Topological valley currents via ballistic edge modes in graphene superlattices near the primary Dirac point

Graphene on hexagonal boron nitride (hBN) can exhibit a topological phase via mutual crystallographic alignment. Recent measurements of nonlocal resistance ($R_{nl}$) near the secondary Dirac point (SDP) in ballistic graphene/hBN superlattices have been interpreted as arising due to the quantum valley Hall state. We report hBN/graphene/hBN superlattices in which $R_{nl}$ at SDP is negligible, but below 60 K approaches the value of $h/2e^{2}$ in zero magnetic field at the primary Dirac point with a characteristic decay length of 2 ${\mu}$m. Furthermore, nonlocal transport transmission probabilities based on the Landauer-B\"uttiker formalism show evidence for spin-degenerate ballistic valley-helical edge modes, which are key for the development of valleytronics


Introduction
Fabrication and characterization of graphene superlattices. hBN/graphene/hBN Hall bars are fabricated by van der Waals assembly with side-contacts (see Methods section for further details) as shown in Fig. 1a, b. The relative rotation angle (φ) between hBN and graphene is determined using a transfer system with a rotating stage under an optical microscope with an accuracy of better than 1.5º . Local and nonlocal transport measurements are taken across a range of temperatures (8.8-300 K) and magnetic fields (0-2.5 T). We focus on the electronic transport of three types of devices denoted as I, II, and III (for device mobility characterization see Supplementary Note 1). For device I (a field-effect mobility of µ ≈ 220,000 cm 2 V -1 s -1 at 9 K, Fig. 1a (Fig. 1c). In Fig. 1d we show the 2D-peaks of the different structures investigated: FWHM(2D) of device I (27 cm -1 ) is larger than devices II and III (17 cm -1 and 22 cm -1 ) for which φ are 30º and 10º . The Raman 2D-peak of graphene is sensitive to φ, and FWHM(2D) increases by rotating from a misaligned position to an aligned position, which is due to a strain distribution with matching moiré potential periodicity 12 .
Local electrical transport properties. Figure 2a shows local measurements in zero magnetic field at 9 K with a pronounced peak in ρxx at the primary DP. Two additional peaks are symmetrically visible on both sides of the DP in the higher carrier density regime. The appearance of SDP depends on moiré minibands occurring near the edges of the superlattices Brillouin zone 4,9 and the moiré wavelength λ of device I is calculated to be around 10 nm (φ < 1º ) (see Supplementary Note 2) and for devices II and III, there are no SDPs within the gate voltage range investigated (±20 V) as the φ > 10º (requires |VTG−VD| > 100 V). Device I also shows the ballistic character (see Supplementary Note 3). Lorentzian fits for devices I, II and III.
In Fig. 2b, c, when temperature is decreasing, a ν = 0 plateau appears in σxy and a doublepeak structure appears in σxx when the gate voltage (VTG) is close to the DP (see Supplementary Note 4 for detailed discussions). Two different types of conductivity variations are seen in Fig. 2b: one is insulating meaning σxx decreases at lower temperatures when VTG is close to the DP; the other is metallic in which σxx increases at lower temperatures. The critical point separating these two regimes is the crossing point of all the curves measured at different temperatures, where σxx is independent of temperature (T) indicating quantum Hall state transitions. Figure 2d, e show the evolutions of ρxx and σxy with VTG and increasing perpendicular magnetic fields (B). Standard quantum Hall state with plateaux in σxy and zeros in σxx at filling factors ν = ±2 , ±6 , ±10 ... is observed. A striking feature is the insulating region near the DP with increasing B, where ρxx ≥ h/2e 2 .
A quantum Hall effect gap at the DP in hBN/graphene/hBN superlattices occurs due to electron interactions and broken sublattice symmetry 6,13 . From the insulating behavior of σxx at the DP, we fit an Arrhenius function σxx(T -1 ) to estimate the band gap in Fig. 2f. The thermally excited transport exhibits two distinct regimes of behavior, separated by a characteristic temperature, which we define as T*. For T > T*, transport is thermally activated 10 where kB is the Boltzmann constant and Ea is the band gap energy. The band gap is estimated to be 391.2 ± 21.8 K (33.7 ± 1.9 meV) for device I and 210.1 ± 11.2 K (18.1 ± 1.0 meV) for device II.
The larger band gap at the DP for device I can be associated with the commensurate state, because the whole area of graphene, which is aligned to hBN, would have the same crystal structure as hBN and tend to increase the gap. Device II has a smaller Ea due to a suppression of the commensurate state by one of the misaligned hBN layers 7 . For T < 60 K, σxx,min decreases slowly with lower T than the activated transport indicating that in this regime the effect of the thermally activated bulk carriers can be neglected. In Fig. 2b, the temperature dependence of the longitudinal conductivity in 2 T shows the appearance of v = 0 state below T < 60 K. The appearance of the quantum Hall state also requires that the effect of the thermally activated bulk carriers can be neglected.  for Rnl in device I. Rnl decays exponentially with increasing distance at 9 K. L varies from 3 µm to 20 µm and W is 2 µm. Reference data is taken from Ref. [8]. c Rnl vs VTG − VD and magnetic field (B) at 9 K. Color bar shows the Rnl from 0 to 20 kΩ. d Rnl vs B for different VTG − VD.
Nonlocal electrical transport properties. In Fig. 3a, Rnl for device I shows a sharp peak (15.45 kΩ) at the DP and ρxx has a 1/n dependence, which decreases at a slower rate than Rnl over the entire range of VTG investigated. Within achievable VTG (±30 V), we do not observe nonlocal transport at the SDPs which could be due to charge inhomogeneity suppression 8 . Rnl for device II (same geometry as device I) is smaller (60 Ω), consistent with a misalignment between graphene and hBN.
In addition, the improved electronic properties of graphene on hBN enable long-range topological valley currents 8,10 . Rnl exponentially decays as a function of nonlocal distance (L) in graphene [i.e.
Rnl  exp(−L/ξ)] with a characteristic length ξ ≈ 2.0 µm (Fig. 3b). The maxima in Rnl for all values of L investigated are at least an order of magnitude larger than previously reported values in equivalent devices with similar mobility 8 . With B applied, we observe a rapid broadening and increase in the Rnl peak above 0.1 T (Fig. 3c, d) due to contribution from charge-neutral spin currents, which become appreciable with broken time reversal symmetry 8,14 . In addition, Rnl under magnetic field can have contributions from heat current and the quantum Hall effect edge current 8,10 . To confirm the origins of Rnl for T < 60 K, we first measure Rnl using a six-terminal configuration (Fig. 4d, e). In Ref. To investigate the origin of quantum valley Hall state, we measure Rnl systematically using a ten-terminal configuration (Fig. 4e) in order to determine the transmission matrix. Device I is fabricated with eighteen terminals (Fig. 1a), fourteen of which show relatively low contact resistances (Fig. 4e). We select ten terminals located symmetrically to measure. The calculated transmission matrix based on the Landauer-Büttiker formalism (see Supplementary Note 7) does not agree with the minimal model for edge mode transport proposed in Ref. [10]. However, when the ballistic counter-propagating edge modes enter these unused but connected terminals, they interact with a reservoir of states and equilibrate to the chemical potential determined by the voltage at each terminal. Therefore, electrons will be injected backward and forward with equal probability.

