Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle

Significance

Multiple flat bands in high-quality twisted bilayer graphene close to the theoretically predicted second magic angle are observed. These well-isolated flat moiré bands host a nontrivial topology which is evidenced by a connecting multiband Hofstadter butterfly spectrum. This work provides a perspective for understanding the emergent quantum phases (i.e., strong correlation and band topology) in twisted bilayer graphene and the fractal Hofstadter spectra of multiple topological bands.

Abstract

Moiré superlattices in two-dimensional van der Waals heterostructures provide an efficient way to engineer electron band properties. The recent discovery of exotic quantum phases and their interplay in twisted bilayer graphene (tBLG) has made this moiré system one of the most renowned condensed matter platforms. So far studies of tBLG have been mostly focused on the lowest two flat moiré bands at the first magic angle θ_m1 ∼ 1.1°, leaving high-order moiré bands and magic angles largely unexplored. Here we report an observation of multiple well-isolated flat moiré bands in tBLG close to the second magic angle θ_m2 ∼ 0.5°, which cannot be explained without considering electron–election interactions. With high magnetic field magnetotransport measurements we further reveal an energetically unbound Hofstadter butterfly spectrum in which continuously extended quantized Landau level gaps cross all trivial band gaps. The connected Hofstadter butterfly strongly evidences the topologically nontrivial textures of the multiple moiré bands. Overall, our work provides a perspective for understanding the quantum phases in tBLG and the fractal Hofstadter spectra of multiple topological bands.

Twisted bilayer graphene (tBLG) has emerged as a rich platform to study strong correlations (1), superconductivity (2–8), magnetism (9), and band topology (10). Stacking two graphene sheets with a slight rotation by an angle θ creates a moiré superpotential. The renormalized bands have C_6z, C_2x rotational symmetries and time-reversal symmetry T. This leads to decoupled valleys (11) and the formation of a mini moiré Brillouin zone (Fig. 1A) with two Dirac cones with identical helicity (Fig. 1B), which makes the resulting two lowest moiré bands at each valley topologically different from the two $\text{π}$ bands in monolayer graphene in which the total helicity vanishes. They carry a nonzero Dirac helicity $\eta =\pm 2$ (Fig. 1B), which is protected by C_2zT symmetry. These attributes prohibit the construction of local symmetric Wannier functions and define the nontrivial topology of tBLG (12–15).

Fig. 1.

Multiple moiré bands with nonzero helicity in θ = 0.45° tBLG. (A) Mini moiré Brillouin zones are defined by two Dirac cones with a total Dirac helicity $\eta =\pm 2$. (B) Nonzero total Dirac helicity in the lowest two moiré bands inside of one valley. (C) Hartree–Fock band structure calculations of 0.45° tBLG with Coulomb interaction show multiple moiré bands, which are separated at the Fermi energy E_F. Here the high symmetry points of the Brillouin zone are K_M and Γ_M. (D) Schematic of the hBN encapsulated θ = 0.45° tBLG device with a graphite bottom gate. (E) Color plot of the longitudinal resistance R_xx vs. carrier density n and out-of-plane magnetic field B_⊥ (T = 1.5 K). (F) R_xx vs. n at different temperatures and B_⊥ = 300 mT. (Inset) The atomic force microscopy image of a typical tBLG device. (G) Hall carrier density n_H = −B_⊥/(eR_xy) as a function of gate-induced carrier density n measured at B_⊥ = 300 mT and T = 1.5 K.

The magic angles in tBLG are a series of well-defined twist angles for which the moiré bands are predicted to become ultraflat (11). While the tBLG at θ_m1 ∼ 1.1° is already extensively studied, tBLG at well-defined smaller magic angles (i.e., θ_m2 ∼ 0.5°), in which inhomogeneity of moiré patterns is much more sensitive to twist-angle fluctuations (16), has so far rarely been experimentally investigated. Since tBLG at θ_m2 offers an exceedingly large moiré unit cell, it gives rise to a multitude of closely packed flat moiré bands. Crucially, the moiré wavelength for tBLG at θ_m2 of λ ∼ 30 nm is much larger than for θ_m1 tBLG or for graphene/hexagonal boron nitride (hBN) moiré superlattices, and hence inducing one magnetic flux per unit cell, required to obtain a Hofstadter butterfly, can be easily achieved at much lower fields (approximately six times) (17–19). Together these characteristics make tBLG at θ_m2 an ideal platform to explore multiple moiré bands and their Hofstadter butterfly spectra which are rendered with interactions and novel band topology.

