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Lucja Kipczak, Artur O. Slobodeniuk, Tomasz Woźniak, Mukul Bhatnagar, Natalia Zawadzka, Katarzyna Olkowska Pucko, Magdalena Joanna Grzeszczyk, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), A Babinski, Maciej Molas

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[Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe<sub>2</sub>](https://mdr.nims.go.jp/datasets/809be8a0-0739-42ab-8c74-fa5997620d02)

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Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe22D MaterialsPAPER • OPEN ACCESSAnalogy and dissimilarity of excitons in monolayerand bilayer of MoSe2To cite this article: Łucja Kipczak et al 2023 2D Mater. 10 025014 View the article online for updates and enhancements.You may also likePorous carbon nanosheets for oxygenreduction reaction and Zn-air batteriesShahzeb Ali Samad, Ziyu Fang, PengfeiShi et al.-Single- and Few-Layers MoSe2Nanoflowers: Synthesis, Characterization,and Their PiezoresponseMei Hsuan Wu and Jyh Ming Wu-Thermal transport properties of MoS2 andMoSe2 monolayersAli Kandemir, Haluk Yapicioglu, AlperKinaci et al.-This content was downloaded from IP address 144.213.253.16 on 19/03/2023 at 00:03https://doi.org/10.1088/2053-1583/acbc8b/article/10.1088/2053-1583/acbc89/article/10.1088/2053-1583/acbc89/article/10.1149/MA2016-01/25/1288/article/10.1149/MA2016-01/25/1288/article/10.1149/MA2016-01/25/1288/article/10.1088/0957-4484/27/5/055703/article/10.1088/0957-4484/27/5/055703/article/10.1088/0957-4484/27/5/055703/article/10.1088/0957-4484/27/5/055703/article/10.1088/0957-4484/27/5/055703/article/10.1088/0957-4484/27/5/0557032D Mater. 10 (2023) 025014 https://doi.org/10.1088/2053-1583/acbc8bOPEN ACCESSRECEIVED29 November 2022REVISED2 February 2023ACCEPTED FOR PUBLICATION16 February 2023PUBLISHED28 February 2023Original Content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERAnalogy and dissimilarity of excitons in monolayer and bilayer ofMoSe2Łucja Kipczak1,∗, Artur O Slobodeniuk2,∗, Tomasz Woźniak3, Mukul Bhatnagar1,Natalia Zawadzka1, Katarzyna Olkowska-Pucko1, Magdalena Grzeszczyk1,4, Kenji Watanabe5,Takashi Taniguchi6, Adam Babiński1 and Maciej R Molas1,∗1 Institute of Experimental Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland2 Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, CZ-121 16 Prague, Czech Republic3 Department of Semiconductor Materials Engineering, Wrocław University of Science and Technology, 50-370 Wrocław, Poland4 Institute for Functional Intelligent Materials, National University of Singapore, Singapore 117544, Singapore5 Research Center for Functional Materials, National Institute for Materials Science, Tsukuba 305-0044, Japan6 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan∗ Authors to whom any correspondence should be addressed.E-mail: lucja.kipczak@fuw.edu.pl, aslobodeniuk@karlov.mff.cuni.cz andmaciej.molas@fuw.edu.plKeywords:monolayer MoSe2, bilayer MoSe2, excitons, binding energy, DFT calculations, kp approach, g-factorSupplementary material for this article is available onlineAbstractExcitons in thin layers of semiconducting transition metal dichalcogenides are highly subject to thestrongly modified Coulomb electron–hole interaction in these materials. Therefore, they do notfollow the model system of a two-dimensional hydrogen atom. We investigate experimentally andtheoretically excitonic properties in both the monolayer (ML) and the bilayer (BL) of MoSe2encapsulated in hexagonal BN. The measured magnetic field evolutions of the reflectance contrastspectra of the MoSe2 ML and BL allow us to determine g-factors of intralayer A and B excitons, aswell as the g-factor of the interlayer exciton. We explain the dependence of g-factors on the numberof layers and excitation state using first principles calculations. Furthermore, we demonstrate thatthe experimentally measured ladder of excitonic s states in the ML can be reproduced using thek · p approach with the Rytova–Keldysh potential that describes the electron–hole interaction. Incontrast, the analogous calculation for the BL case requires taking into account the out-of-planedielectric response of the MoSe2 BL.1. IntroductionThe optical response of monolayers (MLs) belong-ing to semiconducting transition metal dichalcogen-ides (S-TMDs), such as MoS2, MoSe2, MoTe2, WS2,andWSe2, is dominated by the excitonic emission/ab-sorption even at room temperature [1–4]. This is dueto the binding energies (BEs) of excitons, i.e. boundelectron–hole (e-h) pairs, which are as large as a fewhundred meV [5–9]. The most unconventional prop-erty of these excitons is their non-Rydberg modelspectrum, which cannot be described by the stand-ard two-dimensional (2D) hydrogen atom [10, 11].A typical approach to account for the excitonic spec-tra of S-TMD MLs refers to the numerical solutionsof the Schrödinger equation, in which the e-h attrac-tion is approximated by the Rytova–Keldysh (RK)potential [12, 13]. Although this is a well-knownmethod for the ML, the corresponding analysis ofthe excitonic ladder for the bilayer (BL) has not beenreported so far.Moreover, the electronic bands in a BL and other2 H-stacked multilayers, are known to be effectivelymodified compared to the ML case [4, 14–18]. This,in particular, implies the indirect bandgap in mul-tilayers, which strongly affects their emission spec-tra [1–4]. Instead, more subtle effects of the hybrid-isation of electronic states around the direct bandgapin multilayers are relevant for absorption-type pro-cesses [17, 19, 20]. Recently, it has been shown thatthe absorption resonances due to the interlayer/hy-bridised excitons observed in the bi- and trilayersof S-TMDs can be widely tuned using the externalelectric field applied perpendicularly to the layers© 2023 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2053-1583/acbc8bhttps://crossmark.crossref.org/dialog/?doi=10.1088/2053-1583/acbc8b&domain=pdf&date_stamp=2023-2-28https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0003-1266-0201https://orcid.org/0000-0001-5798-0431https://orcid.org/0000-0002-2290-5738https://orcid.org/0000-0003-3712-371Xhttps://orcid.org/0000-0002-3282-9513https://orcid.org/0000-0002-6036-7096https://orcid.org/0000-0001-6861-3098https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-5591-4825https://orcid.org/0000-0002-5516-9415mailto:lucja.kipczak@fuw.edu.plmailto:aslobodeniuk@karlov.mff.cuni.czmailto:maciej.molas@fuw.edu.plhttps://doi.org/10.1088/2053-1583/acbc8b2D Mater. 