Magnetism of topological boundary states induced by boron substitution in graphene nanoribbons
Niklas Friedrich, Pedro Brandimarte, Jingcheng Li, Shohei Saito, Shigehiro Yamaguchi, Iago Pozo, Diego Peña, Thomas Frederiksen, Aran Garcia-Lekue, Daniel Sánchez-Portal, José Ignacio Pascual
MMagnetism of topological boundary states induced byboron substitution in graphene nanoribbons
Niklas Friedrich, Pedro Brandimarte, Jingcheng Li, Shohei Saito, Shigehiro Yamaguchi, Iago Pozo, DiegoPe˜na, Thomas Frederiksen,
2, 6
Aran Garcia-Lekue,
2, 6
Daniel S´anchez-Portal,
2, 7, ∗ and Jos´e Ignacio Pascual
1, 6, † CIC nanoGUNE, 20018 Donostia-San Sebasti´an, Spain Donostia International Physics Center (DIPC), 20018 Donostia-San Sebasti´an, Spain Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan CiQUS, Centro Singular de Investigaci´on en Qu´ımica Biol´oxica e Materiais Moleculares, 15705 Santiago de Compostela, Spain Ikerbasque, Basque Foundation for Science, Bilbao, Spain Centro de F´ısica de Materiales CSIC-UPV/EHU, 20018 Donostia-San Sebasti´an, Spain (Dated: April 23, 2020)Graphene nanoribbons (GNRs), low-dimensional platforms for carbon-based electronics, showthe promising perspective to also incorporate spin polarization in their conjugated electron system.However, magnetism in GNRs is generally associated to localized states around zigzag edges, difficultto fabricate and with high reactivity. Here we demonstrate that magnetism can also be induced awayfrom physical GNR zigzag edges through atomically precise engineering topological defects in itsinterior. A pair of substitutional boron atoms inserted in the carbon backbone breaks the conjugationof their topological bands and builds two spin-polarized boundary states around. The spin state wasdetected in electrical transport measurements through boron-substituted GNRs suspended betweentip and sample of a scanning tunneling microscope. First-principle simulations find that boron pairsinduce a spin 1, which is modified by tuning the spacing between pairs. Our results demonstrate aroute to embed spin chains in GNRs, turning them basic elements of spintronic devices.
In spite of being a diamagnetic material, graphenecan develop a special class of magnetism via the polar-ization of its π -electron cloud. Such π -paramagnetismis less localized than the more conventional d - or f -magnetism, and can interact over longer distances. Mag-netic graphene nanostructures thus offer promising per-spectives for a la carte engineering of interacting spinsystems with applications in quantum spintronics de-vices [1–4]. The vision of graphene π -paramagnetismhas been recently boosted by the development of on-surface synthesis (OSS) as a versatile bottom-up route.In OSS, nanoscale graphene flakes with customized shapeand composition are fabricated over a metal substratethrough the steered reactions between designed organicprecursors [5, 6]. Solid evidence of magnetism in flakeswith segments of zigzag edges has been revealed in scan-ning tunneling spectroscopy experiments [7–10].Substituting one carbon atom of the graphene latticeby heteratoms has been commonly identified as a poten-tial route to induce magnetism[12, 13]. A representativecase is the doping of graphene with substitutional boronatoms (Fig. 1), because it can be idealized as the removalof one electron from the conjugated bipartite lattice plusthe energy upshift of a p z state. However, boron atomsdo not induce any spin imbalance around, but simplybehave as a point potential [12]. A prerequisite for theemergence of π -paramagnetism is that the point defectalso causes a sufficiently large rupture of the conjugatedelectron system, for example by completely removing lat-tice sites or saturating p z orbitals [14–16], resulting in thelocalization of radical states. BB (a) (b) P D O S S=1
E-E (eV) F FIG. 1. (a) Lewis structure of the 2B-7AGNR shown over acolour map representing the spin polarization density map,computed by density functional theory simulations ([11])(green represents the boron moiety). (b) Spin-resolved Pro-jected Density of States (PDOS) over carbon atoms aroundthe boron dimer. A net spin polarization of one kind confirmsthe ferromagnetic alignment of the two magnetic moments.
