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Amine Ben Mhenni, Dinh Van Tuan, Leonard Geilen, Marko M. Petrić, Melike Erdi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Seth Ariel Tongay, Kai Müller, Nathan P. Wilson, Jonathan J. Finley, Hanan Dery, Matteo Barbone

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[Breakdown of the Static Dielectric Screening Approximation of Coulomb Interactions in Atomically Thin Semiconductors](https://mdr.nims.go.jp/datasets/18dbf41f-7237-48ef-a5b1-eb272a4e0b34)

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Breakdown of the Static Dielectric Screening Approximation of Coulomb Interactions in Atomically Thin SemiconductorsBreakdown of the Static Dielectric ScreeningApproximation of Coulomb Interactions inAtomically Thin SemiconductorsAmine Ben Mhenni,* Dinh Van Tuan, Leonard Geilen, Marko M. Petric,́ Melike Erdi, Kenji Watanabe,Takashi Taniguchi, Seth Ariel Tongay, Kai Müller, Nathan P. Wilson, Jonathan J. Finley,* Hanan Dery,and Matteo Barbone*Cite This: ACS Nano 2025, 19, 4269−4278 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Coulomb interactions in atomically thin materialsare remarkably sensitive to variations in the dielectric screeningof the environment, which can be used to control exoticquantum many-body phases and engineer exciton potentiallandscapes. For decades, static or frequency-independentapproximations of the dielectric response, where increaseddielectric screening is predicted to cause an energy redshift ofthe exciton resonance, have been sufficient. These approxima-tions were first applied to quantum wells and were morerecently extended with initial success to layered transition metaldichalcogenides (TMDs). Here, we use charge-tunable exciton resonances to investigate screening effects in TMD monolayersembedded in materials with low-frequency dielectric constants ranging from 4 to more than 1000, a range of 2 orders ofmagnitude larger than in previous studies. In contrast to the redshift predicted by static models, we observe a blueshift of theexciton resonance exceeding 30 meV in higher dielectric constant environments. We explain our observations by introducing adynamical screening model based on a solution to the Bethe-Salpeter equation (BSE). When dynamical effects are strong, wefind that the exciton binding energy remains mostly controlled by the low-frequency dielectric response, while the exciton self-energy is dominated by the high-frequency one. Our results supplant the understanding of screening in layered materials andtheir heterostructures, introduce a knob to tune selected many-body effects, and reshape the framework for detecting andcontrolling correlated quantum many-body states and designing optoelectronic and quantum devices.KEYWORDS: transition metal dichalcogenides (TMDs), dielectric screening in 2D semiconductors, Coulomb interaction engineering,dynamical dielectric screening effects, high-dielectric-constant (high-K) materials, excitonic properties in van der Waals heterostructures,bandgap modulationINTRODUCTIONInteractions among particles give rise to collective phenomenadescribed by new fundamental laws beyond simplified single-particle systems.1 This is particularly evident in hetero-structures of two-dimensional (2D) materials, in which awide variety of correlated electronic and excitonic phases havebeen realized, driven by strong Coulomb interactions.2−6 Forinstance, excitonic complexes up to eight particles7−9 andsignatures of Wigner crystals10 have recently been reported inencapsulated, gated monolayer transition metal dichalcoge-nides (TMDs). Hubbard physics,11,12 unconventional super-conductivity,13 and Chern insulators14 have been observed inmoire ́ superlattices.In all such phenomena, Coulomb interactions are heavilyinfluenced by the dielectric response of the environmentbecause the electric field generated by charged quasiparticles ina 2D material extends into the surrounding medium, whichusually provides lower dielectric screening.15−19 This, in turn,leads to large exciton binding energy and single particlebandgap renormalization (BGR) effects. Therefore, Coulombinteraction engineering in atomically thin materials attractedconsiderable interest as a deterministic, scalable, and cleanroute to control many-body states, from exciton localizationReceived: August 21, 2024Revised: December 9, 2024Accepted: December 20, 2024Published: January 21, 2025Articlewww.acsnano.org© 2025 The Authors. Published byAmerican Chemical Society4269https://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−4278This article is licensed under CC-BY 4.0Downloaded via NATL INST FOR MATLS SCIENCE (NIMS) on February 7, 2025 at 04:41:52 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Amine+Ben+Mhenni"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dinh+Van+Tuan"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Leonard+Geilen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Marko+M.+Petric%CC%81"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Melike+Erdi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kenji+Watanabe"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Seth+Ariel+Tongay"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kai+Mu%CC%88ller"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Nathan+P.+Wilson"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Jonathan+J.+Finley"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Hanan+Dery"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Matteo+Barbone"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Matteo+Barbone"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acsnano.4c11563&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/ancac3/19/4?ref=pdfhttps://pubs.acs.org/toc/ancac3/19/4?ref=pdfhttps://pubs.acs.org/toc/ancac3/19/4?ref=pdfhttps://pubs.acs.org/toc/ancac3/19/4?ref=pdfwww.acsnano.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://www.acsnano.org?ref=pdfhttps://www.acsnano.org?ref=pdfhttps://acsopenscience.org/researchers/open-access/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/and transport to tuning many-body interactions in correlatedstates.20−23 To describe screening in semiconductor quantumwells and 2D materials, a common practice is to use aneffective dielectric constant, neglecting frequency dependenceand greatly simplifying the description of interactions betweenquasiparticles. Such description varies significantly dependingon the screening weight attributed to the semiconductor andthe environment layers.24−29 Theoretical investigations ex-plored the impact of dynamical screening effects on the opticalresonances in TMDs due to plasmons30 and optical phonons31from the surrounding environment. However, within the limitof small variations to the dielectric constants (below an orderof magnitude) of the environments and the carrier densitystudied so far, dynamical screening effects were predicted tointroduce corrections to the binding energy and BGR,30,31 butdid not appear to qualitatively alter the description of excitonsin TMD monolayers.18,20,32−36Here, we track gate-tunable exciton resonances in monolayerWSe2 embedded in environments with low-frequency dielectricconstants ε(ω = 0) spanning 3 orders of magnitude but withhigh (optical)-frequency dielectric constants ε(ω = ∞)changing by less than two times. In contrast with the precedingliterature, we surprisingly observe an exciton