Direct probing of energy gaps and bandwidth in gate-tunable flat band graphene systems

Abstract

Moiré systems featuring flat electronic bands exhibit a vast landscape of emergent exotic quantum states, making them one of the resourceful platforms in condensed matter physics. Tuning these systems via twist angle and the electric field greatly enhances our comprehension of their strongly correlated ground states. Here, we report a technique to investigate the nuanced intricacies of band structures in dual-gated multilayer graphene systems. We utilize the Landau levels of a decoupled monolayer graphene to extract the electric field-dependent bilayer graphene charge neutrality point gap. Then, we extend this method to analyze the evolution of the band gap and the flat bandwidth in twisted mono-bilayer graphene. The band gap maximizes at the same displacement field where the flat bandwidth minimizes, concomitant with the emergence of a strongly correlated phase. Moreover, we extract integer and fractional quantum Hall gaps to further demonstrate the strength of this method. Our technique paves the way for improving the understanding of electronic band structures in versatile flat band systems.

Introduction

Understanding the band structure of a system is of fundamental significance in condensed matter physics. For instance, the linear conical energy spectrum of monolayer graphene (MG) gives rise to two in-equivalent K points, which results in the observation of the half-integer quantum Hall effect^1,2,3. A small twist angle between two graphene sheets induces a long-range periodic pattern, resulting in angle-dependent moiré Bloch bands⁴. Particularly, near a “magic angle” (~ 1.1°) of rotation between the layers, the coupling between two graphene sheets is strongly reinforced, and the low-energy moiré bands become very narrow, almost without any dispersion (flat)⁵. This gives rise to exotic quantum phases, including strongly correlated insulating states^{5,6,7,8,9,10,11,12}, unconventional superconductivity^{9,10,11,12,13,14,15}, ferromagnetism^16,17,18, etc.

Due to the flexible carrier density control by dual electrostatic gating, twisted multilayer graphene flat band systems have become an appealing platform for studying strongly correlated quantum phases. For example, orbital Chern insulators at different integer filling factors were observed in twisted monolayer-bilayer graphene (TMBG)^19,20. In contrast, spin-polarized insulating states were observed in twisted double-bilayer graphene (TDBG)^21,22,23, etc.

The dual gate tuning provides a fundamental degree of freedom to understand the rich phases mentioned above. To study these exotic phases, an experimental technique that can accurately measure the response of the device under an applied electric field is crucial. Several single-gated local probe techniques have been developed to enrich the understanding of band structure in two-dimensional materials, like scanning tunneling microscopy/spectroscopy (STM/STS)^{4,8,23,24,25,26,27}, single electron transistor (SET)²⁸, planar tunneling junction²⁹ and nano-SQUID³⁰. In addition to local probes, global measurements, including magneto-transport^{5,6,7,8,9,10,11,12,13,14,15,19,20,21,22,23}, nano angle-resolved photoemission spectroscopy (Nano-ARPES)^31,32, electronic compressibility³³, nano-infrared imaging³⁴, and Fourier transform infrared (FTIR) spectroscopy^35,36 are widely adopted. However, the limiting factor in most of these aforementioned techniques, like STM/STS and nano-ARPES, is that the samples are designed only with one gate.

Although electronic compressibility measurements, nano-infrared imaging, Nano-SQUID-on-tip microscopy, and FTIR spectroscopy can investigate dual gate devices, these measurements each have their drawbacks^{30,31,32,34,35,36}. For example, the large beam spot (~1 mm) used in FTIR spectroscopy, compared with the size of typical devices, makes the experiments substantially challenging. While conventional transport measurements for extracting the energy gap, such as resistance vs. temperature (R−T) measurements, have been widely adopted, the thermally activated gap overshadows other details. In addition, this method cannot extract the bandwidth.

Recently, the unique band structure in twisted trilayer graphene has enabled the dissociation of intertwined bands and the quantification of energy gaps¹⁵. Similarly, using decoupled monolayer graphene has facilitated the uncovering of spin ordering in twisted bilayer graphene³⁷. These advancements inspire further investigation into the electronic band structures of electric field-tunable systems, including the direct probing of energy gaps and bandwidths in gate-tunable flat band graphene systems.

