Gapped Ferromagnetic Graphene Nanoribbons
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Gapped Ferromagnetic Graphene Nanoribbons
D. Hou,
1, 2
J. H. Wei, ∗ and S. J. Xie † School of Physics, National Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China Department of Physics, Renmin University of China, Beijing 100872, China (Dated: October 17, 2018)We theoretically design a graphene-based all-organic ferromagnetic semiconductor by terminatingzigzag graphene nanoribbons (ZGNRs) with organic magnets. A large spin-split gap with 100% spinpolarized density of states near the Fermi energy is obtained, which is of potential application in spintransistors. The interplays among electron, spin and lattice degrees of freedom are studied usingthe first-principles calculations combined with fundamental model analysis. All of the calculationsconsistently demonstrate that although no d electrons existing, the antiferromagnetic π − π exchangetogether with the strong spin-lattice interactions between organic magnets and ZGNRs make theground state ferromagnetic. The fundamental physics makes it possible to optimally select theorganic magnets towards practical applications. PACS numbers: 73.22.Pr, 75.75.-c, 71.15.Mb
Introduction .—Since the experimental discovery ofgraphene in 2004 [1], graphene-based transistors haverapidly developed and are considered good candidatesfor post-silicon electronics [2]. However, they encounteran outstanding challenge of how to open a sizable andwell-defined band gap in graphene to control the ON-and OFF-states of the transistors. One realistic solu-tion is constraining large-area graphene in one dimensionto form graphene nanoribbons (GNRs) which have bandgaps approximately inversely proportional to their widths[3]. In recent years, this scheme has been experimentallyproved rather practical and thus motivated a new frontierarea called "graphene nanoribbon electronics" [4–6].Besides the gapped semiconductor properties suitablefor electronics, graphene nanoribbons possess fascinat-ing magnetic properties for spintronics. Zigzag graphenenanoribbons (ZGNRs) have spatial spin-polarized groundstates with spin moments coupled ferromagnetically(FM) on the same edge and antiferromagnetically (AFM)between different edges [7, 8], which makes them poten-tial materials for carbon-based spintronic devices avoid-ing heavy magnetic atoms. By means of electric field con-trol, Boron/Nitron doping or edge termination, ZGNRscan exhibit half-metallicity that may be used for spininjection and filtration in metallic spintronics[9–12].On the other side, semiconductor spintronics has beenthe subject of many recent studies [13]. The central issueis how to develop spin transistors with appropriate ferro-magnetic semiconductor materials for non-volatile mem-ory applications [14]. By virtue of the unique electronicand magnetic properties mentioned above, GNRs may beideal materials for spin transistors. However, the diffi-culty is that the ferromagnetic state of ZGNRs is not theground state, and even worse for transistors, it is a gap-less state. Therefore, the organic integration of electronicand magnetic properties of graphene for spin transistoris still an open question.In this paper, we design a graphene nanoribbon ter- minated with organic magnets to give a possible solu-tion for the above issue. In order to convert the groundstate of ZGNRs from the AFM state to a gapped FMone, we use edge passivation of graphene nanoribbons byorganic ferromagnetic radicals (one kind of organic mag-nets, see Fig.1). The physics is that the magnetic orderof the host material will be changed when it exchange-couples to the localized spin of attached organic mag-nets [15]. Comparing with ZGNRs adsorbing magnetictransition metal atoms on their surfaces in literatures[16, 17], these carbon-based materials have the advan-tages of much weaker spin-orbit and hyperfine interac-tions, which should induce much longer spin coherentlength. Furthermore, owing to the planar structure of theselected organic radical, above mentioned exchange cou-pling is tunable by manually rotating the radical plane toaffect the spin polarization near the Fermi energy. Ourresults suggest that the controllable graphene-based spintransistors can be expected.
