Ab Initio Study of the Electronic and Vibrational Properties of 1-nm-Diameter Single-Walled Nanotubes

Jesús Marquina, Chrystian Power, Jesús González, Jean-Marc Broto

Centro de Estudios Avanzado en óptica, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela.
Centro de Estudios de Semiconductores, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela.
LNCMI, Paul Sabatier University, Toulouse, France.
Malta Consolider Team, CITIMAC, Facultad de Ciencias, Universidad de Cantabria, Santander, Espa?a.

DOI: 10.4236/ampc.2013.32025                

Abstract

The electronic structure, band gap, density of states of the (8,8), (14,0) and (12,3) single-walled carbon nanotubes by the SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) method in the framework density-functional theory (DFT) with the generalized gradients approximation (GGA) were studied. Also, we studied the vibrational properties of the (8,8) and (14,0) nanotubes. Only the calculated relaxed geometry for (12,3) nanotube show significant deviations from the ideal rolled graphene sheet configuration. The electronic transition energies of van Hove singularities were studied and compared with previous results. The calculated band structures, density of states and dispersion curves for all tubes were in good agreement with theoretical and experimental results.

Keywords

Carbon Nanotubes; Electronic Structure; Dispersion Curve

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Lijima, “Helical Microtubules of Graphitic Carbon,” Nature, Vol. 354, 1991, pp. 56-58. doi:10.1038/354056a0
[2]	N. Hamada, S.-I. Sawada and A. Oshiyama, “New OneDimensional Conductors: Graphitic Microtubules,” Physical Review Letters, Vol. 68, No. 10, 1992, pp. 15791581. doi:10.1103/PhysRevLett.68.1579
[3]	R. Saito, M. Fujita, G. Dresselhaus and M. S. Dresselhaus, “Electronic Structure of Chiral Graphene Tubules,” Applied Physics Letters, Vol. 60, No. 18, 1992, pp. 22042206. doi:10.1063/1.107080
[4]	R. Saito, K. Sato, Y. Oyama, J. Jiang, G. G. Samsonidze, G. Dresselhaus and M. S. Dresselhaus, “Cutting Lines near the Fermi Energy of Single-Wall Carbon Nanotubes,” Physical Review B, Vol. 72, No. 15, 2005, Article ID: 153413. doi:10.1103/PhysRevB.72.153413
[5]	M. S. Dresselhaus, G. Dresselhaus and P. C. Ecklund, “Science of Fullerene and Carbon Nanotubes,” Academic Press, San Diego, 1996.
[6]	P. J. F. Harris, “Carbon Nanotube Science,” Cambridge University Press, London, 2009. doi:10.1017/CBO9780511609701
[7]	R. Saito, M. Fujita, G. Dresselhaus and M. S. Dresselhaus, “Electronic Structure of Graphene Tubules Based on C60,” Physical Review B, Vol. 46, No. 3, 1992, pp. 1804-1811. doi:10.1103/PhysRevB.46.1804
[8]	J. W. Mintmire and B. I. Dunlap, C. T. White. “Are Fullerene Tubules Metallic?” Physical Review Letters, Vol. 68, No. 5, 1992, pp. 631-634. doi:10.1103/PhysRevLett.68.631
[9]	R. Saito, T. Takeya, T. Kimura, G. Dresselhaus and M. S. Dresselhaus, “Raman Intensity of Single-Wall Carbon Nanotubes,” Physical Review B, Vol. 57, No. 7, 1998, pp. 4145-4153. doi:10.1103/PhysRevB.57.4145
[10]	R. A. Jishi, L. Venkataraman, M. S. Dresselhaus and G. Dresselhaus, “Symmetry Properties of Chiral Carbon Nanotubes,” Physical Review B, Vol. 51, No. 16, 1995, pp. 11176-11179. doi:10.1103/PhysRevB.51.11176
