Eigenvectors of Gamma-phonons of single-walled carbon nanotubes

Eigenvectors of Γ-phonons of single-walled carbon nanotubes

General characteristics

The eigenvectors of the phonons of the different single-walled carbon nanotubes can be categorized in four groups by their similarity with the four types of atomic displacements in the phonons of graphene (shown in the columns of the table below). The eigenvectors of mode of symmetry species E_n have 2n nodes around the tube circumference. Modes of A_1,2 symmetry have no nodes and modes of B_1,2 symmetry have the maximum possible number of nodes N.

Apart from the transverse (T) modes shown in the table below, there are axially-polarized (or longitudinal, L) modes of types "in-phase in-plane" and "out-of-phase in-plane" (not shown). The T and L modes of any tube are in total 6A_1,2+6B_1,2+6E₁+…+6E_N/2-1. In the case of achiral tubes, there are additional indices u and g (omitted here).

Each tube has four acoustic phonons:

- an "in-phase in-plane" A₁ -- a longitudinal (LA) mode (not shown);

- an "in-phase in-plane" A₁ -- a twist (TW) mode (shown in green in the table below);

- two "in-phase out-of-plane" modes of E₁ species -- transverse (TA) modes (shown in green).

Among the optical phonons of each tube, the Raman-active ones are most important, the most intense ones being:

- an "in-phase out-of-plane" A₁ mode -- the radial breathing mode (RBM) -- with frequency between 100 and 450 cm^-1 (shown in red);

- six "out-of-phase in-plane" 2A₁, 2E₁, and 2E₂ modes -- the G modes -- with frequency between 1500 and 1600 cm^-1. The set of the six modes splits into a set A₁+E₁+E₂ of T modes (shown in red) and a set A₁+E₁+E₂ of L modes; only A_1g(T)+E_1g(L)+E_2g(T) / A_1g(L)+E_1g(T)+E_2g(L) are Raman-active in armchair/zigzag tubes.