Abstract
Magic-angle twisted trilayer graphene (MATTG) exhibits a range of strongly correlated electronic phases that spontaneously break its underlying symmetries1,2. Here we investigate the correlated phases of MATTG using scanning tunnelling microscopy and identify marked signatures of interaction-driven spatial symmetry breaking. In low-strain samples, over a filling range of about two to three electrons or holes per moiré unit cell, we observe atomic-scale reconstruction of the graphene lattice that accompanies a correlated gap in the tunnelling spectrum. This short-scale restructuring appears as a Kekulé supercell—implying spontaneous inter-valley coherence between electrons—and persists in a wide range of magnetic fields and temperatures that coincide with the development of the gap. Large-scale maps covering several moiré unit cells further reveal a slow evolution of the Kekulé pattern, indicating that atomic-scale reconstruction coexists with translation symmetry breaking at a much longer moiré scale. We use auto-correlation and Fourier analyses to extract the intrinsic periodicity of these phases and find that they are consistent with the theoretically proposed incommensurate Kekulé spiral order3,4. Moreover, we find that the wavelength characterizing moiré-scale modulations monotonically decreases with hole doping away from half-filling of the bands and depends weakly on the magnetic field. Our results provide essential insights into the nature of the correlated phases of MATTG in the presence of strain and indicate that superconductivity can emerge from an inter-valley coherent parent state.
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Data availability
The raw data shown in the main figures are available at Zenodo (https://doi.org/10.5281/zenodo.8317363). Other data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank N. Bultinck, S. Parameswaran, A. Pasupathy, A. Vishwanath, S. Todadri and A. Yazdani for the discussions. We are grateful in particular to T. Soejima and M. Zaletel for pointing out subtleties regarding the extraction of qIKS through Fourier analysis. This work has been primarily supported by the National Science Foundation (grant no. DMR-2005129); the Office of Naval Research (grant no. N142112635); and the Army Research Office (grant award W911NF17-1-0323). S.N.-P. acknowledges support from the Sloan Foundation. J.A. and S.N.-P. also acknowledge support from the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (grant no. GBMF1250); É.L.-H. and C.L. acknowledge support from the EPiQS Initiative of the Gordon and Betty Moore Foundation (grant no. GBMF8682) (at Caltech). C.L. acknowledges start-up funds from the Florida State University and the National High Magnetic Field Laboratory. The National High Magnetic Field Laboratory is supported by the National Science Foundation through NSF/DMR-1644779 and the state of Florida. A.T. and J.A. are grateful for the support of the Walter Burke Institute for Theoretical Physics at Caltech. H.K. acknowledges support from the Kwanjeong fellowship. L.K. acknowledges support from an IQIM-AWS Quantum postdoctoral fellowship. The primary support for sample fabrication efforts at UCSB was provided by the US Department of Energy (award no. DE-SC0020305). This work used facilities supported by the UC Santa Barbara NSF Quantum Foundry funded by the Q-AMASE-i programme under award DMR-1906325.
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H.K. and Y.C. fabricated samples with the help of Y.Z., H.Z. and L.H. and performed STM measurements. H.K., Y.C. and S.N.-P. analysed the data with the help of L.K. and E.B. É.L.-H., C.L. and A.T. provided the theoretical analysis supervised by J.A. S.N.-P. supervised the project. K.W. and T.T. provided the hBN crystals and A.F.Y. supervised the device fabrication efforts. H.K., Y.C., É.L.-H., C.L., A.T., J.A. and S.N.-P. wrote the paper with input from other authors.
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Extended data figures and tables
Extended Data Fig. 1 Sample fabrication.
a–f, Gold coated PDMS assisted flipping. A stack with PC film is peeled off from a PDMS block (a). A separate PDMS block is Ti/Au coated (b). Then the stack is put down to the gold coated PDMS (c). PC film is dissoved by NMP (d), before the stack is dropped down to a substrate (e,f). g–i, A gold stamp is prepared. A mould is defined by photolithography (g). PDMS is poured on the mould (h), and peeled off. Au is deposited on the stamp (i). j,k, The gold stamp is pressed down onto a desired area of the sample (j), leaving a gold strip that connects the sample and a pre-patterned electrode on the substrate (k).
