Abstract
Atomically precise graphene nanoribbons promise to combine the outstanding electronic properties of graphene with an electronic bandgap that is sufficiently large for digital-logic applications at room temperature. After a brief introduction to nanoribbon nomenclature, experimental synthesis, and characterization, this chapter guides through the theoretical toolbox available for modeling their electronic structure, starting from simple empirical models and ending with state-of-the-art ab initio methods.
This is a preview of subscription content, log in via an institution to check access.
References
Bennett PB, Pedramrazi Z, Madani A, Chen YC, de Oteyza DG, Chen C, Fischer FR, Crommie MF, Bokor J (2013) Bottom-up graphene nanoribbon field-effect transistors. Appl Phys Lett 103(25):253114. https://doi.org/10.1063/1.4855116
Berger C, Song Z, Li X, Wu X, Brown N, Naud C, Mayou D, Li T, Hass J, Marchenkov AN, Conrad EH, First PN, de Heer WA (2006) Electronic confinement and coherence in patterned epitaxial graphene. Science 312(5777):1191–1196. https://doi.org/10.1126/science.1125925
Berland K, Cooper VR, Lee K, Schröder E, Thonhauser T, Hyldgaard P, Lundqvist BI (2015) Van der Waals forces in density functional theory: a review of the vdW-DF method. Rep Prog Phys 78(6):066501. https://doi.org/10.1088/0034-4885/78/6/066501
Biller A, Tamblyn I, Neaton JB, Kronik L (2011) Electronic level alignment at a metal-molecule interface from a short-range hybrid functional. J Chem Phys 135(16):164706. https://doi.org/10.1063/1.3655357
Born M, Huang K (1954) Dynamical theory of crystal lattices. Clarendon press, Oxford
Born M, Oppenheimer JR (1927) Zur Quantentheorie der Molekeln. Ann Phys 389(20):457–84. https://doi.org/10.1002/andp.19273892002
Bostwick A, Ohta T, Seyller T, Horn K, Rotenberg E (2006) Experimental determination of the spectral function of graphene. arXiv:cond-mat/0609660
Boykin TB, Luisier M, Klimeck G, Jiang X, Kharche N, Zhou Y, Nayak SK (2011) Accurate six-band nearest-neighbor tight-binding model for the π-bands of bulk graphene and graphene nanoribbons. J Appl Phys 109(10):104304. https://doi.org/10.1063/1.3582136
Brinkman WF, Rice TM (1970) Application of Gutzwiller’s variational method to the metal-insulator transition. Phys Rev B 2(10):4302–4304. https://doi.org/10.1103/PhysRevB.2.4302
Cai J, Ruffieux P, Jaafar R, Bieri M, Braun T, Blankenburg S, Muoth M, Seitsonen AP, Saleh M, Feng X, Müllen K, Fasel R (2010) Atomically precise bottom-up fabrication of graphene nanoribbons. Nature 466(7305):470–3. https://doi.org/10.1038/nature09211
Cannuccia E, Marini A (2012) Zero point motion effect on the electronic properties of diamond, trans-polyacetylene and polyethylene. Eur Phys JB 85(9):320
Cardona M, Thewalt MLW (2005) Isotope effects on the optical spectra of semiconductors. Rev Mod Phys 77(4):1173–1224. https://doi.org/10.1103/RevModPhys.77.1173
Chen Z, Zhang W, Palma CA, Lodi Rizzini A, Liu B, Abbas A, Richter N, Martini L, Wang XY, Cavani N, Lu H, Mishra N, Coletti C, Berger R, Klappenberger F, Kläui M, Candini A, Affronte M, Zhou C, De Renzi V, del Pennino U, Barth JV, Räder HJ, Narita A, Feng X, Müllen K (2016) Synthesis of graphene nanoribbons by ambient-pressure chemical vapor deposition and device integration. J Am Chem Soc 138(47):15488–15496. https://doi.org/10.1021/jacs.6b10374
