Nanoribbons are model systems for studying nanoscale effects in graphene. For ribbons with zigzag edges, tunable bandgaps have been predicted due to coupling of spin-polarized edge states, which have yet to be systematically demonstrated experimentally. Here we synthesize zigzag nanoribbons using Fe nanoparticle-assisted 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.
Graphene, one atomic layer of sp2-bonded carbon atoms closely packed in a honeycomb lattice, is a zero-gap semiconductor with symmetry-dictated linear dispersion at the K-point at low carrier energies, the same as a massless relativistic Dirac fermion1. This leads to the novel electronic and physical properties of graphene, with potential applications in electronics, spin-, and valley-tronics2,3. Patterning graphene into one-dimensional nanostructures, that is, graphene nanoribbons (GNRs), introduces new properties such as tunable band gaps and spin-polarized 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 edges9,10,11,12, or through edge reconstruction by the incorporation of topological defects such as pentagon and heptagon pairs13,14.
Bandgap engineering is of particular interest, as it opens a myriad of new opportunities for graphene electronics. For GNRs with ac edges, earlier tight-binding calculations show a complete absence of edges states5. 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 width-dependent semiconductor to metal oscillation5. In density functional theory (DFT) calculations, however, all ac GNRs are found to be semiconducting, where quantum confinement is primarily responsible for the gap opening5,6. In the case of GNRs with zz-edges (ZGNRs), on the other hand, tight-binding calculations reveal edge states that form a flat band at the Fermi level (EF) for k ∈ (2π/3a; π/a) (a=0.25 nm is the unit cell of the zz edge)4,5. Spin-polarized DFT calculations show a splitting of the flat band and the opening of gaps of ~0.5 eV at the X-point, and ~0.1 eV at ~3/4 Г-X, a direct result of the antiferromagnetic ordering of the edge states6. Nevertheless, as bandgaps calculated by DFT are typically underestimated, significantly larger gaps have been predicted in more recent calculations that included quasiparticle corrections7. For a 1.1-nm ribbon, gaps of 1.9 and 1.3 eV are predicted at X and at 3/4 Г-X (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 width15,16,17,18,19. However, these GNRs are usually terminated by chiral edges20,21,22 and their widths are typically one order of magnitude larger than those considered in calculations6,7. The band gap measured in transport experiments to date is typically below 0.5 eV (refs 15, 16). Recent advances in bottom-up chemical synthesis have produced GNRs of a few nanometres in width and gaps larger than 1 eV, but they are normally terminated with ac edges23,24,25.
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 studies20,21,22,23,24,25,26, 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 hydrogen-rich environment and, therefore, are expected to be terminated with hydrogen atoms9. 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 width-dependent energy gaps within the spatially varying local density of states (LDOS) of the ZGNRs. For ribbons larger than 3 nm, we find relatively width-independent gaps of 0.12±0.02 and 0.39±0.02 eV, in excellent agreement with standard, that is, without quasiparticle corrections6, DFT calculations. Below the 3-nm 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 1.1-nm 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 ZGNRs7, and determine a critical width of 3 nm for the onset of such corrections.
GNRs by Fe NP assisted H-etching
Measurements were carried out on epitaxial graphene grown on 6H-SiC(0001) in UHV, which typically consists of up to three graphene layers on top of a warped interface layer27,28. In addition, strain-induced ridges are often observed29. Tunnelling conductance dI/dV spectra taken on the monolayer (ML) exhibit two characteristic minima at zero bias (EF) and at −0.41 eV (Fig. 1b), attributed to phonon-assisted inelastic tunnelling30, and the Dirac point, respectively. For the bilayer (BL) and trilayer (TL), the Dirac point shifts closer to EF and appears at −0.35 and −0.12 eV. These observations indicate the epitaxial graphene is n-type doped due to the Si dangling bonds at the graphene–SiC interface, consistent with earlier angle-resolved photoemission and tunnelling spectroscopy measurements31,32.
