Abstract
Nanoribbons are model systems for studying nanoscale effects in graphene. For ribbons with zigzag edges, tunable bandgaps have been predicted due to coupling of spinpolarized edge states, which have yet to be systematically demonstrated experimentally. Here we synthesize zigzag nanoribbons using Fe nanoparticleassisted hydrogen etching of epitaxial graphene/SiC(0001) in ultrahigh vacuum. We observe two gaps in their local density of states by scanning tunnelling spectroscopy. For ribbons wider than 3 nm, gaps up to 0.39 eV are found independent of width, consistent with standard density functional theory calculations. Ribbons narrower than 3 nm, however, exhibit much larger gaps that scale inversely with width, supporting quasiparticle corrections to the calculated gap. These results provide direct experimental confirmation of electron–electron interactions in gap opening in zigzag nanoribbons, and reveal a critical width of 3 nm for its onset. Our findings demonstrate that practical tunable bandgaps can be realized experimentally in zigzag nanoribbons.
Introduction
Graphene, one atomic layer of sp^{2}bonded carbon atoms closely packed in a honeycomb lattice, is a zerogap semiconductor with symmetrydictated linear dispersion at the Kpoint at low carrier energies, the same as a massless relativistic Dirac fermion^{1}. This leads to the novel electronic and physical properties of graphene, with potential applications in electronics, spin, and valleytronics^{2,3}. Patterning graphene into onedimensional nanostructures, that is, graphene nanoribbons (GNRs), introduces new properties such as tunable band gaps and spinpolarized edge states, depending on whether their edges are armchair (ac) or zigzag (zz)^{4,5,6,7,8}. These GNRs can be further functionalized by the adsorption of atoms such as H and O at their edges^{9,10,11,12}, or through edge reconstruction by the incorporation of topological defects such as pentagon and heptagon pairs^{13,14}.
Bandgap engineering is of particular interest, as it opens a myriad of new opportunities for graphene electronics. For GNRs with ac edges, earlier tightbinding calculations show a complete absence of edges states^{5}. However, metallic ribbons are predicted for N=3M−1 (where M is an integer, and N is the number of C–C units across the ribbon), and semiconducting for other widths, thus exhibiting a widthdependent semiconductor to metal oscillation^{5}. In density functional theory (DFT) calculations, however, all ac GNRs are found to be semiconducting, where quantum confinement is primarily responsible for the gap opening^{5,6}. In the case of GNRs with zzedges (ZGNRs), on the other hand, tightbinding calculations reveal edge states that form a flat band at the Fermi level (E_{F}) for k ∈ (2π/3a; π/a) (a=0.25 nm is the unit cell of the zz edge)^{4,5}. Spinpolarized DFT calculations show a splitting of the flat band and the opening of gaps of ~0.5 eV at the Xpoint, and ~0.1 eV at ~3/4 ГX, a direct result of the antiferromagnetic ordering of the edge states^{6}. Nevertheless, as bandgaps calculated by DFT are typically underestimated, significantly larger gaps have been predicted in more recent calculations that included quasiparticle corrections^{7}. For a 1.1nm ribbon, gaps of 1.9 and 1.3 eV are predicted at X and at 3/4 ГX (ref. 7), respectively, values that are much more attractive for realistic device applications.
The experimental realization of these much larger gaps, however, remains elusive, largely due to challenges producing ZGNRs just a few nanometres in width. Transport gaps measured in GNRs synthesized by nanofabrication techniques and unzipping of carbon nanotubes have indeed exhibited an inverse relationship with the width^{15,16,17,18,19}. However, these GNRs are usually terminated by chiral edges^{20,21,22} and their widths are typically one order of magnitude larger than those considered in calculations^{6,7}. The band gap measured in transport experiments to date is typically below 0.5 eV (refs 15, 16). Recent advances in bottomup chemical synthesis have produced GNRs of a few nanometres in width and gaps larger than 1 eV, but they are normally terminated with ac edges^{23,24,25}.
