Surface effects in nanoscale materials are more persistent than in conventional materials and often are computationally explored with the Green function. Obtaining the Green function essentially requires inversion of the Hamiltonian matrix, which becomes impractical for the elongated systems of interest. By assuming the system satisfies a common principal layer approximation, the Hamiltonian becomes block tridiagonal and nearly block Toeplitz. In this talk I discuss how to exploit this algebraic structure to obtain the Green function via a constant-scaling algorithm (which does not depend on the size of the system). Finally, as a proof-of-concept example, I investigate the decay of surface effects in an armchair graphene nanoribbon, demonstrating the persistence of surface effects hundreds of atomic layers (approximately 0.5 micrometers) away from a surface.
