Abstract
Magicangle twisted bilayer graphene exhibits intriguing quantum phase transitions triggered by enhanced electron–electron interactions when its flat bands are partially filled. However, the phases themselves and their connection to the putative nontrivial topology of the flat bands are largely unexplored. Here we report transport measurements revealing a succession of dopinginduced Lifshitz transitions that are accompanied by van Hove singularities, which facilitate the emergence of correlationinduced gaps and topologically nontrivial subbands. In the presence of a magnetic field, wellquantized Hall plateaus at a filling of 1,2,3 carriers per moiré cell reveal the subband topology and signal the emergence of Chern insulators with Chern numbers, C = 3,2,1, respectively. Surprisingly, for magnetic fields exceeding 5 T we observe a van Hove singularity at a filling of 3.5, suggesting the possibility of a fractional Chern insulator. This van Hove singularity is accompanied by a crossover from lowtemperature metallic, to hightemperature insulating behaviour, characteristic of entropically driven Pomeranchuklike transitions.
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Data availability
The data that support the findings of this work are available from the corresponding author upon reasonable request.
Change history
01 June 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41563021009972
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Acknowledgements
We thank M. Xie, A. H. MacDonald, M. Gershenson, S. Kivelson, A. Chbukov, G. Goldstein, G. Kotliar, T. Senthil and A. Bernevig for useful discussions and J. Liu for insightful comments and discussions. Support from DOEFG0299ER45742 and from the Gordon and Betty Moore Foundation GBMF9453 is gratefully acknowledged.
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S.W., Z.Z. and E.Y.A. conceived and designed the experiment, carried out lowtemperature transport measurements and analysed the data. S.W. and Z.Z. fabricated the twisted bilayer graphene devices. K.W. and T.T. synthesized the hBN crystals. S.W., Z.Z. and E.Y.A. wrote the manuscript.
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Extended data
Extended Data Fig. 1 Moiré pattern, Brillouin zone, and band structure of MATBG.
a, Left panel: a moiré pattern with periodicity L_{M} forms by introducing a twist angle θ between the crystallographic axes for two superposed graphene layers. Right panel: The moiré miniBrillouin zone consists of two hexagons (gray and yellow) constructed from \(K_s = \left {\overrightarrow {K_t}  \overrightarrow {K_b} } \right\) and \(K_s^\prime = \left {\overrightarrow {K_t^\prime }  \overrightarrow {K_b^\prime } } \right\) where \(\overrightarrow {K_{t,b}}\), \(\overrightarrow {K_{t,b}^\prime }\) are the wavevectors corresponding to the Brillouin zones corners of the top (t) and bottom (b) graphene layers. b, Schematic lowenergy band structure of MATBG shows the flat bands near the CNP and the gaps separating them from the remote bands.
Extended Data Fig. 2 Signature of superconductivity near n/n_{0}=−2.
a, R(T) measurement mapping the SC dome versus doping. b, R(T) at n/n_{0}=−2.4. The critical temperature, T_{c}=3.5K is marked. The driving current is 10 nA. Inset, Optical micrograph of the device (the browncolored Hall bar is edge contacted (yellow) with metal electrodes (black)) c, Comparison of IV curves at n/n_{0}=−2.4 at zero and finite magnetic field shows suppression of critical current from 12 nA down to zero. d, R(B) curve measured with a 10 nA current, indicates a critical field of B_{c}~0.02T. Signatures of superconductivity (SC) emerge in this device below 5 K. The R(T) measurement mapping the SC dome versus doping is shown in Extended Data Fig. 2a. Here we focus on the moiréfilling range n/n_{0}=−2.4. In Extended Data Fig. 2b, a sharp resistance drop from 10 kΩ to 200Ω is observed within the temperature range 8K0.8 K. The resistance then remains nearly constant from 0.8 K to 0.3 K. The temperature at which the resistance drops to half its value in the normal state, defined here as the critical temperature, is T_{c}~3.5K. In Extended Data Fig. 2c the nonlinear currentvoltage (IV) characteristics in zero field, indicates a critical current of ~ 12 nA. In the presence of a 0.2 T magnetic field at T=0.3K Superconductivity is suppressed as shown by the linear IV. The finite resistance value (~200Ω) at base temperature (0.3 K) and the voltage fluctuations were cause by a nonideal voltage leads which often affect fourterminal measurements of microscopic superconducting samples as reported previously^{39,40}. From Extended Data Fig. 2d the critical field is estimated at B_{c}~0.02T.