Here we fabricate ZGNRs by Fe nanoparticle (NP)assisted hydrogen etching of epitaxial graphene/SiC(0001) in ultrahigh vacuum (UHV), and systematically study their atomic and electronic structures by scanning tunnelling microscopy (STM) and scanning tunnelling spectroscopy. Unlike earlier studies^{20,21,22,23,24,25,26}, these GNRs are also supported by a graphene layer, and interactions with the substrates are expected to be minimal, as confirmed by our DFT calculations. In addition, the ZGNRs are synthesized in a hydrogenrich environment and, therefore, are expected to be terminated with hydrogen atoms^{9}. By carrying out ribbon fabrication, sample transfer, and measurements strictly under in situ and UHV conditions, contamination that may occur in ambient environments are averted.
Using tunnelling spectroscopy, we determine the widthdependent energy gaps within the spatially varying local density of states (LDOS) of the ZGNRs. For ribbons larger than 3 nm, we find relatively widthindependent gaps of 0.12±0.02 and 0.39±0.02 eV, in excellent agreement with standard, that is, without quasiparticle corrections^{6}, DFT calculations. Below the 3nm threshold, the gaps are found to scale with (w) as α/(w+β), where α and β are constants. A gap up to 1.6 eV is observed for a 1.1nm ribbon, the largest measured to date for a ZGNR. These results provide the first direct experimental confirmation of the many body e–e interaction corrections to the calculated bandgap of ZGNRs^{7}, and determine a critical width of 3 nm for the onset of such corrections.
Results
GNRs by Fe NP assisted Hetching
Measurements were carried out on epitaxial graphene grown on 6HSiC(0001) in UHV, which typically consists of up to three graphene layers on top of a warped interface layer^{27,28}. In addition, straininduced ridges are often observed^{29}. Tunnelling conductance dI/dV spectra taken on the monolayer (ML) exhibit two characteristic minima at zero bias (E_{F}) and at −0.41 eV (Fig. 1b), attributed to phononassisted inelastic tunnelling^{30}, and the Dirac point, respectively. For the bilayer (BL) and trilayer (TL), the Dirac point shifts closer to E_{F} and appears at −0.35 and −0.12 eV. These observations indicate the epitaxial graphene is ntype doped due to the Si dangling bonds at the graphene–SiC interface, consistent with earlier angleresolved photoemission and tunnelling spectroscopy measurements^{31,32}.
When deposited on the epitaxial graphene at room temperature, Fe forms NPs (Fig. 1a), which are randomly distributed on the surface, with no preferences for regions of ML, BL and TL. Based on the Gaussian fit of their distribution (Fig. 1a inset), an average size of 6 nm is found for these Fe NPs.
At elevated temperatures in 10^{−3} Torr H_{2}, the Fe NPs diffuse on the graphene surface, leaving behind trenches that are uniform in depth, but uneven in width depending on their sizes and motion (Fig. 1c). This ‘cutting’ of graphene is likely to be a result of the catalytic formation of methane from carbon and hydrogen^{33,34,35}, producing graphene nanostructures of various sizes and shapes, including GNRs with varying widths (one marked by the arrow in Fig. 1c). Compared with earlier work done in atmosphericpressure hydrogen, where the preferred directions of cutting are spaced 30° apart^{33,34,35}, a diverse distribution is found in our case with the predominant types being 90° and 30°, indicating the graphene nanostructures are either hydrogenterminated ac or zz edges (Supplementary Figs 1 and 2, and Supplementary Note 1).
Further analysis of line profiles across various trenches (Fig. 1d) indicates that only one atomic layer of graphene is removed in the tracks of the Fe NPs. The resulting GNRs are therefore single layer, supported either by a BL or by a ML graphene. As a result, the E_{F} of the GNRs depends on whether the supporting graphene is ML or BL, allowing the study of doping on the electronic properties of GNRs.
