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## Creator

[Raul Perea-Causin](https://orcid.org/0000-0002-2229-0147), [Samuel Brem](https://orcid.org/0000-0001-8823-1302), Fabian Buchner, Yao Lu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), John M. Lupton, [Kai-Qiang Lin](https://orcid.org/0000-0001-9609-749X), [Ermin Malic](https://orcid.org/0000-0003-1434-9003)

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[Electrically tunable layer-hybridized trions in doped WSe2 bilayers](https://mdr.nims.go.jp/datasets/10321fbb-2174-4123-85b8-86244fceb962)

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Electrically tunable layer-hybridized trions in doped WSe2 bilayersArticle https://doi.org/10.1038/s41467-024-50834-8Electrically tunable layer-hybridized trions indoped WSe2 bilayersRaul Perea-Causin 1,2 , Samuel Brem 3, Fabian Buchner4, Yao Lu5,Kenji Watanabe 6, Takashi Taniguchi 7, John M. Lupton4,Kai-Qiang Lin 4,5 & Ermin Malic 3Doped van der Waals heterostructures host layer-hybridized trions, i.e.charged excitons with layer-delocalized constituents holding promise forhighly controllable optoelectronics. Combining a microscopic theory withphotoluminescence (PL) experiments, wedemonstrate the electrical tunabilityof the trion energy landscape in naturally stackedWSe2 bilayers. We show thatan out-of-plane electric field modifies the energetic ordering of the lowestlying trion states, which consist of layer-hybridized Λ-point electrons andlayer-localized K-point holes. At small fields, intralayer-like trions yield distinctPL signatures in opposite doping regimes characterized by weak Stark shifts inboth cases. Above a doping-asymmetric critical field, interlayer-like species areenergetically favored and produce PL peaks with a pronounced Stark red-shiftand a counter-intuitively large intensity arising from efficient phonon-assistedrecombination. Our work presents an important step forward in the micro-scopic understanding of layer-hybridized trions in van der Waals hetero-structures and paves the way towards optoelectronic applications based onelectrically controllable atomically-thin semiconductors.Optical and transport properties of atomically thin semiconductorsare governed by tightly bound electron-hole complexes1–4. In theneutral regime, a weak or moderate photoexcitation generatesexcitons5,6—bound electron-hole pairs—while in doped materials eachof these quasi-particles binds to an additional charge forming trions(charged excitons)7–10. Remarkably, the properties of excitons andtrions can be tailored by stacking monolayer semiconductors into vander Waals structures11–14. In particular, naturally stacked WSe2 bilayershost excitons that are hybrids of intra- and interlayer excitonspecies15–19, combining the oscillator strength of the former20 with thepermanent dipole moment of the latter21–23. Moreover, the degree oflayer hybridization and thus the dipole moment can be tuned by anexternalout-of-plane electricfield23–28. Theelectrical tunability of layer-hybridized electron-hole complexes thus offers a platform to tailorinteractions, photoluminescence (PL), and transport properties of vander Waals heterostructures.While the physics of layer-hybridized excitons is rather wellestablished, the influence of doping has not been thoroughly under-stood yet. Initial efforts have focused on resolving the optical polar-ization of PL peaks29–32, tuning exciton PL resonances with a gatevoltage23, calculating interlayer trion binding energies33, and exploringthe realization of Feshbach resonances34. Most of these studies wereperformed directly on twisted bilayers, whose complex properties aredetermined by the moiré superlattice consisting of regions with dif-ferent atomic stacking35 and involving atomic reconstruction36. Theelectrical tunability of trions in naturally stacked bilayers has beenapproached just recently, however, focusing only on high-energystates involving the higher-lying conduction band37.Received: 7 March 2024Accepted: 16 July 2024Check for updates1Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. 2Department of Physics, Stockholm University, Stockholm, Sweden.3Department of Physics, Philipps-Universität Marburg, Marburg, Germany. 4Department of Physics, University of Regensburg, Regensburg, Germany. 5StateKey Laboratory of Physical Chemistry of SolidSurfaces, CollegeofChemistry andChemical Engineering, XiamenUniversity, Xiamen,China. 