Detecting the spin-polarization of edge states in graphene nanoribbons

Low dimensional carbon-based materials can show intrinsic magnetism associated to p-electrons in open-shell π-conjugated systems. Chemical design provides atomically precise control of the π-electron cloud, which makes them promising for nanoscale magnetic devices. However, direct verification of their spatially resolved spin-moment remains elusive. Here, we report the spin-polarization of chiral graphene nanoribbons (one-dimensional strips of graphene with alternating zig-zag and arm-chair boundaries), obtained by means of spin-polarized scanning tunnelling microscopy. We extract the energy-dependent spin-moment distribution of spatially extended edge states with π-orbital character, thus beyond localized magnetic moments at radical or defective carbon sites. Guided by mean-field Hubbard calculations, we demonstrate that electron correlations are responsible for the spin-splitting of the electronic structure. Our versatile platform utilizes a ferromagnetic substrate that stabilizes the organic magnetic moments against thermal and quantum fluctuations, while being fully compatible with on-surface synthesis of the rapidly growing class of nanographenes.

advances in the field combining indirect experimental fingerprints of localized magnetic moments with theoretical models.In spite of that, a characterization of their spatial-and energy-resolved spin-moment has so far remained elusive.
To obtain this information, we present an approach based on the stabilization of the magnetization of π-orbitals by virtue of a supporting substrate with ferromagnetic ground state.Remarkably, we go beyond localized magnetic moments in radical or faulty carbon sites: In our study, energy-dependent spin-moment distributions have been extracted from spatially extended onedimensional edge states of chiral graphene nanoribbons.This method can be generalized to other nanographene structures, representing an essential validation of these materials for their use in spintronics and quantum technologies.
Magnetism is at the heart of a vast amount of applications and conventionally relies on the spin of unpaired dor f -shell electrons.Instead, carbon-based magnetism stems from pshell electrons.(1,2) The vision of exploiting them in new types of applications involving spin-polarized currents and spin-based quantum information (1)(2)(3)(4)(5) is inspired in two distinct properties: weak spin-orbit and hyperfine coupling (two of the main channels responsible for the relaxation and decoherence of electron spins), (1)(2)(3)6,7) and delocalization in π-orbitals with high spin-wave stiffness.(1, 2, 8) However, the experimental realization of the associated openshell molecular structures is challenging.Therefore, it is only recently that they are becoming accessible by on-surface synthesis under vacuum conditions, (9) rendering their characterization of utmost interest.
Within the wide range of carbon nanostructures predicted to display π-magnetism, (9) graphene nanoribbons (GNRs, i.e., one-dimensional stripes of graphene) are probably among the most interesting for potential applications due to their intrinsic length, (10,11) which facilitates contacting and integration into device structures.(12,13) Aside from the presence of vacan-cies or heteroatoms, for GNRs to exhibit magnetic properties they must display zigzag edges, whether continuously throughout the ribbon (zGNRs) or in periodic alternation with armchair segments (chiral or chGNRs).(10,14,15) In either case, the ribbons develop edge states that decay exponentially towards the ribbon's interior.These electronic states are predicted to be spin-polarized.(10,14,15) Both, zGNRs and chGNRs, have been synthesized with atomic precision on Au(111) substrates.(16)(17)(18)(19) Whereas the presence of the edge states has been confirmed for both cases, (16,19), a direct experimental proof of their spin polarization is still lacking.In the more general case of open-shell carbon nanostructures, spatially resolved magnetic signals have only been observed in a few cases involving magnetic field-dependent Zeeman splittings (20)(21)(22).Otherwise, only indirect hints of the magnetism have been obtained, whether from an analysis of the frontier orbital's density of states (23,24), Kondo resonances, (20,21,25) inelastic spin-flip excitations (22,25) or Coulomb gaps.(16,26) In any case, the detection of an intrinsic remanent spin-polarization of π-orbitals has never been obtained to date for any carbon-based material.Actually, the weak spin-orbit coupling (3), a central advantage of carbon-based magnetism, imposes at the same time the main drawback to resolve a stationary spin moment in these systems: the practically null magnetic anisotropy of sp 2 carbon atoms.
For the characterization of an unexplored magnetic ground state by means of SP-STM, it is convenient to arrange the sample under study in coexistence with a surface that has a well known spin-resolved electronic structure.A GdAu 2 monolayer on Au(111) ( 27) is an excellent candidate.Its ability to catalyze nanographene polymerization via Ullmann coupling (28) has been already proven (29,30) and, at the same time, it orders ferromagnetically below 19 K (31) with a large easy-plane magnetic anisotropy, thereby showing strong in-plane contrast in SP-STM measurements (32).