The edges of the ZGNRs produced by the FeNPassisted hydrogen etching method are often defective due to the formation of vacancies and kinks (c.f., Supplementary Fig. 1). The presence of defects makes determination of the atomic structures of the edges challenging. In this work, in addition to atomic resolution STM imaging, we have also analysed the edge orientation with respect to the supporting graphene layer (Supplementary Figs 1 and 2, and Supplementary Note 1), as well as the presence of edge states in the tunnelling spectra. Near the edges, these atomicscale defects typically produce a (√3 × √3) pattern because of the symmetry breaking intervalley K to K’ scattering^{36,37}. As a result, the electronic properties of graphene are strongly modified at the edges of ZGNRs, as demonstrated for the ribbon shown in Fig. 1e. Although line profiles (Fig. 1f) indicate a z_{1} (hydrogenterminated 1 × zz) structure along the edges (Supplementary Fig. 1), the defectinduced intervalley scattering leads to the formation of various standing wave patterns near both edges.
The ZGNRs often exhibit segments of varying width, connected by kinks (for example, bottom left edge in Fig. 1e), further allowing the study of widthdependent electronic properties of the same ribbon. In this work, we systematically investigate a variety of ZGNRs ranging from 1 to 14 nm in width supported on either ML or BL graphene. Before discussing our experimental results, we first address the impact of supporting graphene layer on the electronic properties of the ZGNRs by DFT calculations.
Calculations of the impact of supporting graphene layer
Figure 2a is a ballandstick model for a 5nm ribbon supported by ML graphene. Placing a zz ribbon on top of a graphene layer results in two different configurations for the opposite edges: one top site (left) and one hollow site (right). The calculated bands for the supported and freestanding ribbon are shown Fig. 2b, both indicate edge states starting at k_{s} (~2/3 along ΓX, which corresponds to the projection of the bulk Kpoint) and extending to the zone boundary at X. Because of the structural asymmetry between the left and right edges for the supported ribbon, the two spins are not strictly degenerate as they are for the freestanding ribbon, but are still localized at the two edges (c.f., Fig. 2c). As one moves away from the edge, the intensity of the edge states at X decrease relative to that of at k_{s}, indicative of the kdependent decay of the edge states. In the middle of the ribbon, the bands are essentially BL graphene with negligible edge state contribution at k_{s}. For both the supported and freestanding ribbons, the gaps are ~0.5 eV at the Xpoint and 0.1 eV at k_{s} (Fig. 2b). A small reduction in both gaps (for example, ~30 meV at k_{s}) is found for the supported ribbon.
Figure 2d plots spatially resolved LDOS in regions 1–3 as marked in Fig. 2a. The peaks at 0.20 and −0.22 eV for region 1 arise from edge states at the Xpoint, while the shoulder at 0.02 eV (E_{s}) below E_{F} is due to the band edge (van Hove singularity) at k_{s}. Moving away from the edge, the contribution from X decrease rapidly, so that the contribution from E_{s} becomes dominant both above and below the E_{F}. Above E_{F}, as one moves away from the edge (region 2–3), the range of k that contributes narrows so that the peak moves towards E_{F}.
Figure 2e shows the decay of the wave functions for the conduction band at k_{s} and X. At X, the wave function is strongly localized at the edge, whereas the states at k_{s} are more extended, similar to earlier calculations for freestanding ribbons^{6,7}. This is consistent with these edge states being largely built up from the bulk states at Kpoint. The supported ribbon, however, shows different decay rates for the top and hollow site due to the interaction with the supporting graphene, with the later exhibiting more localized states and hence a more rapid decay.
Edge states of ZGNRs by tunnelling spectroscopy
The electronic structure of ZGNRs are revealed by spatially resolved dI/dV tunnelling spectroscopy that reflects the energyresolved LDOS. Figure 3a shows an STM image of a 14nm ZGNR supported by ML graphene (Supplementary Fig. 3b). Tunnelling spectra were taken at the green dots marked in Fig. 3a from the centre of the ribbon to the edge. At the centre, the spectrum (X_{1}) exhibits a minimum at E_{F} and a local minimum at −0.30 eV (Fig. 3b), as expected from a BL graphene (Figs 1b and 2c). Compared with the asgrown graphene, E_{D} is slightly shifted by 50 meV towards the E_{F}, suggesting the intercalation of hydrogen at the graphene–SiC interface during the cutting of the ribbon^{38,39}. Approaching the edge, a new resonant peak (E_{3}) at −0.03 eV appears (spectrum X_{7}), whose intensity increases closer to the edge (X_{11}). Plot of this peak intensity as a function of distance from the edge is shown in Fig. 3c. A decay length (λ) of 2.43 nm is found by fitting the data with a function e^{−x/λ}. To rule out the possibility that this state is caused by local defects or adsorbates, tunnelling spectra were taken near defects. The results show that this peak is actually suppressed (Supplementary Fig. 3d and Supplementary Note 2), confirming earlier predictions^{40,41}.
