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Michele Masseroni, Isaac Soltero, James G. McHugh, Igor Rozhansky, Xue Li, Alexander Schmidhuber, Markus Niese, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Vladimir I. Fal’ko, Thomas Ihn, Klaus Ensslin

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[Gate-Tunable Band Edge in Few-Layer MoS<sub>2</sub>](https://mdr.nims.go.jp/datasets/cd0f85de-f2ca-4963-850b-cd5b78793d7a)

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Gate-Tunable Band Edge in Few-Layer MoS2Gate-Tunable Band Edge in Few-Layer MoS2Michele Masseroni, Isaac Soltero, James G. McHugh, Igor Rozhansky, Xue Li, Alexander Schmidhuber,Markus Niese, Takashi Taniguchi, Kenji Watanabe, Vladimir I. Fal’ko, Thomas Ihn, and Klaus Ensslin*Cite This: Nano Lett. 2025, 25, 10472−10477 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Transition metal dichalcogenides (TMDs) havegarnered significant research interest due to the variation in bandedge locations within the hexagonal Brillouin zone between single-layer and bulk configurations. In monolayers, the conduction bandminima are centered at the K points, whereas in multilayers, they shiftto the Q points, midway between the Γ and K points. In this study, weconduct magnetotransport experiments to measure the occupation inthe Q and K valleys in four-layer molybdenum disulfide (MoS2). Wedemonstrate electrostatic tunability of the conduction band edge bycombining our experimental results with a hybrid k·p tight-bindingmodel that accounts for interlayer screening effects in a self-consistentmanner. Furthermore, we extend our model to bilayer and trilayerMoS2, reconciling prior experimental results and quantifying thetunable range of band edges in atomically thin TMDs.KEYWORDS: MoS2, Shubnikov−de Haas oscillations, field effect, band edge alignment, layer polarization, interlayer screeningTransition metal dichalcogenides (TMDs) display uniqueelectronic1−3 and optical properties.4,5 A key feature ofTMDs is the presence of multiple valleys in their bandstructure, with the conduction band hosting valleys at both theK points and the Q points (located midway between the Γ andK points) of the first Brillouin zone, while the valence bandexhibits valleys at the K points and the Γ point.6 The relativeenergy levels of these valleys are strongly influenced by thenumber of layers, leading to layer-dependent modifications inthe band structure.7,8In our previous studies, we systematically investigated theconduction band of monolayer,9 bilayer,10 and trilayer MoS2.11We found that electron transport predominantly occurred viathe K valleys of the conduction band, while the occupation ofQ valleys, predicted by theory,12−17 was not detected in ourmeasurements. In these studies, we employed electrostaticgating to overcome the Schottky barrier that typically forms atthe metal−MoS2 interface. However, the influence of the gate-induced electric field on the band structure, particularly therelative energy shift of different valleys,18 was not considered.Here, we investigate four-layer MoS2, where we observe agate-dependent band edge transition in the conduction bandminima, shifting from the Q valleys at low gate voltages to theK valleys at higher voltages. This transition reveals thesensitivity of the conduction band to external electric fields,which we attribute to the differing atomic orbital compositionsof the Q and K valleys as well as interlayer screening effects.Our findings not only demonstrate the coexistence of both Qand K valleys in biased four-layer MoS2 but also show howelectrostatic gating can effectively tune the band structure inTMD-based devices. These results offer deeper insights intothe electronic properties of multilayer MoS2 and presentpotential avenues for valley-selective device applications.In this study, we use dual-gated, multiterminal devices, asshown in Figure 1a. The devices are heterostructuresconsisting of four-layer MoS2 encapsulated in hexagonalboron nitride (hBN), all obtained by mechanical exfoliation(fabrication details in Supplementary Note 1). We conductedthe experiments on two samples: sample A, featuring a graphitebottom gate and a metallic top gate, and sample B, withmetallic gates on both sides. The results presented in the maintext are from sample A, which exhibits higher electron mobilityand more pronounced Shubnikov�de Haas oscillations(SdHO). Similar results obtained for sample B are providedin the Supporting Information.One of the challenges of electronic transport experiments inMoS2 devices is achieving ohmic contacts. This issue iseffectively addressed by employing a sample geometry withgated metallic contacts.2,9,19−21 In such samples, finiteconductance at cryogenic temperatures is achieved only atReceived: April 2, 2025Revised: May 23, 2025Accepted: May 27, 2025Published: June 22, 2025Letterpubs.acs.org/NanoLett© 2025 The Authors. Published byAmerican Chemical Society10472https://doi.org/10.1021/acs.nanolett.5c01998Nano Lett. 2025, 25, 10472−10477This article is licensed under CC-BY 4.0Downloaded via NATL INST FOR MATLS SCIENCE (NIMS) on July 9, 2025 at 03:18:57 (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="Michele+Masseroni"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Isaac+Soltero"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="James+G.