Methods .—Towards the practical applications, wefirstly choose trimethylenemethane (TMM) as the or-ganic magnet, for the reason that it has been widely in-vestigated and designed for kinds of possible spintronicapplications due to its high magnetic moments [18]. How-ever, as will be demonstrated later, the dependence of thephysics on this specific choice is quite weak. TMM con-tains 4 carbon and 6 hydrogen atoms with π -conjugatedstructure and has a ground state of triplet biradical, asshown in Fig.1b. Two unpaired electrons in TMM pro-vide 2 µ B spin moments in total distributing via molecu-lar orbital over all the carbon atoms. At its spin tripletground states, three outer carbon atoms have the samekind of net spin density, while the inner one has theopposite (see Fig.1d). When attached to one of theZGNRs’ edges already possessing spin moments, TMMwill strongly modify the properties of ZGNRs.6-ZGNR terminated by hydrogen atoms is used as pris-tine ZGNR as shown in Fig.1a. The unit cell of 6-ZGNR Figure 1: (color online) Top view structures of (a) pristine6-ZGNR, (b) TMM, (c) TMM terminated 6-ZGNRs, and (d)the net spin density of TMM. The yellow balls are carbonatoms and the cyanic ones are hydrogen atoms. The red/bluearrow indicates the up/down spin component. contains 12 carbon atoms and 2 hydrogen atoms (see thefigure inside the green rectangle in Fig.1a). The lengthof periodic direction of the unit cell is 2.46 Å (experi-ment value), while the repeated images of the molecularare separated by vacuum more than 8 Å thick along theother two directions. To avoid direct interaction betweenimages of terminated TMM, the zigzag direction of theunit cell is doubled (see the region inside the red rect-angle in Fig.1a). In this supercell, TMM replaces one ofthe four hydrogen atoms, making the system terminatedby TMM and hydrogen alternately on one edge (e.g. theright edge). The top view of one possible relaxed struc-ture (that will be proved the ground state) is shown inFig.1c. Please be noted that the TMM biradical is notcoplanar with ZGNR after atomic optimization.The first-principles calculations are performed usingDensity Functional Theory (DFT) method implementedwith
SIESTA code[19] using the Perdew-Burke-Ernzerhofexchange-correlation functional[20]. Double- ζ plus polar-ization function (DZP) basis set is used, together with amesh cutoff of 200 Ry and norm-conserving pseudopo-tentials. All the atoms in the supercell are fully relaxedto fulfill the energy and force convergence of 10 − eVand 0.01 eV/Å respectively. The optimized structureparameters with different initial spin configurations arecrosschecked using a plane-wave method implemented in Quantum Espresso code[21], with the same convergencecriteria and a plane wave kinetic energy cutoff of 30 Ry.The identical results ensure the validity of the simula-tion parameters used, and other data are handled using
SIESTA code solely.
Results. —In what follows, we use " ++ " and " +- " to re- spectively denote the FM and AFM spin configurations,where the first symbol denotes the left edge spin momentwhile the second denotes the right one of the grapheneribbon. Without losing generality, we designate " + /-"to represent up/down spin in what follows. The con-vergence of the simulation parameters is firstly checkedby electronic property calculations of a pristine 6-ZGNRwith a 1 × × +- " order is confirmed tobe the ground state with a net spin moments of 0.26 µ B localized on each edge atom, and its total energy is 26meV/supercell lower than the " ++ " one. These values areconsistent with previously reported results and used asreference data [7, 8].The TMM units are then attached to the right sideof the ZGNR. Since the spin moments on TMM andZGNRs can align either parallel or anti-parallel, the " ++ "and " +- " spin configurations will extend to four possibleones: " +-+ ", " ++- "," +-- " and " +++ ", with the rightmostsymbol representing the net spin moments on TMM. Fullatomic relaxation calculations are done separately usingthese four possible spin