Probing Excitonic Dark States in Single-layer Tungsten Disulfide

Transition metal dichalcogenide (TMDC) monolayer has recently emerged as an important two-dimensional semiconductor with promising potentials for electronic and optoelectronic devices. Unlike semi-metallic graphene, layered TMDC has a sizable band gap. More interestingly, when thinned down to a monolayer, TMDC transforms from an indirect bandgap to a direct bandgap semiconductor, exhibiting a number of intriguing optical phenomena such as valley selective circular dichroism, doping dependent charged excitons, and strong photocurrent responses. However, the fundamental mechanism underlying such a strong light-matter interaction is still under intensive investigation. The observed optical resonance was initially considered to be band-to-band transitions. In contrast, first-principle calculations predicted a much larger quasiparticle band gap size and an optical response that is dominated by excitonic effects. Here, we report experimental evidence of the exciton dominance mechanism by discovering a series of excitonic dark states in single-layer WS2 using two-photon excitation spectroscopy. In combination with GW-BSE theory, we find the excitons are Wannier excitons in nature but possess extraordinarily large binding energy (~0.7 eV), leading to a quasiparticle band gap of 2.7 eV. These strongly bound exciton states are observed stable even at room temperature. We reveal an exciton series in significant deviation from hydrogen models, with a novel inverse energy dependence on the orbital angular momentum. These excitonic energy levels are experimentally found robust against environmental perturbations. The discovery of excitonic dark states and exceptionally large binding energy not only sheds light on the importance of many-electron effects in this two-dimensional gapped system, but also holds exciting potentials for the device application of TMDC monolayers and their heterostructures.

attraction between the two quasiparticles 21 . This bound state often plays an important role in the optical properties of low dimensional materials 22 , such as carbon nanotubes 23,24 and quantum dots 25 , owing to their strong spatial confinement and reduced screening effect compared to bulk solids. In a 2D gapped system with dipole-allowed interband transitions, the optical absorption spectrum in the non-interacting limit exhibits a step function. Strong electron-hole interaction redshifts a large amount of the spectral weight, resulting in a qualitatively different spectrum with a series of new excitonic levels below the quasiparticle band gap. In quasi-2D quantum wells, the electron-hole interaction is weak 26 . Therefore, by measuring the energy difference between the first excitonic peak and band-edge absorption step, the exciton binding energy can be unambiguously determined, which usually has an energy of 10s of meV and is vulnerable to the environment screening and temperature broadening. However, recent experiments on the single-layer TMDC found no absorption step. Instead, two absorption peaks from spinorbit splitting were detected 4,5 around the Kohn-Sham band gap energy given by density functional theory (DFT) within the local density approximation. The peaks were initially interpreted as direct band edge transitions. In sharp contrast, more accurate firstprinciples calculations using the GW method 27,28 predicted a quasiparticle band gap that is larger than the initial experimental reported value by nearly one electron volt [15][16][17][18] . This energy gap discrepancy is computed through first-principles GW-BSE theory 29 to be originated from strong excitonic effects [15][16][17][18] . It is therefore critical to uncover on firm grounds the underlying physics of the strong light-matter interaction in such a 2D system.
We probed the excitonic effects in TMDC using the two-photon excitation spectroscopy 24,30,31 . At the simplest level, if electron and hole interact through a central attractive Coulomb potential, the electron-hole pair forms a series of excitonic Rydberglike states with definite parity, similar to the hydrogen model. For WS 2 , the breaking of rotational and inversion symmetry owing to the crystal structure and the spatialdependence of screening will modified the energy and symmetry of the states from those of the 2D Rydberg series. However, for exciton states with an electron-hole wavefunction that is large compared to the unit cell size (as shown below for WS 2 ), specific parity may still be assigned to each excitonic state. Incident photons can excite the electronic system from the ground state to one of these excitonic states ( Fig. 1(a)). In addition to energy conservation, the selection rule of such a transition depends on the symmetry of the final state: for systems with dipole-allowed interband transitions (which is the case for WS 2 ), one-photon transitions can only reach excitonic states with even parity, while two-photon transitions reach states with odd parity. The two-photon resonances are also known as excitonic dark states as they do not appear in the linear optical spectrum. These dark states are good gauges for excitonic effects, since there is little impurity and bandgap absorption background in the two-photon spectrum. Owing to the direct band gap in TMDC monolayer, we monitor the two-photon absorption induced luminescence (TPL) with a high signal-to-noise ratio. The luminescence results from the radiative recombination of the excitonic ground state, following the rapid non-radiative relaxation from the two-photon excited excitonic dark states to the ground state ( Fig.   1(a)). By scanning the excitation laser energy, we obtain a complete two-photon spectrum, assuming the relaxation and emission efficiency are independent of the excitation energy 24 .