Cheng R, Bai J, Liao L, Zhou H, Chen Y, Liu L, Lin YC, Jiang S, Huang Y, Duan X (2012) High-frequency self-aligned graphene transistors with transferred gate stacks. Proc Nat Acad Sci 109(29):11588–11592. https://doi.org/10.1073/pnas.1205696109
Chou CP, Witek HA (2012) An algorithm and FORTRAN program for automatic computation of the zhang – zhang polynomial of benzenoids. MATCH Commun Math Comput Chem 68:3–30
Chou CP, Witek HA (2014) ZZDecomposer: a graphical toolkit for analyzing the zhang-zhang polynomials of benzenoid structures. MATCH Commun Math Comput Chem 71:741–764
Clar E (1972) The aromatic sextet. Wiley, London
Coh S, Vanderbilt D (2017) Python Tight Binding (PythTB). http://www.physics.rutgers.edu/pythtb
Corso M, Carbonell-Sanromà E, de Oteyza DG (2017) Bottom-up fabrication of atomically precise graphene nanoribbons. arXiv:171103434 [cond-mat]
Denk R, Hohage M, Zeppenfeld P, Cai J, Pignedoli CA, Söde H, Fasel R, Feng X, Müllen K, Wang S, Prezzi D, Ferretti A, Ruini A, Molinari E, Ruffieux P (2014) Exciton-dominated optical response of ultra-narrow graphene nanoribbons. Nat Commun 5(May):4253. https://doi.org/10.1038/ncomms5253
Dvorak M, Wei SH, Wu Z (2013) Origin of the variation of exciton binding energy in semiconductors. Phys Rev Lett 110(1):016402. https://doi.org/10.1103/PhysRevLett.110.016402
Engel E, Dreizler RM (2011) Density functional theory. Springer, Berlin/Heidelberg
Essler FHL, Frahm H, Göhmann F, Klümper A, Korepin VE (2005) The one-dimensional hubbard model. Cambridge University Press, Cambridge
Fajtlowicz S, John PE, Sachs H (2005) On maximum matchings and eigenvalues of benzenoid graphs. Croat Chem Acta 78(2):195–201
Feldner H, Meng ZY, Honecker A, Cabra D, Wessel S, Assaad FF (2010) Magnetism of finite graphene samples: mean-field theory compared with exact diagonalization and quantum Monte Carlo simulations. Phys Rev B 81(11):115416. https://doi.org/10.1103/PhysRevB.81.115416
Foulkes WMC (2016) Tight-binding models and coulomb interactions for s, p, and d electrons. In: Quantum materials: experiments and theory: lecture notes of the autumn school on correlated electrons, Forschungszentrum Jülich, oCLC: 974567924
Foulkes WMC, Haydock R (1989) Tight-binding models and density-functional theory. Phys Rev B 39(17):12520–12536. https://doi.org/10.1103/PhysRevB.39.12520
Fujii S, Enoki T (2012) Clar’s aromatic sextet and π-electron distribution in nanographene. Angew Chem 124(29):7348–7353. https://doi.org/10.1002/ange.201202560
Fujita M, Wakabayashi K, Nakada K, Kusakabe K (1996) Peculiar localized state at zigzag graphite edge. J Phys Soc Japan 65(7):1920–1923. https://doi.org/10.1143/JPSJ.65.1920
Garcia-Lastra JM, Thygesen KS (2011) Renormalization of optical excitations in molecules near a metal surface. Phys Rev Lett 106(18):187402. https://doi.org/10.1103/PhysRevLett.106.187402
Godby RW, Schlüter M, Sham LJ (1986) Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron. Phys Rev Lett 56(22):2415–2418. https://doi.org/10.1103/PhysRevLett.56.2415
