Excitonic Complexes in n-Doped WS2 Monolayer

We investigate the origin of emission lines apparent in the low-temperature photoluminescence spectra of n-doped WS2 monolayer embedded in hexagonal BN layers using external magnetic fields and first-principles calculations. Apart from the neutral A exciton line, all observed emission lines are related to the negatively charged excitons. Consequently, we identify emissions due to both the bright (singlet and triplet) and dark (spin- and momentum-forbidden) negative trions as well as the phonon replicas of the latter optically inactive complexes. The semidark trions and negative biexcitons are distinguished. On the basis of their experimentally extracted and theoretically calculated g-factors, we identify three distinct families of emissions due to exciton complexes in WS2: bright, intravalley, and intervalley dark. The g-factors of the spin-split subbands in both the conduction and valence bands are also determined.


S1. SAMPLE
The studied sample is composed of WS 2 ML encapsulated in hBN flakes and supported by a bare Si substrate. The structure was obtained by two-stage polydimethylsiloxane (PDMS)-based 1 mechanical exfoliation of WS 2 and hBN bulk crystals. A bottom layer of hBN in the hBN/WS 2 /hBN heterostructure was created in the course of a non-deterministic exfoliation. The assembly of the hBN/WS 2 /hBN heterostructure was realized via successive dry transfers of WS 2 ML and capping hBN flake from PDMS stamps onto the bottom hBN layer.

S2. EXPERIMENTAL SETUPS
Low-temperature micro-magneto-PL experiments were performed in the Voigt, Faraday and tilted geometries, i.e. magnetic field oriented parallel, perpendicular, and 45 • with respect to ML's plane. Measurements (spatial resolution ∼2 µm) were carried out with the aid of two systems: a split-coil superconducting magnet and a resistive solenoid producing fields up to 10 T and 30 T using a free-beam-optics arrangement and an optical-fiber-based insert, respectively. The sample was placed on top of a x-y-z piezo-stage kept at T =10 K or T =4.2 K and was excited using a laser diode with 532 nm or 515 nm wavelength (2.33 eV or 2.41 eV photon energy). The emitted light was dispersed with a 0.5 m long monochromator and detected with a charge coupled device (CCD) camera. For measurements up to 10 T, the circular polarizations of the emissions were analyzed using a set of polarizers and a λ/4 waveplate placed directly in front of the spectrometer. In tilted-field configurations in high magnetic fields, the combination of a quarter wave plate and a linear polarizer placed in the insert were used to analyse the circular polarization of signals (the measurements were performed with a fixed circular polarization, whereas reversing the direction of magnetic field yields the information corresponding to the other polarization component due to time-reversal symmetry). Note that the excitation power for experiments performed in magnetic fields up to 30 T was adjusted based on the comparison of the measured PL spectrum and the one obtained under excitation of laser with 532 nm.

S3. THEORETICAL CALCULATIONS
First principles calculations were performed with the use of Vienna Ab Initio Simulation Package (VASP) 2 and the Projector Augmented Wave method 3 . After testing several exchange-correlation functionals, van der Waals corrections and functionals, the parametrization of Perdew-Burke-Ernzerhof revised for solids (PBEsol) 4 was used for geometry optimization, as yielding the best agreement with experimental lattice constant and layer thickness 5 . Atomic positions and lattice constants were optimized with 10 −5 eV/Å and 0.1 kbar precision. Energy cutoff of 400 eV, a 12 × 12 × 1 kmesh and 15 Å of vacuum in the vertical direction of unit cell were chosen after careful convergence tests. Phonon band structure calculations were performed using Phonopy package 6 , which implements the finite displacement method to obtain the interatomic force constants 7 . A 3 × 3 × 1 supercell was found sufficient to yield converged phonon energies at Γ and K points. The orbital angular momenta of bands were obtained from the wave function derivatives that are calculated within density functional perturbation theory 8 .

S4. Q-K-VALLEY MOMENTUM-INDIRECT EXCITONS
The issue of indirect-momentum excitons formed by a hole in K valley and an electron in Q valley recently reported for ML WS 2 9 and WSe 2 10 should also be addressed. For WS 2 ML, the energy separation between the bright excitons (X B ) at K point and the Q-K-valley momentum-indirect excitons was reported in the range of 70 to 100 meV 11,12 . This suggests that the contribution of the latter one to our measured low-temperature PL spectra may be apparent. We evaluate the magnitude of g-factors of the Q-K-valley momentum-indirect excitons to confirm the validity of our assignment of the excitonic lines. From our calculations the spin and orbital angular momenta of the Q ± valley in CB are ±1 and ±0.006. As the precise selection rules for indirect-momentum excitons are not yet known, we can estimate the absolute value of Q-K excitons to be |g| ≈ |2g K ± v | = 8.46, assuming spin-direct excitons. Surprisingly the calculated g-factor value is nearly the same as the extracted experimentally absolute value of the g-factor for the spinforbidden dark trion (T D ) (8.9). However, the identification of the T D is straightforward, as the brightening effect of the in-plane magnetic field is well established. The resulting mixing of the bright and dark excitons, becomes apparent in the optical activation of spin-forbidden dark complexes [13][14][15][16][17][18][19] . The phonon replica line (T D E"(Γ) ) is characterized by the g-factor of 8.9, but its assignment is evident because of the same value of its g-factor as of T D and of the T D -T D

