Band gap formation theory: An alternative to the Bragg diffraction model

TL;DR

This paper proposes an alternative to the traditional Bragg diffraction model for explaining band gap formation, using a sampling perspective based on the discrete nature of the crystal lattice and the Schrödinger equation.

Contribution

It introduces a novel theory that explains band gaps as a sampling effect due to lattice discreteness, avoiding the limitations of Bragg diffraction especially in one-dimensional systems.

Findings

Band gaps emerge from sampling effects related to lattice discreteness.

The energy diagram shows band gaps as a result of wavenumber changes and sampling constraints.

The theory extends naturally to higher-dimensional systems.

Abstract

The band gap, a key concept in solid-state physics, is traditionally explained by the Bragg diffraction of electron waves in the periodic potential of a crystal. Although widely accepted, this framework raises fundamental issues in one-dimensional systems, where Bragg diffraction-which requires multidirectional wave interactions-reduces to simple interference, thus failing to explain band gap formation. In this paper, we introduce an alternative theory that does not rely on Bragg reflection. Using the Schr\"{o}dinger equation for Bloch waves, we consider the crystal lattice as a discrete set of observation points. This discreteness introduces a sampling-like constraint analogous to the Nyquist frequency in signal processing. We show that when the electron wavenumber changes under a periodic potential while the lattice spacing remains fixed, a band gap naturally emerges as a sampling effect. By constructing an energy diagram that incorporates this effect, we reveal that the band gap originates from both the wavenumber change and the role of the lattice as discrete samplers, leading the energy curve to exhibit translational and mirror symmetries with respect to the Nyquist wavenumber. This approach provides a novel, physically grounded explanation of band gap formation and can be naturally extended to higher dimensions.