Giant All-Optical Modulation of Second-Harmonic Generation Mediated by Dark Excitons

All-optical control of nonlinear photonic processes in nanomaterials is of significant interest from a fundamental viewpoint and with regard to applications ranging from ultrafast data processing to spectroscopy and quantum technology. However, these applications rely on a high degree of control over the nonlinear response, which still remains elusive. Here, we demonstrate giant and broadband all-optical ultrafast modulation of second-harmonic generation (SHG) in monolayer transition-metal dichalcogenides mediated by the modified excitonic oscillation strength produced upon optical pumping. We reveal a dominant role of dark excitons to enhance SHG by up to a factor of ∼386 at room temperature, 2 orders of magnitude larger than the current state-of-the-art all-optical modulation results. The amplitude and sign of the observed SHG modulation can be adjusted over a broad spectral range spanning a few electronvolts with ultrafast response down to the sub-picosecond scale via different carrier dynamics. Our results not only introduce an efficient method to study intriguing exciton dynamics, but also reveal a new mechanism involving dark excitons to regulate all-optical nonlinear photonics.


Experimental setup
. Scheme of the experimental setup. The femtosecond laser source is Solstice Ace from Spectra-Physics. The TOPAS system is used for frequency conversion to produce the seed light with a wavelength range of 820-1600 nm. The control light wavelengths are either 400 nm or 800 nm. The pulse duration is ~230 fs and the repetition rate is 2 KHz. BBO: Beta barium borate; WL: white light for optical imaging of the sample; DM: dichroic mirror; BS: beam splitter; PMT: photomultiplier.

Monolayer MoS 2 sample characterization
The MoS 2 sample is prepared by chemical vapor deposition (CVD). Figure S2a shows a typical triangular flake deposited on a SiO 2 /Si substrate. The Raman spectrum in Fig. S2b shows peaks at ~383 and 402 cm -1 , with a difference of 19 cm -1 identifying monolayer MoS 2 . As shown in Fig. S2c, photoluminescence (PL) mapping shows that the MoS 2 layer is grown well and uniformly. Furthermore, the measured PL spectrum reveals two strong peaks at 620 nm and 670 nm (fitted by Lorentzians), corresponding to B and A excitons, respectively, as shown in Fig.   S4 S2d. Photoluminescence and Raman spectra are measured under excitation with a continuouswave laser of 532 nm wavelength and 0.8 W incident power focused with a NA of 0.75.

Time-resolved P SHG spectra
In Figure S7, we plot results from SHG modulation spectroscopy at different time delays in the spectral range from ~2.16 to 2.88 eV. The spectra show very clear dynamics when  varies from -1 to 10 ps ( Fig. S7a-f). We observe that the enhancement peak positions remain unchanged when introducing control light, indicating that bandgap renormalization does not affect the sign of the change in SHG modulation. Furthermore, detailed spectra near the P SHG =0 condition are measured and shown in Fig. S7g, which shows blue shifts due to the evolution from the suppression region to the enhancement region. We attribute the larger SHG responses observed at higher photon energies (e.g., at 2.8 eV) to the C-exciton resonance. S10

MoS 2 field-effect transistor and electrically tunable all-optical modulation of SHG
It has been previously reported that exciton resonances and electrical doping can change the SHG response in monolayer TMDs. 1 We therefore investigate the possibility of electrical tunability of our observation of all optical SHG modulation. To this end, we have fabricated a MoS 2 field-effect transistor (Fig. S9a) for the experiments. The two electrodes are first patterned using electron-beam lithography (Vistec 5000+ES, Germany) and then covered with Ti (10 nm)/Au (60 nm) using an electron beam evaporation (OHMIKER-50B, Taiwan). The electrical performance of this device is shown in Fig. S9b. In particular, P SHG at the 1s A exciton (1.89 eV energy) with different gate voltages (V g ) is shown in Fig. S9c. Figure S9d shows that the maximum P SHG  varies from 43% to 25% at constant delay  150 fs when V g increases ≈

Preservation of MoS 2 symmetry with control light
We perform circularly-polarized SHG measurements in which circularly-polarized seed light is generated by combining a linear polarizer and a quarter-wave plate, focused on the monolayer MoS 2 with a sapphire substrate. The generated SHG is collimated with an objective lens and converted into linear polarization after going through another quarter-wave plate. Finally, a linear polarizer is employed to measure the polarization angle of the converted SHG signal. 2 S12

Theoretical calculation of the change in second-order nonlinear susceptibility
The modulation of the SHG signal before and after the arrival of a control light pulse can be explained as the change in the second-order nonlinear susceptibility of the system . The MoS 2 at the seed wavelength with and without control light, respectively. S14 From here, the relative change in second-order nonlinear susceptibility can be expressed as In the enhancement region, we find a maximum relative change in second.order nonlinear susceptibility of ~19.

Supplementary theoretical elements in the analysis of SHG modulation
We use the GW-BSE method to calculate the absorption spectrum of monolayer MoS 2 . This method yields reliable predictions for the excited-state properties of ultra-thin transition metal dichalcogenides. 4, 5 In Fig. S12a, we represent the electronic band structure calculated within the GGA and G 0 W 0 approximations for monolayer MoS 2 , showing that inclusion of quasiparticle energy corrections to the Kohn-Sham eigenvalues (i.e., when moving from GGA to G 0 W 0 ) leads to a direct band gap of 2.77 eV at the K point.
We obtain the absorbance spectrum of monolayer MoS 2 from the imaginary part of the dielectric function, as shown in Fig. S12b, which reveals several excitonic features located below the band gap of the material with high oscillator strengths. A comparison of the spectral peak positions of the experimentally observed and theoretically predicted bright excitonic states is presented in Table S2.

SHG modulation with monolayer WS 2
A typical WS 2 sample characterization is shown in Fig. S14. Inset: Raman map of the 336 cm -1 shifted peak.

S17
Our measurements of SHG modulation in WS 2 are carried out with the same system employed for MoS 2 using control light of 400 nm wavelength (3.1-eV photon energy). Figure S15a shows the dependence of SHG enhancement  on the power of control and seed light (1170 nm, 1.05 eV). Clearly, the SHG enhancement factor  increases with increasing control light power, whereas it decreases with increasing seed light power. Figure S15b shows  as a function of control light power when the seed light power is 1 W, from which we conclude that  can be as high as ~76 when the control light power is 0.3 W.
The normalized P SHG is shown in Fig. S16