Fine structure of K -excitons in multilayers of transition metal dichalcogenides
A. O. Slobodeniuk, Ł. Bala, M. Koperski, M. R. Molas, P. Kossacki, K. Nogajewski, M. Bartos, K. Watanabe, T. Taniguchi, C. Faugeras, M. Potemski
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Fine structure of K -excitons in multilayers of transition metal dichalcogenides A. O. Slobodeniuk, ∗ L. Bala,
1, 2
M. Koperski,
1, 3, 4
M. R. Molas,
1, 2
P. Kossacki, K. Nogajewski,
1, 2
M. Bartos, K. Watanabe, T. Taniguchi, C. Faugeras, and M. Potemski
1, 2, † Laboratoire National des Champs Magn´etiques Intenses,CNRS-UGA-UPS-INSA-EMFL, 25 avenue des Martyrs, 38042 Grenoble, France Institute of Experimental Physics, Faculty of Physics,University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK National Graphene Institute, University of Manchester, Oxford Road, Manchester, M13 9PL, UK National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
Reflectance and magneto-reflectance experiments together with theoretical modelling based onthe k · p approach have been employed to study the evolution of direct bandgap excitons in MoS layers with a thickness ranging from mono- to trilayer. The extra excitonic resonances observed inMoS multilayers emerge as a result of the hybridization of Bloch states of each sub-layer due to theinterlayer coupling. The properties of such excitons in bi- and trilayers are classified by the symmetryof corresponding crystals. The inter- and intralayer character of the reported excitonic resonances isfingerprinted with the magneto-optical measurements: the excitonic g -factors of opposite sign andof different amplitude are revealed for these two types of resonances. The parameters describing thestrength of the spin-orbit interaction are estimated for bi- and trilayer MoS . I. INTRODUCTION
Scientific curiosity to uncover the properties of newmaterials and to demonstrate their possible novelfunctionalities drive the research efforts focused onatomically-thin matter, and, in particular, on thin layersof semiconducting transition metal dichalcogenides (S-TMD)[1–4]. Intense works have been devoted to studiesof S-TMD monolayers which appeared to be the efficientlight emitters, the two-dimensional semiconductors witha direct bandgap positioned at the K ± points of their1-st hexagonal Brillouin zone (BZ) [5–7]. New and richpossibilities of tuning the band structure, the strengthof Coulomb interaction, and thus the optical properties,are opened when stacking the S-TMD monolayers intoa form of multilayers and/or hetero-layers [8–16]. Theproperties of the archetypes of S-TMD stacks which arethe thermodynamically stable 2H-stacked multilayers areto be well understood first.In 2H stacks of N monolayers ( N ML), the electronicbands are known to be effectively modified, with N , inthe range outside the K ± points of the BZ [17–23]. This,in particular, implies the indirect bandgap in N MLswhen
N >
1, what strongly affects the emission spectraof these multilayers [24–29]. Instead, more subtle effectsof the hybridization of electronic states around the directbandgap which appears at K ± points of the BZ in any N ML are relevant for the absorption-type processes [30–32]. Understanding the absorption response of S-TMDmultilayers might be of special importance for their po-tential applications in photo-sensing or photo-voltaic de-vices [33–36]. ∗ [email protected] † [email protected] In this paper we present the theoretical outline(based on k · p approach [17, 27, 30, 37]) and the ex-perimental data (results of reflectance and magneto-reflectance measurements), which, in a consistent man-ner, unveil the nature of direct bandgap (K ± points)excitonic transitions in 2H-stacked S-TMDs multilay-ers. The geometry of 2H stacking, wherein each sub-sequent layer is 180 ◦ rotated around previous one, in-duces interaction and hybridization of the Bloch statesin K + / K − / K + . . . (K − / K + / K − . . . ) points of subsequentmonolayers, which form the K + (K − ) valleys of the mul-tilayer. Such non-trivial interlayer coupling delocalizesthe electron states in the out-of-plane direction, whichleads to the formation of the new type of excitons, as-sociated with K ± valleys of multilayers as well as bulkcrystals [27, 38, 39]. We classify such excitons, associatethem with the symmetry of the crystal and describe theirproperties in terms of the simple theoretical model. No-tably, the intra- or interlayer character of the excitonictransitions is fingerprinted with, correspondingly, posi-tive or negative sign of the g -factor associated to thesetransitions.Due to the different symmetries of the conduction andvalence band orbitals, the hybridization of K ± -electronicstates in N ML occurs predominantly in the valence bandand is more effective when the spin-orbit splitting ∆ v in the valence band is small. MoS crystals, with thesmallest ∆ v among all other S-TMDs, have been there-fore chosen for the investigations. In the experiments, wehave largely profited of the significantly improved opti-cal quality of MoS layers when they are encapsulated inbetween the hexagonal boron nitride (hBN)[11, 40–44].We consider the bi- and trilayer systems as the simplestmultilayer representatives with different spatial symme-try. Comparing the experimental data with the theoret-ical expectation on a more quantitative level, we discussthe characteristic parameters which reflect the effects ofspin-orbit interaction in our bi- and tri-layer MoS . In-triguingly, we estimate/speculate that in these multilay-ers the spin-orbit splitting in the conduction band is quitelarge ( ∼
50 meV) whereas the spin-orbit coupling param-eter for the valence band ( ∼ monolayer.The paper is organized as follows. The section II in-troduces the theoretical description of the interlayer cou-pling and optically active transitions in 2H stacked mul-tilayers. In the section III we present the experimentaldata for excitonic resonances in bi- and trilayer of MoS .In the section IV we outline the properties of multilay-ers and the possible applications of such materials. Inthe Appendix A the samples, instrumentation and ex-perimental details are presented. The Appendices B andC contain the derivation and discussion of the exciton g -factors in bi- and trilayer of MoS respectively. II. THEORETICAL DESCRIPTION
