Self-consistent calculations of strain-induced band gap changes in semiconducting $\left(n,0\right)$ carbon nanotubes

First-principles density-functional calculations of the electronic structure, energy band gaps $\left(E_g\right)$, and strain-induced band gap changes in moderate-gap single-walled $\left(n,0\right)$ carbon nanotubes (SWNTs) are presented. It is confirmed that $\left(n,0\right)$ SWNTs fall into two classes depending upon $n\text{mod}3=1$ or 2. $E_g$ is always lower for “mod 1” than for “mod 2” SWNTs of similar diameter. For $n<10$, strong curvature effects dominate $E_g$; from $n=10$ to 17, the $E_g$ oscillations, amplified due to $\sigma \text{−}\pi$ mixing, decrease and can be explained very well with a tight-binding model which includes trigonal warping. Under strain, the two families of semiconducting SWNTs are distinguished by equal and opposite energy shifts for these gaps. For (10,0) and (20,0) tubes, the potential surface and band gap changes are explored up to approximately $\pm 6\%$ strain or compression. For each strain value, full internal geometry relaxation is allowed. The calculated band gap changes are $\pm \left(115\pm 10\right)\text{meV}$ per 1% strain, positive for the mod 1 and negative for the mod 2 family, about 10% larger than the tight-binding result of $\pm 97\text{meV}$ and twice as large as the shift predicted from a tight-binding model that includes internal sublattice relaxation.

Figure 1

(Color online) Energy band gap $\left(E_g\right)$ as a function of the diameter for $\left(n,0\right)$ zigzag nanotubes; the line with solid circles shows the predictions from DFT (GGA) and the line with solid squares represents predictions from a tight-binding model with a parameter $\gamma =0.43$.

Figure 2

Energy-band structure of three representative SWNTs (13,0), (14,0), and (16,0). The dotted line at zero represents the Fermi energy level.

Figure 3

(Color online) Band decomposed charge density at the $\mathbf{\Gamma }$ point for (13,0), (14,0), and (16,0).

Figure 4

(Color online) Energy band gap $\left(E_g\right)$ as a function of the lattice parameter along the $z$ axis of the SWNT.

Figure 5

(Color online) (a) Deformation potential (relative SCF binding energy) as a function of strain for the (20,0) SWNT (black squares). A parabola (red line online) is an excellent fit for the SCF energies with only small deviations occurring for strains larger than 5%, i.e., Hooke’s law is fulfilled over a large range of strains. The minimum of the potential $\left(\sigma =0\right)$ is found for $z=4.27\text{Å}$. (b) Calculated SCF band gap changes for the (10,0) and (20,0) SWNTs as a function of strain. For (20,0) SWNT, the SCF gap change is compared to the tight-binding changes predicted by Eq. (2). The SCF slope $d\Delta E/d\sigma$ for (20,0) SWNT is $\pm \left(115\pm 10\right)\text{meV}$, about 10% larger than the tight-binding result of $\pm 97\text{meV}$ per 1% strain.

Figure 6

Two of the available states consistent with the boundary condition for a $\left(n,0\right)$ zigzag nanotube with (a) $n\text{mod}3=1$ and (b) $n\text{mod}3=2$ in the zone-folding picture around the $\mathbf{K}$ point of the graphene Brillouin zone.