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Zhiheng Huang, Yunfei Bai, Yanchong Zhao, Le Liu, Xuan Zhao, Jiangbin Wu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Wei Yang, Dongxia Shi, Yang Xu, Tiantian Zhang, Qingming Zhang, Ping-Heng Tan, Zhipei Sun, Sheng Meng, Yaxian Wang, Luojun Du, Guangyu Zhang

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Observation of phonon Stark effectArticle https://doi.org/10.1038/s41467-024-48992-wObservation of phonon Stark effectZhiheng Huang1,2,10, Yunfei Bai1,2,10, Yanchong Zhao1,2, Le Liu1,2, Xuan Zhao 1,2,Jiangbin Wu 3, Kenji Watanabe 4, Takashi Taniguchi 5, Wei Yang 1,2,Dongxia Shi1,2, Yang Xu 1,2, Tiantian Zhang 6, Qingming Zhang1,2,7,Ping-Heng Tan 3, Zhipei Sun 8, Sheng Meng 1,2,9, Yaxian Wang 1 ,Luojun Du 1,2 & Guangyu Zhang 1,2,9Stark effect, the electric-field analogue of magnetic Zeeman effect, is one ofthe celebrated phenomena in modern physics and appealing for emergentapplications in electronics, optoelectronics, as well as quantum technologies.While in condensed matter it has prospered only for excitons, whether othercollective excitations can display Stark effect remains elusive. Here, we reportthe observation of phonon Stark effect in a two-dimensional quantum systemof bilayer 2H-MoS2. The longitudinal acoustic phonon red-shifts linearly withapplied electric fields and can be tuned over ~1 THz, evidencing giant Starkeffect of phonons. Togetherwithmany-body ab initio calculations, we uncoverthat the observed phonon Stark effect originates fundamentally from thestrong couplingbetweenphonons and interlayer excitons (IXs). In addition, IX-mediated electro-phonon intensity modulation up to ~1200% is discovered forinfrared-active phonon A2u. Our results unveil the exotic phonon Stark effectand effective phonon engineering by IX-mediatedmechanism, promising for aplethora of exciting many-body physics and potential technologicalinnovations.Stark effect, one of the renowned phenomena in modern physics,describes the energy shifting or splitting of spectra lines induced byexternal electric fields. It was first discovered in hydrogen atoms byJohannes Stark in 1913 and soon awarded with the Nobel Prize inPhysics in 1919 for its remarkable contributions to quantumtheory1,2. In condensed-matter physics, Stark effect has beendemonstrated for excitons (i.e., bound pairs of electrons and holes)in various solid-state quantum systems, such as quantum dots,quantumwells and van derWaals heterostructures2–15. The emergingexciton Stark effect not only opens up innovative paradigms tocontrol the material’s properties and quantum states in a precise,high-speed, reversible and efficient manner, but also createsunprecedented possibilities to underpin new physics and to intro-duce a rich variety of technological applications, such as on-chipelectro-optical modulators4,16, tunable quantum light sources17,18,nanoscale spin rectifier control19, and compact spectrometers20.Although notable progress has been witnessed in exciton Starkeffect, the Stark effects of other solid-state collective excitations,such as phonons (i.e., the quantized vibrational excitations of acrystal lattice) that are essential for plenty of emergent physicsand innovative applications (e.g., superconductivity, ultrafastcarrier dynamics, nonequilibrium phenomena, ultrafast control ofReceived: 3 January 2024Accepted: 15 May 2024Check for updates1Beijing National Laboratory for Condensed Matter Physics; Key Laboratory for Nanoscale Physics and Devices, Institute of Physics, Chinese Academy ofSciences, Beijing 100190, China. 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China. 3State Key Laboratory ofSuperlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China. 4Research Center for FunctionalMaterials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5International Center for Materials Nanoarchitectonics, NationalInstitute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, Beijing 100190, China. 7School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China. 8QTF Centre ofExcellence, Department of Electronics and Nanoengineering, Aalto University, Tietotie 3, FI-02150 Espoo, Finland. 9Songshan Lake Materials Laboratory,Dongguan, Guangdong Province 523808, China. 