Govoni M, Galli G (2015) Large scale GW calculations. J Chem Theory Comput 11(6):2680–2696. https://doi.org/10.1021/ct500958p
Grimme S (2011) Density functional theory with London dispersion corrections. Wiley Interdiscip Rev Comput Mol Sci 1(2):211–228. https://doi.org/10.1002/wcms.30
Grüneis A, Attaccalite C, Pichler T, Zabolotnyy V, Shiozawa H, Molodtsov SL, Inosov D, Koitzsch A, Knupfer M, Schiessling J, Follath R, Weber R, Rudolf P, Wirtz L, Rubio A (2008) Electron-electron correlation in graphite: a combined angle-resolved photoemission and first-principles study. Phys Rev Lett 100(3):037601. https://doi.org/10.1103/PhysRevLett.100.037601
Gunlycke D, White CT (2008) Tight-binding energy dispersions of armchair-edge graphene nanostrips. Phys Rev B 77(11):115116. https://doi.org/10.1103/PhysRevB.77.115116
Gunlycke D, Areshkin DA, White CT (2007) Semiconducting graphene nanostrips with edge disorder. Appl Phys Lett 90(14):142104. https://doi.org/10.1063/1.2718515
Günthard HH, Primas H (1956) Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen. Helv Chim Acta 39(6):1645–1653. https://doi.org/10.1002/hlca.19560390623
Gutzwiller MC (1965) Correlation of electrons in a narrow s band. Phys Rev 137(6A):A1726–A1735. https://doi.org/10.1103/PhysRev.137.A1726
Hanke F (2011) Sensitivity analysis and uncertainty calculation for dispersion corrected density functional theory. J Comput Chem 32:1424–30. https://doi.org/10.1002/jcc.21724
Hedin L (1965) New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys Rev 139(3A):A796–A823. https://doi.org/10.1103/PhysRev.139.A796
Hedin L, Lundqvist S (1969) Effects of electron-electron and electron-phonon interactions on the one-electron states of solids. In: Frederick Seitz DT, Ehrenreich H (eds) Solid state physics, vol 23. Academic Press, New York and London, pp 1–181
Huang L, Ogut S (2018) Modeling excited states of confined systems. In: Handbook of materials modeling. Springer, Cham.
Hubbard J (1963) Electron correlations in narrow energy bands. Proc R Soc London A Math Phys Eng Sci 276(1365):238–257. https://doi.org/10.1098/rspa.1963.0204
Hückel E (1933) Die freien Radikale der organischen Chemie. Z Phys 83(9–10):632–668. https://doi.org/10.1007/BF01330865
Huertas-Hernando D, Guinea F, Brataas A (2006) Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Phys Rev B 74(15):155426. https://doi.org/10.1103/PhysRevB.74.155426
Hwang WS, Zhao P, Tahy K, Nyakiti LO, Wheeler VD, Myers-Ward RL, Jr CRE, Gaskill DK, Robinson JA, Haensch W, Xing HG, Seabaugh A, Jena D (2015) Graphene nanoribbon field-effect transistors on wafer-scale epitaxial graphene on SiC substrates a. APL Mater 3(1):011101. https://doi.org/10.1063/1.4905155
Hybertsen MS, Louie SG (1985) First-principle theory of quasiparticles: calculation of band gaps in semiconductors ans insulators. Phys Rev Lett 55(13):1418
Hybertsen MS, Louie SG (1986) Electron correlation in semiconductors and insulators: band gaps and quasiparticle energies. Phys Rev B 34(8):5390–5413. https://doi.org/10.1103/PhysRevB.34.5390
