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Suman Chatterjee, Medha Dandu, Pushkar Dasika, Rabindra Biswas, Sarthak Das, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Varun Raghunathan, Kausik Majumdar

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[Harmonic to anharmonic tuning of moiré potential leading to unconventional Stark effect and giant dipolar repulsion in WS2/WSe2 heterobilayer](https://mdr.nims.go.jp/datasets/677fea34-d832-4548-bb20-ac200690166a)

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Harmonic to anharmonic tuning of moiré potential leading to unconventional Stark effect and giant dipolar repulsion in WS2/WSe2 heterobilayerArticle https://doi.org/10.1038/s41467-023-40329-3Harmonic to anharmonic tuning of moirépotential leading to unconventional Starkeffect and giant dipolar repulsion in WS2/WSe2 heterobilayerSuman Chatterjee 1,6, Medha Dandu1,4,6, Pushkar Dasika1,6, Rabindra Biswas1,Sarthak Das1,5, Kenji Watanabe 2, Takashi Taniguchi 3, Varun Raghunathan1 &Kausik Majumdar 1Excitonic states trapped in harmonic moiré wells of twisted heterobilayers isan intriguing testbed for exploring many-body physics. However, the moirépotential is primarily governed by the twist angle, and its dynamic tuningremains a challenge. Here we demonstrate anharmonic tuning of moirépotential in aWS2/WSe2 heterobilayer through gate voltage and optical power.A gate voltage can result in a local in-plane perturbing field with odd parityaround the high-symmetry points. This allows us to simultaneously observethe first (linear) and second (parabolic) order Stark shift for the ground stateand first excited state, respectively, of the moiré trapped exciton - an effectopposite to conventional quantum-confined Stark shift. Depending on thedegree of confinement, these excitons exhibit up to twenty-fold gate-tun-ability in the lifetime (100 to 5 ns). Also, exciton localization dependent dipolarrepulsion leads to an optical power-induced blueshift of ~ 1 meV/μW - a five-fold enhancement over previous reports.Interlayer van der Waals interaction allows us to stack layers of tran-sition metal dichalcogenides (TMDCs) onto each other with an arbi-trary latticemismatch1–3. This leads to an additional degreeof freedom,the twist angle (θ) between two successive layers, that governs themoiré pattern arising in the corresponding superlattice4–7. The latticeconstant of themoiré superlattice is given by aM≈affiffiffiffiffiffiffiffiffiffiffiθ2 + δ2p where δ is thelattice constant difference between the constituent monolayers and abeing the average lattice constant6,8,9. Different atomic registries pre-sent in this moiré superlattice (Fig. 1a) form a periodic potential fluc-tuation [VM(r)] resulting from local strain and interlayer coupling10,11.Varying twist angle can dramatically change the material properties,drawing attention from the researchers in the recent past8,12–14. Moirésuperlattice in TMDC heterobilayer has been widely explored includ-ing observation of neutral moiré exciton4,15,16, moiré trion17–19, singlephoton emission20,21, and correlated states5,22,23.Due to type-II band alignment, WS2/WSe2 heterobilayer supportsan ultrafast charge transfer24,25 with electrons staying in the WS2 con-duction band, and holes in the WSe2 valance band, forming interlayerexciton (ILE)8,9 under optical excitation (Fig. 1b). The moiré wellsbehave as two-dimensional harmonic traps for the ILEs4,26,27.The depth of the exciton moiré potential is determined by thetwist angle and the degree of lattice mismatch between the two het-erobilayers. Hence, dynamic tuning of moiré potential remains aReceived: 6 April 2023Accepted: 24 July 2023Check for updates1Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, India. 2Research Center for Functional Materials,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-044, Japan. 3International Center for Materials Nanoarchitectonics, National Institute forMaterials Science, 1-1 Namiki, Tsukuba 305-044, Japan. 4Present address: Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,USA. 5Present address: Institute of Materials Research and Engineering (IMRE), Agency for Science, Technology and Research (A*STAR), Singapore 138634,Republic of Singapore. 