Interplay between sublattice and spin symmetry breaking in graphene D. Soriano (1,3), J. Ferna´ndez-Rossier (2,3) (1) CIN2 (ICN-CSIC) and Universitat Autonoma de Barcelona, Catalan Institute of Nanotechnology, Campus UAB, 08193 Bellaterra (Barcelona), Spain (2) International Iberian Nanotechnology Laboratory, Av. Mestre Jos´e Veiga, 4715-330 Braga, Portugal (3) Departamento de F`ısica Aplicada, Universidad de Alicante, San Vicente del Raspeig, Spain (Dated: January 18, 2012) 2 We study the effect of sublattice symmetry breaking on the electronic, magnetic and transport 1 properties of two dimensional graphene as well as zigzag terminated one and zero dimensional 0 graphene nanostructures. The systems are described with the Hubbard model within the collinear 2 mean field approximation. We prove that for the non-interacting bipartite lattice with unequal n number of atoms in each sublattice midgap states still exist in the presence of a staggered on-site a potential±∆/2. Wecomputethephasediagramofboth2Dand1Dgraphenewithzigzagedges,at J half-filling,definedbythenormalizedinteractionstrengthU/tand∆/t,wheretisthefirstneighbor 7 hopping. Inthecaseof2DwefindthatthesystemisalwaysinsulatingandwefindtheUc(∆)curve 1 above which the system goes antiferromagnetic. In 1D we find that the system undergoes a phase transitionfromnon-magneticinsulatorforU <Uc(∆)toaphasewithferromagneticedgeorderand ] antiferromagnetic inter-edgecoupling. Theconduction propertiesof themagnetic phasedependon l l ∆ and can be insulating, conducting and even half-metallic, yet the total magnetic moment in the a system is zero. We compute the transport properties of a heterojunction with two non-magnetic h - graphene ribbon electrodes connected to a finite length armchair ribbon and we find a strong spin s filtereffect. e m PACSnumbers: . t a I. INTRODUCTION consideredwithinthemeanfieldHubbardmodel14. Den- m sity functional calculations confirmed the scenario16,17, - showing that the long-range Coulomb interactions and d Themostsalientelectronicpropertiesofgrapheneand n the other atomic orbitals, absent in the Hubbard model, its nanostructures are linked to the bipartite nature of o do not play a major role in this system. Both the mean the honeycomblattice whichis formedbytwointerpene- c field Hubbard model14,15,18–20,22 and DFT calculations [ trating identical triangular sublattices1. It is customary 2 to refer to the sublattice as a pseudospin degree of free- sghaopw, wthhaicththoepemnasgdnueetictophinatseer-weditghezceorroretolattailonsps1in9.haTshae dom. In this language, the first neighbor hopping is de- v fabrication of graphene ribbons with ultrasmooth edges scribedinterms ofa pseudo-spinflipoperator,whichre- 4 is now possible by unzipping carbon nanotubes23–25. In- sultsinthewellstudiedelectron-holesymmetricbandsin 3 direct evidence of magnetic order in the edges of zigzag 3 graphene, whose wave-functions are sublattice unpolar- ribbons is providedby ScanningTunneling Spectroscopy 6 ized. Thepseudospinsymmetrybecomesachiralsymme- (STS) that can be accounted for within the mean field . tryinthecontinuumlimitinwhichelectronsingraphene 2 Hubbard model26. 1 aredescribedwithaDiracHamiltonian2andaccountsfor 1 thelackofbackscattering3,thesocalledchiraltunneling4 The sublattice degree of freedom plays a central role 1 and the absence of an energy gap in two dimensional inthemagneticpropertiesofbipartitelattices27–29. Very v: graphene. muchlikeanexternalmagneticfieldfavorsonespinorien- i Sublattice symmetry breaking in graphene could tationandsplitsthespinstates,anexternalperturbation X arise spontaneously, due to some electronic phase favorsonesublatticewithrespecttothe otherandopens ar transition5–7,orduetothe couplingofgraphenetosome a gap in the band structure of graphene2,30. When this substrate,likeSiliconCarbide8,9 andBoronNitride10–12. happens, it is not obvious a priori what happens to the Sublattice symmetry breaking would make it