Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Long-range ballistic transport of Brown-Zak fermions in graphene superlattices


In quantizing magnetic fields, graphene superlattices exhibit a complex fractal spectrum often referred to as the Hofstadter butterfly. It can be viewed as a collection of Landau levels that arise from quantization of Brown-Zak minibands recurring at rational (p/q) fractions of the magnetic flux quantum per superlattice unit cell. Here we show that, in graphene-on-boron-nitride superlattices, Brown-Zak fermions can exhibit mobilities above 106 cm2 V−1 s−1 and the mean free path exceeding several micrometers. The exceptional quality of our devices allows us to show that Brown-Zak minibands are 4q times degenerate and all the degeneracies (spin, valley and mini-valley) can be lifted by exchange interactions below 1 K. We also found negative bend resistance at 1/q fractions for electrical probes placed as far as several micrometers apart. The latter observation highlights the fact that Brown-Zak fermions are Bloch quasiparticles propagating in high fields along straight trajectories, just like electrons in zero field.


Van der Waals assembly offers a possibility to create materials by stacking atomically-thin layers of different crystals1,2,3. One of the simplest and most studied van der Waals heterostructures is graphene encapsulated between two hexagonal boron nitride (hBN) crystals. The encapsulation protects graphene from extrinsic disorder4,5, allowing ultra-high electronic quality and micrometer-scale ballistic transport often limited only by edge scattering6,7. A special case of encapsulated graphene heterostructures is graphene superlattices, where crystallographic axes of graphene and hBN are intentionally aligned. A small (1.8%) mismatch between graphene and hBN crystal lattices results in a periodic moiré potential acting on charge carriers in graphene and leading to the formation of electronic minibands1,2,3,8,9,10,11,12,13,14,15,16,17.

A relatively large (14 nm) periodicity of graphene-on-hBN superlattices has also made it possible to study the regime of Hofstadter butterflies, which requires the magnetic flux ϕ per superlattice unit cell to be comparable to the flux quantum ϕ0 in experimentally-accessible magnetic fields B. At B = Bp/q corresponding to ϕ = ϕ0 p/q, where p and q are integer, the translational symmetry of the electronic system is restored (despite the presence of a quantizing magnetic field) and the superlattice’s electronic spectrum can again be described in terms of Bloch states18,19,20,21,22,23,24,25,26,27,28,29, just like in B = 0. These high-field Bloch states are characterized by their own miniband spectra25,26,27,28,29 different from the zero-B spectrum of graphene-on-hBN superlattices. The associated quasiparticles are referred to as Brown-Zak (BZ) fermions. According to the group-theory analysis, an electronic spectrum for each realization of BZ fermions should have an additional q-fold degeneracy18,19,20,21,22,23,24,25,26,27,30,31 (that is, contains q equivalent mini-valleys). This degeneracy is additional to the 4-fold spin and valley degeneracy of graphene’s original spectrum. Importantly, BZ fermions are Bloch quasiparticles, like electrons in solids or Dirac fermions, and, at B = Bp/q, they move through the superlattice as if the applied field is zero. Away from these exact values, BZ fermions experience an effective magnetic field Beff = B − Bp/q (refs. 18,19,20,21,22,23,24,25,26,27,30,31).


Experimental devices and measurement setup

We report the electronic properties of BZ fermions with different p and q using high-quality graphene superlattices. These devices were fabricated using the standard dry transfer procedures, where the studied graphene crystal was carefully aligned with one of the encapsulating hBN crystals using the crystallographic edges11. The alignment was verified by Raman spectroscopy32 prior to encapsulation with the second hBN crystal. The latter was intentionally misaligned to avoid competing moiré patterns33,34,35. The assembled stacks were placed on an oxidized Si wafer, which allowed us to apply the back-gate voltage Vg to control the carrier density n. We studied six devices that were shaped into the multiterminal Hall bar geometry and had the main channel widths W ranging from 2 to 17 μm (see Fig. 1a and Supplementary Note 1). The devices were first characterized by measuring their longitudinal resistivity ρxx in zero B and Hall resistivity ρxy in small non-quantising B below 0.1 T. The latter enabled us to find the n(Vg) dependences, except for gate voltages close to the neutrality points (NPs) and van Hove singularities (vHS), where ρxy reversed its sign and could no longer be described by the standard dependence ρxy = B/ne (e is the electron charge). All our devices exhibited very high carrier mobilities μ = (ρxxne)−1 of the order of 106 cm2 V−1 s−1, which were still somewhat reduced by edge scattering because of finite W (Fig. 1b). We corroborated the high quality of our devices using transverse magnetic focusing measurements (Supplementary Note 2). In the main text, we focus on two large-width devices (D1 and D2) exhibiting highest μ.