6These authors contributed equally: Suman Chatterjee, Medha Dandu, Pushkar Dasika. e-mail: kausikm@iisc.ac.inNature Communications |         (2023) 14:4679 11234567890():,;1234567890():,;http://orcid.org/0000-0001-5825-8176http://orcid.org/0000-0001-5825-8176http://orcid.org/0000-0001-5825-8176http://orcid.org/0000-0001-5825-8176http://orcid.org/0000-0001-5825-8176http://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/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-40329-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-40329-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-40329-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-40329-3&domain=pdfmailto:kausikm@iisc.ac.inchallenge, which, if realised, will be of great importance for both sci-entific exploration and applications. One could perturb the moirépotential by external stimulus, however, the perturbing potential maynot necessarily be harmonic, breaking down the usual harmonicpotential approximation for moiré well. In this work, we explore twosuch anharmonic perturbations to the WS2/WSe2 moiré potential well:the first one through a gate voltage which introduces anharmonicperturbation through screening at highdoping regime; and the secondone is through optical excitation which introduces the perturbingpotential through ILE dipolar repulsion. In both cases, the harmonic toanharmonic switching of the moiré potential manifests through acorresponding change from an equal to unequal inter-excitonicspectral separation. In such a scenario, we explore several intriguingfeatures of the moiré excitons, including giant lifetime tunability,anomalous Stark shift, and dipolar repulsion induced large spectralblueshift.Results and discussionWe prepare hBN-capped ~ 59° twisted (confirmed by second harmonicgeneration (SHG) spectroscopy in Supplementary Note 1 and Fig. 1)WS2/WSe2 heterobilayer (sample D1) with a back gate (see “Methods”section for sample preparation). The schematic and the optical imageof sample D1 are illustrated in Fig. 1c and d. This twist angle creates amoiré superlattice with a lattice constant ~7.3 nm. Figure 1e shows arepresentative photoluminescence (PL) spectrum from the samplewith 532 nm excitation at 4 K. The emission spectrum exhibits threeseparate, strong interlayer moiré excitonic resonances28 X0, X1, and X2at ≈ 1.392, 1.418, and 1.442 eV, respectively (marked by black dashedline). The peaks exhibit alternating sign of the degree of circularpolarization (DOCP) (Supplementary Fig. 2), indicating the existenceof moiré superlattice4,6,29.The near-equal inter-excitonic separation suggests that the threeexciton resonances appear from excitonic states in the harmonicmoiré potential well (Fig. 1f)4,6,26,27. This inter-excitonic separation canbe tuned by varying the twist angle, which regulates the depth of themoiré potential well4,30. We verified this by measuring twist angledependent PL spectra from three samples [D1 (~59°), D2 (~54°), and D3(large anglemisalignment)] in Supplementary Fig. 3. The time-resolvedPL (TRPL) measurement (see “Methods” section) from sample D1 inFig. 1g shows that the lifetimeof the three species (τ0 = 100 ns, τ1 = 15:3ns, and τ2 = 9 ns) increases significantly with stronger confinement.Accordingly, their PL intensity also exhibits significantly differentpower lawwith varying optical power (P): I / Pαi with α0 = 0.34 ±0.02,α1 = 0.59 ± 0.03, and α2 = 1.1 ± 0.11 (Fig. 1h). The corresponding spectraat three different P values are shown in Fig. 1i. At lowpower (30 nW), X0emission is the dominant one, with negligible emission from X2.However, at higher power (5.95μW), three peaks are clearly discern-able, and the fractional contribution of X0 reduces, while X2 emissionbecomes appreciable. All these observations indicate that the threedifferent excitonic species correspond to moiré trapped excitonicstates with varying degrees of localization (Fig. 1f). From the spectralseparation between the quantized states, we calculate peak-to-peakmoiré potential fluctuation of ≈150meV (see Supplementary Note 2),as shown in Fig. 2c. Possible alternative explanations, such as phonon-sidebands and defect-bound excitons, are unlikely in our samplesbased on the observations including alternating signs of the DOCP andsystematic tuning of the ILE peak separation with twist angle, doping,and optical power (discussed