The optical properties of transition metal dichalcogenide monolayers such as the two-dimensional semiconductors MoS2 and WSe2 are dominated by excitons, Coulomb bound electron-hole pairs. The light emission yield depends on whether the electron-hole transitions are optically allowed (bright) or forbidden (dark). By solving the Bethe Salpeter Equation on top of GW wave functions in density functional theory calculations, we determine the sign and amplitude of the splitting between bright and dark exciton states. We evaluate the influence of the spin-orbit coupling on the optical spectra and clearly demonstrate the strong impact of the intra-valley Coulomb exchange term on the dark-bright exciton fine structure splitting. PACS numbers:Introduction.-Transition metal dichalcogenide (TMDC) monolayers (MLs), with chemical formula MX 2 , with M=Mo or W and X=S, Se or Te, are semiconductors with a direct bandgap in the visible region situated at the K-point of the Brillouin zone [1,2]. In these 2D systems the broken inversion symmetry of the crystal lattice allows for optical control of the valley degree of freedom [3][4][5][6]. Combined with strong spin-orbit coupling (SOC) this leads to valley-spin locking [7], opening exciting avenues for original applications in optoelectronics and spintronics. Tightly bound excitons [8][9][10][11], with binding energies around 0.5 eV, originate from the direct term of the electron-hole (e-h) Coulomb interaction, enhanced by the strong 2D quantum confinement, the large effective masses, and the reduced dielectric screening in 2D systems. While the energy spectra of these excitons have been intensively studied by combining both experimental and theoretical technics in the last years [12][13][14][15][16][17], little is known about their fine structure.For optoelectronics based on TMDC MLs it is important to clarify if the lowest energy transition is optically bright or optically dark (i.e. spin forbidden) [18]. This bright-dark exciton fine structure splitting governs the optical properties also of 2D semiconductor nanostructures such as III-V and II-VI quantum wells [19][20][21][22][23] as well as CdSe nanocrystals [24]. In particular, the measured low photo-or electro-luminescence yield and its increase with the temperature in monolayer WSe 2 has been interpreted recently in terms of dark excitons lying at lower energy compared to the bright ones [25][26][27][28][29]. This will also affect the efficiency of recently demonstrated WSe 2 or WS 2 light emitting devices [30,31]. For fundamental physics experiments an optically dark ground state can be an advantage, as it allows to study exciton quantum fluids in Bose-Einstein exciton condensates [32]. It is thus crucial to determine both the amplitude and sign of this bright-dark exciton splitting in TMDC MLs.

MoTe 2 belongs to the semiconducting transition-metal dichalcogenide family with certain properties differing from the other well-studied members (Mo,W)(S,Se) 2. The optical band gap is in the near-infrared region, and both monolayers and bilayers may have a direct optical band gap. We first simulate the single-particle band structure of both monolayer and bilayer MoTe 2 with density-functional-theory-GW calculations. We find a direct (indirect) electronic band gap for the monolayer (bilayer). By solving in addition the Bethe-Salpeter equation, we find similar energies for the direct exciton transitions in monolayers and bilayers. We then study the optical properties by means of photoluminescence (PL) excitation, reflectivity, time-resolved PL, and power-dependent PL spectroscopy. With differential reflectivity, we find a similar oscillator strength for the optical transition observed in PL in both monolayers and bilayers suggesting a direct transition in both cases. We identify the same energy for the B-exciton state in the monolayer and the bilayer. Following circularly polarized excitation, we do not find any exciton polarization for a large range of excitation energies. At low temperatures (T = 10 K), we measure similar PL decay times on the order of 4 ps for both monolayer and bilayer excitons with a slightly longer one for the bilayer. Finally, we observe a reduction of the exciton-exciton annihilation contribution to the nonradiative recombination in bilayers.