Fig. 1: High-quality graphene superlattices and their transport properties.
figure 1

a Optical micrograph of one of our devices (D1; twist angle θ between graphene and hBN of about 0.4°). The Hall bar is seen in violet with golden electrical contacts. b Mobility and mean free path for D1 measured at zero B and 10 mK. Semitransparent vertical strips indicate the doping regions around NPs and vHS where n could not be extracted directly from Hall measurements and charge inhomogeneity also plays a role. To calculate μ and l within the shaded regions, we assumed a constant gate capacitance and linearly extrapolated the n(Vg) dependences found sufficiently far from NPs and vHS (Supplementary Note 3). The noisy behavior at large values of μ and l arises from ρxx becoming small (1 Ohm, about 4 orders of magnitude smaller than that at the NPs). The horizontal black line indicates the device width W. c σxx(Vg, B) measured by sweeping Vg and varying B in small steps of 40 mT. T = 10 and 250 mK below and above 14 T, respectively. Indigo-to-yellow colors: Log scale truncated between 38 nS and 16 mS for B < 14 T and between 4 nS and 0.4 mS above 14 T. White rectangles: these regions are shown in finer detail in Figs. 3 and 4. d Same as in (b) but for ϕ/ϕ0 = 1/2 (B ≈ 15 T); T = 250 mK. In addition to NPs and vHS, the gray strips also cover a wide region of the quantum Hall regime (|Vg | < 20 V), which is dominated by large cyclotron gaps in the main graphene spectrum. The transport data used to calculate μ and l in (b) and (d) are shown in Supplementary Note 3.

Figure 1c shows a map of the longitudinal conductivity \(\sigma _{{\mathrm{xx}}} = \rho _{{\mathrm{xx}}}/(\rho _{{\mathrm{xx}}}^2 + \rho _{{\mathrm{xy}}}^2)\) as a function of Vg and B in fields up to 18 T. The main features on such maps have been well understood in terms of the Hofstadter spectrum for Dirac fermions in moiré superlattices11,12,13,14,15,16,17,22. One can see numerous Landau levels (LLs) fanning out from the main and secondary NPs (miniband edges) which are located at Vg near 0 and ±45 V, respectively. There are also pronounced funneling features near van Hove singularities at Vg ≈ −60, −35, +40 and +55 V in both hole (−) and electron (+) parts of the spectrum. Another important attribute of the map is horizontal yellow streaks that occur at ϕ/ϕ0 p/q (Fig. 1c). If temperature T is increased above 100 K so that Landau quantization is strongly suppressed, the horizontal streaks become the only dominant feature on such transport maps28,29. In the high-T regime, the streaks represent oscillations in both ρxx (B) and ρxx (B) at a constant Vg. Their 1/B frequency is independent of n, and they were named BZ oscillations28. The horizontal streaks are seen in Fig. 1c correspond to maxima in σxx and zeros in ρxx and, as explained in the introduction, reflect the recovery of translational symmetry (p flux quanta penetrate through q superlattice unit cells) and the emergence of Bloch states experiencing zero Beff. Along each horizontal streak, one can find numerous NPs and vHS, which reflect different realizations of BZ fermions at each Bp/q. Landau mini-fans radiate from these NPs in both directions along the B axis (Fig. 1c), representing Landau quantization of BZ electronic spectra by non-zero Beff (see below)18,19,20,21,22,23,24,25,26,27,28,29. Another notable feature of the shown σxx map is a repetitive triangular-like pattern seen most clearly between 2 and 12 T, especially for positive Vg. The yellow triangles are made of horizontal streaks at zero Beff, vertical streaks emerging from NPs for BZ fermions and slanted streaks originating from vHS.