Ito T, Shirakawa H, Ikeda S (1974) Simultaneous polymerization and formation of polyacetylene film on the surface of concentrated soluble Ziegler-type catalyst solution. J Polym Sci Polym Chem Ed 12(1):11–20. https://doi.org/10.1002/pol.1974.170120102
ITRS (2013) Emergent research materials. Technical report ITRS
Jiang X, Kharche N, Kohl P, Boykin TB, Klimeck G, Luisier M, Ajayan PM, Nayak SK (2013) Giant quasiparticle bandgap modulation in graphene nanoribbons supported on weakly interacting surfaces. Appl Phys Lett 103(13):133107. https://doi.org/10.1063/1.4822427
Kawai S (2013) Non-contact AFM image of chevron-type GNR (unpublished)
Kawai S, Saito S, Osumi S, Yamaguchi S, Foster AS, Spijker P, Meyer E (2015) Atomically controlled substitutional boron-doping of graphene nanoribbons. Nat Commun 6:8098. https://doi.org/10.1038/ncomms9098
Kharche N, Meunier V (2016) Width and crystal orientation dependent band gap renormalization in substrate-supported graphene nanoribbons. J Phys Chem Lett 7(8):1526–1533. https://doi.org/10.1021/acs.jpclett.6b00422
Kim K, Choi JY, Kim T, Cho SH, Chung HJ (2011) A role for graphene in silicon-based semiconductor devices. Nature 479(7373):338. https://doi.org/10.1038/nature10680
Kimouche A, Ervasti MM, Drost R, Halonen S, Harju A, Joensuu PM, Sainio J, Liljeroth P (2015) Ultra-narrow metallic armchair graphene nanoribbons. Nat Commun 6:10177. https://doi.org/10.1038/ncomms10177
Klein DJ (1994) Graphitic polymer strips with edge states. Chem Phys Lett 217(3):261–265. https://doi.org/10.1016/0009-2614(93)E1378-T
Knight JW, Wang X, Gallandi L, Dolgounitcheva O, Ren X, Ortiz JV, Rinke P, Körzdörfer T, Marom N (2016) Accurate ionization potentials and electron affinities of acceptor molecules III: a benchmark of GW methods. J Chem Theory Comput 12(2):615–626. https://doi.org/10.1021/acs.jctc.5b00871
Lam SC, Needs RJ (1993) First-principles calculations of the screening of electric fields at the aluminium (111) and (110) surfaces. J Phys Condens Matter 5(14):2101. https://doi.org/10.1088/0953-8984/5/14/009
Lejaeghere K, Bihlmayer G, Björkman T, Blaha P, Blügel S, Blum V, Caliste D, Castelli IE, Clark SJ, Corso AD, de Gironcoli S, Deutsch T, Dewhurst JK, Marco ID, Draxl C, Dułak M, Eriksson O, Flores-Livas JA, Garrity KF, Genovese L, Giannozzi P, Giantomassi M, Goedecker S, Gonze X, Grånäs O, Gross EKU, Gulans A, Gygi F, Hamann DR, Hasnip PJ, Holzwarth NaW, Iuşan D, Jochym DB, Jollet F, Jones D, Kresse G, Koepernik K, Küçükbenli E, Kvashnin YO, Locht ILM, Lubeck S, Marsman M, Marzari N, Nitzsche U, Nordström L, Ozaki T, Paulatto L, Pickard CJ, Poelmans W, Probert MIJ, Refson K, Richter M, Rignanese GM, Saha S, Scheffler M, Schlipf M, Schwarz K, Sharma S, Tavazza F, Thunström P, Tkatchenko A, Torrent M, Vanderbilt D, van Setten MJ, Speybroeck VV, Wills JM, Yates JR, Zhang GX, Cottenier S (2016) Reproducibility in density functional theory calculations of solids. Science 351(6280):aad3000. https://doi.org/10.1126/science.aad3000
Li X, Shi Y, Gutman I (2012) The chemical connection. In: Li X, Shi Y, Gutman I (eds) Graph energy. Springer, New York, pp 11–17
Li Y, Zhang W, Morgenstern M, Mazzarello R (2013) Electronic and magnetic properties of zigzag graphene nanoribbons on the (111) surface of Cu, Ag, and Au. Phys Rev Lett 110(21):216804. https://doi.org/10.1103/PhysRevLett.110.216804
