The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics

TL;DR

This paper introduces a numerical matrix mechanics method to extend the Kronig-Penney model for arbitrary confining potentials, enabling detailed study of band structures and electron-hole mass asymmetries in more realistic systems.

Contribution

It presents a straightforward numerical approach to solve for eigenvalues and eigenstates of arbitrary potentials in periodic systems, generalizing the classic Kronig-Penney model.

Findings

Numerical solutions for eigenvalues and eigenstates for arbitrary potentials.

Realistic potentials lead to increased electron-hole mass asymmetry.

Method simplifies studying band structure effects in complex potentials.

Abstract

The Kronig-Penney model describes what happens to electron states when a confining potential is repeated indefinitely. This model uses a square well potential; the energies and eigenstates can be obtained analytically for a the single well, and then Bloch's Theorem allows one to extend these solutions to the periodically repeating square well potential. In this work we describe how to obtain simple numerical solutions for the eigenvalues and eigenstates for any confining potential within a unit cell, and then extend this procedure, with virtually no extra effort, to the case of periodically repeating potentials. In this way one can study the band structure effects which arise from differently-shaped potentials. One of these effects is the electron-hole mass asymmetry. More realistic unit cell potentials generally give rise to higher electron-hole mass asymmetries.