10These authors contributed equally: Zhiheng Huang, Yunfei Bai. e-mail: yaxianw@iphy.ac.cn;luojun.du@iphy.ac.cn; gyzhang@iphy.ac.cnNature Communications |         (2024) 15:4586 11234567890():,;1234567890():,;http://orcid.org/0000-0002-6004-2988http://orcid.org/0000-0002-6004-2988http://orcid.org/0000-0002-6004-2988http://orcid.org/0000-0002-6004-2988http://orcid.org/0000-0002-6004-2988http://orcid.org/0000-0002-8751-7082http://orcid.org/0000-0002-8751-7082http://orcid.org/0000-0002-8751-7082http://orcid.org/0000-0002-8751-7082http://orcid.org/0000-0002-8751-7082http://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/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0002-3925-0352http://orcid.org/0000-0003-4223-8677http://orcid.org/0000-0003-4223-8677http://orcid.org/0000-0003-4223-8677http://orcid.org/0000-0003-4223-8677http://orcid.org/0000-0003-4223-8677http://orcid.org/0000-0002-8591-2652http://orcid.org/0000-0002-8591-2652http://orcid.org/0000-0002-8591-2652http://orcid.org/0000-0002-8591-2652http://orcid.org/0000-0002-8591-2652http://orcid.org/0000-0001-6575-1516http://orcid.org/0000-0001-6575-1516http://orcid.org/0000-0001-6575-1516http://orcid.org/0000-0001-6575-1516http://orcid.org/0000-0001-6575-1516http://orcid.org/0000-0002-9771-5293http://orcid.org/0000-0002-9771-5293http://orcid.org/0000-0002-9771-5293http://orcid.org/0000-0002-9771-5293http://orcid.org/0000-0002-9771-5293http://orcid.org/0000-0002-1553-1432http://orcid.org/0000-0002-1553-1432http://orcid.org/0000-0002-1553-1432http://orcid.org/0000-0002-1553-1432http://orcid.org/0000-0002-1553-1432http://orcid.org/0000-0003-4790-2880http://orcid.org/0000-0003-4790-2880http://orcid.org/0000-0003-4790-2880http://orcid.org/0000-0003-4790-2880http://orcid.org/0000-0003-4790-2880http://orcid.org/0000-0002-2420-8258http://orcid.org/0000-0002-2420-8258http://orcid.org/0000-0002-2420-8258http://orcid.org/0000-0002-2420-8258http://orcid.org/0000-0002-2420-8258http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://orcid.org/0000-0002-1242-4391http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48992-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48992-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48992-w&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48992-w&domain=pdfmailto:yaxianw@iphy.ac.cnmailto:luojun.du@iphy.ac.cnmailto:gyzhang@iphy.ac.cnmagnetism, and thermal transistors)21–25, though highly desired,have thus far remained elusive.Herein, we report the first observation of the Stark effect forphonons in a two-dimensional (2D) quantum solid of bilayer 2H-MoS2.Specifically, the longitudinal acoustic (LA) phononmode in bilayer 2H-MoS2 undergoes a linear redshift with external electric fields when theinterlayer exciton (IX) energy is tuned across its emission line, evi-dencing the first-order (also dubbed as linear) Stark effect of phonons.Remarkably, the observed phonon Stark effect in bilayer 2H-MoS2 isgiant and can reach an extremely large frequency change up to~33 cm−1 (~1 THz). We remark that although the control of phononswith gating has been reported inmany 2D systems, such asmonolayer/bilayer/trilayer graphene26–30, monolayer transition metal dichalco-genides (TMDs)31,32, and black phosphorus33, the electrostatic dopingeffects, rather than electric field, are typically focused, and the phononenergy modulation is generally nonlinear and within 10 cm−1. Guidedby many-body first principles calculations, we pinpoint the underlyingmicroscopic origin of the observed giant phonon Stark effect to be thestrong coupling between phonons and highly tunable IXs. Further-more, we demonstrate the electro-phonon modulation of emissionintensity mediated by IXs reaching as large as ~1200% for the infrared-active phonon mode A2u. Our results demonstrate the emerging giantphonon Stark effect and effective electric control of phonon states byIX-mediatedmechanism, holding a great promise for a rich diversity ofemergent quantum phenomena and potential technological applica-tions, such as electric-field-tunable phonon laser, dynamical control ofheat transport and THz acoustic-electronic/optic devices.ResultsLinear Stark effect of IXs in bilayer 2H-MoS2High-quality, hexagonal boron nitride (h-BN) encapsulated dual-gate2H-phase bilayer MoS2 devices are fabricated by a van der Waalsmediated dry layer-by-layer transfer-and-stack technique34,35. Figure 1aillustrates schematically the typical device structure (also see Supple-mentary Fig. 1 for the optical microscope images). Few-layer grapheneis used as both the bottom and top gate electrodes to independentlytune the out-of-plane electric field Fz and carrier density n0. HereFz = (CbVbg � CtVtg)/2ε0εBL and n0 = ðCbVbg +CtV tgÞ=e, where e isthe elementary charge, ε0 denotes the vacuum permittivity, and εBL isthe out-of-plane dielectric constant of bilayer MoS2. CbðVbgÞ and CtadebfcA1g&A2uLAFig. 