Non-linear Fourier analysis of aperiodic structures: Mass-density-waves in adsorbed monolayers

The adsorption of a "solid-like" monolayer on a crystalline substrate can produce distortions in the monolayer which are describable as mass-density-waves. These mass-density-waves can lead to a fixed alignment between the monolayer and the substrate crystal axes that may not be aligned with any high-symmetry direction of the substrate surface, a phenomenon known as rotational (orientational, Moiré) epitaxy. The driver of rotational epitaxy has been attributed to the partial matching of the symmetry of the substrate surface, to that of the over-layer (i.e., high-order commensurate phases), and to the competition between longitudinal strains (compressions) verses transverse strains (shears) in the mass-density-wave distortions caused by the mismatch between these two symmetries (i.e., the rotated incommensurate phase). In this work, we will examine both these scenarios within a new formalism, one using a fully non-linear approach to the calculation of the Fourier amplitudes that describe these distortions. The effects on the monolayer due to the substrate are found to be well-described by this formalism, which also has important pedagogical and calculational aspects. Within our proposed framework, the high-order commensurate phase shows up (using the language of structural phase transitions) as a partial regaining of the symmetry though a "lock-in transformation." This work builds upon and extends the earlier work of Novaco and McTague. Both classical and quantum regimes are explored, and the limits of this new approach are tested against other calculations. The extension and application of this framework to other systems, such as two-dimensional van der Waals hetero-structures, is briefly discussed.