energeti- edgestates,evenatthesingleparticlelevel,andtheasso- cally favorable for the electrons to stay in one of the ciatedmagnetism. DFTcalculationsindicatethatBoron sublattices, resulting in pseudo-spin order (either spon- Nitride zigzag ribbons with the edge atoms passivated taneous, or induced). The purpose of this work is to with hydrogen are non-magnetic31,32 whereas graphene understand the interplay between induced pseudo-spin ribbons, deposited on Boron Nitride (BN) whose lattice order and real spin order in graphene. Magnetic order parameter is shifted to match graphene , indicate that is expected to take place in monohidrogenated graphene edgemagnetismsurvives33. TheseDFTcalculationssug- zigzag edges. Within the standard one-orbital tight- gest that as the sublattice symmetry breaking potential bindingmodelofgraphene,theseedgesgiverisetoalarge ∆increases,aphasetransitionmustoccurfrommagnetic density of states at the Fermi energy13 which is prone to tonon-magneticedges. Hereweaddressthisproblemus- a ferromagnetic inestability when Coulomb repulsion is ingamuchsimplerdescriptionoftheelectron-electronin- 2 teractions,namely, the mean field approximationfor the Thus, the chiral symmetry ensures that the spectrum of Hubbardmodel,inthespiritofearlierworkforgraphene h has electron hole symmetry. In addition, since ψ~ and 0 withthe fullsublattice symmetryandonrecentworkfor φ~ areeigenvectorsofthesameHamiltonianwithdifferent graphene zigzag ribbons without inversion symmetry34. eigenvalues they must be orthogonal. This leads to: The rest of this paper is organized as follows. In sec- tion II we present some general theorems regarding the 0=ψ~∗·φ~ =hψ~|σ |ψ~i (4) z propertiesofthesingleparticlestatesofthetight-binding model for graphene with a staggered potential. In sec- or more explicitly tion III we study the interacting model for the case of two dimensional graphene and study how the staggered 0= |ψA(i)|2− |ψB(j)|2 (5) potential affects the non-magnetic to antiferromagnetic Xi∈A jX∈B transition. In section IV we study the interplay of mag- Thus, this lead us directly that eigenstates of h have neticandpseudo-spinorderinthecaseofzigzagribbons. 0 equal total weight on the two sublattices. In a pseu- We find that magnetic order and sub lattice symmetry dospin language, they have a zero expectation value of breaking give rise to spin polarization of the bands, and the σ pseudospin operator, since the Hamiltonian has insomeinstanceswefindhalf-metallicantiferromagnetic z the pseudomagneticfield(the hopping)inthe x,y plane. order. The spin filter properties of this case are stud- Wenowturnourattentiontotheeigenstatesofatight- ied in section V, where we consider quantum transport binding hamiltonian with first neighbour hoppings de- between two half-metallic zigzag ribbons separated by a fined in a bipartite lattice with a sublattice-dependent non-magnetic armchair central region. In section VI we potential which is both homogeneous and traceless, as summarize our main findings. defined by equation (1). We are going to show that they also have electron-hole symmetry. For that matter, we II. SINGLE PARTICLE STATES OF A represent H0 in the the subspace defined for a pair of BIPARTITE LATTICE WITH A STAGGERED eigenstates of h , ψ~ and φ~ = σ ψ~, with energies E and 0 z POTENTIAL −E respectively. We readily obtain E 0 ∆ 0 1 In this section we consider some quite general proper- H = + (6) ties of the single particle states of the Hamiltonian of a 0 (cid:18) 0 −E (cid:19) 2 (cid:18)1 0(cid:19) bipartitesub-latticewithaconstantsublatticesymmetry breaking term: whose eigenvalues are ǫ ≡ ± E2+ ∆2 with corre- ± 4 q 0 h ∆ 1 0 sponding eigenvectors~v± given by: H = AB + =h +V (1) 0 (cid:18)hBA 0 (cid:19) 2 (cid:18) 0 −1(cid:19) 0 θ θ wherehAB,hBA and1arematriceswithdimensiongiven ~v+ =Cos2ψ~+Sin2φ~ (7) by the number of atoms in sublattice A and B. and θ θ A. Null sublattice imbalance ~v =Sin ψ~−Cos φ~ (8) − 2 2 We consider first the case of a bipartite lattice with- where