Long periodic oscillation of electronic properties in capped finite-length armchair carbon nanotubes

Confinement boundary effects, coupled with quantum size effects, on the electronic properties of capped armchair single-walled carbon nanotubes are investigated by use of the tight-binding method. We observe a long periodic oscillation (LPO) behavior of energy gap, except for the conventional short periodic oscillation (SPO) behavior, with the length of carbon nanotube. Different from the intrinsic SPO, the period of LPO is deeply dependent on the topological structure of the closure, but is less related to the tube diameter. It is shown that the LPO behavior results from a closure-induced shift of allowed wave vectors perpendicular to the tube axis.

Figure 1

Top view along the ${\mathrm{C}}_5$ axis and side view of (a) ${\mathrm{C}}_{60+10}$, which is generated from ${\mathrm{C}}_{60}$ by adding 10 carbon atom belt, (b) ${\mathrm{C}}_{40+10}$, which is generated from ${\mathrm{C}}_{40}$ by adding 10 carbon atom belt, and (c) ${\mathrm{C}}_{220+20}$, which is generated from ${\mathrm{C}}_{220}$ by adding 20 carbon atom belt. Half of ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{40}$, and ${\mathrm{C}}_{220}$ will be used as the closures of the SWNTs, and, together with the adding belt of 10 or 20 carbon atoms, can schematically show the connection between the closure and the body of SWNT.

Figure 2

(a) Calculated energy gap $E_g$ as a function of the tube length for the armchair (5,5) SWNT capped with half of ${\mathrm{C}}_{60}$ fullerene. The inset gives the LPO behaviors of $E_g$ with respect to the tubes with $(3q-2)$, $(3q-1)$, and $\left(3q\right)$ sections. (b) Energy gap obtained from Eq. (3) with $\mathrm{\Delta }k=0.02$.

Figure 3

LPO behaviors of energy gap $E_g$ of (a) the (5,5) SWNT capped with half of ${\mathrm{C}}_{40}$ fullerene, (b) the (10,10) SWNT capped with half of ${\mathrm{C}}_{220}$ fullerene, and (c) the (5,5) SWNT terminated by hydrogen atoms as the functions of the tube length with respect to $(3q-2)$, $(3q-1)$, and $\left(3q\right)$ sections.