The exciton is an electron hole bound state in semiconductors and plays an important role in opto-electronic devices such as light-emitting diode. Therefore, a desire exists to control the exciton formation energy, which is closely related to characteristics of optoelectronic applications. Here, we report formation energy of ground exciton states in a MoS$_2$ twisted bilayer measured by using photoluminescence spectroscopy. By varying the twist angle, we found that the exciton formation energy was tunable within the energy range roughly between 1.87 eV at about 0$^{\circ}$ and 60$^{\circ}$ twist angles and 1.90 eV at about a 30$^{\circ}$ twist angle. The exciton formation energy is directly related to the band gap energy, and the band gap of a bilayer MoS$_2$ becomes smaller than that of a MoS$_2$ monolayer due to interlayer coupling. Our results can be explained by the fact that the interlayer distance is smaller and the interlayer coupling is larger at 0$^{\circ}$ and 60$^{\circ}$ twist angles than at 30$^{\circ}$ twist angle. The twist angle dependence of the exciton formation energy can also be qualitatively explained by the twist-angle-dependent direct band-gap energy from first-principles calculations results.

Keywords: Twisted bilayer MoS$_2$, Raman spectroscopy, Exciton, Photoluminescence spectroscopy