Three additional features appear for spectra 10–11, as marked by dashed lines (E_{1}, E_{2}, and E_{4}) in Fig. 3b. Compared with the calculated spatially resolved LDOS (Fig. 2d), these spectroscopic features can be attributed to edge states at k_{s} and Xpoint, respectively. The asymmetry in peak intensity is consistent with states near the E_{F} contributing most to tunnelling. All four features appear below E_{F}, likely a result of heavy ntype doping of the ML graphene (E_{D} at 0.42 eV below E_{F}) supporting the ZGNR.
Next, dI/dV spectra were obtained at positions 1–7 along both edges of a 7nm ZGNR supported by a BL graphene (Fig. 3d) where the E_{D} is closer to E_{F} (0.3 eV below). The spectrum taken at position 3 on the right edge is shown in Fig. 3e together with that from BL graphene as a reference. Again, four characteristic peaks, as indicated by dotted lines, are clearly seen. After normalization of the dI/dV spectrum by I/V, all four peaks become more robust (Supplementary Fig. 4). They are additionally shifted by 0.15 eV towards positive energy with E_{3} now ~0.15 eV above E_{F}, consistent with the difference in Dirac point between BL and TL graphene (c.f., Figs 1b and 3e). The two peaks centred at E_{1} and E_{2} are separated by Δ_{0}(E_{1}−E_{2})=0.15±0.02 eV, and those centred at E_{3} and E_{4} separated by Δ_{1}(E_{3}−E_{4})=0.39±0.02 eV. Overall, these results indicate that the two sharp peaks E_{3} and E_{4} can be attributed to edge states at the Xpoint, and E_{1} and E_{2} to the Van Hove singularity of the band edge at k_{s}, consistent with our DFT calculations (c.f., Fig. 2) and earlier work^{6,7}.
Although all four peaks are observable along the edges, their separations fluctuate, as shown in Fig. 2f: for the left edge, Δ_{0} varies between 0.10 and 0.15 eV, and Δ_{1} between 0.34 and 0.42 eV; for the right edge, Δ_{0} between 0.15 and 0.17 eV, and Δ_{1} between 0.33 and 0.39 eV.
Bandgaps of ZGNRs narrower than 3 nm by tunnelling spectroscopy
ZGNRs comprising segments of varying widths are often observed, as shown in Fig. 4a for a ribbon supported on a BL graphene. Line profiles taken across the ribbon at the marked locations from top to bottom reveal widths of 1.3, 1.1 and 2.0 nm, consistent with 7, 6 and 10ZGNRs (Supplementary Fig. 5), where the integers indicate the number of zz units across the ribbon^{5}.
The four peaks at E_{1}−E_{4} in the dI/dV spectra are now more pronounced with larger separations (Fig. 4b). Moreover, all spectra exhibit a highly suppressed (~0) LDOS between E_{1} and E_{2}, indicating that these ZGNRs are gaped semiconductors. The peaks are also symmetric about the E_{F}, and exhibit variations (even total suppression) in intensity at different positions. For example, the tunnelling spectrum exhibits all four E_{1}−E_{4} peaks at position 1. At position 2, however, although E_{1} and E_{2} are pronounced, E_{3} and E_{4} are completely suppressed, and the situation is reversed for position 3. These fluctuations in the peak intensity are similar to those of the 7nm ribbon (c.f., Fig. 3f) and are related to variations in ribbon width and local defect structures (Supplementary Fig. 6 and Supplementary Note 2). Overall, the separations between the peaks give rise to energy gaps much larger than 0.5 eV (refs 15, 16), and for the 1.1 nm ZGNR, Δ_{0} and Δ_{1} are found to be 1.1±0.1 and 1.6±0.1 eV, respectively, the largest value measured for zz nanoribbons.
