Multiple-peak resonance of optical second harmonic generation arising from band nesting in monolayer transition metal dichalcogenides TX2 on SiO2/Si(001) substrates (T=Mo,W;X=S,Se)

The nonlinear optical response of monolayer transition-metal dichalcogenides $T{X}_{2}\phantom{\rule{0.28em}{0ex}}(T=\mathrm{Mo},\mathrm{W};\phantom{\rule{0.28em}{0ex}}X=\mathrm{S},\phantom{\rule{0.28em}{0ex}}\mathrm{Se})$ on ${\mathrm{SiO}}_{2}/\mathrm{Si}(001)$ substrates was examined using second-harmonic generation (SHG) spectroscopy in combination with differential reflectance (DR) spectroscopy. In the SHG spectroscopy, the second-harmonic intensity corrected by a crystalline quartz reference was obtained for the wavelength of the incident femtosecond laser varied from 750 to 1040 nm. The SHG spectra of all the four ${TX}_{2}$ were dominated by two-photon resonance around the C peaks in the DR spectra that was previously assigned to interband transitions at critical points formed as a result of band nesting. Multiple peaks were observed in each resonance spectrum, which was then decomposed into a few components. The overall feature of the main components---denoted as $\overline{\mathrm{C}1}$ and $\overline{\mathrm{C}2}$---was common among the four ${TX}_{2}$ materials. The optical transitions contributing to these components were concluded to occur at different $k$ points in a ring-shaped area around the \cyrchar\CYRG{} point with the aid of the published results of optical spectrum calculations and the SHG spectrum of bilayer ${\mathrm{MoS}}_{2}$. In addition, one-photon resonance at the $A$ exciton and two-photon resonance at the ${A}^{\ensuremath{'}}$ exciton were observed in ${\mathrm{MoSe}}_{2}$ and ${\mathrm{WSe}}_{2}$, respectively, and their intensities were proven to be significantly lower than those of the resonances around the C peaks.