resonanceblueshif t for larger dielectric constant environments, incompat-ible with the established theoretical understanding. We explainthis behavior by introducing a model that accounts for thedynamic screening of electron−hole bound states, which showsthat when dynamic effects are strong, the exciton bindingenergy primarily responds to ε(0), while the self-energy (theenergy accounting for all interactions) of the bound stateprimarily depends on ε(∞). Crucially, the free-particlebandgap remains dependent on the low-frequency dielectricconstant and manifests its inadequacy to describe boundelectron−hole pairs under more extreme screening. Our resultsreveal conditions under which the frequency-independentdielectric screening approximation breaks down, and dynam-ical effects become a key factor in determining excitonicbehavior. Furthermore, they indicate the necessity of includingboth dynamical screening effects and a bound-state descriptionof excitons to fully capture screening effects in 2D systems.Materials with strong frequency-dependent dielectric functionsallow the selective tuning of exciton binding energy and self-energy in atomically thin semiconductors, providing a knob tocontrol quantum many-body states and their interactions andto design dielectrically engineered optoelectronic and quantumdevices.RESULTS AND DISCUSSIONEffect of the Dielectric Screening on the OpticalSpectrum of Monolayer TMDs. Figure 1a shows theschematic of an exciton in an atomically thin semiconductorembedded in environments with two different effective ε(0)and ε′(0), where ε(0) < ε′(0). Exciton states manifest asdiscrete optical resonances below the renormalized free-particle bandgap energy, as shown in Figure 1b for the excitonground state. The dielectric environment affects the excitonresonance energy through changes to both its binding energyand the electronic bandgap formed of the free electron andhole in the respective electronic bands, the latter being a BGReffect. For increasing effective ε(0), the bandgap reduces,inducing a redshif t of the exciton resonance. At the same time,the binding energy also decreases, thereby inducing a blueshif tof the exciton resonance. Scanning tunneling spectroscopyexperiments, which measure the free-particle bandgap, andoptical absorption, revealed the two effects to be of the sameorder of magnitude in TMDs (up to ∼ hundreds of meV),almost canceling each other.19,20 However, in the staticapproximation, the former is expected to be always slightlyFigure 1. Dielectric screening in monolayer semiconductors and device configurations. (a) Schematic of an exciton and the electric field linesbetween its electron and hole when an atomically thin semiconductor is embedded in a weak (strong) screening environment ε(0) (ε′(0)).(b) Sketches of the absorption spectra expected from (a), with EX being the energy resonance of the exciton ground state (n = 1), Eb thebinding energy, and Eg the continuum single-particle bandgap energy (exciton energy in the limit n = ∞). (c) Schematics of the gate-tunabledevices employed in this study. In each device, a monolayer WSe2 is placed between hBN and a bottom dielectric, which is either Gr, hBN,TiO2, or SrTiO3. (d) Reflection contrast spectrum of an hBN (black) and of a Gr device (red) at 5 K. The resonance energy of X0 redshiftswith increasing ε(0) of the environment. (e) Normalized PL spectra of the hBN, TiO2, and SrTiO3 devices at 8 K. Opposite to (d), theresonance energy of X0 blueshifts with increasing ε(0) of the environment.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784270https://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig1&ref=pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asstronger than the latter by up to a few tens of meV.29,37 Whencalculating the BGR, the Coulomb potential ΔV(r) isevaluated at a distance r → 0, whereas the binding energy isevaluated at a finite distance. Since the difference between theCoulomb potentials in two dielectric environments is greatestat r = 0, the net effect should always be a redshift of the excitonresonance with increasing static dielectric constant.29,37Importantly, this picture also implies that static screeningalone does not allow independent tuning of binding energyand bandgap. To date, applications using dielectric engineeringto control quasiparticles and their interactions as well as todesign devices have rested on this understanding.We fabricate charge-tunable devices based on monolayerWSe2 using van der Waals fabrication techniques (Methods).In this study, we use WSe2 as a prototypical TMD materialsince it offers a larger exciton Bohr radius than Mo-basedTMDs,35,38 amplifying its sensitivity to the dielectric environ-ment and because it does not display significant Fermi levelpinning.7,39 Figure 1c shows the device configuration.Monolayer WSe2 is sandwiched between a top layer ofhexagonal boron nitride (hBN) and a bottom layer withvarying ε(0), either hBN, few-layer graphene (Gr), TiO2, orSrTiO3. Throughout this work, we refer to the differentdielectric screening configurations by their bottom layer. Attemperatures ≤10 K, the effective ε(0) of these configurationsgoes from ∼3.5 for the hBN10 sample, to ∼7 for Gr,40 to ∼75for the TiO2 sample,41 and >1000 for the SrTiO3 sample,42spanning a range more than 2 orders of magnitude wider thanprevious studies.18,20,32,35 Gr is also used as a gate. Figure 1dcompares the reflection contrast ΔR/R0 at charge neutrality forthe Gr and hBN samples. Consistent with the conventionalunderstanding previously discussed, the neutral exciton X0energy redshifts about 15 meV from hBN to Gr due to theincreasing ε(0).20 Figure 1e shows the low-temperatureFigure 2. Gate-dependent optical response in ultrahigh ε(0) environments. (a−c) Gate-dependent d(ΔR/R0)/dE of monolayer WSe2 in thehBN (a), TiO2 (b), and SrTiO3 (c) dielectric configurations. The voltage corresponding to charge neutrality is indicated by a dashedhorizontal line. (d−f) Gate-dependent PL spectra of monolayer WSe2 in the hBN (d), TiO2 (e), and SrTiO3 (f) dielectric configurations. Inboth measurements, at charge neutrality X0 blueshifts with increasing effective ε(0) of the environment.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784271https://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig2&ref=pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asphotoluminescence (PL) spectra for hBN, TiO2, and SrTiO3near charge neutrality, evidenced by the high ratio between theX0 and negative trion X− intensities. In contrast to the Gr caseas well as previous reports,20,33,34 the X0 energy surprisinglyblueshifts with increasing effective ε(0), from 1.731 eV in thehBN device to 1.743 eV in the TiO2 device, and further to1.764 eV in the SrTiO3 device. These findings are not limitedto selected WSe2 samples, but we observe consistent blueshiftsacross more than 12 samples embedded in the same dielectricenvironments, also when replacing monolayer WSe2 withMoSe2 and WS2 (Supporting Information Figure S1).In contrast with past studies, we measure the gate-dependentoptical response of monolayer WSe2 in the different dielectricconfigurations to exclude possible contributions to the excitonresonance shift from charge doping.43 Figure 2a−c comparesthe reflection contrast derivative d(ΔR/R0)/dE from the hBN,TiO2, and SrTiO3 samples. In all cases, we extract the X0energy by fitting a dispersive Lorentzian at the