Topological valley transport at the curved boundary of a folded bilayer graphene

The development of valleytronics demands long-range electronic transport with preserved valley index, a degree of freedom similar to electron spin. A promising structure for this end is a topological one-dimensional (1D) channel formed in bilayer graphene (BLG) under special electrostatic conditions or specific stacking configuration, called domain wall (DW). In these 1D channels, the valley-index defines the propagation direction of the charge carriers and the chiral edge states (kink states) are robust over many kinds of disorder. However, the fabrication of DWs is challenging, requiring the design of complex multi-gate structures or have been producing on rough substrates, showing a limited mean free path. Here, we report on a high-quality DW formed at the curved boundary of folded bilayer graphene (folded-BLG). At such 1D conducting channel we measured a two-terminal resistance close to the quantum resistance $R = e^2/4h$ at zero magnetic field, a signature of kink states. Our experiments reveal a long-range ballistic transport regime that occurs only at the DW of the folded-BLG, while the other regions behave like semiconductors with tunable band gap.

free path. Here, we report on a high-quality DW formed at the curved boundary of folded bilayer graphene (folded-BLG). At such 1D conducting channel we measured a two-terminal resistance close to the quantum resistance R = e 2 /4h at zero magnetic field, a signature of kink states. Our experiments reveal a long-range ballistic transport regime that occurs only at the DW of the folded-BLG, while the other regions behave like semiconductors with tunable band gap.  [1,10,11,12]. One interesting 2D crystal with useful features for valleytronics is BLG. The material is a tunable semiconductor and contains low lattice defects that prevents inter-valley scattering. Moreover, it holds topological properties when its inversion symmetry is broken by the application of transverse electric field. In this condition, a topological invariant is defined, the integer index called Chern number, with important implications on the quantum properties of BLG. For instance, it gives rise to the observation of the valley Hall Effect in graphene [13,14,15], which is a topological phase where gapless edge states labelled by opposite valley-indices counter-propagate at the boundaries of the insulating bulk [16]. One important aspect of the Chern number in BLG is that its sign depends either on the valley-index as well as on the sign of the band gap (interlayer energy difference), which can be changed by inverting the electric field direction or by inverting the stacking order of the material [16,1,2,3,5].
Such control of the band gap of BLG allows the design of topological 1D interfaces between regions with opposite Chern numbers -a domain wall -where strongly confined edge states called kink states are predicted [1,2,3,4,5]. The kink states have several useful features for valleytronics. There are two spin-degenerate kink states per valley and they are chiral, meaning that the propagation direction in the DW is defined by the valley-index. Such chiral edge states are robust for almost any kind of boundary configurations of the domains (except perfect armchair) and the topological protection inhibits backscattering from smooth disorder potentials [3]. If valley-mixing is suppressed in the DW, such as by reducing the short-range disorder like edge defects and substrate corrugation, a dissipationless electrical conduction with conserved valley-index is expected. Figure 1. Valleytronic device based on a folded bilayer graphene. a, Components of the folded-BLG valleytronic device. The folded-BLG is sandwiched by hBN crystals that separate the material from the metallic gates. Under a transverse electric field, the bulk behaves likely semiconductor with band gap and a topological 1D conducting channel forms at the curved boundary, where the valley-index defines the direction of propagation. b, Optical image of a folded-BLG transferred on top of the bottom hBN flake and a false-colour AFM image of device 1 before the transference of the top hBN. Dashed lines indicate the position of the electric terminals. The AFM measurement reveals that the curved boundary is free from contamination of fabrication processes. Scale bar: 1 µm. c, Electrostatic potential energy (∆/2) of the BLG layers, calculated relative to the centre of each BLG. The layer energies reverse sign from the bottom BLG to the top BLG and vanish across the curved boundary, where the electric field is parallel to the layers. This variation of the electrostatic potential energy enable the formation of a domain wall at the curved boundary. d, Illustration of the pair of kink states localized at inequivalent points K and K' in the BZ of BLG, having opposite group-velocities for different valleys. In the domain wall, these edge states propagate in opposite one-dimensional directions due their chiral nature.