Lieb EH (1989) Two theorems on the hubbard model. Phys Rev Lett 62(16):1927–1927. https://doi.org/10.1103/PhysRevLett.62.1927.5
Liu P, Kaltak M, Klimeš J, Kresse G (2016) Cubic scaling $GW$: towards fast quasiparticle calculations. Phys Rev B 94(16):165109. https://doi.org/10.1103/PhysRevB.94.165109
Llinas JP, Fairbrother A, Barin GB, Shi W, Lee K, Wu S, Choi BY, Braganza R, Lear J, Kau N, Choi W, Chen C, Pedramrazi Z, Dumslaff T, Narita A, Feng X, Müllen K, Fischer F, Zettl A, Ruffieux P, Yablonovitch E, Crommie M, Fasel R, Bokor J (2017) Short-channel field-effect transistors with 9-atom and 13-atom wide graphene nanoribbons. Nat Commun 8(1):633. https://doi.org/10.1038/s41467-017-00734-x
Lowe JP, Kafafi SA, LaFemina JP (1986) Qualitative MO theory of some ring and ladder polymers. J Phys Chem 90(25):6602–6610. https://doi.org/10.1021/j100283a004
Maggio E, Kresse G (2017) $GW$ vertex corrected calculations for molecular systems. arXiv:170601815 [cond-mat]
Marinopoulos AG, Reining L, Rubio A, Vast N (2003) Optical and loss spectra of carbon nanotubes: depolarization effects and intertube interactions. Phys Rev Lett 91(4):046402. https://doi.org/10.1103/PhysRevLett.91.046402
Martin RM (2004) Electronic structure: basic theory and practical methods. Cambridge University Press, Cambridge
Martin RM, Reining L, Ceperley DM (2016) Interacting electrons: theory and computational approaches. Cambridge University Press, New York
Mermin N, Wagner H (1966) Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models. Phys Rev Lett 17(22):1133–1136
Metzner W, Vollhardt D (1989) Correlated lattice fermions in d = ∞ dimensions. Phys Rev Lett 62(3):324–327. https://doi.org/10.1103/PhysRevLett.62.324
Nakada K, Fujita M, Dresselhaus G, Dresselhaus MS (1996) Edge state in graphene ribbons: nanometer size effect and edge shape dependence. Phys Rev B 54(24):17954–17961. https://doi.org/10.1103/PhysRevB.54.17954
Narita A, Feng X, Hernandez Y, Jensen SA, Bonn M, Yang H, Verzhbitskiy IA, Casiraghi C, Hansen MR, Koch AHR, Fytas G, Ivasenko O, Li B, Mali KS, Balandina T, Mahesh S, De Feyter S, Müllen K (2014) Synthesis of structurally well-defined and liquid-phase-processable graphene nanoribbons. Nat Chem 6(2):126–132. https://doi.org/10.1038/nchem.1819
Narita A, Feng X, Müllen K (2015) Bottom-up synthesis of chemically precise graphene nanoribbons. Chem Rec 15(1):295–309. https://doi.org/10.1002/tcr.201402082
Novoselov KS, Geim AK, Morozov SV, Dubonos SV, Zhang Y, Jiang D (2004) Room-temperature electric field effect and carrier-type inversion in graphene films. arXiv preprint cond-mat
Onida G, Schmidt W, Pulci O, Palummo M, Marini a, Hogan C, Del Sole R (2001) Theory for modeling the optical properties of surfaces. Phys Status Solidi (a) 188(4):1233–1242. https://doi.org/10.1002/1521-396X(200112)188:4<1233::AID-PSSA1233>3.0.CO;2-W
Pauling L, Brockway LO, Beach JY (1935) The dependence of interatomic distance on single bond-double bond resonance1. J Am Chem Soc 57(12):2705–9. https://doi.org/10.1021/ja01315a105
