Effect of intravalley and intervalley electron-hole exchange on the nonlinear optical response of monolayer MoSe$_2$

The coherent third-order nonlinear response of monolayer transition-metal dichalcogenide (TMD) semiconductors, such as MoSe$_2$ is dominated by the nonlinear exciton response, as well as biexciton and trion resonances. The fact that these resonances may be spectrally close together makes identification of the signatures, for example in differential transmission (DT), challenging. Instead of focusing on explaining a given set of experimental data, a systematic study aimed at elucidating the roles of intravalley and intervalley electron-hole (e-h) exchange on the DT spectra is presented. Previous works have shown that the e-h exchange introduces a linear leading-order term in the exciton dispersion. Based on a generalized Lippmann-Schwinger equation, we show that the presence of this linear dispersion term can reduce the biexciton binding energy to zero, contrary to the conventional situation of quadratic dispersion where an arbitrarily weak (well-behaved) attractive interaction always supports bound state(s). The effects of spin-scattering and the spin-orbit interaction caused by e-h exchange is also clarified, and the DT lineshape at the exciton and trion resonance are studied as a function of e-h exchange strength. In particular, as the exciton lineshape is determined by the interplay of linear exciton susceptibility and the bound-state two-exciton resonance in the T-matrix, the lineshape at the trion is similarly determined by the interplay of the linear trion susceptibility and the bound-state exciton-trion resonance in the T-matrix.