We achieve long (3,1,8)-chGNRs from DBQA precursor molecules (Supplementary Fig. S1) on GdAu 2 using a very similar procedure as the one previously reported in the case of Au(111) (19) (see Supplementary Experimental Methods).Fig. 1A shows an overview image of (3,1,8)-chGNRs on GdAu 2 .The moiré superlattice caused by the superposition of the GdAu 2 lattice (hexagonal unit cell of 5.41±0.03Å) and the underlying Au(111) lattice (27,33) is clearly visible.In the GdAu 2 lattice, each Gd atom is sixfold coordinated with Au atoms, which appear in STM images as dark and bright spots, respectively (Fig. 1B).The edges in (3,1,8)-chGNRs are composed of alternating segments of zigzag and armchair graphene paths (see molecular scheme in Fig. 1B).The GNRs grow preferentially with their longitudinal axis along high symmetry directions of the Gd atomic lattice, either the [110] (e.g. the central ribbon in Fig. 1B) or the [211] directions of the Au(111) substrate.
In addition to enabling the formation of high quality (3,1,8)-chGNRs, the GdAu 2 surface exhibits different domains of the in-plane magnetization, as revealed in the spin-resolved dI/dV maps at V b = 3 V shown in Figs.1C-D, taken with bulk Cr-tips (see also Supplementary Fig. S2) (32).The magnetic coupling across crystallographic antiphase domain boundaries (APB) is such that the magnetization components along the tip sensitivity direction at either side are antiparallel.This provides atomically sharp boundaries between GdAu 2 regions with different magnetization (Figs.1E-F) that serves as a control of the unaltered tip's spin sensitivity throughout the measurement process.We select target GNRs located in a magnetic domain with homogeneous magnetization and close to an APB.Ramping the external out-of-plane field to ±3 Tesla and back to zero allows us to repeatedly switch the remanent state of the substrate underneath the target GNR (see Supplementary Note 2), as shown in Figs.1D-F and sketched in Fig. 1G.By convention, we refer to magnetic states with high and low differential conductance at 3 V as parallel (P ) and antiparallel (AP ) states to the spin direction of the tip.
After setting the magnetic state of the substrate, we proceed with the characterization of the target GNR (Fig. 1H, the characterization region is marked in Fig. 1C, lying on the left domain of Figs.1E-F).The density of states (DoS) around the Fermi level can be by retrieved from low-bias dI/dV maps.The result for a ribbon with N = 17 precursor units (Fig. 1H), measured with metallic tips (Cr or W), is representative of DoS obtained for other GNRs with lengths varying from 10 to 23 precursor units.The internal structure observed in the central region corresponds to the GdAu 2 lattice, and the moiré contrast is visible through the ribbon.
Importantly, we resolve the predicted high DoS at the ribbons's edges (14,15,19).The wellknown short decay length of edge states into the vacuum (34) suppresses their characteristic intensity in constant height DoS images taken with metallic tips (Supplementary Fig. S4).
However, the expected DoS distribution is readily recovered when using CO-functionalized Wtips, as shown in Fig. 2A for a N = 21 chGNR, including the structure of the wave function extending into the central part of the ribbon.The pronounced edge states are, in most cases, further modulated by the moiré pattern underneath the edge (See Supplementary Note 1), rather than displaying pure quantum well states caused by the finite length (18,35).SP-STM experiments require a metallic Cr-tip, and thus we cannot resort to CO functionalization.To recover the sensitivity to the molecular states with this kind of tips it is necessary to acquire grids of dI/dV spectra over the chGNR regulating the current at each pixel (see Methods).Selected point spectra are shown in Fig. 2C, exhibiting some peaks between -6 mV and 28 mV, and a first fully occupied state at -40 mV.Fig. 2D shows dI/dV slices at constant V b for these states, where a certain repetition period along the edge can be discerned.By fast Fourier transform these periodicities are turned into characteristic wave vectors for each molecular state, (18,34) which permits to assign them to discretized states emerging from the conduction and valence bands of the infinitely long GNRs (See Supplementary Fig. S4).The peak at -40 mV arises from the valence band, and thus corresponds to the Highest Occupied Molecular Orbital (HOMO) of the charge-neutral ribbon.The peaks crossing the Fermi level (V b = 0 mV) and above (V b ∼ 20 mV) relate to the conduction band and are the Lowest Unoccupied Molecular Orbitals (LUMO, LUMO+1) of the charge-neutral ribbon.The appearance of the LUMO right at the Fermi level on GdAu 2 /Au(111) is a distinct feature of charge transfer from the substrate to the GNR.This is a consequence of the reduced work function of GdAu 2 /Au(111) with respect to Au(111) (30), which promotes a slight electron doping of the GNR and drives the LUMO states stemming from the bottom of the conduction band below the Fermi level.