chargeneutrality point identified from the X0 absorption maximum(Supporting Information Figure S2). In the hBN sample(Figure 2a), the energy of X0 is 1.731 eV. The spectrum of X0exhibits a pronounced broadening and an energy blueshiftgreater than 15 meV from charge neutrality to higher chargedoping. This highlights the importance of evaluating excitonicenergies at charge neutrality in such studies. The negativeexchange-split trions7 (Xintra− and Xinter− ) appear in the electron-doped regime (positive VG). In contrast, the positively chargedtrion7 (X+) becomes visible in the hole-doped regime (negativeVG). The TiO2 sample (Figure 2b) shows the X0 at 1.740 eV, 9meV blueshifted with respect to X0 in the hBN sample. Evenmore, the SrTiO3 sample (Figure 2c) shows an X0 energy of1.762 eV, 31 meV blueshifted with respect to that in the hBNsample.Figure 3. Modeling dynamical dielectric screening effects. (a) Comparison of the energy shift of X0 in monolayer WSe2 as a function of r2from the optical experiments and from the theoretical calculations for both static and dynamical models. Static models always lead toredshift, with the slab model (red) diverging by almost 180 meV from the experimental results (blue), by almost 170 meV from thedynamical 3χ (green), and by almost 130 meV from the static 3χ (orange). (b) Exciton binding energy calculated from the absorptionspectrum of monolayer WSe2 on SrTiO3 by neglecting the self-energy terms in the BSE equation and using ε(0) (orange), ε(∞) (brown),and ε(ω) (green). (c) Calculated real part of the single-particle self-energy Σc(k, z) of conduction band electrons as a function of Matsubarafrequencies for the SrTiO3 sample. Solid lines indicate dynamical calculations for two relevant electron energies Ek, while dashed lines arethe results of the single-particle BGR calculated with ε(∞) (brown) and ε(0) (orange). Zero energy is set to the calculated static self-energyof the hBN sample for ε(∞). (d) Absorption spectra corresponding to the samples measured experimentally calculated with dynamicalscreening (ε(ω)) by including both the dynamical potential and self-energies of the electron and hole in the exciton. The spectra are plottedrelative to E 1.9ghBN = eV. In the inset, schematic of the absorption spectrum of a monolayer TMD highlighting the exciton resonanceblueshift for r2 increasing from ∼1 to ≫1.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784272https://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig3&ref=pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asTo further corroborate our findings, we inspect the opticalresponse of WSe2 via gate-dependent PL spectroscopy andextract the energy of X0 at charge neutrality. Figure 2d−f showsthe gate-dependent PL spectra of the hBN, TiO2, and SrTiO3samples. In the hBN sample (Figure 2d), the energy of X0 is1.731 eV, accompanied by a line width below 2 meV,consistent with the highest quality samples reported in theliterature,7,39,44,45 and blueshifts due to charge doping by up to5 meV before disappearing (Supporting Information FigureS3). The excited states 2s, 3s, 4s, and (2s)+ are well-resolved inthe PL spectra (Supporting Information Figure S4), furthertestifying to the high sample quality.44,45 In the TiO2 sample(Figure 2e), X0 has a line width of less than 5 meV and appearsat 1.743 eV, blueshifted by ∼12 meV compared to that in thehBN sample. In the SrTiO3 sample (Figure 2f), X0 has a linewidth of ∼6 meV and arises at 1.764 eV, blueshifted by ∼33meV compared to that in the hBN sample. The line widthbroadening may be a consequence of the coupling of excitonswith the lower energy optical phonons of the substrates.31Overall, the PL measurements are in good agreement with thereflection contrast data.To exclude any contribution to the exciton energy shiftsfrom uncontrolled strain fields46 or other spatially dependenteffects, we study the X0 energy distribution over large areas onmultiple samples for each dielectric configuration (SupportingInformation Figure S5). We observe a narrow distributionbelow 3 meV, reflecting the high homogeneity of the samplesand the repeatability of the observations.Since εSrTiO3 (0) increases over 1 order of magnitudebetween 100 and 5 K,42 we also look at the temperaturedependence of the exciton resonance in the SrTiO3 device(Supporting Information Figure S7). Going from 80 K downto 20 K, X0 exhibits a blueshift (∼23 meV), which is more thantwice as large as the blueshift in the hBN sample (∼9 meV).This observation is consistent with X0 blueshifting due toincreasing ε(0) of the environment. Moreover, it unveils a newpathway to control X0 on the same device by tuning the ε(0)of SrTiO3 via temperature or via electric fields.42Fully Dynamical Description of Coulomb Screening.We compare our experimental results with the theoreticalpredictions from two models employed to describe theinfluence of the environment on exciton resonances inTMDs in the static screening approximation, the “3χ”model28 and the “slab” model,29 and track the predictedexciton resonance shift with varying screening r2 = ε(0)/ε(∞)from the reference point of a top and bottom hBNenvironment. Figure 3a shows that with increasing r2, excitonresonances according to the 3χ and the slab model areexpected to redshift from the hBN reference up to about 16and 145 meV respectively, or about 45−170 meV lower inenergy than our experimental results. The large differencebetween the two models stems from the lower screeningweight attributed to the surrounding environment by the 3χ.For a homogeneous strain field to be the source of such a shift,that would amount to a compressive strain ∼1 to 4%,47 whichhas never been reported even for externally applieddeformation, while the adhesion energy of WSe2 to thesubstrate would only support a planar strain well below 0.1%before delamination.48 Also, the energy of the ground state(1s) exciton resonance is less sensitive to strain than otherestablished experimental routes, such as the relative energybetween the 1s and the 2s exciton,46 which is employed as amore direct measurement of the binding energy and theelectronic bandgap.20 Having excluded other potential sourcesof blueshift, we conclude that X0 blueshifts with an increasingstatic ε(0) of the environment. This implies that thecorresponding reduction in the exciton binding energy mustbe greater than the BGR. Hence, the static approximation ofCoulomb interactions is not sufficient to describe the dielectricscreening in atomically thin semiconductors.To reconcile the contradiction between our results and thetheory of screened many-body interactions, we turn toexamining the role of frequency dependence in dielectricscreening. The response of a dielectric material to an electricfield comes from its valence electrons and, if the material ispolar, from field-induced lattice vibrations that induce a netatomic polarization.49 The electron and hole are notindependent entities; instead, they move with respect to eachother with kinetic energy commensurate with the excitonbinding energy as dictated by the virial theorem, resulting in avarying electric field.28,37 Consequently, we lift the assumptionthat the atoms of the encapsulating layers either perfectly trace(ε(0)) or completely ignore (ε(∞)) the variation of theelectric field. If the dielectric layer adjacent to the monolayersemiconductor is a polar material, we can approximate itsresponse to the electric