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Volumn 66, Issue 23, 2002, Pages 1-12

Vibrational properties of single-wall nanotubes and monolayers of hexagonal BN

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EID: 84864480824     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.66.235415     Document Type: Article
Times cited : (4)

References (74)
  • 21
    • 0003431555 scopus 로고    scopus 로고
    • New Materials for the Twenty-First Century) (Cambridge University Press, Cambridge, England
    • P. J. Harris, Carbon Nanotubes and Related Structures (New Materials for the Twenty-First Century) (Cambridge University Press, Cambridge, England, 2001).
    • (2001) Carbon Nanotubes and Related Structures
    • Harris, P.J.1
  • 50
    • 85038960516 scopus 로고    scopus 로고
    • For the construction of the Born effective charge tensor in the case of the tube geometry we have identified purely radial and tangential displacements with, respectively, out-of-plane and in-plane movements in the planar sheet. Therefore, for an atom i, which belongs to a tube of radius r and axis (formula presented) and occupies the position (formula presented) (formula presented) (formula presented) the Born effective charge tensor takes the form (formula presented)
    • For the construction of the Born effective charge tensor in the case of the tube geometry we have identified purely radial and tangential displacements with, respectively, out-of-plane and in-plane movements in the planar sheet. Therefore, for an atom i, which belongs to a tube of radius r and axis (formula presented) and occupies the position (formula presented) (formula presented) (formula presented) the Born effective charge tensor takes the form (formula presented)s
  • 54
    • 85038904709 scopus 로고    scopus 로고
    • Ref. 24 some evidence was presented that, while for the small and moderate radii the strain energy is clearly smaller in the case of BN compared to C nanotubes, the opposite could be true for larger tubes (formula presented) However, this seems quite unlikely if, as pointed out here, a smaller flexular rigidity of BN can already be deduced from the behavior of the ZA phonon band of the flat sheet. Unfortunately, the parameter (formula presented) is quite sensitive to the fit procedure and to numerical uncertainities, and the values for C and BN are close enough so that no definitive conclusion can be given here
    • In Ref. 24 some evidence was presented that, while for the small and moderate radii the strain energy is clearly smaller in the case of BN compared to C nanotubes, the opposite could be true for larger tubes (formula presented) However, this seems quite unlikely if, as pointed out here, a smaller flexular rigidity of BN can already be deduced from the behavior of the ZA phonon band of the flat sheet. Unfortunately, the parameter (formula presented) is quite sensitive to the fit procedure and to numerical uncertainities, and the values for C and BN are close enough so that no definitive conclusion can be given here.
  • 57
    • 85038937682 scopus 로고    scopus 로고
    • fact, the in-plane stiffness C reported in Ref. 24 should be compared with our Young modulus Y (Table I). The elastic constant (formula presented) is, in the notation of that reference, given by (formula presented) with a value of 0.284 TPa nm and (formula presented) being the Poisson ratio
    • In fact, the in-plane stiffness C reported in Ref. 24 should be compared with our Young modulus Y (Table I). The elastic constant (formula presented) is, in the notation of that reference, given by (formula presented) with a value of 0.284 TPa nm and (formula presented) being the Poisson ratio.
  • 68
    • 85038960140 scopus 로고    scopus 로고
    • See EPAPS Document No. E-PRBMDO-66-235415 for the tables containing the numerical values of the vibrational frequencies at (formula presented) including the Raman- and IR-active modes, for all the tubes studied in this work. The frequencies have been classified according to their symmetries. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage () or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information
    • See EPAPS Document No. E-PRBMDO-66-235415 for the tables containing the numerical values of the vibrational frequencies at (formula presented) including the Raman- and IR-active modes, for all the tubes studied in this work. The frequencies have been classified according to their symmetries. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information.
  • 69
    • 85038889760 scopus 로고    scopus 로고
    • Similarly to the case of the monolayer, the main changes on the phonon band structure of the single-walled tubes induced by the long-range dipolar interactions are relatively small and restricted to the high-frequency modes: the in-plane optical modes extend over a somewhat larger energy range, and some of the longitudinal modes (with atomic displacements parallel to the tube axis) exhibit a larger dipersion with (formula presented) These changes are directly related to the larger overbending of the LO band of the BN monolayer when the electrostatic interactions are included
    • Similarly to the case of the monolayer, the main changes on the phonon band structure of the single-walled tubes induced by the long-range dipolar interactions are relatively small and restricted to the high-frequency modes: the in-plane optical modes extend over a somewhat larger energy range, and some of the longitudinal modes (with atomic displacements parallel to the tube axis) exhibit a larger dipersion with (formula presented) These changes are directly related to the larger overbending of the LO band of the BN monolayer when the electrostatic interactions are included.


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