10 (2023) 025014 Ł Kipczak et alplane [18, 21–26], which understanding may be alsoof importance in potential applications.The external out-of-planemagnetic field is widelyapplied to study the thin layers of S-TMDs due totheir significant Zeeman response [27, 28]. Con-sequently, the measured emission/absorption reson-ances split into two circularly polarised componentswith the magnitude of the Zeeman splitting denotedby the g-factor. Because of the g-factor values foundin S-TMD MLs, different types of transition can beidentified: bright (g-factor about−4), spin- (−8) andmomentum-forbidden dark (−12) [29–32]. In thecase of BL, the found g-factors are more scattered.As for intralayer excitons in S-TMD BLs, the foundg-factors are also about −4 [27, 28, 33], the corres-ponding g-factors for interlayer excitons are around8 [17]. Then, the g-factor not only gives us inform-ation concerning the Zeeman effect, but can also bea method to identify different excitonic complexes inthin layers of S-TMDs.In this work, we investigate experimentally andtheoretically excitonic properties in high-quality MLand BL of molybdenum diselenide (MoSe2) encap-sulated in hexagonal boron nitride (hBN). The low-temperature (T= 10 K) reflectance contrast (RC)spectra are measured in external magnetic fields upto 10 T, applied in the out-of-plane configuration.Resonances, related to both neutral A and B excitons,are identified in ML and BL. Moreover, the transitionassociated with the interlayer exciton (IL) was recog-nised in the RC spectra of BL. The excitation ladder ofA excitons in the ML and BL limits is modelled basedon the k · p approach using the modified RK poten-tial. Furthermore, the experimentally obtained Landég-factors of the excitonic resonances are explained byfirst principles calculations. Apart from the Introduc-tion and Summary sections, our paper is composed ofthree (2–4) main sections completed by the Methodssection and the supplemental material (SM). First,in section 2, we focus on the analysis of the mag-netic field evolutions of the RC spectra measured onthe MoSe2 ML and BL. The theoretical approach forexcitation spectra of excitons in the S-TMD ML andBL using the k · p method is presented in section 3.Section 4 is dedicated to calculations of the excitonicg-factors in the MoSe2 ML and BL.2. Experimental resultsFigures 1(a) and (b) present the RC spectra measuredon the ML and BL of MoSe2 encapsulated in hBN atthe selected values of applied out-of-plane magneticfields. The twomain resonances, labelled correspond-ingly 1sA and 1sB in the figure, are associated with theabsorption processes of the ground s states of the int-ralayer A and B excitons [3, 6]. The appearance of theA and B excitons is due to a relatively large spin-orbitsplitting in the MoSe2 VB [6]. For the ML, the firstexcited s state of the A exciton (2sA) is also observedwith a substantially lower intensity compared to themain 1sA and 1sB resonances. There are two less pro-nounced transitions apparent in the RC spectrameas-ured on the BL, denoted 2sA and IL, which we attrib-ute correspondingly to the first excited state of the Aexciton and to the interlayer exciton. The identific-ation of the 1sA, 1sB, and 2sA resonances, observedin the RC spectrum of the MoSe2 ML, is straightfor-ward and is in accordance withmany other studies onMoSe2MLs encapsulated in hBN [8, 9, 34]. Themeas-ured low-temperature RC spectrum of the MoSe2 BLis similar to that reported in [19].As can be appreciated from figure 1, all theobserved resonances split into two circularly polar-ised components under the applied out-of-planemagnetic field due to the Zeeman effect [27]. Theintralayer A- and B-related resonances are charac-terised by the same sign of the splitting (σ+ energyis lower than the σ− one), but with different mag-nitude. At the same time, the splitting of the IL trans-ition is opposite (σ− energy is lower than σ+ one). Inorder to investigate in detail the magnetic field evolu-tion of the observed resonances, we fitted them usingthe Fano-type function. As we have not performedthe analysis within the framework of the transfermatrix method combined with the Lorentz oscillatormodel [4], the extracted energy evolutions as a func-tion of magnetic field are biased, particularly, for theresonances with small intensity (2sA, IL). A detaileddescription of the A, B and IL excitons, and theiroptical selection rules in terms of the electronic bandstructure can be found in the SM.The field evolution for the σ± components of theinvestigated transitions is presented in figure 2. Uponapplication of an out-of-plane magnetic field (B), themagnetic-field dependences of the σ± energies (Eσ±)can be defined as Eσ±(B) = E0 ± 1/2gµBB, where E0is the transition energy at zero field, g denotes theg-factor of the considered resonance, µB is the Bohrmagneton. The fitted curves are presented in figure 2.The E0 energies and the g-factors for all the stud-ied transitions are summarised in table 1. To verifythe obtained values of the g-factors, we also calcu-late theoretically the corresponding g-factors usingthe density functional theory (DFT) method, seesection 4 for details. The experimentally determinedg-factors for the ground states of the A (B) excitons,i.e. 1sA (1sB), in the ML and BL of MoSe2 equalto −3.91± 0.10 (−4.27± 0.17) and −2.94± 0.22(−2.77± 0.34), respectively. These values agree verywell with those theoretically calculated (see table 1)and those previously reported for the intralayer A andB excitons in the MoSe2 ML and BL [9, 27, 28, 33].The g-factors found for the first excited states (2sA)of the A exciton in both ML and BL are−4.99± 0.34and −4.32± 0.49, respectively. It is interesting thatthe magnitudes of the 2sg-factors are significantly22D Mater. 