Hydrogenated Graphene Nanoribbons for Spintronics
D. Soriano, F. Muñoz-Rojas, J. Fernández-Rossier, J. J. Palacios
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Hydrogenated Graphene Nanoribbons for Spintronics
D. Soriano, F. Mu˜noz-Rojas,
1, 2
J. Fern´andez-Rossier, and J. J. Palacios
1, 3 Departamento de F´ısica Aplicada, Universidad de Alicante, San Vicente del Raspeig, Alicante 03690, Spain Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea Departamento de F´ısica de la Materia Condensada,Universidad Aut´onoma de Madrid, Cantoblanco, Madrid 28049, Spain (Dated: October 24, 2018)We show how hydrogenation of graphene nanoribbons at small concentrations can open newvenues towards carbon-based spintronics applications regardless of any especific edge terminationor passivation of the nanoribbons. Density functional theory calculations show that an adsorbed Hatom induces a spin density on the surrounding π orbitals whose symmetry and degree of localizationdepends on the distance to the edges of the nanoribbon. As expected for graphene-based systems,these induced magnetic moments interact ferromagnetically or antiferromagnetically depending onthe relative adsorption graphene sublattice, but the magnitude of the interactions are found tostrongly vary with the position of the H atoms relative to the edges. We also calculate, with thehelp of the Hubbard model, the transport properties of hydrogenated armchair semiconductinggraphene nanoribbons in the diluted regime and show how the exchange coupling between H atomscan be exploited in the design of novel magnetoresistive devices. I. INTRODUCTION
There is a widespread consensus on the large pote-tial of graphene for electronic applications . Theoret-ically, graphene also holds promise for a vast range ofapplications in spintronics, although clear evidence ofmagnetic graphene is, however, elusive to date. In abroad sense, two factors may account for this elusive-ness. First, the fact that hydrocarbons of high spin areknown to be highly reactive and, unless fabricated or syn-thesized under very clean and controled conditions, theywill likely bind surrounding species with the concomit-tant disappearance of magnetism . A second reason re-lates to the fact that the ground state of graphene is nearan interaction-driven phase transition into an insulatingantiferromagnetic state . This underlying antiferromag-netic correlations prevent the magnetic moments, even ifthey develop, from ordering ferromagnetically and pre-clude the possibility of observing hysteresis in standardmagnetic measurements. Notwithstanding, a few reportsof magnetic graphite and graphene can be found in re-cent literature.Most of recent theoretical ideas for graphene basedspintronics applications are rooted on the magnetic prop-erties of nanoribbons or nanographenes withzigzag edges. All these proposals assume a very par-ticular edge hydrogenation where H atoms passivatethe σ dangling bonds, leaving all the π orbitals un-saturated and carrying the magnetic moments. How-ever, this is just one out of many possible edge realiza-tions which range from H-free self-passivation to fullH passivation . According to the work of Wassmann etal. , relatively low H concentations at room tempera-ture suffice to completely passivate the edges, includingthe edge π orbitals responsible for the magnetic order.The self-passivated or reconstructed zigzag edges are,in fact, among the least energetically favorable of all,although, interestingly, have been recently observed by transmission electron microscopy .In the light of the present controversy on the actualpossibilities of ever encounter zigzag magnetic edges, wepropose in this work an alternative to edge-related spin-tronics in which to exploit the recently shown controledhydrogenation of graphene . The key factor here is thatadsorption of atomic H in the bulk of graphene is ac-companied by the appearance of a magnetic momentof 1 µ B localized on the π orbitals surrounding each Hatom . These magnetic moments interact with one an-other ferromagnetically or antiferromagnetically, depend-ing on whether their respective adsorption sublattices(usually labeled A and B) are the same or not . Sta-tistically speaking, a sublattice compensated H coverageis expected unless adsorption on one sublattice is privi-leged, e.g., by the substrate. To date, however, there isno evidence that such an uncompensated coverage can beachieved. In the more likely compensated case an overallantiferromagnetic alignment with a total spin S = 0 isthus energetically favored over a ferromagnetic one with S >