later).Fig. 1 | Moiré trapped interlayer exciton. aDifferent atomic registries in a twistedWS2/WSe2 bilayer with high symmetry points marked by colored circles. b Type-IIheterojunction of WS2/WSe2 bilayer resulting in interlayer exciton. c Schematic ofthe heterobilayerwith back gate connection.dOptical image of a fabricated device.The dotted colored lines indicate different flake boundaries. Scale bar is 10μm.e Representative PL spectrum (using 532 nm CW laser) in the ILE regime (blacksymbols) and fitting (red trace) showing three clear ILE resonances denoted by X0(brown),X1 (green), andX2 (blue) atVg =0V and P =0.675 μW.Black arrows indicatenear equal spacing. f Schematic representation of three ILE states in a harmonicmoiré potential well with varying degree of localization. g Raw TRPL spectra alongwith IRF for the three ILE resonances showing varying decay time scales at Vg =0 V(P = 13.45 μW), namely 100, 15, and 9.3 ns for X0, X1, and X2, respectively. h Opticalpower dependent intensity plot (symbols) of the three ILEs in log-log scale fol-lowing different power-laws (fitted by solid lines). i Evolution of power-dependentPL spectra (black symbols) at three different optical powers, along with fitting (redsolid trace).Article https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 2Gate tunabilityFigure 2a shows a color plot of the interlayer exciton emission spectraas a function of gate voltage (Vg). The estimated n-doping density atthe highest applied Vg (=5 V) is <1.5 × 1012 cm−2 (see SupplementaryFig. 4). This is well below themoiré trap density (NM) ≈ 2 × 1012 cm−2 foraM ~ 7.3 nm. The fitted peak positions are shown in the left panel ofFig. 2b (see individual spectra in Supplementary Fig. 5). While theVg <0V region is nearly featureless, Vg >0V (n-doping) region hasthree conspicuous features: (a) there is a reduction in emissionintensity for all the three ILE peaks, with X0 disappearing at high Vg; (b)there is a large and unequal redshift for the peaks forVg > 0; and (c) theinter-excitonic separation changes at higher Vg, indicating inducedanharmonicity. The reduction in emission intensity with an increase inVg rules out the charged excitonic (trion) nature of any of the threepeaks. Figure 2b (right panel) schematically explains the origin of thestrong redshift with Vg. At positive Vg, theWS2 layer becomes n-doped.Due to small thermal energy at 4 K, the wave function of the inducedelectrons remains primarily in the WS2 layer, with a fraction of itextends into the WSe2 bandgap as an evanescent state with imaginarywave vector. Such a wave function distribution creates a screening ofthe gate field, and in turn a relative potential difference between WS2and WSe2 layers, reducing the interlayer bandgap. Note that, the pre-sence of the charge density from the evanescent state in WSe2 isessential to create such relative potential difference between the twolayers, else dictated by the self-consistent electrostatics, a zeroinduced charge density in WSe2 layer would result in pinning of theWSe2 potential with that of WS2, and no relative interlayer bandgapchange would be allowed.Unconventional Stark effectInterestingly, the average slope (indicated by black dashed line inFig. 2b) of the redshift of X2 is almost similar (about 5meV/V) to that ofthe intra-layer WS2 trion (X−) or charged (XX−) biexciton31 (See Sup-plementary Fig. 6), but the average slope is higher for X1 (~7meV/V)and X0 (~15meV/V). The redshift of the intra-layer WS2 trion emissionpeak with Vg is directly related to the enhanced trion dissociationenergy due to the extra energy required to place the remaining elec-tron into the increasingly filled conduction band. Hence it can becorrelated with the change in the Fermi energy due to doping31–33. Thischange is nearly equal to the shift in the WS2 conduction band withrespect to theWSe2 valence band,making the average slopes of X2 andWS2 trion shift similar. This also is in agreement with the weak con-finement of X2.However, the enhancement in the slope of the redshift for X1 andX0 cannot be explained from doping dependent interlayer bandgapreduction and suggests a strong additional effect of localization. Tounderstand this further, we solve the Poisson equation to obtain themovement of bands with Vg (see Supplementary Note 3 for the