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

TL;DR

This paper demonstrates the formation of topological valley transport along a curved boundary in folded bilayer graphene, revealing long-range ballistic conduction through kink states in a high-quality domain wall.

Contribution

It introduces a novel method to create a topological domain wall in folded bilayer graphene, enabling robust valley-polarized transport without complex fabrication.

Findings

Measured resistance close to quantum resistance, indicating kink states.

Observed long-range ballistic transport at the domain wall.

Other regions exhibit tunable semiconducting behavior.

Abstract

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.

Full text

This is a pre-print of an article published in Communications Physics. The final authenticated version is available online at: https://doi.org/10.1038/s42005-018-0106-4

Electric charge and spin are intrinsic quantum properties of electrons and so far, are the basis of electronics. Likewise, charge carriers in two-dimensional (2D) hexagonal crystals have an additional electronic degree of freedom, the valley-index, associated to degenerate bands at the inequivalent K and K’ points in the Brillouin Zone (BZ). The valleytronic field proposes the creation of a new class of dissipationless electronic devices based on the manipulation of the valley-indices like valley filters and valley valves[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.

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${}_{\text{2}}$ 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 $\mu$m 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${}_{\text{2}}$/Si${}_{\text{++}}$ wafer, such that the Si${}_{\text{++}}$ is a highly p-type doped silicon used as a backgate. The top hBN is covered by a metallic gate composed by Cr/Au. To illustrate our fabrication process, in the top image of Fig. 1b we show a typical heterostructure of a naturally folded-BLG on top of a flat hBN crystal, before patterning the metallic contacts (fabrication details are discussed in Methods and Supplementary Section 2). The dashed lines in this picture indicate the position of such electric terminals: two of them stand on the curved boundary and two of them are placed on the etched edges. The bottom image of Fig. 1b shows a false-colour AFM topography measurement of device 1. 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 ($\Delta$)[19] defined as the layer energy difference between the top graphene layer and the bottom graphene layer of BLG. At the bottom BLG sheet, the top graphene layer acquires a relative energy $+\Delta/2$, calculated relative to the centre of the sheet, and the bottom layer acquires a relative energy $-\Delta/2$. This electrostatic energy distribution inverts in the other BLG sheet on top. The bottom graphene layer (former top layer) now acquires a relative energy $-\Delta/2$ and the top layer (former bottom layer) acquires a relative energy $+\Delta/2$. Then, the band gap inverts its sign from the bottom to top BLG and, consequently, a valley at these different BLGs holds opposite Chern numbers. In this condition, the curved boundary of the folded-BLG transforms into a DW. We use the model proposed by Martin et al.[1] to demonstrate the emergence of kink states in such topological folded structure (Supplementary Section 1). In the Fig. 1d we illustrate the pair of spin-degenerate kink states localized at the points K and K’ of the BZ, having opposite group-velocities at different valleys. These chiral edge states propagate in the curved boundary along a 1D direction defined by the valley-index, as illustrated in Fig 1a.

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 $R=6.45$ k$\Omega$ (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 ($V_{\text{BG}}$) and the topgate voltage ($V_{\text{TG}}$) 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 $R^{\text{EE,CNP}}$ and for the electric measurements realized on the curved boundary it will be called $R^{\text{CB,CNP}}$. A better comparison of $R^{\text{EE,CNP}}$ and $R^{\text{CB,CNP}}$ 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: $D=(C_{\text{TG}}V_{\text{TG}}-C_{\text{BG}}V_{\text{BG}})/\epsilon_{0}$, where $C_{\text{TG}}$ and $C_{\text{BG}}$ are, respectively, the capacitances per unit of area and charge of the top capacitor and bottom capacitor, and $\epsilon_{0}$ is the vacuum permittivity. From data present in Fig. 2c we note that the monotonic increasing of $R^{\text{EE,CNP}}$ with $D$ reflects a tunable band gap caused by the broken inversion symmetry of BLGs[20, 21]. In contrast, the $R^{\text{CB,CNP}}$ 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] $R=R_{\text{0}}(1+L/L_{\text{MFP}})$ to calculate the MFP of the ballistic channel in our two devices. Here, $L_{\text{MFP}}$ is the MFP, $R_{\text{0}}=h/4e^{2}$ is the quantum resistance and $L$ is the length of the channels: $L=1$ $\mu$m for device 1 and $L=1.75$ $\mu$m for device 2 (Supplementary Section 4). Since the experiment shows a resistance close to $R=e^{2}$/$4h$, we neglected any other residual resistances in this channel. The calculated MFP of the channels are: $L_{\text{MFP}}\sim 20$ $\mu$m for device 1 and $L_{\text{MFP}}\sim 17$ $\mu$m 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${}_{\text{2}}$[8].

