Main

In a low-dimensional quantum conductor, charge transport often exhibits discrete characteristics, sometimes appearing as staircases of plateaux under varying magnetic or electric fields, in units of either the superconducting flux quantum, defined as \({\varPhi }_{0}=\frac{h}{2e}\), or conductance quantum defined as \({G}_{0}=\frac{{e}^{2}}{h}\). However, up to now, only a few candidates in solid states can manifest quantized physical measurables, including the conductance of quantum point contacts1,2,3,4, Shapiro voltage steps in Josephson junctions5,6 and the conductance of topologically non-trivial electronic phases such as quantum Hall or Chern insulators7,8,9,10. These simple yet elegant kinds of quantization have found key applications in modern quantum technologies such as quantum metric standards11,12,13,14, as well as quantum information processing15. Exploring previously unexplored systems that may exhibit quantities with quantization (or quantized jumps) is thus of fundamental importance and holds great promise for applications based on exotic quantum states.

Among the reported low-dimensional systems, interlayer charge transfer in large-angle (LA) twisted graphene in the quantum Hall regime has attracted interest. In this system, as the layer-dependent chemical potential is tuned by varying the vertical electric field at a fixed filling fraction ν, charges from the bottom (top) graphene layer migrate to the top (bottom) layer16,17,18,19. Nevertheless, previous experimental observations have been limited to relatively low magnetic fields, with few details explored within the Landau-level (LL) crossings. As a result, potentially sophisticated features may have been missed, since the spin and valley degeneracies remain unresolved in this regime16,17,18,20. Recently, twisted graphene systems with both small and moderate angles have been investigated17,19,21,22,23,24,25,26,27. In twisted bilayer graphene (TBLG) with twist angles close to the so-called magic angle of 1.1°, flat-band physics is observed21,22,24,25, whereas twisted double bilayer graphene systems with moderate twist angles of approximately 2–10° do not exhibit flat bands but instead display tunable excitonic insulating phases17,19,26,27. Electronic transport in TBLG with large twist angles, such as 30°, has also been reported in chemical vapour deposition samples20,28,29.

In this work, we show that, by stacking two layers of graphene with a large twist angle (20–30°), specific equal-sized checkerboard patterns can be observed in the quantum Hall regime at the LL crossings throughout the D–n space, when all spin- and valley-flavour degeneracies are lifted. Our experimental and theoretical investigations reveal that this quantized 4 × 4 checkerboard feature originates from the critical electric displacement field D, which bridges the phase transition of two different charge filling states in the TBLG at LL crossings. This phase transition results from competition between the layer polarization driven by the D field and the capacitance coupling energy, while the charge transfer at LL crossings, driven by vertical electrical fields, corresponds to exactly one elementary charge transferred between the top and bottom Landau orbits. Therefore, at a fixed ν, varying D will lead to LL crossings equally spaced by an interval of δD = \(\frac{e}{2{{\uppi }}{l}_{B}^{2}}=B{e}^{2}/h\). We can thus define a measurable, the ratio of D/B, which displays quantization of e2/h in the LL-crossing checkerboard. Our findings suggest a new paradigm of magnetic sensing, as the displacement-to-magnetic field ratio in such LL-crossing quantized checkerboards can be scaled in a linear B–δD relation. A prototype of this magnetometer is demonstrated, which could be further developed into sensor arrays for high-spatial-resolution surface magnetometry in the future.

Results and discussion

Fabrications and characterizations of LA-TBLG devices

Monolayered graphene and few-layered hexagonal boron nitride (h-BN) flakes were exfoliated from bulk crystals. The graphene was cut by a conductive atomic force microscope tip, using the anode-oxidation technique30,31. The graphene twin flakes, cut using an atomic force microscope, were stacked with a twist angle of 20° or 30° via the dry transfer method32 and then encapsulated by top and bottom h-BN flakes. Detailed fabrication processes can be seen in Supplementary Figs. 1–3. The devices were equipped with dual gates and electrodes of Ti/Au via standard lithography and electron-beam evaporation (for fabrication details, see Methods), as illustrated in Fig. 1a. Figure 1b shows the optical micrograph of a typical LA-TBLG device (Sample-S15, 30°-twisted), with the corresponding fabrication flow shown in Supplementary Fig. 2. Before further analysis, we define the two-gate-induced displacement field as D = (CtgVtg − CbgVbg)/2 − Dr, and the total carrier density as ntot = (CtgVtg + CbgVbg)/e − nr, as commonly used in dual-gated graphene devices33,34. Here, Ctg and Cbg are the top and bottom gate capacitances per area, respectively, and Vtg and Vbg are the top and bottom gate voltages, respectively. nr and Dr are the residual doping and residual displacement field, respectively. Figure 1c shows a mapping of longitudinal resistance Rxx of Sample-S15 in the D–n space, when cooled down to T = 1.5 K at B = 0 T. It is noteworthy that, along the charge neutral line (carrier density n = 0), Rxx decreases when D is departing from zero. This behaviour reflects the weak interlayer coupling16, which markedly distinguishes it from conventional strongly coupled Bernal-stacked bilayer graphene systems. In the latter, Rxx at charge neutrality monotonously increases upon increasing the absolute value of D, due to the gap opening at the Dirac point35,36. Data from a control sample of Bernal-stacked bilayer graphene are presented for comparison in a side-by-side format in Supplementary Fig. 4.