Extended Data Fig. 2 AFM images of samples A and B.
a, Optical microscope image of Sample A. b, AC tapping mode AFM image after the contact mode cleaning the 2 × 2μm2 area around the centre of the image. No sign of residue is found. c, cAFM image showing moiré pattern of MATTG. d,e, AC tapping mode AFM images of Sample B before (d) and after (e) cleaning. The residue boundaries after the cleaning indicate significant amount of residues on the surface. f, cAFM image. Scale bars : 10μm (a), 2μm (b,d,e), 100nm (c,f).
Extended Data Fig. 3 VBias dependent mapping of the lattice tripling order on MATTG.
a, Conductance at VGate = −10V taken along spatial points and for range of VBias. While at large positive VBias, dI/dV shows periodic modulation that corresponds to graphene lattices, at low VBias, additional periodic pattern that triples the graphene lattice periodicity is apparent (black dashed lines). Upper inset shows two linecuts taken at above (VBias = 2mV) and below (VBias = −2mV) EF stressing lattice tripling. b, VBias spectroscopy extracted from Extended Data Fig. 3a that compares total dI/dV obtained by summing up along spatial coordinates and FT filtered Kekulé dI/dV which is a result of FT filtering on Extended Data Fig. 3a along spatial direction to extract Kekulé signal. c,d, dI/dV map measured at fixed VGate = −10V at negative VBias = −2mV (c) and positive VBias = 2mV (d). Red and blue dashed line shows the spatial positions where Extended Data Fig. 3a is measured. Measurements are taken at T = 400mK.
Extended Data Fig. 4 VGate dependent evolution of the lattice tripling order at higher temperatures.
a, VGate dependent dI/dV spectroscopy measured at T = 5K where ν = 2 correlated gap survives but gaps around ν = −2 is greatly suppressed. Red (positive VBias) and blue (negative VBias) dots marks the position where we measured 2D dI/dV maps. b, Intensity of the peak at Kekulé reciprocal lattice vector normalized by the intensity of the peak at graphene reciprocal lattice vector as a function of VGate. c,e, Real space dI/dV map at VGate = −9.5V (c) and VGate = 7.8V (e). d,f, Fourier transformation of Extended Data Fig. 4c,e. g, VGate dependent dI/dV spectroscopy measured at T = 7.5K where ν = 2 correlated gap survives but gaps around ν = −2 are greatly suppressed. Red (positive VBias) and Blue (negative VBias) dots mark the position where we measured 2D dI/dV maps. h, Intensity of the peak at Kekulé reciprocal lattice vector normalized by the intensity of the peak at graphene reciprocal lattice vector as a function of VGate. i,j, Real space dI/dV map at VGate = −9.5V (i) and VGate = 7.8V (j). k,l, Fourier transformation of Extended Data Fig. 4k,l.
Extended Data Fig. 5 Out-of-plane magnetic field dependence of lattice tripling order.
a, LDOS Landau fan diagram measured on an moiré ABA site. The lower panel shows intensity of the lattice tripling signal as in Fig. 2. Black circles marked on Landau fan indicate VGate and B, where we observe Kekulé peaks in FT. The black cross indicate values of VGate and B where we measured dI/dV map but could not observe Kekulé peaks in FT. The green polygon is an eye guide covering black circles and roughly denotes where we observed lattice tripling. b, c, VGate dependent dI/dV spectroscopy measured at B = 4T (b) and B = 8T (c). d, e, Real space dI/dV map (d) and corresponding Fourier transformation (e) showing the absence of Kekulé FT peaks around CNP at B = 8T. f, g, Real space dI/dV map (f) and corresponding Fourier transformation (g) taken at B = 8T that shows Kekulé order at VGate = −11V.
