Valley depolarization due to inter- and intra-valley electron-hole exchange interactions in monolayer MoS$_{2}$

We investigate the valley depolarization due to the electron-hole exchange interaction in monolayer MoS$_{2}$. Both the long- and short-range parts of the intra- and inter-valley electron-hole exchange interactions are calculated. We find that both the long- and short-range exchange interactions can cause the inter- and intra-valley bright exciton transitions. With the intra-valley bright exciton transition channel nearly forbidden due to the large splitting of the valence bands, the inter-valley channel due to the exchange interaction can cause the valley depolarization efficiently by the Maialle-Silva-Sham mechanism [Phys. Rev. B {\bf 47}, 15776 (1993)]. With only the long-range exchange interaction, the calculations show good agreement with the recent valley polarization experiments, including the time-resolved valley polarization measurement, the pump-probe experiment and the steady-state PL polarization measurement. We further show that for the A-exciton with large (small) center-of-mass momentum, the long-range exchange interaction can cause the {\em fast} ({\em slow}) inter-valley exciton transition.