[11]	D. Sánchez-Portal, E. Artacho, J. M. Soler, A. Rubio and P. Ordejón, “Ab Initio Structural, Elastic, and Vibrational Properties of Carbon Nanotubes,” Physical Review B, Vol. 59, No. 19, 1999, pp. 12678-12688. doi:10.1103/PhysRevB.59.12678
[12]	M. Machón, S. Reich, C. Thomsen, D. Sánchez-Portal and P. Ordejon, “Ab Initio Calculations of the Optical Properties of 4-Å-Diameter Single-Walled Nanotubes,” Physical Review B, Vol. 66, No. 15, 2002, Article ID: 155410. doi:10.1103/PhysRevB.66.155410
[13]	K. Kanamitsu and S. Saito, “Geometries, Electronic Properties, and Energetics of Isolated Single Walled Carbon Nanotubes,” Journal of the Physical Society of Japan, Vol. 71, 2002, pp. 483-486. doi:10.1143/JPSJ.71.483
[14]	V. N. Popov, “Curvature Effects on the Structural, Electronic and Optical Properties of Isolated Single-Walled Carbon Nanotubes within a Symmetry-Adapted NonOrthogonal Tight-Binding Model,” New Journal of Physics, Vol. 6, 2004, pp. 1-17. doi:10.1088/1367-2630/6/1/017
[15]	S. Rols, Z. Benes, E. Anglaret, J. L. Sauvajol, P. Papanek, J. E. Fischer, G. Coddens, H. Schober and A. J. Dianoux, “Phonon Density of States of Single-Wall Carbon Nanotubes,” Physical Review Letters, Vol. 85, No. 24, 2000, pp. 5222-5225. doi:10.1103/PhysRevLett.85.5222
[16]	C. Thomsen, “Second-Order Raman Spectra of Single and Multiwalled Carbon Nanotubes,” Physical Review B, Vol. 61, No. 7, 2000, pp. 4542-4544. doi:10.1103/PhysRevB.61.4542
[17]	J. Maultzsch, S. Reich and C. Thomsen, “Chirality-Selective Raman Scattering of the D Mode in Carbon Nanotubes,” Physical Review B, Vol. 64, No. 12, 2001, Article ID: 121407. doi:10.1103/PhysRevB.64.121407
[18]	J. Maultzsch, J. S. Reich and S. C. Thomsen, “Raman Scattering in Carbon Nanotubes Revisited,” Physical Review B, Vol. 65, No. 23, 2002, Article ID: 233402. doi:10.1103/PhysRevB.65.233402
[19]	V. Zólyomi and J. Kiirti, “Calculating the Discrepancy between the Stokes and Anti-Stokes Raman D Band of Carbon Nanotubes Using Double Resonance Theory,” Physical Review B, Vol. 66, No. 7, 2002, Article ID: 073418. doi:10.1103/PhysRevB.66.073418
[20]	G. G. Samsonidze, R. Saito, A. Jorio, A. G. S. Filho, A. Griineis, M. P. Pimenta, G. Dresselhaus and M. S. Dresselhaus, “Phonon Trigonal Warping Effect in Graphite and Carbon Nanotubes,” Physical Review Letters, Vol. 90, No. 2, 2003, Article ID: 027403. doi:10.1103/PhysRevLett.90.027403
[21]	G. L. Kane, E. J. Mele, R. S. Lee, J. E. Fischer, P. Petit, H. Dai, A. Thess, R. E. Smalley, A. R. M. Verschueren, S. J. Tans and G. Dekker, “Temperature-Dependent Resistivity of Single-Wall Carbon Nanotubes,” EPL (Europhysics Letters), Vol. 41, No. 6, 1998, pp. 683-689. doi:10.1209/epl/i1998-00214-6
[22]	Z. Yao, G. L. Kane and G. Dekker, “High-Field Electrical Transport in Single-Wall Carbon Nanotubes,” Physical Review Letters, Vol. 84, No. 13, 2000, pp. 2941-2944. doi:10.1103/PhysRevLett.84.2941
[23]	H. Suzuura and T. Ando, “Phonons and Electron-Phonon Scattering in Carbon Nanotubes,” Physical Review B, Vol. 65, No. 23, 2002, Article ID: 235412. doi:10.1103/PhysRevB.65.235412
[24]	A. Sédéki, L. G. Caron and G. Bourbonnais, “ElectronPhonon Coupling and Peierls Transition in Metallic Carbon Nanotubes,” Physical Review B, Vol. 62, No. 11, 2000, pp. 6975-6978. doi:10.1103/PhysRevB.62.6975