detailsof the calculation). The results are summarized in Fig. 2d. At smallpositive Vg, the bands shift downward in energy (middle panel,Vg =0.5 V). However, at larger positive Vg, the central part of region I(right panel, Vg > 0.5 V) of the conduction band moiré well beingVg(V)Energy (eV)X0X2X1a e-1 0 1-101-75-50-250255075c VM (meV)x/ My/Mψ12ψ02ΔVRRfdR (nm)Energy(eV)0 20 400 20 40-1.20.00.30 20 40CBWS2VBWSe2Vg = 0.5 VRegion I Region IIM = 7.3 nmEFRegion I Region II-1.4-0.20.0EX0′ < EX0-1.2-1.00.20.4Energy(eV)EX0 Stark shift(meV)Vg (V)0 2 4-12-8-404Vg = 0.1 VVg = 0 VX1X0PL intensity (counts)Vg = 1 VWS2WSe2VgVg = 0 V)Ve(ygrenEVg (V)bX2X1X05 meV/V7 meV/V15 meV/V1.461.441.421.401.38-6 -3 0 3Fig. 2 | Gate-tunable moiré potential and unconventional Stark effect. a Colorplot of Vg dependent PL spectra showing X0, X1, and X2 resonances. b Left panel:Fitted peak positions showing the gradual redshift of the three ILE peaks with Vg.The black dashed lines indicate guide-to-eye in the Vg >0 regime. Right panel:interlayer bandgap reduction is shown schematically with increasing Vg. c 2Dprojectionof the variation of the calculatedmoirépotential.dToppanel: Simulatedconduction and valance band profile at three different Vg (0, 0.1, and 0.5 V) valuesobtained by solving the Poisson equation with the moiré potential fluctuation (seeSupplementary Note 3 for details). For simulation, the thickness of the gatedielectric (hBN) is assumed to be 20 nm. Region I (II) in the top left panel denotesthe minimum (maximum) energy of the WS2 conduction band due to moirépotential induced spatial energy fluctuation. At lower Vg (top middle panel), theconduction band gradually comes down in energy towards the Fermi level (reddashed line) maintaining the same degree of fluctuation. At higher Vg (top rightpanel), when the conduction band is close to the Fermi level, it starts flattening dueto screening. This also results in an enhancement in the valence band fluctuation.Bottom panel: Zoomed-in Region I at Vg =0V (in left) and Vg =0.5 V (in right). Thetransition energy for X0 (EX0, shown by arrow) decreases at higher Vg. (e) ∣ψi∣2(i =0, 1) plotted along with the in-plane perturbing potential ΔV indicating strongoverlap (non-overlap) between ΔV and ∣ψ0∣2 (∣ψ1∣2) due to different parity of thewave functions. f Stark shift of X0 (δX0) and X1 (δX 1) plotted with Vg. δX0(δX 1) showsa linear(parabolic) Stark shift fitting (solid traces).Article https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 3energetically closer to the Fermi energy supports more electron den-sity than region II. Accordingly, due to the screening by the inducedcarrier density, region I starts moving down slower than region II. Thenet effect is a suppression in the local moiré fluctuation of the con-duction band. Interestingly, the self-consistent electrostatics forces anamplification in the moiré potential fluctuation in the valence band ofWSe2: The suppressedmovement ofWS2bands in region I also reducesthemovement of bands in WSe2, while the stronger movement of WS2bands in region II (with relatively less carrier density) also pushes theWSe2 bands more downward. The net result is a flattening of theelectron moiré well in the WS2 conduction band, causing a delocali-zation of the electron state, coupled with a deeper hole moiré well inthe WSe2 valence band, resulting in an enhanced localization of thehole state (zoomed in Fig. 2d, bottom panel). This modification of themoiré trapping potential, in turn, causes a reduction in the energy ofthe trapped electron state and an enhancement in the energy of thetrapped hole state. The negative net change gives rise to an additionalredshift in the localized exciton resonance (X0 and X1).This results in an in-plane perturbation potential (ΔV) with evenparity about the high-symmetry points (Fig. 2e). ΔV is maximum at thecenter of the moiré well and reduces symmetrically away from thecenter. On the other hand, the wave function (ψ) has an even and oddparity for the ground (X0) and first excited (X1) states, respectively.This, in turn, results in a large (small) value of ∣ψ0∣2 (∣ψ1∣2) around thecenter of the trap for X0 (X1), as shown in Fig. 2e. Due to such a strongoverlap (non-overlap) of ΔV and ∣ψ0∣2 (∣ψ1∣2), the first-order Stark effect( ψ�∣ΔV ∣ψ�) is nonzero (negligible) for X0 (X1). Accordingly, we expectX0 and X1 to exhibit linear and parabolic Stark shift, respectively, withthe in-plane local electric field (ξ), and hence with Vg, since our simu-lation suggests that ξ is approximately linearly dependent on Vg (seeSupplementary Figure 7). Such local field effect will cancel out for theless-localized X2 state. In Fig. 2f, the respective Stark shifts[δX0,1ðVg Þ � δX0,1ðVg =0Þ where δX0= EX2� EX0and δX 1= EX2� EX 1]exhibit linear andparabolic variationwithVg (reproduced in sampleD4as well, see Supplementary Fig. 8), in excellent agreement with theabove analysis. We note that such Stark effect is unconventional sincethe usual quantum-confined Stark effect (QCSE) in quantum wells,where the applied vertical electric field is uniform, results in a per-turbing potential having odd parity. Thus the first-order QCSE (linear)is usually negligible, and we only observe a parabolic shift in theemission energy due to the second-order correction34–38.Gate tunable exciton lifetimeFigure 3a shows the peak-resolved (spectral resolution of 0.8meV)TRPL spectra (see “Methods” section) for X0, X1, and X2, at Vg =0 and3 V, suggesting a faster decay at higher Vg for all the ILE peaks. Thetransient response is captured well (solid black lines in Fig. 3a) by a setof rate equations and Gaussian formation model (see “Methods” sec-tion, Eqs. (3)–(5)). The extracted decay (τi) and formation time (τfi) areplotted for the exciton Xi, i = 0, 1, 2 in Fig. 3b, c. Around Vg =0 V, thedecay time varies over 10-fold from X0 ( ~ 100 ns) to X2 (~9 ns). How-ever, at large Vg, all the three ILEs show similar decay time (4–6 ns). Onthe other hand, the formation times are relatively weaker function ofVg and reduce slightly with increasing Vg.The kinetics can be understood by the cascaded processesschematically depicted in Fig. 3d. At small Vg, the respective netlifetimes follow the trend τ0≫ τ1 > τ2 (Fig. 3b), which is readilyunderstood due to the additional non-radiative decay paths γ20 andγ21 for X2, and γ10 for X1. The order of the respective formation times(τf0 = 5.6 ns, τf1 = 3.6 ns, and τf2 = 1.1 ns) in Fig. 3c, also supports themodel of cascaded formation. In addition, a longer lifetime wouldmean the state is blocked for a longer duration, increasing theformation time.The strong gate dependence of the ILE lifetime is capturedthrough a simple model where the gate dependent non-radiativeprocess is considered as proportional to induced carrier density (seeEqs. (6)–(7) in “Methods” section):τiðVg Þ=1τiðVg =0Þ+CiðeαVg � 1Þ" #�1ð1ÞFig. 3 | Gate induced lifetimemodulationofmoiré exciton. aPeak-resolvedTRPLspectra (symbols) alongwithmodel (described inMethods) predictedfitting (blacktrace) at Vg =0 and 3 V for X0, X1, and X2. The IRF is shown in the left panel.b Extracted decay time (symbols) for different moiré ILEs as a function of Vg. Solidtraces represent themodel (Eq. (1)) prediction. c Extracted formation times plottedas a function of Vg. d Cascaded formation process for different ILEs, showingradiative (γr,i) channels for the exciton Xi (i =0, 1, 2), and inter-excitonic non-radiative paths (γij) between excitons Xi and Xj.Article https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 4The model (solid traces in Fig. 3b) accurately reproduces the Vgdependent lifetime values (symbols) by using α and Ci as fitting para-meters. We observe a Vg-modulation of τ0 by more than 20-fold from100 to 5 ns (Fig. 3b), which correlates well with the PL intensityreduction of X0 with Vg, in Fig. 2a. This is a direct evidence of the gate-inducednon-radiative process due to thedelocalizationof the electronin the flattened conductionband (Fig. 2d). X0 being the ground state ofthe well, the inter-excitonic transfer-related non-radiative decaychannels (Fig. 3d) are suppressed. On the other hand, At low Vg, τ1,and τ2 are dominated by the (gate independent) non-radiative decaychannels to other lower energy states (that is, γ10, γ20, and γ21), henceremain nearly unchanged up to Vg = 2 V (Fig. 3b). The Vg-dependentnon-radiative decay rate starts dominating only at large Vg for X1 andX2, resulting in a reduction of τ1 and τ2.Optical power induced anharmonicityWenow vary P over nearly two decades using a pulsed laser (531 nm) atVg =0 V and plot the ILE peak positions