field at frequency ω by the dielectricfunction:( ) ( )jjj,LO2 2,TO2 2=(1)The ratio between the low- and high-frequency dielectricc o n s t a n t s i s t h e L y d d a n e - S a c h s - T e l l e r r e l a t i o nr (0)/ ( ) /j j j2 ,LO2,TO2= = .49 The index j runs overthe optical phonon modes, where ωj,LO/TO is the associatedfrequency of the longitudinal/transverse optical latticevibration in the dielectric layers. In the following, we use the3χ formulation of the Coulomb potential28 and introducedynamical dielectric functions to model the response of topand bottom dielectrics. We calculate the dynamical self-energies of the electron and hole in the exciton from thesolution of the Dyson equation and use them for the BSEwhich we solve by an iterative method to obtain the absorptionspectrum50 (Supporting Information).Figure 3a shows the summary of the results for energy shiftΔEx of the exciton resonance. Consistent with our measure-ments, X0 blueshifts because the binding energy blueshiftcontribution ΔEb is larger than the redshift contribution ΔEgfor a higher r2. To understand the physical reasons behind suchresults, we consider individually each contribution.Figure 3b shows the binding energy of the SrTiO3 samplecalculated by neglecting the self-energy terms from the excitonGreen function. The large difference between ε(0) and ε(∞)leads to a significant difference between the binding energycalculated employing ε(0), which assumes that atoms canreadily trace the varying electric field of the electron and hole,and that calculated with ε(∞), which only considers theelectronic contribution to the screening, with the former ∼80meV smaller than the latter. Calculating the binding energy byusing the dynamical dielectric function ε(ω) in the effectiveBSE,51 we obtain results closer to ε(0), indicating that inmaterials with large r2 like SrTiO3 the binding energy is mostlyinfluenced by the low-frequency static dielectric response,while the opposite is true when r2 is close to 1.52 Figure S5shows the 2s energy peak measured from the PL spectrum ofACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784273https://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asthe SrTiO3 sample. The energy separation between the 1s and2s excitons is lower than the corresponding value in the hBNsample, which implies a lower binding energy, consistent withthe calculations.Figure 3c shows the real part of the dynamic single-particleself-energy Σc(k, z) of conduction band electrons in theSrTiO3 sample as a function of imaginary Matsubarafrequencies. The reference point at 0 meV is set to the staticself-energy calculated for the hBN sample with ε(∞). The self-energies calculated for the SrTiO3 sample at ε(0) and ε(∞)are ∼40 meV apart. We calculate the dynamical self-energy fortwo relevant electron energies E = ℏ2k2/2mc of 0 and 200 meV.Across the whole Matsubara frequency spectrum, and inparticular, for high frequencies, the self-energies remain closeto the value calculated with ε(∞), indicating that thedynamical self-energy is mainly influenced by the high-frequency dielectric response. The calculation is performedwith the bandgap Eg = 1.9 eV. Additional calculations withdifferent bandgaps show that decreasing Eg makes thedynamical self-energy come closer to the low-frequencyBGR, indicating that Eg plays an important role in the self-energies of the exciton components.52Figure 3d presents the calculated absorption spectra of X0 byincluding both self-energy and dynamical potential in the BSEfor all of the dielectric configurations considered in ourexperiments. The results show a net blueshift with increasingε(0), in agreement with the experimental findings. To bridgethe parameter gap between hBN and TiO2 and provide furtherguidelines to dielectric engineering efforts, we also calculate theoptical resonance for an intermediate screening materialsystem, hBN/WSe2/HfO2, having r2 ∼ 3,53,54 which weinclude in Figure 3a. The blueshift from the hBN referenceenergy is present, but it is found to be small, ∼1 meV. Weprovide all material parameters employed in the calculations inSupporting Information Table S1.Despite the qualitative agreement of our theoretical andexperimental results, we notice a lower shift in the calculations,possibly due to underestimation of the environmentalscreening in the 3χ model,28 as well as a possible smallerdifference of the ε(∞) values (i.e. εSrTiO3(∞) − εhBN(∞) <3.2). We also underline that we are unable to obtain a blueshiftfrom the slab model even including dynamical screening: themuch larger weight attributed to the environmental screeningbeyond the TMD layer always results in a dominant BGR term.Our understanding also consistently bridges graphene withstronger dielectric screening materials. Unlike the case of TiO2and SrTiO3, where strongly polar oxides result in extremelyhigh ε(0), graphene is a nonpolar material, which is equivalentto having r2 → 1 (or ε(0) ≈ ε(∞)), with the large carriermobility in graphene resulting in a very effective electronicscreening. Our dynamic formalism based on the Lyddane-Sachs-Teller relation cannot be extended to metallic environ-ments such as graphene, however, at the charge dopingdensities investigated in our work, dynamical effects due toplasmons are not expected to visibly affect the excitonic opticalresonances.30 When dynamical effects become negligible, orequivalently r2 → 1, r2 ceases to be the primary parameteraffecting exciton screening, supplanted by ε(0). Thus, weexpect TMD excitons screened by Gr to redshift from the hBNreference energy, which we observed in Figure 1d. The value ofthe optical resonance of the Gr sample is included in Figure 3a.We stress that our findings stem from the effect of dynamicaldielectric screening on the bound exciton: the self-energy of abound exciton is not the self-energy of a free electron in theconduction band, plus that of a free hole in the valence band.In a bound pair, the bandgap energy introduces a relative phaseexp(iEgt/ℏ) between the electron and hole components, andtherefore, at least one of these components is influenced by thehigh-frequency dielectric response (Supporting InformationTheoretical Methods). This subtle but key detail is lost if oneconsiders only the self-energy of a free particle, which isinfluenced by ε(0) because the reference energy level, in thiscase, is the edge of the relevant energy band. This hasimportant experimental consequences: if ε(0) is very differentfrom ε(∞), single-particle electronic bandgap measurementssuch as ARPES or scanning tunneling spectroscopy19 provideincorrect results to derive the self-energy of a bound electron−hole pair. Our results also indicate that the exciton self-energyand the exciton binding energy can be individually controlledby selecting screening materials with different r2 values.Achieving the highest exciton energy difference at a dielectricheterojunction requires maximizing the Δr2 values among thedifferent dielectric materials and employing a low r2 materialwith highest ε(0).Effect of the Dielectric Screening on Short-RangeCoulomb Interactions. To understand the dielectric screen-ing effects on many-body complexes beyond excitons, we alsoexperimentally investigate the behavior of the trion. Figure 4ashows the PL spectra of monolayer WSe2 for the hBN, TiO2,and SrTiO3 samples in the electron doping regime, but close tocharging neutrality (the X0 and negative trions intensities arecomparable) to minimize energy shifts from charge doping.The exciton resonance X0 of each sample is taken as the originof the energy axis to allow for a direct comparison of the trionbinding energies across the different dielectric configurations.The negatively charged intravalley trion Xintra− shows only aweak dependence on r2. Its binding energy starts at ∼30 meVin the hBN sample, drops to ∼24 meV in the TiO2 sample, andrises to ∼27 meV in the SrTiO3 sample. The nonmonotonicFigure 4. Effect of the dielectric environment on the trion bindingenergy. (a) PL spectra of monolayer WSe2 on hBN, TiO2, andSrTiO3 in the electron doping regime. EX is taken as the origin ofthe energy axis. In WSe2, the negative trions are exchange-split. Ineach spectrum, Xintra− is indicated by a black arrow. (b) PL spectraof monolayer MoSe2 on hBN, TiO2, and SrTiO3 in the electrondoping regime. EX is taken as the origin of the energy axis and isindicated on the plots. X− is indicated by a black arrow in eachspectrum.