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Simon Raiber, Paulo E. Faria Junior, Dennis Falter, Simon Feldl, Petter Marzena, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jaroslav Fabian, Christian Schüller

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[Ultrafast pseudospin quantum beats in multilayer WSe2 and MoSe2](https://mdr.nims.go.jp/datasets/69a69206-79f8-4737-989a-c49228f84c2c)

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Ultrafast pseudospin quantum beats in multilayer WSe2 and MoSe2nature communicationsArticle https://doi.org/10.1038/s41467-022-32534-3Ultrafast pseudospin quantum beats inmultilayer WSe2 and MoSe2SimonRaiber1, PauloE. Faria Junior2, Dennis Falter1, SimonFeldl1, PetterMarzena1,Kenji Watanabe 3, Takashi Taniguchi 4, Jaroslav Fabian 2 &Christian Schüller 1Layered van-der-Waals materials with hexagonal symmetry offer an extradegree of freedom to their electrons, the so-called valley index or valleypseudospin, which behaves conceptually like the electron spin. Here, wepresent investigations of excitonic transitions in mono- and multilayer WSe2and MoSe2 materials by time-resolved Faraday ellipticity (TRFE) with in-planemagnetic fields, B∥, of up to 9 T. In monolayer samples, the measured TRFEtime traces are almost independent of B∥, which confirms a close to zero in-plane exciton g factor g∥, consistent with first-principles calculations. In con-trast, we observe pronounced temporal oscillations in multilayer samples forB∥ > 0. Our first-principles calculations confirm the presence of a non-zero g∥for the multilayer samples. We propose that the oscillatory TRFE signal in themultilayer samples is caused by pseudospin quantum beats of excitons, whichis a manifestation of spin- and pseudospin layer locking in the multilayersamples.The semiconducting transition-metal dichalcogenides (TMDCs) holdgreat promise for optoelectronic applications, since they form directbandgap semiconductors in the monolayer limit. Their optical prop-erties are governed by excitons, i.e., Coulomb-bound electron-holepairs1,2, even at room temperature, due to extraordinarily large excitonbinding energies. For high-quality encapsulated MoSe2 monolayers,superior optical quality with exciton linewidths approaching the life-time limit has been demonstrated3,4. Furthermore, anomalous, non-classical diffusion behavior of excitons has been detected5,6 andcalculated7 for TMDCmonolayers. Starting from bilayers, the bandgapbecomes indirect. Nevertheless, going from a single layer to multi-layers, the direct interband transitions at the K points of the Brillouinzone still dominate the optical absorption8. Another property ofmonolayer material is the strong spin-orbit coupling in combinationwith inversion asymmetry, which lead to large valley-selective spin-orbit splittings of the band edges, culminating in the so-called spin-valley locking. This peculiarity is appreciated by the introduction of apseudospin index, which conceptually behaves like the electron spin,and is connected to theoccupationof the twonon-equivalent K+ andK−valleys of the first Brillouin zone. Interestingly, the spin-valley lockingof a single layer transforms into a spin- or pseudospin-layer locking formultilayers9. For TMDC bilayers it has even been suggested that thespin-layer locking can be exploited for the design of spin quantumgates10.Interlayer excitons (IX),where electron andhole reside in adjacentlayers, werefirst detected in heterobilayers11. There, the characteristicsof IX depend crucially on the material combination12–14. Recently, evenvalley-polarized currents of IX in heterobilayers have beendemonstrated15. While in heterobilayers the oscillator strength of IX isweak, the situation can be different for homobilayers or multilayers16.In MoS2 bilayers, strong absorption by IX up to room temperature wasreported17–22. In MoSe2, the situation is similar to MoS2, though theoscillator strength of the IX is smaller23. Nevertheless, IX have beenreported in H-stacked MoTe224 and MoSe223,25 multilayers. In contrastto Mo-based multilayers, the momentum-space direct IX in W-basedmaterials has so far not been observed. It should be noted that forReceived: 4 May 2022Accepted: 4 August 2022Check for updates1Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany. 2Institut für Theoretische Physik, UniversitätRegensburg, D-93040Regensburg,Germany. 3ResearchCenter for FunctionalMaterials, National Institute forMaterials Science, Tsukuba, Ibaraki 305-0044,Japan. 