Ballistic transport

The transport properties of BZ fermions can be analyzed in the same way as in Fig. 1b for Dirac fermions. The results are plotted in Fig. 1d for the case of ϕ/ϕ0 = 1/2 and show that, away from NPs and vHS, BZ fermions in our devices exhibit μ reaching a few 106 cm2 V−1 s−1. This is comparable to μ of Dirac fermions in zero B. Another important characteristic of charge carrier transport is the mean free path l. It can be evaluated for both Dirac and BZ fermions, using the standard expression \(\sigma _{{\mathrm{xx}}} = g\frac{{e^2}}{h}\left( {\frac{{k_{\mathrm{F}}l}}{2}} \right)\) where h is the Planck constant, kF is the Fermi momentum and g is the degeneracy. In zero B, g = 4 because of the spin and valley degeneracy of Dirac fermions. BZ fermions are expected to have an additional, mini-valley degeneracy18,19,20,21,22,23,24,25,26,27,30,31, which is equal to q (that is, g = 8 for the case of Fig. 1d). Using these g, we calculated l as shown in the bottom panels of Fig. 1b, d. The mean free path in zero B reached > 20 μm, implying that ballistic transport in our devices is limited by W rather than impurity scattering. For some realizations of BZ fermions, their mean free path also exceeds 10 μm, marginally smaller than l for Dirac fermions (cf. Fig. 1b, d). Supplementary Note 4 provides similar analysis for μ and l at other values of q.

We corroborate the existence of ballistic BZ fermions using so-called bend resistance geometry5,36 sketched in Fig. 2a. The geometry allows one to detect if charge carriers can move ballistically over the entire channel width W, along straight trajectories connecting current and voltage contacts (see caption of Fig. 2a). If this is the case, ballistic transport gives rise to the negative sign of the bend resistance Rb (ref. 36) in contrast to its conventional, positive sign for diffusive (ohmic) transport. As expected from very long l of Dirac fermions, Rb was negative in zero B everywhere away from NPs and vHS (Fig. 2b), confirming further the high quality of our superlattice devices. Finite B bend Dirac fermion trajectories and, as expected, Rb rapidly reversed its sign36 with increasing B (inset of Fig. 2b). Remarkably, our devices exhibited negative Rb also in high B = Bp/q, thus revealing straight trajectories over distances of several μm (Fig. 2c, d and Supplementary Note 5). For comparison, the corresponding map of ρxx is provided in Supplementary Note 6, which shows that the measured longitudinal resistance always remained positive. The profound negative pockets in Rb(Vg,B) appeared only around B corresponding to ϕ/ϕ0 = 1/2, 1/3, 1/4 and 1/5. This is the regime where the existence of BZ fermions, experiencing zero Beff, was previously inferred28,29 from maxima in σxx and zeros in ρxy. The possibility of ballistic transport of BZ fermions was also suggested using numerical simulations15. The observed negative Rb prove the previous conjectures unequivocally. The negative pockets in Fig. 2c are located between NPs and vHS for BZ fermions, similar to the case of Dirac fermions. Away from B = Bp/q, non-zero Beff bends the BZ fermion trajectories and the negative signal disappeared (inset of Fig. 2d), again like in the case of other Bloch quasiparticles (electrons and Dirac fermions). Our devices exhibited long-range ballistic transport of BZ fermions only for unit fractions, ϕ0/q. For example, Fig. 2c reveals a pronounced set of negative Rb pockets at ϕ/ϕ0 = 1/5 but no negative signal was observed for 2/5 and 3/5. This behavior probably stems from lower μ of BZ fermions with p > 1, which can be attributed to their larger effective masses29.