Results

In this article we present magnetotransport measurements of tBLG devices, with θ close to the predicted second magic angle θ_m2 ∼ 0.5°. Our tBLG device is encapsulated with crystallographically nonaligned insulating layers of hBN and its carrier concentration n is capacitively controlled by an underlying graphite layer (Fig. 1D) and calibrated with quantum oscillations in out-of-plane magnetic field B_⊥ (SI Appendix). Fig. 1 F and G show the evolution of the longitudinal resistance R_xx and the Hall density n_H as a function of n. R_xx exhibits peaks and n_H sign changes at equally spaced, integer multiples of $n_s,$ $n=s{n}_s$, and marks the transitions between the individual bands. (We attribute the Hall density sign changes at noninteger multiples of $n_s$ that are not accompanied by resistance maxima to van Hove singularities within the bands.) The resistance peaks are strongly enhanced by a small B_⊥-field (Fig. 1E) and develop thermally activated, gapped behavior (SI Appendix, Figs. S2–S4) but are insensitive to parallel B-field (SI Appendix, Fig. S5). This suggests an orbital origin and agrees with the fact that small B_⊥-fields can gap out the C_2zT protected Dirac nodes or increase a small band gap.

The noninteracting band structure of tBLG around θ_m2 contains eight strongly convoluted 4-fold spin-valley degenerate low-energy flat bands (SI Appendix, Fig. S10B). To account for interaction effects in the flat bands, we perform Hartree–Fock calculations including Coulomb interactions with random seeds, which allow for spontaneous symmetry breaking (SI Appendix, section 5). These show that Coulomb interactions spontaneously break C_3z, while preserving the C_2zT symmetry. As a direct consequence, the moiré bands separate above and below the Fermi energy E_F and become almost nonoverlapping except for point-like connections.

This is seen in Fig. 1C (for E_F at the charge neutrality point [CNP], $s=0$) and in SI Appendix, Fig. S10A (for E_F at integer band fillings $s=1,2,3,4$), which show the Hartree–Fock moiré bands in one valley of θ = 0.45° tBLG. Here, each separate band remains 4-fold degenerate and can intake $n_s=4/{\text{Ω}}_m$ carriers, where ${\text{Ω}}_m$ is the area of the moiré unit cell. The carrier density can be expressed by integer multiples of band fillings $s{n}_s$, which define the integer band filling factor $s\in \mathrm{\mathbb{Z}}$. For $s=0,\pm 1,\pm 3,\pm 4$, the band structure is semimetallic with two Dirac points per spin per valley at E_F, while for filling $s=\pm 2$ a small indirect gap occurs. In particular, we find that the helical Dirac nodes at the CNP, which are responsible for the nontrivial topology, are quite robust for both interacting and noninteracting calculations, although they may deviate from high symmetry points due to C_3z symmetry breaking.

We compare the Hartree–Fock spectrum with transport measurements of a tBLG device with θ ∼ 0.45° and find good agreement with its nonoverlapping bands picture. The experimentally resolved peaks in R_xx (equally spaced at $n=s{n}_s$) and the sharp interband transitions in n_H indicate that the bands are nonoverlapping, either semimetallic or separated by small gaps (SI Appendix, Fig. S2A), in good agreement with the Hartree–Fock calculations. Since the studied devices have θ close to the predicted second magic angle θ_m2 ∼ 0.5°, these findings support the existence of flat bands and strong electronic interactions. However, we do not yet observe correlation-driven insulating states as in the case of tBLG at θ_m1. This may be due to suppression of the on-site Coulomb energy for a larger moiré wavelength ($\sim 10$meV for dielectric constant $\mathit{ϵ}\sim 5$), the larger noninteracting bandwidths ($\sim 10$meV), and the close energetic proximity of the flat bands.

When the B_⊥-field is increased further (Fig. 2A), Landau levels (LL) develop within the individual bands and form gaps with nonzero Chern numbers ($C$ = 4, 8, 12, … due to spin/valley degeneracy), which manifest themselves as dips in R_xx (blue). These dips trace back to the band edges at integer fillings $n/n_s=s$ at zero B_⊥ = 0T and follow a well-defined slope dn/dB_⊥ = $C$e/h in the n-B_⊥ phase space, where e is the electron charge and h is Planck’s constant. Above B_⊥ > 8 T robust quantum Hall states develop, with R_xx ∼ 0 Ω, and with quantized plateaus in the Hall resistance of R_xy ∼ h/Ce² (Fig. 2C). In contrast, all trivial band gaps have $C$ = 0 and emerge from B_⊥ = 0 T without a slope in dn/dB_⊥ = 0. These gaps are identified by R_xx peaks (red) that follow straight vertical lines in the n-B_⊥ phase space. For clarity, we highlight the most pronounced gaps from Fig. 2A in Fig. 2B.

