# Fileset [q11l-9jy1.pdf](https://mdr.nims.go.jp/filesets/381c5172-ff76-42fc-a17c-992a06dd4a42/download) ## Creator Yujun Deng, William Holtzmann, Ziyan Zhu, Timothy Zaklama, Paulina Majchrzak, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Makoto Hashimoto, Donghui Lu, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Liang Fu, Thomas P. Devereaux, Xiaodong Xu, Zhi-Xun Shen ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Nonmonotonic Band Flattening near the Magic Angle of Twisted Bilayer

{MoTe}_{2}

](https://mdr.nims.go.jp/datasets/f4a66480-e151-40d0-b23f-f2373eef78d2) ## Fulltext Nonmonotonic Band Flattening near the Magic Angle of Twisted Bilayer MoTe2Nonmonotonic Band Flattening near the Magic Angle of Twisted Bilayer MoTe2Yujun Deng ,1,* William Holtzmann,2 Ziyan Zhu,1,3 Timothy Zaklama ,4 Paulina Majchrzak,5 Takashi Taniguchi ,6Kenji Watanabe ,7 Makoto Hashimoto ,8 Donghui Lu ,8 Chris Jozwiak ,9 Aaron Bostwick ,9 Eli Rotenberg,9Liang Fu,4 Thomas P. Devereaux ,1,10 Xiaodong Xu ,2,11 and Zhi-Xun Shen1,5,12,13,†1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,Menlo Park, California 94025, USA2Department of Physics, University of Washington, Seattle, Washington 98195, USA3Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA4Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA5Department of Applied Physics, Stanford University, Stanford, California 94305, USA6Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan7Research Center for Electronic and Optical Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan8Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory,Menlo Park, California 94025, USA9Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA10Department of Materials Science and Engineering, Stanford University,Stanford, California 94305, USA11Department of Materials Science and Engineering, University of Washington,Seattle, Washington 98195, USA12Department of Physics, Stanford University, Stanford, California 94305, USA13Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA(Received 21 July 2025; revised 16 September 2025; accepted 29 October 2025; published 5 December 2025)Twisted bilayer MoTe2 (tMoTe2) is an emergent platform for exploring exotic quantum phases drivenby the interplay between nontrivial band topology and strong electron correlations. Direct experimentalaccess to its momentum-resolved electronic structure is essential for uncovering the microscopic origins ofthe correlated topological phases therein. Here, we report angle-resolved photoemission spectroscopymeasurements of tMoTe2, revealing pronounced twist-angle-dependent band reconstruction shaped byorbital character, interlayer coupling, and moiré potential modulation. Density functional theory capturesthe qualitative evolution, yet underestimates key energy scales across twist angles, highlighting theimportance of electronic correlations. Notably, the hole effective mass at the K point exhibits anonmonotonic dependence on twist angle, peaking near 2°, consistent with band flattening at the magicangle predicted by continuum models. Via electrostatic gating and surface dosing, we further visualize theevolution of electronic structure versus doping, enabling direct observation of the conduction bandminimum and confirm tMoTe2 as a direct band gap semiconductor. These results establish a spectroscopicfoundation for modeling and engineering emergent quantum phases in this moiré platform.DOI: 10.1103/q11l-9jy1 Subject Areas: Condensed Matter Physics,Materials ScienceI. INTRODUCTIONTwisted van der Waals heterostructures have emerged asa powerful platform for engineering novel quantum phasesof matter [1–4]. Stacking two-dimensional materials withcontrolled rotational misalignment results in moiré super-lattices that profoundly alter the electronic structure,enabling correlated insulators, superconductivity, and avariety of topological phases, such as quantum anomalousHall (QAH), fractional QAH (FQAH), and fractionalquantum spin Hall (FQSH) states [5–12]. Among these*Contact author: yjdeng@stanford.edu†Contact author: zxshen@stanford.eduPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 15, 041043 (2025)2160-3308=25=15(4)=041043(11) 041043-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-7407-2603https://orcid.org/0009-0005-9324-5983https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-1689-8997https://orcid.org/0000-0002-9708-0443https://orcid.org/0000-0002-0980-3753https://orcid.org/0000-0002-9008-2980https://orcid.org/0000-0001-8072-9237https://orcid.org/0000-0003-0348-2095https://ror.org/05gzmn429https://ror.org/00cvxb145https://ror.org/02n2fzt79https://ror.org/042nb2s44https://ror.org/00f54p054https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/02vzbm991https://ror.org/05gzmn429https://ror.org/00319zh75https://ror.org/00f54p054https://ror.org/00cvxb145https://ror.org/00f54p054https://ror.org/00f54p054https://crossmark.crossref.org/dialog/?doi=10.1103/q11l-9jy1&domain=pdf&date_stamp=2025-12-05https://doi.org/10.1103/q11l-9jy1https://doi.org/10.1103/q11l-9jy1https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/systems, tMoTe2 