Here we show that inserting a pair of boron atoms inthe carbon lattice of graphene nanoribbons (GNRs) en-ables a magnetic ground state. Density Functional The-ory (DFT) simulations (Fig. 1) show that, while mag-netism is completely absent around a pair of such sub-stitutional B atoms in different sublattices of extendedgraphene, in a 7-carbon-wide armchair GNR (7AGNR)the boron pair builds up a net magnetic moment of 2 µ B (two Bohr magnetons). The spin polarization, shown inFig. 1(a), decays towards the pristine segments with thecharacteristic shape of the 7AGNR end states [17] (seeSupplementary Information (SI), [11] for a comparison).In fact, the spin cloud emerges from the rupture of theconjugated system imposed by the 2B-doped ring andthe two neighbouring Clar sextets (green in Fig. 1(a)). a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Br Br Br Br
BB BB + (a) app r oa c h r e t r a c t
10 2.4 2.6 2.8 3.0 C u rr en t ( n A ) z p -1 -1 -2 -3 Tip retraction (nm)(b) + (c) FIG. 2. (a) Organic precursors mixed in the experiments.(b) STM constant current topography image of a 2B-7AGNR( V b = −
300 mV, I = 30 pA). The green cross indicatesthe position from where the GNR is lifted. (right) Constantheight current scan ( V b = 2 mV) using a CO-functionalizedtip [23] of the region indicated by the dashed rectangle. (c)Tunneling current I at V b = 25 mV as a function of z for aborylated (red) and a pristine (orange) GNR, for comparison.The grey region indicates where spectra in Fig. 3(b) was mea-sured. The background shows results of atomistic simulationsof a retraction stage shown in Fig. 4, for illustration. This moiety behaves as a highly reflective barrier for va-lence band electrons [18, 19], thus inducing localized endstates associated with the termination of the topological7AGNR valence band [20]. This striking result offers thevision of combining band topology of nanoribbons [20–22]and heteroatoms for shaping spin textures in grapheneribbons.In our experiments, we substitutionally inserted boronpairs (2B) inside 7GNRs (2B-7AGNRs) by adding a smallfraction of 2B-doped trianthracene organic precursors ( in Fig. 2(a)) [18, 19, 24–27] during the OSS of 7AGNRsusing precursor [5] (as schematically shown in Fig. 2(a),see Methods in SI [11]). Scanning tunneling microscopy(STM) images of the fabricated ribbons (Fig. 2(b)) re-solved the 2B unit as a topography depression at varyingpositions inside the GNR [18, 24, 25]. Tunneling spectrashowed no fingerprint of magnetism around the 2B moi-eties due to the strong interaction between boron andmetal states [19, 27], which quenches the eventual mag-netic ground state. Therefore, to detect their intrinsicmagnetic state, the 2B moieties had to be removed fromthe metal substrate.We used the STM tip to pick individual 2B-7AGNRsfrom one end (cross in Fig. 2(b)) and lift them off tolie free-standing between tip and sample [7, 29]. In this d I/ d V ( n S ) Bias (mV) d I/ d V ( n S ) T i p r e t r a c t i on ( n m ) Bias (mV) (a)(b)
FIG. 3. (a) Spectrum over a larger bias interval, taken at z = 2 .