Results

Gate-tunable CNP gap in BG

To benchmark our technique, we perform electrical transport measurements on a dual-gated Hall bar fabricated on a monolayer graphene (MG) stacked on a bilayer graphene (BG) with a large twist angle (see Fig. 1a, b). The calculated band structures of TMBG indicate that a large twist angle θ (≥ 10°) between MG and BG is necessary to effectively decouple the two layers (Supplementary Figs. 1, 2)³⁸.

Fig. 1: Twisted decoupled monolayer graphene (DMG) with bilayer graphene (BG).

a The schematic of the device and measurement configuration. b The schematic of DMG stacked on top of the BG. The blue arrow pointing to the DMG indicates the positive direction of the applied displacement field D. c The optical image of the DMG + BG (Device_A1) device. d Longitudinal resistance R_xx as a function of the total carrier density (n_tot) and D at B  = 0.5 T. The red (black) arrow indicates the direction of the bottom gate (top gate). The Dirac cone of the DMG splits into many Dirac Landau levels along the top gate (V_tg) direction. e A colored schematic diagram of the main features of n−D mapping in Fig. 1d. Magenta anti-S-shaped region represents the CNP of the BG (Δ_CNP). The two boundaries of the CNP gap of the BG are indicated by faint dashed lines intersecting at α, where D = 0.13 V/nm. A and B are edges of the CNP gap of the BG for the same displacement field (D = 0 V/nm). A series of Dirac LLs of DMG (red dashed line along the diagonal) shuffle along the top gate direction. ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ is the Nth Dirac Landau level of DMG. The letters inside the parentheses represent the carrier types for BG (first) and DMG (second). “e” stands for electrons, and “h” stands for holes. f Landau fan diagram of R_xx near the CNP of the BG at D = −0.5 V/nm. White dots A and B represent the band crossing between the third Dirac LL of the DMG and the CNP of the BG. g A schematic diagram of extracting the CNP gap of BG. The insets schematize the third LL of DMG located at the edges of the CNP gap of the BG. The gap is extracted from the change in the chemical potential of LL. The chemical potential is calculated from ${\mu }_{N}={\mu }_{N=0}+{v}_{F}\cdot \sqrt{2e\hslash N \, sgn(N)B}$.

We trace the CNP of this decoupled MG (DMG), which is denoted by the red dashed line in phase diagrams of Fig. 1d, e, ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N=0}^{D}$ and N is the Landau level index. In our device configuration, the BG is closer to the bottom gate. Hence, the bottom gate cannot effectively tune the DMG due to the screening from the BG, which is evident from the trace of the CNP line of the DMG as it is almost parallel to the bottom gate direction. Moreover, we observe multiple Landau levels splitting under a finite magnetic field (Supplementary Fig. 3). These Dirac Landau levels of the DMG (${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$) shuffle along the V_tg direction when V_bg is fixed.

Using Landau level spectroscopy, we estimated the Δ_CNP value (see Methods). Since the DMG is decoupled from the BG, we can directly estimate the chemical potential extracted from the ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$. Figure 1f shows the band crossings between the ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ and the CNP gap edges of the BG denoted by A and B. In order to estimate Δ_CNP, we take ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N=3}^{D}$ (Fig. 1g) as an example. μ_N,A, μ_N,B denote chemical potentials of the left and right CNP edges (indicated by two horizontal black dashed lines and inset schematics), following ${\mu }_{N}={\mu }_{N=0}+{v}_{F}\cdot \sqrt{2e\hslash N \, sgn(N)B}$, where Fermi velocity v_F = 1 × 10⁶ m/s^15,39,40,41 and ℏ is the reduced Planck constant. The chemical potential difference Δ_N = ∣μ_N,A − μ_N,B∣ represents the energy gap size of the bilayer graphene CNP.