6ResearchCenterfor Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan. 7Research Center for Materials Nanoarchitectonics, NationalInstitute for Materials Science, Tsukuba, Japan. e-mail: causin@chalmers.se; kaiqiang.lin@xmu.edu.cn; ermin.malic@uni-marburg.deNature Communications |         (2024) 15:6713 11234567890():,;1234567890():,;http://orcid.org/0000-0002-2229-0147http://orcid.org/0000-0002-2229-0147http://orcid.org/0000-0002-2229-0147http://orcid.org/0000-0002-2229-0147http://orcid.org/0000-0002-2229-0147http://orcid.org/0000-0001-8823-1302http://orcid.org/0000-0001-8823-1302http://orcid.org/0000-0001-8823-1302http://orcid.org/0000-0001-8823-1302http://orcid.org/0000-0001-8823-1302http://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/0000-0001-9609-749Xhttp://orcid.org/0000-0001-9609-749Xhttp://orcid.org/0000-0001-9609-749Xhttp://orcid.org/0000-0001-9609-749Xhttp://orcid.org/0000-0001-9609-749Xhttp://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50834-8&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50834-8&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50834-8&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50834-8&domain=pdfmailto:causin@chalmers.semailto:kaiqiang.lin@xmu.edu.cnmailto:ermin.malic@uni-marburg.deIn this theory-experiment collaboration, we provide a micro-scopic picture of the trion landscape in naturally stacked doped WSe2bilayers and demonstrate the electrical tunability of the trion groundstate. Based on a variational solution of the trion Schrödinger equa-tion, we find themost energetically favorable trion states to be formedby layer-hybridized electrons at the Λð0Þ point (also denoted as Q or Σpoint in the literature16,38,39) of the Brillouin zone and layer-localizedholes at the Kð0Þ point, see Fig. 1. By combining experimental PL mea-surements with a microscopic many-body description of trionrecombination,we reveal the tunability of the PL spectraby anexternalout-of-plane electric field. The latter serves as a tuning knob to controlthe layer configuration of the trion ground state (Fig. 1a, b), favoringinterlayer-like trions at large electric fields. Through comparisonbetween experiment and theory, we identify the origin of the PL sig-natures of p- and n-type trions, which are particularly distinct at lowfields due to intricate differences in the energetic landscape of the two.In contrast, in both doping regimes, the PL at large fields is governedby interlayer-like trions, yielding a pronounced Stark red-shift of PLresonances and intense peaks owing to the efficient recombinationassisted by M-point phonons (Fig. 1c). These fundamental insightsserve as a guide for future studies aiming to exploit the electricaltunability of doped bilayer semiconductors.ResultsLayer-hybridized trion statesWe consider a system of interacting electrons and holes that cantunnel between two layers in a multi-valley band structure with band-edge energies, effective masses, and tunneling strengths based onmaterial-specific ab initio calculations38,40. In the regime of equally lowphotoexcitation and doping, the system is well described by aneffective trion Hamiltonian41 Ht,0 (see Supplementary Information, SI,for details). The eigenstates of Ht,0 are hybrid trions with the energyEtνQ = Etν0 + _2Q2=2Mν , where Mν is the hybrid trion effective mass,ν = {νh, νe1, νe2} the spin-valley configuration (for negatively-chargedtrions) andQ the center-of-massmomentum. The trion energy Etν0 andthe wave function Ψνl(r1, r2), with the relative electron-hole coordi-nates r1, r2 and the compound layer index l= {lh, le1, le2}, are obtainedbysolving the trion Schrödinger equation42–44 generalized to describelayer hybridization,Xl0Hνll0 ðr1, r2ÞΨνl0 ðr1,r2Þ= Etν0Ψνlðr1, r2Þ: ð1ÞThe Hamiltonian Hνll0 ðr1, r2Þ contains the single-particle band-edgeenergies and kinetic terms, the Coulomb interaction between thetrion’s constituent particles, and interlayer tunneling processes (seeMethods). The potential Vli ,ljðrÞ for intra- (li = lj) and interlayer (li ≠ lj)interactions is modeled following the generalization of the Rytova-Keldysh potential45,46 to bilayer systems47 with dielectric constantsfrom refs. 48,49. We focus on the qualitative description of the trionlandscape and thereforedisregard the impact of exchange interactionsresulting in the small splitting of degenerate states44,50. The Schrö-dinger equation for hybrid trions, Eq. (1), is solved via a variationalapproach considering the wave function ansatz41Ψνlðr1, r2Þ=wlN le�jr1 ja1,l�jr2 ja2,l +Cle�jr1 jb1,l�jr2 jb2,l� �, ð2Þwith the variational parameterswl, a1/2,l, b1/2,l, Cl and the normalizationfactor N l depending implicitly on the spin-valley index ν. This wavefunction allows to consider an imbalance in the effective mass of thetwo electrons (mle1≠mle2) or in the electron-hole interaction(Vlh,le1≠Vlh,le2), resulting in the reduction of the trion binding energy41.The general theory outlined above is applied here to naturallystacked WSe2 bilayers, which provide an ideal platform to exploit theelectrical tunability of hybrid electron-hole complexes23,25,31,37. Theband structure in this system is characterized by valence bandmaximawith a large spin-orbit splitting at theK(’) point of the Brillouin zone andconduction band minima with a small and large spin-orbit splitting atthe K(’) and Λ(’) points, respectively38 (Fig. 1c). The ordering of the spin-split bands is reversed in opposite layers due to their 180∘ relativeorientation. Importantly, the efficient Λ-point tunneling couples elec-tron states in opposite layers with the same spin, whereas K-pointtunneling is suppressed at the conduction band and weak at thevalence band40. Exploiting these properties of the band structure, weintroduce a concise notation to identify different trion states. Theholes are located in the upper valence band at the Kð0Þ point either inthe top or bottom layer, hence their only relevant degree of freedom isthe layer index (lh = t, b, i.e. top or bottom). For electrons, the valleyindex (νe =Λ,Λ0, K, K 0) is sufficient: Λ(Λ0)-point electrons are layer-hybridized and live mostly in the top (bottom) layer, while for K(K’)-point electrons we consider those in the upper conduction band at thetop (bottom) layer (Fig. 1). Kð0Þ-point electrons in the lower conductionband form high-energy dark trions which we have found to be irrele-vant in PL. We note that additional degenerate states with oppositespin-valley configuration have also been taken into account but areomitted from the discussion for simplicity. With these considerations,the valley and layer configuration of p(n)-type trions is denoted in thesubindex of X+νe lh1lh2(X�lhνe1νe2).We now solve the trion Schrödinger equation, Eq. (1), for an hBN-encapsulated WSe2 bilayer and obtain the trion energy landscapeshown in Fig. 2a–b (and further summarized in Table S1 in the SI). Inanalogy to excitons16,23,25,40, the strong Λð0Þ-point tunneling results in alarge hybridization-induced red-shift of trions with Λð0Þ electrons51,making these states the energetically lowest ones by far (cf. the arrowin Fig. 2a). The lowest p-type trion, X+Λ tt , contains two top-layer holesand a Λ electron that is mostly (82%) in the top layer (Fig. 2c). A fewmeV above lies X+Λ tb , with one hole in the bottom layer, whereas X+Λ0 tt ,with a Λ0 electron that is mostly (65%) in the bottom layer, is 50 meVhigher. The states X+Λ tt , X+Λ tb , and X+Λ0 tt have degenerate partners withFig. 1 | Electrical control of the trionground state. a,b Layer configuration of thetrion ground state in p- and n-dopedWSe2 bilayer for different ranges of the out-of-plane electric field ε. The subindices in X+νe lh1lh2(X�lhνe1νe2) denote the electron valleyνe(1,2) and hole layer lh(1,2). The blue ellipse surrounding the electrons illustrates thelayer hybridization. c Reduced band structure of theWSe2 bilayer. Spin-up (-down)bands from the top/bottom layer are denoted by black/gray solid (dashed) lines.Distinct PL signatures from X+Λ tt (X+Λ0 tt ) trions arise from the scattering with Λ (M)phonons to the virtual bright state followed by radiative recombination.Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 2opposite layer configuration, namely X+Λ0 bb , X+Λ0 tb , and X+Λbb , respec-tively. For n-type trions, the lowest state, X�tΛΛ, is formed by a top-layerhole and two Λ electrons that are mostly in the top layer (cf.Figure 2b, d). X�tΛΛ0 lies 10 meV above and is formed by Λ and Λ0 elec-trons that are mostly in the top and bottom layer, respectively, whileX�tΛ0Λ0 , where the two electrons are mostly in the bottom layer, is 50meV higher than X�tΛΛ. The respective degenerate states with oppositelayer configuration are X�bΛ0Λ0 , X�bΛΛ0 , and X�bΛΛ. Importantly, all theselow-lying trion species are momentum-dark, i.e. they cannot directlyrecombine due to themomentummismatch betweenKð0Þ holes andΛð0Þelectrons. Bright states, with an electron and a hole located in bandswith the same valley, spin, and layer indices, lie 130-140meV above thelowest state (orange shaded lines in Fig. 2a, b), and are thereforeexpected to have a marginal contribution to the PL spectra at thermalequilibrium even at moderate temperatures.In Fig. 2a–b we also show the onset energies for different exciton-electron continua (blue- and orange-shaded), which are obtained byminimizing the energy of a non-interacting exciton-electron com-pound with a variational hydrogenic wave function42 generalized tobilayer systems (see SI for details). The energetic offset between thetrion and the lowest exciton-electron state determines the trionbinding energy. Importantly, an imbalance in the effective mass of theequal charges or in the two possible electron-hole interactions (e.g.when the two equal charges are in different layers) results in the pre-ferential binding of one electron-hole pair and the weaker binding ofthe remaining charge, overall reducing the trion stability41. Accord-ingly, we observe that states with a mass and interaction balance (andwhere, in addition, all particles are mostly in the same layer, e.g. X+Λ tt ,X+Ktt, X�tΛΛ) have binding energies of up to 12 meV, while imbalancedstates (X+Λ tb , X�tΛΛ0 , X�tKΛ, X�tKΛ0 ) have lower binding energies or are evenunbound as is the case for X�tΛΛ0 . Our variational approach captures thelower stability52 or even unbound nature33 of states where the twoequal charges are in separate