Embedding on to a one-dimensional crystal

TL;DR

This paper derives a new analytical expression for the band structure and embedding potential of a one-dimensional crystal, enabling efficient modeling of surface electronic properties with a simplified approach.

Contribution

It introduces a novel formula linking solutions of the Schrödinger equation to the band structure and embedding potential in one-dimensional periodic systems.

Findings

Derived a simple expression for the band structure.

Developed a new embedding potential formula.

Validated results with aluminum surface calculations.

Abstract

A simple expression is derived for the band structure of a one-dimensional periodic potential in terms of two solutions of the Schroedinger equation within the unit cell, one with a zero-derivative boundary condition on the left-hand end of the cell and the other with zero derivative on the right-hand end. From this starting point, a new expression is derived for the embedding potential - this can be added to the Hamiltonian for the surface region of a crystal to replace the semi-infinite substrate, in a one-dimensional approximation. The results are demonstrated in calculations of the band structure and embedding potential for Al in the [001] direction, and the surface electronic structure of the Al(001) surface.