Large spin-orbit splitting of deep in-gap defect states of engineered sulfur vacancies in monolayer \ceWs2
Structural defects in 2D materials offer an effective way to engineer new material functionalities beyond conventional doping in semiconductors 1; 2; 3; 4; 5. Specifically, deep in-gap defect states of chalcogen vacancies have been associated with intriguing phenomena in monolayer transition metal dichalcogenides (TMDs) 2; 1; 6; 7; 8; 9; 10. Here, we report the direct experimental correlation of the atomic and electronic structure of a sulfur vacancy in monolayer \ceWS2 by a combination of CO-tip noncontact atomic force microscopy (nc-AFM) and scanning tunneling microscopy (STM).
Sulfur vacancies, which are absent in as-grown samples, were deliberately created by annealing in vacuum. Two energetically narrow unoccupied defect states of the vacancy provide a unique fingerprint of this defect. Direct imaging of the defect orbitals by STM and state-of-the-art GW calculations reveal that the large splitting of 252 meV between these defect states is induced by spin-orbit coupling.
The controllable incorporation and potential decoration of chalcogen vacancies provide a new route to tailor the optical, catalytic and magnetic properties of TMDs.
Transition metal dichalcogenides (TMDs) and other layered materials have recently attracted considerable interest because of their unique properties arising from the combination of quantum confinement, reduced screening and lack of inversion symmetry in the monolayer limit.
The strong confinement, however, also causes TMD properties to be particularly sensitive to defects.
Structural defects in TMDs are thought to substantially modify optoelectronic properties and induce catalytic functionality to the otherwise inert surface.
Particularly, chalcogen vacancies have been attributed to a variety of phenomena including single-photon emission 6, defect-bound excitons 2; 7; 9; 10, catalytic activity 8 and hopping transport 1. In most of these studies, the chalcogen vacancy functionality was only indirectly inferred by the presence of this defect in transmission electron microscopy (TEM) 11; 1; 12; 13; 14; 5. Moreover, TMD monolayers are known to be electron beam sensitive and vacancy defects can be created in-situ by knock-on or radiolysis effects 11; 1; 12; 13; 14; 5, as reflected in the very high reported defect densities on the order of cm, even in exfoliated samples, 14 with an estimated vacancy generation rate of about cms 11; 1.
A decisive factor for the functionality of defects is the creation of defect states in the band gap of the semiconductor.
While TEM can routinely resolve the atomic lattice, the electronic structure around the Fermi level is not easily accessible by TEM.
Conversely, scanning tunneling microscopy (STM) can probe the electronic structure of even single defects 15; 16; 17; 18; 19; 10; 20; 21, but the defect assignment is not straightforward because their STM contrast is dominated by their electronic states, and tip-dependent contrast inversion makes it difficult to assign lattice sites. Both of these complications have lead to recent contradictory defect identification in TMDs by STM 18; 10; 21.
In this study, we use a combination of CO-tip noncontact atomic force microscopy (nc-AFM), scanning tunneling microscopy/spectroscopy (STM/STS) and ab initio GW calculations to unambiguously identify and characterize the chalcogen vacancy in \ceWS2.
We find that chalcogen vacancies are largely absent in as-grown TMD samples under ambient conditions. Chalcogen vacancies were, however, deliberately created by in vacuo annealing at elevated temperatures. In STS, the sulfur vacancy in WS exhibits a characteristic fingerprint with two narrow unoccupied defect states accompanied by vibronic satellite peaks. The observed splitting between the two defect peaks is caused by extraordinarily strong spin-orbit coupling. Our results challenge the current perception of the presence and functionality of the most discussed defect in TMDs. It also opens up new avenues for defect engineering in the context of valleytronics, solitary dopant optoelectronics and catalysis.
The \ceWS2 samples are grown using chemical vapor deposition (CVD) on graphitized SiC substrates as describe in the Methods section. As argued in a recent paper 20, chalcogen site defects are abundant but they can be identified as oxygen substituents rather than chalcogen vacancies with a radically different electronic structure 22; 20. Undecorated sulfur vacancies can, however, be readily generated by annealing or ion bombardment in vacuum as reported previously 11; 2; 23; 24. Calculations also showed that in vacuum, the chalcogen vacancy has the lowest formation energy of any intrinsic defect in several TMD materials 12; 14. Note that a recent study found that this is not necessarily the case when other molecules are present 20.