has attracted particular interest due to itsexperimental realization of both integer and fractionaltopological phases [10,11,13–18].To date, experimental studies of tMoTe2 have predomi-nantly focused on transport measurements [10,11,14,17,18],unveiling a rich phase diagram featuring the aforementionedstates. However, direct momentum-resolved probes of itselectronic structure are crucial for a deep understanding ofthe underlying mechanisms. Theoretical studies based onDFT [19,20] and continuum models [20–23] predict sig-nificant twist-angle-dependent band reconstructions drivenby the modulation of interlayer tunneling and moiré poten-tial. Yet, key experimental benchmarks to validate thesepredictions have been lacking.Here, we employ nano angle-resolved photoemissionspectroscopy (nanoARPES) to systematically investigatethe twist-angle-dependent electronic band structure oftMoTe2. The obtained band structure information enablesextraction of key band structure parameters includingintravalley band splittings, the energy difference betweenthe valence band maxima (VBM) at Γ and K, and effectivemasses, which reflect the intricate interplay between orbitalcharacter, interlayer hybridization, and moiré potentialmodulation. By comparing them with theoretical models,we provide direct spectroscopic insights into the bandstructure evolution in this system. Furthermore, weinvestigate the doping evolution of electronic structurevia two complementary approaches: electrostatic gatingand surface potassium dosing. These methods enabledirect access to both valence and conduction bands,confirming a direct band gap in tMoTe2. By revealingand tuning the momentum-resolved electronic structureacross twist angles, we lay the groundwork for advancingboth the theoretical understanding and experimentalengineering of correlation- and topology-driven phasesin twisted TMD moiré materials.II. RESULTSA schematic of a typical tMoTe2 device used in ourARPES measurements is shown in Fig. 1(a). From top tobottom, the device consists of a graphene layer, tMoTe2bilayers, a hexagonal boron nitride (hBN) dielectric layer,and a graphite back gate. The top graphene layer, connectedto a metal pad, serves both as a protective layer and areliable electrical ground during measurements. The backgate allows for tuning of the gate voltage (Vg) to controldoping in the tMoTe2. The entire structure is fabricatedon a SiO2=Si substrate. In one of the samples (with twistangle of 3.93°, data included in Figs. 2 and 3), the tMoTe2is encapsulated by graphene on one-half and hBN on theother, enabling a direct comparison of the band structureunder different dielectric screening environments. ARPESmeasurements on three graphene-encapsulated areas[Figs. S1(a)–S1(c) in Supplemental Material [24] ] andone hBN-encapsulated area (Fig. S1(d) in [24]) showcomparable band structure parameters (Fig. S1(e) in [24]),indicating good spatial uniformity and suggesting thatencapsulation with graphene or hBN has minimal impacton the electronic structure at this twist angle, despite a smallrigid band shift [25].An optical image of a representative sample, illustratingthe device geometry, is shown in Fig. 1(b) (left panel).During ARPES measurements, the integrated photoemis-sion map near the Fermi level (EF) clearly resolves sharpfeatures from electrical contact and graphite, locating thesample [shown in Fig. 1(b), middle panel]. Samplesprepared using standard premeasurement treatments,including immersion in organic solvents and annealingin ultrahigh vacuum, already show clear photoemissionsignal from graphene. However, a critical step for acquiringhigh-quality spectra from the underlying tMoTe2 is theremoval of interface residue using an atomic force micro-scope (AFM) tip in contact mode (details can be found inthe Appendix and Fig. S2 in Supplemental Material [24].This process significantly enhances the ARPES spectra inthe AFM-cleaned region [marked by the white dashed linein Fig. 1(b)]. Both photoemission intensity maps integratednear EF and the Te core level [Fig. 1(b), middle and rightpanel] reveal notably stronger intensity in this region,which is the primary area used for our measurements.The twist angles were determined either from the moiréwavelengths aM observed using piezoforce microscopy(PFM) [Fig. 1(e), see also the Appendix] both prior to andafter the ARPES measurements, or from the moiré wavevectors GM extracted from the constant energy contoursnear the graphene K point [Fig. 1(d), see also theAppendix] during the measurements. Figure 1(c) presentsa schematic of the Brillouin zones for tMoTe2 withgraphene. The relative angle between tMoTe2 and graphenevaries across samples, as the two were not intentionallyaligned during fabrication. moiré