To date, there are two routes to investigate kink states in BLG flakes. One exploits DWs formed along stacking faults (AB-BA boundaries). However, so far, such DWs have been only produced in BLG placed on top of rough SiO 2 substrates [8,9] showing limited mean free path. The other possible way is by designing complex gate-controlled topological channels, which requires a very precise alignment of the bottom and top gates [6,7]. Here, we observe strong evidence of kink states in high-quality DW formed at the curved boundary of a folded-BLG. Such compact geometry provides several advantages: the DW is atomically narrow, a variety of techniques enable the controlled production of such folded structures [17,18] and this architecture simplify the fabrication of valley-filters and valley-valves using fewer metallic gates. Moreover, we show that the topologically protected electronic transport is robust up to room temperature and shows a mean free path (MFP) up to the length of 20 µm at low temperatures, which is one of the longest MFP ever reported in a DW.
To introduce our valleytronic device, in the Fig. 1a we show a cartoon with some typical components of the device such as the folded-BLG, the metallic gates and the dielectrics. The folded-BLG is encapsulated in between two hexagonal boron nitride (hBN) crystals. The bottom hBN is placed on top of a SiO 2 /Si ++ wafer, such that the Si ++ is a highly p-type Here we present device 1 after the cleaning process and before encapsulating with a top hBN crystal. From this measurement, we see the high quality and cleanness of the device, which prevents short-range scattering along the 1D channel. These good conditions are provided either by the flat surface of the hBN crystals as well as by the efficacy of the mechanical cleaning method to remove contamination from the fabrication process (Method Section).
In the Fig. 1c we present a scheme that describes the electrostatic conditions imposed to the folded-BLG by application of gate potential. The transverse electric field breaks the inversion symmetry of each BLG (bottom sheet and top sheet of the folded-BLG), which leads to an energy band gap (∆) [19]  One of the main achievements of this work is the measurement of a quantization of the two-terminal resistance (R) along the curved boundary near of the quantum resistance R = 6.45 kΩ (or R = e 2 /4h) at B = 0 T and T = 1.2 K. Such result is a remarkable evidence of kink states, since the conductance of the 1D channel is governed by a ballistic transport regime related to a pair of chiral edge states spin-degenerated [1,2,3]. This result is presented in Fig. 2, which shows raw data of R as a function of the backgate voltage (V BG ) and the topgate voltage (V TG ) measured at electric contacts placed along the etched edge ( Fig. 2a) and along the curved boundary (Fig. 2b). Both measurements exhibit a diagonal line that shows a strong dependence of resistance with gate potentials. Along these diagonal lines, the electrostatic condition defined by the gate potentials set zero charge in the BLGs, called the charge neutrality point (CNP). The resistance at such diagonal lines are defined as follows: for the electric measurements realized on the etched edge it will be called R EE,CNP and for the electric measurements realized on the curved boundary it will be called R CB,CNP . A better comparison of R EE,CNP and R CB,CNP is present in Fig. 2c, where we plot R as a function of the displacement field D, obtained from the data showed in Fig. 2a and Fig. 2b. We converted the gate potentials to displacement field with the following formula: where C TG and C BG are, respectively, the capacitances per unit of area and charge of the top capacitor and bottom capacitor, and 0 is the vacuum permittivity. From data present in Fig. 2c we note that the monotonic increasing of R EE,CNP with D reflects a tunable band gap caused by the broken inversion symmetry of BLGs [20,21].
In contrast, the R CB,CNP saturates near of the quantum resistance R = h/4e 2 for |D| > 1.6 V/nm. This saturation of the resistance reveals that the DW formed at the curved boundary becomes electric isolated from the bulk of the folded-BLG and a ballistic transport regime governs the carrier motion at this 1D region. The quantization of resistance close to the quantum resistance show a robust valley transport, with backscattering strongly inhibited by the lack of short-range disorder along the channel.