In Fig. 2a, at both points A and B, the chemical potential extracted through the different ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ changes as $\sqrt{B}$, implying that A and B both track the same DMG Landau levels. For different ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D},{\mbox{ the }} {\Delta }_{{{{\mathrm{CNP}}}}}$ values remain consistent, with small variation (Fig. 2c).

Fig. 2: Electric field-dependent energy gap extraction from DMG Landau level spectroscopy.

a Zoom-in Landau fan diagrams of the white dashed rectangular regions in Fig. 1f. Clear Dirac LLs of the DMG are indicated by red and blue dashed lines and ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$. The interval between LLs changes linearly with $\sqrt{B}$. b The error bars of the chemical potential are derived from fitting the full-width half maximum (FWHM) of the high longitudinal resistance R_xx peak at points A and B, which are fitted to a Gaussian function. The  ± σ are error edges of the corresponding chemical potentials. c The CNP gap (Δ_CNP) of the BG extracted from different LLs of DMG. Δ_CNP is the change of chemical potential along the gap. Since the error bars are derived from the broadening of Landau levels (LLs), the resolution of the experiments can be improved by using LLs with narrower bandwidths. The top axis B_LL−B tracks the magnetic field of point B for every LL with index N. d Gap evolution as a function of D for  Device_A1 (22°) and Device_A2 (27°). The overlap of blue and red dots demonstrates the consistency of the gap extracted from different LLs, which increases with increasing $\left\vert D\right\vert$. The difference in Coulomb potential between two layers of BG induces a small gap at zero displacement field. e–g Landau Level spectroscopy at different D. The yellow dashed line indicates the kink of the first LL of the DMG. The gap completely closes at a small positive electric field D₃.

We then extend our analysis to include various displacement fields (D), allowing us to observe the electric field dependence of Δ_CNP in BG of two devices (Fig. 2d). The Δ_CNP in BG increases significantly with increasing $\left\vert D\right\vert$ as a result of the difference in Coulomb potential between the layers. The estimated Δ_CNP at D = −0.5 V/nm is 83.2 ± 2.3 meV of Device_A1, is in agreement with previous findings^42,43.

Next, we take Device_A1 as the main device, and we observe a finite built-in gap (17.2 ± 1.3 meV) at the CNP at D = 0 V/nm, which results from layer polarization of the valence and conduction bands^44,45. Furthermore, we observe the evolution of the CNP gap closing and reopening with increasing positive D (Fig. 2e–g and Supplementary Figs. 4–6). Particularly, at D₃ = 0.13 V/nm, the gap is completely closed by a sign change of Coulomb potential difference⁴⁶, which is indicated by the α point in Fig. 1e. This behavior results from the asymmetric response of the displacement field and layer polarization. The weaker response of the CNP gap at positive D (pointing to MG, D = 0.25 V/nm, Δ_CNP is 14.3 ± 2.7 meV) compared to negative D (pointing to BG, D = −0.25 V/nm, Δ_CNP is 55.3 ± 4.0 meV) supports this observation. In addition, we observe that the CNP gap does not fully close within the moderate negative D range (D₂ to D₁). Further investigation is needed into the band structure, particularly considering the overlap between the CNP of the DMG and the BG.

Bandwidth and bandgap of TMBG

To establish the robustness of this technique, we study a more complicated electric field-tunable flat band system-TMBG. The DMG was placed on top of the monolayer graphene of the small twist angle TMBG (1.14°  ± 0.02°) (Fig. 3a). The schematic compares the moiré patterns of TMBG and DMG + TMBG, the moiré length is equal to  ~13 nm, and does not change significantly with the addition of the DMG. A n−D map at B = 1 T is shown in Fig. 3d (Supplementary Fig. 7). We observe similar multiple Landau levels ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ splitting as in Fig. 1d. Then the strongly correlated states are observed at all integer filling factors and are found to vary with the D.

