Abstract
Precise control over the size and shape of graphene nanostructures allows engineering spinpolarized edge and topological states, representing a novel source of nonconventional πmagnetism with promising applications in quantum spintronics. A prerequisite for their emergence is the existence of robust gapped phases, which are difficult to find in extended graphene systems. Here we show that semimetallic chiral GNRs (chGNRs) narrowed down to nanometer widths undergo a topological phase transition. We fabricated atomically precise chGNRs of different chirality and size by on surface synthesis using predesigned molecular precursors. Combining scanning tunneling microscopy (STM) measurements and theory simulations, we follow the evolution of topological properties and bulk band gap depending on the width, length, and chirality of chGNRs. Our findings represent a new platform for producing topologically protected spin states and demonstrate the potential of connecting chiral edge and defect structure with band engineering.
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Introduction
Band topological classification of materials has been successfully applied to predict and explain the emergence of exotic states of matter such as Quantum Spin Hall (QSH) edge states in topological insulators^{1,2,3} or topological superconductivity^{4}. The potential of this classification relies on the protection of the topological order by a symmetry, that can undergo a topological phase transition when the symmetry is changed. Symmetry Protected Topological (SPT) phase transitions were observed in artificial semiconducting systems such as twodimensional quantum wells of HgTe^{5} and onedimensional organic polymers^{6}. The key element is the existence of two gapped SPT phases, the traditional (trivial) band insulator and the nontrivial topological insulating state, separated by a metallic state.
In spite of being a semimetal, graphene has the potential to build up SPT phases by opening an energy gap around the Fermi level and endowing the lattice with an additional chiral symmetric interaction^{7}. For example, onedimensional SPT phases were engineered inside the bandgap of armchair GNRs by modelling a onedimensional SuSchriefferHeeger (SSH) chain^{8} with edge moieties containing localized ingap states^{9,10,11,12,13}. In zigzag GNRs (ZGNRs), however, the existence of zeroenergy edge bands^{14,15} prevents the appearance of gapped topological phases. As proposed by Kane and Mele^{3}, the presence of spin–orbit interaction can open a gap in bulk graphene and turn the zeroenergy modes into QSH edge states. However, the expected gap induced by spin–orbit interaction in graphene is very small, and this effect could only be present at very low temperatures^{16}.
Here, we demonstrate that sizeable topological insulating phases emerge in narrow chiral GNRs driven by the interaction between the opposing edges. The term chiral GNRs (chGNRs) refers to the large set of ribbons extending along lowsymmetry crystallographic directions (n, m) of graphene. The genuine zeroenergy edge bands of ZGNRs persist in chGNRs via the accumulation of states around zero energy over zigzag sites, including their spin polarization in the presence of Coulomb electron–electron interactions^{17,18}. However, chiral ribbons are more sensitive to a reduced width than ZGNRs, allowing to easily produce gapped GNRs with inherited chirality from the edge reconstruction.
Results
Prediction of a SPT phase transition
We consider a family of chGNRs customized from a basic rectangular aromatic block of length z (number of zigzag unit cells) and width w (number of carbon atoms across), blueshadowed in Fig. 1a. Chiral GNRs along any chiral direction (n, m) can be obtained simply by repeating and shifting these blocks along their armchair edges an amount of a − 1/2 armchair unit cells, and connecting them with C–C bonds. The edges of the resulting chGNR alternate z zigzag and a1/2 armchair sites to accommodate its orientation to the chiral vector, such that (n, m) = (z + 1 − a, 2a − 1) (see Supplementary Note 1). This edge reconstruction promotes the localization of zeroenergy edge states at the z zigzag segments, while the perpendicular armchair spacers act as potential barriers between them^{19}. For example, our tightbinding (TB) simulations in Fig. 1a, b for wide chGNRs show the presence of zeroenergy bands localized at the zigzag edge segments, reminiscent of the edge bands in the zigzag edges of graphene, which decay towards the center of the ribbon. Therefore, these ribbons lie in a metallic phase, with electron mobility that can be described as hopping between zigzag segments along the chGNR edge.