+McHugh"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Igor+Rozhansky"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Xue+Li"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Alexander+Schmidhuber"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Markus+Niese"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Markus+Niese"&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="Kenji+Watanabe"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Vladimir+I.+Fal%E2%80%99ko"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Thomas+Ihn"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Klaus+Ensslin"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.nanolett.5c01998&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/nalefd/25/26?ref=pdfhttps://pubs.acs.org/toc/nalefd/25/26?ref=pdfhttps://pubs.acs.org/toc/nalefd/25/26?ref=pdfhttps://pubs.acs.org/toc/nalefd/25/26?ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.nanolett.5c01998?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/NanoLett?ref=pdfhttps://pubs.acs.org/NanoLett?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/relatively large top gate voltages (VTG), which are required toovercome the Schottky barrier at the metal−MoS2 interface(see also Supplementary Note 2).For this reason, in our experiments, we fix the top gatevoltage and tune the density with the bottom gate voltage(VBG), which does not affect the electron density in the regionof the contacts due to the screening provided by the metalliccontact pads. This enables us to maintain low-resistance (∼1kΩ) ohmic contacts for all densities in the Hall bar. However,this sample geometry restricts the parameter space spanned bythe gates. As a consequence, the samples are typically operatedunder a finite electric displacement field (D), which affects theelectron distribution across layers in multilayer MoS2,potentially affecting their electronic properties.In Figure 1b, we present the resistivity (blue) as a functionof VBG measured at VTG = 12 V and T ≈ 100 mK. Theresistivity reveals a nonmonotonic dependence on VBG, whichis also evident in the transport mobility (black). The mobilityincreases roughly linearly in the voltage range VBG < −2 V(regime A in the figure), reaching a peak value of μpeak ≈ 1.5 ×104 cm2 V−1 s−1 at an electron density of approximately 1 ×1013 cm−2. Further increasing VBG results in a significantdecrease in mobility, occurring close to the boundary betweenregimes A and B. We attribute this decrease to the occupationof additional bands in regime B, as determined by the densityanalysis presented below. We note that the boundariesbetween the various regimes do not necessarily coincide withthe onset of mobility reduction, which begins already withinregime A. This discrepancy arises because the mobility isinfluenced not only by band population but also by additionalscattering mechanisms�such as those induced by defectstates�that may become active even before mobile states inthe second band are occupied.To investigate the contribution of the different bands infour-layer MoS2, we examine the data obtained at finitemagnetic fields. In Figure 2a, we present the magnetoresistanceΔρxx/ρ0 = [ρxx(B) − ρxx(0)]/ρxx(0) as a function of VBGmeasured at VTG = 12 V. This measurement reveals an intricatepattern that originates from overlapping Landau fans. TheFigure 1. (a) Schematic view (left) and optical image of the sample(right). In the right panel, the MoS2 layer is outlined with a blackdashed line. The gate defines a conducting channel of width W = 2μm. The distance between adjacent contacts is LC−C = 3 μm. (b)Four-terminal resistivity (blue) and mobility (black) as a function ofthe bottom gate voltage at VTG = 12 V. The vertical dashed lines markthe voltage at which additional bands are populated. This measure-ment was performed at a temperature of T ≈ 100 mK.Figure 2. (a) Δρxx/ρ0 = [ρxx(B) − ρxx(0)]/ρxx(0) plotted as afunction of VBG and B at VTG = 12 V and T ≈ 100 mK. The whitedashed frame highlight the magnified image displayed in panel c. (b)Electron densities versus VBG: total density ntot (black dashed), Halldensity nH (red), SdHO-derived K valley densities nK,1 and nK,2 (blueand green, respectively), and residual density nres = ntot − nK,1 − nK,2(pink). Note that nH is extracted from the high-field linear regime ofRxy(B > 1 T), where multiband effects are suppressed. SeeSupplementary Note 4 for details on the Hall effect analysis. Theinset shows the Brillouin zone of MoS2 with colored pocketsindicating valley occupation. (c) Close-up of a portion of panel a,showing the Landau fan (pink lines) for nres with a degeneracy g = 12.Horizontal dashed lines in regime C indicate valley density saturation.