## Abstract

Monolayer transition metal dichalcogenides (TMDCs) are promising two-dimensional (2D) semiconductors for application in optoelectronics. Their optical properties are dominated by two series of photo-excited exciton states—A (XA) and B (XB)^{1,2}—that are derived from direct interband transitions near the band extrema. These exciton states have large binding energies and strong optical absorption^{3,4,5,6}, and form an ideal system to investigate many-body effects in low dimensions. Because spin–orbit coupling causes a large splitting between bands of opposite spins, XA and XB are usually treated as spin-polarized Ising excitons, each arising from interactions within a specific set of states induced by interband transitions between pairs of either spin-up or spin-down bands (TA or TB). Here, by using monolayer MoS_{2} as a prototypical system and solving the first-principles Bethe–Salpeter equations, we demonstrate a strong intravalley exchange interaction between TA and TB, indicating that XA and XB are mixed states instead of pure Ising excitons. Using 2D electronic spectroscopy, we observe that an optical excitation of the lower-energy TA induces a population of the higher-energy TB, manifesting the intravalley exchange interaction. This work elucidates the dynamics of exciton formation in monolayer TMDCs, and sheds light on many-body effects in 2D materials.

## Main

Among the various types of many-body effect, the electron–hole exchange interaction is known to be an important mechanism for defining exciton landscape^{7,8} and for exciton spin relaxation in quantum wells based on III–V semiconductors^{9,10,11}. This effect also contributes to valley depolarization^{12}, valley decoherence^{13,14}, valley–orbit coupling for excitons^{15} and the shaping of the exciton dispersion relation^{16,17} in monolayer transition metal dichalcogenides (TMDCs). Despite the extensive studies of many-body effects among excitons in monolayer TMDCs^{18,19,20}, understanding of the excitonic exchange interaction within the same valley is incomplete due to the insufficiency of existing first-principles methods, and the restricted resolution of the coupling between photo-excited states with traditional linear optical spectroscopy. In this work, using newly developed first-principles many-body perturbation methods with two-component spinor wavefunctions and state-of-the-art two-dimensional (2D) electronic spectroscopy, we numerically demonstrate and experimentally verify the role of the strong intravalley exchange interaction in forming XA and XB in monolayer MoS_{2}. This exchange interaction is significant because of the reduced dielectric screening and enhanced wavefunction overlap between the electrons and the holes in a monoatomic layer and is manifested in the transient optical response.

An exciton is an electron–hole pair, bound by Coulomb interactions. The interaction between the electron and the hole contains two terms, a direct screened Coulomb interaction and an exchange bare Coulomb interaction. The attractive direct interaction corresponds to a classical picture of Coulomb attraction between the two oppositely charged quasiparticles, producing a series of hydrogenic-like electron–hole states, of which the envelope wavefunctions have specific nodal structure and angular momentum quantum numbers (1*s*, 2*s*, 2*p* and so on). The diagrams labelled TA and TB in Fig. 1a schematically show such electron–hole states in the K valley of monolayer TMDCs, which are bound by the direct interaction only. The states TA and TB are derived from interband transitions occurring between the spin-up and the spin-down band-pairs, respectively, and are therefore referred to as Ising excitons. The repulsive exchange interaction involving the bare Coulomb interaction, on the other hand, arises from the exchange symmetry of the many-fermion wavefunctions, and leads to exciton eigenstates consisting of electron–hole states with mixed spins as also illustrated in diagrams labelled XA and XB in Fig. 1a. We employ the first-principles GW method (the self-energy operator is approximated by the product of the Green's function *G* and screened interaction *W*) and solve the Bethe–Salpeter equations (BSE) with the calculated quasiparticle states—that is, using the GW-BSE approach to fully incorporate the exchange interaction, particularly the intravalley exchange interaction between TA and TB—and demonstrate that the true exciton eigenstates, XA and XB, are mixed states of TA and TB. Fig. 1d–i shows the *k*-space wavefunction amplitudes of the bright exciton eigenstates XA-1*s*, XB-1*s* and XA-2*s* projected onto TA (Fig. 1d,f,h) and TB subspaces (Fig. 1e,g,i). TA and TB are illustrated in Fig. 1b,c, respectively, for clarity. We label the principal quantum number of an exciton eigenstate by the nodal structure of its envelope wavefunction in its major subspace where the amplitude is the largest. The 1*s* state of XA (XB) primarily includes TA (TB), but also mixes in 3.6% of TB (5.3% of TA) by the intravalley exchange interaction. Moreover, the mixing does not only occur between states of the same principal quantum number. For example, the *k*-space wavefunction amplitude of XA-2*s* has components with one node in TA subspace, but shows components with no node (a 1*s*-like character) in TB subspace. Details of the calculations are presented in Supplementary Section 1.

