Research ArticleMATERIALS SCIENCE

Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle

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Science Advances  27 Sep 2019:
Vol. 5, no. 9, eaaw9770
DOI: 10.1126/sciadv.aaw9770

Abstract

The emergence of flat bands and correlated behaviors in “magic angle” twisted bilayer graphene (tBLG) has sparked tremendous interest, though its many aspects are under intense debate. Here we report observation of both superconductivity and the Mott-like insulating state in a tBLG device with a twist angle of ~0.93°, which is smaller than the magic angle by 15%. At an electron concentration of ±5 electrons/moiré unit cell, we observe a narrow resistance peak with an activation energy gap ~0.1 meV. This indicates additional correlated insulating state, and is consistent with theory predicting a high-energy flat band. At doping of ±12 electrons/moiré unit cell we observe resistance peaks arising from the Dirac points in the spectrum. Our results reveal that the “magic” range of tBLG is in fact larger than what is previously expected, and provide a wealth of new information to help decipher the strongly correlated phenomena observed in tBLG.

INTRODUCTION

Twistronics (17), the use of the relative twist angle between adjacent van der Waals layers to produce a moiré superlattice and flat bands, has emerged as a new and uniquely suitable approach to markedly alter and tailor the properties of devices based on two-dimensional materials. The marked effect of twistronics is exemplified by the recent groundbreaking works that demonstrated the emergence of extremely flat bands when two monolayer graphene layers are stacked at a “magic” twist angle, θ = 1.1 ± 0.1° (8, 9). In these twisted bilayer graphene (tBLG) devices at the magic angle, an insulating phase is observed at half filling of the superlattice’s first miniband, identified to be a Mott-like insulator, and superconductivity at slightly higher and lower doping. The phase diagram is reminiscent of high-temperature superconductors (10) for the high ratio between the superconducting transition temperature Tc to the Fermi temperature TF, the relatively small Fermi surface, and the vicinity of the superconducting phase to an insulating state with an apparent magnetic ordering (9). These works have immediately sparked tremendous interest and ignited intense theoretical debate on nearly every aspect of this system, including low-energy band structure, band topology, irreducible symmetries, nature of the correlated insulating state, pairing mechanism and symmetry of the superconducting phase, the exact phase diagram, and additional magic angles (1135). In contrast, apart from the initial reports, experimental studies are scarce and just starting to emerge (3539).

Here, we report transport measurements of a magic angle tBLG device that exhibits both correlated insulating and superconducting states. Unexpectedly, the twist angle is measured to be 0.93° ± 0.01°, which is 15% smaller than the already established magic angles (79) and is the smallest reported to date that exhibits superconductivity. A correlated insulating state is observed at half filling nm = 2, where nm is the number of charges per moiré unit cell, and superconductivity at nm ≈ 2.7. Extending measurements to carrier densities |nm| > 4, we observe previously unreported, narrow resistance peaks at nm = ±5, each of which displays an activation gap of ~0.1 meV. This behavior is consistent with the emergence of additional correlated insulating states when the next electron or hole band beyond the low-energy moiré Dirac bands is quarterly filled. Theoretical calculations indicate that these two high-energy bands have energetically flat regions in the moiré Brillouin zone with substantially large densities of states (DOS). Prominent resistance peaks also appear at nm = ±12, which can be accounted for by the existence of a pair of Dirac points in the high-energy spectrum. Our results indicate that new correlated states can emerge in tBLG below the primary magic angle and beyond the first miniband, provided that the bands are sufficiently flat.

RESULTS AND DISCUSSION

Devices are fabricated using the “tear and stack” approach, encapsulated between hexagonal BN layers (8, 9), patterned into a Hall bar geometry with multiple leads, and coupled to Cr/Au edge contacts (40). The entire device is fabricated on top of a graphite layer that serves as the back gate [e.g., see (41)]. Figure 1 (A and B) shows a schematic diagram of the layer stack and the moiré superlattice, respectively. The devices are then measured in pumped He4 and He3 cryostats using standard dc and ac lock-in techniques.

Fig. 1 Device geometry and magneto-transport data.

(A) Schematic diagram of device geometry. (B) Schematic diagram of moiré superlattice formed by the twisted graphene layers. (C) Rxx versus magnetic field B and gate voltage Vg showing a Landau fan pattern. The top axis labels nm, the number of charges per superlattice cell. (D). Rxx(Vg) at different temperatures. Inset: Optical image of a tBLG device with a scale bar of 10 μm.

