Closed-form Green functions, surface effects, and the importance of dimensionality in tight-binding metals

Closed-form expressions for all elements of a d -dimensional tight-binding metal's Green function matrix are presented and used to explore edge effects of a surface. We find that, when moving from the surface into the bulk, the number of layers passed before the surfaced substrate behaves like the bulk decreases with dimensionality. In particular, the surface of a one-dimensional substrate becomes indistinguishable from the bulk after O (10¹ - 10 ²) layers, a two-dimensional substrate after O (10¹) layers, and a three-dimensional substrate after O (10⁰) layers. Finally, the effects of substrate dimensionality on molecule-substrate interactions are discussed.