Unusual electronic properties of clean and disordered zigzag graphene nanoribbons
Abstract
We revisit the problem of electron transport in clean and disordered zigzag graphene nanoribbons, and expose numerous hitherto unknown peculiar properties of these systems at zero energy, where both sublattices decouple because of chiral symmetry. For clean ribbons, we give a quantitative description of the unusual power-law dispersion of the central energy bands and of its main consequences, including the strong divergence of the density of states near zero energy, and the vanishing of the transverse localization length of the corresponding edge states. In the presence of a weak off-diagonal disorder of strength $w$, which respects the lattice chiral symmetry, all zero-energy localization properties are found to be anomalous. An exact translation of the problem in terms of Brownian motions enables us to derive numerous analytical results. The typical value of the conductance $g_N$ of a disordered sample of width $N$ and length $L$ is shown to decay as $\exp(-C_Nw\sqrt{L})$, while the relative variance of $\ln g_N$ approaches a non-trivial constant $K_N$. The dependence of the constants $C_N$ and $K_N$ on the ribbon width $N$ is predicted. From the mere viewpoint of the transfer-matrix formalism, zigzag ribbons provide a case study with many unusual features. The transfer matrix describing propagation through one unit cell of a clean ribbon is not diagonalizable at zero energy. In the disordered case, we encounter non-trivial random matrix products such that all Lyapunov exponents vanish identically.