layers. However, since the energiesobtained from the variational approach represent an upper bound, wegenerally underestimate the quantitative value for trion bindingenergies21,53–55, and whether X�tΛΛ0 is unbound remains an openquestion.The rich trion landscape described above has direct implicationsfor the temperature dependence of PL spectra. To investigate this, wederive a PL formula for layer-hybridized trions based on a many-bodyequation-of-motion approach41,56 considering the direct recombina-tion of bright trions via electron recoil57,58, the phonon-assistedrecombination of dark trions, and phonon-induced spectralbroadening15,59 (see Methods and SI for details). The trion-phononinteraction is modeled in a spin-conserving deformation potentialapproach considering effective acoustic and optical modes with inputparameters from ab initio calculations60. We evaluate and plot the PLspectra in Fig. 3 considering only the energetically lowest bound trionstate for each spin-valley configuration ν that is accessed in our var-iational approach. The impact of the exciton-electron continuum isexpected to become relevant at marginal doping values and hightemperatures61.At 5 K, the PL spectrum is dominated by twopeaks (yellow areas inFig. 3) appearing 160-180 meV below the intralayer bright excitonresonance (X0), closely resembling experimental observations23,62. Thetwo peaks arise from the recombination of the energetically lowesttrion X+Λ tt (X�tΛΛ) via the transition into the virtual bright state X+Ktt(X�tKΛ) that is assisted by Λ phonons with energies close to 13 meV and30meV60 (Fig. 1c). The different electron-phonon coupling strength ofspecific phononmodes results in the unequal intensities of the two PLpeaks. In particular, the higher peak involves acoustic phonons, whichhave a larger coupling than the optical phonons that are responsiblefor the low-energy peak. When the temperature is increased, higher-lying states become thermally occupied and give rise to new spectralfeatures. At 150K, the PL spectra for p-type doping have a large (small)contribution from X+Λ tb (X+Λ0 tt ), while for n-type doping the X�tΛ0Λ0 trionleads to sizable signatures between -80 meV and -140 meV (orangeareas in Fig. 3). Additional contributions from X�tΛΛ0 could be expectedbut are not captured by our model as we only consider the PL arisingfrom bound trions. Finally, bright trion PL signatures only becomevisible above 150 K (red areas in Fig. 3) due to their large energeticoffset with respect to the trion ground state (cf. Fig. 2a, b), and dom-inate the spectra at room temperature. Note that in some experiments,significant PL signatures from bright states are observed even atcryogenic temperatures due to the non-equilibrium distribution cre-ated during continuous-wave high-energy optical excitation23,25,29,62.Electrical control of trion photoluminescenceAfter having determined the valley configuration and layer mixing ofthe lowest lying trions aswell as their PL signatures, we investigate howthese states can be controlled by an out-of-plane electric field. Weshow a direct comparison between theoretically predicted andexperimentally measured field-dependent PL.First, we fabricated a dual-gate natural WSe2 bilayer device tocontrol both the doping and the out-of-plane electric field in the WSe2layers37 (see Fig. S1 in the SI).Wedemonstrate thedoping control of theenergetically lowest trion states bymeasuring the doping dependenceof the PL from the WSe2 bilayer at 5 K (Fig. S2). After confirming thedoping density needed to form trions, we measure the dependence ofthe trion PL signatures on theout-of-plane electricfieldwhile keeping aconstant doping density. The out-of-plane electric field is determinedfrom the applied bottom (top) gate voltage Vbg(tg) viaε= ϵhBNðVbg � V tgÞ=ðϵWSe2ðdbg +dtgÞÞ, where dbg(tg) is the thickness ofthe bottom (top) hBN insulator layers and ϵhBN ≈ 3.4 and ϵWSe2≈7:2 arethe out-of-plane dielectric constants of hBN63,64 and WSe265. InFig. 4a–b we show the PL spectra for p- and n-type trions, which weremeasured at Vtg + Vbg = − 3 and 0V, respectively.Similar to the low-temperature theoretical predictions in Fig. 3,the PL in the absence of an electric field is dominated by two peakslocated 140-160 meV below the intralayer bright exciton resonanceFig. 2 | Trion landscape. a, b Energetic trion landscape for p- and n-type trions inbilayer WSe2 (relative to the lowest state). The valley and layer configuration ofp(n)-type trion states is described in the subindex of X+νe lh1 lh2(X�lhνe1νe2). Dark (bright)bound trion states are denoted with blue (orange) lines, while the correspondingexciton–electron continua are shaded in light-blue (orange). The large tunneling-induced energy shift at the Λð0Þ point (denoted by an arrow in a) makes trionscontaining Λ electrons the energetically lowest states. c, d Schematic illustration ofthe valley and layer configuration of different trion species.Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 3(Fig. 4a, b). For p-type doping, as the electric field is switched on, thetwo peaks undergo a red-shift due to the Stark effect (Fig. 4a),reflecting the dipole length d of the recombining electron-hole pair,which we extract to be d ≈ 0.14–0.15 nm (see Fig. S4). In addition, afaint blue-shifting peak emerges at -135 meV. At larger fields above0.09 V/nm, the PL becomes dominated by two intense peaks exhi-biting a more pronounced red-shift (corresponding to d ≈ 0.39 nm).In the case of n-type doping and a small electric field, the two initialpeaks split into branches exhibiting opposite Stark shifts (Fig. 4b). Inthis regime, we can only unambiguously extract the dipole lengthd ≈ −0.13 nm for the higher energy peak. At intermediate fieldsbetween 0.03 V/nm and 0.08 V/nm, the PL shows several signaturesexhibiting distinct Stark shifts. In analogy to the p-type case, at largerfields the PL is dominated by two intense peaks with a more pro-nounced red-shift (d ≈ 0.42 nm). For negative electric fields, weobserve the same behavior for trion species with the opposite layerconfiguration (i.e. reversed dipole moment), see Fig. S5. We note thatthe field tunability of the trion PL is qualitatively similar for differentdoping densities in the range of ~ 1011 − 1012 cm−2 (Fig. S5). We alsoemphasize that the doping densities for two layers can differdepending on the out-of-plane electric field, which however is notexpected to change the main trends of the Stark shifts37. While ourobservations are similar to the previously observed field-inducedswitching between low- and high-dipole regimes for excitons23,25,27,28,here we demonstrate the analogous effect for trions and revealed theasymmetric tunability of opposite doping regimes.Now, we make use of our microscopic model to understand theunderlying nature of these experimental observations. We obtain thetrion energies andwave functions by solving Eq. (1) for different valuesof the electric field ε, which is incorporated into the theory via the shiftof the single-particle band-edge energies. This allows us to computethe field-dependent PL spectra shown in Fig. 4c, d. Furthermore, weextract the slope of the field-induced shift of trion PL resonances,which corresponds to the dipolemoment of the recombining electron-hole pair and is hence a two-body quantity (details in the SI). Weemphasize that, despite trions being three-body objects, trion PLinvolves the recombination of one electron with one hole—therefore,the Stark shift of the PL resonance is well described by the afore-mentioned dipole moment. In the following, we consider a trion (lat-tice) temperature of 30 (4) K to simulate the experimentally realisticscenario where a high-energy continuous excitation results in a hotnon-equilibrium trion distribution58,59.In the case of p-type trions, our calculations show that X+Λ ttbecomes the energetically lowest state when an electric field is applied(Fig. 4e), giving rise to the two red-shifting peaks in PLwith d = 0.13 nm(Fig. 4c). While X+Λ tt is intralayer-like (i.e. all charges aremostly locatedin the same layer, cf. Fig. 1a), the finite probability that electrons andholes are in opposite layers results in the small but sizable dipolemoment and Stark shift. The faint blue-shifting signatures (withd = − 0.11 nm) that appear at small fields arise from the recombinationof the Λ0 electron with the bottom-layer hole in X+Λ0 tb , which liesenergetically close to the trion ground state and thus has a sizablethermal occupation. In this case, the dipole formed by the recombin-ing electron-hole pair points in the direction opposite to the electricfield leading to the observed blue-shift. In contrast, the recombinationwith the top-layer hole corresponds to a dipole (d = 0.53 nm) alignedwith the field, resulting in a red-shifting signature that is hidden underthe dominating X+Λ tt peaks in Fig. 4c but slightly visible in the experi-ment (Fig. 4a). At ε > 0.15 V/nm, X+Λ0 tt becomes energetically favorable(Fig. 4e) as the electron-hole separation in the direction of the electricfield is maximized (Fig. 1a), and dominates the PL with two intensepeaks displaying a strong red-shift (d = 0.50 nm). The counter-intuitively larger PL intensity ofX+Λ0 tt despite its interlayer character is adirect consequence of the strong coupling of M-point phonons thatgovern the Λ0 ! K transition, whereas the PL arising from X+Λ tt isinstead governedbyΛ-point phonons (Fig. 1c) characterizedbyweakerdeformation potentials60. The theoretically predicted field-dependentPL (Fig. 4c) is in excellent qualitative agreement with the measuredPL (Fig. 4a).The electrical tunability of the ground state for n-type trions ischaracterized by the three distinct regimes sketched in Fig. 1b. First, atsmall electric fields, X�bΛ0Λ0 becomes the trion ground state (Fig. 4f),while X�tΛΛ is less favorable as the energy of Λ electrons that aremostlyin the