4International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan.e-mail: christian.schueller@ur.deNature Communications |         (2022) 13:4997 11234567890():,;1234567890():,;http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-3009-4525http://orcid.org/0000-0002-3009-4525http://orcid.org/0000-0002-3009-4525http://orcid.org/0000-0002-3009-4525http://orcid.org/0000-0002-3009-4525http://orcid.org/0000-0002-6043-5263http://orcid.org/0000-0002-6043-5263http://orcid.org/0000-0002-6043-5263http://orcid.org/0000-0002-6043-5263http://orcid.org/0000-0002-6043-5263http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32534-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32534-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32534-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32534-3&domain=pdfmailto:christian.schueller@ur.deWSe2 homobilayers, IX due to momentum-indirect transitions belowthe optical bandgap were reported26,27.Whilemonolayer TMDCshavebeenquite intensely investigated inout-of-plane magnetic fields, investigations on multilayer samples arequite rare. The out-of-plane g factor, g⊥, of the intralayer A excitons isin MoSe2 and WSe2 multilayers smaller in magnitude than in singlelayers25,28. So far, there are, however, no experimental investigations onthe in-plane g factor, g∥, in TMDC multilayers available. In-plane mag-netic fields, B∥, have been applied to TMDC monolayers for thebrightening of dark excitonic states viamixing of the spin levels by thein-plane field29–32. In this work, we present time-resolved Faradayellipticity (TRFE) experiments on WSe2 and MoSe2 mono- and multi-layers in in-planemagnetic fields.Whilewe do not observe a significantinfluence of in-plane fields of up to 9 T in experiments on monolayers,pronounced temporal oscillations are observed in theTRFE time tracesof multilayers for B∥ > 0. Remarkably, the derived in-plane exciton gfactors, ∣g∥∣, are close to reported ∣g⊥∣ values of the same materials25.Results and discussionSample characterizationWe start the discussion with reflectance-contrast (RC) experiments ofthe investigated samples, in order to characterize the excitonic tran-sitions in thematerials. Simplified schematic drawings of thefirst twoAexcitons, A1s and A2s, in a multilayer sample are plotted in Fig. 1e.Figure 1a shows an overview of RC spectra of the four samples,investigated in the main body of the manuscript. Excitonic transitionsare marked by small vertical arrows as derived from fitting the RCspectra with a transfer-matrix model, assuming complex Lorentzoscillators for the excitonic transitions (see Supplementary Informa-tion). The schematic drawings in the inset of Fig. 1a depict the samples,which are MoSe2 and WSe2 mono- and multilayers (for more details,see the methods section). Themonolayer samples are encapsulated inhBN to protect them from environmental influences and to provide ahomogeneous dielectric environment. The WSe2 multilayer constistsof about 14 layers, the MoSe2 multilayer is much thicker, countingabout 84 layers, as determined by atomic-force microscopy. In bothmonolayer samples, the intralayer A1s excitons show up as distinct andsharp features in the RC spectra in Fig. 1a. In theMoSe2monolayer alsothe B1s exciton can be detected, while for the WSe2 monolayer it isoutside the displayed energy range. For clarity, the transitions at the Kpoints of the first Brillouin zone, which lead to the excitonic A and Bresonances are sketched for both materials in Fig. 1b, c (see layer 1,only, for the monolayer case). Interlayer transitions are omitted in theschematic pictures, since they play no role in our experiments. Goingfrom themonolayer tomultilayers, the intralayer excitonic resonancesshow a redshift, and the energetic separation between A1s and A2sdecreases because of the stronger dielectric screening24,25. In agree-ment with published results25, we observe in Fig. 1a in the WSe2 mul-tilayer two features, which can be attributed to the A1s and A2sintralayer excitons. This assignment is supported by our excitoniccalculations (see methods section): From the derived effective massesfor electron and hole we calculate an energetic separation of the A1sFig. 1 | Reflectance-contrast and TRFE experiments, intralayer transitions, andpseudospin-layer locking. a White-light reflectance-contrast experiments of theinvestigated samples: MoSe2 and WSe2 monolayers, encapsulated in hBN, andmultilayer samples of both materials. The corresponding zero lines are given asdashed lines of the same color. All samples are prepared on transparent sapphiresubstrates. The substrate temperature in all RCmeasurements was T ~ 20 K, exceptfor the MoSe2 multilayer, where it was < 10 K, as derived from the intensity ratio ofruby lines from the substrate (see methods section). Excitonic transitions, asderived from a transfer-matrix-model fit, are indicated by small vertical arrows. Wenote that the TRFE experiments are all performed at T ~ 5 K. b Schematic picture ofmomentum- and spin-allowed transitions in anH-typeWSe2 bilayer. c Sameasbbutfor anMoSe2 bilayer. d Schematic picture of the layer Brillouin zones in a four-layerstructure. Due to the 180∘ rotation between neighboring layers in an H-typestructure, K+ and K− valleys are alternating. An interlayer pseudospin τ = +1 is con-nected to the K+ valleys of the individual layers (marked by black circles), whileτ = −1 corresponds to the K− valleys (indicated