Rydberg series of dark excitons and the conduction band spin-orbit splitting in monolayer WSe$_2$

Strong Coulomb correlations together with multi-valley electronic bands in the presence of spin-orbit interaction and possible new optoelectronic applications are at the heart of studies of the rich physics of excitons in semiconductor structures made of monolayers of transition metal dichalcogenides (TMD). In intrinsic TMD monolayers the basic, intravalley excitons are formed by a hole from the top of the valence band and an electron either from the lower or upper spin-orbit-split conduction band subbands: one of these excitons is optically active, the second one is"dark", although possibly observed under special conditions. Here we demonstrate the s-series of Rydberg dark exciton states in monolayer WSe$_2$, which appears in addition to a conventional bright exciton series in photoluminescence spectra measured in high in-plane magnetic fields. The comparison of energy ladders of bright and dark Rydberg excitons is shown to be a method to experimentally evaluate one of the missing band parameters in TMD monolayers: the amplitude of the spin-orbit splitting of the conduction band.


Introduction
In most of low-dimensional, direct-gap semiconductor systems (quantum wells and monolayers of transition metal dichalcogenides, quantum wires and dots), the near band edge electron-hole excitations, the excitonic states, can be classified into two categories depending whether their associated projection of the total angular momentum is j z =±1 or it is j z =±2 or 0. According to standard selection rules for optical transitions, the excitons with j z =±1 can be excited by light and/or can recombine by emitting photons and are therefore referred to as bright excitons. Instead, excitons with j z =±2 or 0 are termed dark as they do not efficiently couple to the radiation field. Although dark, the j z =±2 or j z =0 excitons can be brightened under special conditions, e.g., under application of the magnetic field in an appropriate configuration 1,2 and thus possibly initiating the photoluminescence signal (though still hardly observed in absorption-type spectra). Dark excitons can largely alter the dynamics of optical excitations and, as long-lived bosonic quasiparticles, are possible candidates to form a condensate exciton phase 3 . Those states have been the object of vast investigations in semiconductor quantum wells, dots and wires, in carbon nanotubes and more recently in monolayer semiconductors [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] . All these studies have been however largely focused on excitonic ground states whereas the physics of dark excitons associated with the excited excitonic states is a little explored area [20][21][22] . This, in particular, applies to two-dimensional semiconductors which often display a characteristic series of Rydberg s states of bright excitons [23][24][25][26] whereas a demonstration of the corresponding series of dark excitonic states is missing so far.
In this paper, we report on low-temperature magneto-optical study of Rydberg series of excitonic states as they appear in a WSe 2 monolayer. Notably, in this representative example of a 2D semiconductor, the rich spectrum of excited excitonic states can be relatively easily traced with luminescence experiments 25,[27][28][29] . Profiting of this fact we brighten dark excitons in our sample by applying an in-plane magnetic field and uncover their s ( =1, up to 4) Rydberg series which appears in addition to a Rydberg spectrum of bright excitons 24,25 . By comparing energy diagrams of dark and bright exciton series, we experimentally derive the single particle separation between spin-orbit-split conduction band subbands in a WSe 2 monolayer. Intriguingly, this separation is found to be of about two times smaller than that commonly assumed from theoretical modelling of electronic bands of monolayer semiconductors. The focus of our study can be clearly read from Fig.1. Fig.1a illustrates a simple generic scheme of the band structure of a WSe 2 monolayer, as it is found in a close vicinity of the direct bandgap at the two K + and Kpoints of the Brillouin zone. We consider only three relevant bands (per each K + and Kpoint): two spin-orbit split subbands in the conduction band (weakly separated in energy, by ∆ ~ tens of meVs) but only one, top valence subband (neglecting the lower one separated by hundreds of meVs from the upper subband). As generally accepted and shown in Fig.1a, the interband, intravalley transitions which involve the electronic states of the upper conduction subband are optically allowed whereas those associated with lower subband are characterized by j z =0 and thus are not optically active under conventional conditions/configurations 15,30 . The j z =0 transitions are weakly allowed only in the configuration for which the electromagnetic wave has its k-vector aligned along the monolayer plane and non-zero out of plane component of the electric field, or in the presence of an in-plane magnetic field. Please, note, that in the case of TMD monolayers, the j z =±1 selection rules for optical transitions coincide with the ansatz of conservation of electronic spins in optical transitions. The band to band image of optical transitions is not, however, sufficient to describe the optical response of semiconductors, and in particular that of the investigated structure: the electron-hole Coulomb interaction and hence the excitonic effects have to be taken into account. As shown in Fig. 1b, one may expect the appearance of two distinct s Rydberg series of near band edge excitons in our WSe 2 monolayer: the routinely observed bright exciton series 23-26 associated with the upper conduction band subband, but also the dark exciton series associated with the lower conduction band subband, revealed in the present report. In the limit of large , the energy difference between bright and dark exciton states provides a measure of the spin-orbit splitting of the conduction band ∆ = − , where and denote, correspondingly, the single particle bandgaps associated with upper and lower conduction band subbands. It is worth noting, that the energy difference ∆ between the ground states of dark and bright excitons depends also on their respective, and , binding energies: ∆ = ∆ + ( − ). Since and are a priori different, measuring the ∆ parameter alone does not infer the amplitude of ∆ .