10 (2023) 025014 Ł Kipczak et alFigure 1. Helicity-resolved RC spectra of (a) monolayer and (b) bilayer of MoSe2 encapsulated in hBN for selected values ofmagnetic field measured at T = 10 K. The red and blue curves correspond to σ+ and σ− polarizations of reflected light inmagnetic fields applied perpendicularly to the layers plane, respectively. The spectra are vertically shifted for clarity.Figure 2. Obtained excitonic energies of the σ±components of the transitions measured on (a) monolayerand (b) bilayer of MoSe2 encapsulated in hBN as a functionof the out-of-plane magnetic field. The red and blue pointscorrespond to σ+ and σ− polarizations, respectively. Thesolid black lines represent fits according to the equationdescribed in the text.larger (of about 30%− 50%) than those of the 1sAstates. In contrast, the theoretically calculated val-ues of the g-factors for the 1s and 2s states of the Aexcitons are larger by about a few percent. Both pos-sible changes (i.e. increase or decrease) of the corres-ponding g-factors of the s states are reported in the lit-erature [8, 9, 35–38]. In our opinion, the discrepancybetween the experimental and theoretical values ofthe 2sAg-factor requires amore sophisticated analysis,which goes beyond the scope of this work. The valueof g-factor for the IL transition of 8.62± 0.42, whichhas not been reported so far, is in very good agree-ment with the theoretically calculated values of 8.71Table 1. Experimentally determined values of the E0 energies andthe g-factors for all the studied transitions. The gcalc valuescorrespond to the theoretically calculated parameters using DFTmethod. Notably, the experimental values of the E0 energy and theg-factor are assumed to be obtained for the ILA exciton.E0 g gcalcmonolayer1sA 1.640± 0.001 −3.91± 0.10 −3.692sA 1.793± 0.005 −4.99± 0.34 −3.901sB 1.849± 0.003 −4.27± 0.17 −3.75bilayer1sA 1.622± 0.001 −2.94± 0.22 −3.002sA 1.741± 0.006 −4.32± 0.49 −3.191sB 1.852± 0.003 −2.77± 0.34 −3.16ILA/B 1.708± 0.006 +8.62± 0.42 +8.71/+ 8.91and 8.91 for the ILA and ILB excitons, respectively.As the ILA resonance is reported mainly in MoS2and MoSe2 BLs [17–20, 26, 39, 40], we ascribe theobserved IL resonance to the ILA. Further discussionon the interlayer excitons can be found in the SM.3. Excitonic ladder inML and BL:theoretical approachA common approach to account for the excitationspectra of excitons in S-TMD MLs refers to thenumerical solutions of the Schrödinger equation,in which the e-h attraction is approximated byRK potential [12, 13]. Although this numericalmethod gives very good results for Mo- and W-basedMLs [9, 35, 41], there is a lack of analogous calcula-tions of the excitation spectra of excitons inmultilayersystems (particularly, in a BL). In the following, the32D Mater. 10 (2023) 025014 Ł Kipczak et alcalculations of the excitonic ladders of the s states inboth the ML and the BL of the S-TMDs are presen-ted. In order to verify our theoretical results, we com-pare them with the experimentally found energy sep-aration between the 1s and 2s states,∆Eexp12 , obtainedusing the values shown in table 1. Our comprehensiveapproach to the theoretical analysis of the excitationspectra of excitons in S-TMDML and BL is describedin detail in the SM.First, we calculate a spectrum of the intralayerexcitons in the ML and BL with the help of the effect-ive two-body hydrogen-like problem. We derive sucha problem from the band Hamiltonians of the con-sidered 2D systemswithin the k · p approximation. Todo this, we first consider the basic electronic proper-ties of S-TMDML. TheML crystal is a direct bandgapsemiconductor. The extrema of the valence (VB) andconduction (CB) bands are located at theK± points ofthe Brillouin zone (BZ). Due to the strong spin-orbitinteraction, both bands are spin-split. The values ofthe splittings in the VB (∆v) and in the CB (∆c) arehundreds and tens of meV, respectively [42]. There-fore, the Bloch states at the K points can be presen-ted as a tensor product of the spin {| ↓⟩, | ↑⟩} and thespinless band states. The spinless VB and CB states atK± points aremade predominantly from dx2−y2 ± idxyand dz2 orbitals of transition metal atoms, respect-ively [42, 43]. Such a structure of Bloch states inopposite K points defines the optical selection rulesin the ML and is a consequence of the system’s time-reversal symmetry (TRS). The TRS also dictates thatBloch states with the same band index (c or v) butwith opposite spins in opposite valleys have equal dis-persion laws, i.e. the same band structure. Therefore,we restrict our consideration of the conduction andvalence bands to the K+ point for brevity. All conclu-sions for the K− point can be done by analogy.The A excitons at the K+ point of the ML areformed from an electron from the bottom spin-upCB and a hole from the top spin-up VB. The cor-responding spinless Bloch functions are |Ψc⟩ and|Ψv⟩. The two-band fully-diagonalized Hamiltonianof these bands, written in the corresponding basis{|Ψc⟩⊗ | ↑⟩, |Ψv⟩⊗ | ↑⟩}, can be presented in theform [44]HML =[Eg +ℏ2k22me00 −ℏ2k22mh]. (1)Here, Eg is the bandgap energy parameter, k=kxex + kyey is the in-plane momentum of the quasi-particles in theML, where ex,ey are unit vectors in thex and y directions, respectively. me,mh > 0 are cor-respondingly the electron and hole effectivemasses inthe ML. The Rydberg-type spectrum of e-h pairs forsuch a band structure can be found from the solutionof the corresponding eigenvalues problem [13, 45]{− ℏ22µ∇2∥ +VRK(ρ)}ψ(ρ) = Eψ(ρ), (2)where µ=memh/(me +mh) is the reduced excitonmass,∇∥ = ex∂x + ey∂y is the 2D nabla operator, ρ isthe in-plane distance between