in Fig. 4a. While X0 exhibits astrong blueshift ( ≈ 1 meV/μW), the shift for X1 and X2 is negligible.Hence, the inter-excitonic separations (δE21 and δE10) do not remainequal at higher P, suggesting departure fromharmonic behavior. Suchanharmonicity and power-dependent blueshift can be understood bythe perturbing potential (Udd) arising from ILE dipolar repulsion39,40:Udd =ZnUðrÞd2r =nq2dϵ0ϵrð2ÞwhereUðrÞ= q22πϵ0ϵrð1r � 1ffiffiffiffiffiffiffiffiffiffir2 +d2p Þ is the repulsion between two ILEdipolesplaced at a distance r (schematically shown in Fig. 4b, left panel). ϵ0 isthe vacuum permittivity, ϵr is the effective relative permittivity of theheterojunction, n is the effective concentration of exciton dipoles, andd is the interlayer separation. Due to this induced anharmonicity, it isexpected to observe a lifting of degeneracy for X1 and X2, as shownschematically in Fig. 4b (right panel). Since the lifetime of X0 is sig-nificantly larger than that of X1 and X2, the steady-state density(generation rate × lifetime) of ILE dipoles is dominated by thepopulation of X0 (n0). Since IX0ð/ n0Þ / P0:34 (see Fig. 1h), Eq. (2)indicates that the blueshift (Edd) of X0 should follow Edd∝ P0.34, ingood agreement with the linear fit in Fig. 4c. From Eq. (2), n0 isestimated to be ≈ 9.5 × 1011 cm−2 (which is less thanNM/2) at the highestoptical power used (17.7μW).To the best of our knowledge, the observed average rate of theblueshift with power for X0 (≈1meV/μW) is the highest reported valuefor ILE to date39,41–43, indicating a strong inter-excitonic interaction. Thestrong confinement ofX0 does not allow it to drift out of themoiré trapin the presence of such dipole–dipole repulsion, resulting in a largeblueshift. On the other hand, weaker confinement of X1 and X2 allowsthem to drift away under such dipolar repulsion, resulting in a sup-pressed blueshift in this small power regime.Figure 4d, top panel (open symbols) shows the optical powerdependent lifetime of X0, X1, and X2. We notice that the lifetime for allthe three species is a weak function of P. This is in stark contrast withintra-layer free exciton where Auger effect drastically reduces the life-time at higher P44,45. Such a weak dependence of lifetime on P is a resultFig. 4 | Optical power dependent anharmonic tuning of moiré potential. a PLpeak position for X0, X1, and X2, plotted against optical power (P). X0 exhibits astrong blueshift (1meV/μW)with P. The inter-excitonic peak separation is similar atlow P, but becomes different at high P. b Left panel: Schematic representation ofthe interlayer excitonic dipole repulsion model. Right panel: Lifting of degeneracyfor X2 and X1 in a two-dimensional harmonic oscillator shown schematically athigher P. Dipole repulsion results in blueshift of the states (dotted line), which ishighest for X0 (shown by a black arrow). c Peak position of X0 (symbols) plottedagainstP0.34(∝ n0), showingexcellent linearfit.dToppanel: Extracted lifetimeofX0,X1, and X2 (in open symbols) plotted with optical power, showing a weakdependence due to suppressed Auger process. The solid blue symbols (τa) indicateadditional decay path of X2 due to anharmonicity induced degeneracy lifting athigher P. Bottom panel: Percentage change in the inter-exciton peak separationwith P, indicating the degree of anharmonicity induced by P. The Regions 1 (har-monic) and 2 (anharmonic) are separated by a dashedblack line, and correlateswellwith the appearance of τa in X2. e, f The top and bottom panels show the TRPLspectra forX0 and X2, at (e) P = 2.32 and (f) 9.45μW, respectively. X2 decay becomesbi-exponential with a fast (≈1 ns) τa at higher P, while X0 decay remains mono-exponential all through.Article https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 5of protection fromAuger-induced exciton-exciton annihilation due to acombined effect of moiré trapping and strong dipolar repulsion.For a perfect two-dimensional harmonic well, X0, X1, and X2 areexpected to exhibit a degeneracy of 1, 2, and 3, respectively. Throughthe optically induced anharmonicity, we expect the degeneracy of X1and X2 to be lifted (Fig. 4b, right panel). However, our simulationsuggests only < 2 meV fine-splitting, and the inhomogeneous broad-ening of the peaks does not allow us to observe such small splitting inthe emission