Experimental results and discussion
The active part of the structure used for the experiments is a WSe 2 monolayer extracted from a commercially available tungsten disulfide crystal. The monolayer was encapsulated in between hBN flakes and deposited on a Si substrate, following the conventional methods of mechanical exfoliation and deterministic dry transfer techniques. Experiments consisted of low temperature (4.2 K) magnetophotoluminescence measurements carried out in magnetic fields up to 30 T, applied along the monolayer plane. As sketched in the inset to Fig. 2, we used the "back scattering" geometry where both the excitation and the collected light beams were quasi-perpendicular to the sample/monolayer. More details on sample preparation and experimental details can be found in SI. Representative photoluminescence spectra of the investigated WSe 2 monolayer are presented in Fig.2. Inspecting the low spectral range (left panel of Fig.2), one recognizes a characteristic set of multiple photoluminescence transitions, typical of WSe 2 monolayers 31-33 . Possible formations of many different excitonic states (including nontrivial indirect/inter valley excitons and excitonic complexes such as trions and biexcitons) account for the spectrum complexity which we disregard here, focusing our attention on neutral, bright and dark intravalley excitons. First, we examine the photoluminescence (PL) spectrum measured in the absence of magnetic field (B), and, in accordance to previous reports, recognize the characteristic sequence of emission peaks related to the Rydberg series of 1s , 2s , 3s , up to 4s states of bright excitons 24,25,[27][28][29] . The PL spectrum measured at B=0 shows also a weak transition related to the ground state, 1s , of dark exciton [14][15][16] . This is because of a legitimate emission from this state in the direction along the monolayer plane and therefore its weak visibility in our spectrum which we collected using the objective with relatively large numerical aperture 15,16 . Please also note, that we employ here a simplified picture of the dark exciton and view it as raising only a single resonance, i.e., we ignore its doublet, "dark-grey" components split by ~0.7 meV 14,16 , an energy which is negligibly small in the context of our further considerations. Central for our report is the study of the evolution of the PL spectra measured as a function of the in-plane magnetic field ( ‖ ). Such studies have been already successfully applied to unveil the ground (1s ) states of dark excitons in most of TMD monolayers [16][17][18][19]34 . The application of an in-plane magnetic field induces the Zeeman effect, and thus the mixing of spin-orbit split conduction subbands; dark excitons acquire an oscillator strength (~ ‖ ) for the emission in the direction perpendicular to the monolayer plane and become apparent in the PL spectra. The efficient "magnetic brightening" of the ground state of the dark exciton in WSe 2 monolayer is confirmed by the present experiments. As can be seen in the left panel of Fig.2, the intensity of the 1s emission grows progressively as a function of ‖ , and eventually dominates the PL spectrum in the limit of high magnetic fields. Importantly, new transitions driven by the application of the in-plane magnetic field appear also in the higher spectral range (see right panel of Fig.2) in the vicinity of the PL peaks associated to the excited states of bright excitons. These new transitions, labeled as 2s , 3s and 4s in Fig.2, are attributed to the excited Rydberg states of the dark exciton.