Fig. 2: Ballistic transport of BZ fermions over micrometer distances.
figure 2

a Schematic of bend resistance measurements. Current I is applied between contacts 3 and 4, and voltage Vb is measured between 2 and 1, yielding the bend resistance, Rb = Vb/I. The voltage is positive for diffusive transport but becomes negative, if charge carriers move directly from current injecting contact 3 into voltage probe 1 (as shown by the red arrow). b Bend resistance for Dirac fermions in zero B (device D2 with W = 4 μm and θ ≈ 0°). Inset: Rb(B) taken at the minimum indicated by the arrow in the main plot. c Map Rb(Vg,B) for the same device. B was changed in steps of 50 mT. Pockets of negative Rb appear along ϕ/ϕ0 = 1/q and are seen in magenta. d Cross-sections from (c) for q = 2 and 3. The inset shows sign reversals in Rb plotted as a function of Beff = B – Bp/q for the minima marked by the color-coded arrows. T = 2 K for all the plots.

Lifting degeneracy in BZ minibands

The high mobility of BZ fermions resulted in their Landau quantization in small Beff < 1 T, which allowed us to find experimentally the spectral degeneracy g for different p/q. To this end, Fig. 3a shows a high-resolution map of σxx between ϕ/ϕ0 = 1/2 and 1/4 (part of Fig. 1c). One can clearly see many Landau mini-fans originating from NPs for different realizations of BZ fermions. For example, there are three profound mini-fans spreading from B ≈ 9.7 T (ϕ/ϕ0 = 1/3) for both negative and positive Beff. For clarity, the conductivity map in Fig. 3a is replotted in Fig. 3b by tracing all well-defined minima in σxx. Each minimum can be described by its integer filling factor ν, which we calculated from the minimum’s slope. This representation of the Hofstadter spectrum, where LLs are plotted as a function of n or Vg rather than energy, is usually referred to as the Wannier diagram11,12,13,14,15,16,17,21,22. In the diagram of Fig. 3b, one can identify LLs for BZ-fermion realizations at q = 2, 3, 4, 5, 7, 8, 9, and 11, and for p = 1, 2, 3, 4, and 5. The difference in ν between the nearest LLs yields directly their g. For example, all LLs found for ϕ/ϕ0 = 1/2 were separated by Δν = 2 whereas those at ϕ/ϕ0 = 1/3 and 1/4 by Δν = 3 and 4, respectively (see Fig. 3b). Therefore, the observed degeneracies were equal to q. Note that the measured Hall conductance also exhibited quantized values in steps of qe2/h (Supplementary Note 7). Examining the Wannier diagram further, we find that Δν was equal to q, independently of numerator p and for all LLs stemming from the same Bp/q. This observation agrees with the theoretical expectation that there should be q equivalent mini-valleys for each realization of BZ fermions18,19,20,21,22,23,24,25,26,27,30,31. The agreement takes into account that both spin and valley degeneracies for BZ fermions were lifted by exchange interactions14, as in the case of the main Dirac spectrum with its clearly lifted degeneracies (see the LLs marked in black in Fig. 3b). Therefore, the total degeneracy for BZ minibands in Fig. 3 was g = 4q. This is further supported by our measurements at a relatively high T of 2 K that suppressed LLs with lifted spin and valley degeneracies (Supplementary Note 6). The same behavior (Landau mini-fans exhibiting the sequence Δν = q) was also observed in other parts of the Wannier diagram for both electron and hole doping (see, e.g., Fig. 4).

Fig. 3: Landau quantization in BZ minibands at 10 mK.
figure 3

a High-resolution map σxx(Vg, B) for the electron-doped region indicated in Fig. 1c by the white rectangle (device D1). To better resolve LLs around q = 3, B was changed in steps of 10 to 20 mT whereas data in other regions were acquired using 40 to 80 mT steps. This has resulted in the contrast discontinuities seen in the map. Log color scale: indigo (230 nS) to yellow (7.8 mS) for, the entire map. b Minima from (a) are shown schematically. The color-coded numbers are the filling factors for the corresponding LLs. Thick black lines correspond to the main sequence of LLs for graphene’s Dirac spectrum (spin and valley degeneracy lifted). The green, red, navy, blue, orange, magenta, pink, and yellow lines correspond to q = 2, 3, 4, 5, 7, 8, 9, and 11, respectively. Dashed red lines: minima due to lifted mini-valley degeneracy.