Perdew JP, Levy M (1983) Physical content of the exact Kohn-Sham orbital energies: band gaps and derivative discontinuities. Phys Rev Lett 51(6):1884–1887. https://doi.org/10.1103/PhysRevLett.51.1884
Pisana S, Lazzeri M, Casiraghi C, Novoselov KS, Geim AK, Ferrari AC, Mauri F (2007) Breakdown of the adiabatic born–oppenheimer approximation in graphene. Nat Mater 6(3):198–201. https://doi.org/10.1038/nmat1846
Poncé S, Antonius G, Boulanger P, Cannuccia E, Marini a, Côté M, Gonze X (2014) Verification of first-principles codes: comparison of total energies, phonon frequencies, electron–phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo. Comput Mater Sci 83:341–348. https://doi.org/10.1016/j.commatsci.2013.11.031
Prezzi D, Varsano D, Ruini A, Marini A, Molinari E (2008) Optical properties of graphene nanoribbons: the role of many-body effects. Phys Rev B 77(4):041404. https://doi.org/10.1103/PhysRevB.77.041404
Reich S, Maultzsch J, Thomsen C, Ordejón P (2002) Tight-binding description of graphene. Phys Rev B 66(3):035412. https://doi.org/10.1103/PhysRevB.66.035412
Ruffieux P, Wang S, Yang B, Sánchez-Sánchez C, Liu J, Dienel T, Talirz L, Shinde P, Pignedoli CA, Passerone D, Dumslaff T, Feng X, Müllen K, Fasel R (2016) On-surface synthesis of graphene nanoribbons with zigzag edge topology. Nature 531(7595):489–492. https://doi.org/10.1038/nature17151
Rydberg H, Dion M, Jacobson N, Schröder E, Hyldgaard P, Simak S, Langreth D, Lundqvist BI (2003) Van der waals density functional for layered structures. Phys Rev Lett 91(12):126402. https://doi.org/10.1103/PhysRevLett.91.126402
Saito R, Dresselhaus G, Dresselhaus MS (1998) Tight binding calculation of molecules and solids. In: Physical properties of carbon nanotubes. Imperial College Press, London, pp 17–33
Schwierz F (2010) Graphene transistors. Nat Nanotechnol 5(7):487–96. https://doi.org/10.1038/nnano.2010.89
Schwierz F (2013) Graphene transistors: status, prospects, and problems. Proc IEEE 101(7):1567–1584. https://doi.org/10.1109/JPROC.2013.2257633
Son YW, Cohen ML, Louie SG (2006) Energy gaps in graphene nanoribbons. Phys Rev Lett 97(21):216803. https://doi.org/10.1103/PhysRevLett.97.216803
Sorella S, Tosatti E (1992) Semi-metal-insulator transition of the hubbard model in the honeycomb lattice. EPL (Europhys Lett) 19(8):699. https://doi.org/10.1209/0295-5075/19/8/007
Spataru CD (2013) Electronic and optical gap renormalization in carbon nanotubes near a metallic surface. Phys Rev B 88(12):125412. https://doi.org/10.1103/PhysRevB.88.125412
Stampfer C, Güttinger J, Hellmüller S, Molitor F, Ensslin K, Ihn T (2009) Energy gaps in etched graphene nanoribbons. Phys Rev Lett 102(5):056403. https://doi.org/10.1103/PhysRevLett.102.056403
Stein T, Eisenberg H, Kronik L, Baer R (2010) Fundamental gaps in finite systems from eigenvalues of a generalized Kohn-Sham method. Phys Rev Lett 105(26):266802. https://doi.org/10.1103/PhysRevLett.105.266802
Talirz L, Söde H, Cai J, Ruffieux P, Blankenburg S, Jafaar R, Berger R, Feng X, Müllen K, Passerone D, Fasel R, Pignedoli CA (2013) Termini of bottom-up fabricated graphene nanoribbons. J Am Chem Soc 135(6):2060–2063. https://doi.org/10.1021/ja311099k
