Electronic band structure and magnetic states of zigzag graphene nanoribbons: quantum chemical calculations

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1 Electronic band structure and magnetic states of zigzag graphene nanoribbons: quantum chemical calculations Nikolai A. Poklonski Eugene F. Kislyakov Sergey A. Vyrko Oleg N. Bubel Sergey V. Ratkevich

2 Electronic band structure and magnetic states of zigzag graphene nanoribbons: quantum chemical calculations Nikolai A. Poklonski, Eugene F. Kislyakov, Sergey A. Vyrko, Oleg N. Bubel, and Sergey V. Ratkevich Belarusian State University, pr. Nezavisimosti 4, Minsk , Belarus Abstract. Quantum-chemical semi-empirical molecular-orbital calculations of zigzag graphene nanoribbons (nzgnrs) were done for the number of zigzag carbon chains n ¼ 4 and 10. The antiferromagnetic (AFM) nature of zgnrs ground state was confirmed. The energy difference between AFM and ferromagnetic (FM) states was calculated, and dimerization patterns of their chemical bond lengths were elucidated. The electron energy band structure calculations show that narrow nanoribbon (4 zgnr) is semiconducting in both AFM and FM states. For wider nanoribbon (10 zgnr), the AFM state is semiconducting ( 0.1 ev band gap), whereas the FM state is half-metallic (electrical conduction with only one spin orientation) Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: /1.JNP ] Keywords: graphene nanoribbons; electron energy band structure; magnetic states. Paper 12108SS received Aug. 6, 2012; revised manuscript received Oct. 3, 2012; accepted for publication Oct. 9, 2012; published online Oct. 29, Introduction Recently discovered graphene, 1,2 a single atomic layer of graphite, is a promising 3 5 material for flat and flexible electronics. Termination of infinite graphene layer to finite (nanosized) structures gives new properties to graphene nanostructures, determined by their boundaries. Two kinds of straight regular graphene boundaries are possible: zigzag and armchair (see Fig. 1). An armchair boundary is formed by two different types (A and B) of carbon atoms of twodimensional (2-D) graphene lattice, whereas a zigzag boundary contains only one type of carbon atoms. This leads to formation of localized π-electron edge states with flat bands near the Fermi level in the case of a zigzag boundary; however, they are absent for an armchair boundary. 6 The singularity in local density of states at the Fermi level causes instability that leads to ferromagnetic ordering of π-electrons at zigzag boundary. This is an example of Stoner ferromagnetism. 7 The peak in the local density of states near the Fermi level, corresponding to edge states of a zigzag boundary, has been observed experimentally. 8 Termination of dangling bonds of graphene edge atoms by σ-bonds of hydrogen atoms does not influence graphene edge state due to its π-electron nature. Another cause of difference in the properties of graphene nanostructures from 2-D infinite graphene is a lateral confinement of their π-electron wave functions. Zigzag graphene nanoribbons (zgnr), see Fig. 2, are one of the most interesting graphene nanostructures from the fundamental point of view 9 due to their unique and peculiar edge states. It is an example of pure carbon ferromagnetism. 10 Their ground state, according to Lieb s theorem, 11 is antiferromagnetic (spins at the two boundaries belong to different types of carbon atoms and have opposite directions). However, ferromagnetic (FM) state (spins at opposite boundaries have the same directions) is not very high in energy and can be achieved by external fields. 12 This is promising for applications, but a thorough quantitative information about zgnr structure is needed /2012/$ SPIE Journal of Nanophotonics Vol. 6, 2012

3 Fig. 1 Two types of edges in graphene nanoribbons: (a) zigzag edge and (b) armchair edge. The edge is indicated by bold line. The light and dark circles denote the A and B-site carbon atoms, respectively. Fig. 2 Scheme of nzgnr built up of n zigzag carbon chains connected by C C bonds. Big circles are carbon atoms, small circles are hydrogen atoms, and l t is the translational period along a zigzag chain. Unit cell is shown by dotted line. Many theoretical studies of zgnrs exist, as well as less experimental ones (see Refs. 3 and 4, and references therein). The calculations of finite 13 and infinite zgnrs with different types of edges, 14,15 on different substrates, 16,17 and with some types of defects 18 and mechanical bending 19 have been done. Simple tight-binding calculations 6,20 are insufficient for quantitative characterization of zgnrs. For example, they give zero gap in electronic band structure of all zgnrs, but they are semiconductors in experiments. On the other hand, first principles calculations with full geometry optimization of sufficiently large zgnrs are too expensive, and DFT (density functional theory) calculations underestimate band gaps. Therefore, the question arises: how well can a relatively low cost semi-empirical quantum-chemical method catch the main physical properties of zgnrs? To obtain this information, we performed quantum chemical calculations of zgnrs using the semi-empirical molecular-orbital PM3 method. 21 Our calculations were done for 1-D periodic infinite nanoribbons of different width using Born von Karman boundary conditions along the nanoribbon axis. We calculated antiferromagnetic (AFM) and FM ground states of zgnrs for different widths (n ¼ 4 and 10) to elucidate the dependence of their properties on nanoribbon width. The geometry, energies, and electronic energy band structures were calculated. The half-metallicity of 10 zgnr in FM ground state was found. 2 Methodology Semi-empirical molecular-orbital PM3 method 21 is a comparatively low-cost computational method which provides the opportunity to investigate sufficiently large carbon systems with acceptable accuracy. It takes into account four valence electrons for every carbon atom, i.e., Journal of Nanophotonics Vol. 6, 2012

