Evolution of graphene mediated magnetic coupling betweenFe-chains

S. V. Ong, R. Robles, S. N. Khanna

Department of Physics, 701 W. Grace St., P.O. Box 842000, Virginia Commonwealth University,

Richmond, VA 23284-2000, USA

Abstract

First principles electronic structure studies on the structure, stability, and magnetic couplingof Fe-chains at the zigzag edges of graphene ribbons have been carried out within a gradient-corrected density functional formalism. Our studies indicate that Fe sites in the chainundergo a structural rearrangement into Fe-Fe pairs. The Fe sites within a given chain areferromagnetically aligned, but chains across the ribbon have an antiferromagnetic groundstate. Studies on ribbons of increasing width show that the energy difference betweenmagnetic states decreases with increasing width attaining a value of only 8.9 meV per unitcell for chains separated by 2 nm.

Keywords: Fe-chains, Graphene, Nano-Ribbons, Magnetic Coupling

1. Introduction

The possibility of experimentally isolat-ing single sheets of graphene and investi-gating the electronic, magnetic, optical, andchemical behavior of this two-dimensionalarrangement of carbon atoms has gener-ated considerable excitement [1–5]. The sin-gle carbon sheet is a zero gap semiconduc-tor, where the electrons behave like DiracFermions with linear energy dispersion andelectron hole symmetry offering unusualtransport properties [6].These include a dis-play of quantum Hall effect at room temper-ature, ballistic transport at sub-micrometerscale, and finite conductivity at zero car-

Email addresses: ongsv@vcu.edu (S. V. Ong),snkhanna@vcu.edu (S. N. Khanna)

rier concentrations [7]. These novel fea-tures are accompanied by developmentsin experimental techniques [8–13] allowingfabrication of graphene based nanostruc-tures including ribbons, dots and graphenemolecules and clusters. The properties ofthese nanostructures can be very differentfrom those of graphene largely due to thesurface and edges [11, 14–22]. For exam-ple, graphene ribbons have been found tobe semiconducting with variable band gapor even metallic [23–26].

One of the properties that have drawnconsiderable attention is the possibility oflocalized magnetic moments associated withedges in graphene-based nanostructures.While bulk graphite is diamagnetic [27], theelectron-electron interaction can stabilizemagnetic moments at the edges in graphene

Preprint submitted to Chemical Physics Letters May 21, 2010

based nanostructures. Pure graphene isa zero band gap semiconductor where theFermi energy lies at the intersection of thefilled and unfilled states and hence there isa zero density of states at the Fermi energy.For nanostructures with zigzag edges, aswas pointed out by Fujita et al. [14], thereis a large density of states at the Fermi en-ergy due to states localized at the edges thatcan lead to a magnetic instability even fora small electron-electron interaction. Thenature of the magnetic solution is driven bythe symmetry of the underlying nanostruc-ture and both ferromagnetic (FM) or anti-ferromagnetic (AFM) states can be stabi-lized by controlling the underlying nanos-tructure. For example, graphene can bedescribed as a superposition of two atomicsublattices A and B. For a finite nanorib-bon, the two edges are antiferromagneti-cally coupled if they lie on different sublat-tices, while they are ferromagnetically cou-pled if they lie on the same sublattice [4].Geometric rules for graphene-based nanos-tructures including nanodots and nanoholeshave also been proposed [4].The presence of localized magnetic sites

at the edges, and the ability to control theircoupling by modulating the underlying lat-tice opens new intriguing possibilities. Forexample, is it possible to control the cou-pling of chains of magnetic elements by at-taching them at the zigzag edges? More im-portantly, can the strength of the couplingbe dependent on the physical characteristicsof the underlying structure, for example theribbon width? The purpose of this Letter isto explore this intriguing possibility by con-sidering Fe atoms attached to the edges ofzigzag carbon ribbons. Our studies, basedon first principles electronic structure cal-culations within a gradient corrected den-sity functional, first examine the nature of

the ground state of the Fe atoms occupy-ing edge locations. We then investigate thecoupling between these chains as the widthof the underlying ribbon is changed. Ex-perimentally, it is possible to generate rib-bons with edges occupied by H atoms. Acomparison of the coupling between Fe orH edge atoms can then provide informationif the coupling across the chains is mediatedby the underlying lattice. In Section 2, wedescribe the theoretical approach while Sec-tion 3 presents the results and a discussionof results. Section 4 is dedicated to finalconclusions.