Fig. 1: Equal-sized checkerboard cells at LL crossings in the D–n space in LA-TBLG.
Fig. 1: Equal-sized checkerboard cells at LL crossings in the D–n space in LA-TBLG.
Full size image

a, A schematic illustration of the device. b, An optical image of a typical LA-TBLG device (Sample-S15, 30°-TBLG). Notice that a larger top gate is applied to diminish the parasitic contact doping effects. c,d, Rxx mapping in the D–n space of Sample-S15 at T = 1.5 K and B = 0 T (c) and T = 1.6 K and B = 5 T (d) (in a log scale for visual clarity). e,f, Simulated (e) and experimental (f) checkerboard patterns at LL crossings in the weak layer coupling regime of TBLG at B = 12 T in the D–n space. Notice that a background of simulated patterns is overlaid on top of experimental data in f, as a visual guide. The dashed red box indicates LL crossings of [Nb, Nt] = [1, 2]. Typical filling factions of ν = ±4, 8 and 12 are labelled. Rxx is truncated at a maximum value of 100 Ω, appearing white at the charge neutrality point. g–j, The evolution from the single-dotted to 4 × 4 checkerboard-like LL crossings at [Nb, Nt] = [1, 2], at B = 2 (g), 3 (h), 5 (i) and 8 (j) T. Data were obtained at T = 1.6 K.

Source data

Figure 1d illustrates the mapping of Rxx (in a log scale for visual clarity) in the D–n space within the same range of Fig. 1c at B = 5 T. Resistive states (circular-shaped dots) are clearly observed at each LL crossing on both the electron and hole sides, with each dot exhibiting a non-uniform size distribution. This is consistent with previous observations in a weakly coupled TBLG sample, for which the twist angle was not reported16. Note that all LL crossings here exhibit fourfold degeneracy, as can be seen in the transverse conductance σxy in Supplementary Fig. 5, when plotted in the D–n space. Hence, all the crossing points can be marked by [Nb, Nt], where the Nb and Nt represent the LL indices for the bottom and top graphene layers, as shown in Supplementary Fig. 5b.

When further increasing the perpendicular magnetic field, spin and valley degeneracy start to be lifted, and the single-dot-shaped resistive states in the D–n space at low B develop into 4 × 4 matrices. The newly developed LL crossings can be labelled by (νb, νt), within a given [Nb, Nt], where νb,t is the filling fraction of each layer. Surprisingly, it is observed that all 4 × 4 checkerboard cells exhibit the same uniform size throughout the D–n space, regardless of their parent index [Nb, Nt], which is the central finding of this work, as shown in Fig. 1e,f. Here, Fig. 1e is the theoretical simulation that agrees well with the experimental data in Fig. 1f. The x and y axes of Fig. 1e,f are linearly transformed from n to ν, and from D to D/B, respectively. We notice that some of the checkerboards are less clearly observed at certain areas of n and D, which is due to contact issue of the as-fabricated devices. More theoretical details will be discussed in the coming sections. In another typical LA-TBLG device (Sample-S37, 20°-twisted), we carried out high-resolution mapping of Rxx in the D-ν space, at [Nb, Nt] = [1, 2] (Fig. 1f, dashed red box), for different magnetic fields. As shown in Fig. 1g–j, the evolution from the single-dotted to 4 × 4 checkerboard-like LL crossings can be clearly seen, from 2 T to 8 T, which is consistent with the observations in Sample-S15, as shown in Supplementary Fig. 6. We show in Supplementary Figs. 7–9 other tested LA-TBLG samples.

Quantized D/B jumps

In this study, we mainly focus on the physical origin of the equally sized checkerboard patterns at the LL crossing, taking the [Nb, Nt] = [1, 2] checkerboard as an example for examination. The displacement field used in all the figures and discussions has been corrected by using the decoupled interlayer model, taking into account the sample-dependent interlayer quantum capacitance CGG (which largely influences the calculated D) of the TBLG16. In Supplementary Note 1 and Supplementary Fig. 10, we show that, using the decoupled TBLG model16, CGG can be fitted to be 6.30 μF cm−2 for Sample-S37. As a comparison, CGG of Sample-S15 is also calibrated in Supplementary Fig. 10. Figure 2a,b plots the line profiles of σxy and Rxx along dashed black and red lines in Fig. 2c, while the same colour codes are used for the line profiles. At D/ε0 = 0.3493 V nm−1 (outside the checkerboard), spin and valley degeneracies are fully lifted, with all integer quantum Hall states from filling 8 to 16 observed as quantized conductance plateau in σxy, and corresponding resistance minima in Rxx. Meanwhile, at D/ε0 = 0.4726 V nm−1 (centre of the checkerboard), only even-integer quantum Hall states are seen in σxy. The odd integer fillings are occupied by the resistive states at LL crossings.