Extended Data Fig. 3 Evolution of quantum Hall plateaus in a magnetic field.
a, \(\sigma _{xy}(n/n_0)\) at 1.5 T displays quantum Hall plateaus, \(\sigma _{xy} = \nu \frac{{e^2}}{h}\), and concomitant minima in \(\sigma _{xx}(n/n_0)\) (gray bars) near the CNP. The Landau sequence \(\nu = \pm 4, \pm 8\) indicates the 4fold degeneracy. bc, Same as panel (a) at 4.5 T and 7 T shows a new sequence with \(\nu = \pm 2, \pm 4\) and concomitant minima, indicating that either spin or valley degeneracy is lifted by the field. d, Filling \(R_{xy}(B)\) shows the emergence of Cherninsulators in the higher order branches. All data are taken at T=0.3 K.
Extended Data Fig. 4 Calculating the Hall density from the Hall resistance measurements.
a, Doping dependence of the Hall density, \(n_H =  B/(eR_{xy})\), at B=0.8T b, Linear fits of \(R_{xy}(B)\) at fixed moiré fillings, n/n_{0}, as indicated in the legend. c, Filling dependence of \(dR_{xy}/dB\) obtained by fitting R_{xy}(B) curves as illustrated in panel (b). Linear fits of \(R_{xy}(B)\) at fixed moiré fillings, n/n_{0}, as indicated in the legend. The doping dependence of the Hall density, \(n_H =  B/(eR_{xy})\) is shown in Extended Data Fig. 4a. To better resolve its intrinsic features we use the slope \(dR_{xy}/dB\) of obtained from a linear fit of the measured \(R_{xy}(B)\) curves at fixed n/n_{0}, as shown in Extended Data Fig. 4b,c. This was used to obtain the doping dependence of \(n_H =  (1/e)(dR_{xy}/dB)^{  1}\) shown in Fig. 2b of the main text.
Extended Data Fig. 5 Estimating ω_{c}τ.
\(\left {\rho _{xy}/\rho _{xx}} \right\)=\(\omega _c\tau\) as a function of n/n_{0} at several fields as marked. The expression for the logarithmic divergence of n_{H} near a VHS is valid in the lowfield limit^{22} \(\omega _c\tau \ll 1\) where \(\omega _c = \frac{{eB}}{m}\) is the cyclotron frequency and τ the scattering time. To ensure the validity of the fit in the main text we estimated the value of \(\omega _c\tau\) as a function of density and field. Within the Drude model, \(\rho _{xx} = \frac{m}{{ne^2\tau }}\), \(\rho _{xy} = \frac{B}{{ne}}\), we estimate \(\omega _c\tau\) =\(\left {\rho _{xy}/\rho _{xx}} \right\) shown in Extended Data Fig. 5. Clearly all the data taken near the putative VHSs is in the low field limit.
Extended Data Fig. 6 Divergent Hall density and VHS near n/n_{0}=3.5.
a, \(\omega _c\tau\) around n/n_{0}=3.5 obtained from \(\left {\rho _{xy}/\rho _{xx}} \right\) at several Bfields shows that the low field limit \(\omega _c\tau \ll 1\) is valid in this regime. b, Evolution of Hall density with field near n/n_{0}=3.5. The divergent n_{H} behavior is clearly resolved after the gap opens on the s = 3 branch with \(n_H = n  3n_0\). c, Hall density around n/n_{0}=3.5 fits the logarithmic divergence (solid blue line) expected for a VHS as discussed in the main text. With the gap opening at n/n_{0}=3, we observe a divergent dependence of n_{H} on carrier density at n/n_{0}=3.5. Based on the estimate of \(\omega _c\tau \ll 1\) around n/n_{0}=3.5 (Extended Data Fig. 6a), the expression for the logarithmic divergence of VHS in the lowfield limit, \(n_H \approx \alpha + \beta \left {n  n_c} \right\ln \left {(n  n_c)/n_0} \right\) described in the main text was used in fitting the density dependence of n_{H} in this regime, as shown in Extended Data Fig. 6c.
Extended Data Fig. 7 Fieldinduced gaps and Chern insulators.
ab, Field and density dependence of \(d^2R_{xx}/dn^2\), at T=0.3K reveals the emergence of halfLandau fans on the s = 2,3 branches (a) and on the s = 1, 2 branches (b)marked by black lines.
