Pinning and switching of magnetic moments in bilayer graphene
Eduardo V. Castro, María P. López-Sancho, María A. H. Vozmediano
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Pinning and swit hing of magneti moments in bilayergraphene Eduardo V Castro , , M P López-San ho , and M A H Vozmediano Instituto de Cien ia de Materiales de Madrid, CSIC, Cantoblan o, E-28049 Madrid, Spain Centro de Físi a do Porto, Rua do Campo Alegre 687, P-4169-007 Porto, PortugalE-mail: ev astroi mm. si .es, pilari mm. si .es, vozmedianoi mm. si .esAbstra t. We examine the magneti properties of the lo alized states indu ed by latti eva an ies in bilayer graphene with an unrestri ted Hartree-Fo k al ulation. We show thatwith realisti values of the parameters and for experimentally a essible gate voltages we anhave a magneti swit hing between an unpolarized and a fully polarized system.Keywords: Bilayer graphene, magneti momentsPACS numbers: 75.30.-m, 75.70.Ak, 75.75.+a, 81.05.Uw Pinning and swit hing of magneti moments in bilayer graphene1. Introdu tionAfter the explosion of publi ations on graphene following the experimental synthesis [1, 2℄ thepresent attention is entered on the experimental advan es aiming to generate better samples forele troni devi es. One of the major problems preventing appli ations of single layer graphene(SLG) is the di(cid:30) ulty to open and ontrol a gap in the samples. To this respe t bilayer graphene(BLG) and multilayer samples are more promising [3℄. One of the potentially most interestingaspe ts of graphene for the appli ations and that remains up to now partially unexplored on ernsthe magneti properties. Ferromagneti order enhan ed by proton irradiation has been observedin graphite samples [4℄ and demonstrated to be due to the arbon atoms by di hroism experiments[5℄. By now it is lear that the underlying me hanism leading to ferromagnetism in these arbonstru tures is the existen e of unpaired spins at defe ts indu ed by a hange in the oordination ofthe arbon atoms (va an ies, edges or related defe ts) [6℄. Very re ent experiments on thin (cid:28)lmsin irradiated graphite show that the main e(cid:27)e t of proton irradiation is to produ e va an ies onthe outer layers of the samples. For thin enough (cid:28)lms of a few thousands angstroms the protonsgo through the samples leaving some va an ies behind. These samples show an enhan ed lo alferromagnetism and also a better ondu tivity than the untreated samples with less defe ts [7℄.Va an ies an play a major role on these magneti and transport properties and are lately beenre ognized as one of the most important s attering enters in SLG and BLG [8℄.The existen e and nature of lo alized states arising from va an ies in BLG have beenanalyzed in a re ent paper [9℄. It was found that the two di(cid:27)erent types of va an ies that an be present in the BLG system (cid:21) depending on the sublatti e they belong to (cid:21) give rise totwo di(cid:27)erent types of states: quasi-lo alized states, de aying as /r for r → ∞ , similar to thesefound in the SLG ase [10℄, and truly delo alized states, going to a onstant as r → ∞ . Whena gap is indu ed by the ele tri (cid:28)eld e(cid:27)e t quasi-lo alized states give rise to resonan es at thegap edges while the delo alized ones be ome truly lo alized inside the gap. These (cid:28)ndings arevery important in understanding the magneti properties of the graphiti samples sin e theselo alized states arry magneti moments. In this paper we study the magneti properties of thelo alized states found in [9℄ using an unrestri ted Hartree-Fo k al ulation. The most interesting ase arises in the presen e of a gate (perpendi ular ele tri (cid:28)eld E = E z ˆ e z ) opening a gap when onsidering two va an ies of the same sublatti e lo ated at di(cid:27)erent layers. We will show thatwith realisti values of the parameters and for experimentally a essible