Oscillations of local density of states in defective carbon nanotubes

We have made a numerical calculation on the spatial oscillation of the local density of states (LDOS) for a single walled carbon nanotube with a Stone-Wale defect, which induces some localized defect states, decaying rapidly away from the defect. More importantly, it is found that even far from the defect region, there still exist some small regular LDOS oscillations due to the quantum interference, which become much intensified, especially, when the energy slightly exceeds a van Hove singularity. All the oscillations can be classified into two types, depending on the geometrical symmetry of the defective tube, and can be qualitatively explained by the quantum scattering theory of a quasi-one-dimensional system. The LDOS oscillation in energy at a given position far from the defect is also found, whose period becomes shorter and shorter when the position goes away from the defect farther and farther.

Figure 1

A schematic diagram of the defective armchair (7,7) SWNTs with a Stone-Wale defect. The structure (a) is symmetrical with respect to a mirror plane containing the tube axis, while (b) is unsymmetrical under this operation.

Figure 2

(a) plots the LDOS as a function of the position $x$ and energy $E$ for the structure shown in Fig. 1a, where the $x$ axis is along the line denoted by the larger balls in Fig. 1, which is parallel to the tube axis, while $x=0$ is taken at the symmetrical plane of the defect perpendicular to the tube axis. (b) and (c) show the two-dimensional LDOS of two quasibound defect states at energy $E=0.6\mathrm{eV}$ and $-1.07\mathrm{eV}$, respectively, where the $x$ axis is parallel to the tube axis while the $y$ axis corresponds to the circumference, and the origin is taken at the symmetrical center of the defect. (d) gives the reflection coefficients for both $\pi$ and ${\pi }^{*}$ bands.

Figure 3

(a) The cross section plots of Fig. 2a at different energies along the $x$ axis. (b) The Fourier transformation of the LDOS in (a). The unit of the wave vector is $\pi ∕a_0$, where $a_0$ is the length of a unit cell of the tube, $K_1=2\pi ∕a_0$. (c) The $E$ vs $k$ dispersion relation near the Fermi energy. (d) and (e) show the LDOS oscillations at $E=-1.20\mathrm{eV}$, and its Fourier amplitude, respectively.

Figure 4

LDOS plots as a function of energy $E$ at two different positions, $x=30a_0$ and $x^{\prime }=60a_0$, where $a_0=0.246\mathrm{nm}$ is the lattice constant along the tube axis. As a reference, the dashed line shows the LDOS at the “same” position but without the defect.

Figure 5

The calculated results for the asymmetrical defect structure shown in Fig. 1b. (a) The spatial LDOS oscillation at $E=0.60\mathrm{eV}$. (b) Its Fourier transformation. In addition to the components of $2k_{\pi }$ and $2k_{{\pi }^{*}}$, there is a peak at $k_{{\pi }^{*}}-k_{\pi }$.