an electron and a holeof the exciton, E is the exciton energy, and VRK(ρ) isthe Rytova–Keldysh potential [12, 13]VRK(ρ) =−πe22r0[H0(ρεr0)−Y0(ρεr0)]. (3)Here H0(x) and Y0(x) correspond to the Struveand Bessel functions of the second kind, r0 = 2πχ2Drepresents the screening length, χ2D is the ML 2Dpolarizability [41, 46]. ε denotes the dielectric con-stant of the surroundingmedium (hBN in our study).Let us consider the band structure of the BL atthe K+ point. To do this, we first define the Blochstates of the VB and CB at the K+ point of the BL,by constructing them from theML Bloch states of thetop and bottom layers of the BL. Namely, we intro-duce the states |Ψ(m)n ⟩⊗ |s⟩, where m= 1,2 is a layerindex (for bottom and top layers, respectively), n=v, c is a band index (for VB and CB) and s=↑,↓ spe-cifies the spin degree of freedom. The bottom (first)layer states |Ψ(1)v ⟩ and |Ψ(1)c ⟩ aremade predominantlyfrom dx2−y2 + idxy and dz2 orbitals of transition metalatoms, respectively [42, 43]. They coincide with MLspinless states |Ψv⟩ and |Ψc⟩, considered above. Thetop (second) layer states |Ψ(2)v ⟩ and |Ψ(2)c ⟩ are madefrom dx2−y2 − idxy and dz2 orbitals and coincide withspinless states in the K− point of theML. In our study,we suppose the orthogonality of the basis states fromdifferent layers and bands ⟨Ψ(m)n |Ψ(m ′)n ′ ⟩= δnn ′δmm ′ .The symmetry analysis of the BL sys-tem [17, 28, 40] demonstrates that the electronexcitations of the CB of the different layers donot interact with each other in the leading order,i.e. the electron states of the BL are localised eitherin the bottom or in the top layer. On the con-trary, the VB of the different layers of the BL inter-act with each other forming the new VB with theBloch states delocalised in the out-of-plane direc-tion. These states can be found by diagonalizingthe VB part of the BL Hamiltonian, written onthe basis {|Ψ(1)v ⟩⊗ | ↑⟩, |Ψ(2)v ⟩⊗ | ↑⟩, |Ψ(2)v ⟩⊗ | ↓⟩,|Ψ(1)v ⟩⊗ | ↓⟩}HVBBL =−ℏ2k22mht 0 0t −∆v − ℏ2k22mh0 00 0 −ℏ2k22mht0 0 t −∆v − ℏ2k22mh,(4)where t is the interlayer hopping term. The spectrumof this Hamiltonian is doubly degenerated by spin (infull accordance with the TRS and inverse symmetryof the BL)E±VB =−ℏ2k22mh− ∆v2±√∆2v4+ t2. (5)42D Mater. 10 (2023) 025014 Ł Kipczak et alThe eigenstates that correspond to the upper-energy E+VB bands are|Φ+v↑⟩=[cosθ|Ψ(1)v ⟩+ sinθ|Ψ(2)v ⟩]⊗ | ↑⟩, (6)|Φ+v↓⟩=[sinθ|Ψ(1)v ⟩+ cosθ|Ψ(2)v ⟩]⊗ | ↓⟩, (7)where we introduced cos(2θ) = ∆v/√∆2v + 4t2. It isimportant to point out that the new Bloch statesdescribe the delocalised in the out-of-plane directionVB excitations. For example, the state |Φ+v↑⟩ describesthe VB excitation, which can be found with prob-abilities P(1) = cos2 θ and P(2) = sin2 θ in the first(bottom) and second (top) layers, respectively. Theeigenstates |Φ−v↑⟩ and |Φ−v↓⟩, which correspond to thelower-energy E−VB bands, can be derived from the firstones by replacing cosθ→− sinθ, sinθ→ cosθ.The optical transitions in the K+ point of the BL,which form the intralayerA-excitons in the BL, coupleeither {|Ψ(1)c ⟩⊗ | ↑⟩, |Φ+v↑⟩} or {|Ψ(2)c ⟩⊗ | ↓⟩, |Φ+v↓⟩}group of the bands. The transitions between the firstpair of the bands are active in σ+ polarised light,while the transitions between the second pair of thebands are active in σ− polarised light. The first andsecond groups of the bands are described by the sameHamiltonianHBL =[Eg +ℏ2k22me00 −ℏ2k22mh− ∆v2 +√∆2v4 + t2],(8)which looks similar to the two-band Hamiltonianof the ML, compare with equation (1). The effect-ive electron and hole masses of these bands coincidewith the correspondingmasses in theML (in the lead-ing order of the k · p approximation). Therefore, theintralayer A-excitons are characterised by the samereduced mass µ, as in the ML. On the other hand,the delocalisation of the VB Bloch state in the out-of-plane direction leads to the modification of theCoulomb interaction between such a hole excitationand an electron excitation, which remains localisedin one of the layers. The modified Coulomb interac-tion in the BL Vbil(ρ) is derived in the SM. In sum-mary, we conclude that the spectrum of the intralayerA-excitons can be derived from equation (2) by repla-cing VRK(ρ) with Vbil(ρ).3.1. MLWe derive the spectrum of excitons in theML by solv-ing equation (2) with Rytova–Keldysh potential (3).Following, we introduce the dimensionless paramet-ers ξ = ρε/r0 and ϵ= E/Ry∗, with Ry∗ = µe4/2ℏ2ε2and rewrite equation (2) for the case of s-type excitonsin the following form{b21ξddξ(ξddξ)+ bvRK(ξ)+ ϵ}ψ(ξ) = 0, (9)Figure 3. Energy spectrum of s excitonic state of the MoSe2ML and BL encapsulated in hBN calculated using the k · papproach. The ϵ⊥ parameter denotes the out-of-planedielectric constant used in calculations.where b= ℏ2ε2/µe2r0 and vRK(ξ) = π[H0(ξ)−Y0(ξ)]. The parameter Ry∗ defines the natural energyscale for the considered excitonic problem, see [8] fordetails. The dimensionless parameter b is the ratioof the reduced Bohr radius a∗0 = ℏ2ε/µe2 and thereduced screening length r∗0 = r0/ε in the system.We use ε= 4.5 [47], r0 = 51.7 Å, µ= 0.44m0 forour calculations, yielding b≈ 0.47, see details in theSM. The numerical solution of the equation for thisvalue of b gives the energies of the ground (1s) andthe first four excited (2s, 3s, 4s, and 5s) states equalto E1 =−214meV, E2 =−63meV, E3 =−30meV,E4 =−17meV, E5 =−11meV, respectively. Thecalculated energy positions of the excitonic statesin ML are presented in figure 3. It is difficult topredict the absolute energy of a given ns excitonicstate, En,ex = Eg + En, since the bandgap energy (Eg)is renormalised by the Coulomb interaction andalso depends on the dielectric constant ε. Calcu-lating such a bandgap shift requires an additionalmore resource-demanding numerical investigation.To verify our theoretical calculations, we determ-ine the energy distance between 1s and 2s emissionlines ∆E12 = E1,ex − E2,ex = (Eg + E1)− (Eg + E2).Therefore, we found |∆E12|= 151meV, which isnearly perfectly consistent with our experimentalvalue |∆Eexp12 |= 153± 5meV as well as the previouslyreported value of about 153 meV obtained from pho-toluminescence experiment [8].3.2. BLThe computation of the excitonic spectrum in the BLof S-TMDs is a muchmore sophisticated task. For theML case, the charges and wavefunctions of an elec-tron and a hole are confined within the ML plane.The situation with the electron and hole electronicexcitations in the BL is more complex. The hybrid-isation of the VB states leads to the charge redistri-bution of hole quasiparticles between layers in theBL [4, 17, 40]. Using the properties of the VB states in52D Mater. 