by red circles). e Sketch of intralayerexcitons A1s and A2s. f TRFE traces of the encapsulated WSe2 monolayer for σ+(green solid line) and σ− (orange solid line) pump helicities, measured in resonancewith the A1s exciton. The dashed lines are biexponential fits to the data.Article https://doi.org/10.1038/s41467-022-32534-3Nature Communications |         (2022) 13:4997 2and A2s intralayer excitons of ~20.8meV, which is very close to theexperimental value of ~19.2meV. In agreement with the reports in ref.25, we also do not find a feature, related to the IX inWSe2multilayer inour RC experiments in Fig. 1a. Also for theMoSe2multilayer, we do notobserve a spectral feature, related to the IX. Similar to ref. 33, we findspectral features related to the A1s and A2s intralayer excitons in theMoSe2 multilayer. Again, this assignment in Fig. 1a is corroborated byour computed energy separation of A1s and A2s excitons of ~31.8meV,which is close to the experimental value of ~28.6meV.Figure 1 d is a sketch of the individual first Brillouin zones of anH-type four-layer structure. In H-type structure, subsequent layers arerotated by 180∘. Therefore, in momentum space, K+ and K− valleys ofthe individual layers are alternating, which is called spin-layerlocking9,10. A pseudospin quantum number τ = +1( − 1) can be attrib-uted to the K+ (K−) valley, leading to a pseudospin-layer locking.Figure 1 f shows typical TRFE time traces, recorded on the WSe2monolayer at zero magnetic field under resonant excitation of the A1sexciton. All experiments presented in this manuscript are in the exci-tonic regime, i.e., the exciton densities are below theMott density (seemethods section). The light green line shows a trace with σ+-polarizedpump pulses, which create a K+ valley polarization at time Δt = 0. Theorange line is an analogousmeasurement but with σ− pumppulses, i.e.,a K− valley polarization is initialized. The dashed lines represent biex-ponentialfits to thedata. Bothmeasurement curves canbenicelyfittedby a biexponential decaywith a short time constant of τr ~ 0.15 ps and alonger decay time of τv ~ 7.0 ps. There are a couple of different pro-cesses, which can contribute to the fast decay at short times. Amongthem is the direct radiative decay of excitons, which are created insidethe light cone, and which directly decay radiatively before any scat-tering event can take place. Ourmeasured τr of ~0.15 ps is in very goodagreement with previous measurements of the radiative lifetime ofexcitons inWSe2monolayers34. Therefore, it is likely that the fast initialdecay of the TRFE signal is influenced by direct radiative recombina-tion of part of the exciton population, created inside the light cone. Asignificant part of the excitonic population is, however, scattered outof the light cone, e.g., by phonons, and contributes to the valleypolarization over a longer time period. We note that also excitonlocalization in traps anddiffusion, as, e.g., observed for semiconductornanoplatelets35, may contribute to a prolonged exciton lifetime. Themain mechanism leading to valley relaxation in WSe2 monolayers, isthe long-range exchange mechanism between electron and hole,which is proportional to the center-of-mass momentum of theexciton36–38. The valley-polarization decay time of τv ~ 7.0 ps, extractedfrom the TRFE traces of the hBN encapsulated WSe2 monolayer inFig. 1f, is in very good agreement with the reported decay time of 6.0ps, measured on a bare WSe2 monolayer on a SiO2 substrate in ref. 36,andwith calculations, based on the long-range exchangemechanism37.TRFE experiments in in-plane magnetic fieldsWe now move on to the central point of the investigations in thismanuscript: experiments in in-plane magnetic fields, B∥. Figure 2ashows a comparison of TRFE traces of all four investigated samples atB∥ =0, where the laser was tuned in resonance with the A1s excitonicresonances in the respective materials, as marked by arrows in the RCmeasurements in Fig. 1a. The trace of the WSe2 monolayer is the sameas shown in Fig. 1f (σ+ pump). Comparing the twomonolayer samples inFig. 2a, one can recognize the much faster valley depolarization inMoSe2. Themeasured decay time is here ~1 ps, as compared to ~7 ps forthe WSe2 monolayer (see discussion above). The much faster valleydepolarization in the MoSe2 monolayer is reminiscent of a close tozero valley polarization,measured in cw polarized photoluminescenceon this material39. Surprisingly, while the valley depolarization time iscomparable for the WSe2 monolayer and multilayer, it is much longerin the MoSe2 multilayer, as compared to the monolayer. This may berelated to the fact that in MoSe2 monolayers the lowest energy state isa bright state, which is different in all other samples, however, we notehere that this is not the focus of this work. In Fig. 2b, the same mea-surements are shown, now for an in-plane field of B∥ = 9 T. While theFig. 