Electro-mechanical Properties of Carbon Nanotubes: Effect of Small Tensile and Torsional Strains

We present a simple picture to calculate the bandgap ($E_g$) of carbon nanotubes (CNT) in the presence of uniform torsional and tensile strain ($\sigma$). We find that under tensile strain, $ absolute value of dE_g/d\sigma$ of zig-zag tubes is approximately equal to $3t_0$, where $t_0$ is the hopping parameter. Further, $ absolute value of dE_g/d\sigma$ decreases as the chirality changes to armchair, where it takes the value zero. The sign of $dE_g/d\sigma$ follows the $(N_x-N_y) *mod 3$(equal to - 1, 0 and +1) rule. In contrast to the above, we show that under torsional strain, $absolute value of dE_g/d\sigma$ of armchair tubes is approximately equal to $3t_0$ and continually decreases as the chirality changes to zig-zag, where is takes a small value. The sign of $dE_g/d\sigma$ again follows the $(N_x-N_y)*mod 3$ rule. Finally, we predict a change in the sign of $dE_g/d\sigma$ as function of strain, corresponding to a change in the value of $q$ that corresponds to the bandgap minimum.