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Matteo Barbone, Alejandro R-P. Montblanch, Dhiren Kara, Carmen Palacios-Berraquero, Alisson Cadore, Domenico De Fazio, Benjamin Pingault, Elaheh Mostaani, Han Li, Bin Chen, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Sefaattin Tongay, Gang Wang, Andrea Ferrari, Mete Atature

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[Charge-tuneable biexciton complexes in monolayer WSe2](https://mdr.nims.go.jp/datasets/9e53450c-3189-49ad-a10c-537dd9de491f)

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Charge-tuneable biexciton complexes in monolayer WSe2ARTICLECharge-tuneable biexciton complexes in monolayerWSe2Matteo Barbone1,2, Alejandro R.-P. Montblanch1, Dhiren M. Kara1, Carmen Palacios-Berraquero1,Alisson R. Cadore2, Domenico De Fazio2, Benjamin Pingault1, Elaheh Mostaani2, Han Li3, Bin Chen3,Kenji Watanabe 4, Takashi Taniguchi4, Sefaattin Tongay3, Gang Wang2, Andrea C. Ferrari2 & Mete Atatüre1Monolayer transition metal dichalcogenides have strong Coulomb-mediated many-bodyinteractions. Theoretical studies have predicted the existence of numerous multi-particleexcitonic states. Two-particle excitons and three-particle trions have been identified by theiroptical signatures. However, more complex states such as biexcitons have been elusive dueto limited spectral quality of the optical emission. Here, we report direct evidence of twobiexciton complexes in monolayer tungsten diselenide: the four-particle neutral biexciton andthe five-particle negatively charged biexciton. We distinguish these states by power-dependent photoluminescence and demonstrate full electrical switching between them. Wedetermine the band states of the elementary particles comprising the biexcitons throughmagneto-optical spectroscopy. We also resolve a splitting of 2.5 meV for the neutral biex-citon, which we attribute to the fine structure, providing reference for subsequent studies.Our results unveil the nature of multi-exciton complexes in transitionmetal dichalcogenidesand offer direct routes towards deterministic control in many-body quantum phenomena.DOI: 10.1038/s41467-018-05632-4 OPEN1 Cavendish Laboratory, University of Cambridge, JJ Thomson Ave., Cambridge CB3 0HE, UK. 2 Cambridge Graphene Centre, University of Cambridge,Cambridge CB3 0FA, UK. 3 School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA. 4National Institute forMaterials Science, Tsukuba, Ibaraki 305-0034, Japan. These authors contributed equally: Matteo Barbone, Alejandro R.-P. Montblanch. Correspondence andrequests for materials should be addressed to A.C.F. (email: acf26@cam.ac.uk) or to M.Aür. (email: ma424@cam.ac.uk)NATURE COMMUNICATIONS |  (2018) 9:3721 | DOI: 10.1038/s41467-018-05632-4 |www.nature.com/naturecommunications 11234567890():,;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/0000-0003-3701-8119mailto:acf26@cam.ac.ukmailto:ma424@cam.ac.ukwww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsIn monolayer (1L) transitionmetal dichalcogenides (TMDs), thethree-atom thickness of the material reduces the dielectricscreening with respect to their bulk counterparts1,2. As a resultof this and of their large effective mass, excitons (quasi-particlestates formed of electrons and holes via Coulomb interaction) havebinding energies of hundreds of meV1,2 and are stable at roomtemperature. The physics of light–matter interaction is also enri-ched by two inequivalent valleys having opposite spin-locked valleyindices3 at the K points of the Brillouin zone, in which radiativerecombination generates photons carrying opposite angularmomenta4,5. These properties motivated the exploration of excitonand polariton6 condensation7,8 and superfluidity9, and the exploi-tation of the spin and valley degrees of freedom as means to carryand manipulate information in quantum optoelectronic devices3,10.In the limit of quantum-confined