However, this metallic phase vanishes in narrow ribbons due to hybridization of bands at opposing edges. Interestingly, the emergent gapped phase corresponds to a SPT insulating phase characterized by an invariant \({{\mathbb{Z}}}_{2}=1\), as obtained from the computed Zak phase γ_{z} = π^{20} of the occupied bands, using a rectangular unit cell enclosing the blue block in Fig. 1 (see Supplementary Note 3). The nontrivial topological class of this phase turns out to be a global property of narrow chGNRs of this family, protected by the symmetry of a chiral hybridization pattern between edges.
Our simulations also find that the topological phase vanishes upon further reducing the width of the ribbon. The gap closes and reopens again as a trivial band insulator, characterized by \({{\mathbb{Z}}}_{2}=0\). In agreement with the properties of SPT phases, this new trivial state is connected with a symmetry change in the interaction pattern between the edges. Figure 1c exemplifies the transition for ribbons with (n = 3, m = 1) and (2,3) chiral vectors, by plotting their evolution of the energy gap with the width w. First, a sizable energy gap opens up as the ribbons are narrowed down, which corresponds to a onedimensional topological insulating phase, and at a critical width of w ~ 6, the gap closes, and reopens, now in a trivial phase.
Fabrication of chGNRs
To demonstrate the predicted topological phase transition described above, we fabricated several members of this chGNR family with different chiral vectors and widths (Fig. 2) though a combination of customized organic precursors and onsurface synthesis (OSS) over a gold (111) surface^{21,22}. Our strategy started with the synthesis of poli[n’]anthracene precursor molecules 1, 2, 3, and 4 shown in Fig. 2a–d fo producing chGNRs with z = 3 edge structure. Their customized structures, composed of an increasing number of anthacene units and different Br functionalization sites a, were designed for obtaining (3,1) chGNRs with increasing width (using 1, 2, 3) and chiral angle (e.g., (2,3) chGNR, using 4) through a sequence of OSS steps. For a fixed z, the Brsubstitution site, labelled a in Supplementary Note 1, determines the (n, m) chiral vector by steering the Ullmannlike connection between molecular precursors with a shift of a − 1/2 armchair unit lengths (see Fig. 1a). The number of anthracene units [n’] determines the width w = 2[n’] of the ribbon. Hence, in the following we label the ribbons of this family as z,a,wchGNR (Supplementary Note 1).
The GNR precursors were prepared in solution, as shown in Fig. 2a–d. Compound 1, which is formed by the linking of two anthracenes and constitutes the molecular precursor of 3,1,4chGNR, was obtained by Znpromoted reductive coupling of bromoanthrone 5, followed by dehydration^{23}. The trisanthracene 2, precursor of 3,1,6chGNRs, was obtained in one pot from dibromoanthraquinone 6, by addition of two equivalents of 9anthracenyl lithium (7) followed by reduction with a mixture of HI and H_{3}PO_{2}. Similarly, the tetrakisanthracene 3, precursor of 3,1,8chGNRs, was synthesized by reaction of compound 8 with organolithium 7, followed by a reduction step. Finally, compound 4, precursor of 3,2,8chGNRs, was prepared from derivative 9 following a similar procedure (see Methods for further details). In the OSS step, each precursor was independently sublimated onto a clean Au(111) surface held at room temperature and stepwisely annealed to T_{1} to induce their Ullmannlike polymerization. Subsequently, a further annealing to T_{2} activated the cyclodehydrogenation (CDH) of the polymers into the targeted chiral graphene nanoribbons with chiral vectors (3,1) and (2,3) and with different widths (Fig. 2e–h). STM images of the resulting structures (Fig. 2i–l) show the characteristic straight, and planar shape of the ribbons, confirming their successful synthesis. The STM images also shown that chGNRs’ length scales inversely with the size of the precursor. However, the overall length of the ribbons can be increased by adjusting the annealing parameters.
