Helical and rotational symmetries of nanoscale graphitic tubules

White, C. T.; Robertson, Daniel H.; Mintmire, J. W.

doi:10.1103/physrevb.47.5485

Cited by 627 publications

(448 citation statements)

References 9 publications

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“…As for the C tubes both are used in literature (see e.g. [3,6] for km and [7,8,9,10] forkm classification), while for the Z and A tubes only the former one, as it has been done in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…The method is applied to find the energy bands and corresponding eigen functions of the carbon nanotubes [2]. Although the bands dispersion relations have been already calculated in the literature both for km [3,6], andkm quantum numbers [7,8,9,10], only the helical part of the symmetry group has been used, neglecting the parities. Such incomplete band assignation can produce errors in studying various processes, since the selection rules incorporating parities are more severe, forbidding some otherwise allowed interband transitions.…”

Section: Discussionmentioning

confidence: 99%

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Modified group projectors: tight-binding method

Damnjanović

Vuković

Milošević

2000

J. Phys. A: Math. Gen.

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Abstract. Modified group projector technique for induced representations is a powerful tool for calculation and symmetry quantum numbers assignation of a tight binding Hamiltonian energy bands of crystals. Namely, the induced type structure of such a Hamiltonian enables efficient application of the procedure: only the interior representations of the orbit stabilizers are to be considered. Then the generalized Bloch eigen functions are obtained naturally by the expansion to the whole state space. The method is applied to the electronic π-bands of the single wall nanotubes: together with dispersion relations, their complete symmetry assignation by the full symmetry (line) groups and the corresponding symmetry-adapted eigen function are found.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Modified group projectors: tight-binding method

Damnjanović

Vuković

Milošević

2000

J. Phys. A: Math. Gen.

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“…Nevertheless, for these nanotubes the isospin quantum number is protected by a screw axis of the order N S ͑n 1 , n 2 ͒ defined by a Diophantine equation, see Ref. 42. For the k t = 0 point, screw rotations are equivalent to spatial rotations, hence, it follows from Eq.…”

Section: ͑1͒mentioning

confidence: 99%

Spin-orbit effects in carbon-nanotube double quantum dots

Weiss¹,

Rashba²,

Kuemmeth³

et al. 2010

Phys. Rev. B

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We study the energy spectrum of symmetric double quantum dots in narrow-gap carbon nanotubes with one and two electrostatically confined electrons in the presence of spin-orbit and Coulomb interactions. Compared to GaAs quantum dots, the spectrum exhibits a much richer structure because of the spin-orbit interaction that couples the electron's isospin to its real spin through two independent coupling constants. In a single dot, both constants combine to split the spectrum into two Kramers doublets while the antisymmetric constant solely controls the difference in the tunneling rates of the Kramers doublets between the dots. For the two-electron regime, the detailed structure of the spin-orbit split energy spectrum is investigated as a function of detuning between the quantum dots in a 22-dimensional Hilbert space within the framework of a single-longitudinalmode model. We find a competing effect of the tunneling and Coulomb interaction. The former favors a left-right symmetric two-particle ground state while in the regime where the Coulomb interaction dominates over tunneling, a left-right antisymmetric ground state is found. As a result, ground states on both sides of the ͑11͒-͑02͒ degeneracy point may possess opposite left-right symmetry, and the electron dynamics when tuning the system from one side of the ͑11͒-͑02͒ degeneracy point to the other is controlled by three selection rules ͑in spin, isospin, and left-right symmetry͒. We discuss implications for the spin-dephasing and Pauli blockade experiments.

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“…For all of them k = 0, causing that only direct processes are encountered and now m is also a conserved quantum number (since k i = k f yields K = 0). This means that their symmetry properties are related to the isogonal groups (3).…”

Section: Selection Rulesmentioning

confidence: 99%

Carbon nanotubes band assignation, topology, Bloch states, and selection rules

2002

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Various properties of the energy band structures (electronic, phonon, etc.), including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the singlewall carbon nanotubes are established. The complete set of quantum numbers consists of quasi momenta (angular and linear or helical) and parities with respect to the z-reversal symmetries and, for achiral tubes, the vertical plane. The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are characterized by the same set of quantum numbers. All this enables application of the presented exhaustive selection rules. The results are discussed by some examples, e.g. allowed interband transitions, conductivity, Raman tensor, etc.

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Helical and rotational symmetries of nanoscale graphitic tubules

Cited by 627 publications

References 9 publications

Modified group projectors: tight-binding method

Modified group projectors: tight-binding method

Spin-orbit effects in carbon-nanotube double quantum dots

Carbon nanotubes band assignation, topology, Bloch states, and selection rules

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