[01] G. R. Ahmed Jamal, Department of Electrical and Electronic Engineering, University of Asia Pacific, Dhaka, Bangladesh. [02] S. M. Mominuzzaman, Department of Electrical and Electronic Engineering, Bangaldesh University of Engineering and Technology, Dhaka, Bangladesh.

In this work, an empirical model of nearest neighbor hopping parameter (γ0) in tight binding (TB) model of single wall carbon nanotubes is proposed in order to calculate first and second optical transition energies of semiconducting SWCNTs. A highly systematic and nearly linear pattern is observed when the γ0, as calculated from experimental optical transition energies of semiconducting SWCNTs, were scaled by a chirality combination term (2n-m) and plotted against tube diameters. Based on this observation, two empirical expressions of γ0 are formulated for mod 1 and mod 2 type semiconducting SWCNTs. In this model of γ0, observations from various optical spectroscopic experiments are incorporated. First and second optical transition energies (E11 and E22) for all semiconducting SWCNTs within diameter range of 0.4 to 3 nm are calculated using this empirical γ0. Calculated values showed excellent agreement with experimental values for all type of chiralities over the full diameter range and precisely reflected the chirality effect on transition energies. The proposed empirical γ0 highly improved the calculation from simplest tight binding model and enables it to give almost accurate qualitative and quantitative prediction of first two transition energies of semiconducting SWCNTs.

Nanotube, SWCNT, Optical Transition Energy, Chiral Index, Hopping Parameter, Empirical

[01] S. Iijima, “Helical microtubules of graphitic carbon”, Nature 354, pp.56 – 58, 1991. [02] S. Iijima and T. Ichihashi, “Single-shell carbon nanotubes of 1-nm diameter”, Nature 363, pp.603 – 605, 1993. [03] A. P. Graham, G.S. Duesberg, W. Hoenlein, F. Kreupl, M. Liebau, R. martin, B. Rajasekharan, W. Pamler, R. Seidel, W. Steinhoegl, E. Unger, “How Do Carbon Nanotubes Fit into the Semiconductor Roadmap?”, Appl. Phys. A, 80, pp.1141–1151, 2005. [04] Kahng, B. Andre , “Scaling: More than Moore's law”, Design and Test of Computers, IEEE, Vol. 27, Issue 3, pp. 86 – 87, 2010.V. N. Popov, “Carbon nanotubes: properties and application”, Materials Science and Engineering R, 43, pp. 61–102, 2004. [05] V. N. Popov, “Carbon nanotubes : properties and application”, Materials Science and Engineering R, 43, pp. 61–102, 2004. [06] T. W. Odom, J. L. Huang, P. Kim, and C. M. Lieber, “Structure and Electronic Properties of Carbon Nanotubes”, J. Phys. Chem. B, 104, pp.2794-2809, 2000. [07] N. Hamada, S. Sawada, and A. Oshiyama, “New one-dimensional conductors: graphitic microtubules,” Phys. Rev. Lett., Vol.68, No.10, pp.1579-1581, 1992. [08] J. W. Mintmire and C. T. White, “Universal density of states for carbon nanotubes”, Phys. Rev. Lett., Vol. 81, No.12, 1998. [09] S. Reich and C. Thomsen, “Chirality dependence of the density-of-states singularities in carbon nanotubes”, Phys. Rev. B, Vol 62, No. 7, 2000. [10] M.S. Dresselhausa, G. Dresselhausc, A. Jorio, A.G. Souza Filho, R. Saito, “Raman spectroscopy on isolated single wall carbon nanotubes”, Carbon, 40, pp.2043–2061, 2002. [11] J. D. Correa, A. J. R. da Silva, and M. Pacheco, “Tight-binding model for carbon nanotubes from ab initio calculations,” J. Phys.: Condens. Matter, Vol.22, No.7, 275503, 2010. [12] R. Kundu, “Tight binding parameters for graphene”, Modern Physics Letters B, Vol. 25, No. 3, pp.163-173, 