2 | TRFEexperiments in in-planemagneticfields. Comparisonof TRFE tracesof all samples at a B∥ =0, and, b B∥ = 9 T, excited at the A1s excitonic resonances. Inthemultilayer samples, strong temporal oscillations are observed in the time tracesatB∥ = 9T, in contrast to themonolayer samples,which shownooscillations. cTRFEtraces of the WSe2 multilayer for different in-plane magnetic fields. The red solidline represents an exponentially-damped cosine fit to the data. The dashed graylines are guides to the eye.d TRFEmeasurements of theWSe2multilayer atfixed in-plane field B∥ = 6 T for different center energies ELaser of the laser pulses, as given inthe figure. A clear resonance behavior of the signal can be observed. The signalamplitudeΔ, as determined for all curves, is indicated. e Plot of the extracted signalamplitudes Δ (blue solid bullets), as indicated in d, versus central laser energy. Thedashed lines representGaussianfits, while the solid orange line is the sumofbothfitcurves. Resonances with the A1s and A2s excitons are indicated by arrows. For allmeasurements the temperature was T ~ 5 K.Article https://doi.org/10.1038/s41467-022-32534-3Nature Communications |         (2022) 13:4997 3TRFE time traces for themonolayer samples are essentially unchangedwhen compared to B∥ =0, they are significantly different for the mul-tilayer samples. Strong and pronounced oscillations can be observed.The oscillation period of the MoSe2 multilayer is slightly longer thanfor theWSe2multilayer. As a guide to the eye, vertical dashed lines areplotted in Fig. 2b, which mark the maxima of the oscillations of theWSe2multilayer. Figure 2c shows a full data set for theWSe2multilayerfrom B∥ = 0 T to 9 T. A full data set of theMoSe2multilayer is plotted inthe supplementary Fig. S2. Thegray dashed lines in Fig. 2c are guides tothe eye and mark the oscillation maxima, which correspond to thesame oscillation period. To test the resonance behavior of the TRFEmeasurements, we plot in Fig. 2d TRFE traces of theWSe2multilayer atfixed in-plane field of B∥ = 6 T for different central energies of the laserpulses. The central energies are given in Fig. 2d, the spectral widths ofthe pulses is ~ 16meV.We extract the amplitudes of the oscillations,Δ,as indicated in Fig. 2d, and plot them versus central laser energy inFig. 2e. The amplitudes show a clear resonance behavior. The dashedlines in Fig. 2e are Gaussian fits, and the solid orange line is the sum ofthe two Gaussian fit curves. The two maxima can be attributed toresonances with the A1s and A2s intralayer excitons (cf. with the reso-nance features in the RC experiments in Fig. 1a). We note that the A1sresonance position is shifted by about 16 meV to lower energies incomparison to the white-light RC measurements in Fig. 1a, which canbe due to bandgap-renormalization effects40 and/or a temperatureincrease under pulsed excitation. If the redshift would be causedentirely by a temperature increase, the temperature in the TRFEexperiments on the WSe2 multilayer could be up to T ~ 100 K41 as anupper limit. A full dataset of TRFE traces in resonance with the A2sexciton from 0 to 9 T can be found in supplementary Fig. S3 (same forthe MoSe2 multilayer in Fig. S4). It should be emphasized that we donot observe oscillations, i.e., an excitonic resonance, at energies abovethe A1s and A2s excitons in theMoSe2multilayer, in the spectral region,where in ref. 25 an IX was reported in RCmeasurements. From that weconclude that for our observed temporal oscillations only the intra-layer A excitons are relevant.Clearly, the oscillations in the TRFE traces resemble coherentprecession of amagneticmoment about the in-planemagneticfield, asknown from, e.g., electron spins in n-doped GaAs bulk42, hole spins inGaAs quantum wells43, or, localized background charge carriers inMoS2 and WS244, among many other examples. We have fitted allexperimental curves for B∥ > 0 with an exponentially-damped cosinefunction Sðν,τvÞ / expð�Δt=τvÞ cosð2πνΔtÞ for delay times Δt wellabove the fast initial decay of the TRFE signals, as exemplarily shownby the red solid line in Fig. 2c for the 9 T trace. An important result isthat the oscillations with frequency ν at B∥ >0 decay with approxi-mately the same decay time τv as the excitonic signal at B∥ = 0, and nolong-lived oscillatory signal is developed. From that we conclude thatthe oscillations stem from a Larmor precession of the exciton mag-neticmoment, and not from the spin of background charge carriers, asobserved for localized electrons in MoS2 and WS2 monolayers44. Fur-thermore, the approximate independence of the decay time τv from B∥shows that g factor fluctuations do not play a role. Otherwise, a 1/B∥dependence of τv would be expected45,46. Figure 3a shows a summaryof all oscillation frequencies ν, extracted by this procedure, versus B∥.Clearly, a linear, Zeeman-like dependence can be recognized. Thedetermined ∣g∥∣ are given in the legend of Fig. 3a. The experimentalerror margins for these values are about ±0.2. It should be noted thatwith