Fig. 2: Quantized D/B jumps of interlayer charge transfer phase transition at fixed ν in the LL-crossing area of [Nb, Nt] = [1, 2].
Fig. 2: Quantized D/B jumps of interlayer charge transfer phase transition at fixed ν in the LL-crossing area of [Nb, Nt] = [1, 2].
Full size image

a,b, Line profiles of σxy (a) and Rxx (b) along dashed black and red lines in c. The same colour codes are used for the solid lines in a and b, and the dashed lines in c. c, The y axis of the colour mapping of Rxx is plotted in both raw D (right; corrected according to Supplementary Fig. 10 and Supplementary Note 1) and (D − D0)/B (left), where D0 is defined as the central D of the checkerboard. When plotted as (D − D0)/B, adjacent LL crossings are observed to be separated by a quantized value of e2/h when ν is fixed. d, Line profiles along ν = 12, that is, the dashed orange line in c, of Rxx and σxy, respectively. e, A drawing of the quantized 4 × 4 checkerboard at the LL crossing of [Nb, Nt] = [1, 2]. The fillings of Landau bands for (2, 6), (3, 6) and (2, 7) are schematically illustrated on the left. f, A cartoon illustration for the event of charge transfer at critical D for any phase boundaries (marked as white boxes) along fixed ν in e.

Source data

Interestingly, when scaling the displacement field into D/B, all the resistive states at LL crossings are found to be located at a quantized value in the (D − D0)/B axis with respect to the centre of the checkerboard, in units of e2/2h. This effect can be more pronounced when plotted as line profiles along ν = 12 of both Rxx and σxy, with the y axes shifted to the centre of the checkerboard, noted as (D − D0)/B, shown in Fig. 2d. We also rescaled all the y axes into D/B for Fig. 1e–f, or (D − D0)/B for Fig. 1g–j, respectively.

In the LL-crossing checkerboard plotted in the parameter space of (D − D0)/B and filling fraction ν, two types of quantized properties can be observed, with one of them being the well-known quantum Hall effect-originated quantization of filling fractions and the other being the interval in (D − D0)/B for Rxx peaks, as indicated by the arrows in Fig. 2c. We now provide a theoretical explanation for this observation. For low-energy electrons, our current system can be approximated by two layers that are decoupled at the single-particle level because of the large twist angle, but capacitively coupled by Coulomb interaction. Due to the spin and valley degeneracies within each graphene layer, the filling factor νi and LL index Ni have the following relation: \({\nu }_{i}=4{N}_{i}-2+{\mathop{\nu }\limits^{ \sim }}_{i}\), where i is the layer index, \({\mathop{\nu }\limits^{ \sim }}_{i}\) is the filling factor of the partially filled LL with index Ni, and \(0\le {\mathop{\nu }\limits^{ \sim }}_{i}\le 4\). Hence, the filling factor within the LL-crossing checkerboard indexed by [Nb, Nt] can be an integer number between 4(Nb + Nt) ± 4, also written as \(\nu ={\nu }_{{\rm{b}}}+{\nu }_{{\rm{t}}}=4({N}_{{\rm{b}}}+{N}_{{\rm{t}}}-1)+\mathop{\nu }\limits^{ \sim }\), where \(\mathop{\nu }\limits^{ \sim }={\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}+{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\). We construct the total energy per area (defined as Etot), which comprises three terms: the single-particle energy of the occupied LLs, the classical electrostatic energy (including the layer potential difference generated by the D field and the capacitance energy) and the intralayer exchange energy, as described in Methods. Theoretically, we assume that the integer quantum Hall insulators are formed within each layer, neglecting the interlayer exchange interaction and Zeeman energy (the subtle effects of spin ordering and Zeeman effect are discussed in Methods).

In the LL-crossing checkerboard indexed by [Nb, Nt], the states at a fixed ν can change from one phase \(\{{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}};{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\}\) to another adjacent one \(\{{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}-1;{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}+1\}\) driven by the displacement field, leading to a phase transition that yields discrete phase boundaries (corresponding to experimentally observed resistive peaks at a given ν in the checkerboards). The critical displacement field can then be found at the conditions when the energies of the above two adjacent filling phases become degenerate, as shown in Methods. Solving them, one can obtain all the critical field \(\mathop{D}\limits^{ \sim }=\frac{D}{e{n}_{0}}=\frac{D}{B}/(\frac{{e}^{2}}{h})\) in each checkerboard to be 0, ±1/2, ±1 and ±3/2 measured relative to the centre of the checkerboard, as given in the third column in Table 1. At this stage, it is clear that δD/B between each adjacent phase boundary at fixed ν is quantized to e2/h. As our theory shows, the value of δD/B within a checkerboard is fully determined by the competition between the D-field-driven layer potential difference and the Coulomb-driven capacitance energy, where the former (latter) favours layer polarization (equal charge distribution between the two layers), which makes δD/B independent of the parent index [Nb, Nt]. As integer quantum Hall insulators are incompressible with an integer number of electrons per flux quantum, the charge transfer at the crossings is exactly one charge per Landau orbit of area \(2{{\uppi }}{l}_{B}^{2}\), making δD quantized to \(\frac{e}{2{{\uppi }}{l}_{B}^{2}}\). Using the experimentally extracted CGG, we plot the full map of checkerboards in the space of D/B and ν in Fig. 1e (and the background in Fig. 1f), which agrees well with experimental results.

