Finite length nanotubes: Ground state degeneracy and single-electron spectrum

Abstract

By means of the Rutherford method it is found that in zigzag carbon nanotubes with an even number n=2j of hexagons in a tube ring the gap between occupied and empty single-electron levels disappears. The frontier level is shown to be at least doubly degenerate. The reason for the loss of the zero level (for n=2j) upon reaching the infinite length limit is discussed. The existence of a lot of states, specific for finite nanotubes, with the energy very close to the Fermi level is shown, and the number of these states is estimated. The existence of a narrow (of order 10(-4) eV) conduction band, formed by edge states, is demonstrated in tight binding approximation. If the Coulomb interaction is taken into account, the half-filled degenerate pi-electronic level continues to exist. The open shell Hartree-Fock theory demonstrates the degenerate configuration splitting onto three equidistant terms: two singlets with the seniority numbers 0 and 1, and the lowest triplet in accordance with the Hund rule. The behavior of the single-electron gap is alike for all three terms: the gap is of the order 0.2-0.01 eV. The Rutherford method gives analytical results for armchair nanotubes as well, which explains the gap oscillations as a function of the tube length with a period of three rings. Application of the degenerate open shell Hartree-Fock theory supports this explanation.