0. As a proof of principle and since we are inter-ested in the diluted regime, we consider in this work thefundamental problem of two H atoms. These are cova-lently bonded to the surface of a semiconducting arm-chair graphene nanoribbon (AGNR) on different sublat-tices [see Fig. 1(a)]. Here the σ bonds of the edges arefully passivated with H so that they are irrelevant at lowenergies.As shown in Sec. III, after a brief introduction tothe theoretical basics presented in Sec. II, the magnetic-field driven ferromagnetic (F) state, where the H-inducedmagnetic moments are aligned by the field [as shown inFig. 2(b)], can present a different resistance from thatof the natural antiferromagnetic (AF) zero-field state [asthat in Fig. 2(a)]. Two different cases are discussed: Infi-nite semiconducting AGNR’s and finite ones connected toconductive graphene [see schematic piture in Fig. 1(d)].The differences in conductance between the F and the AF FIG. 1: (a) Armchair graphene nanoribbon of width N (where N is the number of dimer lines) with two H atoms (shown ingreen) adsorbed in the middle of the ribbon at a distance d from each other. The inset shows a detail of the adsorption ge-ometry. (b) Pictorical representation of the one-orbital tight-binding model where the presence of H atoms is simulated byremoving sites in a head-to-head configuration. (c) Same asin (b), but for the opposite ordering (tail-to-tail). (d) Sameas in (b), but for a finite semiconducting AGNR connected tometallic nanoribbons at the edges. states can be substantial and translate into a magnetore-sistive response as large as 100% for distances betweenthe H atoms of the order of few nanometers. Practicalimplications of these results are discussed in Sec. IV anda brief summary is presented in Sec. V II. COMPUTATIONAL DETAILS
For the calculation of the electronic structure of hydro-genated AGNR’s we use both ab initio techniques withinthe local spin density approximation (LSDA), aided bythe CRYSTAL03 package , and a one-orbital ( π ) first-neighbor tight-binding model where the electronic repul-sion is treated by means of a Hubbard-like interaction U in the mean field approximation:ˆ H = X i,j t ˆ c † i ˆ c j + U X i (ˆ n i ↑ h ˆ n i ↓ i +ˆ n i ↓ h ˆ n i ↑ i ) − U X i h ˆ n i ↓ ih ˆ n i ↑ i . (1)The first term represents the kinetic energy with firstneighbors hopping t between π orbitals. The remain-ing terms account for the electronic interactions where FIG. 2: Pictorical view of the antiferromagnetic state (a) andthe magnetic-field driven ferromagnetic state (b) where themagnetic moments localized around two hydrogen atoms aredepicted by red arrows (the orientation of the arrows withrespect to the graphene plane is arbitrary since spin-orbitcoupling is neglected here). ˆ n iη = ˆ c + iη ˆ c iη are the number operators associated toeach π orbital with spin η . These two different lev-els of approximation to the electronic structure havebeen shown to yield similar results for nanoribbons andnanographenes where a full passivation by H of the σ bonds is assumed.The bulk adsorption geometry of a H atom has beenobtained by relaxing the C atom bonded to the H and thenearest C atoms until the characteristic sp hybridizationis obtained [see inset in Fig. 1(a)]. The bonding betweena H atom and a C atom results in the effective removalof the π orbital from the low energy sector, so that theH adsorption is simulated by simply removing a site inthe one-orbital mean-field Hubbard model [see Fig. 1(b-d)]. Our results, as shown below, provide further supportto the use of the Hubbard model in graphene systems,as an alternative to the computationally more demand-ing LSDA and extend the range of applicability of Lieb’stheorem to a wider set of situations.To calculate the transport properties we use thestandard Green’s function partitioning method as im-plemented, e.g., in the quantum transport packageALACANT . To this purpose, the infinite system isdivided into three parts, namely a central region (C),containing the H atoms, which is connected to the right(R) and left (L) semi-infinite clean leads. The Hamil-tonian matrix describing the whole system is then givenby H = H C + H R + H L + V LC + V RC (2)where H C , H L and H R are the Hamiltonian matrices ofthe central region, the left and the right lead, respec-tively. V LC and V RC represent the coupling betweenthe central region and the leads. In general, the non-orthogonality of the basis set must be taken into accountwhen writing the Green’s function of the central region: G C ( E ) = [ ES C − H C − Σ L ( E ) − Σ R ( E )] − . (3)The self-energies of the left (Σ L ) and right (Σ R ) leadsaccount for the influence of these on the electronic struc-ture of the central part and S C is the overlap matrix inthis region. For the calculation of the conductance weuse the Landauer formula, G ( E ) = e h T ( E ). The trans-mission function T can, in turn, be obtained from theexpression: T ( E ) = X η Tr h G † C ( E )Γ R ( E ) G C ( E )Γ L ( E ) i η , (4)with Γ R ( L ) = i (cid:16) Σ R ( L ) − Σ † R ( L ) (cid:17) . Notice that since weare interested in collinear magnetic solutions, all the ma-trices carry the spin index η , which we have not madeexplicit in previous equations. All the terms in Eq. 2must be obtained self-consistently either from a periodicboundary condition calculation, e.g., using CRYSTAL03in the case of the LSDA calculations, or following themethodology in Ref. in the case of the Hubbard model.The Fermi energy is set to zero and, in both cases, globalcharge neutrality in all regions is imposed by shifting theonsite energies as necessary. III. RESULTSA. Energetics