10 (2023) 025014 Ł Kipczak et althe BL, we obtain the following values for the chargesof the hole excitation, which belong to the same (Qin)and opposite (Qopp) layers as an electron,Qin/opp =|e|2(1± ∆v√∆2v + 4t2)≈ 0.932|e|/0.068|e|,(10)where ‘+’ and ‘−’ signs correspond to the same(in) and opposite (opp) layers of the BL, respect-ively. Here we used the numbers ∆v = 182meV andt= 53meV [43]. Subsequently, the redistribution ofthe hole charge in the out of-plane-direction modi-fies the Coulomb potential between electron and holeexcitations in the BL. The derivation of the corres-ponding potentialVbil(ρ) as a function of the in-planedistance, ρ between an electron and hole excitation ispresented in the SM. The potentialVbil(ρ) depends onthe distance L between the layers in the BL, the screen-ing length r0, the dielectric constant of the mediumsurrounding the BL ε, and finally the out-of-planedielectric constant ϵ⊥ of the BL. The case ϵ⊥ = 1 cor-responds to the limit situationwhen the BL can not bepolarised by an electric field in an out-of-plane dir-ection. In the real situation the BL, however, may becharacterised by the out-of-plane dielectric constantϵ⊥, which is different from unity ϵ⊥ = ϵbil⊥ > 1. In thefollowing, we demonstrate that the first case can notexplain the experimental observables, and hence, itconfirms that the BL is polarised in an out-of-planedirection.Let us consider the case ϵ⊥ = 1. Then, theexcitonic spectrum of the s states can be calculatedusing the dimensionless eigenvalue equation (withthe same notations for ε and b from the previoussection){b21ξddξ(ξddξ)+ bvbil(ξ)+ ϵ}ψ(ξ) = 0, (11)with the dimensionless electrostatic potentialbetween an electron and a hole in the BLvbil(ξ) = 2εˆ ∞0dxJ0(xξ)0.932(1− δ)(e2xl[(1− δ)εx+ 1]− [εx(1− δ)+ δ])+(0.0681− δ)2e xle2xl[(1− δ)εx+ 1]2 − [δ+ εx(1− δ)]2. (12)Here, J0(x) is the Bessel function of the first kind,δ = (ε− 1)/(ε+ 1)≈ 0.64, and l= Lε/r0 ≈ 0.56. Inthe latter, we used L= 6.44Å for MoSe2 BL from HQGraphene. The vbil(ξ) potential is composed of twocomponents: intra- (the term with 0.932 multiplier)and interlayer (the term with 0.068 multiplier). Onecan see that the intralayer term is dominant at a largel distance between the layers, whereas the contribu-tion of the interlayer term decays exponentially withthis distance ∝ exp(−xl). Using numerical solutionof the eigenvalue problem with potential vbil(ξ), weobtain E1 =−235meV, E2 =−68 meV, E3 =−31meV, E4 =−18 meV, and E5 =−11 meV. The cal-culated energy positions of the excitonic states in theBL are presented in figure 3. The obtained BEs ofthe consecutive s excitons in the BL are slightly lar-ger compared to their ML counterparts. This can beexplained by the fact that the effective dielectric con-stant for the BL is smaller than that for the ML case.However, the calculated difference between the ener-gies of the 1s and 2s excitons, |∆E12|= 167meV, issignificantly larger as compared to the experimentalvalue, |∆Eexp12 |= 119± 6meV. Therefore, the modelwith an out-of-plane dielectric constant, ϵ⊥ = 1 failsin the BL.Consequently, we calculate the excitonic spec-trum in the BL assuming that ϵ⊥ = 7.7 [48], which isthe value that describes the static dielectric constantin the MoSe2 BL [48]. The resulting eigenvalueequation in dimensionless coordinate ξ = ρε√ϵ⊥/r0becomes{b2ϵ⊥1ξddξ(ξddξ)+ bṽbil(ξ)+ ϵ}ψ(ξ) = 0, (13)with the new electrostatic potential, ṽbil(ξ), whichis obtained from equation (12) by replacing δ→δ̃ = (ε−√ϵ⊥)/(ε+√ϵ⊥)≈ 0.24. The dimension-less energy parameter remains the same as in theprevious case. The eigenvalues of this new equationprovide the following spectrum of excitons E1 =−177meV, E2 =−53 meV, E3 =−26 meV, E4 =−17 meV, and E5 =−12 meV. The calculated energypositions of the excitonic states in the BL withϵ⊥ = 7.7 are presented in figure 3. Consequently,the new energy separation |∆E12|= 124meV agreesnicely with the experimental value (|∆Eexp12 |= 119±6meV).To conclude, we demonstrate that the excita-tion spectrum of excitons in the MoSe2 ML can beproperly reproduced using the RK potential withthe approach of infinitely thin ML. However, theRytova–Keldysh potential can not be applied todescribe the spectrum of the intralayer excitons in theBL as a result of the more complex structure of the BLcrystal. We derive the electrostatic potential in the BLby taking into account the geometry and the dielec-tric constant ϵ⊥ of the BL perpendicular to the layers’planes. The spectrum of excitons, based on the new62D Mater. 