Emergence of edge bands in wide 3,1,wchGNRs
We compare first the effect of increasing the width on the electronic structure of 3,1,wchGNRs. Bondresolved STM images shown in Fig. 3a–c (obtained by measuring constant height current maps at V = 2 mV using a COterminated tip^{24,25}) reproduce the hexagonal ring patterns of the different ribbons, in agreement with the chemical structures in Fig. 2g–i. However, the wider 3,1,6 and 3,1,8chGNRs show, on top of the ring structure, a characteristic current increase over the edges, which is absent in the 3,1,4chGNR. These brighter edges unveil a larger density of states (DOS) around the Fermi energy, this being an experimental evidence for the emergence of edge bands in the wider ribbons. This is further corroborated by comparing differential conductance spectra (dI/dV) on the different ribbons, as shown in Fig. 3d. The spectral plots 2 and 3, measured at the edges of 3,1,6 and 3,1,8chGNRs, respectively, show a pronounced increase of dI/dV signal around zero bias, with a peculiar substructure (Fig. 3e), that is absent over the central part of the ribbons (spectral plot 4). In contrast, a wide bandgap of ~0.7 eV with no DOS enhancement around the Fermi level is found all over the 3,1,4chGNRs (plot 1 in Fig. 3d)^{26}.
The emergence of edge bands close to zero energy in 3,1,6 and 3,1,8chGNRs is reproduced by our TB (Fig. 3f) and density functional theory simulations (Supplementary Note 4) of the band structure of infinitely long 3,1,wchGNRs. The relatively large bandgap (E_{g} = 0.26 eV) of the 3,1,4chGNR closes abruptly for the wider ribbons, whose valence and conduction bands (VB and CB) apparently merge at zero energy and flatten, being these the edgelocalized states resolved in the experiments. However, as we show in the inset of Fig. 3f, the frontier bands of 3,1,6 and 3,1,8chGNRs do not overlap at zero, but remain gapped. Contrary to a monotonous gap closing, the theoretical energy gap is very small for 3,1,6chGNRs (~8 meV), and opens again for the wider 3,1,8chGNRs (~29 meV). Only for w ≥ 12 the gap closes definitively (see Fig. 1c).
The origin of the minigap reopening from w = 6 to w = 8 is connected with a gap inversion due to a change in valence band’s topology with the width. This can be deduced by comparing maps of the wavefunction amplitude and phase distribution at k = 0 and at k = π/a, shown in Fig. 3f. The VB of 3,1,4chGNRs (Supplementary Note 2) maintains an odd inversion symmetry with respect to the center of the unit cell in all kspace, whereas for 3,1,8chGNRs it changes parity (from even, at k = 0, to odd at k = π/a), revealing a band inversion at k = 0. As a consequence, the wavefunction acquires a net phase as it disperses along the Brillouin space. To connect these differences in parity with topological classes, we computed the Zak phase γ_{z}^{20} of the occupied band structure for every ribbon and obtained their \({{\mathbb{Z}}}_{2}\) invariant, as described in Supplementary Note 3. The 3,1,4chGNRs accumulate a global Zak phase of γ_{z} = 0 and, hence, are in a trivial topological phase, i.e., \({{\mathbb{Z}}}_{2}\) = 0. The intermediate case, 3,1,6chGNRs, has a very small gap that changes topological phase depending on details of the simulation (Supplementary Note 2) and thus we consider here as the transition metallic case. For the wider ribbon, 3,1,8chGNR, we obtain γ_{z} = π in accordance with a nontrivial SPT phase (\({{\mathbb{Z}}}_{2}=1\)), thus accounting for the gap reopening found in the simulations.