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784274https://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?fig=fig4&ref=pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asbehavior may be attributed to residual energy shifts frominconsistencies in charge doping among samples. Toinvestigate the same effect in a material with spectrally well-separated resonances, we also study the X− binding energy inmonolayer MoSe2. Figure 4b shows the PL spectra ofmonolayer MoSe2 for the hBN, TiO2, and SrTiO3 samples inthe electron doping regime. We first note that X0 in MoSe2 alsoexperiences a blueshift of up to 13 meV at higher values of r2.However, we do not observe any meaningful X− bindingenergy dependence on r2, with the change being of the order ofonly a few meV.The weak sensitivity of the trion binding energy in WSe2 andMoSe2 to r2, together with the conservation of many of theexcitonic features at extreme r2 values, suggests that theformation of trions and other excitonic few-body complexes isonly weakly affected by static ε(0). At large distances, theinteraction between a neutral exciton and an extra charge isdipolar in nature and, thus, has a relatively fast decay (V(r) ∼1/r2). Consequently, the binding energy of few-bodycomplexes such as the trion is governed by short-rangeinterparticle interactions, which are not sensitive to low-frequency screening.28CONCLUSIONSCoulomb interactions in atomically thin semiconductorscoupled to polar oxides require a physical description beyondthe static dielectric constant approximation, breaking themonolithic picture of exciton binding energy and BGR aseffects governed by the same type of screening and revealing anuanced interplay of phenomena with a distinct frequencydependence. Our results offer new avenues to study andmanipulate many-body interactions and provide the necessaryphysical understanding to predict exciton behavior whenintegrating TMDs and functional oxides. A natural conse-quence of our work would be to couple states with built-inelectrical dipoles with polar oxides, such as Janus TMDs,55,56or to tune interlayer and moire ́ excitons via the dielectricenvironment. Using excitonic resonances as sensors for chargeordering could provide deeper insights into correlated states.An exciting direction would be to explore the tuning of long-range interactions in strongly correlated systems, for example,in systems realizing the extended Hubbard model. This mayallow the realization of currently inaccessible many-bodyphases, including interaction-induced Chern insulators andquantum spin liquids.57,58 Finally, enabling the deterministicfabrication of dielectric superlattices could unlock the study ofstrongly correlated physics in artificial solid-state crystals andquasicrystals.21METHODS AND EXPERIMENTAL SECTIONSample Preparation. All of the TMD, Gr, and hBN flakes weremechanically exfoliated from bulk crystals on SiO2 substrates. Theflakes were selected based on their optical contrast, shape, andcleanliness. Single-crystal substrates of (001) TiO2 (Rutile phase) and(100) SrTiO3 were acquired from Shinkosha Co., Ltd. Oxide single-crystals have a purity >99.98% and RT ε(0) measured at 1 MHz of113 and 300, respectively. The devices were assembled via dry-transfer technique using polycarbonate films59 for the hBN and TiO2devices and polypropylene carbonate60 for the SrTiO3 devices.Contacts to the respective layers were patterned using opticallithography and electron beam evaporation (Cr/Au 5/100 nm).Optical Spectroscopy. Optical measurements were performed ina variable-temperature helium flow cryostat with a confocalmicroscope in a reflection geometry. For the PL measurements, 633nm/532 nm continuous wave laser sources were used for theexcitation. The laser beam was focused onto the sample using anobjective with a numerical aperture of 0.75, yielding an excitation spotsize of around 1 μm. A pinhole was used as a spatial filter to obtain adiffraction-limited collection spot. The collected light is dispersedusing a grating monochromator and detected on a CCD sensor array.The laser light was filtered by using a 650/550 nm short-pass filter.For reflection contrast spectroscopy, thermal light from a tungstenhalogen light source was used for excitation. The gate voltage in thegate-tunable measurements was controlled by using a Keithley 2400source meter. Unless otherwise specified, all measurements presentedhere were performed at 10 K. A close-cycle optical cryostat inreflection geometry (Attocube, attoDRY800) with variable-temper-ature capability was used to perform the temperature-dependentmeasurements presented in the Supporting Information.ASSOCIATED CONTENTData Availability StatementThe data sets generated and analyzed during the current studyare available from the corresponding authors upon reasonablerequest.*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acsnano.4c11563.Effect of the dielectric screening on X0 in MoSe2 andWS2; insights in the reflection contrast and PL dataanalysis; Effect of charge doping on EX for monolayerWSe2 in the hBN sample; PL of the exciton Rydbergseries of monolayer WSe2 in the hBN sample; Differ-ential reflectance of the exciton Rydberg series ofmonolayer WSe2 in the SrTiO3 sample; X0 excitonenergy distribution from PL sample maps; temperature-dependence of the SrTiO3 sample; and theoreticalmethods (PDF)AUTHOR INFORMATIONCorresponding AuthorsAmine Ben Mhenni − Walter Schottky Institute and TUMSchool of Natural Sciences, Technical University of Munich,85748 Garching, Germany; Munich Center for QuantumScience and Technology (MCQST), 80799 Munich,Germany; Email: amine.ben-mhenni@tum.deJonathan J. Finley − Walter Schottky Institute and TUMSchool of Natural Sciences, Technical University of Munich,85748 Garching, Germany; Munich Center for QuantumScience and Technology (MCQST), 80799 Munich,Germany; Email: jj.finley@tum.deMatteo Barbone −Walter Schottky Institute and TUM Schoolof Computation, Information and Technology, TechnicalUniversity of Munich, 85748 Garching, Germany; MunichCenter for Quantum Science and Technology (MCQST),80799 Munich, Germany; orcid.org/0000-0003-3553-7281; Email: matteo.barbone@wsi.tum.deAuthorsDinh Van Tuan − Department of Electrical and ComputerEngineering, University of Rochester, Rochester, New York14627, United StatesLeonard Geilen − Walter Schottky Institute and TUM Schoolof Natural Sciences, Technical University of Munich, 85748Garching, Germany; Munich Center for Quantum Scienceand Technology (MCQST), 80799 Munich, GermanyMarko M. Petric ́ − Walter Schottky Institute and TUMSchool of Computation, Information and Technology,ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784275https://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acsnano.4c11563/suppl_file/nn4c11563_si_001.pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Amine+Ben+Mhenni"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfmailto:amine.ben-mhenni@tum.dehttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Jonathan+J.