Calculation of 2D electronic band structure using matrix mechanics

TL;DR

This paper extends matrix mechanics methods to calculate 2D electronic band structures for various lattice types, making complex calculations accessible for undergraduate students and aiding research in materials like graphene.

Contribution

It introduces a straightforward extension of matrix mechanics to 2D systems and complex unit cells, facilitating easier band structure calculations for students and researchers.

Findings

Generated band structures for square lattice, muffin-tin, and Gaussian wells.

Applied method to hexagonal lattices like graphene.

Demonstrated educational utility of the approach.

Abstract

We extend previous work applying elementary matrix mechanics to one-dimensional periodic arrays (to generate energy bands) to two-dimensional arrays. We generate band structures for the square lattice "2D Kronig-Penney model" (square wells), muffin-tin potential (cylindrical wells), and Gaussian wells. We then apply the method to periodic arrays of more than one atomic site in a unit cell, in particular, the case of materials with hexagonal lattices like graphene. These straightforward extensions of undergraduate-level calculations allow students to readily determine band structures of current research interest.