The mixing of TA and TB produces an important change in the ratio of the optical oscillator strengths of XA and XB in the calculations. This large asymmetry in the oscillator strengths has been observed in experiments but has not been well explained^{21}. The calculated oscillator strengths of optical transitions (in atomic units) to the different exciton states are presented in Table 1; these should be proportional to the product of photon frequency and absorbance at normal incidence. The definition of the oscillator strength used in Table 1 can be found in Supplementary Section 1. If the exchange interaction between TA and TB is excluded in the calculations, the two Ising exciton 1*s* states have nearly equal oscillator strengths, consistent with previous theoretical works on monolayer TMDCs, which have all ignored this interaction. With inclusion of the intravalley exchange interaction (the intervalley exchange interaction vanishes for excitons with zero centre-of-mass momentum), the oscillator strength of XA-1*s* decreases whereas that of XB-1*s* increases, leading to a factor of two change in their ratio. This asymmetric change in the oscillator strengths originates from the destructive and the constructive interference between TA and TB due to the intravalley spin-resolved exchange interaction according to our first-principles GW-BSE calculations.

An important role of the exchange interaction in exciton spin dynamics was demonstrated both theoretically and experimentally in III–V quantum wells, where the exchange interaction leads to simultaneous spin flip of the electron and the hole after an ultrafast resonant excitation of excitons^{9,10,11}. Therefore, in monolayer MoS_{2}, when the excitation pulse duration is short enough, we expect an evolution from the transient Ising exciton states (which are directly launched by the pulse) towards the eigenstates XA and XB with mixed spin polarization. The evolution dynamics are mainly driven by the intravalley exchange interaction. Based on these considerations, 2D electronic spectroscopy (an ultrafast four-wave mixing spectroscopy technique) is employed to demonstrate the intravalley exchange interaction unambiguously in both time and frequency domains. This method has been utilized to map excitonic coupling and energy transport in various systems including photosynthetic complexes^{22,23} and semiconductors^{14,19,20,24,25,26,27}, with high resolution in both the time and frequency domains.

Monolayer MoS_{2} with continuous size (~10 mm^{2}, polycrystalline) grown on a sapphire substrate by chemical vapour deposition is used for our experiments. The broadband laser in our experiments, covering the resonance energies of both XA-1*s* and XB-1*s* in monolayer MoS_{2} as shown in Fig. 2a, is generated from an optical parametric amplifier driven by a Ti:sapphire femtosecond laser amplifier (Astrella, Coherent). During the experiments, the sample was cooled to 40 K by liquid helium. In this work, 2D electronic spectroscopy was performed using the boxcar geometry in the rephasing scheme with co-circularly polarized light, as shown in Fig. 2b. Three input laser fields interact with the sample and induce a third-order polarization, which emits a photon echo signal in the phase-matching direction. The signal is acquired as a function of the coherence time *τ* and the waiting time *T*. The signal field *Ɛ*_{s}(*E*_{excitation}, *T*, *E*_{emission}), as a 2D function of the excitation energy *E*_{excitation} and the emission energy *E*_{emission} at a specific *T*, is extracted by Fourier transform with respect to *τ*. Presenting *Ɛ*_{s} in 2D diagrams reveals the correlation between the excitation and the emission energies with *T* as the delay between the excitation and the emission events, thus tracking the energy transport landscape. More details of sample preparation and the experiments are given in Supplementary Sections 2 and 3, respectively.