Table 1 Critical displacement fields for \(\mathop{{{\nu }}}\limits^{{{ \sim }}}={1},{2},{3},{4}\)
Full size table

Figure 2e illustrates the theoretical picture discussed above, taking the LL crossing indexed by [Nb, Nt] = [1, 2] as an example. White boxes denote the phase boundaries at which the states change along the vertical direction (that is, the fixed ν), driven by the displacement field. For the three filling states (νb, νt) = (2, 6), (3, 6) and (2, 7), LLs alignments for each layer are illustrated in the circled schematics (blue and pink colour denote LLs for bottom and top layers, respectively). Those white boxes form an evenly distributed matrix with 4 × 4 elements, due to the quantization of both ν in the x axis, and D/B in the y axis, respectively. The latter quantization of the measurable D/B within a checkerboard at LL crossing was not quantitatively established before, although similar matrices of resistive peaks in LL crossing of twisted graphene have been reported elsewhere19. Charge transfer between LLs in a two-subband system of double quantum wells was also investigated; however, the quantization signature reported here was not observed there37. In the experiment, the two solid blue and pink arrows in Fig. 2e indicate the sweeping direction in terms of bottom and top gates, respectively (see also Supplementary Fig. 6d). The quantization of δD/B is thus a manifestation of the well-defined interlayer distance in LA-TBLG with well-developed quantum Hall ferromagnetism of spin and valley ordering, studied in the current work. Figure 2f illustrates a cartoon drawing that depicts the observed phenomena using a simplified representation of charge transfer between LLs. The density per Landau orbit area is written as δn = eB/h, while the charge change is also defined according to the capacitance coupling by eδn = δD. Taking the above two relations, one obtains δD/B = e2/h.

We further investigate the B and T dependencies of D/B quantizations at fixed ν within one quantized checkerboard. Figure 3a depicts the line profiles of Rxx for ν from 9 to 15, for the checkerboard at LL crossing of [Nb, Nt] = [1, 2] measured in Sample-S37 at T = 1.6 K and B = 12 T (the same analysis of Sample-S15 can be found in Supplementary Fig. 11). These resistive peaks at critical D can be fitted using the Gaussian distribution, and hence the distance between each pair of adjacent peaks can be determined as δD/B. We plot the Rxx at ν = 12 as a function of magnetic fields, as shown in Fig. 3b. It can be seen that, at relatively low magnetic fields, the LL crossings still appear as a single dotted resistive peak, as no degeneracy lifting has occurred. Above about 3 T, a clear four-peak feature can be observed in the line profile, and peaks are well developed at about 5 T. Figure 3c illustrates the measured δD/B (in units of e2/h) as a function of magnetic field, and the experimental data distribute around the unity, which is expected according to our theoretical modelling. The data in Fig. 3c are derived from the statistical analysis of nine δD/B values, as detailed in Supplementary Tables 1 and 2. More data on the temperature dependence of (D − D0)/B are given in Supplementary Fig. 12.

Fig. 3: Magnetic field dependence of D/B quantizations at fixed ν in the LL-crossing area of [Nb, Nt] = [1, 2].
Fig. 3: Magnetic field dependence of D/B quantizations at fixed ν in the LL-crossing area of [Nb, Nt] = [1, 2].
Full size image

a, The line profiles of Rxx at 1.6 K with 12 T magnetic field at filling factors ν from 9 to 15, which were fitted using the Gaussian method to extract the precise crossing Rxx peak positions in the checkerboard. b, Rxx at ν = 12 as a function of (D − D0)/B at different magnetic fields from 1 T to 12 T. Data are shifted on the y axis for visual clarity. c, The statistics of δ(D/B) as a function of magnetic field. Quantization at e2/h of δ(D/B) is clearly seen in b. Error bars in c are defined as the standard error of the mean of nine values of δ(D/B) at each magnetic field.

Source data

Tuning the LL-crossing checkerboard with Zeeman energy

At this stage, it is noticed that the uniform 4 × 4 matrix checkerboard patterns were observed in only a small fraction of (fewer than 1/10) the as-fabricated samples, including Sample-S15 and Sample-S37. As shown in Supplementary Figs. 9–11, the checkerboard patterns in other samples were distorted, where these resistive states (crossing points in the 4 × 4 matrix) are divided into four subgroups. Our theory described in Methods, which assumes spin conservation in the charge transfer, is found to match well with the data for uniform 4 × 4 matrix checkerboards. However, different samples may exhibit different spin orderings due to the competition between Zeeman energy and the atomic-scale interactions. To examine the Zeeman effect, we performed tilted-field magnetotransport of Sample-S37. As illustrated in Supplementary Fig. 13a, the LA-TBLG sample is subjected to a tilted field, while the total magnetic field (Btot) is increased with B⊥ held constant. The in-plane projection B∥ = \({B}_{{\rm{t}}{\rm{o}}{\rm{t}}}\sin \varphi\) can thus effectively tune the Zeeman energy in the tested system, where φ is the angle of tilt.