We consider the optically active transitions at the K ± points in 2H-stacked multilayer S-TMD crystals encap-sulated in hBN. Our investigation is based on k · p ap-proximation. We focus on the optical properties of suchcrystals at K + point for brevity [27]. The results for K − point can be obtained by analogy.We briefly remind the features of S-TMD monolayer which will be used in the subsequent description of themultilayer structures. Namely, a single layer crystal isa direct band gap semiconductor. The maximum of va-lence (VB) and minimum of conduction (CB) bands arelocated at the K ± points of BZ. Due to strong spin-orbitinteraction both bands are spin-split (from hundreds ofmeV in VB, up to tens meV in CB). Hence, Bloch statesin the corresponding points can be presented as a tensorproduct of spin and spinless states. The spinless valence | Ψ v i and conduction | Ψ c i band states at the K + point aremade predominantly from d x − y + id xy and d z orbitalsof transition metal atoms respectively [45, 46]. Due totime-reversal symmetry (TRS) the analogous states atthe K − point are complex conjugated to the previousones, i.e. they are made from d x − y − id xy and d z orbitals accordingly. The TRS also dictates that Blochstates with the same band index ( c or v ) but with op-posite spins in different valleys have equal energies. Thecrystal spatial symmetry together with TRS define theoptical properties of monolayers — only σ ± polarizedlight can be absorbed or emitted at the K ± points re-spectively. Since the VB and CB are split, there are twopossible optical transitions (which conserve spin) at theK ± points — the lower-energy T A and the higher-energy T B ones. All the described features are depicted on Fig. 1for the case of MoS .A bilayer crystal can be presented as two monolayersseparated by a distance l ∼ FIG. 1. Sketch of band positions and optical transitions atthe K + and K − points of the Brillouin zone in a monolayerof MoS . Green and red bump structures represent conduc-tion and valence band states associated with transitions activecorrespondingly in the σ + and σ − polarizations. Solid wavyarrows denote all possible ( T A and T B ) optical transitions. | ∆ c | and ∆ v define the absolute values of spin-orbit splittingin the CB and VB, respectively. E g is the single-particle bandgap. ◦ . We arrange them in z = l/ z = − l/ I with an inversion centerin the z = 0 plane. The bilayer embedded in between twoflakes of the same material (in our case, hBN), thicknessand size conserves this symmetry. Therefore, one canextend the k · p model proposed in [30] also for this case.Let us examine the Bloch states of the valence and con-duction bands at the K + point of bilayer. We constructthem from the Bloch states of top and bottom layersconsidering them separately. Namely, we introduce thestates | Ψ ( m ) n i ⊗ | s i , where m = 1 , n = v, c is a bandindex (for VB and CB) and s = ↑ , ↓ specifies spin degreeof freedom. The bottom (first) layer states | Ψ (1) v i and | Ψ (1) c i are made predominantly from d x − y + id xy and d z orbitals of transition metal atoms respectively [45, 46].They coincide with monolayer spinless states | Ψ v i and | Ψ c i mentioned before. The top (second) layer states | Ψ (2) v i and | Ψ (2) c i are made from d x − y − id xy and d z orbitals and coincide with spinless states at the K − pointof monolayer. The top and bottom states are connectedby the relation | Ψ (2) n i = K I | Ψ (1) n i , where K and I are the complex conjugation and central inversion op-erators respectively. Finally, we suppose the orthogonal-ity of the basis states from different layers and bands FIG. 2. The bilayer crystal embedded in between two thickhBN flakes, side view. The origin of all red arrows representsthe inversion center of the crystal. The arrows, which layon the same line, depict the inversion symmetry of the crys-tal. Namely, the tips of the corresponding arrows indicate thepair of atoms, which positions transform into each other afterinversion procedure. h Ψ ( m ) n | Ψ ( m ′ ) n ′ i = δ nn ′ δ mm ′ .The initial basis states in a given layer are affectedby crystal fields of another layer and surrounding hBNmedium. Such fields being considered as a perturbationin k · p model produce intra- and interlayer correctionsto the bilayer Hamiltonian. Namely, the intralayer onesrenormalize the band gap E g and spin-splittings ∆ c , ∆ v for the electron excitations in the considered layer. Dueto the symmetry of the system, the parameters of theother layer get the same modifications. The interlayercorrections link the states from different layers. The sym-metry analysis of such terms demonstrates i) the strongcoupling between VB basis states with the same spins; ii) the quasi-momentum dependent coupling between CBbasis states with the same spins; iii) the coupling betweenVB and CB basis states of the opposite spins, which ap-pears due to spin-orbit interaction. The latter term issupposed to be small and is omitted from our study. Inthis case, the spin-up and spin-down states of bilayer aredecoupled and can be considered separately.As a result, the VB Hamiltonian written in the basis {| Ψ (1) v i ⊗ | s i , | Ψ (2) v i ⊗ | s i} takes the form H (2) vs = (cid:20) σ s ∆ v tt − σ s ∆ v (cid:21) , (1)where σ s = +1( −
1) for s = ↑ ( ↓ ). The parameter t ∼ −
70 meV [30] defines the coupling between valence bandsfrom different layers. The CB Hamiltonian, written inthe basis {| Ψ (1) c i ⊗ | s i , | Ψ (2) c i ⊗ | s i} is H (2) cs = (cid:20) E g + σ s ∆ c uk + uk − E g − σ s ∆ c (cid:21) , (2)where E g is the band gap of bilayer and k ± = k x ± ik y .Both Hamiltonians are written up to O ( k ). FIG. 3. Sketch of the bands positions and optical transitionsat the K + of the BZ in a bilayer of MoS . Green and red bumpstructures represent conduction and valence band states asso-ciated with transitions active in the σ + and σ − polarizationsrespectively. Solid (dashed) wavy arrows denote optical tran-sitions due to the intralayer (interlayer) A and B excitons. | ∆ c | and √ ∆ v + 4 t denote the splitting in the CB and VB,respectively. E g is the single particle band gap. The bilayer VB have the energies E ± v = ± p ∆ v / t .The corresponding upper-energy eigenstates are | Φ + v ↑ i = h cos θ | Ψ (1) v i + sin θ | Ψ (2) v i i ⊗ | ↑i , | Φ + v ↓ i = h sin θ | Ψ (1) v i + cos θ | Ψ (2) v i i ⊗ | ↓i , (3)where we introduced cos(2 θ ) = ∆ v / p ∆ v + 4 t . The low-energy eigenstates | Φ − v ↑ i and | Φ − v ↓ i can be derived fromthe first ones by replacing cos θ → − sin θ, sin θ → cos θ .The new VB are doubly degenerated by spin.The CB states do not interact with each other in K + valley ( i.e. in k x , k y → E g ± ∆ c /
