Splitting between bright and dark excitons in transition metal dichalcogenide monolayers

Abstract

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.

Probing the Spin-Polarized Electronic Band Structure in Monolayer Transition Metal Dichalcogenides by Optical Spectroscopy

“…Similarly, we can estimate the spinsplitting energies of the bands, particularly, . These values are also consistent with ab initio calculations [24][25][26] . We note that the two spin-split conduction bands in monolayer MoSe 2 are predicted to cross at large momentums from the K/K' point due to their spin-dependent masses 24,25 .…”

“…≈ 480 meV. These values are in good agreement with ab initio calculations [24][25][26] . A similar value for ∆ !…”

“…Recent first principles calculations including higherorder effects have predicted a finite ∆ ! with opposite sign for Mo-and W-based TMDs [24][25][26] [i.e. order of spin-up and spin-down bands, Fig.…”

“…We note, however, that the electronic transitions probed by optical absorption and PL are modified by the Coulomb interaction effects, which are not considered in the simple density functional theory (DFT) calculations 24,25 . A more accurate description of our results and determination of the band spin splitting requires a many-body theory that takes into account the interaction effects 26,35,36 . Figure 1(d) illustrates the schematic structure of dual-gate field-effect devices of monolayer TMDs employed in this study.…”

“…The physical origin of the distinct spin polarization in the conduction bands of these materials has been revealed by ab initio calculations [24][25][26] . As mentioned earlier, the conduction bands mainly arise from the !…”