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Effect of Curvature-Induced Superlattice Structures on Energy Band Structures of Helically Coiled Carbon Nanotubes

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Abstract

We studied electronic properties of a two-dimensional electron gas confined on the surface of helical curved tube, specifically, curvature induced by the geometry of helically coiled carbon nanotubes. The curvature generates two types of effective couplings that are (i) Penney-Kronig potential and (ii) a periodic spin-orbit potential. The energy-momentum dispersion relations are theoretically calculated using new variables to explain the electron spiraling around the circular helix line in counterclockwise and clockwise directions. Then we have to set up the formalism for dealing with a periodic potential as Bloch’s theorem. For calculation, we rewrite the Dirac equation in curved space-time in the form of a block matrix equation. Our numerical results demonstrate that there are pseudo-Landau quantization levels and the motion of an electron in circumference direction coupling with the arc length motion coupling. In addition, the one-dimensional dispersion relation lines present characteristics of time reversal symmetry and inversion symmetry. Finally, we also investigated the differential energy of band structures for the case where the periodic spin-orbit potential increased with respect to the increase of curvature. We found that the non-linearly increase of the differential energy was due to the periodic spin-orbit potential, corresponding to Zeeman effect.

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References

  1. Saini D, Behlow H, Podila R, Dickel D, Pillai B, Skove MJ, Serkiz SM, Rao AM (2014) Mechanical resonances of helically coiled carbon nanowires. Scientific Reports 4:5542

    Article  CAS  Google Scholar 

  2.  Popovic’ ZP, Damnjanovic’ M, Milos’evic I (2014) Phonon transport in helically coiled carbon nanotubes. Carbon 77:281-288

  3. Li D, Pan L, Wu Y, Peng W (2012) The effect of changes in synthesis temperature and acetylene supply on the morphology of carbon nanocoils. Carbon 50:2571-2580

    Article  CAS  Google Scholar 

  4. Novakovic B, Akis R, Knezevic I (2011) Transport in curved nanoribbons in a magnetic field. Phys Rev B 84:195419

    Article  Google Scholar 

  5. Liu WC, Lin HK, Chen YL, Lee CY, Chiu HT (2010) Growth of carbon nanocoils from K and Ag cooperative bicatalyst assisted thermal decomposition of acetylene. ACSNANO 4:4149-4157

  6. Goldstone J, Jaffe RL (1992) Bound states in twisting tubes. Phys Rev B 45:14100

    Article  CAS  Google Scholar 

  7. Encinosa M, Mott L (2003) Curvature-induced toroidal bound states. Phys Rev A 68:014102

    Article  Google Scholar 

  8. Jensen B (2009) Electronic states on the helicoidal surface. Phys Rev A 80:022101

    Article  Google Scholar 

  9. da Costa RCT (1982) Constraints in quantum mechanics. Phys Rev A 25:2893

    Article  Google Scholar 

  10. Burgess M, Jensen B (1993) Fermions near two-dimensional surfaces. Phys Rev A 48:1861

    Article  CAS  Google Scholar 

  11. Entin MV, Magarill LI (2001) Spin-orbit interaction of electrons on a curved surface. Phys Rev B 64:085330

    Article  Google Scholar 

  12. Itoh S, Ihara S, Kitakami J (1993) Toroidal form of carbon C 360. Phys Rev B 47:1703

    Article  CAS  Google Scholar 

  13. Ihara S, Itoh Si, Kitakami J (1993) Helically coiled cage forms of graphitic carbon. Phys Rev B 48:5643

    Article  CAS  Google Scholar 

  14. Dunlap BI (1992) Connecting carbon tubules. Phys Rev B 46:1933

    Article  CAS  Google Scholar 

  15. Zhang XB, Zhang XF, Bernaerts D, van Tendeloo G, Amelinckx S, van Landuyt J, Ivanov V, Nagy JB, Lambin P, Lucas AA (1994) The texture of catalytically grown coil-shaped carbon nanotubules. Europhys Lett 27:141

    Article  CAS  Google Scholar 

  16. Hernadi K, Thibȇn-Nga L, Forr L (2001) Growth and microstructure of catalytically produced coiled carbon nanotubes. J Phys Chem B 105:12464-12468