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.5c01998Nano Lett. 2025, 25, 10472−1047710473https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig2&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.5c01998?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asobservation of multiple Landau fans confirms the existence ofmultiple electronic bands.Each of those bands has a corresponding electron densitythat determines the frequency of the SdHO. We extract thedensities by computing the fast Fourier transform (FFT) ofΔρxx(B−1) at each value of VBG (details in Supplementary Note3) and plot the resulting densities in Figure 2b. In the voltagerange VBG < −1.7 V (regime A), the Fourier analysis revealstwo electron densities, nK,1 and nK,2 (blue and green curves,respectively), both with a similar gate voltage dependence. Thedensities are calculated using the relationship ni = gief i/h,where f i is the frequency, gi is the degeneracy of band i, e is theelementary charge, and h is Plank’s constant. By comparing thesum nK,1 + nK,2 (gray curve) to total density nH obtained fromthe Hall effect measurements (red curve), we determine adegeneracy of g = 2 for both bands. This degeneracy suggeststhat the bands are centered at the K points of the Brillouinzone.The density difference, nK,1 − nK,2, arises from the spin−orbitsplitting of the K valley. By linear extrapolation, we estimatenK,1 ≈ 3.2 × 1012 cm−2 at the onset of density nK,2, which alignswith the density required to populate the upper spin−orbitsplit band at the K points observed in previous studies inmonolayer,9 bilayer,10 and trilayer11 MoS2. This furthersupports our interpretation.Now, we turn our attention to regime B in Figure 2b, wherean additional band begins to be populated. This is evident inthree distinct aspects. The first is the emergence of anotherLandau fan in Figure 2a (highlighted by the white dashedframe), with its origin around VBG ≈ −1.7 V. At the same gatevoltage, the Hall density exhibits a plateau (marked by anarrow), which we attribute to the filling of defect states at thebottom of the newly occupied band. This results in thelocalization of electrons at the defects, preventing them fromcontributing to the Hall effect (see also Supplementary Note 4for further discussion). Third, the sum nK,1 + nK,2 is no longerequal to the Hall density, implying the presence of an extraelectron pocket.We determine the degeneracy of the newly occupied bandby calculating the residual density using the relation nres = ntot− nK,1 − nK,2 (depicted by the pink curve in Figure 2b) andusing it to predict the Landau fan according to the equationB Vhn Vegm( )( )m BGres BG=(1)where m is an integer representing the Landau level index andg is the band degeneracy. In Figure 2c, we plot the calculatedLandau fan where we match the minima of Δρxx/ρ0 byassuming g = 12. This large degeneracy suggests that theresidual electron density originates from the Q valley, whichhas a 6-fold valley degeneracy. To achieve a total degeneracy of12, the bands must be spin degenerate, implying inversionsymmetry. However, while inversion symmetry is preserved inunbiased MoS2 samples with an even number of layers, thelarge displacement field in regime B (D ∼ 1 V/nm) is expectedto break this symmetry.To gain further insight, we implement a model to analyzethe layer-resolved valley densities in four-layer MoS2 undervarious displacement fields and total electron densities. Thismodel incorporates self-consistent screening effects producedby charge accumulation in each layer, allowing us to capturethe impact of interlayer screening on the electron distribution.We construct hybrid k·p tight-binding Hamiltonians tosimulate the electronic structure of the K and Q valleys.These Hamiltonians were initially parametrized using densityfunctional theory (DFT) calculations (see SupplementaryNotes 8 and 9) and included an on-site term to account forelectrostatic band bending due to external fields. Notably, DFTresults confirmed the absence of hybridization between the Kvalleys of different layers, consistent with prior results.10According to the findings in ref 22, the spin−orbit splittingbetween K bands, ΔSOK , and the monolayer energy offsetbetween K and Q valleys, EKQ, undergo significant renormal-ization due to electron−electron exchange interactions. In linewith this framework, we considered both of these quantities tobe dependent on layer density (see Supplementary Note 10).The predictive accuracy of the model is based on twoessential parameters: t0, the interlayer tunnel coupling for Qvalleys, and EKQ. However, DFT-derived parameters exhibit apronounced sensitivity to subtle variations in the MoS2interlayer spacing and lattice constants (see Figure 11 inSupplementary