Abstract
Quasicrystals are characterized by atomic arrangements having long-range order without periodicity. Van der Waals bilayers provide an opportunity to controllably vary the atomic alignment between two layers from a periodic moiré crystal to an aperiodic quasicrystal. Here we reveal that in a dodecagonal WSe2 quasicrystal, two separate valleys in separate layers are brought arbitrarily close in momentum space through higher-order Umklapp scatterings. A modest perpendicular electric field is then sufficient to induce strong interlayer valley hybridization, manifested as another hybrid excitonic doublet. Concurrently, we observe the disappearance of the trion that exists at low field, which we attribute to a modified spatial distribution of the wavefunction due to the quasicrystal potential. This is possibly a precursor to localization. Our findings highlight the ability of incommensurate systems to bring any pair of momenta into close proximity, thereby introducing opportunities for valley engineering.
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Data availability
The data that support the Supplementary Information are available from the corresponding authors upon request. Other data are available via figshare at https://figshare.com/articles/dataset/Raw_data_for_Figure_2a_and_Figure_4a/30153736?file=58055668 (ref. 46). Source data are provided with this paper.
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Acknowledgements
This study is primarily supported by the National Science Foundation (NSF) through the Center for Dynamics and Control of Materials, an NSF MRSEC (Cooperative Agreement DMR 2308817) (Z.L., Q.G., Y.L., D.S.K., G.E., M.H.N., C.-K.S., E.K. and X. Li). C.-K.S. acknowledges support from the NSF (Grant No. DMR-2219610) and the Welch Foundation (Grant No. F-2164). Y.N. and X. Liu gratefully acknowledge support from the Department of Energy, Office of Basic Energy Sciences, for device fabrication (Grant No. DE-SC0019398). H.A. is supported by a Welch Foundation Chair (F-0014) and the NSF (Grant No. ECCS-2130552) for his work on spectral analyses. G.E. and M.H.N. acknowledge the Texas Advanced Computing Center at the University of Texas at Austin for providing computational resources that have contributed to the research results reported within this paper. This work also used computational resources from Stampede3 at the University of Texas at Austin through allocation DMR24024 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services and Support (ACCESS) programme, which is supported by the NSF (Grant Nos. 2138259, 2138286, 2138307, 2137603 and 2138296). C.S. and Y.H. acknowledge support from NSF CMMI (Grant No. 2239545) and the Welch Foundation (Grant No. C-2065). K.W. and T.T. acknowledge support from the JSPS (KAKENHI Grant Nos. 20H00354 and 23H02052) and the World Premier International Research Center Initiative, MEXT, Japan.
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Z.L., K.S., E.K. and X. Li conceived the project. Z.L. fabricated the samples with assistance from Y.N., X. Liu and M.M. Z.L. analysed the data with contributions from Y.L., D.S.K., H.A. and X. Liu. Y.L. and F.Z. conducted the STM measurements under the supervision of C.-K.S. Y.L. fabricated the TEM sample with the help of Z.L. C.S. conducted the TEM and electron ptychography under the supervision of Y.H. K.W. and T.T. synthesized the bulk crystals of hexagonal boron nitride. Q.G. and E.K. proposed the theoretical model. G.E. and M.H.N. performed the first-principles calculations. Z.L., Q.G., E.K. and X. Li wrote the first draft of the paper. All authors contributed to discussions.
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Extended data
Extended Data Fig. 1 Calculated spectral function.
Spectral intensity corresponding to conduction band minimum states arising from the six Qb valleys in 29.84° twisted bilayer WSe2 superlattice. (a) shows the lower order scatterings, with the circles highlighting the coupling between Qb + Gb and Kt + Gt. (b) shows the higher-order scatterings in the Brillouin zone in addition to those shown in (a). These scatterings have a much weaker spectral intensity.
Extended Data Fig. 2 Scanning tunneling spectroscopy measurement of the quasicrystal conduction bands at zero E-field at 4 K.
(a) Typical scanning tunneling microscopy topography image of 30° WSe2 bilayer. (Vbias = -0.5 V, I = 30 pA) (b) The spatially averaged scanning tunneling spectroscopy (STS) taken from a pristine region. The red (blue) dashed line indicates the energy position of the lowest K (second lowest Q) valley in the conduction band(Vbias = -2.0V, I = 100 pA) (c-e) Variable-Z tunneling spectroscopy spectra measured at the same region. (c) The constant current STS (∂I/∂V), (d) tip sample distance (Z) versus bias voltage measurement (Z(V)) and (e) the experimental tunneling decay constant (κ). These measurements complement those in panel b to accurately identify the energies of different valleys. From these measurements, we conclude that K valley is higher in energy than Q valley by ~ 110 meV, a value taken into account in our calculation.
Extended Data Fig. 3 Reflectivity spectra of quasicrystal D1 and 2D map of 1S exciton intensity.