Fig. 3: Evolution of bandwidth and bandgap with displacement field in twisted mono-bilayer graphene.

a Schematics of TMBG and DMG with TMBG. The left panel is a moiré pattern of the TMBG (1.14° ±  0.02°). The right panel is DMG on top of the TMBG, indicating the same moiré length as TMBG. b Optical image of Device_B3. The black scale bar is 10 μm. c The configuration of dual gate measurements. The blue arrow pointing to the DMG indicates the positive direction of the applied displacement field D. d R_xx as a function of the n_tot and D at B = 1 T. The asymmetric phase dependent on D is similar to a phase of pure TMBG, indicating that the monolayer graphene is fully decoupled from BG. The correlated states appear at all integer filling factors at positive D. e Zoom-in of the n−D map presented in Fig. 3d, at B = 0 T. f The TMBG flat bandwidth and bandgap evolution at different D. The minimum of the bandwidth and the maximum of the bandgap at D = 0.53 V/nm (β), where exactly the ν = 1 correlated state emerges in Fig. 3e.

Figure 3f shows the extracted flat bandwidth and bandgap as a function of D (see Methods and Supplementary Figs. 8–14). When the correlated state appears at the same D as point α (D = 0.33 V/nm) with filling factor 3, the flat bandwidth is estimated to be about 73.5 ± 1.8 meV, which is in good agreement with the experimental work in TMBG (70 ± 10 meV)⁴⁷. Furthermore, due to the narrow width of the ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ (Fig. 2a), the experimental error is smaller (<few meV) than the reported values, showcasing the high precision of this method. From point α to point β (D = 0.53 V/nm), the flat bandwidth decreases to 64.5 ± 0.6 meV as D increases, which shows that D greatly suppresses the width of the flat band. Simultaneously, the bandgap varies inversely with the D. Particularly, at point β, the flat bandwidth reaches a minimum while the bandgap maximizes, and a correlated state appears at filling factor ν = 1 (Fig. 3e). This might be explained by the fact that the strongest correlation occurs where the flat band is at a minimum, which is accompanied by the weakest correlated insulator at ν = 1. However, it is not true that strong electron-electron interactions can lead to counterintuitive effects, such as an enhancement in the bandwidth. Specifically, the effective potential landscape that the carriers experience could induce fluctuations, resulting in a broadening of the bandwidth. Despite this, by extracting the bandwidth at the exact points where correlated states are generated, we can observe how the bandwidth varies with the occurring states and extract the energy change in the bandwidth corresponding to the correlation where two correlated states transform. This provides valuable insights into the model and parameters of correlation, which will help refine future studies. From β to γ, the flat bandwidth broadens with increasing D and decreasing bandgap, reflecting weaker electronic correlations. According to the previous works^19,20, the flat band gradually touches the remote dispersive band as  D  increases.

For TMBG, the spatial inversion symmetry is broken; and it exhibits a rich phase diagram similar to the one of twisted bilayer graphene (TBG) when a D is applied in the direction of MG (TBG side). The TMBG also exhibits a phase diagram similar to that of twisted double-bilayer graphene (TDBG) when the D is inverted^19,20.

Thus, in addition to the four DMG/TMBG devices (Supplementary Fig. 7), where the DMG was placed approximately on the monolayer graphene side of the TMBG, in contrast, for Device_B5, the DMG was placed on the bilayer graphene side of the TMBG (Supplementary Fig. 15). When carriers are pushed toward the monolayer graphene side of TMBG under positive (negative) displacement fields, the gap in Device_B5 opens larger than in Device_B3, highlighting the screening effect from the DMG (Supplementary Figs. 16,17). This provides a robust experimental method to support theoretical efforts in understanding the complex phase diagrams of these multilayer graphene systems in the future.

