Polarizabilities of carbon nanotubes: Importance of the crystalline orbitals relaxation in presence of an electric field

Abstract

ABSTRACT:The static polarizabilities of a series of zigzag single walled carbon nanotubes [(m, 0) with m = 3n for m ¼ 7-25] were computed by the coupled perturbed Kohn-Sham (CPKS) coupled method recently implemented in the periodic CRYSTAL code. This method makes it possible to compute both the longitudinal and transverse polarizabilities with the same approach including the local field depolarization effects for the whole series of the investigated nanotubes. To quantify these effects, the unscreened longitudinal and transverse polarizabilities were also calculated within the linear response theory using the Sum Over States (SOS) method. Our results confirmed the inverse-square dependence on the bandgap of the longitudinal polarizabilities which are found weakly affected by the crystalline orbital relaxation. On the other hand, the comparison between the screened (CPKS) and unscreened (Sum Over States) transverse polarizabilities permit to calibrate more precisely the classical correction of Benedict et al. (Phys Rev B, 1995, 52, 8541) which must be applied to include the local field effect at the SOS level.

Properties of Carbon Nanotubes: An ab Initio Study Using Large Gaussian Basis Sets and Various DFT Functionals

“…It is also linear in 1/ E g (in this case the numerical noise is larger, see Figure c), with very different slopes according to the more or less metallic character of the nanotube family. This means that α yy / a increases with r 2 in the interval explored here, confirming the results of previous works for tubes with r up to 10 Å. ,,, …”

“…Due to the linear relation between r and 1/ E g discussed above, the r / E g law can also be formulated as r 2 or ( r / E g 2 ) 2/3 . As a cross-check, our α xx / a data have been fitted vs ( r / E g 2 ) w providing w = 0.63 (quite close to 2/3 ≃ 0.66), in agreement with the value obtained in a recent work ( w = 0.64) where tubes with r < 10 Å were considered. The above discussion applies to large r values, and not to the very small ones, as the tube at this limit collapses with discontinuity.…”

Ab initio electron energy‐loss spectra and depolarization effects: Application to carbon nanotubes

“…In a previous work22, we have used the CPKS method to compute both in‐plane and out‐of‐plane static polarizabilities of the ( m , 0) SWCN series with m ≠ 3 n for m = 7–25. As stated in previous studies, the comparison with SOS values obtained in the same computational conditions clearly confirms that the in‐plane static polarizabilities are slightly sensitive to the polarization effect.…”

“…The number of k points in the first irreducible Brillouin zone (BZ) in which the DFT Hamiltonian matrix is diagonalized, is 133 for h‐BN, 61 for the BN slabs, and 27 for the carbon nanotubes. All electron basis sets already optimized in early studies (7‐321G(d) for N27, 6‐211G(d) for B28 and 6‐21G(d) for C22) were used for orbital expansion solving the KS equation iteratively. We set the total energy tolerance to 10 −7 hartree and eigenvalue tolerance to 10 −6 hartree in the iterative solution of the KS equation.…”

“…Interestingly, the SOS estimate of the static polarizability turns out to be the zero step of the iterative CPKS process. Studies on either zigzag SWCN (m, 0) (with m ≠ 3 n )21, 22 or boton nitride (BN) slabs23 have shown that the perpendicular polarizability value (parallel to the nonperiodic direction) calculated when orbital relaxation effects are taken into account as in CPKS is very different from the unrelaxed SOS one. In both cases, the authors have shown that this important relaxation effect can be modeled by the Clausius‐Mossotti formula relating the SOS and CPKS polarizability values with a depolarization factor equal to 2π (1D system) or 4π (2D system) when the field is perpendicular to the tube or the slab, respectively.…”