As shown in Supplementary Fig. 13b–d, at a fixed index of [Nb, Nt] = [1, 2] and a constant B⊥ = 5 T at T = 1.6 K, the 4 × 4 matrix checkerboard is tuned from its initial uniform distribution (Supplementary Fig. 13b, φ = 0°) into gradually distorted patterns (Supplementary Fig. 13c, φ = 45°). For the case of φ = 65° in Supplementary Fig. 13d, the checkerboard is clearly divided into four subgroups, with the differences between adjacent Rxx in the axis of (D − D0)/B, defined as δD/B, developing into two typical values, marked as δ1 and δ2, respectively, as indicated in Supplementary Fig. 13e. Line profiles of Rxx against (D − D0)/B at ν = 12 for the case of tilted field angle φ = 0° (red line) and φ = 65° (green line) are plotted in Supplementary Fig. 13e, with the Rxx value shifted for visual clarity. With the in-plane magnetic field B∥ varied from zero (red) to 10.72 T (φ = 65°, green), the value of δ2 is almost doubled. Supplementary Fig. 13f further plots δD/B as a function of φ. While δ2 increases drastically with the tilt angle, δ1 remains remarkably quantized at e2/h. This is direct evidence that the checkerboard pattern can be effectively influenced by tuning the Zeeman energy in the system. Despite this influence, the quantization of δD/B in e2/h can still be manifested, for example, through δ1. Further detailed theory that takes into account different spin orders will be needed to quantitatively describe such subtle behaviours.

Finally, we show a potential application using the phenomenon of uniform 4 × 4 matrices of quantized LL-crossing checkerboards for LA-TBLG devices. Magnetometers can be classified into weak-field sensors (superconducting quantum interference device, nitrogen–vacancy centres, fluxgates and so on) that maximize fT-pT/\(\sqrt{{\rm{H}}{\rm{z}}}\) sensitivity in near-zero backgrounds—often needing shielding and offering limited dynamic range—and strong-field sensors (vibrating sample magnetometer, Hall sensors, nuclear magnetic resonance pickup coils and so on) that operate inside multi-tesla static or pulsed magnets, prioritizing range and robustness over ultimate sensitivity38,39. In our study, the proposed sensor uniquely combines scalable on-chip integration and micrometre-scale spatial resolution for sensing ultrahigh B-fields (can be above 30 T) under cryogenic conditions.

When plotting the magnetic field B in tesla against δD with a unit of C m−2, the slope should correspond to the von Klitzing constant h/e2. Intriguingly, such a linear dependence of B–δD can, in principle, be further used as a kind of quantum magnetometer. Interestingly, these micrometre-sized magnetometric sensors (Fig. 4a), incorporating LA-TBLG, can be readily integrated into an on-chip sensor arrays, as illustrated in Fig. 4b. This further enables surface magnetic field mapping at millimetre-to-centimetre scales (Fig. 4c). One potential application is the spatial calibration of high magnetic fields, such as within the spherical region of a superconducting coil. Naturally, implementing the wiring to connect each Hall bar in such an array would require algorithms and data acquisition hardware, which represents a challenging engineering task beyond the scope of the current study and is left for future work. Albeit challenging, recent studies suggest that twisted-graphene films obtained by chemical vapour deposition23 would be a possible route to realize large-scale arrays of LA-TBLG devices illustrated in Fig. 4a.

Fig. 4: Quantized LL-crossing checkerboards as cryogenic magnetometric sensors.
Fig. 4: Quantized LL-crossing checkerboards as cryogenic magnetometric sensors.
Full size image

a, Schematic of a typical LA-TBLG sample, with the twist angle illustrated in a side inset. b, The proposed magnetometric sensor consisting an array of LA-TBLG devices (as shown in a), scalable from millimetres to centimetres, operating under a perpendicular magnetic field B. c, Schematic representation of magnetic field mapping using cryogenic magnetometric sensor arrays (as in b), with equipotential lines schematically drawn, indicating the potential capability of surface mapping of magnetic fields. d, The magnetic field B plotted against δD (in C m−2). The slope is then equal to the von Klitzing constant. This linear dependence of B–δD, once calibrated using experimental data at relatively low B, can then be used as cryogenic magnetometric sensors to measure unknown B at high B and low T, by reading the δD in the quantized LL-crossing checkerboard. Error bars in d are defined as the standard error of the mean of δD. The error bars for the 3–12 T are obtained from 9 δD of Sample-S37 at each field, and the error bars for 20 T and 30 T are estimated from the 14 and 6 δD of Sample-S24, respectively.

Source data

As a proof of concept, we calibrated a typical B-to-δD slope using experimental data at relatively low B, as shown in Fig. 4d. Such a calibrated device can then be utilized as a sensor for measuring unknown B at high B and low T, by simply reading the δD in the quantized LL-crossing checkerboard. Regarding the sensitivity of magnetic field strength, we provide an initial estimation based on noise measurements of our device under fixed experimental conditions. As shown in Supplementary Figs. 14–17 and Supplementary Table 3, the uncertainty in our device is estimated to be 0.68% at 20 T, and 1.06% at 30 T, respectively. In the language of weak-field sensing, the measured noise floor allows us to infer a magnetic field sensitivity at the order of approximately 0.1 – \(\text{0.4}\,{\rm{T}}/\sqrt{{\rm{H}}{\rm{z}}}\) for B ranging from 3 T to 30 T. This level of sensitivity, although not fully optimized, demonstrates the potential of our device for applications in high-field magnetometry.