2. In further we focus on MoS bilayer with ∆ c <
0. For this case, the upper and lowerenergy CB states are {| Ψ (1) c i ⊗ | ↓i , | Ψ (2) c i ⊗ | ↑i} and {| Ψ (1) c i ⊗ | ↑i , | Ψ (2) c i ⊗ | ↓i} , respectively.All new energy states of bilayer system are depictedgraphically in Fig. 3. We divide them on spin-up (left)and spin-down (right) subsets for clarity. The single anddoubled bumps in the figure represent the new CB andVB states. The size of the bump describes the proba-bility to observe the new composite quasiparticle in thebottom (green part) or the top (red part) layers. Notethat the valence band states can be found in both lay-ers. That makes these states optically active in bothpolarizations. Namely, the transitions from any valenceband state into | Ψ (1) c i and | Ψ (2) c i states are possible in σ + and σ − circularly polarised light, respectively. In conse-quence, the system demonstrates four types of opticallyactive exciton resonances — two intralayer and two inter- FIG. 4. The trilayer crystal embedded in between two thickhBN flakes, side view. The mirror symmetry plane z = 0 isrepresented by thin rectangular. The ends of red curvy arrowscouple the atoms, which coordinates transform to each otherafter mirror symmetry transformation operation. layer ones. The half of all transitions — T A ( T B ) betweenspin-up(spin-down) bands are presented in Fig. 3.There are two intralayer exciton transitions in bilayer T A : ( | Φ + v ↑ i → | Ψ (1) c i ⊗ | ↑i , | Φ + v ↓ i → | Ψ (2) c i ⊗ | ↓i ; (4) T B : ( | Φ − v ↑ i → | Ψ (2) c i ⊗ | ↑i , | Φ − v ↓ i → | Ψ (1) c i ⊗ | ↓i . (5)All of them have the same intensity I = I cos θ , where I is the intensity of exciton line in monolayer.There are two interlayer exciton transitions in bilayer T ′ A : ( | Φ + v ↑ i → | Ψ (2) c i ⊗ | ↑i , | Φ + v ↓ i → | Ψ (1) c i ⊗ | ↓i ; (6) T ′ B : ( | Φ − v ↑ i → | Ψ (1) c i ⊗ | ↑i , | Φ − v ↓ i → | Ψ (2) c i ⊗ | ↓i . (7)They have the intensities I ′ = I sin θ . Finally, sincethe intra- and inter-layer exciton transitions are activein opposite circular polarization at a given K point,they should have the opposite signs of g -factors (see Ap-pendix B).For a trilayer case, a crystal can be presented as threemonolayers each separated by distance l , with the mid-dle layer is 180 ◦ rotated around the external ones (seeFig. 4). Similarly to the bilayer case, we focus on prop-erties of quasiparticles at the K + point. The basis states of a trilayer are {| Ψ (1) n i ⊗ | s i , | Ψ (2) n i ⊗ | s i , | Ψ (3) n i ⊗ | s i} .They belong to z = − l , z = 0 and z = l layers, respec-tively. In this case, the crystal has mirror symmetry σ h ,with the mirror plane z = 0. It determines the symme-try relations | Ψ (3) n i = σ h | Ψ (1) n i , | Ψ (2) n i = σ h | Ψ (2) n i . Theproperties of | Ψ (1) n i and | Ψ (2) n i states are the same as inbilayer discussed in the main text. The properties of | Ψ (3) n i states coincide with | Ψ (1) n i according to the crystalsymmetry. Hence, in K + point, the 1-st and 3-d lay-ers absorb only σ + polarised light, while the 2-nd oneis active in σ − polarization. Note that the trilayer en-capsulated in between to equal hBN flakes conserves themirror symmetry.We derive the effective k · p Hamiltonian for consid-ering system within several approximations: i) the in-terlayer spin-orbit coupling between CB and VB stateswith opposite spins from different layers is neglected,like in bilayer; ii) the intralayer crystal field correctionsare equal for each layer, i.e. the bands of each layerare characterised by the same band-gap E g and spin-splitting ∆ v , ∆ c parameters, the magnitudes of which,however, can deviate from their bilayer analogs; iii) weneglect the coupling between the states of the 1-st and3-d layers. In this approach, like in bilayer case, thespin-up and spin-down states of trilayer are decoupled.The Hamiltonian for VB states, written in the basis {| Ψ (1) v i ⊗ | s i , | Ψ (2) v i ⊗ | s i , | Ψ (3) v i ⊗ | s i} , is H (3) vs = σ s ∆ v t t − σ s ∆ v t t σ s ∆ v . (8)The Hamiltonian for conduction bands, written in thebasis {| Ψ (1) c i ⊗ | s i , | Ψ (2) c i ⊗ | s i , | Ψ (3) c i ⊗ | s i} , reads H (3) cs = E g + σ s ∆ c uk + uk − E g − σ s ∆ c uk − uk + E g + σ s ∆ c . (9)The system possesses a mirror symmetry. Hence, it isconvenient to introduce the new basis states which areeven | Υ (1) n i = √ (cid:2) | Ψ (1) n i + | Ψ (3) n i (cid:3) , | Υ (2) n i = | Ψ (2) n i and odd | Υ (3) n i = √ (cid:2) | Ψ (1) n i − | Ψ (3) n i (cid:3) under σ h transformation.The trilayer Hamiltonians have a block-diagonal form inthe basis {| Υ (1) n i ⊗ | s i , | Υ (2) n i ⊗ | s i , | Υ (3) n i ⊗ | s i}H (3) vs = σ s ∆ v √ t √ t − σ s ∆ v
00 0 σ s ∆ v , (10) H (3) cs = E g + σ s ∆ c √ uk + √ uk − E g − σ s ∆ c
00 0 E g + σ s ∆ c . (11)The 2 × t → √ t, u → √ u . Consequently, there are doubly de-generate valence and conduction bands, which define theintensive T A and T B and weak T ′ A and T ′ B groups ex-citonic transitions, which are active in opposite polar-izations of light (see Fig. 5(a)). The 1 × T oA and T oB transitions between odd states of tri-layer. They are active in the same polarization as T A and T B transitions of monolayer (see Fig. 5(b)). Summaris-ing, the trilayer possesses two types of ”even” intralayerexcitons — T A and T B , two types of ”odd”intralayer ones— T oA and T oB . All of them are active in σ + polarizationat the K + point and have the g -factors of the same sign.The interlayer ”even” excitons — T ′ A and T ′ B , converselyare active in σ − polarization at the K + point and havean opposite sign to intralayer excitons g -factors. Thedetailed description of trilayer’s g -factors is presented inAppendix C.Note that the current theoretical description predictsthe number of exciton resonances and their polarizationproperties. However, the relative position of T A and T oA exciton remains an open question. Indeed, both lines areactive in the same polarization and have similar g -factors,that makes impossible to identify each of them from theexperiment. The analysis of the interference effects to-gether with the consideration of the Coulomb interactionbetween new quasiparticles answers this question, whichis, however, beyond the scope of the current paper. In ourcase, we took the position of exciton resonances followingthe position of the valence and conduction bands, derivedin single-particle k · p approximation, for definiteness.Our single-particle considerations are summarized inFig. 6. This picture exhibits the position and numberof optically active transitions in mono-, bi- and trilayerS-TMD crystals. III. ABSORPTION RESONANCES IN MOS MULTILAYERS