Cosθ = E . Thus, if we know (half of) the out sublattice imbalance, so that the number of atoms qE2+∆42 in sublattice A equals those in lattice B: N = N . spectrum and the eigenstate of the sublattice symmetric A B In that case, hAB, hBA and 1 are all matrices of range problem h0, we can easily build the spectrum and the N = N . It can be easily seen that the sub-lattice eigenfunctions for the same lattice when a homogeneous A B symmetrichamiltonian,h (orunperturbedhamiltonian) traceless sublattice Zeeman term is added to the Hamil- 0 anti-conmuteswiththesublatticeimbalanceoperatorσ : tonian. z 0 h 1 0 [h ,σ ] ≡ AB , =0 (2) 0 z + (cid:20)(cid:18) hBA 0 (cid:19) (cid:18)0 −1(cid:19)(cid:21)+ B. System with sublattice imbalance Since σz2 = 1, it is said that the graphene Hamiltonian Theresultsoftheprevioussectionneedtobeexamined has a chiral symmetry. As a result, if ψ~ ≡ ψ~A is an with care in the special case that E = 0. This certainly (cid:18)ψ~B (cid:19) happenswhenweconsiderasystemwithNA =NB+NZ, eigenstate of h0 with energy E, we automatically have where NZ > 0 is a positive integer. In that case the that φ~ ≡σ ψ~ is also eigenstate with energy −E dimension of the A and B subspaces is not the same z and the results of the previous section do not hold in hψ~ =Eψ~, →hφ~ =−Eφ~ (3) general35. In particular, it has been shown that h has 0 3 33 III. ELECTRONIC PROPERTIES OF TWO DIMENSIONAL GRAPHENE WITH A STAGGERED POTENTIAL 22 x2 We now study the interplay between Coulomb repul- 11 x3 sion and sublattice symmetry breaking. We model the interaction using a Hubbard model in the mean field t approximation. For symmetric graphene this approxi- / 00 E mation is known to predict a phase transition from the non-magneticgaplessstate toanantiferromagneticinsu- --11 x3 lating state when37 U > Uc = 2.2t. As usual, the mean field approximation underestimates the critical U nec- essary for the Mott transition. Quantum Monte Carlo --22 calculations indicate37 that the transition takes place at x2 U ≃5.3t. Inaddition,recentworkindicates thatmight c be a third phase with spin-liquid properties separating --33 --33 --22 --11 00 11 22 33 the non-magnetic state from the magnetically ordered ∆/t phase38. In spite of its limitations, the mean field de- scription of the Hubbard model can shed some light on the possible ordered phases and their electronic proper- FIG. 1: (Color online). (a) Symbols: energy levels for tri- ties. angulenewithN =13atomscalculatedbydiagonalization of the single-particle model. Lines: energy levels obtained from equation (9). The degeneracies are indicated in thefigure. A. Hubbard model and a mean field approximation NZ eigenstates ψ~Z with E = 0 that are are sublattice The extended Hubbard model reads: ~z polarized in the majority sublattice, ψ~ = A . This ∆ are the so called midgapstates and play a(cid:18)cru0cia(cid:19)lrole in H =tXii′,sc†isci′s+ 2 Xi,s τz(i)c†iscis+ the emergence of magnetism in graphene zigzag edges14 and graphene with chemisorbed hydrogen36. +U ni↑ni↓ =H0+U ni↑ni↓ (10) It can be inmediately seen that if ψ~ is a zero energy Xi Xi Z eigenstate of h it is also an eigenstate of H h +V with = 0 where c† creates an electron in atomic site i with spin eigenvalue ǫ = +∆. Conversely, if h presents a zero is 2 0 s =↑,↓, i′ stand for the first neighbors of i, τ (i) = +1 energy state sublattice polarized in B, then that state is z if i belongs to the A sublattice and −1 otherwise. We also eigenvector of H with energy −∆. 