Integer and fractional quantum Hall gaps

To further warrant the accuracy of this technique, we demonstrate how this method can be used to extract integer and fractional quantum Hall gaps for the TMBG flat band. The four-fold degeneracy of LLs for both DMG and the flat band is lifted (Supplementary Figs. 8–10). In Fig. 4a, we clearly see four mini-bands ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l}^{D}$ formed from ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N=-1}^{D}$ splitting, indicated by red and purple dashed lines. These DMG Landau levels with different filling factors induce additional Chern number offsets (-l) (Supplementary Figs. 18–20). The total Chern number C (indicated by the black numbers) is the summation of C_f (indicated by the yellow numbers) and C_D, where C_f and C_D are Chern numbers of the TMBG flat band and DMG Dirac band, respectively. Multiple As and Bs represent gap edges of ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l=-2}^{f}$ and the estimated gap △_N = ∣μ_A − μ_B∣ is averaged through different ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l}^{D}$ (Fig. 4b). The extracted values of the gap measured by different LLs are consistent with each other, and the average value of the gap is given by 4.1 ± 0.9 meV.

Fig. 4: Flat-band integer Landau level gap.

a Longitudinal resistance R_xx and Hall resistance R_xy as a function of n_tot and B. The Dirac LLs degeneracy is lifted, and ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N=-1}^{D}$ splits into four mini-bands ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l}^{D}$ indicated by the purple and red dashed lines. The black dashed lines indicate the flat band LLs. As DMG LLs offer Chern number offset, the corresponding Chern number C (indicated by black number) equals C_f (indicated by yellow number) + C_D, C_f and C_D are Chern numbers of the TMBG flat band and DMG Dirac band respectively. b The flat band integer Landau level gap (${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l=-2}^{f}$). Multiple As, Bs represent gap edges of ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l=-2}^{f}$ and the estimated gap △_N = ∣μ_A − μ_B∣ = 4.1 ± 0.9 meV is averaged through different ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l}^{D}$.

We extend a similar analysis to a fractional quantum hall gap. Figure 5a, b show R_xx and R_xy as a function of B and n_tot near the CNP region. The schematic of the corresponding Landau level crossings is shown in Fig. 5c. The N = −1 Dirac Landau level splits into four mini bands, indicated by the green (Landau level filling factor l = −3), blue (l = −4), purple (l = −5), and pink (l = −6) lines, respectively. Thus, the Chern number cascade was observed when crossed with the flat band Landau levels. Figure 5d exhibits near zero R_xx (blue line) and corresponding well-quantized R_xy plateaus (red line); these are line cuts taken along the blue and red dash line in Fig. 5a, b. A difference of neighboring ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N}^{D}$ filling factor is one, indicating that the degeneracy of the system is fully lifted.

Fig. 5: Chern number cascade and fractional quantum Hall gaps.

a, b Longitudinal resistance R_xx and Hall resistance R_xy as a function of n_tot and B. c Schematic of quantum oscillation of Fig. 5a. LL^T is TMBG flat band LLs composed with DMG LLs Chern number offset. d Filling factor dependence of longitudinal resistance (blue dash line in Fig. 5a) and Hall resistance plateaus (red dash line in Fig. 5b). e zoomed Landau fan taken from yellow rectangular in Fig. 5a, b. The solid green line indicates ν = −11 integer quantum Hall state. The green dashed line denotes a fractional quantum Hall state of ν = −11 + 1/3. A, B are band edges of the fractional quantum Hall gap of ν_f = −6 + 1/3. The estimated gap △_N = ∣μ_A −  μ_B∣ = 0.56 ± 0.14 meV. f Evidence of a fractional quantum Hall state. The data of R_xx and R_xy are taken along the blue and red dash lines in Fig. 5e.

According to the band edges A and B shown in Fig. 5e, a fractional quantum hall gap can be extracted from the band crossing between the fractional quantum hall flat band ν_f = −6 + 1/3 and the Dirac band ${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{l=-5}^{D}$ (C_D = −5). The magnetic fields in Fig. 5e at A and B are 4.310 ± 0.005 and 4.270 ± 0.005 T, respectively. Thus, the extracted flat band fractional quantum hall gap is △_N = ∣μ_A − μ_B∣ = 0.56 ±  0.14 meV at around 4.3 T. At filling factors ν = −11 and ν = −11 + 1/3, R_xx exhibits a minimum (blue line), while R_xy shows a kink (red line), as illustrated in Fig. 5f. This guides us to define the edges of the FQHE in Fig. 5e. Our fractional quantum hall gap results are comparable to results from the reported studies, which yielded △_1/3 ~ 1.4 to 1.8 meV at 12 T^48,49,50,51.