In the experiments, we have investigated a set of vander Waals heterostructures with optically active partsconsisting of MoS mono- (1ML), bi- (2ML) and trilayer(3ML). The structures were prepared on ultraflat unoxi-dized silicon substrates by combination of exfoliation anddry transfer techniques [48]. In order to achieve the high-est possible quality of our samples, as well as to preservethe characteristic inverse and mirror symmetries of, re-spectively, 2H-stacked bi- and trilayer, the microcleavedMoS flakes were sandwiched between two hBN flakesexfoliated from high-quality bulk crystals grown underhigh-pressure conditions [49]. Further information on thepreparation of samples and their characterization is pro-vided in Appendix A.Experiments consisted of measurements of the re- flectance contrast spectra, which are defined as RC ( E ) = R ( E ) − R ( E ) R ( E ) + R ( E ) × , (12)where R ( E ) is the reflectance spectrum when the light isfocused on the MoS flake and R ( E ) is the reflectancefrom the region outside the flake. Two different micro-optical setups have been used for measurements: onesetup for measurements in the absence of magnetic fieldsand the second one for measurements in magnetic fieldssupported by either a 14 T superconducting magnet ora 29 T resistive magnet. A spatial resolution of about1 µm (light spot diameter) is characteristic of our free-beam-optics setup used for experiments at zero magneticfield. Instead, the fiber-optics-based arrangement appliedfor experiments in magnetic fields provides a poorer spa-tial resolution, of about 10 µm . A spectral resolution of0.32 nm has been assured for both setups. A combina-tion of a quarter wave plate and a polarizer are used toanalyse the circular polarization of signals. The measure-ments are performed with a fixed circular polarization,whereas reversing the direction of magnetic field yieldsthe information corresponding to the other polarizationcomponent due to time-reversal symmetry. More on ex-perimental details can be found in Appendix A.The collection of low temperature ( ∼ B = 0 setup, measured in the spectralrange of the optical gap (onset of strong absorption) ofthe MoS layers is presented in Fig. 7. The observed,more or less pronounced resonances, show the dispersivespectral shape, correspond to typically, strongly bounddirect bandgap excitons in S-TMD layers. Although ourtheoretical considerations neglect the effects of Coulombbinding and account only for interband transitions, a dis-tinct resemblance between the measured spectral evolu-tion (Fig. 7) and that theoretically concluded in Fig. 6can be recognized.In the following we assume that each interband tran-sition T X is associated with the corresponding excitonicresonance X and we proceed with the assignment of theresonances observed in the experiment. We suppose thatthe resonating excitons are shifted in energy, by theirbinding energies, with respect to the associated inter-band transitions: E ( X ) = E ( T X ) − E b ( X ). Binding en-ergies might be, however, different for different excitonsand this fact must be taken into account when comparingthe energy position of X -resonances and T X -transitions.Important for the assignment of the observed resonancesare the results of the polarization resolved measurementscarried out in magnetic fields (see Figs. 8 and 9). Asdiscussed above, we expect that our multilayers host twodifferent types of intra- and interlayer excitons, each ofthem being distinguishable by their different polarizationproperties ( g -factors of opposite sign and various mag-nitudes). At this point we must admit that all subtlespectral features which are well visible in the RC spectrameasured at B = 0 with high spatial resolution are some-what less pronounced in the magneto-optical measure- FIG. 5. Sketch of bands positions and optical transitions between (a) even and (b) odd states at the K + of the BZ in abilayer of MoS . Green and red bump structures represent conduction and valence band states associated with transitionsactive correspondingly in the σ + and σ − polarizations. Solid (dashed) wavy arrows denote optical transitions due to theintralayer (interlayer) A and B excitons. | ∆ c | and √ ∆ v + 8 t (∆ v ) denote the splitting in the CB and even (odd) states of VB,respectively. E g is the single particle band gap. ments which imply worse spatial resolution. Certain de-gree of sample inhomogeneity is an obvious cause of thisdrawback. In consequence, in the case of weak and/orbroad resonances the information about their g -factorsis not easily extractable from the raw magneto-RC data.This information becomes more apparent if we inspectthe RC-polarization spectra. These spectra have beenconstructed as a difference between the RC-spectra mea-sured at the same strength but for two opposite directionof the magnetic field (that mimic, due to the time rever-sal symmetry, the spectra corresponding to σ + and σ − circular polarization of the reflected light [39, 50]). Ascan be deduced from the data shown in Figs. 8 and 9,the apparent dips in our RC polarization spectra cor-respond to excitonic resonances with negative g -factorswhereas the characteristic upswings mark the resonanceswith positive g -factors.Very first classification of the observed resonance takesinto account a relatively large spin-orbit splitting in theMoS valence band, rising two groups of excitons asso-ciated with well-separated upper ( A -excitons) and lower( B -excitons) valence band subbands. The 1ML spectrumshown in Fig. 7, resembles that previously reported for ahigh quality 1ML MoS encapsulated in hBN [44, 51]. Itdepicts three resonances: one well-separated resonancedue A -exciton (ground state), X A , and two other su-perimposed resonances, due