0 2 only consider the half-filling case, where the number of Thus,wecannowpredicttheevolutionofthespectrum electrons equals the number of sites in the lattice. For of a given system described by H that, at ∆ = 0 has 0 a given filling, the ground state properties of the model midgap states at zero energy as well as pairs of electron- depend on two dimensionless parameters ∆/t and U/t. holesymmetricstateswithfiniteenergies±E . Themid n We explore the properties for a spin-collinear mean field gapstates will split with energy ±∆, depending ontheir approximation, where the U term is approximated by: sublattice polarization, and the finite energy states will evolve as V =+U n hn i+hn in (11) MF i↑ i↓ i↑ i↓ ∆2 Xi ǫ (∆)=± E2 + . (9) ± r n 4 where hn i is the average of the occupation operator of is siteiwithspins,calculatedwiththemany-bodyground In order to illustrate this result, we have computed the state of the mean field Hamiltonian: single particle spectrum of a triangulene29 with N = A 7 and N = 6 atoms. The evolution of the spectrum B hn i= f hα|n |αi (12) is α is as a function of ∆ is shown in figure(1). We compare Xα the result of the numerical diagonalization with those extrapolated from the spectrum of h0 and, expectedly, where fα = 0,1 is the occupation of the single particle find perfect agreement. states |αi that diagonalize calculated with the ground Thiscalculationshowsthat,instructureswithalarger state of the mean field Hamiltonian H +V . Since 0 MF number of midgap states, the midgap shell will remain the potential V depends on the eigenstates of H + MF half-full (when counting the spin), and interactions are V , both the potential and the eigenstates need to be MF expected to favor large spin configurations. computed self-consistently. We do this by iteration. 4 B. Mean field approximation for 2D graphene We now describe the electronic properties of Hubbard model for the two dimensional honeycomb lattice with a staggered potential within the mean field approxima- tion. In this case, we can take a minimal unit cell with 2 atoms, A and B and assume that the mean field in all unit cells is identical, which permits to use Bloch theo- remto representthe meanfieldHamiltonianin the basis set A↑,B ↑,A↓,B ↓: H 0 H = ↑ (13) (cid:18) 0 H↓ (cid:19) FIG. 2: (Color online). Phase diagrams for 2D-graphene (left) and a zigzag graphene nanoribbon with N =48 atoms where each element is a 2 by 2 matrix: in the unit cell (right) with stagger potential (∆) using a mean field Hubbard model at half filling. The dark region H = ∆2 +UhnA↓i f(~k) (14) with∆MIT(U)>∆(U)>∆c(U)correspondtothespinhalf- ↑ (cid:18) f∗(~k) −∆ +Uhn i(cid:19) metallic phase in the graphene ribbon. In the case of 2D- 2 B↓ graphene,thisregion isreducedintoasinglecriticallinesep- and aratingnon-magneticandantiferromagneticinsulatingstates. ∆ +Uhn i f(~k) H = 2 A↑ (15) ↓ (cid:18) f∗(~k) −∆ +Uhn i(cid:19) 2 B↑ and f(k)=t 1+ei~k·~a1 +ei~k·~a2 (16) (cid:16) (cid:17) accounts for the first neighbour particle hopping. In our numerical determination of the self consistent occupa- tions hn i we have taken an unit cell of 4 atoms. We is have verified that our mean field solutions in this ex- tended unit cell do not present inter-cell modulations of the charge density. We have explored the phase diagram defined by U/t and ∆/t and we find 3 types of solution, shown in figure (2): FIG.3: (Coloronline). Bandstructureof2D-graphenewith 1. For U < U (∆) the system is non magnetic and, stagger potential (∆) in the mean field extended Hubbard c except for ∆ = 0, a band insulator. The case of approximation for U >Uc(∆) ∆=0andU <U isthewellstudiedparamagnetic c semimetal phase. insulator transition. Given the large values of U (∆)/t c 2. For U > Uc(∆) the system is an antiferromag- thisorderedelectronicphaseisnotexpectedingraphene. netic insulator. The lack of inversion symmetry The predictions of this theory should be tested in cold produced by the sublattice symmetry breaking re- atomicgasesconfinedinopticallattices39orinartificially sults in a splitting of the spin bands, in contrast paternedhoneycomblatticesintwodimensionalelectron with the standard ∆ = 0 case. This is shown in gases40 figure (3) 3. For U = U (∆) the system is a half-semimetallic c IV. ELECTRONIC PROPERTIES OF antiferromagnet. For one spin channel the system GRAPHENE ZIGZAG RIBBONS WITH A is insulating and for the other is semimetallic. STAGGERED POTENTIAL It is apparent that, as ∆ increases, the critical U in- c