Abstract
The identification and characterization of spontaneous symmetry breaking is central to our understanding of strongly correlated two-dimensional materials. In this work, we utilize the angle-resolved measurements of transport non-reciprocity to investigate spontaneous symmetry breaking in twisted trilayer graphene. By analysing the angular dependence of non-reciprocity in both longitudinal and transverse channels, we are able to identify the symmetry axis associated with the underlying electronic order. We report that a hysteretic rotation in the mirror axis can be induced by thermal cycles and a large current bias, supporting the spontaneous breaking of rotational symmetry. Moreover, the onset of non-reciprocity with decreasing temperature coincides with the emergence of orbital ferromagnetism. Combined with the angular dependence of the superconducting diode effect, our findings uncover a direct link between rotational and time-reversal symmetry breaking. These symmetry requirements point towards exchange-driven instabilities in momentum space as a possible origin for transport non-reciprocity in twisted trilayer graphene.
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Data presented in this work are attached. Source data are provided with this paper. Additional data are available from the corresponding author upon request.
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Acknowledgements
J.I.A.L. wishes to thank O. Vafek and D. Feldman for stimulating discussions. L.F. thanks the organizers of the workshop ‘Superconducting diode effects’ on Virtual Science Forum, which ignited the collaboration. This material is based on the work supported by the Air Force Office of Scientific Research under award no. FA9550-23-1-0482. N.J.Z. acknowledges support from the Jun-Qi fellowship and the Air Force Office of Scientific Research. J.-X.L. and J.I.A.L. acknowledge funding from NSF DMR-2143384. Device fabrication was performed in the Institute for Molecular and Nanoscale Innovation at Brown University. D.V.C. acknowledges financial support from the National High Magnetic Field Laboratory through a Dirac Fellowship, which is funded by the National Science Foundation (grant 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 no. JPMXP0112101001), and JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233). The work at the Massachusetts Institute of Technology was supported by a Simons Investigator Award from the Simons Foundation.
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N.J.Z. and J.I.A.L. conceived the project. N.J.Z. and Y.W. fabricated the device. N.J.Z., J.-X.L. and Y.W. performed the measurement. D.V.C. and L.F. provided theoretical inputs. K.W. and T.T. provided the material. N.J.Z., L.F. and J.I.A.L. wrote the manuscript.
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Extended data
Extended Data Fig. 1 Measurement setup.
(a-h) Schematic diagram showing the measurement configuration with current flowing is different azimuth directions ϕ across the ‘sunflower’ sample. For each ϕ, the longitudinal transport response is defined as the voltage difference across two contacts (each contact resembles a petal of the ‘sunflower’), which are aligned parallel to the direction of current flow. Panels (a) to (h) show measurement configurations for 8 azimuth directions. Along the same vein, the transverse response is measured across two contacts aligned perpendicular to the current flow direction, as shown in panel (i). We use V∥ (V⊥) to denote the voltage difference across two contacts that are parallel (perpendicular) to the current flow direction (Fig. 1a). Instead of applying current bias at the source contact and short to ground at the drain contact, we apply a positive current bias to the source contact, and a negative bias to the drain contact. This ensures that the center of the sample remains at zero electric potential, thus suppressing the potential influence of capacitive coupling and thermal-electric effects associated with the contact resistance.
Extended Data Fig. 2 Current-induced Hysteresis in nonreciprocity.
(a) The angular dependence of nonreciprocity measured at the same moiré band filling ν = − 0.3 and different current bias. At a small current bias, the angular dependence is best described by a one-fold symmetric cosine function. With increasing current, the angular dependence starts to evolves into a mixture between one- and three-fold at I > 200 nA. Further increasing the current bias gives rise to angular dependence at I = 400 nA that is best captured by \(\cos 3\phi\). (b) The angular dependence of nonreciprocity at ν = − 2.2 before (top panel) and after (bottom panel) the application of a large current bias. The large current bias induces a hysteretic rotation in the underlying mirror axis, which is marked by the green solid line in the polar coordinate plots. (c) \({R}_{\perp }^{2\omega }\), defined as \({V}_{\perp }^{2\omega }\)/I, as a function of current bias. As the current bias is swept back and forth, \({R}_{\perp }^{2\omega }\) exhibits a hysteresis loop. The angular dependence of \({V}_{\perp }^{2\omega }\) in panel (b) is measured before and after this hysteresis loop at a fixed current bias of I = 100 nA. Inset shows the schematic angular dependence for the data shown in panel (b).