excitons, the presence of localisedsingle-photon emitters that can be induced deterministically11,12and generated by electroluminescence13, makes TMDs a promisingplatform for the field of quantum photonics. Contrary to theexciton14,15 and trion16,17 states, optical studies of biexcitoncomplexes18,19 in 1L-TMDs have been challenging20–26: inhomo-geneous broadening27 and defect bands28 have limited theirunambiguous identification and control. As a consequence, pre-vious experimental findings20–23,25,26 assigned neutral biexcitonsa larger binding energy than trions, in contrast to theoretical pre-dictions29–33, whereas ref. 24 observed a peak in 1L-molybdenumdiselenide (MoSe2) in the expected energy range, which theylabelled as the neutral biexciton.Here, we use continuous wave photoluminescence (PL) mea-surements at cryogenic temperature combined with electricalgating and magnetic field to identify the four-particle neutralbiexciton (XX0) and the five-particle negatively charged biexciton,the quinton29 (XX−) in 1L-tungsten diselenide (WSe2). We alsoobserve a splitting in XX0, which we attribute to its fine structure.Our results demonstrate tuneable access to multi-exciton com-plexes in TMDs and provide new ways to study and controlmulti-exciton phenomena.ResultsDesign and optical characterisation of heterostructures. We userecent advances in material and device processing27,34 to suppressthe effects that degrade the optical quality of 1L-WSe2. To reducethe PL spectral linewidths27 we place a layered material hetero-structure (LMH) formed of 1L-WSe2 encapsulated between twoflakes of multilayer hexagonal boron nitride (ML-hBN) on a Si/SiO2 substrate. To suppress the effect of SiO2 charge traps weplace a few-layer graphene (FLG) crystal below the bottom ML-hBN. The inset of Fig. 1a shows a schematic of the LMH (seeMethods, and Supplementary Notes 1 and 2 for sample pre-paration and characterisation).We illuminate the LMH with continuous laser excitation at 658nm and collect its optical emission at 4 K (see Methods for furtherdetails on the optical measurements): Fig. 1a is a representative PLspectrum. Consistent with previous reports, we identify the brightneutral exciton10, X0, at ~1.728 eV (width ~5meV), the negativelycharged intervalley trion35, X−inter, at ~1.699 eV, the negativelycharged intravalley trion35, X−intra, at ~1.692 eV, and the darkneutral exciton36,37, X0dark, at ~1.685 eV. Here, bright refers toexcitons with in-plane dipole and spin-allowed radiativerecombination2,36,37, whereas dark refers to excitons with out-of-plane dipole and spin-forbidden radiative recombination2,36,37,for which emission only occurs in plane but is captured partiallyby our high numerical aperture objective. The peak at ~1.711 eV,~4meV wide, is a good candidate for XX0, as it appears in thetheoretically predicted energy range29–32. The peak at ~1.679 eV,~6meV wide, was previously labelled as neutral biexcitonemission20, although it appears in the energy range predicted29,31for XX−. In the top part of Fig. 1a, we include the emissionenergies of single- and multi-exciton species in ML-WSe2calculated via diffusion Quantum Monte Carlo29 combined withenvironment screening (See Methods for details).Unveiling the presence and nature of biexcitons. Figure 1bdisplays the PL intensity I, defined as peak area, as a function ofexcitation power P (with I ∝ Pα) for X0 (filled black circles), XX0(filled red circles) and XX− (filled blue circles). For reference, weplot solid curves corresponding to a linear (α= 1, black) andquadratic (α= 2, red) behaviour. We expect superlinear beha-viour for biexcitons reaching α= 2 in thermodynamic equili-brium18,19. The power dependence of XX0 follows the quadraticcurve, while that of XX−is superlinear with fitted α ~ 1.55 ± 0.03(dashed blue curve). Both trends of XX0 and XX−are thereforeconsistent with a biexcitonic origin