To gain further insights into the widthdependent energy gaps of narrower ZGNRs, spatially resolved tunnelling spectra along the ribbon are taken at the central positions marked in Fig. 4a. The LDOS spectrum evolves from the Ushaped (B_{1}) to completely insulating (B_{3}–B_{12}), where the gap is again clearly evident (Fig. 4c). A twodimensional plot of the continuous evolution of the dI/dV spectra as a function of width is presented in Fig. 4d, where the edge of bandgap is highlighted with a bright line.
The measured energy gaps (Δ_{0} and Δ_{1}) of ZGNRs for a broad range of width ranging from 1 to 14 nm (Supplementary Fig. 7) are summarized in Fig. 5. The results clearly exhibit two regions, with the boundary at a ribbon width of 3 nm. For widths larger than 3 nm, the Δ_{0} and Δ_{1} are ~0.39 and 0.12 eV, and relatively independent of the width. Below 3 nm, the gaps strongly depend on width, which can be fitted with a function α/(w+β), with the fitting parameters α=1.68 eV nm and β=0.47 nm for Δ_{0}, and α=3.90 eV nm and β=1.48 nm for Δ_{1}.
Discussion
The spectroscopic features in our dI/dV tunnelling spectra are closely related to the width and energydependent spinpolarized edge states predicted in our own and earlier calculations for ZGNRs^{6,7}. Our findings further indicate a critical threshold of 3 nm, above which bandgaps up to 0.39 eV are relatively independent of width, consistent with standard DFT calculations^{6}. For ribbons smaller than 3 nm, the trend of the Δ_{0} energy gap versus width agrees with the calculations that take into account the quasiparticle correction to the energy gap^{7}, although the size of the gaps are smaller than those calculated. This may be attributed in part to the fact that the ZGNRs are supported by a graphene layer in our experiment, while the calculations assume freestanding ZGNRs^{6,7}. A small reduction of ~30 meV in the Δ_{0} gaps compared with freestanding ribbons is seen in our DFT calculations for the supported 5 nm ribbon (c.f., Fig. 2b).
In addition, the finding of a critical threshold for the onset of the quasiparticle electron–electron correction to the gap—and lack of them for wide ribbons—can be rationalized by noting that the interior of ribbons wider than 5 nm is nominally graphene, a zero gap semiconductor. Therefore, quasiparticle corrections to the gap are not important, as there is no gap to begin with.
This can be quantitatively seen in additional calculations as shown in Fig. 6. Although the detailed dispersion of the edge states as a function of ribbon width requires calculations and measurements for those systems, the trends and many of the properties already can be obtained from consideration of the (bilayer) graphene bands. As is well known from surface calculations, the bands for a finite slab are related to the (three or twodimensional) bulk bands at given perpendicular momenta k_{p} determined by the Born–von Karman boundary conditions, that is, the bands of an Nlayer slab consist of the N bulk bands uniformly spaced in k_{p}. The surface/edge states ‘pop off’ at the band extrema in both k_{p} and k_{} (along the surface)^{42,43}. Normally, these surface states are predominantly composed of states corresponding to k_{p} and k_{} of the band extremum and those nearby, with their locations in the gap depending on the curvature of the bands at the extremum, with their dispersion typically following the bulk bands, although with smaller magnitude. For smaller slabs, these ‘nearby’ states can be fairly far away in both energy and k_{}, which will alter the energies of the surface states.
This correspondence between the bulk and slab bands describes the ribbon bands quite well. Figure 6a presents the bulk bands for BL graphene along ΓX corresponding to an effective zz ribbon width of w=4.3 nm. The red dots give the k_{p}=0 band bands, which includes the bulk Kpoint at 2/3 of the way along ΓX; this band will be included for any w. (For the armchair direction, the Kpoint is included only for widths corresponding to cells that are multiples of three.) Comparison of these projected bands with the full calculations in Fig. 2b shows excellent agreement for the overall dispersion and, importantly, how the edge states arise from the (red) k_{p}=0 bulk state, with the band edges defined by the bulk bands closest in k_{p} (black dots).