to B -exciton ( X B ) and thefirst excited (2s) state of A -exciton ( X sA ). X A and X B resonances are associated with T A and T B transi-tions sketched in Fig. 1; the excited excitonic states arebeyond the frame of our single particle theoretical ap-proach. As expected, the X A , X B and X sA excitonicresonances display similar polarization properties in the magneto-optical experiments (see Fig. 8). We estimate g X A ≈ − g X B and g X sA is more cumbersome though both these values arealso negative.The RC spectrum of the 2ML (Fig. 7) shows 4 reso-nances [38] which, in reference to our theoretical expec-tations (see Figs. 3 and 6), are assigned to the pair ofintra- ( X A ) and interlayer ( X ′ A ) A -excitons, and to theanalogous pair of intra- ( X B ) and interlayer ( X ′ B ) B -excitons. We use the signs and magnitude of the exciton g -factors as fingerprints the intra- and interlayer natureof the X A - and X ′ A -excitons respectively [39]. Namely,we conclude that g X A ≈ −
4, while g X ′ A ≈
8. Again, theexact values for the g -factor amplitudes of B -excitonsare hard to be precisely estimated. Nonetheless it israther clear that g X B is negative whereas g X ′ B is posi-tive. Whereas our theoretical model meets the experi-mental data at the qualitative level, the energy ladderof the observed resonances requests further comments.Characteristic for the theoretical modelling is the factthat the energy distance, ∆( T ′ A , T A ), between the T ′ A and T A transitions is the same as the energy separation,∆( T B , T ′ B ) between the T B and T ′ B transitions, both dif-ferences being determined by the amplitude ∆ c of thespin orbit splitting in the conduction band of the 2ML:∆( T ′ A , T A ) = ∆( T B , T ′ B ) = | ∆ c | . This property is notseen in the experiment: we estimate that ∆( X ′ A , X A ) ≃
70 meV, whereas ∆( X B , X ′ B ) ≃
30 meV. The inconsis-tency between theory and experiment is likely due todifferent binding energies of excitons associated with dif-ferent interband transitions: E ( X ) = E ( T X ) − E b ( X ).In the first approximation we assume that binding en- FIG. 6. Sketch of optical transitions in (a) mono-, (b) bi- and(c) trilayer S-TMD crystals. The position and size of blackthick segments represent the energies and intensities of thecorresponding transitions. The values of parameters ∆ c and∆ v for bilayer and trilayer can deviate from their monolayeranalogs. ergies of our inter- and intralayer excitons are indeeddifferent but that the excitons within each pair of in-direct and direct resonances are the same: E b ( intra ) = E b ( X A ) = E b ( X B ) = E b ( X ′ A ) = E b ( X ′ B ) = E b ( inter ).Then, comparing the theoretical prediction with the ex-perimental data one obtains that | ∆ c | = ∆( T ′ A , T A ) =∆( T B , T ′ B ) = [∆( X ′ A , X A ) + ∆( X B , X ′ B )] / ≃
50 meVand that E b ( intra ) = E b ( inter ) + 20 meV. Larger bind-ing energies of intralayer excitons with respect to those ofinterlayer excitons are logically expected since Coulombattraction should be indeed stronger/weaker when theelectron and hole charges are localized in the same or inthe neighboring layers, as for the case of intra- or inter-layer excitons, respectively. At first sight, the estimatedamplitude of the spin orbit splitting in the conductionband of the MoS bilayer appears to be surprisingly large.In the case of 1L MoS , the commonly accepted resultsof the DFT calculations predict ∆ c in the range of fewmeV, though recent experimental works point out to-wards much higher values of about 15 meV [55]. Accord-ing to our theoretical approach, the spin orbit interactionis sensitive to interlayer coupling and it is also affectedby the interaction with the surrounding material (hBN).Thus the amplitudes of the ∆ c as well as ∆ v parameters " X B " X ’B X ’A X A X oA R e f l ec t a n ce c on t r a s t ( % ) Energy (eV) X B X FIG. 7. Reflectance contrast spectra of MoS layers encapsu-lated in hBN measured at T =5 K. The spectra are verticallyshifted for clarity. are expected to evolve with the numbers of the stackedMoS layers. Keeping our assumption about equal bind-ing energies for X A and X B excitons we note that theexpected energy separation between these excitonic res-onance is given by | ∆ c | + p ∆ v + 4 t (see Figs. 3 and 6),to be compared to the value of 170 meV estimated fromthe experiment. With the theoretically estimated valueof t ≃
40 meV [30] and the derived above | ∆ c | ≃
50 meVwe find ∆ v ≃
90 meV. Visibly, however, the “effective”spin orbit splitting in the valence band of the MoS bi-layer is larger, as given by p ∆ v + 4 t ≃
120 meV (seeFig. 3).Focusing now on the trilayer spectra (see bottom panelof Fig. 7) we observe a characteristic triplet of A -excitonresonances, which is in accordance with our theoreti-cal expectations (see Fig. 6c). As expected, two strong X A and X oA resonances are characterised by negative g -factors whereas the g -factor of the interlayer X ′ A ex-citon is pretty much positive. The theoretically antic-ipated triplet structure of the ′′ X ′′ B -exciton is not re-solved in the experiment but likely hidden within theobserved broad spectrum of the B -exciton. Even withunresolved fine structure of the B -exciton for our trilayerMoS , a rough estimation of ∆ c and ∆ v parameters canbe done. Expecting that the energy separation between X ′ A and X A is ∆( X ′ A , X A ) = | ∆ c | + E b ( X A ) − E b ( X ′ A )and reading ∆( X ′ A , X A ) ≃
70 meV from the experimentwe conclude that | ∆ c | ≃
50 meV if the difference be-tween binding energies of the intralayer X A and inter-layer X ′ A excitons is E b ( X A ) − E b ( X ′ A ) ≃
20 meV, i.e. the same as we found for intra- and interlayer exci-tons in 2ML MoS . On the other hand, supposing that X sA <0 <0<0
29 T20 T10 T0 T X A X B R e f l ec t a n ce c on t r a s t ( % ) Energy (eV) g<0 <0>0>0 X ’B X ’A X A Energy (eV) X B
29 T20 T10 T0 T10%5% >0<0<0 X ’A X A X oA Energy (eV) " X B "
14 T10 T5 T0 T10%1%
FIG. 8. Helicity-resolved reflectance contrast spectra of MoS layers encapsulated in hBN for selected values of magnetic fieldmeasured at T =4.2 K. The red and blue curves correspond to σ + polarization of reflected light in B and − B configurations ofmagnetic field (applied perpendicularly to the layers plane), respectively. The spectra are vertically shifted for clarity. X sA
