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

Abstract

A closed-form algorithm for computing subsurface Green functions-the blocks of a material's Green function between the surface and the bulk-is presented, where we assume the system satisfies a common principal-layer approximation. By exploiting the block tridiagonal and nearly block Toeplitz structure of the Hamiltonian and overlap matrices, this method scales independently of the system size (constant scaling), allowing studies of large systems. As a proof-of-concept example, we investigate the decay of surface effects in an armchair graphene nanoribbon, demonstrating the persistence of surface effects hundreds of atomic layers (∼0.5 μm) away from a surface. We finally compare the surface-to-bulk transitions of finite and semi-infinite systems, finding that finite systems exhibit amplified surface effects.

Emergent Properties in Locally Ordered Molecular Materials

“…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.…”

“…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.…”

Crystal Growth and Characterization of (Bi0.5Na0.5)TiO3–BaTiO3 Single Crystals Obtained by a Top-Seeded Solution Growth Method under High-Pressure Oxygen Atmosphere

“…, tridiagonal/Toeplitz [5], block tridiagonal [4,[6][7][8][9][10][11][12], block Toeplitz [13] and block tridiagonal/block Toeplitz [14,15] matrices. Moreover, some applications, such as the electronic structure of materials [15][16][17][18], require only specific elements of the inverse matrix (as opposed to the entire matrix), resulting in additional efficiency [10,15,19].…”

“…Moreover, some applications, such as the electronic structure of materials [15][16][17][18], require only specific elements of the inverse matrix (as opposed to the entire matrix), resulting in additional efficiency [10,15,19].…”

“…block tridiagonal, block Toeplitz matrices with a small number of deviations from the purely block Toeplitz structure (hereafter called deviations), have recently appeared when investigating surface effects in materials [15,18]. Physically, these studies examined the persistence of surface effects into the material and, mathematically, confined deviations to the surfaces (upper left and/or bottom right matrix corners).…”