Attractive Kronig-Penney Band Structures and Wave Functions

TL;DR

This paper investigates the band structures and wave functions of attractive potential wells in the Kronig-Penney model, revealing negative-energy bands, bound states, and differences from the traditional repulsive model.

Contribution

It extends the Kronig-Penney model to include attractive potentials, deriving band structures, wave functions, and bound states for both square-well and Dirac-comb potentials.

Findings

Negative-energy bands appear in the attractive model.

Wave functions of negative-energy states differ from positive-energy states.

High-degeneracy bound states are identified at negative energies.

Abstract

The repulsive-potential Kronig-Penney (KP) model for a one-dimensional band structure is well known. However, real metals contain positively-charged ions resulting in attractive potential wells seen by the metallic electrons. Here we consider the latter case in detail. The square-well version of the KP model is considered first, for which the band structure and wave functions for different potential-well depths are derived. Then an extended treatment of the attractive Dirac-comb version of the KP model is presented. For the nearly-free-electron case, the band structure exhibits a negative-energy band in addition to positive-energy bands. The wave functions, electron densities of states, effective masses, and group velocities are derived for the positive-energy band states. The wave functions of the negative-energy band states are also calculated and found to be quite different from the sinusoidal wave functions for the positive-energy band and band-gap states. High-degeneracy bound states are found at negative energies and their wave functions are derived.