Figure 3. Intensities of dark states relative to the intensity of bright states as a function of the in-plane magnetic field for a the 1s states (blue squares), b the 2s states (red circles) and c the 3s states (green triangles) along with quadratic fits (black lines) of the form +
. Note that in the case of 1s states only data points from 0 T to 20 T were considered for the fitting, as for higher fields, the relative intensity clearly starts to deviate from quadratic dependence.
As demonstrated in Fig. 3, the magnetic brightening of dark exciton states, i.e. the increase of the relative intensity ( )/ ( ) of dark s emission peak with respect to its bright s counterpart, roughly follows the expected ( )/ ( ) = ! ‖ rule (see SI for more details on data analysis).
This, however, is not true in the case of ground states for the in-plane fields ∥ > 20 T, where the relative intensity deviates from the quadratic dependence, possibly due to the shortened (by optical activation) dark exciton lifetime leading to the decrease of the dark exciton population. Hence, only the datapoints for ∥ < 20 T were used for the estimation of ' . Whereas the extracted values of ! are in reasonable agreement with the theoretical estimations for > 1, the observed brightening of the ground dark 1s is by far more efficient ( ' ~ 10 ( ! , for > 1) (see SI for more details). This might be due to the expected imbalance between populations of dark and bright exciton ground states, but the factor of ' / ! ~ 10 ( seems to be surprisingly large what points towards highly non-thermal population of optical excitations in a WSe 2 monolayer. where / is the "dark/bright" bandgap associated to the lower/upper conduction band subband and, 89 / should be, in the first approximation, identified with the effective Rydberg energy for the dark/bright exciton; 89 / = 13.6 eV · > / /(? 0 @ ), where > / is the reduced effective mass embracing the valence band hole and electron from upper/lower conduction subbands, ? is the dielectric constant of the surrounding hBN material and 0 @ stands for the electron mass. ) parameter is found to be ) = −0.09, which is notably common for dark and bright series and well matches the previously reported value (−0.08 for the bright exciton series 25 ). The extracted values of "dark" and "bright" (single particle) bandgaps are, correspondingly, On the other hand, one may expect that the difference between and arises not only from the direct Coulomb term but also from the exchange term, as bright and dark excitons consist of parallel or antiparallel spin configurations, respectively [38][39][40] . Whereas we suggest here that the exchange term might be relatively small, this claim has be to taken with caution, since our estimations refer to theoretically calculated effective masses which perhaps are not sufficiently accurate (as inaccurate as the amplitude of the spin orbit splitting). Nonetheless, our findings once again raise a question about the importance of the exchange interaction for the energy difference between the bright and dark excitons binding energies in TMD monolayers.

Conclusions
In this report we demonstrated the magnetic brightening of the Rydberg s-series of dark excitons, up to = 4, in a WSe 2 monolayer encapsulated in hBN. The analysis of the bright and dark excitons series allowed us to determine one of the missing band parameters, the amplitude of the spin-orbit splitting in the conduction band. Its derived value, ∆ = 14 meV, is significantly lower than commonly assumed, what calls for revision of theoretical calculations of electronic bands in TMD monolayers . Moreover, our results suggest that the difference between the binding energies of bright and dark excitons can be fully explained by the difference in the masses of electrons in the two spin-orbit-split conduction bands, without referring to exchange interactions.