[25]	Z. J. Q. Ding, Q. Jiang and D. Tian, “Vibrational Spectrum of Carbon Nanotubes,” Fullerene Science and Technology, Vol. 4, No. 1, 1996, pp. 31-37. doi:10.1080/10641229608001535
[26]	A. Charlier, E. McRae, M.-F. Charlier, A. Spire and S. Forster, “Lattice Dynamics Study of Zigzag and Armchair Carbon Nanotubes,” Physical Review B, Vol. 57, No. 11, 1998, pp. 6689-6696. doi:10.1103/PhysRevB.57.6689
[27]	R. A. Jishi, L. Venkataraman, M. S. Dresselhaus and G. Dresselhaus, “Phonon Modes in Carbon Nanotubules,” Chemical Physics Letters, Vol. 209, No. 1-2, 1993, pp. 77-82. doi:10.1016/0009-2614(93)87205-H
[28]	O. Dubay and G. Kresse, “Accurate Density Functional Calculations for the Phonon Dispersion Relations of Graphite Layer and Carbon Nanotubes,” Physical Review B, Vol. 67, No. 3, 2003, Article ID: 035401. doi:10.1103/PhysRevB.67.035401
[29]	L. H. Ye, B. G. Liu, D. S. Wang and R. Han, “Ab Initio Phonon Dispersions of Single-Wall Carbon Nanotubes,” Physical Review B, Vol. 69, No. 23, 2004, Article ID: 235409. doi:10.1103/PhysRevB.69.235409
[30]	E. Belandria, M. Millot, J. M. Broto, E. Flahaut, F. Rodriguez, R. Valiente and J. González, “Pressure Dependence of Raman Modes in Double Wall Carbon Nanotubes Filled with 1D Tellurium,” Carbon, Vol. 48, No. 9, 2010, pp. 2566-2572. doi:10.1016/j.carbon.2010.03.036
[31]	J. Marquina, E. Flahaut and J. González, “High-Pressure Optical Absorption Studies of Double-Walled Carbon Nanotubes,” Physica Status Solidi (RRL)—Rapid Research Letters, Vol. 6, No. 9-10, 2012, pp. 382-384. doi:10.1002/pssr.201206280
[32]	D. Sánchez-Portal, P. Ordejón, E. Artachoa and J. M. Soler, “Density-Functional Method for Very Large Systems with LCAO Basis Sets,” International Journal of Quantum Chemistry, Vol. 65, No. 5, 1997, pp. 453-461. doi:10.1002/(SICI)1097-461X(1997)65:5<453::AID-QUA9>3.0.CO;2-V
[33]	J. P. Perdew, K. Burke and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Physical Review Letters, Vol. 77, No. 18, 1996, pp. 3865-3868. doi:10.1103/PhysRevLett.77.3865
[34]	N. Troullier and J. L. Martins, “Efficient Pseudopotentials for Plane-Wave Calculations,” Physical Review B, Vol. 43, No. 3, 1991, pp. 1993-2006. doi:10.1103/PhysRevB.43.1993
[35]	J. Junquera, O. Paz, D. Sánchez-Portal and E. Artacho, “Numerical Atomic Orbitals for Linear-Scaling Calculations,” Physical Review B, Vol. 64, No. 23, 2001, Article ID: 235111. doi:10.1103/PhysRevB.64.235111
[36]	J. Maultzsch, S. Reich, C. Thomsen, H. Requardt and P. Ordejón, “Phonon Dispersion in Graphite,” Physical Review Letters, Vol. 92, No. 7, 2004, Article ID: 075501. doi:10.1103/PhysRevLett.92.075501
[37]	M. Cardona, “In Electrons and Photons in Solids,” Scuola Normale, Pisa, 2001.
[38]	P. Pavone, R. Bauer, K. Karch, O. Schtt, S. Vent, W. Windl, D. Strauch, S. Baroni and S. de Gironcoli, “Ab Initio Phonon Calculations in Solids,” Physica B: Condensed Matter, Vol. 219-220, 1996, pp. 439-441. doi:10.1016/0921-4526(95)00771-7
[39]	J. Marquina, Ch. Power and J. González, “Espectroscopia Raman del Grafeno Monocapa y el Grafito: Acoplamiento Electrón Fonón y Efectos no Adiabáticos,” Revista Tumbaga, Vol. 1, No. 5, 2010, pp. 183-194.