70 nm. The dotted grey line is a fitted Frota function[28] with HWHM = 6 . ± . V b and z . The corresponding heightinterval is the gray region indicated in Fig.2(c). A zero-bias,narrow resonance is observed for 2 . < z < . configuration, we measured the (two-terminal) transportthrough the suspended 2B-7AGNR as a function of tipretraction distance z to confirm the presence of a mag-netic state. At the initial stages of suspension (2B unitstill on the surface), the current passing through the rib-bon showed a weak exponential decrease with retractiondistance z (Fig. 2(c)), indicating a small tunneling decayconstant through the elevated pristine GNR [29]. How-ever, at a certain retraction length z p the current ex-hibited a pronounced peak, returning afterwards to theprevious exponential decay. The current peak and itsvertical position z p were reproduced for several retrac-tion/approach cycles of the same ribbon, and appearedin all ribbons with a single 2B pair studied. In every case,the value of z p correlated with the distance between the2B site and the contacted GNR-end (see SI [11]), thusindicating that the current peaks were caused by the de-tachment of a 2B moiety from the surface.To find out the origin of the anomalous current in-crease, we measured current-voltage ( I − V ) character-istics at z positions around z p . The differential con-ductance ( dI/dV ) spectra (Fig. 3) show the suddenappearance of a narrow zero-bias dI/dV resonance at z p = 2 . z > . z range observed, and its narrow linewidthreached a maximum value of Γ HWHM ≈ . ± . z p (Fig. 3(a)). The resonance vanished abruptly whenthe tip was approached below z p , but it was recovered byincreasing z back above the z p = 2 . -0.5 0 0.5 P D O S -0.5 0 0.5 E-E F (eV) -1 -101 -0.5 0 0.5-0.5 0 0.5 E-E F (eV) -101-101 -0.5 0 0.5-0.5 0 0.5 E-E F (eV) -101-101
12 34 56 xyz xyz xyz (a) 21
S=0
65 S=143 S=1/2(b) (c) P D O S P D O S freefree - on Auon Au - P D O S P D O S P D O S FIG. 4. (a-c) Relaxed structures of three different configurations of a 2B-7AGNR bridging a gold tip and a Au(111) surface(red arrows indicate the position of the B heteroatoms). Constant spin density isosurfaces are shown over the atomic structure(1 . × − e / ˚A , spin up in blue and down in red). The insets compare the spin PDOS over C atoms within the boxed regionsaround each boron atom for each bridge geometry (solid lines), with the corresponding one of a free 2B-7AGNR (dashed lines).The GNR zigzag termination on the surface holds a spin-polarized radical state, absent at the contacted end due the bondformed with the tip’s apex [7]. narrow line shape and fixed zero-bias alignment, we con-clude that the resonance is a manifestation of the Kondoeffect [31–33]. A Kondo-derived resonance appears in dI/dV spectra when a spin polarized state weakly in-teracts with the conduction electrons of an underlyingmetal [34]. Thus, its observation here shows that the 2Bmoieties abruptly develop a net spin around the point oftheir detachment from the surface.To correlate these observations with the 2B-inducedspin polarization predicted in Fig. 1, we performed DFTsimulations of a finite 2B-7AGNR suspended between amodel gold tip and the surface of a Au(111) slab (seeMethods). Figure 4 shows the relaxed atomic structuresof the GNR-junctions before, while, and after detachmentof a 2B unit, and includes the computed constant spindensity isosurfaces. Before 2B-detachment from the sur-face ( z < z p , Fig. 4(a)), the intrinsic magnetism aroundthe 2B units is quenched: the PDOS in the regions 1and 2 around each boron atom is broad and spin un-polarized, contrasting with the clear spin polarization offree ribbons (shown as dashed plots). This is caused bythe strong hybridization of the B atoms with the goldsurface [19, 24], which appear 0 . S = 1 / S = 1 state of the iso-lated 2B-7AGNR (Fig. 1(b)). Since the Kondo effect ismsot common for a spin 1/2 near a metal substrate, themost probable scenario is the one pictured in Fig. 4(b).There, the Kondo effect is caused by the spin 1/2 of re-gion 4 interacting weakly with the surface when the firstboron atom is detached, and vanishes when the secondboron is cleaved ∼ S = 1 state of the free rib-bon remains, and is clearly favored over an anti-parallelalignment by ∼
