Phonon Dispersions of a Single-Wall (8,0) Carbon Nanotube: Effects of the Rotational Acoustic Sum Rule and of Surface At
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HH3.34.1
Phonon Dispersions of a Single-Wall (8,0) Carbon Nanotube: Effects of the Rotational Acoustic Sum Rule and of Surface Attachment Nicolas Mounet and Nicola Marzari Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. ABSTRACT The lattice dynamics of single-walled carbon nanotubes (SWCNT) is studied from firstprinciples using density-functional perturbation theory (DFPT) at the GGA-PBE level. The phonon dispersions of a pristine, infinite zigzag (8,0) SWCNT are obtained and the effect of applying the rotational acoustic sum rule on vibrational properties is discussed. Finally we study the effects of covalent functionalizations on the SWCNT phonon frequencies by selectively increasing the effective mass of the carbon atoms that would link to the functional groups. INTRODUCTION Since their discovery by S. Ijima in 1991, carbon nanotubes (especially in their single-wall form) have attracted wide attention for their mechanical and transport properties and their possible use as nanoelectronics components. In this latter respect, the knowledge of both finitetemperature thermodynamic properties and electron-phonon scattering is of great importance, and the study of their vibrational properties is therefore of crucial relevance. Another motivation is the possibility to characterize or identify nanotubes with spectroscopic techniques paired to abinitio calculations [1]. Numerous investigations of SWCNT lattice dynamics have already been performed, both from first-principles [2-4] and experimentally [5]. A detailed review of both theoretical and experimental considerations can also be found in [6]. We present here an accurate DFPT study of a semiconducting zigzag (8,0) nanotube, and augment this study with some additional considerations. First, we analyze the effects of the rotational acoustic sum rule on the interatomic force constants and the phonon dispersions. Second, we simulate the attachment of functional groups to the nanotube and their influence on the vibrational spectrum – this is obtained in first-approximation by simply increasing the effective masses of selected carbon atoms. METHODS Phonon frequencies are derived from the knowledge of the interatomic force constants, i.e. the second derivatives of the total crystal energy versus displacements of the ions:
∂2E Cα i , β j (R − R ') = ∂uα i (R )∂u β j (R ')
(1) equilibriu m
Here R (R’) is a Bravais lattice vector, i (j) indicates the ith (jth) atom of the unit cell, and α (β) represents the Cartesian components.
HH3.34.2
In DFPT, the self-consistent calculation of the linear response [7] to a periodic perturbation of wave-vector q allows a direct determination of the dynamical matrix at any wavevector q; the dynamical matrix is related to the interatomic force constants via the Fourier transform
~ Dα i ,β j (q ) =
1 MiM j
∑ Cα β (R )e i, j
− iq ⋅ R
(2)
R
where Mi (Mj) represents the mass of the ith (jth) atom. Phonon frequencies are given by the eigenvalues of the dynamical matrix
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