minibands are theoreti-cally anticipated to emerge near the K points of MoTe2(Refs. [21,26]).Figure 2 presents the ARPES spectra of pristine mono-layer and bilayer 2H-MoTe2, along with tMoTe2 at varioustwist angles. The upper row shows the raw ARPES spectraalong the Γ-K high-symmetry direction, while the lowerrow displays the corresponding second-derivative spectrawith respect to energy, enhancing the visibility of subtlefeatures. For visual comparison, the energy scales arereferenced to the VBM at the K point (E − EVBM),compensating for sample-dependent Fermi level pinning.Before diving into the band structure of tMoTe2, we firstexamine the band structure of monolayer and bilayer2H-MoTe2 (corresponding to a 60° twist angle), shownin Fig. 2(f) and Fig. 2(a), as benchmarks for understandingthe modifications introduced by moiré effects. 2H bilayersample was exfoliated and measured in situ under ultrahighvacuum on gold substrate, whereas the monolayer MoTe2was measured in situ on a twisted device. The topmostYUJUN DENG et al. PHYS. REV. X 15, 041043 (2025)041043-2valence bands at the K point primarily originate from Modxy and dx2−y2 orbitals, with minor contributions from Te pxand px orbitals. In contrast, the topmost valence band at theΓ point arises from the hybridization of Mo dz2 and Te pzorbitals [27,28]. These distinct orbital characters withdifferent symmetries lead to different manifestations ofspin-orbit coupling at K and Γ, ultimately leading tocontrasting electronic structures at the two valleys. Atthe K point, broken inversion symmetry and strong spin-orbit coupling result in two spin-split bands [Fig. 2(f)]. Inthe H-stacked bilayer, this splitting persists; however, therestored inversion symmetry enforces overall spin degen-eracy, rendering each pair of bands spin degenerate[Fig. 2(a)]. At the Γ point, monolayer exhibits a nearlyflat dispersion [Fig. 2(f)], whereas in the bilayer, stronginterlayer hybridization of orbitals with out-of-plane char-acter leads to a clear band splitting [Fig. 2(a)] [27,28].Now we turn to the band structure of tMoTe2 withvarying twist angles [Figs. 2(b)–2(e), 2(h)–2(k) and S3 inSupplemental Material [24] ]. In tMoTe2, the valence bandstructure resembles that of the 2H bilayer, with distinctband splitting at both the K and Γ points. Acrossall configurations—monolayer, 2H bilayer, and twistedbilayer—the global VBM is always located at the K point,allowing moiré states to be energetically isolated in twistedbilayers. In twisted bilayers, the energy separation betweenVBM at K and Γ becomes more pronounced comparedto the 2H bilayer configuration [Figs. 2(a) and 2(g)], withVBM at Γ point further lowered as the twist angle increases[Figs. 2(b)–2(e) and 2(h)–2(k)]. Additionally, a systematicflattening near the valance band top at the Γ point isobserved with increasing twist angles. This trend indicatesa suppression of interlayer hybridization as the systemapproaches the monolayer limit, resulting in a simplifiedband structure characteristic of fully decoupled layers[Figs. 2(f) and 2(l)].While the qualitative differences in band structurealready reveal twist-angle-dependent band reconstructions,FIG. 1. Device structure and characterization of tMoTe2 heterostructure. (a) Schematic of a tMoTe2 device designed for ARPESmeasurements. (b) Optical image (left), integrated photoemission intensity map near the Femi level (middle), and near the Te 4dcore level (right) of the same region of a tMoTe2 device (twist angle: 1.98°). The AFM-cleaned region is outlined by a whitedashed line. Scale bar, 10 μm. (c) Schematic of the Brillouin zones of tMoTe2 with graphene on top (graphene: gray, top MoTe2:red, and bottom MoTe2: blue). (d) Twist angle determination by ARPES constant-energy contour measurements. Top panel:Constant-energy contours near the graphene K points (Kgraphene), showing multiple replica Dirac cones separated by the moiréwave vectors of tMoTe2. Scale bar, 0.5 Å−1. Bottom-left panel: Band dispersion along the black dashed line in the top panel.Bottom-right panel: Momentum distribution curve (MDC) at the Dirac points, extracted from the dispersion in the bottom-leftpanel. The extracted moiré wavevector GM is 0.21 Å−1, corresponding to a twist angle of 5.89°. (e) Twist angle determination byPFM measurements. Top panel: PFM amplitude image of graphene-protected tMoTe2. Scale bar, 50 nm. Bottom panel: FastFourier transform (FFT) analysis of top panel. Scale bar, 0.2 nm−1. The extracted moiré wavelength aM is 10.36 nm,corresponding to a twist angle of 1.98°.NONMONOTONIC BAND FLATTENING NEAR THE MAGIC … PHYS. REV. X 15, 041043 (2025)041043-3a deeper understanding requires a more systematic andquantitative analysis. In particular, variations in Fermi levelpinning across different twist angles complicate directcomparisons of band structure. We therefore extractedthree key valence-band parameters: energy splittingbetween the topmost two valence