Extended Data Fig. 6 High-order graphene reciprocal lattice vector peaks from FT map at ν = −2.3.
a, Fourier transformation of the real space dI/dV map at VGate = − 9V and VBias = −2mV showing larger momentum range compared to Fig. 4a. b–e, Zoom-in of the Extended Data Fig. 6a around 2G1 (b), 3G1 (c), 2G2 (d), 3G2 (e) that are marked by the violet rectangles in Extended Data Fig. 6a. Position of NGi that is determined from the positions of G1 and G2 from Fig. 4b,c is plotted as yellow circles which matches well with the FT peaks.
Extended Data Fig. 7 42 nm by 42 nm size dI/dV map with Fourier transformation at ν = −2.3 showing sash features.
a,b, Real space dI/dV map (a) and corresponding Fourier transformation (b) taken at VBias = −2mV. White arrows point to the ‘sash’ features.
Extended Data Fig. 8 Lattice tripling order observed in sample B.
a, Real space dI/dV map taken at VGate = 15V and VBias = −4.8mV that includes seven moiré AAA sites. The area has twist angle of θ = 1.59° and heterostrain of ϵ = 0.28%. b, Kekulé auto-correlation map created from real space dI/dV map in Extended Data Fig. 8a. Neighbouring AAA sites are mapped with different colours, exhibiting the change in Kekulé patterns. c, VGate dependent dI/dV spectroscopy measured on sample B. d, VGate dependence of the Kekulé peak intensity in FT images normalized by the graphene lattice peak. Red (blue) dot corresponds to positive (negative) VBias, and is marked in Extended Data Fig. 8c. e, LDOS Landau fan diagram measured on sample B. Landau level degeneracies at each insulating dip is written in white numbers. f, g, Real space dI/dV map (f) and Fourier transformation (g) taken at VGate = −10V showing lattice tripling. h, i, Real space dI/dV map (h) and Fourier transformation (i) taken at VGate = 19V. Measurements are taken at T = 2K.
Extended Data Fig. 9 Lattice tripling order observed in sample C.
a, Real space dI/dV map on one moiré AAA site measured at VGate = −7.8V and VBias = 5mV which is at ν = −2. The area has twist angle of θ = 1.48° and heterostrain of ϵ = 0.08%. b, Fourier transformation of Extended Data Fig. 9a exhibiting prominent Kekulé FT peaks. c, VGate dependent dI/dV spectroscopy focusing on the correlated gaps at ν = −2 ~ −3. d, VGate dependence of the Kekulé peak intensity in FT images normalized by the graphene lattice peak. Measurements are taken at T = 400mK. In this sample, in the same area, we have previously established the presence of superconductivity13.
Extended Data Fig. 10 Nematic semimetal phase at charge neutrality.
a, Atomic resolution dI/dV map (background filtered) at an AAA site. b–d, Signatures of nematic seminetallic (NSM) ground state. The intensity of one bond is stronger than the other two bonds near the centre of the AAA site (c). The pattern slowly evolves into ‘A’ sublattice polarized state around the violet boxed region (b), while ‘B’ sublattice polarized state is dominant around the blue boxed region (d), which agrees with the prediction in ref. 25. e, (Unprocessed) dI/dV map at VGate = 0V (CNP) on the hole side (VBias = − 8mV) where a–d are taken. Other AAA sites in this map also show similar behaviour as a–d. This map and the map in Fig. 3 are taken at the same area, only having different gate and bias voltages. Scale bar : 10 nm.
Supplementary information
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This file contains experimental figures (Supplementary Figs. 1 and 2), supplementary text, four additional figures (Supplementary Figs. 3–6) that describe theoretical analysis and supplementary references.
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Kim, H., Choi, Y., Lantagne-Hurtubise, É. et al. Imaging inter-valley coherent order in magic-angle twisted trilayer graphene. Nature 623, 942–948 (2023). https://doi.org/10.1038/s41586-023-06663-8
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DOI: https://doi.org/10.1038/s41586-023-06663-8
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