Extended Data Fig. 8 Quantized Hall resistance R_{xy}= h/e^{2} in the s = −3 branch.
a, Hall resistance in the s = −3 branch, saturates at a quantized value, R_{xy}= h/e^{2} indicating the emergence of a C = −1 Cherninsulator. The corresponding R_{xx} curves are shown in dashed lines. b, R_{xy} at fixed carrier density (\(n =  3.28n_0\)) as a function of magnetic field shows the onset of the emergent Cherninsulator at ~6.5 T. In the s = −3 branch, quantized \(R_{xy} = h/e^2\) is observed for fields above 7.2 T indicating the emergence of a C = −1 Cherninsulator as shown in Extended Data Fig. 8.
Extended Data Fig. 9 Thermal activation gaps at integer fillings and in finite magnetic fields.
a, Evolution of the resistance peak around n/n_{0}=2 with temperature measured at B=0T. The subtracted background is marked as dashed line. The net resistance (R^{*}) is marked by a red line with arrows. b, Temperature dependence of resistance at n/n_{0}=2 with and without subtraction of the background. c, Arrhenius fits of the temperature dependence of R^{*} are used to calculate the thermal activation gaps at integer fillings at B=0T: ∆_{0}=7.38±0.08 meV, ∆_{2}=2.5±0.15 meV, and ∆_{−2}=1.7±0.15cmeV, for n/n_{0}=0,2,−2 moiré fillings, respectively. The deviation of R^{*} at low temperatures from the exponential divergence expected for activated transport is attributed to variable range hopping^{42}, which is most pronounced at n/n_{0}=0 where the carrier density is lowest. d, Evolution of temperature dependence of the net resistance, R^{*}, with inplane field amplitude, B_{}, at n/n_{0} = 3. It is noted that spectroscopic gaps (for example at \(\left {n/n_0} \right = 2\)) obtained from local measurements such as STS or electronic compressibility, 7.5 meV^{12}; 48 meV^{14}; 3.9 meV^{41}; are larger than thermally activated gaps obtained from in transport, 0.31 meV^{7}; 1.5 meV^{39}; 0.37 meV^{10}. Such discrepancies are expected when comparing local to global probe measurements, because in the presence of gap inhomogeneity the latter are necessarily dominated by the smallest gaps^{41}. In MATBG, linear Tresistivity behavior is unique and prominent thus playing a crucial role in carrier resistivity. As shown in SI, the longitudinal resistance has a linear in temperature background that is observed at all temperatures and fillings. It is found that with decreasing temperature from 60 K, even though the overall resistivity decreases linearly, the resistive humps at integer fillings start showing up which is consistent with previous reports^{7,10}. This indicates the coexistence of the Tlinear behavior and onset of the correlation gap. At high temperature, phononscattered (thermally excited) carriers would short out the correlation gap. In order to access the thermally activated part of the resistivity, R*, we subtract the linear in T background. The thermal activation gap, ∆, is estimated by fitting to the Arrhenius dependence, \(R^ \ast \sim {\mathrm{exp}}(  {{\Delta }}/2k_BT)\), in the temperature range 7K15K, where k_{B} is Boltzmann’s constant. Extended Data Fig. 9a,b show the results at n/n_{0}=2 with and without background subtraction where the value of the calculated gap is ∆_{2}=2.5meV and 0.1 meV, respectively. The value obtained after background subtraction, 2.5 meV, matches the temperature range where the resistance peak starts showing up, and is comparable to the energy scale of the spectroscopic gaps. In addition, the opening of the correlation gap at \(\left {n/n_0} \right = 3\) with increasing magnetic field is also supported by the Hall density measurements. The linear Zeeman splitting relation is consistent with the spin response of the correlated states at integer fillings.
Extended Data Fig. 10 Evolution with inplane field amplitude of R^{*}(T) at n/n_{0}=3.5.
a, Doping dependence of resistance at several inplane fields at T=0.3K. b, Evolution of R^{*}(T) at n/n_{0}=3.5 with inplane field amplitude, B_{}. The miniband created by the gap opening for B>4T at moiré filling n/n_{0}=3 leads to a divergent Hall density at moiré filling n/n_{0}=3.5 that reflects the emergence of a VHS, as discussed in the main text. Another signature of this VHS is the appearance of a peak in the net resistance R^{*}. The initially positive slope of R^{*}(T) at low temperatures, indicating metallic behavior, steadily decreases with increasing field, and becomes slightly insulating at 8T (Extended Data Fig. 10).
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Wu, S., Zhang, Z., Watanabe, K. et al. Chern insulators, van Hove singularities and topological flat bands in magicangle twisted bilayer graphene. Nat. Mater. 20, 488–494 (2021). https://doi.org/10.1038/s41563020009112
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DOI: https://doi.org/10.1038/s41563020009112
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