4 σ-bonds as well as π-bonds. Previously we tested it on many kinds of carbon nanostructures. It gives excellent agreement with precise experimental data on bond length difference in fullerene C 60 (Ref. 22) and qualitatively true results for carbon nanotubes Here, we employ realization of PM3 method in computer program complex MOPAC 26 for computing zgnrs at zero temperature. We calculated infinitely long hydrogen terminated zgnrs (see Fig. 2) using periodic Born von Karman boundary conditions along the nanoribbon axis. 27 The one-dimensional (1-D) period of zgnr (unit cell) is shown in Fig. 2. A computational cell consists of several unit cells. We tried different numbers of unit cells in a computational cell to ensure results independent from computational cell size. Because we were interested in the spin degree of freedom, we used unrestricted Hartree Fock (UHF) method with separate calculations of spin-up and spin-down molecular orbitals. We used the UHF method also for singlet state calculations to compare its energy with the energies of other multiplicity states. In this case, spin-up and spin-down orbitals coincide, but their energies differ significantly from the results of restricted Hartree Fock (RHF) calculations of the same state. According to the theory, the RHF calculations are appropriate for singlet states only. For given multiplicity, we carried calculations with full geometry optimization (without any symmetry constraints), which always resulted in flat structures. 3 Results 3.1 Geometry and Energy of zgnrs The results of the PM3 calculations of the ground AFM and FM states of 4 zgnr (consisting of four zigzag carbon chains) are shown in Fig. 3. The energy difference between these states is 0.2 ev atom. So, energetically they are well separated (even at room temperatures). Dimerization of chemical bond lengths in such narrow nanoribbons is evident from Fig. 3. The patterns of this dimerization are different for AFM and FM states. In AFM state, short bonds are directed along nanoribbon axis, whereas in FM state they are perpendicular to it. Bonds in edge chains are shortened due to hydrogen termination. Both AFM and FM states have mirror symmetry plane σ, containing C 2 symmetry axis. For wider nanoribbon (10 zgnr), in ground AFM state the bond length difference diminishes (bond lengths approach infinite graphene limit nm), but FM state is asymmetric Fig. 3 Computational cells for 4 zgnr: (a) singlet (AFM) state (l ¼ nm) and (b) nonet (FM) state (l ¼ nm). Big circles are carbon atoms (C), small circles are hydrogen atoms (H), l ¼ 4l t is the length of the computational cell, and l t is the translational period along a zigzag chain. Bond lengths are indicated in ångströms. Dashed line shows both mirror plane σ and C 2 axis. Journal of Nanophotonics Vol. 6, 2012

5 Fig. 4 Computational cells for 10 zgnr: (a) singlet (AFM) state (l ¼ nm) and (b) nonet (FM) state (l ¼ nm). Notations are the same as Fig. 3. (see Fig. 4). The energy difference between AFM and FM states also diminishes and is equal to 0.1 ev atom. 3.2 Electron Energy Band Structures of zgnrs in AFM and FM States In tight binding approximation, taking into account only nearest-neighbor interactions and ignoring possible bond length dimerization, electron energy band structure of nzgnrs has the form shown in Fig. 5 (e.g., see Ref. 9). Note, that independently of the ribbon width n, the highest occupied and lowest unoccupied bands merge at the Brillouin zone boundary and correspond here to totally edge localized π-electron states. The results of PM3 calculations of electron energy band structures of 4 zgnr and 10 zgnr in AFM ground states computed by RHF method are shown in Fig. 6. The number of bands are equal to the number of valence electrons in a unit cell. It is seen that even for 10 zgnr, a band gap of 0.1 ev remains. This is consistent with first principles calculations of Ref. 12 and others. The PM3 calculated band structures for spin-up and spin-down electrons of 4 zgnr in its FM ground state computed by the UHF method are shown in Fig. 7. In the FM state of 4 zgnr, one of the electrons of the highest occupied bands of its ground AFM state changes spin orientation. According to the Pauli principle within this band two electrons cannot be in states with the identical orientation. Therefore, one electron occupies the nearest empty conduction band, which shifts down and becomes a valence band. Simultaneously, for the electrons with the opposite spin Journal of Nanophotonics Vol. 6, 2012