The LUMO peaks centered at zero bias exhibit an additional fine splitting of approximately 12 mV across the Fermi level at some locations (e.g.curves 2 and 3 in Fig. 2C).This peak substructure is observed whenever a sizable DoS centered at zero bias is present at chiral edges.To elucidate the origin of the edge electronic structure, we solved the tight binding Hamiltonian for the π-electron system including a MFH term to account for electronic correlations (see Methods).For non-interacting electrons (U = 0), the addition of two electrons to the charge-neutral ribbon shifts the simulated DoS spectra at the chiral edge by about -8 meV, (Fig. 2E).The original LUMO+1 at 34 meV sits now at 26 meV, and the zero-energy end state of topological origin (19) located at the termini shifts to -8 meV.We introduce electron-electron (e-e) interactions via an on-site Coulomb repulsion term U = 0.With U = 1 eV we are able to reproduce our experimental dI/dV spectra (top row in Fig. 2E).First, the main intensity of the first fully occupied states clusters around -40 meV.Second, the structure around the Fermi level splits into two peaks separated by 9 meV, very close to our experimentally determined faint gap (Fig. 2C).
This feature arises as a consequence of the Coulomb repulsion.The end state splits at U = 0.8 eV into a singly occupied and a singly unoccupied orbital, while the low-energy LUMOs are shifted to more negative energies, mixing with each other in the range of U ∼ 1 eV and forming two molecular orbitals that are hybrids of the non-interacting ones (see Supplemental Fig. S3).Fig. 2B shows the theoretical DoS distribution of the occupied state enclosed by the grey box in Fig. 2E, which is in excellent agreement with the experimental DoS in Fig. 2A.
Localized one-dimensional (1-D) states at the zigzag edges of graphene nanostructures (36) have been predicted to become spin-polarized by MFH models (14,15) and first-principles calculations.(37)(38)(39)(40) The driving mechanism is the energy gain of the system obtained by depleting a spin-degenerate doubly-occupied state near the Fermi level, for which the Coulomb repulsion would otherwise introduce a much higher internal energy.This phenomenon is prone to occur in localized electronic states, because the Coulomb repulsion energy grows as electrons get confined in a more reduced space.
Although our (3,1,8)-chGNRs do not have pure zigzag edges (see Fig. 1B), its periodic zigzag segments are known to retain intense localized edge states stemming from flat electronic bands near the Γ point of the 1-D Brillouin zone (see Figs. 2A and 2D, and Supplemental Fig. S4B) (14,19,40,41).If the two edges are far enough as to be considered independent, the edge states are expected to be metallic in (3,1,c)-GNRs (14).However, in narrower ribbons, the coupling between both edge states can open a hybridization gap (14,18,19).In the case of c = 8, this gap amounts to approximately 20 meV, and is centered at the Fermi level (19).On (3,1,8)-chGNRs/GdAu 2 , the slight electron doping causes an increase of the chemical potential in the ribbon, and instead of a gap at the Fermi level we find a partially filled state, at ∼ 35 mV above the HOMO (see Fig. 2C).This state is still close to the conduction band minimum (Supplementary Fig. S4), and therefore it will display a similar degree of localization at the edge as in the metallic case of wider ribbons (c > 8) (14,40,41).This is illustrated in the aforementioned constant height scans with CO-tips (Fig. 2A and Supplemental Figs.S4 and S5) or in grids with current regulation for each pixel (Fig. 2D).Thus, they are excellent candidates to display magnetic instabilities associated to e-e correlations.
The splitting induced by e-e interactions endows different spin quantum numbers to the singly occupied/unoccupied states.From this, an inversion of the spin polarization at both sides of Fermi level follows necessarily.(14,15,37) In the following we provide evidence of such sign inversion in the spin polarization, setting the experimental hallmark for itinerant magnetism in edge states of nanographenes.
