The two classes of low-energy spectra in finite carbon nanotubes

Abstract

We study the spectra of finite-length carbon nanotubes (CNTs) of arbitrary chirality. They divide into two classes, which arise because of different rotational symmetries of the low-energy eigenstates. In one of them (the zigzag class), the spinless spectrum is doubly degenerate and the two states can be assigned to different values of the valley degree of freedom. In the other (armchair class), the valley degeneracy is removed and the eigenstates are combinations of both valley states. Recent experimental observation of the valley mixing in ultraclean CNT quantum dots is consistent with our theory for armchair-class CNTs.

Majorana quasiparticles in semiconducting carbon nanotubes

“…The band crossing at k = 0 is protected by symmetry since the crossing bands belong to different valleys K and K , i.e. in this CNT to different angular momenta [26,27], and the magnetic field can- not hybridize them. The presence of a superconducting substrate plays here a double role.…”

Topology and zero energy edge states in carbon nanotubes with superconducting pairing

“…It is well known that the SWNTs are metallic when mod(2n + m, 3) = 0 and semiconducting if mod(2n + m, 3) = 1, 2 [41]. Recent studies [14,15,18,42] have revealed that the SWNTs can be alternatively classified into two classes according to the angular momentum of the two valleys, denoted in the following K and K : (i) zigzag class, which includes metal-1 (metallic SWNTs with d R = d) and semiconducting SWNTs with d ≥ 4, in which the two valleys have different angular momenta, where d R = gcd(2n + m, 2m + n); (ii) armchair class, which includes metal-2 (metallic SWNTs with d R = 3d) and semiconducting SWNTs with d ≤ 2, in which the two valleys have the same angular momentum. Here the angular momentum µ τ of valley τ (= K, K ) is defined as follows.…”

“…In recent studies, the emphasis has been put on the bound-state spectrum which naturally arises due to the finiteness of the SWNT length. It has been shown that the valley degeneracy of the bound states is not only lifted by the curvature-induced spin-orbit interaction [2][3][4][5][6][7][8][9][10][11][12][13], but also by a valley mixing from the edges [14,15]. Furthermore, open-ended SWNTs commonly host edge states whose energies lie in the bulk band gap [14,16,17].…”

Spin-orbit coupling and proximity effects in metallic carbon nanotubes

“…In a recent work [22], we have showed that it is possible to distinguish a class of tubes defined by a condition (n − m)/gdc(m,n)mod3 = 0, that have two pairs of Fermi points located around K ⊥ = 0, K ≈ 0, that is similar to the zigzag CNT. In Ref.…”

“…In Ref. [22], we considered an infinitely sharp, local chemical potential, an extra term in the Hamiltonian ∼μ 0 δx − x 0 ρ(x) with μ 0 → ∞ and ρ(x) is an electronic density, a Fourier transform of k c † k c k+q . For the zigzaglike tubes, a response to such potential is a reflection matrix that is strictly diagonal in the valley space.…”