Multiscale real-space quantum-mechanical tight-binding calculations of electronic structure in crystals with defects using perfectly matched layers

Abstract

We consider the scattering of incident plane-wave electrons from a defect in a crystal modeled by the time-harmonic Schrodinger equation. While the defect potential is localized, the far-field potential is periodic, unlike standard free-space scattering problems. Previous work on the Schrodinger equation has been almost entirely in free-space conditions; a few works on crystals have been in onedimension. We construct absorbing boundary conditions for this problem using perfectly matched layers in a tight-binding formulation. Using the example of a point defect in graphene, we examine the efficiency and convergence of the proposed absorbing boundary condition.

Symmetry-adapted tight-binding electronic structure analysis of carbon nanotubes with defects, kinks, twist, and stretch

“…The real smooth function σ ( x ) is chosen to be 0 outside the PML, and chosen to have the form [23]…”

“…The Bloch wave analogs developed in Section 3.2 are restricted to infinite nanotubes that are uniformly deformed but without defects. To enable the study of defects, we build on the Bloch wave analogs and use the PML approach, following [23], using the complex coordinate stretching approach [49]. This approach enables us to focus the computational effort in the neighborhood of the defect and truncate at some distance away, with the perfect matching preventing the formation of spurious reflections and size effects, owing to the finite domain.…”

“…While this has been studied by other groups, notably including [17][18][19][20][21][22], we build further on the symmetry-adapted approach to develop a method to study defects, such as vacancies, Stone-Wales defects, and geometric defects, such as localized bending or kinking. Our approach to studying defects in a computationally tractable manner uses perfectly matched layers (PMLs) [23], which have been used to truncate the computational domain for wave equations posed on large or unbounded domains in other contexts, such as elastodynamics [24] and electromagnetism [25]. Specifically, PMLs provide a strategy to truncate the domain by using dissipative layers near the boundary of the truncated domain.…”

Symmetry-Adapted Tight-Binding Electronic Structure Analysis of Carbon Nanotubes with Defects, Kinks, Twist, and Stretch

“…The real smooth function σ(x) is chosen to be 0 outside the PML, and chosen to have the form [PD16]:…”

“…While this has been studied by other groups, notably including [ZJD09b, ZD08, NDJD14, KS20, AWHS17, ZAGS17], we build further on the symmetry-adapted approach to develop a method to study defects such as vacancies, Stone-Wales defects, and geometric defects such as localized bending or kinking. Our approach to study defects in a computationally tractable manner uses perfectly-matched layers (PML) [PD16], which have been used to truncate the computational domain for wave equations posed on large or unbounded domains in other contexts such as elastodynamics [BC04] and electromagnetism [Ber94]. Specifically, PML provides a strategy to truncate the domain by using dissipative layers near the boundary of the truncated domain.…”