Figure 3A shows a N = 15 chGNR, scanned under constant-height conditions for several magnetic P or AP states of the underlying GdAu 2 with homogeneous magnetization in the characterization region (see Supplementary Fig. S6 for a description of the magnetic history).
The spin asymmetry in Fig. 3B, calculated from the dI/dV maps at the energies of interest as S a = 100 × [(dI/dV ) AP − (dI/dV ) P ]/[(dI/dV ) AP + (dI/dV ) P ], is proportional to the spin polarization of the system right at the measurement energy.Figure 3D shows spectral lines along the chiral edges of the spin averaged DoS, i.e. [dI/dV AP + dI/dV P ]/2.Here, the previously discussed splitting for N = 21 manifests again in the regions of larger DoS as two peaks at V b = −7 mV and at V b = +5 mV, noticeable at the bottom of the left edge (region 1 in Fig. 3A) and at the center of the right edge (region 3 in Fig. 3A).This intensity distribution results from the combined effects of the LUMO confinement pattern at the edge and the influence of the moiré periodicity (Supplemental Fig. S4 and S5).
Regions 1 and 3 are also the positions with clear spin contrast in Fig. 3B.Furthermore, we find experimentally a change of sign in the spin polarization across the Fermi level of ±8 %, in neat agreement with the predicted magnetic state driven by e-e correlations.Therefore, the 12 mV gap observed around the Fermi level (Figs.2C and 3D) can be safely attributed to a spin splitting that emerges to accommodate e-e correlations in the partially occupied LUMO edge state.If these two peaks were to correspond to different quantum-well states of the conduction band, they would not have any different spin polarization than the inner region of the ribbon, and certainly their spin asymmetry would not be changing sign across the Fermi level.Note that the spin polarization of the underlying GdAu 2 obtained with the same kind of bulk Cr-tips is very small (< 4 %), and energy independent in this bias regime (Supplementary Fig. S2).
The energy dependence of the spin polarization is better visualized in single point dI/dV spectra (Fig. 3E).As for the constant height images (Fig. 3B), the spin asymmetry is obtained as S a (eV b ) = (AP − P )/(AP + P ), and represented by the green curves.All positions other than the ribbon periphery are characterized by featureless spectra whose spin asymmetry between AP and P states is below our experimental confidence (≤ 1.5 %).Region 1 and 3 exhibit the canonical behavior discussed above for a correlations splitting with S a = +7 % at V b = −10 mV, and S a = −6 % at V b = +6 mV.Region 2, with a much lower intensity of the edge state (see Fig. 3D), only displays a small signal of S a = −3.3% at V b = +4 mV, barely above our experimental uncertainty, and S a 0 at the energy of the negative bias peak.
Figure 3C displays the spin polarization obtained from MFH calculations with the set of parameters determined earlier (U = 1 eV, 2 electrons added, see Methods), which is in good qualitative agreement with the experimental results (see Fig. 3B).This comparison allows us to understand the detected spin polarization.On the one hand, U = 1 eV is smaller than the expected theoretical value for free standing nanoribbons, namely around 2-3 eV, (38,42,43) which in our case is justified by the hybridization of the GNR p z orbitals with the metallic substrate.
The low U value induces a smaller splitting (∼ 10 meV) than the FWHM of the molecular orbitals (∼ 30 meV, see Fig. 2C), leading to the weakening of the spin polarization.On the other hand, the spin polarization of one edge is not homogeneous, contrary to the expectation for the ground state of isolated nanoribbons.This is ascribed to the hybrid character of the edge states, a consequence of electron doping and the orbital mixing associated to the U -term (Supplemental Fig. S3).Both of them favour the concentration of unoccupied quantum well states of the charge-neutral ribbon towards the Fermi level.As a result, the edge spin polarization displays oscillations (Figs.3B-C).
The spatial integration of the spin polarization in Fig. 3B yields a non-zero total spin for occupied states.Furthermore, the magnetization of both edges does not appear fully correlated as in the theoretical simulation shown in Fig. 3C.As discussed in Supplementery Note 3, the prevailing mechanism that distorts the ideal magnetic ground state is the local exchange interaction with the substrate that, at T = 1.2 K, overcomes the antiferromagnetic coupling between the edges (15, 44,45) -mediated by the overlap of the decaying edge states towards the interior-.To corroborate this idea, and using ribbons over GdAu 2 regions with inhomogeneous magnetization, we demonstrate the independent switching of the spin direction in a portion of the ribbon while the rest remains unchanged (Supplementary Fig. S8).This is controlled by the exchange coupling with the local magnetization of the GdAu 2 , showing that this interaction is responsible for stabilizing the magnetic moment of the edge states against spin fluctuations and therefore, plays a fundamental role to gain access to the edge's spin polarization.
