A classical analog for defects in quantum band formation

Abstract

When many individual atoms come together to form a solid, their interaction splits their electronic energy levels to form continuous bands separated by forbidden energy ranges known as band gaps. Introducing defects in a solid results in new electron energy levels that may lie inside the bandgaps. The presence of these defect levels is the heart of the semiconductor-based devices that play a significant role in the modern world. Quantum mechanics provides the best description of interacting atoms. However, band formation is not unique to small-scale atomic interactions but rather is a result of the wave-nature of Schrödinger's equation, which governs quantum mechanics. Using oscillations in a mass-spring system, we present a table-top, classical analog to the quantum system illustrating how defects in a one-dimensional lattice produce changes to the band structure. A pair of masses connected by a spring plays the role of a single atom. Interactions between “atoms” are introduced with weak coupling springs producing two distinct frequency bands from the translational and fundamental modes. Defects are introduced by altering an oscillator pair's total mass or internal spring constant. We provide the theoretical groundwork and experimental verification of the model along with a discussion of the value and limitations of the model as a macroscopic tool to visualize the microscopic world.