Fig. 4: Anomalous behavior of Landau levels in BZ minibands.
figure 4

a High-resolution σxx (Vg, B) for the hole-doped region marked in Fig. 1c (device D1). The measurements below B = 12.5 T were carried out at 10 mK, and at 250 mK in the fields above. B was applied in steps of 16 and 40 mT above and below 12.5 T, respectively, except for the low left corner where the data were acquired at higher resolution of 10 mT (seen as discontinuities in the contrast). Log scale: indigo (310 nS) to yellow (0.3 mS). b Schematics for the conductance minima found in (a). The same color coding as in Fig. 3b. The solid lines indicate LLs evolving as expected, linearly in B and Vg. The thin curves indicate the anomalous bending whereas the dotted curves show the staircase-like evolution observed for some LLs. All the anomalous features were highly reproducible and, for example, did not depend on the step size in B.

There is, however, one notable exception, which was observed for ϕ/ϕ0 = 1/3 (left part of the Wannier diagram in Fig. 3b; dashed lines). In this case, LLs of BZ fermions with q = 3 are separated only by Δν = 1 so that all consecutive LLs from 3 to 12 are present on the fan diagram. The indicated minima in σxx were rather fragile, being rapidly smeared by T or excitation current (Supplementary Note 7). Landau levels with Δν = 1 at ϕ/ϕ0 = 1/q do not exist within the single-particle Hofstadter-Wannier model16,17,21,22. Such ‘extra’ LLs have been observed previously and referred to as the fractional Bloch quantum Hall effect (FBQHE)16 or symmetry-broken Chern insulator (SBCI)17. Theoretically, the many-body states can arise from interaction effects that lift the mini-valley degeneracy of BZ fermions. This may happen via mini-valley mixing due to the formation of charge-density waves (commensurate with the magnetic superlattice composed of q unit cells of the underlying moiré pattern) or a Wigner crystal37 (states localized in a part of the magnetic supercell). Alternatively, the fully lifted degeneracy may occur through spontaneous mini-valley polarization of BZ LLs, analogous to spin/valley ferromagnetism in graphene14. In the latter case, the BZ fermion states localize in the momentum space around one of the mini-valleys but remain delocalized across the magnetic supercell.


Some Landau mini-fans exhibited highly anomalous behavior at low T, which cannot be understood either within the Hofstadter-Wannier model or by considering LLs of non-interacting BZ fermions and, to the best of our knowledge, was never reported before. From either perspective, individual LLs must evolve linearly in Vg and B, as indeed seen in most cases (e.g., Fig. 3). This is because the density of states on each BZ-fermion LL is proportional to Beff = B −Bp/q. Also, the long (300 nm) distance from graphene to the gate in our devices makes quantum capacitance corrections38,39 negligible, leading to the nVg dependence. Unexpectedly, we found that some Landau fans of BZ fermions exhibited bending and staircase-like features. Figure 4 shows examples of such behavior for the case of hole doping. Both bending and staircases appear away from NPs, in the regions close to BZ fermions’ vHS. The observed bending towards the gate-voltage axis indicates that, in addition to the visible LLs, there can be some other electronic states that are populated in parallel. Because electron-electron interactions clearly play a significant role under the reported experimental conditions (e.g., they lift all the spectral degeneracies), it is reasonable to expect that interactions are also involved in the described anomalies. However, the usual suspects, such as negative compressibility cannot possibly explain our findings (in the latter case, LLs would bend toward the B axis). The anomalous behavior is possibly due to an interplay of BZ-fermion LLs with quantized states originating from nearby vHS, which would lead to redistribution of charge carriers between states with the light and heavy effective masses. One possible scenario is localization of electrons within some parts of the magnetic supercell, because its size becomes notably larger than the magnetic length at B > 10 T (Wigner crystallization mentioned above). Unfortunately, we could not find any additional features that would enable us to decipher origins of the described anomalies and, therefore, have to leave them for further investigation.