Fig. 2.

Unbounded and connected Hofstadter butterfly spectrum. (A) Hofstadter spectrum of the θ = 0.45° tBLG device, revealed by the color plot of R_xx vs. carrier density n and B_⊥. (B) Schematic identifying visible LL gaps from A. Vertical solid lines indicate single-particle band gaps ($0,s$) between the moiré bands, and diagonal solid lines indicate Hofstadter LL gaps ($C,s$), which emerge from the band edges. (C) Longitudinal conductance ${\sigma }_{xx}$ and Hall conductance $\left|{\sigma }_{xy}\right|$ vs. $n$, measured at $B_{\perp }=15$T, show robust quantum Hall states with a nonmonotonic evolution with n, as expected from the Diophantine relation.

These findings indicate the formation of a fractal Hofstadter spectrum. The trajectories of the trivial band gaps and the topological LL gaps have a dense set of intersections at which only one of the gaps survives. The intersections occur at well-defined magnetic fields B_⊥ ∼ 2.25 T ($\varphi \sim {\varphi }_0/2$), B_⊥ ∼ 4.5 T ($\varphi \sim {\varphi }_0$), and B_⊥ ∼ 9 T ($\varphi \sim 2{\varphi }_0$), where $\varphi =B_{\perp }{\text{Ω}}_m$ is the magnetic flux per moiré unit cell and ${\varphi }_0$ = h/e is the magnetic flux quantum. The LL gaps in the n-B_⊥ diagram occur along lines that are described by the Diophantine relation, $n/n_s=C\varphi /4{\varphi }_0+s$. This is further confirmed by R_xy versus n measurements in the quantum Hall regime at B_⊥ = 15 T (Fig. 2C), which show a nonmonotonic evolution of the plateaus, consistent with the development of Hofstadter gaps within different moiré bands.

We denote the features of the Hofstadter gaps by ($C,s$), which have Chern number $C$ and emerge from the band edges at filling $n=s{n}_s$. The fan diagram is dominated by a series of zero-field single-particle band gaps ($0,\text{s}$) and by Hofstadter gaps ($\pm 4,s$), ($\pm 8,s$), which are strikingly not confined within the moiré band from which they emerge (at B_⊥ = 0) but continuously extend into the higher-lying moiré bands at high B_⊥ and cross all ($0,s$) band gaps. For example, the ($\pm 4,0$) gaps that are emerging from the CNP (blue diagonal lines) extend indomitably through several higher moiré bands, interrupting all band gaps ($0,s$) at carrier densities $n=\pm s{n}_s$ (red vertical lines). The observed Hofstadter spectrum is qualitatively distinct from that of topologically trivial bands.

We highlight the differences by comparing typical Hofstadter spectra of topologically trivial and nontrivial bands (Fig. 3 A and B). For separated topologically trivial bands, the Hofstadter spectrum is confined within the energy bandwidth of each band. As B_⊥ increases, LLs from the band top and band bottom move toward the middle of the band and are annihilated around its center. This evolution of LLs in B_⊥ field defines an energetically bounded Hofstadter butterfly, where the LL spectra in one band are not connected to those of other bands (while one can devise trivial band structures with a connected spectrum, they are not generic). This is in stark contrast to the Hofstadter butterfly which is formed from topologically nontrivial bands (Fig. 3B). In all generality, the Hofstadter spectrum of a topological band is not energetically confined to the bandwidth of the band and can propagate until it connects to the Hofstadter spectrum of another band, which trivializes the total band topology (14, 20).

Fig. 3.

Hofstadter butterfly with nontrivial topology. (A and B) Examples of Hofstadter spectra (Left) and corresponding gap schematics (Right) of two topologically trivial (A) and nontrivial $C=\pm 1$ (B) bands. The topological Hofstadter spectrum shows distinct features, as its bands are connected at $\varphi /{\varphi }_0=1$, and the $C=0$ band gaps are closed. Both A and B are calculated for tight-binding models (see SI Appendix, section 9). (C) Zoom-in on the Hofstadter spectrum in Fig. 2 A and B. The ($\pm 4,0$) LL gaps continuously extend into the higher-lying moiré bands and interrupt the ($0,\pm 1$) and ($0,\pm 2$) band gaps, showing the main signatures of the topological Hofstadter spectrum. (D) Calculated Hofstadter spectrum of θ = 0.45° tBLG shows good agreement with experiment (C).

