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One-dimensional Luttinger liquids in a two-dimensional moiré lattice

Abstract

The Luttinger liquid (LL) model of one-dimensional (1D) electronic systems provides a powerful tool for understanding strongly correlated physics, including phenomena such as spin–charge separation1. Substantial theoretical efforts have attempted to extend the LL phenomenology to two dimensions, especially in models of closely packed arrays of 1D quantum wires2,3,4,5,6,7,8,9,10,11,12,13, each being described as a LL. Such coupled-wire models have been successfully used to construct two-dimensional (2D) anisotropic non-Fermi liquids2,3,4,5,6, quantum Hall states7,8,9, topological phases10,11 and quantum spin liquids12,13. However, an experimental demonstration of high-quality arrays of 1D LLs suitable for realizing these models remains absent. Here we report the experimental realization of 2D arrays of 1D LLs with crystalline quality in a moiré superlattice made of twisted bilayer tungsten ditelluride (tWTe2). Originating from the anisotropic lattice of the monolayer, the moiré pattern of tWTe2 hosts identical, parallel 1D electronic channels, separated by a fixed nanoscale distance, which is tuneable by the interlayer twist angle. At a twist angle of approximately 5 degrees, we find that hole-doped tWTe2 exhibits exceptionally large transport anisotropy with a resistance ratio of around 1,000 between two orthogonal in-plane directions. The across-wire conductance exhibits power-law scaling behaviours, consistent with the formation of a 2D anisotropic phase that resembles an array of LLs. Our results open the door for realizing a variety of correlated and topological quantum phases based on coupled-wire models and LL physics.

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Fig. 1: Small-angle tWTe2 moiré lattices and large transport anisotropy.
Fig. 2: Luttinger liquid behaviours observed in the tWTe2 devices.
Fig. 3: Gate-tuned power laws and anisotropy cross-over.
Fig. 4: Theoretical modelling and the emergence of quasi-1D moiré bands at the single-particle level.

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Data availability

The data that support the findings of this study are available at Harvard Dataverse (https://doi.org/10.7910/DVN/PFXXSZ) or from the corresponding author upon reasonable request.

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Acknowledgements

We acknowledge discussions with N. P. Ong, A. Yazdani, B. A. Bernevig, F. H. L. Essler, B. Lian, C. Kane, K. Yang, A. J. Uzan, Y. Werman and J. Zhang. We thank S. H. Simon for discussions and for pointing out ref. 45 to us. This research was supported by NSF through a CAREER award to S.W. (DMR-1942942) and the Princeton University Materials Research Science and Engineering Center (DMR-2011750) through support to S.W., R.J.C. and L.M.S. Device characterization and data analysis were partially supported by ONR through a Young Investigator Award (N00014-21-1-2804) to S.W. S.W. and L.M.S. acknowledge support from the Eric and Wendy Schmidt Transformative Technology Fund at Princeton. Early measurements were performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement no. DMR-1644779 and the State of Florida. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant number JPMXP0112101001, JSPS KAKENHI grant number JP20H00354 and the CREST(JPMJCR15F3), JST. L.M.S. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS initiative through grant no. GBMF9064 to L.M.S, the David and Lucile Packard Foundation, the Sloan Foundation and Princeton’s catalysis initiative. Y.H.K. and S.A.P. acknowledge support from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme via grant agreement no. 804213-TMCS. S.L.S. was supported by the Gordon and Betty Moore Foundation through grant no. GBMF8685 towards the Princeton theory program and by a Leverhulme International Professorship at Oxford.

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Authors

Contributions

S.W. conceived and designed the project. P.W. and G.Y. fabricated devices, performed measurement and analysed data, assisted by Y.J. and supervised by S.W. Y.H.K, T.D., S.L.S. and S.A.P. performed theoretical modelling. F.A.C., R.J.C., S.L., S.K., R.S. and L.M.S. grew and characterized bulk WTe2 crystals. K.W. and T.T. provided hBN crystals. S.W., P.W., S.A.P. and Y.H.K. wrote the paper with input from all authors.