Figure 1C displays the device’s longitudinal resistance Rxx versus an extended gate voltage Vg range and magnetic field B at a temperature T = 1.7 K, while the inset to Fig. 1D shows an optical image of the device. The main resistance peak at Vg = 0 corresponds to the charge neutrality point with nm = 0. From the Landau fan emanating from Vg = B = 0, we find the back gate capacitance to be Cb = 374 nF/cm2. Notably, two additional prominent peaks appear at Vg = 0.83 V and Vg = −0.87 V, each accompanied by a set of Landau fans. The small electron-hole asymmetry may be intrinsic to tBLG and has been observed in previous reports (8, 9). We therefore take the features at Vg = ±0.85 V as the satellite peaks that occur when the low-energy moiré bands are filled at densities nm = ±4 (8, 9). Covering an unprecedentedly large range of carrier density up to nm = ±14, the data exhibit Landau fans that converge to nm = 4m, where m is an integer. Given the spin and valley degeneracies in graphene, this is consistent with the Wannier theory (42) that generally predicts that spectral gaps in the Hofstadter butterfly for spinless electrons in a single band follow the Diophantine equation nm = tϕ/ϕ0 + s, where s and t are integers, ϕ is the flux per moiré unit cell, and ϕ0 is the flux quantum. Using 4n08θ23a2, where a = 0.246 nm is the lattice constant of graphene and n0 is the average density corresponding to nm = 1, we estimate that θ = 0.93°, which corresponds to a moiré unit cell area A of 200 nm2 and moiré lattice constant of 15.2 nm. We note that this is the smallest twist angle value reported to date for tBLG devices exhibiting superconductivity.

A close examination of the Landau fan in Fig. 1C reveals a number of salient features. We first focus on the low- to moderate-density regime, where |nm| < 4. At Vg = 0.43 V or nm = +2, a resistance peak appears, from which an accompanying set of Landau levels emanates, with a degeneracy of two. This peak at half filling and the twofold degeneracy of Landau levels is consistent with the previous observation of a Mott-like correlated insulating state (8). They indicate the breaking of approximate spin-valley SU(4) symmetry and the formation of a new quasi-particle Fermi surface, although the details require more delicate examination. Figure 1D displays Rxx(Vg) at B = 0 and T ranging from 5.2 K down to 0.28 K, where the resistance peaks at nm = 0, ±4, and +2 are visible. At T = 280 mK, for 0.51 < Vg < 0.65 or, equivalently, 2.4 < nm < 3.1, Rxx is zero within our measurement error, indicating the emergence of superconductivity (9). This emergence of superconductivity and possibly percolating superconducting regions may be responsible for the increasing Rxx with increasing T at nm = 2, similar to that observed in (9).

To determine the critical temperature Tc of the superconducting phase, we measure ρ(T) at Vg = 0.53 V or nm ≈ 2.5 (Fig. 2A). As T decreases, ρ drops to zero, undergoing two successive steep descents at T ~ 1.5 and T ~ 0.3 K. Such a two-step transition has been observed in other magic angle tBLG device (36) and may be related to non-Planckian dissipation of the strange metal state. Alternatively, it may arise from spatial or structural inhomogeneity of our device or from the presence of domains that host competing superconducting states of different pairing symmetries and critical temperatures. We note that these data were not taken at optimal doping for superconductivity in this sample, and Tc could be as high as ~0.5 K. However, the device became nonfunctional before clear data could be obtained.

Fig. 2 Data from the superconducting state.

(A) ρ versus temperature when the density is tuned to the superconducting phase (Vg ~ 0.53 V or nm ~ 2.5). (B) Differential resistance dV/dI versus bias current and gate in the superconducting phase at base temperature (280 mK). Color scale is in units of kilohms. (C) Voltage-current characteristics at T = 280 mK and Vg = 0.50 V (blue) and 0.58 V (red), respectively. (D) V-I curves at different parallel magnetic fields.

To further investigate the superconducting phase, we measure the device’s four-terminal voltage-current (V-I) characteristics at different carrier densities. Figure 2B displays resistance, which is obtained by numerically differentiating the V-I curves, as a function of Vg and bias current I. V-I characteristics at two representative densities, Vg = 0.50 V and V = 0.58 V, are shown in Fig. 2C. Supercurrent is observed for an extended range of density. with critical current Ic ranging from ~ 1 to 15 nA; at Vg ~ 0.58 V, the maximum value of Ic is observed. We therefore take Vg ~ 0.58 V (or nm ~ 2.7) to be the optimal doping. Upon application of a parallel magnetic field H, the supercurrent is suppressed, with a critical field of H||c ~ 0.5 T (Fig. 2D), consistent with previous work (9).