+Finley"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfmailto:jj.finley@tum.dehttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Matteo+Barbone"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-3553-7281https://orcid.org/0000-0003-3553-7281mailto:matteo.barbone@wsi.tum.dehttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dinh+Van+Tuan"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Leonard+Geilen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Marko+M.+Petric%CC%81"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfwww.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asTechnical University of Munich, 85748 Garching, Germany;Munich Center for Quantum Science and Technology(MCQST), 80799 Munich, Germany; orcid.org/0000-0002-6688-326XMelike Erdi − School for Engineering of Matter, Transportand Energy, Arizona State University, Tempe, Arizona85287, United StatesKenji Watanabe − Research Center for Electronic and OpticalMaterials, National Institute for Materials Science, Tsukuba305-0044, Japan; orcid.org/0000-0003-3701-8119Takashi Taniguchi − Research Center for MaterialsNanoarchitectonics, National Institute for Materials Science,Tsukuba 305-0044, Japan; orcid.org/0000-0002-1467-3105Seth Ariel Tongay − School for Engineering of Matter,Transport and Energy, Arizona State University, Tempe,Arizona 85287, United StatesKai Müller − Walter Schottky Institute and TUM School ofComputation, Information and Technology, TechnicalUniversity of Munich, 85748 Garching, Germany; MunichCenter for Quantum Science and Technology (MCQST),80799 Munich, GermanyNathan P. Wilson − Walter Schottky Institute and TUMSchool of Natural Sciences, Technical University of Munich,85748 Garching, Germany; Munich Center for QuantumScience and Technology (MCQST), 80799 Munich,GermanyHanan Dery − Department of Electrical and ComputerEngineering and Department of Physics and Astronomy,University of Rochester, Rochester, New York 14627, UnitedStatesComplete contact information is available at:https://pubs.acs.org/10.1021/acsnano.4c11563Author ContributionsA.B.M. and M.B. conceived and managed the research. A.B.M.and L.G. fabricated the devices. A.B.M., L.G., M.M.P., andM.B. performed the optical measurements. A.B.M. and M.B.analyzed the results. J.J.F. and K.M. obtained third-partyfunding and provided experimental and nanofabricationinfrastructure. D.V.T. and H.D. developed the theoreticalmodels and performed the calculations. M.E. and S.T. grewbulk WSe2, WS2, and MoSe2 crystals. K.W. and T.T. grew bulkhBN crystals. All authors discussed the results and contributedto the writing of the paper.NotesThe authors declare no competing financial interest.An early version of this manuscript was submitted as preprint:Ben Mhenni, A.; Van Tuan, D.; Geilen, L.; Petric,́ M. M.; Erdi,M.; Watanabe, K.; Taniguchi, T.; Tongay, S.; Müller, K.;Wilson, N. P.; Finley, J. J.; Dery, H.; Barbone, M. Breakdownof the Static Dielectric Screening Approximation of CoulombInteractions in Atomically Thin Semiconductors. 2024,arXiv:2402.18639. ArXiv e-prints. https://doi.org/10.48550/arXiv.2402.18639 (28 Feb 2024).ACKNOWLEDGMENTSWe thank E. Zubizarreta-Casalengua and F. Menzel for usefuldiscussions. A.B.M. acknowledges funding from the Interna-tional Max Planck Research School for Quantum Science andTechnology (IMPRS-QST). M.B. acknowledges funding fromthe Alexander von Humboldt Foundation. We gratefullyacknowledge funding from the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) via Germany’sExcellence Strategy (MCQST, EXC-2111/390814868, and e-conversion, EXC-2089/1-390776260). J.J.F. also acknowledgesthe BMBF for funding via projects 16K15Q027, 13N15760,13N16214, as well as the DFG via INST 95/1719-1, FI 947/6-1, INST 95/1496-1, FI 947/5-1, FI 947/8-1, DI 2013/5-1 andDI 2013/5-2. K.M. also acknowledges the DFG via the projectPQET (INST 95/1654-1). Furthermore, we acknowledge theBavarian Science Ministry for funding via the MunichQuantum Valley, Nequs, and IQ-Sense projects. Work at theUniversity of Rochester was supported by the Department ofEnergy, Basic Energy Sciences, Division of Materials Sciencesand Engineering under Award No. DE-SC0014349. S.T.acknowledges primary support from DOE-SC0020653 (mate-rials synthesis), DMR 2111812 (optical testing), and CMMI2129412 (scaling). S.T. also acknowledges support fromLawrence Semiconductors and Applied Materials Inc. K.W.and T.T. acknowledge support from the JSPS KAKENHI(Grant Numbers 20H00354 and 23H02052) and WorldPremier International Research Center Initiative (WPI),MEXT, Japan.REFERENCES(1) Snoke, D. W. Solid State Physics: Essential Concepts, 2nd ed.;Cambridge University Press: Cambridge, United Kingdom; NewYork, NY, 2019.(2) Yankowitz, M.; Ma, Q.; Jarillo-Herrero, P.; LeRoy, B. J. van derWaals heterostructures combining graphene and hexagonal boronnitride. Nat. Rev. Phys. 2019, 1, 112−125.(3) Mak, K. F.; Shan, J. Semiconductor moire ́ materials. Nat.Nanotechnol. 2022, 17, 686−695.(4) Wilson, N. P.; Yao, W.; Shan, J.; Xu, X. Excitons and emergentquantum phenomena in stacked 2D semiconductors. Nature 2021,599, 383−392.(5) Regan, E. C.; Wang, D.; Paik, E. Y.; Zeng, Y.; Zhang, L.; Zhu, J.;MacDonald, A. H.; Deng, H.; Wang, F. Emerging exciton physics intransition metal dichalcogenide heterobilayers. Nat. Rev. Mater. 2022,7, 778−795.(6) Montblanch, A. R.-P.; Barbone, M.; Aharonovich, I.; Atatüre, M.;Ferrari, A. C. Layered materials as a platform for quantumtechnologies. Nat. Nanotechnol. 2023, 18, 555−571.(7) Barbone, M.; et al. Charge-tuneable biexciton complexes inmonolayer WSe2. Nat. Commun. 2018, 9, 3721.(8) Sidler, M.; Back, P.; Cotlet, O.; Srivastava, A.; Fink, T.; Kroner,M.; Demler, E.; Imamoglu, A. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys. 2017, 13, 255−261.(9) Van Tuan, D.; Shi, S.-F.; Xu, X.; Crooker, S. A.; Dery, H. Six-Body and Eight-Body Exciton States in Monolayer WSe. Phys. Rev.Lett. 2022, 129, No. 076801.(10) Smolenśki, T.; Dolgirev, P. E.; Kuhlenkamp, C.; Popert, A.;Shimazaki, Y.; Back, P.; Lu, X.; Kroner, M.; Watanabe, K.; Taniguchi,T.; Esterlis, I.; Demler, E.; Imamoğlu, A. Signatures of Wigner crystalof electrons in a monolayer semiconductor. Nature 2021, 595, 53−57.(11) Tang, Y.; Li, L.; Li, T.; Xu, Y.; Liu, S.; Barmak, K.; Watanabe,K.; Taniguchi, T.; MacDonald, A. H.; Shan, J.; Mak, K. F. Simulationof Hubbard model physics in WSe2/WS2 moire ́ superlattices. Nature2020, 579, 353−358.(12) Xu, Y.; Liu, S.; Rhodes, D. A.; Watanabe, K.; Taniguchi, T.;Hone, J.; Elser, V.; Mak, K. F.; Shan, J. Correlated insulating states atfractional fillings of moire ́ superlattices. Nature 2020, 587, 214−218.(13) Cao, Y.; Fatemi, V.; Fang, S.; Watanabe, K.; Taniguchi, T.;Kaxiras, E.; Jarillo-Herrero, P. Unconventional superconductivity inmagic-angle graphene superlattices. Nature 2018, 556, 43−50.