Imprinting Tunable π-Magnetism in Graphene Nanoribbons via Edge Extensions
IImprinting tunable π -magnetism in graphene nanoribbons via edge extensions Michele Pizzochero ∗ and Efthimios Kaxiras School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA (Dated: February 11, 2021)Magnetic carbon nanostructures are currently under scrutiny for a wide spectrum of applications.Here, we theoretically investigate armchair graphene nanoribbons patterned with asymmetric edgeextensions consisting of laterally fused naphtho groups, as recently fabricated via on-surface syn-thesis. We show that an individual edge extension acts as a spin- center and develops a sizablespin-polarization of the conductance around the band edges. The Heisenberg exchange couplingbetween a pair of edge extensions is dictated by the position of the second naphtho group in thecarbon backbone, thus enabling ferromagnetic, antiferromagnetic, or non-magnetic states. The pe-riodic arrangement of edge extensions yields full spin-polarization at the band extrema, and theaccompanying ferromagnetic ground state can be driven into non-magnetic or antiferromagneticphases through external stimuli. Overall, our work reveals precise tunability of the π -magnetismin graphene nanoribbons induced by naphtho groups, thereby establishing these one-dimensionalarchitectures as suitable platforms for logic spintronics. Although graphene exhibits a number of unique elec-tronic properties [1], including massless Dirac fermionsand ballistic charge transport over microscopic lengthscales [2, 3], the vanishing band gap [3] and strong dia-magnetic character [4] have hindered its deployment inspin logic operations. In the quest of expanding the func-tionalities of graphene via quantum confinement effects[5], on-surface synthesis has proven successful to achievegraphene nanoribbons (GNRs) with desired atomic struc-tures [6, 7]. Within this bottom-up route, precursororganic molecules are self-assembled on a metal sur-face, yielding target GNRs that feature atomically pre-cise edges [6]. By tailor-making the initial precursormonomer, which encodes the topology of the final prod-uct, several GNRs with diverse geometries [6, 8–10] andwidths [11, 12] have been fabricated, thus stimulating theemergence of complex quantum phenomena [13–16] andnovel concepts for nanoscale devices [17–22].In this vein, certain graphene nanoribbons have beenenvisaged as promising candidates for prospective logicspintronic components [23–27] by virtue of the favor-able combination of weak spin-orbit and hyperfine in-teractions that ensure long spin lifetimes [28, 29] withfine-tunable band-gaps that confer switching capabilities[30, 31]. However, actual examples of intrinsic magnetismin these systems – a key requirement in the context, forinstance, of spin injection – remain quite scarce. Theyare mainly restricted to either zigzag sites [8, 32], endstates [33], or heteroatom insertion [13] in GNRs, in-variably suffering from a limited control over the en-suing magnetism [34, 35]. Desigllen extended carbonnanoarchitectures that offer versatile magnetic propertiesis an essential step toward the development of graphene-based switchable spintronics. In this Letter, we investi-gate, from a theoretical point of view, armchair graphenenanoribbons functionalized with edge extensions, a classof structures that was recently experimentally realizedvia surface-assisted synthesis [36, 37]. We demonstrate that a variety of magnetic states can be imprinted ingraphene nanoribbons upon the incorporation of the edgeextensions and subsequently manipulated through exter-nal stimuli, hence paving the way for all-carbon logicspintronics.The nanostructures investigated in this work are 7-atom wide armchair graphene nanoribbons (7-AGNR)with edge extensions, as recently fabricated by Sun etal. in Ref. [36]. Each edge extension consists of a naph-tho group, that is, a pair of fused aromatic rings. Fivedistinct configurations have been identified in experi-ments. An overview of their scanning-tunneling