top layer is pushed up by the electric field. Nevertheless, the twostates remain energetically close to each other and are similarlypopulated at the trion temperature of 30 K considered, resulting in thesplitting of the initial PL signatures into blue- and red-shifting peakswith d = −0.11, + 0.13 nmcorresponding toX�bΛ0Λ0 andX�tΛΛ, respectively(Fig. 4d). The blue- (red)-shift of the X�bΛ0Λ0 (X�tΛΛ) signatures reflects thedipole orientation of the recombining electron-hole pair opposite to(aligned with) the electric field. Interestingly, at intermediate fieldsbetween 0.05 and 0.13 V/nm the unbound X�tΛΛ0 becomes the loweststate. Since our theory only captures PL arising from bound trionstates, we have replaced this regime by a white box in Fig. 4d. Never-theless, we expect the presence of peaks arising from the recombi-nation of the top-layer hole with Λ (Λ0) electrons displaying red-shiftswith d ≈ 0.1 (0.5) nm, similar to the experimental observation.For ε > 0.13 V/nm, X�tΛ0Λ0 becomes the ground state and dominatesthe PL with intense peaks displaying a pronounced red-shift char-acterized by the large dipole length d = 0.49 nm. The critical electricfield at which the trion with larger interlayer character (i.e. X�tΛ0Λ0 andX+Λ0 tt ) becomes the ground state is thus smaller for n-type than forp-type doping. The reason for this is the smaller energy separationbetween the competing X�tΛΛ0 and X�tΛ0Λ0 states (compared to thatbetween X+Λ tt and X�Λ0 tt ) that must be overcome by the field-inducedshift (Fig. 4e-f). In addition, the PL intensity ratio between intra- andinterlayer-like trions (at small and high fields, respectively) is larger forn-type trions (Fig. 4c-d) due to the internal structure of the virtualbright states involved in the process. Specifically, the recombiningelectron-hole pair in X�tKΛ0 (the virtual state for the recombination ofX�tΛ0Λ0 ) is more tightly bound than in X�bK0Λ0 (the virtual state for therecombination of X�bΛ0Λ0 ), resulting in a larger trion-photon matrixelement—which scales with the probability that the recombiningelectron and hole are at the same position (see SI for more details).Overall, our theoretical model reproduces the main experimentalobservation, i.e. the ground state transition from intralayer-like tointerlayer-like trion species at sufficiently large electric fields. In par-ticular, the extracted dipole moments from the Stark shift of PLresonances in the experiment are in good agreement with the theo-retical predictions (cf. Table 1). In addition, we capture the asymmetricFig. 3 | Temperature-dependent trion photoluminescence. a Calculated trion PLspectra in p- and (b) n-type bilayer WSe2 at 5 K, 150 K, and 300 K with disentangledcontributions from different trion states. The energy is offset with respect to theintralayer bright exciton (X0) resonance, and the intensity is normalized withrespect to themaximum for each temperature. Note that the twopeaks in a andb at5 K correspond to phonon sidebands arising from the same state (X+Λ tt for p-typeand X�tΛΛ for n-type doping).Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 4behavior of p- and n-type trion PL, including the initial peak splitting,larger intensity ratio, and lower critical field for n-type doping. Thequantitative discrepancies between theory and experiment withrespect to the critical electric field values for the different regimescould arise fromuncertainties in the spin-orbit splitting of the Λ valley,additional screening effects from the gates, or the experimental esti-mation of the electric field strength.DiscussionWehave provided amicroscopic description of the electrically tunabletrion energy landscape in naturally stacked WSe2 bilayers. We haveshown that the lowest lying trions are composed of layer-hybridizedelectrons at theΛ andΛ0 points and layer-localized holes at theK andK’points, and dominate the PL spectra via phonon-assisted recombina-tion. By combining our microscopic theory with experimental mea-surements we have demonstrated the doping-asymmetric tunability ofthe trion ground state via an electric field, which is manifested in dis-tinct PL signatures. At low fields, the PL for p-type doping is governedby intralayer-like X+Λ tt trions giving rise to two red-shifting peaks. Incontrast, for n-type doping both intralayer-like X�bΛ0Λ0 and X�tΛΛ statesare similarly populated resulting in blue- and red-shifting peaks due totheir opposite layer configuration. Furthermore, we have shown thatabove a doping-asymmetric critical field the PL is dominated byinterlayer-like trions, resulting in peaks with pronounced red-shiftsand high intensity reflecting the large electron-hole separation and theefficient scattering with M-point phonons, respectively.The electrical tunability of the trion luminescence energy and PLintensity demonstrated here opens up newpossibilities for integratingatomically thin semiconductors in optoelectronic deviceswhere trionsact as charge