Altogether, by synthesizing chiral graphene nanoribbons on top of a magnetic GdAu 2 monolayer, we have been able to access with exquisite spatial and energy resolution the spin-polarization of its edge states by SP-STM/STS.Doing so, a long standing challenge has been resolved that not only reveals important differences with respect to the originally expected spin-polarization of charge neutral ribbons, but also sets the stage to similarly characterize the immense amount of magnetic carbon-based materials that are being synthesized lately.surface by sublimating Gd using an e-beam source at a rate of approximately 0.5 ML (referred to the Au(111) lattice) in 9 minutes while the substrate is held at 320 • C. As shown in Fig. S1C, GdAu 2 forms a moiré superlattice caused by the superposition of its hexagonal unit cell (lattice parameter 5.41±0.03Å along the [211] Au direction of the substrate) and the underlying Au(111) lattice (27,33).In the GdAu 2 lattice, each Gd atom is sixfold coordinated with Au atoms that are visualized as dark and bright spots respectively at V b = 1 V (see Fig. S2A and Fig. 1B of the article).The moiré pattern can be approximated by 4 GdAu 2 unit cells in the [110] Au direction on 13 Au(111) lattice constants, with a period of d m = 37.9 ± 1 Å, although our own atomically resolved images indicate that the moirée is not commensurate.
The reactant 2",3'-dibromo-9,9':10',9":10",9"'-quateranthracene (DBQA) is then deposited on GdAu 2 from a home made resistive evaporator.Subsequent on-surface synthesis of (3,1,8)-chGNRs takes place in a two steps reaction as in the case of Au(111) (19), which is depicted in Supplementary Fig. S1.First, the temperature is slowly ramped up to 290 • C and then mantained for 10 minutes to obtain long polymers via Ullmann coupling.Spin polarized STM (SP-STM) has been carried out using bulk Cr tips.Tips are prepared by electrochemical etching of elongated pure Cr flakes and subsequent field emission cleaning (120 V, 1 µA, 1 hour) at the STM head.GdAu 2 exhibit atomically sharp structural antiphase boundaries (APB) between crystallographic domains that are shifted by half a GdAu 2 lattice vector with respect to each other, as the one shown in Fig. S2A.In agreement with a previous SP-STM study of GdAu 2 (32), we find that a vast majority of these APBs induce a local antiferromagnetic coupling of the neighboring domains, as evidenced by the spin polarized map in Fig. S2B.We make use of this contrast to calibrate the spin sensitivity direction and spin polarization of the Cr tips, as desccribed later in Supplementary Note 2. The tips are submitted to voltage pulses until the expected in-plane spin contrast of 20 % or more at V b = 3 V shown in Fig. S2B is obtained.dI/dV point spectra at equivalent locations of the domains with high (bright, B) and low (dark, D) differential conductance are presented in Fig. S2D.They provide the quantitatively spin resolved electronic structure in the range of -0.5 to 3.5 eV around the Fermi level.In Fig. S2E the spin asymmetry is calculated from the dI/dV spectra of representative moiré regions as %(B − D)/(B + D).Here it can be appreciated how the spin polarization at V b = 3 V is not homogeneous, but always positive and ranging 15 % in average.At V b = 2.6 V there is an inversion of the spin polarization.These findings are further supported by the constant height maps of the electronic density and spin polarization shown in Fig. S2C.In contrast, the spin polarization around the Fermi level is much smaller, of about 4 %.In addition, it is constant in a broad energy window of ±0.25 eV and spatially homogeneous, as evidenced by the spin polarization map in Fig. S2C.This fact corroborates that the highly localized spin contrast and the spin polarization inversion discussed for chGNR's edge states in the main text are intrinsic to the ribbon, and not some kind of signal induced by the substrate.Note that, in addition, the ribbon plane probed by the tip in open feedback is ∼1.7 Å above the substrate, and therefore with a much smaller contribution than the carbon atoms themselves.

