Direct Observation of Intravalley Phonon Scattering of 2s Excitons in MoSe$_2$ and WSe$_2$ Monolayers

We present a high-resolution resonance Raman study of hBN encapsulated MoSe$_2$ and WSe$_2$ monolayers at 4 K using excitation energies from 1.6 eV to 2.25 eV. We report resonances with the WSe$_2$ A2s and MoSe$_2$ A2s and B2s excited Rydberg states despite their low oscillator strength. When resonant with the 2s states we identify new Raman peaks which are associated with intravalley scattering between different Rydberg states via optical phonons. By calibrating the Raman scattering efficiency and separately constraining the electric dipole matrix elements, we reveal that the scattering rates for k=0 optical phonons are comparable for both 1s and 2s states despite differences in the envelope functions. We also observe multiple new dispersive Raman peaks including a peak at the WSe$_2$ A2s resonance that demonstrates non-linear dispersion and peak-splitting behavior that suggests that the dispersion relations for dark excitonic states at energies near the 2s state are extremely complex.


Main Body
Many of the proposed applications of the monolayer TMDCs and their heterostructures are based upon their strongly bound, high oscillator strength excitonic and trionic states 1,2 . Understanding the physics of these states is central to delivering these applications. Valleytronics relies on the suppression of intervalley scattering 3 . Polaritonics 4,5 requires understanding the separation of the dipole and phonon components of the transition linewidths 6 . Quantum computing could take advantage of the ability to generate coherent populations using Raman processes 7 . In addition, understanding the behavior of heterostructures, in which intralayer excitons at different energies interact with each other and interlayer excitons 8 , will require a thorough understanding of excitonic physics in the constituent monolayers.
There has been significant progress in our understanding of excitons in TMDCs in the last few years 3,[9][10][11][12] . The existence and energies of a wide range of excitonic states are now well established 7,11,13,14 . After starting with the three main excitonic bands, A, B and C 9,15 , attention is now focused on the excited excitonic states including both the bright s states 12,16 and the two photon accessible p states 17,18. The effect of the envelope function on the oscillator strength of the excitons has been found to be in line with expectation 19,20 . There are predictions that the B excitons should have enhanced oscillator strength relative to the A excitons because of exchange interactions 21 although only limited experimental validation 22 . Attention is starting to turn to understanding the scattering of the excited state excitons with measurements of the temperature dependence of the transition linewidths and attempts to model the total phonon scattering using simplified excitonic band structures 23,24 . The importance of a wide variety of dark excitonic states is now well established 25 however the difficulties of experimentally accessing these states and calculating full excitonic dispersion relations is hampering fully understanding them. 3 Raman scattering, and particularly resonance Raman spectroscopy, has a proven track record in elucidating the physics of excitons and phonons and their interaction in the TMDCs [26][27][28][29] . Its ability to resolve the phonons involved in excitonic scattering has led to discoveries such as the coupling of substrate phonons to excitons in WSe2 monolayers 30 . The ability to directly probe dark states via multi-phonon processes has elucidated the importance of intervalley scattering and allowed mechanisms for exciton-trion scattering via dark states to be explored 31,32 . However, Raman scattering still has a lot more to give to our understanding of the fundamental physics of excitons in TMDCs. In particular, the only reports of resonance Raman with excited states in monolayer TMDCS are via hBN encapsulation phonons 30 and there are no reports of resonance with B2s excitons.
In this paper we discuss the results of resonance Raman experiments performed on high quality hBN encapsulated monolayers of WSe2 and MoSe2 at 4 K along with photoluminescence and reflectivity spectra taken at the same location on the samples. Whilst the data presented in the main body of this paper is from the best quality samples all key observations are supported by measurements on at least one other monolayer (see SI). As shown in Fig 1, we observe Raman resonances associated with the A1s, A2s, B1s and B2s excitonic states in MoSe2 and the A1s, A2s, B1s excitonic states in WSe2; the WSe2 B2s is too high in energy to be accessed with our laser sources. Due to the quality of the samples the optical transitions are sufficiently narrow that it is possible to see clearly separated incoming and outgoing resonances. The Raman scattering has been calibrated using the strength of the silicon 520 cm -1 Raman peak, taking into account its excitation photon energy dependence 33   In order to discuss the resonance profiles we first need to identify and assign the various Raman peaks. Despite the fact that most of the Raman peaks can be associated with multi-phonon or defect 5 allowed scattering process we obtain a reasonable fit using multiple Lorentzian lineshapes 31 ; 18 in MoSe2 and 16 in WSe2. The majority of these peaks have been previously reported and assigned [35][36][37][38] . However as shown in the supplementary information (see Tables S3 & S4) there are multiple possible assignments for most of these peaks beyond those already proposed in the literature.
Furthermore, comparison of 4 K and room temperature spectra (see Fig S5) demonstrates that a peak 147.3 cm -1 , previously assigned to LA(M) phonon scattering is associated with a process involving both emission and absorption of phonons. Whilst there is some uncertainty concerning the assignment of several Raman peaks, we can be confident that the 249.3 cm -1 peak in WSe2 is due to single phonon scattering with the A1ʹ(Γ) and Eʹ(Γ) phonons and that the 241.0 cm -1 peak in MoSe2 is also due to single A1ʹ(Γ) phonon scattering.
The resonance profiles of the A1ʹ/Eʹ peaks, presented in Fig 2, should be the simplest to understand as they are due to a single phonon scattering event. We have fitted these resonance profiles using the standard perturbation prediction (see SI) for the Raman scattering probability assuming the underlying process where an incoming photon generates a bright exciton which is scattered by the A1ʹ phonon to itself or another bright exciton and recombines. Apart from the case of the A2s/B1s MoSe2 transition, for which we use two excitons and allow for interstate scattering, we have assumed that only one exciton is responsible for each resonance. As can be seen in Fig 2, the fits between this simplest theory and the data are remarkably good in all cases apart from the WSe2 B1s resonance. In the case of the WSe2 B1s resonance the profile is clearly asymmetric with the outgoing resonance stronger than the incoming resonance. In addition, there is significant Raman scattering at energies above the resonance but no comparable scattering at the low energy side of the resonance. Both observations can be explained by a series of different models including 6 at least two discrete optically-active states, including the B1s exciton, and, optionally, the onset of a resonance with the C excitonic band whose effects are observed in many optical measurements 37,39 to extend significantly below the energy at which they peak. Whilst it is possible to obtain reasonable fits to the data with plausible models (see SI) it is not possible to select between these models and so we chose not to present a fit. resonance we were unable to determine which of a series of possible models was the best fit therefore we do not present a fit. Error bars shown are a standard deviation determined from fitting the Raman spectra.
