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

Abstract

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(10(1)-10(2)) layers, a two-dimensional substrate after O(10(1)) layers, and a three-dimensional substrate after O(10(0)) layers. Finally, the effects of substrate dimensionality on molecule-substrate interactions are discussed.

Emergent Properties in Locally Ordered Molecular Materials

“…Model systems similar to those in Refs. [81][82][83][84] are used; both metallic electrodes [85] are identical with a single band centered at the energy origin. a) Transmission per channel through n 2 identical molecular wires (configured in a square) connected in parallel.…”

“…Some answers to this question lie in the decay of edge effects within a material. [85,137,138] Topologically, a single molecular constituent epitomizes edge effects, whereas a bulk molecular crystal is devoid of them. As ordered aggregates increase in size, edge effects persist at the boundaries, but decay towards the interior of the aggregate until bulk-like crystalline behavior is realized.…”

“…We recently examined the decay rates of edge effects for various systems, [85,137,138] finding that the rate strongly depends on the dimensionality of the material and only weakly on the specific chemistry of the molecule. In general, edge effects are very persistent in one-dimensional materials, they are considerably less persistent in two-dimensional materials; and they decay rapidly in three-dimensional materials.…”

Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions

“…3 In the context of condensed-matter physics, 2,4 these single-particle GFs are useful for their applicability to both bulk and surfaced systems, to band structures, and to response properties (e.g., electron transport 5 or magnetoresistance 6,7 ). Since surface effects rapidly decay in three-dimensional materials (∼5 atomic layers), [8][9][10] most work has historically focused on the surface and bulk GFs, 2,4 that is, the blocks of G(E) at and infinitely far from a surface, respectively. The contemporary interest in low-dimensional materials, however, exposes a need for "subsurface GFs" [the other blocks of G(E)] due to the pervasion of surface effects deep into such materials ( 50 atomic layers).…”

“…The contemporary interest in low-dimensional materials, however, exposes a need for "subsurface GFs" [the other blocks of G(E)] due to the pervasion of surface effects deep into such materials ( 50 atomic layers). [10][11][12][13] Understanding the length scales and characteristics of these slow progressions from surfacelike to bulklike environments is important for designing novel devices that incorporate, for example, carbon nanotubes or graphene nanoribbons.…”