creases. Expectedly, the magnetic orderhas to overcome We now study the case of zigzag graphene ribbons for the single-particle gap opened by the staggered poten- which we find that magnetic order could happen at low tial. Interestingly,themeanfieldapproximationdescribe valuesofU/tevenforfinite ∆. The widthofthe ribbons a magnetic transition between two insulating states, the is characterized by N, the number of atoms in the unit non-magnetic insulator and the antiferromagnetic insu- cell. Importantly, one of the edges is formed with A lating phase, which can be interpreted as an excitonic atomsonly,theotherbeingmadeofB atomsonly. Thus, 5 pseudospin polarization implies charge accumulation in 0.4 0.4 one edge and depletion in the other, ie, the formation of (a) (b) an electric dipole. 0.2 0.2 The electronic structure of graphene ribbons has been widelystudiedinthe∆=0limit,bothfortheU =013,41 k)/t 0 k)/t 0 andthefiniteU cases14–20,22,43. Themostprominentfea- E( E( tureoftheirelectronicstructureisgivenbytheflatbands -0.2 -0.2 associatedtoedgestates. At∆=U =0,thesebandsare locatedattheFermienergy,givingrisetoalargedensity -0.4 -0.4 0 2 4 6 0 2 4 6 ofstates atthe Fermienergy. Not surprisingly,Coulomb ka ka repulsionresultsinamagneticinestability14 correspond- 0.4 0.4 ing to the formation of magnetic moments in both edges (c) (d) while the bulk-atoms remainalmostspin unpolarized. It 0.2 0.2 turnsoutthattheinter-edgespincorrelationsareantifer- romagnetic,as expected fromthe Lieb theorem27. Thus, k)/t 0 k)/t 0 foragivenspinorientation,thereischargeaccumulation E( E( in one of the edges and chargedepletion in the opposite. -0.2 -0.2 This results in a spin-resolved pseudo-spin polarization, or spin-dipole19. Here we are interested in the interplay -0.4 -0.4 between pseudo-spin polarization, driven by the ∆ term 0 2 4 6 0 2 4 6 in the Hamiltonian, and the spin polarization,which en- ka ka tails a spin-resolved pseudo-spin polarization, driven by the Coulomb repulsion U. FIG.4: (Coloronline). Electronicstructureofzigzagribbons with U =0 , for ∆= 0 and ∆ =0.2t (left and right panels) for2differentribbonwidths,N =40(top)and,N =80(bot- tom). Forthesakeofclarity,weonlyplotthe2higherenergy A. Non interacting bands valencebands and the2 lowest energy conduction bands. We firstreviewthe effectofthe staggeredpotentialon the non-interacting bands, studied by Qiao et al.45. At merically found solutions present magnetization at both ∆=0twoalmostflatbands,associatedtoedgestates,lie edges equal in magnitude and opposite in sign. Thus, atthe Fermienergy. As ∆ becomesfinite (andpositive), there are two equivalent ground states: m =−m >0 A B the bands at B edge are red-shifted and those at B edge and m = −m < 0. The corresponding energy bands A B areblue-shifted,resultinginaband-gapopening. Thisis for the ribbon with N = 48, and ∆ = 0 and U = t seeninfigure(4)fortworibbonswithN =40andN =80 are shown in figure (5a). The magnetic order results in atoms, for ∆ = 0 (left columns) and ∆ = 0.2t (right a band-gap opening. The spin ↑ and ↓ bands are de- column). We also notice the low energy bands are quite generate. The magnetic moment at the edge atoms is similar for the N =40 and N =80 ribbons, whereas the m = ±0.13. The charge per atom is the same all over gap between the higher energy bands is reduced for the the unit cell, 1 electron per atom. widerribbon. Thisisconsistentwiththefactthatlowest When the sublattice-symmetry breaking potential is energybandsareedgestates,relativelyinsensitivetothe finiteandbelowacriticalvalue∆ (U),westillfindmag- c widthoftheribbon,incontrastwithhigherenergybands netic orderinthe edgeswithantiferromagneticcoupling, made of quantum confined bulk states41. and zero total moment, even if the charge is no longer Thus, for finite ∆ and U = 0 graphene zigzag rib- thesameforbothedges. Theelectronicpropertiesofthe bons are band insulators with pseudospin polarization magneticribbonwithpseudo-spinpolarizationarediffer- thatfeaturestwoflatbandscorrespondingtothe