mi-croscopy (STM) images is presented in Fig. 1. In orderto understand their electronic and magnetic structures,we combine first-principles and non-equilibrium Green’sfunction calculations. We rely on the generalized gradi-ent approximation to density-functional theory devisedby Perdew, Burke, and Ernzerhof [38], as implemented inthe widely used siesta [39] and transiesta [40] codes.Further computational details are given in Supplemen-tary Note 1.Fig. 1(a) shows the atomic structure of a singleedge-extension in 7-AGNR. The functionalization of thenanoribbon with the naphtho group yields an asymmet-ric extension that features a zigzag-edged [8] head andcove-edged [9] tail (see Supplementary Fig. S1), expand-ing the armchair backbone by five carbon atoms. Theaddition of a second naphtho group in the vicinity of thefirst one occurs only in the three distinct dimer config-urations labeled D [Fig. 1(b)], D [Fig. 1(c)], and D [Fig. 1(d)]. Even though in all the three dimers the 7-AGNR scaffold is expanded by ten carbon atoms, thetopology of the resulting nanoribbons differs in two im-portant aspects. First, the relative distance between thepair of edge extensions, i.e., 1.50 nm, 1.64 nm, and 1.37for the D , D , and D configurations, respectively. Sec-ond, the mutual orientation (viz., the edge geometry ofthe extension-tail with respect to that of the adjacent a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b abcd e AC Z Z C FIG. 1. Constant-current STM images (left panels) andatomic structures (right panels) of 7-AGNR incorporating a(a) single edge-extension, double edge-extension in the (b) D , (c) D , and (d) D dimer configuration, along withthe (e) periodic sequence of such extensions, yielding saw-tooth graphene nanoribbon. The edge-extension atoms arehighlighted in green. In panel (a), the zigzag-edged (ZZ)extension-head and coved-edged (C) extension-tail are indi-cated. Scale bars in STM images correspond to 1 nm. STMimages are reproduced with permission from Ref. [36], Amer-ican Chemical Society. extension-head) of the naphtho groups, namely, cove-edged to zigzag-edged in the D , cove-edged to cove-edged in the D , and zigzag-edged to zigzag-edged inthe D dimer configuration. The arrangement of the D dimer in a periodic fashion leads to the array of edgeextensions shown in Fig. 1(e). The on-surface synthesisof this latter structural motif has been reported first inRef. 36 and later by Rizzo et al . in Ref. 37, who namedit “sawtooth” graphene nanoribbon.We begin our electronic structure investigation by con-sidering the individual edge-extension in 7-AGNR shownFig. 1(a). The introduction of the naphtho group breaksthe sublattice symmetry, since the number of carbonatoms in the majority sublattice ( n A ) exceeds that ofthe minority sublattice ( n B ) by one unit. This sublat-tice imbalance translates to a spin imbalance, sparkinga magnetic moment of 1 µ B . Such a magnetic solutionlies 94 meV lower in energy than the non-magnetic one,hence indicating that the introduction of a single edge-extension drives a spin- ground state in otherwise non-magnetic 7-AGNR. These findings agree with Lieb’s theo-rem for the repulsive Hubbard model of a bipartite latticeand a half-filled band [41], which states that the magneticground state has spin S = | n A − n B | . Although in thefollowing we focus on 7-AGNR, the S = ground stateinduced by the naphtho group is robust against the widthof the hosting armchair nanoribbon, as confirmed by ourfirst-principles calculations.In Fig. 2(a), we show the electronic density of states to ρ ( e V - ) -2 -1 0 1 2E (eV)202 G ( G ) -1.5 -1 -0.5 0 0.5 1 1.5E (eV) 020406080100 σ ( % ) a bc d E E