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Correspondence to Sanfeng Wu.

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Extended data figures and tables

Extended Data Fig. 1 Conductive atomic force microscope (cAFM) measurements of tWTe2 at room temperature.

a, Cartoon illustrations of the cAFM measurement on a few-layer hBN/tWTe2/hBN stack on a Pd metal film pre-patterned on a SiO2/Si chip. The inset illustrates the cross-section of the stack. A relatively thick (around 39 nm) hBN was used to mitigate the roughness of the metal surface. b, A cAFM image taken from a θ ≈ 5° tWTe2 device, directly visualizing the moiré structure. The dashed-dot square locates a zoom-in scan, as shown in c. We comment on three aspects of the observations. (1) The measurement was taken at room temperature, where the transport shows no significant anisotropy. This is consistent with the fact the measured local conductance G varies only by a small amount at different tip locations in the map. (2) As shown in b and c, the small variations already allow us to clearly image the underlying moiré structure. (3) Our experimental resolution does not allow us to identify which one is the AA stripe or the AB stripe, but the map is clearly consistent with the pattern shown in Fig 1b, except with lattice relaxations. The relatively low and high conductance regions develop into stripes, with an inter-stripe distance approximately 7.1 nm, consistent with the expectation for a θ ≈ 5° tWTe2 stack. (4) As WTe2 is air sensitive, we have to use a few-layer hBN as a protecting layer and a sample fabrication process that minimizes the time for the sample exposed to air. The top surface of the thin hBN is left behind with polymer residues etc, which we believe could be the main source of the residue-like features in b and c. Our transport devices (devices no. 1 & no. 2 in the main text) use a top graphite gate that serves as a screening layer and hence the tWTe2 channel is of much higher quality. Other details about the cAFM measurements can be found in the Methods.

Extended Data Fig. 2 Sample fabrication process.

a, Cartoon illustrations of a prepared top graphite/hBN stack (top) and a flake of monolayer WTe2 on a SiO2/Si chip (bottom). Their corresponding optical images are also shown to the right. The distance between the aligned graphite edge and hBN edge is carefully optimized during transfer to be within 500 nm. b & c, Tear the monolayer WTe2 by the hBN into two separate pieces, labelled as top and bottom WTe2. d, Rotate the bottom WTe2 flake, i.e., the SiO2/Si chip, counterclockwise by θ. e, Pick up the bottom WTe2 to create a tWTe2 stack. The optical image of the tWTe2 stack shown to the right was taken after flipping the stamp upside down. The tWTe2 stack is highlighted by the red dashed line. No visible bubbles were observed. f, The pre-patterned bottom part (thin hBN/Pd electrodes/hBN dielectric/graphite) with etched holes in the thin hBN layer to expose the tips of the Pd electrodes (see ref. 16,17). An atomic force microscope image shows a clean surface of a prepared bottom stack. g, The final stack of a tWTe2 device, with an optical image of a final device (no. 1). The thickness of flakes is 7.6 nm graphite/8.9 nm top hBN/WTe2/2.0 nm thin hBN/5.5 nm bottom hBN/ 7.0 nm graphite for device no. 1 and 4.8 nm graphite/10.6 nm top hBN/WTe2/4.2 nm thin hBN/15.0 nm bottom hBN/ 6.9 nm graphite for device no. 2. All scale bars are 3 μm.