Note 9). This sensitivity introduces consid-erable uncertainty into their precise values, and in fact,theoretical values alone did not align with our experimentalresults. To overcome this limitation, we varied t0 and EKQsemiempirically to achieve a better fit to the observed layer andvalley densities. Remarkably, we found that the distribution ofelectron densities across layers and valleys is highly sensitive tovariations in t0 and EKQ (see Supplementary Note 11). Thischaracteristic suggests a novel experimental approach fordetermining interlayer tunnel coupling, which may also beextended to other TMDs. The optimized parameters for four-layer MoS2 determined through this process are t0 = 0.16 eVand EKQ = 0.21 eV. These values are physically reasonable, liewithin the range suggested by DFT, and provide significantlybetter agreement with the experimental gate-dependent valleyand layer density evolution.The band densities obtained from our model are shown inFigure 3a as a function of VBG for VTG = 12 V, allowing a directcomparison with the experimental densities in Figure 2b. Themodel identifies three distinct operating regimes controlled bythe bottom gate voltage, reflecting the experimental observa-tions. In regime A, where the voltage is applied asymmetricallybetween the top and bottom gates, the density is concentratedentirely in the K valleys of the top layer (see the layerpolarization shown in Figure 3b). In regime B, where thebottom gate voltage is close to zero, the Q valleys begin to fillwith electrons, and charge is transferred from the K valley tothe newly occupied band. Since the Q valley states arehybridized between layers, the density is distributed acrosslayers, favoring layers with a lower potential energy. In regimeC, the K valleys in the bottom layers are filled, leading tosaturation of the K (top layer) and Q valley densities, asobserved in the experiment.This model successfully captures the gate voltage depend-ence of the valley-specific densities and provides insights intothe layer distribution of electron densities. In regime B, themodel predicts an asymmetric density distribution across thelayers, which results in a spin degeneracy lifting of Q valleystates when D ≠ 0, due to SOC (see the dotted lines in Figure3a). This prediction aligns with a detailed analysis of the FFTspectrum (see Supplementary Note 5), where we observe asplitting of the SdHO frequency corresponding to the residualdensity.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.5c01998Nano Lett. 2025, 25, 10472−1047710474https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.5c01998?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asThe experimentally observed splitting of the Q valley isnotably less pronounced than that predicted by our model.This discrepancy may arise from crystal field effects at theinterface with the hBN dielectrics.23 In the SupportingInformation (see Supplementary Note 12), we account forthe influence of hBN by introducing an on-site potentialenergy difference, ΔhBN, between the outer layers (whichinterface directly with hBN) and the inner layers. Incorporat-ing this offset reduces the level of hybridization between the Qvalleys of the outer and inner layers. This adjustment results ina more symmetric layer distribution of the Q valley electrons,thereby mitigating inversion symmetry breaking induced bythe displacement field and leading to a weaker spin−orbitsplitting. Consequently, the model’s prediction of spin−orbitsplitting becomes more consistent with the experimentalobservations.Next, we examine the effect of the top gate voltage, focusingon the densities in the Q and K valleys of the top layer. Totrack the top gate dependence of these densities, we presentthe FFT of Δρxx(B−1) at VBG = 3.5 V for various VTG values inFigure 4a. In this regime, the finite VBG allows us to observeoscillations originating from both Q and K valley states of thetop layer (highlighted in pink and blue, respectively) withoutinducing charges in the K valleys of the bottom layer. Figure 4bshows that the K valley frequency of the top layer increaseswith VTG, while the Q valley frequency is nearly constant dueto interlayer screening. These data suggest that the top gatevoltage induces a transition of the band edge from the Qvalleys at low gate voltages to the K valleys at higher voltages.In Figure 4c, we compare valley densities predicted by themodel (solid lines) with experimental densities (squaremarkers), showing good agreement. Due to high contactresistance, the low-VTG regime is experimentally inaccessible,so we rely on model predictions for this range. According tothe model, at low bias (VTG < 4.5 V), the Q valley remains theonly occupied band, confirming that the conduction bandminimum in low-bias four-layer MoS2 is located at the Q pointand demonstrating the presence of a gate-induced Q−K bandedge transition.Having established the relevance of the gate