T-matrix analysis

The absence of localization in the single-channel region can be understood from the Dirac equation including the impurity potential term U ˆ imp with armchair edge boundary. To consider the amplitude of backward scattering, we introduce the T -matrix defined as

Here it should be noted that the phase structure in equation (36) is different between K ₊ and

K − states, and these internal phase structures are critical for the scattering matrix elements of armchair nanoribbons, as we discuss in the following. Using the above expression, we can obtain the scattering matrix element

h n, k, s V n, k , (38)

with

L

V n, k d y e

. (39)

It should be emphasized that equation (38) has the same form as that obtained for carbon nanotubes without inter-valley scattering (u ⁰ _X (r) = 0)[5]. Interestingly, in spite of the fact that armchair nanoribbons inevitably suffer from inter-valley scattering due to the armchair edges

0), we can express the matrix element for backward scattering as equation (38) by including u ⁰ _X (r) into V (n, k; n ⁰, k ⁰) in equation (39). This is due to the different phase structure between K ₊ and K − in equation (36).

We focus on the single-channel regime where only the lowest subband with n = 0 crosses the Fermi level. From equation (38), the scattering amplitude from the propagating state |0, k, s i to its backward state |0,− k, s i in the single-channel mode becomes identically zero, i.e.

. (40)

Thus, since the lowest backward scattering matrix element of T -matrix vanishes, the decay of h g i in the single-channel energy regime is extremely slow as a function of the ribbon length as we have seen in figure 8. However, the back-scattering amplitude in the second and much higher order does not vanish. Hence the single-channel conduction is not exactly perfect like carbon nanotubes [5], but nearly perfect in armchair nanoribbons.