In Fig. 1 STM and nc-AFM maps of a substituted oxygen defect (O) and a sulfur vacancy (V) are shown in both the top and bottom sulfur layer (facing the tip or the underlying graphene) of the monolayer \ceWS2. At moderate annealing temperatures of around 250 C only the O top and bottom defects are observed along with tungsten substitutions. When the sample is annealed to 600 C, pristine, undecorated sulfur vacancies were also detected in addition to all other point defects that were present before. Substitutional oxygen and sulfur vacancies can be clearly distinguished from each other in STM (c.f. Fig. 1a,c and Fig. 1b,d). Note that a similar STM contrast like in Fig. 1b,d that we assign to O has been reported for \ceMoS2 and \ceWSe2 and was ascribed either to a sulfur vacancy 18 or a tungsten vacancy 10; 21, revealing the difficulty of defect structure identification based on STM alone.
Our defect structure assignment is based on the CO-tip nc-AFM images that are in excellent agreement with simulations based on the probe particle model 25 (see Fig. S3) as well as the distinct defect electronic structure as discussed in detail below.
While the CO-tip in nc-AFM is exceptionally sensitive to the outermost surface layer, it is difficult to distinguish between vacancies and substituted oxygens, which are located slightly below the surface sulfur plane. For both types of defects the surrounding surface sulfur atoms relax similarly and the oxygen atom of O binds closer to the tungsten plane, the defect in the top sulfur layer appears as a missing S atom, and the defect in the bottom sulfur layer appears as a S atom that is protruding from the surface for both V and O (c.f. Fig. 1e,f and Fig. 1g,h). In direct comparison (i.e. when measured in the same image), O appears slightly more attractive than V (see Fig. S2). We would like to point out that TEM would not be able to discriminate an oxygen substituent from a chalcogen vacancy because the sulfur atom on the opposite side of the layer masks the presence of the low atomic number O atom.
Electronically, however, the sulfur vacancy and the O substituent are fundamentally different. While O does not feature defect states in the \ceWS2 band gap 20, V does have pronounced deep in-gap defect states, which will be discussed next.
In Fig. 2a,b STS spectra of a single V are shown.
It is important to note that both the top and bottom V are electronically equivalent, indicating the negligible influence of the graphene substrate (see Fig. S6).
The defect introduces a series of sharp resonances in the unoccupied spectrum at positive sample bias and a single resonance at negative bias. The occupied defect state resonance is located about below the valence band maximum (VBM) overlapping with delocalized bulk states, similar to the defect resonance observed for O. Strikingly, we find two unoccupied defect states at and below the conduction band minimum (CBM), deep in the band gap. Each of these defect state resonances is accompanied by vibronic satellite peaks that stem from inelastically tunneling electrons commonly observed in transport spectroscopy of molecules 26.
Using the Franck-Condon model we can estimate the electron-phonon coupling strength and the dominant phonon mode that is excited when transiently charging the defect by tunneling electrons. From the relative peak intensities and the vibronic peak separation we estimate a Huang-Rhys factor of with a phonon mode of meV for the lower energy defect state and with meV for the higher energy defect state. Therefore, the electron-phonon coupling strength is in the intermediate regime (). The initial broadening of meV (full width at half maximum) increases for higher phonon replicas.
The V defect’s purely electronic states at the lowest excitation energy are denoted zero-phonon line (ZPL) in Fig. 2b, following the conventions used in absorption/emission spectroscopy. Most notably, both defect resonances exhibit the same spatial electron distribution as evident from the dI/dV maps shown in Fig. 2c. We attribute this observation to a lifted degeneracy induced by spin-orbit coupling (SOC). Hence, the energy separation between the two elastic excitations quantifies the spin-orbit interaction as , which is exceptionally large. As we will see below, each of the two peaks is composed of two degenerate defect states. In dI/dV maps these states are imaged as a superposition of the charge densities of the degenerate orbitals.
The defect states appear different for the top and bottom V since the defect and its orbital are not mirror-symmetric with respect to the tungsten plane.