and contrast the linearbehaviour of X0. The deviation of XX− from α= 2 possibly stemsfrom the competition of electron capture from other opticallyinduced excitons. Remaining peaks of Fig. 1a follow anapproximately linear power dependence.To differentiate the charged and neutral biexciton XX0 andXX−, we fabricate a charge-tuneable device starting from a newLMH analogous to the first one but with the addition of oneelectrode to the FLG and of a second electrode to an uncovered1L-WSe2 portion (see Methods). Figure 2a, b shows the schematicand the optical image of the device, respectively. Figure 2cdisplays how the PL spectrum is modified as a function of voltageV. The charging regime modifies the optical signatures ofIntensity (a.u.)1.64 1.66 1.68 1.70 1.72 1.74101102103104101100 Intensity (a.u.)a bEnergy (eV)X0XX0XX–SiO2GraphitehBNWSe21.7281.6991.6921.6851.6791.711� = 1� = 24 KPower (μW)X0XX0XX–Fig. 1 PL spectrum and power dependence of encapsulated 1L-WSe2 at 4 K. a PL spectrum (black curve, linear scale) of encapsulated 1L-WSe2. Excitationwavelength: 658 nm. The top part of the figure lists the calculated spectral locations of X0 (grey), XX0 (red) and XX− (blue) in the presence of a screeningenvironment. b Double logarithmic plot of PL intensity as a function of excitation power for X0 (black filled circles), XX0 (red filled circles) and XX−(bluefilled circles). The solid curves represent I ∝ Pα for a quadratic (α= 2, red) and linear (α= 1, black) behaviour. The dashed blue curve is a fit to PL intensity,yielding an α of 1.55. For clarity of display, we multiply XX0 by 4 and X0 by 0.4ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05632-42 NATURE COMMUNICATIONS |  (2018) 9:3721 | DOI: 10.1038/s41467-018-05632-4 | www.nature.com/naturecommunicationswww.nature.com/naturecommunications1L-WSe2 strongly. The presence of X0 and X0dark at V ~ 0 Vshows that the material has a negligible intrinsic charge doping.At the same bias, Fig. 2c also shows emission from XX0. In theelectron-charged regime (V > 0) fluorescence from X0, XX0 andX0dark vanishes, while emission from X−inter, X−intra andXX−arises. Around 2 V the X− emission switches to a new peakat ~1.681 eV, likely the next charging state of the trion, X−−. Thispeak was previously assigned to the fine structure of X− inexperiments on bare material10. Negative bias is the hole-chargedregime, where only X0 and the positively charged trion X+ arevisible (refs. 10,35). The voltage-dependence of our PL measure-ments thus clarifies the difference between the two biexcitonspecies: the presence of XX0 only at charge neutrality confirmsthis is the charge-neutral biexciton, and the appearance of XX−only in the electron-charged regime proves it to be the negativelycharged biexciton.We then analyse the correlation between excitation andemission polarisations in the different charging regimes (Fig. 2d).We plot the degree of circular polarisation [DoP= (Ico.- Icross.) /(Ico.+ Icross.)4] where Ico. (Icross.) is the intensity of the circularlypolarised light with the same (opposite) helicity in the excitationand detection paths. We refer to the two orthogonal helicities asσ− and σ+. At 0 V, XX0 has DoP > 80%, while X0dark shows nocircular polarisation36,37, as expected. At 0.8 V, X−inter has DoP >90%, X−intra has DoP < 10% and XX− has DoP ~ 55%. Thecircular polarisation of photons from both XX0 and XX− thusimplies that dissociation occurs with the recombination of abright exciton, as a dark exciton would emit linearly polarisedlight36,37. The DoP of XX− is close to the average of the DoP ofX−inter and X−intra, suggesting that the recombination mechan-isms of both X−inter and X−intra contribute38 to that of XX−.Behaviour of exciton complexes in magnetic field. The electronsand holes comprising the biexcitons can occupy multiple com-binations of band states. To identify them, we resort to the var-iation