For our samples, we exfoliate flakes of WS 2 , which has a higher quantum efficiency than other TMDC monolayers 14 , onto a fused quartz substrate from a synthetic WS 2 crystal. A typical light emission spectrum is shown in Fig. 1(b), excited by the ultrafast laser (190 fs) at 990 nm (1.25 eV) at 10K. The two peaks observed at 2.0 and 2.04 eV correspond to the exciton and trion emissions from the direct band gap at K and K' valleys in the Brillouin zone, consistent with the absorption peaks in the reflectance spectrum. The emitted photon energies of both peaks are much higher than those of the excitation photon, and therefore, they can only originate from the two-photon absorption induced luminescence. The two-photon origin of these emissions is further confirmed in the inset to Fig. 1(b). Both the TPL and the SHG signal show a quadratic power dependence, suggesting that the emission is indeed induced by two-photon absorption. The TPL saturates at higher power as a consequence of heating or exciton-exciton annihilation effects 32,33 . For the rest of the experiments, we limit the excitation power to the unsaturated regime. The trion peak amplitude is selected as our TPL signals, since it is stronger than the neutral ground-state exciton emission at 10K.
We observed two important resonances of similar linewidths in the two-photon spectrum, occurring at 2.28 and 2.48 eV, corresponding to two excitonic dark excited states (Fig. 2).
The absorption spectrum of WS 2 monolayer is plotted for comparison, where the A exciton (the 1s state) and its trion result in two absorption peaks at 2.04 and 2 eV, respectively. Near these one-photon resonances, TPL is negligible, consistent with the 1s nature of these states. On the other hand, no significant one-photon absorption is observed near the excitonic dark states, except for the B exciton (the other 1s state) at 2.45 eV resulted from the spin-orbit splitting in the valence band. Such a complimentary feature reflects the symmetry of the observed excitonic states. Hence, we label the TPL peaks to be the 2p and 3p state of the A exciton series. Accordingly, the 1s-2p and 1s-3p separations are 0.24 eV and 0.44 eV respectively. The extraordinary large 1s-np (n=2, 3) separations suggests that the exciton binding energy, defined as the separation between the 1s exciton ground state and the conduction band edge, is larger than 0.44 eV, which also indicates a significant self-energy contribution to the quasi-particle band gap. Our discovery demonstrates that the previously claimed band-to-band transition mechanism in monolayer TMDC's optical response is qualitatively incorrect, which as we now show is dominated by excitonic states within the band gap, in agreement with the GW-BSE calculation in MoS 2 18 . The real quasiparticle band gap is much larger than previously reported. This finding is expected be general for other TMDC monolayer of similar structures.
We used the ab initio GW method 28 to calculate the quasiparticle band structure and the ab initio GW-BSE approach 29 to calculate the excitonic states and optical spectrum of a WS 2 monolayer ( Fig. 3 A), employing the BerkeleyGW package 34  Wannier nature with their in-plane radii much larger than the unit cell dimension.
In spite of its Wannier character, we found the exciton series in monolayer WS 2 deviates significantly from a 2D hydrogen model, which has also been predicted in recent GW-BSE calculations 18,35 . The ratio between 1s-2p and 1s-3p separations is 27/32 and 25/27 in 2D and 3D hydrogen models, respectively; neither of which is close to our experimental results or the GW-BSE results (approximately 6/11). In addition, in a hydrogen atom, orbitals with the same principal quantum number are degenerate.