The position and dispersion of the band edges change significantly as a function of effective ribbon width (Fig. 6b). For small ribbons, the bulk edge extrema shift away from K to larger k_{}, and also change their energy and shape (curvature). Note that these bulk calculations have no gap in the spectrum, and that the overall band topology is determined by the structure and symmetry so that the trends here are independent of details such as choice of exchange correlation. The changes in the properties of the band edges, which will affect the position and dispersion of the edge states, are quantified in Fig. 6c–e and all show a break in behaviour at ~3 nm; for ribbons larger than that, the interactions between the edge state and the band edges are such (same k_{}, and E_{0}(k_{min})=0) that the Δ_{0} gap is expected to be small and not vary significantly with width, consistent with the experimental observation.
These expectations are consistent with calculations for the supported ribbons shown in Fig. 6f,g. For both cases, the Dirac state of the underlying graphene layer is evident, as are the connections between the edge and the bulk states. For the ribbon corresponding to w=2.6 nm, the valence and conduction states separate, forming a semiconductor for which e–e gap corrections are appropriate. For the wider ribbon (w=4.3 nm), the band of edge states follows the dispersion of the valence band and has clear BL graphene and metallic behaviour. The difference in behaviour between the two cases is also seen when considering the difference between top/hollow configurations: for the small ribbons, the two are basically degenerate, whereas for the wider ones, the bands for the two edges are shifted by 0.08 eV, reflecting the difference in the character of the states. For singlelayer graphene, calculations (Fig. 6h) show similar trends, but not with the sharp breaks as in Fig. 6c–e, suggesting that experiments on unsupported graphene may exhibit a more gradual transition. Electron correlations, including dispersion effects (for example, Fig. 6h), may also contribute to and enhance the observed transition. Thus, these calculations suggest that e–e correlations included in GW or HeydScuseriaErnzerhof (HSE) calculations are needed to correctly describe the semiconducting behaviour for ribbon narrower than ~3 nm, but are not needed (or appropriate) for wide ribbons.
Although these arguments explain the width dependence of the Δ_{0} gap, they do not directly address the Δ_{1} gap. The experimentally measured Δ_{0} and Δ_{1} gaps both show variations with the width for ribbons smaller than 3 nm, while a Δ_{1} gap independent of width is predicted at the Xpoint^{7}, with a value (1.45 eV in our HSE calculations) that is significantly larger than experiment for wide ribbons when e–e corrections are included. In all of these calculations, as well as in standard DFT calculations, the gap at X is basically determined by the exchange splitting of the localized edge state (c.f., Fig. 2) and thus does not vary with ribbon width. However, the correlations included via GW or HSE are calculated within the molecularorbital picture, and do not include the Heitler–Londontype correlations. To estimate these latter correlations, we have performed a series of local density approximation (LDA)+Utype calculations. The results show widthdependent and increased gaps compared with the standard DFT calculations (comparable to HSE calculations). More significantly, they indicate that the Δ_{1} gap also exhibits a similar, albeit slightly larger, decrease with width as the Δ_{0} gap, in agreement with the experimental observations. Thus, these results suggest that e–e correlations beyond those included in the GW and/or HSEtype band approaches are important in the narrow ribbon limit.
In conclusion, we have systematically studied the atomic and electronic structures of Hterminated graphene ZGNRs. We find a critical width of 3 nm where the e–e interaction correction to the bandgap takes place. Above this width, constant gaps of up to 0.39 eV are found, in excellent agreement with DFT calculations^{6}. Below this threshold, the gap scales with the width as α/(w+β). These results provide the first direct experimental confirmation of the e–e interactions that is key in gap opening in ZGNRs^{7} and determine a critical width of 3 nm for its onset.