10 (2023) 025014 Ł Kipczak et alpotential in the MoSe2 BL is in very good agreementwith the experimental data. Our results indicate thatthe study of excitons in thin layers of S-TMDs, beyondtheML limit, is muchmore complicated and requirestaking into account a realistic thickness of the BL andits dielectric response.4. g-factors of exciton ns statesThe g-factors of excitonic ns states, gXns, can be calcu-lated using the dispersions of the band-to-band trans-itions, gX(k), in the vicinity of the K+ point, alongwith the exciton wave functions, |ψXns(k)| (obtainedfrom our aforementioned k · p calculations), follow-ing [36]gXns =ˆk|ψXns(k)|2gX(k)dk (14)withgX(k) =±2(gc(+1)(k)− gv(−1)(k)), (15)where the sign is defined by the optical selectionrules at K± points. gc(+1)(k) and gv(−1)(k) are the g-factors of the spin-split subbands (Bloch states), |mk⟩,involved in the excitonic transition, evaluated asgm(k) = Lzm(k)+ Szm(k). (16)The z component of the orbital angularmomentum of a Bloch state is calculated from thebands–summation formula [49]Lzm(k) =1im0N∑l=1,l ̸=mpxml(k)pylm(k)− p yml(k)pxlm(k)εm(k)− εl(k),(17)where m0 is the free electron mass, px,yml (k) are com-ponents of themomentum operatormatrix elements,εm(k) are the band energies, and the summationruns over all N states in the basis set. The elementspx,yml (k) were obtained from density perturbation the-ory calculations [50]. In order to converge Lzm(k) upto 0.1, 480 bands per formula unit were taken intoaccount. The spin angular momentum Szm(k) =±1for the considered bands. We would like to emphas-ize that the calculated values of the band g-factors forspin-orbit split subbands at the K± points are dis-cussed in the SM.The calculated exciton wave functions squared(see the SM for details) and g-factor dispersions in k-space around K+ point, that enter equation (14), arepresented in figure 4. The widths of the wave func-tions of the A andB excitons are similar inML andBL,with a greater spread of |ψB1s(k)|2 than |ψA1s(k)|2, dueto a larger effective mass of the B exciton. The oppos-ite is observed for the interlayer excitons. The gX(k)dependence has a positive curvature in both the MLFigure 4. Normalised squared moduli of exciton wavefunctions calculated from k · pmodel for (a) ML and(b) BL MoSe2. Dispersions of exciton g-factors around K+point from DFT calculations for (c) ML and (d) BL.and BL structures, with a smaller |gX(K)| in BL. Thisis caused by the reduction of the bands g-factors in BLversus ML (see the SM). As a result, the magnitudesof gXns in BL are reduced with respect to ML. Further-more, a stronger localisation of the 2sA state aroundthe K point leads to the increase of their g-factormag-nitudes, in agreement with previous theoretical andexperimental findings [36, 38]. Due to the differentsigns of valence and conduction Bloch states g-factorsinvolved in ILA and ILB excitons, their g-factors arepositive and exhibit a negative curvature.The resulting trends of the calculated g-factors arein good agreement with the experimental values, aspresented in table 1. Particularly, the experimentallyobserved large increase of the g-factors of 2sA excitonsin ML and BL can not be explained either by theunderestimation of the band gap by DFT [49], or bythe in-plane mechanical strain that might be presentin the samples [51]. Further theoretical investiga-tions are required, which are beyond the scope of thisstudy.5. SummaryThe excitonic properties in high-quality ML and BLof molybdenum diselenide (MoSe2) encapsulated inhBN flakes were investigated both theoretically andexperimentally. We determined the g-factors of theintralayer A and B excitons and of the interlayerexcitons in MoSe2 ML and BL using the RC experi-ment performed in out-of-plane magnetic fields upto 10 T and first-principle calculations. The exper-imental ladder of excitonic s states in the ML wasreproduced using the k · p model with the Rytova–Keldysh potential. Furthermore, we demonstratedthat analogous calculations for the BL require takinginto account the out-of-plane dielectric response ofthe MoSe2 BL, which is neglected for the ML. Finally,we have explained the values and signs of all observed72D Mater. 10 (2023) 025014 Ł Kipczak et alg-factors using a combined k · p and DFT approach.Our results manifest that the excitonic physics in S-TMD BLs is more complex than that in the ML casebecause of the presence of VB hybridization and thenon-infinite size of the BL.6. Methods6.1. Sample and experimental setupThe investigated MoSe2 thin layers and hBN flakeswere fabricated by two-stage PDMS-based mechan-ical exfoliation of the bulk crystal. Initially, the hBNthin flakes were exfoliated onto a 90 nm SiO2/Si sub-strate and annealed at 200◦C. That non-deterministicapproach provides the best quality of the substratesurface. Subsequent layers were transferred determ-inistically using a microscopic system equipped witha x-y-z motorised positioners. The assembled struc-tures were annealed at 160◦C for 1.5 hour in orderto ensure the best layer-to-layer and layer-to-substrateadhesion and to eliminate a substantial portion of airpockets apparent at the interfaces between the con-stituent layers.Low-temperature micro-magneto-experimentsof RC were performed in Faraday geometry, i.e. mag-netic field oriented perpendicularly to the layersplane. Measurements (spatial resolution∼3µm)were carried out with the aid of a superconductingcoil in magnetic fields up to 10 T using an opticalfibre arrangement. The sample was placed on topof a x-y-z piezo-stage kept at T = 10 K and wasilluminated using a 100 W tungsten halogen lamp.The reflectance signal was dispersed with a 0.75 mfocal length monochromator and detected with aliquid-nitrogen-cooled Si-CCD. The combination ofa quarter-wave plate, a linear polariser, and a Wol-lastom prism was used to analyse the circular polar-isation of signals (the σ±-polarized light was meas-ured simultaneously). We define the RC spectrumas RC(E) = [R(E)−R0(E)]/[R(E)+R0(E)]× 100%,where R(E) and R0(E) are the reflectance of thesample and of the same structure without the MLor BL of MoSe2, respectively.6.2. DFT calculationsFirst-principles calculations within the DFTwere car-ried out in the Vienna ab-initio simulation pack-age [52]. The ionic potentials were described usingthe projector augmented wave technique [53]. Weemployed the generalised gradient approximation ofthe exchange correlation-functional within Perdew–Burke–Ernzerhof parametrization [54]. A cut-offenergy for the plane-wave basis and a Monkhorst-Pack k-grid for the BZ sampling were set to 500 eVand 12× 12× 1, respectively. The geometrical struc-tures ofML andBLwere defined using the parametersfrom [46] with a vacuum region of 20 Å in order toavoid spurious interactions between the periodicallyrepeated layers. Spin–orbit