2. Theoretical method

The theoretical studies on the zigzag rib-bons decorated with Fe or H atoms were car-ried out within the density functional the-ory using the Vienna ab initio stimulationpackage (VASP) code [28]. The projectoraugmented wave (PAW) approach [29, 30]was used to describe the electron-ion inter-action. The exchange correlation interac-tions were included through the functionalproposed by Perdew et al. [31]. Bril-louin zone integrations were carried out onMonkhorst-Pack k-point grids. Since theenergy difference between different mag-netic solutions (FM or AFM) is small, onehas to be careful about the quality of thecalculations to ensure the accuracy of quan-titative results. To this effect, we studiedthe variations in the energy of both puregraphene and the Fe-terminated nanorib-bon as a function of the number of k-pointsand as a function of the energy cut-off.For both cases, the results as a functionof k-points indicated that the energy fluc-tuates up until 30x1x1 k-points. To havethe converged results, consequently, all thenanoribbon calculations were carried out

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using 35x1x1 k-points. In concurrence withtesting the k-points, a study of the total en-ergy as a function of energy cut-off showedthat reasonably converged values require akinetic energy cut-off of 450 eV for the planewave basis. The Fe and C atom were de-scribed by using the PAW potentials [Ne]3s2 3p6 4s2 3d6 and [He] 2s2 2p2, respec-tively. The states shown in bold were al-lowed to relax. During the determination ofground state configuration, the atomic po-sitions were optimized by moving atoms inthe direction of forces till the forces wereless than 10 meV/A.

3. Results and discussion

In this work we have investigated rib-bons with widths, as described by Fujita etal. [14], of N=3-10, which correspond tothe edges separated by about 5-20 A. Onlyzigzag terminations were included. Twocomplementary studies were carried out. Inone case, the edge atoms were decoratedwith H atoms with one hydrogen atom at-tached to each edge C atom. In the othercase, an Fe atom was attached to each car-bon atom at the edge. The ground statesearch involved relaxing the C-C, C-H/C-Fe, and Fe-Fe bond lengths. All the cal-culations were carried out using a supercellthat is periodic in the x-direction. Two unitcells were included in the supercell to allowfor Fe-Fe interaction with a vacuum layer of∼12 A in the y- and z- directions to ensureno interaction between neighboring ribbons.By including two unit cells, neighboring Featoms were allowed to form pairs reminis-cent of a Peierls distortion. Our studies in-dicate that such a distortion has a stabiliza-tion energy of around 200 meV per unit cell,irrespective of the ribbon width.

To assess the accuracy of the calcula-tions, we first calculated the ground stateof a free Fe2 molecule. The geometry andthe spin configuration of two Fe atoms wereoptimized. Our studies showed a groundstate with a Fe-Fe bond length of 2.01 A inagreement with the experimental value of2.02 A [32]. The calculated atomizationenergy of 2.77 eV is higher than the esti-mated experimental value of 1.14 eV [33],but is close to the previous calculated valueof 2.08 eV [34]. The calculated total spinmoment of 6µB was also in agreement withprevious reported values. A free Fe atomhas four unpaired electrons while the Fesites have a spin magnetic moment of 3.0µB

per atom in the ground state of Fe2. As theFe sites are attached to the edges, our stud-ies indicate that the moment is reduced. Anintegration of the spin density around the Fesites indicated that each Fe atom carries amoment of 2.3µB. This magnitude remainssame for all widths of both AFM and FMcoupled ribbon edges.

Figure 1: (Color online) Ground state geometryand bond lengths for Fe-terminated edges of a N=4graphene nanoribbon. The large, dark (red) spheresrepresent iron atoms, while carbon is represented bysmall, lighter (yellow) spheres. Supercell width iscontained between the dotted lines.

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We considered the graphene ribbons withFe atoms absorbed at the edge C sites asshown in Fig. 1. As pointed out above, theFe-Fe distance in an isolated Fe2 moleculeis 2.01 A while the distance between C-sitesin a row is 2.46 A. When attached to thegraphene nanoribbon, the neighboring Featoms form pairs separated by 2.77 A. Wealso examined the case where the edges wereterminated by H atoms. The ground stateC-C, C-H/C-Fe, and Fe-Fe separations aremarked in Fig. 1.In addition to geometry we examined the

energetic stability of the ribbon decoratedwith Fe atoms. Note that an FeC moleculehas a binding energy of 3.95 eV comparedto 1.1 eV for a Fe2 molecule indicating thatFeC bonds are stronger. Our recent stud-ies on pure Fen clusters containing up toeight atoms indicate that the binding en-ergy approaches a value of 3.1 eV/atom [35].Hence configurations where Fe sites havemuch higher binding energy should be sta-ble against clustering.An analysis of the energy required to sep-

arate Fe pairs from the edge sites using theequation

BE = −EFe−GNR + EGNR + 4EFe, (1)

indicates that each Fe has a binding energyof 4.23 eV. Here, EFe−GNR is the total en-ergy calculated for the Fe terminated rib-bon, EGNR for the pristine ribbon, and EFe

for Fe atoms. As the binding energy (4.23eV) is higher than the energy per Fe-atom inclusters, the formation of nanoribbons dec-orated with Fe sites is stable against clus-tering of Fe-atoms. Another issue relates tothe formation of Fe-decorated ribbons start-ing from edges with H-atoms. Consideringthat synthetic routes to producing graphenenanoribbons have been under a hydrogen

atmosphere [8, 36], we also analyzed H2 re-moval energy in H-passivated ribbons by theequation

BE = −EH−GNR + EGNR + 2EH2 , (2)

where EH2 represents the total energy cal-culated for the H2 molecule. The calculatedenergy was 2.66 eV per H atom. The cor-responding calculations with formation ofFe2 pairs showed that the pairs are morestrongly bound to the C sites and favoredover H2 by 0.19 eV per atom.