The rephasing amplitude 2D spectra (Fig. 2c–h) feature two diagonal peaks (AA and BB) and two cross peaks (AB and BA). The diagonal peaks indicate ground-state bleaching and stimulated emission due to excitation of each individual transient state TA or TB. The cross peaks correspond to excitation resonant with TA (TB) and photon echo emission resonant with TB (TA). In previous works on semiconductors^{19,20,24}, the presence of cross peaks in 2D spectra was explained by an excitation-induced shift of the resonance energy and excitation-induced dephasing of the electronic polarization caused by many-body effects, which offset the contributions from ground-state bleaching and excited-state absorption (Supplementary Section 4). In this scenario, the amount of peak shift and broadening is proportional to the excited exciton population. Therefore, the cross-peak amplitude should be maximized at small *T*, when the exciton population is maximal, and then decay together with the diagonal peak amplitude, which is proportional to the population of the corresponding state. However, our data deviate from the prediction of the excitation-induced shift/dephasing model (simulation results are shown in Supplementary Fig. 9). As shown in Fig. 2c–h, the cross peak AB is dark for *T* = 61 fs and becomes brighter relative to the diagonal peak AA as *T* increases. This indicates a conversion from the excited lower-energy state TA to the higher-energy state TB, which is qualitatively consistent with our theoretical analysis considering the intravalley exchange interaction. The system, which is initialized into the transient state TA by resonant excitation, evolves towards the eigenstate XA, which mixes TA and TB and therefore populates TB. Phonon-assisted population upconversion^{20,28}, which was previously observed between the trion and the exciton in monolayer MoSe_{2} and WSe_{2}, is excluded here since it can hardly surmount the energy gap of 150 meV between XA-1*s* and XB-1*s*. The small contribution of the Auger effect (by which two XA-1*s* interact and induce one XB-1*s*) to the cross peak AB is confirmed by the nearly constant ratio of the peak amplitude AB/AA versus fluence (Supplementary Fig. 6). Spin flipping of the electron component of TA by carriers and phonons could also bleach the TB transition. However, it has been shown by time-resolved Kerr rotation that the spin lifetime of electrons in TA is as long as 200 ps at 40 K^{29}, whereas the growth of the peak ratio AB/AA occurs within 1 ps. Spin flipping of the hole component of TA takes longer than this timeframe due to the much larger energy splitting in the valence band. Therefore, we attribute the emergence of the TB feature to the excitation of TA to the excitonic exchange interactions.

Considering the intravalley exchange interaction as the mechanism for the population transfer, we computed the 2D spectra. We model the coupling between TA and TB using a four-level diamond system constructed from a Hilbert space transformation of two independent two-level systems. The rephasing amplitude 2D spectra are calculated by perturbatively solving the optical Bloch equations to the third order, which dictates the dynamics of the density matrix. The photon echo signal can be divided into contributions from 14 Liouville pathways, each represented by a double-sided Feynman diagram. Two Feynman diagrams shown in Fig. 3a are employed to incorporate the intravalley exchange interaction between TA and TB; these include the population transfer between TA and TB and contribute to the cross peaks. The other 12 pathways are described in Supplementary Fig. 8. The initial condition for AB (BA) peak dynamics is set to be the pure population of the Ising excitons with amplitude proportional to the initial AA (BB) peak intensity from experiments. The details of the simulations are thoroughly discussed in Supplementary Sections 4 and 5. The simulated rephasing amplitude 2D spectra are shown in Fig. 3c–h at different waiting times, and reproduce the experimental results (Fig. 2c–h) reasonably well. The ratios of the cross-peak amplitudes to the corresponding diagonal-peak amplitudes from the simulations are plotted in Fig. 3b together with the experimental data. The slight overestimation of the ratio AB/AA at *T* > 500 fs is probably due to neglecting phonon-assisted population down-conversion from TB to TA, which is ignored in the simulations. For the same reason, the simulations show a slow rise in BA/BB whereas the experiment captured a fast rise within 200 fs.

## Methods

### First-principles GW and GW-BSE calculations

We performed first-principles GW and GW-BSE calculations for monolayer MoS_{2} including the spin–orbit coupling effect with the BerkeleyGW package^{30}. The quasiparticle band structure is shown in Supplementary Fig. 1. Density functional theory calculations were performed at the local-density approximation level with the Quantum ESPRESSO package^{31}, explicitly including Mo 4*s* and 4*p* semicore states in the pseudopotentials^{3}. A slab model measuring 20 Å along the *z* direction (~16 Å vacuum layer) was employed. The structure was fully relaxed until the force on each atom was less than 0.002 eV Å^{−1} and the in-plane lattice constant was 3.15 Å. The ab initio GW calculations were performed at the G_{0}W_{0} level, and the dynamical screening effects were described by the Hybertsen–Louie generalized plasmon-pole model^{32}. The wavefunctions were constructed with a plane-wave energy cutoff of 120 Ry. Moreover, we achieved convergence of the quasiparticle bandgap within 0.05 eV with a relatively small computational cost by employing the slab Coulomb truncation and the non-uniform sampling grid technique^{33} (6 × 6 × 1 coarse grid and six subsampled points); the dielectric cutoff was 25 Ry. Using the static remainder approach^{34}, we also summed up to 4,000 empty bands to calculate the self-energy correction. The resulting quasiparticle band structure was interpolated with spinor Wannier functions, using the Wannier90 package^{35}. The exciton energies and the wavefunctions in a semiconductor were obtained from the solutions of the BSE of the interacting two-particle Green’s function^{36,37}. For excitons in monolayer MoS_{2}, we included two valence bands and two conduction bands on a 120 × 120 × 1 *k* grid to diagonalize the BSE matrix.