The spectra shown in Fig. 2a,b are measured on \ceWS2(1ML) on bilayer graphene. On a monolayer graphene substrate, the filled-state spectrum is qualitatively different. We find an additional major resonance around and the valence band is pushed upwards (see Fig. S4). This additional feature is identified as a charging peak (see Fig. S5). The different behavior of mono- vs bilayer graphene substrate can be explained by the observed energetic shift of the V\textsubscriptS defect states and conduction band towards lower energies of about . The shift is induced by a combination of screening effects and shift in work function between mono- and bilayer graphene 27. Since the V\textsubscriptS defect states shift closer to the Fermi level on the monolayer graphene substrate, tip-induced band bending is sufficient to pull the lower V\textsubscriptS state below the Fermi level, hence negatively charging the defect in the vicinity of the tip at sufficiently large electric fields. Using the charging peak, a field-induced shift of 11% of the applied bias voltage was estimated at the chosen tunneling set point. Note that the defect state energies stated above were corrected for this field-induced shift. For a detailed discussion about the tip-induced charging we refer to the SI.
To verify our interpretation of the spin-orbit split defect states of the sulfur vacancy, we calculated the electronic structure of a WS monolayer with chalcogen vacancy point defects using the ab initio GW approach. We constructed a supercell consisting of 5 unit cells along each crystalline-axis direction of the monolayer plane (namely 50 S atoms and 25 W atoms) and then removed a single chalcogen atom 4. We accounted for spin-orbit coupling via a fully-relativistic, noncollinear density functional theory (DFT) starting point as implemented in Quantum Espresso and built one-shot GW energy corrections on top of it, as implemented in the BerkeleyGW package (see Methods section for full computational details). Fig. 3 shows the resulting DFT band structures computed within the local density approximation (LDA) and GW energy levels. At both the DFT and GW level, the sulfur vacancy introduces four unoccupied in-gap states, which form two pairs of nearly-degenerate flat bands in the gap, corresponding to the two deep in-gap resonances in the dI/dV. The nearly-degenerate state are time-reversal pairs whose degeneracy has been lifted slightly due to interaction between periodic images of the 5x5 supercell. The charge distribution of the two in-gap states are highly localized around the S vacancy, as shown in Fig. 3a. The calculated defect orbitals for the top and bottom V are in excellent agreement with the corresponding dI/dV maps shown in Fig. 2c. In addition, an occupied doubly-degenerate defect-localized resonant state appears in the valence region, in agreement with experimental observation (black line in Fig. 3c). Fig. 3c shows the resulting energy gaps. The one-shot GW correction opens the VBM-CBM gap to 2.8 eV (compared to the calculated DFT gap of 1.6 eV), comparing well with the experimental value of 2.5 eV. The error bar of the bandgap within our calculation is estimated to be 150 meV - as a result of the sensitivity of the GW approach to the DFT starting point and to structural effects. Note that screening effects from the graphene substrate are not included in the calculations but can be expected to reduce the gap by a few hundred meV 28; 29. Importantly, the GW gap between the highest in-gap defect state and the conduction band is 0.6 eV, in reasonable agreement with the measured value of 0.52 eV in experiment. The spin-orbit energy splitting between the two doubly-degenerate in-gap states is 180 meV from our calculations.
Our theoretical calculations also shed light on the spin-orbit splitting and character of the in-gap states. Although many theoretical studies predicted the chalcogen vacancy to form in-gap states2; 12; 13; 14; 19, only a few explicitly consider the effect of SOC 30; 31.
We find that each in-gap peak in the dI/dV corresponds to two degenerate states belonging to a time-reversal pair. The character of the in-gap states consists primarily of W d-states, which are responsible for the large magnitude of the spin-orbit splitting, with some smaller contributions from the S 3p and W 6p states (see SI). The lower-energy in-gap states have a larger contribution from J = 3/2 states, and the higher energy in-gap states have a larger contribution from J = 5/2 states. The close correspondence between the in-gap states in theory and experiment are a clear indication of the presence of sulfur vacancies. Importantly, the hybridization between defect-localized in-gap states and delocalized, pristine-like states can lead to significant valley depolarization 4 and suggests a path to control spin-valley selectivity through defect engineering.