highest ent on several counts. First, the bands are spin split, as occupied and lowest un-occupied bands. For ∆ >0, the a natural consequence of the lack of both time-reversal bandscorrespondingtoboth↑and↓spinsinthe B edge and inversion symmetries. The evolution of the energy are occupied, whereas those in edge A are empty. As we bands, as we increase ∆, is shown in figure (5) for the shownow,thesebandsarepronetomagneticinestability, N = 48 ribbon with U = t. This figure can be under- not-unlike in the case with ∆=0. stood as follows. The solution has m = −m > 0 so A B that the ↑ band is occupied (empty) in the A (B) edge. Conversely,the ↓ bandis occupied (empty) in the B (A) B. Effect of Coulomb repulsion edge. As∆isturnedon,theB bandsarered-shiftedand A bands are blue shifted. For the ↑ bands, this implies We study now the interplay between pseudospin po- thatthebandgapcloses,sincethevalenceAbandsmove larization and Coulomb repulsion. For that matter, we upwards and the conduction B bands move downwards. use againthe mean field approximationfor the Hubbard Conversely,the gap opens in the ↓ channel. model, as described in previous work14,15,19,20,22,34. Nu- As shown in figure (6) and discussed below, as ∆ in- 6 0.2 n0.15 pi s e 0.1 g d E0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 n o B edge ati1.2 p u c 1 c o ge 0.8 A edge d e 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.2 0.15 p/t a 0.1 g 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ∆/t FIG. 6: (Color online). Properties of N = 48 ribbon with U = t as a function of ∆. Top panel: edge magnetization. Middle panel: edge charges. Bottom panel: gap for spin ↑ FIG. 5: (Color online). Lowest energy bands for for zigzag and ↓. ribbonwithN =48atomsinunitcellandU =t,fordifferent values of ∆: (a) ∆ = 0, (b) ∆ = 0.05t, (c) ∆ = 0.2t (d) ∆=0.3t. Onlythe 2 highest energy occupied bandsand the 2 lowest energy empty bands, per spin channel, are shown. sults are very similar for ribbons with different widths. Blue (red) stands for ↑ (↓) bands. In contrast with the 2D case, for ∆ = 0 the critical U for the edge is zero. This makes zigzag graphene rib- bons suitable systems for the observation of magnetism creases the magnetic moment at the edges are depleted in graphene and the possible effect of sublattice sym- and, eventually disappear when ∆ > ∆ (U). Remark- metry breaking more relevant. Expectedly, the critical c ably, the gap in the ↑ channel closes for ∆ (U) < ∆ (U) is an increasing function of U, or in other words: MIT c ∆ (U), yet the gap is finite in the ↓ channel. Thus, the larger the single particle gap, the strongest the in- c the combination of pseudo-spin polarization and antifer- teraction U required to drive the magnetic inestability. romagnetic order makes the system a half-metallic anti- Spin polarization requires promotion of electrons across ferromagnetinaregionofthe∆,U phasespace(seedark- thesingleparticlegap,fromtheoccupiedB totheempty violet regionin figure (2)right panel). Notice that this is A edge, ie, the formation of a magnetic exciton conden- different from the ferromagnetic half-metallic phase pre- sate. The difference with the ∆ = 0 case stands on the dicted for graphene ribbons in the presence of a trans- size of the the single particle gap, which is vanishingly verse electric field17, for which the total magnetic mo- small (but not zero) in finite width ribbons19. Interest- ment is different from zero. ingly,themagneticexcitoncondensationscenarioalready Finally,wenote thatthe gapofthe non-magneticcase takesplaceinthecaseapparentlyconducting∆=U =0 in figure (5)d for ∆ > ∆ is significantly smaller than ribbon, when U is turned on. c ∆. This is due to the renormalization of the bands due Forafixedvalueof∆,asthevalueofU isincreasedthe to Coulomb repulsion. Basically, the occupied bands are systemundergoesaphasetransitionfromanon-magnetic blue shifted with respect to the empty bands, reducing band insulator, to a