E. Dobardžić

Phonons in MoS2 and WS2 Nanotubes

Phonon of the single-wall MS2 (M = Mo,W) nanotubes with diameters up to 50 nm are systematically calculated using the full symmetry implemented valence-force-field model. Infrared and Raman active modes are singled out by the symmetry, and their applications in the experimental sample characterization are discussed. Raman measurement of the multiwall tubes and bulk has been performed. Obtained results are interpreted by simple coupled-spring like model concerning the same symmetry modes of walls.

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.

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.

Generalized Bloch theorem and topological characterization

The Bloch theorem enables reduction of the eigenvalue problem of the single-particle Hamiltonian that commutes with translational group. Based on a group theory analysis we present generalization of the Bloch theorem that incorporates all additional symmetries of a crystal. The generalized Bloch theorem constrains the form of the Hamiltonian which becomes manifestly invariant under additional symmetries. In the case of isotropic interactions the generalized Bloch theorem gives a unique Hamiltonian. This Hamiltonian coincides with the Hamiltonian in the periodic gauge. In the case of anisotropic interactions the generalized Bloch theorem allows a family of Hamiltonians. Due to the continuity argument we expect that even in this case the Hamiltonian in the periodic gauge defines observables, such as Berry curvature, in the inverse space. For both cases we present examples and demonstrate that the average of the Berry curvatures of all possible Hamiltonians in the Bloch gauge is the Berry curvature in the periodic gauge.

ELECTRON-PHONON COUPLING IN GRAPHENE

Numerical and symmetry based analysis of the electron-phonon coupling in graphene have been performed. Symmetry alone predicts the Kohn anomalies, their positions in the Brillouin zone and all quantum numbers of the Kohn modes and the vectors of the Kohn displacements. Dispersion of kink phonons is calculated and found to be similar to the classical phonon spectrum. Interestingly, most of the Γ (including all the acoustic phonons as well) and K modes are shown to be uncoupled to electrons.

Raman Spectra Of Double‐Walled Carbon Nanotubes

Using nonresonant bond‐polarization theory, Raman spectra of periodic double‐walled carbon nanotubes (DWCNTs) are calculated. Due to the lower symmetry of DWCNT, the number of Raman active modes is much larger compared to those of its layers. Complete frequency range of the tubes spectra has been analyzed for large number of tubes. We found that only modes whose frequencies are below 800 cm−1 have noticeable up shifts compared to those of isolated layers. Special attention is given to radial breathing modes (RBMs) and G‐band region since these modes are used for the identification of singe‐walled carbon nanotubes. In case of breathing like modes (BLMs), 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.

Phonons in MS2 (M=Mo, W) Nanotubes

Using full symmetry implemented valence‐force‐field model, we have calculated phonon energies of all the single‐wall MS2 nanotubes with diameter range 1–20 nm. In particular, we extracted infrared and Raman active modes in order to clarify possible application of these methods in the experimental sample characterization. A prescription for experimental distinction of zig‐zag, armchair and chiral tubes, based on their different symmetry and resulting selection rules, is proposed. Furthermore, we noticed that displacements significantly depend on the chiral angle only for two MoS2 modes, namely twisting in‐phase (rigid layer) and breathing out‐of‐phase, and none of WS2 modes. Also, we found that heat capacity of the nanotubes and corresponding layer are similar.

Optical absorption in molybdenum disulfide nanotubes

Symmetry based calculations of the polarized optical absorption in single-wall MoS2 nanotubes are presented. Optical conductivity tensor for the individual tubes, using line group symmetry implemented POLSym code and DFTB-calculated Slater type orbital functions and Hamiltonian/overlap matrix elements as input data is numerically evaluated. This minimal, full symmetry implementing algorithm enabled calculations of the optical response functions very efficiently and addressing the large diameter tubes and highly chiral tubes (which have huge number of atoms within a unit cell) as well. The absorption spectra dependence on the diameter and chiral angle of the nanotubes is investigated. The results obtained are related to the previously reported measured spectra.

Commensurate Double‐wall Nanotubes: Symmetry and Phonons

For translationally periodic double‐wall carbon nanotubes stable configurations and full symmetry groups are determined. Using this, the phonon dispersions and eigenvectors are calculated and assigned by the complete set of conserved quantum numbers. Breathing‐like modes, friction modes and acoustic modes are studied. The latter are found to be linear in wave vector k.

Optical and Vibrational Spectra of Narrow Nanotubes: A Symmetry Based Approach

By use of the tight binding method for induced representations (based on the line group symmetry) polarized optical conductivity, radial breathing and high energy vibrational mode frequencies of the narrow single‐wall carbon nanotubes are calculated. The absorption spectra features are assigned by the complete set of conserved quantum numbers and the results obtained are discussed in relation to the previously reported calculations and measurements.