In summary, we created and identified individual sulfur vacancies in monolayer \ceWS2 by a combination of atomic-resolution nc-AFM, STS and ab initio GW calculations. We show that a sulfur vacancy gives rise to two unoccupied in-gap defect states that appear as sharp resonances followed by vibronic satellite peaks in STS. The deep in-gap states act as a strong atom trap, which explains why undecorated chalcogen vacancies are largely absent in as-grown TMD samples under ambient conditions. Remarkably, the degeneracy between the four V defect orbitals is lifted by spin-orbit interaction into two pairs of degenerate orbitals as revealed by direct STM orbital imaging and state-of-the-art DFT and GW calculations. The exceptionally large spin-orbit splitting between the sulfur vacancy states was measured to be , consistent with our theoretical predictions. These results suggest that the controllable introduction of chalcogen vacancies in vacuum could be used to tune the spin-valley polarization in TMDs and may be potentially used as single-photon emitters. Moreover, the reactive vacancy sites are expected to trap diffusing adatoms, hence embedding arbitrary dopants into the 2D TMD matrix. This concept could be particularly interesting to study the interaction of magnetic impurities in a highly-correlated material or to catalytically activate the inert basal plane of TMDs.
.1 \ceWS2/MLG/SiC CVD growth\ce
WS2 few-layer islands were grown by a modified chemical vapor deposition process 32 on graphitized (6H)-SiC(0001) substrates. We used WO\textsubscript2.9 powder ( Mo impurity concentration) and \ceH2S gas as the metal and chalcogen precursors, respectively. The growth temperature was 900 C and the growth time was . This process lead to the formation of predominantly monolayer \ceWS2 islands on the mixed single and bilayer graphene substrate (\ceWS2/MLG/SiC). The sample synthesis is explained in detail in Ref. 33.
.2 Scanning probe measurements
The experiments were performed using a CreaTec low-temperature (), ultra-high vacuum () combined STM and nc-AFM. The sensor was based on a qPlus 34 quartz-crystal cantilever design operated in the frequency-modulation mode 35 (resonance frequency , spring constant , quality factor , and oscillation amplitude ). The voltage was applied to the sample. STM images were taken in constant-current mode. STS spectra are performed in constant-height mode with a lock-in amplifier running at and . The STS spectra has been characterized on Au(111) to ensure a relatively flat tip density of states. Nc-AFM measurements were acquired in constant-height mode at . The known contrast mechanism of CO-tip nc-AFM imaging enabled the unambiguous lattice site identification. 25; 36; 37
Sample and tip preparation.
The CVD grown \ceWS2/MLG/SiC was annealed in vacuo either at about or for . At the lower annealing temperatures the only defects at sulfur sites were O\textsubscriptS defects. At the higher annealing temperature, however, pristine sulfur vacancies were found as well along with O\textsubscriptS defects (see Figs. S1,2).
.3 Density-functional theory (DFT) and GW calculations
We first performed a density functional theory (DFT) calculation in the local density approximation (LDA) 40, with the Quantum Espresso package 41, to obtain the mean-field starting point of our GW calculation. The calculations were done with a 55 supercell arrangement of the WS monolayer with one chalcogen vacancy per cell. We use a plane-wave basis and norm-conserving pseudopotentials with a 70 Ry wave function cutoff. We included the W 5s, 5p, and 5d states as valence states. The distance between repeated supercells in the out-of-plane direction was 15 Å. We fully relaxed the atomic coordinates while constraining the lattice constant to the experimental values of 3.15. A 16161 k-point grid was used to calculate the self-consistent charge density.
We performed GW calculations with the BerkeleyGW code 42. Our main results are obtained within the Hybertsen-Louie generalized plasmon-pole (HL-GPP) model 43; 44; 42. We used an energy cutoff of 25 Ry for the reciprocal lattice components of the dielectric matrix and included 2000 states in the summation over unoccupied states (convergence was tested with up to 20,000 states). We employed the nonuniform neck subsampling (NNS) scheme 45 to sample the Brillouin zone and to speed up the convergence with respect to k-point sampling. In this scheme, we use a 331 uniform q-grid and include an additional 10 q-points in the voronoi cell around q=0, such that the smallest q-vector corresponds to â¼1/1150th of a reciprocal lattice. This corresponds to an effective q-grid of more than 100010001 uniform q-points and converges the QP energies within better than 0.01 eV. A truncated Coulomb interaction was used to prevent spurious interactions between periodic images of the 2D sheet 46.