14 meV per isolated 2B pair. The pres-ence of a triplet state is striking; the two spin clouds ateach side of the 2B center extend symmetrically over op-posite sub-lattices of the 7AGNR, what usually favors anantiparallel kinetic exchange [16]. A detailed analysis re-
B B B B B -10 0 10 Bias (mV) T i p r e t r a c t i on ( n m ) Bias (mV) d I/ d V ( n S ) x50 z p1 z p2 Tip retraction (nm) C u rr en t ( n A ) xy zxyz + (d)(e) (f) z=1.79 nmz=3.18 nm HWHM=2.4 mVHWHM=1.5 mV no r m a li z ed d I/ d V ( a r b . un i t s ) (a)(b)(c) BB BB FIG. 5. (a) (Left) Constant current STM image of a borylated GNR ( V b = −
300 mV, I = 30 pA). (Right) Constantheight current image of the marked rectangular region ( V b = 2 mV) using a CO-functionalized tip. (b) DFT simulation ofthe magnetization of a (2B) -7AGNR. (c) Cotunneling current I vs. z through the ribbon in (a) suspended between tip andsample. (d) Normalized differential conductance of the suspended (2B) -7AGNR as a function of V b and z (see SI [11]). Twozero-bias resonances are observed. (e) Representative dI/dV spectra measured at the indicated z positions, with fits (dashed)using Frota functions [28]. (f) DFT relaxed structure of a suspended (2B) -7AGNR. Constant spin-density isosurfaces areshown over the atomic structure; red arrows indicate the position of the B atoms. The inset shows the indicated region of thesuspended GNR from a different angle. veals that the hopping matrix elements between the twolocalized states at the sides of one 2B unit are very small( t intra ∼
18 meV, see SI [11]). Consequently, the moietyformed by a 2B-doped ring surrounded by two Clar sex-tets is a very stable element that blocks conjugated elec-trons from hopping across. This explains the presence ofa magnetic state because the borylated element acts asa barrier for valence band electrons of the 7AGNR seg-ments [18], and induces spin polarized boundary statesdue to the non-trivial topology of this band [20]. Ad-ditionally, the 2B barrier also disconnects the boundarystates at each side, and hinders the (anti-parallel) kineticexchange between them. The stabilization of the tripletconfiguration is then the result of the weak direct overlapbetween both spin-polarized boundary states through the2B barrier, which, due to the tiny hopping between them,dominates the exchange interaction and induces the fer-romagnetic alignment of the spins according to Hund’srule.The synthetic route to produce 2B-7AGNRs also al-lows incorporating multiple 2B units in one GNR toexplore their interactions. We studied GNRs with twoconsecutive 2B moieties like in Fig. 5(a,b), spaced by1 . z , but now showing two peak features at retraction dis-tances z p1 ≈ .
60 nm and z p2 ≈ .
05 nm. These val- ues are related to the positions of the 2B units (nom-inally ∼ . ∼ . z (Fig. 5(d)) shows thatboth current features are also caused by narrow zero-bias dI/dV resonances appearing at ranges z < . . < z < . -7AGNRs (Fig. 5(b)) find that thetwo singly-occupied boundary states between neighbor-ing 2B elements interact strongly and open a large hy-bridization gap [19], forcing them into a closed shell con-figuration. As a consequence, the spin polarization van-ishes between two 2B sites, but persists outside this re-gion as two uncompensated spin 1/2 clouds (Fig. 5(b))with barely no preferred relative spin alignment. Fromour electronic structure calculations [11, 19] we can char-acterize this hybridization by a relatively large effectivehopping term t inter between boundary states of neigh-boring 2B units, which contrasts with the weak hop-ping t intra across each 2B unit. In fact, the electronicstructure close to the Fermi level of a sequence of bory-lated units can be mapped onto the Su-Schrieffer-Heeger(SSH) model [35], characterized by two alternating hop-pings along a 1D wire. Since t inter > t intra , an alternativeway to understand the spin-polarized states in Fig. 5(b)is as zero-energy topological modes of a very short SSHchain. These simulations allows us to predict that a S=1spin chain will emerge for larger inter-2B spacing, whenboth hopping terms become smaller than the Coulombcharging energy U of the boundary states [11].We also simulated a (2B) -7AGNR suspended betweentip and sample to explore how these interactions survivein the experimental geometry. While detaching the first2B site from the surface is equivalent to the single 2Bcase of Fig. 4, the spin polarization between the two 2Bmoieties is absent even