Free electron theory for thin metal films

Quantum free electrons, i.e., plane waves, with wavevector k→, and occupancy constrained by the Pauli exclusion principle, are explained in all solid state physics texts. Although overly simplified, free electron theory works surprisingly well for many properties of simple metals. For bulk materials, it is assumed that the sample is effectively infinite and that surfaces are irrelevant. Over the past 30 years, experiments that visualize surfaces and enable the study of 2D materials have revolutionized solid state physics, stimulating new experiment, theory, and applications. Modified free electron models, adapted to films, have enabled modeling of electronic properties of films. This paper analyzes three such models: periodic boundary conditions, hard wall boundary conditions, and soft wall (SW) boundary conditions, in order of increasing realism. The SW case is illustrated for an aluminum film consisting of six atomic layers, comparing SW free electron theory with a scanning tunneling spectroscopy experiment.