coupling was taken intoaccount during the calculations.Data availability statementThe data cannot be made publicly available uponpublication because they are not available in a formatthat is sufficiently accessible or reusable by otherresearchers. The data that support the findings of thisstudy are available upon reasonable request from theauthors.AcknowledgmentWe thank Paulo E Faria Junior for fruitful discussions.The work has been supported by the National ScienceCentre, Poland (Grant No. 2017/27/B/ST3/00205and 2018/31/B/ST3/02111). A O S acknowledgesthe support by Czech Science Foundation (projectGA23-06369S). T W acknowledges support fromthe National Science Centre, Poland (Grant No.2021/41/N/ST3/04516). K W and T T acknow-ledge support from JSPS KAKENHI (Grant Numbers19H05790, 20H00354 and 21H05233). DFT calcula-tions were performed with the support of Center forInformation Services and High Performance Com-puting (ZIH) at TU Dresden and in part by PLGridInfrastructure.Author ContributionsŁ K, M B, N Z, K O-P, A B, and M R M performedthe experiments. A O S performed theoretical calcu-lations based on the k · p approximation. TW carriedout DFT calculations. M G fabricated the sample KW and T T grew the hBN crystals. M R M supervisedthe project. Ł K, A O S, T W, and M R M wrote themanuscript with inputs from the all co-authors.Conflict of interestThere are no conflicts to declare.ORCID iDsŁucja Kipczak https://orcid.org/0000-0003-1266-0201Artur O Slobodeniuk https://orcid.org/0000-0001-5798-0431Tomasz Woźniak https://orcid.org/0000-0002-2290-5738Mukul Bhatnagar https://orcid.org/0000-0003-3712-371XNatalia Zawadzka https://orcid.org/0000-0002-3282-9513Katarzyna Olkowska-Puckohttps://orcid.org/0000-0002-6036-7096Magdalena Grzeszczyk https://orcid.org/0000-0001-6861-30988https://orcid.org/0000-0003-1266-0201https://orcid.org/0000-0003-1266-0201https://orcid.org/0000-0003-1266-0201https://orcid.org/0000-0001-5798-0431https://orcid.org/0000-0001-5798-0431https://orcid.org/0000-0001-5798-0431https://orcid.org/0000-0002-2290-5738https://orcid.org/0000-0002-2290-5738https://orcid.org/0000-0002-2290-5738https://orcid.org/0000-0003-3712-371Xhttps://orcid.org/0000-0003-3712-371Xhttps://orcid.org/0000-0003-3712-371Xhttps://orcid.org/0000-0002-3282-9513https://orcid.org/0000-0002-3282-9513https://orcid.org/0000-0002-3282-9513https://orcid.org/0000-0002-6036-7096https://orcid.org/0000-0002-6036-7096https://orcid.org/0000-0001-6861-3098https://orcid.org/0000-0001-6861-3098https://orcid.org/0000-0001-6861-30982D Mater. 10 (2023) 025014 Ł Kipczak et alKenji Watanabe https://orcid.org/0000-0003-3701-8119Takashi Taniguchi https://orcid.org/0000-0002-1467-3105Adam Babiński https://orcid.org/0000-0002-5591-4825Maciej R Molas https://orcid.org/0000-0002-5516-9415References[1] Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Phys.Rev. Lett. 105 136805[2] Arora A, Koperski M, Nogajewski K, Marcus J, Faugeras Cand Potemski M 2015 Nanoscale 7 10421[3] Arora A, Nogajewski K, Molas M, Koperski M andPotemski M 2015 Nanoscale 7 20769[4] Molas M R, Nogajewski K, Slobodeniuk A O, Binder J,Bartos M and Potemski M 2017 Nanoscale 9 13128[5] Chernikov A, Berkelbach T C, Hill H M, Rigosi A, Li Y,Aslan O B, Reichman D R, Hybertsen M S and Heinz T F2014 Phys. Rev. Lett. 113 076802[6] Koperski M, Molas M R, Arora A, Nogajewski K,Slobodeniuk A O, Faugeras C and Potemski M 2017Nanophotonics 6 1289[7] Wang G, Chernikov A, Glazov MM, Heinz T F, Marie X,Amand T and Urbaszek B 2018 Rev. Mod. Phys. 90 021001[8] Molas M R, Slobodeniuk A O, Nogajewski K, Bartos M,Bala L, Babiński A, Watanabe K, Taniguchi T, Faugeras C andPotemski M 2019 Phys. Rev. Lett. 123 136801[9] Goryca M et al 2019 Nat. Commun. 10 4172[10] MacDonald A H and Ritchie D S 1986 Phys. Rev. B33 8336[11] Koteles E S and Chi J Y 1988 Phys. Rev. B 37 6332[12] Rytova N S 1967Mosc. Univ. Phys. Bull. 3 30[13] Keldysh L V 1979 JETP Lett. 29 716[14] Fang S, Kuate Defo R, Shirodkar S N, Lieu S, Tritsaris G Aand Kaxiras E 2015 Phys. Rev. B 92 205108[15] Sun Y, Wang D and Shuai Z 2016 J. Phys. Chem. C120 21866[16] Ruiz-Tijerina D A, Danovich M, Yelgel C, Zólyomi V andFal’ko V I 2018 Phys. Rev. B 98 035411[17] Slobodeniuk A O et al 2019 2D Mater. 6 025026[18] Leisgang N et al 2020 Nat. Nanotechnol. 15 901[19] Horng J, Stroucken T, Zhang L, Paik E Y, Deng H andKoch S W 2018 Phys. Rev. B 97 241404[20] Paradisanos I et al 2020 Nat. Commun. 11 2391[21] Das S, Dandu M, Gupta G, Murali K, Abraham N, Kallatt S,Watanabe K, Taniguchi T and Majumdar K 2020 ACSPhotonics 7 3386[22] Lorchat E, Selig M, Katsch F, Yumigeta K, Tongay S, Knorr A,Schneider C and Höfling S 2021 Phys. Rev. Lett. 126 037401[23] Zhao Y et al 2022 Phys. Rev. B 105 L041411[24] Altaiary MM, Liu E, Liang C-T, Hsiao F-C, van Baren J,Taniguchi T, Watanabe K, Gabor N M, Chang Y-C andLui C H 2022 Nano Lett. 22 1829[25] Hagel J, Brem S and Malic E 2022 2D Mater. 10 014013[26] Feng S, Campbell A, Brotons-Gisbert M, Baek H,Watanabe K, Taniguchi T, Urbaszek B, Gerber I C andGerardot B D 2022 arxiv:2212.14338[27] Koperski M, Molas M R, Arora A, Nogajewski K, Bartos M,Wyzula J, Vaclavkova D, Kossacki P and Potemski M 2019 2DMater. 6 015001[28] Arora A, Koperski M, Slobodeniuk A, Nogajewski K,Schmidt R, Schneider R, Molas M R, de Vasconcellos S M,Bratschitsch R and Potemski M 2018 2D Mater. 6 015010[29] Liu E, van Baren J, Liang C-T, Taniguchi T, Watanabe K,Gabor N M, Chang Y-C and Lui C H 2020 Phys. Rev. Lett.124 196802[30] He M et al 2020 Nat. Commun. 11 618[31] Zinkiewicz M et al 2021 Nano Lett. 21 2519[32] Pucko K O et al 2022 2D Mater. 10 015018[33] Li Y et al 2014 Phys. Rev. Lett. 113 266804[34] Xiao K, Yan T, Liu Q, Yang S, Kan C, Duan R, Liu Z andCui X 2021 J. Phys. Chem. Lett. 12 2555[35] Stier A V, Wilson N P, Clark G, Xu X and Crooker S A 2016Nano Lett. 16 7054[36] Chen S-Y et al 2019 Nano Lett. 19 2464[37] Delhomme A et al 2019 Appl. Phys. Lett. 114 232104[38] Raiber S, Faria Junior P E, Falter D, Feldl S, Marzena P,Watanabe K, Taniguchi T, Fabian J and Schüller C 2022 Nat.Commun. 