Epitaxial zigzag GNR

The geometry of a SiC facet with a GNR is depicted in Fig. 1a. Density functional theory (DFT) and transmission electron microscopy (TEM) have revealed that graphene growth is seeded at trenches close to the lower edge of the SiC facet structure, while the top of the ribbon merges into the buffer layer above the mesa19,20,21,22. For mesa structures running along the [1\({\bar{1}}\)00]-direction and with trench depths of around 20 nm, SiC(11\({\bar{2}}\)n) facets approximately 40 nm wide with an inclination of 25–30° are formed during annealing. However, it should be noted that the SiC-sidewalls easily refacet, i.e. forming smaller faceted subareas, at temperatures where Si sublimation and graphene growth sets in refs. 7,23. Recent optimization of growth conditions allows these energy-driven instabilities of the SiC(11\({\bar{2}}\)n) facets to be suppressed24 (see Supplementary Figure 1). The scanning tunneling microscopy (STM) images in Fig. 1b, c show the SiC facet to be almost completely overgrown by graphene in zigzag orientation, with only the top part revealing signs of step-bunching effects (see also Supplementary Information, SI). In situ two-point probe (2pp) transport measurements were used to characterise the long-ranged quantum-transport properties in detail at room temperature in ultrahigh vacuum. The characteristic value of R = h/e2 ≈ 26 kΩ is measured when both probes are located on the ribbon with a separation of 2 μm, as shown in Fig. 1d and in full agreement with prior reports1,3,18.

Fig. 1 Ballistic transport in graphene sidewall ribbons on SiC mesa structures. Graphene nanoribbons (GNRs) can be selectively grown on SiC sidewalls as sketched in a. b Sequence of STM measurements performed at room temperature show the entirely overgrown SiC facet areas (+2 V, 0.5 nA, semi-insulating SiC). c High-resolution STM showing the overgrowth of the SiC facet and zigzag orientation (inset). The scale bars indicate a length of 2 nm (blue) and 0.4 nm (green). d The IV-curves measured in a two point probe assembly (2pp) clearly reveal a resistance of h/e2 on the GNR for a probe distance of 2 μm. The GNR can be easily seen also in SEM (inset, doped-SiC, scale bar, 1 μm). By means of a 4-tip STM the ribbons are contacted for in situ transport measurements Full size image

Spatially resolved transport measurements

To gain insight into the electronic structure variations across the ribbon width, we have performed spatially-resolved in situ transport experiments using a STM/scanning electron microscopy (SEM) system with two probes in ohmic contact. One tip was blunt and covered the entire ribbon width, whereas a second, sharper tip was moved transversely across the ribbon at a fixed probe-to-probe distance (Fig. 2a, b) as small as 70 nm. The correlated microscopy with SEM and STM enabled us to measure reliably the transport with ultra-small probe spacings (see Supplementary Figure 2). Figure 2c shows a conductance of e2/h when the mobile tip connects to the lower edge of the ribbon, corresponding to transport only through the exceptional edge channel. As the tip moves from edge to the bulk, two higher conductance plateaus appear, whose values correspond closely to step sizes of 4e2/h suggesting transport through additional four-fold degenerate ballistic channels. The corresponding IV-curves taken at these three distinct sites are given in the inset of Fig. 2c. The sequential appearance and disappearance of the additional channels is robust and reproducible, as demonstrated in Fig. 2d, which shows repeated sweeps with the mobile probe in both directions. The mean free path lengths of bulk channels in confined graphene nanostructures of this kind are of the order of 100 nm (ref. 25, see Supplementary Note 1). This prevented previous studies, with probe separations greater than 100 nm, from discerning the novel ballistic characteristics of higher order channels in sidewall ribbons. We note that the sharp tip still has a radius of the order of 40 nm, so that a transverse sweep of the tip across the ribbon from the bottom edge captures a cumulative effect as first a single edge channel, and then additional bulk channels, are contacted by the tip.

Fig. 2 Spatially resolved 2pp transport measurements. a SEM image of a ballistic ribbon with an overlaid schematic showing a blunt and sharp tip with a scale bar of 300 nm. Inset: STM topography taken after transport measurements (scale bar, 100 nm). b Line scans across the ribbon, directions are indicated in the inset of a. c Conductance G measured for a fixed distance L = 70 nm, while the sharp tip was moved across the ribbon starting from the lower edge (U = 200 mV). Inset: IV-curves measured at bottom, middle and top of the GNR. d The sequence of the channels can be reversibly measured by moving the ohmically contacted tip forward and backward Full size image

The sharp onset of the single-channel conductance with first a contact between tip and ribbon unambiguously demonstrates the location of the exceptional channel at the lower edge of the GNR, consistent with the previous characterisation of nanoconstrictions18. Its degeneracy and location are also consistent with a fully spin-polarised zigzag edge state26,27,28. The 4e2/h conductance steps, on the other hand, are suggestive of transport through spin-degenerate and valley-degenerate confinement-induced sub-bands, such as those expected for pristine zigzag nanoribbons. The presence of two such steps indicates either contributions from two sub-bands, or that significant band-bending occurs to allow a single sub-band to cross the Fermi level multiple times. We further note that nanoribbon sub-bands are normally expected to be delocalised across the entire ribbon width, so that an increase of the contact area between the tip and ribbon should lead to a steady increase in conductance without significant step features. Quantised conductance plateaus are generally only expected when the electron density is varied to change the number of bands crossing the Fermi level. However, from Fig. 1b it is clear that the upper edge of the ribbon merges into a buffer layer structure present on the flat SiC(0001) parts. Significant charge transfer at this interface, analogous to the n-type doping of epitaxial graphene29, can result in an inhomogeneous potential across the ribbon width, corresponding to a strong effective transverse electric field and thus leading to band-bending effects. A similar effect has also recently been observed at lateral WSe 2 -MoS 2 heterojunctions30. We will demonstrate below that band-bending can account for both the segregation and degeneracy of the bulk transport channels observed in our measurements.