FIG. 2. (a) Electronic density of states ρ of 7-AGNR with(colored lines and areas) and without (black line) a singleedge-extension, see Fig. 1(a). (b) Spin density in the vicin-ity of the edge extension. (c) Conductance spectrum G of7-AGNR with (colored lines and areas) and without (blackline) a single edge-extension. Red and blue colors in pan-els (b-d) indicate spin-majority and spin-minority channels,respectively. (d) Spin-polarization σ of the conductance in7-AGNR upon the addition of a single edge-extension. elucidate the origin of the magnetism induced by the edgeextension in 7-AGNR. The addition of the naphtho grouplargely preserves the electronic structure of 7-AGNR, ex-cept for the emergence of a pair of in-gap spin-split statesthat are symmetrically located around the Fermi leveland separated in energy by 0.38 eV. The magnetic mo-ment that is encompassed in the singly occupied in-gapstate primarily resides on the zigzag edge of the naph-tho group and localizes on the majority sublattice in thevicinity of the extension, as displayed in Fig. 2(b). Fur-thermore, the fused naphtho group largely disrupts theelectronic transport across 7-AGNR, as we show in theconductance spectrum in Fig. 2(c). This effect is partic-ularly marked in the unoccupied states, where the con-ductance is found to be fully suppressed up to ∼ σ ( E )] to arise. As is cus-tomary, we quantify σ ( E ) as σ ( E ) = (cid:12)(cid:12)(cid:12)(cid:12) G ↑ ( E ) − G ↓ ( E ) G ↑ ( E ) + G ↓ ( E ) (cid:12)(cid:12)(cid:12)(cid:12) % , (1)with G ↑ ( E ) [ G ↓ ( E )] being the conductance spectrum ofthe majority [minority] spin channel. The result givenin Fig. 2(d) demonstrates that a spin-polarization of ap-proximately 60% is achieved in the vicinity of the bandedges. This value can be largely increased (e.g., to morethan 90% at energy E = − .
14 eV) or decreased (e.g.,to less than 2% at E = 1.12 eV) through an appropriate ρ ( e V - ) G ( G ) -2 -1 0 1 2E (eV) -2 -1 0 1 2E (eV)-2 -1 0 1 2E (eV)0255075100 σ ( % ) a D D D bcd E EE
FIG. 3. (a) Spin density in the vicinity of double edge-extensions in 7-AGNR in the D , D , and D dimer configu-rations, see Fig. 1(b-d). (b) Electronic density of states ρ of7-AGNR with a double (colored lines and areas) and without(black lines) edge-extension. (c) Conductance spectrum G of7-AGNR with a single (grey area), double (colored lines andareas), and without (black lines) edge-extension. Red andblue colors in panels (a-c) indicate spin-majority and spin-minority channels, respectively. (d) Spin-polarization σ ofthe conductance in 7-AGNR upon the addition of a doubleedge-extension in the D dimer configuration. modulation of the carrier density. Overall, these findingshint at potential spintronic applications of functionalized7-AGNR even at a low density of edge extensions.Next, we study pairs of edge extensions. The atomicstructures are shown in Fig. 1(b-d). A detailed analy-sis of their relative stability and aggregation tendency isprovided in Supplementary Note 2. Of these three struc-tures obtained in experiments [36], only that contain-ing the D dimer exhibits a sublattice imbalance, whichin turn leads to a magnetic moment of 2 µ B , that is, 1 µ B per naphtho group. This triplet state lies 182 meVlower in energy than the non-magnetic one. We assessthe nature of the magnetic interactions between the twoedge-extensions by estimating their Heisenberg exchangecoupling, J = E FM −E AFM , where E FM and E AFM are thetotal energies of the investigated dimer in the ferromag-netic and antiferromagnetic configurations, respectively.We obtain J = − D dimer,in line with the aforementioned Lieb’s theorem [41]. Thedensity of states given in Fig. 3(b) reveals that, simi-larly to the single edge-extension case discussed above,the magnetism associated with the D dimer stems froma pair of (doubly occupied) spin-split states centered atthe Fermi level, again 0.38 eV apart in energy. As faras the charge transport is concerned, the introduction of a second naphtho group in the D configuration furtherdecreases the conductance of the functionalized 7-AGNRwith respect to an individual edge-extension, see Fig.3(c). The doubling of the magnetic moment upon theformation of the D dimer enhances the spin-polarizationaround the band extrema, as we show in Fig. 3(d). This isespecially true at the conduction band edge, where σ ( E )attains a value of over 80%.Although the incorporation of two