Curvature-induced polarization in carbon nanoshells

We investigate the normal polarization induced by bending of graphite shells, which microscopically occurs because of a shift in sp hybridization at each atomic site. Based on a rehybridization model we analytically estimate the dependence on curvature of electronic charge spill into convex region. We further performed DFT calculations and extract the radial atomic dipole by direct electronic charge density integration. A continuum analysis assigns a linear susceptibility tensor to the curvature-induced polarization and shows that this tensor is isotropic for hexagonal lattice, in agreement with microscopic model and computations. The intrinsic polarization effect can be important for nanoscale electronics. 2002 Elsevier Science B.V. All rights reserved. Due to potential applications, there has been a constant interest in the electronic structure of carbon nanotubes (CNTs), graphite cylinders of diameters as small as a few angstroms and lengths of fraction of microns. An increasing attention is directed to the detailed electronic spatial structure of CNT, which prompted a series of theoretical studies, and ultimately direct experimental observations of the electronic charge density [1]. By bending a graphite sheet as in a CNT one introduces an asymmetry in the p-orbital overlap, which brings into a closer proximity the segments of p-orbitals situated inside cavity and spaces apart the outer wings. Therefore, Coulomb repulsion inside cavity increases with curvature and leads to a redistribution (rehybridization) of the porbitals from the sp of graphite to something intermediate between sp and sp. This results into an electronic charge transfer from the concave to the convex region. So far, there have been no selfconsistent calculations that would focus on the magnitude of the charge redistribution in singlewall CNTs due to curvature. Here we investigate the curvature-induced polarization into the rolled graphite sheet by calculating the magnitude of the induced normal atomic dipole at each atomic site. Rehybridization in the CNT network is always expected, but earlier electronic band structure studies estimated that such band mixing effect can be neglected for most nanotubes. However, firstprinciple calculations revealed that curvature effects become dominant in zigzag nanotubes with very small diameters [2]. More recent analytical calculations found that rehybridization can play 3 July 2002 Chemical Physics Letters 360 (2002) 182–188 www.elsevier.com/locate/cplett * Corresponding author. Fax: +1-713-348-5423. E-mail addresses: [email protected] (T. Dumitric a), landis @rice.edu (C.M. Landis), [email protected] (B.I. Yakobson). 0009-2614/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0009 -2614 (02 )00820 -5 an important role in the CNT band structure even at intermediate radii [3]. Recent experiments [4] suggest that curvatureinduced charge redistribution is present, and results into a genuine property of single and multiwall CNTs, non-existent in plane graphite: there is a build-in electrostatic field perpendicular to CNT’s surface, solely due to curved geometry. The surface electric field is believed to cause antilocalization, whose signature is seen in the reported positive magnetoresistance measurements in CNTs under low magnetic field. An additional motivation involves the recent field of chemistry inside CNT. Known to be less chemically reactive than graphite, CNT cavities serve as nanosize test tubes for chemical reactions in confined geometries. Reactions energetics inside could be significantly altered under the influence of electrostatic potential produced by the charge distribution in the wall. In addition, when CNT capillary filling is dominated by van der Waals forces, knowledge of the intrinsic surface polarizationmay serve in understanding the observed size dependent wetting [5]. We begin by recognizing that carbon atoms in the CNT network are not planar and instead are pyramidalized: the three ri-bonds, directed toward positions of nearest neighbors, are now tilted down by the angles ai relative to the tangential plane, as shown in Fig. 1. To capture the curvature effect in an orbital model we adopt the p-orbital axis vector (POAV1) construction [6,7], previously used in studies of CNT reactivity [8] and straindriven chemistry [9]. POAV1 extends the graphite r–p separability to non-planar case, where it is still possible to divide the wavefunction into localized r–p hybrids. The geometrical tilting of r-bonds is accomplished by introducing a degree of pz atomic orbital mixing into graphite’s r network. To preserve orthogonality, a ‘fractional hybridization’ of hp orbitals is further necessary, and the p-hybrids are now represented by a pand s-orbital mixing, as hp 1⁄4 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ k p ðsþ kpzÞ: ð1Þ Parameter k depends directly on curvature through the pyramidalization angle Hrp [6]: k 1⁄4 1 3 cos 2 Hrp 2 cos2 Hrp : ð2Þ Eq. (2) assumes an equal average r-bond hybridization in a C3v symmetry, an approximation adequate for most applications [7]. As shown in Fig. 1b, Hrp is the angle between the r and p hybrids and has the well-known values of 109 280 and 90 for sp and sp hybridization, respectively. The pyramidalization angle deserves special attention since it can be measured exactly from experimental or theoretical configurations and only nearest-neighbor atomic coordinates are needed [6]. Hrp provides a simple way to characterizes the changes in the p-orbital character and is also a measure of local strain and reactivity [8]. We obtain next an analytic description of Hrp for various (n;m) CNTs, to first order in a=R. Here a 1⁄4 1:42 A denotes the distance between two C atoms and R is the CNT radius. For this, it is more convenient to describe CNT atoms in the nanotube coordinate ðĉ; t̂Þ, where ĉ stands for the circumferential direction and t̂ indicates the translational direction along the CNTs axis. If we denote by bi (i 1⁄4 1; 3) the angles made by the rbond angles of the unrolled graphite sheet with the c direction, the CNT’s chirality angle h is given by (a) (b)