29 T 25 T20 T15 T10 T5 T0 T X A X B D i ff e r e n ce s p ec t r u m ( % ) Energy (eV) X ’B X ’A X A Energy (eV) X B
29 T 25 T20 T15 T10 T5 T0 T5%20% X ’A X A X oA Energy (eV) " X B "
14 T 10 T5 T0 T2% 10%
FIG. 9. Zero-field RC spectra (black) with difference spectra [ RC ( σ + , B ) − RC ( σ + , − B )] at selected values of B field for 1ML(left panel), 2ML (middle panel) and 3ML (right panel) MoS encapsulated in hBN. A deep in the difference spectra indicatea negative g -factor, while a peak indicates a positive g -factor. binding energies of two X A and X oA intralayer excitonsare the same we expect that they are separated in en-ergy by ∆( X oA , X A ) = ( p ∆ v + 8 t − ∆ v ) / v ≃
140 meV when reading ∆( X oA , X A ) ≃
20 meV fromthe experiments and assuming again that t ≃
40 meV. With the above estimation, the band-edge structure ofthe 3L MoS at the K ± points of the Brillouin zone isconcluded to consist of two conduction band subbandssplit by 50 meV and 4 valence band subbands with outertwo subbands split by 180 meV and the inner two sub-bands split by 140 meV. IV. CONCLUSION
We have performed magnetooptical µ -reflectance mea-surements along with k · p theory based modelling onfew-layer MoS encapsulated in hBN structures, reveal-ing the intralayer and interlayer nature of the newly dis-covered transitions. Such resonances form due to hybridi-sation of valence and conduction bands states when oneadds new layers to the system.The experiment i) manifests the new exciton reso-nances and their number in bi- and trilayers; ii) demon-strates that g -factors of intralayer and interlayer transi-tions have opposite signs; iii) the g -factor values of thelatter ones are much larger (by magnitude) than of theformer ones.The symmetry based k · p description of the bi- andtrilayers gives the natural explanation of the aforemen-tioned experimental observations. Moreover, from thegeneral symmetry point of view the theoretical modelalso predicts i) the renormalization of the band gap E g and spin splittings ∆ c , ∆ v ; ii) the deviation of the bi- andtrilayer exciton g -factors from their monolayer analogs.These two phenomena appear as a result of the couplingbetween different layer of the system and effects of di-electric screening induced by hBN.Finally, we point out some unique properties of theexciton states of the trilayer. First, we mention the ex-istence of two groups of exciton resonances — “even”and “odd”, which do not interact with each other be-cause they belong to different irreps of the in-plane mir-ror symmetry of the system. However, electric field, ap-plied perpendicularly to the crystal’s plane, violates thissymmetry and causes the controllable coupling of thesestates. Such feature can be used in future exciton basedapplications. We also note that the electrical positive andnegative charges of interlayer excitons are separated be-tween different layers. Such a property can be interestingin photovoltaic applications of S-TMD systems. ACKNOWLEDGEMENTS
The work has been supported by the ATOMOPTOproject (TEAM programme of the Foundation forPoli sh Science co-financed by the EU within theERDFund), the EC Graphene Flagship project (no.604391), the National Science Centre (grants no. DEC-2013/10/M/ST3/00791, UMO-2017/24/C/ST3/00119),the Nanofab facility of the Institut N´eel, CNRS UGAand LNCMI-CNRS, a member of the European MagneticField Laboratory (EMFL).
Appendix A: Samples and experimental setups
In order to provide even and charged-defect-free sup-port to van der Waals heterostructures studied in thiswork we assembled them on ultraflat unoxidized sili-con (Si) substrates from Ted Pella Inc., whose surfaceroughness amounts to about one-tenth of that of stan-dard Si/SiO wafers. Each heterostructure consisted ofthree flakes obtained by means of mechanical exfoliationand successively stacked one on top of another using drytransfer techniques: a 60-70 nm thick bottom hBN flake,a central MoS flake of 1ML, 2ML or 3ML thickness, anda 10-15 nm thick top hBN flake.The optimum thicknesses of hBN flakes were estimatedon the basis of transfer-matrix simulations of reflectancecontrast spectra of complete heterostructures, aimingat maximizing the visibility of absorption resonances inMoS flakes while keeping the thickness of bottom hBNflakes below 100 nm (a value up to which our exfoliationmethod yields a rich harvest of flat, defect-free flakes withlarge-area terraces of constant thickness).Using high-purity ELP BT-150E-CM tape from NittoDenko, bottom hBN flakes were exfoliated from high-quality bulk crystals grown under high-pressure condi-tions [49] and then transferred in a non-deterministic wayonto clean and freshly annealed at 200 ◦ C Si substrates.The MoS flakes were obtained by means of two-stage,tape- and polydimethylsiloxane (PDMS)-based exfolia-tion of a bulk crystal purchased from HQ Graphene andthen stacked on the bottom hBN flakes using an all-drydeterministic stamping technique [48]. The same pro-cedure was applied also to the top hBN flakes. Whileperforming deterministic transfers, special attention waspaid to mutual angular alignment of the flakes and toimmediate 10-minute-long after-transfer annealing of thesamples at 180 ◦ C on a hot plate kept in clean ambientatmosphere.Optical microscope images of investigated heterostruc-tures which were fabricated in this way are shown inFig. 10 (a)-(c). The spots of brown-to-blue colour vis-ible in all images are up to 60-nm-high bubbles of airand/or water vapour (possibly with some amount of hy-drocarbons) trapped between either the MoS and thetop hBN flake or the two hBN flakes. Their appearancedirectly results from the corrugation of thin, soft andflexible top hBN flakes supported by a PDMS stamp.Importantly, the areas between the bubbles, whose sizereaches up to 5 by 10 micrometers, exhibit very flat andhigh-quality intefraces between the constitutent flakes,as revealed with optical measurements and atomic forcemicroscopy (AFM) characterization.An example of tapping-mode AFM imaging performedon finished heterostructures with the use of Digital In-struments Dimension 3100 microscope is shown in Fig. 10(d)-(f). The false colour AFM images correspond to re-spective locations marked in panels (a)-(c) with orangecircles. The grey arrows represent the line profiles drawnin Fig. 10 (g) and labelled with the same numbers