We first examine the energetics of the F ( S = 1) andAF ( S = 0) states as a function of the mutual distance d between H atoms. We first choose a semiconductingAGNR of width N = 9, where N is the number of dimerlines across the ribbon, and restrict ourselves to the caseof H atoms placed on different sublattices [see Fig. 1(a)].The reason for this choice is three-fold. The AB (or BA)configurations are always energetically preferred to theAA or BB configurations for similar distances between Hatoms . Second, and most importantly for the purposeof this work, the magnetic state of the AB (or BA) config-uration can be tuned by a magnetic field. Furthermore,as briefly mentioned in the introduction, even if energetic FIG. 3: (a) Total energy referred to E [lowest energy in case(b)] as a function of the distance between H atoms for the fer-romagnetic (dashed) and the antiferromagnetic (solid) statewhen placed in the middle of the ribbon. Upper inset: Pic-ture of a nanoribbon with two H atoms. Lower inset: Energydifference between both states and extrapolation to large dis-tances (solid line). The vertical line in the inset denotes thedistance above which the energy difference becomes less than1 meV. (b) Same as in (a), but for both H atoms near thesame edge. (c) Same as in (a), but for H atoms placed on op-posite edges. Here the width of the ribbon is larger ( N = 13)and E is the minimum energy among all the AF solutions. considerations are left aside, the AB (or BA) configura-tion represents the simplest case of a random ensembleof H atoms which, in average, will equally populate thetwo graphene sublattices.In Fig. 3(a) we plot the LSDA total energy for Hatoms in the middle of the AGNR as a function of d .The S = 0 state is always the ground state. This impliesantiferromagnetic coupling, except at short distancesfor which the local magnetization, quantified throughΣ = pP i h m i i , vanishes altoghether [see Fig. 4(c)].The quenching of the magnetization is easily understoodin terms of the formation of a spin singlet . At theminimum distance for which Σ = 0, the energy differ-ence presents a maximum and decays exponentially forlarger d [see inset in Fig. 3(a)]. As d → ∞ the spinclouds do not interact anymore and both F and AF so-lutions tend to have the same energy. In summary, forany distance between H atoms the ground state presents S = 0, following Lieb’s theorem , but the overall spintexture strongly depends on their mutual distance. Wenote that it also depends on the ordering (AB or BA)of the H atoms. Whereas in bulk the spin cloud associ-ated to the H atom would be invariant under rotationsof 120 degrees, in a nanoribbon there is a preferentialdirection along the ribbon axis which is different for Hatoms located on A and B sublattices [see Fig. 1(b-c)].We refer to the preferential direction as the head and thetail to the opposite one. Thus, a head-to-head coupling(AB) is expected to be much stronger than a tail-to-tailcoupling (BA). The magnetization clouds are shown foran AB (or head-to-head) case in Fig. 4(a-b). It is easyto appreciate the strong directionality just alluded to.As a consequence, when reversing the ordering of theH atoms to a BA (or tail-to-tail) configuration [see Fig.1(c)], these do not couple magnetically at any distance,except when in very close proximity for which Σ = 0. Allthese results are similar to the ones reported using theone-orbital mean-field Hubbard model .In Fig. 3(b) we present the LSDA total energy of theF and AF states when the H atoms are placed near oneof the edges. Again the ground state is the AF state forany distance. The proximity to the edge decreases thelocalization length of the spin texture, increasing Σ [seeFig. 4(d-e)], and thereby, decreasing the critical distancebelow which the magnetization disappears [see Fig. 4(f)].The perturbation of the edge modifies the spin texturearound each atom and, contrary to the previous case,when reversing the ordering to a tail-to-tail configura-tion, the H atoms couple magnetically in a finite rangeof distances (see below). All energies, including those de-picted in Fig. 3(a), are referred to E , the lowest energysolution for H atoms close to the edge, correspondingto the smallest distance. Note that for the same distancebetween H atoms the total energy is always smaller whenthese are closer to the edge, which reflects that the bind-ing energy is higher there by approximately 1 eV .Finally we present in Fig. 3(c) the case where theH atoms are placed on opposite edges for an N = 13 FIG. 4: Magnetic moments (whose magnitude is representedby the size of the circles) on individual C atoms when theH atoms are placed head-to-head in the middle of an N = 9armchair graphene nanoribbon at a distance d = 32 . d = 15 . semiconducting AGNR. Placing the atoms on differentedges allows us now to explore the energetics of differ-ent magnetic coupling orientations (tail-to-tail for d > d <
0) at a reasonable computa-tional cost ( d is now the longitudinal distance betweenH atoms). As expected, due to the strong anisotropyof the spin texture, the magnetic coupling strongly de-pends on the ordering of the atoms and not only on theirmutual distance. While for d < d > d for the values considered. This asymmetryis easily understood since for d > d <
0) the spin tex-tures approximately couple in a tail-to-tail(head-to-head)manner. As shown below, this may have important ex-perimental consequences.