13 4997[39] Gerber I C et al 2019 Phys. Rev. B 99 035443[40] Grzeszczyk M et al 2021 Sci. Rep. 11 17037[41] Cudazzo P, Tokatly I V and Rubio A 2011 Phys. Rev. B84 085406[42] Kormányos A, Burkard G, Gmitra M, Fabian V, Zólyomi J,Drummond V and Fal’ko N D 2015 2D Mater. 2 022001[43] Gong Z, Liu G-B, Yu H, Xiao D, Cui X, Xu X and Yao W2013 Nat. Commun. 4 2053[44] Slobodeniuk A O, Koutenský P, Bartǒs M, Trojánek F, Maĺy P,Novotný T and Kozák M 2022 Phys. Rev. B 106 235304[45] Wannier G H 1937 Phys. Rev. 52 191[46] Berkelbach T C, Hybertsen M S and Reichman D R 2013Phys. Rev. B 88 045318[47] Geick R, Perry C H and Rupprecht G 1966 Phys. Rev. 146 543[48] Laturia, A, Van de Put M L and Vandenberghe W G 2018 npj2D Mater. Appl. 2 6[49] Woźniak T, Faria Junior P E, Seifert G, Chaves A andKunstmann J 2020 Phys. Rev. B 101 235408[50] Gajdǒs M, Hummer K, Kresse G, Furthmüller J andBechstedt F 2006 Phys. Rev. B 73 045112[51] Junior P E F, Zollner K, Woźniak T, Kurpas M, Gmitra Mand Fabian J 2022 New J. Phys. 24 083004[52] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169[53] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758[54] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett.77 38659https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-5591-4825https://orcid.org/0000-0002-5591-4825https://orcid.org/0000-0002-5591-4825https://orcid.org/0000-0002-5516-9415https://orcid.org/0000-0002-5516-9415https://orcid.org/0000-0002-5516-9415https://doi.org/10.1103/PhysRevLett.105.136805https://doi.org/10.1103/PhysRevLett.105.136805https://doi.org/10.1039/C5NR01536Ghttps://doi.org/10.1039/C5NR01536Ghttps://doi.org/10.1039/C5NR06782Khttps://doi.org/10.1039/C5NR06782Khttps://doi.org/10.1039/C7NR04672Chttps://doi.org/10.1039/C7NR04672Chttps://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1515/nanoph-2016-0165https://doi.org/10.1515/nanoph-2016-0165https://doi.org/10.1103/RevModPhys.90.021001https://doi.org/10.1103/RevModPhys.90.021001https://doi.org/10.1103/PhysRevLett.123.136801https://doi.org/10.1103/PhysRevLett.123.136801https://doi.org/10.1038/s41467-019-12180-yhttps://doi.org/10.1038/s41467-019-12180-yhttps://doi.org/10.1103/PhysRevB.33.8336https://doi.org/10.1103/PhysRevB.33.8336https://doi.org/10.1103/PhysRevB.37.6332https://doi.org/10.1103/PhysRevB.37.6332https://doi.org/10.48550/arXiv.1806.00976https://doi.org/10.48550/arXiv.1806.00976https://doi.org/10.1103/PhysRevB.92.205108https://doi.org/10.1103/PhysRevB.92.205108https://doi.org/10.1021/acs.jpcc.6b08748https://doi.org/10.1021/acs.jpcc.6b08748https://doi.org/10.1103/PhysRevB.98.035411https://doi.org/10.1103/PhysRevB.98.035411https://doi.org/10.1088/2053-1583/ab0776https://doi.org/10.1088/2053-1583/ab0776https://doi.org/10.1038/s41565-020-0750-1https://doi.org/10.1038/s41565-020-0750-1https://doi.org/10.1103/PhysRevB.97.241404https://doi.org/10.1103/PhysRevB.97.241404https://doi.org/10.1038/s41467-020-16023-zhttps://doi.org/10.1038/s41467-020-16023-zhttps://doi.org/10.1021/acsphotonics.0c01159https://doi.org/10.1021/acsphotonics.0c01159https://doi.org/10.1103/PhysRevLett.126.037401https://doi.org/10.1103/PhysRevLett.126.037401https://doi.org/10.1103/PhysRevB.105.L041411https://doi.org/10.1103/PhysRevB.105.L041411https://doi.org/10.1021/acs.nanolett.1c01590https://doi.org/10.1021/acs.nanolett.1c01590https://doi.org/10.1088/2053-1583/aca916https://doi.org/10.1088/2053-1583/aca916https://doi.org/10.1088/2053-1583/aae14bhttps://doi.org/10.1088/2053-1583/aae14bhttps://doi.org/10.1088/2053-1583/aae7e5https://doi.org/10.1088/2053-1583/aae7e5https://doi.org/10.1103/PhysRevLett.124.196802https://doi.org/10.1103/PhysRevLett.124.196802https://doi.org/10.1038/s41467-020-14472-0https://doi.org/10.1038/s41467-020-14472-0https://doi.org/10.1021/acs.nanolett.0c05021https://doi.org/10.1021/acs.nanolett.0c05021https://doi.org/10.1088/2053-1583/aca915https://doi.org/10.1088/2053-1583/aca915https://doi.org/10.1103/PhysRevLett.113.266804https://doi.org/10.1103/PhysRevLett.113.266804https://doi.org/10.1021/acs.jpclett.1c00320https://doi.org/10.1021/acs.jpclett.1c00320https://doi.org/10.1021/acs.nanolett.6b03276https://doi.org/10.1021/acs.nanolett.6b03276https://doi.org/10.1021/acs.nanolett.9b00029https://doi.org/10.1021/acs.nanolett.9b00029https://doi.org/10.1063/1.5095573https://doi.org/10.1063/1.5095573https://doi.org/10.1038/s41467-022-32534-3https://doi.org/10.1038/s41467-022-32534-3https://doi.org/10.1103/PhysRevB.99.035443https://doi.org/10.1103/PhysRevB.99.035443https://doi.org/10.1038/s41598-021-95700-5https://doi.org/10.1038/s41598-021-95700-5https://doi.org/10.1103/PhysRevB.84.085406https://doi.org/10.1103/PhysRevB.84.085406https://doi.org/10.1088/2053-1583/2/2/022001https://doi.org/10.1088/2053-1583/2/2/022001https://doi.org/10.1038/ncomms3053https://doi.org/10.1038/ncomms3053https://doi.org/10.1103/PhysRevB.106.235304https://doi.org/10.1103/PhysRevB.106.235304https://doi.org/10.1103/PhysRev.52.191https://doi.org/10.1103/PhysRev.52.191https://doi.org/10.1103/PhysRevB.88.045318https://doi.org/10.1103/PhysRevB.88.045318https://doi.org/10.1103/PhysRev.146.543https://doi.org/10.1103/PhysRev.146.543https://doi.org/10.1038/s41699-018-0050-xhttps://doi.org/10.1038/s41699-018-0050-xhttps://doi.org/10.1103/PhysRevB.101.235408https://doi.org/10.1103/PhysRevB.101.235408https://doi.org/10.1103/PhysRevB.73.045112https://doi.org/10.1103/PhysRevB.73.045112https://doi.org/10.1088/1367-2630/ac7e21https://doi.org/10.1088/1367-2630/ac7e21https://doi.org/10.1103/PhysRevB.54.11169https://doi.org/10.1103/PhysRevB.54.11169https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/PhysRevLett.77.3865 Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe2 1. Introduction 2. Experimental results 3. Excitonic ladder in ML and BL: theoretical approach 3.1. ML 3.2. BL 4. g-factors of exciton ns states 5. Summary 6. Methods 6.1. Sample and experimental setup 6.2. DFT calculations References