bands at the Γ and Kpoints (denoted asΔΓ andΔK, respectively), and the energydifference between VBM at the K and Γ points (ΔK−Γ).Their evolution with twist angle, shown in Fig. 3, providesfurther insight into the interplay between interlayer cou-pling and moiré modulation.As shown in Fig. 3(a), ΔΓ decreases monotonically withincreasing twist angle. This trend reflects the weakeninginterlayer hybridization between out-of-plane orbitals(Mo dz2 and Te pz), driven by reduced atomic registryand diminished orbital overlap. At small twist angles,strong lattice relaxation promotes the formation of triangu-lar AB/BA stacking domains with reduced interlayerspacing, leading to strong interlayer coupling. Since nomoiré minibands form at the Γ point, the observed ΔΓprovides a direct measure of twist-angle-dependent inter-layer coupling.In contrast, Fig. 3(b) shows that ΔK remains relativelyconstant across twist angles, exhibiting only minor mod-ulations before decreasing toward the monolayer limit. Thisweak dependence stems from the nature of the states livingat the K point, which originate primarily from in-planeorbitals (Mo dxy, dx2−y2 and Te px, py) and are dominatedby spin-orbit coupling within individual layers, makingthem relatively insensitive to changes in vertical interlayertunneling. The slightly larger ΔK observed in bilayers thanin monolayers is attributed to enhanced interlayer coupling.As a result, ΔK−Γ increases systematically with twistangle, as shown in Fig. 3(c). This behavior arises from thecontrasting evolution of ΔΓ and ΔK: while VBM at the Γpoint shifts downward due to suppressed interlayerhybridization, VBM at the K point remains nearlyunchanged owing to its predominantly in-plane orbitalcharacter and weaker sensitivity to interlayer coupling.The resulting increase in energy separation underscoresthe differing responses of the two valleys to twist-inducedinterlayer effects.We performed DFT calculations to capture the evolutionof ΔK−Γ with twist angle, incorporating lattice relaxationand spin-orbit coupling. To assess the reliability of differentapproaches, we tested various van der Waals (vdW) func-tionals. While atom-wise vdW functionals such as DFT-D2,DFT-D3, and dDsC failed to reproduce even thecorrect trend for untwisted bilayers, nonlocal van derWaals functionals including vdW-DF2, vdW-DF3-opt2,SCANþ rVV10, and r2SCANþ rVV10 (see Fig. S5 inSupplemental Material [24], see also the Appendix) quali-tatively reproduce the overall trend of increasing ΔK− Γwith twist angle. Among them, the vdW-DF2 producesresults closest to the experiment for untwisted systems,which we adopt here [Figs. 3(a)–3(c)] and for the rest of theLetter. The corresponding calculated band structures areshown in Fig. S6 of Supplemental Material [24]. However,FIG. 2. Electronic band structure of 2H bilayer, twisted bilayer and monolayer MoTe2. (a)–(f) ARPES band dispersion along theΓ − K high-symmetry direction in 2H bilayer, twisted bilayers at representative twist angles, and monolayer MoTe2. Overlaid red curvesshow DFT-calculated band structures using vdW-DF2 functional, energy shifted to align with experimental features. Scale bar, 0.2 Å−1.(g)–(l) Corresponding second derivative spectra of (a)–(f) with respect to energy. To enable direct comparison, the VBM at theK point isset to zero energy [black dashed lines in (a)–(f), white dashed lines in (g)–(l)].YUJUN DENG et al. PHYS. REV. X 15, 041043 (2025)041043-4the calculated ΔK−Γ is expected to deviate further from theexperimental trend at twist angles below 5°, based on areasonable extrapolation of the DFT results [Fig. 3(f)]. Thisdiscrepancy suggests that mean-field approximations inher-ent in standard DFT are insufficient to fully capture theelectronic structure of tMoTe2, possibly due to enhancedelectron-electron correlation effects beyond the DFT level.To further assess the influence of interlayer coupling andmoiré modulation on band flattening, we extracted holeeffective masses at the Γ and K points (m�Γ and m�K) fromARPES measurements across a range of twist angles. Theeffective masses were obtained by fitting the curvature ofthe topmost valence band near the Γ and K points within a0.2 Å−1 momentum window using the parabolic approxi-mation EðkÞ ¼ ℏ2k2=2m�i , where i ¼ Γ or K. The extractedvalues are shown in Figs. 3(d) and 3(e).As shown in Fig. 3(d), m�Γ at small twist angles (∼0°) isclose to the 2H bilayer value, which serves a reference.With increasing twist angle,m�Γ rises significantly, reaching8.2 m0 near 5°, where m0 is the free electron mass, andapproaching the heavy monolayer limit value of ∼13 m0.This trend reflects a reduction in both interlayer hybridi-zation and lattice relaxation with increasing twist angle,leading to flatter valence bands at Γ. In contrast, m�Kremains relatively constant across most twist angles, con-sistent with the dominant in-plane orbital character of theK-point states and their weak sensitivity to vertical inter-layer coupling. However, a distinct peak appears near 2°,where m�K reaches a local maximum. This nonmonotonicbehavior suggests maximum band flattening at this criticalangle. Although individual moiré minibands are notresolved in the spectra, the effective mass reflects anFIG. 