Devices fabrication

The reported graphene-on-hBN superlattices were assembled using the standard dry transfer procedures and polydimethylsiloxane (PDMS) stamps coated with a polypropylcarbonate (PPC) layer7. The thicknesses of the used hBN crystals was between 20 and 70 nm. After the trilayer stack was assembled, we used the standard electron beam lithography and reactive ion etching to define trenches for electrical contacts. Cr/Au films were evaporated into the trenches. Finally, using ion etching again, the trilayer stack was shaped into a Hall bar mesa.

Electrical measurements

For electrical measurements, the standard low-frequency lock-in technique was employed. At temperatures below 100 mK, we used excitation currents of 10 nA, and between 0.1 to 1 μA at higher T. The carrier concentration was controlled by applying a DC gate voltage between graphene and the silicon substrate.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


  1. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).

    CAS  Article  Google Scholar 

  2. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

    CAS  Article  PubMed  Google Scholar 

  3. Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).

    CAS  Article  Google Scholar 

  4. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotech. 5, 722–726 (2010).

    ADS  CAS  Article  Google Scholar 

  5. Mayorov, A. S. et al. Micrometer-scale ballistic transport in encapsulated graphene at room temperature. Nano Lett. 11, 2396–2399 (2011).

    ADS  CAS  Article  PubMed  Google Scholar 

  6. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

    ADS  CAS  Article  PubMed  Google Scholar 

  7. Kumaravadivel, P. et al. Strong magnetophonon oscillations in extra-large graphene. Nat. Commun. 10, 3334 (2019).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  8. Chen, S., Kraft, R., Danneau, R., Richter, K. & Liu, M.-H. Electrostatic superlattices on scaled graphene lattices. Commun. Phys. 3, 71 (2020).

    Article  Google Scholar 

  9. Wallbank, J. R., Patel, A. A., Mucha-Kruczyński, M., Geim, A. K. & Fal’ko, V. I. Generic miniband structure of graphene on a hexagonal substrate. Phys. Rev. B 87, 245408 (2013).

    ADS  Article  Google Scholar 

  10. Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

    CAS  Article  Google Scholar 

  11. Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

    ADS  CAS  Article  PubMed  Google Scholar 

  12. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

    ADS  CAS  Article  PubMed  Google Scholar 

  13. Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

    ADS  CAS  Article  PubMed  Google Scholar 

  14. Yu, G. L. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nat. Phys. 10, 525–529 (2014).

    CAS  Article  Google Scholar 

  15. Diez, M., Dahlhaus, J. P., Wimmer, M. & Beenakker, C. W. J. Emergence of massless Dirac Fermions in Graphene’s Hofstadter butterfly at switches of the quantum hall phase connectivity. Phys. Rev. Lett. 112, 196602 (2014).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  16. Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).

    ADS  MathSciNet  CAS  Article  PubMed  PubMed Central  Google Scholar 

  17. Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

    ADS  CAS  Article  Google Scholar 

  18. Brown, E. Bloch electrons in a uniform magnetic field. Phys. Rev. 133, A1038–A1044 (1964).

    ADS  MathSciNet  Article  Google Scholar 

  19. Zak, J. Magnetic translation group. Phys. Rev. 134, A1602–A1606 (1964).

    ADS  Article  Google Scholar 

  20. Azbel, M. Y. Energy spectrum of a conduction electron in a magnetic field. Sov. Phys. JETP 19, 634–645 (1964).

    Google Scholar 

  21. Wannier, G. H. A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi 88, 757–765 (1978).

    Article  Google Scholar 

  22. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    ADS  CAS  Article  Google Scholar 

  23. Streda, P. Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C. Solid State Phys. 15, L1299–L1303 (1982).

    ADS  CAS  Article  Google Scholar 

  24. Rhim, J.-W. & Park, K. Self-similar occurrence of massless Dirac particles in graphene under a magnetic field. Phys. Rev. B 86, 235411 (2012).

