TY - JOUR

T1 - Degeneracy breaking and intervalley scattering due to short-ranged impurities in finite single-wall carbon nanotubes.

AU - McCann, Edward

AU - Fal’ko, Vladimir I.

N1 - © 2005 The American Physical Society

PY - 2005/2/15

Y1 - 2005/2/15

N2 - We present a theoretical study of degeneracy breaking due to short-ranged impurities in finite, single-wall, metallic carbon nanotubes. The effective mass model is used to describe the slowly varying spatial envelope wavefunctions of spinless electrons near the Fermi level at two inequivalent valleys (K-points) in terms of the four component Dirac equation for massless fermions, with the role of spin assumed by pseudospin due to the relative amplitude of the wave function on the sublattice atoms (“A” and “B”). Using boundary conditions at the ends of the tube that neither break valley degeneracy nor mix pseudospin eigenvectors, we use degenerate perturbation theory to show that the presence of impurities has two effects. Firstly, the position of the impurity with respect to the spatial variation of the envelope standing waves results in a sinusoidal oscillation of energy level shift as a function of energy. Secondly, the position of the impurity within the hexagonal graphite unit cell produces a particular 4 × 4 matrix structure of the corresponding effective Hamiltonian. The symmetry of this Hamiltonian with respect to pseudospin flip is related to degeneracy breaking and, for an armchair tube, the symmetry with respect to mirror reflection in the nanotube axis is related to pseudospin mixing.

AB - We present a theoretical study of degeneracy breaking due to short-ranged impurities in finite, single-wall, metallic carbon nanotubes. The effective mass model is used to describe the slowly varying spatial envelope wavefunctions of spinless electrons near the Fermi level at two inequivalent valleys (K-points) in terms of the four component Dirac equation for massless fermions, with the role of spin assumed by pseudospin due to the relative amplitude of the wave function on the sublattice atoms (“A” and “B”). Using boundary conditions at the ends of the tube that neither break valley degeneracy nor mix pseudospin eigenvectors, we use degenerate perturbation theory to show that the presence of impurities has two effects. Firstly, the position of the impurity with respect to the spatial variation of the envelope standing waves results in a sinusoidal oscillation of energy level shift as a function of energy. Secondly, the position of the impurity within the hexagonal graphite unit cell produces a particular 4 × 4 matrix structure of the corresponding effective Hamiltonian. The symmetry of this Hamiltonian with respect to pseudospin flip is related to degeneracy breaking and, for an armchair tube, the symmetry with respect to mirror reflection in the nanotube axis is related to pseudospin mixing.

U2 - 10.1103/PhysRevB.71.085415

DO - 10.1103/PhysRevB.71.085415

M3 - Journal article

VL - 71

SP - 085415

JO - Physical review B

JF - Physical review B

SN - 1550-235X

IS - 8

ER -