when the second 2B remains par-tially hybridized with the surface (see Fig. 5(f)). Onlywhen the last boron atom is detached, the terminal spinshows a 1/2 spin cloud outwards, being this the state re-sponsible for the more extended Kondo effect observed inthe experiment above z p2 . These results confirm the spinpolarization predicted for boundary states appearing atthe interface between (2B) chains and pristine 7AGNRsegments [20], which in essence are zero-energy modes ofthe 7AGNR valence band of similar nature than thosecreated by a single 2B unit at every side.The peculiar spin polarization of single 2B units anddimers is a remarkable consequence of the large and long-range exchange interactions present in GNRs [7, 20]. Ourresults indicate that the spins should survive in free-standing GNRs for sufficiently low concentrations of 2Bunits forming either a chain of S = 1 spins, or for suf-ficiently dense doping S = 1 / ∗ [email protected] † [email protected][1] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu,and J. R. Petta, Science , 1174 (2013).[2] M. 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2, 6
AranGarcia-Lekue,
2, 6
Daniel S´anchez-Portal,
2, 7, ∗ and Jos´e Ignacio Pascual
1, 6, † CIC nanoGUNE, 20018 Donostia-San Sebasti´an, Spain Donostia International Physics Center (DIPC),20018 Donostia-San Sebasti´an, Spain Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan CiQUS, Centro Singular de Investigaci´on en Qu´ımica Biol´oxica e Materiais Moleculares,15705 Santiago de Compostela, Spain Ikerbasque, Basque Foundation for Science, Bilbao, Spain Centro de F´ısica de Materiales CSIC-UPV/EHU,20018 Donostia-San Sebasti´an, Spain (Dated: April 23, 2020)
CONTENTS
Materials and Methods 2Sample preparation 2Lifting procedure 2Normalization of differential conductance spectra 2Details on ab-initio calculations 3Supplementary Text 4Lift of GNRs with 2B sites at different positions 4Density Functional Theory simulations of the edge states and boron-induced inner states in7AGNR 5Minimal model of an isolated 2B center in 7AGNR 5Electronic and magnetic interactions between neighbouring 2B centers 6References 8 a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r ATERIALS AND METHODSSample preparation
To prepare the Au(111) substrate we perform a cleaning cycle by sputtering the crystalwith Neon ions for 10 minutes and afterwards annealing the crystal in UHV conditions( p < − mbar) at T = 740 K for 15 minutes. We prepare our samples by simultaneouslyevaporating DBBA and B -DBTA molecular precursors on the clean Au(111) surface [1–3]. Afterwards an Ullmann reaction and cyclodhydrogenation are induced by successiveannealing the sample to 200 ◦ C for few minutes and flashing to 400 ◦ C for 30 seconds.The samples are prepared in situ and transferred directly into a homebuild STM keptwith liquid Helium at ≈ Lifting procedure
The lifting is performed as following. After stabilization of the tip above the clean Au(111)surface with a fixed set of parameters ( I = 30 pA, V b = − V b = 25 mV. At some point during theapproach a sudden increase in the current is observed. This corresponds to the formation ofa tip-molecule bond and defines z = 0 during each experiment. After the tip-molecule bondis formed we are able to partially lift the GNR from the surface.Custom cleaving trajectories are employed for the different molecules, depending on thestrength of the tip-molecule bond. In the most robust cases we cleave the molecule by simpleretraction along the z -axis. In less strongly bound cases, we need to move additionally inthe xy -plane, following a trajectory along the direction of the GNR-backbone on the surface.However, we find that variations in the lifting trajectory do not influence at which z weobserve the peak in I and the Kondo-resonance in the differential conductance spectra. Normalization of differential conductance spectra