In comparison with the conventional R-Tactivation method, the ν = 1/3 fractional quantum Hall gap for two devices in GaAs quantum wells is reported as (8.7 ± 0.1) and (8.0 ± 0.4) K^52,53, corresponding to 0.69 and 0.75 meV, respectively. In graphene, the ν = 1/3 plateau persists up to T  ~ 10 K, equivalent to 0.86 meV^54,55. Particularly, the fractional quantum hall gaps for LLs with higher filling factor is smaller than normal ν = 1/3 (<10 K, 0.86 meV)⁵¹. These results are quite comparable to our data, where the extracted fractional quantum hall gap is 0.56 ± 0.14 meV at around 4.3 T.

To summarize, we have used the Landau levels of decoupled monolayer graphene to measure the chemical potential of dual-gated multilayer graphene devices. We measured the electric field-tunable CNP gap of bilayer graphene. Then, we extracted the electric field-tunable flat bandwidth and bandgap in twisted mono-bilayer graphene. Moreover, the measurements of the flat band integer and fractional quantum Hall gaps provide a promising avenue to investigate nuanced band structure.

This technique has far-reaching consequences for studying strongly correlated states^37,56,57. For example, the superconducting phase diagram and the ground state can be understood by studying adjacent correlated states (Supplementary Fig. 21). Currently, there is a scarcity of techniques that establish a connection between the displacement field-tunable flat bandwidth and strong electron-electron correlations. Our work can encourage more theoretical works to understand the complicated phase diagrams of these multilayer graphene systems. Furthermore, it could be extended in the future to other similar moiré systems, such as transition metal dichalcogenides systems.

Methods

Device fabrication

The devices are fabricated using an advanced technique known as “cut and stack”¹². Pristine materials such as monolayer graphene, bilayer graphene, hBN (10–50 nm), and graphite flakes (3–15 nm) were mechanically exfoliated on an oxygen plasma-etched SiO₂ (285 nm) surface. Next, we used atomic force microscopes(AFM) to pre-cut monolayer graphene and bilayer graphene. High-quality homogeneous poly (bisphenol A carbonate) (PC)/polydimethylsiloxane (PDMS) was then stacked on the glass slide used to transfer the 2D materials flakes to the alignment marker chip. The transfer stage precisely controls the twisted angle between two 2D materials to within 0.1° resolution. The graphite top gate is then fabricated, followed by the electrodes by electron beam lithography and metal evaporation. Here, we use the conventional etching method to define Hall bars. We etch graphite and hbN with O₂ and SF₆ gases, respectively. Optimization of the etching parameters is important to obtain 1D edge contacts with the Cr/Au (5/50 nm) electrodes⁵⁸.

Measurements

Transport measurements were conducted in the cryostat (Oxford Instruments, Heliux), with a base temperature of ~ 275 mK. Standard lock-in techniques were employed using the Stanford Research SR860, with an excitation frequency of f = 17.7777 Hz and an AC excitation current (I_s) of less than 10 nA to avoid sample heating and minimize bias effects, we used NF voltage pre-amplifiers with an impedance of 100 MΩ before feeding signals into the lock-in amplifiers (Stanford Research SR860). The transport measurements are conducted in a four-terminal geometry. The device was biased along the major axis of the Hall bar to ensure a uniform current flow, while both longitudinal and transverse voltage drops were simultaneously recorded. Gate voltages were applied using source meters (Yokogawa GS100), and an additional global gate was implemented to enhance the contact quality. We can extract n_tot and the displacement field D using the following equation n_tot = V_bgC_bg/e + V_tgC_tg/e, D = ∣V_bgC_bg − V_tgC_tg∣/(2ε₀). C_bg and C_tg are the capacitances between the top (V_tg) and bottom (V_bg) gates and the DMG + TMBG, e is the electron charge, and ε₀ is the vacuum permittivity.