SuShriefferHeeger model prediction of a SPT phase transition
The presence of band gaps in narrow chGNRs and the SPT phase transition can be explained using the modified SuShriefferHeeger (mSSH) model^{8} depicted in Fig. 4 and in Supplementary Note 5. We can describe the 3,1,wchGNR as a chain of singly occupied states localized at zigzag edge sites^{27}, with hopping matrix elements t along the edge, and widthdependent hopping terms t_{i}, \({t}_{i}^{\prime}\), and \({t}_{i}^{^{\prime\prime} }\) between states at opposing edges. For very wide ribbons only the edge hopping term t is relevant, and the ribbon’s edges enclose metallic onedimensional bands, as pictured in Fig. 1b. To simulate the emergence of gapped SPT phases during chGNR narrowing, we fitted their VB and CB obtained from 3NN TB simulations using the mSSH model (Supplementary Note 5), and obtained that a gap opens when the three elements representing the hoping between opposing edges becomes sizable (Fig. 4b). Initially, interactions between the diagonal neighbours \({t}_{i}^{\prime}\) and \({t}_{i}^{^{\prime\prime} }\) (i.e., intercell hopping between edges) dominate over confronted zigzag elements (intracell hopping, t_{i}). This chiral interaction pattern causes a gapped phase with negative sign, defined from the \({{\mathbb{Z}}}_{2}\) invariant as \({(1)}^{{{\mathbb{Z}}}_{2}}\), and explains the nontrivial band topology of the 3,1,8chGNR. However, the interaction pattern reverses for narrower ribbons, and the intracell hopping element t_{i} dominates over the others, leading now to a gapped phase with positive sign (\({{\mathbb{Z}}}_{2}=0\)) (Fig. 4b), with a SPT phase transition close to the w = 6 case.
Inspired by the mSSH model, we performed TB simulations for the z,a,wchGNR family. We computed the band structure and the total Zak phase γ_{z} of occupied bands for the set 0 < z ≤ 10, a < 6, and w ≤ 12, comprising ribbons with chiral angle from 4.5° to 80° (Supplementary Note 1). The resulting bandgap values E_{g} and sign \({(1)}^{{Z}_{2}}\) are represented in Fig. 4c. The results show that the SPT phase transition found for the 3,1,wchGNRs is a global property of the z, a, wchGNR family. All chiral ribbons show a similar trend with the width: the gapless phase of wide ribbons, with edge states as in Fig. 1a, transforms first into a onedimensional topological insulating phase and then into a trivial phase below a critical width.
SPT boundary states
To experimentally confirm the existence of different gapped SPT phases in this family of chGNRs, we analyze the origin of the persisting lowbias substructure in dI/dV spectra appearing over the zerobias peaks along the edge (shown in Fig. 3e). It is expected that a nontrivial bulkboundary correspondence in topological chGNRs of finite length leads to ingap states localized at the GNR termini and associated to SPT boundary states. Correspondingly, our TB simulations for finite ribbons reproduce sharp zeroenergy states distributed around the ends of a 3,1,8chGNR (Fig. 5a, b), which are absent in the narrower ribbons, with opposite topological class (Supplementary Note 6). The experimental dI/dV maps measured at low sample bias, like in Fig. 5c, confirm the presence of these boundary states in 3,1,8chGNRs. They appear as a peculiar signal enhancement over the edge’s termini, with symmetry and extension similar to the simulated LDOS in Fig. 5a, b. Additionally, dI/dV spectra over these brighter regions show a sharp peak centered at 2 meV (Fig. 5d), and slowly decaying towards the interior of the GNR edge (Supplementary Note 7). In the middle of the ribbon, the VB and CB onsets appear as two peaks at ~±10 meV, delimiting a bandgap of barely 20 ± 4 meV.