Each individual spectrum taken at one z was normalized following G norm ( V, z ) = G ( V,z ) − G min, z G max, z − G min, z , where G ( V, z ) is the measured dI/dV -signal at z and G min, z ( G max, z ) isthe minimal (maximal) value of G ( V, z ) in the analysed bias interval. The normalization isonly applied to the data presented in Fig. 4D.2 etails on ab-initio calculations First-principles electronic structure calculations are performed using density functionaltheory as implemented in the SIESTA package [4, 5], where the valence-electron wave func-tions are expanded using a linear combination of numerical atomic-orbitals as a basis setand the core electrons are replaced by norm-conserving Troullier-Martins pseudopotentials[6]. A double- ζ plus polarization (DZP) basis set is adopted for the surface Au atoms, wherean extended basis is considered for the top atomic layer optimized for the description of theAu(111) surface [7]. The tip model is defined by a gold rod with its axis along the Au(100)crystalline direction and its valence-electrons described using a DZP basis set for s -orbitalsand single- ζ non-polarized (SZ) basis for d -orbitals (i.e. DZP-SZd). Such geometry and ba-sis set has been shown to provide a reasonable description of the contact, with a smoothlocal density of states at the tip apex around the Fermi level [8]. For the atoms definingthe borylated ribbons a double- ζ non-polarized (DZ) basis set is employed, where the or-bital radii are defined using a 30 meV energy shift [5]. Dispersive interactions between theborylated ribbon and the metallic surface are considered using the van der Waals densityfunctional by Dion et al. [9] with the modified exchange by Klimeˇs, Bowler and Michaelides[10]. The real-space grid for integrations is defined using a 300 Ry energy cutoff [5]. Thesmearing of the electronic occupations is defined by an electronic temperature of 300 K witha Fermi-Dirac distribution. The self-consistency cycle is stopped when variations on theelements of the density matrix are lower than 10 − as well as 10 − eV for the Hamiltonianmatrix elements.The freestanding 2B-GNRs, as the one shown in Fig. 2A in the main text, are computedwithin a simulation cell that includes 20 and 40 ˚A of empty space in the directions along andperpendicular to the ribbon, respectively, to avoid interactions with the periodic replicas.Geometry optimizations are performed using the conjugate gradient (CG) method until allforces are lower than 20 meV / ˚A. For the calculations involving the suspended borylatedGNRs, the Au(111) surface is represented by a 3-layer thick slab within a simulation cell ofdimensions 46 . × .
98 ˚A along the periodic directions. A H passivation is employed atthe reverse side of the slab in order to prevent spurious effects due to interaction betweensurface states belonging to the top and bottom surfaces of the slab [11]. A 1 × k -point meshis used to sample the bidimensional Brillouin zone. The slab defined in this way comprises640 atoms (7680 orbitals), where the atoms from top Au layer are allowed to relax with theCG method until forces are lower than 20 meV / ˚A (the H passivation and the two bottomAu layers are kept fixed). 3 UPPLEMENTARY TEXTLift of GNRs with 2B sites at different positions
In Fig. 1 we present two further examples of I - z retraction curves of two 2B-GNRs withthe 2B site at 2 . . z p = 1 . z p = 2 . Tip retraction (nm) C u rr en t ( n A ) GNR 1GNR 2
200 100 0 100 200 )Vm( saiB d I/ d V ( µ S )
200 100 0 100 200
Bias (mV) d I/ d V ( n S ) z = 2.4 nmz = 1.4 nmGNR 1 GNR 2 (a) approachretract + + (c) (b) (d)(e) FIG. 1.
Fingerprint of magnetism in borylated GNRs: A,B
STM constant current topog-raphy scan of two borylated 7-AGNR ( V b = −
300 mV, I = 30 pA) with the doping position atdifferent distances from the lifting position. The green cross indicates the position from where theGNR is lifted. Scale bar is 3 nm. C Cotunnelling current I as a function of z for the GNRs depictedin A and B . The value of z at which the current peak is observed correlates with the position ofthe doping site in the ribbon. The lifting experiment is performed at V b = 25 mV. D,E
Spectrumover a larger bias window, taken at z = 1 . z = 2 . . ± . D , GNR 1)and HWHM = 5 . ± . E , GNR 2). ensity Functional Theory simulations of the edge states and boron-induced innerstates in 7AGNR The boundary states induced by the substitutional boron pairs exhibit the same shape,symmetry, and decay towards the pristine segments as the 7AGNRs end states, localizedin the zigzag terminations. Fig. 2 below presents a comparison between two 7AGNRs endstates facing each other (shown in the top) and the 2B-induced bonding and anti-bondingstates (bottom). Notice that, interestingly, for 2B centers the “bonding” state is higher inenergy than the “anti-bonding” linear combination. (a) (b)
FIG. 2.