Theoretical Methods
To theoretically describe GNRs and their spin physics we employ the Hubbard Hamiltonian within the mean-field (MFH) approximation (47), where t 1,2,3 are the hopping terms corresponding to the interaction between first, second and third nearest neighbors (3NN), respectively.For these parameters we use the numerical values For the Coulomb repulsion term, we use U = 1 eV.The reason for using this 'small' value compared to other related works (20,25), is because we see in this case a large interaction between the samples and the substrate, which leads to a screening of the electron-electron interactions.Numerically, we solve the Schrödinger equation Eq. 1 using our custom implemented Python package HUBBARD (48).Here, the average number operators We compute the spin polarization for different U values and electron energies, calculated as where b σni is the coefficient of the wave function projected on the 2p z -orbital located at site i for state n and spin orientation σ at energy E nσ .f (E nσ ) is the Fermi function with k B T = 10 −5 eV.We also plot this quantity spanned in a real space grid, where the carbon 2p z basis orbital φ(r − R i ), centered at position R i , is described by the radial function exp (−3|r − R i |/ Å).
Similarly, we obtain the LDOS as, where The Lorentzian broadening, set to γ = 10 meV for simulated spectra and γ = 30 meV for simulated images, is introduced to account for mix from energetically closely-spaced orbitals.
In all cases we slice the grid at a height of z = 3.5 Å above the chGNR.We plot these quantities using our HUBBARD python package (48).
As a consequence of electron doping, the original zero-energy Symmetry Protected Topological (SPT) state ( 19) of the neutral system is shifted to -8 meV after adding 2 electrons to the system (Fig. 2E main text).For increasing on-site Coulomb repulsion, this SPT state becomes hybridized with other orbitals at the following rate (color coded in Fig. S3) where the Ψ 0 SPT state corresponds to the unoccupied SPT state and Ψ U n in the n eigenstate for varying U after addition of 2 electrons to the charge neutral ribbon.This can be explained in terms of coherent rotation of the in-plane sample magnetization to become almost parallel to the field direction (while the tip spin sensitivity direction remains in-plane or slightly canted).In all field sweeps, the depinning field of the domain wall between 3-2 is larger than the depinning field of the wall traveling through domain 1 (∼ ±0.85 T and ∼ ±0.5 T respectively).
The first magnetization ramp exhibits a somewhat different behaviour than the successive ones: during the sweep AP 1 → AP 2 the domain wall in RCA of the virgin state disappears, and the magnetization of the region between the ribbon an the TCA gets homogeneous, as evidenced in large area scans (not shown) and Fig. S6B.Afterwards, the magnetization of the RCA is homogeneous all the way to domain 1 and opposite to that of domain 2 in the TCA.In this way, the switch from AP 2 → P is induced by sweeping the field down to -2.8 T and then back to zero, and the switch from P → AP 3 by the same procedure but up to positive field of +2.8 T.
In order to extract the magnetic contrast discussed in the main text (Fig. 3) we have used AP 3 and P states.The result is reproduced in Fig. S7.Here, we plot the direct current difference (AP 3 − P ) and the dI/dV spin asymmetry (AP 3 − P )/(AP 3 + P ).From this procedure, only the magnetic contrast associated to the change P → AP 3 remains in the signal.
Identical results are obtained when AP 1 and P are compared.In contrast, when two images of equivalent magnetic states (like AP 3 and AP 1) are compared, there is not any intensity left in neither the current difference AP 3 − AP 1 nor the dI/dV spin asymmetry (AP 3 − The intrinsic spin moment of the chGNR could couple with the external magnetic field or undergo exchange interactions with the substrate.To rule out the influence of the external magnetic field we performed all measurements at very small magnetic fields (B < 200 mT), exploiting opposite remanent states of the substrate (see Fig. 1).Zeeman energy at these fields is orders of magnitude smaller than the thermal fluctuations at the experimental temperature of ∼ 1 K (0.08 meV).