B. Magnetoresistance
We now turn our discussion to the implications theseresults may have on the electrical transport. Under theinfluence of a magnetic field, the hydrogenated AGNRbehaves like a diluted paramagnetic semiconductor forsmall concentrations of H. At large concentrations, whenthe spin density is zero everywhere, the influence of thefield can only give rise to a minor diamagnetic response.At intermediate concentrations, where the magnetizationclouds induced by the H atoms interact with each other,one can switch from the AF to the F state by apply-ing a sufficiently strong magnetic field. In analogy withthe H molecule, where switching from the singlet to thetriplet state modifies the orbital part of the wavefunc- FIG. 5: Spin resolved transmission as a function of energy forthe ferromagnetic state of an armchair graphene nanoribbonof width N = 9 with two H atoms adsorbed in the middle ata mutual distance d = 32 ˚A calculated in the (a) local spindensity approximation and (d) with a one-orbital mean-fieldHubbard model. (b) and (e) panels show the same, but forthe antiferromagnetic state. The resulting magnetoresistancein both approximations is shown in (c) and (f). tion, here the electronic structure will be indirectly af-fected by the field even if its direct influence on the or-bital wave function is neglected. This change reflects inthe spin-resolved conductance as shown in Fig. 5(a-b)for d = 32 ˚A (the dashed line represents the conduc-tance of the clean AGNR). The different total transmis-sion for the F and AF solutions, resulting from addingthe two spin channels, results in a positive magnetoresis-tance (MR), M R = G F − G AF /G F + G AF , at energiesnear the bottom and top of the conduction and valencebands, respectively [see Fig. 5(c)]. Similar results areobtained (not shown) for different intermediate distancesbetween H atoms. The right panels in Fig. 5 show theresults obtained from the one-orbital mean-field Hubbardmodel for U = t = 3 eV. Apart from the recovery of theparticle-hole symmetry, the results are remarkably simi-lar, validating the use of the latter model for transportcalculations in hydrogenated graphene.One should note, however, that since the chemical po-tential lies in the middle of the gap, the energy ranges atwhich MR could manifest itself are not relevant in linearresponse transport for infinite AGNR’s. A finite bias cal-culation may reveal the MR obtained at those energies,but this is a non-trivial task beyond the scope of thiswork. The application of a gate voltage is not a prac-tical alternative either since it implies a deviation fromcharge neutrality which would fill up or empty the local-ized states hosting the unpaired spins and kill the mag-netization. Instead, we propose to explore the possibilityof MR at zero bias by considering finite AGNR’s con-nected at the ends to conductive graphene. This is donein our calculations by considering a metallic AGNR witha narrower section in the middle of appropriate width
FIG. 6: Tunneling magnetoresistance for four different atomicH configurations. Two H atoms in a head-to-head configura-tion located in the middle of an armchair graphene nanorib-bon of length L = 73 . N = 9 (a) and N = 15 (b), and two H atoms located near one edge in ahead-to-head (c) and in a tail-to-tail (d) configuration for aribbon width of N = 9 and length of L = 57 . [see Fig. 1(d)]. (Note that in the one-orbital mean-fieldHubbard approximation AGNR’s of width N = 3 m − m is an integer, are metallic, being semiconduct-ing otherwise). In what follows and in the light of theprevious results, we restrict ourselves to the one-orbitalmean-field Hubbard model.In our proposed AGNR heterostructure the differencein the zero-bias tunnel conductance between the F andAF states is now responsible for the appearance of tun-neling MR (TMR), as shown in Fig. 6. Notice thatunlike conventional TMR, where the magnetic elementsare in the electrodes, in our proposal magnetism is in thebarrier. Panels (a) and (b) in Fig. 6 correspond to Hatoms placed in the middle of a semiconducting AGNRof length L = 73 . N = 9 and N = 15,respectively. Both cases refer to head-to-head configura-tions. The obtained TMR changes sign with d , but itis always negligibly small. On the contrary, when placednear the edge [Fig. 6(c)], the TMR is positive and reachesmuch larger values. This result is for an N = 9, L = 57 . -50-40-30-20-100 T M R ( % ) N=15-10-8-6-4-20 T M R ( % ) N=21-5 0 5 10 15 20d (Å)-2-1.5-1-0.50 T M R ( % ) N=27