To include the spin-orbit interactions we then performed a fully-relativistic noncollinear DFT calculation within Quantum Espresso, using the relaxed structure from the non-relativistic calculation. A wave function cutoff of 150 Ry was used in the fully-relativistic calculation. Then, the spin-orbit corrections at the DFT level were combined with the GW corrections following Ref. 47.
- (1) Qiu, H. et al. Hopping transport through defect-induced localized states in molybdenum disulphide. Nat. Commun. 4, 2642 (2013).
- (2) Tongay, S. et al. Defects activated photoluminescence in two-dimensional semiconductors: interplay between bound, charged, and free excitons. Sci. Rep. 3, 2657 (2013).
- (3) Lin, Z. et al. Defect engineering of two-dimensional transition metal dichalcogenides. 2D Mater. 3, 022002 (2016).
- (4) Refaely-Abramson, S., Qiu, D. Y., Louie, S. G. & Neaton, J. B. Defect-induced modification of low-lying excitons and valley selectivity in monolayer transition metal dichalcogenides. Phys. Rev. Lett. (in press) (2018).
- (5) Wang, S., Robertson, A. & Warner, J. H. Atomic structure of defects and dopants in 2D layered transition metal dichalcogenides. Chem. Soc. Rev. 47, 6764–6794 (2018).
- (6) Srivastava, A. et al. Optically active quantum dots in monolayer WSe. Nat. Nanotechnol. 10, 491–496 (2015).
- (7) Chow, P. K. et al. Defect-induced photoluminescence in monolayer semiconducting transition metal dichalcogenides. ACS Nano 9, 1520–1527 (2015).
- (8) Li, H. et al. Activating and optimizing MoS basal planes for hydrogen evolution through the formation of strained sulphur vacancies. Nat. Mater. 15, 48 (2016).
- (9) Carozo, V. et al. Optical identification of sulfur vacancies: Bound excitons at the edges of monolayer tungsten disulfide. Sci. Adv. 3, e1602813 (2017).
- (10) Zhang, S. et al. Defect structure of localized excitons in a WSe monolayer. Phys. Rev. Lett. 119, 046101 (2017).
- (11) Komsa, H.-P. et al. Two-dimensional transition metal dichalcogenides under electron irradiation: defect production and doping. Phys. Rev. Lett. 109, 035503 (2012).
- (12) Zhou, W. et al. Intrinsic structural defects in monolayer molybdenum disulfide. Nano Lett. 13, 2615–2622 (2013).
- (13) Komsa, H.-P. & Krasheninnikov, A. V. Native defects in bulk and monolayer MoS from first principles. Phys. Rev. B 91, 125304 (2015).
- (14) Hong, J. et al. Exploring atomic defects in molybdenum disulphide monolayers. Nat. Commun. 6, 6293 (2015).
- (15) Lu, C.-P., Li, G., Mao, J., Wang, L.-M. & Andrei, E. Y. Bandgap, mid-gap states, and gating effects in MoS. Nano Lett. 14, 4628–4633 (2014).
- (16) Hildebrand, B. et al. Doping nature of native defects in 1T-TiSe. Phys. Rev. Lett. 112, 197001 (2014).
- (17) Peng, J.-P. et al. Molecular beam epitaxy growth and scanning tunneling microscopy study of TiSe ultrathin films. Phys. Rev. B 91, 121113 (2015).
- (18) Liu, X., Balla, I., Bergeron, H. & Hersam, M. C. Point defects and grain boundaries in rotationally commensurate MoS on epitaxial graphene. J. Phys. Chem. C. 120, 20798–20805 (2016).
- (19) Vancsó, P. et al. The intrinsic defect structure of exfoliated MoS single layers revealed by scanning tunneling microscopy. Sci. Rep. 6, 29726 (2016).
- (20) Barja, S. et al. Identifying substitutional oxygen as a prolific point defect in monolayer transition metal dichalcogenides with experiment and theory. (submitted) (2018).
- (21) Lin, Y.-C. et al. Realizing large-scale, electronic-grade two-dimensional semiconductors. ACS Nano 12, 965–975 (2018).
- (22) Lu, J. et al. Atomic healing of defects in transition metal dichalcogenides. Nano Lett. 15, 3524–3532 (2015).
- (23) Klein, J. et al. Robust valley polarization of helium ion modified atomically thin MoS. 2D Mater. 5, 011007 (2017).
- (24) Liu, M. et al. Temperature-triggered sulfur vacancy evolution in monolayer MoS/graphene heterostructures. Small 13, 1602967 (2017).