(a)Typical reflectivity spectra from the quasicrystal with no doping (top), with electron (middle), and with hole doping (bottom). Red dashed lines are fitting results of 1s exciton resonance. We use Lorentzian function with a phase shift as the fitting function. The grey vertical strip indicated the energy range used for 1s exciton intensity map in Fig. S3b. (b) A 2D reflectivity map as a function of top and bottom gates identifies the conditions under which the top (bottom) layer exhibits intrinsic (i), electron-doped (n), or hole-doped (p) as indicated by the first (second) index in the parenthesis. The two diagonal directions, marked by red arrows, allow independent control of doping density and an electric field to be perpendicular to the 2D layers. The color bar represents the 1S exciton intensity in reflectivity spectra.
Extended Data Fig. 4 Reflectivity spectra of quasicrystal D2 as a function of doping with different E-fields.
(a) The reflection spectra are plotted as a function of doping density in the absence of an electric field. Negative trion \({T}_{t,b}^{-}\) and positive trion \({T}_{t,b}^{+}\) are marked by black dashed lines. The blue-shifted exciton Xt,b is marked as a white dashed line. The trion signals come from both layers. (b) Reflection spectra are displayed as a function of doping density at a constant electric field of E=0.03 V nm−1 which below the critical hybridization E field (Ec). Negative trion \({T}_{b}^{-}\) and positive trion \({T}_{t}^{+}\) are marked by black dashed lines. The blue-shifted exciton Xt is marked by the white dashed line. The doped holes reside in the top layer below doping concentration − 5 × 1012cm−2 and in both layers above it. The positive trion signal from both layers is marked as \({T}_{t,b}^{+}\).
Extended Data Fig. 5 The reflectivity spectra of different position of D1 and D2 plotted as a function of electric field at a constant hole doping levels.
Optical microscopy picture of D1 (a) and D2 (b). The scale bar is 10 um. The blue and green dots show the position where the reflectivity spectra are taken. The data from blue dots are shown in the Fig. 2a while the data from green dots are shown in (c). Hybrid exciton doublet at higher electric field is more visible in region III. (d) Reflectivity spectra from D2 with electric field at hole doping − 1.15 × 1012 cm−2 from the marked black dot. Disappearance of the trion and hybrid exciton Xh are observed, similar in Fig. 2a in the main text.
Extended Data Fig. 6 Reflectivity spectra 21.8° moiré crystal at a constant electron doping.
Reflectivity spectra at a few E-fields at a constant electron doping 1.9 × 1012cm−2. Orange dots show negative trion resonance \({T}_{b}^{-}\) and red dots show exciton resonance Xt. No abrupt changes are observed.
Extended Data Fig. 7 Intensity map of 1s exciton in the high electric field region and reflectivity spectra of the hole-doped quasicrystal.
(a) The dashed black line shows the doping-dependent reflectivity spectra in Fig. 4a at a constant E-field 0.1V nm−1. Along this line, the doping density is varied from -5.4 to 3.6 × 1012 cm−2. In the region with red (blue) color bar, minimal (strong) trion resonance is observed. (b) Optical reflectivity spectra from the quasicrystal D2 at several hole doping densities along the black dashed line presented in panel a.
Extended Data Fig. 8 Electric field dependence absorption in the charge neutrality region.
The yellow dashed rectangle highlights the hybrid exciton, which appears in the large electric field and deviates from the prominent A exciton absorption features.
Extended Data Fig. 9 Wavefunction redistribution under K − Q coupling.
(a)Wavefunction at Kt in real space with K − Q coupling switched off. (b) Hybridized wavefunction at Kt in real space with K − Q coupling switched on. The wavefunctions are simulated using simple plane wave expansion that respects the corresponding crystalline and quasi-crystalline structures. A 12-fold pattern is obvious in the layer-hybridized wavefunction shown in panel (b).
Extended Data Fig. 10 STEM performed on a 30-Degree WSe2 bilayer.
(a) Atomic-resolution ADF-STEM image of 30-degree twisted bilayer WSe2. The white particles visible are polymer residues on the top surface of the bilayer, which do not influence the twist angle or the structure. The image shows a filtered quasicrystal structure (b) FFT of the image in (a), revealing a twist angle of 30.1 degrees. (c) 4D-STEM angle measurements over a 600 nm field of view of the twisted sample. The uniformity of the map indicates a consistent angle distribution across this region. The angle is calculated from diffraction patterns collected at each pixel, with the averaged diffraction pattern shown in the inset. (d) Histogram of angles measured from the uniform region, showing a peak value of 30. 1°, consistent with the angle measured from the atomic resolution image in panel a. A standard deviation of ± 0.25 degrees demonstrates the high precision of the measurements, with the peak position representing the most accurate angle for this twisted sample.
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Liu, Z., Gao, Q., Li, Y. et al. Field-tunable valley coupling in a dodecagonal semiconductor quasicrystal. Nat. Phys. 22, 33–38 (2026). https://doi.org/10.1038/s41567-025-03080-8
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DOI: https://doi.org/10.1038/s41567-025-03080-8