1 | Quantum-confined Stark effect of IXs in bilayer 2H-MoS2. a Schematicimage of h-BN encapsulated dual-gate bilayer MoS2 devices. Few-layer graphene(FLG) is used as both the bottom and top gate electrodes to tune the Fz .b Schematic of IX configurations with a strong intralayer B exciton componentfrom lower layer L2. c, Schematic of coexistence of IXs and phonons. d Contourplot of the PL spectra of a typical device H1 as a function of photon energy (bottomaxis) and Fz (left axis). Doping density remains unchanged. IXs (Raman peaks) arehighlighted by the dashed (dotted) lines. e Extracted emission energies of IX1(green disks) and IX2 (blue disks) from (d) as a function of Fz . The green and bluedashed lines are linear-fits of IX1 and IX2, respectively. Insets: schematic of the IXconfigurations in real-space. The directions of the dipole moment denoted by redarrows depend on the location of the constituent electron, either in the bottom ortop layer. f Normalized linecuts of PL spectra at selected Fz from (d). Offset is setvertically for clarity and the corresponding Fz of each curve is labelled. Green andblue dashed lines respectively represent IX1 and IX2 to guide for the eye. Greydotted lines denote Raman modes.Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 2ðV tgÞ are the geometrical capacitances per area (applied voltages) forthe bottom and top gates, respectively (seeMethods formore details).In contrast to the monolayer case, bilayer 2H-MoS2 shows layerdegree of freedom and can host IXs whose constituent electrons andholes are spatially displaced and thus are highly electric-field tunableby the first-order Stark effect15,35–38. It is noteworthy that because ofthe layer-hybridized hole states, IXs in 2H-stacked MoS2 bilayersinteract strongly with intralayer B excitons and acquire appreciableoscillator strengths (Fig. 1b)15,35. This is in stark contrast to IXs in typicalTMD heterobilayers, where their coupling to light is substantiallyreduced5,39. Such a powerful combination of strong light-matterinteraction and highly efficient electric tunability enables IXs to cou-ple with other elementary excitations e.g., lattice vibrations (Fig. 1c),and potentially yields new hybrid excited states18,39–41.To capture the unique IXs in bilayer 2H-MoS2, we perform thephotoluminescence (PL)measurements as a function of the out-of-planeelectric field Fz , while keeping the carrier density n0 unchanged. Unlessotherwise noted, all measurements (including both PL and Ramanspectroscopy) are carried out in a high vacuum at 10K with an on-resonance 633nm laser excitation. Figure 1d depicts the colour plot ofPL spectra against Fz for a typical bilayer 2H-MoS2 device H1 through agrating of 600 gr/mm. Clear features of IX emissions whose energiesshift linearly with increasing Fz can be unequivocally identified (dashedlines in Fig. 1d) and further confirmed by the linecuts at different Fz(dashed lines in Fig. 1f). The linear IX energy shift with Fz suggests thefirst-order Stark effect caused by the out-of-plane static electric dipolemoments across the bilayer 2H-MoS215,35. Notably, there are two well-separated IX branches with opposite static electric dipole moments, i.e.,IX1 (IX2) species with positive (negative) electric dipole moment. Thiscan be well understood as the layer degeneracy of band structure inbilayer 2H-MoS2, which gives rise to one IX with electron localized in thelower layer (i.e., IX1) and the other with electron restrained in the upperlayer (i.e., IX2), as schematically depicted in the inset of Fig. 1e15,35–37. TheIX1 and IX2 transition energies against applied electrical fields extractedby Voigt function fitting are respectively plotted as green and blue disksin Fig. 1e. The linear dependences (dashed lines in Fig. 1e) give theelectric dipole moments of IX1 and IX2 to be μ IX1� �= 0:612±0:003ð Þe �nm and μ IX2� �= � 0:608±0:003ð Þe � nm, respectively. This is ingood agreement with the calculated value (e.g., ±0.606 e ·nm) based onhybridized hole model (Supplementary Note 2) and previousresults15,35–37,42.Observation of phonon Stark effectApart from the electrically tunable IXs, several narrow peaks in theenergy range from ~1.939 to ~1.90 eV, corresponding to the Raman shiftfrom ~166 to~480 cm−1, are also noticeable in Fig. 1d, f (grey dotedlines). In the light of prior work43–46, these peaks can be recognized asthe Raman phonon signals in bilayer 2H-MoS2 (e.g., LA around230 cm−1, and A2u&A1g around 406 cm−1). Strikingly, the IXs can betuned to cross these phonon lines by the electric fields. This mayenable the resonant coupling between these phonons and IXs, givingrise to the formation of exciton-dressed phonon states with a non-vanishing electric dipole moment and thus exotic Stark effect ofphonons. To better distinguish the fine features of these phonon linesand demystify the phonon Stark effect, we perform the Fz-dependentRaman measurements with an improved