FIG. 7: Tunneling magnetoresistance as a function of the lon-gitudinal distance between H atoms for three different arm-chair graphene nanoribbon widths. The H atoms are locatedon opposite edges of the ribbon. on the opposite edges of an L = 57 . IV. PRACTICAL CONSIDERATIONS
A critical assesment of the practical consequences ofthe results presented is due at this point: • First, the TMR results presented above have beenobtained for a particular type of AGNR het-erostructure and rely on the existence of metal-lic AGNR’s. Calculations where the relaxation ofthe atomic structure on the edges is considered re-veal that a gap always opens even for the nom-inally metallic armchair nanoribbons , compro-mising the applicability of these nanoribbons aselectrodes. Graphene-based metallic leads, how-ever, can be found in recent literature. For in-stance, zigzag graphene nanoribbons with passi-vated edges are metallic . The physics describedin this work does not rely on the edge terminationof the nanoribbons and could be reproduced in fullypassivated zigzag nanoribbons as well. Anotheralternative can be based on using partially un- zipped metallic carbon nanotubes , as suggestedin Ref. . A third possibility consists in gatingselectively a semiconducting AGNR as previouslydone for nanotubes . • From the LSDA calculations we note that the en-ergy difference between the F and AF states, orexchange coupling energy, can be as large as tensof meV for short distances, particularly for head-to-head configurations. However, as far as magnetore-sistive properties is concerned, exchange couplingenergies above 1 meV are of no practical use sincethe Zeeman energy gain per spin in a magnetic field B is 0 . gµ B B meV T − and fields higher than 10T are hardly accesible in the lab. The AF-F transi-tion is therefore practically forbidden below d ≈ d ≈
40 ˚A in the examples shown in Fig. 3(a)and (b), respectively. At larger distances the fieldsnecessary to induce the AF-F transition can be assmall as needed, but also is the associated MR asshown in Fig. 6(a) and (c). This is not a problem,however, when the H atoms are coupled tail-to-tailas, e.g., in the cases where they are located on op-posite edges. There, as shown in Fig. 7, a TMR ashigh as ≈
50% can be obtained at reasonably smallexchange coupling energies [see inset in Fig. 3(c)]. • The third caveat relates to the stability of theatomic configurations. Given the tendency forH atoms to form lowest-energy non-magneticaggregates , the existence of magnetically activedilute ensembles of adsorbed H atoms requires cer-tain conditions. For instance, working at reason-able low temperatures prevents H atoms from di-fusing after adsorption . Another possibility is oneintrinsic to AGNR’s: The binding energy is largerfor H atoms close to the edge than in the bulk of thenanoribbon. Related to this is the fact that the mo-bility of H decreases significantly near the edge, re-ducing the possiblity of formation of lowest-energynon-magnetic aggregates near the edge . As wehave shown, it is precisely the hydrogenation nearthe edge that favors the appearance of sizable MRand thus an actual experimental verification. V. SUMMARY
In summary, we have shown, as a proof of princi-ple, that hydrogenated AGNR’s at small concentrationscan exhibit magnetoresistive properties without invok-ing purely edge-related physics. Our results, which aredeeply rooted in Lieb’s theorem, provide further evidencefor the wide applicability of this theorem beyond thebipartite lattice Hubbard model for which was demon-strated. We have also shown that hydrogenation nearthe edge presents advantages with respect to bulk hydro-genation both from energetic and electronic standpoints.This aspect, especific to nanoribbons, might favor the useof these ones for spintronics applications, as opposed tousing large flakes of graphene. Although our conclusionsare based on the simplest case of two H atoms, noth-ing seems to prevent magnetoresistance from occuring foremsembles of many H atoms. This, however, still needsto be supported by a more extensive statistical analysiswhich is beyond the scope of this work.
Acknowledgments
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