Strain dependence of second harmonic generation in transition metal dichalcogenide monolayers and the fine structure of the C exciton

First-principles density functional theory with the time-dependent Bethe-Salpeter Equation (BSE) is used to calculate the second harmonic generation (SHG) spectra of five trigonal prismatic ($2H$-phase) transition metal dichalcogenide (TMD) monolayers, $\mathrm{Mo}{\mathrm{S}}_{2}, \mathrm{MoS}{\mathrm{e}}_{2}, \mathrm{W}{\mathrm{S}}_{2}, \mathrm{WS}{\mathrm{e}}_{2}$, and $\mathrm{MoT}{\mathrm{e}}_{2}$, as a function of biaxial and uniaxial strain. It is shown that it is important to take excitonic effects into account when studying the effects of strain on SHG. All the TMDs exhibit very strong SHG, with hexagonal monolayer $\mathrm{MoT}{\mathrm{e}}_{2}$ dominating over the other TMDs in the 0.5--1.4 eV region. These results are shown to agree with experimental data. While $\mathrm{W}{\mathrm{S}}_{2}$ appears to be a good candidate for strain-tolerant applications because the SHG is insensitive to strain, $\mathrm{WS}{\mathrm{e}}_{2}$ exhibits a strong, consistent strain response that points to potential strain-sensitive applications. Distinct subpeaks of the $C$ exciton are observed which, when compared with the electronic structure, are attributed to specific transitions in the Brillouin zone, with the most important contributions coming from the $K({v}_{1}{\ensuremath{\rightarrow}}_{}{c}_{2}),Q({v}_{1}{\ensuremath{\rightarrow}}_{}{c}_{1}),$ and $\mathrm{\ensuremath{\Gamma}}({v}_{1}\ensuremath{\rightarrow}{c}_{1})$ points. These findings open avenues of exploration for potential applications of these materials in strain-sensitive devices and improve our understanding of the $C$ exciton in TMDs.