The small bandgap E_{g} of these chGNRs accounts for the slow decay of the end states inside the ribbon (Supplementary Note 6). In short ribbons, end states from opposing termini may overlap and open a hybridization gap E_{Δ}^{28} that can hinder the observation of end states when E_{Δ} > E_{g}. In spite of the large spatial extension of the end states in 3,1,8chGNRs, and the very small gap E_{g}, we found that SPT boundary states survive for ribbons with only four precursor units (PUs) length, while vanish completely in chGNRs with three PUs or less, whose spectra is fully gapped (Fig. 5e). The survival of end states in short ribbons is confirmed by our TB simulations, which also find the opening of an unusually small hybridization gap (Supplementary Note 6). The weak interaction between end states is due to their peculiar distribution in the chiral backbone, where each end state lays at opposing ribbon’s edges (as expected from the mSSH model), thus reducing their overlap in short ribbons.
Contrasting with the topological character of 3,1,8chGNR, end states are absent in narrower ribbons, proving their trivial insulating phase and confirming the existence of a width controlled topological phase transition in this family of chGNRs. For 3,1,6chGNRs, lowbias peaks are spaced by tens of meVs, with a central one pinned close above the Fermi level (Fig. 3e). As we show in Supplementary Note 8, these peaks correspond to valence and conducting bands, discretized in quantumwell states due to the finite length of the ribbon^{26,29,30}. The peak pinned above the Fermi level coincides with the VB onset, partially depopulated in response to the large electron affinity of the Au(111) substrate^{10,31,32}. The first peak above the VB corresponds to the CB onset, and no subgap features neither signal at the chGNR ends is observed, in agreement with their trivial semiconducting character.
Topological insulating phase of 3,2,8chGNR
The simulations from Fig. 4c also illustrate that the size of the chGNR band gaps increase with the chiral angle: E_{g} varies from just a few tens of meVs for lowerangle ribbons to almost one electronvolt for some orientations. This property allows engineering robust topological chGNRs with wider gaps than for the 3,1,8chGNR. For example, we note that the theoretical gap of the 3,2,8chGNR in Fig. 4 amounts to 199 meV and is inverted (i.e., \({{\mathbb{Z}}}_{2}=1\)). Correspondingly, the simulations for finite ribbons of this kind reveal zeroenergy topological modes at the termini (Fig. 5f, g).
To confirm this topological insulating state, we studied 3,2,8chGNRs fabricated using the modified precursor 4. As depicted in Fig. 2, the modified halogen substitution of this precursor (at a = 2 sites) steers the formation of 3,2,8chGNRs on a Au(111) surface at elevated temperatures. These ribbons are oriented along a (2,3) vector of the graphene lattice, and alternate three zigzag with one and a half armchair sites along the edges. The bondresolved STM current image in Fig. 5h confirms the successful OSS of 3,2,8chGNRs by revealing its characteristic carbonring structure over the bulk part of the ribbon. However, the STM image also reproduces over the edges a characteristic signal enhancement the resembles the SPT boundary states in the TB LDOS maps of Fig. 5f. Furthermore, dI/dV spectra measured over the chGNR ends show sharp peaks pinned at zero bias, while over the bulk region of the ribbon a bare ~300 meV bandgap is found (Fig. 5i). The resonances at the ends correspond to the SPT boundary states predicted by our TB simulations, thus confirming the \({{\mathbb{Z}}}_{2}=1\) topological class of this ribbon as well. The resonances’ line width is ~25 mV, much broader than Kondo resonances observed in openshell nanographenes on Au(111)^{33,34,35}, and lie pinned slightly above E_{F}. This indicates that the SPT boundary states are partially depopulated due to electron transfer to the substrate, as found in 3,1,8chGNRs. However, due to the larger bandgap of this family of ribbons, these end states are more localized at the terminations (Supplementary Note 6) and, hence, they are readily detected in even shorter ribbons, with only two precursors units (Fig. 5j).
