T. Vuković

FULL SYMMETRY, OPTICAL ACTIVITY, AND POTENTIALS OF SINGLE-WALL AND MULTIWALL NANOTUBES

The full symmetry groups for all single- and multi-wall carbon nanotubes are found. As for the single-wall tubes, the symmetries form nonabelian nonsymorphic line groups, enlarging the groups reported in literature. In the multi-wall case, any type of the line and the axial point groups can be obtained, depending on single-wall constituents and their relative position. Several other consequences are discussed: quantum numbers and related selection rules, electronic and phonon bands, and their degeneracy, application to tensor properties.

Carbon nanotubes band assignation, topology, Bloch states, and selection rules

Various properties of the energy band structures ~electronic, phonon, etc.!, including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the single-wall carbon nanotubes are established. The complete set of quantum numbers consists of the angular and linear quasimomenta and parities with respect to the U axis and, for achiral tubes, the mirror planes. The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are characterized by the quantum numbers. All this enables application of the presented exhaustive selection rules. The results are discussed by some examples, e.g., allowed interband transitions, conductivity, Raman tensor, etc.

Super-slippery carbon nanotubes

Abstract:The friction between the walls of multi-wall carbon nanotubes is shown to be extremely low in general, with important details related to the specific choice of the walls. This is governed by a simple expression revealing that the phenomenon is a profound consequence of the specific symmetry breaking: super-slippery sliding of the incommensurate walls is a Goldstone mode. Three universal principles of tribology, offering a recipe for lubricant selection are emphasized.

Symmetry Based Fundamentals of Carbon Nanotubes

Due to pioneering work of Eugen Wigner, Symmetry is well recognized as a clue to understanding physical processes. However, except in Particle Physics, role of Symmetry is mostly conceptual, to give a posteriori profound interpretation of the obtained results. For instance, in Solid State Physics, more than a century known crystal symmetries are applied to describe phase transitions, selection rules, tensor shapes. Probably the only exception is the Bloch theorem, which gives a priori general forms of the electronic quantum states and ionic displacements. However, this theorem treats only the translational symmetry, reducing the calculations to the elementary cell, which is, in many cases, far bellow the maximal possible reduction.

Diffraction intensity and symmetry of single-wall carbon nanotubes

We present a thorough full symmetry-based analysis of the diffraction space of single-wall carbon nanotubes. This enables direct insight into various effects of the tubes symmetry on intensity distribution, among which are the patterns symmetry and orientational dependence. Also, some particular features of diffraction intensity are singled out. These are directly linked to a tubes symmetry parameters and therefore may give new valuable information about the tube itself. Finally, we discuss how these parameters can be determined from diffraction intensity distribution.

Mechanical coupling in homogeneously deformed single-wall carbon nanotubes.

We developed a full symmetry implementing method, based on classical molecular dynamics with the Brenner-Tersoff potential, for determining the optimal configuration of homogeneously deformed infinitely long single-wall carbon nanotubes. All nanotubes with diameter less than 1.3 nm are studied. Systematic analysis of the coupling between different types of deformation shows that, in the tubes with chiral angle close to zero and 30°, the torsion-induced circumferential strain is opposite to the induced axial strain. Also, torsion in zigzag tubes induces larger circumferential strain, but in armchair tubes it yields a larger axial strain. The influence of diameter on axial-stretching-induced torsion becomes significant only for narrow tubes. Finally, significant deviation from the linear dependence of axial-strain-induced radial deformation is found for zigzag tubes. The presented results are important for the calculation of electro-optical properties of nanotubes under strain.

Raman spectra of commensurate double-walled carbon nanotubes

Using nonresonant bond-polarization theory, Raman spectra of periodic double-walled carbon nanotubes are calculated. Although the symmetry of these tubes is highly reduced in comparison with the symmetry of the layers, the intensities of Raman active modes do not show significant changes. In ZZ polarization, modes related to radial breathing and high energy modes of individual layers remain the most intensive ones, even though the number of totally symmetric modes could be quite high. The shift of all Raman active modes with notable intensity is discussed and special attention is given to the possibility of tube identification from Raman data. In the case of breathing-like modes, the frequency of the out of phase mode is found to be chirality dependent, while the in phase one remains only diameter dependent as in the case of individual layers.

Extremely compact formulas for the Fourier transform of a product of two-centre Slater-type orbitals

A compact formula for the Fourier transform of a product of Slater-type orbitals on different centres is derived. The integral is reduced to a finite one-dimensional integration over non-oscillatory hypergeometric functions of type . The formula is valid for all quantum numbers and does not involve the reduced Bessel functions that are usually used to evaluate these integrals. Reduced formulas are calculated for some special directions in the reciprocal space. Also, some useful identities for the Fourier transforms of a product of Slater-type orbitals with correlated sets of parameters are obtained. In order to illustrate simple and efficient use of the presented results, we have applied them to graphene.

Symmetry breaking breaks friction

The harmonic analysis of the symmetry breaking is used to study the interaction and inter-layer motion of double-wall carbon nanotubes shells. The tremendous total symmetry reduction with respect to the high symmetry of the isolated layers results in low interaction: its decreasing with the symmetry breaking rate explains the observed small friction.

Wigner-Eckart Theorem in the Inductive Spaces

By use of the modified Wigner group projectors the reduced matrix element of Wigner-Eckart theorem is found to be an ordinary matrix element of the invariants built of the involved states and operator. This is used to derive the general form of Wigner-Eckart theorem for the inductive spaces and to propose a symmetry based procedure of the matrix elements calculations.