bias indetermining the relative band alignment in multilayer MoS2,we extended our model to bilayer and trilayer systems (seeSupplementary Note 13) to address discrepancies betweenexperiments and DFT-calculated band structures. Usingconsistent model parameters, EKQ = 0.21 eV and t0 = 0.16eV, we qualitatively reproduce the experimental trends acrossdifferent layer numbers.Figure 3. (a) Band densities obtained from the theoretical modelusing the parameters EQK = 0.21 eV and t0 = 0.16 eV. The densitiesare plotted as a function of VBG at VTG = 12 V, enabling a directcomparison with Figure 2b. The densities in the K valleys of the toplayer (layer 1), nK,1,t and nK,2,t, are shown as solid blue and green lines,respectively. The K valley densities in the bottom layer (layer 4) aredepicted with dashed lines following the same color code. The totaldensity in the Q valleys (nQ,1 + nQ,2 for a direct comparison with theexperiment) is represented by the pink solid line, while the individualcomponents are shown by the dotted lines. (b) Percentages of thetotal density distributed across the four MoS2 layers with the differentbands encoded by the colors (consistent with panel a). As arepresentation of the three regimes, we selected specific gate voltageconfigurations: (VTG = 12 V, VBG = −6 V) for regime A, (VTG = 12 V,VBG = 0 V) for regime B, and (VTG = 12 V, VBG = 8 V) for regime C.Figure 4. (a) FFT of Δρxx(B−1) at VBG = 3.5 V and various VTGvalues. The peaks associated with the Q and K1 valleys are highlightedby the vertical pink and blue lines, respectively. Note that the low-frequency component visible in the FFT arises from imperfectbackground subtraction and does not correspond to a physical valleypopulation. (b) Frequencies obtained from panel a plotted as afunction of VTG. (c) Comparison between the densities obtained fromthe model (solid lines) and the densities obtained from theexperiment (square markers). The model parameters are the sameas those of Figure 3.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.5c01998Nano Lett. 2025, 25, 10472−1047710475https://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig4&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?fig=fig4&ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.5c01998?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asFor biased bilayer MoS2, the model predicts that the Kvalleys in top and bottom layers are occupied bands while theQ valleys are not, as observed in ref 10. In our earlier study ontrilayer MoS2,11 we observed electron transport predominantlythrough the K valleys in the outer layers, although the bandcontributing to the electrons in the middle layer remainedunidentified. Our model reveals the missing element in ourprevious work, showing that both K and Q valleys werepopulated, as observed in the four-layer case.In conclusion, our study provides a comprehensive under-standing of the gate-tunable band structure and valleyoccupation in four-layer MoS2 through a combination ofmagnetotransport experiments and a self-consistent hybrid k·ptight-binding model. At low gate voltages, we confirm that theconduction band minimum is centered at the Q point, which isin line with theoretical expectations for multilayer MoS2.However, as the gate voltage increases, charge redistributiontoward the layer closest to the positive gate electrode induces atransition of the band edge from the Q valleys to the K valleys.By extending our model to bilayer and trilayer MoS2, wesuccessfully bridge previous discrepancies between experimen-tal observations10,11 and density functional theory predic-tions,12−17 underscoring the importance of interlayer screeningand layer-specific charge accumulation in determining valleyoccupation. These findings highlight the intricate interplaybetween the K and Q valleys in multilayer MoS2, paving theway for future valleytronic and electronic applications thatexploit layer-specific charge and valley control.■ ASSOCIATED CONTENT*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998.Details of device fabrication, additional magnetotran-sport characterization, Fourier analysis of Shubnikov−deHaas oscillations, comparison between different samples,and theoretical model description with parametricdependence analysis (PDF)■ AUTHOR INFORMATIONCorresponding AuthorKlaus Ensslin − Solid State Physics Laboratory and QuantumCenter, ETH Zürich, 8093 Zürich, Switzerland;orcid.org/0000-0001-7007-6949; Email: ensslin@phys.ethz.chAuthorsMichele Masseroni − Solid State Physics Laboratory, ETHZürich, 8093 Zürich, Switzerland; orcid.org/0000-0003-1663-8239Isaac Soltero − Department of Physics and Astronomy,University of Manchester, Manchester M13 9PL, UnitedKingdom; National Graphene Institute, University ofManchester, Manchester M13 9PL, United Kingdom;orcid.org/0000-0002-2593-0891James G. McHugh − Department of Physics and Astronomy,University of Manchester, Manchester M13 9PL, UnitedKingdom; National