To conclude, we have devised a system of LA (20°–30°) TBLG, in which equal-sized 4 × 4 checkerboards are seen at each interlayer LL-crossing point throughout the D–n parameter space, when all spin- and valley-flavour degeneracies are lifted. When looking into one of such checkerboards in the D–n (or D/B–ν) space, in addition to the well-known quantum Hall effect-driven quantized properties on the ν axis, varying D at a fixed integer filling fraction ν will also yield quantized distance between resistance peaks in the D/B axis in a unit of e2/h. Such quantization of δD/B originates from the charge quantization per Landau orbit of integer quantum Hall states. The intriguing linear relationship between magnetic field B and intervals of critical displacement fields δD, with a slope equal to the conductance quantum, in LA-TBLG-based devices may be exploited as a cryogenic magnetometer beyond conventional approaches. For example, it can be scaled up into sensor arrays to map in-plane magnetic field distributions with micrometre-scale spatial resolution, limited in principle by the device size. The self-calibrating nature of the LL-crossing checkerboards enables low-field calibration, as well as high-field sensing through a δD readout process.

Methods

Sample fabrication

Van der Waals few-layer structures of the h-BN/graphene/h-BN sandwich were obtained by mechanically exfoliating high-quality bulk crystals. The vertical assembly of these van der Waals layered compounds was fabricated using a dry-transfer method under ambient conditions. The h-BN/graphene/h-BN sandwich was then transferred onto the prefabricated h-BN/Au substrate. Hall bars of the devices were achieved by reactive ion etching. During the fabrication processes, electron beam lithography was performed using a Zeiss Sigma 300 SEM with a Raith Elphy Quantum graphic writer. One-dimensional edge contacts were achieved using electron beam evaporation. After atomic layer deposition of about 30 nm Al2O3, a big top gate was deposited to form the complete dual-gated h-BN-encapsulated LA-TBLG devices as shown in Fig. 1a,b. Gate and contact electrodes were fabricated by electron beam evaporation, with typical Au/Ti thicknesses of ≈30/5 nm and ≈50/5 nm, respectively.

a.c. electrical measurements

During measurements, the graphene layers were fed with an a.c. Ibias of about 50–500 nA. The longitudinal and Hall voltages were recorded using low-frequency SR830 lock-in amplifiers. Four-probe measurements were used throughout the transport measurements under a high magnetic field and at low temperatures in an Oxford TeslaTron cryostat. Gate voltages on the as-prepared Hall bar devices were maintained by a Keithley 2400 source meter.

High-magnetic-field facility

A water-cooled magnet with maximum of 30 T magnetic field and base temperature of 1.7 K was used. The facility is equipped with a water-cooling system and is maintained by the Steady High Magnetic Field Facilities at the High Magnetic Field Laboratory, Chinese Academy of Sciences.

Theory of LL-crossing checkerboards

Energetics

We formulate the energy of the LA-TBLG in a strong magnetic field. Because of the large twist angle, the low-energy electronic states of the two layers can be described by decoupled Dirac fermions at the single-particle level. In the presence of the magnetic field, the Dirac fermions form quantized LLs. We denote the LL filling factor of the i layer as νi, where i = b and t, respectively, for the bottom (b) and top (t) layer. We use Ni to label the index of the partially filled LL nearby the Fermi energy in the i layer. Given the spin and valley degrees of freedom within each graphene layer, νi and Ni have the following relation:

$${\nu }_{i}=4{N}_{i}-2+{\mathop{\nu }\limits^{ \sim }}_{i},$$
(1)

where \({\mathop{\nu }\limits^{ \sim }}_{i}\) is the filling factor of the partially filled LL with index Ni, and \(0\le {\mathop{\nu }\limits^{ \sim }}_{i}\le 4\).

The electron density of the i layer is ni = νin0. Here, n0 is the electron density per LL given by

$${n}_{0}=\frac{1}{2{{\uppi }}{l}_{B}^{2}}=\frac{eB}{h},$$
(2)

where \({l}_{B}=\sqrt{\hslash /(eB)}\) is the magnetic length.

The total filling factor is

$${\nu }_{{\rm{t}}{\rm{o}}{\rm{t}}}={\nu }_{{\rm{b}}}+{\nu }_{{\rm{t}}}=4({N}_{{\rm{b}}}+{N}_{{\rm{t}}}-1)+\mathop{\nu }\limits^{ \sim },$$
(3)

where \(\mathop{\nu }\limits^{ \sim }={\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}+{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\).

In our theory, the system can be described using four indices \(\{{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}};{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\}\) in the quantum Hall regime, which determines the filling factor of each layer and the total filling factor.