    ADS  Article  Google Scholar 

  25. Rammal, R. Landau level spectrum of Bloch electrons in a honeycomb lattice. J. Phys. 46, 1345–1354 (1985).

    MathSciNet  Article  Google Scholar 

  26. Delplace, P. & Montambaux, G. Semi-Dirac point in the Hofstadter spectrum. Phys. Rev. B 82, 035438 (2010).

    ADS  Article  Google Scholar 

  27. Chen, X. et al. Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices. Phys. Rev. B 89, 075401 (2014).

    ADS  Article  Google Scholar 

  28. Krishna Kumar, R. et al. High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices. Science 357, 181–184 (2017).

    ADS  CAS  Article  PubMed  Google Scholar 

  29. Krishna Kumar, R. et al. High-order fractal states in graphene superlattices. Proc. Natl Acad. Sci. USA 115, 5135–5139 (2018).

    ADS  CAS  Article  PubMed  Google Scholar 

  30. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    ADS  CAS  Article  Google Scholar 

  31. Landau, L. D. & Lifschitz, E. M. Statistical Physics Part 2—The theory of the condensed state. in Course of Theoretical Physics (Pergamon Press Ltd., 1980).

  32. Woods, C. R. et al. Commensurate–incommensurate transition in graphene on hexagonal boron nitride. Nat. Phys. 10, 451–456 (2014).

    CAS  Article  Google Scholar 

  33. Wang, L. et al. New generation of moiré superlattices in doubly aligned hBN/graphene/hBN heterostructures. Nano Lett. 19, 2371–2376 (2019).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  34. Wang, Z. et al. Composite super-moiré lattices in double-aligned graphene heterostructures. Sci. Adv. 5, eaay8897 (2019).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  35. Finney, N. R. et al. Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices. Nat. Nanotech. 14, 1029–1034 (2019).

    ADS  CAS  Article  Google Scholar 

  36. Beenakker, C. W. J. & van Houten, H. Quantum transport in semiconductor nanostructures. Solid State Phys. - Adv. Res. Appl. 44, 1–228 (1991).

    Google Scholar 

  37. DaSilva, A. M., Jung, J. & MacDonald, A. H. Fractional Hofstadter states in graphene on hexagonal boron nitride. Phys. Rev. Lett. 117, 036802 (2016).

    ADS  Article  PubMed  Google Scholar 

  38. Chen, Z. & Appenzeller, J. Mobility extraction and quantum capacitance impact in high performance graphene field-effect transistor devices. In 2008 IEEE International Electron Devices Meeting 1–4 (IEEE, 2008).

  39. Yu, G. L. et al. Interaction phenomena in graphene seen through quantum capacitance. Proc. Natl Acad. Sci. USA 110, 3282–3286 (2013).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

Download references


This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grand Challenges (EP/N010345), Lloyd’s Register Foundation, Graphene Flagship, the Royal Society and the European Research Council (grant Vander). A.I.B. acknowledges support from the Graphene NOWNANO Doctoral Training Centre.

Author information

Authors and Affiliations



P.K., N.X., and M.H. fabricated devices. J.B., R.K.K., L.A.P., C.M., M.K., and A.I.B. performed transport measurements with help from A.M, M.D.T., and J.R.P. J.B., P.K., R.K.K., L.A.P., V.I.F, A.K.G., and A.I.B. performed data analysis. T.T. and K.W. provided the hBN crystals. J.B., P.K., V.I.F., A.I.B., and A.K.G. wrote the paper. All authors contributed to the discussions.

Corresponding authors

Correspondence to A. K. Geim or A. I. Berdyugin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barrier, J., Kumaravadivel, P., Krishna Kumar, R. et al. Long-range ballistic transport of Brown-Zak fermions in graphene superlattices. Nat Commun 11, 5756 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing