By computing spin-polarized electronic transport across a finite zigzag
graphene ribbon bridging two metallic graphene electrodes, we demonstrate, as a
proof of principle, that devices featuring 100% magnetoresistance can be built
entirely out of carbon. In the ground state a short zig-zag ribbon is an
antiferromagnetic insulator which, when connecting two metallic electrodes,
acts as a tunnel barrier that suppresses the conductance. Application of a
magnetic field turns the ribbon ferromagnetic and conducting, increasing
dramatically the current between electrodes. We predict large magnetoresistance
in this system at liquid nitrogen temperature and 10 Tesla or at liquid helium
temperature and 300 Gauss.Comment: 4 pages, 4 figure

Fernandez-Rossier, J.

Muñoz-Rojas, F.

Palacios, J. J.

English

arXiv

F. Muñoz-Rojas

J. Fernández-Rossier

J. J. Palacios

Crossref

Giant Magnetoresistance in Ultrasmall Graphene Based Devices

By computing spin-polarized electronic transport across a finite zigzag graphene ribbon bridging two metallic graphene electrodes, we demonstrate, as a proof of principle, that devices featuring 100% magnetoresistance can be built entirely out of carbon. In the ground state a short zigzag ribbon is an antiferromagnetic insulator which, when connecting two metallic electrodes, acts as a tunnel barrier that suppresses the conductance. The application of a magnetic field makes the ribbon ferromagnetic and conductive, increasing dramatically the current between electrodes. We predict large magnetoresistance in this system at liquid nitrogen temperature and 10 T or at liquid helium temperature and 300 G.This work has been financially supported by MEC-Spain (Grants No. MAT07-67845, No. MAT07-31099-E, and CONSOLIDER CSD2007-0010)