(14) Chen, G.; Sharpe, A. L.; Fox, E. J.; Zhang, Y.-H.; Wang, S.;Jiang, L.; Lyu, B.; Li, H.; Watanabe, K.; Taniguchi, T.; Shi, Z.; Senthil,ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784276https://orcid.org/0000-0002-6688-326Xhttps://orcid.org/0000-0002-6688-326Xhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Melike+Erdi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kenji+Watanabe"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-3701-8119https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Takashi+Taniguchi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1467-3105https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Seth+Ariel+Tongay"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kai+Mu%CC%88ller"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Nathan+P.+Wilson"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Hanan+Dery"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/doi/10.1021/acsnano.4c11563?ref=pdfhttps://doi.org/10.1038/s42254-018-0016-0https://doi.org/10.1038/s42254-018-0016-0https://doi.org/10.1038/s42254-018-0016-0https://doi.org/10.1038/s41565-022-01165-6https://doi.org/10.1038/s41586-021-03979-1https://doi.org/10.1038/s41586-021-03979-1https://doi.org/10.1038/s41578-022-00440-1https://doi.org/10.1038/s41578-022-00440-1https://doi.org/10.1038/s41565-023-01354-xhttps://doi.org/10.1038/s41565-023-01354-xhttps://doi.org/10.1038/s41467-018-05632-4https://doi.org/10.1038/s41467-018-05632-4https://doi.org/10.1038/nphys3949https://doi.org/10.1038/nphys3949https://doi.org/10.1103/PhysRevLett.129.076801https://doi.org/10.1103/PhysRevLett.129.076801https://doi.org/10.1038/s41586-021-03590-4https://doi.org/10.1038/s41586-021-03590-4https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160www.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asT.; Goldhaber-Gordon, D.; Zhang, Y.; Wang, F. Tunable correlatedChern insulator and ferromagnetism in a moire ́ superlattice. Nature2020, 579, 56−61.(15) Rytova, N. S. Screened potential of a point charge in a thin film.Proc. MSU Phys., Astron. 1967, 3, 18.(16) Keldysh, L. V. Coulomb interaction in thin semiconductor andsemimetal films. JETP Lett. 1979, 29, 658.(17) Cudazzo, P.; Tokatly, I. V.; Rubio, A. Dielectric screening intwo-dimensional insulators: Implications for excitonic and impuritystates in graphane. Phys. Rev. B 2011, 84, No. 085406.(18) Chernikov, A.; Berkelbach, T. C.; Hill, H. M.; Rigosi, A.; Li, Y.;Aslan, B.; Reichman, D. R.; Hybertsen, M. S.; Heinz, T. F. ExcitonBinding Energy and Nonhydrogenic Rydberg Series in MonolayerWS2. Phys. Rev. Lett. 2014, 113, No. 076802.(19) Ugeda, M. M.; Bradley, A. J.; Shi, S.-F.; da Jornada, F. H.;Zhang, Y.; Qiu, D. Y.; Ruan, W.; Mo, S.-K.; Hussain, Z.; Shen, Z.-X.;Wang, F.; Louie, S. G.; Crommie, M. F. Giant bandgaprenormalization and excitonic effects in a monolayer transitionmetal dichalcogenide semiconductor. Nat. Mater. 2014, 13, 1091−1095.(20) Raja, A.; et al. Coulomb engineering of the bandgap andexcitons in two-dimensional materials. Nat. Commun. 2017, 8, 15251.(21) Forsythe, C.; Zhou, X.; Watanabe, K.; Taniguchi, T.;Pasupathy, A.; Moon, P.; Koshino, M.; Kim, P.; Dean, C. R. Bandstructure engineering of 2D materials using patterned dielectricsuperlattices. Nat. Nanotechnol. 2018, 13, 566−571.(22) Steinke, C.; Wehling, T. O.; Rösner, M. Coulomb-engineeredheterojunctions and dynamical screening in transition metaldichalcogenide monolayers. Phys. Rev. B 2020, 102, No. 115111.(23) Steinhoff, A.; Florian, M.; Rösner, M.; Schönhoff, G.; Wehling,T. O.; Jahnke, F. Exciton fission in monolayer transition metaldichalcogenide semiconductors. Nat. Commun. 2017, 8, 1166.(24) Rösner, M.; Şasi̧oğlu, E.; Friedrich, C.; Blügel, S.; Wehling, T.O. Wannier function approach to realistic Coulomb interactions inlayered materials and heterostructures. Phys. Rev. B 2015, 92,No. 085102.(25) Trolle, M. L.; Pedersen, T. G.; Véniard, V. Model dielectricfunction for 2D semiconductors including substrate screening. Sci.Rep. 2017, 7, 39844.(26) Andersen, K.; Latini, S.; Thygesen, K. S. Dielectric Genome ofvan der Waals Heterostructures. Nano Lett. 2015, 15, 4616−4621.(27) Riis-Jensen, A. C.; Gjerding, M. N.; Russo, S.; Thygesen, K. S.Anomalous exciton Rydberg series in two-dimensional semiconduc-tors on high-κ dielectric substrates. Phys. Rev. B 2020, 102,No. 201402.(28) Van Tuan, D.; Yang, M.; Dery, H. Coulomb interaction inmonolayer transition-metal dichalcogenides. Phys. Rev. B 2018, 98,No. 125308.(29) Cho, Y.; Berkelbach, T. C. Environmentally sensitive theory ofelectronic and optical transitions in atomically thin semiconductors.Phys. Rev. B 2018, 97, No. 041409.(30) Steinhoff, A.; Wehling, T. O.; Rösner, M. Frequency-dependentsubstrate screening of excitons in atomically thin transition metaldichalcogenide semiconductors. Phys. Rev. B 2018, 98, No. 045304.(31) Steinhoff, A.; Florian, M.; Jahnke, F. Dynamical screeningeffects of substrate phonons on two-dimensional excitons. Phys. Rev. B2020, 101, No. 045411.(32) Raja, A.; Waldecker, L.; Zipfel, J.; Cho, Y.; Brem, S.; Ziegler, J.D.; Kulig, M.; Taniguchi, T.; Watanabe, K.; Malic, E.; Heinz, T. F.;Berkelbach, T. C.; Chernikov, A. Dielectric disorder in two-dimensional materials. Nat. Nanotechnol. 2019, 14, 832−837.(33) Stier, A. V.; Wilson, N. P.; Clark, G.; Xu, X.; Crooker, S. A.Probing the Influence of Dielectric Environment on Excitons inMonolayer WSe2: Insight from High Magnetic Fields. Nano Lett.2016, 16, 7054−7060.(34) Tebbe, D.; Schütte, M.; Watanabe, K.; Taniguchi, T.; Stampfer,C.; Beschoten, B.; Waldecker, L. Tailoring the dielectric screening inWS2−graphene heterostructures. npj 2D Mater. Appl. 2023, 7, 29.(35) Goryca, M.; Li, J.; Stier, A. V.; Taniguchi, T.; Watanabe, K.;Courtade, E.; Shree, S.; Robert, C.; Urbaszek, B.; Marie, X.; Crooker,S. A. Revealing exciton masses and dielectric properties of monolayersemiconductors with high magnetic fields. Nat. Commun. 2019, 10,4172.(36) Peimyoo, N.; Wu, H.-Y.; Escolar, J.; De Sanctis, A.; Prando, G.;Vollmer, F.; Withers, F.; Riis-Jensen, A. C.; Craciun, M. F.; Thygesen,K. S.; Russo, S. Engineering Dielectric Screening for Potential-wellArrays of Excitons in 2D Materials. ACS Appl. Mater. Interfaces 2020,12, 55134−55140.(37) Scharf, B.; Tuan, D. V.; Žutic,́ I.; Dery, H. Dynamical screeningin monolayer transition-metal dichalcogenides and its manifestationsin the exciton spectrum. J. Phys.: Cond. Matt. 2019, 31, 203001.(38) Stier, A.; Wilson, N.; Velizhanin, K.; Kono, J.; Xu, X.; Crooker,S. Magnetooptics of Exciton Rydberg States in a MonolayerSemiconductor. Phys. Rev. Lett. 2018, 120, No. 057405.(39) He, M.; Rivera, P.; Van Tuan, D.; Wilson, N. P.; Yang, M.;Taniguchi, T.; Watanabe, K.; Yan, J.; Mandrus, D. G.; Yu, H.; Dery,H.; Yao, W.; Xu, X. Valley phonons and exciton complexes in amonolayer semiconductor. Nat. Commun. 2020, 11, 618.(40) Taft, E. A.; Philipp, H. R. Optical Properties of Graphite. Phys.Rev. 1965, 138, A197−A202.(41) Schöche, S.; Hofmann, T.; Korlacki, R.; Tiwald, T. E.;Schubert, M. Infrared dielectric anisotropy and phonon modes ofrutile TiO2. J. Appl. Phys. 2013, 113, 164102.(42) Neville, R. C.; Hoeneisen, B.; Mead, C. A. Permittivity ofStrontium Titanate. J. Appl. Phys. 1972, 43, 2124−2131.(43) Van Tuan, D.; Scharf, B.; Wang, Z.; Shan, J.; Mak, K. F.; Žutic,́I.; Dery, H. Probing many-body interactions in monolayer transition-metal dichalcogenides. Phys. Rev. B 2019, 99, No. 085301.