Extended Data Fig. 3 More analysis on the transport anisotropy on device no. 1.

a, Two-probe resistance between neighbouring electrodes as a function of ng in cooldown no. 1. Inset shows the contact configurations for each measurement, where the estimated hard direction (stripe direction) is indicated by the grey lines (not to scale). R2-3 and R6-7 display larger values than all others in the hole-doped regime, signifying the hard direction, whereas R4-5 shows the lowest value. Contact 1 was broken during the fabrication. The contact resistance plays a significant role here. After the easy and hard directions were identified, we performed four-probe measurements, as shown in Fig. 1 in the main text.). b, Two-probe resistance across (R2-3, R6-7, and R5–8) and along (R4-5, R2–7, and R3–6) the stripes as a function of ng in cooldown no. 2. Inset shows the contact configurations for each measurement, where the easy direction (along stripes) is indicated by the grey lines (not to scale). c, Four-probe resistance across (“Isd: 5–8; Vxx: 6-7” and “Isd: 5–8; Vxx: 3-2”) and along (“Isd: 3–6; Vxx: 4-5”, “Isd: 3–6; Vxx: 2–7”, and “Isd: 2–7; Vxx: 4-5”) the stripes as a function of ng in cooldown no. 2.

Extended Data Fig. 4 ng dependent transport anisotropy for devices no. 1 and no. 2.

Four-probe resistance Rxx as a function of ng measured with an excitation current applied along hard and easy directions in linear plots. a, data taken for device no. 1 at 1.8 K (the same data as Fig. 1i). b-d, Rhard, Reasy, and Rhard/Reasy as a function of ng at different temperatures taken from device no. 1. e-h, The same plots for device no. 2.

Extended Data Fig. 5 Electrostatic simulation for the four-probe contact configuration.

a & b, Electric potential distribution for contact arrangements corresponding to Rhard and Reasy four-probe measurements respectively (see Methods). Black dots indicate current contacts that source/sink current. Red dots indicate the placement of voltage contacts. c, Predicted four-probe anisotropy β4pRhard/Reasy as a function of the intrinsic sheet resistivity anisotropy βbulk. For β4p ≈ 1,000, we estimate βbulk ≈ 50.

Extended Data Fig. 6 Dual-gate-dependent transport along hard and easy directions (device no. 1, cooldown no. 1).

The four-probe resistance taken at 1.8 K (200 K) along the hard and easy directions were shown in a (d) and b (e), respectively. Rhard/Reasy at 1.8 K (200 K) is shown in c (f).

Extended Data Fig. 7 Fitting to the differential conductance data based on the universal scaling formula.

a, Raw data for the differential conductance measurements taken in device no. 1 (replotted from Fig. 2d). b, 2D map of calculated root-mean-square error (r.m.s.e.) as a function of the fitting parameters, α and γ (see Methods for details). The best fit is obtained by finding the minimal value of r.m.s.e. in this plot, i.e., α = 0.94 and 1/γ = 9. c & d, Scaled conductance as a function of scaled excitation by assigning α = 0.94 in a log–log plot and log-linear plot. The dashed line indicates the fitting result given by the universal formula defined in the Method section. e-h, The same fitting plots for device no. 2, using the same raw data shown in Fig. 2f.

Extended Data Fig. 8 Comparison between along-wire and across-wire transport.

a, Illustration of tWTe2 moiré stripes on the electrodes (top view). b, Illustration of transport along wires. At low T, the along-wire transport is dominated by contact resistance, i.e., tunnelling from the metal (FL) to the moiré wires (LL). c, Illustration of the across-wire transport, where the dominant resistance is due to interwire tunnelling in the stripe regime (i.e., LL to LL tunnelling). d, Along-wire two-probe conductance G as a function of T, plotted in log–log scale at a selected gate parameter. A power-law fit (solid line) to the low T data is shown. e, Differential conductance dI/dV taken under the along-wire transport configuration as a function of d.c. bias V at different T. The dashed line indicates a power-law trend. The dot-dash line indicates a deviation from the trend at high bias. Note that distortions, strain, unintentional doing and other interface effects occur at the moiré in the contact regime, which could cause the deviation. f & g, the same plot for data taken from the across-wire transport (the same data as Fig. 2c, d), exhibiting a more robust power-law behaviour to higher bias and T. This can be understood as the dominant resistance in the across-wire transport comes from the tWTe2 channel regime, which is more uniform compared to the contact regime. Data were taken from device no. 1 in cooldown no. 1.

Extended Data Fig. 9 Comparison of two-probe and four-probe measurements across the wires.