Muñoz Rojas, Federico

Fernández-Rossier, Joaquín

Palacios Burgos, Juan José

Repositorio Institucional de la Universidad de Alicante

Giant Magnetoresistance in Ultrasmall Graphene Based DevicesF. Mun˜oz-Rojas, J. Ferna´ndez-Rossier,* and J. J. PalaciosDepartamento de Fı´sica Aplicada, Universidad de Alicante, San Vicente del Raspeig, E-03690 Alicante, Spain(Received 13 November 2008; published 3 April 2009)By computing spin-polarized electronic transport across a finite zigzag graphene ribbon bridging twometallic graphene electrodes, we demonstrate, as a proof of principle, that devices featuring 100%magnetoresistance can be built entirely out of carbon. In the ground state a short zigzag ribbon is anantiferromagnetic insulator which, when connecting two metallic electrodes, acts as a tunnel barrier thatsuppresses the conductance. The application of a magnetic field makes the ribbon ferromagnetic andconductive, increasing dramatically the current between electrodes. We predict large magnetoresistance inthis system at liquid nitrogen temperature and 10 T or at liquid helium temperature and 300 G.DOI: 10.1103/PhysRevLett.102.136810 PACS numbers: 81.05.Uw, 73.20.r, 85.75.dThe term spintronics has been coined to refer to theinterplay between spin polarization and electrical conduc-tance and is one of the major themes today in condensedmatter and applied physics. A central concept in spin-tronics is that of giant magnetoresistance (GMR), discov-ered originally in layered structures alternating magneticand nonmagnetic transition metals [1]. The resistance ofthe whole structure undergoes a large increase when therelative orientation of the magnetization in adjacent layersgoes from parallel to antiparallel, provided that the non-magnetic layers are thinner than the spin relaxation length.This phenomenon represents the paradigm under whichcommercial devices such as magnetic reading heads oper-ate nowadays.Here we propose a new type of magnetoresistive devicewhich makes use of the remarkable electronic properties ofzigzag graphene ribbons [2,3]. Its operational principle issimilar to that found in conventional GMR layers, with thedifference that is entirely based on carbon. Our proposal ismotivated by the spectacular progress in the fabrication ofhigh mobility graphene based field effect transistors [4–6]and by the recent developments in the fabrication of gra-phene ribbons [7–10] with top-down techniques, as well asin the synthesis of graphene ribbons [11].The electronic structure of infinite graphene ribbons hasbeen studied thoroughly. Idealized graphene ribbons fallinto two categories: armchair and zigzag. Within the sim-plest one orbital tight-binding description [2], armchairribbons can be either semiconducting or metallic, depend-ing on their width, whereas in the case of zigzag ribbonstwo flat bands, associated with edge states, lie at the Fermienergy. These edge flat bands favor the appearance ofmagnetization on the edges when electron-electron repul-sion is included in the calculation, either with a Hubbardmodel [3,12,13] or with density functional theory (DFT)[14–16]. In the ground state the respective magnetizationdirection of the edges is antiparallel, and a gap opens in theband structure [12,14–16]. This is the ground state.Slightly above in energy, the parallel magnetic configura-tion is conducting. Application of either a magnetic field ora transverse electric field [15] can make the ferromagneticconfiguration more stable. Here we explore the former.Both for conceptual and practical reasons our proposal isbased on finite length L graphene ribbons. For L ¼ 1, thelong range order predicted by mean-field calculations isnot robust, due to the proliferation of spin wave excitationsof energy Ln, with n ¼ 1 or n ¼ 2 for both antiferromag-netic and ferromagnetic alignments, respectively [17]. Inshort ribbons, in contrast, there is a gap for spin waves.Quantum fluctuations between the manifold of equivalentmean-field ground states would not alter the conductingproperties of the system. Additionally, recent DFT calcu-lations [18] show that monohydrogenated zigzag ribbonsare not the most stable edge configuration unless the hydro-gen density is very small. Whereas this poses a severeproblem for the chemical stability of infinite monohydro-genated zigzag ribbons, short magnetic ribbons, as well asother magnetic polycyclic aromatic hydrocarbons [19],might be more stable.Here we show that the spin driven metal insulator tran-sition, predicted so far for infinite ribbons, is still present inFIG. 