Abstract
One-dimensional (1D) interacting electrons are often described as a Luttinger liquid1,2,3,4 having properties that are intrinsically different from those of Fermi liquids in higher dimensions5,6. In materials systems, 1D electrons exhibit exotic quantum phenomena that can be tuned by both intra- and inter-1D-chain electronic interactions, but their experimental characterization can be challenging. Here we demonstrate that layer-stacking domain walls (DWs) in van der Waals heterostructures form a broadly tunable Luttinger liquid system, including both isolated and coupled arrays. We have imaged the evolution of DW Luttinger liquids under different interaction regimes tuned by electron density using scanning tunnelling microscopy. Single DWs at low carrier density are highly susceptible to Wigner crystallization consistent with a spin-incoherent Luttinger liquid, whereas at intermediate densities dimerized Wigner crystals form because of an enhanced magneto-elastic coupling. Periodic arrays of DWs exhibit an interplay between intra- and inter-chain interactions that gives rise to new quantum phases. At low electron densities, inter-chain interactions are dominant and induce a 2D electron crystal composed of phased-locked 1D Wigner crystal in a staggered configuration. Increased electron density causes intra-chain fluctuation potentials to dominate, leading to an electronic smectic liquid crystal phase in which electrons are ordered with algebraical correlation decay along the chain direction but disordered between chains. Our work shows that layer-stacking DWs in 2D heterostructures provides opportunities to explore Luttinger liquid physics.
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Data availability
The data supporting the findings of this study can be found at GitHub (https://github.com/HongyuanLiCMP/Imaging-Tunable-Luttinger-Liquid-Systems-in-van-der-Waals-Heterostructures) and are also available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was primarily funded by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 within the van der Waals heterostructure programme KCFW16 (device fabrication, STM spectroscopy, theoretical analyses and computations). Support was also provided by the National Science Foundation Award DMR-2221750 (surface preparation). This research used the Lawrencium computational cluster provided by the Lawrence Berkeley National Laboratory (supported by the US Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC02-05-CH11231). S.T. acknowledges primary support from the US Department of Energy SC0020653 (materials synthesis), NSF CMMI1825594 (NMR and TEM studies on crystals), NSF DMR-1955889 (magnetic measurements on crystals), NSF ECCS2052527 (bulk electrical tests), DMR 2111812 and CMMI 2129412 (optical tests on bulk crystals). K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant nos. 21H05233 and 23H02052) and the World Premier International Research Center Initiative (WPI), MEXT, Japan. H.L. acknowledges support from Kavli Energy Nano Sciences Institute graduate student fellowship. We acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing high-performance computing resources. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy, Office of Science User Facility, located at Lawrence Berkeley National Laboratory, operated under contract no. DE-AC02-05CH11231.
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H.L., M.P.Z., M.F.C. and F.W. conceived the project. H.L., Z.X., J.N. and S.L. fabricated the WS2 heterostructure device. H.L. and Z.X. performed the STM/STS measurement of the WS2 device. T.W. and M.P.Z. performed the DMRG calculations of the 1D interacting electrons. M.H.N., W.K. and S.G.L. performed the ab initio calculations of the DW structures. Z.G. and Z.H. performed the measurement of the QPI in bilayer MoSe2. H.L., Z.X., A.Z., M.F.C. and F.W. discussed the experimental design and analysed the experimental data. Y.O., R.B. and S.T. grew the WS2 crystals. K.W. and T.T. grew the hBN single crystal. All authors discussed the results and wrote the paper.
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Extended data figures and tables
Extended Data Fig. 1 Sketch of the atomic structure for a shear-type stacking DW.
The left and right regions are AB stacked while the center shows a vertically aligned DW. The positions of W atoms in the bottom layer (orange dots) and S atoms in the top layer (blue dots) are highlighted along a linecut across the DW. For shear-type DWs, the two AB stacking regions have an interlayer unit-vector shift parallel to the DW.
Extended Data Fig. 2 Conduction band edge tunnel current measurement.
a. Schematic energy diagram for the conduction band edge (CBE) tunnel current measurement of electron-doped WS2. The WS2 chemical potential \({\mu }_{W{S}_{2}}\) lies above the CBE. When the tip chemical potential \({\mu }_{{tip}}\) (controlled by Vbias) is the aligned within the band gap of the WS2, the tunnel current arises from the doped electrons at the conduction band edge. b,c. Tunnel current I-V characteristics as a function of VBG measured at the DW center for electron-doped WS2 with (b) a large (Vbias = −3.30 V, Isp = 20 pA, htip = −50 pm) and (c) small (Vbias = −2.70 V, Isp = 20 pA, htip = −100 pm) tip-sample separation. The current is plotted on a log scale with the positive and negative branches using different colormaps. The CBE and valence band edge (VBE) are marked with white dashed lines. For small tip-sample separation a negative CBE tunnel current can be seen in the WS2 gap.
Extended Data Fig. 3 Analysis of the Wigner-Friedel crossover.
a. Calculated charge phonon energy \(\hbar {\omega }_{0}\) and exchange interaction energy \({E}_{J}\) as a function of 1D chain electron density. The experimental temperature energy scale is labeled with a green dashed line. b. Schematic illustration of 1D electron chain with decreasing interaction strength (from bottom to top) shows three regimes: Wigner crystal, dimerized crystal, and Friedel oscillation. c. DMRG calculation of the charge density \(n(x)\) and entanglement entropy \({S}_{{EE}}\) across each site of 1D electron chain for n = 0.3 nm−1 (see SI section 6 for details). SEE reflects the degree of entanglement (and therefore singlet formation) between neighboring electrons in a dimerized Wigner crystal. Vertical lines label the boundary (solid) and center (dashed) of singlet pairs.
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Li, H., Xiang, Z., Wang, T. et al. Imaging tunable Luttinger liquid systems in van der Waals heterostructures. Nature 631, 765–770 (2024). https://doi.org/10.1038/s41586-024-07596-6
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DOI: https://doi.org/10.1038/s41586-024-07596-6
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