Discussion
The good agreement of our experimental results with the band structure obtained by TB simulations indicates that manifestations of Coulomb interactions are not very prominent in these measurements, probably due to the charge doped state of the ribbons and by their charge screening on a metallic substrate. However, interesting scenarios can be expected in the presence of electron–electron interactions such as for chGNRs on insulating layers^{36} or free standing^{37}. In Supplementary Note 9 we show results of meanfield Hubbard simulations of free chGNRs exploring the effect of a finite onsite Coulomb interaction U on their band structure and spin polarization, both in the neutral and in the charged state. In the neutral case, SPT boundary states of 3,1,8 and 3,2,8chGNRs split and develop a correlation gap already for small U. Band states of the chGNRs, on the contrary, are less sensible to Coulomb interactions because of their lower localization. They only open up when split SPT states mixes with them (e.g., see Supplementary Figure 16), causing that one cannot associate them with a SPT class. In the even doped state claimed in our experiments (+2e state), the effect of finite U on the SPT end states and bulk band structure is very small, and barely consists in a shift instead of a split because they are depopulated (Supplementary Figure 28). This justifies the use of TB models in our interpretation of experimental results.
Coulomb interactions in the neutral case also produces the build up of net spin density (Supplementary Figure 17). Narrow bandgap ribbons such as the 3,1,6 or 3,1,8chGNRs can develop spinpolarized edge bands^{17} for finite U, similar to the expected behavior in zGNRs^{17,38}. The effect of electron–electron interactions on the “bulk” band structure is smaller in the 3,2,8chGNR because of its wider gap^{39}, and their edge bands show weaker or no spin polarization. In wide ribbons, however, the larger degree of localization of their SPT end states augment their potential to build spinpolarized end states^{37,39,40} in the charge undoped state.
Our results thus demonstrate that endowing graphene with a chiral interaction is an effective method to induce gapped phases with exotic properties^{41}. The generalized behavior described here for this family of chiral GNRs represents a novel route to manufacture graphene ribbons with metallic edge bands and to transform them into topological states in graphene platforms. We envision that this method could be extended not only to other chiral geometries in onedimensional nanoribbons^{42}, but also to twodimensional porous graphene networks^{43}, or moiré 2D systems, in which combination of flat bands with chiral symmetries might lead to novel SPT phases.
Methods
General methods for the synthesis of the precursors
All reactions for the synthesis of the precursors were carried out under argon using ovendried glassware. TLC was performed on Merck silica gel 60 F254; chromatograms were visualized with UV light (254 and 360 nm). Flash column chromatography was performed on Merck silica gel 60 (ASTM 230–400 mesh). ^{1}H and ^{13}C NMR spectra were recorded at 300 and 75 MHz or 500 and 125 MHz (Varian Mercury 300 or Bruker DPX500 instruments), respectively. Lowresolution electron impact mass spectra were determined at 70 eV on a HP5988A instrument. Highresolution mass spectra (HRMS) were obtained on a Micromass Autospec spectrometer. NALDITOF spectra were determined on a Bruker Autoflex instrument. Experimental details for the synthesis of the precursors and spectroscopic data can be found in Supplementary Methods.
Sample preparation and details of STM measurements
The experiments were performed on a home made ultrahigh vacuum (UHV) scanning tunneling microscope (STM) operating at 5 K. The Au(111) single crystal was cleaned in UHV by repeated cycles of Ne^{+} ion sputtering and subsequent annealing to 730 K. The three molecular precursors were separately sublimed from Knusden cells onto a clean Au(111) substrate kept at room temperature. The sublimation temperatures of molecular precursors 1–4 in Fig. 2a–d are 173, 260, 312, and 333 °C, respectively. Each sample was then stepwisely annealed at elevated temperatures to induce the polymerization (200 °C for all the precursors) and cyclodehydrogenation (250 °C for 1 and 300 °C for 2–4.) of molecular precursors. The annealing time for each step is 10 min for all the precursors. A tungsten tip was used in the experiment. Highresolution images were constantheight current maps acquired with a COfunctionalized tip at very small voltages, and junction resistances of typically 20 MΩ. The dI/dV signal was recorded using a lockin amplifier with a bias modulation of V_{rms} = 4 mV (spectra in Figs. 3d and 5i, j) and 0.4 mV (spectra in Figs. 3e and 5d, e) at 760 Hz, respectively. All STM images were processed with the software WSxM^{44}.