gate voltages we anhave a magneti swit hing between unpolarized and fully polarized system.2. The ele troni stru ture of bilayer grapheneThe latti e stru ture of a BLG is shown in (cid:28)gure 1. In this work we onsider only AB -Bernalsta king, where the top layer has its A sublatti e on top of sublatti e B of the bottom layer. Weuse indi es 1 and 2 to label the top and bottom layer, respe tively.In the tight-binding approximation, the in-plane hopping energy, t , and the inter-layerhopping energy, γ , de(cid:28)ne the most relevant energy s ales (see (cid:28)gure 1). The simplest tight-binging Hamiltonian des ribing non-intera ting π − ele trons in BLG reads [11, 12, 13℄: H T B = X i =1 H i + γ X R ,σ (cid:2) a † ,σ ( R ) b ,σ ( R ) + h. . (cid:3) , (1)with H i being the SLG Hamiltonian H i = − t X R ,σ (cid:2) a † i,σ ( R ) b i,σ ( R )+ a † i,σ ( R ) b i,σ ( R − a )+ a † i,σ ( R ) b i,σ ( R − a )+ h. (cid:3) , (2)inning and swit hing of magneti moments in bilayer graphene 3 -3 0 3 E/t D O S -0.2 0 0.2 E/t γ = 0 γ ≠ 0 E z ≠ 0 (a) (b)(c)(d) B1 A2 B2 A1 Figure 1. (a) Total DOS of bilayer graphene. The inset shows the latti e stru ture. (b)-(d) DOS zoom at low energies for the minimal model, for the model with γ , and the modelwith (cid:28)nite gap due to perpendi ular ele tri (cid:28)eld E z , respe tively.where a i,σ ( R ) [ b i,σ ( R ) ℄ is the annihilation operator for ele trons at position R in sublatti e Ai ( Bi ), i = 1 , , and spin σ . The basis ve tors may be written as a = a ˆe x and a = a (ˆe x −√ y ) / ,where a = 0 .
246 nm . The estimated values of the parameters for this minimal model are: t ≈ eV, γ ≈ . eV ∼ t/ [14℄.The main additional tight binding parameters are the inter-layer se ond-nearest-neighborhoppings γ and γ shown in (cid:28)gure 1 whi h play an important role in what follows. γ onne tsdi(cid:27)erent sublatti es ( B − A ) and γ onne ts atoms of the same sublatti es ( A − A and B − B ). Their values are less well known but we an assume that the following relationbetween parameters holds, γ ∼ γ ∼ γ / ∼ t/ .For the analysis of the bound states and asso iated magneti moments of the present worka summary of the most relevant issues of the BLG ele troni stru ture is the following: • The minimal model with only γ has ele tron-hole symmetry and it is a bipartite latti ealthough not all the A and B atoms are equivalent sin e some have (have not) a hoppingto the other layer. It has two degenerate stable Fermi points similar to SLG [15℄ but thedispersion relation around them is quadrati and the density of states (DOS) at the Fermipoints is (cid:28)nite ((cid:28)gure 1(b)). Opening of a gap gives rise to the DOS shown in (cid:28)gure 1(d)with the hara teristi double minimum shape [16℄. The value of γ sets a bound on themaximal value of the gap. • In lusion of a γ together with γ oupling lifts the degenera y of the Fermi points that areshifted in momentum spa e. The dispersion relation around the Fermi points is linear andthe DOS is zero very mu h like in the SLG ase ((cid:28)gure 1( )). The latti e is still bipartitein the sense that, generi ally, atoms of type A are only linked to atoms of type B althoughthe layer index and ouplings make some di(cid:27)eren es between di(cid:27)erent A ( B ) atoms. • The ombination γ - γ breaks the bipartite nature of the latti e. On the ele troni pointof view it indu es an ele tron-hole asymmetry but the DOS at the Fermi point does not hange. This oupling is important for the magnetism of the samples. Pinning and swit hing of magneti moments in bilayer graphene U/t m AF U c Figure 2. Sublatti e magnetization m = | n Γ i, ↑ − n Γ i, ↓ | vs U for bilayer graphene in theHartree-Fo k approximation, where γ = 0 . t and Γ i = A , B , A , B .3. The model and the magneti properties of the perfe t latti eIn order to study the magneti behaviour of BLG in the presen e of va an ies and/or topologi aldefe