1 (color online). Atomic structure of the zigzag ribbonwith length Nx ¼ 12 and width Ny ¼ 6 attached to semi-infiniteelectrodes. The unit cell of a zigzag ribbon is highlighted.PRL 102, 136810 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 20090031-9007=09=102(13)=136810(4) 136810-1  2009 The American Physical Societyshort ribbons attached to conducting electrodes. Using awell-established methodology [20] extended to account forelectron-electron interactions in a Hubbard model, westudy both the magnetic and transport properties of asystem in which two conducting graphene electrodes arecoupled through a finite length graphene ribbon.Statement of the problem and methodology.—We studythe system depicted in Fig. 1: A short zigzag grapheneribbon attached to two conducting graphene electrodes. Inthe case pictured in the figure we have chosen the elec-trodes to be semi-infinite metallic armchair graphene rib-bons of width W. Both the device and the electrodes aredescribed with a Hubbard model for the z orbitals withfirst-neighbor hopping t and on site repulsionU, solved at acollinear mean-field level [3,12,13,21–23]. This approachis known to capture the main features of the ab initiocalculations, both for finite [21] and infinite [12] graphenesystems. Thus, the mean-field Hamiltonian readsH ¼ XI;I0;tcyIcI0 þUXIðnI;"hnI;#i þ nI;#hnI;"iÞ; (1)where cyI creates an electron at the z orbital of atom Iwith spin , nI; ¼ cyIcI is the occupation operator.Since the mean fields hnI;i depend on the eigenstates ofthe mean-field Hamiltonian, this defines a self-consistentproblem which is solved by numerical iteration.In this work we have to solve the self-consistent problemfor an infinite system without translational invariance. Thisis done using the partition method [20]. We split the systeminto three regions, the left and right electrodes and thecentral region. The Hamiltonian of Eq. (1) reads:H ¼H L þH R þH C þV LC þV RC; (2)where H L, H R and H C are the mean-field HubbardHamiltonians of the left, right, and central regions, respec-tively. On the other side, V LC and V RC describe thehopping between the central and the left and right regions,respectively. Sufficiently away from the central region, themean-field Hamiltonian of the electrodes is identical to thatof an infinite ribbon. In the case of the armchair electrodes,the effect of the Hubbard interactions in the charge neutral-ity point is a rigid shift of the bands without the appearanceof magnetic moment and keeping their metallic character.We first determine the surface Green function of the semi-infinite electrodes, gL and gR [20]. This requires the solu-tion of a self-consistent Dyson equation. Second, the Greenfunction of the central region is evaluated:GCðEÞ  ½EI H C LðEÞ RðEÞ1 (3)with LðRÞðEÞ ¼V LðRÞgLðRÞðEÞV yLðRÞ.This expression is a functional of the expectation valueshnI;i, through the central Hamiltonian H C. The Greenfunction yields the density of states projected over theorbital z with spin  sitting in the atom I in the device,is given by:ðE; I; Þ   1ImTrðhIjGCðEÞjIiÞ: (4)Here hIjGCðEÞjIi is a matrix of size 2N, with N beingthe number of atoms of the central region. In turn, theexpectation values of the spin density are given byhnI;i ¼Z EF1ðE; I; ÞdE: (5)The spin density in a given atom I is defined as mI ¼hnI;"ihnI;#i2 .Equations (3) and (5) define a self-consistent problemwhich is solved by numerical iteration. Within this formal-ism the Landauer conductance is given byG ¼ e2h TrTðEFÞwith:TðEÞ ¼ TrðLðEÞGCRðEÞGyCðEÞÞ (6)with LðRÞðEÞ ¼ i½LðRÞðEÞ  yLðRÞðEÞ.Results.