half-metallic antiferromagnet and the size of the gap. then to an insulating antiferromagnet. The fact that interactions can drive the system from insulating from metallicisquiteexoticanddiffersfromtheusualMottin- C. Phase Diagram sulatorscenario,inwhichinteractionsdriveabandmetal insulating. In the phase diagram we mark the insulator In figure (2) we show the phase diagram defined by to metal transition when the smallest energy gap, at a U/tand∆/t foraribbonwithN =48atoms,calculated given spin channel, is 100 times smaller than the gap at within the mean field approximation at half-filling. Ear- ∆ = 0. As shown in figure (6) for U = t, the gap at lierwork34hasaddressedthephasediagramdefinedby∆ ∆ = 0 is 0.07t. Thus, for that particular value of U, we andtheelectrondensity. The diagraminfigure(2)has2 declare the system conducting when the gap is 7 10−4t. phases regarding the magnetic order: non-magnetic, for Variationsuponthiscriteriayieldquantitativechangesin ∆ > ∆ (U) and antiferromagnetic otherwise. The re- themetaltoinsulatortransitionline,butitisalwaysthe c 7 case that ∆ (U) runs along and below the magnetic trices containing the information about the coupling of MIT phase transition line ∆ (U). the central region to the leads. c In Fig. 8(a-f), we have computed the spin-polarized conductance for the three systems shown in Fig. 7. The V. GRAPHENE ZIGZAG RIBBONS WITH top and middle panels correspond to the cases with and STAGGERED POTENTIAL AS AN IDEAL SPIN withoutstaggerpotentialrespectively. Thebottompanel INJECTOR of Fig. 8 shows the conductance polarization P = G − ↑ G /G +G ×100at different energies with (violet line) ↓ ↑ ↓ Interestingly, the predicted conducting phase for and without (green line) stagger potential. ∆ (U) < ∆ < ∆ (U) is a half-metallic antiferromag- The results obtained for the spin conductance can be MIT c net. In this section we study the spin transport proper- easily inferred by looking at the band structures of the ties of a tunnel junction where the electrodes are made three regions in Fig. 7. The two antiferromagnetic cases of such half-metallic antiferromagnets and the barrier is without stagger show a large gap in the conductance made of semiconducting armchair graphene ribbon. due to the semiconducting behavior of the three regions. In figure(7) we consider three possible situations, all For the ferromagnetic case, there is a finite conductance of them with ∆ = 0.25t , U = t. The first and second near the Fermi energy due to the metallic nature of the cases feature two antiferromagnetic electrodes with mu- electrodes. In this case, the evanescent modes coming tually parallel and antiparallel magnetizations, respec- from both electrodes penetrate into the central region tively. Basedonthe bandstructuresoftheseinfinite rib- and overlap due to the short length of the armchair rib- bons, shownin figure, we expect the tunnel conductance bon. tobecompletelydepletedwhenthemagneticmomentsof When both electrodes are antiferromagnetic with mu- the different electrodes are anti-parallel. In the last case tuallyparallelmagnetization,thesystemtransformsinto we consider two ferromagnetic electrodes. The bands of a spin-half metal where both valence and conduction the infinite ribbon with ferromagnetic coupling between bands show the same spin polarization. In this case, the themagneticedgesreflecttheconductingandspinunpo- polarizationoftheconductanceisthesameatbothsides larized character of the system at the Fermi energy (set of the Fermi energy. The case featuring antiferromag- at E=0 eV in the three cases studied). neticleadswithmutuallyantiparallelmagnetizationalso The conductance of the system is calculated within shows a spin half-metallic phase in both leads but with the LandauerformalismandtheGreen’sfuntionmethod opposite spin polarization around the Fermi level. Thus in a system consisting of an armchair semiconducting the spin polarized current injected from one electrode is graphene nanoribbon with an