2011. [13] S. Reich, J. Maultzsch, and C. Thomsen, “Tight-binding description of graphene,” Phys. Rev. B, Vol.66, No.3, pp.035412, 2002. [14] Y. Lian, Y. Maeda, T. Wakahara, T. Akasaka, S. Kazaoui, N. Minami, N. Choi and H. Tokumoto, “Assignment of the Fine Structure in the Optical Absorption Spectra of Soluble Single-Walled Carbon Nanotubes”, J. Phys. Chem. B, 107, 12082-12087, 2003. [15] H. Kataura, Y. Kumazawa, Y. Maniwa, I. Umezu, S. Suzuki, Y. Ohtsuka, and Y.Achiba, “Optical properties of single-wall carbon nanotubes,” Synthetic Met., Vol.103, pp. 2555, 1999. [16] R. B. Weisman and S. M. Bachilo, “Dependence of optical transition energies on structure for single-walled carbon nanotubes in aqueous suspension: an empirical kataura plot,” Nano Lett., Vol.3, No.9, pp.1235-1238, 2003. [17] M. Y Sfeir., T.Beetz, F Wang, L.Huang, X. M. H Huang., M.Huang, J. Hone, S. O’Brien, J. A Misewich, T. F.Heinz, L.Wu, Y.Zhu, L. E. Brus, “ Optical Spectroscopy of Individual Single-Walled Carbon Nanotubes of Defined Chiral Structure”, Science, Vol. 312, April 2006. [18] Bachilo S. M., Strano M. S., Kittrell C., Hauge R. H., Smalley R. E., Weisman R. B., “Structure-Assigned Optical Spectra of Single-Walled Carbon Nanotubes”, Science, Vol 298 No. 5602, pp.2361, 2002. [19] V. Zólyomi and J. Kürti, “First-principles calculations for the electronic band structures of small diameter single-wall carbon nanotubes”, Phys. Rev. B 70, 085403, 2004. [20] V. N. Popov, “Curvature effects on the structural, electronic and optical properties of isolated single-walled carbon nanotubes within a symmetry-adapted non-orthogonal tight-binding model”, New Journal of Physics, Vol. 6, 2004. [21] H. Zeng, H. F. Hu, J. W. Wei, Z. Y. Wang, L. Wang, and P. Peng, “Curvature effects on electronic properties of small radius nanotube,” Appl. Phys. Lett., Vol.91, No.3, pp.033102, 2007. [22] O. Gulseren, T. Yildirim and S. Ciraci, “A systematic ab-initio study of curvature effects in carbon nanotubes”, Phys. Rev. B 65, 153405, 2002. [23] J W Ding, X H Yan, J X Cao, “Analytical relation of band gaps to both chirality and diameter of single-wall carbon nanotubes”, Phys. Rev. B , Vol. 66, Issue 7, Pages: 2-5, 2002. [24] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, “Trigonal warping effect of carbon nanotubes,” Phys. Rev. B, Vol.61, No.4, pp.2981-2990, 2000. [25] Kane C. L. and Mele E. J., “The Ratio Problem in Single Carbon Nanotube Fluorescence Spectroscopy”, Phys. Rev. Lett. 90, 207401, 2003. [26] E.J. Mele, C.L. Kane, “Many body effects in carbon nanotube fluorescence spectroscopy”, Solid State Communications 135, pp. 527–531, 2005. [27] H. Lin, J. Lagoute, V. Repain, C. Chacon, Y. Girard, J.-S. Lauret, F. Ducastelle, A. Loiseau S. Rousset , “Many-body effects in electronic bandgaps of carbon nanotubes measured by scanning tunnelling spectroscopy”, Nature Materials 9, 235–238, 2010. [28] C. D. Spataru, S. I. Beigi, L. X. Benedict and S. G. Louie, “Excitonic Effects and Optical Spectra of Single-Walled Carbon Nanotubes”, AIP Conf. Proc., vol 772, p. 1061-1062, 2004. [29] H. Zhao, S. Mazumdar, “Excitons in semiconducting single-walled carbon nanotubes”, Synthetic Metals, 155, p.250–253, 2005. [30] G. Dukovic, F. Wang, D. Song, M. Y. Sfeir, T. F. Heinz, and L. E. Brus, “Structural dependence of excitonic optical transitions and band-gap energies in carbon nanotubes,” Nano Lett., Vol.5, No.11, pp.2314-2318, 2005. [31] J. X. Cao, X. H. Yan, J. W. Ding and D. L. Wang, “Band structures of carbon nanotubes: the sp3s* tight-binding