To gain insight into this behavior, we calculate the moiré band structure for the 0.93° tBLG using the Bistritzer-MacDonald model (7) with refined parameter values that take into account lattice relaxation: tAA = 79.7 meV, tAB = 97.5 meV, and vF = 7.98 × 105 m/s (43). Here, tAA and tAB are the electron interlayer tunneling amplitudes between the same and different sublattices, respectively, and vF is the Fermi velocity. In sharp contrast to those cases near and above the magic angle, our calculation of the 0.93° tBLG shows that the low-energy moiré Dirac bands (|nm| < 4) are not energetically isolated from the high-energy bands. While the single-particle superlattice gaps vanish at the complete filling of the low-energy moiré Dirac bands, new Dirac points exist at Γs, as shown in Fig. 3A. This explains why our Rxx peaks at nm = ±4 are narrower compared to a previous report (8) and comparable to the one at the charge neutrality. The fact that the Rxx peak is relatively stronger at nm = 4, together with the fact that superconductivity emerges only at the electron side, suggests that the electron-hole asymmetry is substantially enhanced by the electron-electron interactions. The emergence of both the Mott-like correlated insulating state at half-filling and the superconductivity at slightly higher doping indicates that, although the twist angle of 0.93° is ~15% smaller than the magic angle (79), the device hosts strongly correlated physics. This unexpected yet desirable behavior can also be understood by the calculated moiré bands and DOS of the 0.93° tBLG in Fig. 3. The low-energy moiré Dirac bands (|nm| < 4) are bounded by the aforementioned multiple high-energy Dirac points near −4.68 and 5.96 meV. The narrow bandwidth (~11 meV), comparable to that of the magic angle tBLG (8, 9) and much smaller than the Coulomb interaction strength, produces a sharp DOS peak for |nm| < 4.

Fig. 3 Calculations of electronic band structures of 0.93° tBLG.

(A) Energy dispersion. (B) DOS. In obtaining the DOS from the band structure, 1 meV was used for the energy interval, and the spin-valley degeneracy was considered.

We now turn to the behavior at large density (|nm| > 4). At the lowest temperature, additional narrow resistance peaks are observed at nm = 5.08 and −5.03 (Fig. 1C). Figure 4A shows a zoomed-in plot of the data for the nm = 5 peaks. The peaks are almost indiscernible as T is increased above ~5 K. Plotting the resistance on an Arrhenius plot as shown in Fig. 4B yields an energy gap of ~1 K. These resistance peaks at nm = ±5 have not been previously reported. Conceivably, it may arise from the presence of another domain with a slightly larger angle. Domains are likely to be present (35, 38, 39). However, the sharpness of the peaks and their small energy scale are very different from the behavior expected for a superlattice gap (8, 9), which are much broader and show very little low-temperature dependence. Thus, the peaks at nm ≈ ±5 are unlikely to originate from angular disorder. They are also unlikely to arise from a single-particle gap due to an alignment between hBN and graphene, because such a gap is expected to be ~100 K in magnitude (44, 45). We thus tentatively attribute these features to the emergence of a new correlated insulating state when the lowest (highest) high-energy conduction (valence) band is quarterly filled with electrons (holes). Evidently, in Fig. 3, our calculation reveals that these two bands are nearly flat in a large region of the moiré Brillouin zone, and that the corresponding DOS peaks are substantially large. From a background-subtracted Rxx(Vg, B) plot, several Landau levels can be observed emanating from the two peaks and disperse toward the larger density sides, with degeneracy estimated to be 10 ± 2, where the relatively large error bar of ±2 arises from the limited range in magnetic field at which the features extrapolate to nm = ±5 and B = 0. Hence, within error bars, the degeneracy measured is larger than the band degeneracy of 4. This unusual feature suggests that the fermionic quasi-particles not only respect the approximate spin-valley SU(4) symmetry but also might even enjoy an emergent new symmetry. More delicate future studies are required to examine the symmetries of these exotic insulating states and to determine whether or not they are quantum spin liquids.

Fig. 4 Behavior of resistance peak near density nm = 5.

(A) Temperature dependence of the resistance peak. (B) Arrhenius plot of resistance showing a gap of ~0.1 meV.

