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.

Studied devices. We studied six different superlattice devices as summarized in Supplementary Table  1. Here W refers to the device width, Cg is its gate capacitance and θ is the angle between the crystallographic axes of hBN and graphene. The latter was calculated from the measured period of BZ oscillations 1 .
Supplementary Table 1 Device W (µm) q (°) Cg (µF/m 2 ) Supplementary Note 2 Magnetic focusing in graphene superlattices. As a further confirmation of the devices' high quality, we report transverse magnetic focusing (TMF) experiments at low B ( Supplementary Fig. 1). The observation of resistance oscillations due to TMF confirms that Dirac fermions travel ballistically across the device 2,3 , forming skipping orbits extending over hundreds of superlattice unit cells. TMF measurements are also known to provide information about the Fermi surface topography in clean metals, including graphene superlattices 2,3 . Our TMF results are in agreement with those reported previously 2,4 . ⁄ where %&' is the carrier density of BZ fermions and ## = 1 ## ⁄ . Note that the latter expression is exact at = -p/q (that is, it does not contain #. because the effective magnetic field Beff acting on BZ fermions is zero). To determine %&' for a given Vg, we first used Hall measurements at small fields B £ 0.1 T to determine the geometrical capacitance. Then, using longitudinal conductivity maps around the p/q fractions, we identified positions of the neutrality points (NPs) as Vg into which Landau mini-fans converged (see Fig. 1c and Fig. 3a of the main text). Finally, vHS were identified from Hall effect measurements as S3 Vg where #. changed its sign without exhibiting mini-fans ( Supplementary Fig. 2b). As %&' varies linearly across NPs and exhibits jumps at vHS, the known geometrical capacitance allowed us to reconstruct %&' (Vg) as shown in Supplementary Fig. 2a.

Supplementary
The mean free path was calculated using the standard formula ## = 1 2 3 ( 5 6

Supplementary Note 5 Additional examples of ballistic transfer of BZ fermions.
In the main text, we have emphasized that, at fields B = Bp/q, BZ fermions move through the superlattice as if the applied field were zero. The effective mass of BZ fermions depends on p/q because electronic spectra differ in different magnetic minibands. Away from the exact Bp/q values, BZ fermions are expected 1 to experience an effective magnetic field Beff = B -Bp/q and, therefore, replicate magneto-transport effects known for charge carriers in conventional 2D electronic systems. This includes the negative bend resistance that is one of the most distinct, qualitative signatures of ballistic transport of charge carriers [5][6][7] . The effect can be understood as follows. With reference to Fig. 2a of the main text, let us for simplicity consider positive charge carriers (hole-doping regime). If holes injected from contact 3 can travel ballistically over a distance exceeding W (that is, can reach contact 1 without scattering), an extra positive charge would be accumulated near contact 1. As a result, the voltage difference 8> = 8 − > should be negative (see Fig. 2a of the main text). In contrast, if the transport is conventional (diffusive), holes from contact 3 travel along lines of the electric field and accumulate at contact 4. Accordingly, the sign of 8> should be conventional (that is, positive). The same consideration for 8> is valid for electrons. Therefore, the negative sign of Rb signifies ballistic transport over distances larger than W. Negative Rb was reported in Fig. 2 of the main text for one of our devices (D2). Supplementary Fig. 5 provides further examples of ballistic transport of BZ fermions using two other superlattices (devices D3 and D4). Pronounced pockets of negative Rb are seen in Supplementary Fig. 5 at unit fractions of with q from 2 to 5. Despite relatively small W » 3 µm, no evidence for ballistic transfer was observed for high-order BZ states (p > 1), in agreement with the results reported in the main text. Note that occasionally we observed negative bend resistance away from -⁄ = 1/ (see, e.g., the vertical magenta stripe close to zero Vg in Supplementary Fig. 5a). Unlike the ballistic transfer resistance at unit flux fractions, negative signals away from the unit fractions were not reproducible in different contact configurations. Such 'extra' negative signals are not surprising in our experimental geometry and well known to appear in the quantum Hall effect regime using narrow (mesoscopic) devices 8 .

