Contrasting interedge superexchange interactions of graphene nanoribbons embedded in h-BN and graphane
Sun-Woo Kim, Hyun-Jung Kim, Jin-Ho Choi, Ralph H. Scheicher, Jun-Hyung Cho
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Contrasting interedge superexchange interactions of graphene nanoribbons embedded in h -BN andgraphane Sun-Woo Kim, Hyun-Jung Kim, Jin-Ho Choi, , Ralph H. Scheicher, and Jun-Hyung Cho ∗ Department of Physics and Research Institute for Natural Sciences,Hanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Korea Hefei National Laboratory for Physical Sciences at the Microscale and Department of Physics,University of Science and Technology of China, 96 JinZhai Road, Hefei, Anhui 230026, China Division of Materials Theory, Department of Physics and Astronomy,˚Angstr¨om Laboratory, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden (Dated: August 14, 2018)Based on first-principles density-functional theory calculations, we present a comparative study of the elec-tronic structures of ultranarrow zigzag graphene nanoribbons (ZGNRs) embedded in hexagonal boron nitride(BN) sheet and fully hydrogenated graphene (graphane) as a function of their width N (the number of zigzagC chains composing the ZGNRs). We find that ZGNRs/BN have the nonmagnetic ground state except at N = 5 and 6 with a weakly stabilized half-semimetallic state, whereas ZGNRs/graphane with N ≥ PACS numbers: 73.21.Hb, 73.22.Pr, 75.75.-c
Graphene nanoribbons (GNRs) have been regarded as oneof the most important classes of carbon-based nanomaterialsdue to their unique electronic and magnetic properties.
Fab-rication of GNRs with different widths and edges has beenachieved by lithographic patterning, bottom-up fabrication, and chemical unzipping of carbon nanotubes. It is knownthat the electronic and magnetic properties of GNRs vary withthe ribbon width and the edge geometry, thereby being uti-lized to design novel electronic and spintronic devices. For in-stance, the band gap of GNRs depends on the ribbon width, and the zigzag-edged graphene nanoribbons (ZGNRs) havepeculiar localized electronic states at both edges while theGNRs with armchair edges do not have such localized edgestates. Here, the localized edge states of ZGNRs are ferro-magnetically ordered at each edge with an opposite spin orien-tation, forming an antiferromagnetic (AFM) spin order. Inter-estingly, it was predicted that such an AFM ordered ZGNRcan have a half-metallic property if in-plane homogeneouselectric field is applied across the edges of the ZGNR. How-ever, the applied electric field is practically too high to re-alize half-metallic ZGNRs, and therefore various alterna-tive approaches have been proposed. Most of the alter-natives focus essentially on the same conceptual basis that thehalf-metallicity of ZGNRs can be enabled by the modifica-tion of edge states: e.g., the edge modification of ZGNRswith two different functional groups can produce the half-metallic property even in the absence of an external electricfield.
Such an asymmetric edge modification can also beachieved when ZGNRs are embedded in a hexagonal boronnitride ( h -BN) sheet. These embedded ZGNRs (hereafterdesignated as ZGNRs/BN) have a C − B interface at one edgeand a C − N interface at the opposite edge [see Fig. 1(a)]. Anumber of density functional theory (DFT) calculations re- ported the presence of half-metallicity or half-semimetallicityin ZGNRs/BN.
On the experimental side, the fabricationof graphene-BN hybrid structures was recently reported, withgraphene strips as narrow as tens of nanometers.