In our system, there is another possible magnetic interaction: the exchange coupling with the substrate's magnetization.To investigate its influence on the spin distribution of the supported chGNRs, we designed an experiment involving an intermediate state with in-homogeneous magnetization under the ribbon (i.e., a natural domain wall of the GdAu 2 across the ribbon is formed) , which is summarized in Fig. S8.Here, as shown in panel A for a N = 22 GNR, the magnetization of the GdAu 2 between P and AP states is opposite underneath the ribbon, but in the domain wall (DW ) state the top part of the ribbon lies in a region with the same magnetization as the P state, whereas the bottom part lies in a region with the same magnetization as the AP state.In this way, we detect a small spin asymmetry S a > 0 at Fermi level extracted from the difference (P − DW ) only in the bottom part of the left edge (Figs S8C,D).For the (AP − DW ) case, S a is zero in the bottom part, and S a < 0 in a small spot of the right edge at the top.Thereby, it can be concluded that the spin polarization does not behave like a strongly correlated spin density of the overall molecule, but it rather couples locally to the magnetic moments of the substrate.Uncorrelated inter-edge spin configurations are those for which there is no energy difference between AFM or ferromagnetic alignment.In the case of zigzag GNRs with c = 8 the energy gain of the AFM arrangement has been estimated to be ∼ 2 meV (45,53) per edge atom.The exchange interaction in chGNRs is even weaker (14), so this number is just an upper threshold.At the same time, although all SP-STM data discussed so far have been taken at T = 1.2K, we have obtained a similar S a distribution to the one shown in Fig. S8 at T = 4.4 K for a N = 26 (3,1,8)-chGNR.This suggest that the average exchange interaction between carbon atoms and the substrate must be significantly higher than the thermal energy of ∼0.4 meV, which leads to the conclusion that it determines the inter-edge spin alignment at a greater extent than small perturbations of the already weak internal AFM coupling.
More importantly to our work, this interaction with the substrate is responsible for stabilizing the magnetic moment of the edge states against thermal (54) and quantum fluctuations (55).
Available estimates of the spin-orbit coupling strength in graphene provide the figure of 15 µeV (3), which in a uniaxial approximation for the magnetic anisotropy would provide a energy barrier of the order of only 0.2 K.This means that the thermal fluctuations at any temperature near 0.2 K would suffice to destabilize spin moments in nanographenes, or in other words, the spin correlation lengths would be smaller than the characteristic dimensions of the ribbbons (8).
In our experimental approach, however, this is not happening thanks to the exchange coupling to the GdAu 2 .For ribbons positioned over homogeneous magnetization regions, as corresponds to the model case discussed in the main text, this magnetic interaction plays, therefore, a fundamental role to gain access to the edge's spin polarization.It enables the access to the spin density by means of slow SP-STM scans under different configurations of the tip-sample magnetization, and the demonstration of magnetic remanence in the extended 1-D edge states of GNRs with high density of zig-zag segments.

Figure 2 :
Figure 2: Spin averaged electronic structure of (3,1,8)-chGNRs on GdAu2.(T = 4.3K, B = 0 T, W tip). A) Constant height current with CO functionalized W-tip at V b = −3.7 mV of a N = 21 ribbon (open feedback at ribbon's center with current SP of 50 pA).B) Simulation of the GNR DoS within the Mean Field Hubbard (MFH) model at the energy window enclosed by the grey box in E, U = 1 eV and two electrond added (see also Methods).C) High resolution dI/dV spectra (SP: -100 mV, 200 pA, V mod =0.7 mV) taken at the positions given by the colored circles in A. D) Constant current dI/dV maps at the energy values marked by dashed lines in C with the same color code (SP: -150mV, 125pA, V mod = 10 mV).E) Projected DoS from MFH calculations (see Theoretical Methods) at the atomic sites enclosed by the transparent rectangle in B. Bottom panel corresponds to the charge-neutral case without e-e interactions, top panel corresponds to the charged system with 2 electrons for U = 0 and U = 1 eV.Dashed lines in C and E indicate the energy positions of the HOMO (green), LUMO (black) and LUMO+1 (blue) named after their equivalent states in the charge-neutral specimen.