Further details about the mechanisms of conduction in the folded-BLG are obtained by using a four-terminal electronic configuration. In the Fig. 3, we show the raw data of the longitudinal resistance ($R_{\text{xx}}$) as a function of the $V_{\text{BG}}$ and $V_{\text{TG}}$ measured at contacts placed along the etched edge (Fig. 3a) and the curved boundary (Fig. 3b). Again, both measurements show distinct diagonal lines where the resistance strongly depends on the gate potential. These dependence are better visualized in Fig. 3c, where we plot only the longitudinal resistances along the diagonal line as a function of $D$. Such resistances are called $R_{\text{xx}}^{\text{EE,CNP}}$ for the electric measurements realized at the etched edge and $R_{\text{xx}}^{\text{CB,CNP}}$ for the electric measurements performed at the curved boundary. We note that similarly to the two-terminal measurements, $R_{\text{xx}}^{\text{EE,CNP}}$ continuously increase with the displacement field. It changes from $R_{\text{xx}}^{\text{EE,CNP}}$ $\sim 1$ k$\Omega$ up to $R_{\text{xx}}^{\text{EE,CNP}}$ $\sim 12$ k$\Omega$, an expected behavior related to the tunable semiconducting nature of the BLGs by the action of the transverse electric field. On the other hand, we measure a different trend at the curved boundary channel. We observe that $R_{\text{xx}}^{\text{CB,CNP}}$ decreases with $D$, changing from $R_{\text{xx}}^{\text{CB,CNP}}$ $\sim 400$ $\Omega$ down to $R_{\text{xx}}^{\text{CB,CNP}}$ $\sim 60$ $\Omega$ for high $D$. Such small longitudinal resistance is a distinct feature of a ballistic transport regime in a channel with quasi-absence of backscattering. It provides another important evidence that a ballistic charge motion was achieved at the DW in the curved boundary.

Next, we investigate the effect of temperature on the electric conduction at the different regions of the folded-BLG. In Fig. 3d and Fig. 3e we show measurements of $R_{\text{xx}}$ as function of $V_{\text{TG}}$, with the backgate voltage fixed at $V_{\text{BG}}=-86$ V, while we changed the temperature of the system from $T=1.2$ K up to room temperature ($T=300$ K). At low temperatures, the electric conduction in the curved boundary is governed by a ballistic transport regime, revealed by the measurement of a small longitudinal resistance. At same electrostatic condition the BLGs at the etched edge reveal a semiconducting regime. As showed in Fig. 3d, the elevation of temperature in the system let to a decreasing of $R_{\text{xx}}$ measured along the etched edge. Clearly, it is the expected behavior of a semiconducting regime dominated by thermally activated processes[21]. A different feature is observed at the curved boundary, as illustrated in Fig. 3e. The longitudinal resistance of such region increases when temperature goes up. Such behavior revels that temperature promotes valley-mixing by phonon scattering and enhances the scattering of edge states localized at the curved boundary channel to BLG states that may conduct due to charge inhomogeneity. These measurements on different temperatures give a third evidence that chiral edge states at the curved boundary let to a different electronic transport regime when compared to the other regions of the folded-BLG.

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${}_{\text{2}}$/Si${}_{\text{++}}$, where Si${}_{\text{++}}$ 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 ($V_{\text{bias}}=1$ 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 ($V_{\text{xx}}$) is measured in between the other electric terminals on the opposite side. The longitudinal resistance is calculated by using the Ohm’s law $R_{\text{xx}}$=$V_{\text{xx}}/I$.

Acknowledgements

The authors thank Javier D. Sanchez-Yamagishi, Hadar Steinberg and Marcos H. D. Guimaraes for the fruitful discussions and the paper revision. The authors acknowledge the support of LabNS and LabNano for the Raman and AFM measurements. E.M., A.R.C. and L.C.C. acknowledge the support of the Brazilian agencies: Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq/MCTI), Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (Capes), Fundacao de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST.

Author contributions

L.C.C. conceived the main idea of the work. E.M. and A.R.C. fabricated the devices. K. W. and T.T. supplied the high-quality hBN flakes. E.M. and L.C.C. planned, realize the measurements, analyzed the data and wrote the manuscript.

Competing Interests

The authors declare no competing interests.