energy resolution through agrating of 1800gr/mm. Figure 2a presents the colour plot of the first-order derivative of Raman intensity I over phonon frequencyω (∂I=∂ω)for deviceH1 (please refer to Supplementary Fig. 3 for the intensity plotof raw Raman spectra). Surprisingly, when IXs (navy dashed lines) aretuned by the electric field Fz across the phonon line around 230 cm−1,an exotic phonon mode (labelled as Stark phonon, SP) emerges andredshifts linearly with increasing Fz (black dotted lines). In closeresemblance to the linear Stark effect of atom spectra lines and exci-tons in solid-state quantum systems1,2, such a linear modulation ofphonon energies by external electric fields offers the hallmark of first-order phonon Stark effect. We highlight that although vibrational Starkeffect has been previously uncovered in molecule systems47, this is thefirst achievement of linear Stark effect of phonon collective vibrationsin condensed matter solids. Note that according to previous results43and our calculated results (Fig. 3) which we shall come to shortly, theintriguing SP mode with energy around 230 cm−1 can be discerned as aLA phonon with finite momentum around M point.To further verify the linear Stark effect of LA(M) phonon in bilayer2H-MoS2, we carry out Fz -dependent Raman spectroscopy measure-ments on other two h-BN-encapsulated bilayer 2H-MoS2 devices(labelled as H2 and H3). Figure 2b, c respectively show the colour plotof the raw Raman data and ∂I=∂ω for device H2 ranging from 166 cm−1to 270 cm−1 (refer to Supplementary Note 4 for the results of deviceH3). Similar to what is observed in device H1 (Fig. 2a), an unconven-tional SP mode also emerges in devices H2 and H3 and displays linearenergymodulation with Fz (highlighted by the dotted lines in Fig. 2b, cand Supplementary Fig. 4), evidencing the first-order phonon Starkeffect. Figure 2e presents the Raman linecuts at different Fz where thelinear redshift of the SP mode on Fz is illustrated by the bold blackdashed line. Fitting the Raman spectra of device H2 with the Voigtfunction gives the energies of SP mode against Fz, as shown in Fig. 2d(red squares). The perfect linear dependence of the energies of SPmode on the strength of electric field quantitatively confirms the first-order phonon Stark effect. We extract, via linear fitting (black dottedline in Fig. 2d), that the slope of the phonon Stark effect, which can beviewed as the ‘Stark tuning rate’ or ‘phonon dipole’, is 289 ± 3 cm−1/(V/nm). Importantly, the observed ‘phonon dipole’ in bilayer 2H-MoS2 isgiant, more than an order ofmagnitude larger than the state-of-the-artresults [~20 cm−1/(V/nm)] for vibrational modes of molecule systems47.Benefiting from such a colossal ‘phonon dipole’, the electric modula-tion of phonon energies by the Stark effect can reach a frequencyshift up to ~33 cm−1 (~1 THz) within the maximum electric fields per-mitted by our experiments (i.e., ~1.587MV/cm).We highlight that sucha phonon Stark shift of ~1 THz is fairly large, competitive to the bestphonon energymodulation by othermechanisms (typically < 0.3 THz),such as phase control48, symmetry engineering49, Kohn anomalymechanism26,29, strain deformation50, and optical/magnetic control51,52.Furthermore, a much larger phonon Stark shift can be anticipated bydesigning new device architectures, for example double ionic gatedtransistors which can enable the application of electric fields of morethan one order of magnitude stronger (~30MV/cm)53.Underlying mechanism of phonon Stark effectThe observation of linear phonon Stark effect in bilayer 2H-MoS2 isfairly surprising, considering that the direct coupling between pho-nons and electric fields is generally ignorable. We notice that the SPphononmode begins to redshift linearly with the applied electric fieldFz when the IX emission line is tuned to resonate with it (Fig. 2a–c).This strongly indicates that the observed phonon Stark effect could bemediated by the electrically tunable IX states. To confirm the role ofIXs in the phonon Stark effect, we perform Fz -dependent Ramanspectroscopy measurements on a high-quality, encapsulated, dual-gate bilayer 3R-MoS2with the samedevice architecture shown in Fig. 1a(see Supplementary Fig. 10 for the optical image). Bilayer 3R-MoS2exhibits similar electronic structure and phonon dispersion to thebilayer 2H counterpart, but lacks the hybridized IXs as a result of layer-dependent Berry phase winding54,55. Remarkably, no apparent phononStark effect is observed in bilayer 3R-MoS2 (please see SupplementaryNote 10 for more details). Such a strong contrast between the bilayer2H- and 3R-MoS2 manifests the pivotal role played by IXs.To explicitly unravel themicroscopic