Wavefunction of edge states and 2B-induced states in 2B-7AGNRs:
Bonding(a) and anti-bonding (b) boundary states localized around the boron dimer (bottom) as comparedto two 7AGNR edge states (top). Red arrows indicate the position of the boron atoms.
Minimal model of an isolated 2B center in 7AGNR
A single 2B moiety in a 7-armchair graphene nanoribbon (7AGNR) is found to build upa total spin S=1 due to the predominant ferromagnetic interaction between the two singlyoccupied boundary states (B-states) localized at each side. The purpose of this section isto formulate the minimal model (depicted in Fig. 3) capable of explaining the low-energyelectronic structure and the magnetism of isolated 2B pairs.The effective Hamiltonian describing an isolated 2B pair also can be seen in Fig. 3 in thelimit U (cid:29) t intra . Exact and mean field solutions of this model are readily obtained and canbe compared with the relative energies and the band structures of different DFT solutions.In particular, we use information from DFT simulations (both non-spin-polarized and spin-polarized with either ferromagnetic or anti-ferromagnetic spin alignment) to estimate thefollowing values of the effective parameters describing the system according to the modelof Fig. 3: the charging energy of each boundary state U ∼
260 meV, the effective hoppingmatrix element between both boundary states t intra ∼
18 meV, and the Coulomb exchangeinteraction J intra ∼
17 meV.While the Coulomb exchange J intra favours parallel alignment, the kinetic exchange (KE)term, given by 4( t U ) (Fig. 3) tends to align spins antiferromagnetically. The small valueof t intra relative to U translates into a small KE interaction, of the order of ∼ IG. 3.
Minimal model of an isolated 2B center in a 7AGNR:
Two boundary states L andR, one at each side of the 2B pair and characterized by a charging energy U (Hubbard parameter),are connected by a hopping matrix element t intra . We also add a Coulomb exchange interactionJ intra between electrons in different levels. The corresponding effective Hamiltonian, expressed interms of spin ˆ ~S and occupation ˆ n operators at each site, is shown below in the relevant limit here, U (cid:29) t intra . ground state of an isolated 2B pair corresponds to a total spin S=1 as expected from Hund’srules. Electronic and magnetic interactions between neighbouring 2B centers
We now consider the case of periodic doping of 7AGNRs with 2B centers. The distancebetween two 2B sites is given by the number n of 7AGNR unit cells defining the periodicityof the system. Fig. 4 compares the computed hopping matrix elements between B-states ateach side of a given 2B pair ( t intra ) and those between two neighbour 2B pairs ( t inter ). Thesevalues are obtained by fitting to the Su-Schrieffer-Heeger (SSH) model [12] the B-derivedin-gap states appearing in the non-spin-polarized DFT band structures in Fig. 5 of Ref.[3]for 2B-(7AGNR) n with n = 3 , . . . ,
13. The fitting is rather accurate for n >
3, while for n=3there is a somewhat larger uncertainty due to the interaction between the lower B-derivedstate and the valence band of the 7AGNR near the Γ-point. As mentioned above we obtaina very small effective hopping parameter between the boundary states across the diboronmoiety t intra , which amounts to 18 meV for n ≥ t inter is much larger than t intra for n <
7, and decreases exponentially with the separation, resulting residual above n ≥ n =3. The inter-diboron hopping is very large in this case ( ∼
350 meV > U), resultingin a closed-shell state with no spin polarization in the spacing between them. The smallelectron hopping across a 2B site protects the outer B states, which maintain each a netspin polarization of S=1/2, as detected in Fig. 3.The larger hopping between neighbour diboron sites than inside one, i.e., t inter > t intra ,6esemble a non-trivial bulk-boundary correspondence, as described by the SSH model. Ac-cording to this model, for the smallest spacings between diborons ( n =3, 5) boundary statesare expected at the terminations since t intra < t inter . However, if we consider Coulomb cor-relations, only the case n =3 shows end states because it fulfills U < t inter . The n =5 case,for which already U > t inter , results in a chain with a spin texture formed by S=1 spinslocalized at the 2B sites and aligned antiferromagnetically along the chain due to the largekinetic exchange associated with t inter .