In a dif-ferent way, the ultranarrow ZGNRs can be fabricated by re-moving hydrogen atoms from a fully hydrogenated graphene(viz. graphane): see Fig. 1(b). Such embedded ZGNRs L R
BCN (a)
L R CH (b) x y FIG. 1: (Color online) Top view and side view of the optimized struc-tures of (a) the ZGNR/BN and (b) the ZGNR/graphane with N = 5.The x and y axes are taken to be parallel and perpendicular to theedges of ZGNR, respectively. L and R represent the left and rightedges, respectively. Typeset by REVTEX C N C B N E F B (a) N 4= N 5= N 8=N 3=N 2=N 1=h-BNGraphane N 4= N 5= N 8=N 3=N 2=N 1= (b)
FIG. 2: (Color online) Calculated band structures of (a) ZGNRs/BN and (b) ZGNRs/graphane as a function of N . The results for the h -BNsheet and graphane are also given. The charge character of edge states is shown with an isosurface of 0.01 electrons/ ˚A . In the schematicdiagram, C B (C N ) indicates the edge C atom bonding to a B (N) edge atom. The energy zero represents the Fermi level. The direction of G − X line is parallel to the edges. For distinction, C atoms composing the ZGNR with N = 1 are drawn with circles in different brightness. (hereafter designated as ZGNRs/graphane) were predicted toexhibit the insulating AFM ground state, similar to iso-lated ZGNRs. Note that ZGNRs/graphane have the symmet-ric interface structure with an identical C − CH interface onboth edges, differing from the asymmetric interface structureof ZGNRs/BN. Therefore, it is very interesting to explore theroles of interface structure in determining the drastically dif-ferent electronic and magnetic properties of ZGNRs/BN andZGNRs/graphane.In this paper, we perform first-principles DFT calcula-tions to investigate the electronic structures of ultranarrowZGNRs/BN and ZGNRs/graphane as a function of N from1 to 8. We find that, in contrast to ZGNRs/graphane wherethe nonmagnetic (NM) ground state at N = 1 is converted tothe AFM ground state for N ≥
2, ZGNRs/BN have the NMground state except at N = 5 and 6 with the half-semimetallicstate. Such different behaviors of the ZGNRs embedded in h -BN sheet and graphane can be traced to the contrastingfeatures of edge states due to their different interface struc-tures. The asymmetric interface structure of ZGNRs/BN pro-duces the inequivalent edge states originating from the B − Cand N − C interfaces, giving rise to a relatively short-rangeand weak superexchange interaction between the two edgestates. On the other hand, the symmetric interface structureof ZGNRs/graphane produces the identical edge states withpartially flat bands, leading not only to a magnetic instabilitydue to the enhanced density of states (DOS) at the Fermi level( E F ) but also to a long-range, strong interedge superexchangeinteraction.The present first-principles DFT calculations were per- formed using the Fritz-Haber-Institute ab -initio molecularsimulations (FHI-aims) code for an accurate, all-electrondescription based on numeric atom-centered orbitals, with“tight” computational settings and accurate tier-2 basis sets.For the exchange-correlation energy, we employed the gen-eralized gradient approximation functional of Perdew-Burke-Ernzerhof. The embedded ZGNRs were simulated using aperiodic supercell with a constant in-plane unit cell length of ∼
46 ˚A (changing the width of BN or graphane from ∼ ∼
41 ˚A with respect to N ) and a vacuum spacing of ∼ × × k -points in the surface Brillouinzone. All of the atoms were allowed to relax along the cal-culated forces until all the residual force components are lessthan 0.02 eV/ ˚A. The optimized lattice constant and band gapof h -BN sheet (graphane) are found to be 2.51 (2.54) ˚A and4.66 (3.62) eV, respectively, in good agreement with previousDFT calculations. We begin to determine the atomic and electronic structuresof ZGNRs/BN and ZGNRs/graphane using spin-unpolarizedcalculations. The calculated band structures of ZGNRs/BNand ZGNRS/graphane are displayed as a function of N inFig. 