Figure 3 :
Figure3: Spin polarized edge states in chiral graphene nanoribbons.(T = 1.2 K, in-plane sensitive bulk Cr-tip, N = 15 precursors).Control tests of the obtained magnetic contrast are provided at Supplementary Note 2. All images are 3.9 × 15.8 nm 2 and taken at constant height after opening the feedback over the ribbon's center (SP: 20 mV and 100 pA, V mod = 0.5 mV).dI/dV point spectroscopies taken with set point of 100 mV and 250 pA (V mod = 0.5 mV).A) Tunneling current image of the ribbon at V b = −8 mV.B) dI/dV spin asymmetry calculated from the constant height differential conductance images as %(AP − P )/(AP + P ) at V b = −8 mV (left) and V b = +5 mV (right).Note the sign change of the spin polarization when crossing the Fermi level.C) Spin polarization of the GNR summed over occupied states, obtained with U = 1 eV from the MFH model (see Methods).The spin polarization is spanned in a real space grid and the plot is obtained by slicing 3.5 Å above the molecular plane.D) Stack plot of the spin averaged point spectroscopies, (AP + P )/2, taken along the green dashed lines in A. The position/diameter of the white circles represent the center/area of Gaussian functions used to fit each individual spectrum together with a baseline.E) Spin resolved point spectroscopy at the positions indicated by the yellow numbers in A for the P (red curves,) and AP (blue curves) states.The resulting energy resolved spin polarization is given by the green curves in the bottom panel of each graph.Light grey lines represent the background instrumental asymmetry obtained over GdAu2 or the ribbon center (both are identical).Error bars are the standard deviation with confidence of 68 % obtained from 20 individual curves measured at each point.17 Fig. S1B shows the characteristic corrugated topography of these kind of polymers, surrounded by Br atoms coming from the dehalogenation of the DBQA that are still on the surface.Second, the inner hydrogens of the polymer are cleaved by further heating up to 340 • C to form the C-C bonds marked in red, resulting in the final fully planarized chGNR shown in Fig. S1C.Most of the GNRs grow with their longitudinal edge along high symmetry directions of the Gd atomic lattice ([110] Au and [211] Au substrate directions), although there are as well about 20 % of them oriented at 13 • from these directions (the case of the marked ribbon in Fig. 1C main text), which corresponds to having the internal graphene lattice aligned with the high symmetry directions of the Gd mesh.
each spin component σ = {↑, ↓}, are calculated by summing the eigenvectors (b n σ ) resulting from the diagonalization of the MFH Hamiltonian, up to the last occupied nth molecular orbital (which depends on the present number of electrons), as encoded in the Fermi function f .

2 E 2 Supplementary Figure 4 :
− E midgap[eV ]    This effect can be partially suppressed working in constant current conditions, in which, for each pixel, the tunneling resistance is kept constant by regulating at a fixed set point of 150 mV and 125 pA.The resulting dI/dV signal as a function of V b and longitudinal position is shown in Fig. S4B.1.Line-wise Fast Fourier Transform (FFT) of energy and spatially resolved edge DoS (panel B.1) provides the 1D reciprocal space representation of the quasiparticle wave function modulus.In finite 1D ribbons, quantum confinement of the allowed k-vectors is expected to discretize the conduction and valence bands of the infinite counter parts, which have been reported elsewhere for (3,1,8)-chGNRs (14, 19).For long enough ribbons, a collection of allowed k-vectors and energies E at which they appear, allows us the reconstruction of the dispersion relation, and thereby the identification of the band which the peaks of interest in spectroscopy (Figs.2C and 3D main text) come from.This study is summarized in Figs.S4B.2-4.The E(k) spots above 10 meV clearly follow a parabollic dispersion with positive effective mass (pink guidelines in S4B).For 0 < E < 10 meV, two additional spots (at k 1 and k 2 ) appear, which we ascribe to the moiré periodicity discussed earlier (see Figs. S4B.2 and S4C).On top of them, a distinct intensity at −10 < E < −6 meV with k = k 1 is identified, which we associate to the onset of the conduction band because it matches the quadratic fit to the E(k) dispersion at higher energies.The next experimental spots are found at E =-40 meV, and together with those at lower energies, fit well another quadratic E(k) dispersion with negative effective mass.As a consequence, we ascribe the peaks at E ≤ −40 meV to confined states belonging to the valence band.This analysis justifies the assignment of the observed peaks in Fig.2Cof the main text to HOMO (-40 meV), LUMO (Fermi level) and LUMO+1 (∼20 meV) of the charge neutral ribbon.The straightforward conclusion is that the chGNRs are slightly electron doped on GdAu 2 /Au(111).Owing to the sizable Coulomb repulsion (U = 1 eV, see eq. (1)) required A) Left panel is a dI/dV constant current map of the N = 21 precursor units However, we can unambiguously rule out any change in the tip's probe spin because, when this reversal takes place, the magnetization of domain 3 remains constant (pinned by other distant APBs).The saturation contrast of all three domains (see images at ±2.8 T in panel C) is also illustrative in this respect.dI/dV in all domains experiences a slow monotonous evolution with increasing |B| towards an intermediate value between those of maximum contrast in remanence.