3. Evolution of band structure parameters and effective masses in 2H bilayer, twisted bilayers, and monolayer MoTe2.(a)–(c) Twist-angle dependence of key valence band parameters: ΔΓ (a), ΔK (b), and ΔK−Γ (c). For ΔK−Γ we compare the Γ-pointVBM in both monolayer and bilayers, but bilayer splitting makes it a different band, so the monolayer value is shown as a reference.(d) and (e) Twist-angle dependence of mΓ� and mK�. The continuum model (green line) uses a monolayer effective mass of 0.70 m0(obtained from experimental fitting) and interlayer coupling parameters V ¼ −11.2 and ω ¼ 13.3 meV as inputs. Both experimentsand theory show a peak near 2°, indicating maximal band flattening at this critical angle. Error bars on the effective mass representfitting uncertainties. (f) ΔK−Γ plotted over an extended twist-angle range, complementing panel (c). In all panels, experimental dataare shown as blue dots; DFT results using the vdW-DF2 functional are shown as red crosses. The DFT value at 0° twist angle isobtained from a calculation of a rhombohedral-stacked bilayer. Green and yellow shaded regions represent the 2H bilayer (60°) andmonolayer limits, respectively. Twist angle uncertainties are derived from PFM or ARPES measurements. Error bars smaller thanthe marker size are not shown.NONMONOTONIC BAND FLATTENING NEAR THE MAGIC … PHYS. REV. X 15, 041043 (2025)041043-5average over these dispersions. Such a feature is indicativeof emerging interaction-driven phenomena in tMoTe2.Having established how twist angle modulates thevalence band structure and hole effective masses intMoTe2, we next explore how external doping, imple-mented through electrostatic gating and surface potassiumdosing, enables tunability of the Fermi level and access tothe conduction band.In a 1.98° tMoTe2 device, we shifted the Fermi leveltoward the hole-doped regime by applying negative gatevoltages. As shown in Figs. 4(a)–4(c), increasingly neg-ative Vg shifts the K-point VBM upward, confirmingprogressive hole doping, while the underlying band dis-persions and key band structure parameters remain essen-tially unchanged up to Vg ¼ − 2.5 V [Fig. S9(e) inSupplemental Material [24] ]. We also gated the samedevice towards the electron side, up to a conservativeþ2.5 V to avoid sample damage, the conduction band wasnot observed [see Figs. S9(d) and 9(i) in [24] ]. This mayresult from the limited bias as well as the reduced gatingefficiency in the presence of the graphene capping layer[29]. Additionally, the spectra become substantially broad-ened under positive gating, likely due to local charge andpotential inhomogeneity, consistent with previous obser-vations under similar gating conditions [30].In a separate 3.44° tMoTe2 device, efficient electron-side doping was achieved via in situ potassium dosing[Figs. 4(d)–4(f)]. As dosing proceeds, the Fermi level shiftsupward, and the spectral features sharpen noticeably, whichcould be attributed to enhanced screening of the Coulombinteraction and a subsequent reduction in electron-electronscattering. After 620 sec of dosing, the conduction bandminimum (CBM) became clearly resolvable, revealing adirect band gap of 1.14 eV [Fig. 4(f)]. Importantly, bothmonolayer and 2H bilayer MoTe2 are known to host directband gaps [31,32], supporting that the direct band gapobserved in tMoTe2 reflects the intrinsic band structure ofMoTe2. Our DFT calculations (Figs. S6 and S10 in theSupplemental Material [24]), conducted across a broad rangeof twist angles, consistently show that tMoTe2 remains adirect-gap semiconductor with a nearly constant band gap, inexcellent agreement with our experimental value.These results demonstrate how electrostatic gating andpotassium dosing, as complementary modulation methods,enable carrier-density control in moiré materials, high-lighting their potential for device-level engineering.III. DISCUSSION AND CONCLUSIONAlthough moiré minibands are expected to appear nearthe K valley in tMoTe2 [21,26], our ARPES measurementsdid not observe such features. Photon energy-dependentmeasurements across the K valley (Fig. S4 in theSupplemental Material [24]) show no evidence of mini-bands, suggesting that matrix element effects are unlikelyto account for their absence. Nevertheless, the presence of amoiré potential in our tMoTe2 samples, indicated by eitherreplica bands near the K valley of graphene or moirépatterns detected by PFM, suggests that the lack ofobservable minibands stems from other limitations. Themeasurement temperature of 20 K exceeds the reportedcritical temperatures of the Chern insulating states intMoTe2, so the associated gaps may not have opened [12].The finite instrumental resolution of nanoARPES (about20 meV in energy and 0.02 Å−1 in momentum), relative tothe small expected moiré band gap [26] makes suchfeatures difficult to resolve. In addition, sample inhomo-geneity, including twist-angle variations and domain for-mation within the beam spot may further broaden thespectral features beyond the miniband scale. Finally, themoiré potential at the K point is inherently weaker,FIG. 4. Doping evolution of the electronic structure in tMoTe2. (a)–(c) ARPES band dispersion along the Γ − K high-symmetrydirection of 1.98° tMoTe2 under different backgate voltages (Vg: −2.5, −1.5, and 0 V), showing systematic upward shifts of the valenceband edge with increasing hole doping. Scale bar, 0.2 Å−1. (d)–(f) ARPES band dispersion along the Γ − K high-symmetry direction of3.44° tMoTe2 before (d) and after in situ potassium surface dosing for 440 (e) and 620 s (f), illustrating progressive downward bandshifts corresponding to electron doping. In (e) and (f), the data near the K point close to EF is shown with an adjusted color scale toenhance contrast. While no CBM is visible in (e), it becomes clearly resolved in (f), revealing a direct band gap of 1.14 eV.YUJUN DENG et al. PHYS. REV. X 15, 041043 (2025)041043-6less than one-fifth of that at the Γ point according tocontinuum model calculations [26,33,34], due to the weakinterlayer coupling of the relevant orbitals (Mo dxy, dx2−y2and Te px, py) at K.To gain further insight into the origin and implications ofthe observed band flattening, we now discuss the theoreticalmodeling results in light of the experimental trends. Forsmall twist angles, direct DFT calculations become practi-cally inaccessible due to the large moiré supercells, makingthe continuum model essential for capturing the bandstructure evolution. Our continuum model calculations,using the experimentally measured monolayer effective mass(0.70 m0) as input, predict maximal band flattening near 2°,in good agreement with our experimental data [Fig. 3(e), seealso the Appendix and Figs. S7 and S8 in SupplementalMaterial [24], and consistent with earlier continuum modelstudies [21,23]. It is worth noting that while the continuummodel computes the effective mass of the topmost moiréminiband, the experimentally extracted values reflect anaverage over unresolved minibands. Nevertheless, the modelcaptures the overall trend and identifies the critical twistangle. Although lattice relaxation is not explicitly includedin our continuum Hamiltonian, its effects are effectivelyincorporated through DFT-informed parameters, enablingquantitatively accurate modeling (see the Appendix). Finally,we note DFT calculations underestimate the effective massesfor monolayer, 2H bilayer, and 0° tMoTe2 at the K point,again underscoring the necessity of beyond-DFT treatmentsto capture correlation effects in this system.In conclusion, our combined experimental and theoreticalinvestigation of tMoTe2 reveals a systematic evolution of thevalence band structure and hole effective mass with twistangle. ARPES measurements uncover pronounced bandreconstruction at the Γ point and nonmonotonic band flat-tening at the K point, identifying ∼2° as a magic angle,consistent with continuum model calculations. Additionally,we establish electrostatic gating and surface dosing aseffective methods for modulating the Fermi level andaccessing conduction states. These findings position twistedMoTe2 as a tunable platform for engineering correlatedelectronic states.Looking forward, our results motivate future studies atlower temperatures and higher energy resolution, as well asgate-tunable ARPES experiments, to resolve the elusivemoiré minibands and explore interaction-driven phasesnear the magic angle.Note added—During the preparation of this manuscript, webecame aware of a related ARPES study on MoTe2 [35].ACKNOWLEDGMENTSWe thank Y. Chen, C. Chen and S. He for helpfuldiscussions. We thank K.-J. Xu, C. Lin, and E. Corbae forassistance with ARPES measurements; G. Zaborski,K. Crust, and A. Khandelwal for support with PFMmeasurements; M. Pendharkar and S. Tran for help withtorsional force microscopy measurements; P. Nguyen, H.Park, N. Wang, and Y. Yu for assistance with samplefabrication; and A. Reddy for providing code used to buildthe initial continuum model. Y. D., P. M., M. H., D. L., andZ.