2(a) and 2(b), respectively, together with those of h -BN sheet and graphane. For the ZGNR/graphane with N =1, we obtain a bond-alternated structure with two differentC − C bond lengths, d C − C = 1.42 and 1.45 ˚A [see the insetof Fig. 2(b)], indicating a Peierls distortion of D d = ± A . This Peierls distortion accompanies a band-gap opening of0.22 eV between the p and p ∗ bands. On the other hand, forthe ZGNR/BN with N = 1, such a bond alternation does not TABLE I: Calculated energy difference (in meV/unit cell) between the NM and half-semimetallic (HS) or AFM configurations for ZGNRs/BNand ZGNRs/graphane as a function of N . The band gap in each system is also given in the unit of eV. N = N = N = N = N = N = N = N = D E NM − HS − − − − − − E g D E NM − AFM − E g occur with an equal C − C bond length of d C − C = 1.43 ˚A, andthe charge character of the p ( p ∗ ) state at the X point repre-sents the hybridization between C and B (N) atoms [see theinset of Fig. 2(a)], giving rise to a large band-gap opening of1.71 eV. As N increases, the C − C bond lengths in ZGNRs/BNand ZGNRs/graphane are close to each other as 1.44 and 1.45˚A, respectively. Figure 2(a) shows that, as N of ZGNRs/BNincreases, the band gap ( E g ) decreases, almost being closedfrom N = 4. However, we note that the breaking of the sublat-tice symmetry in ZGNRs/BN, due to their asymmetric inter-face structure, avoids the crossing of p and p ∗ bands at E F [see the inset of N = 8 in Fig. 2(a)]. On the other hand,for ZGNRs/graphane with identical edge interfaces, the p and p ∗ bands cross the Fermi level with increasing N , formimg apartially two-fold degenerate flat band at a sufficiently widerwidth [see Fig. 2(b)].It is noteworthy that the charge characters of p and p ∗ statesin ZGNRs/BN represent asymmetric edge states localized atthe C − B and C − N interfaces, respectively [see the inset for N = 8 in Fig. 2(a)]. Here, the schematic diagram of fron-tier orbital interactions shows that the highest occupied p andlowest unoccupied p ∗ states are characterized as the C − Bbonding and C − N antibonding orbitals, respectively. On theother hand, the ZGNR/graphane with N = 8 shows symmet-ric p and p ∗ edge states localized at both sides [see the in-set for N = 8 in Fig. 2(b)]. These different features of edgestates between ZGNRs/BN and ZGNRs/graphane may influ-ence the range and strength of the interaction between twoedges. In order to compare the effects of the interedge in-teraction on the half-semimetallicity of ZGNRs/BN and theAFM order of ZGNRs/graphane, we perform spin-polarizedcalculations for the two systems as a function of N . It isknown that the electric field created by different electrostaticpotentials at the C − B and C − N interfaces is associated withhalf-semimetallicity in ZGNRs/BN, while the flat-band-likecharacter in the edge states of ZGNRs/graphane induces amagnetic instability due to the enhanced DOS at E F . Thecalculated stabilization energies of half-semimetallicity andAFM order in ZGNRs/BN and ZGNRS/graphane relative tothe corresponding NM configuration are given as a functionof N in Table I. In ZGNRs/BN, we obtain the NM groundstate for N ≤ N ≥
7, while the half-semimetallic groundstate at N = 5 and 6. This trend showing that wide-(or ex-tremely narrow) and intermediate-width ZGNRs are stabilizedas the NM and half-semimetallic configurations, respectively, is consistent with previous DFT study. Note that the half-semimetallic configuration at N = 5 and 6 is only a few meVlower in energy than the corresponding NM configuration(see Table I), indicating that the interedge interaction produc-ing half-semimetallicity in ZGNRs/BN is very weak. On theother hand, in ZGNRs/graphane, we obtain the AFM groundstate for N ≥
2, where the total-energy difference D E NM − AFM between the NM and AFM configurations monotonically in-creases as N increases, reaching ∼
73 meV at N = 8 (see TableI). These results obviously indicate that the interedge interac-tion in ZGNRs/graphane is long-range and strong compared tothat in ZGNRs/BN. It is remarkable that the geometric sym- L RL R (a)(b)