  17. Amelinckx S, Zhang XB, Bernaerts D, Zhang XF, Ivanov V, Nagy JB (1994) Formation mechanism for catalytically grown helix-shaped graphite nanotubes. Science 265:635-639

  18. Iijima S, Ajayan PM, Ichihashi T (1992) Growth model for carbon nanotubes. Phys Rev Lett 69:3100

    Article  CAS  Google Scholar 

  19. lijima S, Ichihashi T, Ando Y (1992) Pentagons, heptagons and negative curvature in graphite microtubule growth. Nature 356:776-778

  20. Dandoloff R, Truong TT (2004) Quantum Hall-like effect on strips due to geometry. Phys Lett A 325:233-236

  21. Perfetto E, Gonzlez J, Guinea F, Bellucci S, Onorato P (2007) Quantum Hall effect in carbon nanotubes and curved graphene strips. Phys Rev B 76:125430

    Article  Google Scholar 

  22. Guinea F, Katsnelson MI, Geim AK (2010) Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nat Phys 6:30-33

  23. Guimaraes MHD, Shevtsov O, Waintal X, van Wees BJ (2012) From quantum confinement to quantum Hall effect in graphene nanostructures. Phys Rev B 85:075424

    Article  Google Scholar 

  24. Atanasov V, Dandoloff R (2008) Curvature-induced quantum behaviour on a helical nanotube. Phys Lett A 372:6141

    Article  CAS  Google Scholar 

  25. Birrell ND, Davies PCW (1982) Quantum fields in curved space. Cambridge University Press, New York

    Book  Google Scholar 

  26. Atanasov V, Saxena A (2011) Electronic properties of corrugated graphene: the Heisenberg principle and wormhole geometry in the solid state. J Phys Condens Matter 23:175301

    Article  Google Scholar 

  27. Michalski PJ, Mele EJ (2008) Carbon nanotubes in helically modulated potentials. Phys Rev B 77:085429

    Article  Google Scholar 

  28. Park CH, Tan LZ, Steven GL (2011) Theory of the electronic and transport properties of graphene under a periodic electric or magnetic field. Phys E 43:651-656

  29. Park CH, Yang L, Son YW, Marvin Cohen L, Steven GL (2008) New generation of massless Dirac fermions in graphene under external periodic potentials. Phys Rev Lett 101:126804

    Article  Google Scholar 

  30. Lenz L, Bercioux D (2011) Dirac-Weyl electrons in a periodic spin-orbit potential. Europhys Lett 96:27006

    Article  Google Scholar 

  31. Feng C, Liew KM (2011) Structural stability of carbon nanosprings. Carbon 49:4688-4694

  32. Barbier M, Vasilopoulos P, Peeters FM (2010) Extra Dirac points in the energy spectrum for superlattices on single-layer graphene. Phys Rev B 81:075438

    Article  Google Scholar 

  33. Vozmediano MAH, Katsnelson MI, Guinea F (2010) Gauge fields in graphene. Phys Rep 496:109-148

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Acknowledgments

We wish to thank the Department of Physics, Faculty of Science, Kasetsart University, and Department of Physics, Faculty of Science, King Mongkut’s University of Technology Thonburi, for partial support.

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Authors and Affiliations

  1. Department of Physics, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand

    Thiti Thitapura, Watchara Liewrian & Tula Jutarosaga

  2. Department of Physics, Kasetsart University, Bangkok, 10900, Thailand

    Sutee Boonchui

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  1. Thiti Thitapura
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  2. Watchara Liewrian
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  3. Tula Jutarosaga
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  4. Sutee Boonchui
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Correspondence to Sutee Boonchui.

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Thitapura, T., Liewrian, W., Jutarosaga, T. et al. Effect of Curvature-Induced Superlattice Structures on Energy Band Structures of Helically Coiled Carbon Nanotubes. Plasmonics 12, 1439–1447 (2017). https://doi.org/10.1007/s11468-016-0404-1

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  • DOI: https://doi.org/10.1007/s11468-016-0404-1

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