-X. S. acknowledge the support of the U.S. Departmentof Energy, Office of Science, Office of Basic EnergySciences, Division of Material Sciences and Engineering,under Contract No. DE-AC02-76SF00515. The samplefabrication and characterization at U. Washington (W. H.and X. X.) are supported by Department of Energy, BasicEnergy Sciences, Materials Sciences and EngineeringDivision (DE-SC0012509). Y. D. acknowledges partialsupport from a Stanford Q-FARM Bloch PostdoctoralFellowship. Z. Z. acknowledges support from a StanfordScience fellowship. T. Z. acknowledges support from theMIT Dean of Science Graduate Student Fellowship. L. F.acknowledges partial support from a Simons InvestigatorAward from the Simons Foundation. P. M. acknowledgessupport from a Stanford Energy Postdoctoral Fellowshipthrough contributions from the Dai and Li Family StanfordSustainability Postdoctoral Fellow Program Fund, PrecourtInstitute for Energy, Bits & Watts Initiative, StorageXInitiative, and TomKat Center for Sustainable Energy.K.W. and T. T. acknowledge support from the JSPSKAKENHI (Grants No. 21H05233 and No. 23H02052),the CREST (JPMJCR24A5), JST and World PremierInternational Research Center Initiative (WPI), MEXT,Japan. Use of the Advanced Light Source, LawrenceBerkeley National Laboratory, is supported by theU.S. Department of Energy, Office of Science underContract No. DE-AC02-05CH11231. Use of theStanford Synchrotron Radiation Lightsource, SLACNational Accelerator Laboratory, is supported by theU.S. Department of Energy, Office of Science, Office ofBasic Energy Sciences under Contract No. DE-AC02-76SF00515. Use of the Stanford Nano Shared Facilities(SNSF), is supported by the National Science Foundationunder Grant No. ECCS-2026822.Y. D. and Z.-X. S. conceived and supervised the project.Y. D. and W.H. fabricated the samples. Y. D. and P. M.conducted ARPES measurements at ALS beamline 7,while Y. D. performed additional ARPES measurementsat SSRL beamline 5. Y. D. analyzed the ARPES data. T. T.and K.W. synthesized the high-quality hBN crystals. C. J.,A. B., and E. R. maintained ALS beamline 7, while M. H.and D. L. maintained SSRL beamline 5. Z. Z. and T. P. D.provided DFT calculations. T. Z. and L. F. provided con-tinuum model calculations. Y. D. wrote the Letter withcontributions from all authors.DATA AVAILABILITYThe data supporting the findings of this article are openlyavailable [36], embargo periods may apply.NONMONOTONIC BAND FLATTENING NEAR THE MAGIC … PHYS. REV. X 15, 041043 (2025)041043-7APPENDIX: METHODS1. Sample fabricationtMoTe2 devices were fabricated inside a glove box usingtwo methods. Most samples were prepared using method 1,while the 1.98° sample was prepared using method 2. hBN,graphene, and MoTe2 flakes were prepared by mechanicalexfoliation onto O2 plasma-treated SiO2=Si substrates.a. Method 1The tMoTe2 device was fabricated following theapproach of Ref. [37] with minor modifications. ThehBN, which eventually serves as the bottom gate dielectric,was first picked up using a polycarbonate (PC) film on adome-shaped polydimethylsiloxane (PDMS) block. Thenwe employed the tear-and-stack method [38] to sequen-tially assemble two halves of exfoliated MoTe2 at thedesired twist angle, followed by a graphene flake, all at90 °C. The hBN=tMoTe2=graphene stack was transferredonto a gold-coated PDMS block by melting the PC at180 °C. The stack was cleaned by sequential rinsing inN-methyl-2-pyrrolidone and isopropanol, blow with anitrogen gas gun and then transferred onto a graphite flakeon a SiO2=Si substrate or a highly conductive siliconsubstrate. Au=Ti (20 nm=3 nm) electrodes for groundingor gating, were deposited through stencil masks inside aglove box.b. Method 2The hBN bottom gate dielectric layer was pretransferredonto the graphite gate using a PC film on a PDMS block.Pt=Ti (30 nm=5 nm) electrodes were pre-patterned on thesubstrate via electron-beam lithography, with a smallelectrode contacting the graphite gate and a larger groundedelectrode surrounding the sample to minimize electrostaticdistortion during gating. A graphene flake and two halvesof exfoliated tMoTe2 were sequentially picked up with PCand transferred onto the hBN bottom gate by melting thePC at 180 °C, followed by rinsing in chloroform and dryingwith a nitrogen gas gun.Additionally, the 2H bilayer sample was prepared byin situ exfoliation of bulk single crystals onto gold-coatedSiO2=Si substrates.2. AFM measurementsMechanical cleaning of polymer residues at sampleinterfaces was performed using an Oxford InstrumentsAsylum Research Cypher AFM in contact mode withBudgetSensors Tap300-G tips (force constant: 40 N=m,tip radius: <10 nm). The setpoint voltage was adjustedbetween 0–0.3 V to accommodate tip wear duringscanning, with a scan rate of 0.1–0.15 Hz over a10 μm× 10 μm scan area.Subsequent twist-angle determination was conductedvia vector PFM using Ir=Pt-coated conductive tips(Nanosensor PPP-EFM, force constant: 2.8 N=m, tipradius: <25 nm). An ac bias of 1–3 V was appliedduring scanning. Typical resonance frequencies were300–350 kHz for vertical and 750–850 kHz for lateralPFM. The setpoint voltage was adjusted between 0–0.4 Vduring the scan. FFT analysis of the PFM amplitude imageyielded the moiré wavelength aM, allowing calculation ofthe twist angle θ using θ ¼ 2 arcsin ða0=2aMÞ, where theMoTe2 lattice constant a0 is taken as 0.355 nm.3. ARPES measurementsARPES measurements of tMoTe2 devices were con-ducted at Beamline 7.0.2 (MAESTRO) of the AdvancedLight Source (ALS) at Lawrence