origin of the observed linearphonon Stark effect, we carry out many-body first principles GW cal-culations (GW, one-body Green’s function G and the dynamicallyscreened Coulomb interaction W) in combination with the Bethe-Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 3Salpeter equation (BSE) as well as density functional perturbationtheory (DFPT). By expanding the total energy of the coupled exciton-phonon system in a perturbation theory-based formula and applyingthe variational principles, we can self-consistently solve the eigen-functions in the exciton and phonon basis, and obtain the mode- andmomentum-resolved exciton-phonon coupling matrix element (seeMethods and Supplementary Note 14 for more details). Figure 3ashows the calculated exciton absorption spectrum of bilayer 2H-MoS2,where the well-defined intralayer exciton transitions (1 s/2 s state ofintralayer A exciton: ~1.92/2.08 eV; 1 s state of intralayer B exciton:~2.12 eV) are nicely captured. Importantly, a prominent peak around2.00 eV (labelled by the grey arrow), corresponding to the IX transi-tion, can be unequivocally identified, in agreement with our mea-surements and previous studies15,36,54. The real-space wave functions ofthe constituent electrons (upper panel) and holes (lower panel) for aselected IX species are presented in Fig. 3b. Interestingly, the wavefunction of its constituent electron is completely localized in theupperlayer, while the constituent hole mainly distributes in the lower layer.Such spatially displaced wave functions of the constituent electronsand holes confirms the IX nature.Figure 3c displays the calculated phonon dispersion of bilayer 2H-MoS2, consistent with previous results43. In total, there are 18 phononmodes, which are labelled asmodes 1–18 in order of increasing energy.Their coupling strength with the IXs, eGex�ph, are calculated and pre-sented in Fig. 3d, for selected wavevectors, namely at the high sym-metry points of the Brillouin zone (Γ, K and M). Strikingly, the giantcoupling strength eGex�ph is found between IXs and the mode 5/6 atzone boundary M point, i.e., the nearly degenerate LA(M) modesaround 230 cm−1 marked by the blue box in Fig. 3c (the correspondingnormal displacements are shown in the upper panel of Fig. 3e). Thelower panel of Fig. 3e displays the calculated eGex�ph distribution of LAphonon in the momentum space, further confirming the strong cou-pling between LA(M) phonon mode and IXs. Such gigantic couplingcould lead to intriguing collective excitations described by hybridphonon-IX excited states, in which the phonon is ‘dressed’ with anexciton cloud56. This exciton-dressed phonon elementary excitation-0.423eabcd-0.212-0.265-0.317-0.370-0.476-0.529-0.582-0.635-0.688-0.741-0.794-0.847-0.900-0.952-1.058-1.164-1.270-1.376-1.482-1.587Fig. 2 | Observation of phonon Stark effect. a Colour plot of the first orderderivative of phonon emission intensity I over phonon energy ω (∂I=∂ω) for deviceH1. Black dotted lines highlight the SPmode to guide for the eye. Navy dashed linesrespectively represent the IXs.b, Contour plot of the Raman spectraof deviceH2asa function of phonon energy (bottom axis) and electric field Fz (left axis). c First-order derivative of (b). d Extracted phonon energy of SP mode (red squares) from(b) as a function of Fz . The error bars are dereved from the fitting. Dotted lines inb–d represent the linear-fitting of SPmode. Navy dashed lines in b–d represent thefitted IX in device H2. e Normalized linecuts of Raman spectra at selected electricfield Fz from (b). Offset is set vertically for clarity and the corresponding Fz of eachcurve is labelled. Black dashed line traces SP mode to guide for the eye.Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 4may inherit a fraction of the static dipole moment from the IXs andtherefore be continuously tuned through out-of-plane electric fields,giving rise to the linear phonon Stark effect. This is in good agreementwithour experimental results that a SPmodeemerges andundergoes alinear shift with applied electric fields when IXs are electrically tunedacross its emission line (Fig. 2).We highlight that exciton-phonon coupling has been one ofthe research frontiers of condensed matter physics since itsfoundation in the 1950s57,58. In 2D systems, exciton-phonon cou-pling has been extensively studied and is believed to underliemany intriguing physics, including but not limited to phonon-assisted dark-exciton formation59,60, phonon-mediated valleydepolarization61,62 and activation of optically silent phonon63,64.However, previous work mainly focusses on the interactionbetween phonons and intralayer excitons. By contrast, our workreveals the exciton-phonon coupling involving highly tunable IXs,and uncovers the exotic phonon Stark effect.Electric control of phonon intensity mediated by IXsBesides the phonon mode 5/6 at M point, our theoretical calculationsshow that mode 15/16 at Γ point (marked by the red box in Fig. 3c),corresponding to the infrared active A2u(Γ)/Raman active A1g(Γ) pho-nons with normal displacements shown in the upper panels ofFig. 