Tamm states in zigzag carbon nanotubes: Analytic estimates in the Hückel approximation

The frontier molecular orbitals of [Cn]q hydrocarbons, fragments of (n, 0) nanotubes, were considered. Asymptotic estimates for two different sets of frontier molecular orbitals were obtained in the Hückel approximation. The problem was shown to be equivalent to the search for model polyene levels with alternating resonance parameters (β and β′ = γβ). The γ parameter was determined by the symmetry type of molecular orbitals and took on values from 0 to 2. The problem was reduced to an analysis of a regular linear polyene with a modified terminal atom by taking alternant symmetry into account. The states of the system were classified according to quasi-momentum Θ values, which could be complex. Solutions with a complex Θ value were localized at tube ends and could be interpreted as Tamm states. The criterion of the appearance of Tamm solutions for the problem under consideration was formulated, γ < 1 − 1/(q + 2). The conclusion was drawn that these states for an arbitrary [Cn]q fragment lied close to the Fermi level, and their energies were described by the asymptotic equation α ± βρ(q + 1). Delocalized levels were always situated farther from the Fermi level than Tamm states.