model”, J. Phys.: Condens. Matter 13, L271–L275, 2001. [32] A. Hagen and T. Hertel, “Quantitative Analysis of Optical Spectra from Individual Single-Wall Carbon Nanotubes”, Nano letters, Vol. 3, No. 3, pp. 383-388, 2003. [33] Z. Wang, H. Zhao, and S. Mazumdar, “Quantitative calculations of the excitonic energy spectra of semiconducting single-walled carbon nanotubes within a π-electron model”, Phys. Rev. B 74, 195406, 2006. [34] J. Maultzsch, H. Telg, S. Reich, and C. Thomsen, “Radial breathing mode of single-walled carbon nanotubes Optical transition energies and chiral-index assignment”, Phys. Rev. B 72, 205438, 2005. [35] H. Yorikawa, S. Muramatsu, “Electronic structure characteristic of carbon nanotubules”, Z. Phys. B, Condensed Matt., vol. 104, pp. 71–76, 1997. [36] A. Jorio, C. Fantini, M. A. Pimenta, R. B. Capaz ,Ge. G. Samsonidze, G. Dresselhaus, M. S. Dresselhaus, J. Jiang, N. Kobayashi, A. Grüneis and R. Saito, “Resonance Raman spectroscopy (n,m)-dependent effects in small-diameter single-wall carbon nanotubes”, Phys. Rev. B 71, 075401, 2005. [37] J. Lefebvre, S. Maruyama and P. Finnie, “Photoluminescence: science and applications”, Topics in Applied Physics, Vol. 111, pp.287-319, 2008. [38] P. K. Valavala, D. Banyai, M. Seel, and R. Pati, “Self-consistent calculations of strain-induced band gap changes in semiconducting (n,0) carbon nanotubes,” Phys. Rev. B, Vol.78, No.23, pp.235430, 2008. [39] A. Jorio, P. Araujo, S. K. Doorn, S. Maruyama, H. Chacham, and M. A. Pimenta, “The Kataura plot over broad energy and diameter ranges,” Phys. Stat. Sol. (b), Vol.243, No.13, pp.3117-3121, 2006. [40] H. Yorikawa and S. Muramatsu, “Energy gaps of semiconducting nanotubles,” Phys. Rev. B, Vol.52, No.4, pp.2723-2727, 1995. [41] H. Yorikawa and S. Muramatsu , “Chirality-dependence of energy gaps of semiconducting nanotubules” , Solid State Communications, Vol. 94, Issue 6, Pages 435-437, 1995. [42] G. Lanzani, L. Luer, “Carbon Nanotubes: Electronic Structure and Spectroscopy”, Comprehensive Nanoscience and Technology, Vol. 1, pp. 23–39, 2011. [43] Ge. G. Samsonidze, R. Saito, N. Kobayashi, A. Grüneis, J. Jiang, A. Jorio, S. G. Chou, G. Dresselhaus and M. S. Dresselhaus, “Family behavior of the optical transition energies in single-wall carbon nanotubes of smaller diameters, Appl. Phys. Lett., Vol 85, No. 23, 2004. [44] Francois Leonard, “The physics of carbon nanotube devices”, Ed: Jeremy Ramsden, William Andrew Inc. Norwich, New York. 2009. [45] Ana Dergan, “Electronic and transport properties of carbon nanotubes”, seminar paper, Department of physics, University of Ljubljana, October 2010. [46] C. Thomsen, H. Telg, J. Maultzsch and S. Reich, “Chirality assignments in carbon nanotubes based on resonant Raman scattering”, phys. stat. sol. (b) 242, No. 9, pp.1802–1806, 2005. [47] M. S. Strano, S. K. Doorn, E. H. Haroz, C. Kittrell, R. H. Hauge and R. E. Smalley, “Assignment of (n, m) Raman and Optical Features of Metallic Single-Walled Carbon Nanotubes”, Nano Lett., Vol.3, No.8, pp.1091-1096, 2003. [48] J. E. Herrera, L. Balzano, F. Pompeo and D. E. Resasco, “Raman characterizatiuon of Single wall nanotubes of various diameters obtained by catalytic disproportionation of CO”, J. Nanosci. Nanotech., Vol. 3, No. 1, 2003. [49] S.K. doorn, D.A. Heller, P.W. Barone, M.L. Usrey, M.S. Strano, “Resonant Raman excitation profiles of individually dispersed single walled carbon nanotubes in solution”, Appl. Phys. A 78, p.1147–1155, 2004. [50] Z. Yu and L. E. Brus, “(n, m) Structural Assignments and