The strong suppression of backscattering in the DW formed along the curved boundary leads to a long MFP. We use the Landauer-Büttiker formula [22]  µm for device 2. Such long MFP show that the DW formed at the folded-BLG is comparable to the best topological channels created by gate-confinement [6,7] and at least, two-orders higher than the MFP reported in a DW of BLG on SiO 2 [8].  In summary, our findings show the existence of topological chiral edge states in a domain wall formed at the curved boundary of a folded-BLG. We observe a strong suppression of valley scattering at this high-quality 1D channel that leads to a long-range ballistic conduction at zero-magnetic fields. Such novel platform contains elements to promote the development of dissipationless valleytronic devices and provides a new route to investigate graphene-based superconducting effects [23,24] as well as Luttinger liquid interactions [25].

Methods
The heterostructures of folded-BLG sandwiched between hBN crystals are prepared with the following steps: we first employed the mechanical cleavage method to separate few layers of graphene from graphite flakes on top of a polymeric film of methyl methacrylate (MMA 495 C4). Next, we selected self-folded BLG samples and we transferred such flakes to top of clean hBN crystals supported on a 285-nm thick SiO 2 /Si ++ , where Si ++ is a highly doped Si wafer used as a metallic backgate. The fabrication of devices is divided into three main steps. First, we fabricated the electric terminals by using conventional electron beam lithography and thermal metalization of Cr/Au (1 nm/40 nm). We also used electron beam lithography and etching processes with oxygen plasma to define and shape our devices. Next, we used a mechanical cleaning method with an AFM probe [26] to remove any contamination of fabrication processes from the surface of the folded-BLG. We finished the fabrication by covering the device with another hBN flake and patterning a metallic top-gate. The electronic measurements are realized inside a cryogen system that enables the application of magnetic field up to B = 7 T. In our electronic measurements we normally operated at T = 1.2 K and we performed the measurements using a low-frequency (f = 17 Hz) Lock-in technique.
In the two-terminal measurements we applied a constant bias (V bias = 1 mV) between the contacts and we measured it electric current. The conductance is calculated by using the formula G = I/V . In the four-terminal measurements, a constant electric current (I = 100 nA) is applied between two electric terminals and a longitudinal voltage (V xx ) is measured in between the other electric terminals on the opposite side. The longitudinal resistance is calculated by using the Ohm's law R xx =V xx /I.

Supplementary information
Topological valley transport at the curved boundary of a folded bilayer graphene of its layers change in space (kink potential). Equation (1) shows the simplified twocomponent Hamiltonian [2] acting in the space of wave functions related to non-dimer states Ψ(x) = (ψ A2 (x), ψ B1 (x)), where ξ is +1 for valley K and -1 for valley K'. The potential energy of each layer are al. [1], a set of second-order differential equations is the result of applying the Hamiltonian of equation (1) in the wave functions, where we define u(x) = m∆(x), the energy E is normalized (E = 2mE) and we choose we expect that kink states are confined in the DW formed at the curved boundary.
To finish, let us discuss some important points. In our experiments we do not observe any evidence of the formation of superlattices and, therefore, we treat the folded bilayer graphene far from the curved boundary using a naive approximation of two independent sheets of BLG. As expected in our approximation, the BLGs behave like a semiconductor with tunable band gap. However, this band gap estimated from electrical measurements is smaller than reported in BLG [3,4]. Finally, in our naive approximation we do not consider any quantum effect that could be caused by the curvature of the curved boundary, which may be a subject of future theoretical studies. S3: Raman Characterization Figure S3. Determination of the number of layers of graphene by using the Raman spectroscopy. a, and b, show, respectively, the measured Raman 2D peak of the graphene flake of device 1 and device 2. The measurements are realized near of the folded region, using an excitation laser with the wavelength of 532 nm. For this wavelength, the shape of both measured 2D Raman peaks is quite similar to the expected shape of a BLG [6]. It confirms that the folded region is composed by two BLG.