[40]	Y. Miyamoto, M. L. Cohen and S. G. Louie, “Ab Initio Calculation of Phonon Spectra for Graphite, BN, and BC2N Sheets,” Physical Review B, Vol. 52, No. 20, 1995, pp. 14971-14975. doi:10.1103/PhysRevB.52.14971
[41]	A. N. Imtani and V. Jindal, “Structure of Chiral SingleWalled Carbon Nanotubes under Hydrostatic Pressure,” Computational Materials Science, Vol. 46, No. 2, 2009, pp. 297-302. doi:10.1016/j.commatsci.2009.03.002
[42]	H. Jiang, P. Zhang, B. Liu, Y. Huang, P. H. Geubelle, H. Gao and K. C. Hwang, “The Effect of Nanotube Radius on the Constitutive Model for Carbon Nanotubes,” Computational Materials Science, Vol. 28, No. 3-4, 2003, pp. 429-442. doi:10.1016/j.commatsci.2003.08.004
[43]	M. Ouyang, J. L. Huang, C. L. Cheung and C. M. Lieber, “Energy Gaps in ‘Metallic’ Single-Walled Carbon Nanotubes,” Science, Vol. 292, No. 5517, 2001, pp. 702-705. doi:10.1126/science.1058853
[44]	H. S. Khadigeh, “Ab Initio Study of Electronic Properties of a Armchair (7,7) Carbon Nanotube,” Advances in Materials Physics and Chemistry, Vol. 2, 2012, pp. 159162. doi:10.4236/ampc.2012.23024
[45]	T. Ito, K. Nishidate, M. Baba and M. Hasegawa, “First Principles Calculations for Electronic Band Structure of Single-Walled Carbon Nanotube under Uniaxial Strain,” Surface Science, Vol. 514, No. 1-3, 2002, pp. 222-226. doi:10.1016/S0039-6028(02)01633-3
[46]	M. Hulman, R. Pfeiffer and H. Kuzmany, “Raman Spectroscopy of Small-Diameter Nanotubes,” New Journal of Physics, Vol. 6, 2004, pp. 1-17. doi:10.1088/1367-2630/6/1/001
[47]	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,” Physical. Review Letters, Vol. 93, No. 14, 2004, Article ID: 147406. doi:10.1103/PhysRevLett.93.147406
[48]	O. E. Alon, “Number of Ramanand Infrared-Active Vibrations in Single-Walled Carbon Nanotubes,” Physical Review B, Vol. 63, No. 20, 2001, Article ID: 201403. doi:10.1103/PhysRevB.63.201403
[49]	A. M. Rao, E. Richter, S. Bandow, B. Chase, P. C. Eklund, K. A. Williams, S. Fang, K. R. Subbaswamy, M. Menon, A. Thess, R. E. Smalley, G. Dresselhaus and M. S. Dresselhaus, “Diameter-Selective Raman Scattering from Vibrational Modes in Carbon Nanotubes,” Science, Vol. 275, No. 5297, 1997, pp. 187-191. doi:10.1126/science.275.5297.187
[50]	H. D. Sun, Z. K. Tang, J. Chen and G. Li, “Polarized Raman Spectra of Single-Wall Carbon Nanotubes MonoDispersed in Channels of ALPO4-5 Single Crystals,” Solid State Communications, Vol. 109, No. 6, 1999, pp. 365-369. doi:10.1016/S0038-1098(98)00588-2
[51]	C. Thomsen, “Second-Order Raman Spectra of Single and Multiwalled Carbon Nanotubes,” Physical Review B, Vol. 61, No. 7, 2000, pp. 4542-4544. doi:10.1103/PhysRevB.61.4542
[52]	A. Jorio, A. G. Souza Filho, G. Dresselhaus, M. S. Dresselhaus, A. K. Swan, M. S. ünlü, B. B. Goldberg, M. A. Pimenta, J. H. Hafner, C. M. Lieber and R. Saito, “GBand Resonant Raman Study of 62 Isolated Single-Wall Carbon Nanotubes,” Physical Review B, Vol. 65, No. 15, 2002, Article ID: 155412. doi:10.1103/PhysRevB.65.155412
[53]	M. Dresselhaus, G. Dresselhaus, R. Saito and A. Jorio, “Raman Spectroscopy of Carbon Nanotubes,” Physics Reports, Vol. 409, No. 2, 2005, pp. 47-99. doi:10.1016/j.physrep.2004.10.006
[54]	D. Kahn and J. P. Lu, “Vibrational Modes of Carbon Nanotubes and Nanoropes,” Physical Review B, Vol. 60, No. 9, 1999, pp. 6535-6540. doi:10.1103/PhysRevB.60.6535