FIG. 3: (Color online) Calculated AFM band structures of (a)ZGNR/BN and (b) ZGNR/graphane for N = 5. The energy zerorepresents the Fermi level. The spin densities of ZGNR/BN andZGNR/graphane are also given. The inset in (a) magnifies the bandgap of the spin-down bands. The spin densities are drawn with anisosurface of 0.02 ( − . metry of two edges in ZGNRs embedded in either h -BN sheetor graphane plays an important role in determining the rangeand strength of interedge interaction.Figure 3(a) and 3(b) show the comparison of thespin-polarized band structures of the ZGNR/BN andZGNR/graphane with N = 5. In the former system, the spin-up and spin-down bands open a gap of 0.14 and 0.02 eV[see Fig. 3(a)], respectively. These values of gap openingare much smaller compared to the ZGNR/graphane systemwhere the spin-up and spin-down bands open an identicalband gap of 0.51 eV [see Fig. 3(b)]. Here, the much smallerband gap in the half-semimetallic ZGNR/BN compared to theAFM ZGNR/graphane gives rise to a much smaller value of D E NM − HS = 1.7 meV than D E NM − AFM = 59.5 meV (see TableI). It is interesting to notice that there is a subtle difference ofthe spin characters between the half-semimetallic ZGNR/BNand the AFM ZGNR/graphane. As shown in Fig. 3(a) and3(b), the spin density of the former system is relatively well-localized around the two edges, whereas that of the latter sys-tem shows some extension up to the middle of the ribbon.This reflects relatively short-range (weak) versus long-range(strong) interedge spin-spin interactions in ZGNRs/BN andZGNRs/graphane.To understand the microscopic mechanism for thehalf-semimetallicity and AFM order in ZGNRs/BN andZGNRs/graphane, we plot, in Fig. 4(a) and 4(b), the spin-polarized local DOS projected onto the two edge C atoms [inthe left (L) or right (R) edge site in Fig. 1] together with theirspin characters. For the ZGNR/BN with N = 5, it is seen thatthe occupied (unoccupied) spin-up and spin-down edge statesare localized at the L (R) edge [see Fig. 4(a)]. However, forthe ZGNR/graphane with N = 5, the occupied (unoccupied)spin-up and spin-down edge states are localized at the L (R)and R (L) edges, respectively [see Fig. 4(b)]. Since electronicstates with the same spin direction can hybridize with eachother, the hybridization occurs between the occupied and un-occupied spin-up or spin-down states localized at the L andR (or R and L) edges. This kind of exchange interaction be-tween the occupied and unoccupied states is characterized asa superexchange mechanism. Such an interedge superex-change interaction leads to a relatively long-range, stronginteredge spin-spin interaction in ZGNRs/graphane, therebygiving rise to a large energy gain in D E NM − AFM as well as alarge gap opening, as shown in Table I.In summary, using first-principles DFT calculations, wehave performed a comparative study of the electronic struc-tures of ultranarrow ZGNRs/BN and ZGNRs/graphane. Suchembedded ZGNRs in h -BN sheet and graphane are foundto exhibit drastically different electronic characteristics. Un-like ZGNRs/graphane, whose NM configuration exhibits par-tially flat bands at E F as N increases, ZGNRs/BN do nothave such a flat-band-like character. Consequently, the for-mer ZGNRs show a magnetic instability due to the enhanced DOS at E F , whereas the latter ZGNRs preserve the NMground state except at N = 5 and 6 with a half-semimetallicstate. We revealed that the disparate magnetic properties ofthe two classes of ZGNRs can be traced to the different fea-tures of their interface structures: i.e., unlike the symmetric -2 0 2DOS (states/eV) -2 0 2DOS (states/eV) E ne r g y ( e V ) (a) (b)L R L R FIG. 4: (Color online) The spin-polarized local DOS projected ontothe two edge C atoms [in the left (L) or right (R) edge site in Fig. 1]of (a) ZGNR/BN and (b) ZGNR/graphane with N = 5. The energyzero represents Fermi level. The charge characters of the spin-up andspin-down states for the occupied and the unoccupied band are takenat the X point with an isosurface of 0.04 ( − interface structure of ZGNRs/graphane, the asymmetric inter-face structure of ZGNRs/BN produces the inequivalent edgestates on both sides of the nanoribbons, giving rise to a rela-tively short-range, weak interedge superexchange interaction.The resulting different electronic and magnetic properties ofZGNRs/BN and ZGNRs/graphane may be utilized for the ap-plication of nano-scale electronic devices such as conductingwires or field-effect transistors. Acknowledgement.