S4: Evidence of kink states in device 2
We fabricate a second valleytronic device (device 2) and performed electronic measurements at the curved boundary of a folded-BLG. At this device we observed evidence of kink states, again revealed by the quantization of resistance close to the quantum resistance of R = 6.45 kΩ at zero-magnetic field. We present this device in Fig. S4a. We note that R CB,CNP saturates close to the quantum resistance R = h/4e 2 for |D| > 1.6 V/nm, which indicates that a DW forms at the 1D curved boundary and the charge motion is governed by a ballistic transport regime. Using the Landauer-Büttiker formula [7] R = R 0 (1 + L/L MFP ) we calculate the mean-free path of the channel: L MFP = 17.5 µm.
We next study the influence of a magnetic field B on the electronic properties of the curved boundary channel. We chose the electrostatic condition (D ∼ 1.4 V/nm) such that the resistance is saturated close to the ballistic resistance (see the arrow in Fig. S4c), and we ramped the magnetic field in the interval 0 T< B < 6.9 T. Fig. S4d shows such resistance R CB,CNP as function of B. We note that R CB,CNP remains close to R = e 2 /4h for all B. At such conditions, the increase of the magneto resistance on the bulk lead to a suppression of backscattering, promoting the protection of the chiral edge states in the curved boundary. Figure S4. Electronic transport properties of device 2. a, Top image: Optical picture of a folded-BLG deposited on top of a hBN crystal. Bottom image: false-color AFM topographic measurement of device 2 after the cleaning process and before it was covered with a top boron nitride flake. The curved boundary and its surroundings are free from contamination of the fabrication process. Scale bar: 1 µm. b, Raw data of the two-terminal R vs V BG vs V TG measured along the curved boundary at T = 1.2 K and B = 0 T. The inset show the electronic measurement configuration. c, R CNP vs D of the data presented in Fig. S4b. The two-terminal resistance saturates near of the quantum resistance R = e 2 /4h for D < −0.8 V/nm. A similar behavior was observed in device 1 and is another important evidence of the kink states at the curved boundary of a folded-BLG. d, The R CNP at the saturation, indicated by the arrow of Fig. S4c, as function of B. For B ∼ 0.5 T and B ∼ 3.5 T the resistance of the channel approaches the quantum resistance, indicating that backscattering are inhibit in the channel.
S5: Influence of temperature on the kink states Figure S5. Two-terminal electronic measurements as function of temperature. a, Two-terminal resistance as function of temperature measured on device 1, along the etched edge (red square), R EE,CNP , and curved boundary (blue circle), R CB,CNP . b, Two-terminal resistance R CB,CNP measured in the curved boundary of device 2 for temperatures in the range of 1.2 K< T < 300 K.
In this section, we describe the effects of temperature on the electronic transport properties of kink states at the domain wall in the curved boundary of the folded-BLG. In The measurements are done for temperatures ramping from T = 1.2 K up to T = 100 K.
We observe different mechanisms of conduction at the curved boundary and at the etched edge. The resistance R EE,CNP rapidly decreases when temperature goes up and show a small decreasing rate for higher temperatures. Such behavior is related to thermally activated conducting processes -typical for semiconducting materials. On the other hand, R CB,CNP slowly decreases with temperature, remaining near of the quantum resistance R = e 2 /4h even at T = 100 K. In the Fig. S5b, we show R CB,CNP measured along the electric contacts at the curved boundary of device 2. Here, we ramp the temperature of the system from 1.2 K up to room temperature (T = 300 K). Again, we observe a slow decrease of R CB,CNP with temperature, which remains on the order of the quantum resistance even at room temperature. Such robust feature of topological valley transport at high temperatures is similar to the non-local signal of the valley Hall Effect [8,9]. We believe that a large band gap in the bulk of the folded-BLG will provide a better protection of the chiral edge states in DW, that may promote a stable operation of valleytronic devices at ambient conditions.