Analytical Solution to the Kronig-Penney Model with Harmonic Oscillator Wells: Insights to Tight-Binding

TL;DR

This paper analytically solves a modified Kronig-Penney model using harmonic oscillator wells, providing insights into energy dispersion, wave functions, and tunneling amplitudes relevant to tight-binding approximations.

Contribution

It introduces an analytical solution to a Kronig-Penney model with harmonic oscillator wells, linking potential parameters to tunneling amplitudes.

Findings

Derived energy dispersion and wave functions for the model.

Established a form of the governing equation similar to tight-binding.

Expressed tunneling amplitude in terms of harmonic oscillator parameters.

Abstract

The celebrated Kronig-Penney model traditionally has been formulated with square well potentials representing atomic centres. Here, we use a slightly more realistic potential, the truncated harmonic oscillator, in lieu of square well potentials, and solve the model analytically. We derive the energy dispersion and wave functions for this model. This configuration has some important similarities and differences compared to the usual model. In particular, we write the governing equation in a form suggestive of the tight-binding approximation, as can be done for the usual model. In this way, it is straightforward to derive an expression for the tunneling amplitude used in tight-binding in terms of the harmonic oscillator potential parameters.