naphtho groups inthe D and D dimer configurations preserves the sublat-tice symmetry in the hosting 7-AGNR, strikingly differ-ent magnetic ground states arise in the resulting nanorib-bons. On the one hand, the addition of the D dimerleaves the non-magnetic ground state of 7-AGNR unaf-fected [Fig. 3(a)], being the triplet state 425 meV higherin energy. On the other hand, the addition of D dimerdevelops a magnetic moment of 2 µ B . For this latter con-figuration, however, we find a Heisenberg exchange cou-pling between the two naphtho groups J = 4 meV. Themagnetic solution is 168 meV lower in energy than thenon-magnetic one, thereby pointing to an antiferromag-netic (spin-0) ground state [Fig. 3(a)], in full compliancewith Lieb’s theorem [41]. Thus, spin couplings are inter-twined with the positioning the naphtho groups in theseAGNR. Despite these differences in the magnetic groundstates of 7-AGNR upon the functionalization with the D and D dimers, similar changes in the electronic densityof states are observed in Fig. 3(b). In both cases, twopairs of spin-degenerate in-gap localized states emerge,whose separation in energy is 0.38 eV and 0.68 eV forthe former and latter dimer, respectively. In Fig. 3(c),we present the conductance spectra of 7-AGNR contain-ing the D and D dimers, along with that of the singleedge-extension. Compared to the single edge-extension,the formation of a second naphtho group in the D ( D )configuration retains (reduces) the conductance at theband edges.Finally, we investigate the sawtooth graphene nanorib-bon (sGNR) shown in Fig. 1(e). As in the case ofthe D dimer configuration, sGNR possesses a S = 1ground state that lies 180 meV lower in energy than thenon-magnetic one. The edge extensions are coupled by J = −
13 meV, effectively acting as a ferromagnetic spinchain. This value is much larger than that reported forone-dimensional transition-metal chains [42]. The elec-tronic structure of sGNR is given in Fig. 4(a). The fron-tier bands feature a narrow bandwidth, which gives riseto a peaked density of states around the Fermi level. Asthe spin-majority channel encodes the valence band andthe spin-minority the conduction band, a complete spin-polarization is achieved at both band edges, i.e. σ ( E )= 100%. This characteristic is insensitive to the width(see Supplementary Fig. S2) and edge passivation (seeSupplementary Fig. S5) of the hosting nanoribbon, ren-dering sGNR a promising candidate for switchable spin-tronic components.
20 10 0 ρ (eV -1 )-101 E ( e V ) Γ X Γ X ) 0 0.1 0.2 0.3 0.4E (V / Å)-1501530 J ( m e V ) -1 -0.5 0 0.5 1n (e / nm)-20-15-10-50 J ( m e V ) Γ X -0.4-0.20.00.20.4 E ( e V ) Γ X Γ X Γ X Γ X Γ X -0.4-0.20.00.20.4 E ( e V ) Γ X Γ X Γ X bc dea EEE
FIG. 4. (a) Electronic structure of sGNR shown in Fig. 1(e), comprising the electronic density of states ρ (left panel), bandstructure (middle panel), and conductance spectrum G (right panel). (b) Evolution of the Heisenberg exchange coupling J withthe charge doping n . Light red and light green backgrounds define the range of stability of the ferromagnetic and non-magneticphases, respectively. (c) Evolution of the Heisenberg exchange coupling J with the electric field E applied in the tranversedirection. Light red and light blue backgrounds define the range of stability of the ferromagnetic and antiferromagnetic phases,respectively. (d) Spin density in the ferromagnetic phase of sGNR, along with the evolution of the band structure with increasingstrength of the electric field. (e) Spin density in the antiferromagnetic phase of sGNR, along with the evolution of the bandstructure with increasing strength of the electric field. Red and blue colors in panels (a, d, e) indicate spin-majority andspin-minority channels, respectively. With the knowledge of the basic properties of sGNRat hand, we then explore its response to two externalperturbations, i.e., charge doping and electric field [43].Fig. 4(b) presents the evolution of the Heisenberg ex-change coupling in sGNR with both n -type and p -typedoping. Two important effects take place. Firstly, wenotice that a certain amount of extra charge appreciablystrengthens J , which increases in magnitude from − −