Talirz L, Ruffieux P, Fasel R (2016a) On-surface synthesis of atomically precise graphene nanoribbons. Adv Mater 28(29):6222–6231. https://doi.org/10.1002/adma.201505738
Talirz L, Shinde P, Passerone D, Pignedoli CA (2016b) Synthesis of atomically precise graphene-based nanostructures: a simulation point of view. In: Gourdon A (ed) On-surface synthesis, advances in atom and single molecule machines. Springer International Publishing, Cham
Talirz L, Söde H, Dumslaff T, Wang S, Sanchez-Valencia JR, Liu J, Shinde P, Pignedoli CA, Liang L, Meunier V, Plumb NC, Shi M, Feng X, Narita A, Müllen K, Fasel R, Ruffieux P (2017) On-surface synthesis and characterization of 9-atom wide armchair graphene nanoribbons. ACS Nano. https://doi.org/10.1021/acsnano.6b06405
Tamblyn I, Darancet P, Quek SY, Bonev SA, Neaton JB (2011) Electronic energy level alignment at metal-molecule interfaces with a GW approach. Phys Rev B 84(20):201402. https://doi.org/10.1103/PhysRevB.84.201402
Tanaka K, Yamabe T (1985) Electronic structure of conductive conjugated systems and their physicochemical properties. In: Löwdin PO (ed) Advances in quantum chemistry, vol 17. Academic Press, pp 251–284. https://doi.org/10.1016/S0065-3276(08)60304-4
Tanaka K, Yamashita S, Yamabe H, Yamabe T (1987) Electronic properties of one-dimensional graphite family. Synth Met 17(1–3):143–148. https://doi.org/10.1016/0379-6779(87)90729-6
Tersoff JD (1989) Method for the calculation of scanning tunneling microscope images and spectra. Phys Rev B 40(17):11990–11993. https://doi.org/10.1103/PhysRevB.40.11990
Thomann H, Dalton LK, Grabowski M, Clarke TC (1985) Direct observation of Coulomb correlation effects in polyacetylene. Phys Rev B 31(5):3141–3143. https://doi.org/10.1103/PhysRevB.31.3141
Trevisanutto PE, Giorgetti C, Reining L, Ladisa M, Olevano V (2008) Ab initio GW many-body effects in graphene. Phys Rev Lett 101(22):226405. https://doi.org/10.1103/PhysRevLett.101.226405
van Schilfgaarde M, Kotani T, Faleev S (2006) Quasiparticle self-consistent GW theory. Phys Rev Lett 96(22):226402. https://doi.org/10.1103/PhysRevLett.96.226402
van Setten MJ, Caruso F, Sharifzadeh S, Ren X, Scheffler M, Liu F, Lischner J, Lin L, Deslippe JR, Louie SG, Yang C, Weigend F, Neaton JB, Evers F, Rinke P (2015) GW100: benchmarking G0W0 for molecular systems. J Chem Theory Comput 11(12):5665–5687. https://doi.org/10.1021/acs.jctc.5b00453
Vo TH, Shekhirev M, Kunkel DA, Morton MD, Berglund E, Kong L, Wilson PM, Dowben PA, Enders A, Sinitskii A (2014) Large-scale solution synthesis of narrow graphene nanoribbons. Nat Commun 5:3189. https://doi.org/10.1038/ncomms4189
Volpilhac G, Hoarau J (1978) Effect of the interlayer interactions on the diamagnetism of graphitic ribbons. Phys Rev B 17(3):1445–1449. https://doi.org/10.1103/PhysRevB.17.1445
Wakabayashi K, Sasaki Ki, Nakanishi T, Enoki T (2010) Electronic states of graphene nanoribbons and analytical solutions. Sci Technol Adv Mater 11(5):054504. https://doi.org/10.1088/1468-6996/11/5/054504
Wallace PR (1947) The band theory of graphite. Phys Rev 71(9):622–634. https://doi.org/10.1103/PhysRev.71.622