—The size of the zigzag ribbons is defined bytwo integer numbers, Nx, the number of unit cells of theribbon, and Ny the number of zigzag rows in the unit cell.Thus, the length of the ribbon is L ¼ Nxa, with a ¼2:42 A and the width of the ribbon is given by W ¼ﬃﬃ3p2 Nya.00.20.40.60.81DOS([eV.UC]-1 )-0.4 -0.2 0 0.2 0.4E(eV)00.20.40.60.8DOS([eV.UC]-1 )-0.4 -0.2 0 0.2 0.4E(eV)1.5 2 2.5 3ka-1-0.500.51E-E F(eV)1.5 2 2.5 3ka(a) (b)(c) (d)(e) (f)FIG. 2 (color online). (a),(b) Bands for infinite ribbon withNy ¼ 12 with AF (a) and FM (b) ground states. (c),(d) Corresponding DOS of the AF (c) and FM (d) infiniteribbons. (e) DOS of the AF finite ribbon. (f) DOS of the FMfinite ribbon.PRL 102, 136810 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009136810-2As in the case of both isolated finite ribbons [24] andinfinite ribbons [15], we obtain two kind of solutions,ferromagnetic and antiferromagnetic, depending on theinitial guess in the iterative self-consistent procedure. Wesee how only the edge atoms of the zigzag ribbon aremagnetic and how the size of the edge moments is largeraway from the interface. In the case of the AF solution, thelargest local spin density ismI ¼ 0:13 forU ¼ t ¼ 2:7 eValmost identical to the result of infinite ribbons. In Fig. 2we compare the electronic structure of the finite size con-nected zigzag ribbons with the case of infinite ribbons, bothfor ferromagnetic (FM) and antiferromagnetic (AF) con-figurations. The electronic structure of the infinite ribbonsis calculated taking advantage of crystal invariance andmaking use of the Bloch theorem [12]. Whereas AF infiniteribbons have a gap 0 [12,14] and zero density of states(DOS) at EF (hereon set to zero), two bands cross EF forthe FM ribbons, resulting in a finite density of states at EF.The same trend is observed in finite ribbons. The DOS ofthe AF configurations, summing over the atoms of a unitcell (see Fig. 2), presents a pseudogap. The DOS becomesmore similar to the one of the infinite case both for longerribbons and for unit cells in the middle of the ribbon. Thesmall DOS inside the gap arises from the coupling to theelectrodes. In contrast to the AF solution, the DOS of theFM solutions, again summing over the atoms of a unit cell(see Fig. 2), is that of a conducting system.From the DOS we anticipate a strong dependence of theconductance on the magnetic configuration. This is con-firmed by our calculations. In Fig. 3 we show the conduc-tance curves for the Nx ¼ 12, Ny ¼ 6 ribbon. In the leftpanel we show how the transmission of the AF solution forE ¼ EF is highly reduced, compared to the FM solution,shown in the right panel. This is the central result of ourwork: The conductance is strongly dependent on the rela-tive orientation of the magnetization of the edges of thezigzag ribbon bridging the electrodes. In the ground statethe ribbon is antiferromagnetic and the conductance issmall or vanishes for sufficiently large Nx. In Fig. 4 weplot the conductance at EF, both for AF and FM configu-rations, as a function of the ribbon length L ¼ aNx for twowidths. The conductance of the FM state is always close to2e2=h, the value associated with the two bands, one perspin channel, that cross the Fermi energy. This conduc-tance value is sensitive to the actual form of the grapheneelectrodes, but the vanishingly small values for the AF stateare not.As in standard GMR devices, the application of a mag-netic field can force the ferromagnetic solution to be theground state, resulting in a dramatic increase of the con-ductance. We define the magnetoresistance (MR) of eachdevice asMR  RAF  RFMRAF þ RFM  100 ¼GFM GAFGFM þGAF  100; (7)where R ¼ 1G is the resistance and G ¼ e2@TðEFÞ is theconductance calculated with the Landauer formula. InFig. 4 we plot the MR as a function of the ribbon lengthfor ribbons of width Ny ¼ 6 and Ny ¼ 3. We see that,except for ribbons shorter than 1 nm, the MR saturates to100%. The origin of the MR proposed here is differentfrom that discussed by Brey and Fertig [25] and that ofKim and Kim [26] which require ferromagnetic electrodes.Our proposal is more similar to the original GMR experi-ments with current in the plane in exchanged coupledmultilayers [1].We now discuss the operational limits of our proposeddevice. There are three relevant magnitudes to be consid-ered. First, the energy difference between the AF and theFIG. 