ts we use the Hubbard model, treated in the Hartree-Fo k approximation. The totalHamiltonian then reads H = H T B + H U , where the on-site Coulomb part is given by H U = U X R ,ι [ n aι ↑ ( R ) n a ι ↓ ( R ) + n bι ↑ ( R ) n bι ↓ ( R )] , (3)where n xισ ( R ) = x † ισ ( R ) x ισ ( R ) , with x = a, b , ι = 1 , and σ = ↑ , ↓ . We use (cid:28)nite lusters withperiodi boundary onditions at half-(cid:28)lling (one ele tron per atom).It is well known that the Hartree-Fo k-RPA approximation for SLG produ es a phasetransition at the riti al Hubbard intera tion U c ≈ . t , above whi h the staggered magnetizationbe omes (cid:28)nite [17, 18℄.The antiferromagneti transition in the BLG ase has been analyzed in [19℄. Figure 2 showsthe sublatti e magnetization as a fun tion of the Hubbard repulsion U . Throughout this workwe will explore the magneti behaviour of the system with va an ies for values of U ≤ t deepinto the region of U where the sublatti e magnetization is exponentially suppressed so that we an attribute any magneti moment to the presen e of defe ts.4. Va an ies in bilayer grapheneUnlike the ase of lean undoped SLG where the DOS at the Fermi level is zero and there is nogap, in the BLG ase and depending on the more relevant tight binding parameters (see (cid:28)gure1) we an have either a onstant DOS (cid:21) minimal model with only t and γ (cid:21) or a zero DOS inthe presen e of γ . Moreover a gap an be easily generated by an ele tri (cid:28)eld perpendi ular tothe plane, as mentioned before. The DOS is ru ial for the study of lo alized states. In the SLG ase, single va an ies indu e quasi-lo alized states around the defe t, de aying as /r [10, 20℄.Due to the absen e of a gap, true bound states do not exist in the thermodynami limit.In the BLG ase there are two types of va an ies beta and alpha for sites onne ted (or not)to the other layer. As shown in [9℄ asso iated to the presen e of va an ies and to the existen e ofa gap in the spe trum generated by an ele tri (cid:28)eld E z three di(cid:27)erent types of va an y-indu edstates are found:(i) For E z = 0 a β − va an y indu es a resonan e for γ = 0 (delo alized state) and a quasi-lo alized state ( /r behaviour) for a (cid:28)nite value of γ .(ii) For E z = 0 an α − va an y indu es always a resonan e irrespe tive of γ .inning and swit hing of magneti moments in bilayer graphene 5 -3 0 3 E/t L D O S -3 0 3 E/t I P R (c) (d) (b)(a) Figure 3. (a)-(b) Zero-energy eigenstates in a minimal model bilayer graphene luster with γ = 0 . t ontaining a single β − (a) and α − va an y (b). We show only the region aroundthe va an y and the layer where the va an y is lo ated. ( )-(d) Lo al density of states (rightaxis) and inverse partition ratio (left axis) for a luster with a single β − ( ) and α − va an y(d) in luding γ = 0 . t .(iii) For E z = 0 a β − va an y produ es a resonan e inside the ontinuum near the band edgewhile an α − va an y gives rise to a truly lo alized state inside the gap. This is the mostinteresting state for the magneti impli ations.In (cid:28)gure 3(a) and 3(b) we illustrate the nature of the lo alized states by plotting thenumeri al wavefun tion for zero-energy eigenstates in a BLG luster with a single β − and α − va an y, respe tively. The luster ontains × sites. The image shows only the layerwhere the va an ies are lo ated and the region around the va an ies. It an be seen that a quasi-lo alized state exists for a β − va an y, and for an α − va an y the zero-energy mode also appearsquasi-lo alized over the diluted layer. The redu ed amplitude of the zero mode in (cid:28)gure 3(b) forthe α − va an y is due to the presen e of a delo alized omponent on the underlaying layer (notshown).The above results are on(cid:28)rmed by the enhan ed lo al DOS and enhan ed inverse partitionratio at zero energy for a β − va an y (quasi-lo alized state), as shown in (cid:28)gure 