Wang W, Yazyev O, Meng S, Kaxiras E (2009) Topological frustration in graphene nanoflakes: magnetic order and spin logic devices. Phys Rev Lett 102(15):157201. https://doi.org/10.1103/PhysRevLett.102.157201
Wang S, Talirz L, Pignedoli CA, Feng X, Müllen K, Fasel R, Ruffieux P (2016a) Giant edge state splitting at atomically precise graphene zigzag edges. Nat Commun 7:11507. https://doi.org/10.1038/ncomms11507
Wang Z, Ki DK, Khoo JY, Mauro D, Berger H, Levitov LS, Morpurgo AF (2016b) Origin and magnitude of ‘designer’ spin-orbit interaction in graphene on semiconducting transition metal dichalcogenides. Phys Rev X 6(4):041020. https://doi.org/10.1103/PhysRevX.6.041020
Wassmann T, Seitsonen AP, Saitta AM, Lazzeri M, Mauri F (2010) Clar’s theory, π-electron distribution, and geometry of graphene nanoribbons. J Am Chem Soc 132(10):3440–3451. https://doi.org/10.1021/ja909234y
White CT, Li J, Gunlycke D, Mintmire JW (2007) Hidden one-electron interactions in carbon nanotubes revealed in graphene nanostrips. Nano Lett 7(3):825–830. https://doi.org/10.1021/nl0627745
Wilhelm J, Golze D, Talirz L, Hutter J, Pignedoli CA (2018) Toward GW calculations on thousands of atoms. J Phys Chem Lett 9(2):306–312. https://doi.org/10.1021/acs.jpclett.7b02740
Yamabe T, Tanaka K, Ohzeki K, Yata S (1982) Electronic structure of polyacenacene. A one-dimensional graphite. Solid State Commun 44(6):823–825. https://doi.org/10.1016/0038-1098(82)90282-4
Yang L, Park CH, Son YW, Cohen M, Louie S (2007) Quasiparticle energies and band gaps in graphene nanoribbons. Phys Rev Lett 99(18):186801. https://doi.org/10.1103/PhysRevLett.99.186801
Yang X, Dou X, Rouhanipour A, Zhi L, Räder HJ, Müllen K (2008) Two-dimensional graphene nanoribbons. J Am Chem Soc 130(13):4216–4217. https://doi.org/10.1021/ja710234t
Yazyev OV (2010) Emergence of magnetism in graphene materials and nanostructures. Rep Prog Phys 73(5):056501. https://doi.org/10.1088/0034-4885/73/5/056501
Yazyev O, Katsnelson M (2008) Magnetic correlations at graphene edges: basis for novel spintronics devices. Phys Rev Lett 100(4):047209. https://doi.org/10.1103/PhysRevLett.100.047209
Zhu X, Su H (2011) Scaling of excitons in graphene nanoribbons with armchair shaped edges. J Phys Chem A 115(43):11998–12003. https://doi.org/10.1021/jp202787h
Acknowledgements
This work was supported by the National Center of Competence in Research MARVEL founded by the Swiss National Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this entry
Cite this entry
Talirz, L., Pignedoli, C.A. (2019). Electronic Structure of Atomically Precise Graphene Nanoribbons. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-50257-1_41-2
Download citation
DOI: https://doi.org/10.1007/978-3-319-50257-1_41-2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50257-1
Online ISBN: 978-3-319-50257-1
eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics
Publish with us
Chapter history
-
Latest
Electronic Structure of Atomically Precise Graphene Nanoribbons- Published:
- 26 June 2019
DOI: https://doi.org/10.1007/978-3-319-50257-1_41-2
-
Original
Electronic Structure of Atomically Precise Graphene Nanoribbons- Published:
- 13 July 2018
DOI: https://doi.org/10.1007/978-3-319-50257-1_41-1