6 Fig. 5 π-electron band structure of 8 zgnr in the tight-binding approximation (e.g., see Ref. 9). Here and in figures below, the horizontal dashed line shows Fermi level E F for π-electrons. The highest occupied band for oppositely oriented π-electrons is denoted with arrows. (All C C bond lengths are nm.) Fig. 6 Band structures of the ground AFM state of 4 zgnr (a) and 10 zgnr (b) calculated by RHF PM3: (a) l t ¼ nm, and (b) l t ¼ nm. Here and in figures below π and σ-bands close to Fermi level E F are shown. The highest occupied band for oppositely oriented π-electrons is denoted with arrows. orientation, the former highest valence band loses one electron, shifts up, and becomes a conduction band merged with the rest of the conduction bands at the Brillouine zone boundary. For 4 zgnr in the FM state, band gaps exist for both spin orientations. The results of UHF PM3 calculations for 10 zgnr FM ground state are shown in Fig. 8. In this case, the system is a typical half-metal as defined in Refs. 12 and 28, that is, it is insulating for one spin orientation, and conducting for another. The band gap for insulating spin orientation is narrower in 10 zgnr ( 2.1 ev), than in 4 zgnr ( 5.5 ev). Therefore, half-metallicity of zgnr FM state is a property that is dependent on nanoribbon width. There may be a finite range of zgnr widths that correspond to half-metallicity. The problem of correct determination of this range should be further investigated. Journal of Nanophotonics Vol. 6, 2012

7 Fig. 7 Band structure and density of states for 4 zgnr in FM state calculated by UHF PM3: (a) bands for spin-up states and (b) bands for spin-down states; l t ¼ nm. For numerical evaluation of DOS, a Gaussian broadening of energy bands of 0.05 ev was assumed. The highest occupied band for each spin orientation is denoted with an arrow. Fig. 8 Band structure and density of states for 10 zgnr in FM state calculated by UHF PM3: (a) bands for spin-up states and (b) bands for spin-down states; l t ¼ nm. For numerical evaluation of DOS, a Gaussian energy band broadening of 0.05 ev was assumed. Arrows indicate: (a) the highest occupied band for spin-up orientation, and (b) the partly occupied bands for spin-down orientation. 4 Conclusions Quantum-chemical semi-empirical molecular-orbital PM3 calculations confirm the AFM ground state and half-metallic nature of FM state for nzgnrs with small n (of about 10). The energy of AFM state of 4 zgnr was found to be lower by 0.2 ev atom than the FM state (for 10 zgnr, by 0.1 ev atom). These states differ in their geometrical structures considerably. Calculations of electron energy band structures of zgnr FM and AFM states show that the AFM state is semiconducting, with the band gap decreasing with the number n of zigzag carbon chains in nzgnr (from 1.8 ev for 4 zgnr to 0.1 ev for 10 zgnr), whereas the FM state is halfmetallic for sufficiently large n. The conductivity type change due to AFM FM transition of zgnr can be used for example in spin switches and filters. Journal of Nanophotonics Vol. 6, 2012