(44) Liu, E.; van Baren, J.; Taniguchi, T.; Watanabe, K.; Chang, Y.-C.; Lui, C. H. Magnetophotoluminescence of exciton Rydberg statesin monolayer WSe2. Phys. Rev. B 2019, 99, No. 205420.(45) Liu, E.; van Baren, J.; Lu, Z.; Taniguchi, T.; Watanabe, K.;Smirnov, D.; Chang, Y.-C.; Lui, C. H. Exciton-polaron Rydberg statesin monolayer MoSe2 and WSe2. Nat. Commun. 2021, 12, 6131.(46) Aslan, B.; Deng, M.; Heinz, T. F. Strain tuning of excitons inmonolayer WSe2. Phys. Rev. B 2018, 98, No. 115308.(47) Schmidt, R.; Niehues, I.; Schneider, R.; Drüppel, M.; Deilmann,T.; Rohlfing, M.; Vasconcellos, S. M. D.; Castellanos-Gomez, A.;Bratschitsch, R. Reversible uniaxial strain tuning in atomically thinWSe2. 2D Mater. 2016, 3, No. 021011.(48) Blundo, E.; Yildirim, T.; Pettinari, G.; Polimeni, A.Experimental Adhesion Energy in van der Waals Crystals andHeterostructures from Atomically Thin Bubbles. Phys. Rev. Lett.2021, 127, No. 046101.(49) Lyddane, R. H.; Sachs, R. G.; Teller, E. On the Polar Vibrationsof Alkali Halides. Phys. Rev. 1941, 59, 673−676.(50) Dyson, F. J. The S Matrix in Quantum Electrodynamics. Phys.Rev. 1949, 75, 1736−1755.(51) Salpeter, E. E.; Bethe, H. A. A Relativistic Equation for Bound-State Problems. Phys. Rev. 1951, 84, 1232−1242.(52) Van Tuan, D.; Dery, H. Effects of dynamical dielectricscreening on the excitonic spectrum of monolayer semiconductors.Phys. Rev. B 2024, 110, No. 125301.(53) Peimyoo, N.; Barnes, M. D.; Mehew, J. D.; De Sanctis, A.;Amit, I.; Escolar, J.; Anastasiou, K.; Rooney, A. P.; Haigh, S. J.; Russo,S.; Craciun, M. F.; Withers, F. Laser-writable high-k dielectric for vander Waals nanoelectronics. Sci. Adv. 2019, 5, No. eaau0906.(54) Dou, H.; Strkalj, N.; Zhang, Y.; MacManus-Driscoll, J. L.; Jia,Q.; Wang, H. Optical dielectric properties of HfO2-based films. J. Vac.Sci. Technol. A 2022, 40, No. 033412.(55) Petric,́ M. M.; Villafañe, V.; Herrmann, P.; Ben Mhenni, A.;Qin, Y.; Sayyad, Y.; Shen, Y.; Tongay, S.; Müller, K.; Soavi, G.; Finley,J. J.; Barbone, M. Nonlinear Dispersion Relation and Out-of-PlaneSecond Harmonic Generation in MoSSe and WSSe JanusMonolayers. Adv. Opti. Mater. 2023, 11, No. 2300958.(56) Petric,́ M. M.; Kremser, M.; Barbone, M.; Qin, Y.; Sayyad, Y.;Shen, Y.; Tongay, S.; Finley, J. J.; Botello-Méndez, A. R.; Müller, K.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784277https://doi.org/10.1038/s41586-020-2049-7https://doi.org/10.1038/s41586-020-2049-7https://doi.org/10.1103/PhysRevB.84.085406https://doi.org/10.1103/PhysRevB.84.085406https://doi.org/10.1103/PhysRevB.84.085406https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1038/nmat4061https://doi.org/10.1038/nmat4061https://doi.org/10.1038/nmat4061https://doi.org/10.1038/ncomms15251https://doi.org/10.1038/ncomms15251https://doi.org/10.1038/s41565-018-0138-7https://doi.org/10.1038/s41565-018-0138-7https://doi.org/10.1038/s41565-018-0138-7https://doi.org/10.1103/PhysRevB.102.115111https://doi.org/10.1103/PhysRevB.102.115111https://doi.org/10.1103/PhysRevB.102.115111https://doi.org/10.1038/s41467-017-01298-6https://doi.org/10.1038/s41467-017-01298-6https://doi.org/10.1103/PhysRevB.92.085102https://doi.org/10.1103/PhysRevB.92.085102https://doi.org/10.1038/srep39844https://doi.org/10.1038/srep39844https://doi.org/10.1021/acs.nanolett.5b01251?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.nanolett.5b01251?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/PhysRevB.102.201402https://doi.org/10.1103/PhysRevB.102.201402https://doi.org/10.1103/PhysRevB.98.125308https://doi.org/10.1103/PhysRevB.98.125308https://doi.org/10.1103/PhysRevB.97.041409https://doi.org/10.1103/PhysRevB.97.041409https://doi.org/10.1103/PhysRevB.98.045304https://doi.org/10.1103/PhysRevB.98.045304https://doi.org/10.1103/PhysRevB.98.045304https://doi.org/10.1103/PhysRevB.101.045411https://doi.org/10.1103/PhysRevB.101.045411https://doi.org/10.1038/s41565-019-0520-0https://doi.org/10.1038/s41565-019-0520-0https://doi.org/10.1021/acs.nanolett.6b03276?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.nanolett.6b03276?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1038/s41699-023-00394-0https://doi.org/10.1038/s41699-023-00394-0https://doi.org/10.1038/s41467-019-12180-yhttps://doi.org/10.1038/s41467-019-12180-yhttps://doi.org/10.1021/acsami.0c14696?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acsami.0c14696?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1088/1361-648X/ab071fhttps://doi.org/10.1088/1361-648X/ab071fhttps://doi.org/10.1088/1361-648X/ab071fhttps://doi.org/10.1103/PhysRevLett.120.057405https://doi.org/10.1103/PhysRevLett.120.057405https://doi.org/10.1038/s41467-020-14472-0https://doi.org/10.1038/s41467-020-14472-0https://doi.org/10.1103/PhysRev.138.A197https://doi.org/10.1063/1.4802715https://doi.org/10.1063/1.4802715https://doi.org/10.1063/1.1661463https://doi.org/10.1063/1.1661463https://doi.org/10.1103/PhysRevB.99.085301https://doi.org/10.1103/PhysRevB.99.085301https://doi.org/10.1103/PhysRevB.99.205420https://doi.org/10.1103/PhysRevB.99.205420https://doi.org/10.1038/s41467-021-26304-whttps://doi.org/10.1038/s41467-021-26304-whttps://doi.org/10.1103/PhysRevB.98.115308https://doi.org/10.1103/PhysRevB.98.115308https://doi.org/10.1088/2053-1583/3/2/021011https://doi.org/10.1088/2053-1583/3/2/021011https://doi.org/10.1103/PhysRevLett.127.046101https://doi.org/10.1103/PhysRevLett.127.046101https://doi.org/10.1103/PhysRev.59.673https://doi.org/10.1103/PhysRev.59.673https://doi.org/10.1103/PhysRev.75.1736https://doi.org/10.1103/PhysRev.84.1232https://doi.org/10.1103/PhysRev.84.1232https://doi.org/10.1103/PhysRevB.110.125301https://doi.org/10.1103/PhysRevB.110.125301https://doi.org/10.1126/sciadv.aau0906https://doi.org/10.1126/sciadv.aau0906https://doi.org/10.1116/6.0001651https://doi.org/10.1002/adom.202300958https://doi.org/10.1002/adom.202300958https://doi.org/10.1002/adom.202300958www.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asRaman spectrum of Janus transition metal dichalcogenide monolayersWSSe and MoSSe. Phys. Rev. B 2021, 103, No. 035414.(57) Wu, F.; Lovorn, T.; Tutuc, E.; MacDonald, A. H. HubbardModel Physics in Transition Metal Dichalcogenide Moire ́ Bands.Phys. Rev. Lett. 2018, 121, No. 026402.(58) Pan, H.; Wu, F.; Das Sarma, S. Quantum phase diagram of aMoire-́Hubbard model. Phys. Rev. B 2020, 102, No. 201104.(59) Purdie, D. G.; Pugno, N. M.; Taniguchi, T.; Watanabe, K.;Ferrari, A. C.; Lombardo, A. Cleaning interfaces in layered materialsheterostructures. Nat. Commun. 2018, 9, 5387.(60) Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.;Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.;Kim, P.; Hone, J.; Shepard, K. L.; Dean, C. R. One-DimensionalElectrical Contact to a Two-Dimensional Material. Science 2013, 342,614−617.ACS Nano www.acsnano.org Articlehttps://doi.org/10.1021/acsnano.4c11563ACS Nano 2025, 19, 4269−42784278https://doi.org/10.1103/PhysRevB.103.035414https://doi.org/10.1103/PhysRevB.103.035414https://doi.org/10.1103/PhysRevLett.121.026402https://doi.org/10.1103/PhysRevLett.121.026402https://doi.org/10.1103/PhysRevB.102.201104https://doi.org/10.1103/PhysRevB.102.201104https://doi.org/10.1038/s41467-018-07558-3https://doi.org/10.1038/s41467-018-07558-3https://doi.org/10.1126/science.1244358https://doi.org/10.1126/science.1244358www.acsnano.org?ref=pdfhttps://doi.org/10.1021/acsnano.4c11563?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as