Such a topologically nontrivial, unbound, and connected Hofstadter butterfly spectrum is clearly present in the demonstrated θ = 0.45° tBLG device, as well as in several other devices with θ = 0.3° to 0.5° (SI Appendix, section 3). Fig. 3C displays a zoom-in of the dashed region in Fig. 2A and highlights the interplay between the trivial band gaps ($0,s$) (red) and the Hofstadter gaps ($\pm 4,0$) (blue), where ($0,\pm 2$) are clearly interrupted by ($\pm 4,0$) at B_⊥ ∼ 9 T ($\varphi /{\varphi }_0=2$). Similarly ($0,\pm 1$) are interrupted by ($\pm 4,0$) at B_⊥ ∼ 4.5 T ($\varphi /{\varphi }_0=1$). These interruptions indicate the closing of the ($0,\pm s$) band gaps between two neighboring bands, when their fillings coincide with that of the ($\pm 4,0$) LL gaps. This connects the Hofstadter spectra of the lowest two moiré bands with the spectra of all higher moiré bands, which is direct evidence of the nontrivial topology of tBLG and is in agreement with the theoretical predictions in refs. 14 and 20.

The experimentally obtained unbounded and connected Hofstadter spectrum is in good agreement with the theoretically calculated Hofstadter spectrum for low n ($|n/n_s|<3$) (Fig. 3D and SI Appendix, Fig. S12B). It is calculated with a tBLG continuum model at θ = 0.45°, which includes broken C_3z symmetry due to strain or Hartree–Fock mean fields (SI Appendix, section 6), to realistically mimic a strain-reconstructed tBLG device. We note that C_3z breaking could also be induced by angle disorders, which is however difficult to model and will not be discussed here (21). We find that the calculated ($\pm 4,0$) gaps continuously extend to the higher bands as expected. Moreover, experimentally we observe reemerged ($\mathrm{0,0}$) and ($0,\pm 1$) gaps at $\varphi >{\varphi }_0$ and (0, ±2) gaps in the range ${\varphi }_0<\varphi <2{\varphi }_0$, which match the calculations only if C_3z breaking is considered (SI Appendix, Fig. S12). However, the observed Hofstadter spectra at high doping densities ($|n/n_s|>3$) show deviations from the theoretical calculations. Theoretical calculations predict a series of ($4,s$) and ($-4,-s$) gaps ($s=\mathrm{1,2,3},\dots$) at $\varphi >{\varphi }_0$ in addition to the ($\pm \mathrm{4,0}$) gaps, while experimentally this region is dominated by the ($\mathrm{8,1}$) and ($-8,-1$) gaps, and only a ($\mathrm{4,1}$) gap occurs at $\varphi >3{\varphi }_0$. This discrepancy may have its origins in the nonnegligible role of interactions in the flat bands, which may considerably affect the Hofstadter spectrum of tBLG at large doping densities. In SI Appendix, section 6 we show that the inclusion of an electron-doping-induced Hartree term in the Hofstadter spectrum calculation suppresses the ($4,s$) gaps, which suggests that interactions may be responsible for their experimental absence.

Discussion

One possible origin of the nontrivial unbound Hofstadter butterfly is the single-particle fragile topology hosted by the lowest two moiré bands of each spin and valley in tBLG (theoretically predicted and characterized by the C_2zT-protected nonzero Dirac helicity $\eta =\pm 2$ at the CNP) (12–14). This fragile topology is further enhanced into a stable topology if the particle–hole symmetry of tBLG is preserved, which is robust against adding trivial particle–hole symmetric pairs of bands (15). It is theoretically shown that the Hofstadter spectrum of the lowest two topological moiré bands of tBLG is always connected with the spectrum of the higher moiré bands at sufficiently strong magnetic fields (up to infinity) (14) [although not universally protected by C_2zT (20); see SI Appendix, section 4]. For a wide range of twist angles (including those studied in this paper), the $(\pm \mathrm{4,0})$ LL gaps are predicted to extend from the lowest two moiré bands to all the higher bands, forcing their Hofstadter spectra to be connected at $\varphi /{\varphi }_0=1$ (14). Accordingly, the band gap between two bands with connected Hofstadter spectra will close when the connection happens. We note that the flat moiré bands in tBLG can also develop correlated states of different stable topologies breaking C_2zT, for instance Chern insulators at the first magic angle, which are driven by interactions and stabilized by finite magnetic fields (22–27). At this point our results cannot radically rule out the possibility that either type of topology, C_2zT-preserved fragile topology or C_2zT-broken stable topology, is responsible for the connected Hofstadter spectra. The absence of correlated states in our experiment and the unbroken C_2zT symmetry in our Hartree–Fock calculations, however, suggest the former is more likely.