configurations to find the groundstate. It is found that both the bond length and the dihe-dral angle between the ZGNR and TMM are varied withdifferent spin configurations ( we will discuss this point indetails later). The relative total energy ( taking E +-+ asreference), converged net spin moments and optimizedbond structures are summarized in Tab.I. Crosscheck-ing with Quantum Espresso presents highly consistentresults, as shown in Tab.I.As presented in Tab.I, the " +-+ " state is the groundstate. It has a total energy of 36 meV lower than thatof " ++- " state, 578 meV than that of " +-- " state and612 meV than that of " +++ " state, respectively. Thesedata indicate that, in the ground state of the radical ter-minated graphene nanoribbon, the net spin moments ofTMM radical antiferromagnetically couples to the near-est edge spin of ZGNR. As the energy difference betweenpristine " +- "" ++ " ZGNR is 26 meV, the present couplingis much stronger than the interedge superexchange ofZGNR[22]. A relative result was reported just recentlyby Atodiresei et al. focusing on the spin polarization ofnonmagnetic benzene, cyclopentadienyl radical and cy-clooctatetraene molecule adsorbed onto a ferromagnetic2 ML Fe/W(110) surface. It is found that at the organicmolecule site an inversion of the spin polarization occurswith respect to the ferromagnetic surface, resulting fromthe antiferromagnetic coupling between π -electrons of themolecule and the d -electrons of Fe atoms[23]. While inthe present case, the strong antiferromagnetic couplingpresented here results from the π - π interactions.Comparing the corresponding structures of the fourspin configurations manifests the close connection be-tween the spin coupling and the variation of the ZGNR-TMM bond length (or dihedral angle). It can be seenfrom Tab.I that the changes of bond lengths and spinmoments among the four possible spin configurations Table I: Relative total energy, converged net spin moments on selected carbon atoms, optimized bond lengths and spin anglesfor all possible spin configurations of TMM terminated 6ZGNR. The meanings of the symbols for the carbon atoms are depictedin Fig.1c. The values in brackets are corresponding results obtained by
Quantum Espresso code.Label E-E +-+ (meV) Net Spin Moments ( µ B ) Bond Lengths (Å) Spin AngleC g C g C t C t C t C t C g -C t C t -C t C t -C t C t -C t C g -C t +-+ o ++-
36 0.08 -0.08 -0.24 0.11 -0.60 -0.51 1.39(1.38) 1.48(1.48) 1.42(1.41) 1.40(1.40) 180 o +--
578 -0.24 0.02 -0.63 0.11 -0.68 -0.68 1.48(1.48) 1.43(1.43) 1.42(1.42) 1.42(1.42) 70 o +++
612 0.23 -0.02 0.63 -0.11 0.68 0.68 1.48(1.48) 1.43(1.43) 1.42(1.42) 1.42(1.42) 70 o mainly take place around the C g -C t bond. When TMMand ZGNR are coupled antiferromagnetically (" +-+ " and" ++- " states), the length of the C g -C t bond is short-ened about 0.1Å relative to those of " +-- " and " +++ "states. It is reasonable to attribute such a large distortionto the softness of the all-organic system and the strongspin-lattice interaction. It is further found that, if thedihedral angle is disturbed, the bond length will changescorrespondingly. As the π electron orbital trends to beperpendicular to the σ bonds surface, the dihedral anglereflects the spin angle of π - π electrons on C g and C t .Therefore, the calculation reveals that the π - π spin cou-pling depends not only on the bond length d but also onthe spin angle θ . We thus propose a spin-lattice (spin-phonon) coupling model to illustrate the physics as fol-lows, H T = H c + J (1 − αu ) ~S g · ~S t + k u (1)where H c denotes the invariant part (except the spin-spincoupling between C g and C t ) before and after spin-lattice interaction involved. The second term is the π - π spin interaction modulated by the bond distortion u between C g and C t , where J is the coupling constantat u = 0 , α the spin-lattice coupling constant ( α > ).The third one is the elastic potential energy caused bythe bond distortion. As the magnetism of the terminatedradical is robust, we take a classic approach with ~S g · ~S t = S g S t cos θ (2)By taking the differential of the Hamiltonian (1) withrespect to the bond distortion u , we can obtain the stablestate under certain fixed angle θ , u = αJ S g S t cos θk = u cos θ (3)which indicates that the optimal bond distortion u willchange with the spin angle θ . We then obtain the optimalspin angel θ and bond distortion u by minimizing thetotal energy, which is given as ( θ = πu = − u < (4) It derives that the ground state is characterized by an-tiferromagnetic exchange interaction and a shorter C g -C t bond length, which is consistent with our first-principles calculations as shown in Tab. I.In order to elucidate the robust physics of the spin-lattice coupling, we substitute the TMM radicals withCH , the simplest carbon based organic radical hold-ing the sp hybridization. In the first principle calcu-lation, the optimal bond length d between C g and C t for each θ is obtained by minimizing the total energy.The calculated d − θ relation is shown in Fig. 2, whichis well fitted with a cosine type line as deduced from ourmodel. Therefore, in our proposed all-organic ferromag-netic graphene structure, the ferromagnetism is providedby the π − π spin interaction, while the spin-lattice cou-pling is vital for its stability. Figure 2: (color online) d − θ relation of the total energyminimum state for given spin angle θ . The black squares arecalculated using DFT, and the blue line is the fitted line usingequation d = d + u = d + u cos θ , where the fitting parameters d =1.470Å and u =0.093Å. Besides the all-organic ferromagnetic ground statementioned above, we find that a large spin-split bandgap and the 100% spin polarization near the Fermi en-ergy are specific characters of TMM and CH terminatedZGNR. Fig.3a and b respectively plot their spin-resolvedband structures and density of states (DOS). From thesefigures, TMM and CH terminated ZGNRs exhibit thefollowing similar features: 1) both of the spin up anddown subbands manifest semiconductor character withenergy gaps ∼ eV; and 2) the spin splitting betweenup and down subbands near the Fermi energy ( E F ) isabout 0.5 eV, which induces a gapped spin-split DOSwith 100% spin polarization within a wide energy regionnear E F . The highest occupied and lowest unoccupiedenergy level are proved to be extended states which canserve a large current in graphene-based transistors. InFig.3c, the local density of states (LDOS, with energyregion E F → E F − . eV) of the highest (spin up) oc-cupied energy level of CH terminated ZGNR is shown.Towards the practical transistors, those features providewell-defined conducting ON- and insulating OFF-states,and also the high spin-polarized current at the ON-state.By applying a positive or negative gate voltage, one canselectively shift the (purely spin up) occupied highest en-ergy level or the lowest (purely spin down) unoccupiedone towards the Fermi energy. A 100% spin polarizedcurrent is thus produced via the gate voltage control.For comparison, we make some comments on thenonmagnetic side group terminated ZGNRs in theliterature[12]. For those materials, the sizable energygap and 100% spin polarization in large energy rangenear E F can not obtained simultaneously. For example,in the NH , OH, and COOH terminated ZGNRs, the en-ergy gap is about 0.3 eV, but the spin splitting of thestates near the E F is very small. In the NO terminatedZGNR, although the spin splitting is relatively enlarged,the energy gap reduced to about 0.1 eV, too small to beused in transistors. Therefore, the present design has anadvantage in realizing a large energy gap and a high spinpolarization near the Fermi energy simultaneously.In spintronics, the spin polarization P ( E ) may be de-fined as P ( E ) = DOS ↑ ( E ) − DOS ↓ ( E ) DOS ↑ ( E ) + DOS ↓ ( E ) , (5)which can be tuned by changing the spin polarized DOSnear E F . For this purpose, we study the evolution of thespin polarization near E F with manually varying the spinangle θ in CH terminated ZGNR. The result is demon-strated in Fig. 3d. As mentioned