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

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