Twisted angle determination

Due to the decoupling of the DMG from the TMBG, the carrier density n_flat−band observed by ν = ±4 = ±n_s of band insulating states. This value can be determined by analyzing quantum oscillations in the Landau fan diagram or by making observations in an n−D diagram. n as a function of the twisted angle following equation ${n}_{s}=8{\theta }^{2}/\sqrt{3}{a}^{2}$ determines twisted angle of our TMBG (1.14° ± 0.02°),  a = 0.246 nm is the lattice constant of graphene. The same rule applies to determining small-angle TMBG.

Extracting the gaps and bandwidths

We focus on the CNP gap of BG to demonstrate the method of extracting the energy scale of gaps and bandwidths. By measuring the chemical potential jump across at the intersections between Dirac Landau levels (LLs) and the CNP gap, the chemical potentials at the two edges of the CNP can be determined by: ${\mu }_{N}={\mu }_{N=0}+{v}_{F}\cdot \sqrt{2e\hslash N \, sgn(N)B}$ as mentioned in the main text. The Fermi velocity (v_F) remains nearly constant with respect to the Landau level index (N), magnetic field (B), large twist angle (θ > 10°), and intermediate displacement field (D)^15,39,40,41, where the Fermi velocity v_F = 1 × 10⁶ m/s and ℏ is reduced Planck constant. The steps involved in this observation are outlined below:

• (i) Identify the gap edge of BG. Since in the gap of BLG, there’s no DOS, and the charging electrons will accumulate in LL of DMG shown in Fig. 1f, g. We obtain the exact CNP gap edge points A and B for every LL index N by tracking the crossing state that corresponds to partial filling of Dirac LLs at the Fermi level, which is marked by an R_xx peak, indicating the edge of incompressible gaps of Dirac LLs. The R_xx peak represents a general mark of extracting the energy scale without considering the lifting of LLs degeneracy. This process is illustrated by the red dashed line and the blue dashed line in the top panel, Region I, of Supplementary Fig. 8c and the right panel of Supplementary Fig. 9c.

• (ii) Identify the Landau level index of DMG.

• (iii) To determine the magnetic field corresponding to the CNP gap edge points A and B at the gap boundary, we refer to Fig. 1f and g. The magnetic field dictates the chemical potential at each point for a given Nth Landau level (LL). By fitting the high-resistance states R_xx peak at the gap edge, we extracted the values of B₁ and B₂ along with their respective uncertainties (σ). Owing to the exceptional quality of our devices, these measurements offer remarkably high energy resolution.

• (iv) Using formula ${\mu }_{N}={\mu }_{N=0}+{v}_{F}\cdot \sqrt{2e\hslash N \, sgn(N)B}$ to extract the chemical potential of the edges and then get the CNP gap value.

We observed that the degeneracy of Landau levels (LLs) is lifted by applying higher magnetic fields, altering the energy spectrum of massless Dirac fermions. Consequently, we introduce a generalized energy spectrum for graphene Landau levels that incorporates the effects of degeneracy lifting, such as Zeeman splitting and valley polarization.

The modified equation is shown as below, ${E}_{N,s,\sigma }={v}_{F}\sqrt{2e\hslash \left\vert N\right\vert B}+\frac{1}{2}{g}_{s}{\mu }_{B}Bs+{\Delta }_{\sigma }K$^{3,41,59,60,61}, where N is Landau level index, ℏ is Reduced Planck constant, g_s is spin g-factor (typically  ~ 2 in graphene), μ_B is Bohr magneton, s is the spin state (+1 for up, −1 for down), Δ_σ is Valley splitting energy, representing energy differences between the K and $K'$ valleys, K is Valley index (±1).

