[Paper Review] Analytical Tight-binding Approach for Ballistic Transport through Armchair Graphene Ribbons: Exact Solutions for Propagation through Step-like and Barrier-like Potentials

Yu. O. Klymenko, Oleksii Shevtsov|ArXiv.org|Jun 27, 2008

Graphene research and applications3 citations

TL;DR

This paper presents an exact analytical tight-binding model for electron transport in armchair graphene ribbons, deriving closed-form solutions for wavefunctions and propagation velocities. It reveals transmission resonances in semiconducting ribbons and shows that backscattering for gapless modes is possible—contrary to the Klein paradox—with backscattering probabilities scaling as the square of the potential in the low-energy limit.

ABSTRACT

Based on a tight-binding approximation, we present analytical solutions for the wavefunction and propagation velocity of an electron in armchair graphene ribbons. The derived expressions are used for computing the transmission coefficients through step-like and barrier-like potentials. Our analytical solutions predict a new kind of transmission resonances for one-mode propagation in semiconducting ribbons. Contrary to the Klein paradox in graphene, this approach shows that backscattering for gapless mode is possible. In consistence with a higher order $\bf{k\cdot p}$ method, the backscattering probabilities vary with the square of the applied potential in the low-energy limit. We also demonstrate that gapless-mode propagation through a potential step in armchair ribbons can be described by the same through-step relation as that for an undimerized 1D chain of identical atoms.

Motivation & Objective

• To develop an exact analytical framework for electron transport in armchair graphene ribbons using the tight-binding model.
• To compute transmission coefficients through step-like and barrier-like potentials in a closed-form analytical manner.
• To investigate deviations from the Klein paradox in graphene, particularly in the context of backscattering for gapless modes.
• To compare analytical results from the tight-binding model with those from the relativistic Dirac (k·p) approximation.
• To establish the validity limits of the k·p method for describing transport in armchair ribbons, especially near the Fermi level.

Proposed method

• Derives exact analytical expressions for normalized wavefunctions and propagation velocity in infinite armchair graphene ribbons using nearest-neighbor tight-binding approximation.
• Solves scattering problems for step-like and barrier-like potentials by formulating and solving a system of linear equations for scattering amplitudes.
• Computes transmission coefficients in closed form using the derived wavefunction solutions and flux conservation principles.
• Compares analytical results with the relativistic Dirac model, particularly focusing on low-energy behavior and backscattering probabilities.
• Uses a 1D linear chain model as a reference to show that through-step transmission in armchair ribbons follows the same relation as in undimerized chains.
• Applies exact solutions from the relativistic Dirac formalism (Katsnelson et al.) for comparison, validating the tight-binding results.

Experimental results

Research questions

• RQ1 How do transmission coefficients behave for electron transport through step-like and barrier-like potentials in armchair graphene ribbons?
• RQ2 To what extent does the tight-binding model predict backscattering for gapless modes in armchair ribbons, contradicting the Klein paradox?
• RQ3 What is the dependence of backscattering probability on the applied potential in the low-energy limit?
• RQ4 How do the analytical results from the tight-binding model compare with those from the relativistic k·p approximation?
• RQ5 Is the transmission through a potential step in armchair ribbons described by the same relation as in a 1D chain of identical atoms?

Key findings

• The analytical tight-binding model yields exact closed-form solutions for wavefunctions and propagation velocity in armchair graphene ribbons.
• Transmission resonances are predicted for one-mode propagation in semiconducting ribbons, indicating non-trivial transport behavior.
• Backscattering for gapless modes is possible, contradicting the perfect transmission predicted by the Klein paradox in graphene.
• Backscattering probabilities scale with the square of the applied potential in the low-energy limit, consistent with higher-order k·p methods.
• Gapless-mode transmission through a potential step in armchair ribbons follows the same through-step relation as in a 1D undimerized chain.
• The relativistic Dirac model overestimates the validity of perfect transmission; the tight-binding model reveals deviations due to trigonal warping effects.