Another unusual feature of the device behavior at high density is the presence of resistance peaks at nm = ±12, where Landau fans with degeneracy of 4 emanate on both sides of each density. In addition, their maximum resistivities yield minimum conductivities ~ 20e2/h. These features are reminiscent of the presence of Dirac points at the charge neutrality. Our moiré band structure calculation of the 0.93° tBLG reveals that a pair of Dirac points appears at Ks and Ks near nm = ±12 (Fig. 3). Consistently, the DOS calculation also shows local minima at the corresponding energies (Fig. 3).

CONCLUSION

In summary, in a twisted bilayer device at a small twist angle of 0.93°, superconductivity is observed near a Mott-like insulating state, with a critical temperature of 0.3 to 0.5 K. A gap is observed at a filling of ±5 electrons per moiré unit cell. Theoretical calculations predict no band gap at this filling, indicating correlated insulating behavior for the quarter-filled high-energy band. Dirac points at a filling of ±12 electrons per moiré unit cell lead to conductivity minima. Our work shows that electron correlations can markedly influence the properties of moiré superlattices even at small angles and high densities. Future work will focus on the spin-valley ordering of the insulating phases and investigations at lower temperatures to search for new superconducting phases, as well as theoretical efforts to understand the origin of these behaviors.

MATERIALS AND METHODS

Monolayer graphene, 5 to 10 nm of graphite, and 10 to 30 nm of hBN flakes were mechanically exfoliated onto SiO2/Si wafers. Clean and homogeneously flat flakes were carefully identified. A dry transfer technique using PPC (polypropylene carbonate) on a PDMS (polydimethylsiloxane) point-contact stamp was used to transfer the flakes (40). A tear-and-stack method was used to twist the graphene layers relative to each other (2). After assembling the flakes, the stack was dropped onto a clean SiO2/Si chip at 70° to 90°C to allow bubbles to flow away during the final lamination. Last, the temperature was raised to 120°C to melt the PPC onto the chip, which was then dissolved with acetone.

The stack was then shaped into a hall bar (channel width, ~1 μm; leads separation, ~1.5 μm) using electron beam lithography and reactive ion etching (RIE) with SF6 and O2 gases. One-dimensional electrical contact was made by patterning leads with electron beam lithography and depositing 2/50 nm of Cr/Au with electron beam deposition (40). A short O2 RIE etch was done before deposition to improve contact (40).

Measurements were performed in pumped He4 and He3 refrigerators. Data were collected via standard ac and dc measurement techniques using SR830 lock-in amplifiers, Keithley 2400 source meters, and SR560 preamplifiers.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaaw9770/DC1

A. Theory of 0.93° Twisted Bilayer Graphene

B. Supplementary Data

Fig. S1. Extended electronic structures of the 0.93° tBLG.

Fig. S2. DOS versus density (n) for the 0.93° tBLG in fig. S1.

Fig. S3. Moiré band structures of the 0.93° tBLG in other models.

Fig. S4. Superconductivity response to magnetic field.

Fig. S5. Temperature dependence of the nm = −5 resistance peak.

Fig. S6. Landau fan for nm = +5.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: F.Z. is grateful to F. Wu, A. Po, and F. Yang for valuable discussions. We thank the groups of J. Hone and C. Dean for experimental advice on device fabrication. Funding: The experiments are supported by the DOE BES Division under grant no. ER 46940-DE-SC0010597. S.C. acknowledges partial support from the Center for Emergent Materials: an NSF MRSEC under award number DMR-1420451 for device fabrication in NSL. The theoretical works (Q.W. and F.Z.) are supported by the Army Research Office under grant number W911NF-18-1-0416. Growth of hBN crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and a Grant-in-Aid for Scientific Research on Innovative Areas “Science of Atomic Layers” from JSPS. Author contributions: M.B. and C.N.L. conceived the experiment. T.T. and K.W. synthesized the hBN crystals. E.C. fabricated the samples. R.K assisted with sample preparation. E.C. and S.C. performed the transport measurements. H.T., R.L., and S.T. assisted with fabrication and measurements. Q.W. and F.Z. performed the theoretical calculations. E.C., Q.W., S.C., F.Z., M.B., and C.N.L. analyzed and interpreted the data. E.C., F.Z., M.B., and C.N.L. wrote the manuscript. All authors discussed and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. The devices can be provided by Lau and Bockrath groups pending scientific review and a completed material transfer agreement and subject to availability. Requests for devices should be submitted to lau.232{at}osu.edu or bockrath.31{at}osu.edu.
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