The quantum structure of carbon tori

Abstract

Carbon nanotubes are very promising new materials that fit perfectly well into the miniaturisation of technology, but at the same time show a fascinating quantum structure. When the tube is closed to form a torus the structure acquires discrete molecular characteristics. In the present paper the electronic levels of tori are obtained by a double folding of the Brillouin zone of graphitic carbon. An illustrative example of an hypothetical giant torus with molecular symmetry is fully analysed, with special emphasis on its pictorial representation.

Visualization of higher genus carbon nanomaterials: free energy, persistent current, and entanglement entropy

“…4, we can calculate the energy eigenvalues of the TCNT. Equivalently, we can also calculate the energy eigenvalues of TCNT mathematically using the dispersion relation [1] where k = k xx +k yŷ is the Bloch wave-vector, γ 0 = 2.7eV is the nearest-neighbor hopping integral for electron in graphene, φ 1 and φ 2 are the magnetic flux. The expression to calculate the free energy is given below,…”

Symmetry groups associated with tilings on a flat torus

“…These motions are not distance preserving and hence are not elements of G. Nevertheless, we can regard them as four-dimensional symmetries in G Ã acting on H Ã with analogous effects as threedimensional non-rigid motions acting on the nanotorus. In related literature (Bovin et al, 2001;Arezoomand & Taeri, 2009;Zhao et al, 2012), these are also considered as symmetry elements.…”

“…It is analogous to the approach used by De Las Peñ as et al (2014) to determine the line-group type of the symmetry group of a single-wall nanotube. Another advantage of the approach is that it allows the relevant non-rigid motions of a nanotorus considered in the literature (Bovin et al, 2001;Arezoomand & Taeri, 2009;Zhao et al, 2012) to be derived and characterized as four-dimensional symmetries. This offers a rigorous treatment of these motions as Euclidean symmetries.…”

Isospectral but physically distinct: Modular symmetries and their implications for carbon nanotori

“…We shall choose a Euclidean coordinate system such that a 1 = (1, 0) and a 2 = ( 1 2 , − 1 2 √ 3) in units of √ 3R cc , where R cc is the fundamental carbon-carbon bond length. As always, the band structure associated with a given lattice can be described in terms of a dispersion relation E( k) which relates an electron wavevector k to its energy E. For the graphene sheet, and in the tight-binding (or Hückel) approximation in which the only significant overlap integrals are those between the 2p z orbitals associated with nearest-neighbor carbon atoms, this dispersion relation is given by [13][14][15]…”

“…In other words, nanotori built from zigzag or armchair nanotubes might have spectral properties which are identical to those of nanotori built from non-zigzag or non-armchair nanotubes; moreover, these spectral properties might or might not correspond to the special zigzag or armchair angles β = 0, π/6. As an example, the (6, 0, −17, −6) nanotorus is built from a zigzag nanotube while the (9, 9, −40, −44) nanotorus is built from an armchair nanotube and the (3,15,13,53) and (4,8,21,33) nanotori are built from chiral nanotubes with different chiralities. Yet all four nanotori have identical spectral properties which are the same as those of a chiral nanotorus with N hex = 36, τ = 12(1 + 3 2 √ 3i)/31, and tan β = 5 √ 3/7 (or β ≈ 51.05 • ).…”