3f, g, also display colossal coupling strength eGex�ph with IXs(Fig. 3d and lower panels of Fig. 3f, g). Such strong interactionsbetween A2u(Γ)/A1g(Γ) and highly tunable IXs may trigger the renor-malization of Raman scattering cross-section and activity, leading toelectric control of these two phonon states. Indeed, we observe anefficient electro-phonon modulation of emission intensity of A2u(Γ)/A1g(Γ)mediatedby IXs inbilayer 2H-MoS2. Figure 4adisplays the colourplot of Raman spectra against the electricfield Fz in the range from320to 480 cm−1 for the high-quality device H2. Around 406 cm−1, there aretwo distinguishable phonon modes in bilayer 2H-MoS2 because of theDavydov splitting43, namely one Raman active mode A1g(Γ) at~406.8 cm−1 and one infrared active A2u(Γ) mode at ~402.6 cm−1. SinceA2u(Γ) mode is infrared active, its emission intensity should be extre-mely weak compared to the Raman active A1g(Γ) mode. This is indeedthe case under small electric fields where the IX emission line is lyingbelow theA2u(Γ) phonon line. By contrast, when IXs (navy dashed lines,Fig. 4a) are tuned to resonate with A2u(Γ) phonon line by the electricfield Fz , the emission intensity of A2u(Γ) phonon dramatically increases(see Supplementary Fig. 9 for the linecuts at different Fz).M K ΓΓdcabgfeMKΓMKΓMKΓFig. 3 | Strong coupling between phonon and IXs. a Calculated exciton absorp-tion spectra of bilayer 2H-MoS2. Four main transitions are observed: 1 s/2 s state ofintralayer A exciton at ~ 1.92/2.08 eV, 1 s state of intralayer B exciton at ~ 2.12 eV, andIX transition around2.0 eV (labelled by grey arrow).bThe real-spacedistributionofwave functions of the constituent electrons (upper panel) and holes (lower panel)for a selected IX species. c Calculated phonon dispersion of bilayer 2H-MoS2. Thephonon modes at M (Γ) point with the largest coupling to the IXs are marked byblue (red) box, corresponding to LA(M) mode [A1g(Γ) and A2u(Γ) modes].d Calculated coupling strength eGex�ph between IXs and all the 18 phononmodes athigh symmetry points of Γ, M and K with order numbers in sequence of increasingenergies. e–g Lower panel: calculated eGex�ph distributions inmomentum space forselected modes 5 (e), 15 (f) and 16 (g). Upper panel: the normal displacements ofLA(M) (e), A2u(Γ) (f) and A1g(Γ) phonon modes (g).Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 5The upper panel of Fig. 4b shows the fitted emission intensities ofinfrared active A2u(Γ) phonon (pink spheres) and Raman active A1g(Γ)mode (violet spheres) as a function of Fz . Clearly, the emission inten-sities of infrared active mode A2u(Γ) experience a sharp enhancementaround the specific electric fields F0 = ±0:715MV=cm (grey verticallines) under which the IX emission line coincides with the A2u(Γ) pho-non line. To quantitatively weigh their intensity change, we define theelectro-phonon modulation depth as ρ= I�I0I0, where I (I0) representsthe phonon emission intensity at finite (zero) Fz . The lower panel ofFig. 4b presents the electro-phonon modulation depth ρ of infraredactive A2u(Γ) phonon (pink spheres) and Raman active A1g(Γ) mode(violet spheres) against Fz . The modulation depth ρ of infrared activeA2u(Γ) mode mediated by IXs is strikingly dependent on the appliedelectric fields and can be tuned in a wide range from ~0 to ~1200%. Bycontrast, there is no noticeable tuning of A2u(Γ) emission intensity inbilayer 3R-MoS2 (Supplementary Note 10), again confirming the keyrole of IXs. In comparison, the maximummodulation depth of Ramanactive A1g(Γ) mode is only ~60%, 20 times smaller than that of infraredactive A2u(Γ) phonon. Themuch larger modulation depth ρ of infraredactiveA2u(Γ) than Raman activeA1g(Γ) can beunderstood as that owingto the antiphase displacements in adjacent layers (upper panel ofFig. 