Cartoon illustration of (a) two-probe (G2p) and (b) four-probe (Gxx) configurations used for the measurements. c and d, G2p and Gxx as a function of temperature taken in the hole-doped region (ng = −5.5 × 1012 cm−2 and ng = −13.5 × 1012 cm−2, respectively). At low T (1.8 K ~ 25 K) the trends of G2p and Gxx both follow a power law and match well, demonstrating that the power law is intrinsic to the tWTe2 channel. At high T, the two trends of G2p and Gxx deviate from each other, which can be understood as G2p saturates due to contact resistances whereas Gxx is strongly affected by the temperature induced changes of anisotropy. The effective geometry factor, important for determining Gxx, changes as the sample is tuned from a strongly anisotropic phase at low T to an isotropic phase at high T. The main analyses in this paper are focused on the low T regime. The measurements were performed on device no. 1 in cooldown no. 2.

Extended Data Fig. 10 Gate-tuned anisotropy cross-over.

a, The across-wire two-probe conductance G(T) displays a power-law relation (G Tα) for a wide range of doping for device no. 1 (cooldown no. 2). The colour of the data points encodes ng, as shown in the colour bar. The solid lines are the power-law fittings, where the extracted exponent α is shown in the inset. The grey line replots the anisotropy ratio. b, The same plots for device no. 2 (cooldown no. 1). The grey line replots the anisotropy ratio shown in the inset of Fig. 1h. c, The same plots for device no. 2 (cooldown no. 2). Note that data taken from two different cooldowns from device no. 2 shows qualitatively consistent results with only minor quantitative differences (dashed line in the inset of c is the exponent α replotted from the inset of b for comparison). Arrows to the insets in a and c indicate the selected ng, at which the scaling analysis of the differential conductance is performed in Extended Data Figs. 11 and 12, respectively.

Extended Data Fig. 11 Additional power-law scaling analysis for device no. 1 (cooldown no. 2).

The corresponding ng for each data set is indicated in the inset of Extended Data Fig. 10a. a, Temperature dependent across-wire two-probe conductance G (T) taken at the indicated ng. The solid line is the power low fit. a’, Bias dependent differential conductance taken at the same ng under different T. The dashed line indicates the power-law trend with the same exponent α extracted in a. a’’, the same data in a’, but replotted as scaled differential conductance (dI/dV)/Tα v.s. scaled bias eV/kBT. Other panels are the same plots for different ng. As seen in the plots, in the hole side (a-c) the data generally follows a power law very well, whereas near charge neutrality (d) and in the electron side (e), deviations start to develop at high bias. In the highly electron-doped region (f), dI/dV and G vary only a little bit (α ≈ 0) with changing both V and T, hence the behaviour is approximately ohmic. Data used for Figs. 3b-d are indicated in the lowest panel.

Extended Data Fig. 12 Additional power-law scaling analysis for device no. 2 (cooldown no. 2).

The corresponding ng for each data set is indicated in the inset of Extended Data Fig. 10c. a, Temperature dependent across-wire two-probe conductance G (T) taken at the indicated ng. The solid line is the power low fit. a’, Bias dependent differential conductance taken at the same ng under different T. The dashed line indicates the power-law trend with the same exponent α extracted in a. a’’, the same data in a’, but replotted as scaled differential conductance (dI/dV)/Tα v.s. scaled bias eV/kBT. Other panels are the same plots for different ng. As seen in the plots, in the hole side (a-d) the data generally follows a power law very well, whereas near charge neutrality (e), deviations start to develop at high bias. In the highly electron-doped region (f), dI/dV and G vary only a little bit (α ≈ 0) with changing both V and T, hence the behaviour is approximately ohmic. Data used for Figs. 3f–h are indicated in the lowest panel.

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Wang, P., Yu, G., Kwan, Y.H. et al. One-dimensional Luttinger liquids in a two-dimensional moiré lattice. Nature 605, 57–62 (2022). https://doi.org/10.1038/s41586-022-04514-6

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