Conductive-AFM measurements

The spatial distribution of the various transport channels across the GNR, suggested by 2pp measurements, is further confirmed by conductive atomic force microscopy (c-AFM), which gives access to a direct real-space imaging of the transport channels. The AFM topography (Fig. 3a, b) once more uncovers a uniformly overgrown and smooth facet structure. Moreover, the simultaneously measured current image reveals multiple extended conductive channels parallel to the ribbon edge (Fig. 3a). A cross section (Fig. 3b) shows that a large local current flows at the lower edge with smaller currents across the rest of the ribbon. As evidenced by multiple IV-measurements, summarised by the histogram shown in Fig. 3d, the quantised conductance of the edge channel is once more reproduced (cf. Fig. 3c, d). The measurement of the characteristic quantum conductance value e2/h at large probe spacings under ambient conditions strongly underlines the robustness of the exceptional edge channel.

Fig. 3 Direct imaging of the current channels in ballistic GNRs. a Top: Topographic AFM image of a GNR recorded with a conductive Pt tip. Bottom: the simultaneously recorded current image (sample bias 30 mV), demonstrating that the bottom of the ribbon is significantly more conductive than the top. The scale bar corresponds to a length of 50 nm. b Current and topography cross sections measured across the GNR indicated with the white line in a. c IV-curves recorded in contact mode at the locations indicated at the inset. d The histogram of the resistance values measured on the ribbon of the inset of c. The AFM measurements were performed under ambient conditions at 300 K Full size image

Tight-binding calculations

To analyse the exceptional transport features and understand the exact origin of the various modes, we have performed full-scale quantum-transport simulations of zigzag-edged nanoribbons. Excellent agreement with experimental measurements is obtained (Fig. 4a) when these calculations account for both edge magnetism and a spatial segregation of the bulk eigenmodes induced by asymmetries between the lower and upper edges of the ribbon, which connect to SiC(0001) and the buffer layer, respectively. Previous studies support the formation of a spin-polarised state at a zigzag edge27,31 and its robustness at the graphene/SiC(0001) interface32. In our model, we restrict the presence of edge magnetism to the lower edge of the ribbon, as strong doping effects and the lack of a sharp zigzag interface are expected to quench such behaviour at the top edge. To account, in a general way, for inhomogeneous potentials that arise due to the merging of the upper edge with the buffer region, and consequent charge transfer at this interface, we include a transverse electric field term which shifts the Fermi energy of the upper edge by approximately 0.5 eV relative to the lower edge. Monolayer graphene on SiC has been shown to be n-type doped by unsaturated bonds at the SiC interface29,33. The merging of the top edge of the ribbon into the buffer layer should result in a similar local doping scenario. In addition, ab initio studies32 of narrow nanoribbons with both edges bonded to the Si-face of SiC(0001) reveal that hybridization quenches states other than the magnetic edge state, leading to secondary gap opening near the Fermi energy. Only the lower edge in our system has such a bond, so we reproduce the effect by adding a sublattice-dependent gap-opening term only at sites near this edge (see Methods).

Fig. 4 Tight-binding model of the edge and bulk channels. a Simulated two-point conductance as a function of the width of the mobile probe in contact with the ribbon, as shown schematically in the upper inset, capturing the characteristic stepwise features from experiment (see Fig. 2c). Lower inset: Band structure for spin-up (grey) and spin-down (red, dashed) electrons for a 188-ZGNR with edge magnetism and asymmetric potential terms. The dashed horizontal line shows the Fermi energy considered in the other panels, with the band crossings highlighted. b Schematic of the GNR. c Real-space projections of the states contributing to transmission (e.g. those at the corresponding crossing points in a) across the ribbon width. The blue and purple states are spin-degenerate, whereas the red state corresponds to the spin-polarised channel at the lower edge of the ribbon Full size image

The experimental three-plateau feature is accurately captured by our simulations once all the terms discussed above are included (Fig. 4a). Spin polarisation is required for an edge-localised state contributing e2/h to the conductance. The additional terms impose a spatial segregation of channels leading to step-like transitions as a function of probe position. The gap-opening term isolates the magnetic edge state from the bulk channels so that it can be resolved separately, whereas the effective electric field breaks the uniform distribution of bulk states across the ribbon width. For a small field, this term segregates valence and conduction band states towards opposite edges of the ribbon, but, for larger values, bands near the Fermi energy contain of an admixture of states with both conduction and valence band characteristics. This leads to a distinctive W-shape bending of the low-energy bands, as evidenced by the band structure in the lower inset of Fig. 4a and further analysed in Supplementary Note 3. Within this energy region, current-carrying states from the same band can be localised at opposite edges of the ribbon (Fig. 4c), belying a mix of conduction and valence band characteristics. We note that the spatial segregation and degeneracies of the bulk experimental transport channels are entirely consistent with a single bent sub-band with spin and valley degeneracies. They are however not consistent with transport through multiple sub-bands since such a scenario tends to cluster states entirely along one edge. We note that a wide-range of gap-opening and transverse field parameters give rise to spatially-separated channels such as those reported here, supporting the robustness of these transport signatures (see Supplementary Figure 5). Furthermore this behaviour persists over a wide range of Fermi energies near the Dirac point (see Supplementary Figure 4).