The total energy per area Etot of the system in the quantum Hall regime can be decomposed into the following contributions:

$${E}_{{\rm{t}}{\rm{o}}{\rm{t}}}[{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}};{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}]={E}_{{\rm{L}}{\rm{L}}}+{E}_{{\rm{C}}}+{E}_{{\rm{X}}}.$$
(4)

First, ELL accounts for the single-particle energies of the occupied LLs,

$${E}_{{\rm{L}}{\rm{L}}}=\mathop{\sum }\limits_{i=\rm{b,t}}\left\{\mathop{\sum }\limits_{N=-\infty }^{{N}_{i}-1}4{{\mathcal{E}}}_{\mathrm{LL}}(N){n}_{0}+{{\mathcal{E}}}_{\mathrm{LL}}({N}_{i}){\mathop{\nu }\limits^{ \sim }}_{i}{n}_{0}\right\},$$
(5)

where \({{\mathcal{E}}}_{{\rm{L}}{\rm{L}}}(N)={\rm{s}}{\rm{g}}{\rm{n}}(N\,){v}_{{\rm{F}}}\sqrt{2\hslash eB| N| }\) is the energy of the Nth LL, with vF being the velocity of the Dirac fermion.

Second, EC describes the classical electrostatic energy for the charge distribution of the bilayer

$${E}_{{\rm{C}}}=\frac{U}{2}({n}_{{\rm{b}}}-{n}_{{\rm{t}}})-\frac{{e}^{2}{d}_{{\rm{G}}{\rm{G}}}}{2{\varepsilon }_{{\rm{G}}{\rm{G}}}}{n}_{{\rm{t}}}{n}_{{\rm{b}}}={E}_{0}[\mathop{D}\limits^{ \sim }({\nu }_{{\rm{b}}}-{\nu }_{{\rm{t}}})-{\nu }_{{\rm{t}}}{\nu }_{{\rm{b}}}],$$
(6)

where U = eDdGG/εGG is the layer potential difference generated by the displacement field D, e > 0 is the elementary charge, dGG is the interlayer distance and εGG is the dielectric constant. The second term in EC arises from the capacitor energy of the charged bilayer, where the capacitance per area is CGG = εGG/dGG. In equation (6), \({E}_{0}={e}^{2}{n}_{0}^{2}/(2{C}_{{\rm{G}}{\rm{G}}})\) sets a scale for the electrostatic energy, and \(\mathop{D}\limits^{ \sim }\) is given by

$$\mathop{D}\limits^{ \sim }=\frac{D}{e{n}_{0}}=\frac{D}{B}/\left(\frac{{e}^{2}}{h}\right),$$
(7)

where \(\mathop{D}\limits^{ \sim }\) is dimensionless.

Finally, EX is the intralayer exchange energy of the filled LLs. We assume that EX takes the following form:

$${E}_{{\rm{X}}}=\mathop{\sum }\limits_{i=\rm{b,t}}\left\{(4-{\mathop{\nu }\limits^{ \sim }}_{i}){{\mathcal{E}}}_{{\rm{X}}}({N}_{i}-1)+{\mathop{\nu }\limits^{ \sim }}_{i}{{\mathcal{E}}}_{{\rm{X}}}({N}_{i})\right\}.$$
(8)

The assumption is as follows. Because \({\mathop{\nu }\limits^{ \sim }}_{i}\) flavours occupy LLs with index N ≤ Ni, they contribute exchange energy of \({\mathop{\nu }\limits^{ \sim }}_{i}{{\mathcal{E}}}_{{\rm{X}}}({N}_{i})\), with \({{\mathcal{E}}}_{{\rm{X}}}({N}_{i})\) being the exchange energy per flavour. A similar reasoning leads to the other term \((4-{\mathop{\nu }\limits^{ \sim }}_{i}){{\mathcal{E}}}_{{\rm{X}}}({N}_{i}-1)\). We assume that the exchange energy is additive regarding the flavour degree of freedom, which is reasonable because of the approximate SU(4) symmetry of graphene in the quantum Hall regime.

The atomic-scale interactions in graphene are known to break the SU(4) symmetry in the spin and valley flavour space and act as perturbations to lift degeneracies between states that would otherwise be related by SU(4) symmetry. However, their energy scale compared with the dominant SU(4) symmetric long-range Coulomb interaction is smaller by a factor of a0/lB, where a0 is the monolayer graphene lattice constant and lB is the magnetic length40,41. For a typical magnetic length of 10 T, a0/lB ≈ 0.03. Therefore, the total energy is mainly contributed by the SU(4) symmetric long-range Coulomb interaction. This is why the exchange energy is approximately additive with respect to the flavour degree of freedom, as the long-range Coulomb interaction dominates.

Motivated by the experimental observation, we focus on the case that both νb and νt are integers. Our experiment shows no evidence of an interlayer coherent state. Accordingly, we theoretically consider quantum Hall states that preserve the layer U(1) symmetry, so that the particle numbers in each layer are conserved. For this class of states, the absence of interlayer coherence causes the interlayer exchange interaction to vanish. Meanwhile, EC represents the overall effect of intralayer and interlayer electrostatic energies.

We note that the Zeeman energy is not included in equation (4). We discuss the effect of the spin ordering and Zeeman energy separately.