In summary, we have reported a systematic magnetotransport study of tBLG close to θ_m2. Our results show 1) multiple well-isolated flat moiré bands in tBLG at θ_m2, which cannot be explained without considering interactions, 2) that tBLG is a highly tunable platform to study the Hofstadter butterfly, where the band topology can manifest itself qualitatively (trivial or nontrivial), and 3) that interaction effects play a clear role in tBLG Hofstadter spectra that merits further exploration.

Materials and Methods

Device Fabrication.

The hBN/tBLG/hBN/graphite stacks were exfoliated and assembled using a van der Waals assembly technique. Monolayer graphene, thin graphite, and hBN flakes (∼10 nm thick) were first exfoliated on an SiO₂ (∼300 nm)/Si substrate, followed by the “tear and stack” technique with a polycarbonate/polydimethylsiloxane stamp to obtain the final hBN/tBLG/hBN/graphite stack. The separated graphene pieces were rotated manually by the twist angle ∼1°. We purposefully chose a larger twist angle during the heterostructure assembly due to the high risk of relaxation of the twist angle to the random lower values. To increase the structural homogeneity, we further carried out a mechanical cleaning process to squeeze the trapped blister out and release the local strain. We did not perform subsequent high-temperature annealing to avoid twist-angle relaxation. We further patterned the stacks with polymethyl methacrylate resist and CHF₃+O₂ plasma and exposed the edges of graphene, which was subsequently contacted by Cr/Au (5/50 nm) metal leads using electron-beam evaporation (Cr) and thermal evaporation (Au).

Measurement.

Transport measurements were carried out in a refrigerator with a base temperature of 1.3 K and up to 16-T magnetic field. All the data without specific notification was measured at the base temperature. Standard low-frequency lock-in techniques were used to measure the resistance R_xx and R_xy with an excitation current of ∼10 nA at a frequency of 19.111 Hz. A global gate voltage (+20 V) through Si/SiO₂ (∼300 nm) is applied to reduce the contact resistance by tuning the charge carrier density separately in the device leads.

Twist Angle Extraction.

The twist angle $\theta$ is deduced from the area of the moiré unit ${\text{Ω}}_m$ given by ${\text{Ω}}_m=\sqrt{3}{a}^2/4\left(1-\cos \theta \right)$, where $a$ is the lattice constant of graphene. ${\text{Ω}}_m$ is extracted with two independent methods.

• Method 1: Gate-induced carrier density is first calibrated with LLs. The carrier density of LL fanning out from CNP with total Chern number $C$ strictly follows $n=C\text{eB}/\text{h}$. Here we use LL with index $C$ = 8 to calibrate the carrier density. The area of moiré superlattice ${\text{Ω}}_m=4/n_s$, where $n_s$ is the carrier density of a completely filled moiré band.

• Method 2: The Hofstadter butterfly is also used to extract ${\text{Ω}}_m$. When the magnetic flux per moiré unit cell given by $\varphi =B{\text{Ω}}_m$ equals to the magnetic flux quantum ${\varphi }_0=h/e$ ($h$, Planck’s constant; e, magnitude of the electron charge), the Hofstadter spectrum will exhibit fractal signature (i.e., B $\sim$ 2.25 T, B $\sim$ 4.5 T, and B $\sim$ 9 T corresponding to $\varphi ={\varphi }_0/2$, $\varphi ={\varphi }_0$, and $\varphi =2{\varphi }_0$, respectively). The moiré area ${\text{Ω}}_m$ can be described by ${\text{Ω}}_m={\varphi }_0/B_0$ ($B_0$ is the magnetic field corresponding to $\varphi ={\varphi }_0$) where the straight lines of LL gaps in the n-B plot cross each other at integer band fillings.

Data Availability

All study data are included in the article and/or SI Appendix.

Change History

November 24, 2021: The affiliations have been updated.