8 Acknowledgments This work was partially supported by the Belarusian Republican Foundation for Fundamental Research (Grants F11V-001 and F12R-178) and Scientific Program Convergence. Author contributions: N.A.P. set the problem of magnetism of infinite narrow zgnrs. O.N.B., S.A.V., and S.V.R. performed the PM3 calculations. N.A.P., E.F.K., S.A.V., and S.V.R. contributed to the analysis of the half-metallicity of zigzag nanoribbons and wrote the original manuscript. All authors commented on the paper and approved the final paper. References 1. A. K. Geim, Nobel lecture: random walk to graphene, Rev. Mod. Phys. 83(3), (2011), 2. K. S. Novoselov, Nobel lecture: graphene: materials in the Flatland, Rev. Mod. Phys. 83(3), (2011), 3. S. K. Pati, T. Enoki, and C. N. R. Rao, Eds., Graphene and Its Fascinating Attributes, p. 270, World Scientific, New Jersey (2011). 4. A. V. Rozhkov et al., Electronic properties of mesoscopic graphene structures: charge confinement and control of spin and charge transport, Phys. Rep. 503(2 3), (2011), 5. S. V. Morozov, New effects in graphene with high carrier mobility, Phys. Usp. 55(4), (2012), 6. M. Fujita et al., Peculiar localized state at zigzag graphite edge, J. Phys. Soc. Jpn. 65(7), (1996), 7. E. C. Stoner, Collective electron ferromagnetism, Proc. R. Soc. Lond. A 165(922), (1938), 8. Y. Niimi et al., Scanning tunneling microscopy and spectroscopy of the electronic local density of states of graphite surfaces near monoatomic step edges, Phys. Rev. B 73(8), (2006), 9. S. M.-M. Dubois et al., Electronic properties and quantum transport in graphene-based nanostructures, Eur. Phys. J. B 72(1), 1 24 (2009), e O. V. Yazyev and M. I. Katsnelson, Magnetic correlations at graphene edges: basis for novel spintronics devices, Phys. Rev. Lett. 100(4), (2008), E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62(10), (1989), Y.-W. Son, M. L. Cohen, and S. G. Louie, Half-metallic graphene nanoribbons, Nature 444(7117), (2006), X. H. Zheng and W. Duley, First-principles study of edge chemical modifications in graphene nanodots, Phys. Rev. B 78(4), (2008), J. Kang, F. Wu, and J. Li, Doping induced spin filtering effect in zigzag graphene nanoribbons with asymmetric edge hydrogenation, Appl. Phys. Lett. 98(8), (2011), B. Xu et al., Electronic and magnetic properties of zigzag graphene nanoribbon with one edge saturated, Appl. Phys. Lett. 96(16), (2010), K. Sawada, F. Ishii, and M. Saito, Magnetism in graphene nanoribbons on Ni(111): firstprinciples density functional study, Phys. Rev. B 82(24), (2010), /PhysRevB Z. Zhang et al., Tuning the magnetic and electronic properties of bilayer graphene nanoribbons on Si(001) by bias voltage, Phys. Rev. B 81(15), (2010), /PhysRevB M. Topsakal et al., First-principles approach to monitoring the band gap and magnetic state of a graphene nanoribbon via its vacancies, Phys. Rev. B 78(23), (2008), Journal of Nanophotonics Vol. 6, 2012

9 19. X. Hu, L. Sun, and A. V. Krasheninnikov, Tuning electronic and magnetic properties of zigzag graphene nanoribbons by large-scale bending, Appl. Phys. Lett. 100(26), (2012), J. J. Palacios et al., Electronic and magnetic structure of graphene nanoribbons, Semicond. Sci. Technol. 25(3), (2010), 3/ J. J. P. Stewart, Optimization of parameters for semiempirical methods. I. Method, J. Comp. Chem. 10(2), (1989), O. N. Bubel et al., Totally symmetric vibrational modes of fullerene C 60, JETP Lett. 71(12), (2000), N. A. Poklonski et al., Uniaxially deformed (5,5) carbon nanotube: structural transitions, Chem. Phys. Lett. 464(4 6), (2008), N. A. Poklonski et al., Totally symmetric vibrations of armchair carbon nanotubes, Comput. Mater. Sci. 49(4 Suppl. 1), 231S 234S (2010), N. A. Poklonski et al., Structural phase transition and band gap of uniaxially deformed (6,0) carbon nanotube, Chem. Phys. Lett. 545, (2012), j.cplett J. J. P. Stewart, MOPAC2009, Stewart Computational Chemistry, Colorado Springs, CO, (2009). 27. J. J. P. Stewart, MNDO cluster model calculations on organic polymers, New Polym. Mater. 1(1), (1987). 28. M. I. Katsnelson et al., Half-metallic ferromagnets: from band structure to many-body effects, Rev. Mod. Phys. 80(2), (2008), Nikolai A. Poklonski is a professor at Belarusian State University. He received his Dr. Sci. degree in semiconductor physics from Belarusian State University in He is the author of more than 175 journal papers. His current research interests include semiconductor physics and electronics, radiospectroscopy of diamond, and physics of carbon nanosystems. Eugene F. Kislyakov is a senior scientific researcher at Belarusian State University. He received degrees in physics from Moscow State University (PhD in nuclear physics 1982). He is the author of more than 50 journal papers. His current research interests include carbon nanosystems. Sergey A. Vyrko is a senior scientific researcher at Belarusian State University. He received his PhD in semiconductor physics from Belarusian State University in He is the author of more than 25 journal papers. His current research interests include physics of phase transitions in semiconductors, hopping conductivity, and physics of carbon nanostructures. Journal of Nanophotonics Vol. 6, 2012

10 Oleg N. Bubel is an associate professor at Belarusian State University. He received his PhD in organic chemistry from Belarusian State University in He is the author of more than 170 journal papers. He is an expert in quantum chemistry. His current research interests include carbon nanosystems. Sergey V. Ratkevich is a leading engineer at Belarusian State University and assistant professor at the Belarusian State University of Informatics and Radioelectronics. He received his specialist diploma (1999) in theoretical physics from Belarusian State University. He is the author of 5 journal papers. His current research interests include carbon nanosystems. Journal of Nanophotonics Vol. 6, 2012