This work was supported by the NationalResearch Foundation of Korea (NRF) grant funded by the Ko-rea Government (MSIP) (Grant No. 2014M2B2A9032247)and the Korea − Sweden Research Cooperation Programme ofthe NRF (Grant No. 2011-0031286) and the Swedish Founda-tion for International Cooperation in Research and Higher Ed-ucation (STINT, Grant No. 2011/036). R.H.S. acknowledgessupport from the Swedish Research Council (VR, Grant No.621-2009-3628). The calculations were performed by KISTIsupercomputing center through the strategic support program(KSC-2014-C3-049) for the supercomputing application re-search. ∗ Corresponding author: [email protected] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys.Soc. Jpn. , 1920 (1996). K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus,Phys. Rev. B , 17954 (1996). Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. ,216803 (2006). Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature (London) ,347 (2006). M. Y. Han, B. ¨Ozyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. , 206805 (2007). J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg,M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. M¨ullen, and R.Fasel, Nature (London) , 470 (2010). D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda,A. Dimiev, B. K. Price, and J. M. Tour, Nature (London) , 872(2009). L. Jiao, X. Wang, G. Diankov, H. Wang, and H. Dai, Nat. Nan-otech. , 321 (2010). C. Tao, L. Jiao, O. V. Yazyev, Y.-C. Chen, J. Feng, X. Zhang, R.B. Capaz, J. M. Tour, A. Zettl, S. G. Louie, H. Dai, and M. F.Crommie, Nat. Phys. , 616 (2011). E. J. Kan, Z. Li, J. Yang, and J. G. Hou, Appl. Phys. Lett. ,243116 (2007). O. Hod, V. Barone, J. E. Peralta, and G. E. Scuseria, Nano Lett. , 2295 (2007). D. Gunlycke, J. Li, J. W. Mintmire, and C. T. White, Appl. Phys.Lett. , 112108 (2007). E. J. Kan, Z. Li, J. Yang, and J. G. Hou, J. Am. Chem. Soc. ,4224 (2008). M. Wu, X. Wu, Y. Gao, and X. C. Zeng, Appl. Phys. Lett. ,223111 (2009). Y. Ding, Y. Wang, and J. Ni, Appl. Phys. Lett. , 123105 (2009). J. M. Pruneda, Phys. Rev. B , 161409(R) (2010). S. Jungthawan, S. Limpijumnong, and J.-L. Kuo, Phys. Rev. B , 235424 (2011). Z. Liu, L. Ma, G. Shi, W. Zhou, Y. Gong, S. Lei, X. Yang, J.Zhang, J. Yu, K. P. Hackenberg, A. Babakhani, J. C. Idrobo, R.Vajtai, J. Lou, and P. M. Ajayan, Nat. Nanotech. , 119 (2013). L. Liu, J. Park, D. A. Siegel, K. F. Mccarty, K. W. Clark, W.Deng, L. Basile, J. C. Idrobo, A.-P. Li, and G. Gu, Science ,163 (2014). Y. Wang, X. Xu, J. Lu, M. Lin, Q. Bao, B. ¨Ozyilmaz, and K. P.Loh, ACS Nano , 6146 (2010). J. O. Sofo, A. S. Chaudhari, and G. D. Barber, Phys. Rev. B ,153401 (2007). A. K. Singh and B. I. Yakobson, Nano Lett. , 1540 (2009). J.-H. Lee and J. C. Grossman, Appl. Phys. Lett. , 133102(2010). H.-J. Kim, S. Oh, C. Zeng, and J.-H. Cho, J. Phys. Chem. C ,13795 (2012). V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K.Reuter, and M. Scheffler, Comput. Phys. Commun. , 2175(2009). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. ,3865 (1996); , 1396(E) (1997). M. Topsakal, E. Akt¨urk, and S. Ciraci, Phys. Rev. B , 115442(2009). J. B. Goodenough, Phys. Rev. , 564 (1955). J. Kanamori, J. Phys. Chem. Solids , 87 (1959). K. Sato, L. Bergqvist, J. Kudrnovsk´y, P. H. Dederichs, O. Eriks-son, I. Turek, B. Sanyal,G. Bouzerar,H. Katayama-Yoshida, V. A.Dinh, T. Fukushima, H. Kizaki, and R. Zeller, Rev. Mod. Phys.82