Perturbation calculations on interlayer transmission rates from symmetric to antisymmetric channels in parallel armchair nanotube junctions

Partially overlapping two parallel armchair nanotubes are investigated theoretically with the $\pi$ orbital tight bonding model. Considering the interlayer Hamiltonian as perturbation, we obtain approximate analytical formulas of the interlayer transmission rates $T_{\sigma',\sigma}$ from channel $\sigma$ to $\sigma'$ for all the four combinations $(\sigma',\sigma)=(\pm,\pm)$ and $(\pm,\mp)$, where suffixes $+$ and $-$ represent symmetric and anti-symmetric channels, respectively, with respect to the mirror plane of each tube. Landauer's formula conductance is equal to the sum of them in units of $2e^2/h$. According to the perturbation calculation, the interlayer Hamiltonian is transformed into the parameter $w_{\sigma',\sigma}$ that determines the analytical formula of $T_{\sigma',\sigma}$. By comparison with the exact numerical results, the effective range of the analytical formulas is discussed. In the telescoped coaxial contact, the off-diagonal part $T_{-,+}+T_{+,-}$ is very small compared to the diagonal part $T_{+,+}+T_{-,-}$. In the side contact, on the other hand, the off-diagonal part is more significant than the diagonal part in the zero energy peak of the conductance.