG and THG for FL-MoS (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). 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Hetero2D, Nokia Foundation, EPSRC Grants EP/K01711X/1, EP/K017144/1, EP/ NATURE COMMUNICATIONS 8: 893 DOI: 10.1038/s41467-017-00749-4 www.nature.com/naturecommunications 7 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00749-4 L016087/1, AFOSR COMAS MURI (FA9550-10-1-0558), ONR NECom MURI, CIAN Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in NSF ERC under Grant EEC-0812072, and TRIF Photonics funding from the state of published maps and institutional affiliations. Arizona and the Micronova, Nanofabrication Centre of Aalto University. Open Access This article is licensed under a Creative Commons Author contributions Attribution 4.0 International License, which permits use, sharing, A.S., L.K., A.A., S.M., K.K., and Z.S. carried out the multiphoton experiments and data adaptation, distribution and reproduction in any medium or format, as long as you give analysis. 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