carriers21. The control over the trion layer configurationshould significantly influence trion-trion interactions with importantimplications for charge transport25 and for the stabilization of exoticquantum phases66,67, and should also be relevant for the study ofexciton-electron Bose-Fermi mixtures67,68. Our insights will guidefuture studies exploring and utilizing the electrical tunability of themany-body landscape of electron-hole complexes in van der Waalsheterostructures.MethodsTrion eigenstatesLayer-hybridized trion eigenstates fulfill the Schrödinger equation (1),which is determined by the three-body HamiltonianHll0 ðr1, r2Þ= Hð0Þl ðr1, r2Þ+HðCÞl ðr1, r2Þh iδll0 +HðtunÞll0 :The latter is derived by starting from the electron-hole picture (seedetails in SI) and it implicitly depends on the compound valley index ν.Here, the kinetic term reads Hð0Þl ðr1,r2Þ= ~Etl �_2∇2r12μlh le1� _2∇2r22μlh le2� _2∇r1�∇r2~mlh,where ~Etl is the sum of the band-edge energies of the trion’sconstituents, ~mlhis the effective hole mass, and μ�1lh lnis the reducedelectron-hole mass with ln being the single-particle layer index. Theexternal out-of-plane electric field ε is incorporated into the model viathe shift of the single-particle energies, ~Ee=hln! ~Ee=hln± σlne0dε=2, with +(−) for electrons (holes), σtop = + 1, σbottom =− 1, and the layer separationd = 0.65 nm48. The Coulomb interaction between the trion’sconstituent particles is described by HðCÞl ðr1,r2Þ=Vle1le2ðr2 � r1Þ �Vlh le1ðr1Þ � Vlh le2ðr2Þ, where the potential for intra (Vl = l0 ) and interlayerFig. 4 | Direct theory-experiment comparison of the electrically tunable trionphotoluminescence. a, b Experimentally measured and (c, d) theoretically pre-dicted electric-field tunability of the trion photoluminescence in p- and n-typebilayer WSe2. The energies are offset with respect to the intralayer bright excitonX0. The trion species responsible for different peaks are denoted in (c) and (d). Thewhite box in d corresponds to the regime governed by the unbound X�tΛΛ0 statewhich is not captured by our PL model. e, f Theoretically predicted electric-fielddependence of the trion energies for X+Λ tt /X�tΛΛ (dark yellow), X+Λ0 bb /X�bΛ0Λ0 (yellow),X+Λ0 tt /X�tΛ0Λ0 (red), X+Λ tb /X�tΛΛ0 (green), and X+Λ0 tb /X�bΛΛ0 (turquoise). The shaded areasare colored according to the energetically lowest trion state at different elec-tric fields.Table 1 | Experimentally extracted (dexp) and theoreticallypredicted (dth) dipole lengths of the recombining electron-hole pair in the most relevant trion speciesTrion dexp (nm) dth (nm)X+Λ tt 0.14 − 0.15 0.13X+Λ0 tb – −0.11, 0.53X+Λ0 tt 0.39 0.50X�bΛ0Λ0 −0.13 −0.11X�tΛΛ – 0.13X�tΛ0Λ0 0.42 0.49For X+Λ0 tb two values are given corresponding to the recombination of the electron with thebottom- and top-layer hole, respectively. The exact fitting values anderrors for dexp are shown inthe SI.Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 5(Vl≠l0 ) interactions is modeled by generalizing the Rytova-Keldyshpotential45,46 to bilayer systems47 with dielectric constants fromrefs. 48,49. Finally, tunneling of the trion’s constituents fromone TMD layer to the other is modeled byHðtunÞll0 = teδ�le1,l0e1δle2,l0e2δlh,l0h+ teδle1,l0e1δ�le2,le2δlh,l0h+ thδle1,l0e1δle2,l0e2δ�lh,l0h,where te/h is the tunneling strength for electrons/holes and �ln denotesthe layer opposite to ln. The energetically lowest trion state for eachvalley configuration is obtained by minimizing the energy Etν0 with thewave function ansatz in Eq. (2) via a combination of global and localoptimization algorithms (see SI for more details).Microscopic model of trion luminescenceThe photoluminescence arising from layer-hybridized trions is descri-bed by extending the direct and phonon-assisted trion PL formula fromref. 41 to bilayers with finite tunneling. In particular, we considerinteracting electrons, holes, phonons, andphotons in a semiconductingbilayer, transform the Hamiltonian to trion basis taking only intoaccount the Fock subspace of single trions59, and obtain a formula forthe frequency-dependent PL intensity IPL(ω) (Eq. S2.5 in the SI) byexploiting the Heisenberg equation within the cluster expansion andtruncation scheme56,69. We obtain a contribution describing the directrecombination of bright trions via the electron recoil effect57,58, and asecond contribution accounting for phonon-assisted recombination bywhich dark trions can emit light. More details can be found in the SI.Device fabricationAll materials were mechanically exfoliated from bulk crystals (WSe2from HQ Graphene, hBN from NIMS, graphite from NGS Trading andConsulting) onto siliconwafers with a 285 nm silicon dioxide top layer.At a temperature of 120 °C, a thin polycarbonate film was used tosequentially pick up a cover hBN layer, a few-layer graphene (the topgate), a top hBN layer, a WSe2 bilayer, a bottom hBN layer, and asecond few-layer graphene (the bottom gate). The top and