Berkeley NationalLaboratory. Samples were annealed at 200 °C in vacuumfor 1.5 hours and measured at ∼20 K under a pressurebelow 3 × 10−11 torr. The photon energy was 58 eV forall data shown in the main text. Spectra were collectedusing a Scienta R4000 analyzer with deflectors, and thephoton beam was focused to a 1–2 μm spot using acapillary mirror. The energy and angular resolutions were10–20 meV and 0.2°, respectively.ARPES measurements of 2H bilayer and monolayerMoTe2 were performed at Beamline 5-2 of the StanfordSynchrotron Radiation Lightsource (SSRL) at SLACNational Accelerator Laboratory. The samples were exfo-liated in situ at 200 K and measured at 7 K under a pressurebelow 2 × 10−11 torr. Data were collected using a ScientaDA 30 analyzer with deflectors with 58 eV photons and∼10 meV energy resolution. The beam spot size was∼5 μm × 32 μm.The moiré potential of tMoTe2 modulates the Diracelectrons in the top graphene layer, producing replicaDirac bands arranged hexagonally around the KG point[Fig. 1(c) [39] ]. These replicas are separated by themoiré wavevector GM of tMoTe2, whose magnitudewas extracted from an MDC taken across the Dirac pointsalong the GM direction [Fig. 1(d), bottom-right panel]. Thetwist angle θ was then determined using the relationθ ¼ 2 arcsin½ð ffiffiffi3pGM · a0Þ=ð8πÞ�.4. DFT calculationsThe DFT calculations were performed with the Viennaab initio simulationpackage [40,41].VanderWaals forces areweak, long-ranged interactions that significantly modify themechanical properties and band structures of layered materi-als. We incorporated vdW forces through nonlocal vdWfunctionals that do not rely on empirical fitting or predefinedatom-pairwise dispersion coefficients like the DFT-D2functional [42]. Specifically, we compared four differentnonlocal vdW functionals: vdW-DF2 [43], vdW-DF3-opt2[44], SCANþ rVV10 [45], r2SCANþ rVV10 [46].YUJUN DENG et al. PHYS. REV. X 15, 041043 (2025)041043-8To obtain the DFT band structures, we included bothstructural relaxation and spin-orbit coupling to best matchthe experimental observations. We followed the followingworkflow. First, we relaxed the atomic positions with fixedlattice parameter of 3.55 Å without spin-orbit coupling.With the fully relaxed structure, we performed self-consistent calculations that include spin-orbit coupling.Finally, the band structure was calculated along a high-symmetry line cut using the self-consistent wave functions.For untwisted systems, we relaxed the structure to elec-tronic convergence of 1 × 10−6 eV=Å and force conver-gence of 1 × 10−4 eV=Å. We used k-point sampling of6 × 6 × 1 for the self-consistent calculations. For twistedsupercell calculations, we first constructed the supercellaccording to the following relation:θ ¼ acos�N2 þ 4MN þM22ðN2 þ NM þM2�;where M and N are two integers. We chose M ¼ N þ 1so that the periodic supercell and moiré cell are the same.We used electronic convergence of 1 × 10−4 eV=Å andforce convergence of 1 × 10−3 eV=Å, as well as k-pointsampling of 5 × 5 × 1 for the self-consistent calculations.5. Continuum model calculationsTo investigate the moiré minibands of tMoTe2, weemployed a K-valley continuum model constructed forhole states near the K and K0 points of the Brillouin zone,where spin-orbit coupling leads to a single spin-valleylocked band per layer. The twist introduces a moirésuperlattice potential and modifies interlayer tunneling,both incorporated in the model. This approach is appro-priate for hole-doped samples where the Fermi level liesnear the valence band top at the K point and the Γ-pointstates are far below in energy.The continuum model Hamiltonian was constructedin the basis of layer pseudospin. The coordinate originwas set midway along the rotation axis of the two layers.We focused on the spin-up component (H↑):H↑ ¼0B@ℏ2ð−i∇−κþÞ22m þ V1ðrÞ tðrÞt†ðrÞ ℏ2ð−i∇−κ− Þ22m þ V2ðrÞ1CA:while the spin-down component (H↓) is the time reversalconjugate.The corners of the moiré Brillouin zone are κ� ¼½ð4πÞ=ð3aMÞ�½ðffiffiffi3p=2Þ;� 12�. VlðrÞ and tðrÞ represent theintralayer moiré potential and interlayer tunneling, respec-tively, for layer index l. Incorporating D3 symmetry andretaining only the lowest Fourier harmonics, we expressedthese terms asVlðrÞ ¼ −2V Xi¼1;3;5cosðgi þ ϕlÞ;tðrÞ ¼ ωð1þ e−ig2·r þ e−ig3·rÞ:Here, the moiré reciprocal lattice vectors are gi ¼½ð4πÞ=ð ffiffiffi3paMÞ�( cosf½πði − 1Þ�=3g; sinf½πði − 1Þ=3�g) fori ¼ 1;…; 6, and ϕ1 ¼ −ϕ2 ¼ ϕ.We retained only the valence band sector of the model,consistent with the experimentally hole-doped regime. Theresulting minibands were obtained by diagonalizing thecontinuum Hamiltonian in a plane-wave basis. 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DISCUSSION AND CONCLUSION Note added ACKNOWLEDGMENTS DATA AVAILABILITY APPENDIX: METHODS 1. Sample fabrication a. Method 1 b. Method 2 2. AFM measurements 3. ARPES measurements 4. DFT calculations 5. Continuum model calculations References