Coherent transport of armchair graphene constrictions

The coherent transport properties of armchair graphene nanoconstrictions(GNC) are studied using tight-binding approach and Green's function method. We find a non-bonding state at zero Fermi energy which results in a zero conductance valley, when a single vacancy locates at $y=3n\pm 1$ of a perfect metallic armchair graphene nanoribbon(aGNR). However, the non-bonding state doesn't exist when a vacancy locates at y=3n, and the conductance behavior of lowest conducting channel will not be affected by the vacancy. For the square-shaped armchair GNC consisting of three metallic aGNR segments, resonant tunneling behavior is observed in the single channel energy region. We find that the presence of localized edge state locating at the zigzag boundary can affect the resonant tunneling severely. A simplified one dimensional model is put forward at last, which explains the resonant tunneling behavior of armchair GNC very well.Comment: 6 pages, 11 figure

Nearly Tight Bounds for Sandpile Transience on the Grid

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $\Omega(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $\Omega(n^{3d -2})$ lower bounds.Comment: 36 pages, 4 figure