of PL as a function of an out-of-plane magnetic field.Figure 3a shows the σ− polarised PL of X0 and XX0 under co-polarised (σ−) excitation. We resolve a finite splitting in the XX0emission, with a separation of 2.5 meV between the two peakslabelled XX01 and XX02 (line-cut spectra at different magneticfields are shown in Supplementary Fig. 3). Figure 3b shows the σ+polarised PL of X0 and XX0 under cross-polarised (σ−) excitation.Here, only XX02 remains visible, revealing a different DoP forXX01 and XX02, in analogy to the different DoP between X−interand X−intra.The energy of the splitting excludes one of the peaks to be aphonon replica, and the two peaks reveal different DoP, thus weassign this doublet to fine structure introduced by exchangeinteraction, in analogy to the case of the splitting between X−interand X−intra39. This experimental observation of the XX0 finestructure will set a reference for further computational studies,which otherwise suffer from limitations due to the complextreatment of the exchange interaction. Additionally, the PLintensity of XX0 emission increases when it shifts to higherenergies, in contrast to that of X0. We observe the same behaviourfor XX− in Fig. 3c, where the co-polarised PL from therecombination of the quasi-particle also shows valley-dependentZeeman shift.In Fig. 3d we plot the energies of X0, XX01, XX02 and XX− as afunction of magnetic field. For each multi-exciton species, wecalculate the Landé factor g, defined as ΔE= gμBB, where ΔE=Eσ+−Eσ− is the difference in the emission energy of excitons inopposite valleys, μB= eħ/2me= 58 μeV T−1 is the Bohr magnetonand B is the magnetic field. We derive g ~ -4.44 ± 0.12 for X0consistent with previous observations40, ~ -4.10 ± 0.15 for XX0and ~ -3.86 ± 0.17 for XX−. We note that these values do notrepresent the total g factor of the multi-particle states, but ratherbelong to their optically active components.The emission intensities of XX0 and XX− change dramaticallywith magnetic field, being stronger when shifted to higher energy.Figure 3e displays the Iσ-/σ-/Iσ+/σ+ ratio as a function of magneticfield for XX01+ XX02 and XX−. For comparison, we also includeIσ-/σ-/Iσ+/σ+ for X0. At zero magnetic field Iσ-/σ-/Iσ+/σ+ is ~1 for allpeaks, i.e., the two valleys have the same exciton population.When magnetic field is applied, Iσ-/σ-/Iσ+/σ+ remains unaffectedfor X0. This can be explained by X0 in each valley recombiningbefore reaching thermal equilibrium. In stark contrast, XX0 andXX− display strongly anti-symmetric magnetic-field dependence:for increasing magnetic field, the lower-energy transition isweaker.We can understand the complex behaviour of the magnetic-field dependent PL through the single-particle picture of theenergy bands. Figure 4a, b, c illustrates the effect of B > 0 on the1.68 1.70 1.72a dcSiO2GraphiteAuWSe2AuAuVhBNWSe2–110013.01.50.0–1.5–3.01.68 1.70 1.723.01.50.0–1.5–3.0Bias (V)Bias (V)Energy (eV) Energy (eV)XX0X–inter X–intraX– –X0X+XX0XX–XX–X0darkX0darkX– –X0X+Ico.bX–inter X–intraIco.–Icross.Ico.+Icross.Fig. 2 Charge dependence of PL. a Schematic and b optical image of the charge-tuneable device. The red dashed frame highlights the 1L-WSe2 flake. Thescale bar is 5 μm. c Circular co-polarised PL intensity (Iσ+/σ++ Iσ-/σ-) as a function of applied bias. The dashed lines are a guide to the eye to highlight eachpeak. d DoP of PL as a function of bias and energy in the same range as (c). The colour code is such that blue regions indicate co-polarisation, whereas thered regions indicate counter-polarisation with respect to excitation polarisationNATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05632-4 ARTICLENATURE COMMUNICATIONS |  (2018) 9:3721 | DOI: 10.1038/s41467-018-05632-4 |www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsband structure of 1L-WSe2 around the K and K′ points,considering the contribution of the spin, valley and atomicorbital magnetic moments. The 1L-WSe2 bandgap decreases(increases) in the K (K′) valley as the energies of both hole andelectron experience the same spin upshift (downshift), while thehole experiences a larger orbital upshift (downshift)40,41 withrespect to the electron. Further, the contribution from the valleymagnetic moment results in an additional upshift (downshift) ofall bands in the K (K′) valley40. The applied magnetic fieldinduces unequal bright exciton populations in the two valleys(Fig. 3e). This excludes the possibility that XX0 (Fig. 4a, b) may beformed by two bright or two dark excitons, as both cases wouldresult in equally intense radiative recombination from both K andK′ at all magnetic fields. XX0 is therefore a combination of abright and a dark exciton. Under positive magnetic field, thebright exciton component of XX0 occupies the higher-radiativeenergy transition in the K′ valley (Fig. 3a, b, e) due tothermalisation of the photogenerated electrons. We expect thisto be allowed by a longer lifetime of XX0 compared to X0, inanalogy to XX−, where the lifetime was measured to be ~2–100times longer than single excitons20,25, and also exhibiting similarpolarisation properties. In parallel, the electron of the darkcomponent of XX0 can be either in the opposite (Fig. 4a) or in thesame (Fig. 4b) valley as the bright exciton component, yielding anenergy shift between these two configurations, which is the originof the fine-structure of XX0 observed in Fig. 3d.Figure 4c illustrates the single-particle configuration of XX−.As for XX0, the combination of two bright excitons is excludeddue to different recombination intensities in K and K′. From thesimilar g of XX0 and XX−, we can understand this five-particlecomplex as a bound state of a bright exciton with a dark trion, ora bright trion with a dark exciton. Both configurations wouldshow inequivalent valley population as for XX0 in Fig. 3e.Figure 4d is a qualitative many-body picture for XX0 formed bya bright and a dark exciton component in opposite valleys undermagnetic field. As its total Zeeman splitting depends on both thebright and the dark component, XX0 splits with a reversed energyorder compared to its bright exciton component and dissociatesinto a dark exciton and a photon due to the dark exciton havinglarger g than X0 with opposite sign42. The distribution ofbiexciton states follows the case near thermal equilibrium, whichis the reason behind the inequivalent circularly co-polarisedemission intensity under σ+ or σ−, as shown in Fig. 3.DiscussionWe have discovered the quinton, the five-particle negativelycharged biexciton in 1L-WSe2, unambiguously, as well as theneutral biexciton and its fine structure. Immediate next stepsinclude the unequivocal verification of the X−− state and theidentification of bound states within the lower-energy peaks. Acomplete understanding of multi-exciton complexes is key tostudy coherent many-body phenomena, such as condensation7,8and superfluidity9. Further, the ability to access and manipulatebiexciton complexes in TMD-based heterostructures offers newroutes towards probing other fundamental excitations in thissystem and the interplay between free and localised excitons.Extending our findings to the quantum confined regime will opennew capabilities for cascaded emission of entangled photons andspin-multiphoton interfaces.MethodsSample fabrication and room-temperature characterisation. Bulk WSe2 crystalsare prepared by the flux zone growth method (see Supplementary Note 1). BulkhBN crystals are grown by the temperature-gradient method under high pressureand high temperature. Graphite is sourced from NGS Naturgrafit. All bulk crystalsare exfoliated by micromechanical cleavage43 on Si/SiO2 (oxide thickness 285 nm).1L- and FL-samples are identified by optical contrast44. Selected crystals areassembled within ~5 h into LMHs via