as in0 FIG. 10. Upper row: optical microscope images of hBN/MoS /hBN heterostructures comprising (a) mono-, (b) bi- and (c)trilayer MoS flakes (outlined in white) sandwiched between two hBN flakes – a 60-70 nm thick flake from the bottom (outlinedin yellow) and 10-15 nm thick flake from the top (outlined in red). The spots of brown-to-blue colour visible in all images arebubbles of air and/or water vapour trapped between either the MoS and the top hBN flake or the two hBN flakes. Lower row:(d)-(f) false colour atomic force microscope images taken at respective locations marked in (a)-(c) with orange circles. Thegrey arrows represent the line profiles drawn in panel (g) and labelled with the same numbers as in images (d)-(f). images (d)-(f). As can be seen, the profiles unambigu-ously confirm the mono-, bi-, and trilayer thickness ofMoS flakes incorporated into the heterostructures shownin the upper row of Fig. 10.Compared to based on neutron scattering measure-ments estimation equal to 0.615 nm [56], the value of0.79 nm we got for the monolayer most probably indi-cates that the equilibrium distance between the hBN andMoS layers differs from that between two neighboringMoS layers in a bulk crystal. A smaller single-layer stephight obtained for the second and third layer in the bi-and trilayer MoS flakes may on the other hand implythe existence of small uniaxial compressive strain alongthe c -axis in hBN/MoS /hBN heterostructures.Measurements at zero magnetic field were carried outwith the aid of a continuous flow cryostat mounted on x − y motorized positioners. The samples were placedon a cold finger of the cryostat. The temperature of thesamples was kept at T = 5 K. The excitation light wasfocused by means of a 50x long-working distance objec-tive with a 0.5 numerical aperture producing a spot ofabout 1 µ m. The signal was collected via the same mi-croscope objective, sent through a 0.5 m monochromator,and then detected by a CCD camera.Magneto-optical experiments were performed in theFaraday configuration using an optical-fiber-based insertplaced in a resistive or a superconducting magnet produc-ing magnetic fields up to 29 T or 14 T, respectively. Thesample was mounted on top of an x − y − z piezo-stagekept in gaseous helium at T =4.2 K. The µ -RC experi-ments were performed with the use of 100 W tungstenhalogen lamp. The excitation light was coupled to an op-tical fiber with a core of 50 µ m diameter and focused onthe sample by an aspheric lens (spot diameter around 10 µ m). The signal was collected by the same lens, injectedinto a second optical fiber of the same diameter, and an-alyzed by a 0.5 m long monochromator equipped with aCCD camera. A combination of a quarter wave plate anda polarizer are used to analyse the circular polarizationof signals. Appendix B: Bilayer in magnetic field
The magnetic field correction to the valence bandHamiltonian at the K + point written in the basis {| Ψ (1) v i ⊗ | s i , | Ψ (2) v i ⊗ | s i} is δH (2) vs = (cid:0) G (2) v + σ s (cid:1) µ B B .Here µ B is the Bohr magneton, B is the magnetic field,applied perpendicularly to the bilayer plane, is the unitmatrix and G (2) v = (cid:20) g v − δg v − g v + δg v (cid:21) . (B1)The correction to the conduction band Hamiltonian writ-ten in the basis {| Ψ (1) c i ⊗ | s i , | Ψ (2) c i ⊗ | s i} has the sameform δH (2) c = (cid:0) G (2) c + σ s (cid:1) µ B B , where G (2) c = (cid:20) g c − δg c − g c + δg c (cid:21) µ B B. (B2)The expressions for g v , g c , δg v , δg c are derived in [37].The corrections to the Hamiltonians provide the energyshifts of excitons in magnetic field. The intralayer X A and X B excitons, constructed from the quasiparticles atthe K + point of bilayer, are active in σ ± polarizations.The corresponding energy shifts are linear in magneticfield δE σ ± A,B = ± g (2) A,B µ B B . The g -coefficients of such1exciton transitions are g (2) A = − g u + ( g c − δg c ) − ( g v − δg v ) ∆ v p ∆ v + 4 t , (B3) g (2) B = g u + ( g c − δg c ) − ( g v − δg v ) ∆ v p ∆ v + 4 t . (B4)We introduced g u = 2 m u / ~ ∆ c parameter, which orig-inates from k -dependent admixture of conduction bands.The mixing is negligibly small in the absence of magneticfield, but gives finite correction if B = 0. The inter-layer excitons X ′ A and X ′ B at the bilayer K + point arealso active in σ ± polarizations with the magnetic shifts δE σ ± A ′ ,B ′ = ± g (2) A ′ ,B ′ µ B B respectively. The corresponding g -coefficients are g (2) A ′ = g u + ( g c − δg c ) + ( g v − δg v ) ∆ v p ∆ v + 4 t , (B5) g (2) B ′ = − g u + ( g c − δg c ) + ( g v − δg v ) ∆ v p ∆ v + 4 t . (B6)Note that there are the following relations between g -coefficients of interlayer and intralayer exciton transitions g (2) A + g (2) A ′ = g (2) B + g (2) B ′ = 2( g c − δg c ) , (B7) g (2) A ′ − g (2) B = g (2) B ′ − g (2) A = ( g v − δg v ) 2∆ v p ∆ v + 4 t , (B8) g (2) B − g (2) A = g (2) A ′ − g (2) B ′ = 2 g u . (B9)One can mention that the corresponding transitions atthe K − point in σ ± polarizations are characterized bythe same values ± g (2) A,B and ± g (2) A ′ ,B ′ as in K + point. Thisis the consequences of time reversal and inversion sym-metries of the crystal. As a result, we can restore theabsorption spectra of the bilayer in σ − and σ + polar-izations using our methodology of the experiment — bymeasuring the reflected light in fixed σ + polarization andchanging the orientation of magnetic field from B to − B .The change of the direction of the magnetic field to − B in experiment mimics the transitions in σ − polarizationof the reflected light.We use the formula g = [ E σ + ( B ) − E σ + ( − B )] /µ B B =[ δE σ + ( B ) − δE σ + ( − B )] /µ B B for exciton’s g -factor. Asa result the intralayer and interlayer A, B excitons have2 g (2) A,B and 2 g (2) A ′ ,B ′ g -factors, respectively.The possible signs of intra- and interlayer exciton g -factors can be obtain from the expressions (B3), (B4),(B5) and (B6). Indeed, in the experiment we have2 g (2) A,B ≈ −
4. It means that corrections g u , δg c and δg v are not significally large, and therefore very roughly2 g (2) A,B ≈ g c − g v ) is nothing but the g -factors ofmonolayer A and B excitons, for which we know that g v > g c >
0. Substituting the positive g c and g v intoEqs. (B5) and (B6) one can see that 2 g (2) A ′ ,B ′ >
0, whichis confirmed from the experiment.