bottomhBN layers have the same thicknessof 32 nm,whichwasdeterminedbythe atomic force microscopy. This assembly was then released at ahigher temperature of 170 °C on a SiO2/Si substrate with pre-patternedplatinum electrodes. The thin polycarbonate film was dissolved awayusing chloroform as a solvent and the device was further cleaned withisopropyl alcohol. The schematic diagram and the microscopic imageof the dual-gate device are shown in Supplementary Fig. S1.Optical spectroscopyThe device was placed in a helium-flow microscope cryostat (CRYO-VAC, Konti model) and cooled down to 5 K. We used a microscopeobjective with a numerical aperture of 0.6 (Olympus, LUCPLFLN, 40xmagnification) to focus the laser at 488 nm (Sapphire 488, CoherentGmbH) onto the sample and to collect the reflected signals. A 488 nmlong-pass filter was inserted into the detection beam path to removethe laser line. The photoluminescence (PL) signal was dispersed by adiffraction grating of 1200 grooves per millimeter and detected by acharge-coupled device (CCD) camera (Princeton Instruments, PIXIS100). The bottom and top gates of the device were connected to twoseparate source-measure units (Keithley 2400), and the WSe2 bilayerwas connected to the common ground. Supplementary Fig. S2 showsthe PL spectrum of natural bilayer WSe2 at different gate voltagecombinations for doping dependence and electric-field dependence.Supplementary Fig. S3 shows the PL spectrum of the sample atVtg = Vbg = −0.3 V, i.e. in the undoped regime.Data availabilityAll the data generated in this study are available in the article andsupplementary information or from the corresponding author uponrequest.Code availabilityThe codes used to generate and analyse the data are available from thecorresponding author upon request.References1. Mak, K. F. & Shan, J. Photonics and optoelectronics of 2D semi-conductor transition metal dichalcogenides. Nat. Photonics 10,216–226 (2016).2. Wang, G. et al. Colloquium: excitons in atomically thin transitionmetal dichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).3. Mueller, T. &Malic, E. Excitonphysics anddevice applicationof two-dimensional transition metal dichalcogenide semiconductors. npj2D Mater. Appl. 2, 29 (2018).4. 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Strong-coupling phases of trions and excitons inelectron-hole bilayers at commensurate densities. Phys. Rev. Lett.132, 196202 (2024).67. Qi, R. et al. Electrically controlled interlayer trion fluid in electron-hole bilayers. Preprint at https://arxiv.org/abs/2312.03251 (2023).68. Zerba, C., Kuhlenkamp, C., Imamoğlu, A. & Knap, M. Realizingtopological superconductivity in tunable Bose-Fermi mixtures withtransition metal dichalcogenide heterostructures. Phys. Rev. Lett.133, 056902 (2024).69. Kira, M. & Koch, S. W. Many-body correlations and excitonic effectsin semiconductor spectroscopy. Prog. Quantum Electron. 30,155–296 (2006).AcknowledgementsR.P.-C. acknowledges fruitful discussions with Joakim Hagel and DanielErkensten (Chalmers University of Technology). This work has beenfunded by the Deutsche Forschungsgemeinschaft (DFG) via SFB 1083and the regular project 542873285. K.-Q.L. acknowledges the Funda-mental Research Funds for the Central Universities (20720230009), andfunding support from the DFG via SFB 1277 (B11 314695032) and SPP2244 (443378379). K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) andWorld Premier International Research Center Initiative (WPI),MEXT, Japan.Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 7https://arxiv.org/abs/2310.08729https://arxiv.org/abs/2312.03251Author contributionsR.P.-C., S.B., and E.M. conceived the research and developed the theo-retical model. F.B., Y.L., J.M.L., and K.-Q.L. carried out the experiments.K.W. and T.T. synthesized the hBN crystals. R.P.-C. performed the cal-culations and wrote the manuscript with input from all authors.FundingOpen access funding provided by Chalmers University of Technology.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-50834-8.Correspondence and requests for materials should be addressed toRaul Perea-Causin, Kai-Qiang Lin or Ermin Malic.Peer review information Nature Communications thanks Kyoung-DuckPark, Jiajie Pe and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. 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If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-50834-8Nature Communications |         (2024) 15:6713 8https://doi.org/10.1038/s41467-024-50834-8http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Electrically tunable layer-hybridized trions in doped WSe2 bilayers Results Layer-hybridized trion states Electrical control of trion photoluminescence Discussion Methods Trion eigenstates Microscopic model of trion luminescence Device fabrication Optical spectroscopy Data availability Code availability References Acknowledgements Author contributions Funding Competing interests Additional information