TRFE experiments we can only determine the magnitude of the gfactor but not its sign. Very remarkably, for all excitonic resonances,the determined ∣g∥∣ are very close to out-of-plane g factors, ∣g⊥∣, of thecorresponding materials, reorted in refs. 25, 28, which are for WSe2bulk material ∣g⊥∣ = 3.2 ± 0.2 and 3.3 ± 0.6 for the A1s and A2s intralayerexcitons, respectively25. For MoSe2 bulk, the reported value for A1s is∣g⊥∣ = 2.7 ± 0.125. Hence, we conclude that ∣g∥∣ ~ ∣g⊥∣ for multilayerTMDCs, approaching the bulk limit.Comparison to first-principles calculations and discussionIn the following we will discuss our experimental findings further andcompare them to first-principles calculations (seemethods section). InTable 1, computed spin- and orbital angular momenta for out-of-plane(Siz,Liz) aswell as in-plane (Six, Lix) directions aregiven for themonolayerand multilayer samples. The superscript i stands for CB or VB, i.e., forthe conduction-band or valence-band states, respectively, which arerelevant for the intralayer A excitons in thematerials (cf. Fig. 1b, c). Wenote that the relevant CB states are different for the two materialsbecause of the reverse spin order. The computed g factors for the Aexcitons, which are determined by g?=k = 2ðSCBz=x + LCBz=x � SVBz=x � LVBz=xÞ47are also given. The minus signs in front of the VB angular momentaaccount for the fact that the angular momentum of a hole is justopposite to the angularmomentum of an electron in the VB state. Onecan see that for the monolayers and WSe2 multilayers, the calculatedg⊥ agree well with published experimental values, while for MoSe2multilayers the computed ∣g⊥∣ is somewhat smaller as compared to theexperimental report. The experimental result of g∥ ~ 0 for the mono-layers is confirmed by the calculations, which give exactly g∥ =0 (both,spin and orbital angular momenta contributions are zero due tosymmetry considerations and verified numerically, cf. table 1).For the multilayer samples on the other hand, the calculations dodeliver nonzero g∥, consistent with our experimental finding, though0 2 4 6 8 100.00.10.20.30.40.5C g = 2.5ycneuq erfnoital licso(THz)B|| (T)A2s WSe2   |g| = 3.4A1s WSe2   |g| = 3.1A2s MoSe2 |g| = 2.6A1s MoSe2 |g| = 2.5mul�layersKA1sA2sEBǁ > 0ca|+ ⟩1,|- ⟩1,bFig. 3 | Extracted in-plane g factors, and proposedmechanism. a Experimentallydetected oscillation frequencies for A1s and A2s intralayer excitons (orange andyellow solid bullets) in MoSe2, and, A1s and A2s intralayer excitons (dark green andlight green solid bullets) inWSe2multilayer samples versus in-planemagnetic field.Exemplary error bars for low and high fields are indicated. The extracted absolutevalues ∣g∥∣ are given in the legend. The experimental error margins are about ±0.2.b Energy versus center-of-mass momentum K dispersion of intralayer A excitons atB∥ >0 in a multilayer. For the excitons, the z component of the spin of the electronis symbolized by a small arrow with a single line, while the hole spin is indicated byan arrow with a double line. Energy splittings of the excitons due to finite g∥ aretaken into account. The red double arrows should symbolize the coherentmomentum-allowed oscillation between adjacent layers. c Representation of thepseudospin rotation on a Bloch sphere. The north pole corresponds to τ = +1, whilethe south pole represents the τ = −1 state. The orange arrow would correspond to asystem, excited with linearly-polarized light.Article https://doi.org/10.1038/s41467-022-32534-3Nature Communications |         (2022) 13:4997 4their magnitudes are smaller than the experimental values, which areclose to reported out-of-plane g factors, i.e., ∣g∥(exp.)∣ ~ ∣g⊥(exp.)∣.Interestingly, because of the particular symmetry of the bands (CB ~ Γ9and VB ~ Γ7 in the D3h point group of the K valleys), only the valenceband shows a nonzero value of g∥, while for the conduction band it isstrictly zero, i.e., the orbital, LCBx , and the spin, SCBx , angular momentaare both zero (cf. table 1). This situation is similar to the zero gx (Voigtgeometry) of the heavy-hole valence band in wurtzite materials withhexagonal symmetry48–51. While in the first-principles calculations theinterlayer hybridization of electronic bands is fully taken into account,excitonic correlations are not considered. Since we observe the oscil-lations at excitonic resonances, it is likely that additional hybridizationon the excitonic level contribute to the observed g factor. For instance,in-planemagnetic fields introduce amixing of bright and dark excitonsin monolayer TMDCs30–32. In the bulk case, excitonic correlations mayfacilitate themixing of different exciton channels due to the additionaldegeneracy of the bands. We emphasize that investigations of theseexcitonic correlations in the bulk case are beyond the scope of ourcurrent study but remain an open topic for future investigations. Alsothe experimental observation that for the multilayers the ∣g∥∣ of the 2sexcitons are slightly larger than thoseof the 1s excitons (cf. Fig. 3a)maybe explained by exciton hybridization: The Bohr radius, i.e., the spatialexpansion of the 2s excitons is larger than that of the 1s excitons (seeFig. 