However for the WS 2 excitons, our calculations show that states in the same shell but of higher orbital angular momentums are at lower energies, i.e., E 3d < E 3p < E 3s . Analysis of the theoretical results revealed that these two exotic energy-level behaviors are caused by a strong spatial-dependent dielectric screening: in an atomically thin semiconductor, the screening effect is weaker when the separation between the electron and hole is bigger, which is known as the anti-screening effect in 1D carbon nanotube 36 . Since the wavefunction of excitonic states with higher principal or higher orbital quantum number features a larger nodal structure near the hole (i.e., larger average electron-hole separation), weaker screening at larger separation leads to enhanced Coulomb attraction in the excited states and therefore lowering their excitation energies as compared those of the hydrogen model 36 . Also, because of the degeneracy of the K and K' valleys in TMDC system, each s level has two degenerate states, while each p and d level has 4 states if perfect rotational symmetry is assumed. All of these features are expected to be quite general for 2D TMDC excitons.
The GW quasiparticle band gap is calculated to be ~ 2.7 eV, labeled by the blue arrow in Fig. 3. Comparing it with the 1s exciton energy found in either our experiment or our GW-BSE calculation, we obtain an exciton binding energy of ~0.7 eV. Such an exceptionally large binding energy is more than ten times larger than the excitons in bulk In all capped samples, we observed the 2p and 3p resonances even at room temperature, as expected from the large exciton binding energy. (Fig. 4(a)) We find no significant shift in the excitation energy of either the s or the p states with different capping layers, except for an overall temperature related redshift (0.04 eV) and linewidth broadening compared with measurement at 10k (Fig. 2). For the lowest bound excitonic state, the insensitivity of the emitted photon energy to external dielectric screening can be understood as the environmental screening's opposite effects to the electron self-energy and the exciton binding energy. The same argument may apply to the lower-energy excited states.
Nevertheless, it is interesting that, with different capping, the 1s-2p and 1s-3p energy differences remain roughly unchanged, ~0.2 and 0.5 eV, respectively. This robustness indicates the measured excitation energies for the 2p and 3p states are intrinsic to the monolayer, thus agreeing well with those from ab-initio GW-BSE calculation for the vacuum condition. Together with the TPL signal, SHG is also observed as a slanted straight line in the excitation-emission spectra ( Fig. 4(b)). At room temperature, the exciton-trion separation is no longer distinguishable, but the 2p and 3p absorption peaks remain prominent. A SHG resonance occurs as the TPL and SHG line cross each other, known as the exciton enhanced SHG effect 37 .
SHG signals are originated from the broken inversion symmetry within the WS 2 monolayer 37,38 . As discussed above, strictly speaking, parity is not a good quantum number for any inversion-symmetry

Two-photon excitation spectroscopy:
The excitation spectroscopy is carried out with an optical parametric oscillator (Newport,     At a low excitation level, both of them exhibit quadratic power dependence, confirming the two-photon absorption nature of the luminescence, until the TPL signal saturates at a high excitation level. In the low temperature experiment, the TPL signal represents the peak amplitude of the trion peak. Additionally, the A exciton and trion (1s) absorption peaks are detected consistently with the TPL emission peaks (Fig. 1 b), with a 20 meV Stoke shift, and are marked at 2.04 and 2 eV, respectively. The energy difference between the A exciton 1s state emission peak and the 3p state absorption peak is 0.44 eV which yields the lower bound for the exciton binding energy in monolayer WS 2 . This binding energy is extraordinarily large for a Wannier exciton, and implies a dominating excitonic mechanism for the intense lightmatter interaction in 2D TMDC. The total excitation scan is achieved by tuning an output beam of an optical parametric oscillator over a 600 meV span, with a scanning resolution about 15 meV. (see Methods)  separation is approximately the same as the low-temperature uncapped result (Fig. 2), suggesting the excitation energy of the low-energy exciton levels are relatively insensitive to dielectric environmental and temperature perturbations, as discussed in the main text.
b. Measured emission spectra at different excitation energies of an immersion-oil capped WS 2 monolayer at room temperature. The horizontal line signal is the TPL emission, with two hotspots along the line corresponding to the 2p and 3p two-photon absorption peaks.