16 meV ( −
15 meV)upon moderate electron (hole) doping. Second, we re-mark that carrier concentrations exceeding | n | = 0 . e /nm (that is, two electrons or holes per unit cell) drivea ferromagnetic-to-non-magnetic phase transition, indi-cating that magnetism in sGNR can be switched onand off. The emergence of this doping-induced non-magnetic phase is width- and termination-independent(see Supplementary Fig. S3 and Fig. S5) and can be un-derstood from the approximately symmetric nature ofthe density of states around the Fermi level shown inFig. 4(a). In analogy with previous observations in dis-ordered graphene [44, 45], n -type doping populates theotherwise unoccupied electronic frontier energy level (theconverse applies to p -type doping), decreasing and even-tually quenching the magnetic moment. It is worth notic-ing that undesirable doping may prevent magnetism tobe probed, e.g., if sGNR is placed or grown on a stronglyinteracting substrate. From the experimental point ofview, this effect is of particular relevance in the frame- work of on-surface fabrication, where self-assembling ofprecursor molecules typically occurs on a gold surface,which in turn results in unintentional p -type doping ofthe as-synthesized samples [36, 37].Fig. 4(c) shows the dependence of the Heisenberg ex-change coupling in sawtooth graphene nanoribbon onan electric field ( E ) applied in the transverse direc-tion. As the strength of the field exceeds the criticalvalue of 0.2 V/˚A, the sign of J reverts, thereby sig-naling a ferromagnetic-to-antiferromagnetic phase tran-sition. The spin density pertaining to each of these mag-netic states is displayed in Fig. 4(d) and Fig. 4(e), re-spectively. This field-controlled magnetic transition oc-curs irrespectively of the width of sGNR, albeit the criti-cal value is dependent on the width and edge-passivation,see Supplementary Fig. S4 and Fig. S5. Furthermore, themagnitude of J can be modulated by E to a great extent(up to 30 meV at E = 0 . k B T ln(2) (cid:39)
18 meV [46]. The effectof the transverse electric field is not only restricted to themagnetism, but largely impacts the electronic propertiesas well. The evolution of the band structure of sGNRwith E in both ferro- and antiferro-magnetic phases isgiven in Fig. 4(d-e). In the ferromagnetic phase, in-creasing the electric field reduces the gap, eventuallyreaching a band crossing at the Fermi level for E = 0 . center and causing a considerable spin-polarization ofthe conductance at the band edges. Depending on theexact positioning of the second extension in the scaffoldof the nanoribbon, ferromagnetic ( S = 1), antiferromag-netic, or non-magnetic states ( S = 0) arise. Upon thearrangement in a periodic array, these edge-extensionslead to a full spin-polarization at the band extrema.The accompanying ferromagnetic ground state can beguided into a non-magnetic or antiferromagnetic statethrough the application of charge doping or electric field,respectively. Our results show that unconventional π -magnetism in otherwise non-magnetic graphene nanorib-bons can be induced and engineered `a la carte by pat-terning the carbon skeleton with edge extensions and sub-sequent external stimuli. These nanoarchitectures possi-bly represent the ultimate limit of miniaturization forall-carbon spintronic devices.M.P. gratefully acknowledges Q. Sun (Shanghai Uni-versity), K. ˇCer¸neviˇcs (EPFL), G. Dellaferrera (IBM Re-search Europe), and Q.S. Wu (EPFL) for fruitful interac-tions. M.P. is financially supported by the Swiss NationalScience Foundation through the Early Postdoc.Mobilityprogram (Grant No. P2ELP2-191706). ∗ Electronic address: [email protected] [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.Firsov, Science , 666 (2004).[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, andA. A. Firsov, Nature , 197 (2005).[3] A. H. 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