# Interband transitions in narrow-gap carbon nanotubes and graphene nanoribbons

## TL;DR

This paper investigates interband transitions in narrow-gap carbon nanotubes and graphene nanoribbons, revealing curvature and edge effects that enhance optical transition probabilities and potential for THz radiation applications.

## Contribution

It demonstrates the universal value of the velocity matrix element at the band gap and explores the photon-energy dependence of transition probabilities in these nanostructures.

## Key findings

- Curvature and edge effects open band gaps and enhance optical transition probabilities.
- The velocity matrix element equals the Fermi velocity at the band gap energy.
- Transition matrix element varies with photon energy, affecting THz emission potential.

## Abstract

We use the robust nearest-neighbour tight-binding approximation to study on the same footing interband dipole transitions in narrow-bandgap carbon nanotubes and graphene nanoribbons. It is demonstrated that curvature effects in metallic single-walled carbon nanotubes and edge effects in gapless graphene nanoribbons not only open up bang gaps, which typically correspond to THz frequencies, but also result in a giant enhancement of the probability of optical transitions across these gaps. Moreover, the matrix element of the velocity operator for these transitions has a universal value (equal to the Fermi velocity in graphene) when the photon energy coincides with the band-gap energy. Upon increasing the excitation energy, the transition matrix element first rapidly decreases (for photon energies remaining in the THz range but exceeding two band gap energies it is reduced by three orders of magnitude), and thereafter it starts to increase proportionally to the photon frequency. A similar effect occurs in an armchair carbon nanotube with a band gap opened and controlled by a magnetic field applied along the nanotube axis. There is a direct correspondence between armchair graphene nanoribbons and single-walled zigzag carbon nanotubes. The described sharp photon-energy dependence of the transition matrix element together with the van Hove singularity at the band gap edge of the considered quasi-one-dimensional systems make them promising candidates for active elements of coherent THz radiation emitters. The effect of Pauli blocking of low-energy interband transitions caused by residual doping can be suppressed by creating a population inversion using high-frequency (optical) excitation.