above, the DOS aboveand below the E F are both 100% spin polarized in a wideenergy range at θ = π . However, as θ decreases towards π/ , the high spin polarized region near E F shrinks andeven almost disappears at θ = π/ . As seen in the figure,the large value of DOS near E F during radical rotationcan keep the ON-state under certain gate voltage. Thus,controlling the exchange coupling strength by radical ro-tation is an effective way to tune the spin polarization.Finally, let us comment on the possibility of takingLieb’s theorem on the repulsive Hubbard model [24] asthe mechanism of the gapped ferromagnetic state, for thereason that its deduced Klein’s edge [25] has been appliedto explain the ferromagnetic state of some other edge ter- -2-1012 E - E f ( e V ) K Points Spin up Spin down a a) DOS -2-1012 E - E f ( e V ) K Points Spin up Spin down a DOS b) -2 -1 0 1 2 E-E f (eV) -1.000-0.7500-0.5000-0.250000.25000.50000.75001.000 D en s i t y o f s t a t e s Spin Up Spin Down -2 -1 0 1 2
E-E f (eV) -1.000-0.7500-0.5000-0.250000.25000.50000.75001.000 D en s i t y o f s t a t e s Spin Up Spin Down -2 -1 0 1 2
E-E f (eV) -1.000-0.7500-0.5000-0.250000.25000.50000.75001.000 D en s i t y o f s t a t e s Spin Up Spin Down -2 -1 0 1 2
E-E f (eV) -1.000-0.7500-0.5000-0.250000.25000.50000.75001.000 D en s i t y o f s t a t e s Spin Up Spin Down d ) Figure 3: (color online) (a)(b) Spin-resolved band structure(left panel) and density of states (right panel) of the groundstate of TMM and CH terminated ZGNR, respectively. (c)Local density of states(LDOS) of the ground state of CH terminated ZGNR. Red for up spin. (d) Spin-resolved DOSof CH terminated ZGNRs with spin angle θ equal to π , π/ , π/ , and π/ respectively. The shaded area presents the spinpolarization P(E) defined in Eq. (5). minated GNRs in literatures [7, 26]. The central point ofthe Lieb’s theorem is that the ground state of a bipartitelattice and half-filled band has a net total spin S provid-ing the two sublattice have different number of sites. Bycarefully comparing our results with Lieb’s theorem, weconclude that it can not consistently explain all the fea-tures of the ground state here. The main discrepanciesare summarized as follows: 1) it is the strong spin-latticeinteraction, rather than the Hubbard (electron-electron)correlation, stabilizes the ground state of the present sys-tem; 2) the spatial spin order of the ground state is be-yond Lieb’s theorem that can only deduce a nonzero totalspin; and 3) our DFT and model calculations have bothomitted the quantum fluctuation of electron-electron in-teraction which is the curial part of Hubbard model andLieb’s theorem. Relative to Lieb’s theorem, our mech-anism of the antiferromagnetic exchange together withstrong spin-lattice interactions can explain all of the fea-tures of the gapped ferromagnetic state in a consistentmanner, as already demonstrated above. Summary .—In summary, we designed a graphene-based all-organic ferromagnetic semiconductor by termi-nating ZGNR with organic mangets. A large spin-splitgap with 100% spin polarized DOS near E F is obtained,which is of potential usage in spintronic devices. Combin-ing first-principles calculations with basic model analysis,we conclude that the mechanism for the gapped ferro-magnetic state is the antiferromagnetic π − π exchangetogether with strong spin-lattice interactions between or-ganic magnet and graphene nanoribbons. By controllingthe π − π interaction strength, the spin polarization ofthe DOS near E F can be tuned from 100% to nearly zero.The fundamental physics makes it possible to optimallyselect the organic magnet towards practical applications.The authors thank Prof. Shimin Hou of Peking Univer-sity and Prof. Mingwen Zhao of Shandong University forthe insightful discussions and suggestions. Support fromNational Basic Research Program of China (Grant No.s2007CB925001, 2009CB929204 and 2010CB923402) andNSFC of China (Grant No.s 10874100 and 11074303) aregratefully acknowledged. 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