Extended Data Fig. 3 Angle dependence of η∥ measured at different DC current bias.
Polar-coordinate plot of the angle dependence of η||, measured at B = 0, T = 20 mK and ν = 0.2. With increasing DC current bias, a similar angular dependence that is predominantly one-fold symmetric is observed up to IDC = 100 nA, all pointing towards around 140∘.
Extended Data Fig. 4 Nonreciprocity and nonlinearity.
(a) Current-voltage characteristic measured at ν = − 0.3 with DC current flowing along azimuth angle of ϕ = 0∘. The black dashed line is a linear fit to the portion of the IV curve near zero current bias. (b) The nonreciprocal component of the IV curve, which is extracted by subtracting the ohmic component of the transport response, η/2 = V∥ − IDCR0. R0 denotes the slope of the IV curve at IDC = 0 The solid line is a quadratic fit to the current dependence of η/2.
Extended Data Fig. 5 Second-harmonic nonlinear response and transport nonreciprocity.
Angle dependence measurement of (a) \({V}_{\parallel }^{2\omega }\) and (b) η|| at ν = 0.25 and (c) \({V}_{\parallel }^{2\omega }\) and (d) ηR at ν = 2.14. The remarkable match in the angle dependence between \({V}_{\parallel }^{2\omega }\) and (b) η|| at different densities illustrates their correspondence relation. \({V}_{\parallel }^{2\omega }\) are measured at IAC = 100 nA, ηR are measured at IDC = 100 nA. All measurements are performed at B = 0 and T = 20 mK.
Extended Data Fig. 6 Relative Root Mean Squared Error in the angular fit with large V1.
(a) The angular dependence of nonreciprocity measured at ν = 1.2. (b) Relative root-mean-squared error (RRMSE), defined according to Eq. M2, as a function of V1 (left panel), V3 (middle panel), and β (right panel). (c) The angular dependence of nonreciprocity measured at ν = 2.0. (d) Relative root-mean-squared error (RRMSE), defined according to Eq. M2, as a function of V1 (left panel), V3 (middle panel), and β (right panel).
Extended Data Fig. 7 Relative Root Mean Squared Error in the angular fit with large V3.
(a) The angular dependence of nonreciprocity measured at ν = 2.15. (b) Relative root-mean-squared error (RRMSE), defined according to Eq. M2, as a function of V1 (left panel), V3 (middle panel), and β (right panel).
Extended Data Fig. 8 The ‘sunflower’ model beyond angular fit.
(a) The angular dependence of longitudinal and transverse resistance, R∥ and R⊥, measured at ν = 2. (b) Measurements from different ‘sunflower’ configurations compared to the expected value extracted from a single conductivity matrix13,18. The RRMSE of the fit is 1.74%. (c) The angular dependence of longitudinal and transverse second-harmonic nonlinear transport response, 𝑉2𝜔∥ and 𝑉2𝜔⊥, measured with an AC current of 100 nA at ν = 2.
Extended Data Fig. 9 Angle-resolved nonreciprocity measurement with higher angular resolution.
(a) Schematic diagram of the ‘sunflower’-shaped sample with 10 petals. The increased number of electrical contacts enables higher angular resolution in the nonreciprocity measurement. (b-d) Angle-resolved nonreciprocity measured at different moiré band filling of the tTLG sample, which has a twist angle of θ = 1.34∘. The best angular fit for the angular dependence (black solid line) is captured by Eq. 1, which is extracted by minimizing the RRMSE of nonreciprocity from both longitudinal and transverse channels. With increased angular resolution, the best fit to the angular dependence of nonreciprocity in panel (b) features RRMSE of 0.3%, while other angular dependence shows RRMSE of less than 2%. Beyond the one-fold and three-fold components, the next lowest order angular component is described by \(\cos (5\phi )\) and \(\sin (5\phi )\). While an oscillatory period of 72∘ is well within the measurement resolution, such a component is not observed in our measurement. This is a strong indication that the angular components with N > 3 has small oscillatory amplitude. All measurement performed at T = 20 mK, B = 0, and IAC = 50 nA.
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Zhang, N.J., Lin, JX., Chichinadze, D.V. et al. Angle-resolved transport non-reciprocity and spontaneous symmetry breaking in twisted trilayer graphene. Nat. Mater. 23, 356–362 (2024). https://doi.org/10.1038/s41563-024-01809-z
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DOI: https://doi.org/10.1038/s41563-024-01809-z