Critical displacement fields

At a critical displacement field for the LL crossing, the energies of two different states become degenerate. When the crossing checkerboard is formed, the filling factor within each layer changes by 1 at each critical displacement field. Therefore, we compare the energies of two states described respectively by \(\{{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}-1;{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}+1\}\) and \(\{{N}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}};{N}_{{\rm{t}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\}\),

$$\begin{array}{l} {E}_{{\rm{t}}{\rm{o}}{\rm{t}}}\left[{N}_{{\rm{b}}},{\mathop{\nu}\limits^{\sim}}_{{\rm{b}}}-1;{N}_{{\rm{t}}},{\mathop{\nu}\limits^{\sim}}_{{\rm{t}}}+1\right]-{E}_{{\rm{t}}{\rm{o}}{\rm{t}}}\left[{N}_{{\rm{b}}},{\mathop{\nu}\limits^{\sim}}_{{\rm{b}}};{N}_{{\rm{t}}},{\mathop{\nu }\limits^{\sim}}_{{\rm{t}}}\right]\\ =\left[{{\mathcal{E}}}_{\mathrm{LL}}({N}_{{\rm{t}}})-{{\mathcal{E}}}_{\mathrm{LL}}({N}_{{\rm{b}}})\right]{n}_{0}+{E}_{0}\left[-2\mathop{D}\limits^{ \sim }+1+4({N}_{{\rm{t}}}-{N}_{{\rm{b}}})+{\mathop{\nu }\limits^{\sim}}_{{\rm{t}}}-{\mathop{\nu}\limits^{\sim}}_{{\rm{b}}}\right]\\ +\left.\left. \left[{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{t}}})-{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{t}}}-1)\right)\right]-\left[{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{b}}})-{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{b}}}-1)\right)\right].\end{array}$$
(9)

By setting equation (9) to zero, we obtain the critical displacement field, which is given by

$${\mathop{D}\limits^{ \sim }}_{{N}_{{\rm{b}}},{N}_{{\rm{t}}}}^{(c)}\left({\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}},{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\leftrightarrow {\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}-1,{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}+1\right)={\mathop{D}\limits^{ \sim }}_{{N}_{{\rm{b}}},{N}_{{\rm{t}}}}^{(0)}+\frac{{\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}-{\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}+1}{2},$$
(10)

where \({\mathop{D}\limits^{ \sim }}_{{N}_{{\rm{b}}},{N}_{{\rm{t}}}}^{(0)}\) is independent of \({\mathop{\nu }\limits^{ \sim }}_{{\rm{b}}}\) and \({\mathop{\nu }\limits^{ \sim }}_{{\rm{t}}}\),

$$\begin{array}{rcl}{\mathop{D}\limits^{ \sim }}_{{N}_{{\rm{b}}},{N}_{{\rm{t}}}}^{(0)} & = & \left. \left\{\left[{{\mathcal{E}}}_{\mathrm{LL}}({N}_{{\rm{t}}})-{{\mathcal{E}}}_{\mathrm{LL}}({N}_{{\rm{b}}})\right]{n}_{0}+\left[{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{t}}})-{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{t}}}-1)\right)\right]\right.\\&& \left. \left.-\left[{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{b}}})-{{\mathcal{E}}}_{{\rm{X}}}({N}_{{\rm{b}}}-1)\right)\right]\right\}/2{E}_{0}+2({N}_{{\rm{t}}}-{N}_{{\rm{b}}}).\end{array}$$
(11)

Here, the dimensionless \(\mathop{D}\limits^{ \sim }\) and the physical D are related through equation (7). \({\mathop{D}\limits^{ \sim }}_{{N}_{{\rm{b}}},{N}_{{\rm{t}}}}^{(0)}\) is the displacement field at the centre of the checkerboard labelled by (Nb, Nt). Equation (10) correctly captures the quantized interval of critical displacement fields within each LL-crossing checkerboard, as listed in Table 1.

Spin ordering and Zeeman effect

The effect of Zeeman energy, which is not taken into account in the above derivation, depends on the spin ordering at each filling factor. Meanwhile, the spin and valley ordering in the quantum Hall regime of graphene is determined by microscopic physics down to the atomic scale. Depending on the spin polarization at each filling factor, the Zeeman energy can shift the critical displacement fields and lead to corrections on the quantized interval. However, for Sample-S15 and Sample-S37, the intervals of critical displacement fields within each checkerboard display descent quantized value, as if the Zeeman energy does not play a role. One scenario for this observation is that the charge transfer between the two layers at the critical displacement fields is spin-conserved; the Zeeman energy of the system does not change across the transition, which leads to no correction on the critical displacement fields.

The Zeeman energy should be identical across different samples under the same magnetic field; however, atomic-scale interactions may vary between samples owing to differences in the screening environment, as demonstrated by previous experiments. Specifically, scanning tunnelling microscopy studies of charge-neutral graphene in strong magnetic fields have revealed a Kekulé-distorted ground state42, whereas non-local spin and charge transport measurements on double-encapsulated graphene with a higher effective dielectric constant have identified an antiferromagnetic ground state43. A theoretical mechanism that takes into account LL mixing shows that the atomic-scale interactions are indeed sensitive to the dielectric environment, providing an explanation of the above seemingly conflicting observations44. In our case, the screening environment can vary across samples owing to differences in the thickness of the h-BN-encapsulating layers, which can lead to distinct atomic-scale interactions and spin orderings.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.