3f), infrared active A2u(Γ) mode can possess an interlayer electricdipole and thus a stronger coupling strength with the IXs (Fig. 3d)64.In addition to the intensity modulation, the energy of infraredactive A2u(Γ) phonon and thus the Davydov splitting, i.e., the energydifference between A2u(Γ) and A1g(Γ) phonon modes, can also beengineered by the electric field, as depicted in Fig. 4c. While the A1g(Γ)phonon energy is insensitive to the electric field, the energy of A2uphonon blueshifts gradually as the electric field increases and reachesits maximum around F0 (grey solid lines) at which the IX and A2u(Γ)emission lines resonate with each other (upper panel of Fig. 4c). Thelower panel of Fig. 4c shows the Davydov splitting against the electricfield, which follows the similar electric field-dependent evolution withthe A2u(Γ) phonon energy and can be modulated by ~1.1 cm−1.ab cAA2u A1gFig. 4 | Electric modulation of phonon intensity and Davydov splitting.a Contour plot of the Raman spectra of device H2 as a function of phonon energy(bottom axis) and Fz (left axis). Navy dashed lines represent the IXs that areobtained by fitting. b Extracted phonon emission intensities (upper panel) andelectric-phonon modulation depths (lower panel) as a function of Fz for A2u(Γ)(pink spheres) and A1g(Γ) (violet spheres). Note that the modulation depths ρ ofA1g(Γ) phonon are multiplied by a factor of 10 for better visualization. c Extractedphonon energies (upper panel) of A2u(Γ) (pink spheres)/A1g(Γ) (violet spheres), andtheir corresponding Davydov splitting (lower panel) against Fz . Grey vertical solidlines in (b) and (c) denote the specific electric field F0 where the IX and A2u(Γ)emission lines coincidentally intersect. The error bars are derived from the fitting.Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 6DiscussionIn summary, we demonstrate the observation of linear phonon Starkeffect in bilayer 2H-MoS2 mediated by highly efficient gate-tunable IXs.The LA(M) phonon energy begins to redshift linearly with the appliedelectric fields when the IXs are tuned to resonate with its emission line,and the Stark shift can reach a frequency up to ~1 THz within theexperimentally accessible electric field range. Together withmany-bodyfirst principles calculations, we unveil that the strong coupling betweenthe LA(M) phonons and IXs underlies the observed giant phonon Starkeffect in bilayer 2H-MoS2. In addition, IX-mediated strong renormaliza-tion of phonon emission intensity up to ~1200% is achieved for infrared-active A2u(Γ) phonon mode. Our results demonstrate an IX-mediatedmechanism for emerging phonon Stark effect and phonon engineering,and can also been applied to a wide range of solid-state quantum sys-tems, such as TMD homo- and hetero-structures, promising the pro-spect of fascinating many-body physics and novel applications such asphonon lasers and THz acoustic-electronic devices.MethodsDevice Fabricationh-BN encapsulated dual-gate devices were fabricated by a van derWaals mediated dry transfer technique. In short, bilayer MoS2, few-layer graphene and h-BN, were first mechanically exfoliated from bulkcrystals on 285 nm SiO2/Si substrates. We highlight that all the bilayer2H-MoS2 (3R-MoS2) samples we investigated are directly exfoliatedfrombulk 2H (3R) crystals (acquired fromHQGraphene), and the twistangle between the two constituent layers is a perfect 60° (0°). Flakeswith appropriate size and thickness were then selected based on theiroptical contrast. Few-layer graphene is utilized as ground and gateelectrodes. After stacking and releasing via layer-by-layer dry-transfermethod, the devices were annealed in argon/hydrogen atmosphere at350 °C for 4 h to diminish the influence of strains and bubbles, andimprove quality. The thickness of gate dielectricmaterialh-BN on eachside is determined by atomic force microscope (AFM) beforeFz -dependent opticalmeasurements. Finally,metal contacts to groundand gate electrodes were patterned by the standard micro-fabricationprocesses, including e-beam lithography (EBL), metal evaporation Ti(3 nm)/Au (30nm) and lifting-off.Determination of Fz and n0The dual-gate device structure enables us to independently tune thevertical electric field (Fz) in bilayer MoS2 without changing the dopingdensity n0. A parallel plate capacitor model is used to extract the Fzunder a top/bottom gate V tg=Vbg. In this way, the displacement fieldsDT and DB across the top and bottom h-BN are:DT =CTV tg andDB =CBVbg. CT =ε0εh�BNtTand CB =ε0εh�BNtBrespectively represent thegeometric capacitance per unit area of top and bottom gates. Here, tT(tB) is the thicknesses of the top (bottom) h-BN layer as determined byAFM measurements, ε0 is the vacuum dielectric constant andεh�BN ≈ 3.0 is the out-of-plane dielectric constant of h-BN. The verticalelectric field can be defined as Fz =Dε0 � εBL, where εBL ≈ 6.5 is the out-of-plane dielectric constant of bilayer MoS2 and D= 12 ½DT � DB� is theelectric displacement field in the system. Thus, we can calculate Fz asFz = ðCbVbg � CtV tgÞ=2ε0εBL = εh�BN2 � εBL1tTV tg � tTtBVbgh i. Here positive/negative Fz represents vertical electric field upward/downward.Meanwhile, the doping