S5. EXCITATION POWER EVOLUTIONS OF EXCITONIC EMISSIONS
In order to verify the assignment of different investigated excitonic complexes, we measured excitation power dependency of the PL spectra.  . Upon application of an out-of-plane magnetic field, the excitonic emissions split into two circularly polarized components due to the excitonic Zeeman effect 23 . Their energies evolutions (E(B)) in external out-of-plane magnetic fields (B ⊥ ) can be described as: where E 0 is the energy of the transition at zero field, g denotes the g-factor of the considered excitonic complex and µ B is the Bohr magneton. The fitting results of Eq. 1 to the experimental results are presented in Fig. S2(b). As can be seen, for all the excitonic complexes, the observed evolutions can be described by the aforementioned formula. We found that the obtained values of g-factors could be organized in three groups: ∼4 (black fitted lines), ∼9 (navy fitted lines), and ∼13 (orange fitted lines). The g-factors value for each excitonic complex as well as the origin of three groups are presented in the main article.

S7. SUPPRESSION/SATURATION OF THE T' LINE
As the suppression/saturation of the T' line is observed in Fig. 2 and 5(c) in the main text, we decided to analyse these effects.
The initial state of the spin-(T D ) and momentum-forbidden (T I ) dark excitons and also semi-dark trion (T') is the same [compare Figs. 1 and 5(a) in the main text]. For Fig. 2, there are two presented PL spectra: zero-field and measured in in-plane magnetic field of 10 T. It is well established that the application of the in-plane magnetic field leads to the brightening effect of the spin-forbidden dark excitons in S-TMD MLs, i.e. radiative recombination of these types of complexes. In our case, it manifests in the observation of the emission line due to the T D trion. As a result, the population of the initial state for the T D , T I and T' trions decrease significantly, which causes the suppression of the T' emission line. Similar situation takes place for measurements performed as a function of excitation power, shown in Fig. 5(c) in the main text. In that case, however, the interplay takes place between the T' and XX − complexes. The final state of the negative biexciton and the initial state of the semi-dark trion are the same [compare Figs. 5(a) and (b) in the main text]. As the intensities of the XX − lines increase superlinearly as a function of excitation power, the formation of the negative biexcitons is favoured at the highest excitation powers, while the T' intensity starts to saturate. These two effects suggest that the T' complex cannot recombine radiatively, because the formation of the XX − complexes is much faster.
To summarize, the observed suppression/saturation of the T' line is consistent with assignment of this complex to the semi-dark trion.

S8. CONDUCTION AND VALENCE BANDS g-FACTORS
Recently, a method for determination of single subbands g-factors in the WSe 2 MLs was proposed in Ref. 24. As low temperature PL spectra of both the WSe 2 and WS 2 MLs comprise of several emission lines including so-called phonon replicas, we have adapted this method to obtain g-factors of single conduction and valence subbands in WS 2 ML. Within this approach, we compare energy evolution of spin-forbidden dark trion (T D ) and its phonon replica (T D E"(K) ) in out-of-plane magnetic fields (B ⊥ ). Particularly, the energy difference between their circularly-polarized components (σ ± ) corresponding to emissions from different valleys (K + or K − ) is analysed, see Fig. S3(a). Fig. S3(b) shows the magnetic field evolution of the σ ± components of the T D and T D E"(K) lines. Note that the field dependence of T D emission was measured in the tilted configuration of the sample by 45 o with respect to the applied magnetic field direction, which gives both in-plane and out-of-plane components of magnetic fields, see Ref. 19 for details. There are four energy distances marked in Fig. S3(b) by black vertical lines: ∆σ + , ∆σ − , ∆min and ∆max. With help of schematic illustration shown in Fig. S3(a), it follows that they depend only on single subband g-factors and phonon energy and are equal to: where T D (K ± ) and T D E"(K) (K ± ) represent the magnetic field evolutions of σ ± components of spin forbidden dark trion (T D ) and its E" phonon replica (T D E"(K) ). E"(K) is energy of E" phonon at K point. g c and g v are the g-factors of the lowest conduction band g c and the highest valence band g v , while µ B and B ⊥ are related to the Bohr's magneton and the applied out-of-plane magnetic field. Further transformations of the aforementioned equations lead to establishing equations which allow determination of g c and g v values: Fig. S3(c) demonstrates experimentally obtained evolutions with linear fits marked with solid black lines. Obtained values are equal to 1.1 and 5.5 for g c and g v , respectively. That is in a good agreement with theoretical calculations, which are discussed widely in the main article.
Knowing values of g c and g v , we are able to determine the g-factor of top conduction band subband (g c+1 ) from previously obtained g-factor of the bright A exciton, i.e. X B line, with the aid of formula: g X B = 2(g c+1 − g v ). We extracted that g c+1 value is equal to 3.7.
Finally, taking g-factor of the bright B exciton (g BX B ) from Ref. 25, it is possible to determine the g-factor of the bottom valence subband (g v−1 ) using formula: g BX B = 2(g c − g v−1 ). We found that g v−1 value is equal to 2.9.