Supplementary Figure 5 | Ballistic transport of BZ fermions over micrometer distances.
Fan diagrams obtained in the bend resistance geometry for devices D3 (a) and D4 (b) with W = 3 and 3.2 µm, respectively. T = 2 K. Pockets of negative Rb are highlighted in magenta. Indigo-to-yellow: Log scale truncated between 10 and 2,000 Ω to optimize the contrast.
Ballistic transport of BZ fermions was found to be rather sensitive to T, and the pockets of negative Rb universally disappeared above 30-50 K as shown in Supplementary Fig. 6. This is generally expected because the mean free path of BZ fermions should become shorter at higher T. However, the exact scattering mechanism could be nontrivial (see, e.g. Umklapp electron-electron scattering 9 ) and requires further investigation. Figure 6| Temperature dependence of BZ fermions' ballistic transport. An example of the bend resistance measured at ϕ ϕ -= 1/2 ⁄ using device D5 with W = 2 µm.

Supplementary Note 6 Supporting measurements in the longitudinal geometry.
To crosscheck our conclusions about ballistic transport of BZ fermions, we compare the negative bend resistance measurements shown in Figs. 2c,d of the main text with those made in the conventional longitudinal geometry for the same device D2 (Supplementary Fig. 7a). The longitudinal resistance Rxx for BZ fermions was found positive in all the regions of the map where the negative bend resistance was reported, which corroborates the conclusion in the main text about ballistic transfer of BZ fermions across the device. Let us note here that, according to the group theory of irreducible representations for the group of translations in a magnetic field, an electronic spectrum for each realization of BZ fermions should have an additional q-fold degeneracy. This is prescribed by the fact that a group corresponding to any p/q fraction is non-Abelian (due to Aharonov-Bohm phases acquired upon translations in non-colinear directions) but contains an Abelian subgroup of translations corresponding to a magnetic superlattice with a q times larger supercell. The additional q-fold degeneracy takes the form of q mini-valleys in the magnetic mini Brillouin zone with an area q times smaller than the moiré superlattice Brillouin zone at B = 0. This degeneracy is additional to the 4-fold spin and valley degeneracy of graphene's original spectrum. With this consideration in mind, the measurements in Supplementary Fig. 7 also support our other conclusion that the full degeneracy of BZ fermions is 4q. Indeed, the q-fold degeneracy reported in Fig. 3 of the main text corresponds to the case where both spin and valley degeneracies of both Dirac and BZ fermions were lifted. Supplementary Fig. 7 shows LL fans at 2 K, the temperature much higher than 10 mK for the measurements in Fig. 3. Dirac fermions of the main spectrum exhibit the lifted spin and valley degeneracies by the relatively strong B (thin black lines in Supplementary Fig.  7b). At lower fields B < 3T, these interaction-induced gaps become progressively smeared. As for BZ fermions, their mini-fans visible in Supplementary Fig. 7b reach only the effective field |Beff| < 2T, which does not allow the lifting of spin and valley degeneracies at this temperature. Accordingly, only the main sequence of LLs for BZ fermions could be observed at 2 K, and it corresponds to the 4q-fold degeneracy, as expected and explained in the previous paragraph.

Supplementary Note 7 Lifting mini-valley degeneracy.
In the main text we reported additional quantum Hall effect minima that cannot be explained within the single-particle Hofstadter-Wannier (dashed lines in Fig 3b of  The described lifting of mini-valley degeneracy involves very small energy gaps as witnessed by rapid disappearance of the corresponding features with increasing T. Indeed, the quantized Hall plateaus seen in the above figure and the conductance minima marked by the dashed lines in Fig. 3b of the main text could not be resolved at 2 K. The features also disappeared rapidly with increasing the excitation current. For example, Supplementary Fig. 9 shows a Landau mini-fan around -⁄ = 1/3 for currents of 10 and 100 nA. In the former case ( Supplementary Fig. 9a), there are clear minima associated with to the lifted mini-valley degeneracy. The higher current (100 nA) resulted in complete smearing of these mini-gaps ( Supplementary Fig. 9b), presumably because of an increase in the electronic temperature. Finally, let us draw attention to the rather unusual re-entrant behavior seen for the mini-fan around Vg = 28 V in Fig. 3 and Supplementary Fig. 9. The BZ-fermion gaps for ν = 18 and 21 seem to close within a certain interval B and Vg. We attribute this closure to competition between these BZ states and the ν = 7 state from the main Dirac sequence. The likely mechanism of the suppression of exchange gaps is discussed in ref. 10 .