dry-transfer13,34. The LMH sample used forpower-dependent and magnetic field-dependent PL measurements is formed, fromtop to bottom, of ML-hBN flakes (~5 nm thick as determined by optical contrast),1L-WSe2, and a second ML-hBN flake (~10 nm thick as determined by opticalcontrast) and FLG (~5 layers thick as determined by optical contrast). That usedfor voltage-dependent measurements is prepared in a similar way, but the top ML-hBN does not fully cover the 1L-WSe2 to allow for Cr/Au (5/50 nm) electrodes to1110Magnetic field (T)Magnetic field (T)Magnetic field (T)Magnetic field (T) Magnetic field (T)888444000–4–4–4–8–8–81.71 1.72 1.73 1.680 1.685Energy (eV)Energy (eV)Intensity ratio log [(I �–/�–)/(I�+/�+)]8 8–8 –80 0 44–4 –4a c d1.6801.6851.7101.7151.730Energy (eV)e12 XX02 XX01XX02XX–0.60.40.5X0bI�–/�– I�–/�–I�–/�+X0X0XX02XX01XX–XX0XX–Fig. 3 Magnetic field dependence of PL. a Magnetic field dependent PL of X0 and XX0 in circular co-polarised and b cross-polarised configurations, for σ−excitation. The fine-structure lines are indicated as XX01 and XX02. The emission of XX0 brightens with increasing emission energy. X0 is displayed forreference. c Magnetic field dependent PL of XX− in a circular co-polarised configuration, for σ− excitation. In a, b and c the colour scale is linear. d Zeemanshift in the PL spectrum of X0 (filled black circles), XX0 (filled red and pink circles for the two components of the fine-structure) and XX− (filled bluecircles). The double arrow is a scale bar of 2.5 meV. e PL intensity ratio of circular co-polarisation with opposite helicity I(σ-/σ-)/I(σ+/σ+) for X0, XX01+XX02 and XX− as a function of magnetic fieldARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05632-44 NATURE COMMUNICATIONS |  (2018) 9:3721 | DOI: 10.1038/s41467-018-05632-4 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsdirectly contact it. The second electrode contacts FLG. The electrodes are patternedby e-beam lithography followed by lift-off. The ML-hBN thickness is chosen toisolate the 1L-WSe2 from the environment, smoothen the roughness of SiO2, shieldthe charge-traps of the substrate and avoid tunnelling between FLG and 1L-WSe2,while not compromising the optical contrast under the optical microscope. Ramanspectroscopy (Supplementary Fig. 1) and PL (Supplementary Fig. 2) are performedon the bulk crystals and after the assembly of LMH to characterise the startingmaterial and confirm the 1L-WSe2 thickness45–47. Raman and PL spectra areacquired at room temperature using a Horiba LabRam Evolution (spectral reso-lution ~0.3 cm−1) at 514.5 nm. See Supplementary Note 2 for details on the room-temperature optical characterisation.Optical measurements at 4 K. Power dependent and gate-controlled measure-ments are performed in a variable-temperature Helium flow cryostat (OxfordInstruments Microstat HiRes2) with a home-built confocal microscope at anominal temperature of 4.2 K. The magneto-optical measurements are performedin a close-cycle bath cryostat (Attocube Attodry 1000) equipped with a super-conducting magnet (maximum out-of-plane magnetic field 8 T) at a nominalsample temperature of 3.8 K. In the main text we refer to measurements at 4 K asan average of these two nominal temperatures.Theoretical calculations. We use Mott–Wannier model and quantum MonteCarlo (QMC) as implemented in CASINO48 to calculate the energies of X0, XX0and XX−in ML-WSe229. The full photoemission spectra of ML-WSe2 in vacuumare reported in ref. 29. To consider the effect of the dielectric screening provided byhBN, we use the experimental value of the binding energy of XX0 and use Eq. (48)of ref. 29 to derive the screening parameter r* which is 54 Å. We use the many-bodyGW electron and hole effective masses as 0.29m0 and 0.34m049, respectively, wherem0 is the bare electron mass. We calculate the binding energy of XX− by sub-tracting the total energy of the exciton and trion from the total energy of XX−.Data availability. The datasets generated and analysed during the current studyare available from the corresponding author on reasonable request.Received: 23 May 2018 Accepted: 13 July 2018References1. Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg seriesin monolayer WS2. Phys. Rev. Lett. 113, 076802 (2014).2. Wang, G. et al. Colloquium: excitons in atomically thin transition metaldichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).3. Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valleyphysics in monolayers of MoS2 and other Group-VI dichalcogenides. Phys.Rev. Lett. 108, 196802 (2012).4. Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarizationin monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498(2012).5. Zeng, H., Dai, J., Yao, W., Xiao, D. & Cui, X. 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Lett. 110, 193102 (2017).27. Cadiz, F. et al. Excitonic linewidth approaching the homogeneous limit inMoS2-Based van der Waals heterostructures. Phys. Rev. X 7, 021026 (2017).XX–XX0VacuumIncreasing magnetic fieldXX0K K’No fieldMagnetic fieldNo fieldMagnetic fieldK K’K K’a b c dσ– σ– σ–X0(K)+X0dark(K’)σ–X0(K’)+X0dark(K)X0darkX0 dark(K’)X 0dark (K)Fig. 4 Composition of biexciton species with applied magnetic field. a, b, c Single-particle picture of the internal structure of (a, b) XX0 and (c) XX− for B >0. The eigenstates shift inequivalently in K and K′ (dashed curves indicate no magnetic field, solid curves indicate applied magnetic field, red and bluecolours indicate opposite spin). XX0 comprises a bright exciton with highest radiative energy and a dark exciton with the electron (a) inter- or (b) intra-valley with the bright exciton. d Many-body picture of the magnetic field effect on XX0, comprising a bright and a dark exciton. 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Rev. B 87,155304 (2013).AcknowledgementsThe authors thank Bernhard Urbaszek, Neil D. Drummond, Vladimir I. Fal’ko andIoannis Paradisanos for useful discussions. We acknowledge funding from NSF DMR-1552220, Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST(JPMJCR15F3), JST, EU Graphene Flagship, ERC Grants Hetero2D and PHOENICS,EPSRC Grants EP/509K01711X/1, EP/K017144/1, EP/N010345/1, EP/M507799/ 5101,and EP/L016087/1, Marie Skłodowska-Curie Actions Spin-NANO, Grant No. 676108,Quantum Technology Hub NQIT EP/M013243/1.Author contributionsM.B., A.C.F. and M.A. conceived and managed the project; K.W. and T.T. provided hBNcrystals; H.L., B.C. and S.T. provided WSe2 crystals; M.B., A.R.C. and D.D. fabricated andcharacterised the devices; M.B., A.R.-P.M., D.M.K., C.P.-B., B.P. and M.A. performed thelow-temperature PL measurements and analysed the results; E.M. performed the cal-culations. All authors participated in the discussion of the results and the writing of themanuscript.Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-05632-4.Competing interests: The authors declare no competing interests.Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2018ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05632-46 NATURE COMMUNICATIONS |  (2018) 9:3721 | DOI: 10.1038/s41467-018-05632-4 | www.nature.com/naturecommunicationshttps://doi.org/10.1038/s41467-018-05632-4https://doi.org/10.1038/s41467-018-05632-4http://npg.nature.com/reprintsandpermissions/http://npg.nature.com/reprintsandpermissions/http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Charge-tuneable biexciton complexes in monolayer WSe2 Results Design and optical characterisation of heterostructures Unveiling the presence and nature of biexcitons Behaviour of exciton complexes in magnetic field Discussion Methods Sample fabrication and room-temperature characterisation Optical measurements at 4 K Theoretical calculations Data availability References Acknowledgements Author contributions Competing interests ACKNOWLEDGEMENTS