The excitonic energies, widths and amplitude coefficients obtained from the best fits are presented in Table 1  based on the reflectivity fits with the ratio of the Raman amplitude coefficients, WSe2 A2s/A1s 0.008 ± 0.002 and 0.012 ± 0.002, and MoSe2 B1s/A1s 3.0 ± 0.1 and 4.7 ± 0.7, indicates that most, if not all, of the variation of the strength of the Raman scattering can be explained by the dipole matrix element. Therefore, we deduce that excitonphonon matrix elements are the same for different excitonic states. This is not unexpected, as the underlying deformation potentials are the same 46 and the envelope function should not influence q=0 scattering 6 , but hasn't been shown directly in experiments before. resonances but not the A1s resonance. At the B2s resonance these are the next strongest peaks after the one phonon peaks. At the A2s/B1s resonance these peaks display an unusual resonance behavior. As can be seen in Fig 1 unlike the other peaks seen at this compound resonance, they are strongly resonant at the B1s incoming resonance energy with no strong resonance at the expected B1s outgoing resonance energy. The shifts of these peaks correspond very closely to the A2s-B1s separation energy ~ 55 meV. Therefore, the natural interpretation of the behavior of these peaks is that they are due to B1s to A2s scattering which is strongly resonant at both the initial and final excitonic states. Whilst these peaks can be assigned to multi-phonon processes associated with  In WSe2 we observe an intriguing dispersive mode, between 490 and 500 cm -1 , which is only observed at the A2s (Fig 4) resonance. At the incident resonance the Raman peak appears to be a single feature. As the excitation energy is increased from the incoming resonance the Raman peak 11 at first disappears. However, the peak returns as the excitation is increased through a threshold at about 26 meV (210 cm -1 ) greater than the A2s energy. At this point the peak disperses to lower shift with increasing excitation energy in a non-linear manner. At the outgoing resonance the spectrum changes again with three peaks appearing which disperse relatively quickly; two to higher shift and one to lower shift. As can be seen in Fig 4,   The data also shows important features not associated with the 2s states. For instance, at the MoSe2 A 1s resonance all the peaks, apart from the A1ʹ, have asymmetric resonance profiles with the outgoing resonance between 4 and 6 times as strong as the incoming resonance (see Fig 3). in the two materials is the difference in available excitonic states arising from the conduction band splitting spin splitting. Therefore, we attribute this behavior at the MoSe2 A1s to the sequential two phonon scattering via the intervalley K exciton which is predicted to be 30-40 meV 23,47,48 above the bright A1s exciton. A second clear feature of the data (Fig 3) is that at both the MoSe2 A1s and WSe2 B1s resonances there are two peaks, at 260 and 390 cm -1 in WSe2 and 320 and 460 cm -1 in MoSe2, which are considerably enhanced; going from approximately one fifth of the A1ʹ peak to greater than the A1ʹ. It is interesting to note, that the 390 and 460 cm -1 peaks are both associated with two other peaks to form triplets. In addition, the 260 and 320 cm -1 peaks are both associated with a band of peaks, from 215 to 260 cm -1 inWSe2 and 280 to 320 cm -1 in MoSe2, and in both cases they are the highest shift peak in the band. Taken together, these observations suggest these peaks are associated with the same underlying phonons shifted in frequency due to the 13 different metal masses. It is not obvious why these peaks aren't resonant at other excitons however one thing the MoSe2 A1s and WSe2 B1s have in common is that they both involve the lower lying conduction band state of the spin split pair, and therefore, their equivalent intervalley excitons involve the higher energy conduction band state. A number of other interesting features of the data including not previously reported dispersive modes observed at the MoSe2 A2s/B 1s and the WSe2 A 1s resonances are discussed in the SI (see Fig S5). peaks, which have potential assignments to multiphonon/large wavevector scattering processes, demonstrate significant asymmetry between the incident and outgoing resonance peaks (see SI).
The contrasting resonance behavior for WSe2 and MoSe2 A1s resonance profiles is attributed to 14 the absence of available large wavevector states at the opposite K valley. Error bars shown are a standard deviation determined from fitting the Raman spectra.
In this paper we present the first observation of Raman scattering from A2s and B2s excitonic states in any TMDCs, along with high resolution resonance Raman results at the A1s and B1s.
This allows the differences between phonon scattering processes at different excitonic states to be explored. A quantitative analysis of the absolute Raman scattering probability at various resonances experimentally confirms that the q=0 exciton-phonon scattering matrix elements are the same/ nearly the same for all excitonic states 24 . However, they highlight significant differences between the exciton-phonon scattering of the excitonic states due to the dark states available for scattering. In MoSe2 we observe three peaks at the A2s/B1s and B2s resonances which are clearly associated with intravalley scattering between different Rydberg states. We propose that these peaks can be associated with two phonon scattering by the A1ʹ(Γ) and Eʹ(Γ) phonons and that it is these phonons which dominate intravalley scattering between different Rydberg states. In WSe2 we observe a feature with complex dispersive behavior at the A2s. Whilst it isn't possible to fully explain this peak with the current state of knowledge of the excitonic dispersion relations in these materials, it is clear that the dark excitonic states must have important features at energies close to the A2s resonance.
The data also contains important new features associated with the A1s and B1s resonances.
Asymmetric resonance profiles for all of the multi-phonon peaks at MoSe2 A1s state are a direct consequence of the existence of inter-K-valley states with energies a few tens of meV above the Gamma point excitons 49 . In addition, we observe two analogous peaks in each material that are resonant at the A1s in MoSe2 and B1s in WSe2 and which existing literature suggests are not 15 strongly resonant at the C exciton in either material 35,37 . A possible interpretation of the resonance conditions is that it is associated with the different sign spin-orbit coupling in the conduction bands of the two materials.
Finally, our results demonstrate that whilst Raman scattering has already provided significant insight into monolayers and heterostructures of TMDCs there is still a huge amount more than could be extracted from this technique particularly as sample quality improves.