### Sample preparation and characterization

A continuous MoS_{2} monolayer was grown on a sapphire substrate polished on both sides by chemical vapour deposition^{38}. The sample was characterized at room temperature by the Raman spectrum and photoluminescence as shown in Supplementary Fig. 4, both of which indicate monolayer features^{39,40}. In both measurements, a laser at 473 nm was used for excitation.

### 2D electronic spectroscopy experiments

The laser for 2D electronic spectroscopy in this work was output from a home-built non-collinear optical parametric amplifier driven by a Ti:sapphire femtosecond laser amplifier (Astrella, Coherent, 800 nm, 1 kHz). The design of the non-collinear optical parametric amplifier included careful selection of the nonlinear crystal, alignment of the seed versus the pump^{41} and compensation of the pulse-front tilt^{42,43}, which enabled broadband output in the visible range covering the resonance energies of both exciton A and exciton B for monolayer MoS_{2}. The pulse was compressed by a prism pair and measured to be 28.3 fs (full width at half maximum) at the sample position. All four beams were co-circularly polarized. 2D electronic spectroscopy was conducted in the phase-stabilized boxcar geometry as described in detail in ref. ^{44}. Two choppers were used to suppress the scattered light, with one modulating beam 3 at 100 Hz and the other modulating beam 1 and 2 at 50 Hz before they were separated by the diffractive optics. The residual scattering terms were removed by inverse Fourier transformation and windowing. A microcontroller was used to read the status of each beam (on or off) from the feedback signal of the choppers, which were synchronized with the laser pulses and the CCD (charge-coupled device) acquisition cycles. The configuration of these devices was described in ref. ^{45} using a pump–probe test as a demo, and this design has been used to study coherent phonon dynamics in methylammonium lead iodide hybrid perovskite^{46}. Given the weak signal from the monolayer sample, this design ensured good signal-to-noise ratio with reasonable time cost.

### Code availability

The BerkeleyGW package is available from berkeleygw.org. The MATLAB codes used to simulate the 2D spectra are available from the corresponding authors on reasonable request.

## Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

## Additional information

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## Acknowledgements

We thank G. Moody and K. Hao for helpful discussion. This material is based on work supported by the National Science Foundation under grant no. CHE-1362830, grant no. DMR-1508412 and grant no. EFMA-1542741. D.M.M. received a National Science Foundation Graduate Research Fellowship under grant no. DGE-1106400. Advanced codes were provided by the Center for Computational Study of Excited-State Phenomena in Energy Materials (C2SEPEM) at LBNL, which is funded by the US 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. Computational resources were provided by the DOE at Lawrence Berkeley National Laboratory’s NERSC facility and the NSF through XSEDE resources at NICS. Y.-H. L. acknowledges support from the Ministry of Science and Technology (MoST-106-2119-M-007-023-MY3; MoST-105-2112-M-007-032-MY3), the Frontier Research Center on Fundamental and Applied Sciences of Matters, and the Center for Quantum Technology of National Tsing Hua University.

## Author information

### Author notes

These authors contributed equally: Liang Guo, Meng Wu, Ting Cao, Daniele M. Monahan.

### Affiliations

#### Department of Chemistry, University of California, Berkeley, CA, USA

- Liang Guo
- , Daniele M. Monahan
- & Graham R. Fleming

#### Kavli Energy Nanoscience Institute at Berkeley, Berkeley, CA, USA

- Liang Guo
- , Daniele M. Monahan
- & Graham R. Fleming

#### Department of Physics, University of California, Berkeley, CA, USA

- Meng Wu
- , Ting Cao
- & Steven G. Louie

#### Material Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

- Meng Wu
- , Ting Cao
- & Steven G. Louie

#### Material Sciences and Engineering, National Tsing-Hua University, Hsinchu, Taiwan

- Yi-Hsien Lee

#### Biophysics and Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

- Graham R. Fleming

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### Contributions

L.G., M.W., T.C. and D.M.M. contributed equally to this work. L.G., M.W. and T.C. conceived the concept. Supervised by G.R.F., L.G. and D.M.M. led the experiments by designing the optical system and acquiring the data. M.W. and T.C. performed the theoretical studies under the supervision of S.G.L. Y.-H.L. provided the samples. L.G., M.W. and T.C. wrote the manuscript. All authors discussed the results and commented on the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Steven G. Louie or Graham R. Fleming.

## Supplementary information

### Supplementary Information

Supplementary Figures 1–11; Supplementary Tables 1–4; Supplementary References 1–24

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