1e). Therefore, it is likely that hybridization effects may be slightlymore important for 2s than for 1s excitons.In Fig. 3b, a schematic picture of the excitonic center-of-massdispersion is shown for the relevant excitonic resonances in the twomultilayer materials, namely the A1s and A2s resonances. For the exci-tons, electron and hole spins, Sz, are depicted by single-line anddouble-line arrows, respectively. The energetic splittings, corre-sponding to thenonzero g∥, are taken into account bydashed and solidlines for the center-of-mass parabolas. The helicities are given next tothe spin configurations of the excitons. The bold red double arrowsshould symbolize the coherent oscillations between the excitonicstates, when resonantly excited. Hence, we suggest that the observedoscillations originate from coherent oscillations between excitoniclevels with different pseudospins, i.e., pseudospin quantum beats.These can be visualized on a Bloch sphere, as shown in Fig. 3c: Thenorth pole corresponds to excitons with pseudospin τ = +1. Thismeans, they occupy the K+ valleys of the individual layers (cf. Fig. 1d).Once they are initialized by a σ+ pump pulse, they can coherentlyoscillate to the south pole, which are excitons with pseudospin τ = −1,i.e., which occupy the K− valleys of the individual layers. A question,which we can not answer conclusively so far is, if the coherent oscil-lations are either spin quantum beats of K+ and K− A excitons solelywithin the layers (intralayer oscillations), or between the layers(interlayer oscillations), or, a mixture of both. The experimental find-ing that we do not observe oscillations for the monolayers may favorthe scenario of interlayer spin quantum beats in the multilayer sam-ples. This is, furthermore, corroborated by the fact that the interlayercomponent of the oscillations is momentum-allowed, since, in k spaceK+ and K− valleys are on top of each other in an H-type structure (cf.Fig. 1d). However, it was previously suggested for WSe2 bilayers thatonly holes may exhibit coherent oscillations in in-plane magneticfields9. Presumably, there may be contributions from both, intralayer-and interlayer oscillations. Which part dominates, we can not say sofar. In future investigations this may be further highlighted byexperiments on R-type multilayer samples: In contrast to H-type, inR-type stacking, interlayer oscillations of A excitons are momentumforbidden. This scenario may favor intralayer oscillations. However,such experiments will be technically demanding, since the TMDCselenides do not grow in R-type, so, multilayer samples will have to befabricated manually.DiscussionFinally, we would like to make some notes on the layer numberdependence. In principle, wewould expect the pseudospin oscillationsto occur, starting with symmetric H-type bilayer samples, where thespin degeneracy is restored. To elucidate this in more detail, we havecomputed the g factors for a symmetric WSe2 bilayer (see table SI inthe Supplementary Information). We receive indeed for the bilayer anon-zero g∥, which is in between the values of the monolayer (whereg∥ ~ 0) and the bulk limit. Also, g⊥ of the bilayer is in between thecorresponding values for the monolayer and multilayer (cf. Table SI).Unfortunately, preliminary TRFE experiments on a large-area encap-sulated H-type WSe2 bilayer do not show oscillations for an in-planemagnetic field. These preliminary experiments are shown in Fig. S6 ofthe Supplementary Information, where they are compared to TRFEtraces of a closeby multilayer. We speculate that within our laser spotwith diameter of about 50μmon the large-area sample, theremaybe alarge number of microscopic regions with different asymmetricpotentials, caused by locally varying strain, dielectric environment,etc., due to the hBN encapsulation, where the spin degeneracy is notrestored. This could hinder the development of pseudospin rotationson a large scale. For future experiments, it would be highly desirable tosystematically study series of samples with increasing layer number,starting from the bilayer, possibly with smaller laser-spot sizes.In summary, we have detected ultrafast pseudospin rotations inthe GHz to THz frequency range in TMDC multilayers in in-planemagnetic fields via time-resolved Faraday ellipticity. Surprisingly, themagnitudes of the extracted in-plane g factors are close to reportedvalues of out-of-plane g factors of the same materials. This is in starkcontrast to monolayer samples, which show no temporal oscillationsfor nonzero in-planemagneticfield, andwhich, hence, have an in-planeexciton g factor close to zero. The experimental results are confirmedby first-principles calculations of the g factors. Our study opens thedoor for manipulation of these pseudospins on ultrafast time scales,making TMDC multilayers an interesting platform for pseudospinoperations, possibly putting quantum-gate operations, as suggested inref. 