Appendix C: Trilayer in magnetic field
The magnetic field correction to the valence and con-duction band Hamiltonians written in the basis {| Ψ (1) n i ⊗| s i , | Ψ (2) n i ⊗ | s i , | Ψ (3) n i ⊗ | s i} , is δH (3) ns = (cid:0) G (3) n + σ s (cid:1) µ B B where G (3) n = g n − δg n g n − g n + 2 δg n g n g n − δg n , (C1)Here n = v, c and ¯ g c and ¯ g v are the additional parameterswhich describe the magnetic dependent coupling betweenlayers of the system [37]. In new basis, defined in themain part of the text, the full Hamiltonians H (3) ns + δH (3) ns are reduced to a block-diagonal form. Namely, the G (3) n matrix transforms to G (3) n = g n − δg n + ¯ g n − g n + 2 δg n
00 0 g n − δg n − ¯ g n . (C2)The 1 × | Υ (3) ns i ⊗ | s i with total energies E ons ( B ) = E n + σ s ∆ n g n − δg n − ¯ g n + σ s ) µ B B, (C3)with E v = 0 and E c = E g . The expressions coincidewith the monolayer ones, but with the new g -coefficient g n − δg n + ¯ g n . Moreover, the uk ± terms do not affect theodd states, and therefore such excitons have the samereduced masses as their monolayer analogs. Hence thecorresponding trilayer exciton line has the same opto-electronic properties as it’s monolayer analog. The odd A and B exciton transitions at the K + point are activeonly in σ + polarization. The corresponding energy shiftin magnetic field is δE σ + A o ,B o = [( g c − δg c ) − ( g v − δg v ) − ¯ g c + ¯ g v ] µ B B . The same type of transitions at the K − point are active in σ − polarization and have the energyshift in magnetic field δE σ − A o ,B o = − δE σ + A o ,B o . It immedi-ately gives us g X oA,B = 2( g c − g v ) − g for X oA and X oB exciton g -factors both for K + and K − points of trilayer.Here we introduce the parameter g = 12 (cid:16) δg c + ¯ g c − δg v − ¯ g v (cid:17) . (C4)From the experiment we know that g X oA ≈ − .
5, whichis close to monolayer g X A = 2( g c − g v ) ≈ −
4. Therefore,we can roughly estimate g ≈ . × {| Υ (1) n i ⊗ | s i , | Υ (2) n i ⊗ | s i} . Note that thestructure of this matrix does not coincide with the struc-ture of G (2) n . Therefore the subsystem of even states oftrilayer demonstrates another behavior in magnetic fieldthan it’s bilayer analog. The conduction band states in2K + point are decoupled and have the energies E (1) cs = E g + σ s ∆ c g c − δg c + ¯ g c + σ s − σ s g u ) µ B B, (C5) E (2) cs = E g − σ s ∆ c − ( g c − δg c − σ s − σ s g u ) µ B B. (C6)The conduction band energies correspond to the states {| Υ (1) c i ⊗ | s i , | Υ (2) c i ⊗ | s i} . The valence band energiesof admixed states as a function of magnetic field B arecalculated similarly to the bilayer case.The energy shifts of intralayer A and B excitons in σ ± polarizations at the K + point are δE σ ± A,B = [ ± g (3) A,B + g ] µ B B , where g (3) A = − g u + (cid:16) g c − δg c + 12 ¯ g c (cid:17) −− (cid:16) g v − δg v + 12 ¯ g v (cid:17) ∆ v p ∆ v + 8 t , (C7) g (3) B = 2 g u + (cid:16) g c − δg c + 12 ¯ g c (cid:17) −− (cid:16) g v − δg v + 12 ¯ g v (cid:17) ∆ v p ∆ v + 8 t . (C8)The energy shifts for interlayer A ′ and B ′ excitons in bothpolarizations have the form δE σ ± A ′ ,B ′ = [ ± g (3) A ′ ,B ′ + g ] µ B B , where g (3) A ′ = 2 g u + (cid:16) g c − δg c + 12 ¯ g c (cid:17) ++ (cid:16) g v − δg v + 12 ¯ g v (cid:17) ∆ v p ∆ v + 8 t , (C9) g (3) B ′ = − g u + (cid:16) g c − δg c + 12 ¯ g c (cid:17) ++ (cid:16) g v − δg v + 12 ¯ g v (cid:17) ∆ v p ∆ v + 8 t . (C10)In the absence of g the latter results coincide with thebilayer case up to redefinition of the parameters. Thenon-zero value of g shows remarkable difference betweenpure bilayer and effective bilayer cases. The correspond-ing energy shifts in σ ± polarizations at the K − pointare δE σ ± A,B = [ ± g (3) A,B − g ] µ B B and δE σ ± A ′ ,B ′ = [ ± g (3) A ′ ,B ′ − g ] µ B B .This non-equivalency makes the analysis of the g -factors of the system more complicated. Let us considerthe results of the measurements presented on the Fig. 8more carefully, focusing mainly on X A and X ′ A excitontransitions. Again, according to the methodology of ourexperiment we measure the g -factor using the formula g = [ δE σ + ( B ) − δE σ + ( − B )] /µ B B . Then the g -factorsat the K + point of intralayer X A , X B and interlayer X ′ A , X ′ B excitons are g K + X A,B = 2 g (3) A,B + 2 g and g K + X A ′ ,B ′ =2 g (3) A ′ ,B ′ + 2 g respectively. For K − point transitions weobtain g K − X A,B = 2 g (3) A,B − g and g K − X A ′ ,B ′ = 2 g (3) A ′ ,B ′ − g .Taking into account the relative smallness of g and ab-sence of the results for magnetic field B larger than 14 Twe suppose that the double g -factor structure of X A and X ′ A resonances is indistinguishable. Instead of this, prob-ably, we observe only their average values g X A,B = 2 g (3) A,B and g X A ′ ,B ′ = 2 g (3) A ′ ,B ′ respectively. One can mentionthat these average g -factors surprisingly coincide withthe ones we can get from the standard formula g =[ E σ + ( B ) − E σ − ( B )] /µ B B = [ δE σ + ( B ) − δE σ − ( B )] /µ B B .The analysis of the signs of intra- and interlayer exciton g -factors can be done in the same way as in bilayer case. [1] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth,V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Two-dimensional atomic crystals , Proc. Natl. Acad. Sci.U. S. A. , 10451 (2005)[2] Q. H. Wang, K.-Z. Kourosh, A. Kis, J. N. Coleman,and M. S. Strano,
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