Discussion
These unused terminals in-between the measured terminals effectively reduce the ideal transmission probability by a half. The transmission probabilities are approximately 2.0 between terminals 12 and 14, and reach 1.0 between terminals 5 and 27, 6 and 27, 14 and 28 in the narrow VTG range near the DP, consistent with spin-degenerate ballistic counter-propagating edge mode transport. In addition, it is known that commensurate stacking in aligned van der Waals heterostructure (φ < 1º ) leads to the soliton-like narrow domain walls 25 . One-dimensional conducting channels exist at these domain walls, which can form a network leading to Rnl when bulk graphene/hBN superlattices domains become insulating [26][27][28][29][30] . If edge modes intersect with domain walls, the electrons can go into two different directions at the intersection, and this will lower the transmission probabilities for each given direction. The transmission probabilities for terminals 5, 6, 7 and 8 (Supplementary Table 1) are significantly smaller than expected values based on the ballistic edge mode transport, which perhaps are consistent with the existence of domain walls.
To summarize, we have investigated hBN/graphene/hBN Hall bars with a field-effect mobility of 220,000 cm 2 V -1 s -1 at 9 K and low charge impurities. Alignment between hBN and graphene (φ < 1º ) leads to a 33.7 meV band gap at the DP. In zero magnetic field and 9 K, a ν = 0 state in gapped graphene is demonstrated in σxx and σxy with large Rnl values close to h/2e 2 . Rnl decays nonlocally over distances of 15 μm with a characteristic constant of 2 μm. Nonlocal measurements suggest that, below 60 K a spin-degenerate ballistic counter-propagating edge mode is dominant, and there is a possible secondary contribution from a network of one-dimensional conducting channels appearing at the soliton-like domain walls. A further direct imaging of the edge modes [31][32][33] would be desirable for conclusive determination of the mechanism of nonlocal transport. The valleyhelical ballistic edge modes offer important possibilities for electronic applications beyond quantum spin Hall effect and quantum anomalous Hall effect since quantized resistance can be observed at higher temperature with a tunable energy gap through valley coupling.

Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. wrote the paper. All authors commented on the manuscript.

Competing interests
The authors declare no competing interests.

Additional information
Supporting information is available for this paper, which includes analysis of extrinsic contributions to nonlocal signals, and transmission probability calculation.

Supplementary Note 1. Device characterization
To investigate structural and electronic homogeneity of the graphene on hBN, Raman spectroscopy measurements are performed over entire region at 293 K with a laser excitation at a wavelength of 532 nm, are shown in Fig. 1c and Supplementary Figure 1c   Nonlocal measurement setup.

Supplementary Note 2. Moiré wavelength calculation
hBN may have the same lattice structure as graphene with a 1.8% longer lattice constant. The alignment between the graphene and hBN lattices leads to moiré patterns. The moiré wavelength λ is described as 2 2 (1 ) where δ is the lattice mismatch between hBN and graphene, a is the graphene lattice constant, φ is the relative rotation angle between hBN and graphene. Supplementary Figure 4 plots the λ as a function of φ and shows a maximum value of about 14 nm.
The appearance of SDP depends on moiré minibands, which occur near the edges of the superlattice Brillouin zone and are characterized by the energy 2,3 of SDP, where n is the carrier density related to the SDP, vF is the Fermi velocity, ħ is Planck's constant divided by 2π and λ is moiré wavelength. From Supplementary Equation (S2), the position of SDP corresponds to a carrier density of n = 4π/3λ 2 , and in the case of φ = 0º, λ = 14 nm yields n ≈ 2 × 10 12 cm -2 . Supplementary Fig. 4. Moiré wavelength as a function of the relative rotation angle (φ) between the graphene and hBN.

Supplementary Note 5. Ohmic and thermal contributions to Rnl
Ohmic contribution is described by the van der Pauw formula 6 , where L and W are the channel length and width. In zero magnetic field for device I, L/W = 2 and Rxx = 126 kΩ, and from Supplementary Equation (S3) we find Rnl,Ω ≈150 Ω, which is two orders of magnitude smaller than the measured Rnl in Supplementary Figure 8. In all transport measurements we use a low alternating-current excitation of 10 nA with a frequency 7 Hz. These low current amplitudes are chosen to minimize thermal contributions to the nonlocal transport due to Joule heating and Ettingshausen effects 7 whilst simultaneously maximizing the signal-to-noise ratio of the measured voltages. In zero magnetic field, only Joule heating effect contributes to the second harmonic of nonlocal signal R 2f nl,J , which is less than 1% of Rnl as shown in Supplementary Figure 8. R nl 10xR nl, Joule heating effect

Supplementary Note 6. Band gap calculation from nonlocal transport
From the Arrhenius plot of the Rnl (Supplementary Figure 9), the associated band gap is estimated as 760.4 ± 69.9 K assuming 1/Rnl ∝ exp(−Ea/2kBT) (Ref. [8] and I is the matrix where column describes a given current configuration (current injects from terminal to the device) and V is a matrix where each column describes the corresponding voltage configuration (voltages of different terminals).
These equations can be solved in three ways: (i) If the voltage configuration and Tr p,q are known, we can directly determine the corresponding currents from Supplementary Equation (S5, S6).
(ii) If the current configuration and Tr p,q are known, we can determine the corresponding voltages in different terminals.
(iii) If we measure the ten independent current configuration with corresponding voltage, the conductance matrix can be determined from After determining G, the transmission matrix can be obtained from Supplementary Equation (S6). One possible way of obtaining ten independent current with corresponding voltages is to apply current between terminals p − 1 and p, then cyclically shift the source and drain terminals so that one gets a 10×10 current matrix, 0 1 1 0 0 0 1 1 0 1 where I0 is the magnitude of the current. For each current configuration, one measures the corresponding voltages in different terminals and gets voltage matrix V.
In a simple edge mode transport model proposed in Ref. [8] for quantum valley Hall state, a pair of ballistic edge modes connect terminal. Assuming that all terminals are ordered along the edge so that terminal i is connected to both terminals i-1 and i+1, then Tr ij is given by