Tightbinding simulations
We describe the graphene nanostructures with the following Hamiltonian for the sp^{2} carbon atoms, as implemented in the SISL python package^{45}:
where c_{i} (\({c}_{i}^{{{{\dagger}}} }\)) annihilates (creates) an electron in the p_{z} orbital centered at site i. Equation (1) describes a tightbinding model with hopping amplitudes t_{1}, t_{2}, and t_{3} for the first, second, and thirdnearest neighbor matrix elements defined in terms of interatomic distances d_{1} < 1.6Å < d_{2} < 2.6Å < d_{3} < 3.1 Å. We follow the parametrization of ref. ^{46} and consider the thirdnearest neighbor (3NN) model with t_{1} = 2.7 eV, t_{2} = 0.2 eV, and t_{3} = 0.18 eV that has successfully described other synthesized sp^{2} carbon systems^{33,35}. For completeness, we also have compared the band structures, gaps and SPT phase with simulations with a firstnearest neighbor (1NN) model (i.e., with t_{1} = 2.7 eV and t_{2} = t_{3} = 0, see Supplementary Note 2).
To analyze the effect of the electron–electron interactions in the (n, m, w)chGNRs within the meanfield Hubbard (MFH) model, the Hamiltonian H of Eq. (1) incorporates the onsite Coulomb repulsion term modulated by the U parameter:
as implemented in the HUBBARD python package^{47} (See Supplementary Note 9).
Density functional theory simulations
The optimized geometry and electronic structure of freestanding chiral GNRs were calculated using density functional theory, as implemented in the SIESTA code^{48}. The nanoribbons were relaxed until forces on all atoms were smaller than 0.01 eV/Å, and the dispersion interactions were taken into account by the nonlocal optB88vdW functional^{49}. The basis set consisted of doubleζ plus polarization orbitals for all species, with an energy shift parameter of 0.01 Ry. A 1 × 1 × 101 MonkhorstPack mesh was used for the kpoint sampling of the Brillouin zone, where the 101 kpoints are taken along the direction of the ribbon. A cutoff of 300 Ry was used for the realspace grid integrations.
Data availability
The data that support the findings of this study are available from the authors on reasonable request. The “hubbard v0.1.0 (2021)” package used for TB simulations, created by S. Sanz Wuhl, N. Papior, M. Brandbyge, and T. Frederiksen, is available in https://doi.org/10.5281/zenodo.4748765.
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Acknowledgements
We gratefully acknowledge financial support from the Agencia Estatal de Investigación (AEI) through projects No MAT201678293, PID2019107338RB, and FIS201783780P, and the Maria de Maeztu Units of Excellence Programme MDM20160618, from the Xunta de Galicia (Centro singular de investigación de Galicia, accreditation 2016–2019, ED431G/09), from the University of the Basque Country (Grant IT124619) and the Basque Departamento de Educación (PhD scholarship no. PRE_2019_2_0218 of S.S.), and from the European Regional Development Fund. We also acknowledge funding from the European Union (EU) H2020 program through the ERC (grant agreement No. 635919) and FET Open project SPRING (grant agreement No. 863098).
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D.G.O., T.F., D.P., and J.I.P. devised the experiment. M.V.V. and D.P. synthesized the molecular precursors. J.L. and N.M. realized the experiments with the support of M.C. S.S. and T.F. did the TB simulations. A.G.L. did the DFT simulations. All the authors discussed the results. J.L., S.S., T.F., D.P., and J.I.P. wrote the manuscript.
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Li, J., Sanz, S., MerinoDíez, N. et al. Topological phase transition in chiral graphene nanoribbons: from edge bands to end states. Nat Commun 12, 5538 (2021). https://doi.org/10.1038/s4146702125688z
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DOI: https://doi.org/10.1038/s4146702125688z
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