Graphene Institute, University ofManchester, Manchester M13 9PL, United Kingdom;orcid.org/0000-0001-8509-4883Igor Rozhansky − Department of Physics and Astronomy,University of Manchester, Manchester M13 9PL, UnitedKingdom; National Graphene Institute, University ofManchester, Manchester M13 9PL, United KingdomXue Li − Department of Physics and Astronomy, University ofManchester, Manchester M13 9PL, United Kingdom;National Graphene Institute, University of Manchester,Manchester M13 9PL, United KingdomAlexander Schmidhuber − Solid State Physics Laboratory,ETH Zürich, 8093 Zürich, SwitzerlandMarkus Niese − Solid State Physics Laboratory, ETH Zürich,8093 Zürich, SwitzerlandTakashi Taniguchi − Research Center for MaterialsNanoarchitectonics, National Institute for Materials Science,Tsukuba 305-0044, Japan; orcid.org/0000-0002-1467-3105Kenji Watanabe − Research Center for Electronic and OpticalMaterials, National Institute for Materials Science, Tsukuba305-0044, Japan; orcid.org/0000-0003-3701-8119Vladimir I. Fal’ko − Department of Physics and Astronomy,University of Manchester, Manchester M13 9PL, UnitedKingdom; National Graphene Institute, University ofManchester, Manchester M13 9PL, United KingdomThomas Ihn − Solid State Physics Laboratory and QuantumCenter, ETH Zürich, 8093 Zürich, SwitzerlandComplete contact information is available at:https://pubs.acs.org/10.1021/acs.nanolett.5c01998Author ContributionsM.M. and I.S. contributed equally to this work. M.M., T.I., andK.E. conceived and designed the experiments. M.M. fabricatedthe device with input from T.I. and K.E. M.M. performed themeasurements with input from T.I. and K.E. M.M. and A.S.analyzed the experimental data with input from M.N., T.I., andK.E. I.S. and V.I.F. developed the theoretical model with inputfrom J.G.M., I.R., and X.L. M.M. designed the figures withinput from I.S. T.T. and K.W. supplied the hexagonal boronnitride. M.M. wrote the manuscript with input from I.S.,J.G.M., and I.R. All of the co-authors mentioned above readand commented on the manuscript.NotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSThe authors thank Peter Märki, Thomas Bähler, and theFIRST staff for their technical support. The authors acknowl-edge support from the European Graphene Flagship Core 3Project, Swiss National Science Foundation via NCCRQuantum Science, and H2020 European Research Council(ERC) Synergy Grant under Grant Agreement 95154. Thiswork was supported by EPSRC Grants EP/S030719/1 andEP/V007033/1 and the Lloyd Register Foundation Nano-technology Grant. Collaboration between ETH and theUniversity of Manchester was supported by the InternationalScience Partnerships Fund (UK). I.S. and X.L. acknowledgefinancial support from the University of Manchester’s Dean’sDoctoral Scholarship. I.R. gratefully acknowledges the supportof a CARA Fellowship and BA/Cara/Leverhulme ResearchSupport Grant LTRSF24/100044. K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grants 21H05233 and23H02052), the CREST (JPMJCR24A5), JST, and the WorldPremier International Research Center Initiative (WPI),MEXT, Japan.Nano Letters pubs.acs.org/NanoLett Letterhttps://doi.org/10.1021/acs.nanolett.5c01998Nano Lett. 2025, 25, 10472−1047710476https://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.nanolett.5c01998/suppl_file/nl5c01998_si_001.pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Klaus+Ensslin"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0001-7007-6949https://orcid.org/0000-0001-7007-6949mailto:ensslin@phys.ethz.chmailto:ensslin@phys.ethz.chhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Michele+Masseroni"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-1663-8239https://orcid.org/0000-0003-1663-8239https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Isaac+Soltero"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-2593-0891https://orcid.org/0000-0002-2593-0891https://pubs.acs.org/action/doSearch?field1=Contrib&text1="James+G.+McHugh"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0001-8509-4883https://orcid.org/0000-0001-8509-4883https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Igor+Rozhansky"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Xue+Li"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Alexander+Schmidhuber"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Markus+Niese"&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://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1467-3105https://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="Vladimir+I.+Fal%E2%80%99ko"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Thomas+Ihn"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.nanolett.5c01998?ref=pdfpubs.acs.org/NanoLett?ref=pdfhttps://doi.org/10.1021/acs.nanolett.5c01998?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as■ REFERENCES(1) Kim, S.; Konar, A.; Hwang, W.-S.; Lee, J. 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