The degeneracy-lifted sub-LLs create incompressible Landau level gaps in the partially filled Dirac LLs, as observed in the original R_xx peak (Supplementary Figs. 8, 9). In Supplementary Fig. 8c, from region I to III, we track the crossing state at point A₁, which corresponds to the crossing between the lowest incompressible degeneracy-lifted Dirac sub-LL (l = −6) and the flat band edge. This crossing is marked by a minimum in R_xx and corresponding kinks in R_xy (Supplementary Fig. 8d, e). A similar rule is applied to the isospin-polarized Dirac LLs (${{{\mathrm{L}}}}{{{{\mathrm{L}}}}}_{N=-2}^{f},{A}_{2}$). Further details can be found in Supplementary Fig. 9.

The errors in the energy extraction of gaps or bandwidths arise from three main factors:

• (i) The broadening of Landau levels.

It plays a crucial role in determining the accuracy of energy extraction for gaps or bandwidths. This broadening can result from several factors, including disorder, random strain fields, temperature effects, electron-phonon coupling, and many-body interactions. To achieve a narrower bandwidth for Dirac LLs and reduce these errors, it is essential to use clean and uniform samples.

• (ii) Zeeman splitting, isospin-polarized sub-LLs, and fully-degeneracy lifting sub-LLs.

As the magnetic field increases, the Zeeman splitting effect and the lifting of LL degeneracy become significant and should not be neglected. To accurately extract the chemical potential from the LL spectrum of Dirac fermions, it is necessary to modify the spectrum equation, as shown above.

For the Zeeman splitting term^59,60, $\frac{1}{2}{g}_{s}{\mu }_{B}Bs$, using two points at different magnetic fields can introduce some errors. However, this term is relatively small compared to the gaps or bandwidths we measure. For instance, for a change of 1 T magnetic field, the contribution from pure Zeeman splitting to Dirac LLs is approximately 1 μ_BB ~ 0.058 meV (~0.67 K)⁵⁹, which is negligible in comparison to the CNP gap and bandwidth observed in the main text.

For measuring the fractional quantum Hall gap, the magnetic fields at points A and B in Fig. 5e are 4.310  ± 0.005 and 4.270 ± 0.005 T, respectively. The 0.04 T difference in magnetic field results in a pure Zeeman splitting contribution of 0.04 × 0.058 meV = 0.002 meV, which is much smaller than the gap we measured (0.56 ± 0.14 meV). Therefore, we neglected the Zeeman splitting term in the main text for the decoupled system when measuring the chemical potentials at the two edge points under different magnetic fields.

At high magnetic fields, the Dirac LLs split into two isospin-polarized sub-LLs or four fully lifted degeneracy LLs, as described by the LL filling factor l. To minimize the effects of this splitting, we track the crossings between the lowest incompressible, degeneracy-lifted Dirac sub-LLs (l = −4) and the target state. These crossings are marked by a minimum in R_xx and corresponding kinks in R_xy, (Supplementary Figs. 8d, 9, 10f)

In Supplementary Fig. 9, we specifically discuss the influence on bandgap extraction from two different Dirac LLs: The left panel of the normal case and the right panel for the isospin-polarized case. We found that the difference in the bandwidth extracted for both cases is small, and we averaged the results to reduce errors.

• (iii) Valley splitting

We notice that the energy separation between the Dirac LLs of two sub-LLs (l) is significantly different. In Supplementary Fig. 10d, we clearly observe the A₁ and A₂ sub-LLs of l = −6, −4, with a substantial magnetic field separation (B_A2 − B_A1 = 0.5 T). In contrast, at point B, this separation is much smaller, making it difficult to clearly differentiate between the sub-LLs. This suggests that the observed effects might arise from valley splitting. See more discussion in Supplementary Fig. 10.

We must acknowledge that the valley effect cannot be completely ignored, and further theoretical work is needed to incorporate the valley splitting term into the modification of the Dirac fermion LL spectrum.

We should also note that our technique is particularly suitable for detecting systems whose band structures are not highly sensitive to magnetic fields. A useful approach is to focus on higher-index LLs, as this enables the measurement of the chemical potential at lower magnetic fields, where Landau level splitting is not yet significant. Additionally, from Fig. 2a, we observe that the bandwidth at low magnetic fields is narrow, which provides higher resolution with reduced effects from both Zeeman splitting and valley splitting.