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 H2, 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 hydrogen33,34,35, 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 atmospheric-pressure hydrogen, where the preferred directions of cutting are spaced 30° apart33,34,35, a diverse distribution is found in our case with the predominant types being 90° and 30°, indicating the graphene nanostructures are either hydrogen-terminated 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 EF 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 Fe-NP-assisted 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 atomic-scale defects typically produce a (√3 × √3) pattern because of the symmetry breaking intervalley K to K’ scattering36,37. 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 z1 (hydrogen-terminated 1 × zz) structure along the edges (Supplementary Fig. 1), the defect-induced 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 width-dependent 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 ball-and-stick model for a 5-nm 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 free-standing ribbon are shown Fig. 2b, both indicate edge states starting at ks (~2/3 along Γ-X, which corresponds to the projection of the bulk K-point) 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 ks, indicative of the k-dependent decay of the edge states. In the middle of the ribbon, the bands are essentially BL graphene with negligible edge state contribution at ks. For both the supported and freestanding ribbons, the gaps are ~0.5 eV at the X-point and 0.1 eV at ks (Fig. 2b). A small reduction in both gaps (for example, ~30 meV at ks) 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 X-point, while the shoulder at 0.02 eV (Es) below EF is due to the band edge (van Hove singularity) at ks. Moving away from the edge, the contribution from X decrease rapidly, so that the contribution from Es becomes dominant both above and below the EF. Above EF, as one moves away from the edge (region 2–3), the range of k that contributes narrows so that the peak moves towards EF.
Figure 2e shows the decay of the wave functions for the conduction band at ks and X. At X, the wave function is strongly localized at the edge, whereas the states at ks are more extended, similar to earlier calculations for freestanding ribbons6,7. This is consistent with these edge states being largely built up from the bulk states at K-point. 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 energy-resolved LDOS. Figure 3a shows an STM image of a 14-nm 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 (X1) exhibits a minimum at EF and a local minimum at −0.30 eV (Fig. 3b), as expected from a BL graphene (Figs 1b and 2c). Compared with the as-grown graphene, ED is slightly shifted by 50 meV towards the EF, suggesting the intercalation of hydrogen at the graphene–SiC interface during the cutting of the ribbon38,39. Approaching the edge, a new resonant peak (E3) at −0.03 eV appears (spectrum X7), whose intensity increases closer to the edge (X11). 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 predictions40,41.
Three additional features appear for spectra 10–11, as marked by dashed lines (E1, E2, and E4) in Fig. 3b. Compared with the calculated spatially resolved LDOS (Fig. 2d), these spectroscopic features can be attributed to edge states at ks and X-point, respectively. The asymmetry in peak intensity is consistent with states near the EF contributing most to tunnelling. All four features appear below EF, likely a result of heavy n-type doping of the ML graphene (ED at 0.42 eV below EF) supporting the ZGNR.
Next, dI/dV spectra were obtained at positions 1–7 along both edges of a 7-nm ZGNR supported by a BL graphene (Fig. 3d) where the ED is closer to EF (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 E3 now ~0.15 eV above EF, consistent with the difference in Dirac point between BL and TL graphene (c.f., Figs 1b and 3e). The two peaks centred at E1 and E2 are separated by Δ0(E1−E2)=0.15±0.02 eV, and those centred at E3 and E4 separated by Δ1(E3−E4)=0.39±0.02 eV. Overall, these results indicate that the two sharp peaks E3 and E4 can be attributed to edge states at the X-point, and E1 and E2 to the Van Hove singularity of the band edge at ks, consistent with our DFT calculations (c.f., Fig. 2) and earlier work6,7.
Although all four peaks are observable along the edges, their separations fluctuate, as shown in Fig. 2f: for the left edge, Δ0 varies between 0.10 and 0.15 eV, and Δ1 between 0.34 and 0.42 eV; for the right edge, Δ0 between 0.15 and 0.17 eV, and Δ1 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 10-ZGNRs (Supplementary Fig. 5), where the integers indicate the number of zz units across the ribbon5.
The four peaks at E1−E4 in the dI/dV spectra are now more pronounced with larger separations (Fig. 4b). Moreover, all spectra exhibit a highly suppressed (~0) LDOS between E1 and E2, indicating that these ZGNRs are gaped semiconductors. The peaks are also symmetric about the EF, and exhibit variations (even total suppression) in intensity at different positions. For example, the tunnelling spectrum exhibits all four E1−E4 peaks at position 1. At position 2, however, although E1 and E2 are pronounced, E3 and E4 are completely suppressed, and the situation is reversed for position 3. These fluctuations in the peak intensity are similar to those of the 7-nm 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, Δ0 and Δ1 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 width-dependent 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 U-shaped (B1) to completely insulating (B3–B12), where the gap is again clearly evident (Fig. 4c). A two-dimensional 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 (Δ0 and Δ1) 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 Δ0 and Δ1 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 Δ0, and α=3.90 eV nm and β=1.48 nm for Δ1.
The spectroscopic features in our dI/dV tunnelling spectra are closely related to the width- and energy-dependent spin-polarized edge states predicted in our own and earlier calculations for ZGNRs6,7. 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 calculations6. For ribbons smaller than 3 nm, the trend of the Δ0 energy gap versus width agrees with the calculations that take into account the quasiparticle correction to the energy gap7, 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 free-standing ZGNRs6,7. A small reduction of ~30 meV in the Δ0 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 two-dimensional) bulk bands at given perpendicular momenta kp determined by the Born–von Karman boundary conditions, that is, the bands of an N-layer slab consist of the N bulk bands uniformly spaced in kp. The surface/edge states ‘pop off’ at the band extrema in both kp and k|| (along the surface)42,43. Normally, these surface states are predominantly composed of states corresponding to kp 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 Γ-X corresponding to an effective zz ribbon width of w=4.3 nm. The red dots give the kp=0 band bands, which includes the bulk K-point at 2/3 of the way along Γ-X; this band will be included for any w. (For the armchair direction, the K-point 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) kp=0 bulk state, with the band edges defined by the bulk bands closest in kp (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 E0(kmin)=0) that the Δ0 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 single-layer 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 Heyd-Scuseria-Ernzerhof (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 Δ0 gap, they do not directly address the Δ1 gap. The experimentally measured Δ0 and Δ1 gaps both show variations with the width for ribbons smaller than 3 nm, while a Δ1 gap independent of width is predicted at the X-point7, 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 molecular-orbital picture, and do not include the Heitler–London-type correlations. To estimate these latter correlations, we have performed a series of local density approximation (LDA)+U-type calculations. The results show width-dependent and increased gaps compared with the standard DFT calculations (comparable to HSE calculations). More significantly, they indicate that the Δ1 gap also exhibits a similar, albeit slightly larger, decrease with width as the Δ0 gap, in agreement with the experimental observations. Thus, these results suggest that e–e correlations beyond those included in the GW and/or HSE-type band approaches are important in the narrow ribbon limit.
In conclusion, we have systematically studied the atomic and electronic structures of H-terminated 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 calculations6. 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 ZGNRs7 and determine a critical width of 3 nm for its onset.
Experiments are carried out on epitaxial graphene grown by thermal decomposition of 6H-SiC(0001), which was first prepared by removal of polishing damages in a H2/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 NP-assisted etching of graphene in 10−3 Torr H2 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 lock-in 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 6H-SiC(0001) surface44 (Supplementary Fig. 8).
Calculations were performed using the Vienna Ab initio Simulation Package45,46, the Perdew–Zunger47 parameterization of the exchange-correlation functional and pseudopotentials constructed by Blöchl’s projector augmented wave method48,49. The one-dimensional 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 μB 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 full-potential linearized augmented plane wave code flair50, both for the ‘bulk’ BLs and for the LDA+U. In the case of the LDA+U, the correction was applied to the pz (π) portions of the density matrices responsible for the antiferromagnetism of the ribbons for the carbon atoms near the edges to mimic the Heitler–London-type 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 about-mean 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.)
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This study was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46228.
The authors declare no competing financial interests.
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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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