Quantum Beating Patterns Observed in the Energetics of Pb Film Nanostructures

We have studied the nanoscale structural evolution of Pb films grown at 110 K on a Si(111) substrate as they are annealed to increasingly higher temperatures. Surface x-ray diffraction from a synchrotron source is used to observe the morphology evolve from an initial smooth film through various metastable states before reaching a state of local equilibrium, at which point the coverage of different height Pb structures is analyzed and related to the thickness-dependent surface energy. Rich patterns are seen in the resulting energy landscape similar to the beating patterns heard from the interference of two musical notes of similar pitch. The explanation is, however, very simple, as demonstrated by a model calculation based on the confinement of free electrons to a quantum well.