Chirality Dependence in Single-Wall Carbon Nanotube Raman Scattering”, J. Phys. Chem. B, 105, pp.6831-6837, 2001. [51] H. Telg, J. Maultzsch, S. Reich, F. Hennrich and C. Thomsen, “Chirality Distribution and Transition Energies of Carbon Nanotubes” Phys. Rev. Lett., Vol. 93, No. 17, 2004. [52] A. Jorio, A. P. Santos, H. B. Ribeiro, C. Fantini, M. Souza, J. P. M. Vieira, C. A. Furtado, J. Jiang, R. Saito, L. Balzano, D. E. Resasco and M. A. Pimenta, “Quantifying carbon-nanotube species with resonance Raman scattering” Phys. Rev. B 72, 075207, 2005. [53] H. Telg, J. Maultzsch, S. Reich, F. Hennrich and C. Thomsen, “Raman excitation profiles for the (n1, n2) assignment in carbon nanotubes” AIP Conf. Proceedings, Vol. 723, Issue 1, p.330, 2004. [54] C. Fantini, A. Jorio, M. Souza, M. S. Strano, M. S. Dresselhaus and M. A. Pimenta, “Optical Transition Energies for Carbon Nanotubes from Resonant Raman Spectroscopy: Environment and Temperature Effects”, Phys. Rev. B, Vol. 93, No. 14, 2004. [55] H. Telg, J. Maultzsch, S. Reich and C. Thomsen, “Resonant-Raman intensities and transition energies of the E11 transition in carbon nanotubes”, Phys. Rev. B 74, 115415, 2006. [56] M. Namkung, P. A.Williams, C. D.Mayweather, B. Wincheski, C. Park,; Namkung, S. Juock, “Chirality Characterization of Dispersed Single Wall Carbon Nanotubes” NASA, 2005 MRS Spring Meeting; San Francisco, CA; United States, 28 Mar. - 1 Apr. 2005. [57] S. Berciaud, L. Cognet, P. Poulin, R. Bruce Weisman, and B. Lounisa, “Absorption spectroscopy of individual single-walled carbon nanotubes”, Nano Lett., 7 (5), pp.1203–1207, 2007. [58] R. B. Weisman, “Fluorimetric characterization of single-walled carbon nanotubes”, Anal. Bioanal. Chem. 396, 1015–1023, 2010. [59] M.J. O'Connell, S.M. Bachilo, C.B. Huffman, V.C. Moore, M.S. Strano, E.H. Haroz, K.L. Rialon, P.J. Boul, W.H. Noon, C. Kittrell, J. Ma, R.H. Hauge, R.B. Weisman, and R.E. Smalley, “Band Gap Fluorescence from Individual Single-Walled Carbon Nanotubes”, Science 297, pp.5581-5593, 2002. [60] D. A. Tsyboulski1, J. D. R. Rocha, S. M. Bachilo1, L. Cognet and R. B. Weisman,” Structure-Dependent Fluorescence Efficiencies of Individual Single-Walled Carbon Nanotubes”, Nano Lett., 7(10), pp.3080-5, 2007. [61] M. Jones, C. Engtrakul, W. K. Metzger, R. J. Ellingson, A. J. Nozik, M. J. Heben, and G. Rumbles, “Analysis of photoluminescence from solubilized single-walled carbon nanotubes”, Phys. Rev B, 71, 115426, 2005. [62] Y. Miyauchi, S. Chiashi, Y. Murakami, Y. Hayashida, S. Maruyama, “Fluorescence spectroscopy of single-walled carbon nanotubes synthesized from alcohol”, Chem. Phys. Lett., Vol. 387, Issues 1–3, pp.198–203, 2004. [63] Liu K, Deslippe J, Xiao F, Capaz R B, Hong X, Aloni S, Zettl A, Wang W, Bai X, Louie SG, Wang E, Wang F., “An atlas of carbon nanotube optical transitions”, Nat. Nanotech. 7, 325–329, 2012. [64] Telg H, Duque J G, Staiger M, Tu X, Hennrich F, Kappes M M, Zheng M, Maultzsch J, Thomsen C, Doorn S K., “Chiral index dependence of the G+ and G- Raman modes in semiconducting carbon nanotubes”, ACS Nano. 6(1), pp. 904-11, 2012. [65] Y. Hirana, G. Juhasz, Y. Miyauchi, S. Mouri, K. Matsuda, N. Nakashima, “Empirical Prediction of Electronic Potentials of Single-Walled Carbon Nanotubes With a Specific Chirality (n,m)”, Sci. Rep. 3, Art. No. 2959, 2013. [66] K. Sato, R. Saito, J. Jiang, G. Dresselhaus, M. S. Dresselhaus, “Chirality dependence of many body effects of single wall carbon nanotubes”, Vibrational Spectroscopy: Elsevier, Vol. 45, Issue 2, pp.89–94, 2007.