Effective $\pi$-electron Hamiltonian for small-radius nanotubes: Interpretation of curvature-induced conductivity

A subsystem, even if it strongly interacts with its environment, can be described by an effective Hamiltonian whose eigenvalues are precisely the same as the selected eigenvalues of the total system. We apply this principle to develop an energy-independent partitioning scheme extracting an effective $\pi$-electron model from valence Hamiltonians. The technique is used to explain the metallic band structure of nanotubes with small diameter—e.g., the (5,0) tube—showing that the increased curvature enhances second- and third-neighbor hopping interactions which are directly responsible for the metallic features in these systems.

Figure 1

The band structure of the (5,0) tube at the Fermi level with either (i) the valence-electron Hamiltonian or (ii) the effective $\pi$-electron Hamiltonian. The energies are shifted to set the Fermi energy zero.

Figure 2

The band structure of the (5,0) tube at the Fermi level with the hybridized bare (undressed) quasi-$\pi$ block of the valence-electron Hamiltonian. The energies are shifted to set the Fermi energy zero.

Figure 3

The Hückel band structure of the (5,0) tube at the Fermi level The energies are shifted to set the Fermi energy zero. First-neighbor matrix elements were bond-length dependent (see Ref. 14).

Figure 4

All-valence electron band structure of the (5,0) tube at the Fermi level neglecting all but first-neighbor hopping integrals. The energies are shifted to set the Fermi energy zero.

Figure 5

Hückel band structure of the (5,0) tube at the Fermi level obtained by the effective parameters of the energy-independent partitioning technique. The energies are shifted to set the Fermi energy zero.