Methods
The samples used consist of mechanically exfoliated monolayers of MoSe2 and WSe2 encapsulated between layers of hexagonal boron nitride using a dry transfer technique 50  for PL measurements a 532 nm Coherent Verdi laser was also utilized. The incident polarizations of the lasers were horizontal relative to the optical bench and the Raman scattered light coupled into the spectrometer was analyzed using both horizontal and vertical polarizations. The polarization of the Raman peaks was observed to be strongly co-linear. This allowed unwanted luminescence from the samples to be removed from the Raman spectra by subtraction of the crossed and parallel polarized spectra. Both Pl and Raman spectra were measured using a Princeton Instruments Tri-vista Triple spectrometer equipped with a liquid nitrogen cooled CCD. The Raman peak frequencies were all calibrated using the silicon Raman peak at 520 cm -1 as an internal reference. To allow comparison of the Raman scattering on the MoSe2 and WSe2 samples the spectra were calibrated to absolute Raman scattering probability. This required the normalization of the Raman spectra to the 520 cm -1 Silicon peak intensity, correction of Fabry-Perot interference effects and calibration using the absolute Raman scattering results of Aggarwal et al 33 . To account for the Fabry-Perot interference we made use of reflectivity spectra measured using a Fianium super continuum source and Ocean optics HR4000 spectrometer. For further detail on the experimental methods, data analysis and Raman calibration please see the supplementary information.

Supporting Information
The supplementary information contains additional data on both samples presented in the main body of the paper as well as results of measurements on three repeat samples. These include: photoluminescence; reflectivity spectra; details on fitting Raman spectra; Raman peak assignment tables and analysis; reflectivity fitting details; correction of thin film interference effects; Absolute Raman Scattering calibration; Resonance Raman scattering model details, and additional resonance Raman profiles and analysis for all samples.

Data Availability
The data presented in this paper is openly available from the University of Southampton Repository. DOI:10.5258/SOTON/D1315.

Author Contributions
The experiments were conceived by D.C.S, L.P.M and X.X. Samples were fabricated by P.R. The