3( ). Theequivalent result for a α − va an y (delo alized state with a quasi-lo alized omponent in oneof the layers) is shown in (cid:28)gure 3(d). The lo al DOS is omputed at a site losest to the va an yusing the re ursive Green's fun tion method for a luster with × sites. The inverseparti ipation ratio, de(cid:28)ned as fourth moment of the wavefun tion amplitude, is omputed forthe luster used in (cid:28)gures 3(a) and 3(b).5. Magneti behaviourThe generation and stru ture of the magneti moments asso iated to unpaired atoms in SLGand multilayer graphene is to a great extent determined by the bipartite nature of the underlyinglatti e and hen e by the Lieb's theorem [21℄. The theorem states that the ground state of therepulsive half (cid:28)lled Hubbard model in any bipartite latti e with N = N A + N B sites is uniqueand has total spin S = | N A − N B | . A ording to the Lieb's theorem [21℄ the quasi-lo alizedzero modes indu ed by unpaired atoms in the bipartite latti e be ome spin-polarized in thepresen e of a Hubbard repulsion U and lo al moments appear in the latti e [20, 22, 23, 24, 25℄.In the thermodynami limit the spin polarized modes (no longer at zero energy) merge into the ontinuum and even though Lieb theorem applies equally, the spin polarization is delo alizedand itinerant ferromagnetism appears [21℄.Pinning of magneti moments in lo alized regions, in the thermodynami limit, would be avery interesting possibility for appli ations. In SLG we ould try to open a gap and push thequasi-lo alized modes out of the ontinuum. However, a gap is not easily open in graphene, and Pinning and swit hing of magneti moments in bilayer graphenea mass-gap does not work: the same linear algebra theorem that guarantees the existen e of zeromodes when no diagonal terms exist in the Hamiltonian [26℄ also states that, in the presen e ofa staggered (diagonal) potential, these modes move to the gap edges; this is due to the fa t thatthese modes live only on one sublatti e, the less diluted. In BLG we an easily open a gap byindu ing layer asymmetry through the appli ation of a perpendi ular ele tri (cid:28)eld E z = 0 (ba kgate, for example), whi h is not a staggered potential. In [9℄ it was proven that truly lo alizedstates exist inside the gap indu ed through ele tri (cid:28)eld e(cid:27)e t in BLG.5.1. Single va an yThe magneti properties of a va an y in BLG were studied in [27℄ using spin-polarized densityfun tional theory. It was found that the spin magneti moment lo alized at the va an y is ofthe order of ten per ent smaller than that of SLG for both types of va an ies α and β . Thisredu tion of the spin magneti moment in the bilayer was attributed to the interlayer hargetransfer from the adja ent layer to the layer with the va an y. We have veri(cid:28)ed that both inthe minimal model γ = 0 and for a (cid:28)nite value of γ we obtain the results expe ted for a singlelayer in a ordan e with Lieb theorem. The se ond-nearest-neighbor hopping γ that breaks thebipartite hara ter of the whole latti e does not hange this behaviour, as long as one va an yis onsidered. A (cid:28)nite γ makes the va an y-indu ed state to appear o(cid:27) zero energy, but stillspin degenerate. In luding U lifts spin degenera y and indu es a magneti ground state. Thesituation is similar to the one dis ussed in SLG [25℄ when a pentagon is in luded.5.2. Two va an ies and the e(cid:27)e t of an asymmetry gapRegarding the e(cid:27)e t of two va an ies, we have found rather di(cid:27)erent behaviour depending onwhether an asymmetry gap is present or not, and depending on the ombination layer/sublatti ewhere the two va an ies o ur. As mentioned before, su h an asymmetry gap is indu edby making the two layers asymmetri , for example, by applying a perpendi ular ele tri (cid:28)eldthrough a ba k gate voltage. The resultant ele trostati energy di(cid:27)eren e between layers eE z d ( d = 0 .
