Spin-Orbit Gaps in Armchair Nanotubes Calculated Using the Linear Augmented Cylindrical Wave Method

Int J of Quantum Chemistry

Makaev

2015

Using a linearized augmented cylindrical wave (LACW) approach taking into account the screw and rotational symmetries of carbon nanotubes (CNTs), the first principles technique for the spin-dependent band structure calculations of singlewalled CNTs is developed. The method is applicable to any tubule independent on diameter and chirality. The calculations are based on the two-component relativistic Hamiltonian and muffin-tin and exchange approximations for potentials. As example, the band structures of the three chiral, one armchair, and one zigzag CNTs are calculated and presented as the functions of the screw wave vector and rotational quantum number. The spin-orbit coupling effects appear as splitting of some nonrelativistic electron bands equal to between the 0.01 and 1 meV depending on the CNTs structure, rotational quantum number, and Brillouin zone position. In agreement with previous empirical tight-binding theories, almost perfect polarization of spin is observed in the case of chiral tubules. V C 2015 Wiley Periodicals, Inc.

“…The calculations of integrals

true〈 Ψ_{P_{2} M_{2} N_{2}} (r | k L) χ_{2} false| H_{S - O} false| Ψ_{P_{1} M_{1} N_{1}} (r | k L) χ_{1} true〉

with the H S–O operator Eq. is facilitated by the fact that the basis functions

Ψ_{PMN} (r | k L)

are the products of radial functions and spherical harmonics in the MT regions . The final equation obtained here to calculating these integrals are presented below

\begin{matrix} false〈 Ψ_{P_{2} M_{2} N_{2}} (r | k L) α false| H_{S - O} false| Ψ_{P_{1} M_{1} N_{1}} (r | k L) α false〉 = \frac{n}{2 c^{2} h} {(- 1)}^{n false(M_{1} + M_{2} false)} \sum_{α_{MT} = 1, 2} {false(r_{α_{MT}} false)}^{4} \sum_{l = 0}^{\infty} false(2 l + 1 false) \sum_{m = - l}^{l} m \frac{false(l - false| m false| false)!}{false(l + false| m false| false)!} \times \\ false{ς \end{matrix}

…”

Section: Methods Of Calculationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ab initio spin‐dependent band structures of carbon nanotubes

Int J of Quantum Chemistry

Makaev

2015

“…In the absence of a spin‐orbit (SO) interaction, the double orbital and double spin degeneracy should yield the fourfold degenerate electronic levels. In papers, we calculated the SO gaps in the Fermi energy region for the metallic ( n , n ) tubules and for the metallic and semiconducting linear carbon chains (carbynes). These cases are very interesting, because a formation of gap between the occupied and unoccupied bands is purely relativistic effect here.…”

Section: Spin‐orbit Couplingmentioning

confidence: 99%

“…The

L_{z}

L_{\pm}

operators do not perturb the radial part of the wave function, and the integral

〈 Ψ_{P_{2} M_{2} N_{2}}^{k} true(boldr true) χ_{2} | H_{S - O} | Ψ_{P_{1} M_{1} N_{1}}^{k} true(boldr true) χ_{1} 〉

takes the form of product of the radial and angular parts simplifying calculation of these matrix elements, which are presented in papers …”

Section: Spin‐orbit Couplingmentioning

confidence: 99%

Linear augmented cylindrical wave method for nanotubes electronic structure

Int J of Quantum Chemistry

2015

In this review paper, the main ideas and results of application of the linearized augmented cylindrical wave (LACW) method for the electron properties of the single-walled, double-walled, embedded, and intercalated nanotubes are summarized. We start with the simplest case of the achiral single-walled (n, 0) and (n, n) tubules having small translational unit cells. Then, the electron properties of chiral (n, m) nanotubes having very large translational cells are discussed with account of tubules possessing rotational and screw symmetries. Based on the LACW and Green's function techniques, the ab initio numerical approach to calculating the electron local densities of states of the substitutional impurities in the nanotubes is presented. The relativistic version of LACW theory is described and applied to calculating the effects of spin-orbit coupling on pbands of the cumulenic (C) n and polyynic (C 2 ) n carbynes and armchair tubules. The approach of the cylindrical waves applied for the description of nanotubes permits to reproduce their geometry in an explicit form that offers the great advantages.

“…In terms of the kp- method, it was shown that the curvature of the nanotube surface breaks a symmetry that is present in graphene; this broken symmetry enhances the intrinsic spin–orbit coupling in carbon nanotubes compared with flat graphene, and the spin–orbit term gives rise to the splitting of energy levels and formation of gaps about 0.1 meV. The curvature-induced spin–orbit splitting in the Fermi level region of nanotubes was also studied in terms of the empirical tight-binding models, − in addition to the first-principle projected augmented wave and linear augmented cylindrical wave methods . The spin–orbit effects were also qualitatively studied in the curved graphene, fullerene, and nanotube caps in terms of simple π-electron model and perturbation theory, the importance of curvature of a carbon material surface for larger spin–orbital coupling being indicated again .…”

Section: Introductionmentioning

confidence: 99%

Spin–Orbit Gaps in Carbynes

Zaluev

2014

J. Phys. Chem. C

The effect of spin−orbit interaction on the band structures of the monatomic carbon chains, called the carbynes, is calculated in terms of a linear augmented cylindrical wave method. Because of the cylindrical symmetry of carbynes, the twofold orbitally degenerate π bands correspond to the semiclassical clockwise and anticlockwise rotational motion of electrons around the symmetry axis. In the absence of spin−orbit interaction with the two possible directions of spin, the π bands would be the fourfold degenerate ones. The spin and orbital motion of electrons are coupled, thereby splitting the fourfold degeneracy. Each π sub-band still has the twofold degeneracy, the spin polarization direction between degenerate two bands being opposite to each other. In a cumulenic carbyne with the double bonds (...C C...), the splitting of π band at the Fermi energy region is equal to 2.4 meV, but the metallic character of band structure is not broken by spin−orbit interaction. In the semiconducting polyynic carbyne with alternating single and triple bonds (...−CC− CC−...), the spin−orbit gaps are different for the highest valence band (3.1 meV) and the lowest conduction band (2.1 meV). The spin−orbit gaps in carbyne are about 2 or 3 times smaller than the spin−orbit splitting (6 meV) in the carbon atom. In carbyne, the spin−orbit interaction is larger than that in carbon nanotubes because of the larger curvature of electron orbits encircling the carbyne chains; the larger spin−orbit coupling can be attractive for new experiments and applications.