9, into reach.Table 1 | Computed values of out-of-plane and in-plane spin-, S, and orbital, L, angularmomenta for the conduction-band (CB)and valence-band (VB) states, which are relevant for the A excitons of the investigated materialsMaterial SCBz LCBz SVBz LVBz g⊥ g⊥ (exp.) SCBx LCBx SVBx LVBx ∣g∥∣ ∣g∥∣ (exp.)WSe2 monolayer 0.98 2.97 1.00 5.00 −4.10 −4.38…−1.57a 0.00 0.00 0.00 0.00 0.00 ~0WSe2 multilayer 0.97 2.98 1.00 4.40 −2.89 −3.4…−2.3b 0.00 0.00 0.47 ±0.07 0.80…1.08 3.1 ± 0.2MoSe2 monolayer 1.00 1.81 1.00 3.96 −4.30 −4.4…−3.8c 0.00 0.00 0.00 0.00 0.00 ~0MoSe2 multilayer 1.00 1.76 1.00 2.67 −1.84 −2.7d 0.00 0.00 0.74 ±0.06 1.36…1.60 2.5 ± 0.2For the first-principles calculations, see the methods section. The corresponding theoretical g factors, g⊥ and g∥, for the A excitons are given. For experimental g⊥ of the A1s exciton, we refer toliterature values. The experimental values for g∥ from this work are shown in the last column. Since in the experiments we can only determine the magnitude but not the sign, we denote only themagnitude ∣g∥∣arefs. 28, 61, 67–73brefs. 25, 28, 74crefs. 28, 32, 67, 75–78dref. 25Article https://doi.org/10.1038/s41467-022-32534-3Nature Communications |         (2022) 13:4997 5MethodsSamplesAll investigated TMDC samples are mechanically exfoliated from bulksource material (purchased from HQ Graphene) using nitto tape, andthen transferred onto transparent sapphire substrates by viscoelasticpolymethyldisiloxane stamps52. Large-area MoSe2 and WSe2 mono-layers areprepared andencapsulated in hexagonalBoronnitride (hBN)multilayers for protection against environmental influences. In themain body of the manuscript, results from two multilayer samples arepresented: A WSe2 multilayer, consisting of 14 layers, and a MoSe2multilayer with about 84 layers.Optical experimentsFor sample characterization, reflectance-contrast (RC) measurementsof all samples are conducted in an optical microscope setup. Thesamples are mounted by an elastic organic glue on the cold finger of aHe-flow cryostat and are kept in vacuum, while the sample holder iscooled down to nominally 5 K. The temperature at the sample positionis estimated by the relative intensities of Ruby lines of the sapphiresubstrate. The substrate temperature is typically between aboutT = 10K and 30K. For the RC measurements, a white-light source isused, which is focused by a x60 microscope objective to a spot withdiameter of about 10μm. Reference spectra are recorded at positionsnext to the TMDC sample. Evaluation of the RC spectra, using atransfer-matrixmodel, canbe found in the Supplementary Information(supplementary Fig. S5)A schematic picture of the experimental setup, used for TRFEexperiments, is shown in supplemental Fig. S1. For TRFE experiments, amode-locked Ti:Sapphire laser is used, which produces laser pulseswith a temporal length of about 80 fs at a repetition rate of 80MHz.The laser beam is divided into two pulse trains by a beam splitter. Thetime delay, Δt, between pump and probe pulses is adjusted by a ret-roreflector, which is mounted on a linear stepper stage. Both beamsare focused by a plano convex lens onto the sample surface, wherethey overlap. The laser spot diameter at the sample position is about50μm. The sample is mounted in an optical cryostat with super-conductingmagnet coils (split-coil cryostat) at a temperature of aboutT = 5 K, which is maintained by a constant flow of cold He gas. Bymeasuring the laser pulse length before and after themagnet cryostat,we estimate the pulse length at the sample position to be about 130 fs.The pump pulses are circularly polarized and the laser wavelength istuned to excitonic absorption lines to create a valley polarization in thesample. The temporal dynamics of the valley polarization is thenmeasured by detecting the ellipticity of the linearly-polarized probepulses after transmission of the sample. For measurement of theellipticity, a combination of aWollaston prism, quarter-wave plate andtwo balanced photo diodes is used. The pump beam is mechanicallychopped at a frequency of about 1.6 kHz, and for detection of thephotodiode difference signal, lockin technique is used.Exciton densitiesTo get the most accurate estimate of the exciton densities in theexperiments, we measure the power of the transmitted pump laserbeam for the two cases, when (i) the pump beam is focused on thesample, and (ii) focused next to the sample on the sapphire substrate.The difference in power is the upper limit of the power absorbed bythe sample, since with this approach we neglect the difference inreflectivity of the sapphire substrate versus sapphire substrate withTMDC sample. We then assume that the exciton density n is equal tothe density of absorbed photons nphotons, which is related to theabsorbed power Pabs by Pabs = nphotonsELaserfr2π. ELaser