10 20 30 40 50 H o pp i n g ( m e V )
10 20 30 40 50 H o pp i n g ( m e V ) t inter t intra n=3 n=5 n=7 n=9 n=11 t intra t inter n-3 t inter J intra BB BB t intra t intra a)b) FIG. 4.
Electronic interactions between B-derived states from DFT calculations: (a) Schematic model of a 2B-AGNR with intra-2B and inter-2B hopping terms indicated. J intra represents the direct Coulomb exchange resulting in ferromagnetic alignment of the spins withineach pair. (b) Computed values of intra-2B and inter-2B hopping terms from the analysis of non-spin-polarized DFT band structures of the periodic systems [3]. The inter-2B spacing given by n ,the number of 7AGNR unit cells between 2B sites. The correspondence with the Su-Schrieffer-Heeger (SSH) model is evident. The inset shows the exponential decay of t inter and magnifies thecrossing point where the topology of the non-spin-polarized band structure of the system changes.However, as discussed in the text, the non-polarized solution is not relevant in the limit where t inter < U that clearly holds for n ≥ [email protected] † [email protected][1] S. Kawai, S. Saito, S. Osumi, S. Yamaguchi, A. S. Foster, P. Spijker, and E. Meyer, Nat.Commun. , 1 (2015).[2] E. Carbonell-Sanrom`a, P. Brandimarte, R. Balog, M. Corso, S. Kawai, A. Garcia-Lekue,S. Saito, S. Yamaguchi, E. Meyer, D. S´anchez-Portal, and J. I. Pascual, Nano Lett. , 50(2017).[3] E. Carbonell-Sanrom`a, A. Garcia-Lekue, M. Corso, G. Vasseur, P. Brandimarte, J. Lobo-Checa, D. G. De Oteyza, J. Li, S. Kawai, S. Saito, S. Yamaguchi, J. E. Ortega, D. S´anchez-Portal, and J. I. Pascual, J. Phys. Chem. C , 16092 (2018), 1806.02385.[4] E. Artacho, D. S´anchezPortal, P. Ordej´on, A. Garc´ıa, and J. M. Soler, Phys. Stat. Solidi (b) , 809 (1999).[5] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´on, and D. S´anchez-Portal,J. of Phys.: Cond. Matt. , 2745 (2002).[6] N. Troullier and J. L. Martins, Phys. Rev. B , 1993 (1991).[7] S. Garc´ıa-Gil, A. Garc´ıa, N. Lorente, and P. Ordej´on, Phys. Rev. B , 075441 (2009).[8] M. Kolmer, P. Brandimarte, J. Lis, R. Zuzak, S. Godlewski, H. Kawai, A. Garcia-Lekue,N. Lorente, T. Frederiksen, C. Joachim, D. Sanchez-Portal, and M. Szymonski, Nature Com-munications (2019).[9] M. Dion, H. Rydberg, E. Schr¨oder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. ,599 (2004).[10] J. Klimeˇs, D. R. Bowler, and A. Michaelides, J. Phys.: Condens. Matter , 022201 (2010).[11] N. Gonzalez-Lakunza, I. Fern´andez-Torrente, K. J. Franke, N. Lorente, A. Arnau, and J. I.Pascual, Phys. Rev. Lett. , 156805 (2008).[12] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. , 1698 (1979)., 1698 (1979).