density has a form asn0 = ðCTV tg +CBVbgÞ=e= ε0εh�BNe1tTV tg +tTtBVbgh i. Clearly, by keepingV tg +tTtBVbg as a constant and changing V tg � tTtBVbg, we can tune Fzwhile keeping n0 unchanged.Optic measurementsIn our cryogenic Fz -dependent optic experiments, devices were wire-bonded onto a chip carrier, placed in an optical chamber with a highvacuum and cooled down to 10K by a closed cryocooler (CS-204PF-DMX-20B-OM from ARS). Raman (PL) spectra were obtained using aHORIBA spectrometer (LabRAM HR Evolution) in a confocal back-scattering configuration through a grating of 1800 (600) gr/mm. Lightfrom 633 nm (1.96 eV) continuous laser with a power of 137μW(34μW) for Raman (PL) measurements was focused through a Nikonobjective (N.A. = 0.5, W.D. = 10.6, F.N. = 26.5) onto the sample with aspot diameter of ~2μm. The spectrometer integration time was 30 s(5 s) for Raman (PL) measurements.Many-body theoretical calculationsWe perform ab initio calculations using Quantum Espresso code65combining density functional theory (DFT) and density functionalperturbation theory (DFPT). Fully relativistic optimized norm-conserving Vanderbilt (ONCV) pseudopotentials66 for the PBEexchange-correlation functional67 were used for both Mo and S, whichallowed us to calculate the electronic structure including spin-orbitcoupling (SOC). To correctly account for interlayer coupling, dft-d2correction are included. An 11 × 11 × 1 Gamma centred k-mesh with100Ry cutoff is employed for the Brillouin zone integration and pho-non dispersion is calculated on an 18 × 18 × 1 q-mesh.G0W0 +BSE calculation was carried out with Yambo code68,69 withthe same 18 × 18 × 1 k-mesh in the Brillouin zone. A cutoff of 10 Ry ofstatic screening and a total number of 400 bands are employed for thesingle-shot G0W0 calculation to assure enough unoccupied states. AG0W0 direct gap of 2.807 eV is obtained, in good agreement withprevious results. Four valance bands and four conduction bands areincluded in BSE calculation, from which we obtain the dispersion ofeight exciton bands, and a binding energy of 0.879 eV for the A exci-ton. We further construct themode andmomentum-resolved exciton-phonon coupling matrix on the same grid.Data availabilityThe Source Data underlying the figures of this study have beendeposited in Figshare and are available at https://doi.org/10.6084/m9.figshare.25751568. 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This work is supported bythe National Key R&D Program of China (2023YFA1407000,2021YFA1202900, 2021YFA1400502), the National Science Foundationof China (NSFC) (12274447, 61888102), Guangdong Major Project ofBasic and Applied Basic Research (2021B0301030002), and the Strate-gic Priority Research Program of Chinese Academy of Sciences underthe grant No. XDB0470101. Y. W. and S. M. acknowledge funding sup-port from the Ministry of Science and Technology (No.2021YFA1400201), NSFC (Nos. 12025407, 92250303 and 11934004),and Chinese Academy of Sciences (Nos. YSBR047 and XDB33030100).K.W. and T.T. acknowledge support from the JSPSKAKENHI (20H00354,21H05233 and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Author contributionsL.D. and G.Z. supervised this work; L.D. and Z.H. conceived the projectand designed the experiments; Z.H. fabricated the devices and carriedout the optical measurements; Y.B. conducted the first principles cal-culations under the supervision of Y.W. and S.M.; K.W. and T.T. providedhighqualityh-BNcrystals; X.Z. performedSHGmeasurements under thesupervision of Y.X.; Z.H. and L.D. analysed the data; Y.Z., L.L., W.Y., D.S.,J.W., T.Z.,Q.Z., P.-H. T. andZ.S. helpedwithdata analysis; Z.H., Y.W., L.D.,and G.Z. co-wrote the manuscript. All authors discussed the results andcommented on the paper.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-48992-w.Correspondence and requests for materials should be addressed toYaxian Wang, Luojun Du or Guangyu Zhang.Peer review information Nature Communications thanks ZachariahHennighausen, Mandeep Khatoniar and the other, anonymous, review-er(s) for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-48992-wNature Communications |         (2024) 15:4586 9https://doi.org/10.1038/s41467-024-48992-whttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Observation of phonon Stark�effect Results Linear Stark effect of IXs in bilayer 2H-MoS2 Observation of phonon Stark�effect Underlying mechanism of phonon Stark�effect Electric control of phonon intensity mediated�by IXs Discussion Methods Device Fabrication Determination of FzFz and n0n0 Optic measurements Many-body theoretical calculations Data availability References Acknowledgements Author contributions Competing interests Additional information