spectra.Interestingly, while X2 exhibits a mono-exponential decay at lowpower, its dynamics becomes bi-exponential at higher power(P> 3.9μW) with an additional lifetime of τa ~ 1 ns, as indicated by theblue solid symbols in Fig. 4d (top panel), and the TRPL spectra in the toppanels of Fig. 4e, f. In the bottompanel of Fig. 4d,wequantify thedegreeof anharmonic perturbation by plotting, from Fig. 4a, the relative mag-nitude of the peak separation (δE = δE21�δE10δE21× 100%) with incidentpower (0% corresponding to the harmonic case). The strong correlationbetween the appearance of the faster additional decay (in region 2) andthe strength of the anharmonic perturbation is evident. The fasteradditional decay likely arises from thefine-split higher energy state ofX2,which has reduced confinement into the moiré trap, thus havingenhanced decay rate (schematically shown in Fig. 4b, right panel). Notethat the decay of X0 remains mono-exponential even at higher powersince the ground state is non-degenerate (bottom panels of Fig. 4e, f).In summary, we have shown that the exciton moiré potential inheterobilayer can be dynamically tuned through external stimuli, suchas gate voltage and optical power. The usual harmonic approximationofmoiré potential breaks downunder suchperturbation. The strengthof such tunability is evidenced through moiré excitons exhibiting (a)confinement dependent tuning of features, (b) anomalous Stark shiftwhere parity is reversed with respect to conventional quantum-confined Stark effect, (c) strong modulation of the lifetime and theinter-excitonic separation, and (d) a giant spectral blueshift throughdipolar repulsion. The results will lead to intriguing experiments andapplications exploiting dynamic tuning of moiré potential.MethodsDevice fabricationWe prepared the hBN-capped WS2/WSe2 heterojunctions using asequential dry-transfer method (with micromanipulators) where theindividual layers were exfoliated from bulk crystals (hq graphene) onpolydimethylsiloxane (PDMS) using Scotch tape. For back-gatedsamples, the pre-patterned metal electrodes are prepared using pho-tolithography followed by sputtering of Ni/Au (10/50 nm) and lift-off.The entire stack (for D1 and D4) is gated from the backside (from theWS2 side) through hBN layer (dielectric) and the pre-patterned metalline. TheWS2 layer is contacted to a different electrode (Gr) for carrierinjection. After completion of the transfer process, the devices areannealed inside a vacuum chamber (10−6 mbar) at 250 °C for 5 h forbetter adhesion of the layers and removal of air bubbles. The angle andstacking between WS2/WSe2 layers are confirmed using SHG (seeSupplementary Fig. 1).PL measurementAll the PL measurements on the samples are carried out in a closed-cycle cryostat at 4 K using a × 50 objective (0.5 numerical aperture)lense. The bottom gate voltages are applied using a Keithley 2636Bsource meter (for both PL and TRPL), and then the PL spectra arecollected using a spectrometer with 1800 lines per mm grating andCCD (Renishaw spectrometer). We use the 532 nm CW and 531 nmpulsed lasers to excite the sample. The spot size for both pulsed andCW laser is ~ 1.5μm. All the power values are measured using a siliconphotodetector from Edmund Optics. All the error bars in differentplots in the manuscript indicate mean± standard deviation.TRPL measurementOur custom-built TRPL setup comprises of a 531 nm pulsed laser head(LDH-D-TA-530B from PicoQuant) controlled by the PDL-800D driver,a photon-counting detector (SPD-050-CTC from Micro Photon Devi-ces), and a time-correlated single photon counting (TCSPC) system(PicoHarp 300 from PicoQuant). The pulse width of the laser is 40 ps.For the spectrally resolved TRPL from moiré ILEs, a combination of along pass filter (cut in wavelength of 650 nm) and a wavelength-tunable monochromator (Edmund optics, 2 cm2 Square holographicgratings) with 0.5 nm resolution (corresponding to about 0.8meVresolution in the ILE spectral regime) are placed in front of the SPD.The peak position of the emission from ILEs are simultaneously mea-sured along with TRPL measurement by performing in-situ PL (seeSupplemental Material in ref. 31 for setup schematic). The instrumentresponse function (IRF) has a full-width-at-half-maximum (fwhm)of 52 ps.Exciton formation and decay