Relation between the transmission rates and the wave functions in the carbon nanotube junctions

TL;DR

This paper investigates the relationship between electron transmission rates and wave functions in carbon nanotube junctions, using analytical and numerical methods to understand how defects affect electron transport.

Contribution

It introduces an analytical approach based on the effective mass equation and a numerical transfer matrix method to analyze transmission and wave functions in nanotube junctions with defects.

Findings

Transmission rate is inversely proportional to the squared wave function amplitude.

Good agreement between analytical and tight-binding models away from critical energy.

Discrepancies near critical energy are due to evanescent waves with long decay lengths.

Abstract

Electron transmission and wave functions through junctions with a pair of a pentagonal defect and a heptagonal defect connecting two metallic carbon nanotubes are analyzed by the analytical calculation with the effective mass equation. The energy region $|E| < E_c$ is considered where the channel number is kept to two. Close relation between the transmission rate and the wave function is found; the transmission rate is given by the inverse squared absolute value of the wave function. The dependence of the transmission rates on the energy and on the size of the junction is clearly explained by the nature of the wave function. Though the wave function and the transmission rate calculated by the tight binding model agree well with the corresponding analytical results by the effective mass approximation, the discrepancy becomes considerable when $|E| \simeq E_c$. To study the origin of this discrepancy, an efficient numerical calculation method is developed with a generalized transfer matrix for the tight binding model. Their numerical results are compared with the corresponding analytical ones and the results show that the origin of the discrepancy comes from the evanescent waves with the longest decay length in the tube parts.