34 nm is the interlayer distan e and e the ele tron harge) introdu es an interestingtuning apability to the system sin e all other parameters, in luding the strength of the Hubbardintera tion U , an hardly be tuned in experiments.For the bipartite ase where γ = 0 results are in omplete agreement with Lieb theorem,as expe ted. In parti ular, irrespe tive of the layer index, two va an ies of the same sublatti eindu e a total spin S = 1 , while va an ies in di(cid:27)erent sublatti es give rise to a ground statewith S = 0 . When E z = 0 , sin e there are nonzero diagonal elements in the Hamiltonianmatrix, we no longer have a bipartite system in the Lieb sense [21℄ and di(cid:27)erent behaviour fromthese determined by Lieb theorem might arise. We have found that two va an ies in di(cid:27)erentsublatti es always give S = 0 , irrespe tive of the layer index and of the gap's size, in a ordan ewith Lieb theorem. Also following Lieb, va an ies in the same sublatti e, and belonging to thesame layer, i.e. Ai, Ai or Bi, Bi with i = 1 , , originate S = 1 .The interesting ase arises when the system is gaped and has two va an ies from the samesublatti e in di(cid:27)erent layers, i.e. A , A or B , B . In this ase we have two regimes: for small E z we get S = 1 in agreement with Lieb theorem. When E z in reases we rea h a regime at a riti al value of E z where the ground state has S = 0 . In lusion of the γ hoping goes in the samedire tion as the gap depressing the polarization. The riti al U to maintain the full polarizationof the latti e in reases for bigger values of γ . The riti al line in the E z − U plane is shown in(cid:28)gure 4 for di(cid:27)erent values of γ . The explanation for this behaviour lies in the di(cid:27)erent ways inwhi h the degenera y of the zero modes is lifted with U (cid:21) that splits the degenera y a ordingto the spin (cid:21) and with other ouplings like E z , γ , or in-plane next-nearest-neighbor t ′ [28℄. It isinning and swit hing of magneti moments in bilayer graphene 7 eE z d / γ U /t S = S = γ Figure 4. Transition line between total spin S = 1 and S = 0 in the E z − U plane for theground state of a bilayer system with two va an ies belonging to the same sublatti e lo ated indi(cid:27)erent layers for di(cid:27)erent values of the γ hopping integral. Cir les (blue) , squares (purple),and diamonds (red) stand for γ = 0 , γ / , γ , respe tively. The error bar is of the order ofsymbol size.important to note that the transition between (cid:28)nite and zero magneti polarization of the latti eo urs in the region of experimentally relevant values of the external voltage: < E z d . γ [29℄, whi h makes the realization and observation of su h magneti swit hing apability a realpossibility.6. Con lusions and dis ussionWe have examined the magneti properties of the lo alized states indu ed by latti e va an ies inBLG re ently analyzed in [9℄. We have found that in the presen e of a gap the system supportstwo types of spin polarized lo al states related to the two types of inequivalent va an ies that anexist in a Bernal sta king. Those living inside the gap are truly normalizable bound states what an give rise to fully lo alized large magneti moments if there are several va an ies belonging tothe same graphene layer. This an be related to the measurement of lo al magneti moments inproton bombarded graphite asso iated to the defe ts [30℄ and to the observation of the insulatingnature of the ferromagneti regions [31℄. A density of su h va an ies would give rise to a mid-gapband ontributing to the total ondu tivity of the sample. In su h band many body e(cid:27)e ts willbe important and an drive the system to other kinds of instabilities [32℄. The other type ofva an ies stay at the edge of the gap and give rise to quasi-lo alized magneti moments whosewave fun tion de ays as /r/r