Figure 2: (Color online) Band structure for theAFM (top) and FM (bottom) configurations of theFe-terminated zigzag edged graphene nanoribbonwith width N=4. The Fermi energy (EF ) is setto zero in both plots and is highlighted with thedashed line. Extended from the bandstructure pan-els are the site-projected density of states for Fe d-(solid blue lines), Fe p- (dotted black lines) and Cp- (dashed red lines) states.

We were particularly interested in the rel-ative stability of the ferromagnetic and anti-ferromagnetic states. To this end, we calcu-lated the energy difference between forced

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Figure 3: (Color online) Spin density figures for Fe-terminated edges of a zigzag graphene nanoribbon withwidths N=3-6. An isovalue of 0.03µB/A3 is used. Spin-up and spin-down states are shown in dark (blue)isosurfaces and light (red) isosurfaces, respectively. Transparency is used in isosurfaces to show the large,dark (red) spheres representing Fe atoms and small, light (yellow) spheres representing C atoms.

ferromagnetic and antiferromagnetic solu-tions when the edges are occupied by H orby Fe atoms. For each ribbon width, ourresults indicate that the AFM ordering ismore stable than the FM state. This isin direct agreement with the geometric rulefor nanomagnetism in graphene proposedby Yu et al. [4]. Fig. 2 shows the bandstructure for the AFM and FM configura-tions, and the majority and minority spindensity of states for the case of N=4. Notethat there is significant density of states atthe Fermi energy in both the AFM and FMcase. More importantly, the states at theFermi energy in the FM case are only com-posed of minority electrons. We found sim-ilar results at all ribbon widths and only re-sults for N=4 are shown in this Letter. Thefinite density of states of only one spin com-ponent indicates that such a ribbon couldbe used to generate spin-polarized currents.

This requires stabilization of the FM stateand hence it is important to consider thedifference in energy between the two statesand its evolution with size.

Before we discuss the difference betweenAFM and FM solutions, let us considerhow the underlying graphene stabilizes theAFM solution. We see in Fig. 3 the spincharge density for the majority and minor-ity spin states for N=3-6. The C-sites ontwo sub-lattices show opposite polarizationand hence help stabilize the AFM state asthe Fe chains are far separated to have di-rect exchange interaction. The key issue ishow this evolves with size. With this inmind, we show in Fig. 4, the exchange en-ergy (difference in energy between the AFMand FM solutions) as a function of the widthof the ribbon. One notices two things: (1)That the exchange energies are fairly small.Even at N=3, the difference between the

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Figure 4: (Color online) Magnetic exchange energyper unit cell defined by EFM -EAFM shown in meV.EFM is the energy of the FM ordered graphenenanoribbon, and EAFM for the AFM orderedgraphene nanoribbon. The dark (black) plus sym-bols corresponds to the ribbons with Fe-terminatededges, where the light (red) asterisk symbols cor-responds to the ribbons with H-terminated edges.The lines are shown as guides.

AFM and FM solution is only around 35meV. Since the ground state is AFM, thisshows that the FM state could be stabilizedby applying magnetic fields. (2) The ex-change interaction decreases as a function ofthe width. Since one does not expect signif-icant direct exchange coupling between Fe-chains, this indicates that as the width isincreased, the ribbon becomes softer to sta-bilize the AFM solution. To further showthat this effect is intrinsic to the ribbon andnot related to Fe atoms, we calculated theenergy difference between the AFM and FMsolutions by replacing Fe atoms by H atoms.The corresponding results are shown in Fig.4 and the values are close to those for Fe,indicating that the effect is intrinsic to thegraphene nanoribbon.

4. Conclusions

Our studies indicate that the zigzaggraphene ribbons decorated with Fe atomsat the edges have a ground state wherethe Fe-atoms form dimers. The chains atthe two ends of the ribbon are coupledanti-ferromagnetically where the couplingis mediated by the spin-orbitals localizedat the two sub-lattices. The difference be-tween the AFM and FM solution, howeverdecreases with increasing width and is onlyaround 9 meV for ribbons with N=10. Thesmall energy difference could be overcomeby applying magnetic field that shouldstabilize the FM state. As the states at theFermi energy in FM solution are primarilyminority in character, it may be possibleto have spin-polarized currents in suchribbons.

The authors also acknowledge supportfrom US Department of the Army througha MURI Grant # W911NF-06-1-0280.

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