Methods
Sample preparation
Experiments are carried out on epitaxial graphene grown by thermal decomposition of 6HSiC(0001), which was first prepared by removal of polishing damages in a H_{2}/Ar atmosphere between 1,500 and 1,600 °C. After degassing in UHV, the SiC sample was annealed at ~950 °C for 15 min in a Si flux to produce a (3 × 3) reconstructed surface, then heated to 1,100–1,300 °C for 15 min for the growth of graphene. Consistent with earlier work, this growth procedure produces a mixture of ML, BL and TL graphene on top of a warped buffer layer. GNRs are created by Fe NPassisted etching of graphene in 10^{−3} Torr H_{2} at ~900 °C in an interconnected UHV system, which also houses two STMs.
STM and scanning tunnelling spectroscopy
STM imaging is carried out with electrochemically etched W tips or mechanically sharpened Pt tips at room temperature and liquid nitrogen temperature in UHV, with a base pressure better than 1 × 10^{−10} Torr. The dI/dV tunnelling spectra are acquired using lockin detection at liquid nitrogen temperature by turning off the feedback loop and applying a small ac modulation of 9 mV (r.m.s.) at 860 Hz to the bias voltage. Each single spectrum shown in the manuscript is the average of 10–20 spectra. Tip is calibrated by carrying out dI/dV measurements on Ag(111) films grown on 6HSiC(0001) surface^{44} (Supplementary Fig. 8).
DFT calculations
Calculations were performed using the Vienna Ab initio Simulation Package^{45,46}, the Perdew–Zunger^{47} parameterization of the exchangecorrelation functional and pseudopotentials constructed by Blöchl’s projector augmented wave method^{48,49}. The onedimensional Brillouin zone was sampled by a 1 × 24 × 1 Monkhorst–Pack mesh. GNRs of up to ~5 nm were constructed with zz edges, in which each edge C atom is saturated by one hydrogen atom, supported by a continuous graphene layer, that is, the ribbons are in a BL configuration. The C atoms at the two edges always have different stacking with respect to the substrate, either on top (left edge) or in the hollow (right edge) site. Edge states are assumed to be ferromagnetically coupled along each edge but antiferromagnetically aligned between the two edges, with a magnetic moment of ~0.15 μ_{B} on each edge. (Larger moments are obtained if the core correction to the density is neglected.) To mimic the STM experiments, bands and density of states are weighted by the relative contribution of the wave functions in boxes in the vacuum region starting ~1 Å above the nanoribbon. Additional calculations on small ribbons were done using the fullpotential linearized augmented plane wave code flair^{50}, both for the ‘bulk’ BLs and for the LDA+U. In the case of the LDA+U, the correction was applied to the p_{z} (π) portions of the density matrices responsible for the antiferromagnetism of the ribbons for the carbon atoms near the edges to mimic the Heitler–Londontype correlations. Although the results depend on details of the LDA+U calculations and thus these results are qualitative rather than quantitative, the trends for reasonable parameters support the importance of these correlations and reproduce the behaviour seen experimentally. (The results depend, for example, on the choice of U–J, the form of the density matrix, and fully localized versus aboutmean field, and for some reasonable set of parameters can lead to ferrromagnetic solutions. The U–J values were restricted so that the dispersions of the ‘bulk’ ribbon bands were similar to those in the normal cases.)
Additional information
How to cite this article: Li, Y. Y. et al. Direct experimental determination of onset of electron–electron interactions in gap opening of zigzag graphene nanoribbons. Nat. Commun. 5:4311 doi: 10.1038/ncomms5311 (2014).
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Acknowledgements
This study was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DEFG0207ER46228.
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Y.Y.L. and L.L. designed the experiment. Y.Y.L. carried out the measurements. M.X.C. and M.W. performed the calculations. All authors contributed to the analysis and interpretation of the data. Y.Y.L., M.X.C., M.W. and L.L. wrote the paper.
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Supplementary Figures 18, Supplementary Notes 12 and Supplementary References (PDF 1178 kb)
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Li, Y., Chen, M., Weinert, M. et al. Direct experimental determination of onset of electron–electron interactions in gap opening of zigzag graphene nanoribbons. Nat Commun 5, 4311 (2014). https://doi.org/10.1038/ncomms5311
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