Spectral butterfly and electronic localization in rippled-graphene nanorribons: mapping onto effective one-dimensional chains

TL;DR

This paper maps the electronic structure of rippled graphene nanoribbons onto effective one-dimensional chains, revealing complex spectra with localization phenomena and band gaps influenced by ripple wavelength and orientation.

Contribution

It provides an exact mapping technique for rippled graphene nanoribbons, enabling detailed analysis of spectral and localization effects due to uniaxial ripples.

Findings

Spectrum exhibits Hofstadter butterfly-like structure.

Presence of Fermi level gaps and dispersionless bands.

Localization behavior varies with ripple wavelength and orientation.

Abstract

We report an exact map into one dimensional effective chains, of the tight-binding Hamiltonian for electrons in armchair and zigzag graphene nanoribbons with any uniaxial ripple. This mapping is used for studying the effect of uniaxial periodic ripples, taking into account the relative orientation changes between $\pi$ orbitals. Such effects are important for short wavelength ripples, while for long-wave ones, the system behaves nearly as strained graphene. The spectrum has a complex nature, akin to the Hofstadter butterfly with a rich localization behavior. Gaps at the Fermi level and dispersionless bands were observed, as well. The complex features of the spectrum arise as a consequence of the quasiperiodic or periodic nature of the effective one dimensional system. Some features of these systems are understandable by considering weakly coupled dimers. The eigenenergies of such dimers are highly degenerated and the net effect of the ripple can be seen as a perturbation potential that splits the energy spectrum. Several particular cases were analytically solved to understand such feature.