- (25) Hapala, P. et al. Mechanism of high-resolution STM/AFM imaging with functionalized tips. Phys. Rev. B 90, 085421 (2014).
- (26) Qiu, X., Nazin, G. & Ho, W. Vibronic states in single molecule electron transport. Phys. Rev. Lett. 92, 206102 (2004).
- (27) Mammadov, S. et al. Work function of graphene multilayers on SiC(0001). 2D Mater. 4, 015043 (2017).
- (28) Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nat. Mater. 13, 1091–1095 (2014).
- (29) Bradley, A. J. et al. Probing the role of interlayer coupling and coulomb interactions on electronic structure in few-layer MoSe nanostructures. Nano Lett. 15, 2591–2599 (2015).
- (30) Yuan, S., Roldán, R., Katsnelson, M. & Guinea, F. Effect of point defects on the optical and transport properties of MoS and WS. Phys. Rev. B 90, 041402 (2014).
- (31) Li, W.-F., Fang, C. & van Huis, M. A. Strong spin-orbit splitting and magnetism of point defect states in monolayer WS. Phys. Rev. B 94, 195425 (2016).
- (32) Kastl, C. et al. The important role of water in growth of monolayer transition metal dichalcogenides. 2D Mater. 4, 021024 (2017).
- (33) Kastl, C. et al. Multimodal spectromicroscopy of monolayer WS enabled by ultra-clean van der Waals epitaxy. 2D Mater. 5, 045010 (2018).
- (34) Giessibl, F. J. High-speed force sensor for force microscopy and profilometry utilizing a quartz tuning fork. Appl. Phys. Lett. 73, 3956 (1998).
- (35) Albrecht, T. R., Grütter, P., Horne, D. & Rugar, D. Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity. J. Appl. Phys. 69, 668–673 (1991).
- (36) Hapala, P., Temirov, R., Tautz, F. S. & Jelínek, P. Origin of high-resolution IETS-STM images of organic molecules with functionalized tips. Phys. Rev. Lett. 113, 226101 (2014).
- (37) Barja, S. et al. Charge density wave order in 1D mirror twin boundaries of single-layer MoSe. Nat. Phys. 12, 751–756 (2016).
- (38) Gross, L., Mohn, F., Moll, N., Liljeroth, P. & Meyer, G. The chemical structure of a molecule resolved by atomic force microscopy. Science 325, 1110 (2009).
- (39) Mohn, F., Schuler, B., Gross, L. & Meyer, G. Different tips for high-resolution AFM and STM imaging of single molecules. Appl. Phys. Lett. 102, 073109 (2013).
- (40) Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).
- (41) Giannozzi, P. et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys. Condens. Mat. 29, 465901 (2017).
- (42) Deslippe, J. et al. Berkeleygw: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Comput. Phys. Commun. 183, 1269 – 1289 (2012).
- (43) Hybertsen, M. S. & Louie, S. G. First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators. Phys. Rev. Lett. 55, 1418–1421 (1985).
- (44) Hybertsen, M. S. & Louie, S. G. Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Phys. Rev. B 34, 5390–5413 (1986).
- (45) da Jornada, F. H., Qiu, D. Y. & Louie, S. G. Nonuniform sampling schemes of the brillouin zone for many-electron perturbation-theory calculations in reduced dimensionality. Phys. Rev. B 95, 035109 (2017).
- (46) Ismail-Beigi, S. Truncation of periodic image interactions for confined systems. Phys. Rev. B 73, 233103 (2006).
- (47) Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Optical spectrum of MoS: Many-body effects and diversity of exciton states. Phys. Rev. Lett. 111, 216805 (2013).
We thank Andreas Schmid and Nicholas Borys for discussions. B.S. appreciates support from the Swiss National Science Foundation under project number P2SKP2_171770.
Theoretical work was supported by the Center for Computational Study of Excited State Phenomena in Energy Materials, which is funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05CH11231, as part of the Computational Materials Sciences Program. Work performed at the Molecular Foundry was also supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under the same contract number. S.R.A acknowledges Rothschild and Fulbright fellowships. S.B. acknowledges support by the European Union under FP7-PEOPLE-2012-IOF-327581 and Spanish MINECO (MAT2017-88377-C2-1-R). This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.