Disordered carbon nanotube alloys in the effective-medium supercell approximation

We investigate a disordered single-walled carbon nanotube (SWCNT) in an effective-medium supercell approximation (EMSCA). The first type of disorder that we consider is the presence of vacancies. Our results show that the vacancies induce some bound states on their neighbor host sites, leading to the creation of a band around the Fermi energy in the SWCNT average density of states. The second type of disorder considered is a substitutional ${\mathrm{B}}_{\mathit{cb}}{\mathrm{N}}_{\mathit{cn}}{\mathrm{C}}_{1-\mathit{cb}-\mathit{cn}}$ alloy due to its applications in hetrojunctions. We found that for a fixed boron (nitrogen) concentration, by increasing the nitrogen (boron) concentration the averaged semiconducting gap $E_g$ decreases and disappears at a critical concentration. A consequence of our results for nanoelectronic devices is that by changing the boron (nitrogen) concentration, one can make a semiconductor SWCNT with a predetermined energy gap.

Figure 1

A two-dimensional graphene sheet. The light dashed lines illustrate the Bravais lattice unit cells, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the primitive vectors. Each cell includes two nonequivalent sites, which are denoted by A and B. $L=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2$ is the chiral vector. For an armchair SWCNT $n_1=n$ and $n_2=2n$, while for a zigzag SWCNT $n_1=n$ and $n_2=0$. The heavy dashed line denotes a four-site supercell.

Figure 2

First Brillouin zone and the patches of supercell wave vectors ${\mathbf{K}}_n$ where they are corresponding to a four-sites supercell in real space for: (a) an armchair SWCNT where the supercell wave vectors are ${\mathbf{K}}_1=0$, ${\mathbf{K}}_2=(-{\mathbf{b}}_1+{\mathbf{b}}_2)∕2$, ${\mathbf{K}}_3={\mathbf{b}}_2∕2$, ${\mathbf{K}}_4={\mathbf{b}}_1∕2$, and (b) a zigzag SWCNT where the supercell wave vectors are ${\mathbf{K}}_1=0$, ${\mathbf{K}}_2=-{\mathbf{b}}_2∕2$, ${\mathbf{K}}_3=({\mathbf{b}}_1-{\mathbf{b}}_2)∕2$, and ${\mathbf{K}}_4={\mathbf{b}}_1∕2$. The heavy lines indicate the FBZ and the dashed lines illustrate the patches where belong to each ${\mathbf{K}}_n$.

Figure 3

(a) Comparison of average density of states for a (10,10) armchair SWCNT and (b) for a (10,0) zigzag SWCNT at $\delta =40t$ and different impurity concentration. At high random energies, $\delta ⪢\text{bandwidth}$, we have band splitting and the impurity band is located at higher energies, while remaining sites are the vacancies that induce a band around the Fermi energy. By increasing the vacancy concentration, the height of the peak at the Fermi energy increases and the van Hove singularities are smeared out.

Figure 4

The figure shows the effects of nitrogen doping on semiconducting gap $E_g$ for a (10,0), (20,0), and (40,0) zigzag SWCNT alloy.

Figure 5

The effects of nitrogen doping on a (10,0), (20,0), and (40,0) zigzag SWCNT’s average density of states. At a fixed boron concentration $cb=0.15$, the average density of states for two nitrogen concentrations, low $cn=0.0005$ and high, are compared. For (20,0) and (40,0), at the critical concentration $E_g$ is zero and the van Hove singularities disappear.

Figure 6

Figure shows the effects of boron doping on the semiconducting gap $E_g$ in a (10,0), (20,0), and (40,0) zigzag SWCNT alloy.

Figure 7

Effects of boron doping on a (10,0), (20,0), and (40,0) zigzag SWCNT’s average density of states. At a fixed nitrogen concentration of $cn=0.1$, the average density of states for two boron concentrations are compared. For (20,0), (40,0) at a critical concentration, $E_g$ is closed and the van Hove singularities disappears and a semiconductor semimetal phase transition takes place.