3 (color online). (a) Spin resolved transmission for theAF infinite ribbon with Nx ¼ 12 and Ny ¼ 6. (b) Same forthe FM solution. Insets: self-consistently calculated local spindensity.FIG. 4 (color online). (a) Conductance and (b) magneto-resistance [Eq. (7)] at the Fermi energy as a function of theribbon length for two ribbon widths, both for FM and AFconfigurations.PRL 102, 136810 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009136810-3FM states, which depends both on the width of the ribbonand the length:ðNx; NyÞ ’ Nx30Ny3=2eV: (8)This expression has been obtained from the fitting to DFTcalculations by Pisani et al. [16]. Second, the temperature-dependent spin correlation length ðTÞ, over which mag-netic order is lost [17]. Finally, the critical switchingmagnetic field is defined through the equationgBMTB ¼ Nx; (9)where MT¼PImI is the total spin density of the FM con-figuration, g¼2 for graphene and B¼0:058meVT1.Equation (9) neglects the dependence of mI on B [27].The temperature determines the minimal value minbelow which thermal fluctuations make the ribbon switchbetween the AF and the FM states. It also determines themaximal value for the length of the ribbon Lmax ¼ ðTÞabove which magnetic order is lost. By choosing L ¼Lmax, we guarantee the maximum possible width Wmaxfor the ribbon through Eq. (8). This, in turn, provides theminimum critical switching field B through Eq. (9). Forinstance, at room temperature, min  26 meV andð300Þ ¼ Lmax  0:8 nm. This yields Wmax  1:6 nmand B  200 T. For liquid nitrogen temperatures (75 K)min  6 meV, Lmax  4 nm, Wmax  12 nm, and B 10 T. At He liquid temperatures (4 K) min  0:35 meV,Lmax  80 nm, Wmax  600 nm, and B  0:03 T. Wenote that since ðTÞ< ð0Þ the actual values of B aresomewhat larger. Using the extrapolation formula ofSon et al. [14] for the ribbon transport gaps, a ribbon ofW  600 nm has a gap of 1.55 meV, which is still largerthan kT and standard bias voltages. In summary, there is atrade-off between the operational temperature and thecritical field. This trade-off is controlled by the spin corre-lation length. A high operational temperature requires aprohibitively large B. At liquid He temperatures, thecritical field is bounded within reasonable ranges easilyattainable in the laboratory.In conclusion, we propose an ultrasmall and chemicallysimple magnetoresistive device based on a zigzag ribbonjoining metallic graphene electrodes. The conductionproperties of this device change dramatically as the relativeorientation of the magnetic edges of the ribbon go fromparallel to antiparallel relative orientations upon applica-tion of a magnetic field. From a more fundamental point ofview, low-temperature experiments showing a drasticchange in the resistance on applying strong magnetic fieldswould signal the existence of magnetism in zigzag gra-phene ribbons.We acknowledge fruitful discussions with D. Sorianoand A. S. Nu´n˜ez. This work has been financially supportedby MEC-Spain (Grants No. MAT07-67845, No. MAT07-31099-E, and CONSOLIDER CSD2007-0010).*jfrossier@ua.es[1] M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988);G. Binasch, P. Grunberg, F. 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Louie, Phys. Rev. Lett.97, 216803 (2006).[15] Y.-W. Son, M. L. Cohen, and S.G. Louie, Nature (London)444, 347 (2006).[16] L. Pisani, J. A. Chan, B. Montanari, and N.M. Harrison,Phys. Rev. B 75, 064418 (2007).[17] O. V. Yazyev and M. I. Katsnelson, Phys. Rev. Lett. 100,047209 (2008).[18] T. Wassmann et al., Phys. Rev. Lett. 101, 096402 (2008).[19] J. Wu, W. Pisula, and K. Mullen, Chem. Rev. 107, 718(2007).[20] F. Mun˜oz-Rojas, D. Jacob, J. Ferna´ndez-Rossier, and J. J.Palacios, Phys. Rev. B 74, 195417 (2006).[21] J. Ferna´ndez-Rossier and J. J. Palacios, Phys. Rev. Lett.99, 177204 (2007).[22] J. J. Palacios, J. Ferna´ndez-Rossier, and L. Brey, Phys.Rev. B 77, 195428 (2008).[23] O. V. Yazyev, Phys. Rev. Lett. 101, 037203 (2008).[24] O. Hod, V. Barone, and G. E. Scuseria, Phys. Rev. B 77,035411 (2008).[25] L. Brey and H.A. Fertig, Phys. Rev. B 76, 205435 (2007).[26] W.Y. Kim and K. S. Kim, Nature Nanotech. 3, 408 (2008).[27] This is an excellent approximation, as expected fromRef. [28] where the single-particle diamagnetic responseof zigzag ribbons is calculated and found very small forthe ribbons and fields considered here. Additionally, ourown self-consistent calculations for narrow infinite rib-bons, including the orbital coupling to the field, show thatthe magnetic moment M does not depend on B.[28] K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phys.Rev. B 59, 8271 (1999).PRL 102, 136810 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009136810-4

Giant magnetoresistance in ultrasmall graphene based devices

Giant magnetoresistance in ultra-small Graphene based devices

Giant magnetoresistance in ultra-small Graphene based devices

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