modelTo fit the experimentally obtained TRPL data, we use three differentialequations:dn0ðtÞdt= f 0ðtÞ �n0ðtÞτ0ð3Þdn1ðtÞdt= f 1ðtÞ �n1ðtÞτ1ð4Þdn2ðtÞdt= f 2ðtÞ �n2ðtÞτ2ð5ÞHere ni(t) is the time dependent population density, τi is the net decaytime, and f iðtÞ= 1σiffiffiffiffiffi2πp e�ðt�τf i Þ22σ2i is the Gaussian formation function, and τfiis the formation time measured from the laser excitation time forexciton Xi, i =0, 1, 2. After solving these equations numerically, we fitthemeasured TRPL data from the threemoiré exciton emissions usingτfi, σi, and τi as fitting parameters.Model for gate-voltage dependent lifetimeThe net decay time (τi) measured in TRPL (Fig. 3b), for exciton Xi(i =0, 1, 2) is given by:1τiðVg Þ=1τr,i+1τnr0,i+1τnrg,iðVg Þð6Þwhere τr,i, τnr0,i, and τnrg,i(Vg) represent the radiative lifetime,gate voltage independent non-radiative lifetime, and the gatevoltage-dependent non-radiative lifetime, respectively. FromFig. 3d, 1τnr0,2= γ20 + γ21 + γ02 for X2, and 1τnr0,1= γ10 + γ01 for X1, and1τnr0,0= γ00, where γ0i is the rate of any other unaccounted non-radiative process for exciton Xi. Considering that the rate of thegate-dependent non-radiative process is proportional to inducedcarrier density, which in turn is an exponential function of Vg, wewrite 1τnrg,i=CieαVg , where Ci and α are fitting parameters. By notingthat 1τr,iis relative small (in Eq. (6)) and becomes smaller with anincrease in Vg, we write1τiðVg Þ≈1τiðVg =0Þ+CiðeαVg � 1Þ ð7ÞArticle https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 6Data availabilityThe data that support the findings of this study are available within themain text and Supplementary Information. Any other relevant data areavailable from the corresponding authors upon request.References1. Chiu, M.-H. et al. Determination of band alignment in the single-layer MoS2/WSe2 heterojunction. Nat. Commun. 6, 7666(2015).2. Cheng, R. et al. Electroluminescence and photocurrent generationfromatomically sharpWSe2/MoS2 heterojunction p–ndiodes.NanoLett. 14, 5590–5597 (2014).3. Dandu, M. et al. Electrically tunable localized versus delocalizedintralayer moiré excitons and trions in a twisted MoS2 bilayer. ACSNano 16, 8983–8992 (2022).4. Tran, K. et al. Evidence for moiré excitons in van der waals hetero-structures. Nature 567, 71–75 (2019).5. Mak, K. F. & Shan, J. 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K.W. andT.T. acknowledge support from the JSPS KAKENHI (GrantArticle https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 7https://arxiv.org/abs/2302.01266https://arxiv.org/abs/2302.01266Numbers 19H05790 and 20H00354). K.M. acknowledges thesupport from a grant from Science and Engineering ResearchBoard (SERB) under Core Research Grant, a grant from the IndianSpace Research Organization (ISRO), a grant from MHRD underSTARS, and support from MHRD, MeitY, and DST Nano Missionthrough NNetRA.Author contributionsK.M. designed the experiment.M.D., S.C. andS.D. fabricated thedevicesand conducted the measurements. P.D. conducted the electrostaticsimulation. R.B. and V.R. performed the SHG measurements for allsamples. K.W. and T.T. grew the hBN crystals. S.C., M.D. and K.M. con-ducted the data analysis and wrote the manuscript with inputs fromothers.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-40329-3.Correspondence and requests for materials should be addressed toKausik Majumdar.Peer review information Nature Communications thanks Ying Jiang,Yanping Liu, and theother, anonymous, reviewer(s) for their contributionto the peer review 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) 2023Article https://doi.org/10.1038/s41467-023-40329-3Nature Communications |         (2023) 14:4679 8https://doi.org/10.1038/s41467-023-40329-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Harmonic to anharmonic tuning of moiré potential leading to unconventional Stark effect and giant dipolar repulsion in WS2/WSe2 heterobilayer Results and discussion Gate tunability Unconventional Stark effect Gate tunable exciton lifetime Optical power induced anharmonicity Methods Device fabrication PL measurement TRPL measurement Exciton formation and decay model Model for gate-voltage dependent lifetime Data availability References Acknowledgements Author contributions Competing interests Additional information