on-site Coulomb potential always reflected by the opposite electrode resulting in a connectedtotwospin-polarizedzigzagnanoribbonswith zero conductance around the Fermi energy. If both elec- both stagger and on-site Coulomb potentials, as those trodesarecoupledferromagneticallythesystembecomes studied in the previous section. Except for the atoms metallic and the valence and conduction bands cross at at the interface with the zigzag ribbons, the Hubbard the Fermi level but with a different spin polarization. U term is not able to spin-polarized the amrchair cen- This makes the conductance polarization to change sign tral region, as shown in the figure (7). The short length at each side of the Fermi level. of the tunneling barrier justifies also the neglect of spin relaxation which are expected in longer samples. To compute the transmission function T(E) along the VI. SUMMARY AND CONCLUSIONS armchair nanoribbon we adopt a partitioning method as implemented in the ALACANT(Ant.U)48 transport Our main findings can be summarized as follows: package,wherethesystemisdividedintothreeparts20,21, 1. We have shown that for a bipartite lattice with a namely, the central (C) part which consists on the arm- site-independent pseudo-spin Zeeman term the re- chair ribbon connected to a small part of the electrodes, sulting spectrum has electron hole symmetry, ex- and the left and right semi-infinite staggered zigzag cept for mid-gap states that arise in lattices with nanoribbons (L and R). a different number of sites in the two sublattices. The transmission probability can then be obtained from the Caroli expression49 We haveshownthat, ifthe ∆=0 Hamiltonianhas aneigenstateφ~ withenergyE,ithasalsoaneigen- T (E)=Tr[G†(E)Γ (E)G (E)Γ (E)] (17) state ψ~ with energy −E and the Hamiltonian with σ C R C L σ finite ∆ has two eigenstates, linear combination of where GC,σ(E)=[zI−HC,σ−ΣR,σ(E)−ΣL,σ(E)]−1 is φ~ and ψ~ with energies ± ∆2 +E2 the Green’sfunctionofthe centralregionwhichcontains q 4 allthe informationconcerningthe electronicstructureof 2. Ifφ~localizedintheA(B)sublatticeisaneigenstate thesemi-infiniteleadsthroughtheself-energies(Σ (E) R,σ with energy E = 0 of the ∆ = 0 Hamiltonian, and Σ (E)), and Γ (E) = i[Σ (E) − Σ† (E)], L,σ R,σ L,σ L,σ thenitisalsoaneigenstateoftheHamiltonianwith Γ (E) = i[Σ (E)−Σ† (E)] are the coupling ma- finite ∆ and energy ∆ (−∆) L,σ R,σ R,σ 8 (a) (b) (c) 0.8 h) 2e/ G (0.4 0 (d) (e) (f) 0.8 h) 2e/ G (0.4 0 80 40 %) 0 P( -40 -80 -0.4-0.2 0 0.2 0.4-0.4-0.2 0 0.2 0.4-0.4-0.2 0 0.2 0.4 Energy (eV) Energy (eV) Energy (eV) FIG. 8: (Color online) Spin conductance through a finite armchair ribbon connected to two spin-polarized zigzag rib- bonswith(a-c)andwithout(d-f)staggerpotential. Thebot- tom panel shows the spin conductance polarization for each magnetic ordering at the electrodes in the presence (violet) and absence (green) of stagger potential. 3. Twodimensionalgraphenewithfinite∆undergoes atransitiontoanantiferromagneticstatewithspin- split bands, for U > U (∆). The critical U is an c c increasingfunctionof∆. Atthetransitionbetween thenon-magneticinsulatingstate(U <U (∆))and c the magnetic state U > U , the system is a half c metallic antiferromagnet. 4. Zigzag graphene ribbons with a finite sublat- tice symmetry-breaking potential below a criti- cal value (∆(U) < ∆ (U)) can undergo a transi- c tionfromnon-magneticinsulatorstospin-polarized half-metallic antiferromagnets in the presence of Coulomb interaction. The fact that interactions can drive the system from insulating to metallic differs from the usual Mott insulator scenario in which interactions drive a band metal insulating. 5. Zigzag graphene ribbons with stagger potential ∆ slightly below the ∆ (U) are predicted to be ideal c spin injectors. The spintransportcalculationscar- riedoutinthisworkshowhighspinpolarizationof the conductance (≈ 100%) around the Fermi level when used as spin injectors in a tunnel junction. 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