is the energy ofthe laser photons, f the repetition rate (80 MHz) of the laser, andr = 25μm the laser-spot radius on the sample.With this procedure, we get for the WSe2 monolayer an initialexciton density of n ~ 1.3 × 1012 cm−2 and for the MoSe2 monolayern ~ 1.9 × 1012 cm−2, when in both cases the A1s exciton is excited reso-nantly. Both values are well below the Mott density40,53,54. For themultilayer samples, we devide the total exciton density by the numberof layers to get an estimateof thedensity per layer.Weget for theWSe2multilayer (14 layers) n ~ 3.7 × 1011 cm−2/layer when exciting the A1sexciton resonantly, and, n ~ 2.1 × 1011 cm−2/layer for resonant excitationat the A2s exciton. For the MoSe2 multilayer (80 layers), we haven ~ 1.0 × 1012 cm−2/layer for the A1s exciton resonance, andn ~ 1.4 × 1012 cm−2/layer for the A2s exciton resonance.Theoretical modelingThe first-principles calculations are performed within the densityfunctional theory (DFT) using the full-potential all-electron codeWIEN2k55. We use the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional56, a core-valence separation energy of − 6 Ry,atomic spheres with orbital quantum numbers up to 10 and the plane-wave cutoff multiplied by the smallest atomic radii is set to 9. For theinclusion of spin-orbit coupling, core electrons are considered fullyrelativistically whereas valence electrons are treated in a second var-iational step57. We use a Monkhorst-Pack k-grid of 15 × 15 × 6 (15 × 15)for the bulk (monolayer). The bulk calculations include van der Waalsinteractions via the D3 correction58. Self-consistency convergence wasachived using the criteria of 10−6 e for the charge and 10−6 Ry for theenergy. The bulk lattice parameters, taken from ref. 59, are a = 3.282Å,d = 3.340 Å and c = 12.960Å for WSe2; and a = 3.289Å, d = 3.335Å andc = 12.927Å for MoSe2. Here, the in-plane lattice parameter, a, and thelayer thickness,d, are considered the same for bulk andmonolayers. Inmonolayers, we used a vacuum spacing of 16 Å to avoid interactionamong the periodic replicas whereas in the bulk case the total size ofthe unit cell is the lattice parameter c. The calculations of the orbitalangular momenta Lx and Lz are based on the fully convergedsummation-over-bands approach discussed in refs. 47,60–62.For the calculations of the bulk intralayer excitons we used theeffective Bethe-Salpeter equation63,64. The energy band dispersion nearthe K valley is treated as Eðkx , ky, kz Þ= _22m* k2x + k2y� �+ f ðkzÞ, with m*being the in-plane effective mass and f(kz) models the dispersion fromalong the −H −K −H direction of the bulk first Brillouin zone. The DFTcalculated in-plane effective masses for WSe2 are mCB = 0.29m0 andmVB = 0.36m0, and for MoSe2,mCB = 0.90m0 andmVB = 0.61m0. For thefunction f(kz), we take the numerical values directly from the DFT cal-culations. The electron-hole interaction is mediated by the anisotropicCoulomb potential, with the dielectric constants for WSe2 given byεxx = εyy = 15.75 and εzz = 7.75, and for MoSe2, εxx = εyy = 17.45 and εzz =8.3, taken from ref. 65. Our calculations reveal binding energies of 29.9(9.1)meV for the A1s(2s) exciton inWSe2 and 41.9 (10.1)meV for the A1s(2s)exciton in MoSe2, respectively. 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Funding by the Deutsche For-schungsgemeinschaft (DFG, German Research Foundation) - Project-ID314695032 - SFB 1277 (subprojects B05 (C.S.), B07 and B11 (J.F.)), andprojects SCHU1171/8-1 (C.S.) and SCHU1171/10-1 (SPP 2244) (C.S.) isgratefully acknowledged. K.W. and T.T. acknowledge support from JSPSKAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).Author contributionsS.R., D.F., S.F. and P.M. prepared the samples, performed the experi-ments and analyzed the data. P.E.F.J. and J.F. performed the first-principles and exciton calculations. K.W. and T.T. supported the high-quality hBN material. C.S. conceived the project, analyzed the data andwrote the manuscript. All authors contributed to the discussion ofresults and to the finalization of the manuscript.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-32534-3.Correspondence and requests for materials should be addressed toChristian Schüller.Peer review informationNatureCommunications thanksNikolai Sinitsynand the other, anonymous, reviewer(s) for their contribution to the peerreview of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-32534-3Nature Communications |         (2022) 13:4997 8https://doi.org/10.1038/s41467-022-32534-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Ultrafast pseudospin quantum beats in multilayer�WSe2 and MoSe2 Results and discussion Sample characterization TRFE experiments in in-plane magnetic fields Comparison to first-principles calculations and discussion Discussion Methods Samples Optical experiments Exciton densities Theoretical modeling Data availability References Acknowledgements Author contributions Funding Competing interests Additional information