Spectral approach to transport in a two-dimensional honeycomb lattice with substitutional disorder

Abstract

The transport properties of a disordered two-dimensional (2D) honeycomb lattice are examined numerically using the spectral approach to the quantum percolation problem, characterized by an Andersontype Hamiltonian. In our simulations, substitutional disorder (or doping) is represented by a modified bimodal probability distribution of the on-site energies. The results indicate the existence of extended energy states for nonzero disorder and the emergence of a transition towards localized behavior for critical doping concentration > 0.3%, in agreement with the experimentally observed metal-to-insulator transition in graphene sheet doped with hydrogen.

Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method

“…Since D n s, is a positive, monotonically decreasing sequence, one can construct approximate lower bounds (with respect to the probability distribution) of the limit D s, . This is exactly the approach taken in [14,[19][20][21]23]. In these works, it is noted that the distance values may decay logarithmically, so the authors performed a careful re-scaling of the horizontal axis by an exponent a < 0.…”

“…In this representation, P s (m) is a probability distribution on Z, allowing one to interpret the fractional case in a similar fashion to the classical result given by (21). That is, by (22), it follows that a fractional discrete harmonic function describes a particle which may jump to any point in Z and the probability that the particle jumps from point n to point m is given by P s (n − m). If s ∈ (0, 1), Theorem 1 implies that the probability of jumping from point n to point m is proportional to |n − m| −(1+2s) .…”

“…The spectral method was introduced in [13], where it was also used to numerically confirm the existence of extended states for the two-dimensional discrete random Schrödinger operator for weak disorder. Applications to other underlying geometries, such as the square, hexagonal, triangular lattice in two-dimensions, and the three-dimensional square lattice, were explored in [18][19][20][21][22]. The unperturbed operator used in these papers was the classical Laplacian (s = 1).…”